Optimization in Food Engineering

Optimization in Food Engineering

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Contemporary Food Engineering Series Editor

Professor Da-Wen Sun, Director Food Refrigeration & Computerized Food Technology National University of Ireland, Dublin (University College Dublin) Dublin, Ireland http://www.ucd.ie/sun/

Optimization in Food Engineering, edited by Ferruh Erdoˇgdu (2009) Advances in Food Dehydration, edited by Cristina Ratti (2009) Optical Monitoring of Fresh and Processed Agricultural Crops, edited by Manuela Zude (2009) Food Engineering Aspects of Baking Sweet Goods, edited by Servet Gülüm Sumnu ˛ and Serpil Sahin (2008) Computational Fluid Dynamics in Food Processing, edited by Da-Wen Sun (2007)

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Optimization in Food Engineering

Edited by

Ferruh Erdoˇ gdu

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

ß 2008 by Taylor & Francis Group, LLC.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6141-3 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Dedication To my mother, Aynur for all her love, support, patience and efforts giving me the best education possible, to my wife Belgin and my sister Aylin for their love, support and patience and to the memories of my father, Feran and my uncle, S¸eref.

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Contents Series Editor’s Preface Preface Series Editor Editor Contributors

PART I Chapter 1

Modeling: Significance, Fundamentals, and Methods Significance of Mathematical Modeling and Simulation for Optimization Quang Tri Ho, Hibru Kelemu Mebatsion, Bart Nicolaï, and Pieter Verboven

Chapter 2

Analytical Solutions in Conduction Heat Transfer Problems Ferruh Erdo gdu and Mahir Turhan

Chapter 3

Numerical Solutions: Finite Difference Methods T. Koray Palazo glu and Ferruh Erdogdu

Chapter 4

Numerical Solutions: Finite Element and Finite Volume Methods Rui C. Martins, Vitor V. Lopes, António A. Vicente, and José A. Teixeira

PART II Chapter 5

Optimization Optimization: An Introduction Ferruh Erdo gdu

Chapter 6

Statistical Optimization: Response Surface Methodology . Kun-Nan Chen and Ming-Ju Chen

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Chapter 7

Random-Centroid Optimization Shuryo Nakai, Yasumi Horimoto, Jinglie Dou, and Roxana A. Verdini

Chapter 8

Multi-Objective Optimization in Food Engineering Cheah Keen Seng and Gade Pandu Rangaiah

Chapter 9

Applications of the Minimum Principle of Pontryagin for Solving Optimal Control Problems Andrey V. Kuznetsov

Chapter 10

Neural Networks and Genetic Algorithms Yang Meng and Hosahalli S. Ramaswamy

Chapter 11

Computational Fluid Dynamics for Optimization in Food Processing Ferruh Erdogdu

Chapter 12

Dynamic Optimization J. Ricardo Pérez-Correa, Claudio A. Gelmi, and Lorenz T. Biegler

Chapter 13

Tabu Search: Development, Algorithm, Performance, and Applications Mekapati Srinivas and Gade Pandu Rangaiah

Chapter 14

Eigenvalue Optimization Techniques for Nonlinear Dynamic Analysis and Design Luis G. Matallana, Aníbal M. Blanco, and J. Alberto Bandoni

Chapter 15

Complex Method Optimization Ferruh Erdogdu and Murat O. Balaban

Chapter 16

Mixed Integer Linear Programming Scheduling in the Food Industry Philip Doganis and Haralambos Sarimveis

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Chapter 17

Mixed Integer Nonlinear Programming: Applications to Food Dehydration and Deep Chilling Panagiotis P. Repoussis and Christos T. Kiranoudis

PART III Chapter 18

Optimization Studies for Different Food Processes Optimization and Control Strategy to Improve the Performance of Batch Reactors Iqbal M. Mujtaba

Chapter 19

Pulsed Microwave Heating of Foods: Temperature Measurement and Optimization Sundaram Gunasekaran

Chapter 20

Optimization of Freeze-Drying Process Applied to Food and Biological Products: From Response Surface Methodologies to an Interactive Tool Michèle Marin, Stéphanie Passot, Fernanda Fonseca, and Ioan Cristian Trelea

Chapter 21

Optimization of Spray Drying of Sugar-Rich Foods Vinh Truong

Chapter 22

Structural Optimization Techniques for Developing Beverage Containers Koetsu Yamazaki, Jing Han, and Sadao Nishiyama

Chapter 23

Optimization for Continuous Shortest Paths in Transportation J. Miguel Díaz-Báñez

Chapter 24

Real-Time Nonlinear Optimal Control of Refrigeration Processes Ioan Cristian Trelea

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Chapter 25

Optimization of Apple Juice Extraction María Teresa González and Martín Juan Urbicain

Chapter 26

Optimization of Canned Food Processing Ricardo Simpson and Arthur A. Teixeira

Chapter 27

Optimal Design of Continuous Thermal Processing with Plate Heat Exchangers Jorge Andrey Wilhelms Gut and José Maurício Pinto

Chapter 28

Process Optimization Strategies to Reduce Variability in Thermal Processing of Packaged Foods Kevin Cronin and Philippe Baucour

Chapter 29

Loading Optimization Reinaldo Morabito and Vitória Pureza

Chapter 30

Optimization of the Arrays of Impinging Jets Muhiddin Can and A. Burak Etemo glu

Chapter 31

Optimal Operational Planning in the Fruit Industry Supply Chain Guillermo L. Masini, Aníbal M. Blanco, Noemí C. Petracci, and J. Alberto Bandoni

Chapter 32

Optimizing the Management of Curing Chambers Jose Bon and Antonio Mulet

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Series Editor’s Preface CONTEMPORARY FOOD ENGINEERING Food engineering is the multidisciplinary field of applied physical sciences combined with the knowledge of product properties. Food engineers provide the technological knowledge transfer essential to the cost-effective production and commercialization of food products and services. In particular, food engineers develop and design processes and equipment in order to convert raw agricultural materials and ingredients into safe, convenient, and nutritious consumer food products. However, food engineering topics are continuously undergoing changes to meet diverse consumer demands, and the subject is being rapidly developed to reflect market needs. In the development of food engineering, one of the many challenges is to employ modern tools and knowledge, such as computational materials science and nanotechnology, to develop new products and processes. Simultaneously, improving food quality, safety, and security remain critical issues in food engineering study. New packaging materials and techniques are being developed to provide more protection to foods, and novel preservation technologies are emerging to enhance food security and defense. Additionally, process control and automation regularly appear among the top priorities identified in food engineering. Advanced monitoring and control systems are developed to facilitate automation and flexible food manufacturing. Furthermore, energy saving and minimization of environmental problems continue to be an important food engineering issue and significant progress is being made in waste management, efficient utilization of energy, and reduction of effluents and emissions in food production. The Contemporary Food Engineering series, consisting of edited books, attempts to address some of the recent developments in food engineering. Advances in classical unit operations in engineering applied to food manufacturing are covered as well as such topics as progress in the transport and storage of liquid and solid foods; heating, chilling, and freezing of foods; mass transfer in foods; chemical and biochemical aspects of food engineering and the use of kinetic analysis; dehydration, thermal processing, nonthermal processing, extrusion, liquid food concentration, membrane processes, and applications of membranes in food processing; shelf-life, electronic indicators in inventory management, and sustainable technologies in food processing; and packaging, cleaning, and sanitation. The books are aimed at professional food scientists, academics researching food engineering problems, and graduate level students. The books’ editors are leading engineers and scientists from many parts of the world. All the editors were asked to present their books to address the market need and pinpoint the cutting-edge technologies in food engineering. Furthermore, all contributions are written by internationally renowned experts who have both academic and professional credentials. All authors have attempted to

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provide critical, comprehensive, and readily accessible information on the art and science of a relevant topic in each chapter, with reference lists for further information. Therefore, each book can serve as an essential reference source to students and researchers in universities and research institutions. Da-Wen Sun, Series Editor

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Preface Food engineering has gained more and more significance in the last couple of decades. Mathematical models have been used to better understand and improve food processing operations, and in this concept, various optimization approaches have played a significant role. As a result, there has been a dramatic increase in the efficiency and reliability of optimization methods for different problem categories. Optimization methods can be easily applied in food processing as long as the changes during a process can be predicted mathematically. This case, of course, depends on the presence of mathematical models. Since heat, mass, and momentum transfers are major mechanisms in food processing, mathematical models describing these phenomena are also required for further mathematical-based optimization procedures. Within this context, mathematical optimization plays an important role in optimizing different food processing operations. Excellent text and reference books are available for educational and research purposes in the field of optimization and food processing. It will therefore be quite significant to combine the advantages in this field for further optimization strategies to improve the quality and safety of food processes and optimal operating policies in the food industry. Based on this concept, Optimization in Food Engineering has been divided into the following sections to serve as a reference for professional food scientists, food engineers, academicians, and graduate level students working in the field of food engineering and processing. This book consists of three parts. In the first part, the significance of modeling, fundamentals, and methods are covered for analytical and numerical procedures, since an optimization procedure depends on the presence of an effective mathematical model. It is a known fact that knowledge of mathematical modeling techniques provides significant information for further research and developments in food processing. In addition, the changes predicted by a model in a given process are required if the given process is described other than by trial-and-error physical experiments. In the second part, optimization and different optimization techniques are presented. This part begins with statistical optimization techniques and continues with Pontryagin’s method, multi-objective and dynamic optimization techniques, and mixed integer linear and nonlinear programming methodologies. In addition, limitations and possibilities of using neural networks and genetic algorithms and computational fluid dynamics programming approaches are presented with tabu search, complex method, and Eigenvalue optimization techniques. Finally, in the last part, optimization studies for different food processes are discussed. This part covers a broad area for different processes starting from the optimization strategies to improve the performance of batch reactors to the optimization of conventional thermal processing, microwave heating, freeze drying, spray drying, and refrigeration systems. Different food processing areas are presented for optimization purposes,

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and structural optimization techniques for developing beverage containers are discussed. Loading optimization, optimization approaches for impingement processing, and optimal operational planning methodologies are also covered. In each chapter, the required parameters for the given process are presented in detail along with the optimization procedures that need to be applied. Ferruh Erdogdu

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Series Editor Professor Da-Wen Sun was born in Southern China and is a world authority on food engineering research and education. His main research activities include cooling, drying, and refrigeration processes and systems; quality and safety of food products; bioprocess simulation and optimization; and computer vision technology. His innovative studies on vacuum cooling of cooked meats, pizza quality inspection by computer vision, and edible films for shelf-life extension of fruits and vegetables have been widely reported in national and international media. Results of his work have been published in over 180 peer-reviewed journal papers and more than 200 conference papers. Professor Sun received first class BSc honors and MSc in mechanical engineering, and a PhD in chemical engineering in China before working in various universities in Europe. He became the first Chinese national to be permanently employed in an Irish university when he was appointed college lecturer at National University of Ireland, Dublin (University College Dublin), Ireland, in 1995, and was then continuously promoted in the shortest possible time to senior lecturer, associate professor, and full professor. Sun is now professor of Food and Biosystems Engineering and director of the Food Refrigeration and Computerized Food Technology Research Group in University College Dublin. As a leading educator in food engineering, Sun has contributed significantly to the field of food engineering. He has trained many PhD students, who have made their own contributions to the industry and academia. He has also, on a regular basis, given lectures on advances in food engineering in academic institutions internationally and delivered keynote speeches at international conferences. As a recognized authority in food engineering, he has been conferred adjunct=visiting=consulting professorships from 10 top universities in China including Zhejiang University, Shanghai Jiaotong University, Harbin Institute of Technology, China Agricultural University, South China University of Technology, and Jiangnan University. In recognition of his significant contribution to food engineering worldwide and for his outstanding leadership in the field, the International Commission of Agricultural Engineering (CIGR) awarded him the CIGR Merit Award in 2000 and again in 2006 and the Institution of Mechanical Engineers based in the United Kingdom named him Food Engineer of the Year 2004, in 2008 he was awarded CIGR Recognition Award in recognition of his distinguished achievements as top one percent of Agricultural Engineering scientists around the world.

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He is a fellow of the Institution of Agricultural Engineers. He has also received numerous awards for teaching and research excellence, including the President’s Research Fellowship, and has received the President’s Research Award from University College Dublin on two occasions. He is a member of the CIGR executive board and honorary vice president of CIGR; editor-in-chief of Food and Bioprocess Technology—An International Journal (Springer); series editor of Contemporary Food Engineering (CRC Press=Taylor & Francis); former editor of Journal of Food Engineering (Elsevier); and editorial board member for Journal of Food Engineering (Elsevier), Journal of Food Process Engineering (Blackwell), Sensing and Instrumentation for Food Quality and Safety (Springer), and Czech Journal of Food Sciences. He is also a chartered engineer registered in the U.K. Engineering Council.

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Editor Dr. Ferruh Erdogdu is an associate professor of food engineering at the University of Mersin, Mersin, Turkey. He was born in Eregli, Turkey, and graduated from the Department of Food Engineering at Hacettepe University in Ankara in 1992 with honors and the highest GPA. In 1994, he succeeded in a nationwide exam by the Ministry of National Education of Turkey to pursue masters and PhD degrees in food engineering in the United States. Dr. Erdo gdu received his master of engineering degree in 1996 and PhD in 2000 at the University of Florida, Gainesville, Florida. While working with Dr. Murat O. Balaban at the University of Florida, he maintained a status of distinguished scholar. He received outstanding academic achievement awards from College of Engineering (1997–2000) and College of Agriculture (1999), and won the student paper competition hosted by the food engineering division of the Institute of Food Technologists (IFT). After receiving his PhD, he conducted his postdoctoral work at the University of California, Davis, California, with Dr. R. Paul Singh. In 2001, Dr. Erdo gdu joined the faculty of food engineering at the University of Mersin where he has been teaching undergraduate- and graduate-level courses on topics in food engineering. In 2007, he was appointed holder of a scholarship within the Swedish–Turkish Programme by the Swedish Institute for studies=research work at Lund University, Lund, Sweden. Ferruh is the author or coauthor of more than 30 research papers published in internationally known peer-reviewed journals, 4 book chapters, and more than 50 presentations. He is the coauthor of the books Virtual Experiments in Food Processing, published in 2004, and Industrial Scale Food Freezing Simulation Software published by the World Food Logistics Organization. Dr. Erdo gdu is a professional member of the IFT. He has been serving on the editorial board of the Journal of Food Process Engineering since 2003 and assisting in review processes for Journal of Food Engineering; Journal of Food and Bioprocess Technologies; Journal of Food Technology and Biotechnology; International Journal of Engineering; Computers and Chemical Engineering; and Chemical Process Engineering. His current research interests include mathematical modeling and optimization of heat, mass, and momentum transfer operations in food processing.

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Contributors Murat O. Balaban Fishery Industrial Technology Center University of Alaska Fairbanks Fairbanks, Alaska J. Alberto Bandoni Planta Piloto de Ingeniería Quimica Bahía Blanca, Argentina Philippe Baucour Institut FEMTO-ST, Department of CREST Belfort, France Lorenz T. Biegler Department of Chemical Engineering Carnegie Mellon University Pittsburgh, Pennsylvania Aníbal M. Blanco Planta Piloto de Ingeniería Quimica Bahía Blanca, Argentina Jose Bon Research Group Analysis and Simulation of Agro-food Processes Food Technology Department Polytechnic University of Valencia Valencia, Spain Muhiddin Can Uludag University Faculty of Engineering and Architecture Mechanical Engineering Department Gorukle Campus Bursa, Turkey Kun-Nan Chen Department of Mechanical Engineering Tungnan University Taipei, Taiwan

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Ming-Ju Chen Department of Animal Science and Technology National Taiwan University Taipei, Taiwan Kevin Cronin Department of Process and Chemical Engineering University College Cork Cork, Ireland J. Miguel Díaz-Báñez Universidad de Sevilla Departament de Matemática Aplicada II Escuela Superior de Ingenieros Sevilla, Spain Philip Doganis National Technical University of Athens, School of Chemical Engineering, Zografou Campus, Athens, Greece Jinglie Dou University of British Columbia Food, Nutrition and Health Vancouver, British Columbia, Canada Ferruh Erdogdu Department of Food Engineering University of Mersin Çiftlikköy-Mersin, Turkey A. Burak Etemoglu Uludag University, Faculty of Engineering and Architecture Mechanical Engineering Department Gorukle Campus Bursa, Turkey

Fernanda Fonseca INRA Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France Claudio A. Gelmi Department of Chemical and Bioprocess Engineering Pontificia Universidad Católica de Chile Santiago, Chile María Teresa González Planta Piloto de Ingeniería Química Bahía Blanca, Argentina Sundaram Gunasekaran Biological Systems Engineering Department University of Wisconsin-Madison Madison, Wisconsin Jorge Andrey Wilhelms Gut Department of Chemical Engineering—Escola Politécnica University of São Paulo São Paulo, Brazil Jing Han Universal Can Corporation Shizuoka, Japan Quang Tri Ho BIOSYST-MeBioS, K.U.Leuven Leuven, Belgium Yasumi Horimoto University of British Columbia Food, Nutrition and Health Vancouver, British Columbia, Canada

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Christos T. Kiranoudis School of Chemical Engineering Department of Process Control and Plant Design National Technical University of Athens Athens, Greece Andrey V. Kuznetsov Department of Mechanical and Aerospace Engineering North Carolina State University Raleigh, North Carolina Vitor V. Lopes Institute of Systems and Robotics Technical University of Lisbon Lisbon, Portugal Michèle Marin AgroParisTech, INRA Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France Rui C. Martins BioInformatics—Molecular and Environmental Research Centre University of Minho Braga, Portugal Guillermo L. Masini Facultad de Ingeniería Departamento de Mecánica Aplicada Universidad Nacional del Comahue Neuquén, Argentina Luis G. Matallana Planta Piloto de Ingeniería Química Bahía Blanca, Argentina

Hibru Kelemu Mebatsion BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium Yang Meng Department of Food Science McGill University Macdonald Campus Ste-Anne-de-Bellevue, Quebec, Canada Reinaldo Morabito Department of Production Engineering Universidade Federal de São Carlos São Paulo, Brazil Iqbal M. Mujtaba School of Engineering, Design and Technology University of Bradford Bradford, England Antonio Mulet Research group Analysis and Simulation of Agro-food Processes Food Technology Department Polytechnic University of Valencia Valencia, Spain Shuryo Nakai University of British Columbia Food, Nutrition and Health Vancouver, British Columbia, Canada Bart Nicolaï BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium Sadao Nishiyama Universal Can Corporation Tokyo, Japan

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T. Koray Palazoglu Department of Food Engineering University of Mersin Çiftlikköy-Mersin, Turkey Stéphanie Passot AgroParisTech Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France J. Ricardo Pérez-Correa Department of Chemical and Bioprocess Engineering Pontificia Universidad Católica de Chile Santiago, Chile Noemí C. Petracci Planta Piloto de Ingeniería Química Bahía Blanca, Argentina José Maurício Pinto Advanced Control and Operations Research Technology Group Praxair, Inc. Danbury, Connecticut Vitória Pureza Department of Production Engineering Universidade Federal de São Carlos São Paulo, Brazil Hosahalli S. Ramaswamy Department of Food Science McGill University Macdonald Campus Ste-Anne-de-Bellevue, Québec, Canada

Gade Pandu Rangaiah Department of Chemical and Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore Panagiotis P. Repoussis School of Chemical Engineering Department of Process Control and Plant Design National Technical University of Athens Athens, Greece Haralambos Sarimveis National Technical University of Athens School of Chemical Engineering Athens, Greece Cheah Keen Seng Department of Chemical and Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore Ricardo Simpson Departamento de Procesos Químicos Biotecnológicos, y Ambientales Universidad Técnica Federico Santa María Valparaíso, Chile Mekapati Srinivas Department of Chemical and Biomolecular Engineering National University of Singapore Singapore, Republic of Singapore

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Arthur A. Teixeira Department of Agricultural and Biological Engineering Institute of Food and Agricultural Sciences University of Florida Gainesville, Florida José A. Teixeira Institute for Biotechnology and BioEngineering Centro de Engenharia Biológica, University of Minho Braga, Portugal Ioan Cristian Trelea AgroParisTech Joint Research Unit Génie et Microbiologie des Procédés Alimentaires AgroParisTech, INRA Thiverval–Grignon, France Vinh Truong Department of Chemical Engineering Nong Lam University Ho Chi Minh, Vietnam Mahir Turhan Department of Food Engineering University of Mersin Çiftlikköy-Mersin, Turkey Martín Juan Urbicain (Deceased) Planta Piloto de Ingeniería Química Bahía Blanca, Argentina Pieter Verboven BIOSYST-MeBioS, K.U. Leuven Leuven, Belgium

Roxana A. Verdini Instituto de Desarrollo Technológico para la Industria Química Santa Fe, Argentina António A. Vicente Institute for Biotechnology and BioEngineering Centro de Engenharia Biológica University of Minho Braga, Portugal

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Koetsu Yamazaki Division of Innovative Technology and Science Graduate School of Natural Science and Technology Kanazawa University Kanazawa, Ishikawa, Japan

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Part I Modeling: Significance, Fundamentals, and Methods

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1

Significance of Mathematical Modeling and Simulation for Optimization Quang Tri Ho, Hibru Kelemu Mebatsion, Bart Nicolaï, and Pieter Verboven

CONTENTS 1.1 1.2

Introduction .................................................................................................... 3 Heat and Mass Transfer Modeling ................................................................ 4 1.2.1 General Considerations ....................................................................... 4 1.2.2 Case Study: Permeation–Diffusion–Reaction Model of Gas Exchange in Pear Fruit ............................................................ 6 1.3 Kinetics Modeling.......................................................................................... 7 1.3.1 General Considerations ....................................................................... 7 1.3.2 Case Study: Model for the Respiration of Fruit ................................. 9 1.4 Model Parameters ........................................................................................ 11 1.4.1 Thermophysical Properties ................................................................ 11 1.4.2 Kinetics Parameters ........................................................................... 11 1.5 Solution Methods ......................................................................................... 14 1.5.1 General Considerations ..................................................................... 14 1.5.2 Case Study: Solution of Three-Dimensional Gas Exchange and Respiration in Pear Fruit ............................................................ 14 1.6 Towards Food Process Modeling at Different Scales: Multiscale Modeling .................................................................................... 15 1.7 Conclusion ................................................................................................... 16 Acknowledgments................................................................................................... 16 Nomenclature .......................................................................................................... 16 Greek Letters................................................................................................ 17 References ............................................................................................................... 17

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1.1 INTRODUCTION This chapter introduces fundamental mechanisms of heat–mass transfer and mathematical basics for modeling aspects involving kinetics (such as biochemical changes, quality, and safety). Mathematical aspects of formulating and solving models that describe time-dependent spatial and coupled phenomena in food processes are outlined and explained. Modeling issues related to thermophysical properties and model parameters, such as variability and parameter estimation, are mentioned, and tools for food process simulation at different scales are presented. The merits and limitations of food process simulation are demonstrated by means of an illustrative case study that is explored in all its modeling facets throughout the chapter. We demonstrate modeling to the application of ultra low oxygen storage of pears. Pears are typically stored under a controlled atmosphere with reduced O2 and increased CO2 levels to extend their commercial storage life, which can be as long as 9 months. The exact optimal gas conditions depend on factors such as cultivar, origin, growing conditions, and picking date of the fruit. At too low-oxygen concentration, anoxia may occur eventually leading to cell death and loss of the product. Other fruit such as apples are considerably less sensitive to variations in low oxygen conditions. This is probably related to differences in gas concentration gradients resulting from differences in tissue diffusivity and respiratory activity. There is little information about such gas gradients in fruit. Knowledge on internal gas exchange would be, nevertheless, very valuable to guide commercial storage practices since disorders under controlled atmosphere related to fermentation are a prime cause of concern. Optimal storage conditions of new cultivars are generally determined by tedious experimental trials that should cover several growing years. Modeling will help better understand the processes of gas exchange and kinetics of respiration associated with fruit storage potential and will allow performing numerical experiments to determine optimal storage conditions. This chapter is subdivided as follows. In Section 1.2, mathematical modeling of heat and mass transfer is introduced. In Section 1.3, modeling of kinetics is described. Section 1.4 deals with model parameters. Solution methods are mentioned in Section 1.5 and in Section 1.6 a future perspective is given in terms of multiscale modeling of food processes. Finally some conclusions are drawn in Section 1.7. The chapter is intended to give the reader a framework and flavor for the following detailed chapters.

1.2 HEAT AND MASS TRANSFER MODELING 1.2.1 GENERAL CONSIDERATIONS In general, heat and mass transport occurs by diffusion as well as convective mechanisms. In its physical definition, diffusion is due to the spontaneous net movement of particles from high to low concentration. For heat transfer, the conduction term is more often used and refers to the transfer of thermal energy from a region of higher temperature to a region of lower temperature. However, diffusion is often used as an apparent mechanism encompassing more complex micro- and

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nanoscale phenomena such as pressure driven flow, Knudsen flow, and capillary flow. The driving force behind convective transport is a pressure gradient in the case of forced convection (e.g., due to a pump), or density differences because of, e.g., temperature gradients. For simplicity, the discussion in this chapter is restricted to a single Newtonian system. This means that the materials for which there is a linear relationship between shear stress and velocity gradient, such as water or air will be considered. More complicated fluids such as ketchup, starch solutions etc., are so-called non-Newtonian fluids, and the reader is referred to standard books on rheology for more details. In case of solid materials, convection will not be significant. Applying the conservation principle to a fixed infinitesimal control volume dx1dx2dx3 one obtains the mass continuity, momentum and energy, and mass fraction equations, written in index notation for Cartesian coordinates xi (i ¼ 1, 2, 3 for the x-, y-, and z-direction, respectively), and whenever an index appears twice in any @ru term, summation over the range of that index is implied (for example, @xj j becomes @ru1 @ru2 @ru3 @x1 þ @x2 þ @x3 in reality). This model can be applied to any such system @r @ruj þ ¼0 @t @xj

 @rui @ruj ui @ @ui @uj @ 2 @uj  þ fi þ ¼ h þ pþ h @t @xj @xj @xi @xj @xi 3 @xj
 @rH @ruj H @ @T @p þ þ þQ ¼ k @t @xj @xj @xj @t @rXa @ @ @ þ ruj Xa ¼ rDa X a þ ra @t @xj @xi @xi

(1:1) (1:2) (1:3) (1:4)

For a full derivation of these equations we refer to any text book on fluid mechanics. The system of at least five equations (three equations for the velocity components plus the continuity and the energy equation), and added with a mass fraction equation for each component of interest) contains at least eight variables (u1, u2, u3, p, H, T, Xa, r). Therefore, additional equations to close the system are required. Thermodynamic equation of state gives the relation between density r and pressure p and temperature T. The constitutive equation relates the enthalpy h to the pressure  . There are no concluand the temperature by means of the heat capacity c ¼ @H @T p sive general rules for implementation of boundary conditions for the Navier–Stokes equations to have a well-posed problem because of their complex mathematical nature. For incompressible and weakly compressible flows, it is possible to define Dirichlet boundary conditions (fixed values of the variables mostly for an incoming flow), Neumann boundary conditions (fixed gradients, mostly for an outgoing flow), and wall boundary conditions (a wall function reflecting the system behavior at the solid boundaries at the edge of the system considered). Initial values must be provided for all variables.

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1.2.2 CASE STUDY: PERMEATION–DIFFUSION–REACTION MODEL OF GAS EXCHANGE IN PEAR FRUIT The tissue structure of pear fruit is considered to contain mainly two phases: intracellular liquid phase of the cells and air-filled intercellular space. Assuming local equilibrium at a certain concentration of the gas component i in the gas phase Ci,g (mol m3), concentration of the compound in the liquid phase of fruit tissue normally follows Henry’s law. If the tissue has a porosity «, the volume-averaged concentration Ci,tissue (mol m3) of species i is then defined as Ci,tissue ¼ «  Ci,g þ (1  «)  R  T  Hi  Ci,g

(1:5)

with Hi is Henry’s constant of component i (i is O2, CO2, or N2). From this definition, following expression for the gas capacity (ai) of the component i of the tissue is derived ai ¼ « þ (1  «)  R  T  Hi ¼

Ci,tissue Ci,g

(1:6)

A permeation–diffusion–reaction model was constructed describing the diffusion and permeation processes in pear tissue for the three major atmospheric gases O2, CO2, and N2. Equations for transport of O2, CO2, and N2 were established by Ho et al. (2008) ai

@Ci þ r  (uCi ) ¼ r  Di rCi þ Ri @t

(1:7)

with boundary conditions at the external surface of the pear: Ci ¼ Ci,1

(1:8)

where Ri is the production term of the gas component i related to O2 consumption or CO2 production r (m1) is the gradient operator The index 1 refers to the gas concentration of the ambient atmosphere. The first term in Equation 1.7 represents the accumulation of gas i, the second term permeation transport driven by an overall pressure gradient, the third term molecular diffusion due to a partial pressure gradient, and the last term consumption or production of gas i because of respiration or fermentation. If, for example, oxygen is consumed in the fruit center, it creates a local partial pressure gradient, which drives molecular diffusion. However, if the rates of transport of different gasses are different, overall pressure gradients may build up and cause permeation transport. Permeation through the barrier of tissue by the pressure gradient is described by Darcy’s law (Geankoplis, 1993), which in fact is an apparent form of the momentum equation outlined above

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20

3.5 92

14 12 10 8 6

2.5 2 1.5 1

0.02 x (m)

0

0.02 x (m)

88 86 84 82

0.5

4 0

90

N2 partial pressure (kPa)

16

3

CO2 partial pressure (kPa)

O2 partial pressure (kPa)

18

80 0

0.02 x (m)

0

0.02 x (m)

FIGURE 1.1 Finite element mesh of pear geometry and simulated gas partial pressure distribution in pear intact fruit using permeation–diffusion–reaction model. Simulation was carried out at 18C, 20 kPa O2, 0 kPa CO2 at the ambient atmosphere and applied to an axisymmetrical pear shape. Parameters were taken from Ho, Q.T. et al., PLoS. Comput. Biol., 4, e1000023, 2008.

u¼

K K  R  T X  rP ¼  r Ci m m

(1:9)

The relation between gas concentration and pressure was assumed to follow the ideal gas law (P ¼ CRT). A typical model simulation result for gas exchange in pear tissue is shown in Figure 1.1. An axi-symmetric geometry model was created for the pear. Simulation showed that, due to the respiration of the tissue, the O2 gas partial pressure decreased from surface to the pear center while CO2 decreased in the opposite direction. An increase of N2 from surface to the pear center was also found. The profiles are strongly dependent on the tissue properties. In the past, it was assumed that gas transfer barriers were restricted to the skin layers of fruit. Here, it was demonstrated that the resistance to gas exchange of the fleshy part of fruit is also significant and could lead to oxygen deficiency.

1.3 KINETICS MODELING 1.3.1 GENERAL CONSIDERATIONS Many food processes are associated with a kinetic aspect, an attribute that changes with time (e.g., microbial activity, active components of disinfectants in cool rooms, color changes), and the resulting kinetic reaction must be solved. Therefore, the reactions rates, property changes, and heat releases must be calculated as a part of the solution. Consider the following reaction AþB!C 1

where the reaction rate Rc (mol s ) is defined

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(1:10)

Rc ¼ 

d d d [A] ¼  [B] ¼ [C] ¼ kf [A]n [B]m [C]o  kb [A]p [B]q [C]r dt dt dt

(1:11)

with kf the forward rate constant and kb the backward rate constant. The rate constants can be modeled by the following Arrhenius-like expression E b R



kf ,b ¼ aT e

1 1 T Tref

 (1:12)

where a and b are the empirical constants E is the empirical activation energy Tref is the reference temperature The heat of reaction can be calculated from the heats of formation of the species and depends on temperature. The reaction leads to sources=sinks in the conservation and energy equations. The most widely studied kinetics is that of enzymes. They are involved in many aspects of food quality, catalyzing the complex underlying biochemical reactions that result in quality changes in such attributes as taste, odor, color, and many more. In its simplest form, in an enzymatic reaction the substrate (S) is converted into a product (P) with the help of enzyme (E): E

S ! P

(1:13)

The rate of reaction rP can be expressed in terms of either the change of substrate concentration, CS or the product concentration, CP rP ¼ 

dCS dCP ¼ dt dt

(1:14)

It is important to know how the reaction rate is influenced by reaction conditions such as substrate, product, and enzyme concentration if you want to understand the effectiveness and characteristics of an enzymatic reaction. If the initial reaction at different levels of substrate and enzyme concentrations are measured, we often obtain a series of characteristic curves, where the reaction rate is proportional to the substrate concentration (first-order reaction) at low values of substrate concentration, and does not depend on the substrate concentration (zero-order reaction) at high values of substrate concentration which means the reaction goes gradually from first- to zero-order as the concentration of the substrate is increased. The maximum reaction rate, Vmax is proportional to the enzyme concentration. This was what Henri observed in 1902, and he proposed the following rate equation

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rP ¼

Vmax CS KM þ CS

(1:15)

where Vmax (mol m3 s1) KM (mol m3) are kinetic parameters which need to be experimentally determined KM is the substrate concentration required for an enzyme to reach half of its maximum velocity. This equation describes many experimental results well. A quantitative theory exists to support the observed enzyme kinetics and is still widely used today under the name Michaelis–Menten kinetics.

1.3.2 CASE STUDY: MODEL

FOR THE

RESPIRATION

OF

FRUIT

Respiration is one of the most important processes in fruits. Extended Michaelis– Menten kinetics is widely used as a semiempirical model to describe the relationship of the respiration to the O2 and CO2 concentration, and the whole respiration pathway is assumed to be determined by one rate-limiting enzymatic reaction (Chevillotte, 1973). A noncompetitive inhibition model (Peppelenbos et al., 1996; Chang, 1981; Lammertyn et al., 2001) can be used to describe consumption of O2 by respiration as formulated by RO2 ¼ 

(Km,O2

Vm,O2  PO2   PCO2 þ PO2 )  1 þ Kmn,CO

(1:16)

2

where Vm,O2 (mol m3 s1) is the maximum oxygen consumption rate P (kPa) is the partial pressure for O2 and CO2 Km (kPa) is the Michaelis–Menten constant for O2 consumption and noncompetitive CO2 inhibition RO2 (mol m3 s1) is the O2 consumption rate of the sample The equation for production rate of CO2 consists of an oxidative respiration part and a fermentative part (Peppelenbos et al., 1996) RCO2 ¼ rq,ox  RO2 þ 

Vm, f ,CO2



P

1 þ Km,Of ,O2

(1:17)

2

where Vm,f,CO2 (mol m3 s1) is the maximum fermentative CO2 production rate Km,f,O2 (kPa) is the Michaelis–Menten constant of O2 inhibition on fermentative CO2 production rq,ox is the respiration quotient at high O2 partial pressure RCO2 (mol m3 s1) is the CO2 production rate of the sample

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The effect of temperature was described by Arrhenius’ law (Hertog et al., 1998) 
 Ea,VmO2 1 1  Vm,O2 ¼ Vm,O2 ,ref exp R Tref T 
 Ea,Vmf CO2 1 1  Vm, f ,CO2 ¼ Vm, f ,CO2 ,ref exp R Tref T

(1:18) (1:19)

where Vm,O2,ref and Vm, f,CO2,ref (mol m3s1) are the maximal O2 consumption and maximal fermentative CO2 production rate at Tref ¼ 2938K, respectively Ea,Vm (kJ mol1) is the activation energy for O2 consumption and fermentative CO2 production Typical respiration rates of pear tissue are given in Figure 1.2. The estimated parameters for Vm,O2 and Vm,f,CO2 of cortex tissue were (2.39  0.14)  104 mol m3 s1 and (1.61  0.13)  104 mol m3 s1, respectively. Km,O2, a measure for the saturation of respiration with respect to O2 was relatively small and equal to (1.00  0.23) kPa. A significant but low inhibition effect of CO2 on O2 consumption of pear cortex tissue was found (Kmn,CO2 ¼ 66.4  21.3 kPa). The respiration quotient rq,ox was 0.97  0.04 and showed that the O2 consumption was about the same as the oxidative CO2 production. The value Km,f,O2 is a measure of the extent to which fermentation can be inhibited by O2. The estimated value of 0.28  0.14 kPa implies that fermentation was already inhibited at very low levels of O2 concentration.

⫻10⫺4

RCO2 (mol m⫺3 s⫺1)

RO2 (mol m⫺3 s⫺1)

⫻10⫺4

2

1

2

1

0

0 0

5 O2 partial pressure (kPa)

10

0

2 4 6 8 O2 partial pressure (kPa)

10

FIGURE 1.2 O2 consumption and CO2 production rate in pear tissue disks at 208C; Solid lines (—) and dashed lines (- -) indicate the respiration model at 0 and 10 kPa CO2 while the symbols () and (o) indicate the experiment at 0 and 10 kPa CO2. (Adapted from Ho, Q.T. et al., PLoS. Comput. Biol., 4, e1000023, 2008.)

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1.4 MODEL PARAMETERS 1.4.1 THERMOPHYSICAL PROPERTIES The thermophysical properties k, r, and c may be temperature dependent (due to insufficient problem decomposition, i.e., the underlying physicochemical changes are not modeled explicitly) so that the problem becomes nonlinear. Thermophysical properties of various agricultural and food products are compiled in various reference books (e.g., the compilation by ASHRAE). Further, equations have been published, which relate the thermophysical properties of agricultural products and food materials to their chemical composition. In general, both heat capacity and density can be calculated with sufficient accuracy, but the models for thermal conductivity require some assumptions about the orientation of different main chemical constituents with respect to the direction of heat flow. Determination of material properties, including diffusivity of certain components, is a task that often requires experiments because mostly one is interested in diffusivity as an apparent property for a particular material in particular conditions, and one cannot rely on fundamental equations to calculate the properties. For the case study under consideration here, one should measure gas concentration as a function of time and space, to which the mass transfer model is fitted by optimizing the apparent diffusivity value. This is usually achieved by an iterative least squares procedure. Although one can carefully design experiments to improve the quality of the fitting, in many cases, variability due to the food material composition and structure must be investigated by, e.g., analysis of variance to reveal significant effects (Ho et al., 2006a,b). Estimated diffusivities of pear tissue are given in Table 1.1. High variation of the estimated value was found in the measurement. Lowest diffusivity was reported for the skin, and anisotropic diffusivity was found in the axial and radial directions. The higher diffusivity in the axial direction compared to that along the radial direction is probably due to the fact that vascular bundles may be not fully filled with sap during storage of the fruit and facilitate gas exchange. Higher diffusivity of CO2 compared to O2 and N2 is probably due to the larger solubility of CO2 in water than that of O2 and N2. In addition, while O2 and N2 would be transported mostly through the apoplast, CO2 would also diffuse through the cytoplasm.

1.4.2 KINETICS PARAMETERS The kinetics parameters may be determined by fitting the proposed kinetics model to the experimental data of the observed changes (e.g., quality changes) using a nonlinear regression program (Ho et al., 2008). For example in the case study of respiration of pears, the data on O2 consumption and CO2 production rates are pooled, and the same weight can be attributed to both gases. Accuracy of the estimated kinetics parameters reflecting the variability and experimental error structure can be expressed by confidence intervals, and asymptotic confidence intervals can be calculated from the asymptotic covariance matrix C of the parameters C ¼ (JT  J)1 s2

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(1:20)

TABLE 1.1 Gas Transport Properties of Pear Tissue Estimated Values Based on Tissue Measurement

Diffusivity (m2 s1)

(1.86  0.70)  1010 a (5.06  3.3)  1010 a (1.06  0.29)  1010 b (2.8  1.59)  1010 b (2.32  0.41)  109 a (2.67  1.62)  1010 b (1.10  0.40)  109 b (6.97  2.19)  109 a (1.06  0.66)  109 b

DO2,skin DCO2,skin DN2,skin DO2,r DCO2,r DN2,r DO2,z DCO2,z DN2,z

Note: 95% confidence limits. Indices skin, r, and z refer to the position of the skin, along the radial direction and along the vertical axis of pear, respectively. a Indicate values measured by Ho, Q.T. et al., Post. Biol. Tech., 41, 113, 2006a. b Indicate values measured by Ho, Q.T. et al., J. Exp. Bot., 57, 4215, 2006b.

where J is the Jacobian matrix with respect to the estimated parameters s2 is the mean squared error The asymptotic (1  a)% confidence interval on the ith parameter estimate Pi was calculated from   pffiffiffiffiffiffiffi a Pi  t 1  , n  p  Ci,i (1:21) 2 where t is the Student t-distribution n is the number of measurements p is the number of parameters Ci,i is the ith diagonal element of C Correlation coefficients ri,j of estimated model parameters i and j indicate the strength and direction of the relationship between estimated model parameters i and j. These coefficients can be computed from Ci,j ri, j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ci,i  Cj, j

(1:22)

The correlation coefficients of kinetics parameters of pear tissue respiration from Figure 1.2 are given in Table 1.2. The correlation coefficients are all smaller than 0.71 suggesting that the model is not over-parameterized. ß 2008 by Taylor & Francis Group, LLC.

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TABLE 1.2 Correlation Coefficient Table of Estimated Parameters on Pear Tissue Respiration Parameters

Vm,O2,tissue

Km,O2

Kmn,CO2

rq,ox

Km,f,O2

Vm,f,CO2,tissue

Ea,VmO2

Ea,VmfCO2

Vm,O2 ,tissue Km,O2 Kmn,CO2 rq,ox Km,f,O2 Vm,f ,CO2 ,tissue Ea,VmO2 Ea,VmfCO2

1 0.70 0.50 0.56 0.38 0.02 0.20 0.01

0.70 1 0.23 0.26 0.58 0.03 0.12 0.01

0.50 0.22 1 0.02 0.04 0.00 0.10 0.00

0.56 0.26 0.02 1 0.43 0.05 0.11 0.02

0.38 0.58 0.04 0.43 1 0.30 0.06 0.10

0.02 0.03 0.00 0.05 0.30 1 0.00 0.34

0.20 0.12 0.10 0.11 0.06 0.00 1 0.00

0.01 0.01 0.00 0.02 0.10 0.34 0.00 1

1.5 SOLUTION METHODS 1.5.1 GENERAL CONSIDERATIONS Unless simplifications are made, the models presented in this chapter cannot be solved by analytical means. In many industrial applications, however, simplifications are very well possible. At a first simplification level, the geometry can be simplified. For simple shapes such as cylinders, spheres, and blocks one can find, with certain conditions on the model parameters, (usually they have to be constants!), analytical solutions as a function of time and spatial coordinates. At a second level, overall balances can sometimes be made excluding the spatial dimension. One then typically ends up with ordinary differential equations that can be solved quite efficiently with the latest numerical solvers. At the last simplification level, one is also able to exclude the time dimension and overall balances leading to the algebraic equations. If one does have to rely on numerical means to solve mathematical models, there are a number of efficient methods available. For this purpose the problem is first reduced significantly by requiring a solution only for a discrete number of points (the so-called grid) rather than for each point of the space-time continuum through which the heat and mass transfer proceed. The original governing partial differential equations are accordingly transformed into a system of difference equations and solved by simple mathematical manipulations such as addition, subtraction, multiplication, and division, which can easily be automated using a computer program. However, as a consequence of the discretization, the obtained solution is no longer exact, but only an approximation of the exact solution. Fortunately, the approximation error can be decreased substantially by increasing the number of discretization points at the expense of additional computing time. Various discretization methods have been used in the past for the numerical solution of heat and mass transfer problems arising in food technology. Among the most commonly used are finite difference method, finite element method, and finite volume method. It must be emphasized that, particularly in the case of nonlinear heat transfer problems, the numerical solution must always be validated. It is very well possible that a plausible, convergent but incorrect solution is obtained. At least a grid dependency study must be carried out to verify whether the solution basically remains the same when the computational grid is refined.

1.5.2 CASE STUDY: SOLUTION OF THREE-DIMENSIONAL GAS EXCHANGE AND RESPIRATION IN PEAR FRUIT Numerical solution can be applied to solve the governing partial differential equations of heat and mass transfer using the finite element method. In the axi-symmetric case study, 2719 quadratic finite elements with triangular shape were used and required less than 5 min of CPU time on a desktop PC. For the mass transfer model involving a kinetic reaction term, for example consumption of O2 inside the fruit, mathematical equation of the reaction may not exclude negative concentrations. Numerical problems may then be expected when the concentration approaches zero resulting in nonphysical negative results. Here is an example of two alternative approaches to solve the problem for O2 exchange in intact fruit involving respiration:

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1. To ensure that the O2 concentration cannot become negative due to O2 consumption (RO2  0), the respiration term in the permeation–diffusion reaction model was modified for O2 and CO2 in the solution. If CO2,g < 0 then RO2 ¼ 0 and RCO2 ¼ Vm,f,CO2. If CO2,g  0 then RO2 and RCO2 are described by their original equations above. Analytically, there is no O2 consumption when O2 reaches zero. Therefore, the O2 concentration should never become negative. The solution, therefore, will be physically consistent. 2. Another method was based on the exponential transformation of the variable in the model equations in such a way that the solution is guaranteed to be positive. For example, exponential transformation of the main variable was used to impose positive values for the O2 concentration. CO2 ,g ¼ exp (UO2 )

(1:23)

Hence, the mass transfer equation for O2 transforms into @ UO2 þ r  (u  exp (UO2 )) @t  exp (UO2 ))rUO2 þ RO2

(exp (UO2 )  aO2 ) ¼ r  (DO2

(1:24)

At the boundary UO2,r ¼ ln (CO2,1). Similarly, exponential transformation of the O2 concentration was applied in the other equations. Both methods avoided nonrealistic errors in the computations of gas exchange in fruit (Figure 1.1).

1.6 TOWARDS FOOD PROCESS MODELING AT DIFFERENT SCALES: MULTISCALE MODELING Many problems related to mathematical modeling of foods and food processes is the poor understanding of the microscopic and nanoscopic mechanisms that affect the macroscopic behavior of the food or process that is being modeled. As a consequence, apparent material properties that have to be expressed in complex equations in relation to other variables are used, and sophisticated experiments to find these relationships are required. Hence, more then often the validity range is quite limited, and variability is large. For the case of biological materials like the pear fruit, the macroscopic properties likely depend on various microscopic histological and cellular features such as tissue types, geometric properties of the cell, presence of an adhesive middle lamella between individual cells, cellular water potential, mechanical properties of the cell wall, presence of intercellular spaces, and many more. These features cover a wide range of spatial scales, from nanoscopic (plasmodesmata, plasma membranes), over microscopic (cell wall–middle lamella complex, cell geometry), to macroscopic

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(actual geometry of the material). The material properties of the continuum model, such as diffusion properties incorporate both actual physical material constants such as the diffusivity of water and air but also the microscale geometry of the tissue and intracellular space (Mebatsion et al., 2008). Multiscale models are basically a hierarchy of submodels, which describe the material behavior at different spatial scales in such a way that the submodels are interconnected. As a result, investigation of the microstructure becomes a prerequisite to understand transitional theoretical frameworks and modeling techniques to bridge the gap between length scale extremes. Multiscale modeling may involve challenging physical processes such as transport phenomena. Sometimes it is sufficient to find the solution of the coarser scale by including procedures to construct the equations on the coarser scale that account for the contribution of finer scales. However, this amounts to writing effective equations for the macroscale that account for lower scales, and it is a difficult task. Alternatively, equations for the fine scale itself can be solved. The up-scaling of fine scale solutions to a macroscale solution is known as homogenization. Homogenization has been defined as a collection of methods for extracting or constructing equations for the coarse scale (macroscale) behavior of materials and systems, which incorporate many smaller (nano-, micro- and meso-) scales. The main objective of such an approach is to use simpler fine scale equations that are considerably less expensive to solve, and whose solutions have the same coarse scale properties. While still in its infancy, multiscale modeling in food applications could have a large contribution to a better understanding of the complex food composition and behavior of foods in industrial processes.

1.7 CONCLUSION Basic mathematical models for describing food processes in terms of the transport phenomena and kinetic changes taking place were outlined, and that the main problem for mathematical modeling of food processes is poor understanding of the biochemical and microscopic mechanisms that cause macroscopic changes in appearance and appreciation of foods was demonstrated. As a result, considerable uncertainty in food process simulation and optimization was to be managed.

ACKNOWLEDGMENTS The authors wish to acknowledge financial support by the Flanders Fund for Scientific Research (FWO-Vlaanderen) (project G.0200.02) and the K.U. Leuven (project IDO=00=008 and OT 04=31, IRO PhD scholarship for Q.T. Ho). Pieter Verboven is Fellow of the Industrial Research Fund at the K.U. Leuven.

NOMENCLATURE c fi D

Specific heat External body forces, including the gravitational force Apparent diffusion coefficient, diffusivity

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J kg1 8C1 N m3 m2 s1

H Hi k K P Q ra R Ri T u ui (i ¼ 1, 2, 3) Xa t

Static enthalpy Henry’s constant of component I Thermal conductivity Permeation coefficient Pressure Heat source or sink Source or sink of component a of the material Unversal gas constant Production term of the gas component Temperature Apparent velocity vector Cartesian components of the velocity vector U(u1, u2, u3) Mass fraction of a component a of the material Time

J kg1 mol m3 kPa1 W m1 8C1 m2 Pa W m3 kg m3 s1 8.314 J mol1 K1 mol m3 s1 8C, K m s1 m s1

s

GREEK LETTERS m, h r

Viscosity Density

Pa s kg m3

REFERENCES Chang, R., Physical Chemistry with Applications to Biological Systems, MacMillan Publishers, New York, 1981. Chevillotte, P., Relation between the reaction cytochrome oxidase-oxygen and oxygen uptake in cells in vivo, J. Theor. Biol., 39, 277, 1973. Geankoplis, J.C., Transport Processes and Unit Operations, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1993. Hertog, M.L.A. et al., A dynamic and generic model on the gas exchange of respiring produce: the effects of oxygen, carbon dioxide and temperature, Post. Biol. Tech., 14, 335, 1998. Ho, Q.T. et al., Gas diffusion properties at different positions in the pear, Post. Biol. Tech., 41, 113, 2006a. Ho, Q.T. et al., A permeation–diffusion–reaction model of gas transport in cellular tissue of plant materials, J. Exp. Bot., 57, 4215, 2006b. Ho, Q.T. et al., A continuum model for metabolic gas exchange in pear fruit, PLoS Comput. Biol., 4(3): e1000023, 2008. doi: 10.1371=journal.pcbi.1000023. Lammertyn, J., Comparative study of the O2, CO2 and temperature effect on respiration between ‘‘Conference’’ pear cells in suspension and intact pears, J. Exp. Bot., 52, 1769, 2001. Mebatsion, H.K. et al., Modelling fruit (micro)structures, why and how, Trends Food Sci. Tech., 19, 59, 2008. Peppelenbos, H.W. et al., Modelling oxidative and fermentative carbon dioxide production of fruit and vegetables, Post. Biol. Tech., 9, 283,1996.

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2

Analytical Solutions in Conduction Heat Transfer Problems Ferruh Erdo gdu and Mahir Turhan

CONTENTS 2.1 Introduction .................................................................................................. 19 2.2 Analytical Solutions..................................................................................... 21 2.3 Application and Use of Analytical Solutions .............................................. 26 2.4 Conclusion ................................................................................................... 27 Nomenclature .......................................................................................................... 27 Greek Letters................................................................................................ 28 Subscripts ..................................................................................................... 28 References ............................................................................................................... 28

2.1 INTRODUCTION In process optimization studies, the first step generally would be to have a mathematical model for the given process. Exact solutions of the differential equations describing the process and numerical solutions (finite difference and finite element solutions) are preferred for this objective. Since all the variables affecting the process and the physical–chemical changes occurring in the medium can be defined in a numerical model, applying these solutions in the optimization models would result in longer run-time solutions. Therefore, exact solutions are sometimes preferred for testing the models for convergence of the optimization algorithms. In this chapter, the exact solutions, mostly preferred in the food engineering literature, for the conduction heat transfer problems are reviewed. A general heat transfer problem encountered in food process engineering area is to determine the steady or unsteady (transient) state temperature distribution in solid food products where the initial temperature distribution and the boundary conditions are specified. An unsteady temperature distribution includes not only the temperature variation from point to point in the medium but also with time (Kakac and Yener, 1993). This problem includes finding exact solution of the governing diffusion equation for different geometries or different coordinate systems. The simplest

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case of diffusion equation with constant and isotropic thermophysical properties is given as follows: r2 T ¼

1 @T  a @t

(2:1)

where a (thermal diffusivity) is given by a¼

k r  cp

(2:2)

The Laplacian (r2) of temperature in various coordinate systems is as follows (Kakac and Yener, 1993): Rectangular (Cartesian): r2 T ¼

@2T @2T @2T þ þ @x2 @y2 @z2

(2:3)

Cylindrical:
 1 @ @T 1 @2T @2T r þ 2 2þ 2 rT¼  r @r @r r @f @z 2

(2:4)

Spherical: r2 T ¼


   1 @ 1 @ @T 1 @2T 2 @T 2     )  r þ (1  m þ r 2 @r @r r 2 @m @m r 2  (1  m2 ) @f2

(2:5)

where m ¼ cos u. Physical significance of thermal diffusivity is associated with the speed of heat propagation into the solid product (Ozisik, 1993). The higher the thermal diffusivity the faster the heat transfer rate is the general belief for thermal diffusivity. In a recent publication, Palazoglu (2006) reported that the speed of heat penetration was a function of thermal diffusivity and heat transfer coefficient combination rather than the thermal diffusivity itself. The exact solution, called analytical solution, of Equation 2.1 for Cartesian (for infinite and finite slab shaped geometries), cylindrical (for infinite and finite cylinder geometries), and spherical (for sphere) coordinate systems are generally used in the literature to especially verify numerical solutions and to develop numerical schemes and grid generation methods (Cai et al., 2006). The first step in obtaining the analytical solution is to choose the orthogonal coordinate system of which the coordinate surfaces will be coinciding with the boundary surfaces of the solid product (Ozisik, 1993). For example, Cartesian coordinate system is used for rectangular bodies while cylindrical coordinate system is used for cylinder shapes and spherical coordinate system is used for sphere shapes.

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To simplify the analytical solutions, generally one-dimensional solution representing an infinite slab for Cartesian coordinates, an infinite cylinder for cylindrical coordinates, and sphere for spherical coordinates are applied assuming all the surfaces facing equivalent boundary conditions. An additional solution for the given coordinates requiring the temperature change in only one dimension might be a semi-infinite solution approach. Lumped system solution is another methodology independent of any coordinate system. Even though the situations described by analytical solutions represent a small proportion in heat transfer analysis, these solutions can be applied on the basis leading to check the solutions given, for example, by numerical solutions. Therefore, the objective of this chapter is to summarize the usefulness of analytical solutions with their applications in variety of situations encountered in food processing.

2.2 ANALYTICAL SOLUTIONS The exact solutions for the given cases above are generally called analytical solutions, and they play a significant role in heat transfer simulations for design and optimization purposes where the solid food products can be approximated by regular shapes of slab, cylinder, or sphere (Ramaswamy et al., 1982). The solutions are available in the literature to obtain the transient temperature distribution in such shaped foods. These solutions are obtained using some analytical techniques including Laplace transform and the method of separation of variables. Separation of variables has been widely used in solving the heat conduction problems where the homogeneous equation systems are readily handled (Ozisik, 1993). This method is based on expanding a function in terms of Fourier series. In this method, the dependent variable (temperature, T in the given cases here) is assumed to be the product of independent variables (location, x and time, t). This method is applied when the governing equation and differential equations representing the boundary and initial conditions are homogeneous and linear. For the cases of nonhomogeneous conditions where there is more than one nonhomogeneous condition, the superposition techniques are applied to split the problem into simpler problems (Cengel, 2007). One typical use of analytical solutions is to validate numerical solutions where the constant thermophysical (thermal conductivity, specific heat and density) and constant boundary conditions with uniform initial temperature distribution are used and to develop numerical schemes and grid generation methods (Cai et al., 2006). Analytical solutions give a concise parametric solution in ideal problems of academic interest or to check numerical simulations as explained above. The analytical solutions are generally restricted by the following assumptions: . .

Solid object conforming to a regular geometry (i.e., slab, cylinder, or sphere) with the exception of lumped system analysis Constant thermal properties and physical dimensions

If the initial temperature distribution and surrounding medium temperatures are not constant and are given by certain functions, an analytical solution can still be obtained, but the results might be complex compared to the constant temperature

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cases. This statement is also true when any of the thermal properties is not constant (e.g., when the thermal conductivity is a function of temperature). In a more general format for one-dimensional heat transfer, the Equation 2.1 can be modified for infinite slab (n ¼ 0), infinite cylinder (n ¼ 1), and sphere (n ¼ 2) geometries as follows:
 1 @T 1 @ n @T  ¼ n x  (2:6) a @t x @x @x Equation 2.6 can then be written for a one-dimensional heat transfer in an infinite cylinder and sphere, respectively 1 @T 1 @T @ 2 T  ¼  þ a @t x @x @x2

(2:7)

1 @T 2 @T @ 2 T  ¼  þ a @t x @x @x2

(2:8)

The case of @T @t ¼ 0 is the simplest approach in a heat transfer analysis leading to the steady state condition where the knowledge of two boundary (since a double integration in the space variable is involved) is required. When the steady-state assumption is not valid, integration in time with requirement of an initial condition must be used. Boundary conditions across the surfaces usually encountered in conduction heat transfer are prescribed surface temperature, prescribed heat flux, and convection boundary: . . .

Prescribed surface temperature: Surface temperature as constant or function of space and time Prescribed heat flux: Heat flux across the boundaries specified at constant or as a function of space and time Convection boundary condition: Equivalence of convection over the boundary to the conduction towards the solid geometry  @T  ¼ h  (Tjs  T1 ) (2:9) k  @x s

The analytical solutions are obtained using convection boundary present across the surface boundary and a symmetry condition at the center (special case of the prescribed heat flux where q00 and therefore dT dx is equal to 0) with a constant uniform initial temperature distribution by applying separation of variables solution methodology to Equations 2.3 through 2.5 1 h  T(x,t)  T1 X a  t i ¼ Cn (x) exp m2n 2 Ti  T1 L n¼1

(2:10)

where mn and Cn(x) and are given for regular shaped geometries of slab, cylinder, and sphere, respectively

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Slab: NBi ¼ m  tan (m) Cn (x) ¼

(2:11)

 2  sin (mn ) x  cos mn  mn þ sin (mn )  cos (mn ) L

(2:12)

J1 (m) J0 (m)

(2:13)

 2  J (m ) x 2 1 n 2

 J0 m n  L mi  J0 (mn ) þ J1 (mn )

(2:14)

Cylinder: NBi ¼ m  Cn (x) ¼ Sphere: NBi ¼ 1 

m tan (m)

 2  [ sin (mn )  mn  cos (mn )] sin mn  Lx Cn (x) ¼  mn  Lx mn  sin (mn )  cos (mn )

(2:15) (2:16)

where J0 and J1 are the first kind, 0th and 1st order Bessel functions L is half-thickness for slab and radius for cylinder and sphere x is the distance from the center (0  x  L) NBi is the Biot number These equations were further simplified by Ramaswamy et al. (1982) to easily determine the transient temperature change in solid foods with convective boundary condition at the surface in the Biot number range of 0.02–200 with Fourier number being greater than 0.2. The reference by Carslaw and Jaeger (1959) is suggested for different situations where numerous initial and boundary conditions are applied. The given solutions are further simplified using the first term (C1) of the given  series where Fourier number NFo ¼ aL2t is greater than 0.2. There has been some argument in the literature on limiting value of NFo. McCabe et al. (1987) reported that only the first term of the series analytical solutions is significant and other terms can be neglected when the value of NFo is greater than about 0.1. Kee et al. (2002) stated that analytical solutions at the center can be approximated by an exponential decay when NFo is greater than 0.15. Since all the regular shaped geometries cannot be modeled in a one-dimensional way (an infinite plate, infinite cylinder and sphere), combinations or intersections of the given geometries can be modeled for determining the temperature changes in two- or three-dimensional geometries. For the case of two- (finite cylinder) and

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three-dimensional (finite slab) geometries, given solutions can be combined using the superimposition technique (Newman, 1936) 

T  T1 Ti  T1





finite slab

¼

T  T1 Ti  T1



    T  T1 T  T1   slab,depth Ti  T1 slab,width Ti  T1 slab,height (2:17)



T  T1 Ti  T1





T  T1 ¼ T i  T1 finite cylinder





T  T1  T i  T1 cylinder

 (2:18) slab

For the case of volume average temperature change, the problem becomes to determine the transient volume average temperature change in the solid body. Therefore, integrated throughout the whole  theÐ Vsolution given withÐ xEquation 2.7 is volume V1  0 T(x,t)  dV ¼ V1  0 T(x,t)  A  dx resulting in the Cn values as independent of the location Slab: 2 sin2 (mn )  mn mn þ sin (mn )  cos (mn )

(2:19)

4 2  J12 (mn )  m2n J02 (mn ) þ J12 (mn )

(2:20)

6 [ sin (mn )  mn  cos (mn )]2  m3n mn  sin (mn )  cos (mn )

(2:21)

Cn ¼ Cylinder:

Cn ¼ Sphere: Cn ¼

For this case, it should be noted that the measurement of average transient temperature change becomes a challenging task since the location of the average temperature might not be constant. In addition to these coordinate systems, the elliptical coordinate system is applied for the case of elliptical cylinders. Application of this latter case is rather limited in the literature. McLahlan (1945) first published the equations governing the heat conduction problem in an infinite elliptical cylinder subjected to a medium of infinite heat transfer coefficient. Kirkpartrick and Stokey (1959) described the numerical solution of McLahlan’s solution for required zeros of the modified Mathieu functions d2 y (solution of the differential equation dx 2 þ [a  2  q  cos (2x)]  y ¼ 0 that arises during the solution in elliptical coordinate system) necessary for solution in elliptical coordinate systems for different eccentricities from 0 (indicating an infinite circular

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cylinder) to 1 (indicating an infinite plate). The Laplacian (r2) in elliptical coordinate system is given by
2  1 @ T @2T þ r T¼ 2  @u2 @v2 a  ( sinh2 u þ sin2 v) 2

(2:22)

where x ¼ a  cosh u  cos x y ¼ a  sinh u  sin v For different geometries of anomalous shapes, Smith et al. (1968) presented a unified system of charts and graphs for use over a wide range of conditions. The basis of the system was on the concept of representing a given shape by a geometry index, and equations, based in the first term of the infinite series solutions for transient conduction heat transfer, given to determine these parameters. The analytical solutions for slab, cylinder, and sphere have also been reduced to relatively simple charts (Heisler charts), where the center temperature ratio is plotted as a function of NFo and NBi. These charts, in any heat transfer book, are given in the NFo range of 0–100 and NBi range of 0.01–1. Schneider (1963) also presents a total of 120 charts covering temperature response of different simple geometries (semiinfinite solids, one- and two-dimensional slabs, cylinder, cylindrical cavity and shells, sphere, spherical cavity and shells, and ellipse and ellipsoids) under a variety of boundary conditions and for a wide range of Fourier and Biot numbers. Other than these given analytical solutions, another solution is obtained when a semi-infinite body (a body with infinite depth, width, and length) assumption is held for the heat transfer medium. It is generally accepted that the semi-infinite medium approach would hold with the following criteria is satisfied (Hagen, 1999) L0 

pffiffiffiffiffiffiffiffi at

(2:23)

where L0 is the thickness of the given semi-infinite region. Solution for semi-infinite medium approach with a very high heat transfer coefficient assumption is given as
 T(x,t)  Ts x pffiffiffiffiffiffiffiffi (2:24) ¼ erf Ti  Ts 2 at ‘‘erf’’ is the error function. Under some special circumstances of conduction heat transfer, spatial temperature distribution within the solid during processing can also be ignored. Lumped system approach provides a great simplification in the conduction heat transfer analysis (Ozisik, 1993) even though the range of applicability is restricted. For this approach to hold, thermal conductivity value of the solid must be really high to enable the uniform temperature distribution inside the body


 hL !0 lim NBi ¼ lim k!1 k!1 k

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(2:25)

With this condition, temperature would only vary with time, and this situation is called lumped system methodology. After applying an energy balance for this condition to a given geometry @T hA  (T  T1 ) ¼ @t r  V  cp

(2:26)

For this case, only initial condition is required. When Equation 2.26 is solved by applying the uniform initial temperature condition (T(t ¼ 0) ¼ Ti)   T  T1 hA ¼ exp   t ¼ exp [NBi  NFo ] Ti  T1 r  V  cp

(2:27)

Validity of lumped system solution holds when NBi  0.1 where the characteristic dimension is used as VA . Lumped system analysis is commonly used to experimentally determine the convective heat transfer coefficient (Erdogdu et al., 1998; Erdogdu, 2005). For the cases of nonuniform initial or medium temperature cases, analytical solutions can be obtained using the separation of variables technique, but results will definitely be complicated compared to the given cases. Analytical solutions for infinite slab, infinite cylinder, and sphere were also reduced to simple to use charts where the temperature ratio at the center was plotted as a function of Fourier and Biot numbers. These charts, in any heat transfer book, are given in the Fourier number range of 0–100 and Biot number range of 0.01–1. Unlike conduction, convection heat transfer is extremely difficult to obtain simple to use analytical solutions since simultaneous solution for energy, momentum, and continuity equations are required. Cai and Zhang (2003) gave a detailed explanation for explicit analytical solutions of two-dimensional laminar natural convection along a vertical porous plate and between two vertical plates. Complexity of these solutions obviously increases in cylindrical and spherical coordinates, and generally a computational fluid dynamics (CFD) numerical solution is preferred.

2.3 APPLICATION AND USE OF ANALYTICAL SOLUTIONS A major concern in the use of analytical solutions is to decide on the number of terms of the infinite series to obtain a correct a solution. A general approach is to apply only the  first  term when  the Fourier number is greater than 0.2 where the temperature ratio ln

T(x,t)  T1 T i  T1

is accepted to be linear. As long as thermal diffusivity for the  case of heat transfer, slope of the temperature ratio versus time curve m21 La2 may be easily used to determine the thermal diffusivity value with the knowledge of heat transfer coefficient of the heat transfer medium. The knowledge of heat transfer coefficient (with the thermal conductivity of the product) leads to knowing the m1 value where it can be used to determine the thermal diffusivity. As it can be realized from the slope’s not being a function of location, this method will not require the knowledge of location in the material where temperature change is recorded when

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the heat transfer coefficient is known. At any location, after a certain amount of processing time, slope of the temperature ratio would be the same (Erdogdu, 2005). The heat transfer coefficient, on the other hand, can be determined from the intercept of the temperature ratio curve where the intercept is a function of location and m value. The known m value leads to Biot number (Equations 2.11, 2.13, and 2.15 depending on the geometry) and therefore the heat transfer coefficient assuming thermal conductivity of the solid object is known. Erdogdu (2005) gives a detailed analysis of this situation. Yıldız et al. (2007) determined the effective heat (heat transfer coefficient) and mass transfer (mass transfer and diffusion coefficient) coefficients during frying of two-dimensional finite slab shaped potato slices. In this study, Equation 2.17 was applied (accepting the third dimension is infinitely long, and the temperature or mass ratio is 1) assuming the linear portion of the temperature and mass concentration curves started when the Fourier number was greater than 0.1. The results and conclusions of this study deserved to be mention since the analytical solutions were used when the phase change was also present in the heat transfer medium. Turhan and Erdo gdu (2003, 2004) and Erdogdu and Turhan (2006) used the analytical solutions for finite and infinite slab and cylinders to demonstrate the accumulated errors for assumptions of approximation a finite geometry as an infinite one during unsteady-state heat and mass transfer processes. These assumptions were based on that area of one of the surfaces if too large compared to the area of the other surfaces of a given geometry. The results of these studies concluded the significant error accumulations were functions of Biot and Fourier numbers. The lumped system methodology is another preferred method to determine the heat transfer coefficient using a highly conductive material (e.g., copper, aluminum, etc.) of the same shape with the solid object. In addition, Anderson and Singh (2002) used the semi-infinite medium approach to determine the heat transfer coefficient solving an inverse heat conduction problem.

2.4 CONCLUSION Analytical solutions of the diffusion equation play a significant role in heat transfer simulations for design and optimization purposes where the solid food products can be approximated by a regular shape to obtain the transient temperature distribution in such food products. Another typical use of analytical solutions is to validate the numerical solutions and to develop numerical schemes and grid generation methods. In addition, they give a concise parametric solution in ideal problems of academic interest or to check numerical simulations. They are also useful for applications in the optimization algorithms instead of numerical solutions to test the convergence of the algorithm and to save the processing time.

NOMENCLATURE A cp h J0

Surface area Specific heat Convective heat transfer coefficient 1st kind 0th order Bessel function

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m2 J=kg-K W=m2-K

J1 k NBi NFo L L0 r t T V x

1st kind 1st order Bessel function Thermal conductivity Biot number Fourier number Half-thickness of a slab or radius of a cylinder or sphere Characteristic dimension in lumped system analysis (V=A) Thickness of a semi-infinite region Radial distance, distance from the center Time Temperature Volume Distance from the center

W=m-K

m m m m s 8C, K m3 m

GREEK LETTERS a m f r u

Thermal diffusivity Root of the characteristic equation of infinite slab, infinite cylinder or sphere Azimuthal angle in spherical coordinates Density Radial direction in cylindrical coordinates and polar angle in spherical coordinates

m2=s

kg=m3

SUBSCRIPTS i s s 1

Initial Surface Surface boundary Medium

REFERENCES Anderson, B. and Singh, R.P., Air impingement heat transfer on a cylindrically shaped object. IFT Annual Meeting and Food Expo. Presentation no: 91C-22, 2002. Cai, R., Gou, C., and Li, H., Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction. Int. J. Thermal Sci., 45, 893, 2006. Cai, R. and Zhang, N., Explicit analytical solutions of 2-D laminar natural convection. Int. J. Heat Mass Transfer, 26, 931, 2003. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids. Oxford University Press, London UK, 1959. Cengel, Y., Heat Transfer: A Practical Approach. McGraw Hill Inc., New York, 2007. Erdogdu, F. and Turhan, M., Analysis of dimensional ratios of regular geometries for infintie geometry assumptions in conduction heat transfer problems. J. Food Eng., 77, 818, 2006. Erdogdu, F., Mathematical approaches for use of analytical solutions in experimental determination of heat and mass transfer parameters. J. Food Eng., 68, 233, 2005. Erdogdu, F., Balaban, M.O., and Chau, K.V., Automation of heat transfer coefficient determination: development of a Windows-based software tool. Food Tech. Turkey, 10, 66, 1998.

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Hagen, K.D., Heat Transfer with Applications, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1999. Kakac, S. and Yener, Y., Heat Conduction. Taylor & Francis, Washington DC, 1993. Kee, W.L., Ma, S., and Wilson, D.I., Thermal diffusivity measurements of petfood. Int. J. Food Prop., 5, 145, 2002. Kirkpatrick, E.T. and Stokey, W.F., Transient heat conduction in elliptical plate and cylinders. J. Heat Transfer, 80, 54, 1959. McCabe, W.L., Smith, J.C., and Harriot, P., Unit Operations in Chemical Engineering. 4th edn., p. 268, McGraw-Hill, Inc., New York, 1987. McLahlan, N.W., Heat conduction in elliptical cylinder and an analogous electromagnetic problem, Philosophical Magazine, 36, 600, 1945. Newman, A.B., Heating and cooling rectangular and cylindrical solids, Ind. Eng. Chem., 28, 545, 1936. Ozisik, M.N., Heat Conduction. John Wiley and Sons, Inc. New York, 1993. Palazoglu, T.K., Influence of convective heat transfer coefficient on the heating rate of materials with different thermal diffusivities, J. Food Eng., 73, 290, 2006. Ramaswamy, H.S., Lo, K.V., and Tung, M.A., Simplified equations for transient temperatures in conductive foods with convective heat transfer at the surface, J. Food Sci., 47, 2042, 1982. Schneider, P.J., Temperature Response Charts. John Wiley and Sons, Inc. New York, 1963. Smith, R.E., Nelson, G.L., and Henrickson, R.L., Applications of geometry analysis of anomalous shapes to problems in transient heat transfer. ASAE Transactions, 2, 296, 1968. Turhan, M. and Erdogdu, F., Error associated with assuming a finite regular geometry as an infinite one for modeling of transient heat and mass transfer processes. J. Food Eng., 59, 291, 2003. Turhan, M. and Erdo gdu, F., Errors based on location and volume average changes in infinite geometry assumptions in heat and mass transfer process. J. Food Eng., 64, 199, 2004. Yıldız, A., Palazoglu, T.K., and Erdogdu, F., Determination of heat and mass transfer parameters during frying of potato slices. J. Food Eng., 79, 11, 2007.

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3

Numerical Solutions: Finite Difference Methods T. Koray Palazo glu and Ferruh Erdo gdu

CONTENTS 3.1 3.2

Introduction .................................................................................................. 31 Finite Difference Modeling: Explicit and Implicit Methods ....................... 32 3.2.1 Standard Explicit Method ................................................................. 32 3.2.2 Discretization of the Computational Domain and Node Generation ........................................................................ 33 3.2.3 Initial and Boundary Conditions ....................................................... 34 3.2.4 Energy Balance Equations ................................................................ 35 3.2.5 Simplifying Assumptions .................................................................. 37 3.2.6 Stability and Convergence ................................................................ 39 3.2.7 Numerical versus Analytical Solution: Model Validation ................ 39 3.2.8 Alternating Direction Implicit Method.............................................. 41 3.3 Conclusion ................................................................................................... 47 Nomenclature .......................................................................................................... 47 Greek Letters................................................................................................ 47 References ............................................................................................................... 47

3.1 INTRODUCTION Mathematical modeling is a very useful tool for studying the effect of process variables on the safety and quality-related attributes of food products. Computer simulation of food processing operations has become a popular approach since it allows identifying the critical processing parameters without having to carry out numerous experiments. By mathematical modeling, considerable time, money, and effort can be saved during the stages of process design and optimization. A mathematical model is a representation of the relationship between the parameters of a physical system. Mathematical models may be simple or complex and are always abstractions of the real world leaving out some details. After all, the goal in mathematical modeling is to develop a model that is as simple as possible yet accurate enough to predict the expected behavior of a system. Therefore, a mathematical model is a tool for assessing the effect of critical processing variables.

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Thermal processing is a common method of ensuring microbial safety, inactivating enzymes, and improving eating quality of foods; and it may involve transport phenomena such as heat, mass, and momentum transfer. There is a direct relationship between these transport phenomena and the safety and quality of products. While heat and momentum transfers take place simultaneously in some thermal processes (e.g., retorting, aseptic processing), some thermal processes involve simultaneous heat and mass transfer (e.g., frying, baking, roasting). Modeling of these processes has been performed in the interest of ensuring microbial safety and improving product quality. However, due to the lack of engineering knowledge, some of these processes have been traditionally developed empirically. Mathematical modeling will be better appreciated as a powerful tool for process design and optimization once engineering knowledge about these processes is generated. All the transport phenomena mentioned above are governed by differential equations which can be solved analytically or numerically. However, analytical solutions for heat transfer problems (Chapter 2) are available only for simple geometries (infinite slab, infinite cylinder, and sphere). Although solutions can be obtained for finite slab and finite cylinder geometries by using superimposition, availability of analytical solutions is still very limited. Numerical methods are useful when there is no analytical solution available for a particular problem. They have become popular for their flexibility in handling irregular geometries, complex boundary conditions, temperature dependent properties, phase change, etc. However, it is to be noted that the numerical solutions are not exact but only approximations to exact solutions.

3.2 FINITE DIFFERENCE MODELING: EXPLICIT AND IMPLICIT METHODS Finite difference and finite element methods are widely used for solving partial differential equations of heat, mass, and momentum transfer numerically. Although finite element method provides more flexibility for modeling irregular geometries, finite difference methods are often the method of choice due to their simplicity to formulate when the problem involves a simple (regular) geometry. Finite difference modeling is, therefore, also easier to learn and apply.

3.2.1 STANDARD EXPLICIT METHOD One of the most frequently used finite difference methods is the standard explicit method which is also known as the forward difference method. In this method, temperature change over the time interval from n to n þ 1 is based on the heat flow rates at time step n. Time interval (Dt), however, must be sufficiently small for this assumption to hold, since for large time step values, the standard explicit method violates the second law of thermodynamics. The standard implicit method (backward difference method), on the other hand, is based on the heat flow rates at time step n þ 1, eliminating the possibility of this violation. Although not placing a bound on the time step, the standard implicit method involves simultaneous solution of a set of equations. This significantly increases the number of calculations required per time

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step, especially for two- and three-dimensional problems (Clausing, 1969). The standard explicit, therefore, remains to be the simplest finite difference scheme. In standard explicit method, every new temperature at a new time step (t þ Dt) is computed explicitly from the known values of temperature at a previous time t. However, the explicit scheme becomes unstable if the following condition is not satisfied for each node. rcp Vi X 1  Dt Ri

(3:1)

where Ri represents the resistance to heat transfer for node i. There are two thermal resistance terms for each node in the case of one-dimensional modeling since heat propagates only in one direction. The number of thermal resistance terms for twoand three-dimensional problems are four and six, respectively. To avoid a potential violation of the second law of thermodynamics, the smallest Dt that satisfies the above condition (Equation 3.1) should be regarded as the upper limit for the selection of incremental time step.

3.2.2 DISCRETIZATION OF THE COMPUTATIONAL DOMAIN AND NODE GENERATION The first step in numerical modeling is discretization of the computational domain. The basic concept of domain discretization is the division of the geometric object into differential elements. The number of differential elements is determined based on the required accuracy of numerical solution, and the domain is finally represented by a set of nodes. Although the computation time increases with increasing number of nodes, this has become less of an issue with today’s fast computing capabilities. However, it should be noted that the number of nodes would play a role in selection of the incremental time step. A sphere is the simplest geometry, and thereby was used for illustration purposes in this section. To illustrate the numerical procedure, let us consider heating of a spherical body as a result of a convective boundary condition imposed at the surface. This procedure, however, very well applies to other geometric objects (cube, cylinder, etc.). For spherical objects (one-dimensional), the domain is typically divided into concentric spherical shells. There are two approaches in node generation: capacitance and noncapacitance nodes at the surface. In the capacitance surface node model, all the concentric shells, except for the outermost shell, are of the same thickness (Dr). The thickness of the outermost shell is half of that of the inner shells (Dr=2) making the distances between all nodes equal over the computational domain where the node in each differential element is located at the midpoint between its two surfaces. Discretization of a spherical body into differential elements with a capacitance node at the surface is shown in Figure 3.1, where Dr ¼ i maxR 1. One disadvantage of the capacitance surface node approach was reported by Chau and Gaffney (1990) to be that temperature of the surface node cannot be representative of the temperature of the differential element it is associated with. Another problem arises when heat transfer coefficient increases. For large values of heat

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R ⌬r ⌬r/2

FIGURE 3.1 Discretization of a spherical body into differential elements with a capacitance surface node (shaded volume represents a differential element).

transfer coefficient, time step becomes restrictively small (Chau and Gaffney, 1990). These problems can be minimized by increasing the number of differential elements (thereby reducing the thickness of surface differential element which would provide a better representation of its temperature by the surface node) or by selecting a very small incremental time step with both at the cost of added computational time. Nevertheless, a model that provides a better approximation of surface temperature is needed when the effect of process parameters on surfacerelated phenomena such as texture and color formation are of interest. Alternatively, a more accurate approach to discretizing was proposed by Chau and Gaffney (1990) in which the surface node had no differential volume element associated with it. Since this surface node had no mass it cannot store energy, it was called noncapacitance surface node (Figure 3.2). Noncapacitance surface node approach provides a better approximation of surface temperature with less computation time. Furthermore, the time step is bounded and does not indefinitely decrease with increasing heat transfer coefficient (Chau and Gaffney, 1990). For this case, the distance between the nodes becomes Dr ¼ i maxR0:5.

3.2.3 INITIAL

AND

BOUNDARY CONDITIONS

A boundary condition is said to be the first kind (Dirichlet boundary condition) when the boundary is maintained at a prescribed temperature (e.g., constant surface temperature). The second kind boundary condition, which is called Neumann boundary condition, represents a prescribed heat flux (the simplest case would be constant heat flux) along the boundary. Another boundary condition that is often encountered in heating=cooling applications is called the third kind boundary

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R ⌬r

FIGURE 3.2 Discretization of a spherical body into differential elements with a noncapacitance surface node (shaded volume represents a differential element).

condition (Robin boundary condition), and it involves a convective heat transfer between the surface and the surrounding medium (Luikov, 1968). Such boundary condition exists in many food processing applications. In baking, for example, the product is heated convectionally by the surrounding hot air. Cooling of the baked product also takes place by convection as a result of the product losing heat to the surrounding cooler air. Similarly, in deep-fat frying, heat is transferred by convection from the hot oil to the product. In retort processing, the heat transfer medium is either steam or water, again providing a convective boundary condition at the surface of the packaged product. Thermal processing of solid–liquid mixtures (multiphase foods) also involves exchange of heat by convection between the solid and liquid portions. Heat transfer coefficient (h), associated with the convective boundary condition is a measure of the resistance to heat transfer between the surface and the ambient. It should, however, be noted that for a very large value of heat transfer coefficient (h ! 1) this resistance becomes negligible, and the boundary condition of third kind is reduced to the first kind boundary condition.

3.2.4 ENERGY BALANCE EQUATIONS It is a common practice in numerical modeling to use indices when setting up energy balance equations. In this chapter, the symbol i is used as an index for node numbering while imax specifies the total number of nodes, and the superscript n denotes the time step. Change in internal energy over the time interval from nDt to (n þ 1)Dt can be based on the heat flow rates at time step nDt (standard explicit method). In order for this assumption to be valid, the time step must approach zero.

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Heat transfer in a spherical isotropic body is one-dimensional (in the radial direction) provided that the initial temperature distribution is uniform and the boundary conditions are the same over the entire surface. Energy balance equation for each differential element with a convective boundary condition imposed at the surface is given below. As one can see, the energy balance equations, generated assuming that all the energy is transferred into the center node from the surrounding nodes, for capacitance and noncapacitance surface node approaches are identical, except for the surface node. Capacitance surface node (Figure 3.3):

Tinþ1 max

¼

" T1  Tinmax

(3:2)

# T1  Tinmax Tinmax1  Tinmax Dt þ þ 1 Dr Dr rcp Vi max hAi max þ 2kAint kAi max1

(3:3)

1

þ

Tinmax1  Tinmax

#

Dt rcp Vi max

þ

Tinmax

hAi max

Dr

kAi max1

Noncapacitance surface node (Figure 3.4): "

Tinþ1 max

¼

Tinmax

T⬁ h

R 1 V1 A1

⌬r

i

imax−2 A imax−2 A imax−1 A imax

Vi max−2

⌬r /2

imax−1 i max

Vimax−1 Vimax

Surface node

FIGURE 3.3 Node generation in a spherical body (capacitance surface node).

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T⬁ h R

1 V1 A1

⌬r

i

imax−2 Aimax−2

Vimax−2 Vimax−1

Aimax−1

i max−1 imax

Vimax

Aint Aimax

Surface node

FIGURE 3.4 Node generation in a spherical body (noncapacitance surface node).

Interior nodes: " Tinþ1

¼

Tin

þ

n Tiþ1  Tin Dr kAi

þ

n Ti1  Tin Dr kAi1

#

Dt rcp Vi

(3:4)

Center node: T1nþ1 ¼ T1n þ

(T2n  T1n ) Dr kA1

Dt rcp V1

(3:5)

3.2.5 SIMPLIFYING ASSUMPTIONS The goal in developing a mathematical model is not necessarily to describe your system perfectly since such a model would involve too many parameters to be useful. Rather, the goal is to develop as simple a model as possible that comes close to making accurate predictions of the physical system. Therefore, one must make simplifying assumptions when constructing a numerical model. Assumptions make a problem more managable by allowing working with a less number of parameters. It is, however, to be noted that one should only make the appropriate assumptions to avoid oversimplifying the problem.

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Reducing the problem from multidimensional to a one-dimensional one greatly simplifies the numerical solution. If a three-dimensional regular geometry (e.g., finite plate) can be assumed as infinite in one direction, the problem is reduced to twodimensional. Further simplification is possible if the thickness in two directions is significantly larger than that in the third direction. Then, the problem becomes onedimensional with heat transfer taking place only in one direction. In other words, heat transfer into the body in the directions of larger dimensions through smaller surface areas is assumed negligible. Similarly, assumption of the finite cylinder geometry as an infinite one (a disk or a cylindrical rod depending on whether the diameter or the height is the larger dimension) reduces the heat transfer problem, which generally is two-dimensional (in radial and vertical directions), to a onedimensional one. An important question may arise: What is the criterion for assuming a finite geometry infinite? A detailed discussion on the errors associated with assuming a finite geometry as an infinite one is presented in Turhan and Erdogdu (2003). They constructed a chart that can be used to determine the error imposed by assuming a finite plate or rod as an infinite one for heat and mass transfer processes. Their results indicate that there is no clear-cut value for the ratio of the dimensions for an object to be regarded as infinite, and the error is dependent on Fourier and Biot numbers and geometrical properties. It should be noted that this simplification is not always desirable. For example, if the temperature at or near the whole surface of the object is of interest (in order to model formation of surface-related attributes such as texture, color, etc.), such simplification should be avoided even if the finite object qualifies geometry-wise to be assumed as an infinite one. A very common assumption made during numerical solution of transient heat conduction problems is that the thermophysical properties (density, thermal conductivity, and specific heat) are assumed constant. Thermophysical properties of materials are indeed a function of temperature and expected to change during a thermal process. In some cases, however, although these properties change, when there is negligible change in thermal diffusivity, the outcome of the solution is not affected. Symmetry can also be used when appropriate to reduce the computation time. If all material properties are isotropic and the same boundary condition exists at all surfaces, the symmetry condition can be assumed at the center of geometry, reducing the number of nodes and hence computational work significantly. It would be possible to model only 1=8 of the whole volume cutting the domain in half in all three directions (x, y, and z) for a three-dimensional geometry. In many thermal processes, volume changes lead to shrinkage or expansion of the product. While expansion of the product is desirable in some cases (e.g., baking of cakes, etc.), shrinkage is usually neither desirable nor avoidable (e.g., hot air drying of vegetables, etc.). Volume change makes it difficult to model the thermal process by introducing one more parameter to be considered to the model. Whether it is shrinkage or expansion, if volume change is negligible it may not affect the solution. However, when there is considerable change in volume, it should be taken into account in the model by representing it as a function of temperature, as this would improve the accuracy of the model. For example, Erdogdu et al. (1998) applied the shrinkage in the noncapacitance finite difference model when modeling the heat transfer during shrimp cooking. For this purpose, the equation developed to

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predict the shrinkage was used in the models at each time step after the shrinkage started by regenerating the computation domain with the new physical dimensions. Demirkol et al. (2006), on the other hand, included the volume increases of the cookies in their capacitance finite difference model used to determine the effective heat transfer coefficient values during baking.

3.2.6 STABILITY

AND

CONVERGENCE

Different types of errors are involved in solving differential equations numerically. The main sources of errors in a typical finite difference solution are round-off, truncation, and discretization errors. To understand the significance of the errors associated with finite difference modeling as well as the issues of stability, convergence, and consistency, a brief discussion is presented here. Numerical solution of transient heat transfer problems (heating and cooling) is constructed step-by-step, in a forward-marching manner with respect to time. This introduces the danger of instability with the possibility that the round-off error (numerical error) may grow as solution progresses. Round-off error results from rounding off of real numbers to represent them in ‘‘floating point’’ form. The solution is said to be unstable if the errors introduced at one time step grows unboundedly at later times; conversely, the solution is stable if these errors die out (Clausing, 1969). The error introduced by truncating terms in the finite difference formulation of derivatives with Taylor’s series expansion is called the truncation error. In other words, truncation error is the difference between the exact solution of a differential equation and its finite difference solution without the round-off error (Özıs¸ık, 1994). Discretization error, however, is caused by transforming the continuous problem into a discrete one and represents the errors due to the treatment of boundary conditions plus the truncation error. Consistency is the property that as spacial and time increments approach zero, the truncation error due to the finite difference approximation must also approach zero. A finite difference representation is said to be consistent if the truncation errors tend to zero with increasing number of differential elements. Consistency and stability are necessary conditions for convergence, and a numerical method is convergent if the numerical solution approaches the exact solution as both time and space increments are reduced.

3.2.7 NUMERICAL

VERSUS

ANALYTICAL SOLUTION: MODEL VALIDATION

Although numerical models have the capability of solving complex problems, their performance often needs to be validated against experimental observations or analytical solutions. Experimental validation of numerical procedures is not only time consuming but can also be expensive. Analytical solutions, on the other hand, provide a more convenient and cost-effective means for evaluating the accuracy of numerical models. In order to test the accuracy of the numerical scheme developed in this section, a simple heat transfer problem will be solved both numerically and analytically. Consider a spherical object of which the center and surface temperatures, after imposing a convective boundary condition at the surface, are to be

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TABLE 3.1 Values of Parameters Used in the Example Parameter Radius (m) Density (kg=m3) Thermal conductivity (W=m K) Specific heat (J=kg K) Uniform initial temperature (8C) Heat transfer coefficient (W=m2 K) Ambient temperature (8C) Heating time (s)

Value 0.01 1000 0.6 3600 20 200 100 180

determined at time t. The parameters needed to solve such a heat transfer problem are listed in Table 3.1. The first step will be to divide the spherical object into differential elements with a noncapacitance node at the surface. Then, the problem will be solved by employing first 10 and then 20 nodes to see the effect of the number of differential elements on the outcome of the numerical solution. Since the numerical method used for illustration purposes is an explicit scheme, care needs to be taken in selection of the time step to ensure stability. The upper limit for Dt was determined to be 1.10 and 0.26 s for 10- and 20-node model, respectively. An incremental time step of 0.1 s, therefore, provides a stable solution in both cases. Analytical solution of the problem can be obtained by using the one-dimensional heat conduction equation for a sphere presented by Luikov (1968). " #   1 T(r,t)  T1 X 2  ( sin mn  mn  cos mn ) sin mn  Rr 2 ¼   r   exp (mn  Fo) Ti  T1 mn  sin mn  cos mn mn R n¼1 (3:6) where mn ¼ tan mn(1  Bi). Equation 3.6 reduces to the following for the center of sphere:  1  T(0,t)  T1 X 2  ( sin mn  mn  cos mn ) 2 ¼  exp (mn  Fo) Ti  T1 mn  sin mn  cos mn n¼1

(3:7)

While the above infinite series equation yields the exact solution, for the values of the variables given in Table 3.2, the first six terms of the series provided sufficiently accurate results. Table 3.2 presents the center and surface temperature values obtained numerically and analytically. Close agreement between the numerical and analytical values can be seen from this table. One can also see that numerical results approach the analytical solution as the number of nodes is increased. Although numerical modeling offers the flexibility of handling complex problems, analytical solutions are still valuable for checking the accuracy of numerical

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TABLE 3.2 Comparison of Numerical and Analytical Results Temperature (8C) Location Center Surface a

Analytical

Numerical, imax ¼ 10

Numerical, imax ¼ 20

a

74.79 92.37

74.76 (0.04)a 91.70 (0.73)a

74.53 (0.35) 90.89 (1.60)a

Numbers in parenthesis represent % deviation from analytical solution.

procedures. If analytical solution for a specific problem is not available, then one needs to resort to experimental results for validation of the numerical solution. In Section 3.2.8, alternating direction implicit (ADI) method, an unconditionally stable finite difference method, is presented. In the ADI method, heat flow in one direction is based on one time step whereas heat flow in the other direction is based on the next time step. It, therefore, applies to problems in which heat transfer takes place in more than one direction.

3.2.8 ALTERNATING DIRECTION IMPLICIT METHOD The standard explicit method, explained in Section 3.2.1, is computationally simple since there is only one unknown (the temperature of the given node) at a given time step. At each time step, the unknown node temperature is easily determined as a function of the surrounding nodes’ temperatures at the previous time step. Despite its being computationally stable with properly applying the time step, this time step is restricted by the explained stability considerations as explained above. If the calculations are to be performed over a longer period, the number of calculations becomes large. To alleviate these difficulties, implicit methods that are not restricted by the time step have been developed (Özıs¸ık, 1994). In such implicit schemes, the unknown temperatures become the function of temperatures of the surrounding nodes at the given time step. One such efficient implicit scheme, called Crank–Nicolson method modifies energy balance equations where the unknown temperature at a given time step becomes the function of surrounding nodes’ temperatures at the given and previous time steps. For example, the Equation 3.4 becomes as follows in the Crank–Nicolson implicit scheme without any restriction on the size of the time step Dt for computations (Özıs¸ık, 1994): Tinþ1  Tin 1 1 ¼   Dt 2 rcp Vi þ

"

nþ1 Tiþ1  Tinþ1

n Tiþ1  Tin Dr kAi

Dr kAi

þ

þ

n Ti1  Tin Dr kAi1

nþ1 Ti1  Tinþ1

!#

!

Dr kAi1

(3:8)

As observed in this equation, number of unknowns at a given time step become five for a two-dimensional problem leading to the requirement of the solution of system

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of equations. Crank–Nicolson and other implicit methods are unconditionally stable, but the computations become enormous especially for two- and three-dimensional situations (Özıs¸ık, 1994). For example, a two-dimensional problem with N nodes in each direction has N2 nodes, and N2  N2 matrix solution must be accomplished at each time step. To alleviate this problem and simplify the computations, ADI methods have been developed (Özıs¸ık, 1994). The ADI method was first introduced by Peaceman and Rachford (1955) for solving parabolic and elliptical partial differential equations in two-dimensional geometry. The idea in ADI method is to split the time step into half. In the first half, the conduction term in one direction is treated implicitly and the other explicitly. This allows the solution to be obtained by solving the resulting tridiagonal matrix. In the next time step, the implicit and explicit treatment of the terms reversed. Even though splitting time step introduces some truncation error, the scheme is unconditionally stable for the heat conduction equation. In this section, the ADI method will be presented for a two-dimensional heat transfer problem where the temperature changes in a finite cylinder domain is solved. In fact, the heat transfer problem in a finite cylinder domain is three-dimensional problem. To reduce this problem to a two-dimensional one, the domain will be represented by concentric rings with a known thickness in the radial direction where each ring will be one node (Figure 3.5). With this way, the modeling can be applied in the one half of the cylinder. One can even plan on applying the numerical methodology in one quarter. When the finite difference equations (e.g., Equations 3.9 and 3.10 for an interior node) are derived and arranged for the concentric ring-shaped nodes, a set of simultaneous algebraic equations are obtained (Equation 3.11).

Δr

Δz

L j i R

FIGURE 3.5 Discretization of a finite cylinder into differential elements with noncapacitance surface nodes (shaded volume represents a differential element).

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0 1 nþ1 nþ1 n n Ti1,j  Ti,jnþ1 Tiþ1,j  Ti,jnþ1 Ti,j1 Ti,jnþ1  Ti,jn  Ti,jn Ti,jþ1  Ti,jn 1 A ¼ @ þ þ þ Dri1,j Dri,j Dzi,j1 Dzi,j Dt rcp Vi kAi1,j

kAi,j

kAZi,j1

kAZi,j

(3:9) where Dr and Dz are the distance between the nodes A and AZ are the areas of the nodes in the radial and longitudinal directions, respectively In this equation, explicit scheme is applied in the radial direction while the implicit scheme is in the longitudinal direction. In the next time step, when the explicit and implicit schemes are reversed: 0 1 nþ1 nþ1 nþ1 nþ2 nþ2 Tiþ1,j  Ti,jnþ1 Ti,j1  Ti,jnþ2 Ti,jþ1  Ti,jnþ2 Ti,jnþ2  Ti,jnþ1 1 @Ti1,j  Ti,j A ¼  þ þ þ Dri1,j Dri,j Dzi,j1 Dzi,j Dt rcp Vi kAi1,j

kAi,j

kAZi,j1

kAZi,j

(3:10) In the latter equation for the given time step ((nþ1)  Dt), T n values are known while the T nþ1 values are known for the time step ((nþ2)  Dt) for solving the obtained set of algebraic equations in each direction. The algebraic set of equations, in a simple way, can be given as follows: a11 T1 þ a12 T2 þ    þ a1N TN ¼ b1 a21 T1 þ a22 T2 þ    þ a2N TN ¼ b2  

 

(3:11)

aN1 T1 þ aN2 T2 þ    þ aNN TN ¼ bN When these equations are written in a matrix form: [A] [T] ¼ [B]

(3:12)

where [T] is the unknown temperature matrix (vector) [B] is the constant matrix (vector) [A] is the coefficient matrix The coefficient matrix [A] is a diagonal matrix where the diagonal elements are relatively larger compared to the off-diagonal elements enabling the use of iterative solution methods. For the given two-dimensional finite cylinder problem this diagonal matrix becomes a tridiagonal matrix and the Gauss elimination method can easily be applied

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for the solution of temperatures at each time step. For a given set of linear equations, the Gauss elimination method is applied as follows (Clausing, 1969): Consider the set of linear equations [A][T] ¼ [B] where 2

B1

6 6 A2 6 6 0 6 6 [A] ¼ 6  6 6  6 6 4 0 0

C1

0











B2

C2

0









A3 

B3 

C3 

0 

 

 

 

 

 

 

 

   AN1











 BN1



AN

0

3

7 0 7 7 0 7 7 7  7 7  7 7 7 CN1 5 NN

2

and

b1

3

7 6 6 b2 7 7 6 6  7 7 6 7 6 [B] ¼ 6  7 7 6 6  7 7 6 7 6 4 bN1 5 bN (3:13)

Then, the Gauss elimination method starts by dividing the first equation by B1 to reduce the coefficient of T1 to 1. This equation is then employed to eliminate T1 from the second equation. Next, the second equation is divided by resulting coefficient of T2. By continuing this procedure, the following equations are obtained (Clausing, 1969): C1 T2 ¼ F1 G1 C2 T2 þ T3 ¼ F2 G2 T1 þ

  TN1 þ

CN1 TN ¼ FN1 GN1 TN ¼ FN

where B1 ¼ G 1 B1 G1 CR1 DR ¼ GR1 F1 ¼

GR ¼ BR  AR DR FR ¼

bR  AR FR1 GR

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(3:14)

Proceeding in reverse order, by back-substitution, the following relations are obtained for determining the unknown temperature values starting with TN and proceeding in the reverse order: TN ¼ FN TR1 ¼ FR1  DR1 TR ,

(3:15)

1 1. The corresponding coded variable can be obtained by the following linear transformation: x¼

X  XC d

(6:2)

The resulted coded variable x has five levels as a, 1, 0, þ1, þa. As another example, for a natural variable with three symmetrically spaced levels and with the low level XL, high level XH, and center level XC ¼ (XL þ XH)=2, applying Equation 6.2 with d ¼ (XH  XL)=2 yields three levels for the coded variables: 1, 0, 1. Hereafter, unless otherwise stated, all independent variables are assumed in coded form, and Equation 6.1 can be expressed as y ¼ f (x1 , x2 , . . . , xk ) þ «

(6:3)

The true response function f, which is usually unknown, is frequently approximated by polynomial functions whose coefficients can be determined by the least squares method using data from a chosen set (decided by DOE) of independent variables and measured responses. The most common polynomial models used for response surface analysis are linear, also known as first-order model and the quadratic, also known as second-order model. For linear response functions with k independent variables and in particular k ¼ 2, the first-order models have the following forms: y ¼ b0 þ

k X

bi x i þ «

(6:4)

i¼1

y ¼ b0 þ b1 x1 þ b2 x2 þ «

(6:5)

where the b’s are unknown coefficients often called the regression coefficients to be determined by linear regression analysis. For second-order polynomial expressions with k independent variables and in particular k ¼ 2, the models are

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y ¼ b0 þ

k X

k 1 X k X

(6:6)

y ¼ b0 þ b1 x1 þ b2 x2 þ b12 x1 x2 þ b11 x21 þ b22 x22 þ «

(6:7)

i¼1 j¼iþ1

bij xi xj þ

k X

bii x2i þ «

i¼1

b i xi þ

i ¼1

In Equation 6.6, on the right hand side of the equation, the first two terms are those of the linear model, the third term represents interactions between the input factors, and the fourth term shows the quadratic effects of the independent variables on the model. There are (k þ 1)  (k þ 2)=2 regression coefficients in Equation 6.6, and they can also be determined by the linear regression method since the function is linear with respect to the coefficients.

6.3 DESIGN OF EXPERIMENTS A successful application of the response surface method is greatly dictated by a proper choice of sampling points in design space, i.e., design of experiments. In this subsection, DOE such as the two-level factorial design, central composite designs, face-centered central composite design, and the Box–Behnken design are briefly introduced. Theoretically, all the factors that affect the physicochemical properties of a final product should be included in the experimental design. However, if all the variables are included in the experimental design, the search process may become fastidious. Therefore, the potentially dominant parameters must be identified by a screening process to limit the number of experiments needed to a reasonable extent. The twolevel design (or the 2k design) with many factors, each at a ‘‘low’’ (or in coded variable xi ¼ 1) and a ‘‘high’’ level (or in coded variable xi ¼ 1) can be utilized to detect prominent trends in the input–output relationship and identify the dominant parameters in the process with a minimum number of experiments. As a result, the two-level design is often used at the early stage of experimentation as a screening process to determine potential candidate factors for further detailed investigation. Figure 6.3a and b shows two-level factorial designs with two factors (k ¼ 2) and with three factors (k ¼ 3), respectively. The dots in the figures correspond to the design points. The central composite design (CCD) is a popular class of the secondorder designs. This design involves the use of two-level factorial points, 2k and 2k axial points, and multiple center points, nc, for a total of 2k þ 2k þ nc design points with k being the number of input factors. Figure 6.3c and d illustrates CCDs with two and three factors. All axial points have a distance of a from the center point, i.e., all coded variables are zeros. The rotatability is a desirable property of a design that the variance of the predicted response is constant at all points equally distanced from the center of the The CCD can be made rotatable by setting the axial point pffiffidesign. ffi values as a ¼ k, and this design is often referred to as a spherical design. There are many practical circumstances that specific ranges or limits on the input factors have to be strictly imposed, and these limits cannot be exceeded. The design region constrained by this clause is a cube. A face-centered central composite

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(−1, 1)

(1, 1)

x2

x3 x1

(a)

(−1, −1)

(1, −1)

x2 x1

(b)

(0, a) (−1, 1)

x2

(1, 1)

(0, 0)

(−a, 0)

x1 (c)

(−1, −1)

(0, −a)

(−1, 1)

x2

x1 (e)

(1, −1)

(0, 1)

(0, 0)

(−1, 0)

(a , 0)

x2 x1

(d)

(1, 1)

(1, 0)

(−1, −1) (0, −1) (1, −1)

x3

x3

x3

x2 x1

(f)

x2 x1

(g)

FIGURE 6.3 Several commonly used designs: (a) and (b) two-level factorial designs with k ¼ 2 and k ¼ 3, (c) and (d) central composite designs with k ¼ 2 and k ¼ 3, (e) and (f) facecentered central composite designs with k ¼ 2 and k ¼ 3, and (g) Box–Behnken design with k ¼ 3.

design (FCCD) with its independent variables confined within upper and lower bounds has a cuboidal design region and belongs to the family of central composite designs by setting a ¼ 1. An FCCD also consists of 2k þ 2k þ nc design points. Figure 6.3e and f demonstrates FCCD with two and three independent variables.

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The Box–Behnken design (BBD, Box and Behnken, 1960) is an effective threelevel design based on the construction of balanced incomplete block designs, and is an important alternative to CCD and FCCD. In contrast with the CCD that requires five levels if a 6¼ 1, the BBD needs only three levels for each factor. There is no BBD for k ¼ 2 and Figure 6.3g shows the BBD with k ¼ 3. Since the design points lie on the edges of a cube rather than at the corners like those of FCCD, the BBD is spherical rather than cuboidal. The BBD with k ¼ 4 is known to be rotatable and the other BBDs are nearly rotatable.

6.4 LINEAR REGRESSION FOR BUILDING EMPIRICAL MODELS The unknown parameters, the b’s, in approximated response surfaces can be estimated by linear regression as long as the surfaces are linear functions of the unknowns, even for response functions with nonlinear terms in the input factors, such as Equations 6.6 and 6.7. For example, by setting x3 ¼ x1 x2 , x4 ¼ x21 , x5 ¼ x22 , b3 ¼ b12, b4 ¼ b11, and b5 ¼ b22, Equation 6.7 can be written as y ¼ b 0 þ b1 x 1 þ b 2 x 2 þ b 3 x 3 þ b4 x 4 þ b5 x 5 þ «

(6:8)

which is a linear regression model with five input factors and six unknown parameters. To determine the regression coefficients, the method of least squares is usually adopted. The method requires that the number of observations, denoted by n, on the response variable and the input factors is greater than (or at least equal to) k. Express the first-order model in Equation 6.4 in terms of the observations as follows: y ¼ Ax b þ «

(6:9)

where the bold-faced characters denote vectors or matrices:    yn  T

y ¼ ½ y1 y2 2 1 x11 6 1 x21 6 Ax ¼ 6 .. .. 4. .

x12 x22 .. .

1

xn2

xn1

(6:10) 3

   x1k    x2k 7 7 .. .. 7 . . 5    xnk

b ¼ ½ b0

b1

   b k T

(6:12)

« ¼ ½ «1

«2

   «n  T

(6:13)

where y1, y2, . . . , yn are the observations on response variable xij is the ith observation of the factor xj «1, «2, . . . , «n are the uncorrelated random variables with mean zero

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(6:11)

The length of the random error vector can be expressed as k « k¼ «T « ¼ (y  Ax b)T (y  Ax b) ¼ yT y  2bT ATx y þ bT ATx Ax b

(6:14)

Differentiate the length of the error vector with respect to the vector of regression parameters b, set the result equal to a zero vector of dimension (k þ 1) by 1, and denote the solution of the equation by b, which is also the least squares estimator of the regression coefficients to give 2ATx y þ 2ATx Ax b ¼ 0

(6:15)

which can be rearranged to yield:  b ¼ b^0

b^1

...

b^k

T

 1 ¼ ATx Ax ATx y

(6:16)

The fitted linear regression model can now be used to predict the response observations as ^ y ¼ Ax b

(6:17)

and the vector of residuals with elements e1, e2, . . . , en is defined as: e¼y^ y

(6:18)

The quality of the fitted model is crucially dictated by the choice of the design points, i.e., design of experiments.

6.5 ANALYSIS OF SECOND-ORDER RESPONSE SURFACES The second-order response function is of particular importance since it approximates most functions very well within a small enough domain. Using the least squares regression method to estimate the regression coefficients in a second-order response model gives the fitted model as ^y ¼ b^0 þ

k X

b^i xi þ

k1 X k X

i¼1

b^ij xi xj þ

i¼1 j¼iþ1

k X

b^ii x2i

(6:19)

i¼1

which can be written in matrix form ^y ¼ b^0 þ xT b þ xT B x

(6:20)

where xT ¼ ½ x1

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x2

   xk 

(6:21)

 T b ¼ b^1 b^2    2 b^11 b^12 =2    6  b^22 6 B¼6 .. 4 . symm:

b^k



3 b^1k =2 b^2k =2 7 7 .. 7 . 5 ^ bkk

(6:22)

(6:23)

To obtain the extremum (either maximum or minimum) of the quadratic function, differentiate Equation 6.20 with respect to x and set it to 0 to yield @^y ¼ b þ 2Bx ¼ 0 @x

(6:24)

which leads to the solution for x, denoted by xs, as xs ¼ B1 b=2

(6:25)

The point xs is the stationary point of the quadratic surface in k-dimensional space. The characteristics of the stationary point are determined by the signs of the eigenvalues of the matrix B. Assume that l1, l2, . . . , lk are the eigenvalues of B, then three scenarios exist: 1. If l1, l2, . . . , lk are all positive, the stationary point is a minimum. 2. If l1, l2, . . . , lk are all negative, the stationary point is a maximum. 3. If l1, l2, . . . , lk are of mixed signs, the stationary point is a saddle point, i.e., neither a maximum nor a minimum. For scenarios 1 and 2, the response surface around the stationary point is an elliptical system. Figure 6.4a shows the contour plot of a response surface with a stationary point of maximum response. For scenario 3, the response function around the stationary point is a hyperbolic system, and Figure 6.4b exemplifies such a case. If one of the eigenvalues becomes very small relative to the others, for scenarios 1 and 2, the ellipse-shaped contours are in turn significantly elongated along one of their principal axes. In the limiting case of one of the eigenvalues equal to zero, the surface contours become a system of parallel lines. The result of this limiting case is equally applicable to scenario 3. If the stationary point is inside the experimental region, the system is called a stationary ridge system, and Figure 6.4c illustrates such a system. In fact, the stationary ‘‘point’’ in a stationary ridge system is no longer a point; instead it is a line, along which the same maximum (or minimum) response is attained. If the stationary point of an elliptical system is not inside the experimental region, the response surface is called a rising ridge system for the case when the stationary point is a maximum, while it is called a falling ridge system when the stationary point is a minimum. Figure 6.4d shows a rising ridge system, whose stationary point lies outside the experimental region.

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700 620 740

540 460 380

770

770

−20

800 220 140

60

300 300

220 140

380

650 620 590

680

710

770

(a)

−2060

460 540

300 620

(b)

590 120 130 100 110 140 90

620 590

650

150 140 130 120 110 100 90

620

680

650 680

710

710 740

(c)

770 740

(d)

FIGURE 6.4 Contour plots for second-order response surfaces: (a) maximum surface, (b) saddle system, (c) stationary ridge, and (d) rising ridge.

6.6 ADEQUACY CHECKING FOR REGRESSION MODELS Once a response surface is successfully constructed by a set of design points and measured responses before proceeding to optimization of the function, the response model has to be checked for its adequacy. This section describes some of the most common tests for accessing the suitability of the fitted model. All these tests may be integrated into the analysis of variance (ANOVA) to examine the adequacy of the regression models.

6.6.1 TEST

FOR

SIGNIFICANCE

OF THE

LINEAR REGRESSION MODEL

Whether a linear relationship exists between the response variable and a subset of the input factors can be determined by the test for significance of regression, this test involves whether to reject or accept a null hypothesis stating that none of the regression variables (input factors) contributes significantly to the model, i.e., H0: b1 ¼ b2 ¼    ¼ bk ¼ 0. Rejection of the hypothesis leads to a conclusion that the response is linearly related to at least one of the factors. The total sum of squares (SST), regression sum of squares (SSR), and error sum of squares (SSE), respectively, are defined as

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SST ¼

n X

(yi  y)2

(6:26)

(^yi  y)2

(6:27)

(yi  ^yi )2

(6:28)

i¼1

SSR ¼

n X i¼1

SSE ¼

n X i¼1

where yi is the ith observation y is the average value of all observations ^yi is the predicted response n is the total number of observations The total sum of squares can be partitioned into a regression sum of squares and an error sum of squares: SST ¼ SSR þ SSE. Further, defining a statistic as follows: F0 ¼

SSR=k MSR ¼ SSE=(n  p) MSE

(6:29)

where p denotes the number of the regression coefficients MSR and MSE are called the regression and error mean squares, respectively Denoting g as the significance level of the test, if the computed value of the test statistic in Equation 6.29, f0, is greater than fg,k,np, whose value may be looked up from the tables of the F distribution or calculated by computer software, the hypothesis should be rejected, and a linear relationship between the response function and at least one of the input factors is confirmed.

6.6.2 TEST

FOR

SIGNIFICANCE

ON

SUBSETS

OF THE

REGRESSION COEFFICIENTS

Passing the test for significance of the full regression model mentioned above is often not enough to conclude that the model is appropriate. Testing on the individual terms is required to determine the significance of the linear, interaction, quadratic, and other terms in the regression model. Remember that a response function with nonlinear terms in the input factors can always be rewritten as the form of a standard linear regression model. The coefficient vector b in Equation 6.12 can be partitioned into two parts as follows: b ¼ ½ b1

b2 T

where b1 contains the coefficients of the terms to be tested for significance b2 represents the rest of the terms in the model

ß 2008 by Taylor & Francis Group, LLC.

(6:30)

Suppose b1 is a vector of dimension m  1, then b2 is a vector of dimension (p  m)1, and the terms to be tested involve independent variables x1, x2, . . . , xm. Now, given a model with b2 coefficients, to test the significance of adding x1, x2, . . . , xm terms to the model is to test the hypothesis H0: b1 ¼ 0, and an appropriate test statistic is (Montgomery and Runger, 1999) F0 ¼

SSR(b1 jb2 )=m MSE

(6:31)

where SSR(b1jb2) ¼ SSR(b)  SSR(b2). If the calculated value of the test statistic in Equation 6.31, f0, is greater than fg,m,np, the hypothesis H0 should be rejected. As a result, at least one of the input factors x1, x2, . . . , xm contributes significantly to the regression model. This significance test for a subset of the regression coefficients, also known as the partial F-test, can be used to determine the contribution of each input factor xi by treating it as the last variable to be added to the regression model, i.e., SSR(bijb0, b1, . . . , bi1, biþ1, . . . , bk), i ¼ 1, 2, . . . , k.

6.6.3 LACK-OF-FIT TEST The lack-of-fit test is used to check the integrity of the regression model and to determine if the order of the model is correct. The statistical hypothesis for the lack-of-fit test is that H0: the linear regression model is correct, and we can test this hypothesis by first splitting the error sum of squares into two portions: SSE ¼ SSEP þ SSEL, in which SSEP is the sum of squares due to pure error and SSEL is the sum of squares due to lack of fit of the model. To perform the lack-of-fit test, we must have at least one set of repeated observations on the response. Suppose we have q sets of repeated trials that contain r1, r2, . . . , rq observations. Then, the sum of squares due to pure error can be calculated by SSEP ¼

q X ri X

(yij  yi )2

(6:32)

i¼1 j¼1

where yij is the jth observation in the ith set containing repeated trials yi stands for the average value of all ri repeat observations, and there are n  q degrees of freedoms for SSEP The sum of squares due to lack of fit, having q  2 degrees of freedom, can now be computed by SSEL ¼ SSE  SSEP. Finally, the statistic for the lack-of-fit test is (Montgomery and Runger, 1999): F0 ¼

SSEL =(q  2) MSEL ¼ SSEP =(n  q) MSEP

(6:33)

If the computed value f0 is greater than fg,q2,nq, the hypothesis H0 should be rejected and a conclusion drawn that the regression model fails to adequately fit the

ß 2008 by Taylor & Francis Group, LLC.

data, which means the model should be abandoned and a more suitable model should be sought after. On the other hand, if the computed value leads to the acceptance of the hypothesis H0, the model is probably an appropriate one. However, it is a common practice to incorporate multiple tests, including those that will be discussed later, to strengthen the confidence in the adequacy of the model.

6.6.4 COEFFICIENTS

OF

MULTIPLE DETERMINATION

The coefficient of multiple determination R2 (Myers and Montgomery, 1995) is a measure of the amount of predictability for the response y by the fitted response model ^y, both evaluated using the independent variables x1, x2, . . . , xk, and the coefficient is defined as R2 ¼

SSR SSE ¼1 SST SST

(6:34)

R2 takes on a value between 0 and 1. A larger value of R2 does not necessarily indicate a closer fit of the approximation to the response since adding a variable will always raise the value of R2. The adjusted coefficient of multiple determination R2adj (Myers and Montgomery, 1995), which will not increase if an added variable is not statistically significant and therefore is a better indicator than R2, is defined as R2adj ¼ 1 

  SSE=(n  p) n1 ¼1 (1  R2 ) SST=(n  1) np

(6:35)

Finally, the coefficient of multiple determination for prediction can be defined as (Myers and Montgomery, 1995) R2pred ¼ 1 

PRESS SST

(6:36)

where PRESS ¼

n X i¼1

(yi  ^y(i) )2 ¼

n X

e2(i)

(6:37)

i¼1

and PRESS is the prediction error sum of squares ^y(i) denotes the predicted value of yi using the regression model that is fitted by omitting yi and applying the remaining n  1 observations e(i) is often called the ith PRESS residual The R2pred measures the ability of the regression model to predict new observations.

6.7 MULTIPLE RESPONSES In many instances, two or more responses from the process under investigation are interested and have to be considered simultaneously. Two different actions may be taken:

ß 2008 by Taylor & Francis Group, LLC.

1. Treat only one of the responses as the optimization objective function and the rest as the constraints, and then either use a graphical method that superimposes the contour plots for the objective function and the constraints or apply one of the constrained optimization techniques to find an optimal solution that satisfies all the requirements on the process. 2. Include all or some of the responses and form a single expression as the objective function and set the rest as the constraints. Then, solve the optimization problem by a constrained optimization method. In optimization terminology, the function to be maximized or minimized is called the objective function; while functions, either explicit or implicit, of the input factors or responses that define the feasible region of the optimization problem are known as the constraints. If a process optimization problem has more than two independent variables or more complex response constraints that need to be satisfied, more effective optimization techniques other than the graphical method are probably required. To integrate multiple responses into a single function, the concept of composite functions needs to be addressed. A composite function that is defined as a summation or a mathematical product of several functions has one of the following two forms: fcomp ¼

s X

mi (wi fi )

(6:38)

(wi fi )mi

(6:39)

i¼1

fcomp ¼

s Y i¼1

where s is the number of response functions included mi, i ¼ 1, 2, . . . , s are the weighing factors that satisfy 0  mi  1 and m1 þ m2 þ  þ ms ¼ 1 wi represent the scaling factors to scale all response functions to similar magnitudes fi denote the response functions The weighing factors, mi, can be so chosen that they reflect the relative importance of each response. As to the scaling factors, wi, one way of choosing them is to set wi ¼ 1=( fi)max where ( fi)max is a predicted, approximately, maximum value of fi. A composite function having the form of Equation 6.39 is preferable to Equation 6.38 since any variation in the responses of the mathematical product has more significant effects on the composite function. However, a composite function of the form of Equation 6.39 is in general no longer a simple polynomial function that has only one optimum, but a complex function with multiple optima, which are called local optima and the one of which surpassing the rest is called the global optimum. An alternative approach to apply Equation 6.39 is to replace the responses, fi, by the so-called desirability functions (Myers and Montgomery, 1995). There are three types of desirability functions that are commonly utilized, and they are chosen

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according to the nature of the optimization problem: to maximize the response, to minimize the response, or to attain an optimal target value.

6.8 OPTIMIZATION ON THE RESPONSE SURFACES The same optimization method can be applied to either maximize or minimize the objective function of an optimization problem without extra work since maximizing a response function f multiplied by 1, i.e., f, is equivalent to minimizing f. For a first- or second-order polynomial response surface whose input factors are not bounded by preset upper and lower limits, the optimization of this response can be easily accomplished by applying either the numerical method of steepest ascent or the analytical expression of Equation 6.25. Sometimes, however, predetermined limits on the independent variables have to be strictly satisfied. In this case, a constrained optimization method such as the quadratic programming (QP) method should be employed. Furthermore, if a composite function that consists of several response functions and has multiple local maxima needs to be optimized, a nonlinear constrained optimization technique may be needed. Even with this technique, a local instead of global maximum is more likely to be found. A local maximum usually corresponds to an improvement in the processing conditions, but if further improvement or the best result is to be achieved, a global optimization scheme that is capable of finding the global maximum has to be employed. Among all global optimization methods, the sequential quadratic programming (SQP) technique with multistarts and the genetic algorithms, along with a popular unconstrained method, the steepest ascent search, will be introduced shortly.

6.8.1 STEEPEST ASCENT SEARCH The steepest ascent search is the simplest gradient-based optimization method that utilizes the gradient information to dictate the search direction. The gradient of a function f(x1, x2, . . . , xk) is defined as rf (x1 , x2 , . . . , xk ) ¼ [@f =@x1

@f =@x2



@f =@xk ]T

(6:40)

which is a vector quantity representing the direction of the greatest increase for the function f. In a minimization problem, the search direction is replaced by rf , i.e., the direction of the greatest decrease, and the method is now called the steepest descent. Although the rationale behind the steepest ascent is very sound, the method is inefficient when the function to be maximized has long, narrow summits. For a single-response process optimization problem approximated by a first- or secondorder polynomial, the steepest ascent search is quite enough. The steepest ascent scheme, unless properly modified, can be categorized as one of the local optimization techniques that are capable of finding only local optima. For a process with multiple responses that are integrated to form a composite function, the scheme most often yields a result that can be further improved by a global optimization method. Consider the following function with four local optima, only one of which is the global one:

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5 x2

f (x 1,x 2)

10 12 10 8 6 4 2 0 10

0 –5

5 0 x 2 –5–10

–10

10 0 5 –10 –5 x 1

(a)

–10

–5

0 x1

(b)

5

10

FIGURE 6.5 (a) Three-dimensional surface and (b) contour plot of the four-peak function.

f (x1 , x2 ) ¼ 7eð0:07(x1 3)

2

þ0:07(x2 3)2 Þ

þ 9eð0:05(x1 4)

2

þ 8eð0:06(x1 þ5)

þ0:1(x2 þ7)2 Þ

2

þ0:06(x2 þ5)2 Þ

þ 10eð0:1(x1 þ6)

2

þ0:2(x2 6)2 Þ

(6:41)

Figure 6.5a shows a three-dimensional surface of this four-peak function, and Figure 6.5b plots its contours. The function has the global optimizer at (x1, x2) ¼ (6, 6) with the maximum response value of about 10.0182. The optimized result from a local optimization method is significantly influenced by the choice of the initial search point. Figure 6.6 demonstrates eight sets of searching processes that are produced by applying repeatedly the steepest ascent using eight different initial points, (12, 12), (0, 12), (12, 12), (12, 0), (12, 0), (12, 12), (0, 12), and (12, 12) leading to only once the global optimum on the upper left peak. In obtaining these results, the steepest ascent method was programmed and run under the Matlab environment (The Math Works, 2000). To increase the odds of

The only initial search point leading to the global optimum 10

x2

5 0 –5 –10 –10

–5

0 x1

5

10

FIGURE 6.6 Steepest ascent searching processes on the four-peak function using eight different initial points.

ß 2008 by Taylor & Francis Group, LLC.

locating the global optimum of a composite function, global optimization techniques should be used.

6.8.2 SEQUENTIAL QUADRATIC PROGRAMMING TECHNIQUE

WITH

MULTI-STARTS

A quadratic programming problem is an optimization problem involving a quadratic objective function and linear constraints. The SQP method represents the state of the art in nonlinear programming methods and can be used to solve a series of QP problems approximating the original nonlinear programming problem. The SQP technique is very powerful and efficient, and with some modifications it can also perform global optimizations (Chen, 2003). The basic scheme of an SQP technique can be expressed in the following steps (Reklaintis et al., 1983; Chen, 2003): Step Step Step Step

1: 2: 3: 4:

Set up and solve a QP subproblem, giving a search direction. Test for convergence, stop if it is satisfied. Step forward to a new point along the search direction. Update the Hessian matrix in QP and go to Step 1.

In order to search for the global optimum, the concept of multi-start global optimization procedure (Snyman and Fatti, 1987) may be combined with the SQP method. Let F* denote the global maximum and v be the number of sample points falling within the region of convergence of the current overall maximum F after u points have been sampled. Then, under statistically noninformative prior distribution, the probability that F be equal to F* satisfies the following relationship (Snyman and Fatti, 1987): Prob[F ¼ F*]  1 

(u þ 1)!(2u  n)! (2u þ 1)!(u  n)!

(6:42)

A global optimization program equipped with a multistart SQP technique was coded using the Matlab software to solve for the optimal solution of a practical example in a later section. The modified SQP with the multistart ability, which is capable of reaching the global optimum with great certainty, has been proved to be a very efficient method (Chen et al., 2004). The program generates a series of uniformly distributed random points for initial search, and then the SQP is applied to find the optimum based on each initial point. If the probability of locating the global optimum exceeds a preset value (99.99% in this example), the global optimum is considered to be found. Otherwise, the next random, initial point is generated and the SQP re-executed.

6.8.3 GENETIC ALGORITHMS Genetic algorithms (GAs) provide a very flexible framework and recently have been regarded as not only a global optimization method but also a multiobjective optimization method in various areas. Generally, the algorithms can be described as follows (Goldberg, 1989; Mitchell, 1998):

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1. Encoding: GA works with the coding of the parameters. The methods of parameter coding that have often been used are binary encodings and realvalued encodings. Binary encodings are most commonly used. An l-bit binary variable is used to represent one parameter xi, i ¼ 1, . . . , k, where k is the number of parameters. 2. Initial population: The population consists of N chromosomes (binary encoding GAs). The initial population of chromosomes is randomly generated. A chromosome represents a set of values for the input factors. 3. Selection for reproduction: Selection for reproduction is the operation that couples of chromosomes are selected from the current population. The chromosomes with higher fitness values are more likely to be selected to reproduce. 4. Crossover: Crossover allows us to generate new chromosomes starting from existing ones. This operator randomly chooses a locus and exchanges the subsequences after that locus between two chromosomes to create two offspring. The percentage of the population chromosomes that mates is called the crossover probability. 5. Mutation: Mutation operator alters some of the bits of randomly selected chromosomes with a probability equal to the mutation rate, which is usually very small. 6. New generation: After the mutations take place, a new generation of population has formed. The fitness value associated with each chromosome in this generation is calculated. 7. Termination criteria: The stopping criteria often used are the maximum number of generations, set at the beginning of the optimization process, and that there is no appreciable improvement in the highest fitness for a number of generations. With each generation, the population gets closer to an optimal solution. Steps 3 through 6 are repeated until one of the termination criteria is satisfied. A genetic algorithm was programmed using also the Matlab software to optimize the fourpeak function. Figure 6.7 shows the population distributions of the first, 50th, and 200th generations using the GA program with a population size of 10. At the 200th generation shown in Figure 6.7c, all population has reached the global optimum at

5

5

5

0

0

0

x2

10

x2

10

x2

10

−5

−5

−5

−10

−10

−10

−10 −5 (a)

0 x1

5

−10 −5

10 (b)

0 x1

5

−10 −5

10 (c)

0 x1

5

10

FIGURE 6.7 Population distributions of (a) the first, (b) the 50th, and (c) the 200th generations using the GA program with a population size of 10.

ß 2008 by Taylor & Francis Group, LLC.

(x1, x2) ¼ (6, 6). Although recently there have been researchers proclaimed the advantages of real-value encodings of GAs (Barrios et al., 2000; Bessaou and Siarry, 2001), binary encodings are still more popular for several reasons, one of which is that much of the existing GA theory is based on the assumption of fixed-length binary encodings, including the so-called the fundamental theorem of genetic algorithms or the Schema theorem (Holland, 1975; Goldberg, 1989). In this example, the binary encoding for the GA was adopted.

6.9 BRIEF REVIEW OF LITERATURE USING RSM IN FOOD ENGINEERING RSM has been very popular as a tool for optimization study in food engineering in recent years. This method was successfully employed for applications in developing probiotic candies with maximum viability using the multistart SQP (Chen et al., 2008), in optimizing the pistachio nut roasting process (Kahyaoglu, 2008), in maximizing the thermotolerance of Bifidobacterium bifidum in gellan-alginate microparticles (Chen et al., 2007), in searching for the maximum cell immobilization conditions for the production of palatinose (Mundra et al., 2007), in finding the optimum formulation of cassava cake (Gan et al., 2007), in optimizing the extraction process of crude polysaccharides from boat-fruited sterculia seeds (Wu et al., 2007), in finding the optimal combination of the coating materials for probiotic microcapsules (Chen et al., 2006), in obtaining the optimal manufacturing conditions of dairy tofu (Chen et al., 2005), in maximizing the viability of probiotics in a new fermented milk drink using genetic algorithms (Chen et al., 2003), in developing new edible gels with fibrinogen=plasma protein (Chen and Lin, 2002), and in finding the optimum producing conditions of the dairy product Kou Woan Lao (Weng et al., 2001). A recent review paper (Bas¸ and Boyaci, 2007) discussed the applications and limitations of RSM.

6.10 APPLICATION TO THE OPTIMAL PROCESSING CONDITIONS OF A NEW DAIRY PRODUCT The objective of this section is to demonstrate the application of RSM to optimize the manufacturing conditions of a new dairy product. The entire procedure includes (1) performing screening experiment and experimental design, (2) manufacturing functional cream according to the experimental design, (3) building the response surface model, (4) performing optimization, and (5) verifying the optimal manufacturing conditions. A practical example about developing a new functional fermented cream with high level of conjugated linoleic acids (CLA) by Lactococcus lactis IO-1 is given below.

6.10.1 PERFORMING SCREENING EXPERIMENT

AND

EXPERIMENTAL DESIGN

CLAs are a mixture of octadecadienoic acids that have been recently recognized as anticarcinogens. Preliminary results were obtained after a screening test revealing that the CLA level was affected by viabilities of L. lactis IO-1 during processing.

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TABLE 6.1 Natural and Coded Variables and Levels Levels Factor

Symbol

Coded

Natural

IMO

x1

FOS

x2

b-CD

x3

1 0 þ1 1 0 þ1 1 0 þ1

0.0 1.5 3.0 0.0 1.5 3.0 0.0 5.0 10.0

Note: b-CD, b-cyclodextrin; FOS, fructooligosaccharides; IMO, isomaltooligosaccharides.

Additional cholesterol removing agent b-cyclodextrin (b-CD) and two prebiotics, i.e., fructooligosaccharides (FOS) and isomaltooligosaccharides (IMO) influenced viabilities of L. lactis IO-1. These three concentrations were regarded as the independent variables, and therefore a three-variable BBD with five replicates at the center point (total 17 trials) was selected to build the response surface models. The coded and natural variables and their respective levels are given in Table 6.1. Manufacturing functional cream according to the experimental design: Functional cream were prepared according to the BBD by blending b-CD (0%, 5%, 10%) with cream (18% fat) at 408C under 1200 rpm to reduce the cholesterol level, and then the samples were mixed with prebiotics, FOS (0%  3%) and IMO (0%  3%) and fermented with L. lactis IO-1 at 378C for 12 h to increase the CLA level. The growth rate (log CFU=h) of L. lactis IO-1 was determined and defined as the response. Building the response surface model: Experimental data (Table 6.2) can be utilized to build mathematical models using linear, quadratic, and cubic functions by the least squares regression method after which the fitted functions are tested for adequacy using the ANOVA. Once an appropriate approximating model is derived, it can then be analyzed using various optimization techniques to determine the optimum conditions for the process. Model analysis, the lack-of-fit test, and model summary statistics can be used for selection of adequate models. The model analysis compares the validities of the linear, quadratic, and cubic models for the different responses according to their F values. A model with P values (Prob > F) below 0.05 is regarded as significant and the highest-order polynomial that is significant will be selected. The lack-of-fit test demonstrates if the lack of fit between the experimental values and those predicted by the regression model equations can be explained by the experimental error. The model with no significant lack of fit is an appropriate one for representing the response surface. PRESS is a measure of how a particular model fits each

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TABLE 6.2 Natural Variables, Levels, and the Experimental Data Factor Treatment

Response

IMO (%)

FOS (%)

b-CD (%)

Growth Rate (Log CFU=h)

1.50 1.50 1.50 3.00 1.50 0.00 0.00 1.50 3.00 1.50 0.00 1.50 1.50 3.00 3.00 0.00 1.50

0.00 3.00 1.50 0.00 1.50 1.50 3.00 1.50 1.50 1.50 1.50 3.00 1.50 1.50 3.00 0.00 0.00

10.00 0.00 5.00 5.00 5.00 0.00 5.00 5.00 10.00 5.00 10.00 10.00 5.00 0.00 5.00 5.00 0.00

0.087 0.150 0.086 0.115 0.096 0.135 0.095 0.087 0.093 0.086 0.082 0.096 0.095 0.137 0.091 0.077 0.126

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Note: B-CD, b-cyclodextrin; FOS, isomaltooligosaccharides.

fructooligosaccharides;

IMO,

point in the design. The PRESS value for the chosen model should be small relative to the other models under consideration. In this example, the model analysis results (Table 6.3) show that the following model, which is a quadratic equation, appears to be the most accurate with no significant lack of fit (Table 6.3): ^y ¼ 0:12 þ 8:417  103 x1 þ 6:750  103 x2  0:013x3 þ 3:333  104 x21 þ 1:667  103 x22 þ 8:400  104 x23  4:667  103 x1 x2 þ 3:000  104 x1 x3  5:000  104 x2 x3

(6:43)

where ^y represents the fitted model for the growth rate of L. lactis IO-1. The relationship between the factors and the responses can be investigated by examining the contour plots created by holding constant one of the three independent variables. By fixing the b-CD at different levels, three-dimensional and contour plots of the responses as a function of FOS and IMO can be produced. Figure 6.8 clearly shows strong interactions between IMO and FOS when b-CD is at the constant levels of 0%, 5%, and 10%. Figure 6.8a depicts that a higher level of FOS produces a

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TABLE 6.3 Analysis of Variance for the Variables as Linear, Quadratic, and Cubic Terms and Their Interactions in the Response Surface Model Model Analysis Response Sum of Squares

Degrees of Freedom

Mean Square

0.18 4.880  103 2.493  103 2.473  104 1.020  104 0.18

1 3 6 3 4 17

0.18 1.627  103 4.155  104 8.242  105 2.550  105 0.011

Lack-of-Fit Test Linear 2.740  103 Quadratic 2.473  104 Cubic 0 Pure error 1.020  104

9 3 0 4

3.045  104 8.242  105

Source Mean Linear Quadratic Cubic Residual Total

F Value

Prob > F

7.44 8.33 3.23

0.003a 0.006a 0.146

11.94 3.23

0.146 0.143

2.550  105

Model Summary Statistics

Linear Quadratic a

R-Squared

Adjusted R-Squared

Predicted R-Squared

PRESS

0.632 0.955

0.547 0.897

0.332 0.457

5.157  103 4.110  103

Significant at 5% level.

higher response value when b-CD is 0%; while Figure 6.8c illustrates that a higher IMO concentration leads to a better result when b-CD is 10%.

6.10.2 PERFORMING OPTIMIZATION The Equation 6.43 can be used as the objective function to be maximized in an optimization problem, and the problem can be solved to find the optimal formulation for improving the growth rate of L. lactis IO-1. A modified SQP with the multistart capability was employed to perform the global optimization. A very high probability (>0.9999) in Equation 6.42 was set to ensure the global optimum would be attained. Figure 6.9 shows the evolution of the optimal growth rate of L. lactis IO-1 for a sequence of randomly generated initial searching points used by the multistart SQP. The optimization results clearly show that determination of the optima depends on the initial search points and there are three different local optimal values identified from 29 randomly generated initial points. Of these local optima, the global optimal growth rate of L. lactis IO-1 is 0.155 with 99.99% certainty. The global maximum corresponds to 0.155 log CFU=h of the growth rate of L. lactis IO-1. The highest optimal function value (0.155) was attained for 11 out

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3 0.145 0.145

0.17

0.13

FOS%

0.5

3

2 1 0 0

(a)

0. 12 5

2 1

0

IMO%

0.5

0

0.135

0.12 3

0.1 35

0.13

1

0.14

0.135

1.5

1

0.1 45

FOS%

0.14

0.14

0.14

0.14

2

0.15

0.14

0.16

3 0.1

Growth rate (log CFU/h)

0.15

2.5

1.5

2

2.5

3

IMO% 3 95 0.0

2

0.08

1.5 09 0.

FOS%

0.1 0.09

5 09 0.

0.09

0.0 85

1

0.0 95

0.11

0.095

0.095

0.12

1 0 0

(b)

0

IMO%

0

0.5

0.09

2

1

0.085

FOS%

0.5

3

2

1

1.5

2

2.5

3

IMO% 3

0.11

0.08

1.5

0. 08

0. 09 5

FOS%

0.09

0.0 9

2 0.085

0.1

IMO%

0

0.0 9

0

0.5

1

1.5

2

0. 1

1 0 0

(c)

0.0 95

0.5

0.085

3 2

1

0.08

2 FOS%

0. 1

1

0.07 3

0.09

0. 10 5

0.12

0.09

0.09

0.09

2.5

85 0.0

Growth rate (log CFU/h)

5 10 0.

0.1

3

0.0 95

0.07 08 0.

Growth rate (log CFU/h)

0.1

2.5

2.5

3

IMO%

FIGURE 6.8 Three-dimensional and contour plots of the response surfaces for the growth rate of L. lactis IO-1 showing the effects of FOS and IMO under the conditions of constant levels of (a) 0%, (b) 5%, and (c) 10% b-CD.

of 29 sets and the optimal point consists of the natural variables at X1 ¼ 0, X2 ¼ 3, and X3 ¼ 0, which are on the boundary of the design region. In other words, the optimal growth rate of L. lactis IO-1 is cream blended with 3% FOS. It is interesting to note that although IMO (x1) and b-CD (x3) are important individually to the growth rate of L. lactis IO-1, they did not produce a positive effect when all three factors were combined together. Verifying the optimal manufacturing conditions: After the optimal processing condition is found by the SQP, repeated experiments based on the optimal condition should be conducted to verify the predicted optimum. The verification results can

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0.16 0.15

Response

0.14 0.13 0.12 0.11 0.1 0.09 0

5

10 15 20 Initial searching point set

25

30

FIGURE 6.9 Evolution curve of the optimal growth rate of L. lactis IO-1 for randomly generated initial searching points used by the multistart SQP.

then be analyzed using ANOVA, with Duncan’s multiple range test for significance to detect differences between predicted and observed values. In this example, the optimal production conditions for the functional cream with a high level of CLA derived from the SQP is that cream mixed with 3% FOS and fermented with L. lactis IO-1 at 378C for 12 h. The growth rate of L. lactis IO-1 derived from the verification experiments is all very close to the SQP prediction, with no apparent significant differences (P > 0.05) comparing the two sets.

6.11 SUMMARY This chapter has described the theories, procedures, and applications of the response surface methodology. RSM consists of a collection of mathematical and statistical procedures including design of experiments, model selection and fitting, and optimization on the fitted model. There are two typical ways to apply RSM, one in a sequential mode and the other a direct approach. In the sequential mode, RSM is applied sequentially: In the early stages, a simpler model such as the first-order model can be employed repeatedly to move efficiently toward the vicinity of the optimum; once the general region of the optimum is identified, a higher-order model can be utilized to locate the optimum. In the direct approach, a second- or higher-order response surface model is constructed by experimentation on the design points of a DOE of higher levels, with the experiment region set equal to those bounded by the process variables’ upper and lower limits. If the optimization results on the model are not satisfactory, we can reduce the design region (and the experiment region) by cutting each variable’s range by half, for example, and center the design region at the optimum acquired from the last step. Then, rebuild a response model and optimization are required.

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A successful application of the response surface method is greatly influenced by design of experiments. The two-level factorial design, CCDs, FCCD, and the BBD were introduced. Once a response surface is successfully constructed by DOE and measured responses, the response model has to be checked for adequacy. Some of the most common tests for accessing the suitability of the fitted model, including tests for significance on the regression coefficients, lack-of-fit test, and coefficients of multiple determination, have been described. All these tests may be integrated into the ANOVA to examine the adequacy of the regression models. In many instances, two or more responses from the process under investigation are interested and have to be considered simultaneously. Then, multiple responses may be integrated into a single composite function. A local maximum usually corresponds to an improvement in the processing conditions, but if further improvement or the best result is to be achieved, a global optimization technique has to be employed. Global methods such as the SQP technique with multi-starts and the GAs were also introduced in this chapter. Finally, a practical example that demonstrates the application of RSM to optimize the manufacturing conditions of a new dairy product was presented. The procedure to implement RSM comprises (1) performing screening experiment and experimental design, (2) manufacturing functional cream according to the experimental design, (3) building the RSM, (4) performing optimization, and (5) verifying the optimal manufacturing conditions.

REFERENCES Barrios, D. et al., Real-coded genetic algorithms based on mathematical morphology, LNCS, 1876, 706, 2000. Bas¸, D. and Boyaci, I.H., Modeling and optimization I: Usability of response surface methodology, J. Food Eng., 78, 836, 2007. Bessaou, M. and Siarry, P., A genetic algorithm with real-value coding to optimize multimodel continuous functions, Struct. Multidisc. Optim., 23, 63, 2001. Box, G.E.P. and Behnken, E.W., Some new three level designs for the study of quantitative variables, Technometrics, 2, 455, 1960. Chen, M.J. and Lin, C.W., Factors affecting the water-holding capacity of fibrinogen=plasma protein gels optimized by response surface methodology, J. Food Sci., 67, 2579, 2002. Chen, M.J., Chen, K.N., and Lin, C.W., Optimization of the viability of probiotics in a new fermented milk drink by genetic algorithms for response surface modeling, J. Food Sci., 68, 632, 2003. Chen, M.J., Chen, K.N., and Lin, C.W., Sequential quadratic programming for development of a new probiotic dairy tofu with Glucon-Delta-Lactone, J. Food Sci., 69, 344, 2004. Chen, M.J., Chen, K.N., and Lin, C.W., Optimization on response surface models for the optimal manufacturing conditions of dairy tofu, J. Food Eng., 68, 471, 2005. Chen, M.J., Chen, K.N., and Kuo, Y.T., Optimal thermotolerance of Bifidobacterium bifidum in gellan-alginate microparticles, Biotech. Bioeng., 98, 411, 2007. Chen, K.N., Chen, M.J., and Lin, C.W., Optimal combination of the coating materials for probiotic microcapsules and its experimental verification, J. Food Eng., 76, 313, 2006. Chen, K.N., Chen, M.J., and Shiu, J.S., Development of probiotic candies with maximum viability by using response surface methodology and sequential quadratic programming, Asian-Australasian J. Animal Sci., 21, 896, 2008. Chen, S., Robust design with dynamic characteristics using stochastic sequential quadratic programming, Eng. Optimization, 35, 79, 2003.

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Gan, H.E. et al., Optimization of the basic formulation of a traditional baked cassava cake using response surface methodology, LWT, 40, 611, 2007. Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning, AddisonWesley Publishing Company, Boston, MA, 1989. Holland, J.H., Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence, The MIT Press, Cambridge, MA, 1975. Kahyaoglu, T., Optimization of the pistachio nut roasting process using response surface methodology and gene expression programming, LWT, 41, 26, 2008. Mitchell, M., An Introduction to Genetic Algorithms, The MIT Press, Cambridge, MA, 1998. Montgomery, D.C. and Runger, G.C., Applied Statistics and Probability for Engineers, 2nd edn., John Wiley & Sons, Inc., New York, 1999. Mundra, P., Desai, K., and Lele, K., Application of response surface methodology to cell immobilization for the production of palatinose, Bioresource Tech., 98, 2892, 2007. Myers, R.H. and Montgomery, D.C., Response Surface methodology: Process and Product Optimization using Designed Experiments, John Wiley & Sons, Inc., New York, 1995. Reklaintis, G.V., Ravindran, A., and Ragsdell, K.M., Engineering Optimization: Methods and Applications, John Wiley & Sons, Inc., New York, 1983. Snyman, J.A. and Fatti, L.P., A multi-start global minimization algorithm with dynamic search trajectories, J. Optimization Theory App., 54, 121, 1987. The Math Works Inc., Using MATLAB, The Math Works Inc., Natick, Massachusetts, 2000. Weng, W., Liu, W., and Lin, W., Studies on the optimum models of the dairy product Kou Woan Lao using response surface methodology, Asian-Australasian J. Animal Sci., 14, 1470, 2001. Wu, Y. et al., Optimization of extraction process of crude polysaccharides from boat-fruited sterculia seeds by response surface methodology, Food Chem., 105, 1599, 2007.

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7

Random-Centroid Optimization Shuryo Nakai, Yasumi Horimoto, Jinglie Dou, and Roxana A. Verdini

CONTENTS 7.1 7.2 7.3

Introduction ................................................................................................ 141 RCO Procedure .......................................................................................... 144 Applications of RCO ................................................................................. 148 7.3.1 Mixture Designs .............................................................................. 148 7.3.2 Food Formulation ............................................................................ 148 7.3.3 Food Preparation ............................................................................. 149 7.3.4 Designing Optimization .................................................................. 149 7.3.5 Sensory Analysis ............................................................................. 149 7.4 RCG Procedure .......................................................................................... 149 7.4.1 Peptide QSAR ................................................................................. 150 7.4.2 Application of RCG ........................................................................ 150 7.5 Discussion .................................................................................................. 150 7.6 Conclusion ................................................................................................. 151 7.7 Downloading of RCO Program ................................................................. 151 References ............................................................................................................. 151

7.1 INTRODUCTION Multimodal cases containing many local optima such as shown in Figure 7.1 (Al-Mashikh and Nakai, 1987) are quite infrequent to run into in food research. As a result, the response surface methodology (RSM) and optimization based on it searching for the maximum or the minimum on smooth surfaces is rather popular (Banga et al., 2003). However, the advantage of stochastic approach in global optimization was extensively studied so that there is no need of a priori knowledge on the mechanism of the optimization projects in concern. The number of papers published on the global optimization in chemistry and engineering had dramatically increased since 1900. This recent trend in publications for global optimization may be generated due to the introduction of a new algorithm, i.e., ‘‘genetic algorithm’’ that was a general methodology used to search for a solution space in a manner analogous to the natural selection procedure in the biological evolution (Holland, 1975). This optimization framework is able to provide the global optimum when

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SE

4.0

0

4 pH .25

4

1.

4.5

0

0

1.

2 1. MP SH

FIGURE 7.1 Surface plot of relationship between pH, sodium hexametaphosphate, and separation efficiency of cheese whey treatment. (Reprinted from Al-Mashikh, S.A. and Nakai, S., J. Food Sci., 52, 1237, 1987. With permission.)

conventional gradient-based algorithms have failed. The results of computations thus obtained were comparable to that derived from more recent ‘‘simulated annealing’’ (Androulakis and Venkatasubramanian, 1991). According to Marinari and Parici (1992), the simulated annealing was an efficient heuristic approach that was used to find the absolute minimum (or maximum) of functions along with many local minima (or maxima). It had been introduced independently in the framework of the Monte Carlo approach for assigning discrete variables in the method of Kirkpatrick et al. (1983). According to Yassien (1993), who had employed a ‘‘level-set program’’ for global optimization, it was possible that problems in chemical engineering possessed a number of local optima. He also stated that because of the highly nonlinear nature of modeling equations, which were frequently involved, the aforementioned phenomena were quite common in the engineering system designs. Many other algorithms, e.g., Lipschitz optimization, had also been used for global optimization (Horst et al., 1995). Mathematical modeling can provide process optimization with predictive capacity thereby the process automation accompanied with control capability became popular (Sablani et al., 2007). Since ‘‘physics-based models’’ would incur insight into a process, experimentation may not be practical. On the contrary, ‘‘observationbased models’’ can manipulate plethora of other general cases no matter what are the mathematical principle behind them. These models can be relatively easy to be optimized. Considering that biological projects are usually more complex than engineering problems, it is likely that the global optimization in biology is more acute because of its nature. It may be worth noting that for the peptide QSAR

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(quantitative structure-activity relationship), random-centroid optimization (RCO) for biology (RCG) was separately developed in this study in an additional category as described later. As Schwefel (1981) stated, the most reliable global search is the grid method that is, however, time-consuming and in general costly. The most popular alternative is random (or stochastic) strategy because of mathematical simplicity, flexibility, and resistance to perturbation. However, because of no availability of highly user-friendly technique for explicitly determining search direction, random searches are usually quite slow to approach to the true global optimum. After enduring trials of the application of a variety of iterative optimization techniques such as super modified simplex optimization (SSO) of Routh et al. (1977) and its derivatives, we have proposed a new algorithm, i.e., RCO. The RCO consists of random search, centroid search and mapping, which altogether composed of a search cycle (Nakai, 1990; Dou et al., 1993). Mapping was first introduced into SSO for the purpose of approximation of the search direction (Nakai et al., 1984). Decision making in search direction determination of the move for continuing toward the subsequent cycles based on the maps is extremely critical in the subsequent optimization step since great improvement of the optimization efficiency can be expected during running the search for the optimum. The progress of ‘‘Visual Basic’’ by the MicroSoft in the early 1990s from the previous ‘‘Quick Basic’’ certainly expedited computer programming that was applied to the mapping process in the RCO, thereby, facilitating corroboration of the search direction during progressive cycling of the optimization search. Mutation in biology is a sudden departure from heredity caused by a sudden change in a gene or chromosome. Since the mutations cannot be predicted precisely in any ways, random mutation is customarily adopted in genetic engineering to mimic the Darwinian evolution. However, completely randomized approaches are inefficient in mathematical principle because they would rely solely on luck or chance; thus some regularization is in necessity (Schwefel, 1981). Accordingly, the search cycles of RCO are composed of some regulated random design along with the central search around the best response within the random sequences finally followed by mapping (Nakai, 1990). It could, therefore, be an appropriate algorithm for optimization of site-directed mutagenesis (RCG). The RCG uses the properties (hydrophobicity, charge, bulkiness, tendencies to a-helix and b-strand) of amino acid residues as factors to elucidate the functionality of protein or macropeptide. The range of property index value was entered in the RCG for amino acid residues at a site in sequences of protein or peptide (Nakai et al., 1998b). Furthermore, an extended application of RCG is in progress for finding a short, optimal segment within the sequences of protein or polypeptide. This technique is coined as ‘‘Peptide Quantitative Structure Activity Relationship (QSAR).’’ The importance of sequence of oligopeptide segments in the structure–function relation study of proteins and peptides has been recognized (Nakai et al., 2005). The objectives of this chapter are to illustrate the RCO steps on computers, to monitor with special emphasis on the mapping process for determining the search directions, and to improve the efficiency of RCO as well as RCG later. It is our strong belief that the peptide QSAR could be the most convenient, user-friendly strategy in the proteomics as well as genomics projects in the near future.

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7.2 RCO PROCEDURE Mathematical model used for five-factor maximization was y ¼ 3:66 þ 33:6x1 þ 25x2 þ 34:4x3 þ 22x4 þ 7x5  12x1 x3  10x1 x4  4x1 x5  16x2 x3  12x2 x4  6x2 x5  15x21  18x22  20x23  26x24  10x25

(7:1)

where the values of all x are 0–1.0 and the theoretical maximum y is 32.0 at 0.8, 0.3, 0.5, 0.2, and 0.1 for x1 to x5, respectively (single peak). Figure 7.2A is the cover screen of the RCO package (RCOTNS.exe). By depressing ‘‘Start’’ button on this cover screen, a new menu screen appears in B; then click ‘‘Maximization’’ (or ‘‘Minimization’’ depending on your optimization purpose), ‘‘1st cycle,’’ and finally ‘‘Random11.’’ After pressing the ‘‘Continue’’ button in Figure 7.2B, factor entry begins. The first factor and its limit values are entered as shown in Figure 7.2C; and after ‘‘Enter’’ing for all factors, the entered summary list will appear by clicking the ‘‘List’’ button in the same figure as shown in Figure 7.2D. The factor list can be obtained by pressing the ‘‘Print’’ button in Figure 7.2D. To compute the random design, first click the ‘‘Random’’ button in Figure 7.2D to change to the MDI Parent Window (Figure 7.3E). Thereafter, by

(A)

(B)

(C)

(D)

FIGURE 7.2 (A) Click ‘‘Start.’’ (B) Click ‘‘Maximization,’’ ‘‘1st cycle,’’ ‘‘Random11,’’ then ‘‘Continue.’’ (C) Enter data for factor 1 (X1) and depress ‘‘Enter.’’ After entering all factors, click ‘‘List’’ to change to (D). Clicking ‘‘Random’’ to change to Figure 7.3.

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(E)

(F)

(G)

(H)

FIGURE 7.3 In (E) (blank), depressing File=list yielded the current E for the random design, which was saved in ‘‘NK12-7.RCO’’ (F). This will complete Step 1 (Random11). Step 2 (Centroid12) is initiated by clicking ‘‘Centroid12’’ followed by clicking ‘‘Continue’’ on the Menu screen (G) yielded (H) by File=Open file.

clicking ‘‘List’’ on the drop-down File tab in Figure 7.3E, the random design table will appear as shown in Figure 7.3E. This table can be printed by clicking File=Print in Figure 7.3E for running experiments as given in the random design. This table should be ‘‘Save’’ed (Figure 7.3F) and ‘‘Save as’’ in a file of a diskette if required to back-up. This saving may be important because the random experimental conditions produced here is difficult to reproduce later due to its random nature. After returning to the menu screen (Figure 7.3G), the computation should shift to the ‘‘Centroid12’’ design. ‘‘Continue’’ in Figure 7.3G requests response entry for the random design as shown in Figure 7.3H; File=Openfile along with the Open tab (Figure 7.3H) provides the MDI Parent Window. File=Input response starts response entry to the random design; after repeating the entry by depressing ‘‘OK’’ each time in Figure 7.4I. After completing all entries, File=List centroid gives the centroid design (Figure 7.4J). ‘‘Print’’ and ‘‘Save’’ (and ‘‘Save as’’) can be performed as before, then ‘‘Exit.’’ To move to the last step, flicking Sum=Map13 in the Figure 7.4K screen and File=Openfile reads the centroid design (Figure 7.4L) from the saved file. Then File=Input will ask the number of responses to be added to the RCO table (Figure 7.5M). This number could be two to four in order to avoid unnecessary replication of useless, similar experiments. After reporting responses as in the case of

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(I)

(J)

(K)

(L)

FIGURE 7.4 File=input response in (I) to complete the input of all random design conditions 1 to 10. Then file=list centroid produces centroid design (J) that should be saved. The menu screen should be recalled (K) for Sum=Map13. File=Open gives MDI Parent Window (Summary) in (L).

random design (Figure 7.5N), File=List indicates the summary data of cycle 1 as shown in (Figure 7.5O) that could be printed as before. Some useful operation herein is that addition of run data from other experiments or deletion of useless runs can be done to the summary data in Figure 7.5O screen by pressing File=Edit in Figure 7.4K screen. File=Mapping will further produce the MDI Parent Window (Draw Lines). ‘‘1. Select Factor’’ is for entering the factor number; ‘‘Locate Optimum’’ in Figure 7.6Q moves the cursor on the horizontal scale to the best point on the X coordinate. The result of depressing ‘‘Lin Draw’’ is shown in Figure 7.6Q; and ‘‘DoubleIgnorew’’ illustrates Figure 7.6R for factor 1. The ignored factors are shown in the right-upper corner of each map for different factors. Another useful operation herein is that the scale of axis can be modified by depressing ‘‘Axis’’ in the top bar in Figure 7.6Q. After selecting a new search range of each factor that is narrower than that of cycle 1 from the mapping, opening of the saved program is critical. This process is not automatic as the file name of the saved site is required to open as seen in Figure 7.3H. In the Menu screen (Figure 7.2B), the search cycles can be programmed up to 5th cycle. Through application of RCO to multifactor mathematical models, it was

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(M)

(N)

(O)

(P)

FIGURE 7.5 File=input in L shift the screen to (M) asking how many experiments will be carried out. In (N), the number of responses are supplied; thereafter the summary data (O) will appear by depressing file=list. For mapping (P), click File=Mapping in the pull-down tab.

(Q)

(R)

(SS)

FIGURE 7.6 Clicking ‘‘Select Factor,’’ ‘‘Locate Optimum’’ followed by ‘‘Line Draw’’ generates (Q). Depressing ‘‘Re Plot’’ and click ‘‘SingleIgnore’’ and=or ‘‘DoubleIgnore’’ resulted in (R) (double). (SS) is the first screen to be used in data entry when ‘‘Simult. Shift’’ in the menu screen was clicked. ß 2008 by Taylor & Francis Group, LLC.

found that when the number of factors was large, large number of cycles was required and was effective. However, when the number of factors were few, large number of cycles was ineffective, wasting by very slow progress in optimization. ‘‘Simultaneous Shift’’ (Figure 7.6(SS)) would avoid this problem. An example of global minimization for Woods four-factor model is shown in Nakai et al. (1998a). h n
2
2 y ¼ 100 x2  x21 þ ð1  x1 Þ2 þ 90 x4  x23 þ ð1  x3 Þ2 þ 10:1 ðx2  1Þ2 i. o (7:2) þ ðx4  1Þ2 þ 19:8ðx2  1Þðx4  1Þ 100 þ 10 where the theoretical minimum y is 10.0 at 0.8, 0.2, 0.9, and 0.8 within the ranges of 0–1.0 for the four factors; another local optimum (ffi18) at around 0.3 of x1. Another model equation for global optimization was also computed there. RCO was successfully employed for optimizing a variety of mathematical models including different multimodal (up to 10) equations (Nakai et al., 1998a). When super-simplex optimization (SSO) was applied to a Fletcher-Powell arctangent equation containing two local minima in addition to the deeper global optimum in between, the search was stalled at the local optima five times during optimization runs. Although when SSO did not stall using numbers of different search spaces to start computations, 60–180 iterations were required to reach the global optimum, whereas, less than 50 RCO iterations were adequate to find the global optimum. As a model-based method Kohonen self-organizing artificial network can also be used (Verdini et al., 2007).

7.3 APPLICATIONS OF RCO 7.3.1 MIXTURE DESIGNS The analysis of components of Raman spectra was performed to best fit to the surface hydrophobicity of proteins determined by different fluorescent probes (AlizadehPasdar et al., 2004). The component constraint of SCi ¼ 1.0, where i is the number of component C, was imposed. Using this approach, the best fluorescent-probe method for protein surface hydrophobicity was selected based on quantitative assessment. In comparison to the conventional mixture-design technology, in which experimental designs become more complicated along with enhanced unreliability as the number of components increases, RCO could readily overcome this problem due to evolutionary operation property. This fact may be true when mixture-design is applied for mixing food components or ingredients.

7.3.2 FOOD FORMULATION Factorial analysis, Taguchi method for experimental design, response surface methodology, and constrained simplex optimization were compared (Arteaga et al., 1994). Also, formulation of a food with component constraints was successfully performed using RCO. Frankfurter formulation was optimized when prepared from skinless porkfat, frozen mechanically deboned poultry meat, and lean beef. For more

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complex case was surimi base formulation using surimi base, egg white powder, soy protein isolate, gluten, starch, fat, and water as the difference from the total of 1.0; cooking conditions of preheating temperature and time and extrusion cooking temperature and time were added to the preceding ingredient factors (Dou et al., 1993). To accommodate many structural constraints, penalty function was imposed to experiments, which violate a constraint by intentionally assigning a higher inferior response value.

7.3.3 FOOD PREPARATION Cooking of Indica rice was optimized by changing water soaking conditions, i.e., amount of water against rice, soaking time and temperature. Sensory data after cooking very close to that of control Japanese rice was obtained (Nishimura et al., 1997). Heat-induced breaking strength of carp actomyosin gel was maximized by varying the amount of FeCl2 solution and heating time and temperature (Bouraoui et al., 1998; Nakai et al., 1999). In another experiment, cream puff paste was prepared without failure in puffing-up (expansion) when baked (Nishimura et al., 1998). Flour, shortening, yolk, egg white, and water as ingredients and heating time were optimized in the first experiment and temperature of the heated mixture of water, shortening and flour in addition to the egg solution and incubation time and temperature of the yolk were optimized in the second experiment; altogether, 45 vertices were required. Meanwhile, baking of dietary-fiber bread was optimized to obtain the highest loaf volume (Kobayashi et al., 1997). Separation of chicken sauce during heating was also prevented (Nishimura et al., 2001).

7.3.4 DESIGNING OPTIMIZATION Glass shapes for beer drinking was successfully optimized by changing bottom width, height, top diameter of the hand-holding at the neck, and diameter of the round top. The optimal preference of glass shape for various consumer groups was determined using this RCO shape optimization (Nakai et al., 2002). Objective multifactor optimization is generally in great need for designing, such as shape–color combination. In principle, there is no limit for numbers of factors, search cycles and iterations in the RCO program; all depend to the memory capacity of available computers.

7.3.5 SENSORY ANALYSIS Because the above optimization of glass shape is an example of the optimization of sensory data, RCO may be applicable to other sensory data or for market survey for yielding reliable, objective judgment of survey results.

7.4 RCG PROCEDURE The RCG program was used to select both the position in the sequence of protein or peptide along with amino acid for substitution as two factors in the single-site modification. Therefore, for modification of certain range in sequence, site (factor 1) and range (starting position and finishing position) should be entered. For selection

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of amino acid (factor 2), range of property index value of hydrophobicity (AA), bulkiness (AB) and propensity of helix (AH), and strand (AS) of 20 residues can be entered. Random selection of this range would find the nearest amino acid residue, which is recommended to replace the randomly selected site.

7.4.1 PEPTIDE QSAR There is an extensive area of application of peptide QSAR (Nakai et al., 2002), especially pertinent to human health, such as antihypertensive, anti-inflammatory, and anticancer activities of proteins=peptides, even for antiaging trials, considering almost all of enzymes and hormones would belong to this category. For QSAR study, not only RCG, but also principal component similarity (PCS) and homology similarity analysis (HSA) and its search (HSS) are useful to complete the search process (Nakai and Alizadeh-Pasdar, 2006).

7.4.2 APPLICATION

OF

RCG

The first application of RCG was made in 1998 (Nakai et al., 1998b) to modify single-site of the active helix region of microbial neutral proteinase. Sixteen amino acid residues of the active-site helix (G139–Y154) in B. stearothermophilus neutral protease were genetically modified (Nakai et al., 1998b). Substitution of V143E increased thermostability DT50 by 6.58C from 68.38C of the wild-type enzyme, despite a decrease of enzymatic activity to 32%. It was also found that lower hydrophobicity and bulkiness at the N-terminal of the active site were critical in the enzymatic activity. The RCG was also applied to two-sites simultaneously in mutagenesis of human cystatin C (Ogawa et al., 2002). The entire sequence of the cystatin with 120 residues was modified. G12W=H86V mutant showed a fivefold increase in the papaininhibitory activity over that of recombinant wild-type (WT) cystatin used as a control. Also, P13F mutant exhibited 5.28C of DT50 from 68.28C of WT in addition to a 56% greater activity. The original RCG has been modified to establish the new plan of ‘‘Peptide QSAR’’ as described above to apply multimodal, some times multiresponse projects in biology due to the facts that enzymes and hormones are also all polypeptides. Not only genetically modifying amino acid residues on sites, but also completely new, different combination of the residues in the entire sequences of oligopeptides can be prepared by peptide synthesis without following genetic rules. Therefore, the peptide QSAR may have great future in application in new area of bioresearch and biotechnology.

7.5 DISCUSSION Experimental designs followed by factorial analysis, furthermore prediction using multiple regression analysis or more advanced regression neural networks are a recent trend in data processing techniques. In the biological studies, selection of placebo control in the multifactor projects in biological problems is also important. Because of ready illustration of the effects of variables, RCO=RCG maps can replace

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most of the above computing process in visible demonstration. This could be the most crucial advantage of these new computing methods. Despite possible improvement of optimization efficiency using simultaneous shift by decreasing cycle number in the past, occasionally this did not occur; instead repeating a new cycle for spot search by setting narrower limits based on the maps was effective (Nakai et al., 1998a).

7.6 CONCLUSION Efficient random-centroid optimization, which is applicable to multimodal cases including global optimum, was illustrated in stepwise that will facilitate visual comparison to determine new search ranges for the subsequent search cycles. Broad application of RCO sensory scoring can be anticipated in improving its objectivity and reliability, especially in broad area of biological research, not only for conducting food research, in the future. Advantages of the RCO=RCG approach are in the applicability to multifactor models (up to 10 factors was experimentally validated so far with a temporary maximum capacity of 30 factors in the currently available programs).

7.7 DOWNLOADING OF RCO PROGRAM RCO and RCG can be downloaded from ftp:==ftp.agsci.ubc.ca=foodsci=. The program package includes instructions as well as model equations to practice as well as for training.

REFERENCES Al-Mashikh, S. A. and Nakai, S., Reduction of beta-lactoglobulin content of cheese whey by polyphosphate precipitation, J. Food Sci., 52, 1237, 1987. Alizadeh-Pasdar, N., Li-Chan, E. C. Y., and Nakai, S., FT-Raman spectroscopy, fluorescent probe and solvent accessibility study of egg and milk proteins, J. Ag. Food Chem., 52, 5277, 2004. Androulakis, I. P. and Venkatasuburamanian, V., A genetic algorithmic framework for process design and optimization, Comp. Chem. Eng., 15, 217, 1991. Arteaga, G. E., Li-Chan, E. C. Y., Vazque-Arteaga, M. C., and Nakai, S., Systematic experimental designs for product formula optimization, Trends Food Sci. Tech., 5, 243, 1994. Banga, J. R., Balsa-Canto, E., Moles, C. G., and Alonso, A., Improving food processing using modern optimization methods, Trends Food Sci. Tech., 14, 131, 2003. Bouraoui, M. M., Fichtali, J., Pinder, K. L., Nakai, S., and Bowen, B. D., Viscous properties of salmon surimi paste, J. Food Eng., 34, 441, 1998. Dou, J., Toma, S., and Nakai, S., Random-centroid optimization for food formulation, Food Res. Int., 26, 27, 1993. Holland, J. H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan, 1975. Horst, R., Pardalos, P. M., and Toal, N. V., Introduction to Global Optimization, Kluwer Academic, Dordrecht, the Netherlands, 1995.

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Kirckpatrick, S., Gelatt, C. D., and Vecchi, Jr. M. P., Optimization by simulated annealing, Science, 220, 671, 1983. Kobayashi, Y., Nakai, S., Horimoto, Y., and Chiji, H., Optimum composition in ingredients for high-fiber bread containing wheat bran by random-centroid optimization, Bull. Fuji Womens’ Col., 35, Series II, 31, 1997. Marinari, E. and Paraci, G., Simulated tempering: A new Monte Carlo scheme, Europhys. Lett., 19, 451, 1992. Nakai, S., Computer-aided optimization with potential application in biorheology, J. Jan. Biorheolog. Soc., 4, 143, 1990. Nakai, S. and Alizadeh-Pasdar, N., Rational designing of bioactive peptides. In Nutritional Proteins and Peptides in Health and Disease, Mine, Y. and Shahidi, F., (Eds.), Taylor & Francis, New York, 565, 2006. Nakai, S., Amantea, G., Nakai, H., Ogawa, M., and Kanagawa, S., Definition of outliers using unsupervised principal component similarity analysis for sensory evaluation of foods, Int. J. Food Prop., 5, 289, 2002. Nakai, S., Dou, J., Victor, L., and Christine, H. S., Optimization of site-directed mutagenesis. 1. New random-centroid optimization program for Windows useful in research and development, J. Ag. Chem., 46, 1642, 1998a. Nakai, S., Dou, J., Victor, L., and Christine, H. S., Optimization of site-directed mutagenesis. 2. Application of random-centroid optimization to one-site mutation of B. stearothermophilus neutral protease to improve thermostability, J. Ag. Chem., 46, 1655, 1998b. Nakai, S., Koide, K., and Euguster, L., A new mapping super-simplex optimization for food products and processing development, J. Food Sci., 49, 1143, 1984. Nakai, S., Li-Chan, E. C. Y., and Dou, J., Pattern similarity of functional sites in protein sequences: Lysozyme and cystatins, BMC Biochem., 6, 1, 2005. Nakai, S., Saeki, H., and Nakamura, K., A graphical solution of multimodal optimization to improve food properties, Int. J. Food Prop., 2, 277, 1999. Nishimura, K., Goto, M., Izumiya, N., and Nakai, S., Optimum cooking conditions for Indica type rice by using random-centroid optimization, J. Cook. Sci. Jap., 30, 9, 1997. Nishimura, K., Goto, M., Nakai, S., Kawase, S., and Matsumura, Y., Preventing sauce separation by proteins released from chicken during heating, J. Home Econ. Jap., 52, 699, 2001. Nishimura, K., Imazuya, N., and Nakai, S., Optimum preparative method for storing cream puff paste without deterioration, Food Sci. Tech. Int. Tokyo, 4, 18, 1998. Ogawa, M., Nakamura, S., Scaman, C. H., Jing, H., Kitts, D. D., Dou, J., and Nakai, S., Enhancement of proteinase inhibitory activity of recombinant human cystatin C using random-centroid optimization, Biochim. Biophys. Acta, 1599, 115–124, 2002. Routh, W. W., Swartz, P. A., and Denton, M. B., Performance of the super modified simplex, Anal. Chem., 49, 1422, 1977. Sablani, S. S., Raman, M. S., Datta, A. K., and Mujumdar, A. S., Handbook of Food and Bioprocess Modeling Techniques, CRC Press, Boca Raton, Florida, 2007. Schwefel, H–P., Numerical Optimization of Computer Models, John Wiley & Sons, New York 87, 1981. Verdini, R. A., Zorrilla, S. E., Rubiolo, A. C., and Nakai, S., Multivariate statistical methods for Port Salut Argentino cheese analysis based of ripening time, storage conditions, and sampling sites, Chemomet. Intel. Lab. Sys. 86, 60, 2007. Yassien, H. A., A level set global optimization method for nonlinear engineering problems, PhD Dissertation, University of British Columbia, Vancouver, BC, Canada, 1993.

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8

Multi-Objective Optimization in Food Engineering Cheah Keen Seng and Gade Pandu Rangaiah

CONTENTS 8.1 8.2 8.3 8.4

Introduction ................................................................................................ 153 Evaporator System Design for Multiple Objectives .................................. 155 Overview of MOO Methods...................................................................... 161 MOO in Food Processing=Industry ........................................................... 162 8.4.1 Heat Processing ............................................................................... 172 8.4.2 Fermentation ................................................................................... 173 8.4.3 Separation Processes ....................................................................... 173 8.4.4 Miscellaneous Applications ............................................................ 174 8.5 MOO Software........................................................................................... 174 8.6 Conclusion ................................................................................................. 175 References ............................................................................................................. 176

8.1 INTRODUCTION Optimization is finding the best solution for the chosen objective or criterion. In the real world, decisions have to be made based on more than one objective, and it may not be possible or satisfactory to reduce the objectives into a single objective. For example, in a milk concentration plant, there are many aspects that can be used to measure the performance of the plant. These aspects could include economic measures such as payback period (PBP), net present worth or value (NPW or NPV) and internal rate of return (IRR). Noneconomic aspects could include energy efficiency, product yield, and product quality. It is not meaningful to combine economic and noneconomic measures. The optimization of this milk concentration plant is an example of a type of problem known as a multi-objective problem (MOP). A multi-objective optimization (MOO) problem can be mathematically defined as follows: Maximize Fi (x) where i ¼ 1, 2, . . . , m

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(8:1a)

Minimize fj (x)

(8:1b)

Subject to xL  x  xU

(8:1c)

hk (x) ¼ 0

(8:1d)

gl (x)  0

(8:1e)

where j ¼ 1, 2, . . . , n With respect to x

where k ¼ 1, 2, . . . , o

where l ¼ 1, 2, . . . , p These problems can involve maximization of objectives, minimization of objectives, or be a combination of both. Although there can be many objectives, most MOO problems in food engineering consider two or three objectives at the same time. See Tables 8.4 through 8.7 presented later for the objectives used in MOO applications in food engineering. The decision variables in Equation 8.1 are represented by the vector x, with each of its elements representing a particular decision variable, which can be continuous, integer and=or binary. Continuous and integer variables are often bounded within lower and upper limits (Equation 8.1c) based on practical considerations. One example of a real variable is heater duty with maximum and minimum allowed duties as upper and lower limits, respectively. If the heater is represented either as ‘‘Off’’ or ‘‘On,’’ then the variable is called a binary one. Integer variables would refer to a heater, which has a few levels of operation, such as low, medium, and high. More common examples of integer variables are quantities of inputs that are available only on integer basis such as the number of tubes in heat exchangers, evaporator stages, and batch units (Tables 8.4 through 8.7). Constraints in MOO problems, similar to those in single objective problems, are divided into equality constraints (Equation 8.1d) and inequality constraints (Equation 8.1e). In food engineering, equality constraints usually consist of energy and mass balances whereas inequality constraints arise from physical limitations and safety considerations. The objectives in a MOP (Equation 8.1) may be conflicting, non-conflicting, or partially conflicting (Tan et al., 2005): (1) conflicting objectives refer to situations in which a particular objective can only be improved by compromising the other, (2) nonconflicting objectives share a similar optimal point, and (3) partially conflicting objectives refer to cases in which the objectives are conflicting only within a particular range of values. MOPs with conflicting or partially conflicting objectives yield not just one but many optimal solutions each of which is better than the rest in at least one objective. This implies that one objective is better while at least one other objective is worse when one of these optimal solutions is compared to another. The optimal solutions of an MOP are known as the Pareto-optimal solutions, or less

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commonly, Edgeworth–Pareto optimal solutions after the two economists, Edgeworth and Pareto, who developed the theory of indifference curves in the late nineteenth century. They are also referred to as nondominated, noninferior, efficient, or simply Pareto solutions. Pareto-optimal solutions of an MOP can be represented in two spaces—objective and decision variable spaces. Definitions, techniques, and discussions in MOO mainly focus on the objective space. However, implementation of the selected Pareto-optimal solution will require some consideration of the decision variable values. Further, multiple solution sets in the decision variable space may give the same or comparable objectives in the objective space; in such cases, the engineer can choose the most desirable solution in the decision variable space. See Tarafder et al. (2007) for a study on finding multiple solution sets in MOO of chemical processes. In Section 8.2, an MOO application in food engineering is presented along with its solution by two techniques=programs. It is followed by an overview of MOO methods and then a review of MOO applications in food engineering in the past three decades. Available MOO programs in public domain and their sources are presented before ending this chapter with concluding remarks.

8.2 EVAPORATOR SYSTEM DESIGN FOR MULTIPLE OBJECTIVES A simple evaporation process for milk concentration (Figure 8.1) is used to illustrate the MOP problems and their solution. The milk evaporation process is briefly covered in many dairy technology textbooks such as that by Walstra et al. (2006). This process system consists of three main units: (1) evaporator, (2) multijet condenser, and (3) heat exchanger (preheater in Figure 8.1). The milk feed stream enters the heat exchanger and is preheated by the condensed steam from the evaporator. The preheated feed stream then enters the tubes in the evaporator where it is further heated by ‘‘live’’ steam condensing outside the tubes. Some water in the milk feed is evaporated and exits as vapor. This vapor is then sent to the multijet condenser, which condenses the water vapor and also generates the vacuum; this allows the evaporation to occur at subatmospheric conditions. The MOP for the evaporation system is adopted from Nishitani and Kunugita (1983). The equations are simplified and the cost coefficients are updated to reflect the current prices. In this problem, there are two objectives: the net exergy consumption and the investment cost, and three decision variables: temperature of the pre-heated feed stream entering the evaporator tubes (Ti), temperature of ‘‘live’’ steam (TS), and evaporation=vapor temperature (Tv). Tables 8.1 and 8.2 summarize the design conditions, data, objectives, and model equations for the evaporator system described above. These equations involve many variables, which can be calculated once the values of the three decision variables and the feed conditions are known. The MOO problem is solved using two different methods, the «-constraint method and a multi-objective evolutionary algorithm. The «-constraint method can be developed from the Kuhn-Tucker conditions for noninferior solutions. For simplicity, we consider the MOP in Equation 8.1 with only n minimization objectives and p inequality constraints but no maximization objectives, equality constraints, and

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Water vapor (mv, Tv, Pv, Hv)

Cooling water (mc, Tc, Pc, Hc, ec)

Multijet condenser “Live” steam (ms, Ts, Ps, Hs, es)

Evaporator

Vapor run-down (mc + m v, Tx, P0, Hx, ex)

Preheated feed (mf, xf, Ti, P0, Hi)

Concentrated product (m l, xl, Tv, Pv, H l, e l)

Water run-down (ms, Te, P0, He, ee)

Condensed steam (ms, Ts, Hd) Preheater m P T H

= = = = e =

Feed (mf, xf, Tf, P0, Hf, e f)

Mass flow rate Pressure Temperature Enthalpy Exergy

FIGURE 8.1 Schematic of the evaporator system.

bounds. If a solution x to this MOP is feasible and noninferior, then there exist wl  0, l ¼ 1, 2, . . . , n and mi  0, i ¼ 1, 2, . . . , p such that mi gi (x) ¼ 0

i ¼ 1, 2, . . . , p

TABLE 8.1 Design Conditions and Data for the Milk Concentration Process Feed conditions: Tf ¼ 298.15 K, mf ¼ 2.0 kg=s, xf ¼ 0.1 Solid fraction in the product: xl ¼ 0.4 Specific heat capacity of milk, Cp ¼ 4.184–2.686x kJ=kg=K Specific heat capacity of water, Cw ¼ 4.184 kJ=kg=K Latent heat of vaporization of water, l ¼ 3250–2.68T kJ=kg Overall heat transfer coefficient, U ¼ 0.00567(350–250x) kJ=K=m2=s Evaporation pressure, Pv ¼ 133.3 exp(20.39–5126=Tv) Pa Cooling water pressure and temperature: Pc ¼ 3.039  105 Pa, Tc ¼ 298.15 K Reference pressure and temperature: P0 ¼ 1.013  105 Pa, T0 ¼ 298.15 K

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(8:2)

TABLE 8.2 Objectives, Decision Variables, System Equations, and Thermodynamic Properties for the Evaporator System Shown in Figure 8.1 Objective Functions Net exergy consumption (J=s), f1 ¼ mf «f  ml«l þ ms«s þ mc«c Annualized investment cost ($=Year), f2 ¼ 27500(Ae)0.67 þ 1680(mc)0.6 þ 27500(Ah)0.67 Decision Variables and Their Bounds 383.15 K  Live steam temperature, TS  443.15 K 298.15 K  Feed preheat temperature, Ti  343.15 K 313.15 K  Evaporation temperature, Tv  343.15 K Constraints Feed pre-heat temperature, Ti  Evaporation temperature, Tv System Equations and Thermodynamic Properties Evaporator m f ¼ ml þ mv mfxf ¼ mlxl msHs þ mfHi ¼ msHd þ mlHl þ mvHv ms(Hs – Hd) ¼ AeUexl(Ts – Tv) Multijet condenser mvHv þ mcHc ¼ (mv þ mc)Hx ln(mc=mv) ¼ 17.684  4.817Z þ 0.3829Z2 Z ¼ ln (Pv) – 0.05539 (Tc – T0) Countercurrent heat exchanger msHd þ mfHf ¼ msHe þ mfHi ms(Hd – He) ¼ AhUhxfDT DT ¼ {(Ts – Ti) – (Te – Tf)}=ln{(Ts – Ti)=(Te – Tf)} if mfcp=mscw 6¼ 1 DT ¼ Ts – Tf if mfcp=mscw ¼ 1 Feed, water, and saturated steam H ¼ 4.184(T – T0) þ l « ¼ [T – T0 – T0 ln(T=T0)] þ l [1 – ln(T0=T)]

and n X l¼1

wl rfl (x) 

p X

mi rgi (x) ¼ 0

(8:3)

i ¼1

These are the necessary conditions for a noninferior solution. When all minimization objectives are convex and the feasible region is convex, these conditions are sufficient as well. Equation 8.3 can then be re-written as follows:

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wr rfr (x) þ

n X

wl rfl (x) 

l¼1,l6¼r

p X

mi rgi (x) ¼ 0

(8:4)

i¼1

Thus, only relative values of the weights become of significance. In Equation 8.4, the second term can be interpreted as a weighted sum of the gradients of n  1 upper-bound constraints. Thus, the noninferior solutions can be found by solving: min fr (x) subject to fl (x) < «l gi (x)  0

for l ¼ 1, 2 . . . , n

(8:5a) and

where i ¼ 1, 2, . . . , p

l 6¼ r

(8:5b) (8:5c)

where «l are assumed values of the objective functions that must not be exceeded. Thus, the central idea of the «-constraint method is to minimize or maximize one objective (usually the most preferred) while considering all other objectives as constraints bounded appropriately (i.e., upper or lower bound in case of minimization and maximization objective respectively). The resulting single-objective optimization can be solved by a suitable method for different «-values. Interestingly, a few studies considered MOO and used «-constraint method without mentioning them. For example, Therdthai et al. (2002) optimized the bread oven temperature to minimize the weight loss during baking for several values of baking times. Obviously, it is desirable to reduce the baking time, which is thus the second objective but was considered as a constraint in this study. Before applying the «-constraint method, each objective in the MOP should be optimized individually in order to find its optimal value, which will be useful for selecting suitable «-values. Hence, the evaporator system is first optimized to obtain the minimum exergy consumption. Using the Solver tool in Excel, the optimal solution obtained is 2.29  1013 Joules per year and the optimal values of the decision variables are TS ¼ 383.15 K, Ti ¼ 343.15 K, and Tv ¼ 343.15 K. Thus, for minimizing the exergy consumption, a combination of (1) the lowest temperature of steam, (2) the highest saturated temperature of vapor (corresponding to the least vacuum), and (3) the most heat recovery from the steam condensate are required. Next, the evaporator system is optimized to obtain the minimum possible investment cost. Using the Solver tool again, the minimum annualized investment cost is found to be 2.27  105 $=year. The optimum values of the decision variables are TS ¼ 443.15 K, Ti ¼ 298.15 K, and Tv ¼ 313.15 K. The optimal solution coincides with the expectation that the evaporator should be operating at the maximum temperature difference in order to minimize heat transfer area and that there should be no preheater. Now, the «-constraint method is used to solve the MOP. By fixing the exergy consumption as the primary objective, the problem is solved by converting the investment cost into a constraint. The investment cost is constrained such that it is less than or equal to «. The resulting single-objective optimization is solved for different «-values above the minimum investment cost found. The single-objective

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optimization can be done using the Solver tool in Excel. Several initial guesses should be tried to ensure that the global=correct solution is obtained. Premature convergence, perhaps to a local optimum, can be avoided by using more advanced programs such as the Solver Premium provided by Frontline systems (http:==www.solver.com) or the What’sBest Solver provided by Lindo Systems (http:==lindo.com). The single-objective optimization problems for the evaporator system can be solved with little difficulty. The results obtained for the evaporator system optimization for two objectives by the «-constraint method are given in Table 8.3. The two objectives are found to be conflicting with a decrease in the annual exergy consumption possible only with a corresponding increase in the annual investment cost. From minimum exergy consumption at TS ¼ 383.15 K, Ti ¼ 343.15 K, and Tv ¼ 343.15 K, Ti and Tv decrease quickly as exergy consumption increases and annual investment cost falls. This continues until Ti and Tv reach the respective lower bound of 298.15 K and 313.15 K. Where possible (i.e., before Tv reaches its lower bound of 313.15 K), Ti is equal to Tv. Any further decrease in the annual investment cost is then possible only with an increase in TS. This preference to reduce Ti before increasing TS can be explained by the smaller size of the preheater as compared to the evaporator. A fall in Ti reduces the duty of the preheater, reducing its size. This duty is then shifted over to the evaporator. Due to the large size of the evaporator, the increase in the evaporator cost is more than compensated for by the reduced preheater cost. The MOP for the evaporator system is then solved using a multi-objective genetic algorithm, namely, the Nondominated Sorting Genetic Algorithm II TABLE 8.3 Optimal Results for the Evaporator System Design for Two Objectives Using the «-Constraint Method and the Solver Tool Exergy Consumption (J=year) 3.6369E þ 13 3.4283E þ 13 3.0201E þ 13 2.6906E þ 13 2.4211E þ 13 2.3894E þ 13 2.3820E þ 13 2.3680E þ 13 2.3501E þ 13 2.3303E þ 13 2.3099E þ 13 2.2914E þ 13 a

Investment Cost (Specified «-Value in Brackets) ($=year) a

226946.4 240000.4 (240000) 269999.9 (270000) 299999.8 (300000) 329999.8 (330000) 360000.0 (360000) 390000.3 (390000) 419999.9 (420000) 449999.9 (450000) 480000.4 (480000) 510000.2 (510000) 537171.7a

TS (K)

Ti (K)

Tv (K)

443.15 431.78 411.14 395.86 384.18 383.15 383.15 383.15 383.15 383.15 383.15 383.15

298.15 298.15 298.15 298.15 298.15 314.94 321.59 327.34 332.28 336.54 340.23 343.15

313.15 313.15 313.15 313.15 313.15 314.94 321.59 327.34 332.28 336.54 340.23 343.15

No «-value is used. Optimization is done for minimum investment cost (first row) or for minimum exergy consumption (last row).

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(NSGA-II). This can easily be done using any of the available software described later in this chapter. The results obtained using NSGA-II in the jMetal library (see Section 8.5) are compared with those obtained by the «-constraint method along with the Solver in Figure 8.2. Both methods yield similar results for the evaporator system optimization. An ‘‘elbow’’ is observed when the objectives are compared (Figure 8.2). The steep slope prior to the elbow indicates that the investment cost can be greatly reduced with a less than proportional increase in the exergy consumption.

Investment cost ($/year)

Epsilon constraint

JMetal NSGA-II

600,000 500,000 400,000 300,000 200,000 2E+13

2.5E+13 3.5E+13 3E+13 Exergy consumption (J/s)

4E+13

2.5E+13

4E+13

460 Ts

440 420 400 380 2E+13

3E+13

3.5E+13

Exergy consumption (J/s) 350 Ti

330 310 290 2E+13

2.5E+13

3E+13

3.5E+13

4E+13

Exergy consumption (J/s) 350 Tl

340 330 320 310 2E+13

2.5E+13

3E+13

3.5E+13

4E+13

Exergy consumption (J/s)

FIGURE 8.2 Results of the MOO of the evaporator system: the first plot shows the trade-off between the two objectives, and the remaining three plots depict variation of decision variables with the objective: exergy consumption.

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The steep decline in the investment cost coincides with declines in the evaporation temperature, Tv and the feed preheat temperature, Ti. Beyond the elbow, the steep increase in the exergy consumption coincides with increasing steam temperaure, TS. If traditional optimization for only one objective is used, only two solutions with minimum investment cost and minimum exergy consumption (the first and last row in Table 8.3) would be obtained. On using the MOO methods, many Pareto-optimal solutions shown in Table 8.3 and Figure 8.2 are obtained. The decision maker (DM) can then select one of them knowing fully the trade-off among the objectives and the trends in the decision variables. Although experts may be able to predict the trade-off among some objectives qualitatively, quantitative variation in objectives and decision variables for realistic applications can only be obtained through MOO.

8.3 OVERVIEW OF MOO METHODS MOO involves two stages: optimization of the objectives and the process of deciding what ‘‘trade-offs’’ among the conflicting objectives are appropriate from the perspective of a DM. Note that the DM can be one or more managers or engineers who are responsible for selecting one optimal solution for actual implementation. There are many methods of classifying MOO techniques. One particular classification is based firstly on whether many solutions are generated and then subdivided according to when and how the DM influences the optimization process (Miettinen, 1999; Rangaiah, 2008). Accordingly, MOO methods are divided into two broad categories: generating methods and preference-based methods (Figure 8.3). Generating methods can be subdivided into three categories: no-preference methods, a posteriori methods using scalarization approach, and a posteriori methods

Multi-objective optimization methods

Preferencebased methods

Generating methods

No-preference methods (e.g., global criterion and multi-objective proximal bundle methods)

A posteriori methods using scalarization approach (e.g., weighting method and e-constraint method)

A posteriori methods using multi-objective approach (e.g., non dominated sorting genetic algorithm, multi-objective simulated annealing)

FIGURE 8.3 Classification of MOO methods.

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A priori methods (e.g., value function method, lexicog raphic ordering, goal programming)

Interactive methods (e.g., interactive surrogate worth trade-off and NIMBUS)

using a multi-objective approach. They generate several or many solutions. Nopreference methods such as the global criterion method generate several solutions by minimizing a function that measures the deviation away from an ideal solution. A posteriori methods require no prior knowledge of the relationships between the objectives. In these methods, many Pareto-optimal (or nondominated or efficient) solutions are found, and then one of them is selected based on acceptable trade-offs and other considerations not included in the MOP. The available a posteriori methods can be subdivided into two subtypes—those using scalarization approach and those using multi-objective approach. The scalarization approach, a traditional approach, converts a MOP into a single-objective optimization problem that can be appropriately solved to yield a single Paretooptimal solution. This approach is then repeated several times to obtain other Pareto-optimal solutions. Two common methods using scalarization approach are the «-constraint method (used earlier for optimizing the evaporator system) and the weighting method. A posteriori methods using multi-objective approach generate and rank many trial solutions. Successful iterations usually replace poorer solutions with better ones. These methods have found many applications in food engineering (see Tables 8.4 through 8.7) and in chemical engineering (Masuduzzaman and Rangaiah, 2008). Preference methods are subdivided into two main categories, a priori methods and interactive methods. In these methods, inputs from the DM are considered before or during the optimization process; the DM has to set certain desired goals, which requires prior knowledge regarding the preferences with respect to each objective. Examples of a priori methods are value function methods, lexicographic ordering and goal programming. Interactive methods are also known as progressive preference articulation methods. In these methods, decision making and optimization are intertwined where partial preference information is provided upon which optimization occurs. These techniques normally operate in three stages (Coello Coello et al., 2002): (1) find a nondominated solution, (2) obtain the reaction and input of the DM regarding this solution and its improvements, and (3) repeat steps (1) and (2) until the DM is satisfied or until it is not possible to improve the solution further. Examples of interactive methods include the probabilistic trade-off development method, the sequential MOP solving method, and the NIMBUS. For more details on MOO methods including their relative merits, algorithms and applications, readers are referred to recent books devoted to MOO such as Miettinen (1999), Deb (2001), Coello Coello et al. (2002), and Tan et al. (2005).

8.4 MOO IN FOOD PROCESSING=INDUSTRY MOO has been used in the food industry. The earliest works in the food industry began in the late 1970s, with the bi-objective optimization method applied to determine the optimal flow pattern in a multieffect evaporator system reported by Nishitani and Kunugita (1979). From then until the year 2007, nearly 30 applications of MOO in the food industry have been reported in around 40 journal papers. However, apart from the early works by Nishitani and Kunugita (1979, 1983), further applications of MOO in the food industry were not reported in journals until 1995.

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TABLE 8.4 MOO Applications in Heat Processing of Food Number

Application

Objectives

1

A multieffect evaporator system for milk concentration

Minimization of heat transfer area and steam consumed

2

Single-effect evaporator for milk concentration

Minimization of exergy and total investment cost

3

Processing of apples by dewatering and impregnation (DIS) followed by air drying

Minimization of processing time and energy consumption, and maximization of product quality parameters: reduction of shrinkage, softening of texture, product stability, and preservation of color

Selected Decision Variables Flow pattern, heat transfer area in each effect (assumed to be the same in all effects), and steam consumption Steam temperature, temperature of the preheated feed entering the evaporator, and vapor temperature in the evaporator Solution concentration, solution temperature, soaking time, air relative humidity, and air drying time

Method=Comments

References

Enumeration of all flow patterns followed by selection of noninferior solutions

Nishitani and Kunugita (1979)

«-constraint method along with the max-sensitive method

Nishitani and Kunugita (1983)

Experimental data were modeled using multiple linear regression. MOO was carried out using these models and their visualization in the form of response surfaces

Themelin et al. (1997)

(continued)

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TABLE 8.4 (continued) MOO Applications in Heat Processing of Food Number

Application

Objectives

Selected Decision Variables

Method=Comments First principles model and nopreference method were employed. Both Kiranoudis et al. (1999) and Kiranoudis and Markatos (2000) optimized the design of a conveyorbelt dryer; the former with only the product color deterioration parameters as the objectives whereas the latter included unit product cost as well. Similar studies for a fluidized-bed dryer were reported by Krokida and Kiranoudis (2000a and b) Weighting method was first used to scalarize the problem. The centered finite differences method and the control parameterization method were then used to solve the problem

4

Dryers for sliced potato

Minimization of color deterioration and minimization of unit product cost

Drying air temperature and humidity, and drying air temperature change through the conveyor-belt

5

Optimal control of enzyme drying in a batch fluidized-bed dryer

Maximization of profit (which increases with product quality) and minimization of energy costs

Trajectories of air temperature and humidity

References Kiranoudis et al. (1999); Kiranoudis and Markatos (2000); Krokida and Kiranoudis (2000a and b)

Quirijns et al. (2000)

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6

Rice drying

Maximization of final product quality and minimization of drying time

Air temperature and relative humidity as functions of time

7

Food processing by conduction heating

Minimization of surface cook value and processing time

Variable retort temperature profile represented by sine and exponential functions

8

Food processing by conduction heating

Variable retort temperature profile

9

Blanching-freezing system

Maximization of volumeaverage retention of thiamine for two geometries: spherical and finite cylinders Productivity, costs, quality, and treatment flexibility

Different types of blanchers and freezers

The first principles model was validated and then used for optimization by sequential quadratic programming along with «-constraint method An artificial neural network model was developed based on simulated data from the first principles model, and then used for optimization by a genetic algorithm along with «-constraint method The weighting method and lexicographic ordering were used along with a modified complex method

Olmos et al. (2002)

An analytical hierarchy process was employed for choosing the optimal system

Bevilacqua et al. (2004)

Chen and Ramaswamy (2002)

Erdogdu and Balaban (2003)

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TABLE 8.5 MOO Applications in Fermentation Number

Selected=Independent Decision Variables

Application

Objectives

1

Table olive preparation systems

Minimization of total investment cost and annual product cost, both per ton of prepared olive

2

Estimation of model parameters in ethanol production

Minimization of the mean least squares error in two or more experiments

3

Gluconic acid production

Maximization of productivity and final concentration of gluconic acid, and minimization of the final substrate concentration

Batch time, initial substrate concentration, and initial biomass

4

Selective product enhancement for Aspergillus niger fermentation

Maximization of catalase enzyme and minimization of protease enzyme, and vice versa

Sucrose and nitrogen concentrations in the feed, initial broth volume and trajectories of sucrose, nitrogen and hydrogen peroxide additions

Percentage of olive prepared as green olive, percentage of factory total capacity used for preparation of green olive, quantity of prepared olive per year, and total capacity of the factory Eleven parameters in the kinetic model for ethanol production using a high-ethanol tolerant yeast

Method=Comments

References

The problem, including cost equations and data, was presented in detail. The nonconvex MOO problem was solved using a modified weighting method

Kopsidas (1995)

Weighted min–max method was used to convert the MOO problem into a single-objective problem and then solved by hybrid differential evolution An evolutionary algorithm was used to generate Pareto-optimal solutions, which were then ranked using net flow method. Effect of controlling the overall mass transfer coefficient was studied. Halsall-Whitney and Thibault (2006) studied three algorithms for generating Paretooptimal solutions «-constraint method along with differential evolution was employed for solving the MOO. Simulation results were verified experimentally

Wang and Shue (2000)

Halsall-Whitney et al. (2003); Halsall-Whitney and Thibault (2006)

Mandal et al. (2005)

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5

Fed-batch bioreactors for (a) lysine and (b) protein by recombinant bacteria

6

Batch plant design for the production of four recombinant proteins: insulin, vaccine for Hepatitis B, chymosin (a food-grade protein), and cryophilic protease (a detergent enzyme)

Maximization of lysine productivity and yield Maximization of protein production and minimization of inducer volume added Nutrient (glucose) feeding rate and Inducer feeding rate for protein production Four cases of 2 and 3 objectives from minimization of investment cost, and environmental impact (EI) due to biomass, and EI due to solvent

Feed rate trajectory and final time for lysine production

NSGA-II was used for solving the multi-objective optimal control problem in the two applications, which were studied as singleobjective optimization problems in the earlier studies

Sarkar and Modak (2005)

Number and size of the different equipments, and operating variables used in the process

Dietz et al. (2006)

Three cases: maximization of net present value (NPV); maximization of NPV and minimization of product delay=advance criterion; and maximization of NPV and flexibility criterion, and minimization of product delay=advance criterion

Number and size of the different equipments, and operating variables used in the process

Discrete event simulator for simulating and checking feasibility of the batch plant and the multiobjective GA (MOGA) were used. Both mono- and multiproduct scenarios were considered for each case A fuzzy approach was proposed to account for uncertain demand, in the optimization of batch plant design for multiple objectives by MOGA

Dietz et al. (2008)

(continued)

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TABLE 8.5 (continued) MOO Applications in Fermentation Number

Selected=Independent Decision Variables

Application

Objectives

7

Optimal design of a bioreactor for growing Saccharomyces cerevisiae in sugar cane molasses

Maximization of profit and minimization of fixed capital investment

Height of the fermentor, feed flow rate, concentrated feed flow rate, and specific growth rate

8

Metabolic pathways for ethanol production by Saccharomyces cerevisiae

Maximization of ethanol production and minimization of the five dependent metabolic concentrations

Maximum enzyme activities characterizing different metabolic pathways

9

Glutamine production

Maximization of glutamine concentration and glutamate concentration

Concentrations of glucose and ammonium sulfate in the production medium

Method=Comments NSGA-II, normal boundary intersection (NBI) and normalized normal constraint methods were used, and their performance was compared and discussed. Bifurcation analysis to assess the stability of Pareto-optimal solutions is suggested to aid in the selection of a compromise solution Five different MOO methods: weighted sum method, goal attainment method, NBI method, multi-objective indirect optimization method, and multiobjective evolutionary algorithm were used for the case study, and their relative performance was discussed Response surface methodology combined with desirability function approach (involving weights) was used to solve the MOO problem to determine the optimal medium. The resulting solution has higher product yields and lower costs, compared to the original medium

References Sendin et al. (2006a)

Sendin et al. (2006b)

Li et al. (2007)

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TABLE 8.6 MOO Applications in Separation Processes Number

Selected=Independent Decision Variables

Application

Objectives

1

Dialysis of beer to produce low-alcohol beer using hollow-fiber membrane modules

Two cases: (1) maximization of alcohol removal from beer and minimization of ‘‘taste chemicals or extract’’ removal and (2) maximization of alcohol removal, and minimization of ‘‘taste chemicals or extract’’ removal and cost

Flow rate of pure water on the shell side, inner radius of a single hollow fiber, length of the fiber, fractional free area in the shell, and thickness of the hollow fiber membrane

2

Membrane filtration of wine

Maximization of quality parameters and permeate filtration flux

Membrane pore size, recycle flow rate, and type of pretreatment

3

Glucose-fructose separation using simulated moving bed (SMB) and Varicol processes

Two cases: (1) maximization of fructose productivity and purity and (2) maximization of glucose productivity and fructose productivity

Switching period, raffinate flow rate, and eluent flow rate

Method=Comments NSGA was used for solving bi-objective problems. For the tri-objective problem, «-constraint was first used to reduce it to a bi-objective problem, whose solution by NSGA gave a unique solution for each value of «. The inner radius of the hollow fiber was found to be the most important decision variable in most cases Measured data on objectives and decision variables were regressed. Minimum loss method (similar to Weighting method) was used for solving the MOO problem for three different cases of champagne and wine produced from different sources NSGA method was used for both operation and design optimization. This is one of the three applications presented in Yu et al. (2004)

References Chan et al. (2000)

Gergely et al. (2003)

Subramani et al. (2003); Yu et al. (2004)

(continued)

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TABLE 8.6 (continued) MOO Applications in Separation Processes Number

Application

4

SMB bioreactor for high fructose syrup by glucose isomerization

5

SMB bioreactors for sucrose inversion to produce fructose and glucose

Objectives

Selected=Independent Decision Variables

Maximization of throughput and minimization of desorbent consumption

Operating parameters of the different operating schemes

Maximization of throughput, product purity and recovery of the valuable component recovery, and minimization of solvent consumption in the desorbent stream Maximization of productivity of fructose and minimization of desorbent used

Four zone velocities and step time

Two sets: (1) maximization of concentrated fructose productivity and minimization of solvent consumption and (2) maximization of fructose mass flow rate and minimization of solvent consumption

Three cases with the same objectives but different decision variables from: switching time, volumetric flow rate in zone III, volumetric flow rate of desorbent, separator length, reactor Length, and number of columns in separator zones 2 and 3 Three or more decision variables from: switching time, raffinate flow rate, desorbent flow rate, column length, and configuration

Method=Comments

References

A superstructure optimization problem for SMB process is considered. «-constraint method and an interior point optimizer (IPOPT) for solving single-objective problems were used This study includes more objectives than the previous studies on SMB, where two or three objectives were considered. NIMBUS with IPOPT was employed for solving the MOO problem NSGA-II with jumping genes (NSGA-II-JG) was used for MOO of both operation and design of the SMB bioreactor. The SMB bioreactor used is a separative bioreactor

Kawajiri and Biegler (2006)

NSGA-II-JG method was used for MOO of several cases: operation and design of SMB bioreactors and modified SMB bioreactors including reactive Varicol system

Kurup et al. (2005)

Hakanen et al. (2007)

Zhang et al. (2004)

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TABLE 8.7 Miscellaneous Applications of MOO in Food Engineering Number

Application

Objectives

1

Extrusion process in cattle food granulation

2

Optimization of process conditions for the modification of starch

Minimization of product moisture content, product friability, and process energy consumption Maximization of conversion and product quality

3

Proportional-integral (PI) controller design

Minimization of integral of time weighted absolute error (ITAE), integral of square of manipulated variable changes (ISDU), and settling time of a controller

4

Properties of whey protein-methyl cellulose films

Minimization of water vapor permeability, and maximization of tensile strength and elongation

Selected=Independent Decision Variables Flour temperature and drawplate profile

System temperature, water fraction, starch fraction, molar ratio of NaOH to sodium monochloroacetate (SMCA), molar ratio of SMCA to starch and reaction time PI controller parameters, namely, proportional gain and integral time

Ratio of glycerol to total polymers

Method=Comments

References

Both Massebeuf et al. (1999) and Mokeddem and Khellaf (2007) described a diploid GA and used it for the same application Pareto sets were generated from experimental data using the backward elimination strategy to eliminate factors. Optimization was done using MS Excel For generating Pareto-optimal solutions, single and dual population evolutionary algorithms (SPEA and DPEA) were found to be more efficient than grid search algorithm (GSA) when the optimization problem has many decision variables. DPEA was found to be more robust and faster than the other two methods Complex method was used in combination with the weighting method. The sensitivities of the solution were checked to ensure that the results were not sensitive to the weighting factors selected

Massebeuf et al. (1999); Mokeddem and Khellaf (2007) Tijsen et al. (1999)

Halsall-Whitney and Thibault (2006)

Turhan et al. (2007)

14

Number of journal articles

12 10 8 6 4 2 0 1979 –1995 1996 –1998 1999 –2001 2002–2004 2005 –2007

FIGURE 8.4 Number of journal papers on MOO applications in food engineering since 1979. Note that each bar except the first one is for a 3 year period.

From then, an increasing number of studies on MOO in food engineering can be found in journals (Figure 8.4). In fact, the number of journal papers on MOO in food engineering in the last 3 years has increased to 15 compared to 9 in the previous two 3-year periods. Most applications of MOO in the food industry involve either the optimization of plant operating conditions or the optimization of the plant design. The applications of MOO in the food industry can be grouped into four main categories: (1) heat processing, (2) fermentation, (3) separation processes, and (4) miscellaneous.

8.4.1 HEAT PROCESSING Food processing by application of heat is divided into different classifications based on the type of heating medium used (Fellows, 2000). Evaporation with steam as the heating medium, drying with hot air, and conduction heating are three areas in which MOO has been applied. Most applications of MOO in heat processing consisted of an economic objective and other objectives that include environmental and product quality objectives (Table 8.4). The earliest work of MOO in heat processing is the optimization of the flow pattern in a multi-effect evaporator by Nishitani and Kunugita (1979). Chen and Ramaswamy (2002) and Erdo gdu and Balaban (2003) explored the application of MOO to thermal treatment. The application of MOO to food drying is reported in five applications (Table 8.4): apple drying (Themelin et al., 1997), conveyer-belt dryer design (Kiranoudis and Markotos, 2000), fluidized bed dryer (Krokida and Kiranoudis, 2000a and b), control of a drying process (Quirjns et al., 2000), and batch drying of rice (Olmos et al., 2002). Quirjins et al. (2000) noted the importance of including spatial modeling in

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optimizing the control of the drying process. Olmos, et al. (2002) presented the use of MOO in the batch drying of rice, providing new insights into rice drying, and food quality preservation processes. Bevilacqua et al. (2004) used MOO to determine the best blanching-freezing system for multiple objectives (Table 8.4).

8.4.2 FERMENTATION Fermentation typically refers to the conversion of sugar to alcohol using yeast under anaerobic conditions. Steinkraus (1995) defined fermented foods as the ‘‘food substrates that are invaded or overgrown by edible microorganisms whose enzymes, particularly amylases, proteases, and lipases, hydrolyze the polysaccharides, proteins, and lipids to nontoxic products with flavors, aromas and textures pleasant and attractive to the human consumer. If the products of enzyme activities have unpleasant odors or undesirable, unattractive flavors or the products are toxic or disease producing, the foods are described as spoiled.’’ Fermentation in the food industry is usually represented by kinetic models. The fermentation process itself can often be represented by a mathematical model (Winkler, 1990). Product qualities often expressed in terms of the concentration of a desired acid, production rates, and substrate concentrations are often used as the objectives in MOO problems (Table 8.5). Decision variables include residence time in the bioreactor, concentration of microorganisms and operating temperature. The first application of MOO to fermentation was by Kopsidas (1995) in the design of table olive preparation systems (Table 8.5). Wang and Sheu (2000) used MOO to estimate the kinetic model parameters of yeast with high ethanol tolerance. The production of glutamine and gluconic acid were also optimized for multiple objectives (Halsall-Whitney et al., 2003; Halsall-Whitney and Thibault, 2006; Li et al., 2007). Recently, Sakar and Modak (2005) and Dietz et al. (2006, 2008) used MOO to optimize the design of multistage batch bioreactors (Table 8.5). Sakar and Modak (2005) discussed the usefulness of using MOO methods in the design of fed-batch bioreactors listing various conflicting objectives that are usually encountered in real-world situations. MOO was also used for product enhancement (Mandal et al., 2005), and Sendin et al. (2006a,b) discussed the MOO of fermentation of sugar cane molasses.

8.4.3 SEPARATION PROCESSES A wide range of MOO applications in food processing is related to separation processes (Table 8.6). Separation is done either to improve product quality and to obtain a particular extract or to preserve food. Chan et al. (2000) applied MOO to optimization of membrane separation modules. Mathematical models were developed and tuned using the experimental results. The use of MOO in wine filtration was studied by Gergely et al. (2003). The application of MOO to simulated moving beds (SMBs) for food processing can be found in six different studies (Subramani et al., 2003; Yu et al., 2004; Zhang et al., 2004; Kurup et al., 2005; Kawajiri and Biegler, 2006; Hakanen et al., 2007). MOO is particularly applicable to SMBs due to the conflicting objectives

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that are associated with the design and operation of SMBs. Of the six studies, three of them were accomplished using NSGA (Table 8.6). Owing to the complexities of SMBs, the optimization was computationally demanding. Hakanen et al. (2007) presented the use of NIMBUS to first reduce the initial problem into subproblems before solving, thus allowing the previous complex problem to be solved quickly.

8.4.4 MISCELLANEOUS APPLICATIONS This group covers the applications of MOO that do not fall into any of the above categories (Table 8.7). Tjisen et al. (1999) employed MOO for optimizing the carboxymethylation of potato starch, and Turhan et al. (2007) in the production of whey-protein methyl cellulose films. Halsall-Whitney and Thibault (2006) applied MOO to the design of controllers for a fermentation process. Massebeuf et al. (1999) and Mokeddem and Khellaf (2007) used the diploid genetic algorithm in the MOO of a food extrusion process (Table 8.7).

8.5 MOO SOFTWARE There are many types of software available for MOO. Only academic software, as most of them are freely available from the Web sites (accessed in late 2007), is summarized below in alphabetical order: jMetal library (http:==mallba10.lcc.uma.es=wiki=index.php=jMetal) is developed by Antonio J. Nebro and Juan J. Durillo. jMetal stands for Meta-heuristic Algorithms in Java, and it is an object-oriented Java-based framework aimed at facilitating the development, experimentation and study of meta-heuristics for solving MOPs. jMetal provides a rich set of classes which can be used as the building blocks of multi-objective meta-heuristics. Taking advantage of code reusing, the algorithms share the same base components such as implementation of genetic operators and density estimators thus facilitating the development of new multi-objective algorithms. Algorithms available in jMetal are: NSGA-II, strength Pareto evolutionary algorithm 2 (SPEA2), Pareto archived evolution strategy (PAES), optimized multiobjective particle swarm optimization (OMOPSO), archive-based hYbrid scatter search (AbYSS), multi-objective cellular genetic algorithm (MOCell), and Pareto envelope based selection algorithm (PESA-II). Constraints are handled through the use of penalty functions. jMetal comes with an extensive number of examples including binary, mixed integer, and constrained problems. Adapting the software to new problems is also extremely simple and requires minimal knowledge in Java programming. Multi-Objective Meta-Heuristics Library in Cþþ (MOMHLibCþþ) library (available at http:==www-idss.cs.put.poznan.pl=jaszkiewicz=MOMHLib=), developed by Andrzej Jaszkiewicz, is implemented in standard Cþþ and uses the standard template library (stl), which may not be available in old Cþþ compilers. However, use of stl makes the library portable. MOMHLibCþþ contains an extensive range of meta-heuristics including hybrid genetic algorithms, local search algorithms, simulated annealing, tabu search, and ant colony algorithms (e.g., Pareto simulated annealing, PSA; Serafini’s multi-objective simulated annealing,

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SMOSA; multi-objective genetic local search, MOGLS; Ishibuchi and Murata’s multi-objective genetic local search, IMMOGLS; multi-objective multiple start local search, MOMSLS; nondominated sorting genetic algorithm, NSGA and NSGA-II; and SPEA). Many of the algorithms in the MOMHLibCþþ library store Pareto-optimal solutions as an external nondominated set. Constraints are handled using penalty functions. This penalty is then reflected when individuals are ranked. The example used is the traveling salesman problem and is simple to use. It requires the user to compile the example code using a basic Cþþ programming environment. Extending the code to other MOPs requires some knowledge of Cþþ programming, and extensive tutorials are available from the Web site. NIMBUS (http:==nimbus.mit.jyu.fi=) has been developed at the University of Jyvaskyla, Department of Mathematical Information Technology. This is a fourth version of the World Wide Web interface. NIMBUS is an interactive multi-objective optimization system. It is suitable for both differentiable and nondifferentiable multiand single-objective optimization problems subject to nonlinear and linear constraints with bounds for the variables. The user guides the solution process by indicating which objectives should be improved by compromising which other objectives. NIMBUS can easily be applied to solve simple multi-objective test problems. However, keying in objective functions may be difficult especially if the objectives are complicated. Normal Boundary Intersection Method (NBI) software, available at http:==www. caam.rice.edu=indra=NBIhomepage.html, was developed by Aimin Zhou, Department of Computer Science, University of Essex. NBI is a technique, which is used to determine the portion of @ F, which contains the Pareto optimal points. The NBI software runs on MATLAB. The examples provided are easy to use, and adapting the code to new problems can be easily done easily. Besides the four software described above, several other programs are available on the Internet (Table 8.8); they typically consist of only one of the many algorithms found in either the MOMHLibCþþ or the jMetal library. Readers are also referred to the EMOO (Evolutionary MOO) Web site http:==www.lania. mx=ccoello=EMOO=EMOOsoftware.html maintained by Coello Coello for both algorithms, software and publications related to EMOO. The NSGA-II, MOPSO and the MOPSEA were downloaded from this Web site and successfully tested them. All these three are written in Cþþ and their use requires a basic foundation in C programming.

8.6 CONCLUSION MOO applications in food engineering have centered mainly on finding Paretooptimal solutions for product quality and cost objectives with many of them using MOO methods that generate multiple solutions. These methods are popular probably because they are readily available and effective at generating Pareto-optimal solutions. An increasing number of food engineering processes are modeled using first principles. Coupled with the conflicting objectives often present in the food industry and the availability of computational power, MOO is becoming increasingly attractive. This chapter has defined the MOP, outlined the different methods of solving MOO

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TABLE 8.8 Summary of Selected MOO Software Software GAtoolbox

Kanpur Genetic Algorithms Library

NSGA-II

SGALABbeta

SPEA

Comments=Link The GAtoolbox uses the NSGA-II algorithm written in Cþþ. The tutorial available at the link below is extremely comprehensive. Compiling, running, and creating new problems can be done easily. http:==www.illigal.uiuc.edu=pub=papers=IlliGALs=2007016.pdf The NSGA-II and the related algorithms (e.g., constraint handling) and publications can be found at this Web site. The NSGA-II program is packaged with a plotting tool. http:==www.iitk.ac.in= kangal=codes.shtml The following Web site has the NSGA-II algorithm for MATLab. Running and creating new problems can be done easily. http:==www.mathworks.com=matlabcentral=fileexchange=loadFile.do? objectId¼10351&objectType¼file This software contains a variety of genetic algorithms. http:==www. mathworks.com=matlabcentral=fileexchange=loadFile.do? objectId¼5882&objectType¼file The Strength Pareto Evolutionary Algorithm is written in Cþþ. The source code is available via this link: ftp:==ftp.tik.ee.ethz.ch=pub= people=zitzler=spea.cc

problems, introduced a simple MOO method, namely, the «-constraint method with an example, briefly reviewed various applications of MOO in food engineering, and finally reviewed the available MOO software. All these enable the reader to study MOO for the chosen applications.

REFERENCES Bevilacqua, M., D’Amore, A., and Polonara, F., A multi-criteria decision approach to choosing the optimal blanching-freezing system, J. Food Eng., 63, 253, 2004. Chan, C.Y. et al., Multi-objective optimization of membrane separation modules using genetic algorithm, J. Membrane Sci., 176, 177, 2000. Chen, C.R. and Ramaswamy, H.S., Modeling and optimization of variable retort temperature (VRT) thermal processing using coupled neural networks and genetic algorithms, J. Food Eng., 53, 209, 2002. Coello Coello, C.A., Veldhuizen, D.A., and Lamont, G.B., Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic, New York, 2002. Deb, K., Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, Singapore, 2001. Dietz, A. et al., Multiobjective optimization for multiproduct batch plant design under economic and environmental considerations, Comp. Chem. Eng., 30, 599, 2006. Dietz, A. et al., A fuzzy multiobjective algorithm for multiproduct batch plant: Application to protein production, Comp. Chem. Eng., 32, 292, 2008. Erdogdu, F. and Balaban, M.O., Complex method for nonlinear constrained multi-criteria (multi-objective function) optimization of thermal processing, J. Food Process Eng., 26, 357, 2003.

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Fellows, P., Food Processing Technology: Principles and Practice, Woodhead, Cambridge, 2000. Gergely, S., Bekassy-Molnar, E., and Vatai, G., The use of multiobjective optimization to improve wine filtration, J. Food Eng., 58, 311, 2003. Hakanen, J. et al., Interactive multi-objective optimization for simulated moving bed processes, Control and Cybern., 36, 50, 2007. Halsall-Whitney, H. and Thibault, J., Multi-objective optimization for chemical processes and controller design: Approximating and classifying the Pareto domain, Comp. Chem. Eng., 30, 1155, 2006. Halsall-Whitney, H., Taylor, D., and Thibault, J., Multicriteria optimization of gluconic acid production using net flow, Bioprocess Biosyst. Eng., 25, 299, 2003. Kawajiri, Y. and Biegler, L.T., Nonlinear programming superstructure for optimal dynamic operations of simulated moving bed processes, Ind. Eng. Chem. Res., 45, 8503, 2006. Kiranoudis, C.T. and Markatos, N.C., Pareto design of conveyor-belt dryers, J. Food Eng., 46, 145, 2000. Kiranoudis, C.T., Maroulis, Z.B., and Marinos-Kouris, D., Product quality multi-objective dryer design, Drying Tech., 17, 2251, 1999. Kopsidas, G.C., Multiobjective optimization of table olive preparation systems, Eur. J. Oper. Res., 85, 383, 1995. Krokida, M.K. and Kiranoudis, C.T., Pareto design of fluidized bed dryers, Chem. Eng. J., 79, 1, 2000a. Krokida, M. and Kiranoudis, C.T., Product quality multi-objective optimization of fluidized bed dryers, Drying Tech., 18, 143, 2000b. Kurup, A.S. et al., Optimal design and operation of SMB bioreactor for sucrose inversion, Chem. Eng. J., 108, 19, 2005. Li, J. et al., Medium optimization by combination of response surface methodology and desirability function: An application in glutamine production, Appl. Microbiol. Biotechnol., 74, 563, 2007. Mandal, C.M., Gudi, R.D., and Suraishkumar, G.K., Multi-objective optimization in Aspergillus niger fermentation for selective product enhancement, Bioprocess Biosyst. Eng., 28, 149, 2005. Massebeuf, S. et al., Multicriteria optimization and decision engineering of an extrusion process aided by a diploid genetic algorithm, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, 1, 14, 1999. Masuduzzaman and Rangaiah, G.P., Multi-objective optimization applications in chemical engineering, in Multi-objective Optimization: Techniques and Applications in Chemical Engineering, Rangaiah, G.P., (Ed.), World Scientific, Singapore, 2008. Miettinen, K.M., Nonlinear Multi-objective Optimization, Kluwer Academic Publishers, Boston, MA, 1999. Mokeddem, D. and Khellaf, A., Pareto-optimal solutions for multicriteria optimization of a chemical engineering process using a diploid genetic algorithm, Comp. Chem. Eng., doi:10.1016=j.compchemeng.2006.12.006, 2007. Nishitani, H. and Kunugita, E., Optimal flow-pattern of multiple effect evaporator systems, Comp. Chem. Eng., 3, 261, 1979. Nishitani, H. and Kunugita, E., Multiobjective analysis for energy and resource conservation in an evaporation system, ACS Symposium Series, 333, 1983. Olmos, A. et al., Dynamic optimal control of batch rice drying process, Drying Tech., 20, 1319, 2002. Quirijns, I.E.J. et al., The significance of modelling spatial distributions of quality in optimal control of drying processes, J. A. Benelux Quarterly J. Automatic Contro, 41, 56, 2000. Rangaiah, G.P., Multi-objective Optimization: Techniques and Applications in Chemical Engineering, World Scientific, Singapore, 2008.

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Sarkar, D. and Modak, J.M., Pareto-optimal solutions for multi-objective optimization of fedbatch bioreactors using nondominated sorting genetic algorithm, Chem. Eng. Sci., 60, 481, 2005. Sendin, J.O.H. et al., Improved optimization methods for the multiobjective design of bioprocesses, Ind. Eng. Chem. Res., 45, 8594, 2006a. Sendin, O.H. et al., Model based optimization of biochemical systems using multiple objectives: a comparison of several solution strategies, Mathematical and Computer Modelling of Dynamical Systems, 12, 469, 2006b. Steinkraus, K.H., Handbook of Indigenous Fermented Foods, Marcel Dekker, Inc., New York, 1995. Subramani, H.J., Hidajat, K., and Ray, A., Optimization of simulated moving bed and varicol processes for glucose-fructose separation, Chem. Eng. Res. Design, 81, 549, 2003. Tan, K.C., Khor, E.F., and Lee, T.H., Multiobjective Evolutionary Algorithms and Applications, Springer, London, UK, 2005. Tarafder, A., Rangaiah, G.P., and Ray, A.K., A Study of finding many desirable solutions in multi-objective optimization of chemical processes, Comp. Chem. Eng., 31, 1257, 2007. Themelin, A. et al., Multicriteria optimization of food processing combining soaking prior to air drying, Drying Tech., 15, 2263, 1997. Therdthai, N., Zhou, W., and Adamczak, T., Optimization of the temperature profile in bread baking, J. Food Eng., 55, 41, 2002. Tijsen, C.J. et al., Optimisation of the process conditions for the modification of starch, Chem. Eng. Sci., 54, 2765, 1999. Turhan, K.N. et al., Optimization of glycerol effect on the mechanical properties and water vapor permeability of whey protein-methylcellulose films, J. Food Process Eng., 30, 485, 2007. Walstra, P., Wouters, J.T., and Geurts, T., Dairy Science and Technology, Taylor and Francis, Singapore, 2006. Wang, F.S. and Sheu, J.W., Multiobjective parameter estimation problems of fermentation processes using a high ethanol tolerance yeast, Chem. Eng. Sci., 55, 3685, 2000. Winkler, M., Problems in fermenter design and operation, in Chemical Engineering Problems in Biotechnology, Winkler, M., Ed., Essex: Elsevier Science, 215, 1990. Yu, W. et al., Application of multi-objective optimization in the design of SMB in chemical process industry, J. Chinese Inst. Chem. Eng., 35, 141, 2004. Zhang, Y., Hidajat, K., and Ray, A.K., Optimal design and operation of SMB bioreactor: Production of high fructose syrup by isomerization of glucose, Biochem. Eng. J., 21, 111, 2004.

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9

Applications of the Minimum Principle of Pontryagin for Solving Optimal Control Problems Andrey V. Kuznetsov

CONTENTS 9.1 9.2

Introduction ................................................................................................ 179 Optimal Control Problem for the Bioheat Equation.................................. 181 9.2.1 Mathematical Formulation .............................................................. 181 9.2.2 Thermal Dose Assessment .............................................................. 184 9.2.3 Numerical Results ........................................................................... 184 9.3 Optimization of the Inlet Temperature Distribution in a Porous Packed Bed for Maximizing its Heat Storage Capacity ............................ 186 9.3.1 Mathematical Formulation .............................................................. 187 9.3.2 Optimal Control Problem ................................................................ 189 9.3.3 Numerical Results ........................................................................... 191 9.4 Optimal Control of the Heat Generation Rate for Conductive Heating of a One-Dimensional Plate ......................................................... 192 9.4.1 Mathematical Formulation .............................................................. 192 9.4.2 Optimal Control Problem ................................................................ 193 9.4.3 Numerical Results ........................................................................... 194 References ............................................................................................................. 195

9.1 INTRODUCTION The aim of this chapter is to give a simple and practical introduction to the utilization of the minimum principle of Pontryagin to engineering systems without concentrating too much on extremely sophisticated mathematical tools that are typically used in the infinite dimensional control theory (Yong and Li, 1995). The problems investigated in this chapter are diffusion-type (heat transfer) problems. These problems are

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typically described by parabolic and elliptic partial differential equations and, unlike problems governed by ordinary differential equations (an introduction to such type of problems is given, for example, in Hocking, 1991), are characterized by the infinitely dimensional space framework. There is significant potential in using optimal control methods for improving and optimizing different food processes. An optimal control problem resulting from modeling sterilization of food was considered in Kleis and Sachs (2000). Sterilization is the process of destruction of microorganisms by heating. This process is typically carried out by heating prepackaged food in an autoclave. The temperature in the autoclave (which determines the surface boundary condition for the can) is used as a control variable. The objective of the optimal control problem solved by Kleis and Sachs (2000) was to retain a maximum of vitamins subject to the constraint that a prescribed level of sterility was obtained. Saa et al. (1998) considered two types of optimization problems related to microwave heating of foods: maximum temperature and maximum quality uniformity (the quality uniformity was measured by uniformity of the ‘‘cook’’ C-value, a parameter that was defined as the equivalent number of minutes the product would have to spend at a reference temperature, usually 1008C, to achieve a given effect on the product quality, see Ohlsson, 1980). Banga et al. (2003a,b) reviewed applications of modern optimization methods to bioreactors, dynamic optimization of thermal processing as well as dynamic optimization of the drying process identifying relevant optimization problems and barriers to optimization. Tayeb and Lim (1986) developed an algorithm for determining the optimal glucose feeding rate in fed-batch penicillin fermentation. Glucose has an inhabitation effect on penicillin production, and overfeeding leads to inhabitation of the process. On the other hand, underfeeding also results in the reduction of penicillin production rate. The optimization strategy developed in Tayeb and Lim (1986) was based on determination of necessary optimality conditions for the process. VanImpe et al. (1992) applied the optimal control theory for determining the optimal glucose feed rate for the penicillin-G fed-batch fermentation. The authors used a modified unstructured mathematical model that allowed smooth transition between maintenance and endogenous metabolisms. Cuthrell and Biegler (1987) reviewed optimization strategies for chemical engineering systems described by simultaneous differential and algebraic equations. A numerical method for solving complex differential–algebraic optimization problems developed by Cuthrell and Biegler (1987) was based on discretizing the differential equations obtained using polynomial approximation and orthogonal collocation. The resulting algebraic equations were then solved by standard nonlinear programming methods with a successive quadratic programming technique. Park and Ramirez (1988) utilized the minimum principle of Pontryagin to develop an optimal control policy for the maximization of the secreted protein in a fed-batch bioreactor for a specified final time. It was shown that pumping up of the fresh media at the maximum rate close to a final time increased the amount of secreted protein culture. Tartakovsky et al. (1995) developed an optimal control strategy of fed-batch fermentation for the case of autoinduction of metabolite

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production (when cells, under condition of growth limitation, produced an inducer that initiated the synthesis of a product). The time of induction had to be optimized so that enough biomass was produced prior to the induction while nutrients were available to sustain the synthesis. Konde and Modak (2007) developed a new methodology for optimization of a continuous cascade bioreactor system, which allowed for multilevel manipulation and possessed a potential for increased productivity. The developed method maximized the gradient of the objective function with respect to all decision variables. In this chapter, the minimum principle of Pontryagin is introduced utilizing three heat transfer problems. The first one is an optimal control problem for heating a layer of tissue in order to destroy a cancerous tumor, the second one is the optimization of heat storage in a porous body, and the third one is optimizing heating a one-dimensional plate with one face in perfect thermal contact with a layer of a well-stirred fluid.

9.2 OPTIMAL CONTROL PROBLEM FOR THE BIOHEAT EQUATION Hyperthermia is a noninvasive method of cancer treatment to destroy solid tumors by heating them to temperatures ranging from 558C to 908C. The purpose of the heating is to induce thermal coagulation and necrosis of the tumor (McCann et al., 2003). The volumetric heating, Qr(t), of the tumor region is achieved by applying an ultrasound beam over the tissue domain (Diederich et al., 2000; Nau et al., 2001), using microwave heating (Converse et al., 2004), or utilizing a radiofrequency generator (Takahashi et al., 2000). Loulou and Scott (2002) suggested the utilization of the conjugate gradient method for thermal dose optimization in hyperthermia.

9.2.1 MATHEMATICAL FORMULATION Consider the following optimal control problem (Kuznetsov, 2006). For a given target location in the tissue, x ¼ xtarget, maximize the temperature at this location that results from the exposure to a spatially uniform (but time-dependent) volumetric heat generation, Qr(t):
 T xtarget , tf ! max

(9:1)

It is assumed that the total volumetric heat generation over the duration of the procedure is fixed: ðtf Qr ðt Þdt ¼ Qtotal

(9:2)

0

where Qtotal is a given constant (determined from the condition that the damage to a healthy tissue does not exceed safe limits). Also, Qr(t) cannot be smaller than a certain minimum value (typically zero) and cannot exceed a certain maximum value: Qmin  Qr ðt Þ  Qmax

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(9:3)

The temperature distribution in the tissue, T(x, t), is given by the solution of Pennes bioheat equation: rc

@T @2T ¼ k 2 þ vb rb cb ðTa  T Þ þ Qm þ Qr ðt Þ @t @x

(9:4)

where c is the specific heat of the tissue cb is the specific heat of blood k is the thermal conductivity of the tissue Qm is the metabolic heat generation Qr (t) is the heat generation due to spatial heating (for example, by an ultrasound beam) T is the tissue temperature Ta is the arterial temperature (assumed to be constant) r is the density of the tissue rb is the density of blood vb is the blood perfusion It is assumed that initially the temperature distribution in the tissue, T0(x), corresponds to a steady-state solution of Equation 9.4 with Qr ¼ 0 with the following boundary conditions: x ¼ 0: k

dT0 ¼ h½T1  T0  dx

x ¼ L : T0 ¼ Ta

(9:5a) (9:5b)

where T1 is the ambient temperature of the surrounding air. The solution for T0(x) is (Deng and Liu, 2002):  1=2
1=2  h
1=2  Qm ch A x þ k sh A x Qm vb r b c b A  T0 ð xÞ ¼ Ta þ vb rb cb A1=2 chðA1=2 LÞ þ hk shðA1=2 LÞ     Qm h 1=2 T  T  ð L  xÞ 1 a k vb rb cb sh A þ A1=2 chðA1=2 LÞ þ hk shðA1=2 LÞ

(9:6)

where A ¼ vbrbcb=k sh(ˆ) is a hyperbolic sine of ˆ ch(ˆ) is a hyperbolic cosine of ˆ Once the volumetric heating, Qr(t), is on at t ¼ 0, the transient solution of Equation 9.4 is given by (Deng and Liu, 2002):  vb rb cb t (9:7) T ð x, t Þ ¼ T0 ð xÞ þ W ð x, t Þ exp  rc

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where W(x, t) is given by the following equation: 3  1 v r c b b b t dj5dt W ð x, t Þ ¼ 4Qr ðt Þ G2 ð x, t; j, t Þ exp rc rc ðt

2

ðL

0

(9:8)

0

1 X

G2 ð x, t; j, t Þ ¼

n¼1

h i 2 2 2 b þ ð h=k Þ H ðt  t Þ   n h i exp ab2n ðt  t Þ L b2n þ ðh=kÞ2 þ h=k

 sin½bn ðL  xÞ sin½bn ðL  jÞ

(9:9)

where a ¼ k=(rc) is the thermal diffusivity

of the tissue 0, t < 0 H(t) is the Heaviside function, 1, t  0 bn are the positive roots of the following transcendental equation: bn cotðbn LÞ ¼ h=k

(9:10)

Utilizing the minimum principle of Pontryagin (Pontryagin et al., 1962; Athans and ^ r (t), is determined Falb, 1966; Kirk, 2004), the optimal volumetric heating rate, Q from the following optimality condition: ^ r ðt Þ½l1  Cðt Þ ! min Q

(9:11)

where l1 is the Lagrange multiplier (a constant in this case) C(t) is a function defined by  1 vb rb cb t dj exp Cðt Þ ¼ G2 xtarget , tf ; j, t rc rc ðL




(9:12)

0

Equation 9.11, applied accounting for constraints given by Equations 9.2 and 9.3, ^ r (t): results in the following solution for the optimal volumetric heat generation, Q ^ r ðt Þ ¼ Qmin Q

if l1  Cðt Þ > 0

(9:13a)

^ r ðt Þ ¼ Qmax Q

if l1  Cðt Þ < 0

(9:13b)

To calculate the Lagrange multiplier, l1, transcendental Equation 9.2 is be solved accounting for Equation 9.13a and b. To do this, first an interval that contains l1 is selected. Then, l1 is found by solving Equation 9.2 using an algorithm for finding a root of a transcendental equation.

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9.2.2 THERMAL DOSE ASSESSMENT The thermal dose at a given location, x, is calculated as (Sapareto and Dewey, 1984): 1 Dð xÞ ¼ 60

Nt ðf

RTr T ðx,tÞ dt

(9:14)

0

where tf is the heating duration Tr equals to 438C If N ¼ 1, Equation 9.14 assumes that the thermal dose is accumulated only between 0 and tf and neglects any thermal dose accumulation during the cool-off period. If thermal dose accumulation during the cool-off period is accounted for, N > 1. Equation 9.14 gives the time (in minutes) that would take to accumulate the thermal damage at 438C, which is equivalent to the damage that occurs at temperature T. According to Sapareto and Dewey (1984), Shih et al. (2003), Hahn et al. (1989), and Dewhirst et al. (2003), the onset of thermal damage occurs at approximately Donset ¼ 100 min although there is a difference between different tissues. R in Equation 9.14 is the piecewise constant function given by R¼

0:50 0:25

if T ð x, tÞ  Tr if T ð x, t Þ < Tr

(9:15)

9.2.3 NUMERICAL RESULTS ^ r , versus Figure 9.1a and b displays function  and optimal rate of heat generation, Q time, computed utilizing Equation 9.13a and b for the following parameter values: c ¼ cb ¼ 3,770 J=kg8C, Donset ¼ 100 min, k ¼ 0.50 W=m8C, h ¼ 6 W=(m28C), L ¼ 0.01 m, Qm ¼ 33,800 W=m3, Qmax ¼ 220,000 W=m3, Qmin ¼ 0 W=m3, of blood Qtotal ¼ 6.6  107 J=m3, Ta ¼ 378C, T1 ¼ 258C, Tr ¼ 438C, vb ¼ 1000 mmL 3 of tissues, 3 tf ¼ 600 s, xtarget ¼ 0.005 m, and r ¼ rb ¼ 998 kg=m . The optimality condition here is understood in terms of Equation 9.1, that is in terms of maximizing the temperature at xtarget ¼ L=2 ¼ 0.005 m at t ¼ tf ¼ 600s. Figure 9.2 displays the results for three different choices of controlling the volumetric energy generation: the optimal, displayed in Figure 9.1b (Qr ¼ 0, 0  t  tf=2; Qr ¼ Qmax, tf=2  t  tf), the constant (Qr ¼ Qmax=2, 0  t  tf), and the one when the maximum heat generation is used in the beginning of the process (Qr ¼ Qmax, 0  t  tf=2; Qr ¼ 0, tf=2  t  tf). Figure 9.2 demonstrates that the maximization of temperature is not necessarily equivalent to the maximization of the thermal dose. Figure 9.2a and b displays temperature and thermal dose in the tissue at the end of the process, at t ¼ tf ¼ 600 s. As expected, the utilization of the optimal rate of energy generation results in the highest temperature at t ¼ 600 s. However, as Figure 9.2b shows, the maximum thermal dose at t ¼ 600 s occurs for the case when the maximum

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5e−07 4.5e−07 4e−07 3.5e−07

Ψ

3e−07 2.5e−07 2e−07 1.5e−07 1e−07 5e−08 0 (a)

0

100

200

300 t

400

500

600

0

100

200

300 t

400

500

600

250,000 Qmax

ˆr Q

200,000

150,000

100,000

50,000

0 (b)

^ r (W=m3) FIGURE 9.1 (a) Function C and (b) optimal volumetric rate of heat generation, Q versus time (s). Optimality is understood in terms of Equation 9.1. (Reprinted from Kuznetsov, A.V., Int. Commun. Heat Mass Transfer, 33, 537, 2006. With permission.)

rate of energy generation is used in the beginning of the process. This is explained by thermal dose accumulation during the cool-off period. Indeed, the optimal heat generation and the one that gives the largest thermal dose at t ¼ 600 s are different only by the moment of time when the heat generation is turned on. For the optimal case, the heat generation is turned on 300 s after the beginning of the process while for the case that gives the largest thermal dose at t ¼ 600 s the heat generation is turned on right at the beginning of the process, at t ¼ 0. This suggests the largest thermal dose for the case of Qr ¼ Qmax, 0  t  tf =2; Qr ¼ 0, tf = 2  t  tf is the result of thermal dose accumulation during the last 300 s of the process, when the power is turned off.

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50 Qr = 0, 0 ≤ t ≤ t f /2 Qr = Qmax, t f /2 ≤ t ≤ t f

48 46

T

44 Q r = Qmax /2, 0 ≤ t ≤ t f

42 40

Q r = Qmax, 0 ≤ t ≤ t f /2 Q r = 0, t f /2 ≤ t ≤ t f

38 36 0

0.002

0.004

(a)

x

0.006

0.008

0.01

180 Q r = Qmax, 0 ≤ t ≤ t f /2

160

Q r = 0, t f /2 ≤ t ≤ t f

140 120

Q r = 0, 0 ≤ t ≤ t f /2

80

Q r =Qmax, t f /2 ≤ t ≤ t f

D

100

60 Q r =Qmax /2, 0 ≤ t ≤ t f

40 20 0 (b)

0

0.002

0.004

x

0.006

0.008

0.01

FIGURE 9.2 (a) Temperature, T (8C) and (b) thermal dose, D (min), versus x (m) in the tissue at t ¼ tf ¼ 600 s. (Reprinted from Kuznetsov, A.V., Int. Commun. Heat Mass Transfer, 33, 537, 2006. With permission.)

9.3 OPTIMIZATION OF THE INLET TEMPERATURE DISTRIBUTION IN A POROUS PACKED BED FOR MAXIMIZING ITS HEAT STORAGE CAPACITY This section follows Kuznetsov (1997a, 1998) and considers maximization of the storage capacity of a porous packed bed. Optimization problems in heat and mass

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transfer system have been extensively investigated (Mereu et al., 1993; Bejan, 1993, 1995; Bejan and Morega, 1994); the goal of this section is demonstrating how the minimum principle of Pontryagin can be used to solve this type of problems.

9.3.1 MATHEMATICAL FORMULATION The classical Schumann model of a packed bed (Schumann, 1929) which neglects the longitudinal conduction terms in both the fluid and solid phase energy equations is utilized. Following Carslaw and Jaeger (1959), equations governing the solid and fluid temperature distributions are expressed in the following dimensionless form: @u ¼fu @t L

@f @f þ ¼uf @t @z

(9:16) (9:17)

where the dimensionless temperature of the solid phase is defined as uðz, t Þ ¼

Ts  T1 T2  T1

(9:18)

and the dimensionless temperature of the fluid phase is defined as wðz, t Þ ¼

Tf  T1 T2  T1

(9:19)

where T1 and T2 are suitably chosen reference temperatures and L¼

«rf cf ð1  «Þrs cs

(9:20)

In Equation 9.20, cf is the specific heat of the fluid phase, cs is the specific heat of the solid phase, « is the porosity, rf is the density of the fluid phase, and rs is the density of the solid phase. The dimensionless time and streamwise coordinate in Equations 9.16 and 9.17 are defined as follows: t¼

hsf asf~t ð1  «Þrs cs

(9:21)

hsf asf~z «rf cf vf

(9:22)

and z¼

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where asf is the specific surface area common to the solid and fluid phases hsf is the fluid-to-particle heat transfer coefficient ~t is the time vf is the fluid filtration velocity ~z is the streamwise coordinate The exact solution of Equations 9.16 and 9.17 for the case when the inlet fluid temperature is a function of time and the initial temperature of a packed bed is a function of the space variable are obtained in White and Korpela (1979) and Spiga and Spiga (1981). White and Korpela (1979) utilized the following initial and boundary conditions: uðz, 0Þ ¼ u0 ð0Þ

(9:23)

wð0, t Þ ¼ win ðt Þ

(9:24)

Upon rearrangement, the solution obtained in White and Korpela (1979) can be presented as tLz ð

uðz, t Þ ¼ expðzÞ

h i fin ðt  Lz  t Þ expðt ÞI0 ð4tzÞ1=2 dt

0

"



ðz

þ expðLz  t Þ u0 ðzÞ þ u0 ðz  jÞ expðjÞ 0

t  Lz j

1=2

# 1=2 dj  I1 f4jðt  LzÞg 

(9:25)

ðz

h i fðz, t Þ ¼ expðLz  t Þ u0 ðz  jÞ expðjÞI0 f4jðt  LzÞg1=2 dj "

0 tLz ð

þ expðzÞ fin ðt  LzÞ þ  # 1=2 dt  I1 ð4tzÞ

fin ðt  Lz  t Þ expðt Þ 0

 z 1=2 t (9:26)

Equations 9.25 and 9.26 determine the temperature of the solid and liquid phases at a particular location z after this location has been reached by the temperature front propagating from the inlet boundary with a velocity vf, i.e., when t  Lz. Since the thermal conductivity in both phases is neglected, as long as t < Lz the solid phase temperature at this point equals to the initial temperature given in Equation 9.23.

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9.3.2 OPTIMAL CONTROL PROBLEM The inlet flow temperature, win(t), is considered as the control and the amount of thermal energy stored in the porous bed is utilized as the optimization criterion. In mathematical terms the optimality requirement is expressed as ðL Fðwin Þ ¼ uðz, tf Þdz ! max

(9:27)

0

where tf is a given duration of the process. The dimensionless width of a one~ is defined as dimensional porous bed of width L L¼

~ hsf asf L «rf cf vf

(9:28)

and the function u (z, tf) is determined by Equation 9.25. The functional F (win) must be maximized subject to the following constraints: A given amount of heat is supplied by the incoming fluid flow ðtf win ðt Þdt ¼ E ¼ constant

(9:29)

0

The inlet fluid flow temperature varies between a given minimum value, umin (corresponding to the fluid temperature in the ‘‘cold tap’’) and a given maximum value, umax (corresponding to the fluid temperature in the ‘‘hot tap’’). umin  win ðt Þ  umax

(9:30)

It is assumed that the initial temperature of the porous bed is uniform and constant. The reference temperature T1 in Equations 9.18 and 9.19 is set equal to the initial temperature of the porous bed; therefore, the dimensionless initial temperature of the solid phase is zero. This simplifies Equations 9.25 and 9.26; in this case the second term on the right hand side of Equation 9.25 and the first term on the right hand side of Equation 9.26 are equal to zero. Since there is local thermal nonequilibrium between the fluid and solid phases, the outlet fluid temperature is higher than the outlet solid temperature. Therefore, some heat that could be stored in a porous bed leaves it with the outgoing fluid flow. Satisfying the optimality requirement (Equation 9.27) is equivalent to minimizing this loss of thermal energy. In order to put this problem in the form of an optimal control problem Equation 9.25 is first rearranged by the following change of the integration variable: t* ¼ tf  t  Lz

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(9:31)

Accounting for Equation 9.31 and the assumption that the initial temperature of the porous bed is uniform, Equation 9.25 at the time t ¼ tf can be written as tf Lz ð

uðz, tf Þ ¼ expðzÞ

h i win ðt*Þ expðtf þ t* þ LzÞI0 f4ðtf  t*  LzÞzg1=2 dt*

0

(9:32) For tf < LL, the temperature front has not yet reached the outlet boundary of the porous bed, and no heat has yet been lost to the fluid leaving the porous bed. Therefore, only the case tf  LL is considered. The function C(z, t) is defined as follows: ( Cðz, t Þ ¼

h i expðz  tf þ t þ LzÞI0 f4ðtf  t  LzÞzg1=2 if 0  t  tf  Lz if t > tf  Lz

0

(9:33) Utilizing Equations 9.32 and 9.33, and changing the order of the integration, Equation 9.27 is recast as ðL

ðtf

Fðwin Þ ¼ uðz, tf Þdz ¼ win ðt Þðt Þdt ! max 0

(9:34)

0

where ðL ðt Þ ¼ Cðz, t Þdz

(9:35)

0

The problem given by Equations 9.29, 9.30, and 9.34 is an optimal control problem solved in Kuznetsov (1997a). The utilization of the minimum principle of Pontryagin (Pontryagin et al., 1962; Athans and Falb, 1966; Kuznetsov, 1997b; Kirk, 2004) results in the following optimality requirement: w ^in ðt Þ½l1  ðt Þ ! min

(9:36)

where l1 is the Lagrange multiplier. Utilizing Equation 9.36 together with constraint (Equation 9.30), leads to the following optimal temperature distribution: w ^in ðt Þ ¼ umin

if l1  ðt Þ > 0

w ^in ðt Þ ¼ umax

if l1  ðt Þ < 0

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(9:37)

The value of the Lagrange multiplier l1 is determined by solving transcendental Equation 9.29 accounting for Equation 9.37. The solution is carried out as follows: An interval that contains the desired value of l1 is first selected, and then an algorithm for finding a root of a transcendental equation within a given interval is applied to Equation 9.29.

9.3.3 NUMERICAL RESULTS Optimal inlet fluid temperature distributions satisfying constraint (Equation 9.29) are depicted in Figure 9.3. This figure is computed for the following parameter values: umin ¼ 0, umax ¼ 2, E ¼ tf, L ¼ 1, and L ¼ 0.05. Commutations are performed for three different durations of the process: tf ¼ 0.08, 0.2, and 2. Figure 9.3 shows that for a small duration of heating (tf ¼ 0.08) the optimal fluid inlet temperature w^in(t) first takes its maximum value umax and then its minimum value umin. As the duration of heating is increased (tf ¼ 0.2), a qualitative change in the behavior of the optimal inlet temperature takes place. Now, the optimal inlet temperature first takes its minimum value, then the maximum value, and then again the minimum value. As the heating duration is increased further (tf ¼ 2) this qualitative behavior remains, but 2

tf = 2

0

fˆ in

2

t f = 0.2

0 2

t f = 0.08

0 0

0.5

1

t /t f

FIGURE 9.3 The optimal fluid inlet temperature for different durations of heating as a function of time. (Reprinted from Kuznetsov, A.V., Int. J. Heat Mass Transfer, 40, 1720, 1997a. With permission.)

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the duration of the third segment decreases. Thus, Figure 9.3 shows that as the duration of heating increases, there is a transition from the first type of behavior of the optimal inlet temperature (maximum–minimum) to the second type (minimum– maximum–minimum). To understand qualitatively the reason for this transition two extreme cases, a very short and a very long duration of heating, are considered. For a very short duration of heating the temperature front has just reached the outlet boundary of the porous bed and the maximum–minimum behavior is obviously beneficial. This is because if the hot fluid is supplied in the beginning of the process, the time of contact of the hot fluid and the solid phase is the longest. Consequently, more thermal energy is transferred to the solid phase. Contrarily, for a very long duration of heating the hot fluid should be supplied at the end of the process (otherwise the cold fluid supplied at the end will ‘‘wash out’’ the thermal energy accumulated in the porous bed by the hot fluid supplied in the beginning). For an intermediate duration of heating the hot fluid should be supplied sometime between the beginning and the end of the process. This leads to the minimum– maximum–minimum behavior.

9.4 OPTIMAL CONTROL OF THE HEAT GENERATION RATE FOR CONDUCTIVE HEATING OF A ONE-DIMENSIONAL PLATE This section follows Kuznetsov (1997c) and considers heating a one-dimensional plate with one face in perfect thermal contact with a layer of a well-stirred fluid or a perfect conductor. The other face of the plate is kept at a constant temperature. Heating element in the fluid causes internal energy generation whose intensity can be controlled anywhere between given minimum and maximum values. The other natural constraints for this problem are a fixed amount of heat that can be generated in the fluid and a given duration of heating.

9.4.1 MATHEMATICAL FORMULATION A one-dimensional plate with its face ~x ¼ L in perfect thermal contact with mass Mf per unit area of a well-stirred fluid (or a perfect conductor) of specific heat cf is considered. The rate of thermal energy generation in the fluid per unit contact area with the plate is the control, Q(~t). There is no loss of heat from the fluid except to the plate. The heat balance at ~x ¼ L then results in the following equation (Carslaw and Jaeger, 1959): k

@T @T þ Mf cf ¼ Qð~t Þ @~x @~t

(9:38)

where k is the thermal conductivity of the plate. The boundary ~x ¼ 0 is kept at a constant temperature T0 which is the same as the initial temperature of the fluid. The temperature distribution in the plate at ~t ¼ 0 is also uniform and equal to T0. The solution of this problem is given in Carslaw and Jaeger (1959) as

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 ðt 1 X
 bn sinð bn xÞ exp b2n t

2  qðt Þ exp b2n t dt uð x, t Þ ¼ 2H 2 n¼1 cos bn bn þ H þ H

(9:39)

0

where dimensionless parameters are defined by the following equations: ~x k~t x¼ , t¼ , rs cs L2 L

q¼

QL , kðT1  T0 Þ

u¼

T  T0 , T1  T0

H¼

r s cs L M f cf

(9:40)

(reference temperature T1 is chosen to suitably normalize the problem). bn in Equation 9.39 are different positive roots of the following transcendental equation: bn tan bn ¼ H

(9:41)

9.4.2 OPTIMAL CONTROL PROBLEM The dimensionless rate of heat generation in the fluid, q(t), is considered as control. It is assumed to be a bounded, piecewise continuous function with a minimum value qmin (which can be, for example, zero) and the maximum value qmax (determined by the maximum power of the heat generating element). It is necessary to find the optimal control ^ q(t) that maximizes the amount of heat energy stored in the plate by the end of the process, t ¼ tf: ð1 FðqÞ ¼ uð x, tf Þdx ! max

(9:42)

0

under the following constraints: The amount of heat generated in the fluid over time tf is fixed: ðtf

^ ¼ constant qðt Þdt ¼ E

(9:43)

0

and given minimum and maximum powers of heat generation in the fluid: qmin  qðt Þ  qmax

(9:44)

To bring this problem to a standard form of optimal control problems, Equation 9.39 is used in Equation 9.42 and the integration order is changed: ðtf FðqÞ ¼ qðt ÞCðt Þdt ! max 0

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(9:45)

where Cðt Þ ¼ 2H

1 X n¼1

 1  cos bn

2  exp b2n ðtf  t Þ 2 cos bn bn þ H þ H

(9:46)

Solving the optimal control problem given by Equations 9.43 through 9.45 using the minimum principle of Pontryagin results in the following optimality requirement: ^ qðt Þ ¼ qmin ^ðt Þ ¼ qmax q

if l1  Cðt Þ > 0 if l1  Cðt Þ < 0

(9:47)

The Lagrange multiplier, l1, is found by solving Equation 9.43 for the case when q(t) satisfies the optimality requirement (Equation 9.47).

9.4.3 NUMERICAL RESULTS Computations are performed for the following parameter values: qmin ¼ 0, qmax ¼ 1, H ¼ 1, and Ê ¼ qmaxtf=2. Figure 9.4 shows that for a small duration of heating

1

tf = 2

0 1

qˆ

t f = 0.6

0 1

t f = 0.3

0 0

0.5

1

t/t f

FIGURE 9.4 The optimal energy generation rate for different durations of the process. (Reprinted from Kuznetsov, A.V., ZAMM, 77, 237, 1997c. With permission.)

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(tf ¼ 0.3) one must first use the maximum heating power, qmax, and then the minimum heating power, qmin. As duration of heating is increased (tf ¼ 0.6) there is a qualitative change in the way the optimal process should be carried on. In this case one must first use the minimum heating power, then the maximum power, and again the minimum power. As the duration of the process is increased further (tf ¼ 2), the qualitative behavior of the optimal control remains the same, but the duration of third segment decreases. This behavior of the optimal control is explained by again considering two extremes: a very short and a very long process. For a very short process almost no heat is lost through the boundary ~x ¼ 0, resulting in the optimality of the maximum– minimum behavior because this behavior maximizes the amount of heat transferred to the plate through the boundary ~x ¼ L (more time is available for heat transfer from the fluid to the plate). For a very long process the maximum power should be supplied at the end of the process, otherwise all of the heat energy will be wasted through the boundary ~x ¼ L instead of being accumulated in the plate. For an intermediate process duration the maximum power should be used somewhere between the beginning and the end of the process. This results in optimality of the minimum–maximum–minimum behavior.

REFERENCES Athans, M. and Falb, P.L., Optimal Control: An Introduction to the Theory and Its Applications, McGraw Hill, New York, 1966. Banga, J.R. et al., Improving food processing using modern optimization methods, Trends in Food Sci. Tech., 14, 131, 2003a. Banga, J.R. et al., Dynamic optimization of bioreactors: A review. Proc. Indian Natn. Sci. Acad., 69A, 257, 2003b. Bejan, A., How to distribute a finite amount of insulation on a wall with nonuniform temperature, Int. J. Heat Mass Transfer, 36, 49, 1993. Bejan, A., The optimal spacing for cylinders in crossflow forced convection, ASME J. Heat Transfer, 117, 767, 1995. Bejan, A. and Morega, A.M., The optimal spacing of a stack of plates cooled by turbulent forced convection, Int. J. Heat Mass Transfer, 37, 1045, 1994. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in Solids, Oxford University Press, London, United Kingdom, 1959. Converse, M. et al., Ultrawide-band microwave space-time beamforming for hyperthermia treatment of breast cancer: A computational feasibility study, IEEE Trans. Microwave Theor. Tech., 52(part 2), 1876, 2004. Cuthrell, J.E. and Biegler, L.T., On the optimization of differential-algebraic process systems, AIChE J., 33, 1257, 1987. Deng, Z.-S. and Liu, J., Analytical study of bioheat transfer problems with spatial or transient heating on skin surface or inside biological bodies, ASME J. Heat Transfer, 124, 638, 2002. Dewhirst, M.W. et al., Basic principles of thermal dosimetry and thermal thresholds for tissue damage from hyperthermia, Int. J. Hyperthermia, 19, 267, 2003. Diederich, C.J. et al., Combination of transurethral and interstitial ultrasound applicators for high-temperature prostate thermal therapy, Int. J. Hyperthermia, 16, 385, 2000. Hahn, G.M. et al., A comparison of thermal responses of human and rodent cells, Int. J. Radiat. Biol., 56, 817, 1989.

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Hocking, L.M., Optimal Control: An Introduction to the Theory with Applications, Oxford University Press, London, United Kingdom, 1991. Kirk, D.E., Optimal Control Theory: An Introduction, Dover Publications, New York, 2004. Kleis, D. and Sachs, E.W., Optimal control of sterilization of prepackaged food, SIAM J. Optim., 10, 1180, 2000. Konde, K.S. and Modak, J.M., Optimization of bioreactor using metabolic control analysis approach, Biotechnol. Prog., 23, 370–380, 2007. Kuznetsov, A.V., Optimal control of the heat storage in a porous slab, Int. J. Heat Mass Transfer, 40, 1720, 1997a. Kuznetsov, A.V., Determination of the optimal initial temperature distribution in a porous bed, Acta Mechanica, 120, 61, 1997b. Kuznetsov, A.V., Determination of the optimal rate of heat generation for conductive heating a one-dimensional slab, ZAMM, 77, 237, 1997c. Kuznetsov, A.V., Thermal nonequilibrium forced convection in porous media, in Transport Phenomena in Porous Media, Ingham, D.B. and Pop, I. (Eds.), Oxford: Elsevier, 1998, p. 103. Kuznetsov, A.V., Optimization problems for bioheat equation, Int. Comm. Heat Mass Transfer, 33, 537, 2006. Loulou, T. and Scott, E.P., Thermal dose optimization in hyperthermia treatments by using the conjugate gradient method, Numerical Heat Transfer A, 42, 661, 2002. McCann, C. et al., Feasibility of salvage interstitial microwave thermal therapy for prostate carcinoma following failed brachytherapy: Studies in a tissue equivalent phantom, Phys. Med. Biol., 48, 1041, 2003. Mereu, S., Sciubba, E., and Bejan, A., The optimal cooling of a stack of heat generating boards with fixed pressure drop, flowrate or pumping power, Int. J. Heat Mass Transfer, 36, 3677, 1993. Nau, W.H., Diederich, C.J., and Burdette, E.C., Evaluation of multielement catheter-cooled interstitial ultrasound applicators for high-temperature thermal therapy, Med. Phys., 28, 1525, 2001. Ohlsson, T., Temperature dependence of sensory quality changes during thermal processing, J. Food Sci., 45, 836, 1980. Park, S. and Ramirez, W.F., Optimal production of secreted protein in fed-batch reactors, AIChE J., 34, 1550, 1988. Pontryagin, L.S. et al., The Mathematical Theory of Optimal Processes, Interscience Publishers, New York, 1962. Saa, J., Alonso, A.A., and Banga, J.R., Optimal Control of Microwave Heating Using Mathematical Models of Medium Complexity, ACoFoP IV (Automatic Control of Food & Biological Processes), Göteborg, Sweden, pp. 21–23 September 1998. Sapareto, S.A. and Dewey, W.C., Thermal dose determination in cancer therapy, Int. J. Radiat. Oncol. Biol. Phys., 10, 787, 1984. Schumann, T.E.W., Heat transfer: Liquid flowing through a porous prism, J. Franklin Inst., 208, 405, 1929. Shih, T.-C., Kou, H.-S., and Lin, W.-L., The impact of thermally significant blood vessels in perfused tumor tissue on thermal dose distributions during thermal therapies, Int. Comm. Heat Mass Transfer, 30, 975, 2003. Spiga, G. and Spiga, M., A rigorous solution to a heat transfer two phase model in porous media and packed beds, Int. J. Heat Mass Transfer, 24, 355, 1981. Takahashi, H. et al., Radiofrequency interstitial hyperthermia of malignant brain tumors: Development of heating system, Exp. Oncol., 22, 186, 2000. Tartakovsky, B., Ulitzurt, S., and Sheintuch, M., Optimal control of fed-batch fermentation with autoinduction of metabolite production, Biotechnol. Prog., 11, 80, 1995.

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Tayeb, Y.J. and Lim, H.C., Optimal glucose feed rates for fed-batch penicillin fermentation– An efficient algorithm and computational results, Ann. N Y Acad. Sci., 469, 382, 1986. VanImpe, J.F. et al., Optimal-control of penicillin-G fed-batch fermentation–an analysis of a modified unstructured model, Chem. Eng. Comm., 117, 337, 1992. White, H.C. and Korpela, S.A., On the calculation of the temperature distribution in a packed bed for solar energy applications, Sol. Energy, 23, 141, 1979. Yong, J. and Li, X., Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, MA, 1995.

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10

Neural Networks and Genetic Algorithms Yang Meng and Hosahalli S. Ramaswamy

CONTENTS 10.1

Introduction of Neural Networks ............................................................. 200 10.1.1 Principles of Neural Networks .................................................... 200 10.1.1.1 Neural Network Architecture ....................................... 200 10.1.1.2 Artificial Neurons ......................................................... 202 10.1.1.3 Learning Rules ............................................................. 203 10.1.2 Development of Neural Networks .............................................. 204 10.1.3 Properties of Neural Networks.................................................... 205 10.1.3.1 Advantages of Neural Networks .................................. 205 10.1.3.2 Limitations of Neural Networks................................... 205 10.2 Introduction of Genetic Algorithms......................................................... 206 10.2.1 Principles of Genetic Algorithms................................................ 207 10.2.1.1 Encoding of Chromosomes .......................................... 207 10.2.1.2 Objective Function ....................................................... 207 10.2.1.3 Genetic Operators......................................................... 207 10.2.1.4 Selection ....................................................................... 208 10.2.1.5 Crossover...................................................................... 209 10.2.1.6 Mutation ....................................................................... 210 10.2.1.7 Parameters of Genetic Algorithms ............................... 210 10.2.1.8 Types of Genetic Algorithm......................................... 211 10.2.2 Advantages of Genetic Algorithms over Conventional Optimization Methods .......................................... 212 10.3 Application of Neural Networks and Genetic Algorithms in Modeling and Optimization of Food Processing ................................ 212 10.3.1 Coupling Genetic Algorithms and Neural Networks.................. 213 10.3.2 Neural Network Modeling of Heat Transfer in End-Over-End Thermal Processing of Particulates in Non-Newtonian Fluids............................................................ 213 10.3.3 Optimization of Variable Retort Temperature Thermal Processing Using Genetic Algorithms Coupled with Neural Networks ................................................................. 215 References ............................................................................................................. 217

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10.1 INTRODUCTION OF NEURAL NETWORKS Modeling food processing is important in food industry. For a food manufacturer, an accurate model will ensure a good production control and higher quality product. An appropriate model is also beneficial in the development of new products by saving experimental time and cost. However, establishing a model for a food process is basically difficult because of the complexities associated with physical, chemical, and biological changes during the process. Neural networks (NNs) or artificial neural networks (ANNs) are powerful and efficient tools to model a complex process, especially to represent a nonlinear relationship which is common in food processing. In last decades, several studies have demonstrated the potential of neural networks to be a powerful alternative to conventional modeling in many areas of food processing.

10.1.1 PRINCIPLES

OF

NEURAL NETWORKS

A neural network is a collection of interconnecting computational elements simulated like neurons in biological systems. It has the capability of relating the input and output parameters without any prior knowledge of the relationship between them. Artificial neural network models were originally developed to mimic the function of the human brain. A brain contains billions of nerve cells (neurons) highly interconnected through synapses. A typical biological neuron contains neuronal cell bodies (soma), dendrites, and axons. Each neuron receives electrochemical inputs from other neurons at the dendrites. If the sum of these electrical inputs is sufficiently powerful to activate the neuron, it transmits an electrochemical signal along the axon and passes this signal to the other neurons whose dendrites are attached at any of the axon terminals. These attached neurons may then fire. It is important to note that a neuron fires only if the total signal received at the cell body exceeds a certain level. The entire brain is composed of these interconnected electrochemical transmitting neurons. From a very large number of extremely simple processing units (each performing a weighted sum of its inputs and then firing a binary signal if the total input exceeds a certain level), the brain manages to perform extremely complex tasks. This is the principle on which ANN models are based. However, it should be noted that ANN only represents extremely simplified brain models without actually attempting to model the biological system itself. 10.1.1.1

Neural Network Architecture

Figure 10.1 shows typical NN architecture. Neurons are arranged in layers; there is at least one input layer, one hidden layer, and one output layer in any NN. Each neuron in a layer is linked to neurons in other layers with varying connection weights (W) that represent the strengths of these connections. An NN can be viewed as a ‘‘black box.’’ Input layer receives the information from an external source and passes this information to the network for processing. Hidden layer receives information from the input layer and processes the information. The entire process is hidden from view. Output layer receives processed information from the network and sends the results out to an external receptor. When the input layer receives information from an external source, it becomes ‘‘activated’’ and emits signals to the neurons

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Input layer

Hidden layer

Output layer

Bias

FIGURE 10.1 Typical neural network architecture.

in hidden layer. Each neuron ( j) in hidden layer receives input signals from m neurons in input layer in proportion to their connection weights (Wij). A threshold value (bias Bj) is added to this weighted sum of connection weights and nonlinearly transformed using an activation function ( f) to generate the output signals. The response (O) of each neuron ( j) to the input signals (X) can be mathematically expressed as Oj ¼ f

m X

! Xi Wij þ Bj

(10:1)

i¼1

Accordingly, the output (Y) of each neuron (k) in output layer can be mathematically expressed as (assuming p neurons in hidden layer): Yk ¼ f

p X

! Oj Wjk þ Bk

(10:2)

j¼1

For a neural network with single hidden layer, Trelea et al. (1997) and HernandezPerez et al. (2004) represented the output of the neural network by the following expression: Y ¼ f2 (W2 f1 (W1 X þ B1 ) þ B2 )

(10:3)

where X is the matrix of input variables Y is the matrix of output variables f1 and f2 are the transfer functions in the hidden and output layers W1 and W2 are the matrices of connection weights in the hidden and output layers B1 and B2 are the matrices of biases in the hidden and output layers, respectively

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Determining connection weights is known as learning process in neural networks. Supervised learning is most commonly used for modeling purpose neural networks. In supervised learning, the learning process starts with random initialization of connection weights. During learning, the response of each output neuron is compared with a corresponding desired response. Errors associated with the output neurons are computed. In order to minimize the errors, the weights are adjusted at the end of each learning cycle. This procedure is repeated for a specific number of times. In practice, if a sufficient number of these input=output combinations are used for learning, it should be able to predict the output for new inputs. 10.1.1.2

Artificial Neurons

The artificial neurons are simple processing units similar to the biological neurons: they receive multiple inputs from other neurons but generate only a single output. The generated output may be propagated to several other neurons. Each neuron has two basic functions: gathering information from other neurons in the prior layer and sending signals to the neurons in the next layers. The first artificial neuron model (Figure 10.2) proposed in 1943 by McCulloch and Pitts (1943) is based on the simplified consideration of the biological model. The elementary computing neuron functions as an arithmetic logic-computing element. The binary inputs of the neurons are X1, X2, . . . , Xn. Zero represents absence, and one represents existence. The weight of connection between the ith input Xi and the neuron is represented by Wi. When Wi > 1, the input is excitatory. When Wi < 0, it is inhibitory. The net summation of inputs weighted by the synaptic strength Wi at connection i is net ¼

n X

Wi Xi

i¼1

Input X 1, X 2, ..., Xn Wi Sum up

Transfer function

Y

FIGURE 10.2 McCulloch and Pitts neural model.

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(10:4)

The net value is then mapped through an activation function f of neuron output. The activation function used in the model is a threshold function: y ¼ f (net) f (X) ¼

(10:5)

1, X > u

(10:6)

0, otherwise

where u is the threshold value. The neuron models used in current neural networks are constructed in a more general way. The input and output signals are not limited to binary data and the activation function can be any continuous function other than the threshold function used in the earlier model. The activation function is typically a monotonic nondecreasing nonlinear function. Some of the commonly used activation functions are (where a denotes the parameter, and u denotes the threshold value): Sigmoid function:

f (X) ¼

Hyperbolic function: Linear threshold: Gaussian function: 10.1.1.3

1 1 þ eax

f (X) ¼ tan h(aX) ¼

1, f (x) ¼ X=u, 0,

(10:7) eaX  eaX eaX þ eaX

Xu u>X>0 Xu

f (X) ¼ eaX2

(10:8) (10:9) (10:10)

Learning Rules

In a supervised learning neural network, training involves feeding the network a set of known input=output patterns, and adjusting the connection weights until the output pattern (response) calculated from the given input reflects the desired relationship. A neural network’s knowledge is actually stored within those interneuron connection strengths. Learning rule is a method to adjust the weight factors based on trial and error. Many learning rules have been developed to train neural networks. Back-propagation is a very popular learning algorithm due to its relative simplicity and stability (NeuralWare, Inc. 1996). In a back propagation algorithm, the output error is firstly calculated for the kth neuron on the output layer: «k ¼ dk  ck where «k is the output error dk is the desired output ck is the calculated output

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(10:11)

Then the total square error on the output layer can be calculated as X X E¼ «2 ¼ (dk  ck )2 k

(10:12)

k

The squares of errors must be used; otherwise positive and negative errors may cancel each other out. Connection weights can be adjusted as follows. Connection weights in the output layer Partial derivatives of E with respect to the connection weight Wjk (Equation 10.2) is obtained: @E @ X ¼ (dk  ck )2 @Wjk @Wjk k

(10:13)

The new Wjk is calculated as follows: Wjk (new) ¼ Wjk (old)  C

@E @Wjk

(10:14)

where C is a constant, called learning rate (typically 0 < C  1). Training with a small learning rate will make the progress significantly slow. Large learning rate will proceed much faster, but may result in local optima. Connection weights in hidden layer The same procedure can also be used to adjust the connection weights in the hidden layer. Firstly the partial derivatives of E with respect to the connection weight Wij (Equation 10.1) are calculated and then the new Wij is obtained: Wij (new) ¼ Wij (old)  C

10.1.2 DEVELOPMENT

OF

@E @Wij

(10:15)

NEURAL NETWORKS

Developing a NN generally involves the following steps: . . . .

Determining inputs and outputs Optimizing NN configurations Training or learning Testing or generalization

Inputs and outputs are determined by the problem being investigated. NN can be used to simultaneously produce more than one output unlike traditional models where one model is required for each output. Optimization of NN configurations includes the determination of number of hidden layers, number of neurons in hidden layers, transfer function and learning rule, learning rate and learning runs. Trial and error is normally used to select those parameters. When selecting the number of hidden layers, number of neurons in hidden layers and learning runs, overtraining must be avoided. Generally the more the hidden layers and the neurons in hidden layers, the better the NN will perform (Nguyen and Cripps, 2001). Because with a large number of hidden layers and ß 2008 by Taylor & Francis Group, LLC.

neurons, ANN may memorize the input training samples (Rai et al., 2005) implying that the NN is over trained. As a result, it is less likely to accurately predict new data, and its generalization ability is weakened. Too many learning runs will possibly result in overtraining as well. It is always a good practice to keep the neural network as simple as possible to maintain its generalization potential. In the training or learning phase, a set of known input=output patterns is repeatedly presented to train the network. The weight factors between neurons are adjusted until the specified input yields the desired output. Through these adjustments, the NN learns the correct input=output response behavior. In neural network development, this phase is typically the longest and most time consuming, and it is critical to the success of the network. A well-trained neural network should have the capability to respond to previously unseen data sets as well, which is refereed as generalization. Therefore, after training, the testing of the trained network will proceed to assess its generalization ability. All the available data sets should be assorted into two parts: one for training the NN and the other for testing of the trained network. The trained network will be subjected to input patterns not used in training, but whose outputs are known, and the network’s performance will be evaluated. Root mean square (RSM) is the often used statistical index to evaluate the performance of NNs (Sablani et al., 1997).

10.1.3 PROPERTIES 10.1.3.1

OF

NEURAL NETWORKS

Advantages of Neural Networks

NNs have a number of advantages over other modeling techniques as described below: Learning ability A prior knowledge of the system is not required to construct a neural NN because it can learn the input=output relationship from the given data. This makes NN suitable to problems where the relationships are dynamic or nonlinear, which is difficult for other modeling techniques. Fault tolerance The neural network is more tolerant of noisy and incomplete data because the information is distributed in the massive processing neurons and connections. Partial destruction of the network will not degrade the overall performance significantly. High computational speed Because the NN is an inherently parallel architecture, the result comes from the collective behavior of a large number of simple parallel processing units, which makes it suitable for the on site modeling=controlling. 10.1.3.2

Limitations of Neural Networks

While NNs have many advantages, they still have some limitations as follows: Requirement of large amount of training data NNs should not be considered to model a problem or process only with little training data since they rely heavily on such data. In addition, NNs are not suitable in the situation where large but similar training data exist, which will cause the same

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problems as small training data sets. Thus, broad-based data sets are essential for training a neural network. No guarantee of optimal results Most training techniques are capable of ‘‘tuning’’ the network, but they do not guarantee that the network will operate properly. The training may ‘‘bias’’ the network making it accurate in some operating regions but inaccurate in others. Requirement of good set of input variables Selection of input variables that give the proper input=output mapping is often difficult but required for a good performing neural network. Since it is not always obvious which input variables will give the best result, some trial and error in selecting input variables is often required. Black box instead of clear physical relationship The individual relations between the input variables and the output variables are not developed by engineering judgment so that the model tends to be a black box or input=output table without analytical basis.

10.2 INTRODUCTION OF GENETIC ALGORITHMS Genetic algorithms (GAs) are search algorithms to identify the optimum solution which is based on the mechanics of biological evolution. The natural evolution theory states that a species will, after many generations, adapt to live better in its environment. For example, if a population of an animal lives mainly in a swampy area, they may eventually evolve with webbed feet. The reason is that the members of this population will die if they are poor swimmers because they cannot easily get food and live long enough to reproduce. The offsprings of the good swimmers will probably be good swimmers as well because they will usually carry genetic traits of their parents such as slight webbing between the toes. GAs were developed by John Holland of University of Michigan in 1965 and have been successfully used in a wide variety of problem domains (Goldberg, 1989; Davis, 1991). GAs solve an optimization problem in the same way as nature solves problems in evolution. They start with generating a set of possible solutions to the problem. Each candidate solution is called a chromosome, and the whole set of solutions is called a population. Each solution (chromosome) could consist of several decision variables. For example, a population may have M chromosomes (Equation 10.16), and each chromosome may have N decision variables (Equation 10.17): 2

3 chromosome1 6 chromosome2 7 6 7 6 7  6 7 Population ¼ 6 7  6 7 4 5  chromosomeM Chromosomei ¼ [variablei1 , variablei2 , . . . , variableiN ]

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(10:16)

(10:17)

When an initial population (known as parental generation) is generated, a fitness value is assigned to each solution (chromosome) representing how close it is to solving the problem. Those chromosomes with a higher fitness value are more likely to reproduce offspring. In GAs, offspring are created by using principles of evolution such as crossover and mutation (known as genetic operators). Those operators act on chromosomes (solutions) in the population (a set of solutions) to yield a set of new solutions which is known as the next generation. The algorithm is iterative in the hope to discover better solutions to a problem.

10.2.1 PRINCIPLES 10.2.1.1

OF

GENETIC ALGORITHMS

Encoding of Chromosomes

Encoding of chromosomes is to transform the decision variables into a format that can be operated by GAs. Encoding is very important, which determines how well the algorithm performs on the problem. There are two main coding ways (Morimoto, 2006). The most traditional approach is to use a set of binary strings consisting of 0 and 1 to represent decision variables. Binary strings are commonly used since they are known to perform well with standard crossover and mutation operators. Each variable has one binary string. Each bit in this string can represent some characteristic of the solution, which is called a gene (Morimoto, 2006). For example, a chromosome consisting of N variables can be expressed as follows: Chromosome ¼ [variable1 , variable2 , . . . , variableN ] ¼ [011101, 000101, . . . , 101101] In this case, a chromosome is expressed by 6-bit strings. The 6-bit binary strings provide numerical values between 0 (000000) and 63 (111111). The length of the binary strings depends on the range of the numerical values required. The other coding way is to use integral numbers which might be more effective when the chromosome consists of many variables. There are many other ways of encoding depending on the problem. For example, one can encode directly real values in finding weights for an NN. 10.2.1.2

Objective Function

An objective function relates the decision variables of the problem and assigns a ‘‘fitness’’ value to the solution (chromosome) that determines how good that solution is. Ideally this function will be as monotonic as possible (Keedwell and Narayanan, 2005), and it will vary consistently with decision variable values. The more adapted chromosomes will receive higher fitness values. Chromosomes with high fitness values are more likely to reproduce in each generation during the evolution process. When the GA is used for minimization, a transformation is necessary to derive a maximization problem. 10.2.1.3

Genetic Operators

GAs are started with generating an initial population of chromosomes. This initialization is often achieved at random, but the population may be initialized by

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chromosomes which are already known to perform well. When random initialization of binary chromosomes is used, each bit of the chromosomes is randomly set to 0 or 1 according to a probability which is called the initialization probability. Once the first generation has been created, the genetic operators drive the population to find new, more optimal solutions. Selection, mutation, and crossover are the most important operators. 10.2.1.4

Selection

Selection is the process of picking out a suitable chromosome from the population in order to create a new generation. According to Darwin’s evolution theory, suitable chromosomes are the ones with good fitness values. This operator is the implementation of the principle ‘‘survival of the fittest.’’ However, to make sure that the GA does not converge on a set of solutions too quickly, a random element is usually introduced into the selection procedure (Keedwell and Narayanan, 2005). There are many selection strategies. Some of them are discussed here: Roulette wheel selection (proportional selection) This is the most simple and fundamental selection approach. In this method, the probability for a chromosome to be selected is in proportion to its fitness value, as shown in Table 10.1. The principle is just like a roulette wheel where each chromosome in the population occupies a slice of the wheel, the higher the fitness value, the larger the portion of wheel occupied by that chromosome (Figure 10.3). Therefore, the better the chromosomes are, the more chances they have of being selected. Roulette wheel selection gives more adapted chromosomes better chances to be kept in the next generation. However, there is still a chance for less adapted chromosomes to be selected since the selection procedure depends on a random number which will ensure that the diversity in the population is maintained. Tournament selection In tournament selection, a number of chromosomes (normally 2) are randomly selected from the population and their fitness values are compared. The chromosome with the highest fitness is selected as a parent to generate the next generation. The random selection to the tournament gives a chance that two chromosomes with lower fitness TABLE 10.1 Fitness Values and Probabilities to be Selected Chromosome1 Chromosome2 Chromosome3 Chromosome4 Chromosome5 Chromosome6 Total

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Fitness Value

Probability to be Selected

25 14 56 41 10 5 151

0.165562914 0.092715232 0.370860927 0.271523179 0.066225166 0.033112583 1

0.07

0.03 0.17 Chromosome1 0.09

0.27

Chromosome2 Chromosome3 Chromosome4 Chromosome5 Chromosome6

0.37

FIGURE 10.3 Illustration of roulette wheel selection.

values could be selected at once. In this situation, although the chromosomes are poor with respect to the rest of the population, the better chromosome of the two will be selected. Therefore, chromosomes with low fitness values can still be selected by the tournament selection which ensures that the diversity in the population could be kept. Elitist selection In elitist selection, the best chromosome (or a few best chromosomes) is copied to new generation. The rest of the chromosomes in the new generation are created in a classical way. Since creating new population by crossover and mutation is easy to lose the best chromosome, elitist selection prevents losing the best found chromosomes. 10.2.1.5

Crossover

Crossover is the most important genetic operator. Crossover combines the information from two ‘‘parent’’ to produce two new ‘‘offspring’’ solutions which are different but related to the original solutions. Crossover is performed in the hope that the combination of two well-adapted chromosomes may give two new better adapted ones. There are a number of methods to achieve this purpose, and the most commonly used are single point, multipoint, and uniform crossover. Single point crossover In single point crossover, two chromosomes are involved and one crossover point is selected randomly. The two new offspring are created in such a way that the binary string from the beginning of the chromosome to the crossover point is copied from one parent, and the rest is copied from the second parent. For example, the following two chromosomes have 7 bits each: chromosome1: 0 0 1 0 0 1 1 chromosome2: 1 0 0 0 1 1 0 The chosen crossover position is: 3. After crossover, the created two new chromosomes are offspring1: 0 0 1 0 1 1 0 offspring2: 1 0 0 0 0 1 1

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Multipoint crossover For multipoint crossover, n crossover positions are chosen randomly and sorted into ascending order. Then, the bits between successive crossover points are exchanged between the two parents to produce two new offspring. For example, the following two chromosomes have 12 bits each: chromosome1: 0 0 0 1 1 1 0 1 1 0 1 0 chromosome2: 1 0 1 0 0 1 1 1 0 0 0 1 The chosen crossover positions are 3, 5, and 7. After crossover the created two new chromosomes are offspring1: 0 0 0 0 0 1 0 1 0 0 0 1 offspring2: 1 0 1 1 1 1 1 1 1 0 1 0 Uniform crossover In uniform crossover, for each bit of the first offspring, a probability is calculated separately (mostly p ¼ 0.5) that this bit should come from either the first parent or the second parent, and for the second offspring, the bit in the same position will take the value of the other parent. Uniform crossover combines the feature in each bit no matter where the bit is located, but in single point crossover, the bits towards the center of the chromosome are perturbed more often than those at the edges of the chromosome (Keedwell and Narayanan, 2005). 10.2.1.6

Mutation

After crossover, every offspring undergoes mutation. In binary chromosomes, mutation inverts one or more bits at random from 0 to 1 or vice versa. Mutation helps to keep the genetic diversity from one generation to the next and prevents premature convergence to a local optimum solution (Morimoto, 2006) by preventing the population of chromosomes from becoming too similar to each other. Without mutation, GA can only manipulate the genetic material that is present in parent population. For example, an offspring chromosome after crossover has 8 bits: 0 1 0 0 1 1 0 1. If the second and fourth bits are chosen for mutation at random, the created new chromosome will be 0 0 0 1 1 1 0 1. 10.2.1.7

Parameters of Genetic Algorithms

There are two basic parameters in GAs: crossover probability and mutation probability. Crossover probability tells how often crossover will be performed. If crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation is made from exact copies of chromosomes from old population. Normally, crossover probability is around 0.6–0.8. Mutation probability tells how often mutation will be conducted. If there is no mutation, offspring is taken after crossover without any change. If mutation probability is 100%, whole chromosome is changed. Mutation probability is usually very small (0.01–0.1). Some studies (Back, 1997; Mayer et al., 1999) used high mutation probability (up to 0.4 or 0.6).

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There are two other very important parameters GAs: population size and generation number. Population size tells the amount of chromosomes in a population. If there are too few chromosomes, genetic algorithm only has a small part of search space to explore. However, if there are too many chromosomes, GA will take more time to solve the problem. Research shows that after some limit (which depends mainly on encoding and the problem) it is not useful to increase population size. Maximum generation number must be prespecified as a termination condition of GAs. If the algorithm reaches the maximum generation number, the evolution ends. 10.2.1.8

Types of Genetic Algorithm

There are two types of GAs, using different ways to carry chromosomes from one generation to the next. Generational genetic algorithms completely replace the chromosomes between generations. From the initial population generated, only two chromosomes are selected as parents for reproduction. The two parents create two new offspring by crossover and mutation. This process (selection, crossover, and mutation) is repeated until the number of offspring is the same as that in the parental generation. The produced offspring are then inserted into the population to replace their parents to form a new generation. The new generation will be the new parental generation for the next offspring generation. If the new generation contains a solution that produces an output that is close enough or equal to the desired answer then the problem has been solved. If this is not the case, then the new generation will go through the same process as their parents did. This will continue until a solution or the maximum number of generation is reached. Steady-state genetic algorithms just partially replace the chromosomes between generations. Figure 10.4 shows the flow chart of steady-state GAs. From the initial population generated, two chromosomes are selected to create two new offspring by crossover and mutation replacing the two less adapted chromosomes, which have the two lowest fitness values in the population, by these two new chromosomes to form the new generation. The algorithm stops when the maximum number of generations is reached, or the optimum solution is found.

Chromosome1 Chromosome2

Selection

Crossover mutation ChromosomeN Replacement

FIGURE 10.4 Steady-state genetic algorithms.

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10.2.2 ADVANTAGES OF GENETIC ALGORITHMS OVER CONVENTIONAL OPTIMIZATION METHODS Compared with conventional optimization methods, GAs have many significant advantages. GAs search from a population of points not a single point. In many traditional optimization methods, we move gingerly from a single point in the decision space to the next using some transition rule to determine the next point. This pointto-point method is dangerous because it is a perfect prescription for locating false peaks in multimodal (many-peaked) search spaces. By contrast, GAs work from a rich database of points simultaneously (a population of strings) climbing many peaks in parallel, and the probability of finding a false peak is reduced over the methods going point to point. GAs use payoff (objective function) information, not derivatives or other auxiliary knowledge. Many search techniques require much auxiliary information in order to work properly. For example, gradient techniques need derivatives (calculated analytically or numerically) in order to be able to climb the current peak and other local search procedures like the greedy techniques of combinatorial optimization (Goldberg, 1989) require access to most if not all tabular parameters. By contrast, GAs have no need for all this auxiliary information. To perform an effective search for better and better structures, they only require payoff values (objective function values) associated with individual strings. This characteristic makes the genetic algorithm a more canonical method than many search schemes. Unlike many methods, GAs use probabilistic transition rules, not deterministic rules, to guide their search. The use of probability does not suggest that the method is some simple random search; it uses random choice as a tool to guide a search toward regions of the search space with likely improvement. This makes GAs more robust than other traditional optimization methods.

10.3 APPLICATION OF NEURAL NETWORKS AND GENETIC ALGORITHMS IN MODELING AND OPTIMIZATION OF FOOD PROCESSING Applications of NNs in food processing have covered different areas: modeling, prediction, computation, classification, and optimization as well as process control. If the number of papers published were used to determine the order of relative popularity of application purposes, modeling or prediction was the most popular purpose for applications of NNs in food processing areas; control was listed second; and then classification, optimization, and computation, respectively (Chen, 2001). With the rapid development of computer technologies, GAs have been applied to the optimal control of agricultural and food production systems. Olmos et al. (2004) optimized a pork meat dehydration–impregnation–soaking (DIS) process using GAs. Chtioui et al. (1998) reported the use of a GA application to select for features in seed discrimination. In different studies, GAs were found to be mostly coupled with NNs in food processing area.

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10.3.1 COUPLING GENETIC ALGORITHMS

AND

NEURAL NETWORKS

GAs have been used for artificial NNs in two main ways: to optimize the network architecture and to train the weights of a fixed architecture. For the case of NNs, back-propagation is the traditional technique for learning the weights in an NN. On the basis of trial and error procedure, back-propagation has some limitations. It searches from a single point, and as a point-to-point searching strategy, it is easy to become stuck at a local optimum. Another drawback of back-propagation is that it has to calculate the derivative or gradient for each weight, which makes the process complex and time consuming. GAs overcome those drawbacks and have been successfully used to optimize the connection weights. GA has also been used to optimize the neural network topology parameters, including the number of hidden layers and the number of neurons in hidden layers. Coupling GAs to NNs has been investigated in many publications. Goni et al. (2008) applied genetic algorithm search technique to obtain the initial training parameters of the neural network which improved its generalization capacity. Liu et al. (2007) used GAs to determine the optimal number of neurons in the hidden layer of the NN. Morimoto et al. (1997) used a neural network to predict color change of tomatoes affected by heat treatment, and used a GA to search for the optimal heat treatment condition. Ferentinos (2005) gave a 10-bit binary representation of NN topology and training parameterization for GA optimization. Two example uses of neural network modeling and ANN-GA are detailed below as illustration of the applicability of the models.

10.3.2 NEURAL NETWORK MODELING OF HEAT TRANSFER IN END-OVER-END THERMAL PROCESSING OF PARTICULATES IN NON-NEWTONIAN FLUIDS Establishing processing schedule for particulate liquid canned foods requires better prediction of the process lethality (Fo) obtained by the particle and the liquid because of the large variability associated with the process. Overall heat transfer coefficient (U) from the heating medium to liquids in the can and fluid-to-particle heat transfer coefficient (hfp) from the can liquid to the particle have been widely used to quantify the heat transfer process and predict the product lethality for particulates in low viscosity Newtonian fluids. However, most of fluid foods are non-Newtonian in nature, and for particulates in viscous non-Newtonian fluids, it is difficult to gather hfp=U data (Meng and Ramaswamy, 2005). For these cases, apparent values of heat transfer coefficient hap and Ua were proposed to quantify the heat transfer process and predict the product lethality (Meng and Ramaswamy, 2005). Meng and Ramaswamy (2007) established dimensionless correlations for predicting hap and Ua values in particulate viscous non-Newtonian fluids during end-over-end rotation. However, a good dimensionless correlation depends on the appropriate selection of dimensionless numbers, which requires the prior knowledge of the phenomena under investigation (Sablani et al., 1997). NNs do not require the prior knowledge of the relationship between the input and output variables, and have been shown to provide more accurate models by proper training. Sablani et al. (1997)

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developed NN models for the prediction of hfp and U associated with canned particles in Newtonian liquids subjected to end-over-end rotation. The prediction errors were less than 3% and 5%, respectively, for U and hfp, which were about 50% better than those associated with dimensionless correlations. In this illustration, experimental data (Meng and Ramaswamy, 2007a) of hap and Ua under different experiment conditions were used to train and test an NN model. With the power law model, two parameters (flow behavior index n and consistency coefficient K) were used to describe rheological properties of CMC aqueous solutions. Diameter and height of the can were used to quantify the can size. Rotation radius, can headspace, and particle size were found insignificant to hap and Ua, and therefore were not used for the development of the neural network model. The input layer consisted of eight neurons corresponding six significant variables: rotation speed, retort temperature, can size (two parameters), particle density, liquid viscosity (two parameters), and particle concentration. The output layer comprised of two neurons, one for hap and second for Ua. The standard back-propagation algorithm was used for learning=training of the network. In order to find the optimal configuration, initially, keeping two hidden layers with three neurons in each hidden layer and 50,000 learning runs, the optimal combination of learning rules and transfer functions was determined by a full factorial experiment design. Next, keeping the optimal combination of the learning rule and the transfer function, a full factorial set was used to optimize the number of hidden layers and the number of neurons in each hidden layer. Finally, keeping the parameters selected above, the effect of learning runs was determined. The optimum configuration was decided based on minimizing the difference between the predicted value by the NN and the desired output, which was shown by the root mean square (RMS). The optimized configurations were 1 hidden layer, 6 neurons in hidden layer, Delta rule of learning rule, TanH of transfer function, 50,000 of learning runs. In the optimization process, the full set of data (303 cases) were used. However, when training the final NN model and testing its performance, only partial data sets were used. Three hundred and three cases were also divided into 3 data sets. The combination of data sets used for training and testing included 1 set for learning and 1 set for testing; 1 set for training and 2 sets for testing; 1 set for training and 3 sets for testing; 2 sets for training and 1 set for testing; 2 sets for training and 2 sets for testing; 2 sets for training and 3 sets for testing; all 3 sets for training and all 3 sets for testing. All were chosen randomly. On an average, the magnitude of errors for all the data set combinations was nearly the same as those for the full set (3 data sets for learning and 3 data sets for testing). The prediction accuracies for NNs models were much better compared to those from dimensionless correlations. The trained network was found to predict responses with mean relative error of 4.7%–5.9%, which were 27%–62% lower than those associated with dimensionless correlations. An algebraic equation (Equation 10.18) was applied to represent the trained models, making the trained NN easier to access. Y ¼ W2 sin (W1 X þ B1 ) þ B2

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(10:18)

In the above equation, X is the input matrix expressed in Equation 10.19, in which X1 is particle material (density); X2 is rotation speed; X3 is retort temperature; X4 is the diameter of the can; X5 is the length of the can; X6 is particle concentration; X7 is the consistency coefficient of the liquid; X8 is the flow behavior index of the liquid. 2

3 X1 6 X2 7 6 7 6 X3 7 6 7 6 X4 7 7 X¼6 6 X5 7 6 7 6 X6 7 6 7 4 X7 5 X8

(10:19)

Y is the output matrix expressed in Equation 10.20, in which Y1 is hap, Y2 is Ua. 

Y1 Y¼ Y2

 (10:20)

In Equation 10.18, W1 is the matrix of connection weights in hidden layer, B1 is the matrix of bias in hidden layer, W2 is the matrix of connection weights in output layer, and B2 is the matrix of bias in output layer (Meng, 2006).

10.3.3 OPTIMIZATION OF VARIABLE RETORT TEMPERATURE THERMAL PROCESSING USING GENETIC ALGORITHMS COUPLED WITH NEURAL NETWORKS Variable retort temperature (VRT) thermal processing has been recognized as one of potential techniques in canning industry (Durance, 1997). Although early researches focusing on optimizing the average (volumetric) nutrient retention Qv (Teixeira et al., 1975; Saguy and Karel, 1979; Nardkarni and Hatton, 1985) using VRT thermal processing have shown minimal effects compared with CRT (constant retort temperature) processing, more recent studies (Banga et al., 1991; Almonacid-Merino et al., 1993; Noronha et al., 1993; Durance et al., 1996) have shown that VRT is very promising to improve food surface quality and save process time. Banga et al. (1991) indicated that surface quality was improved by up to 20% under the optimal VRT process and the process time could be reduced by up to 16.5%. Choosing a reasonable (or optimal) VRT function is the key to design a VRT thermal process. Considering that the commonly used VRT functions such as sine, exponential, ramps and steps, all involve more than one function parameters, searching out an optimal VRT function is a multivariable optimization problem. Although many optimization techniques have been developed for solving a variety of optimization problems including multivariable problems, several limitations still exist, such as the robustness of optimal results and calculation speed which are important for practical applications. Fortunately, the development of artificial intelligent technologies has offered a new path to deal with this kind of problems. Chen and Ramaswamy (2002) explored the use of combined ANNs and GAs to develop

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various prediction models of VRT processing and search for the optimal VRT function parameters, respectively. The specific objectives included (1) developing NN models to predict process time (Pt) and surface cook value (Fs), respectively, under different VRT function parameters, (2) searching out the optimal processing temperature profiles using the GA coupled with trained neural network models. A computer simulation program (Chen, 2001) was used to gather data needed for developing the neural network models relating process time (Pt) and surface cook value (Fs), respectively, to VRT function parameters. Two VRT functions, sine and exponential, both involving only two variable parameters were compared in the study. Simulated results showed that VRT processing with an exponential function was more beneficial because the minimum Pt and Fs values were much lower than those obtained with sine functions. Therefore, the optimization concentrated on VRT processing with the exponential function, which is mathematically described as f (t) ¼ A(1  ekt )

(10:21)

where A is the range limitation of exponent function k is the constant related to the increasing rate t is the process time, in minutes The adjustable ranges for each independent variable were determined by sensitivity studies: 14 < A < 31 and 0.005 < k < 0.2. To optimize for A and k, main independent fitness functions as minimum Fs and minimum Pt were used. The additional functions included some different combinations: for minimum Pt, Fs  Fsd and Qv  Qvd; and for minimum Fs, Pt  Ptd and Qv  Qvd, where Fsd, Qvd, and Ptd were desired surface cook value, average quality retention, and process time, respectively. In this study, those based on the best retort temperature (1158C) in CRT processing were used as the constraint conditions. The optimization procedures included first an initial population of 20 types of individuals generated at random. Second, the related neural network model was called to compute the fitness for all initial individuals. Third, the three operators: selection, crossover, and mutation were applied to produce a new generation. The above operations were repeated until the given limitation number N1 of generations was reached or no better results were obtained after the given number N2 of generations. They used N1 ¼ 200, N2 ¼ 50. Usually, the performance of GAs is affected by the size of initial population, crossover rate, and mutation rate. However, the best fitness of process time Pt remained constant under different crossover rates, mutation rates, and initial population sizes. This indicated that the effect of the genetic parameter changes on the performance of genetic algorithm was not significant. Thus, the default values, initial population of 20, crossover rate of 0.9, and mutation rate of 0.01 were adopted for the following optimization searching. The optimal results were all found before 100 generations meaning that the designed number of 200 generations was sufficient to find the optimal VRT results. Results presented (Chen and Ramaswamy, 2002) demonstrated that VRT processing with optimal parameters reduced the process time significantly. For example, when Fs ¼ 66 min, the process time with VRT

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processing was 80.3 min while that with CRT was 105 min resulting in more than 20% improvement in process time. If a Qv of 60% could be used as the constraint condition, the process time could be reduced by 30%–40% using the VRT processing. The VRT process also lowered the surface cook value of the product as compared with the CRT process. The improvement of surface quality Fs was about 7%–10%, somewhat lower when compared with lowering of process time. Nevertheless, it is an improvement in the right direction. These results confirm that VRT processing can reduce process time and improve quality of the canned food.

REFERENCES Almonacid-Merino, S.F., Simpson, R., and Torres, J., Time-variable retort temperature profiles for cylindrical cans: Batch process time, energy consumption, and quality retention model, J. Food Process Eng., 16, 271, 1993. Back, T., Mutation Parameters, Handbook of Evolutionary Computation, Oxford University Press, New York, 1997. Banga, J.R. et al., Optimization of thermal processing of conduction-heated canned foods: study of several objective functions, J. Food Eng., 14, 25, 1991. Chen, C.R. and Ramaswamy, H.S., Modeling and optimization of variable retort temperature (VRT) thermal processing using coupled neural networks and genetic algorithms, J. Food Eng., 53, 209, 2002. Chen, C.R., Application of computer simulation and artificial intelligence technologies for modeling and optimization of food thermal processing, PhD Dissertation, McGill University, Canada, 2001. Chtioui, Y., Bertrand, D., and Barba, D., Feature selection by a genetic algorithm, application to seed discrimination by artificial vision, J. Sci. Food Agriculture, 76, 77, 1998. Davis, L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, New York, 1991. Durance, T.D., Improving canned food quality with variable retort temperature processes, Trends Food Sci. Tech., 8, 113, 1997. Durance, T., Dou, J., and Mazza, J., Selection of variable retort temperature processes for canned salmon, J. Food Process Eng., 20, 65, 1996. Ferentinos, K.P., Biological engineering applications of feedforward neural networks designed and parameterized by genetic algorithms, Neural Networks, 18, 934, 2005. Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning, AddisonWesley, New York, 1989. Goni, S.M. et al., Prediction of foods freezing and thawing times: Artificial neural networks and genetic algorithm approach, J. Food Eng., 1, 164, 2008. Hernandez-Perez, J.A. et al., Neural networks for the heat and mass transfer prediction during drying of cassava and mango, Innovative Food Sci. Emerging Tech., 5, 57, 2004. Keedwell, E. and Narayanan, A., Intelligent Bioinformatics: The Application of Artificial Intelligence Techniques to Bioinformatics Problems, John Wiley & Sons Ltd., New York, 2005. Liu, X. et al., A neural network for predicting moisture content of grain drying process using genetic algorithm, Food Control, 18, 928, 2007. Mayer, D.G., Belward, J.A., and Burrage, K., Survival of the fittest-genetic algorithms versus evolution strategies in the optimization of systems models, Agric. Syst., 60, l13, 1999. McCulloch, W.S. and Pitts, W., A logical calculus of the ideas immanent nervous activity, Bull. Math. Biophys., 5, 115, 1943. Meng, Y., Heat transfer studies on canned particulate viscous fluids during end-over-end rotation, PhD Dissertation, McGill University, Canada, 2006.

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Meng, Y. and Ramaswamy, H.S., Dimensionless heat transfer correlations for high viscosity fluid-particle mixtures in cans during end-over-end rotation, J. Food Eng., 80, 528, 2007. Meng, Y. and Ramaswamy, H.S., Effect of system variables on heat transfer to canned particulate non-Newtonian fluids during end-over-end rotation, Trans. I. Chem. E, Part C, Food and Bioproducts Processing, 85(C1), 34, 2007a. Meng, Y. and Ramaswamy, H.S., Heat transfer coefficients associated with canned particulate= non-Newtonian fluid (CMC) system during end-over-end rotation, Trans. I. Chem. E, Part C, Food and Bioproducts Processing, 83(C3), 1, 2005. Morimoto T., Genetic algorithms, in Handbook of Food and Bioprocess Modeling Techniques, Sablani, S.S., Datta, A.K., Rahman, M.S., and Mujumdar, A.S. (Eds.), CRC Press, Taylor & Francis, Boca Raton, 2006, p. 405. Morimoto, T. et al., Optimization of heat treatment for fruit during storage using neural networks and genetic algorithms, Comput. Electron. Agric., 19, 87, 1997. Nardkarni, M.M. and Hatton, T.A., Optimal nutrient retention during the thermal processing of conduction-heated canned foods: Application of the distributed minimum principle, J. Food Sci., 50, 1312, 1985. NeuralWare, Inc., Using NeuralWorks, NeuralWare, Inc. Technical Publication Group, Pittsburgh, PA, 1996. Noronha, J. et al., Optimization of surface quality retention during the thermal processing of conduction heated foods using variable temperature retort profiles, J. Food Process Preserv., 51, 1297, 1993. Nguyen, N. and Cripps, A., Predicting housing value: A comparison of multiple regression analysis and artificial neural networks, J. Real Estate Res., 22, 313, 2001. Olmos, A. et al., Optimal operating conditions calculation for a pork meat dehydration– impregnation–soaking process, Lebensmittel-Wissenschaft und-Technologie, 37, 763, 2004. Rai, P. et al., Prediction of the viscosity of clarified fruit juice using artificial neural network: a combined effect of concentration and temperature, J. Food Eng., 68, 527, 2005. Sablani, S.S. et al., Neural network modeling of heat transfer to liquid particle mixture in cans subjected to end-over-end processing, Food Res. Int., 30, 105, 1997. Saguy, I. and Karel, M., Optimal retort temperature profile in optimizing thiamin retention in conduction-type heating of canned foods, J. Food Sci., 44, 1485, 1979. Teixeira, A.A., Zinmeister, G.E., and Zahradnik, J.W., Computer simulation of variable retort control and container geometry as a possible means of improving thiamin retention in thermally processed foods, J. Food Sci., 40, 656, 1975. Trelea, I.C., Raoult-Wack, A.L., and Trystram, G., Application of neural network modelling for the control of dewatering and impregnation soaking process (osmotic dehydration), Food Sci. Tech. Int., 3, 459, 1997.

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11

Computational Fluid Dynamics for Optimization in Food Processing Ferruh Erdo gdu

CONTENTS 11.1 11.2 11.3 11.4 11.5

Introduction .............................................................................................. 219 CFD Fundamentals .................................................................................. 220 Process Optimization ............................................................................... 222 CFD and Optimization............................................................................. 223 CFD Optimization for Food Processing Operations................................ 224 11.5.1 Convective Heat Transfer Coefficient......................................... 224 11.5.2 Heat Exchangers.......................................................................... 225 11.5.3 Shape and Geometry ................................................................... 225 11.5.4 Heating Configurations ............................................................... 226 11.6 Conclusion ............................................................................................... 226 References ............................................................................................................. 226

11.1 INTRODUCTION In designing and analyzing the heat and momentum transfer (fluid flow) problems, there are two fundamental approaches: experimental and computational. The computation part involves solution of governing differential equations with applied boundary conditions in a defined computational domain. For the case of conduction heat transfer, relatively simple analytical solutions (consisted of infinite series solutions) can be easily obtained. In addition, numerical methodologies are also easier to apply for this case of heat transfer. However, when the fluid flow is involved with the heat transfer or with just itself, obtaining analytical or numerical solutions get rather difficult since the governing physical laws of fluid motion require all the parameters regarding fluid (e.g., pressure, temperature, and velocity) to be determined (Kumar and Dilber, 2006). For the latter cases, computational fluid dynamics (CFD) packages have been extensively used in the last decades.

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CFD is defined as the numerical solution of fluid flow involving problems based on the solution of the set of Navier–Stokes equations. Navier–Stokes equations are developed from conservation principles on an infinitesimal scale covering single fluid flows for liquids and gases, compressible and incompressible flows, inviscid and creeping flows, laminar and turbulent flows, and Newtonian and non-Newtonian flows (Verboven et al., 2004). In some cases, the fluid flow is coupled with energy equation to determine the temperature changes simultaneously with the fluid flow or with electromagnetic distribution where it is modeled as an internal heat generation in the energy equation. All these are in the capabilities of the newly improved CFD programs. In the view of food engineering, CFD can be defined to be a numerical methodology that has been used for solution of heat, mass, and momentum (fluid flow) transfer equations simultaneously with the given boundary conditions in a given computational domain, and it is widely used to simulate many processes in food industry. Over the last two decades, there has been enormous development of commercial CFD codes to enhance their interaction with sophisticated modeling requirements via the increasing power and memory capacity of computers and development of efficient numerical algorithms for the solution of required governing equations.

11.2 CFD FUNDAMENTALS General solution procedure in the CFD applications starts with composing the computation domain. Then, the grid (or mesh) where the computational domain is divided into small areas (for the case of two-dimensional [2D] problems) and small volumes (for the case of three-dimensional [3D] problems) is prepared. Grid generation is a very significant part of a CFD study since, with an improper mesh, erroneous or nonphysical solutions can also be obtained. For example, in CFD modeling of heat transfer in canned liquid foods, meshing should be implemented with a very fine grid near the walls to accurately resolve the velocities and temperature since velocity and temperature gradients would be relatively higher due to higher temperature difference, especially at the initial steps of the analysis leading to nonphysical solutions. The next step, after completing the grid generation, is to specify the initial and boundary conditions with the thermal and physical properties (sometimes their variation with, for example, temperature also becomes a required parameter) of the computational domain (thermal conductivity, specific heat, density and viscosity). Then, the numerical solution algorithms are selected to start for the solution. Then, the results of a CFD study can be analyzed both numerically and visually. The governing equations for continuity, conservation of momentum and conservation of energy to represent fluid flow and heat transfer can be written in general as follows: Continuity equation: @Uj ¼0 @xj

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(11:1)

Conservation of momentum:    @Ui @Ui Uj 1 @P @ @Ui @Uj D 0 0 E þ ¼  þ n þ  ui uj @t @xj @xj @xi r @xi @xj

(11:2)

Conservation of energy: r  cv 

 D E @T @T @ @T ¼ k  r  cp  u0j T 0 þ r  U j  cp  @t @xj @xj @xi

(11:3)

where U is the average velocity (m=s) u0 is the turbulent component of velocity (m=s) hu0i u0j i is the average value of the fluctuating component of the velocity T is the average temperature (K) T 0 is the fluctuating component of temperature (K) P is the pressure (Pa) r is the density (kg=m3) n is the kinematic viscosity of the fluid (m2=s) cp and cv are the heat capacity (J=kg-K) at constant pressure and volume, respectively and

Uj 

@ @ @ @ , ¼ , @xi @x @y @z

(11:4)

@ @ @ @ þ þ ¼ @xj @x @y @z

(11:5)

@ @ @ @ ¼ Ux  þ Uy  þ Uz  @xj @x @y @z

(11:6)

In turbulent flow, velocity magnitude fluctuates with time, and these fluctuations are known as the turbulence where the velocity in turbulent flow can be divided into average and turbulent components. The decomposition of flow field into average and turbulent (fluctuating) components has isolated the effects of fluctuations on the average flow. However, addition of the turbulence in the Navier–Stoke’s equations, as seen above, results in additional terms, known as Reynolds stresses, leading to a closure problem increasing the number of unknowns to be solved (Erdo gdu et al., 2007). In order to solve this problem, a mathematical path for the calculation of the turbulence quantities must be provided. There are special turbulence models to solve this issue, e.g., k  «, k  v (where k is the turbulence kinetic energy, « is the turbulence energy dissipation rate, and v is the turbulence frequency), Reynolds stress models and many others. Olsson et al. (2004) gives information for comparisons of these models with the experimental data available in the literature.

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The viscous dissipation term that appears in the conservation of energy equation (Equation 11.3) is neglected since it is of significance only for highspeed flows where its magnitude can be comparable to the magnitude of conduction term. For the solution of these equations using CFD, initial and boundary conditions are required after the grid generation is accomplished. Generally, the no-slip condition at the inside can walls was applied for momentum boundary condition with temperature or convective boundary condition of temperature across the outer surfaces. No-slip boundary condition states that the velocity of a fluid, which is in contact with a solid wall, is equal to the velocity of the wall. As the name implies, there is no-slip between the wall and the fluid itself. In other words, the fluid particles adjacent to the wall adheres to the wall and move with the same velocity. For the conditions, when the wall is motionless, then the fluid adjacent to the wall has the zero velocity (Çengel and Cimbala, 2006). Alternatively, wall velocity, free slip (zero shear stress), or a specified wall shear stress can also be set (Verboven et al., 2004). In addition to the no-slip condition, depending on the physical nature of the problem, pressure boundary conditions might also be required to define at the inlet and outlet conditions. As seen in the above given equations, energy equation representing the heat transfer is a scalar equation bringing one extra equation to solve into the continuity and momentum equations. Therefore, the computational requirements are not expected to increase in a high manner compared to the solution of only equations representing the fluid flow (Çengel and Cimbala, 2006). In addition to the given advantages of solving the heat and momentum transfer simultaneously, CFD applications should also be noted that the boundary layer phenomena and corresponding temperature changes are easily solved across the surface of the objects. Therefore, additional heat transfer coefficients for the case of heat transfer and boundary layer models are not required. Reviews on fundamentals of CFD use in food processing have been given by different authors in the literature (Scott and Richardson, 1997; Verboven et al., 1999, 2004; Abdul Ghani and Farid, 2006; Kumar and Dilber, 2006).

11.3 PROCESS OPTIMIZATION Optimization can be defined as the choice of a best alternative from a specified set of alternatives. Achieving optimization, therefore, requires some way of describing the potential alternatives and deciding which of the alternatives is the best (Norback, 1980; Evans, 1982). Formal description of any optimization problem has three parts: a set of variables, called decision variables, which the optimization method can control and use to specify the alternatives (e.g., process temperature profile during thermal processing), a set of requirements or constraints (e.g., the differential equations, boundary conditions, and integral equations specifying the biological material concentration at the end of the process), which the optimization method must achieve or satisfy and a measure of performance to compare one alternative to another (the objective function; e.g., retention of nutrients) (Norback, 1980).

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Objective function, which may be continuous or in some cases discrete, is the function to be optimized (maximized or minimized). This may be obtained from either a mathematical model or by fitting an equation through the experimental points (Saguy, 1983). Optimization problems are divided into continuous and discrete types depending on the objective function. Discrete problems usually have a finite number of variables, each of which assumes exactly one value at an optimal solution. In continuous problems, the optimal variable values are functions of some parameter, and a solution to the problem requires the specification of this function over the parameter set. Generally, the continuous optimization problems require numerical methods for a solution (Norback, 1980; Saguy, 1983). Thermal processing of food products can be given as an example for a continuous problem. In this example, the change in retort temperature is control variable, and the maximization of a nutrient in overall volume or at the surface can be taken as an objective function. The constraints can be given as the lower and higher limits of the process temperature profile-explicit constraints, as well as the coldest point lethality or temperature obtained at the end of the process-implicit constraints (Norback, 1980). On the other hand, the food product systems might contain more than one product, and optimization using any objective function for each product results in a multiobjective optimization problem, which is sometimes referred to as vector optimization since a vector of objectives instead of a single objective is optimized (Deb, 2002). The objective then becomes to maximize each individual objective function simultaneously (Bhaskar et al., 2000). Combination of several objective functions into a scalar function using arbitrary weight factors leads to the optimization problem that could become computationally tractable. However, this scalarization causes the results’ becoming sensitive to the values of weighting factors, which are difficult to pre-assign, and there is a risk of losing some optimal points (Bhaskar et al., 2000). Therefore, the objective functions and constraints are then classified by priority using the Lexicographic ordering approach, and preferred solution is defined which simultaneously optimizes as many of the objective functions as possible (Bhaskar et al., 2000). If there is no conflict between the objective functions and constraints, a solution can be found where each objective function attains its maximum (Miettinen, 1999).

11.4 CFD AND OPTIMIZATION Due to the modeling efforts and required computational resources, CFD was not used with full ability for optimization purposes. Most of the CFD optimization efforts since 1970s have focused on aerospace design problems (El-Sayed and Berry, 2005). Basic elements of CFD optimization are the geometry-grid manipulation via the same CFD package or different software and application of CFD analysis systematically through the optimization algorithm (Figure 11.1). Use of CFD for optimization purposes in food processing is also very limited. So far, there have been some studies reported where the CFD models were applied in a manner of trial and error optimization methodologies in different food processing operations to determine the convective heat transfer coefficient.

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Explicit and implicit constraints

Composing computational domain, grid generation, and specifiying initial and boundary conditions Determine objective function

Optimally set decision variables

CFD solution Check with constraints Adjust decision variables through an optimization algorithm

FIGURE 11.1 Design of a CFD-optimization problem.

11.5 CFD OPTIMIZATION FOR FOOD PROCESSING OPERATIONS 11.5.1 CONVECTIVE HEAT TRANSFER COEFFICIENT Verboven et al. (1997) determined the surface heat transfer coefficient during thermal processing of foods for various heating conditions using a CFD technique. For this purpose, the local heat transfer coefficients were obtained from the local surface flux and surface temperatures at a given time step: q00 ¼ k 

 @T  ¼ h  (Tjs  T1 ) @x s

(11:7)

where q00 is the heat flux (W=m2) h is the local heat transfer coefficient Tjs is the local surface temperature T1 is the medium temperature k is the thermal conductivity of air at the boundary layer @T @x js is the temperature gradient at the boundary Then, integration of the local coefficients over the surface led to the surface average heat transfer coefficient. Denys et al. (2003) also presented a combined CFD and experimental based approach to determine the surface heat transfer coefficient across the shell when processing in-shell eggs. The procedure was based on the comparison of experimental temperature change of the egg at three locations (from somewhere around the center to the surface-under the shell) with the CFD model results. For the CFD model, convective boundary condition was applied across the shell where the heat transfer coefficient was the parameter to include by a temperature change based systematic trial and error methodology. Even though there have been limited work on the CFD applications for optimization purposes in food processing, and the studies on the subject were on determination of heat transfer coefficient, there are certain developments in other engineering disciplines for this subject, and these can be easily applied and adapted

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for food process engineering. The following covers the new applied CFD optimization studies from different areas of engineering applications.

11.5.2 HEAT EXCHANGERS Foli et al. (2006) presented two methods for determining the optimal design parameters for micro-channels in micro heat exchangers. One method was to combine CFD analysis with an analytical method of calculating optimal geometric parameters while the second one involved using multi-objective genetic algorithms with CFD. Hilbert et al. (2006) applied CFD with parallel genetic algorithms for multiobjective shape optimization of a heat exchanger. The objective was to find out the geometry favorable to simultaneously maximize heat transfer, while minimizing the pressure losses. CFD was applied to perform the numerical simulation of fluid flow and heat transfer. Their algorithm consisted of three steps: generation of the profile blades for the exchanger from the design variables, CFD solution, and postprocessing of the results to determine the objective functions with respect to the design variables. Parallelization has been applied with the genetic algorithms to reduce the computing time. These studies showed a possible use of CFD combined with an optimization algorithm for a better design of heat exchangers. Heat exchangers are used for heating and cooling of foods in food processing, and heat exchange through the smooth and corrugated walls are of a greater importance due to the increase in turbulence and resulting heat transfer coefficient. There should be an optimum configuration of the fins and pins inside the heat exchangers (Lee et al., 2001), and heat transfer optimization in heat exchangers (Fabbri, 2000; Rozzi et al., 2007) are important for better processing.

11.5.3 SHAPE

AND

GEOMETRY

There have been also studies reported for shape optimization to accomplish a more uniform fluid flow (El-Sayed and Berry, 2007). In this study, a numerical optimization tool (i.e., VisualDOC=DOT) was integrated with a CFD code for shape optimization of an airfoil and S-shaped duct with an objective of minimizing the pressure losses. Flow of solid–liquid food mixtures in continuous aseptic processing systems are extensively studied in food process engineering. Analysis of flow in different tube configurations is significant due to the changes in the residence time distribution of the food mixtures. Helical tubes, for example, are used to prove mixing under laminar flow conditions as a result of the unbalanced centrifugal forces with leading to more uniform heat treatments and more superior heat transfer performance (Palazoglu and Sandeep, 2002a). Palazoglu and Sandeep (2002b) applied CFD analysis to determine the maximum fluid velocity and pressure losses in the axial and radial directions of different helical tube configurations concluding that a helical holding tube 20% shorter than a straight one with a curvature ratio of 0.143 provided a similar residence time. An optimization study for improved design of helical tubes, similar to the work reported by El-Sayed and Berry (2007) can also be applied for better food processing. For this case, CFD simulation of fluid flow and heat transfer will be required via the applied optimization procedures for better shape of the tubes.

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In addition, it is a known fact that shape, dimensions, and surface properties might significantly affect the surface heat and mass transfer coefficient (Ghisalberti and Kondjoyan, 1999). Based on Ghisalberti and Kondjoyan (1999), effect of flow properties, i.e., velocity, turbulence intensity, is more effective compared to the geometrical properties of the product, but the flow properties can be partly accepted to be a function of the geometry. CFD applications for maximum heat transfer where a fluid flow is included can therefore be applied with further optimization techniques where the objective function can be the modifications in the shape of the product.

11.5.4 HEATING CONFIGURATIONS In addition to the shape optimization, different heating configurations are also significant in food processing. For example, in ohmic heating, configuration of heating elements play significant role in temperature distribution of the products. Recently, Jun and Sastry (2007) developed a 3D model for verification of ohmic heating for sterilization efficacy and stated that the given CFD model can be a tool for further optimization for electrode configurations, resigning pouch configurations, and minimizing the presence of cold spots during the sterilization process. To accomplish this objective, the given CFD model should be used with an optimization algorithm where the objective functions and constraints were stated based on the given purposes.

11.6 CONCLUSION As pointed out, the CFD optimization studies are limited for food processing operations, but it is obvious that, with the increasing demand to the CFD applications in this area, there is a need for these works similar to the ones given from other disciplines. CFD optimization studies are expected to fill a big gap in food processing especially via the ability of CFD programs where the complex geometry foods and food processing equipment can be simulated exactly. The major problem noted so far is the computing time. The computing time really becomes an important issue in the optimization studies involving CFD solutions for heat transfer and fluid flow. In a given CFD optimization study, variations in the design take a significant amount of time to evaluate, and this makes the iterative optimization excessively time consuming since optimization procedure generally lies in nonlinear optimization area with both equality and inequality constraints. Further investigations have been presently conducted to decrease the computing time for CFD solutions.

REFERENCES Abdul Ghani, A.G. Al-Baali, and Farid, M., Fundamentals of computational fluid mechanics, in Sterilization of Food in Retort Pouches, Springer, New York, Chapter 6, 2006. Bhaskar, V., Gupta, S.K., and Ray, A.K., Applications of multiobjective optimization in chemical engineering, Reviews Chem. Eng., 16, 1, 2000. Çengel, Y.A. and Cimbala, J.M., Fluid Mechanics Fundamentals and Applications, McGraw-Hill Companies, Inc., New York, Chapter 15, 2006.

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Deb, K., Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Inc., New York, 2002. Denys, S., Pieters, J.G., and Dewettinck, K., Combined CFD and experimental approach for determination of the surface heat transfer coefficient during thermal processing of eggs, J. Food Sci., 68, 943, 2003. El-Sayed, M. and Berry, T.S., Shape optimization with computational fluid dynamics, Adv. Eng. Software, 36, 607, 2005. Erdogdu, F., Ferrua, M., Singh, S.K., and Singh, R.P., Air-impingement cooling of boiled eggs: Analysis of flow visualization and heat transfer, J. Food Eng., 79, 920, 2007. Evans, L.B., Optimization theory and its application in food processing, Food Tech., 36(7), 88, 1982. Fabbri, G. and Chang, M.H., Heat transfer optimization in corrugated wall channels, Int. J. Heat Mass Transfer, 43, 4299, 2000. Foli, K., Okabe, T., Olhofer, M., Jin, Y., and Sendhoff, B., Optimization of micro heat exchanger: CFD, analytical approach abd multi-objective evolutionary algorithms, Int. J. Heat Mass Transfer, 49, 1090, 2006. Ghisalberti, L. and Kondjoyan, A., Convective heat transfer coefficients between air flow and a short cylinder. Effect of air velocity and turbulence. Effect of body shape, dimensions and position in the flow, J. Food Eng., 42, 33, 1999. Hilbert, R., Janiga, G., Baron, R., and Thevenin, D., Multi-objective optimization of a heat exchanger using parallel genetic algorithms, Int. J. Heat Mass Transfer, 49, 2567, 2006. Jun, S. and Sastry, S., Reusable pouch development for long term space missions: A 3D ohmic model for verification of sterilization efficacy, J. Food Eng., 80, 1199, 2007. Kumar, A. and Dilber, I., Fluid flow and its modeling using computational fluid dynamics, in Handbook of Food and Bioprocess Modeling Techniques, Sablani, S.S., Rahman, M.S., Datta, A.K., and Mujumdar, A.A., (Eds.), CRC Press, Boca Raton, FL, Chapter 3, 2006. Lee, K.S., Kim, W.S., and Si, J.M., Optimal shape and arrangement for staggered pins in the channel of a plate heat exchanger, Int. J. Heat Mass Transfer, 44, 3223, 2001. Miettinen, K.M., Nonlinear Multiobjective Optimization, Kluwer, Norwell, MA, 1999. Norback, J.B., Techniques for optimization of food processes, Food Tech., 34(2), 86, 1980. Olsson, E.E.M., Ahrne, L.M., and Tragardh, A.C., Heat transfer from a slot air jet impinging on a circular cylinder, J. Food Eng., 63, 393, 2004. Palazoglu, T.K. and Sandeep, K.P., Effect of holding tube configuration on the residence time distribution of multiple particles in helical tube flow, J. Food Proc. Eng., 25, 337, 2002a. Palazoglu, T.K. and Sandeep, K.P., Computational fluid dynamics modeling of fluid flow in helical tubes, J. Food Proc. Eng., 25, 141, 2002b. Rozzi, S., Massini, R., Paciello, G., Pagliarini, G., Rainieri, S., and Trifiro, A., Heat treatment of fluid foods in a shell and tube heat exchanger: Comparison between smooth and helically corrugated wall tubes, J. Food Eng., 79, 249, 2007. Saguy, I., Optimization methods and applications, in Computer Aided Techniques in Food Technology, Saguy, I. (Ed.), Marcel Dekker, New York, Chapter 10, pp. 268–320, 1983. Scott, G. and Richardson, P., The application of computational fluid dynamics in the food industry, Trends in Food Sci. and Tech., 8, 119, 1997. Verboven, P., Nicolai, B.M., Scheerlinck, N., and De Baerdemaeker, J., The local surface heat transfer coefficient in food process calculations: A CFD approach, J. Food Eng., 33, 15, 1997. Verboven, P., Scheerlinck, N., De Baerdemaeker, J., and Nicolai, B.M., Possibilities and limitations of computational fluid dynamics for thermal process optimization, in Processing Foods: Quality Optimization and Process Assessment, Oliveira, F.A.R., Oliveira, J.C., Hendrickx, M.E., and Korr, D., (Eds.), CRC press, Boca Raton, FL, Chapter 3, 1999. Verboven, P., De Baerdemaeker, J., and Nicolai, B.M., Using computational fluid dynamics to optimize thermal processes, in Improving the Thermal Processing of Foods, Richardson, P., (Ed.), CRC Press, Boca Raton, FL, Chapter 4, 2004.

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12

Dynamic Optimization J. Ricardo Pérez-Correa, Claudio A. Gelmi, and Lorenz T. Biegler

CONTENTS 12.1 12.2

Introduction .............................................................................................. 229 Comparison of Sequential and Simultaneous Approaches: Batch Rice Drying ................................................................................... 232 12.3 Sequential Approach Example: Optimal Distillation of Young Spirits....................................................................................... 242 12.4 Large Scale Example with the Simultaneous Approach: Calibration of a Wine Fermentation Metabolic Model ........................... 246 12.5 Summary .................................................................................................. 250 References ............................................................................................................. 251

12.1 INTRODUCTION Modern process engineering increasingly relies on model-based methods. Hence, mathematical models are widely used to design and establish optimum operating conditions of processing units or whole processing plants. Most of the advances in this area come from applications to chemical processing plants that operate in continuous (steady state) mode. Consequently, mathematical models are defined by large sets of nonlinear algebraic equations, and optimal design can be formulated as a nonlinear programming (NLP) problem. Powerful tools are available to optimize the design and operation of such plants. By contrast, the large majority of the operations in food processing are batch or semicontinuous, meaning that they are intrinsically dynamic. In these cases, finding optimal designs and operating strategies are much more difficult problems. These problems are called dynamic optimization (DO) or open loop optimal control. In food process engineering, DO can handle a variety of problems, such as optimal design and operation of batch processes, nonlinear model predictive control, parameter estimation of dynamic models, and dynamic data reconciliation. In this chapter we focus on optimal operation of batch processes. DO has been applied to compute optimal operating policies in several discontinuous food processing problems, such as reverse osmosis of cheese whey (Van Boxtel and Otten, 1993), microwave heating (Sanchez et al., 2000), batch drying (Olmos et al., 2002), contact cooking (Zorrilla et al., 2003), freeze drying (Boss et al., 2004), wine distillation (Osorio et al., 2005), batch crystallization (Vu et al., 2006), thermal sterilization (Simpson et al., 2008), and bakery operations (Hadiyanto et al., 2008).

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We consider the DO problem with N dynamic stages (or periods) stated in the following form: min

u(t), p

N X

fk (zk (tk ), uk (tk )),

k¼1

dzk (t) ¼ f (zk (t), yk (t), uk (t), pk ), dt g(zk (t), yk (t), uk (t), pk ) ¼ 0, gk, f (zk (tk )) ¼ 0,

s:t:

zk (tk1 ) ¼ h(zk1 (tk1 )),

zk,L  zk (t)  zk,U , uk, L  uk (t)  uk, U , t 2 [tk1 , tk ], k ¼ 1, . . . , N

(12:1) yk, L  yk (t)  yk, U

where wk is the performance index in stage k ¼ 1, . . . , N zk(t) are the differential state variables yk(t) are the algebraic variables uk(t) are the control variables In addition, pk represents the time-invariant parameters and t 2 [tk1, tk] is the time. The constraints are given by the process model, defined by differential and algebraic equations (DAEs) and by lower and upper bounds in the control, algebraic, and state variables; uk(t) and pk are the decision variables that should be found. Equation 12.1 is sufficiently general to cover a wide range of applications. Moreover, process dynamics in food engineering problems are often described by partial differential equations (PDEs) and efficient discretization methods can be used to transform the PDEs into DAEs (Balsa-Canto et al., 2002a,b; Balsa-Canto et al., 2004). Although many methods have been developed to solve the DO problem defined by Equation 12.1, none of them can be used blindly to solve any of the medium-tohigh complexity problems usually found in food processing (Garcia et al., 2006). This complexity arises from strong nonlinearities, coupled phenomena, and many safety and quality constraints. In the variational or indirect approach, the solution is based on the first-order necessary conditions from Pontryagin’s maximum principle (Pontryagin et al., 1962). For DO problems without inequalities (such as the bounds in Equation 12.1) these conditions can be written as a set of state and adjoint DAEs, which results in a two-point boundary value problem that can be solved using several methods (Cervantes and Biegler, 2000). If inequality constraints are present, the usual case in food processing, finding the correct switching structure as well as suitable initial guesses for state and adjoint variables is often very difficult (Bryson, 1999). Problems with inequality constraints are much easier to solve using the direct approach. Here, the original infinite dimensional DO problem is transformed into a finite dimensional NLP problem through suitable discretization using sequential or simultaneous strategies. In the former scheme, also known as control vector parameterization or single shooting, only the control variables uk(t) are discretized.

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Typically, low order piecewise polynomials are used to represent the control variables and polynomial coefficients are the optimization variables. In turn, the DAEs representing the dynamic process models are solved at each iteration with the control variables defined by the NLP solver. The sequential strategy is easy to apply especially if a reliable process model is already available. On the other hand, for large-scale problems sequential methods may be slow due to the repeated numerical integration of the DAE set. Additionally, these methods cannot deal with open loop instability, and path constraints can be handled only approximately (Cervantes and Biegler, 2000). Instead, for unstable systems multiple shooting, which inherits many of the advantages of sequential approaches, should be applied. Here, the time domain can be partitioned into N smaller time elements, similar to the multistage structure in Equation 12.1, and the DAE models are integrated separately in each element (Bock, 1983; Bock and Plitt, 1984; Leineweber, 1999). Control variables are parameterized as in the sequential approach and gradient information is obtained for both control variables and the initial conditions of the state variables in each element. Finally, equality constraints (Equation 12.1) are added in the NLP to link the elements and ensure that the states are continuous across each element. As with the sequential approach, bound constraints for states and controls are usually imposed only at the grid points of tk. In the simultaneous collocation approach, also known as complete parameterization or direct transcription, both the state and control variables are discretized using collocation of finite elements (Biegler, 2007), say k ¼ 1, . . . , N in Equation 12.1. This approach corresponds to a fully implicit Runge–Kutta method with high-order accuracy and excellent stability properties. The discretization is a desirable way to obtain accurate solutions for boundary value problems and related optimal control problems. Moreover, this approach avoids the repeated solution of the DAEs; the set of DAEs is solved only once at the optimal point avoiding infeasible or hard to converge intermediate solutions. Additionally, the simultaneous approach can deal with instabilities and is able to enforce state and control variable constraints accurately. On the other hand, simultaneous approaches require the solution of large nonlinear optimization problems. For this, efficient large-scale optimization strategies should be applied to realize the benefits of this approach. Fortunately, a number of recent nonlinear programming solvers, including KNITRO (Byrd et al., 1999), LOQO (Vanderbei and Shanno, 1999), SOCS (Betts and Huffman, 1992), and IPOPT (Wächter and Biegler, 2006, see also https:==projects.coin-or.org=Ipopt) are well suited for this task. A particular advantage of the simultaneous approach is that exact first and second derivatives can be obtained very cheaply using optimization tools like AMPL (Fourer et al., 2002) and GAMS (Brooke et al., 1998). Moreover the expensive DAE integration and sensitivity steps are avoided. Also, inequality constraints can be handled efficiently by interior point (i.e., log-barrier) terms and the NLP solvers can be tailored to adapt to different problem structures. This allows the efficient solution of very large NLPs on the order of several million variables, constraints, and even degrees of freedom. In this chapter we present several detailed case studies in order to illustrate the application of some of the ideas and strategies mentioned above and the variety of food engineering problems that can be solved by DO.

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12.2 COMPARISON OF SEQUENTIAL AND SIMULTANEOUS APPROACHES: BATCH RICE DRYING Paddy rice and rough rice are terms used to describe the separate grains of rice obtained directly from the field after harvest. These grains consist of a thick, hard shell, an inside kernel and a bran layer that covers this kernel. Drying of paddy rice causes varying percentages of broken kernels resulting in quality degradation. However, through carefully defined process conditions, the impact of drying on paddy rice quality can be significantly minimized. Olmos et al. (2002) developed a constrained DO procedure to establish optimum inlet air conditions to minimize quality degradation during paddy rice batch drying. The authors applied a simultaneous approach where a collocation method was used to discretize the differential equations, and a sequential quadratic programming (SQP) algorithm was used to solve the optimization problem. The procedure was experimentally validated with a laboratory scale batch dryer. Here, we use the model described in Olmos et al. (2002) to illustrate the characteristics of both the sequential and the simultaneous approaches for dynamic optimization. The model consists of two compartments each representing thin layer of rice and an empirical quality degradation kinetic expression (Olmos et al., 2002). It is assumed that both compartments have different humidities but have the same temperature. Hence, process state variables are moisture content in the inner compartment (x1), moisture content in the outer compartment (x2), grain temperature (Tg), and grain quality (Q). Applying dynamic mass and energy balances, the following dynamic model is obtained: dx1 b1 ¼ (x2  x1 ) dt rg  t 1

(12:2)

dx2 b2  Ssg dx1 t 1 ¼  (pa  pg )  dt rg  t 2 dt t 2

(12:3)

dTg a  Ssg  (Ta  Tg ) þ b2  Ssg  (pa  pg )  Lv ¼ dt rg  (Cpg þ Cpw  xm )

(12:4)

dQ ¼ K  Q2 dt

(12:5)

Here K is the quality degradation rate coefficient, which is given by an Arrheniustype law:   Ea 5 (12:6) K ¼ K0  (x2  x1 )  exp R  (Tg þ 273:1) According to Equation 12.5, rice quality can increase or decrease during the drying process since K can be positive or negative. In practice, grain quality Q cannot improve during drying, therefore the condition dQ dt  0 was imposed to the solution of Equation 12.5.

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The volumetric average moisture content of the grain is x m ¼ x1  t 1 þ x 2  t 2

(12:7)

Mass transfer between compartments 1 and 2 is defined by the coefficient: b1 ¼ B10  exp (B11  xm  Tg )

(12:8)

and that between compartment 2 and the air is given by the coefficient: b2 ¼ B20  exp (B21  Ta )

(12:9)

Heat transfer between the grain surface and the drying air is calculated using a coefficient given by an empirical relationship: a ¼ C5  Lv  b2 The water activity at the grain surface is given by    exp ((C1  x2 )=C2 ) Aw ¼ exp C3  (Tg  C4 )

(12:10)

(12:11)

The partial vapor pressure at the grain surface (pg) and the partial water pressure in the drying air (pa) are modeled using the Antoine equation: pg ¼ Aw  exp(E  F =(G þ (Tg þ 273:1)))  103

(12:12)

pa ¼ Hr  exp(E  F =(G þ (Ta þ 273:1)))  103

(12:13)

Physical constants and parameters of the dynamic model above are presented in Table 12.1. The goal of the optimization problem is to ensure the highest possible final product quality (Q) for a fixed drying time (tf) and final moisture content (xtarget). In order to achieve this goal, the inlet air conditions, air temperature (Ta), and relative humidity (Hr) are manipulated as a function of time. The optimization problem in mathematical form is max Q

Hr (t ),Ta (t )

s:t: model Equations 12:2 through 12:13 xm (tf ) ¼ xtarget Ta min  Ta (t)  Ta max Hr min  Hr (t)  Hr max Initial and imposed conditions and constraints are shown in Table 12.2.

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(12:14)

TABLE 12.1 Constants and Parameters of the Mathematical Model Constants C5 Cpg Cpw R Ssg rg t1 t2 Lv

Parameters 1

65 Pa  8C 1300 J  (kg dry matter)1  8C1 4210 J  (kg water)1  8C1 8.32 J  mol1  K1 2000 m2  m3 1500 (kg dry matter)  m3 0.6 m3  m3 0.4 m3  m3 2.358  106 J  kg1

B10 B11 B20 B21 C1 C2 C3 C4 Ea K0 E F G

0.01316 (kg dry matter)  m3  s1 0.3083 (kg water)1  (kg dry matter)  8C1 2.304  109 (kg water)  m2  Pa1  s1 0.04428C1 0.319 (kg water)  (kg dry matter)1 0.0493 (kg water)  (kg dry matter)1 1.89948C1 2.54578C 1.657  105 J  mol1 1.56  1027 (kg water)5  (kg dry matter)5  %1  s1 38.9974 3985.44 (Antoine parameters) 16.5362

On the basis of the optimization results obtained by Olmos et al. (2002), we apply the sequential approach and approximate the air temperature (Ta) and relative humidity (Hr) profiles by quadratic functions: Ta (t) ¼ a0 þ a1  t þ a2  t 2

(12:15)

Hr (t) ¼ b0 þ b1  t þ b2  t 2

(12:16)

Here, ai and bi are unknown coefficients adjusted by the optimization procedure. TABLE 12.2 Imposed, Initial Conditions and Constraints Initial conditions x1(0) x2(0) Tg(0) Q(0)

0.27 0.27 208C 80%

Bounds for Ta and Hr Hr min Hr max Ta min Ta max

5% 80% 408C 808C

Imposed conditions tf xtarget

7200 s 0.13

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The Optimization Toolbox (ver. 3.0.3) of MATLAB R14SP3 (http:==www.math works.com) was used in an Intel Pentium Dual Core (1.86 GHz) computer with 2 GB of RAM to solve the maximization problem of paddy rice drying. Here, we applied a sequential simulation-optimization approach using fmincon, a constrained nonlinear multivariable SQP code, and fminsearch a multivariable, unconstrained, derivative free code that implements the Nelder–Mead simplex search method. We could not achieve convergence with both codes for the original problem. However, a solution was found if the problem was simplified by fixing Hr at the optimum profile and optimizing only the ai coefficients of Ta. An optimum value of Q ¼ 69.5% was obtained with fmincon after 69 iterations, 303 function evaluations, and 52 s. This is likely due to gradient errors (obtained from finite difference) as well as convergence difficulties due to poor starting points. On the other hand, better results were obtained using the fminsearch solver, as is now explained below. To consider the problem constraints with the unconstrained fminsearch code, two terms were added to the objective function: L  (x1t1 þ x2t2  xtarget)2, where L is an arbitrary large number (5  104 in this case), and a constant penalty of 10 if bounds on Ta were not satisfied. In this case, we also fixed Hr to the optimum profile. To help the reader adapt the program to a similar problem, the corresponding code in MATLAB for this problem is given next. function optimricefinal % This program maximizes the final product quality (Q) for a fixed % drying time (tf) and final moisture content (xtarget). In order to % achieve this goal, the air temperature (Ta) is manipulated as a function % of time. % % Mathematical model taken from the article: % % A. Olmos, I.C. Trelea, F. Courtois, C. Bonazzi, and G. Trystam. % Dynamic optimal control of batch rice drying process. Drying % Technology, 20(7), 1319–1345, 2002. % % Last update: 18=01=2008. %%%%%%%%%%%%%%%% Main Code %%%%%%%%%%%% tic % Optimization function (fminsearch) x0 ¼ [5 5 90]; % Initial condition for the coefficients of Ta 2nd order polynomial options ¼ optimset(‘MaxFunEvals’,1e4,‘MaxIter’,1e5, ‘TolX’,1e-4,‘TolFun’,1e-4,‘Display’,‘iter’);

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[param,fval,exitflag,output] ¼ fminsearch(@of,x0, options) Toc % Values at t ¼ tf [t,x] ¼ ode113(@modelo,[0 7200],[.27 .27 20 80],[],param); vfinales ¼ x(end,:) xm ¼ x(end,1)*0.6þx(end,2)*0.4 % Figures t ¼ t=3600; % time is converted from seconds to hours figure(1) subplot(2,2,1) plot(t,0.6*x(:,1)þ0.4*x(:,2)*0.4) xlabel(‘Time (h)’) ylabel(‘x_m’) subplot(2,2,2) plot(t,x(:,3)) xlabel(‘Time (h)’) ylabel(‘Tg (8C)’) subplot(2,2,3) plot(t,x(:,4)) xlabel(‘Time (h)’) ylabel(‘Q (%)’) figure(2) Ta0 ¼ polyval(x0,t); % Initial profile Ta ¼ param(1)*t.^2þparam(2)*tþparam(3); % Final profile plot(t,[Ta0 Ta]) xlabel(‘Time (h)’) ylabel(‘Temperature profile (8C)’) legend (‘Initial profile’,‘Final profile’,0) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function Qtf ¼ of(param) % The constraint 40 < Ta < 80 is checked t ¼ 0:0.1:2; % 0 to 2 hours Ta ¼ param(1)*t.^2þparam(2)*tþparam(3); aux ¼ 0; if (min(Ta) < 40) j (max(Ta) > 80) aux ¼ 10; end

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% Integration of the ODE model [t,x] ¼ ode113(@modelo,[0 7200],[.27 .27 20 80],[],param); % Objective function is computed after the model is integrated Qtf ¼ x(end,4) þ 5e4*(0.6*x(end,1)þ0.4*x(end,2)0.13) ^2 þ aux; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function dx ¼ modelo(t,x,param) % ODE model % Constants C5 ¼ 65; Cpg ¼ 1300; Cpw ¼ 4210; rog ¼ 1500; Ssg ¼ 2000; tau1 ¼ 0.6; tau2 ¼ 0.4; R ¼ 8.32; % Parameters determined from experimental data B10 ¼ 0.01316; B11 ¼ 0.3083; B20 ¼ 2.304e-9; B21 ¼ 0.0442; C1 ¼ 0.319; C2 ¼ 0.0493; C3 ¼ 1.8994; C4 ¼ 2.5457; K0 ¼ 1.56e27; Ea ¼ 1.657e5; Lv ¼ (2608.8-251.2)*1e3; % Polynomial profiles t1 ¼ t=3600; % time is converted from seconds to hours Ta ¼ param(1)*t1^2þparam(2)*t1þparam(3); % 2nd order polynomial Hr ¼ (103.03*t1^9774.39*t1^8þ2359.2*t1^73717.5*t1^6þ 3126.8*t1^51173.9*t1^465.224*t1^3þ146.36*t1^2þ 23.267*t1þ52.443)=100; % 9th order %polynomial to fit Olmos results better. % Constitutive equations K ¼ -K0*((x(2)-x(1)))^5*exp(-Ea=(R*(x(3)þ273.1))); xm ¼ x(1)*tau1þx(2)*tau2; Beta1 ¼ B10*exp(B11*xm*x(3)); Beta2 ¼ B20*exp(B21*Ta); alpha ¼ C5*Lv*Beta2; Aw ¼ exp(-exp( (C1-x(2))=C2)=(C3*(x(3)-C4))); Pgsat ¼ exp((16.5362-3985.44=(38.9974þ(x(3)þ273.1))))*1e3; pg ¼ Pgsat*Aw;

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pa ¼ Hr*exp((16.5362-3985.44=(38.9974þ(Taþ273.1))))*1e3; % Differential equations dx(1) ¼ Beta1*(x(2)-x(1))=(rog*tau1); % dx1=dt dx(2) ¼ Beta2*Ssg=(rog*tau2)*(pa-pg)-dx(1)*tau1=tau2; % dx2=dt dx(3) ¼ (alpha*Ssg*(Ta-x(3))þBeta2*Ssg*(pa-pg)*Lv)=(rog* (CpgþCpw*xm)); % dTg=dt % Here we impossed that dQ=dt < 0 A ¼ -K*x(4)^2; % dQ=dt if A > 0 A ¼ 0; end dx(4) ¼ A; dx ¼ dx0 ; We were able to find an optimum with this code when the initial temperature profile was very different from the optimum profile. Moreover, the initial profile does not have to satisfy operating variables constraints. Figure 12.1 shows the result of a typical optimization run that took 110 iterations, 258 function evaluations, and 31 s. In summary, the sequential simulation–optimization approach with codes from the Optimization Toolbox from MATLAB can be used to solve the paddy rice batch drying problem although the original problem had to be simplified, i.e., only one operating variable profile could be optimized at a time. This is a serious limitation in this case since both operating variables are important. In turn, using the simultaneous approach, Olmos et al. (2002) were able to optimize both Hr and Ta, achieving a higher Q (76%). For the simultaneous approach we solved the problem given by Equation 12.14 directly, using a complete discretization of the state and control profiles with

Temperature profile (°C)

100 90 80 70 60 50 40

0

0.5

1

1.5

2

Time (h)

FIGURE 12.1 Optimal temperature profile (dark line) and initial profile (dashed line) for batch drying of paddy rice. Optimum Q ¼ 75.2%.

ß 2008 by Taylor & Francis Group, LLC.

Relative humidity

Air temperature (°C)

80 75 70 65 60 55 50 45 40 0

0.5

1

1.5

2

0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05 0

Time (h)

FIGURE 12.2 Q ¼ 77.7%.

0.5

1

1.5

2

Time (h)

Optimal controls for rice drying with simultaneous approach. Optimum

Radau collocation on finite elements (Biegler, 2007), an implicit Runge–Kutta method. In particular, we apply the same level of discretization to the two control profiles (Ta, Hr) as for the four state profiles (x1, x2, Tg, Q). This leads to a large NLP with many degrees of freedom. For instance, with 3 collocation points and 100 finite elements, the discretized counterpart to Equation 12.14 contains 2600 variables and 2401 equality constraints. Programmed in AMPL, the problem can be coded in the AMPL. mod file as shown below. This self-contained file can also be adapted to other optimal control problems. Here, we initialize the NLP with constant control profiles (Ta ¼ 90, Hr ¼ 0.5) which are far from the optimum. Nevertheless, solution of this problem requires only 146 iterations and 10.12 CPU s (Intel Pentium, IV 2.2 GHz, 1.0 GB RAM running Windows XP) with IPOPT. The control profiles are given in Figure 12.2, and their trends compare qualitatively with those obtained in Olmos et al. (2002) and the optimum value for Q (77.7%) is also slightly better than theirs. Moreover, this problem was solved with 600 finite elements (15,600 variables and 14,401 equality constraints). Requiring 156.2 CPU s with IPOPT and also achieving an optimum Q of 77.7%, the solution profiles are virtually identical to the one with 100 finite elements. # # # # # # # # # # #

¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼ This program maximizes the final product quality (Q) for a fixed drying time (tf) and final moisture content (xtarget). In order to achieve this goal, the air temperature (Ta) and relative humidity (Hr) are manipulated as functions of time. Model taken from the article: A. Olmos, I.C. Trelea, F. Courtois, C. Bonazzi, and G. Trystam. Dynamic optimal control of batch rice drying process. Drying Technology, 20(7), 1319–1345, 2002. Last update: 28=01=2008. ¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼¼

# define constants in rice drying model param time :¼ 2; # final time in hours

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param C5 :¼ 65; param Cpg : ¼ 1300 ; param Cpw : ¼ 4210 ; param Ssg :¼ 2000 ; param R : ¼ 8.32 ; param tau1 : ¼ 0.6 ; param tau2 :¼ 0.4 ; param rog : ¼ 1500 ; param tf : ¼ 2.0 ; param B10 :¼ 0.01316*3600; param B11: ¼ 0.3083; param B20 : ¼ 2.304e-9*3600; param B21 :¼ 0.0442; param C1 : ¼ 0.319; param C2 : ¼ 0.0493; param C3:¼ 1.8994; param C4 : ¼ 2.5457; param Ea : ¼ 1.657e5; param K0 :¼ 1.56e27*3600; param Lv : ¼ (2608.8-251.2)*1e3; # initial conditions for differential variables param x1_init :¼ 0.27; param x2_init : ¼ 0.27; param x3_init : ¼ 20; param q_init :¼ 80; param t_init : ¼ 0; # define indices and number of collocation points and finite elements param nfe >¼ 1 integer ; param ncp >¼ 1 integer ; let nfe :¼ 100 ; let ncp :¼ 3 ; set fe :¼ 1..nfe ; # number of finite elements set cp :¼ 1..ncp ; # number of collocation points param h{fe} : ¼ 1=nfe ; # finite element length # coefficients of Radau collocation (implicit Runge-Kutta) matrix param a{cp,cp}; let a[1,1] : ¼ 0.19681547722366; let a[1,2] : ¼ 0.39442431473909; let a[1,3] : ¼ 0.37640306270047; let a[2,1] : ¼ 0.06553542585020; let a[2,2] : ¼ 0.29207341166523; let a[2,3] : ¼ 0.51248582618842; let a[3,1] : ¼ 0.02377097434822; let a[3,2] : ¼ 0.04154875212600; let a[3,3] : ¼ 0.11111111111111; # define discretized profiles for the state variables and time var x1 {fe,cp} >¼ 0 :¼ x1_init ; var x2 {fe,cp} >¼ 0 :¼ x2_init ; var x3 {fe,cp} >¼ 0 :¼ x3_init ; var q {fe,cp} >¼ 0 :¼ q_init ; var t {fe,cp} >¼ 0 :¼ t_init ; # define discretized derivatives for state variables and time var x1dot {fe,cp} ; var x2dot {fe,cp} ;

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var x3dot {fe,cp} ; var qdot {fe,cp} ; var tdot {fe,cp} ; # define weighted moisture content for endpoint constraint var xm {i in fe,j in cp} ¼ x1[i,j]*tau1þx2[i,j]*tau2; # define discretized profiles for the control variables var Ta {fe} >¼ 40, ¼ 0.05, 0 λ2 < 0

Saddle

λ1, λ2 complex, Re(λi ) < 0

Stable focus

λ1, λ2 complex, Re(λi ) > 0

Unstable focus

λ1, λ2 imaginary

Center

Phase Portrait

eigenvalues of matrix A can be translated into a positive definition condition on symmetric matrix P defined by the following equation (Vidyasagar, 1993): AT P þ PA þ I ¼ 0

(14:5)

Equation 14.5 is known as the Lyapunov identity and matrix P as the Lyapunov matrix. Positive definition of matrix P implies that the following condition holds: qT Pq > 0

8 q 6¼ 0

and qT Pq ¼ 0

for

q¼0

(14:6)

The position in the states space and the nature of the equilibrium points of Equation 14.1 depend on parameters u, manipulations u, and design variables d. As u*, u*, and d* change, equilibrium points can appear or disappear or simply change their characterization (stable, unstable, or saddle). Behavior study of the equilibrium points as parameters change is known as bifurcation analysis. Bifurcations occur when the topological structure of the system changes due to a change in the model parameters (Strogatz, 1994).

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There exists an amount of possible bifurcations in nonlinear systems. The saddle node bifurcation, for example, is the basic mechanism by which equilibrium points are created or destroyed. Other types of bifurcation (transcritical, pitchfork, and Hopf bifurcations) involve a change in the stability nature of a particular equilibrium point (Strogatz, 1994). It should be also mentioned that for general nonlinear dynamic systems, stability properties of the equilibrium points are local in nature. This means that, for a certain (asymptotically) stable equilibrium point, there exists a limited portion of the state space where trajectories that converge to such point originate. Such a portion of the state space is a more or less complex region known as the ‘‘domain of attraction’’ of the equilibrium under study. The estimation of domains of attractions is a challenging problem of nonlinear dynamic analysis (Matallana et al., 2007).

14.3 EIGENVALUE OPTIMIZATION As pointed out by Kokossis and Floudas (1994) there exists the impossibility of obtaining explicit mathematical expressions for the eigenvalues of matrices larger than 4  4. This makes it impossible to include eigenvalues within an optimization model as objectives and constraints in a straightforward manner. Another critical difficulty when dealing with eigenvalues in optimization models is that it frequently happens that although the elements of the matrix are themselves differentiable in the optimization variables, the eigenvalues of such matrix may not be differentiable due to ‘‘coalescence’’ occurrence (Overton, 1992). Coalescence is related with eigenvalue multiplicity and gives rise to nonsmooth optimization models unsuitable for gradientbased NLP solvers. In order to overcome the difficulties when eigenvalues are present in optimization models, it is necessary to develop tailored formulations. In this section two eigenvalue optimization problems and their solution strategies are reviewed.

14.3.1 OPTIMIZATION WITH CONSTRAINTS OF THE EIGENVALUES

ON THE

REAL PART

Consider the optimization of a certain objective function subject to the constraint on the real part of the eigenvalues of the general matrix H (n  n) to be negative. min F(y) y

s:t: Reðli (H(x))Þ < 0, i ¼ 1, . . . , n h(y) ¼ 0

(14:7)

g(y)  0 y 2 Y ¼ {yjyl  y  yu } where y denotes the vector of optimization variables F(y) is the scalar objective function h(y) and g(y) are the vectors of equality and inequality constraints, respectively li is the eigenvalues of matrix H(y)

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Equation 14.7 can be straightforwardly formulated for a 2  2 matrix since the characteristic polynomial of such a matrix can be analytically solved for the real part of the eigenvalues. Alternatively, for a 2  2 matrix, the constraints on the real part of the eigenvalues can be replaced by conditions on its trace and determinant (Strogatz, 1994), which are nonlinear functions of the optimization variables. In such cases, Equation 14.7 becomes a simple optimization model with nonlinear constraints. However, for larger matrices, the real part of the eigenvalues cannot be analytically solved and alternative formulations are required. As described in Blanco and Bandoni (2007), Equation 14.7 can be reformulated so that the constraint on the real part of the eigenvalues of H(y) is replaced by the positive definition condition (denoted by symbol  0) on the inverse of the real symmetric matrix P defined by Lyapunov identity Equation 14.5: min F(y) y

s:t: HT (y)P þ PH(y) þ I ¼ 0 P1  0 h(y) ¼ 0

(14:8)

g(y)  0 y 2 Y ¼ {yjyl  y  yu } Sylvester criterion (Noble and Daniel, 1989) is adopted to impose positive definition on P1. Sylvester criterion states that necessary and sufficient conditions for a symmetric matrix to be positive definite are that the determinants of its successive principal minors be positive. Therefore, Equation 14.8 can be reformulated as follows: min F(y) y

s:t: HT (y)P þ PH(y) þ I ¼ 0   det P1 > 0, i ¼ 1, . . . , n i h(y) ¼ 0

(14:9)

g(y)  0

  y 2 Y ¼ yjyl  y  yu

In the context of dynamic systems design and operations, Equations 14.7 or 14.9 can be formulated to optimize a certain objective function, while ensuring steady-state asymptotic stability of the resulting equilibrium point. In such a case matrix H in Equation 14.7 represents the Jacobean of the dynamic system, A. In addition, vectors x, u, and d and vector f(x, u, u, d) of dynamic equations become subsets of optimization variables y and equality constraints h(y), respectively. In other words, in general, vector y comprises vectors x, u, and d and vector h(y) includes the steady-state version of Equation 14.1.

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To illustrate this, approach, consider the following dynamic system: dx1 ¼ x21 þ x22  u1 dt

(14:10a)

dx2 ¼ x21 þ x2  4u1 dt

(14:10b)

x1 and x2 are the state variables of the dynamic system u1 is a manipulated variable u1 is a parameter whose value is 0.9 Two motivating optimization problems are addressed (Table 14.2): Problem 14.1a: Optimization without stability constraints. Problem 14.1b: Optimization with stability constraints. Problem 14.1a was solved as a standard NLP while Problem 14.1b was solved according to formulation 14.9. Results for both problems are reported in Table 14.2, and the corresponding phase portraits are presented in Figure 14.1. It can be observed from Table 14.2 that the objective function of Problem 14.1a is better than the one corresponding to Problem 14.1b. However, an unstable equilibrium results from such formulation as illustrated in Figure 14.1a since eigenvalue l1 is positive (unstable saddle). On the other hand, a stable equilibrium (stable node) is obtained from formulation 14.1b as expected (Figure 14.1b) at the expense of a worse objective function value. This tradeoff between ‘‘economics’’ and ‘‘operability’’ has been discussed in Blanco and Bandoni (2003) and Blanco et al. (2004). TABLE 14.2 Optimization Results for Motivating Example Problem 14.1a min x2 x ,x ,u 2 1

2

s:t: x21 þ x22  u1 ¼ 0 x21 þ x2  4u1 ¼ 0 0  u1  1

u1 x1 x2 x22 l1(A) l2(A)

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Problem 14.1b min x2 x1 ,x2 ,u 2 s:t: x21 þ x22  u1 ¼ 0 x21 þ x2  4u1 ¼ 0 Reðli (A(x))Þ < 0

x1  0

0  u1  1

x2  0

x1 < 0

0.2250 0.9487 0.0000 0.0000 1.0000 1.8974

x2  0 0.2875 0.8042 0.5032 0.2532 0.5910 0.0174

1

0.5

x2

0

−0.5 −1 −1.5 −1.5

−1

−0.5

0

x1

(a)

0.6

x2

0.55

0.5

0.45

(b)

FIGURE 14.1

−0.9 −0.88 −0.86 −0.84 −0.82 −0.8 −0.78 −0.76 −0.74 −0.72 −0.7 x1

Phase portraits for motivating example: (a) Problem 14.1a and (b) Problem 14.1b.

14.3.2 MAXIMIZATION

OF THE

MINIMUM EIGENVALUE

OF A

SYMMETRIC MATRIX

Consider now the problem of maximizing the smallest eigenvalue of a real symmetric matrix G(y): max lmin (G(y)) y

s:t: h(y) ¼ 0 g(y)  0 y2Y

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(14:11)

As with Equation 14.7 the presence of eigenvalues in Equation 14.11 calls for further reformulation to be addressed with standard NLP solvers. A classic reformulation (Blanco and Bandoni, 2007) considers the introduction of an auxiliary variable z, which bounds from below the set of eigenvalues of matrix G: max z y,z

li (G(y))  z,

i ¼ 1, . . . , n

z>0 s:t: h(y) ¼ 0

(14:12)

g(y)  0 y2Y Equation 14.12 can be further expressed in terms of a positive definition constraint (Blanco and Bandoni, 2007). According to the definition of eigenvalue: Gv ¼ lIv. By subtracting zIv to both terms it results: (G  zI)v ¼ (l  z)Iv. Since GzI is a symmetric matrix, conditions li > z and z > 0 implies that G  zI  0, therefore: max z y,z

G(y)  zI  0 z>0 s:t: h(y) ¼ 0

(14:13)

g(y)  0 y2Y Finally, in order to pose a regular NLP problem which can be tackled with standard NLP solvers, the positive definition condition in Equation 14.13 can be expressed in terms of the determinants of the principal minors of matrix G(y)zI as already described in Section 3.1: max z y,z   det ½G(y)  zI i > 0, z>0 s:t: h(y) ¼ 0

i ¼ 1, . . . , n (14:14)

g(y)  0 y2Y In the context of dynamic analysis, Equations 14.12 or 14.14 have been used for design-for-operability studies in Blanco and Bandoni (2004) and for transient

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response optimization in Blanco and Bandoni (2007). This last application motivates the reminder of this section. The minimum eigenvalue of the inverse Lyapunov matrix can be considered a dynamic performance index since it is related with the inverse of the largest time constant of the system in the region of asymptotic stability (Koppel, 1968). Therefore, the maximization of the minimum eigenvalue of P1 constitutes a meaningful problem from a dynamic systems analysis perspective. In such case, matrix P1 takes the place of G in formulations 14.12 through 14.14. Matrix P is defined through the Lyapunov identity (Equation 14.5). As in formulations 14.7 or 14.9, vectors x, u, and d and vector f(x, u, u, d) of dynamic equations become subsets of optimization variables y and equality constraints h(y), respectively. max z y,z

AT P þ PA þ I ¼ 0 n o s:t: det P1  zI i > 0, i ¼ 1, . . . , n z>0 h(y) ¼ 0

(14:15)

g(y)  0 y2Y To illustrate the use of formulation 14.15, consider the controller parameters tuning problem. A simple proportional feedback control law is applied on manipulated variable u1 of system 14.10 to improve the open-loop dynamics around some particular equilibrium points (Equation 14.16). Consider the following case studies: Problem 14.2a: Stabilization of unstable equilibrium (Problem 14.1a). Problem 14.2b: Improvement of the transient response of equilibrium (Problem 14.1b). max z

k1 ,k2 ,z

AT P þ PA þ I ¼ 0 n o s:t: det P1  zI i > 0,

i ¼ 1, . . . , n

z>0 x21 þ x22  u1 ¼ 0 x21 þ x2  4u1 ¼ 0     u1 ¼ k1 x1  x1* þ k2 x2  x2* þ uSP 1 " # * * 2x1 2x2 A¼ 2x1* þ 4k1 1 þ 4k2 SP* x1 ¼ x1*, x2 ¼ x2*, uSP 1 ¼ u1 ,

 10  k2  0

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k1 ¼ 0

(14:16)

TABLE 14.3 Optimization Results for Motivating Example u1

x1

x2

lmin(P1)

l1(A)

l2(A)

0.2250 k1 ¼ 0 k2 ¼ 10

0.9487 0.9487

0.0000 0.0000

N=A 3.7862

1.0000 39

1.8974 1.8974

Problem 14.1a 14.2a

Consider as a first study case that it is desired to stabilize the unstable equilibrium from Problem 14.1a. In such case x1* ¼ 0.9487 and x2* ¼ 0.0000 (Table 14.2). Numerical results for Problems 14.1a and 14.2a are reported in Table 14.3. Simulations from different starting points are presented in Figure 14.2 for the closed-loop solution (Problem 14.2a). It can be observed that for k1 ¼ 0 and k2 ¼ 10 the openloop unstable saddle (Problem 14.1a) (Figure 14.1a) is effectively stabilized by transforming it into a stable node. As second study case, let us consider the problem of improving the transient response of the open-loop stable equilibrium (Problem 14.1b) by applying proper feedback action. In such case x1* ¼ 0.8053 and x2* ¼ 0.5015 (Table 14.2). Results are reported in Table 14.4 and simulations from particular nonzero initial conditions to illustrate the behavior of both systems presented in Figure 14.3. It can be observed that for k2 ¼ 1.9119, the feedback control effectively induces a faster approach to the equilibrium than in the open-loop case.

1

0.5

x2

0

−0.5 −1 −1.5 −1.5

FIGURE 14.2

−1

x1

Phase portrait for Problem 14.2a.

ß 2008 by Taylor & Francis Group, LLC.

−0.5

0

TABLE 14.4 Optimization Results for Motivating Example u1

x1

x2

lmin(P1)

l1(A)

l2(A)

0.2875 k1 ¼ 0 k2 ¼ 1.9119

0.8042 0.8042

0.5032 0.5032

0.0017 3.4212

0.5910 1.9549

0.0174 6.3032

Problem 14.1b 14.2b

−0.6

x1

−0.7 −0.8 −0.9 0

5

10

15

20 t

25

30

35

40

0.65

x2

0.6 0.55 0.5 0.45 0

10

20

30

40

50

60

t

(a) −0.6

x1

−0.7 −0.8 −0.9

0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

0.65

x2

0.6 0.55 0.5 0.45 0 (b)

0.5

1

1.5 t

2

2.5

3

FIGURE 14.3 Transient response of systems 14.1b and 14.2b: (a) Problem 14.1b and (b) Problem 14.2b.

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14.4 SOFTWARE IMPLEMENTATION Equations 14.7 and 14.12 share very similar structures when applied to dynamic analysis (formulations 14.9 and 14.15). A Matlab package was developed to address both problems in a unified framework. General formulation 14.17 was actually implemented. Parameter « is a small greater than zero ‘‘user provided’’ tolerance to ensure strict positivity of the determinants. Formulation 14.17 takes the form of 14.9 if Q(y, z) ¼ F(y) and z is set to zero. Formulation 14.17 takes the form of Equation 14.15 if Q(y, z) ¼ z and z is treated as an optimization variable. Table 14.5 summarizes the correspondence between the different elements in formulations 14.9 and 14.17 and formulations 14.15 and 14.17. It should be mentioned that the vector of states of the dynamic system (x), is a subset of the optimization variables vector (y). As well, the steady-state version of dynamic system (Equation 14.1) is a subset of the vector of equality constraints h(y). The calculation scheme is illustrated in Figure 14.4. min Q(y, z) y,z

AT (y)P þ PA(y) þ I ¼ 0 n o s:t: det P1  zI i > «, i ¼ 1, . . . , n «>0

(14:17)

z>0 h(y) ¼ 0 g(y)  0

  y 2 Y ¼ yjyl  y  yu

The user must provide routines for objective function Q(y, z) and constraints h(y), g(y) evaluation. The first n elements of vector h(y) are the steady-state version of the dynamic system equations f(y). The Jacobean matrix A(y) is symbolically computed from f(y). The following procedure was implemented for the evaluation of the Lyapunov constraints within the constraints evaluation routine: Step 1: Provide a starting point for y and z. Step 2: Evaluate A(y) either analytically or numerically. Step 3: Evaluate P by solving Lyapunov equation. TABLE 14.5 Correspondence between Formulations 14.17

14.9

14.15

Q(y, z) z

F(y) 0

z z

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Θk, Θk •

Give z0, y0

Main program

Routine Fmincom •

Evaluate Θ, Θ •

zk , yk

hk , h-k gk , gk •

Present results Evaluate h,h •

End

Evaluate g, g

Evaluate A

•

Ak Solve for P ATP+PA+I=0

Lyapack package

Pk Evaluate P-1

Pk

-1

Evaluate det(P-1 – z I) det(P-1 – z I) •

FIGURE 14.4 Solution scheme for formulation 14.17.

Step 4: Evaluate P1. Step 5: Evaluate P1 – zI. Step 6: Evaluate det{[P1 – zI]i},

i ¼ 1, . . . , n.

Matrix P is solved from Lyapunov equation using function ‘‘Lyap’’ (LYAPACK Users’ Guide, 1999) and inverse and determinants are computed with standard Matlab functions. Matlab function ‘‘fmincon’’ is used as nonlinear solver. An initial point for the optimization [y0, z0] has to be provided. The solver attempts to find a local optimal solution of Equation 14.17 from the provided starting point. As in any NLP problem, the selection of the starting point is of critical importance. For appropriate performance, a feasible starting point with respect to Lyapunov matrix (P  0) should be provided.

14.5 APPLICATION Fermentation is an important process in the food industry. For example, fermentation of sugary foodstuff is the principal means of manufacturing alcoholic beverages. Bio-ethanol is the most important product of such fermentation processes. Besides its other uses in the food industry, for example as raw material in the production of white vinegar, bio-ethanol has gained importance in the last years as an alternative clean fuel (Ward and Singh, 2002). In microbial fermentation, biomass acts as the catalyst for substrate conversion and is also produced by the process. The bio-catalyzing microorganism Zymomonas mobilis presents attractive features for the industrial production of bio-ethanol (McLellan et al., 1999). Like most anaerobic fermentation processes, ethanol

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Din CS0 CP0 CX0 Ce0

Fermentor CS, CP, CX, Ce

DMin CPm0

Membrane (Cpm)

Dout CS CP CX Ce DMout CPm

FIGURE 14.5 Fermentor schematic configuration.

production by Z. mobilis is subject to end-product inhibition meaning that the produced ethanol alters the enzyme inhibiting its catalytic effect. In order to prevent inhibition and increase productivity and efficiency, ethanol is usually removed from the system while produced. Several industrial solutions exist for this purpose. Recently, a fermentation–diffusion system, which makes use of an ethanol-specific membrane, has been proposed (Mahecha-Botero et al., 2006). The perfectly mixed continuous fermentor is equipped with an ethanol-specific removal membrane as shown in Figure 14.5.  

1 (CS Ce )   mS CX þ Din CS0  Dout CS YSX (KS þ CS )

 dCx (CS Ce ) ¼ þ Din CX0  Dout CX dt (KS þ CS )

  (CS Ce ) dCe  2 ¼ k1  k2 CP þ k3 CP þ Din Ce0  Dout Ce dt (KS þ CS ) dCS ¼ dt

dCP ¼ dt

(14:18a) (14:18b) (14:18c)

  (CS Ce ) AM P þ mP CX þ Din CP0  Dout CP  (CP  CPm ) (KS þ CS ) VF (14:18d)  dCPm AM P ¼ (CP  CPm ) þ DMin CPm0  DMout CPm (14:18e) dt VM



1 YPX



where Dout ¼ Din 

AM P(CP  CPm ) VF r

DMout ¼ DMin 

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AM P(CP  CPm ) VM r

(14:18f) (14:18g)

TABLE 14.6 Model Parameters Parameter AM KS k1 k2 k3 mS mP YSX YPX CP0 Ce0 CPm0 r VM VF DR HR P

Description

Value

Units

Area of membrane Monod constant Empirical constant Empirical constant Empirical constant Maintenance factor based on substrate Maintenance factor based on product Yield factor based on substrate Yield factor based on product Concentration of influent ethanol to the fermentor Concentration of internal key component to the fermentor Concentration of influent ethanol to the membrane Ethanol density Membrane volume Fermentor volume Fermentor diameter Fermentor height Membrane permeability

0.24 0.5 16 0.497 0.00383 2.16 1.1 0.0244498 0.0526315 0 0 0 789 0.0003 0.003 0.1241 0.2482 0.1283

[m2] [kg=m3] [h1] [m3=kg h] [m6=kg2 h] [kg=kg h] [kg=kg h] [kg=kg] [kg=kg] [kg=m3] [kg=m3] [kg=m3] [kg=m3] [m3] [m3] [m] [m] [m=h]

Equation 14.18a through g presents the five states model for the described system. See Mahecha-Botero et al. (2006) for a detailed description of the processes and a sound analysis of its nonlinear behavior. Definition of process parameters and variables are provided in Tables 14.6 and 14.7. In the following, open- and closed-loop studies of different versions of model 14.18 are presented to illustrate the application of general formulation 14.17. For the

TABLE 14.7 Model Variables Variable CS0 DMin Din CX0 CS CX Ce CP CPm DMout Dout

Description

Units

Concentration of substrate inside the fermentor Inlet membrane dilution rate, inlet flow rate=membrane volume Inlet fermentor dilution rate, inlet flow rate=monitor volume Concentration of biomass (microorganisms) inside the fermentor Concentration of substrate inside the fermentor Concentration of biomass inside the fermentor Concentration of key component inside the fermentor Concentration of ethanol inside the fermentor Concentration of ethanol inside the membrane Output membrane dilution rate, output flow rate=membrane volume Output fermentor dilution rate, output flow rate=fermentor volume

[kg=m3] [h1] [h1] [kg=m3] [kg=m3] [kg=m3] [kg=m3] [kg=m3] [kg=m3] [h1] [h1]

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analyzed case studies, the laboratory scale reactor described in Mahecha-Botero et al. (2006) was considered. Therefore, industrial-related issues such as purification of product streams, recycle of unreacted material, and capital and operating costs were not included in the analysis, which mostly focus on the dynamics of the system.

14.5.1 OPEN-LOOP OPERATIVE OPTIMIZATION From an operative point of view, a meaningful objective is to maximize the overall ethanol production [kg=h]. Objective function is therefore: F(y) ¼ DMout Cpm VM þ Dout Cp VF

(14:19)

As optimization variables, it was considered that the following were available for manipulation: Din, DMin, CS0, and CX0 together with the state of the system. In order to illustrate the application of the methodology three cases were addressed: Problem 14.3: Production optimization without stability constraints. Problem 14.4: Production optimization with stability constraints. Problem 14.5: Transient response optimization. Results for these three problems are reported in Table 14.8. It can be observed that the production of ethanol resulting from Problem 14.3 is somewhat larger than

TABLE 14.8 Operative Optimization of the Fermentation Process

CS0 DMin Din CX0 CS CX Ce CP CPm F l1(A) l2(A) l3(A) l4(A) l5(A) lmin(P1) z

Problem 14.3

Problem 14.4

Problem 14.5

max F s.t. Equation 14.18a–g

max F s.t. Equation 14.18a–g, Re(li(A)) < 0

max lmin(P1) s.t. Equation 14.18a–g

61.9381 0.1000 0.2014 5.0000 8.2318 5.0080 0.0000 26.1432 26.1185 0.0165 0.2010 5.1014 116.0680 0.1963 þ 0.0065i 0.1963 – 0.0065i N=A N=A

49.1468 0.0948 0.2188 4.9708 0.0006 4.9772 0.0000 24.0488 24.0273 0.0164 115.8203 0.2185 0.2111 0.2127 þ 0.0067i 0.2127 – 0.0067i 0.0059 N=A

4.0000 0.1000 0.9061 0.5000 0.0320 0.5674 0.9671 1.8828 1.8810 0.0052 113.2982 69.8301 0.9331 0.9596 0.9061 0.8095 0.8095

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that corresponding to Problems 14.4 and 14.5. High productivity is achieved in Problem 14.3 by feeding less substrate (low dilution rate) at a high concentration and operating the process at a high substrate concentration as well. However, such an operating point is unstable as can be concluded from the spectrum of matrix A. Therefore, practical implementation of such a solution could not be achieved without proper control action. On the other hand, the lower production solutions (Problems 14.4 and 14.5) present open-loop asymptotic stability since the real parts of the eigenvalues of the corresponding Jacobean matrices are effectively negative. Stability is obtained in such cases by operating the process with a larger feed-stream dilution rates and lower feed-stream substrate concentrations. Attention is called on the fact that the solutions of Problems 14.3 and 14.4 are quite similar since they share the same productivity objective function. As expected, however, the solution of Problem 14.5 is pretty different from the others since the objective is purely ‘‘dynamic’’ in this case.

14.5.2 OPEN-LOOP DESIGN OPTIMIZATION In order to address the open-loop design problem, reactor dimensions VF, VM, and AF were set free as optimization variables and additional geometrical relationships included in the model assuming a cylindrical reactor (Equation 14.20a through d). The membrane area and volume was considered to follow a linear relationship with the reactor volume.  2 pDR (HR ) VF ¼ 4

(14:20a)

HR ¼ 2DR

(14:20b)

VM ¼ 0:1Vf

(14:20c)

AM ¼

Vf 0:0125[m]

(14:20d)

Again the overall ethanol production (Equation 14.19) was maximized and the reactor diameter was allowed to vary between 0.05 and 0.2 m. The following case studies were addressed to illustrate the open-loop design problem: Problem 14.6: Production optimization without stability constraints. Problem 14.7: Production optimization with stability constraints. Problem 14.8: Transient response optimization. Similar trends regarding the previous studies (Table 14.8) can be observed in these cases (Table 14.9). Problem 14.6 rendered large productivity at the expense of instability. Stability in Problems 14.7 and 14.8 was achieved at larger feed dilution rates and lower feed substrate concentrations with a worsening in the objective function. As expected, to achieve large productivities in Problems 14.6 and 14.7, the largest possible reactor was selected while to optimize dynamic response a smaller reactor was preferred (Problem 14.8).

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TABLE 14.9 Optimal Design of the Fermentation Process

CS0 DMin Din CX0 CS CX Ce CP CPm DR VF VM AM F l1(A) l2(A) l3(A) l4(A) l5(A) lmin(P1) z

Problem 14.6

Problem 14.7

Problem 14.8

max F s.t. Equation 14.18a–g Equation 14.20a–d

max F s.t. Equation 14.18a–g Equation 14.20a–d, Re(li(A)) < 0

max lmin(P1) s.t. Equation 14.18a–g Equation 14.20a–d

61.9446 0.1000 0.2014 5.0000 8.2348 5.0080 0.0000 26.1448 26.1202 0.2000 0.0126 0.0013 1.0053 0.0692 0.2010 5.1011 116.0682 0.1963 þ 0.0065i 0.1963  0.0065i N=A N=A

49.0882 0.0781 0.2185 4.9608 0.0000 4.9661 0.0000 24.1864 24.1686 0.2000 0.0126 0.0013 1.0053 0.0686 115.8221 0.2183 0.2183 0.2118 þ 0.0068i 0.2118  0.0068i 7.4506e  009 N=A

4.9840 0.1000 0.3057 0.7003 0.0105 0.7009 0.0077 2.4522 2.4498 0.1782 0.0089 8.8869e  004 0.7109 0.0069 113.3092 0.4547 þ 0.0988i 0.4547  0.0988i 0.2870 0.3057 0.0803 0.0560

14.5.3 CLOSED-LOOP OPERATIVE OPTIMIZATION Let us now consider the fermentor without membrane (AM ¼ 0 in Equation 14.18). Reactor size was fixed according to the data in Table 14.6. As optimization variables, it was considered that Din, DMin, and CS0 were available for manipulation while CX0 was set to zero in this case. The concentration of ethanol was considered as the objective function to be maximized: F(y) ¼ CP

(14:21)

First, the open-loop operative optimization of the process was performed, with and without stability constraints: Problem 14.9: Open-loop operative optimization without stability constraints. Problem 14.10: Open-loop operative optimization with stability constraints. The corresponding optimization results are reported in Table 14.10. As expected, open-loop stability is achieved at the expense of worsening the objective function.

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TABLE 14.10 Operative Optimization of the Fermentation Process Problem 14.9

CS0 Din CX0 CS CX Ce CP F l1(A) l2(A) l3(A) l4(A) l5(A) lmin(P1) Kc tI a b

Problem 14.10

Problem 14.11

Problem 14.12

max F s.t. Equation 14.18a–d, f, g

max F s.t. Equation 14.19a–d, f, g Re(li(A)) < 0

max lmin(P1) s.t. Equation 14.19a–d, f, g Equation 14.22a and b, x* ¼ xa

max lmin(P1) s.t. Equation 14.19a–d, f, g Equation 14.22a and b, x* ¼ xb

140.9078 0.0090 0.0000 23.5513 0.4178 0.0038 59.0000 59.0000 0.0141 þ 0.0660i 0.0141  0.0660i 0.0090 0.0464 N=A N=A N=A N=A

116.9981 0.0226 0.0000 0.2699 0.8551 0.0551 57.8808 57.8808 1.8422 0.0774 0.0226 0.0226 N=A 0.0007 N=A N=A

140.9078 0.0090 0.0000 23.5513 0.4178 0.0038 59.0000 59.0000 14.1645 1.1048 0.0093 0.0087 0.0090 2.9254e  7 0.2588 0.9760

116.9981 0.0226 0.0000 0.2699 0.8551 0.0551 57.8808 57.8808 15.3923 þ 6.0018i 15.392  6.0018i 0.1428 0.0226 0.0226 0.0011 0.5011 0.1062

Problem 14.9. Problem 14.10.

In order to improve the dynamic behavior of the system, a proportional-integral feedback law was included to control the ethanol concentration CP by manipulating the incoming dilution rate Din. Equation 14.22a and b represents the set of algebraic– differential equations modeling the proposed controller. dj ¼ CPSP  CP dt  SP  Kc Din ¼ DSP j in þ Kc CP  CP þ tI

(14:22a) (14:22b)

Superscript SP denotes set-point value, Kc, and tI are the proportional and integral gains, respectively, and j is the ‘‘integral action state.’’ Two additional problems were addressed: Problem 14.11: Stabilization of the unstable equilibrium of Problem 14.9. Problem 14.12: Improvement of the transient response to equilibrium (Problem 14.10).

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1.5

CX (kg/m3)

CS (kg/m3)

40

20

0 −20

0

500 t (h)

0.5

0

1000

0.2

0

500 t (h)

0

500 t (h)

1000

75 70

0.15

CP (kg/m3)

Ce (kg/m3)

1

0.1 0.05

65 60 55

0

50 0

500 t (h)

1000

1000

FIGURE 14.6 Transient response of Problems 14.9 (solid line) and 14.11 (dotted line).

In Problem 14.11, it is desired to stabilize the unstable operating point from Problem 14.9 by proper feedback action. As well, in Problem 14.12, a faster transient response to the solution of Problem 14.10, which is a stable equilibrium, is pursued. In both cases, the controller gains Kc and tI are the optimization variables. Results for Problems 14.11 and 14.12 are shown in Table 14.10 and illustrated by simulations in Figures 14.6 and 14.7. In Figure 14.6, it can be observed how the unstable equilibrium (limit cycle) is effectively stabilized by proper feedback action. From Figure 14.7, it can be observed how the approach to the equilibrium is faster under feedback action than in the open-loop case.

14.6 CONCLUSION AND FUTURE WORK In this chapter, the fundamentals of linear theory for nonlinear systems analysis have been briefly reviewed with particular emphasis on eigenvalue analysis. Although local in scope, eigenvalue analysis permits the characterization of the equilibriums of dynamic systems regarding stability and transient response quality. Moreover, some eigenvalue optimization formulations can serve to automate certain analysis and design problems. Such formulations have been also reviewed and their scope illustrated through examples. In particular, the design-for-stability problem has been addressed for both, the open-loop and the controlled cases. The main considered case study was a fermentation reactor to obtain ethanol from glucose. Such is an important process for several industries, in particular for the food industry, and also present challenging dynamics for open- and closed-loop studies.

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1.5

20

C X (kg/m3)

CS (kg/m3)

30

10

0

0

100

t (h)

200

0.5

0

300

0

100

0

100

t (h)

200

300

200

300

65

0.2 0.15

CP (kg/m3)

Ce (kg/m3)

1

0.1

60

55

0.05 0

0

100

200 t (h)

300

50

t (h)

FIGURE 14.7 Transient response of Problems 14.10 (solid line) and 14.12 (dotted line).

As a general conclusion for the open-loop case, it was observed that optimal from an economic point of view solutions can be dynamically unstable. The inclusion of open-loop stability constraints renders, of course, stable equilibriums but at the expense of worsening the economic objective function. This issue has to do with the well-known trade-off between economics and dynamic operability. It was also shown that under feedback control, the dynamic performance of ‘‘poor’’ open-loop systems can be greatly improved. For example, it was illustrated how unstable equilibriums can be stabilized by proper feedback action. Such a practice has very important operative implications. As previously commented, from an economic point of view it is usually desirable to operate at unstable operating points. This can be achieved with the help of some control scheme. This issue was thoroughly discussed for example by Flores-Tlacuahuac et al. (2005) and Flores-Tlacuahuac and Biegler (2007). Moreover, stable but ‘‘slow’’ transient responses can be also improved by proper feedback action. The objective in this case is to minimize the largest time constant of the system. Simple proportional and proportional–integral controllers were used in this contribution for the described controller tuning problems. Further research on this line includes the analysis of parameter uncertainty and disturbances on the dynamic behavior of the system. This issue has to do with the quantification of the uncertainty=perturbations space within which the equilibriums remain stable. Such a problem is connected with the evaluation of the size and shape of the domains of attraction of the equilibriums under study.

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REFERENCES Blanco, A.M. and Bandoni, J.A., Interaction between process design and process operability of chemical processes: An eigenvalue optimization approach, Comp. and Chem. Eng., 27, 1291, 2003. Blanco, A.M. and Bandoni, J.A., Eigenvalue optimization-based formulations for nonlinear dynamics and control problems, Chem. Eng. Proc.: Process Intensification, 46, 1192, 2007. Blanco, A.M., Bandoni, J.A., and Biegler, L.T., Re-design of the Tennessee eastman challenge process: An eigenvalue optimization approach, Proc. FOCAPD 2004. Flores-Tlacuahuac, A. and Biegler, L.T., Simultaneous mixed-integer dynamic optimization for integrated design and control, Comp. Chem. Eng., 31, 588, 2007. Flores-Tlacuahuac, A., Biegler, L.T., and Saldivar-Guerra, E., Dynamic optimization of HIPS open-loop unstable polymerization reactors, Ind. Eng. Chem. Res., 44, 2659, 2005. Kahlil, H.K., Nonlinear Systems, Prentice Hall, Upper Saddle River, New Jersey, 1996. Kokossis, A.C. and Floudas, C.A., Stability in optimal design: Synthesis of complex reactor networks, AIChE J., 40, 849, 1994. Koppel, L.B., Introduction to Control Theory with Applications to Process Control, Prentice Hall, Englewood Cliffs, New Jersey, 1968. LYAPACK Users’ Guide: A Matlab Toolbox for Large Scale Lyapunov and Riccati Equations, Model Reduction Problems and Linear-Quadratic Optimal Control Problems, 1999. McLellan, P.J., Daugulis, A.J., and Li, J., The incidence of oscillatory behavior in the continuous fermentation of Zymomonas mobilis, Biotech. Progress, 15, 667, 1999. Mahecha-Botero, A., Garhyan, P., and Elnashaie, S.S.E.H., Non-linear characteristics of a membrane fermentor for ethanol production and their implications, Nonlinear Anal.: Real World Appl., 7, 432, 2006. Matallana L.G., Blanco, A.M., and Bandoni, J.A., Estimation of domains of attraction in epidemiological models with constant removal rates of infected individuals, J. Phys.: Conf. Ser., 90, 2007, doi: 10.1088=1742–6596=90=1=012052. Matlab. MathWorks, Inc., Natick, M.A., The Language of Technical Computing, 2004. Noble, B. and Daniel, J.W., Applied Linear Algebra. Prentice Hall, Englewood Cliffs, New Jersey, 1989. Overton, M., Large Scale Optimization of Eigenvalues, SIAM J. Optim., 2, 88, 1992. Ringertz, U.T., Eigenvalues in optimum structural design, in Proceedings of an IMA Workshop on Large-Scale Optimization. Conn, A.R. et al. (Eds.), Part I, p. 135, 1997. Strogatz, S.H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Westview Press, Boulder, CO, 1994. Vidyasagar, M., Nonlinear Systems Analysis, Prentice Hall, Englewood Cliffs, New Jersey, 1993. Ward, O.P. and Singh, A., Bioethanol technology: Developments and perspectives, Adv. App. Microbiol., 51, 53, 2002.

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15

Complex Method Optimization Ferruh Erdo
gdu and Murat O. Balaban

CONTENTS 15.1 15.2 15.3

Introduction .............................................................................................. 295 Method Development............................................................................... 297 Optimization............................................................................................. 298 15.3.1 Objective Function ...................................................................... 298 15.3.2 Constraints................................................................................... 298 15.3.3 Decision Variable ........................................................................ 299 15.3.4 Algorithm .................................................................................... 299 15.4 Results and Discussion ............................................................................ 300 15.5 Conclusion ............................................................................................... 303 Nomenclature ........................................................................................................ 303 References ............................................................................................................. 304

15.1 INTRODUCTION Complex method is a reliable and simple approach to optimization problems with explicit and implicit constraints (Mishkin et al., 1984). It is based on modifications of the simplex method which uses generation and maintenance of a pattern of search points and application of projections of undesirable points through the centroid (center of the combined feasible points) of remaining points for means of finding new trial points (Ravindran et al., 2006). This method belongs to the family of semistochastic algorithms, and it is a sequential search technique providing a promising method of direct search in finite space problems (Saguy, 1982). Complex method, named from COstrained siMPLEX (Saguy, 1982), first presented by Box (1965), is an efficient and mathematically simple constrained optimization method compared to other optimization methodologies, e.g., Pontryagin’s maximum principle. Box (1965) proposed the random and sequential generation of trial points set using the pseudo-random variables uniformly distributed on the interval of (0, 1) with given upper and lower bounds of the explicit constraints. Each trial point should be tested to satisfy given explicit and implicit constraints which can be handled with a great ease by the complex method (Saguy, 1982), and the total number of points should not be less than the number of unknown points (Ravindran et al., 2006). The basic idea then is to compare the trial points with respect to their resulting objective function values.

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With given trial points, the objective function is evaluated, and the point where a trial point, corresponding to the worst value depending on the objective function value, is rejected by reflecting the rejected point to a certain distance through the centroid. Determination and rejection of the worst trial point, however, might sometimes become a major issue. Since the responses of the simulation models are stochastic, an apparently worst trial point might be a point from where a better improvement can be obtained, and rejecting it might take the search away from the optimum region (Azadivar, 1999). As indicated by Azadivar (1999), the statistical comparison of responses of the trial points might be a way to avoid making a wrong decision regarding to drop the worst trial points. The reflection point would sometimes give a better value compared to the trial points leading to extension toward that direction with possible improvements (Kazmierczak, 1996). If the reflection causes a reverse result, obtaining a trial point with unimproved objective function, then the reflected point is retracted by half the distance towards the previously calculated centroid. This search procedure continues until the trial points shrink so that the centroid of trial points are collapsed into a very narrow range resulting in a not possible way to create a new point using reflection, expansion, or retraction. Shrinking also brings the result of the difference in objective function values’ becoming small enough. The procedure tends to find the global optimum since the initial complex (set of the initial trial points) is randomly scattered through the feasible space (Saguy, 1982). A major assumption in the application of the complex method is that the centroid itself always becomes a feasible point. This might not be the case within a nonconvex region (Figure 15.1), and the method could fail to converge (Ravindran et al., 2006). On the other hand, it is obvious that the complex method does not require any computation of derivatives, and therefore it is regarded to be efficient and easily programmable in a certain algorithm structure. Umeda et al. (1972) presented the use of complex method for solving variational problems with state-variable inequality constraints using a tubular reactor design example and demonstrated its applicability. Adelman and Stevens (1972) determined the method’s speed of solution for determining the optimum solutions via the simultaneous solution of implicit constraints and objective functions for chemical

Trial point I x1

Nonfeasible centroid x

Trial point II x2

Trial point III x3 Feasible region

FIGURE 15.1 Representation of obtaining a nonfeasible centroid in a nonconvex region. (Adapted from Ravindran, A., Ragsdell, K.M., and Reklaitis, G.V., Engineering Optimization, John Wiley & Sons, Hoboken, NJ, 2006.)

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engineering problems. Saguy (1982) applied complex method to maximize the profit of a fermentation process. Kazmierczak (1996) gave an example of a pest management problem illustrating the method as a very efficient optimization algorithm and mathematically very simple. Erdo
gdu and Balaban (2002, 2003a) modified the complex method to apply in a thermal processing problem. In another study, Erdo
gdu and Balaban (2003b) applied this method for a simultaneous multiobjective optimization for thermal processing of different geometrical-shaped food products. Turhan et al. (2007) applied the complex method for optimization of glycerol effect on the mechanical properties and water vapor permeability of whey protein-methylcellulose films. These studies from different disciplines showed that the complex method easily allows the incorporation of different constraints on food quality and food safety as explicit or implicit constraints. Even though it could be easily used in food processing optimization problems with these features, its applicability is not common (Erdo
gdu and Balaban, 2002). The objective of this study was therefore to describe an algorithm for complex method to optimize nutrient retention during thermal processing food products as a function of variable process temperature profiles.

15.2 METHOD DEVELOPMENT The complex method algorithm will be described using a thermal processing problem for a finite cylinder 15 mm in radius and length, k ¼ 0.566 W=m-K, cp ¼ 3660 J=kg-K, r ¼ 1050 kg=m3, hheating medium ¼ 2000 W=m2-K, hcooling medium ¼ 500 W=m2-K, for 18.3 min (1098 s) of processing time. This processing time was obtained for the optimum time for maximizing the given objective function while satisfying the implicit constraints for a processing temperature at 121.18C. The resulting objective function (Section 15.3.1) for this optimum constant PT profile was 85.80%. To calculate transient temperature distribution in finite cylinder geometry, an explicit finite difference model with noncapacitance surface nodes was applied to solve the partial differential equations for the initial and boundary conditions given below: Governing equation:
 1 @ @T(r, l, t) @ 2 T(r, l, t) r  cp @T(r, l, t)  ¼  r þ k r @r @r @z2 @r

(15:1)

Initial condition: T(r, l, 0) ¼ Ti

(15:2)

@T(R, l, t) ¼ h(t)  [T(R, l, t)  T1 (t)] @r @T(r, L, t) k  ¼ h(t)  [T(r, L, t)  T1 (t)] @z @T(0, l, t) ¼0 @r

(15:3)

Boundary conditions: k 

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These equations were solved in a 21  21 number of volume elements where the time step applied was 0.335 s with the distances between the volume elements were 0.769 mm in both radial and longitudinal directions. The numerical model was then used to determine the objective function, and explicit and implicit constraints.

15.3 OPTIMIZATION Choosing the objective function, decision variable, and constraints are important steps in the application of any method used since the global optimum might be difficult to achieve due to the insensitivity of the objective function to the decision variable. Convergence difficulties may also appear due to the high-linear and discontinuous nature of the thermal processing systems (Banga et al., 2001). Variable process temperatures have been suggested as alternatives in the food sterilization processes. It was reported that the benefits may include the improved nutrient nutrition, reduced heat damage to the food surface, lower energy costs, and shorter process times (Durance et al., 1997). Because a large number of variable process temperature profiles are possible for a given product, selection of an optimum process, when the process temperature profiles were chosen to be the decision variable, can be easily found with an optimization search technique (Durance et al., 1997).

15.3.1 OBJECTIVE FUNCTION Volume average retention of nutrient-thiamine (VACN) was used as the objective function (Equation 15.4) and kinetic parameters of thiamine (Banga et al., 1991) (D121.18C ¼ 178.6 min (¼ 10716 s); z ¼ 25.568C) were used. ðV

VACN ¼

2

1  410 V

 D1  ref

Ðt

T(t,v)Tref z

10

0

 3  dt

5  dV

(15:4)

0

This equation was derived using the general first-order kinetics equation.

15.3.2 CONSTRAINTS An explicit constraint was the process temperature (PT) range of 58C–1508C. Another was that the PT had to reach 58C at the end of the process to assure final cooling. As reported by Erdo
gdu and Balaban (2002), in commercial practice, water at ambient temperature is used for cooling, and the processing temperature rarely exceeds 1258C. However, in this study, higher temperature of 1508C to represent the upper limit of the explicit constraint was used to prove the method’s efficiency to find its optimum way towards the generally preferred range in the industry. The lower limit, on the other hand, was specifically chosen as a lower value to quickly quench the quality factor degradation. Lethality at the coldest point of the container, based on the target microorganism, and that a threshold below which the center temperature must reach at the end of the process were used as implicit constraints:

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ðt F0 ¼ 10

Tc (t)Tref z

dt

(15:5)

Tc (tmax )  Tthreshold

(15:6)

0

Bacillus stearothermophilus was taken as the target organism. Its inactivation kinetics were taken as D121.18C ¼ 4 min (¼240 s); z ¼ 108C. Also, F0  8 min was taken. The threshold center temperature to reach at the end of the process was taken as the initial uniform temperature of the product (208C).

15.3.3 DECISION VARIABLE The decision variable for the optimization was PT profile discretized at equidistant time steps (N) throughout the process. These points were connected by lines, and interpolation was applied for intermediate temperatures.

15.3.4 ALGORITHM Step 1 Establishing the initial simplex—For this purpose the variable PT profile was discretized by dividing total process time to N (¼ 10) equidistant points. The PTs at these times were randomized between the high- and low limits of process temperature range: PTi ¼ PTL þ ri  (PTh  PTL ) i ¼ 1, . . . , N  1 PTN ¼ PTL

(15:7)

where PTL ¼ 5 PTh ¼ 150 ri was the pseudo-random number in the interval of [0, 1] In this step, the PT profile was checked for its violation of the first and second implicit constraints. The violation of the first implicit constraint was corrected by moving all the temperatures but the last one at the discretized time step one half way towards the higher limit of the explicit constraint: 1 PTi ¼ (PTi þ PTh ) 2

(15:8)

This was repeated until F0 was higher than (or equal to) 8 min. The correction procedure for the first implicit constraint would possibly cause a violation in the second one since the PT profile was moved towards the higher temperatures. For correction of this problem the process temperatures, starting from point N1, were moved half way towards the lower limit one by one. After each move, the violation of the first implicit constraint was verified. When decreasing

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temperatures this way caused the violation of F0, temperatures were moved up starting from the point where the correction for the second implicit constraint ended until the F0 value was again satisfied. Of course, that would again affect the center temperature, and if there was any violation, the correction step was achieved again as described above. If the adjustment of second implicit constraint reaches all the way back to the first point (N ¼ 1 and t ¼ 0), and still does not satisfy the implicit constraints, the algorithm starts over by going back to determining the initial point and finding a new temperature profile. This step was repeated N þ 1 times finding an initial process temperature profile matrix of (N þ 1, N) with accompanying evaluation of objective function and satisfaction of implicit constraints constituting the initial complex where #11 represents the best and #1 represents the worst PT profile (Table 15.1). Figure 15.2 shows the change in the center temperature of the finite cylinder geometry subjected to these PT profiles achieving the first and second implicit constraints. Step 2 At this point, the conventional complex method started. The vertices (PTs-each process temperature profile being a vertex) were moved based on their objective function values applying the reflection–expansion and retractions steps of the complex method on the worst vertex (indicating the lower objective function value) to improve the objective function (Erdo
gdu and Balaban, 2002). The reflection coefficient of 1.3, expansion coefficient of 2, and retraction coefficient in the interval of [0, 1] were applied in these steps (Erdo
gdu and Balaban, 2002). Step 3 Stopping criterion for the algorithm was VACN1  VACNNþ1  0:01

(15:9)

where subscripts 1 and N þ 1 show the results of the best and worst vertices, respectively.

15.4 RESULTS AND DISCUSSION The best profile was the optimum process temperature profile resulting in the maximum retention of nutrients (VACN) while satisfying the given explicit and implicit constraints (Table 15.1). Figure 15.3 also shows the change in the center temperature, achieving the first and second implicit constraints, with the optimum process temperature profile. It is also important for an optimization algorithm to reproducibly find the optimum for a given problem. The program was run for the given finite cylinder example 10 times. The optimum PT profiles had similar trends but not same of each other. However, the average value for the resulting objective functions was 89.28% with a standard deviation of 0.55%. The small standard deviation is accepted to be the proof of method’s reproducibility. On the basis of this, it might be concluded that the complex method produces unique and reproducible results with respect to the

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TABLE 15.1 Initial Complex and Resulting Optimum Process Temperature Profile with Resulting Objective Function (VACN %) and First Implicit Constraint (F0 min) Values Process Temperature (PT) Profiles (Vertices) Time (s) 0 122 244 366 488 610 732 854 976 1098 VACN (%) F0 (min)

1

2

3

4

5

6

7

8

9

10

11

Optimum PT Profile

124.03 145.90 127.76 143.99 114.7 139.85 63.09 5.83 5.73 5.00 73.96 53.63

115.09 128.53 149.10 119.97 116.42 125.72 65.91 5.54 5.86 5.00 80.50 21.34

132.15 133.25 134.95 130.70 129.60 135.92 59.86 5.96 5.87 5.00 76.54 46.56

146.77 115.57 132.09 130.23 144.52 127.59 65.25 5.54 5.51 5.00 74.78 83.28

109.41 93.23 104.11 143.61 148.60 121.17 57.20 5.71 5.85 5.00 74.79 90.94

146.98 103.82 110.24 93.12 132.29 149.41 72.14 5.94 5.58 5.00 84.96 15.43

123.70 133.26 120.29 127.95 114.85 136.13 67.50 5.53 5.51 5.00 86.30 8.40

114.98 142.48 111.89 115.93 120.27 149.11 71.35 5.99 5.54 5.00 83.17 14.02

6.59 86.70 99.32 91.02 138.49 149.12 40.19 5.83 5.81 5.00 85.86 13.06

100.41 79.32 132.67 142.84 122.75 110.92 63.78 5.60 5.65 5.00 84.79 17.30

102.43 100.23 115.83 137.25 135.64 92.98 46.01 5.77 5.81 5.00 87.07 14.57

99.13 96.58 115.70 126.33 136.07 108.25 51.56 5.75 5.75 5.00 89.65 8.00

150

Temperature (°C)

120 PT-1-worst Tc-1 PT-11-best Tc-11

90 60 30 0 0

122

244

366

488

610

732 854

976 1098

Time (s)

FIGURE 15.2 Center temperature change of the finite cylinder geometry subjected to the best (F0 ¼ 14.57 min) and worst (F0 ¼ 53.63 min) process temperature profile of the initial complex.

objective function while it might not be with respect to the decision variable (PT profiles for the given example). As explained by Erdo
gdu and Balaban (2002), this might be the result of truncation and rounding errors extensively applied in the optimization algorithm for achieving the required numerical finite difference calculations. The improvements were about 4% regarding the objective function compared to a constant PT process (89.28% versus 85.80%). In addition, the variable PT profiles can also be used to reduce the processing time while obtaining a similar value in the nutrient retention. Erdo
gdu and Balaban (2003a) reported a 40% reduction in the processing time for the given finite cylindrical geometry (15 mm in radius and total length) where a similar objective function value can be obtained with 11.5 min process time when the variable PT profiles were used. This reduction was around 6% (90 min compared to 85 min) for somewhat larger size geometry (30 mm in radius and 60 mm in total length).

Temperature (°C)

150 120 90

PT Tc

60 30 0 0

122

244

366

488 610 Time (s)

732

854

976 1098

FIGURE 15.3 Optimum process temperature profile with resulting center temperature change (VACN ¼ 89.65%, F0 ¼ 8.00 min).

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15.5 CONCLUSION Because of the easier mathematical manipulation, the complex method gives an advantage of choosing the explicit and implicit constraints in any restrictive or nonrestrictive way. The implicit constraints can even be chosen in more restrictive ways, e.g., the maximum color degradation at the surface shall not be higher than 25%, or the minimum retention of thiamin shall not be lower than 70% at the surface. In addition, it should find the global maximum rather than the local one via the use of randomly generated trial points composing the initial complex. As indicated by Adelman and Stevens (1972), at least one of the trial points would lie in the vicinity of the global maximum. The reflection and extension features would also lead to enlargement of the initial complex bringing additional advantages searching and scanning in the feasible space. With the algorithm developed, optimization for any shape can be accomplished with minor modifications. In addition, use of variable process temperature profiles can increase the nutrient retention and reduce total processing time compared to constant temperature processes.

NOMENCLATURE cp D121.1 Dl Dr Dt F0 h l L k r r R t PT T Tc Tref V VACN z

Heat capacity Decimal reduction time at the reference temperature Distance between the volume elements in the longitudinal direction Distance between the volume elements in the radial direction Time step used in the numerical calculations Integrated lethality at the slowest heating point Convective heat transfer coefficient Distance from the center in the longitudinal direction Total length of finite cylinder Thermal conductivity Density Distance from the center in the radial direction Radius Time Process temperature profile Temperature Center temperature Reference temperature Volume Volume average quality retention Temperature change needed to reduce the D-value one log cycle

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J=kg-K min m m s min W=m2-K m mm W=m-K kg=m3 m mm s 8C 8C 8C m3 % 8C

REFERENCES Adelman, A. and Stevens, W.F., Process optimization by the ‘‘Complex’’ method, AIChe J., 18, 20, 1972. Azadivar, F., Simulation optimization methodologies, Proceedings of the 1999 Winter Simulation Conference, Farrington, P.A., Nembhards, H.B., Sturrock, D.T., and Evans, G.W. (Eds.), Vol. I, 1999, pp. 93–100. Banga, J.R. et al., Optimization of the thermal processing of conduction-heated canned foods: study of several objective functions, J. Food Eng., 14, 25, 1991. Banga, J.R., Pan, Z., and Singh, R.P., On the optimal control of contact cooking processes, Transactions Ind. Chem. Eng., 79, 145, 2001. Box, M.J., A new method of constrained optimization and comparison with other methods, Comp. J., 8, 42, 1965. Durance, T.D., Dou, J., and Mazza, J., Selection of variable retort temperature processes for canned salmon, J. Food Proc. Eng., 20, 65, 1997. Erdo
gdu, F. and Balaban, M.O., Nonlinear constrained optimization of thermal processing: I. development of a modified algorithm of complex method, J. Food Proc. Eng., 25, 1, 2002. Erdo
gdu, F. and Balaban, M.O., Nonlinear constrained optimization of thermal processing: II. Variable process temperature profiles to reduce process time and to improve nutrient retention in spherical and finite cylindrical geometries, J. Food Proc. Eng., 26, 303, 2003a. Erdo
gdu, F. and Balaban, M.O., Complex method for nonlinear constrained multi-criteria (multi-objective function) optimization of thermal processing, J. Food Proc. Eng., 26, 357, 2003b. Kazmierczak, Jr., R.F., Optimizing complex bioeconomic simulations using an efficient search heuristic. LSU, Res. Report: 704, Baton Rouge, LA, 1996. Mishkin, M., Saguy, I., and Karel, M., Optimization of nutrient retention during processing: Ascorbic acid in potato dehydration, J. Food Sci., 49, 1262, 1984. Ravindran, A., Ragsdell, K.M., and Reklaitis, G.V., Engineering Optimization, John Wiley & Sons, Hoboken, NJ, 2006, Chap. 7. Saguy, I., Utilization of the ‘‘Complex method’’ to optimize a fermentation process, Biotech. Bioeng., 24, 1519, 1982. Turhan, K.N. et al., Optimization of glycerol effect on the mechanical properties and water vapor permeability of whey protein-methylcellulose films, J. Food Proc. Eng., 30, 485, 2007. Umeda, T., Shindo, A., and Ichikawa, A., Complex method for solving variational problems with state-variable inequality constraints, Ind. Eng. Chem. Proc. Design Development, 11, 102, 1972.

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16

Mixed Integer Linear Programming Scheduling in the Food Industry Philip Doganis and Haralambos Sarimveis

CONTENTS 16.1 16.2

Introduction: Literature Review............................................................... 305 Proposed Methodology ............................................................................ 311 16.2.1 Main Features .............................................................................. 311 16.2.2 Problem Definition ...................................................................... 311 16.2.3 Model Description....................................................................... 312 16.2.3.1 Parameters.................................................................... 312 16.2.3.2 Decision Variables....................................................... 312 16.2.3.3 Constraints ................................................................... 313 16.3 Case Study ............................................................................................... 317 16.3.1 Yogurt Production ....................................................................... 317 16.3.2 Computational Results ................................................................ 318 16.4 Conclusion ............................................................................................... 322 Nomenclature ........................................................................................................ 324 Parameters ................................................................................................ 324 Decision Variables ................................................................................... 325 Indices ...................................................................................................... 325 References ............................................................................................................. 326

16.1 INTRODUCTION: LITERATURE REVIEW Food production is one of the oldest process industries. In the beginning of the previous century, it leaped from the small, rural scale to the industrialized national and later international level. This change scaled up production levels from some hundred kilos to millions of tons per year, which are produced using special equipment. At first, companies sought only to raise their profit by expanding their market share, giving limited attention to costs within the factory. Over the years, increased competition narrowed profitability, making cost reduction, flexible production, and prompt reaction an urgent necessity (Tahmassebi, 1996). The benefits offered by scheduling tools (cost reduction, improved management of equipment, time, and

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manpower) made it possible to continue meeting production targets and at the same time achieve significant cost improvements through more efficient planning and scheduling of actions. The role of the first scheduling tools used by the food industry was to implement some simple heuristic rules. The increase in computing power and the evolution of mathematical methodologies facilitated tackling more complicated formulations and including more detailed characteristics of the process. Scheduling systems usually include the following capabilities: assignment of tasks to equipment, sequencing of tasks on machines, and event timing. Pinto and Grossmann (1998), while reviewing assignment and sequencing models in details, noted that many scheduling models follow a Mixed Integer Linear Programming (MILP) formulation. Scheduling in process industries often has to deal with changeovers between products. In some cases, the cost and time involved can be neglected; in many cases however, transitions are associated with significant losses in production time and considerable costs. Such cases are more frequent in the food industry and in chemicals, plastics, printing, and automobile manufacturing. Despite the detrimental influence changeovers, and to a greater extent sequence-dependent changeovers, can have on productivity and even on the feasibility of scheduled processes, this attribute is usually not included in the methodologies proposed in the literature. A review of scheduling research with setup considerations has been conducted by Allahverdi et al. (1999). In their work, lot sizes are fixed in most of the methodologies reported. Many recently published methodologies that include sequence-dependent changeovers take into consideration only setup time ignoring the cost involved (Chen et al., 2002; Gupta and Karimi, 2003; Lim and Karimi, 2003; Giannelos and Georgiadis, 2003; Janak et al., 2004). In all the aforementioned work, the continuous representation of time was followed. Sequence-dependent setup costs are modeled in the methodology presented by Kang et al. (1999). They used a discrete time formulation for the parallel machines scheduling problem where production costs and inventory have been considered but not the timing of actions. Lamba and Karimi (2002) proposed a model for parallel machines scheduling under controlled resources that take into account both sequence-dependent setup times and costs, but there are no due dates for product delivery and the assumption that setup cost is proportional to setup time is made. This can be an adequate approximation when time is the most important factor; however, it is not the case for transitions with short duration and high-incurred cost because of bulk product loss, idling of machinery, and nonproductive use of workforce. Until recently, the tools used in the food industry were adopted from other applications overlooking any particularities of this sector. It was found, however, that food production possessed some particular characteristics and leaving them out of the simulation and optimization procedures led to inadequate solutions. Food processing is firstly characterized by the numerous different products each company offers in order to satisfy divergent customer tastes. This causes organizational problems in the processing stage, which are amplified in the packaging stage since goods are usually offered in packaging of various sizes (single portion, family packs or limited-time promotional packages), so that the items that are due are much more than the flavors or mixes produced (van Dam et al., 1993; Jakeman, 1994). Setups occur in transitions between items and they are divided in three categories

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(van Dam et al., 1993): format changes where the size of the packaging material changes, product changes where the bulk product fed to the machine changes, and end product changes where only the labeling of the packaging changes. When product changes occur, it might be necessary to perform cleaning, so as to maintain high hygienic and quality standards since, in the opposite case, microorganisms could multiply in some parts of the equipment and infect other facilities as well (Flapper et al., 2002). It may also happen that flavor, color, and other ingredients of different products get mixed together giving a mix of doubtful sensorial properties. Setups that depend on the previous and following product are called sequencedependent; it is this kind of setups that is most frequent in food industries (Soman et al., 2004a; van Dam et al., 1993; van Wezel et al., 2006). Because of the time spent for their execution, setups involve losses in time. At the same time, the amount of bulk or final product that has to be discarded, the cost of cleaning materials, and the idling of valuable production, all bring about increased costs for the industry. An account of the environmental impact of some main food product categories from production to consumption and disposal is provided by Foster et al. (2006). The sequence-dependent setup times of high costs and long duration influence production decisions as choosing one sequence over another can result to significant differences in the usable production time. van Dam et al. (1993), who studied packaging lines in process industries, found that in average about half of the available production time was lost either because of disturbances (20%) or due to setups (30%). These figures may vary but demonstrate the potential to improve productivity through better choice of the sequence to be followed. Actions in food production are also influenced by the orders placed by the retailers. The last years have seen a great increase in the market force of retailers (Entrup, 2005), who leverage production to suit their needs best by placing orders of increasingly small volume which must be delivered in ever shorter lead times (Jakeman, 1994; van Wezel et al., 2006; Soman et al., 2006; Gargouri et al., 2002). This combination goes directly against the actions that the industry would choose to increase its effectiveness: larger orders would mean longer runs and fewer tasks to schedule reducing the number of setups needed and leaving room for the scheduler to find an efficient sequence, while longer lead times make more cost-saving actions possible and reduce the strain on production equipment. Short lead times also make it more difficult to group small orders into larger ones and increase the need for storing raw materials, intermediate, and final products. Responding to demand under an MTS (make-to-stock) regime is often impractical in the food industry due to the limited shelf life of its products, which makes keeping large volumes of inventory uneconomical due to spoilage (van Dam et al., 1993); for classifications of deterioration of inventory please refer to Yao and Huang (2005). As customers opt for the freshest product, they choose the expiration date that is further away in the future. Knowing that, retailers often do not accept two subsequent deliveries with the same expiration date even if they expect to sell it well before it expires. Other retailers advertise the freshness of their products and remove them from the shelf significantly sooner than obliged by law. It can therefore happen that some food products may be away from their expiration date, but they cannot be sold and have become commercially obsolete (Soman et al., 2004a).

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These characteristics formulate an environment where capacity utilization is high and essential for a successful response to demand. The presence of sequencedependent setup times, the limited shelf life of end products, combined with the market need for small orders and short lead times for a large variety of products makes planning in food industry a complex task (Nakhla, 1995). Furthermore, a crucial factor that determines the applicability of tools in the actual process is the inclusion of some rules and constraints that are specific to the process or even the production lines utilized (see Nakhla, 1995; Berlin et al., 2007 for examples of such rules). The complexity of such a problem could not be handled by spreadsheet based solutions and as Kondili et al. (1993) point out, the flexibility offered by batch plants for the production of a large number of products in rather small quantities can only be fully exploited through the use of advanced planning tools. Additionally, they provide a direct insight into the consequences of actions on the shop floor, which would otherwise be hardly visible to planning staff. Scheduling literature offers several systems developed specifically for food processing. As can be seen in Table 16.1, the areas covered are diverse and some systems are based on heuristics, using simple rules for scheduling and following predefined steps to allocate production according to certain logic while others follow mathematical programming approaches in order to arrive at a solution. Vaidyanathan et al. (1998) proposed a hybrid methodology based on discrete event simulation and a heuristic to schedule coffee production. Nakhla (1995) presented a heuristic that aims to demonstrate the impact of constraints on the capability of production equipment to respond to demand. It requires interaction with the operator to ensure operational feasibility. Jakeman (1994) used APV company software to schedule a yogurt production facility (processing and packaging) taking into account cleaning operations. Darlington and Rahimifard (2007) implemented a heuristic using the commercial scheduling software supplied by PREACTOR in order to schedule convenience food production. Commercial software was also used by Pinedo (2005) who presented an application of SAP for planning and scheduling of beer production at a large beer producer. Katok and Ott (2000) also studied beer production and used mixed-integer programming to reduce label changes in the packaging stage. In the area of crop scheduling, an early attempt is that of Alpar and Srikanth (1989), who first crafted a mathematical program, which could not be solved with the tools available at the time and then used a spreadsheet approach that nevertheless limited the applicability of solutions. The expert system they formulated delivers nonoptimal solutions but worked better than the other approaches. Jones et al. (2003) studied the same topic many years later and used dynamic programming. Doganis and Sarimveis (2007) presented a methodology for optimal scheduling under food-specific constraints which was later extended to handle systems of parallel machines where actions have to be synchronized across all machines due to a common feeding line (Doganis and Sarimveis, 2008). Yogurt packaging was also studied by Marinelli et al. (2007), who showed an MILP formulation, decomposed the problem into two subproblems in order to employ a heuristic to beat the large integrality gap faced by the implementation of a commercial solver. The approach of decomposition was also followed by Claassen and van Beek (1993) while studying another dairy product, cheese. In their approach, planning

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TABLE 16.1 Food Processing Scheduling Applications System Architecture Heuristic systems

Application

References

Baked goods Coffee Convenience food Cookies Cooking fats and oils Dairy

Franchini et al. (1998) Vaidyanathan et al. (1998) Darlington and Rahimifard (2007) Gargouri et al. (2002) Lagodimos et al. (1996) Berlin et al. (2007) Nakhla (1995) Soman et al. (2006) Akkerman and Van Donk (2007) Sundararajan et al. (1998) Katok and Ott (2000) Alpar and Srinkanth (1989)

Food (various) Mathematical programming

Beer Cereal blending=processing Crop scheduling Dairy

Dehydration of foods Food (various)

Industrial packages

SAP APV Software

Food preserves Fresh and frozen foods Meat processing Poultry Salad dressings Sausages Wine Beer Dairy

van Berlo (1993) Jones et al. (2003) Claassen and van Beek (1993) Doganis and Sarimveis (2008) Doganis and Sarimveis (2007) Entrup (2005) Marinelli et al. (2007) Stefanis et al. (1997) Tarantilis and Kiranoudis (2002) Lagodimos and Leopoulos (2000) Yao and Huang (2005) Tadei et al. (1995) Houghton and Portougal (2001) Chan et al. (1992) Entrup (2005) Brown et al. (2002) Entrup (2005) Berruto et al. (2006) Pinedo (2005) Jakeman (1994)

and scheduling problem are decomposed into a tactical and an operational planning level but the schedules need user review to be applicable to daily industrial floor reality. The idea of leaving room for scheduling decisions to be made by operating managers is followed by Berruto et al. (2006) in the system that they propose for wine bottling scheduling where the week schedules are optimized but the daily schedules are left to the manager.

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A part of research focused on side aspects of production scheduling, like the environmental effects of production tasks and the scheduling of workforce. The minimization of cleaning operations in production scheduling was studied under an environmental perspective by Berlin et al. (2007) and Stefanis et al. (1997). Berlin et al. (2007) employed a heuristic to sequence products in order to minimize the environmental impact of yogurt products throughout their life cycle. Stefanis et al. (1997) studied dairy products and used an MILP mathematical programming formulation so as to explore the effects of environmental impact minimization on the scheduling and design of multipurpose batch plants. The scheduling of workforce in food processing was studied by Franchini et al. (1998), who described a heuristic methodology with a decision support system to analyze the manufacturing system in order to calculate and evaluate the allocation of operators (manpower). An MILP model for manpower shift planning was presented by Lagodimos and Leopoulos (2000) which they solved using greedy heuristic algorithms. Entrup (2005) presented MILP models that integrate shelf life into operational food production planning and scheduling. He recognized that consideration of shelf life is a key factor in the commercial success of a model because product freshness is one of the most important buying criteria for consumers. Surprisingly, Entrup (2005), and also Soman et al. (2004b), who studied a special category of scheduling problems, found only a limited number of papers that considered shelf life of the products although freshness of the product was considered by many authors as a significant addition to scheduling methodologies (Kallrath, 2002; Entrup, 2005; Günther et al., 2006). Entrup (2005) studied the problem of parallel machine scheduling for food processing and achieved near-optimal solutions making use of the block planning approach by Günther et al. to group products. Under this approach, setups occur only in transitions between products that belong to different recipes (blocks). Due dates were modeled and backlogging and variable lot sizing were allowed. However, the group setup time and cost are not sequence dependent, rather they are determined only by the line on which they occur, in the case of setup time, and on the recipe of the product, in the case of setup cost. Inventory is not included in the formulation but it can be easily determined from the produced results. As mentioned previously, sequence-dependent changeovers between products can greatly influence productivity and production cost. In previous works (Doganis and Sarimveis, 2007; Doganis and Sarimveis, 2008), we presented methodologies for optimal scheduling of production lines with single or multiple parallel machines and production sequence limitations. The methodologies incorporated features that allow them to tackle problems, such as multiple intermediate due dates, job mixing and splitting, product-specific machine speed, minimum lot sizes, and sequencedependent changeover times and costs, but did not consider freshness in the formulation of the MILP models. In this chapter, we present a new MILP model that combines the advantages of the aforementioned models. We build on the work on modeling of the special constraints in food production systems by Doganis and Sarimveis (2007, 2008) and extend it to include the factor of product freshness as introduced by Entrup

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(2005). The combined model has the capability to depict the production constraints and management directives present in an industrial food processing environment. At the same time it considers shelf life restrictions and optimizes the balance between the cost-inflicting factors, like setups, machine utilization and storage cost, and the profit-contributing aspects of minimizing time duration between production and delivery of products to the retailers.

16.2 PROPOSED METHODOLOGY 16.2.1 MAIN FEATURES The proposed methodology incorporates various features that are common in the food industry and require special attention. It allows scheduling of parallel machines in a single stage in the presence of sequence limitations where technological restrictions impose a specific production order, which must always be followed, although it is possible to omit one or more products. Moreover, the impact of choosing one production sequence over another is modeled realistically since sequence-dependent setup times and costs are included in the model. Indeed, a product transition needs more time and brings about more cost than transitions where only the labels or the packaging are changed. In the first case, there are increased setup costs due to the product quantity wasted during cleaning and the time, the chemicals and the manhours spent for cleaning operations. Furthermore, the shelf life of products is considered in order to achieve improved freshness in a profitable manner. Shelf life is defined as the duration between producing a product and using it, for which the product remains safe and acceptable to the user. Shelf life is determined by the manufacturer or legislation. Consideration of shelf life is achieved by incorporating it in two levels: on a first level, the shelf life and the least remaining duration until end of shelf life, chosen by an agreement between the retailer and the industry, have been integrated in the model, so that products exiting the factory always abide by these constraints. On the second level, delays between production and delivery are penalized in the objective function, since they correspond to products that are less fresh on delivery. Additionally, the demand for products is fed into the model as production tasks with specific due dates. It is possible to have multiple intermediate due dates, and also to split or join one or more batches, in order to achieve improvements in the objective function. The due dates must always be met although it is possible to fulfill the demand for one day using production from previous days. Thus, earliness is allowed but not tardiness. The number and sizes of the lots are not defined or limited in advance; only a minimum lot size is enforced which is directed by economical and technological restrictions. The total production time of each machine is limited by the available machine time. As it happens in actual industrial practice, production rates are allowed to vary by product.

16.2.2 PROBLEM DEFINITION The methodology proposed is based on a MILP formulation and yields a detailed production schedule given the following: (1) demand for each product in each day

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through the scheduling horizon, (2) time and cost involved in every possible transition between products, (3) restrictions in the sequencing of products, (4) time before delivery date that production can occur and time required for intermediate preparation before shipping, (5) marketability loss due to freshness lost in storage, (6) opening and the finishing inventory of each product, (7) inventory holding cost, (8) machine hours available for a day, (9) cost of machine utilization, (10) quantity limitations on individual products or groups of products on each machine (lower and upper bounds), (11) production speed of each product, and (12) regular and overtime labor cost for all machines. The resulting production schedule provides: (1) quantities of each product produced daily to fully or partially satisfy the demand for each subsequent day, (2) starting and finishing times for the production of each item on each machine, (3) total time of machine utilization (transition times are included), and (4) inventory of each product at the end of the day.

16.2.3 MODEL DESCRIPTION We will now describe the model in detail. Before that, it is helpful to first define the parameters and variables (continuous or binary) of the problem at hand. 16.2.3.1 . . . . . . . . . . . . . .

Parameters

Number of products Scheduling horizon Number of parallel machines Sequencing restrictions Demand for each product in each day Machine speed for each product Setup time and cost for each possible transition Storage cost Time before delivery date that production can start and time required for intermediate preparation before shipping Marketability loss due to days of shelf life lost while in in-house storage Opening inventory Target (final) inventory Upper and lower bounds on the size of lots Regular and overtime labor cost

16.2.3.2

Decision Variables

Binary variables .

.

Binary (0–1) variables (one for each day-product-machine combination) that take the value of 1 when the product is to be produced by the machine in the particular day Binary (0–1) variables (one for each day-product-product-machine combination) that take the value of 1 when the transition between the two products is to take place in the machine in the particular day

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.

Binary (0–1) variables (one for each day-product group combination) that take the value of 1 when the product group is to be produced in the particular day

Continuous variables The optimal values of the following variables are to be determined for each day (all continuous variables are nonnegative): . . . .

Produced quantity of each product that each machine will produce on each day to fully or partially satisfy the demand of each subsequent day Machine time on each machine and on each day that is spent on producing each product together with the exact starting and finishing times of tasks Total operation time of each machine on each day including changeover times and intermediate idle times Inventory level of each product at the end of the day

The nomenclature used in the model is given in the end of the chapter. A presentation of the model constraints and the objective function follows next. 16.2.3.3

Constraints

Production levels N X

prod (i, j, l, iD)  hi(j)  bin (i, j, l)

8i, 8j, 8l

(16:1)

prod (i, j, l, iD)  lo(j)  bin (i, j, l)

8i, 8j, 8l

(16:2)

iD¼i N X iD¼i

The variable prod(i, j, l, iD), which appears in Constraints 16.1 and 16.2, denotes the quantity of product j that is produced by machine l on day i and is designated to fully or partially satisfy the demand of day iD. Constraints 16.1 and 16.2 link the variable prod(i, j, l, iD) to its respective binary variable bin(i, j, l), so that production for any day i is allowed only when bin(i, j, l) ¼ 1. At the same time, upper and lower limits to the production batches are set, represented by parameters hi( j) and lo( j). These constraints limit the production of individual products. When it is necessary to limit the produced quantities of groups of products, Constraints 16.3 and 16.4 are employed: MC X N XX

prod (i, j, l, iD)  bingroup (i, m)  grouphi (m) 8i, 8m

(16:3)

prod (i, j, l, iD)  bingroup (i, m)  grouplo (m) 8i, 8m

(16:4)

j2Pm l¼1 iD¼i MC X N XX j2Pm l¼1 iD¼i

ß 2008 by Taylor & Francis Group, LLC.

In the above constraints, Pm is a subset of the P different products that can be produced on the machines available. The parameters grouphi(m) and grouplo(m) are the upper and lower limits for product group m and the binary variables bingroup (i, m) take the value of 1 when one or more products of product group Pm are produced on day i. Inventory levels and demand fulfillment The next two constraints define the value of inventory across the scheduling horizon. Constraint 16.5 holds only for the first day: according to it, the inventory at the end of the first day equals to the opening inventory, plus the quantities of product j produced on the first day on all machines, minus the demand for the first day. On the remaining days, the inventory of product j at the end of the day is equal to the inventory at the end of the previous day, minus the demand for this day, plus the quantities of product j produced on all machines that are destined for all days as shown in Constraint 16.6: inv(1, j) ¼ openinv( j) þ

MC X N X

prod(1, j, l, iD)  demand(1, j) 8j

(16:5)

l¼1 iD¼1

inv(i, j) ¼ inv(i  1, j) þ

MC X N X

prod(i, j, l, iD)  demand(i, j) i > 1,8j (16:6)

l¼1 iD¼i

There are specific requirements for product freshness that direct demand fulfillment occurs as defined in Constraint 16.7: Products that will cover demand at day iD have to be produced within a specific time window, which starts tf days before iD and ends tp days before iD. The parameter tf refers to the number of days before delivery during which production is allowed. This parameter is defined as the difference between shelf life of the product and the least remaining duration until end of shelf life at the time of delivery, which has been agreed with retailers. The parameter tp is the processing time required for intermediate preparation before shipping, which is the time required for procedures like cooling, secondary packaging in case of family packs, or special packaging as part of a promotional activity. Constraint 16.8 renders production out of the allowed time window impossible. iDt Xp

MC X

prod(i, j, l, iD) ¼ demand(iD, j)

8iD, 8j

(16:7)

i¼iDtf l¼1

prod(i, j, l, iD) ¼ 0, 8j, 8l, 8iD,

(i < iD  tf or i > iD  tp )

(16:8)

Constraint 16.9 sets the inventory at the end of the last day equal to a desired target inventory. A desired inventory level could otherwise be ensured by modifying Constraint 16.9 with an inequality sign in order to set the inventory at the end of the last day to be less or greater than the target value. inv(N, j) ¼ tarinv( j) 8j

ß 2008 by Taylor & Francis Group, LLC.

(16:9)

Timing of events tstart(i, 1, l ) ¼ 0

8i, 8l

(16:10)

time(i, l ) ¼ tfin(i, P, l) 8i, 8l time(i, l)  maxtime N P

tprod(i, j, l ) ¼

8i, 8l

(16:11) (16:12)

prod(i, j, l, iD)

iD¼i

u(j)

8i, 8j, 8l

(16:13)

tfin(i, j, l ) ¼ tstart(i, j, l ) þ tprod(i, j, l ) þ þ

P X

tsetup(j, jj)  binsetup(i, j, jj, l) 8i, 8j, 8l

(16:14)

jj¼jþ1

time(i, l )  8 þ excesstime(i, l) 8i, 8l

(16:15)

In order to ensure correct timing of events in the model, it is necessary to assign a starting time and a finishing time to all products even to those that are not produced. In this case, the starting and finishing times coincide. Constraint 16.10 sets the first product on each machine and each day to start at t ¼ 0. As declared by Constraint 16.11, the finishing time of the last product denotes the total machine utilization time, which must be less than a time limit (Constraint 16.12). While the total machine utilization time includes the time spent on changeovers, tprod(i, l, j) is the net production time devoted to product j on machine l for day i and equals to the quotient of the produced quantity and the production speed of machine l Constraint 16.13 finishing time of product j on machine l on a certain day is defined by Constraint 16.14 and equals to the starting time of product j, plus the net production time tprod(i, l, j), plus the time spent on the changeover from product j to the next product that is actually produced on machine l on that day. If there is no production of j on that machine, then tprod(i, l, j) and all associated binary variables are equal to zero (no setup occurs), so that the starting and the finishing times coincide. The optional use of Constraint 16.15 allows us to determine the hours of overtime work (the time by which the machine utilization exceeds 8 h) on each machine and each day in an indirect manner: the excess time variable that appears in Constraint 16.15 is penalized in the objective function, thus pushing the algorithm to minimize the utilization of overtime, as will be explained later. Constraints on the binary variables binsetup(i, j, jj, l ) ¼ 0 8i, j  jj, 8l

(16:16)

binsetup(i, j, jj, l)  1 þ (1  bin(i, j, l)) þ (1  bin(i, jj, l)) l

jj1 X

bin(i, k, l) 8i, 8j, jj > j, 8l

k¼jþ1

ß 2008 by Taylor & Francis Group, LLC.

(16:17)

binsetup(i, j, jj, l)  bin(i, j, l) þ bin(i, jj, l)  1 

jj1 X

bin(i, k, l)

k¼jþ1

(16:18)

8i, 8j, jj > j, 8l P X

bin(i, j, l) 

j¼1

P X P X

binsetup(i, j, jj, l)  1

8i, 8l

(16:19)

j¼1 jj¼1

binsetup(i, j, jj, l)  bin(i, j, l) 8i, 8j, jj > j, 8l

(16:20)

binsetup(i, j, jj, l)  bin(i, jj, l) 8i, 8j, jj > j, 8l

(16:21)

The group of constraints of binary variables presented here has been designed in order to embed a set of logical rules in the model structure so as to ensure that actions on the shop floor accurately represent industrial reality. Constraint 16.16 ensures that only a specific sequence of products is allowed on the machines that of increasing product index; in such a case only transitions from a product j to product jj where jj is next in the sequence are allowed. Constraints 16.17 through 16.21 (where l is a small number) define the relationship between the variables bin(i, l, j) and binsetup(i, j, jj, l) in such a manner that a setup between two products can only occur when both products are produced and no other product that is between them in the sequence is produced, that is, binsetup(i, j, jj, l) is allowed to take the value 1 if and only if both products j and jj are produced (bin(i, l, j) and bin(i, jj, l) both are equal to one) and at the same time all bin(i, k, l) equals to zero for j < k < jj. According to Constraint 16.19, the number of produced items, minus the number of setups must be less than or equal to 1. When the machine is utilized, this constraint clearly holds. Although it is uncommon in industrial practice for a machine not to be utilized during a day, in that case both the number of produced items and the number of setups are equal to 0, allowing (19) to hold. Objective function N X P X P X MC X

csetup(j, jj)  binsetup(i, j, jj, l )

i¼1 j¼1 jj¼1 l¼1

þ

N X P X

inv(i, j)  storagecost( j)

i¼1 j¼1

þ

N X MC X

time(i, l)  machinecost(l)

i¼1 l¼1

þ

N X MC X

excesstime(i, l)  overtimecost(l)

i¼1 l¼1

þ

N X P X MC X N X

prod (i, j, l, iD)  (iD  i)  cupweight( j)  lostlifecost(iD, j)

i¼1 j¼1 l¼1 iD¼1

(16:22)

ß 2008 by Taylor & Francis Group, LLC.

The objective function to be minimized is comprised of five cost terms: the setup cost, the storage costs, the machine utilization cost, the overtime labor cost, and a term to represent the marketability loss due to reduction of duration until the end of shelf life while in in-house storage. The overtime labor cost term encourages a more uniform spreading of production across the scheduling horizon by penalizing accumulated production on certain days. It should be noted here that the raw material cost does not appear in the objective function, since it does not depend on the production schedule.

16.3 CASE STUDY The proposed methodology was applied on data provided by a leading dairy product manufacturer in Greece and in particular on data concerning yogurt production. Yogurt production possesses characteristics that make it an interesting area to employ such a methodology: It produces a wide variety of products that differ in characteristics like yogurt culture, flavor, fat content, packaging, presence of special ingredients (honey, fruit, chocolate flakes, etc.,) in order to respond to a very competitive market where low lead times and order volumes are becoming increasingly common. Moreover, the sequence-dependent setup costs and times obscure the way to the best production sequence and the large number of special constraints that are present impedes finding feasible solutions. Another characteristic of packaging processes is the large increase in the number of products handled in this phase in comparison to the previous stage, the processing phase. This is especially apparent in consumer products manufacturing because of the many different packages used in that case; industrial goods are usually packaged in bulk containers. The large number of products, combined with the limited possibility to keep stock because of the perishability of dairy products, render packaging a complicated procedure that determines the pace and direction of production. Equivalent structures can be found in many cases in food production. Especially in the case of yogurt, processing prior to packaging can easily be adjusted to follow the scheduling decisions for packaging as it involves only one raw material, milk (whey is added to it later before or during packaging as will be explained below). Therefore, packaging is the dominant stage in yogurt production and modeling it allows us to seize control of the entire facility. The processes involved in making yogurt will be presented next in this section, followed by the computational results.

16.3.1 YOGURT PRODUCTION The production of yogurt begins with the collection of its basic raw material, which in this case is milk. Each day, after being locally collected in dairy farms, it is transferred to factories using refrigerated containers. Should the tests by the factory personnel prove it to be suitable, it is fed into refrigerated silos. A first stage constitutes of a series of processing steps that prepare raw milk for the next stage, which is that of yogurt production. In order of occurrence, the steps of the first stage are (1) Standardization of fat content in order to control and direct it to a lower level. This is achieved through by means of skimming, blending of full-fat and skimmed

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milk, or using standardizing centrifuges. (2) Increase of the solid content either by removing water through evaporation or by adding supplements. This procedure is often followed at the aim of reducing the acidity of the yogurt that will be the final product, while at the same time reducing the danger of syneresis, a process by which excessive coagulation leads to the separation of liquid from the yogurt mixture. (3) De-aeration, which may be employed to enhance the fermentation that is to follow and reduce whey separation brought about by dispersible gases. (4) Homogenization which constitutes the reduction of the size of milk fat globules, reduces the rate of cream separation, and leads to improved flavor and increased firmness of the coagulum that will be made from thus processed milk. (5) Heat treatment, which reduces the population of adventitious bacteria, eliminates milkborne pathogens such as Salmonella and Campylobacter, and it is significant for improving yogurt texture and the augmenting the viscosity of yogurt. It is performed at around 908C for a duration of less than 15 s. The second stage includes addition of the yogurt culture, incubation, cooling, and packaging. After milk has reached the incubation temperature, which in most cases of modern large-scale production is performed through a two-step process around 308–408C, a combination of bacterial cultures in symbiotic proportions that depend on the final product is added. This procedure is called inoculation. If the inoculated milk is incubated in double-jacketed tanks of controlled temperature, stirred-type yogurt is produced. If, on the contrary, incubation takes place in a temperature-controlled space, which could be done in a room in batches or continuously through a heated tunnel, fermentation takes place in the final container and settype yoghurt is produced. Depending on the desired final product, it is possible to add fruit and nuts to stirred-type yoghurt. More details on dairy technology can be found in the works of Kessler (1981) and Varnam and Sutherland (1994). Automated packaging machines are used to transfer yogurt mixture of either two types to the cups using volumetric pumps. There are main categories of packaging machines: (1) Filling machines. Prefabricated plastic or glass containers are used. (2) Chain heat-forming-filling machines. Form-fill and seal-type machines perform heat-forming of containers before filling them with yoghurt. (3) Packaging under aseptic atmosphere. This type of machines is used to pack specific types of yogurt because they allow us to achieve low microbial content and thus extend the salability period, while providing a product with excellent sensorial characteristics (for instance flavor, color, and aroma). Bureau and Multon (1996) provide a detailed account of food packaging technology.

16.3.2 COMPUTATIONAL RESULTS The problem studied is that of two parallel machines used for packaging yogurt and the goal was the optimization of the production schedule over a scheduling horizon of 5 production days given the daily demand of the 25 separate products produced by the machines. Both machines have the capability to produce all 25 products. Only the most important problem parameters will be given in detail in order to be concise. The time required for intermediate preparation before shipping is 1 day,

ß 2008 by Taylor & Francis Group, LLC.

so the production for day 2 has to be ready by the end of day 1 (for that reason, no demand is set for day 1). Opening and final inventory have been set to zero and upper and lower bounds on the production of products and product groups have been imposed. The parameter tf is set equal to three. The production speed of all machines for all products has been set to 9000 cups=h, and the maximum daily machine time is 16 h. Storage cost has been assumed negligible and the regular and overtime (extra) labor cost are 60 and 15 e=h, respectively. The demand for all products over the scheduling horizon is given in Table 16.2. The value of lost life cost that is used for all products and days is 0.15 e=kg=day, which is about 5% of the value of yogurt per kilogram in Greece.

TABLE 16.2 Total Demand Over the Scheduling Horizon (in 1000 cups) Product P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 Total daily demand

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

Total Demand per Product

— — — — — — — — — — — — — — — — — — — — — — — — — —

— — 20,000 20,000 20,000 20,000 20,000 20,000 2,000 — — — — — — — — — — — — — — — — 122,000

— — — — — 21,944 — 27,072 24,816 18,592 — — — — — — 41,864 — 25,512 8,512 4,208 75,200 5,656 28,516 3,316 285,208

— — — — — — 37,664 — — — 8,416 9,208 83,560 18,328 32,960 61,926 — 17,424 — — — — — — 10,476 279,962

50,000 50,000 — — — — — — — — — — — — — — — — — — — — — — — 100,000

— — — — — — — — — — — 20,000 20,000 — — — — — — 300,000 — — — — — 340,000

50,000 50,000 20,000 20,000 20,000 41,944 57,664 47,072 26,816 18,592 8,416 29,208 103,560 18,328 32,960 61,926 41,864 17,424 25,512 308,512 4,208 75,200 5,656 28,516 13,792 —

ß 2008 by Taylor & Francis Group, LLC.

The model consists of 6035 variables, 3285 binary, and 2750 continuous and 38281 constraints. It was solved to the Global Optimum in 2 min on a Pentium IV PC using a common commercial solver. The results are given in Tables 16.3 through 16.7. Table 16.3 shows the additive production schedule for both packaging lines. It must be noted that products can be produced 1 day before delivery at the latest, in order to allow preparation within the factory. Many products are produced close to the delivery date in order to achieve maximum freshness (i.e., products 1 and 2). However, in other cases, like that of products 6 and 13, batches are mixed in order to economize on cleaning operations and lost quantities. It is the relative contribution of the two factors in the objective function that defines the overall tendency towards

TABLE 16.3 Optimal Additive Production Schedule for Both Machines Over the Production Horizon (in 1000 cups) Product P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 Total daily production

Day 1

Day 2

Day 3

Day 4

Day 5

— — 20,000 20,000 20,000 41,944 57,664 47,072 26,816 18,592 — — — — — — — 17,424 — — — — — — — 269,512

— — — — — — — — — — — — — 18,328 32,960 — 41,864 — 25,512 8,512 4,208 75,200 5,656 28,516 13,792 254,548

— — — — — — — — — — 8,416 29,208 103,560 — — 61,926 — — — — — — — — — 203,110

50,000 50,000 — — — — — — — — — — — — — — — — — 144,000 — — — — — 244,000

— — — — — — — — — — — — — — — — — — — 156,000 — — — — — 156,000

ß 2008 by Taylor & Francis Group, LLC.

TABLE 16.4 Optimal Production Schedule for Machine 1 Over the Production Horizon (in 1000 cups) Product P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 Total daily production of machine

Day 1

Day 2

Day 3

Day 4

Day 5

— — — 20,000 — 41,944 57,664 — — 18,592 — — — — — — — — — — — — — — — 138,200

— — — — — — — — — — — — — — — — — — — 8,512 4,208 75,200 5,656 28,516 13,792 135,884

— — — — — — — — — — 8,416 — 103,560 — — — — — — — — — — — — 111,976

— — — — — — — — — — — — — — — — — — — 144,000 — — — — — 144,000

— — — — — — — — — — — — — — — — — — — 72,000 — — — — — 72,000

freshness or setup reduction. It can be observed in the schedules of the two machines that there is an overall tendency to produce each order on one machine completely in order to save setup time and cost. This, however, will not hold in cases of large orders, like the order for product 20 that is produced by both machines. Tables 16.4 and 16.5 show the calculated production schedule for machines 1 and 2, respectively, while a detailed schedule including starting and finishing time for each task is presented in Tables 16.6 and 16.7. It can be observed in these tables, that the schedules for the two machines are not identical. This means that we cannot obtain this optimal solution by considering both machines as one with double the production speed.

ß 2008 by Taylor & Francis Group, LLC.

TABLE 16.5 Optimal Production Schedule for Machine 2 Over the Production Horizon (in 1000 cups) Product P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25 Total daily production of machine

Day 1

Day 2

Day 3

Day 4

Day 5

— — 20,000 — 20,000 — — 47,072 26,816 — — — — — — — — 17,424 — — — — — — — 131,312

— — — — — — — — — — — — — 18,328 32,960 — 41,864 — 25,512 — — — — — — 118,664

— — — — — — — — — — — 29,208 — — — 61,926 — — — — — — — — — 91,134

50,000 50,000 — — — — — — — — — — — — — — — — — — — — — — — 100,000

— — — — — — — — — — — — — — — — — — — 84,000 — — — — — 84,000

16.4 CONCLUSION In this chapter, a methodology for the production scheduling of food processing lines was presented. Attributes that are frequent in the food industry, like sequence-dependent changeovers, product shelf-life limitations, multiple intermediate due dates, batch mixing and splitting, individual machine speeds for products, and minimum batch sizes were included in the model formulation and allow it to tackle the particularities of scheduling problems of the food industry. The method produces production schedules in short computational times and can lead to a more efficient management of production by optimizing the balance

ß 2008 by Taylor & Francis Group, LLC.

TABLE 16.6 Start and End Time of Each Production Task in Machine 1 (in hours) Day 1

Day 2

Day 3

Day 4

Day 5

Product

Start

Finish

Start

Finish

Start

Finish

Start

Finish

Start

Finish

P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25

— — — 0 — 2,4 7,2 — — 13,8 — — — — — — — — — — — — — — —

— — — 2,4 — 7,2 13,8 — — 15,9 — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — — — 0 1,2 1,7 10,1 10,8 14,2

— — — — — — — — — — — — — — — — — — — 1,2 1,7 10,1 10,8 14,2 15,7

— — — — — — — — — — 0 — 1,1 — — — — — — — — — — — —

— — — — — — — — — — 1,1 — 12,6 — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — — — 0 — — — — —

— — — — — — — — — — — — — — — — — — — 16,0 — — — — —

— — — — — — — — — — — — — — — — — — — 0 — — — — —

— — — — — — — — — — — — — — — — — — — 8,0 — — — — —

TABLE 16.7 Start and End Time of Each Production Task in Machine 2 (in hours) Day 1 Product P1 P2 P3 P4 P5 P6 P7

Day 2

Day 3

Day 4

Day 5

Start

Finish

Start

Finish

Start

Finish

Start

Finish

Start

Finish

— — 0 — 2,4 — —

— — 2,4 — 5,4 — —

— — — — — — —

— — — — — — —

— — — — — — —

— — — — — — —

0 5,7 — — — — —

5,7 11,3 — — — — —

— — — — — — —

— — — — — — — (continued)

ß 2008 by Taylor & Francis Group, LLC.

TABLE 16.7 (continued) Start and End Time of Each Production Task in Machine 2 (in hours) Day 1

Day 2

Day 3

Day 4

Day 5

Product

Start

Finish

Start

Finish

Start

Finish

Start

Finish

Start

Finish

P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21 P22 P23 P24 P25

5,4 10,8 — — — — — — — — 13,9 — — — — — — —

10,8 13,9 — — — — — — — — 15,8 — — — — — — —

— — — — — — 0 2,2 — 6,0 — 10,9 — — — — — —

— — — — — — 2,2 6,0 — 10,9 — 13,7 — — — — — —

— — — — 0 — — — 3,4 — — — — — — — — —

— — — — 3,4 — — — 10,3 — — — — — — — — —

— — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — 0 — — — — —

— — — — — — — — — — — — 9,3 — — — — —

between the marketability gains stemming from fresher on-delivery products and the cost reductions achievable within the factory that can be reached via more effective lot sizing and reduced setup times and costs.

NOMENCLATURE PARAMETERS csetup(j, k) cupweight(j) demand(i, j) hi(j), lo(j)

Changeover cost from product j to product k (e) Weight of cup for product j (kg=cup) Demand for product j on day i (1000 cups) Maximum and minimum production lots (1000 cups) for product j grouphi(m), grouplo(m) Maximum and minimum production lots (1000 cups) for product group m lostlifecost(i, j) The marketability loss of product j delivered on day i due to days of shelf life spent while in in-house storage (e=kg=day) MC Number of available machines machinecost(l) Cost of regular utilization of machine l per unit of time (e=h)

ß 2008 by Taylor & Francis Group, LLC.

maxtime N openinv(j), tarinv(j)

overtimecost(l) P Pm storagecost tf tp tsetup(j, k) u(j)

Maximum machine time available daily (h) Scheduling horizon (days) Opening and target inventory level of product j in the beginning and at the end of the demand horizon (1000 cups) Extra cost for overtime utilization of machine l per unit of time (e=h) Number of products Products belonging to group m Inventory holding cost (e=1000 cups=day) Time before delivery date that production can start (days) Time required for intermediate preparation before shipping (days) Changeover time from product j to product k (h) Machine speed for product j (1000 cups=h)

DECISION VARIABLES bin(i, j, l) bingroup(i, m) binsetup(i, j, jj, l) excesstime(i,l) inv(i, j) prod(i, j, l, iD)

tfin(i, j, l) time(i, l) tprod(i, j, l) tstart(i, j, l)

Production of product j on machine l on day i (binary variable) Production of product group m on day i (binary variable) Changeover from product j to product jj on machine l on day i (binary variable) Operating time of machine l on day i beyond the first 8 h shift (h) Inventory level of product j at the end of day i (1000 cups) Produced quantity of product j on machine l on day i designated to satisfy the demand of day iD (fully or partially) (1000 cups) Finishing time for the processing of product j on machine l on day i (h) Total utilization of machine l, including idle time and changeover times on day i (h) Utilization of machine l for product j on day i (h) Starting time for the processing of product j on machine l on day i (h)

INDICES i, ii, iD j, jj, k l, ll m

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Days on which production can occur Products Machines Product group

REFERENCES Akkerman, R. and Van Donk, D.P., Product prioritization in a two-stage food production system with intermediate storage, Int. J. Prod. Econ., 108, 43, 2007. Allahverdi, A., Gupta, J.N.D., and Aldowaisan, T., A review of scheduling research involving setup considerations, Omega, 27, 219, 1999. Alpar, P. and Srikanth, K.N., A comparison of analytic and knowledge-based approaches to closed-shop scheduling, Ann. Oper. Res., 17, 347, 1989. Berlin, J., Sonesson, U., and Tillman, A.M., A life cycle based method to minimise environmental impact of dairy production through product sequencing, J. Cleaner Prod., 15, 347, 2007. Berruto, R., Tortia, C., and Gay, P., Wine bottling scheduling optimization, Trans. ASABE, 49, 291, 2006. Brown, G.G., Dell, R.F., Davis, R.L., and Duff, R.H., Optimizing plant-line schedules and an application at hidden valley manufacturing company, Interfaces, 32, 1, 2002. Bureau, G. and Multon, J.-L., Food Packaging Technology. John Wiley-VCH, New York, 1996. Chan, J.C.M., Hui, Y.Y., and Sculli, D., Computerization of production management information: A case study in a food processing factory, Comp. Ind., 18, 91, 1992. Chen, C., Liu, C., Feng, X., and Shao, H., Optimal short-term scheduling of multiproduct single-stage batch plants with parallel lines, Ind. Eng. Chem. Res., 41, 1249, 2002. Claassen, G.D.H. and van Beek, P., Planning and scheduling packaging lines in food industry, Eur. J. Oper. Res., 70, 150, 1993. Darlington, R. and Rahimifard, S., Hybrid two-stage planning for food industry overproduction waste minimization, Int. J. Prod. Res., 45, 4273, 2007. Doganis, P. and Sarimveis, H., Optimal scheduling in a yogurt production line based on mixed integer linear programming, J. Food Eng., 80, 445, 2007. Doganis, P. and Sarimveis, H., Optimal production scheduling for the dairy industry, Annals of Oper. Res., 159, 315, 2008. Entrup, M.L., Advanced Planning in Fresh Food Industries: Integrating Shelf Life into Production Planning, Mueller, W. and Bihn, M.H., (Eds.), Physica Verlag, Springer, Berlin, Germany, 2005. Flapper, S.D.P., Fransoo, J.C., Broekmeulen, R.A.C.M., and Inderfurth, K., Planning and control of rework in the process industries: A review. Prod. Plan. Control, 13, 26, 2002. Foster, C., Green, K., Bleda, M., Dewick, P., Evans, B., Flynn, A., and Mylan, J., Environmental Impacts of Food Production and Consumption: Final Report to the Department for Environment Food and Rural Affairs, Manchester Business School, Delfra, London, U.K., 2006. Franchini, L., Caillaud, E., Nguyen, P., and Lacoste, G., Planning and scheduling skills: A case study in an agri-food industry, IEEE Int. Conf. Sys. Man Cybernetics, 1, 417, 1998. Gargouri, E., Hammadi, S., and Borne, P., A study of scheduling problem in agro-food manufacturing systems, Math. Comp. Simulation, 60, 277, 2002. Giannelos, N.F. and Georgiadis, M.C., Efficient scheduling of consumer goods manufacturing processes in the continuous time domain, Comput. Oper. Res., 30, 1367, 2003. Gupta, S. and Karimi, I.A., An improved MILP formulation for scheduling multiproduct, multistage batch plants, Ind. Eng. Chem. Res., 42, 2365, 2003. Günther, H.-O., Grunow, M., and Neuhaus, U., Realizing block planning concepts in makeand-pack production using MILP modelling and SAP APO, Int. J. Prod. Res., 44, 3711, 2006. Houghton, E. and Portougal, V., Optimum production planning an analytic framework, Int. J. Oper. Prod. Manag., 21, 1205, 2001. Jakeman, C.M., Scheduling needs of the food processing industry, Food Res. Int., 27, 117, 1994.

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Janak, S.L., Lin, X., and Floudas, C.A., Enhanced continuous-time unit-specific event-based formulation for short-term scheduling of multipurpose batch processes: Resource constraints and mixed storage policies, Ind. Eng. Chem. Res., 43, 2516, 2004. Jones, P.C., Kegler, G., Lowe, T.J., and Traub, R.D., Managing the seed-corn supply chain at Syngenta, Interfaces, 33, 80, 2003. Kallrath, J., Planning and scheduling in the process industry, OR Spectrum, 24, 219, 2002. Kang, S., Malik, K., and Thomas, L.J., Lot sizing and scheduling on parallel machines with sequence dependent setup costs, Manag. Sci., 45, 273, 1999. Katok, E. and Ott, D., Using mixed-integer programming to reduce label changes in the coors aluminum can plant, Interfaces, 30, 1, 2000. Kessler, H.G., Food Engineering and Dairy Technology, Kessler, Munich, Germany, 1981. Kondili, E., Pantelides, C.C., and Sargent, R.W.H., A general algorithm for short-term scheduling of batch operations—I: MILP formulation, Comp. Chem. Eng., 17, 211, 1993. Lagodimos, A.G., Charalambopoulos, A., and Kavgalaki, A., Computer-aided packing shop scheduling in a manufacturing plant, Int. J. Prod. Econ., 46–47, 621, 1996. Lagodimos, A.G. and Leopoulos, V., Greedy heuristic algorithms for manpower shift planning, Int. J. Prod. Econ., 68, 95, 2000. Lamba, N. and Karimi, I.A., Scheduling parallel production lines with resource constraints. 1. Model formulation, Ind. Eng. Chem. Res., 41, 779, 2002. Lim, M.-F. and Karimi, I.A., Resource-constrained scheduling of parallel production lines using asynchronous slots, Ind. Eng. Chem. Res., 42, 6832, 2003. Marinelli, F., Nenni, M.E., and Sforza, A., Capacitated lot sizing and scheduling with parallel machines and shared buffers: A case study in a packaging company, Ann. Oper. Res., 150, 177, 2007. Nakhla, M., Production control in the food processing industry: The need for flexibility in operations scheduling, Int. J. Oper. Prod. Manag., 15, 73, 1995. Pinedo, M.L., Planning and scheduling in manufacturing and services, in Springer Series in Operations Research, Glynn, P.W. and Robinson, S.R., (Eds.), Springer Publishing, New York, pp. 193–198, 2005. Pinto, J.M. and Grossmann, I.E., Assignment and sequencing models for the scheduling of process systems, Ann. Oper. Res., 81, 433, 1998. Soman, C.A., van Donk, D.P., and Gaalman, G.J.C., A basic period approach to the economic lot scheduling problem with shelf life considerations, Int. J. Prod. Res., 42, 1677, 2004a. Soman, C.A., van Donk, D.P., and Gaalman, G., Combined make-to-order and make-to-stock in a food production system, Int. J. Prod. Econ., 90, 223, 2004b. Soman, C.A., van Donk, D.P., and Gaalman, G., Comparison of dynamic scheduling policies for hybrid make-to-order and make-to-stock production systems with stochastic demand, Int. J. Prod. Econ., 104, 441, 2006. Stefanis, S.K., Livingston, A.G., and Pistikopoulos, E.N., Environmental impact considerations in the optimal design and scheduling of batch processes, Comp. Chem. Eng., 21, 1073, 1997. Sundararajan, S., Srinivasan, G., Staehle, W.O., and Zimmers, E.W. Jr., Application of a decision support system for operational decisions, Comp. Ind. Eng., 35, 141, 1998. Tadei, R., Trubian, M., Avendafio, J.L., Della Croce, F., and Menga, G., Aggregate planning and scheduling in the food industry: A case study, Eur. J. Oper. Res., 87, 564, 1995. Tahmassebi, T., Industrial experience with a mathematical-programming based system for factory systems planning=scheduling, Comp. Chem. Eng., 20, 1565, 1996. Tarantilis, C.D. and Kiranoudis, C.T., A modern local search method for operations scheduling of dehydration plants, J. Food Eng., 52, 17, 2002. Vaidyanathan, B.S., Miller, D.M., and Park, Y.H., Application of discrete event simulation in production scheduling, Proceedings of the 1998 Winter Simulation Conference (WSC’98), 2, 965, 1998.

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17

Mixed Integer Nonlinear Programming: Applications to Food Dehydration and Deep Chilling Panagiotis P. Repoussis and Christos T. Kiranoudis

CONTENTS 17.1 17.2

Introduction .............................................................................................. 329 Structural Optimization of Process Flowsheets ....................................... 330 17.2.1 Mathematical and Optimization Models ..................................... 330 17.2.2 Process Synthesis ........................................................................ 331 17.3 Conveyor-Belt Dryers .............................................................................. 333 17.3.1 Structure of Conveyor-Belt Dryers ............................................. 333 17.3.2 Model Formulation...................................................................... 335 17.3.3 Design Problem of Conveyor-Belt Dryers.................................. 341 17.3.3.1 Single Product Single Dryer ........................................ 341 17.3.3.2 Multiple Products Multiple Dryers.............................. 343 17.4 Deep Chilling Tray Tunnels .................................................................... 345 17.4.1 Structure of Tray Tunnels ........................................................... 345 17.4.2 Model Formulation...................................................................... 347 17.4.3 Design Problem of Deep Chilling Tray Tunnels ........................ 351 17.4.3.1 Single Tunnel............................................................... 351 17.4.3.2 System of Parallel Tunnels .......................................... 352 17.5 Conclusion ............................................................................................... 353 References ............................................................................................................. 353

17.1 INTRODUCTION Dehydration and deep chilling operations are important activities of the chemical and food processing industries. The objective of dehydration is the removal of water up to a certain level such that the microbial spoilage is minimized. On the other hand,

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the main effort of chilling food products is to ensure that they have reached certain low temperature thermodynamic conditions in order to prevent the development of pathogenic bacteria that contaminates them. The increasing concern for meeting product quality specifications and conserving energy, the need for a thorough understanding of drying and chilling processes, and the problems related to the design and operation of their related equipment should be emphasized. The process design or synthesis problem typically involves determination and evaluation of the process flowsheet structure, corresponding construction characteristics, and operation conditions when a nominal production level is given for all products processed in the plant. The objective is to minimize the total annual cost (or similarly to maximize the profit) resulting from the construction of a new plant or the operation of an existing one by adding new production lines under structural and varying operating constraints. In most cases, determination of the flowsheet structure refers to the solution of a complex combinatorial mixed integer nonlinear programming (MINLP) problem. This involves numerous continuous and integer decision variables, a large number of space variables, and constraints. In this chapter, the main effort is to examine and analyze two typical real life industrial case studies, namely design of multiproduct conveyor-belt dryers and tray tunnels for deep chilling. In both cases, optimum flowsheet configurations are sought and verified by effective mathematical formulations and modeling approaches of design and optimization strategies and techniques. Process design involves principally a hybrid methodology of combining equipment configuration and operational parameter optimization efforts carried out under certain flowsheet constraints available for the various stages of processing. Towards this end, synthesis of a process flowsheet can be performed through a superstructure optimization in which the problem is formulated as an MINLP. In order to accomplish this task, two major questions need to be addressed. The first one is how to develop the superstructure and the second one is how to effectively model and solve the MINLP for the selected superstructure. The remaining part of the chapter is organized as follows: Section 17.2 illustrates the basic notion behind the structural optimization of process flowsheets. Section 17.3 examines the design problem of a multiproduct dehydration plant of conveyorbelt dryers. Section 17.4 illustrates the design problem of single and parallel tray tunnels for food deep chilling, and finally Section 17.5 concludes the chapter and provides pointers for further research.

17.2 STRUCTURAL OPTIMIZATION OF PROCESS FLOWSHEETS 17.2.1 MATHEMATICAL

AND

OPTIMIZATION MODELS

A mathematical model of a system is a set of mathematical relationships, i.e., equalities, inequalities, and logical conditions representing an abstraction of the system under consideration. A mathematical model of a system typically consists of four key elements: variables, parameters, constraints, and mathematical relations. The variables can be continuous, integer, or a mixed set of integer and continuous. They can take different values that define different states of the system. On the other

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hand, the parameters are fixed to one or multiple specific values. Each fixation defines a different model while constants are fixed quantities by the model statement. The mathematical model equalities typically represent mass and energy balances, equilibrium relations, design relations, and others describing the system. The model inequalities often consist of allowable operating regimes, specifications, bounds on availabilities, performance requirements, and so on. Finally, the logical conditions connect continuous and integer variables (Floudas, 1995). An optimization problem is a mathematical model, as described above, which in addition contains one or multiple performance criteria denoted as objective function. A well-defined optimization problem features a number of variables greater than the number of equality constraints implying that there exist degrees of freedom (Floudas, 1995).

17.2.2 PROCESS SYNTHESIS The rationale behind process synthesis is to systematically develop process flowsheets that transform available raw materials into desired products such that particular performance criteria are met. As shown in Figure 17.1, given specification of the inputs (e.g., feed streams) and outputs (e.g., products and byproducts), the goal is to determine the flowsheet of an overall process system or one of its subsystem. The performance criteria are multiple ranging from economic (e.g., maximum profit or minimum cost) to thermodynamic (e.g., energy efficiency) to operability conditions (e.g., flexibility, controllability, safety, or environmental regulations). The resulting process synthesis problem can be classified as a multi-objective mixed discrete-continuous optimization problem. The solution of this problem (optimal process flowsheet with respect to the imposed performance criteria) provides information on the topology-structure of the process flowsheet (selection of units and their interconnections) and on the optimum values of the design parameters and operating conditions. The major difficulties arising in process synthesis problems are the combinatorial nature and the nonlinear characteristics. Due to the interconnections of process units, the nonlinear models that describe each process unit, and the required search for the optimum operating conditions and sizing of each unit, the resulting model is highly nonlinear and contains nonconvexities. As a result, one has to deal with the solution of the MINLP model and to explore algorithms that allow determining the global optimum solution. To meet the objectives of the process synthesis problem, three main approaches have been proposed: heuristics, thermodynamic targets, and optimization-algorithmic approach (see for details Floudas, 1995).

Inputs

Process flowsheet

Outputs

FIGURE 17.1 Schematic view of a process synthesis problem.

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As described by Biegler et al. (1997), the optimization approach in process synthesis consists of the following main steps: I. Postulate superstructure of alternatives. II. Formulate optimization model for superstructure. III. Solve optimization problem to ‘‘extract’’ optimum configuration. The development of the appropriate superstructure is a difficult task and is of major importance, since the optimum process flowsheets sought will be as good as the postulated representation of alternative structures. As described by Floudas (1995), representation of alternative process flowsheets is conceptually based on elementary graph theory ideas. By representing each unit, input and output of the superstructure as a node and interconnections between these nodes as one- and two-way arcs, we have a bipartite planar graph that represents all options of the superstructure. Biegler et al. (1997) describes three approaches for the development of superstructures. The first is to combine the detailed superstructures for each subsystem, in which each unit performs a single preassigned task. The second is to consider detailed models of units that perform multiple tasks or functions and interconnect the units with all feasible connections. Clearly, advantage of the first approach compared to the second is that all interactions and economic trade-offs are explicitly considered while advantage of the second is the reduced size of resulting MINLP problem. Finally, a third approach is to perform a preliminary screening in order to reduce the number of alternatives. Although, this approach is restrictive in a sense, it provides a systematic framework for analyzing specific alternatives at the level of tasks. Furthermore, it is also possible to treat part of the problem with an aggregated model and the rest of the process with a more detailed superstructure. Given the superstructure, whether at the high level of abstraction or at a relatively detailed level of units, the synthesis problem can be formulated as a mixed integer nonlinear optimization model of the form: min f (x, y) x,y

s:t: h(x, y) ¼ 0 g(x, y)  0 x 2 X  0 as inputs. For any fixed «, the scheme is an approximation algorithm with relative error bound «. Approximation schemes that run in polynomial time with size n of the input instance, are called polynomial-time approximation schemes (PTAS) (Cormen et al., 2001).

23.3 COVERING SHORT PATHS A generic application of covering short path problems is the design of a bilevel transportation structure where the path corresponds to a primary vehicle route, and all points not on that route are within easy reach (Figure 23.2). For example, the problem may be to locate postboxes (vertices of the path) on the domain in such a way that all users are located within reasonable distance from a postbox, and that the cost of a collection route through all postboxes is minimized. The problem has a continuous version where the vertices of the path (traveling stops) are located in a continuous region or a discrete version with selected vertices from a discrete set of points. As a discrete model, the Traveling Circus Problem (Revelle and Laporte, 1993) can be given where the problem is faced by a circus making stops at a number of locations during a season to be always accessible by unvisited populations. In this context, two goals are considered: the cost of the path (length, number of vertices or bends) and the cost of coverage (maximum distance to the population). These cost functions can be considered as objective function or constraints to the following problems.

D

S

FIGURE 23.2 Bilevel transportation system.

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23.3.1 MINIMIZING

THE

COST (LENGTH-VERTICES) OF

THE

PATH

In this model, the goal is to determine a minimum length (vertices) path such that every demand point lies within a fixed distance «  0 from the path. In general, this problem is NP-hard (the hardest problems in the nondeterministic polynomial-time) as it reduces to a Traveling Salesman Problem (TSP) when the distance from the demand points to the route is 0. This is called a direct covering if the path is Hamiltonian, that is, the path passes through all points. An indirect covering, on the other hand, means the maximum distance between the path and the demand points is given by «. Direct covering: Direct covering problems are related to the TSP problem and its variants. This is a classical problem in combinatorial optimization and extensively studied even in continuous domains (Lawler et al., 1985). This problem is NP-hard even for points in the plane (Garey et al., 1976; Papadimitriou, 1977). Many heuristic and approximation algorithms have been proposed for direct covering problems (Arora, 1998). In the quota-driven TSP, each point to visit has an associated integral value, and the salesman has a given integer quota. The objective is, for this case, to find the shortest path with sum of the values for visited points (Awerbuch et al., 1995). The k-TSP takes an integer k as input and requires that the covering shortest path visits some subset of k demand points (Garg, 1996). Some recent studies considered the Max TSP where the objective is to maximize the length of the covering path. In Fekete (1998), the proof on the NP-hardness of the Max TSP is given for Euclidean spaces of three dimensions or higher. However, the complexity still remains open for two dimensions with the Euclidean distance. A polynomial-time algorithm can be done in metrics defined by a convex polytope in Rd for some fixed d (Barvinok et al., 1998). For instance, the algorithm requires O(n2 log n) time for the L1- and L1-metrics. In fact, the main idea here is to solve the problem by using a reduction to a transportation problem with a bounded number of customer locations in an appropriate bipartite graph. The TSP with neighborhoods is a natural generalization of the Euclidean TSP: given a set of k objects in the plane, called neighborhoods, find a shortest path that visits at least one point in each set of k neighborhoods. The neighborhoods can be connected (disks, polygonal regions, etc.) or disconnected sets (set of discrete points, set of polygons, . . .). Since this problem is a generalization of the TSP, it is by nature, NP-hard. The problem was first studied by Arkin and Hassin (1994) where they proposed O(1)-approximation algorithms for some kinds of neighborhoods with running time O(n þ k log k) where n is the total complexity of the k objects. Mata and Mitchell (1995) provided a general framework giving an O(log k) approximation algorithm with time complexity O(n5). This result was later improved to O(n2 log n)-time complexity (Gudmundsson and Levcopoulos, 1999) where a polynomial-time approximation scheme (PTAS) was given for the special case when the tour was ‘‘short’’ compared to the size of the neighborhoods. Recently, a polynomial time method that guaranties a constant factor approximation was devised by de Berg et al. (2005) for disjoint convex set of arbitrary size. Furthermore, in this study, it was proved that a PTAS did not exist for this problem unless P ¼ NP in

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contrast to the standard Euclidean TSP for which there is a PTAS allowing  one Oð1«Þ to compute for any given « > 0, a (1 þ «)-approximation in O n time (Arora, 1998). A related problem is the so-called Errand Scheduling Problem (Slavik, 1997) with a given collection set of points, and the aim is to visit at least one point from each set. The difference from the TSP with Neighborhoods is that each set to be visited has a finite number of points. An interesting version of this problem is the Trip Planning Problem (TPP) (Li and Cheng, 2004): Given a set of n points in the plane and k colors where k is a fixed constant, k  n such that each point is associated with one color, a given starting point s and a destination point d, find the shortest trip that starts at s, passes through at least one point from each color and ends at d. This problem is a generalization of the TSP, and it is NP-hard. Study of heterochromatic minimum substructures in a graph is also related to this topic (Brualdi and Hollingsworth, 2001; Suzuki, 2006; Li and Zhang, 2007). The TPP has its origins in many applications such as: route location (someone is traveling for one place in town to another visiting a bank, a supermarket, a newspaper stand, etc., on the way); advanced internet systems (maps of navigation, planning of tasks in Google, etc.) and computer networks (a computer network and a set of jobs such that each job should be executed only by a specific set of nodes in the network, then the objective is to find the shortest path that visits one node in each category). In a recent study by Díaz-Báñez et al. (2007), it was proved that the problem can be polynomially solved under some restrictions on the path with monotony and a prefixed order of the colors to visit. A polygonal chain is ordered if the visits are in different categories or colors according to the fixed order. Let P ¼ (p1, p2, . . . , pk) be the polygonal chain with vertices p1, p2, . . . , pk ordered by (1, 2, . . . , k). Let u 2 (0, p) be an angle that determines certain orientation. A polygonal chain P is monotone in this orientation if every line with orientation 0 þ p2 intersects P in a connected set. Similarly, it can be said that P is monotone respect to line l if the projections of the points p1, p2, . . . , pk on l are ordered according to the order (1, 2, . . . , k). The polygonal chain for which C is monotone if there exists an orientation u (or a line l(u)) (Figure 23.3). The property of monotony has been widely used in modeling transportation and robotic problems since somehow going back increases the costs to simplify the structure to be determined (Arkin et al., 1989; Díaz-Báñez et al., 2000).

I(0) (a)

I(q) (b)

FIGURE 23.3 (a) Solution for monotony in the x-direction and order (cross, square, disk, circle, box), (b) solution for monotony in other direction.

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The proposed algorithms in Díaz-Báñez et al. (2007) run in time O(n log2 n) and O(n3 log n) when the direction of monotony is fixed or free, respectively. The length-constraint orienteering problem has been considered in the operational research field: Given a set points S in the plane, a starting point s and a length constraint l0, one needs to find a tour starting at s that visits as many points of S as possible and of length not exceeding l0. Some related problems include the prizecollecting traveling salesman problem and the vehicle routing problem. They arise from real world applications such as delivering goods to locations or assigning technicians to maintenance service jobs. A substantial amount of work on heuristics for these problems can be found in the operations research literature (Toth and Vigo, 2002; Arora, 2003). More recently, Chen and Har-Peled (2006) presented a (1  «)1 approximation algorithm for the length-constraint problem that runs in (nO(«) ) time and visits at least (1  «)k* points of S where k* is the number of points visited by the optimal solution. The algorithm was also reported to work in higher dimensions. Indirect covering: Indirect covering path is a path that is not required to pass through the demand points. Some problems for indirect covering path have been studied both for discrete and continuous demand. In the distance-constraint orienteering problem or bank robber problem, the salesman is allowed to travel at most to a distance from the points to cover, and the goal is to maximize the number of visited points subject to distance restrictions. Since it is a generalization of TSP, the problem is clearly NP-hard. The first result for geometric instances was given by Arkin et al. (1998) where a 2-approximation algorithm was obtained. For a continuous demand, i.e., the set of point to cover is not discrete, but a continuous region there is a collection of problems was well studied in the computational geometry. In the lawn mowing and milling problems, the covering is given by motion of a disk of a fixed radius « in a region. The ‘‘cutter’’ can cross the region boundary (lawn mowing) or stay within the region (milling). Both problems are NP-hard and efficient constant-factor approximation algorithms for both problems were given by Arkin et al. (2000). In the transportation area, this problem can be viewed as a irrigation system. Other issues for this problem were discussed by Held (1991). Although the problem arises naturally in the area of automatic tool path generation for pocket machining, it can be considered a generalization of the geometric TSP with ‘‘mobile clients’’: Find the shortest path for a salesman visiting a given set of clients, each of which is willing to travel up to distance in order to meet the salesman. Arkin and Hassin (1994) gives a discrete version of this problem where a discrete set of points must be ‘‘mowed.’’ Another problem related to a particular indirect covering is the watchman route problem: Find a shortest possible path within a polygonal region R such that every point in R is seen by some point in the path. The Euclidean version of the watchman route has been extensively studied (Chin and Ntafos, 1991, 1998; Carlsson et al., 1999; Tan et al., 1999), and it is NP-hard (clearly, from Euclidean TSP). Recently, Arkin et al. (2003) studied the minimum-link watchman tour problem where the objective is to minimize the number of links in a polygonal route rather than its length. Another related problem is the Zookeeper’s problem where the goal is to find a shortest cycle in a simple polygon P (the zoo) through a given vertex

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(the zookeeper’s chair) (Chin and Ntafos, 1992; Jonsson, 2003) such that the cycle visits every set of k disjoint convex polygons (cages), each sharing an edge with P without entering any of the cages. This problem is a special case of the TSP with neighborhoods within a simple polygon. Finally, the Aquarium keeper’s problem by Czyzowicz et al. (1991) where the cycle (path) touches every edge of P is presented as an indirect covering problem.

23.3.2 MINMAX COVERING PROBLEMS The minmax problem, to compute a path P such that the maximum distance to the population is minimized, has as trivial solution for the path passing through all demand points. Thus, some constrained problems are considered to control the cost for building the route. With this criterion, the usual constraints on the route to be constructed arise from two factors. First, number of bends (vertices or corners) of the polygonal path plays an important role for transportation of heavy vehicles or when there is no room to maneuver. On the other hand, in a more general routing scenario, polygonal route length could be more important than the number of vertices. The Bend (Length)-Constrained Minmax Problem can be formulated as follows: min max (pm , P) s:t: P

pm 2S

v(p)  k(l(P)  l0 )

A great number of such problems have been solved from computational geometry viewpoint. In fact, a problem closely related to the search of bend-constrained minmax polygonal routes is the approximation of polygonal curves. In various situations and applications, images of a scene have to be represented at different resolutions. A topic studied in computational geometry and applied to approximate boundaries of complicated figures in cartography, pattern recognition and graphic design (Chin et al., 1992) is to approximate the piecewise linear curves by more simple ones. Among the research in this field (Guibas et al., 1993; Chan and Chin, 1996), the problem of approximating a given polygonal curve by another has been studied. In these studies, the vertices of the new curve were assumed to either have the same abscissas as given vertices in S, or they consisted of a subset of the vertex set of the original polygonal curve. Hence, if the problems for the transportation area were planned, a route with stops either on points with the same abscissas as the demand points or on the given population points should be searched. A general strategy was to divide the problem into subproblems and use one as a subroutine for the other: Min-« problem: Given «  0, find a polygonal route P with minimum number of vertices such that d(p, S) is not greater than «. Min-# problem: Given k, find a polygonal path P minimizing the distance d(p, S) among those with a number of vertices not greater than k. This type of problems admits several variants that arise while imposing constraints on location of the vertices or considering various types of distance. In order to remain focused on transportation, two approximation distances are considered,

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d2(P, S)maxpm2S ¼ d2(pm, P) and dv(P, S)maxpm2S ¼ dv(pm, P) when d2(pm, P) and dv(pm, P) are the distance from the point to polygonal path induced by Euclidean and vertical distances, respectively. From the facility location point of view, the Min-« problem with d2 and dv distances becomes an Euclidean minmax and a Fitting minmax problem, respectively. The idea of using the methods of polygonal approximation area to calculate an optimal route was first proposed by Díaz-Báñez (1998). The approach for solving the Vertices-Constrained Minmax Problem was first presented by Imai and Iri (1986) as follows: first generate a set G of candidate distance values in such a way that the polygonal path P* searched for one of the values as distance d(P, S). To each candidate value d in the set, minimal number of vertices of a polygonal curve P(d) that can be constructed with error at most d (the solution of a Min-# problem) can be associated. Finally, the smallest d* 2 G where associated length is not greater than k searched to get P* ¼ P(d*). The most efficient algorithms for Euclidean distance problems were devised by Chan and Chin (1996). They give an O(n2) and an O(n2logn)-time complexity algorithm for Min-# and Min-« problem, respectively. They further showed the solution of two problems in O(n) time if the points representing the population form a part of a convex polygon. Note that the Euclidean case was only solved when the vertices or bends of the new polygonal path were a subset of the original set of points. This is the called discrete k-bend polygonal route. On the other hand, the problem with respect to vertical distances appears in several and important disciplines as statistics, computer graphics or artificial intelligence. The k-bend constrained minmax problem with vertical distance was first posed by Hakimi and Schmeichel (1991) where they solved two variants of the problem: when P was required to have its vertices on points in S, the discrete problem, and when its bends can be on any point in the plane, it is called free problem. For both, they developed O(n2 log n) time algorithms which did not work in the presence of degeneracies, i.e., they did not admit points with the same x-coordinate. The approach was similar to the general method used in Imai and Iri (1986). With respect to the free problem by Wang et al. (1993), it was shown how to use a clever plane sweep procedure to find a best k-bend approximation under the Chebychev measure of error in O(n2) time. More recently, several applications of the parametric searching technique were proposed by Goodrich (1995) to solve the problem for any such k in O(n log n) time. However, in the context of facility location and transportation, the points of S (potential users) were to be determined in any position. For this reason, a dynamic programming procedure was applied in Díaz-Báñez et al. (1998) to remove nondegeneracy assumption in the discrete case. Besides, a nice observation made here was that the algorithm to solve the discrete k-bend constrained minmax problem could be adapted to find the length constrained minmax problem. Until now, minmax problems where a general polygonal chain should be determined were reviewed. Next, some optimization problems for polygonal chains with a particular configuration will be mentioned. Rectilinear path: A rectilinear path is a chain of consecutive orthogonal segments (links) such that the extreme segments are in fact half-lines with the same slope. A rectilinear path is monotone with respect to a given orientation a if every line with

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(b)

(a)

FIGURE 23.4 (a) Monotone rectilinear chain, and (b) monotone nonrectilinear chain.

slope tana þ p2 intersects the path either in a point or in a segment with slope tana þ p2 (Figure 23.4). The location of this special type of a polygonal chain was considered by Díaz-Báñez (1998). Generally, this kind of path appears in problems involving transportation routing design with applications such as floor planning, manufacturing environment design, robot moving, etc. In Díaz-Báñez and Mesa (2001), the minmax location of a monotone rectilinear route R was studied. The constraints were either the number of vertices v(R) or the length l(R), and the vertical distance was considered: minR maxpi 2S dv (pi , R) s:t: v(R)  (l(R)  l0 ). This problem was first solved in O(n2 log n) time by Díaz-Báñez and Mesa (2001). This result was further improved to O(n2) by Wang (2002) and to min {n2, nk log n} by López and Mayster (2006) where k was the number of line segments (links) forming the chain. Various approximation schemes were also presented in López and Mayster (2006). 2-links polygonal paths: This is a particular case of a very simple polygonal path. Suppose the whole population is required to split into two groups such that every group uses a part of the path. Thus, the path is reduced to a chain consisting of two links. The so-called double-ray center problem is defined as follows. Given the population set (n points S in the plane), a configuration needs to be found, C ¼ (O, r1, r2) consisting of a point O in the plane and two rays, r1, r2 emanating from O such that the Hausdorf distance from S to C is minimized (Figure 23.5). The Hausdorf distance

O

r2 r1

FIGURE 23.5 Double-ray with Euclidean distance.

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a

b

P1

P2

FIGURE 23.6 One-discrete problem with vertical distance.

from S to C is defined by: h(S, C) ¼ maxp2S min½ d2 (p, r1 ), d2 (p, r2 ) where d2(p, r) denotes the Euclidean distance between the point p and the ray r. The distance between p and r starting at point O is defined by the distance between p and the line l through r if the perpendicular line to l through p intersects r and the distance between O and p, otherwise. This problem was solved in time O(n3a(n) log2n) by Glozman et al. (1999) where a(n) was the inverse Ackermann function (Sharir and Agarwal, 1995). The authors applied the parametric search technique (Megiddo, 1983), a popular tool in the computational geometry field. The reader is refereed to Goodman and O’Rourke (1997) for a comprehensive survey of this technique. On the other hand, efficient algorithms for finding 2-links (1-bend) polygonal chains were developed when the vertical distance was used instead of the Euclidean distance. In the study by Díaz-Báñez et al. (2000), the restriction that the chain must start and end at specified anchor points s and t was considered, and it was shown that the algorithms can be extended to deal with nonanchored polygonal paths. They solved two variants of the problem: when C was required to have its corner on points in S, the discrete problem (Figure 23.6), and when the corner can be on any point in the plane, the free problem. An efficient O(n log n) time algorithm for both the 1-bend discrete and free minmax problem were proposed with any of them making degeneracy assumptions. Besides, the 1-bend discrete case can be solved within the same time bound as the problem in which s and t are not fixed but both satisfies a feasible set of linear constraints. Such problems lead to algorithms of quadratic complexity as shown by Díaz-Báñez (1998). However, by using more structure and suitable incremental updating, more efficient solutions are possible. Their procedures help determine the solutions of 1-bend polygonal chain problems with vertical distance in O(n log n) time providing O(n log2 n) versions of the algorithms using more elementary approaches and become easy to implement.

23.4 OBNOXIOUS PATHS PROBLEMS Management of hazardous material (hazmat) is an extremely complex issue environmental, engineering, economic, social and political concerns. There is a general agreement that shipment of these type of materials is sizable and growing. Thus, development of planning criteria for minimization of industrial risks requires application of efficient techniques. In all processes transforming raw materials into final products, by-products are also generated. Some of the by-products are dangerous and have to be removed to special locations to process. An immediate problem one may

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think of is that of finding routes as far as possible from the surrounding cities. Also, in food industries, transportation of dangerous products arises, for instance, in the management of chemical substances required in the production. Large quantities of hazmats are shipped on trucks and consequences of accidents are severe (Abkowitz and Cheng, 1998). Authorities at different levels take measures to mitigate the risks associated with transportation of hazmats (U.S. Department of Transportation, 1998). Minimization of risks for such problems has been extensively studied in a space network. CCPS (1995) gives an excellent resource for hazmat risk assessment. Batta and Chiu (1988), Boffey and Karkazis (1995), Erkut and Newmann (1995), and Erkut et al. (2005), on the other hand, present operational research studies for hazmat transportation in space networks. However, little progress has been reported in continuous spaces. Since accidents involving hazardous materials may occur during transportation, continuous context must be taken into account. In some situations, a risk reduction could pass through a re-definition of the transport system, and new better paths should be constructed. First of all, in a continuous space it is necessary to consider a restriction on the path, otherwise the path to the infinity can be removed. Two problems arise when considering constraints either on spacial situation or length of the polygonal curve: .

.

Region-constrained problem: given a polygonal region R containing the point set S to find a polygonal path P within R maximizing d(S, P), i.e., maxPR minPm 2S d(pm , P). Length-constrained problem: given a positive value l0, to find a polygonal path P with length-bound l0 maximizing d(S; P), i.e., maxP:l(P)l0 minPm 2S d(pm , P).

A related problem with former version was posed in Drezner and Wesolowsky (1989). This problem showed, in a given a polygonal region R, containing point set S, computing a polygonal path within P connecting two points s and t (or two segments of R) such that the minimum distance to S is maximal. The authors provided an approximate algorithm for calculating such an anchored obnoxious route. However, this problem could be easily solved by using the Voronoi diagram, V(S) of the point in S. Aurenhammer (1991) gave a comprehensive survey on Voronoi Diagrams. A simple exact approach that solves this problem was included in this chapter. Since the goal is achieved if the segment giving the minimum distance maintains the same distance to either point, an optimal route walking along the edges of V(S) can be found. Then, optimization problem immediately is reduced to a discrete graph problem: After labeling each edge of V(S) with its minimum distance to its two sites and adding the end points s and t as new vertices to V(S), a breadth first search from s will find a polygonal route to t in V(S) where the minimum label will be a maximum within O(n) time (Cormen et al., 2001). As a consequence, this problem can be solved exactly with a simple exact O(n log n)time algorithm. An approximate algorithm for the second version, the length-constrained problem was proposed recently in Díaz-Báñez et al. (2005). By using several

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S

t

FIGURE 23.7 Finding the shortest path among circles.

geometric techniques as Voronoi diagrams in the Laguerre geometry and shortest paths among obstacles, they have proposed a [O(n) þ O(log(«1))][O(n) þ O(kn)]time algorithm where « was the allowed error. This approach can be adapted to many situations with the idea to use a bisection method for the following decision problem: Place a circle of radius r at each site si, i ¼ 1, . . . , n (Figure 23.7) and compute the shortest path avoiding the circles. The desired path must go among the union of those circles if r is equal to the maximum of minimum distances. Next, compute the union of the arrangements of circles, and compute the contour of the union from that. Such union will be formed by many connected components (the number of components ranges from 1 to n), and the one that contains s and t should then be found. Finally, the shortest path P between s and t among the circular obstacles can be obtained. If l(P) is equal to the bound l0, then a solution of the problem where r is the maximum achieved is computed. These techniques can be useful when demand population is modeled by polygonal regions as well. Also, the problem can be addressed in a similar way for other distances, for example, induced by polyhedral metrics. In the particular case of the L1- or L1-metrics, efficient algorithms can be found (Díaz-Báñez et al., 2005). 2-links polygonal paths: As seen above, placement of an obnoxious path can be modeled by a polygonal chain with one corner (two links). Apart from a spacial constraint (as otherwise the route may be simply removed to infinity), a constraint on the length of the chain was considered in Díaz-Báñez and Hurtado (2006). They also added the restriction that the chain had to start and end at specified anchor points s and t corresponding to given origin and destination. The so-called maximin 1-corner polygonal chain problem is stated as follows: Given a set S of points in the plane and a positive value l0, find a 1-corner polygonal route, P with Euclidean length l(P)  l0, such that min d(pi , P) is maximized among all possible chains fulfilling the p 2S conditions. i Due the geometric nature of the problem, the resolution is addressed from the computational geometry point of view. A boomerang is the area swept by a disk with its center describing the 1-corner route. Thus, in a geometric setting, the problem asks to find the largest empty boomerang anchored at s and t. Let us observe that the restriction on the length implies that the vertex cannot be exterior to an ellipse with focus at s and t. This defines a continuous search space; however, a discrete set of candidate placements can be generated (Figure 23.8). The study by Díaz-Báñez and

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B1

B2 B3

FIGURE 23.8 Empty anchored boomerang and its three parts.

Hurtado (2006) proposes an O(n log n)-time algorithm for finding a 1-corner obnoxious polygonal chain whose length is exactly l0, and an O(n2)-time algorithm when the length of the optimal chain is at most the given bound l0. The placement of others empty geometric objects (circle, rectangle, unbounded rectangular strip, annulus, etc.) of ‘‘maximum measure’’ among a set points have been extensively studied (Cheng, 1996; Houle and Maciel, 1998; Mukhopadhyay and Rao, 2003; Nandy and Bhattacharya, 2003; and Díaz-Báñez et al., 2003). Motivation for computing optimal empty figures comes from a variety of practical problems such as robot manipulation (Houle and Maciel, 1998), computer-aided design (Nandy and Bhattacharya, 2003) and collision-free routing for transport objects through a set of obstacles (points) (Cheng, 1996). It is of particular interest to use empty corridors (Figure 23.9). An L-shaped corridor is the concatenation of two perpendicular links (a link is composed by two parallel rays and one line segment forming an unbounded trapezoid). The angle of a L-shaped corridor C is the angle determined by their rays. A widest empty L-shaped corridor can be calculated in O(n3) time and O(n3) space (Cheng, 1996) when the interior angle is fixed: a(C) ¼ p2 . More recently, Díaz-Báñez et al. (2006) relaxed the angle constraint and allowed a(C) to assume arbitrary values. The proposed algorithm computed a widest empty 1-corner corridor in O(n4 log n) time and O(n) space.

c e

P1

e a

e

e

P2

(a)

(b)

(c)

FIGURE 23.9 (a) Corridor, (b) L-corridor, (c) a-siphon, (d) silo.

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(d)

Another kind of empty corridor, called siphon, was recently addressed in Bereg et al. (2007). A siphon was the locus of points in the plane that were at distance w from 1-corner polygonal path P where w is called the siphon width. Notice that a siphon is the area swept by a disk with its center describing the 1-corner polygonal chain. An a-siphon is a siphon such that the interior angle a of P, called the siphon angle, is fixed. The widest empty siphon problem can be stated as follows: Given a set S of n points in the plane and a fixed value a, 0  a  p, compute the a-siphon with the largest width w such that no points of S lies in its interior. The widest a-siphon gives a ‘‘better’’ solution than the L-shaped corridor of Cheng in the following sense: Suppose that transporting a circular object (a disk) with radius w through a set P of obstacles (points) is of interest. Then, the decision problem becomes: Is there a 1-corner polygonal path for transporting the disk through P without collision? Notice that the noncollision property means disk center (vehicle for transportation) pass at distance at least w from the points (population). The algorithm for L-shaped corridor can produce a negative answer while the a-siphon’s method gives an affirmative answer since the width of the widest a-siphon is always larger than or equal to the width of the widest L-shaped corridor. In fact, every L-shaped corridor contains a siphon of the same width, and the reciprocal is false (Figure 23.10). In Bereg et al. (2007), three variants of the widest a-siphon problem were considered: (1) the widest oriented a-siphon problem where the angle a and orientation of one of the half-lines of 1-corner polygonal path P was known; (2) the widest arbitrarily oriented a-siphon problem where only the angle a was known; and (3) the widest anchored and arbitrarily oriented a-siphon problem where the corner of P was anchored at a given point. An efficient O(n log3n)-time

2e

FIGURE 23.10 Width of the widest a-siphon is larger or equal than the width of the widest L-corridor.

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algorithm for the first problem was given, and the arbitrarily oriented case was solved with an O(n3 log2n)-time algorithm. Finally, the problem of computing the widest anchored and arbitrarily-oriented siphon were solved within an optimal O(n log n)-time algorithm.

23.5 OTHER OPTIMAL PATHS 23.5.1 MULTIPLE OBJECTIVE PROBLEMS Due to the inherent multiobjective nature of many routing problems, there has been a tremendous increase in multiobjective optimization problems in networks (Figueira et al., 2005). For an interesting routing problem, see the minimumcovering=shortest-path in Current et al. (1994) where a two-objective path problem was addressed: minimization of total population was negatively impacted by the path and minimization of the total path length. Typically, many models are designed primarily to generate Pareto-optimal solutions. A path is called Pareto-optimal or efficient if no other path has a better value for one criterion without having a worse value for other criterion. For example, one may wish to compute a short polygonal path with few vertices. Note that a minimumvertices path may be far from optimal with respect to length; similarly, a shortest path may have many vertices while there exists a path connecting source and destination with few links (legs). Experimental studies suggest that average number of Paretooptimal is small in practice although this number can be exponential in theory. In a continuous domain, criteria of interest are length (Euclidean or other metrics), total number of vertices (or links) or distance between path and population. Multi-objective optimization problems tend to be difficult even for two criteria (Arkin et al., 1991): find a path in a polygonal domain whose length is at most l0, and total number of vertices is at most v0 is NP-hard; finding the shortest path constrained to have almost k links is a current open problem, and no exact solution is actually known. A comprehensible text by Ehrgott (2005) on multi-objective problems is also suggested for further reading.

23.5.2 OPTIMAL PATHS

IN

THREE DIMENSIONS

Computing optimal paths in higher dimensional geometric spaces is difficult, thus most effort has been devoted on three-dimensional spaces. In fact, unlike shortest paths among obstacles in the plane, shortest paths in a polyhedral domain do not need to lie on a discrete graph. It was shown in Bajaj (1988) that the algebraic numbers describing optimal path lengths might be exponential. Moreover, the number of combinatorially distinct shortest paths connecting two points may be exponential in the input. This fact was used in Canny and Reif (1987) to prove the NP-hardness of the three-dimensional shortest path problem. In light of the difficulty of a general problem, special cases of shortest paths that can be efficiently solved have been addressed. For instance, paths on a polyhedral terrain (a connected subset of the space where the boundary consists of a union of a finite number of triangles) makes the problem to be two-dimensional, and the continuous Dijkstra’s paradigm leads to

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O(n2)-time algorithms (Chen and Chan, 1996). Har-Peleg (1998) and Mitchell and Sharir (2004) cover the approximation algorithms on polyhedral domains for polynomial-time algorithms for some special cases. For a complete review of threedimensional shortest path problems, Mitchell (2000) is suggested.

23.5.3 ONLINE ALGORITHMS

FOR

OPTIMAL PATH PROBLEMS

The vast majority of published research on shortest paths algorithms deal with static environment, that is, the exact layout of the environment where the vehicle moves is known. The area of geographical positioning system (GPS) navigation proves to be a challenge for current planning engines, and dynamic route-planning systems are necessary. It subsumes several algorithmic issues from computational geometry in general and artificial intelligence search in particular such as autonomous robotics to gather and refine raw data by integrating different input sources as well as algorithms to build and query the graph and known novel search techniques to speed up the short path computations. The general goal is to find a navigation strategy that controls the motion while minimizing the objective function. There has been an increasing interest in dynamic management of transportation networks (Chabini, 1998). This results in a new family of shortest paths problems known as dynamic (time dependent) shortest paths problems. If the underlying space is a continuous environment, the problems are focused from the robotic applications point of view. The robot does not have a prior information about obstacles in the environment. In such cases, it has the information about their current location as well as the location of the goal, and the information about the environment is acquired on-line. Mitchell (2000) gives a complete survey on online algorithms in robotic. Another interesting issue is the class of algorithms that solve global problems and wireless networks by means of local algorithms. A local algorithm is the one in which any node of a network has only information on nodes at distance at most k from itself, for a constant k. Given a set of points S on the plane, the unit distance graph associated to S is the one with its vertex set consisting of the elements of S, two of which are connected if they are at distance at most one (Figure 23.11). Unit distance graphs are used to model various types of wireless networks, including

d=1

FIGURE 23.11

Unit distance graph.

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cellular networks, sensor networks, ad hoc networks and others where the nodes represent broadcast stations with a uniform broadcast range. A recent survey on local algorithms on unit distance networks is given by Urrutia (2007). This area provides a new set of computational problems for optimal paths in a continuous domain and promises future in the research on this kind of problems since overall information is not known in advance due to the temporal changes of the environment.

23.6 CONCLUSION In this chapter, main problems in the literature for various shortest paths problems in a continuous environment were described focusing primarily on transportation with theoretical results. There are numerous issues involved implementing the algorithms for real scenarios. Zhan (1997) gives useful information for researches and practitioners studying in network transportation area. It was concluded that, in a continuous domain, some algorithms may well be implementable and useful; however, in many cases, the implementation is too complex and may have numerous constants behind the asymptotical notation of complexity (big O notation). In addition, the algorithms may be not robust under uncertainties. Montemanni et al. (2004) presented a robust heuristical approach on networks. A future review should address the issues facing practitioners in the implementation of optimization algorithms for continuous shortest paths. As a conclusion, compared to shortest paths problem for transportation on networks, the literature on continuous versions, is very limited. The geometric shortest paths issue has been mostly studied for robotic mainly when the object moves inside a polygonal region. However, there is a vast set of problems to be addressed if a well studied problem on real road networks is considered in a continuous domain. This chapter also identifies potential research areas and weak points in the existing literature in the field of transportation, and it is expected to stimulate and suggest the development of applied research in the field of transportation regarding food products.

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24

Real-Time Nonlinear Optimal Control of Refrigeration Processes Ioan Cristian Trelea

CONTENTS 24.1 24.2 24.3 24.4

Introduction .............................................................................................. 524 Refrigeration Plant and Products ............................................................. 524 Dynamic Process Model .......................................................................... 525 Parameter and State Estimation ............................................................... 526 24.4.1 State Estimation Based on Core Product Temperature Measurement ........................................................ 526 24.4.2 State Estimation Based on Air Temperature Measurement ............................................................................. 528 24.5 Optimal Control Problem Formulation.................................................... 529 24.5.1 Control Vector Parameterization............................................... 529 24.5.2 Global Optimization Criterion .................................................. 530 24.5.3 Temperature Settling ................................................................. 530 24.5.4 Productivity ............................................................................... 531 24.5.5 Energy Consumption ................................................................ 531 24.5.6 Smoothing Out the Control Profile ........................................... 531 24.5.7 Preservation of Product Quality during Refrigeration .............. 532 24.5.8 Product Stability during Subsequent Storage and Process End Time............................................................... 532 24.5.9 Maximum Admissible Mass Loss............................................. 532 24.5.10 Admissible Range for the Air Temperature.............................. 533 24.5.11 Admissible Range for the Remaining Processing Time ........... 533 24.6 Online Optimal Control Experiments ...................................................... 533 24.6.1 State and Parameter Estimation................................................. 533 24.6.2 Real-Time Optimal Control: Typical Results and Parameter Selection ............................................................ 537 24.6.3 Robustness to Measured and Unmeasured Disturbances.......... 540 24.7 Conclusion ............................................................................................... 541 Acknowledgments................................................................................................. 542

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Nomenclature ........................................................................................................ 542 Greek Letters............................................................................................ 543 Subscripts ................................................................................................. 543 References ............................................................................................................. 543

24.1 INTRODUCTION Refrigeration is a widely used stabilization process in food industry. Fruits, vegetables, meat, and dairy undergo chilling or freezing in postharvest and postprocessing periods. Compared to other thermal stabilization processes, refrigeration preserves organoleptic, nutritional, and texture properties quite well. It is a relatively long and energy-consuming process. Moreover, improper temperature management during refrigeration may easily result in significant quality losses. The optimal control of a refrigeration process is thus expected to have significant economic impact, in terms of operation costs, productivity, and product quality preservation. In this chapter, economic objectives and product quality preservation constraints are introduced explicitly resulting in a nonlinear dynamic optimal problem with control and state constraints. The chapter describes a successful application of dynamic optimization for controlling a batch refrigeration process. The originality of this work resides in the use of nonlinear constrained optimization online. Traditionally, nonlinear dynamic optimizations of batch processes are performed off-line, and the precalculated optimal control or state profiles are then regarded as set points and applied without change by low-level plant controllers (Banga et al., 2003). This classical strategy has the advantage of minimizing the online computation load and risks, e.g., risk of divergence or convergence to local optima. It has an obvious disadvantage however: Precalculated control profiles are no longer optimal if disturbances occur, which is always the case in practice. The initial product state, properties, and ambient conditions generally differ from the ‘‘typical’’ ones assumed for off-line calculations. In contrast to this classical approach, in the case study presented here, constrained nonlinear dynamic optimization is periodically used online to reconsider the optimal control policy in the light of newly acquired measurements and of the estimated process state. To overcome the above mentioned difficulties and risks, the optimization algorithm was specifically designed to update the optimal control profiles by searching in a relatively small vicinity of previously successful profiles and to fall back to those profiles in case of numerical difficulties. Thus, the control policy actually applied can be no worse than the precalculated one. In general, however, it is significantly better as illustrated below in pilot-scale experiments.

24.2 REFRIGERATION PLANT AND PRODUCTS The refrigeration plant used in the experiments consisted of a wind tunnel with preconditioned air circulated in closed loop (Alvarez and Trystram, 1995). Air temperature was controlled by suitable heating and cooling devices and velocity by a variable speed fan. Air humidity was measured but not controlled. Process measurements included air temperature, velocity, and humidity as well as product temperatures at critical locations as detailed below. Data logging every 30 s was performed on a personal computer. Numerical calculations for process state estimation

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and online optimal control policy updating were performed by a second computer which was also connected to the process. Numerical calculations were performed with Matlab (The MathWorks Inc., Natick, Massachusetts) equipped with Optimization Toolbox extension and Simulink nonlinear dynamic systems simulator. Routines for communication with the process were written in QuickBasic. Refrigerated spherical fruits (75 mm in diameter) were conditioned in a pallet consisting of 3
5 industrial bins (160
400
600 mm each). The air flow (1 m=s) crossed the bins along the longest edge. Each bin contained 2
5
7 spheres. The bin in the center of the stack was studied and contained real product. The other bins were filled with plastic spheres of the same diameter. Real fruits used in experiments were apples, oranges, and tomatoes calibrated to the indicated diameter. In some cases, fruits were replaced by carrageenan gel spheres to verify repeatability of the experiments. Inside the bins, two critical points, a coldest point and a hottest point were identified in previous experiments (Alvarez and Trystram, 1995). The coldest point has the highest heat transfer coefficient (60 W m2K1), and is situated on the front edge of the bin. The hottest point has the lowest heat transfer coefficient (33 W m2K1), and is situated inside the bin in the center of the 5th range of fruit out of 7. Fruits situated at critical points were equipped with two thermocouples located on the surface at the stationary air flow point and in the center of the fruit. Temperatures at these two critical points were considered in the control problem formulation. The interested reader may consult Trelea et al. (1998) for more details on the experimental setting.

24.3 DYNAMIC PROCESS MODEL Iterative calculation of optimal control strategies requires repeated use of a predictive dynamic process model. Such a model is subject to contradictory requirements. On one hand, it should represent the process with sufficient level of detail and accuracy to predict the evolution of the relevant variables subject to the control policy being tested. On the other hand, it should be simple enough to allow a large number of iterations typically required by optimization algorithms to be performed in reasonable time. This last requirement is particularly stringent for online calculations which should be much faster than the dynamics of the physical process. In the considered case, a typical refrigeration experiment lasts for approximately 2 h. Optimal control policy should thus be recomputed with a period of at most several minutes which gives a maximum order of magnitude of one second for every single model simulation. The various choices and simplifications concerning the refrigeration model adopted should be interpreted with this constraint in mind. Heat and mass transfer during the refrigeration process are classically described by the Fourier and Fick’s laws, respectively, with suitable boundary conditions at the product-air interface. Boundary conditions, in turn, depend on local air flow pattern in the vicinity of each fruit which could be in principle obtained by solving Navier– Stokes equations. Modeling three-dimensional air flow, heat, and mass transfer was out of question for the given simulation time reasons. The following simplifications were adopted and validated by local temperature and mass loss measurements. First, instead of modeling the air flow pattern numerically, experimental measurement of heat transfer coefficients were performed inside the bin at each fruit

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location (Alvarez and Trystram, 1995). A common variation factor with the average air velocity was adopted for all heat transfer coefficients based on usual relationships between dimensionless numbers (Reynolds and Nusselt numbers). Second, spherical symmetry of each individual fruit was assumed reducing three-dimensional expressions for heat transfer to one dimension. The Fourier law in spherical coordinates was spatially discretized in a small (n ¼ 10) number of finite-dimensional ‘‘slices’’ with the finite difference method. For mass transfer, the simplification was even greater since moisture gradient inside fruit was neglected based on an estimation of the Biot number. Third, instead of modeling all 70 fruits in a bin, only fruits situated at the two critical points mentioned previously were actually considered. Full details on the developed model are given in Trelea et al. (1998). With these simplifications, the dynamic process model contained a total of 2(n þ 1) ¼ 22 state variables: n state variables for the temperature distribution and 1 state variable for the water content inside each fruit situated at a critical point in the bin. In other words, model simulation required the integration of 22 coupled nonlinear ordinary differential equations which can be easily handled on a personal computer with adequate simulation software (Matlab=Simulink in this case).

24.4 PARAMETER AND STATE ESTIMATION In actual refrigeration runs, discrepancies between model predictions and process measurements appear due to unavoidable model simplifications, imperfectly known product properties, and air flow pattern. In case of industrial processing, actual fruit size and shape are not known precisely and various unmeasured disturbances occur. To limit the effect of these disturbances and poorly known properties on the model predictions on the control policy calculation, a parameter and state estimation step were designed. Before each periodic online control optimization cycle, recent measurements are used in conjunction with the model to reconstruct unmeasured state, i.e., temperature distribution and moisture content inside the fruit. Simultaneously, some of the model parameters may be estimated from these measurements. The purpose is to introduce feedback: update the model and avoid systematic prediction errors in the control optimization step in case of consistent over- or underestimation of the transfer.

24.4.1 STATE ESTIMATION BASED MEASUREMENT

ON

CORE PRODUCT TEMPERATURE

The measurements selected for the state and parameter estimation are core temperatures for the two fruits located at the critical points. The surface temperatures can thus be used for independent validation of the estimation procedure. After some tests and a sensitivity analysis, a relative heat transfer coefficient was selected as the estimated model parameter. This multiplicative correction coefficient, with unitary nominal value, is applied to all fruits in the pallet. It was found to account quite well for systematically lower or higher cooling rates due to various disturbances studied below. The simultaneous parameter and state estimation was performed according to the so-called receding horizon procedure (Boillereaux and Flaus, 1995). The ‘‘best’’ value of the estimated parameter (relative heat transfer coefficient, k) is the one that

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TABLE 24.1 Product Properties and Experimental Conditions Product Mass (kg) Density (kg m3) Heat capacity (J-kg1 K1) Thermal conductivity (W-m1 K1) Lowest surface temperature Tsc lim (8C) Security margin Tsec (8C) Highest core temperature Tch lim (8C) Sampling time (min) Minimum estimation horizon (min) Maximum estimation horizon (min)

Apples

Oranges

Tomatoes

Carrageenan Gel

0.185 . . . 0.225 754 3800

0.215 . . . 0.225 948 3770

0.195 . . . 0.215 1028 3980

0.205 . . . 0.235 1013 4100

0.40

0.50

0.70

0.95

3

1.5

10

0

1 7

1 5

1 13

1 4

5 15

8 32

8 32

5 15

35

56

56

35

Source: Reprinted from Trelea, I.C. et al., J. Food Process Eng., 21, 1, 1998. With permission.

minimizes the gap between the measured temperatures (at the core of the coldest fruit, Tcc, and at the core of the hottest fruit, Tch) and their respective values predicted by the model. The minimization is performed over the recent measurements available in a specified time window (Table 24.1) called estimation horizon. The gap between the measured and the calculated temperature at the core of the coldest fruit is expressed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X  u Tcc (t )  Tcc calc (t , k) 2 t (k ) ¼ E1 sTcc t 2[ts tp ]

(24:1)

where ts is the starting estimation moment tp is the present moment Tcc(t) is the measured core temperature of the coldest fruit at time t Tcc calc(t, k) is the coldest core temperature calculated by the model with the adjustable parameter set to k sTcc is the standard deviation for the measurement of Tcc (Table 24.2) Small values of the measurement standard deviation give high weight in matching Tcc. The gap between measured and calculated temperatures at the core of the hottest fruit (E2) was expressed in a similar way replacing Tcc by Tch as appropriate. It may happen that the mismatch between measured and calculated core temperatures is due to other causes and cannot be explained by the relative heat transfer

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TABLE 24.2 Control and Estimation Algorithm Settings Conditions Weight

W0

W1

W2

W3

W4

W5

W6

W7

Value

a

0

0.1

0.05

5

10

10

40

10

Weight

W8

W9

W10

Wk

sT cc

sT ch

kmin

kmax

Value

0.1

10

1

20

0.1

0.1

0.33

3

Source: Reprinted from Trelea, I.C. et al., J. Food Process Eng., 21, 1, 1998. With permission. a Except in Figure 24.2b, where W0 ¼ 5.

coefficient. In this case, minimization of E1 and E2 may lead to inadmissible values of the estimated parameter and absurd predictions. The following term prevents this by keeping the estimated parameter between the bounds kmin and kmax: E3 (k) ¼ Wk max{0, kmin  k, k  kmax }

(24:2)

The weight Wk is selected to keep this term dominant if the bounds tend to be violated (Table 24.2). Finally, the receding horizon estimation of the relative heat transfer coefficient is given by   k ¼ arg min E12 þ E22 þ E32

(24:3)

With this value of the relative heat transfer coefficient, model simulations give an estimated process state at the end of the estimation time window. This constitutes the initial state for the control optimization step.

24.4.2 STATE ESTIMATION BASED

ON

AIR TEMPERATURE MEASUREMENT

In real-life applications, core fruit temperature measurement is quite impractical. A much more convenient measurement would be air temperature before and after the product stack. Considering appropriate product-air heat balances, the heat transfer model can be extended to predict air temperature. The state estimation procedure can thus be modified to use air temperature instead of product temperature. Experiments showed, however, that relative heat transfer coefficient estimations based on air temperature are unreliable at least with the considered experimental setting. In order to be useful for estimation, the considered measurement (here, air temperature after the product stack) has to be as sensitive as possible to the estimated model parameter (here, the heat transfer coefficient). It turns out that, at the beginning of the batch, high transfer coefficients lead to large heat fluxes and hence to high air temperatures (58C–158C higher than the inlet air temperature). During this period (10%–20% of the total batch time) estimation of the heat transfer coefficient is possible. Eventually, however, high heat fluxes lead to high cooling rates, which in turn lead to low product temperatures and low air temperatures (less than 58C above

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the inlet air temperature). On the basis of air temperature alone, it is thus virtually impossible to distinguish this case from the opposite situation where the heat transfer coefficient would be low and the product temperature remain higher for a longer time. The situation is worsened by the unsteady air flow through the pallet, leading to 18C or higher fluctuations of the measured air temperature. Mathematically, this translates into unreliable heat transfer coefficient estimations after 20% of the batch duration. This attempt illustrates one of the most common pitfalls in using simultaneous parameter and state estimation. The available measurements might carry insufficient information to reliably estimate the desired model parameters and state. Numerically, this can be checked by assessing the sensitivity of the model predictions with respect to the estimated parameters. A reasonable change of this parameter (i.e., in the usual range) should induce significant change in the predicted values for the measured variables (i.e., significantly higher than measurement accuracy). Often, however, physical considerations can readily indicate which estimations are possible based on which measurements with reasonable signal to noise ratio. In the considered refrigeration process, it is expected that in industrial processing conditions the air flow would cross a higher amount of product that at pilot scale, the air heating would be higher and estimation would remain feasible for a longer time. One might also decide that after the beginning of the processing, the determined value of the heat transfer coefficient might be kept constant and used in subsequent predictions without change. The results presented below are all performed at pilot scale and use core temperature measurements for parameter and state estimation.

24.5 OPTIMAL CONTROL PROBLEM FORMULATION The calculated control profile (air temperature evolution in time) is the result of a compromise between various technical, economical, and product quality requirements. These requirements are introduced via associated components of the global optimization cost function. The weight of each component is selected to reflect the desired compromise.

24.5.1 CONTROL VECTOR PARAMETERIZATION Calculation of an air temperature profile in time is an infinite-dimensional optimization problem. In order to make it tractable in real time, the temperature profile was parameterized by a finite and relatively small number of real variables included in a control vector. The control vector parameterization method was selected based on some practical considerations. To be realized online, the temperature profile has to be smooth. It is also desirable to make temperature values and the remaining processing time appear explicitly in order to check the validity of the profile directly. The adopted solution consists in taking the control profile parameters as the remaining refrigeration time and the temperature values at several equally spaced moments (1–5, depending on the previously estimated remaining time): U ¼ [t f

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Ta1

Ta2



TaN ]T

(24:4)

The temperature variation between these moments is linear. During preliminary offline calculations, the initial air temperature can be optimized, but once the experiment has started, only future values can be manipulated.

24.5.2 GLOBAL OPTIMIZATION CRITERION As stated below, the control optimization problem is a nonlinear one with inequality control and state constraints. The various routines in the available numeric optimization library (Optimization Toolbox in Matlab) were extensively tested in simulation. It was found that the most reliable is the one designed for minimization of nonlinear sums of squares based on a variant of the Levenberg–Marquardt algorithm (Moore, 1977). The cost function for control optimization is thus somewhat artificially expressed as sum of squares, and the constraints are introduced as penalties of the form max {0, . . . }. The optimal control was calculated by minimizing numerically a composite criterion of the form: U ¼ arg min

10 X

Ci2

(24:5)

i¼0

The various components (Ci) of this criterion are detailed below. Introducing constraints as quadratic penalties has several drawbacks. For example, the minimum is achieved when active constraints are, even slightly, violated (Gill et al., 1981). This violation can be reduced by increasing the weight of the constraint, but this in turn degrades numerical conditioning. Critical constraints thus need some additional security margins to prevent any violation. Furthermore, constraint penalties introduce discontinuities in the second derivatives of the cost function which may cause numerical difficulties for algorithms based on second derivatives (e.g., quasi-Newton). The alternative to introducing constraints as penalties in the objective function would be to use true constrained optimization. In the particular software package considered here, the constrained optimization routines exhibited occasional numerical problems making them unsuitable for online use. The situation might be different, however, with different software packages. The optimization problem formulation as a sum of squares was finally retained for its robustness. Moderate accuracy in the optimum location and very slight constraint violation were considered minor drawbacks compared to the advantage of a reliable convergence after a very small number of iterations (10) essential for online operation.

24.5.3 TEMPERATURE SETTLING To minimize time waste and energy consumption between successive batches, it might be desirable to begin processing at a temperature close to the end of the previous batch. This is expressed as C0 ¼ W0 (Ta f  Ta 0 )

ß 2008 by Taylor & Francis Group, LLC.

(24:6)

This component is only useful during the preliminary off-line optimization. After the batch start, the weight W0 should be set to zero, since it is no point in trying to drive Ta f towards a possibly disturbed value.

24.5.4 PRODUCTIVITY A basic economic requirement is to maximize the process throughput by reducing the total processing time: C1 ¼ W1 (tf  tp )

(24:7)

During online operation, tf – tp represents the remaining time up to the end of the batch.

24.5.5 ENERGY CONSUMPTION Another component of production cost is energy consumption. In traditional processing, air temperature is kept constant at the lowest admissible value for a given fruit type, noted Tsc lim. What is actually penalized here is additional energy necessary for cooling the air beyond this value: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðtf u u C2 ¼ W2 t (Ta set (t)  Tsc lim )2 dt

(24:8)

tp

24.5.6 SMOOTHING OUT

THE

CONTROL PROFILE

Smooth control profiles are usually desired in practice. In traditional predictive control, a control move suppressing term is added to the cost function (Garcia et al., 1989) penalizing the control profile derivatives: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðtf  2 u dT u a set (t) dt C3 ¼ W3 t dt

(24:9)

tp

The additional terms C4 and C5 penalize the second and the third derivatives respectively, that is, high curvature and zigzagging: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðtf  2 u d2 T u a set (t) dt C4 ¼ W4 t dt 2

(24:10)

tp

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðtf  2 u d3 T u a set (t) dt C5 ¼ W5 t dt 3 tp

ß 2008 by Taylor & Francis Group, LLC.

(24:11)

It was found that these terms are extremely useful in helping the optimization algorithm converging to the global minimum which is physically expected to lie on a smooth control trajectory. The weights are selected to make their contribution negligible for smooth trajectories but become large for irregular temperature profiles that occasionally appear during numerical optimization. Such irregular profiles usually correspond to local optima in which the optimization can easily be trapped in the absence of smoothing terms.

24.5.7 PRESERVATION

OF

PRODUCT QUALITY

DURING

REFRIGERATION

In the course of the refrigeration process, product temperature must always remain above a given limiting value (Tsc lim) depending on the fruit sort (Table 24.1). In the considered experimental setting, the lowest product temperature is at the fruit surface situated at the coldest point (Tsc). The minimum product temperature constrained was thus introduced in the following way:   (24:12) C6 ¼ W6 max 0, Tsc lim þ Tsec  min Tsc (t) t2(tp ,tf )

Here Tsec is a security margin (Table 24.1) introduced to prevent any violation of this mostly important quality constraint. Indeed, C6 will not become dominant with respect to other cost function components until the constraint is not (even slightly) violated. The weight W6 is chosen dominant with respect to other weights, but a too large value was found to cause numerical difficulties during optimization.

24.5.8 PRODUCT STABILITY DURING SUBSEQUENT STORAGE AND PROCESS END TIME The refrigeration process continues until the product temperature at any point goes below a maximum admissible value (Tch lim) specific to each fruit sort (Table 24.1). This condition insures product preservation during subsequent storage and actually defines the final process time (tf). The highest product temperature occurs at the center of the fruit situated at the hottest point (Tch). Hence the product preservation constraint is C7 ¼ W7 max{0, Tch (tf )  Tch lim }

(24:13)

A security margin is not necessary in this case because this constraint is enforced during run time: refrigeration ends only when it is satisfied. The weight W7 is selected to make the component C7 dominant with respect to economic objectives.

24.5.9 MAXIMUM ADMISSIBLE MASS LOSS To preserve commercial value of fruit, the relative mass loss (Lm) must remain below a specified limit (Lm lim). The component C8 was introduced to express this requirement: C8 ¼ W8 max {0, Lm (tf )  Lm lim }

ß 2008 by Taylor & Francis Group, LLC.

(24:14)

24.5.10 ADMISSIBLE RANGE

FOR THE

AIR TEMPERATURE

Calculated temperature profiles must include values that can actually be attained on the considered refrigeration plant. The component C9 penalizes air temperature values (Ta) that lie outside a specified range (Ta min, Ta max), depending on the physical equipment used: ( ) C9 ¼ W9 max 0, sup Ta min  Ta (t), sup Ta (t)  Ta max t2[tp tf ]

24.5.11 ADMISSIBLE RANGE

FOR THE

t2[tp tf ]

(24:15)

REMAINING PROCESSING TIME

The refrigeration end time is one of the optimized process variables. The component C10 indicates to the optimization algorithm a valid range for the remaining refrigeration time (tf): C10 ¼ W10 max{0, t f min þ tp  tf , tf  tp  t f max }

(24:16)

The lower bound tf min is taken as two sampling times. Prediction is thus performed at least two sampling time intervals ahead even towards the end of the batch. The actual stopping decision is taken based on the measured temperature regardless of the predicted end time. The upper bound tf max is selected to avoid unnecessarily long predictions during the optimization phase but large enough to accommodate all reasonable refrigeration experiments.

24.6 ONLINE OPTIMAL CONTROL EXPERIMENTS The described optimal control algorithm was applied in real time in a series of refrigeration experiments. The discussion of these results begins with the state estimation part of the algorithm. The core of the real-time optimization algorithm and its tuning, via weight selection, is presented next. Finally, the behavior of the algorithm in face of some typical measured and unmeasured process disturbances is analyzed.

24.6.1 STATE AND PARAMETER ESTIMATION As previously mentioned, some of the available process measurements, namely the core temperatures of the two fruits located at the critical points are used for feedback. The parameter estimation algorithm uses these measurements to compute corrections to one of the model parameters, here the heat transfer coefficient, to properly reproduce the observed cooling rate. At the same time, an estimation of the full process state is obtained, i.e., the temperature distribution inside the fruit and the mass loss. It should be emphasized that the parameter and state estimation take place simultaneously. Iterative model simulations with different values of the heat transfer coefficient are performed until calculated core temperature evolutions best fit the measured ones. The temperature distribution inside fruit and the mass loss are

ß 2008 by Taylor & Francis Group, LLC.

byproducts of these simulations. These values are indispensable starting points for the subsequent control optimization process. Additional process measurements, namely the surface temperatures of the fruits located at the critical points, are used as independent validations of the state estimation process. The parameter and state estimation algorithm does not see these measurements. They only serve for a posteriori assessment of the state estimation quality. Moreover, actual fruit mass loss is measured at the end of the batch giving indications on the validity of the mass loss calculation in the model. Three typical experimental results for the state estimation are shown in Figure 24.1, for three sorts of fruits: apples, oranges, and tomatoes. It is recalled that what is actually estimated is a correction to the precomputed heat transfer coefficient. This correction is relative to the value of the heat transfer coefficient obtained from usual relationships between dimensionless Reynolds and Nusselt numbers and is applied to all fruits in the pallet. The nominal value of this relative heat transfer coefficient is thus unity. Higher than unity values indicate higher cooling rates than the ones predicted by theory, and vice versa. It appears from Figure 24.1 that the estimated relative heat transfer coefficient values strongly depend on the considered fruit sort. The most obvious deviations from unity occur for apples (Figure 24.1a). The experiments were repeated several times and this kind of behavior was systematically observed. During the first 15 min of the experiment, no parameter and state estimation is performed. This is required Oranges

Apples °C

°C

Temperatures

40 30

40 30

Tch

20 Tch lim

20

Tch lim

0 T sc lim

Ta 0

−10 40

60

80

100

0

20

40

60

Tsc

10 Ta

0

Time (min)

Estimated relative heat transfer coefficient

(a)

Tch

80 100

Time (min)

Tsc lim

Time (min)

Estimated relative heat transfer coefficient 3 2 1 0

Tch lim

Ta −10 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

Time (min)

3 2 1 0

40

20 Tsc

10

T sc lim

0

Temperatures

30

Tsc 10

Tomatoes °C

Tch

20

−10

Temperatures

0 20 40 60 80 100 120 140

(b)

Time (min)

Estimated relative heat transfer coefficient 3 2 1 0

0 20 40 60 80 100 120 140

(c)

Time (min)

FIGURE 24.1 Real-time optimal control results: (a) apples, (b) oranges, and (c) tomatoes. Temperatures: solid, measured; dotted, simulated; dash-dotted, constraints. Estimated relative heat transfer coefficient: solid, nominal value; dash-dotted, constraints; circles, estimated values. (Reprinted from Trelea, I.C. et al., J. Food Process Eng., 21, 1, 1998. With permission.)

ß 2008 by Taylor & Francis Group, LLC.

because the core fruit temperature does not evolve significantly at the very beginning of the experiment, and using these measurements for estimation would be irrelevant. The relative heat transfer coefficient is fixed to its nominal unitary value during this time. The first two estimations of the relative heat transfer coefficient, at 20 and 25 min, are unrealistically high. They are equal to three times the nominal value and correspond to the imposed bound (Table 24.2). This effect is produced by unexpectedly high core cooling rates compared to the model predictions. The parameter estimation algorithm ‘‘explains’’ these abnormally low core temperatures by the only available degree of freedom, namely the heat transfer coefficient. The most likely physical explanation, though, is the deviation from the sphericity hypothesis adopted in the model. The particular shape of the apples induces preferential heat transfer through the hollow near the stem to the temperature sensors located in the center of the fruit. The initial overestimation of the heat transfer coefficient is much smaller or inexistent for essentially spherical fruits such as oranges and tomatoes (Figure 24.1b and c). In the rare cases when this happens for spherical fruits or even carrageenan spheres, one might invoke slightly eccentrically placed sensors or some initial cooling of the fruits during bin manipulation between the oven and the refrigeration tunnel. The consequence of the overestimated heat transfer coefficient (Figure 24.1a) is the underestimation of the fruit surface temperature. Underestimated surface temperature induces false constraint violation and forces the control algorithm to apply higher air temperatures, unnecessarily increasing the refrigeration time. Several solutions may be considered for this problem. Dropping the sphericity hypothesis for a more realistic description of the apple shape would significantly complicate the heat transfer model. This would strongly increase the simulation time which is not feasible for online optimal control. More realistically, the adjustable model parameter could be constrained to a narrower range or the length of the estimation time horizon could be increased. This last solution was actually applied in some experiments with apples, one of which is shown in Figure 24.2a. The estimation starts at 32 min instead of 15 min and the maximum length of the estimation horizon grows up to 56 min, instead of 35 min (Table 24.1). The averaging effect of a longer estimation horizon has a favorable consequence on the accuracy of the surface temperature estimation, even if some underestimation still occurs during the first 30 min of the run. Moderate underestimation is not a major drawback, however, since the surface temperature constraint is not active at the beginning of the batch. Later on, when the constraint becomes limiting, surface temperature estimation is quite good. It is also apparent in Figure 24.1 that in the second half of the batch, the estimated relative heat transfer coefficient is lower than unity. This kind of behavior was observed systematically. It means that during this part of the process, the actual heat transfer rate is lower than predicted by theory. The very good estimation of both inner and outer fruit temperature during this period supports the fact that this heat transfer rate corresponds to reality. An explanation of lower than expected heat transfer rate may be related to the air flow pattern inside the pallet. Preferential air pathways are likely to exist around the pallet, air flow rate in contact with fruit being in fact lower than the total flow rate measured in the wind tunnel.

ß 2008 by Taylor & Francis Group, LLC.

°C 40

Apples

Tch

30 20

30

Tsc

Tsc

Tch lim

10

Tsc lim

Tch lim

0

Ta

−10

Tsc lim

Ta

−10 0

20

(a)

40

60

80

100 120

0

20

(b)

Time (min)

40

60

80

100 120

Time (min)

°C 40

Carrageenan gel

°C 40

Carrageenan gel

30

30 Tch

20

Tsc 10

Tch lim

Tsc lim

0

0

20

Tch lim Tsc lim

0

Ta

−10

Tch

20

Tsc

10

(c)

Tch

20

10 0

Carrageenan gel

°C 40

Ta

−10 40

60

80

Time (min)

0

100 120 (d)

20

40

60

80

100 120

Time (min)

FIGURE 24.2 Refrigeration experiments in presence of disturbances: (a) Model mismatch: apples with orange model (parametric disturbance), (b) Refrigeration device failure (measured disturbance), W0 ¼ 5, (c) Ventilation device failure with measured air velocity (measured disturbance), (d) Ventilation device failure with unmeasured air velocity (unmeasured disturbance). Temperatures: solid, measured; dotted, simulated; dash-dotted, constraints. (Reprinted from Trelea, I.C. et al., J. Food Process Eng., 21, 1, 1998. With permission.)

To close the analysis of the state and parameter estimation results, let us now turn to the mass loss. Recall that to compute the mass losses, the mass transfer model adopted the usual simplifying assumption of unity water activity at the surface of the fruit. As one might have expected, this assumption turned out to be reasonable for carrageenan spheres but certainly not for the real fruit used in the experiments (apples, oranges, and tomatoes). Indeed, the model based on unity water activity predicts about 2.5% mass loss, while the measured value for the real fruit is about 0.01% for the duration of the refrigeration experiment. This is obviously due to the impermeable peel of the considered fruit. Two solutions to this problem were tested with comparable results.

ß 2008 by Taylor & Francis Group, LLC.

The first solution, tested for tomatoes (Figure 24.1c) was to set the mass transfer coefficient to a small ad hoc value to account for the presence of the peel and for the experimentally observed mass loss. The surface temperature appears overestimated during a significant part of the batch. To explain this, one may consider the fact that the tomato peel is damaged in the vicinity of the temperature sensor. The thermocouple is thus in contact with the wet pulp whose temperature is lower than the major part of the fruit surface due to local water evaporation. Anyway, with the presence of the peel, the mass loss constraint was far from being limiting. One might thus consider dropping it altogether which leads to the second solution presented below. The second solution was to set the mass transfer coefficient at the surface of the fruit to be exactly zero. This was tested in experiments with oranges (Figure 24.1b) with quite good results. Of course, the predicted mass loss becomes zero and the mass loss constraint irrelevant. An important benefit is obtained in terms of model simulation time: The surface temperature can be computed explicitly rather than iteratively, and the model simulations run about two times faster. This in turn allows more optimization iterations to be performed in a sampling period and shorter sampling times. The optimal control algorithm performs more accurate optimizations and reacts faster to possible process disturbances. The surface temperature predictions are quite accurate since negligible water evaporation actually occurs with oranges neither in the vicinity of the temperature sensor nor on the rest of the (intact) fruit surface. It should be noted that setting the mass transfer coefficient to zero in experiments with carrageenan gel spheres would have been inappropriate. Significant water evaporation really occurs producing mass loss and lowering the surface temperature. In summary, it appears from the performed experiments, some of which are shown in Figure 24.1, that the receding horizon state and parameter estimation procedure gives quite satisfactory results for the purpose it was designed for: Providing updated state and parameter values for the main dynamic optimization step of the control algorithm.

24.6.2 REAL-TIME OPTIMAL CONTROL: TYPICAL RESULTS AND PARAMETER SELECTION The behavior of the optimal control algorithm is dictated by the relative importance given to the various criteria and constraints expressed by the user and introduced in the control problem formulation (C0–C10). This relative importance is in turn given by the numerical values of the associated weights (W0–W10) given in Table 24.2. It should be emphasized that in principle, only the relative magnitude of the weights is important, but weights that are globally too low or too high cause numerical difficulties. An example of weight selection procedure is described next. Of course, if the desired optimal control features are different, the most appropriate settings will be different. One of the weights has to be selected arbitrarily. For example, the weight W1 is selected to give a value of order of 100 to the main cost function component (here, C1). This component aims at shortening the refrigeration cycle duration which is the

ß 2008 by Taylor & Francis Group, LLC.

main goal in this study. The selection of the other weights requires some preliminary simulation studies. All these weights (except W1) are initially set to zero and increased as necessary to obtain the desired properties of the optimal solution. If the temperature settling weight (W0) is set to zero, the optimal air temperature profile begins at the lowest possible values compatible with the other requirements and is gradually increased to satisfy the surface temperature constraint (Figure 24.1). Indeed, the fruits are initially hot and the heat flux at the beginning of the batch is sufficient to maintain the surface temperature higher than the allowed lower limit (Tsc lim). Lowering the initial air temperature as much as possible obviously reduces the refrigeration time. On the contrary, if the temperature settling criterion is important, selecting a nonzero value for W0 drives initial air temperature towards the final one. The final air temperature is in turn generally close to Tsc lim, since this is the lowest value compatible with the surface temperature constraint towards the end of the batch (fruit temperature close to the air temperature). Even if W0 6¼ 0, the control algorithm lowers the air temperature in the beginning of the batch in order to minimize the refrigeration time before increasing it later. Such a V-shaped optimal air temperature profile is characteristic of cases with W0 6¼ 0 and results in longer (5% to 10%) refrigeration times than with W0 ¼ 0. Increasing the energy saving weight (W2) tends to drive the air temperature towards its constant traditional value Tsc lim, since the cost function component C2 penalizes additional temperature decrease beyond this limit. In the limiting case when W2 is dominant with respect to W1, refrigeration takes place at constant temperature. In the shown results (Figure 24.1), the energy saving is a secondary objective: W2 is selected to make the contribution of the energy saving (C2) relatively small compared to time minimization (C1). The optimal air temperature is thus significantly lower than Tsc lim, as long as the surface temperature constraint is satisfied. The cost function component C3 has a control move suppressing role as in traditional predictive control. Increasing this weight tends to give flat air temperature profiles. Conversely, if the associated weight is decreased, optimal temperature profiles become very steep and the minimum air temperatures become lower than given in Figure 24.1. Abrupt temperature variations cannot be realized in practice due to the plant inertia, though. The weight W3 is thus selected to reflect the admissible cooling and heating rates of the plant. The components C4 and C5 penalize second and third control profile derivatives respectively, that is high curvature and zigzagging. For smooth temperature profiles, like those in Figure 24.1, the contribution of these components to the total cost function is negligible. The associated weights should not be set to zero, however, because this would lead to likely optimization convergence to local minima, in form of zigzagging temperature profiles. Components C4 and C5 are thus very useful in real-time optimization because they favor robust convergence to smooth control profiles. The associated weights are selected after simulation tests and given the smallest values which eliminate zigzagging. One of the critical product quality constraints, namely the lowest admissible surface temperature requirement, is introduced in the optimization cost function via the component C6. The associated weight (W6) is increased until this constraint is

ß 2008 by Taylor & Francis Group, LLC.

satisfied at any time during the batch. Unnecessarily high values of this weight should be avoided, however, because of possible numerical difficulties. It appears in Figure 24.1 that the surface temperature constraint is limiting during a large part of the refrigeration batch and is the main obstacle to further productivity increase by lowering the air temperature. The control optimization algorithm selects the lowest possible air temperature compatible with this constraint. The air temperature has to be increased as the batch proceeds since the heat flux from the center of the fruit decreases with time and becomes insufficient for maintaining acceptable surface temperature. Duration of the refrigeration process is determined by the moment when the core temperature of the fruit situated at the hottest point in the bin reaches the desired final value Tcc lim. This product stability constraint is achieved by the cost function component C7. This constraint is always active since time and energy minimizations tend to reduce product cooling. The weight W7 is large enough to ensure that the product is always cooled to the desired limit. The cost function component C8 was designed to set an upper limit on the product mass loss during refrigeration. As already mentioned, the real fruits studied in this work have a relatively impermeable peel. The observed mass loss is around 0.01%, far below the imposed 2% limit. The constraint C8 is nonlimiting and plays no role in this case, whatever the value of the weight (W8). For the carrageenan gel, however, the final estimated mass loss is about 2.4% which violates the constraint. It turns out that increasing the weight W8 to enforce the satisfaction of the mass loss constraint violates either the surface temperature constraint (C6) or the final core temperature constraint (C7). These three constraints (C6, C7, and C7) are actually incompatible for the considered mass transfer coefficient value. When the mass loss constraint is made dominant via the weight W8, the only two ways the control algorithm has to satisfy it are either to stop refrigeration too early, i.e., before the core temperature reached the desired value (which violates C7) or to use very low air temperatures to reduce the refrigeration time which violates C6. The optimization problem cannot thus be solved with the considered manipulated variable (air temperature) alone. An additional degree of freedom should be used. For example, increasing air humidity would reduce the mass loss by reducing the driving force for water vapor, i.e., the difference in partial pressures between fruit surface and air. One might argue that from a practical point of view, refrigerating carrageenan gel is of little interest. This discussion is a good illustration, however, of a typical practical situation when a priori imposed requirements are overly stringent and the optimal control problem has no solution which satisfies all the constraints. It also illustrates one of the advantages of using constraint violation penalty in the cost function instead of rigid constraint handling. When using weighted penalties, it readily appears which constraints are incompatible. Trying to satisfy a given constraint by increasing its weight, the user quickly realizes which price has to be paid in terms of other constraint violations. A moment’s thought and some physical insights generally reveal the reason for the incompatibility. Weight selection explicitly allows assigning priorities to various requirements. On the contrary, when using ‘‘true’’ constrained optimization, ranking constraints by their importance is generally not possible. In face of incompatible constraints, various software packages react

ß 2008 by Taylor & Francis Group, LLC.

in rather unpredictable ways. The author’s tests with constrained optimization algorithms showed completely inconsistent behavior. In no way, the solution to incompatible optimization problems reflected the kind of compromise a human operator would try to achieve. The minimum and maximum bounds for air temperature (C9) and maximum refrigeration time (C10) are not reached in normal process operation. To determine appropriate weights, the bounds were temporarily modified to make them limiting and the weights increased until these bounds were satisfied irrespective of the other constraints.

24.6.3 ROBUSTNESS

TO

MEASURED

AND

UNMEASURED DISTURBANCES

One of the primary roles of a control algorithm is to compensate process disturbances that appear in everyday plant use. In case of real-time optimal control, this means selecting the best possible control strategy for the remaining period with respect to the optimality criteria introduced in the objective function and the specified constraints. Of course, the control algorithm can only detect deviations from nominal plant operation via available feedback measurements. The response of the control algorithm to a set of typical disturbances was tested experimentally and the results are presented in Figure 24.2. One of the most common practical situations is model mismatch. Model predictions used for control optimization are thus systematically in error. The experiment shown in Figure 24.2a was carried out with apples while the orange model was used for predictions. Despite significant differences in the thermal properties of the two fruits (Table 24.1), the adopted control policy is quite good and the constraints are satisfied indicating that feedback information (core temperature measurement) compensates systematic prediction errors to a large extent. Careful comparison with Figure 24.1a shows that the refrigeration time is longer, however, due to higher air temperature used. Imperfect model predictions result into a slightly suboptimal control policy. A practical conclusion is that the given control algorithm can be used with approximate knowledge of fruit physical properties, but some performance degradation is likely to occur as one might expect. In the experiment reported in Figure 24.2b, a cooling device failure was artificially set up for 30 min. The door was also kept open during the same time to increase the amplitude of the disturbance. True air temperatures are available to the state estimation algorithm (measured disturbance) but model predictions are always based on the optimal calculated air temperature which is not actually applied until the end of the simulated malfunction. From a control point of view, the process remains in open loop for the time of the failure. Despite this fact, there is no tendency to compute excessive controls because the optimal controller has no explicit integral action. The process smoothly moves to normal operation as soon as the computed air temperature can be applied. Two examples of disturbances concerning the ventilation device are shown in Figure 24.2c and d. In both cases, the fan was disconnected for 35 min. In Figure 24.2c, the actual measured air velocity is available to the control algorithm (measured disturbance) while, in the experiment reported in Figure 24.2d, it is not

ß 2008 by Taylor & Francis Group, LLC.

(unmeasured disturbance). The air velocity is used both in state estimation and prediction, i.e., in calculation of the temperature distribution inside the fruit and prediction of its future evolution. In case of the measured disturbance, surface temperature estimation is always good (Figure 24.2c) since the heat and mass transfer coefficient values are correct. During the failure, the predicted batch end times become extremely long since future air velocity is always assumed equal to the last measured one and the predicted heat transfer is low. As soon as the ventilation is turned back on, this effect disappears. In the case of the unmeasured disturbance, the state estimation and control optimization algorithms always assume the air velocity equal to the nominal one. This introduces serious error in state estimation especially for surface temperature (Figure 24.2d). Fortunately, it does not prevent the control algorithm from acting correctly, because, in this particular case, neither the cost function nor the active constraints is seriously affected.

24.7 CONCLUSION Predictive optimal control is a particularly user-friendly framework for developing and implementing control laws. The user does not have to specify the control law explicitly, but only to state desirable features of the plant control policy, in terms of familiar process variables. Desirable features usually take the form of minimization or maximization criteria and constraints. Ranking among these requirements is expressed using appropriate weights. In realistic situations, the criteria and the constraints are nonlinear functions of the control variables. The difficulty of finding the appropriate time evolution of the control variables which satisfies the generally contradictory requirements relies on the dynamic optimization algorithm. During iteratively testing candidate future control policies, the nonlinear optimization algorithm needs repeated simulations of a predictive dynamic process model. The key to successful implementation of nonlinear optimal control lies in the appropriate selection of the dynamic model and optimization algorithm couple. For example, the model has to be fast enough to allow a sufficient number of optimization iterations to be performed. Similarly, if the optimization algorithm uses derivatives of the objective function and of the constraints up to some order, the model predictions have to be continuous functions of the control variables with continuous derivatives up to the same order. Furthermore, well-designed optimization objectives and constraints may greatly help the optimization algorithm in finding good solutions, e.g., efficiently avoiding local optima. For example, appropriate control vector parameterization and use of control smoothing terms are simple but always efficient techniques. The distinctive feature of the presented application is nonlinear constrained control optimization in real time. Process feedback allows the optimization algorithm to interactively adapt the control policy to the reality of each run. Optimality with respect to given criteria and constraint satisfaction are thus retained as much as possible even in presence of unavoidable disturbances. As a counterpart to adaptive optimality, real-time optimization imposes particularly stringent requirements on model simulation speed and reliability of the optimization algorithm especially in face of unexpected situations such as incompatible constraints.

ß 2008 by Taylor & Francis Group, LLC.

Nonlinear constrained optimization algorithms particularly suitable for real-time dynamic optimization still have to be developed. Nevertheless, working applications can perfectly be designed on basis of existing optimization software as illustrated in this chapter. Performed experiments demonstrate the feasibility of this approach and robustness to various measured and unmeasured disturbances. The described methodology can easily be generalized to different products and other batch and continuous processes provided that the control problem can be described in a similar manner: finding control profiles that optimize operation costs while satisfying technological and product quality constraints.

ACKNOWLEDGMENTS The author expresses his gratitude to Graciela Alvarez and Denis Leducq from CEMAGREF Antony, France, for their essential contribution to modeling the refrigeration process and running the experiments described in this chapter. Their expertise in the field of refrigeration and kind permission and help in using experimental facilities at CEMAGREF made this work possible. I will never thank enough Gilles Trystram from AgroParisTech, France, for initiating and continuously supporting work in the field of optimal control of food processes. He gave me a unique opportunity for developing and testing nonlinear real-time optimal control algorithms. His expertise, assistance, and uninterrupted encouragements are gratefully acknowledged.

NOMENCLATURE C E k Lm n N Ta Ta 0 Ta f Tcc Tch Tsc Tsec t tf tp ts U W

Component of the control optimization cost function Component of the parameter and state estimation cost function Relative heat transfer coefficient Relative mass loss Number of finite-difference virtual slices in a fruit Number of temperature values optimized by the control algorithm Air temperature before the product stack Initial air temperature Final air temperature Temperature at the core of the coldest fruit Temperature at the core of the hottest fruit Temperature at the surface of the coldest fruit Security margin for the temperature at the surface of the coldest fruit Current time Final batch time Present moment in real-time operation Starting moment for parameter and state estimation time windows Vector of optimized control parameters Weight of a component in the cost function

ß 2008 by Taylor & Francis Group, LLC.

8C 8C 8C 8C 8C 8C 8C s s s s

GREEK LETTERS s tf

Standard deviation for temperature measurement Time remaining until the end of the batch

8C s

SUBSCRIPTS calc i lim min max set

Value calculated by the process model Component number in the cost function Limiting admissible value Minimum value Maximum value Set point value

REFERENCES Alvarez, G. and Trystram, G., Design of a new strategy for the control of the refrigeration process: Fruit and vegetables conditioned in a pallet, Food Control, 6, 347, 1995. Banga, J.R. et al., Improving food processing using modern optimization methods, Trends Food Sci. Tech., 14, 131, 2003. Boillereaux, L. and Flaus, J.M., Adaptive receding horizon state estimation for nonlinear processes, Paper presented at the 5th IFAC Symposium ACASP, Budapest, Hungary, 1995. Garcia, C.E., Prett, D.M., and Morari, M., Model predictive control: Theory and practice—a survey, Automatica, 25, 335, 1989. Gill, P.E., Murray, W., and Wright, M.H., Practical Optimization, Academic Press, London, United Kingdom, 1981. Moré, J.J., The Levenberg-Marquardt algorithm: Implementation and theory, in Numerical Analysis, Watson, G.A. (Ed.), Springer Verlag, Berlin, 1977. Trelea, I.C., Alvarez, G., and Trystram, G., Nonlinear predictive optimal control of a batch refrigeration process, J. Food Process Eng., 21, 1, 1998.

ß 2008 by Taylor & Francis Group, LLC.

ß 2008 by Taylor & Francis Group, LLC.

25

Optimization of Apple Juice Extraction María Teresa González and Martín Juan Urbicain

CONTENTS 25.1 Introduction .............................................................................................. 545 25.2 Methods to Improve the Extraction of Juice ........................................... 545 25.3 Material Balances..................................................................................... 547 25.4 Economical Balances ............................................................................... 552 25.5 Optimization............................................................................................. 554 25.6 Example ................................................................................................... 557 Nomenclature ........................................................................................................ 559 Greek Letters............................................................................................ 559 References ............................................................................................................. 560

25.1 INTRODUCTION An apple juice concentrate manufacturing plant is built up of some operating units, usually connected such as no recycles are likely to be found. The final product is essentially concentrated apple juice. In a typical plant flowsheet, apples are classified and sent to the crusher where they are ground, and the resulting pulp is pressed to obtain juice. Then the aroma is stripped, the juice is clarified, and finally concentrated by different procedures. Economical reasons, more than technical, made us focus our attention on pressing sector of the plant. There are different ways of improving the juice yield for a given quality of fruit: One can make more efficient the pressing operation itself or add some water to the pomace in order to promote the diffusion of more soluble solids out of the cellulosic matrix prior to the next pressing stage in the case of a number of presses arranged in series.

25.2 METHODS TO IMPROVE THE EXTRACTION OF JUICE Different methods are reported in the literature that aim to improve the extraction of juice from a given apple quality. Among them, we can cite: Addition of enzymes before the pressing operation, utilization of pressing aids, and pulsed electric fields. Al-Mashat and Zuritz (1993) found that juice yield increased with addition of rice hull (up to 12%), decreasing temperature (up to 8%), and decreasing ram speed (up to 10%).

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They claimed that their study provided a basis for the determination of the properties of apple pomace and could facilitate the design and operation of juice presses. The addition of rice hull before pressing increased adequate drainage channels during pressing which allowed greater juice outflow. This addition of pressing aid is specially recommended for very soft fruit to facilitate juice flow. A decrease in press ram speed produced a 2%–10% increase in juice yield. This increase was more pronounced at 218C compared to the process at 88C. Lanoisellé et al. (1996) developed a general theory for the mechanical expression of agrofood, cellular materials. The model considers liquid transfer within a network of three different volumes: extraparticle, extracellular, and intracellular. It is able to predict the behavior of four different steps in the consolidation stage: the primary deformation and the creep deformation of extraparticle volume and the deformation and deliquoring of both extracellular and intracellular volumes. Enzymatic mash treatment is a well-known modern process for gaining more juice from fruits and vegetables. According to the technique, cell wall and middlelamina pectin of the fruit are degraded by pectinase activities. Pectolytic enzymes degrade pectic substances of several kinds. Pectolytic enzymes are used in the fruitprocessing industry to increase yields as enzymatic hydrolysis of the cell walls increases the extraction yield. The resultant pulp has a lower viscosity, and the quantity of waste pomace is reduced. Demir et al. (2001) investigated the behavior of the immobilized mash treatment on the pectolytic enzyme in carrot puree medium. Bazhal et al. (2001) investigated the influence of a pulsed electric field (PEF) simultaneous to pressure treatment on juice expression from the cut, apple raw material. Dependencies of specific conductivity s, juice yield Y, instantaneous flow rate v, and qualitative juice characteristics at different modes of PEF were discussed in their study. Three main compression phases were observed in a case of mechanical expression. A unified approach was proposed for juice yield data analysis allowing a reduction in data scattering caused by differences in the quality of samples. Simultaneous application of pressure and PEF treatment revealed a passive form of PEF-induced cell plasmolysis. Pressure provoked the damage of defect cells, enhanced diffusion migration of moisture, and depressed the cell resealing processes. PEF application at the moment when the press-cake’s specific electrical conductivity reached a minimum, and the pressure achieved its constant value seemed to be the most optimal. They found that a combination of pressing and PEF treatment gives optimum results and enhances the yield of juice significantly and improves its quality. Lebovka et al. (2004) discussed about the specific influence of PEFs and thermal treatment on the textural properties of apple tissue and apple juice expression. Force relaxation curves were analyzed for different regimes of PEF and thermal pretreatment of apple tissue. The concept of the effective relaxation time t1 was used to characterize the different modes of treatment. The curves of juice yield versus time for nontreated, thermal-treated, PEF-treated, and combined thermal and PEF treated samples led them to the conclusion that the PEF treatment increased the juice yield for both nontreated and thermally pretreated samples. However, enhancing effect of PEF on the juice yield was significantly higher for the thermally pretreated samples. All these methods involve some modifications to be carried out on the material to be pressed and can be considered, combined with the mechanical design of the press itself, a set of factors that finally result in a better efficiency of the pressing stage hj or the diffusion factor Zj.

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On the other hand, another resource exists that can be used with the final purpose of increasing the juice yield when the juice extraction operation is carried out within a train of presses in series. This resource is the hydration of the pomace arising from one press in order to achieve the solubilization of more soluble compounds trapped inside the pomace. They diffuse to the liquid phase and can be recovered in the next pressing stage. There are obvious limits to the amount of hydration arising from the capacity of one press to handle the added liquid phase. Besides, it should be kept in mind that the final product is a concentrated juice, and so all added water should be evaporated at the evaporation section. This situation of trade between ‘‘more hydration-better extraction’’ and ‘‘more hydration-more energy to be spent on evaporation’’ has to be solved via the optimization of an economic equation. We shall now focus our attention on the analysis of the hydration to a series of presses considering that the methods that improve the yield of one press are included into the parameters hj or Zj.

25.3 MATERIAL BALANCES The hydration of the exhausted pomace arising from one press can be regarded as an equilibrium stage in which soluble solids from the pomace diffuse to the added liquid before the mixture is pressed in the next press. The simulation of the pressing sector involves the posing and solution of the equations of mass balance and equilibrium (if hydration is performed) in each unit, such that soluble solids (sugar) concentrations are assessed in each exit stream, and consequently, the flow rate and brix of the final product are determined. Let us first make a material balance in the simple system showed in Figure 25.1 in order to better understand the concept of hydration and how it improves the extraction of soluble solids. In what follows, ‘‘hj’’ is the yield of press ‘‘j,’’ the yield being defined as the mass of juice produced per unit mass of fresh pulp. It must be recalled that this definition is only valid for the expression of fresh pulp, being meaningless if hydrated pomace is being pressed. Let us consider the model to be applied to the set of two presses in Figure 25.1 (Urbicain et al., 1990). The mass balance around the first press can be written as J 1 ¼ h1 W 0

(25:1)

W 0 ¼ J1 þ W 1

(25:2)

H

W0

Press 1 J1

W1

Hydration tank

W 1+ H

Press 2

W2

J2

FIGURE 25.1 Two presses in series with a hydration vessel between them.

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Y1 Y ∗(>Y H) From press 1

W1

+

H

YH X T(=Y T)

X1 (a)

(b)

(c)

X ∗( 0

FIGURE 25.3 Different situations after second pressing stage.

The overall mass balance gives: W1  W2 ¼ (h2  h1 )

(25:8)

J2 ¼ H þ (h2  h1 )

(25:9)

If (h2  h1) ¼ 0 (Figure 25.3b), the actual values X2 and Y2 leaving the second press are the equilibrium ones X* and Y*, respectively, and the values of J2 and W2 are equal to H and W1. If (h2  h1) < 0 (Figure 25.3a), the second press will extract less liquid than the first one, and consequently the equilibrium relationship and the mass balance will lead to Y2 ¼ Y * X2 ¼

W 1 X 1 þ ( H  J2 ) Y * W2

(25:10) (25:11)

It is apparent that these equations have no sense if J2 < 0, and H should be at least greater than the absolute value of (h2  h1). If, on the other side, (h2  h1) > 0, the situation is shown in Figure 25.3c. Press 2 will extract all the added liquid plus some of the liquid entering with the pomace. The resulting concentrations will be X2 ¼ X * Y2 ¼

HY * þ (J2  H )X * J2

(25:12) (25:13)

In a general configuration of N presses in series, each one receives the pomace from the preceding and fresh water or some juice from any of the following. Equations 25.8 through 25.13 have to be generalized according to the resulting layout. In order to make the calculation general, the fresh water supply will be considered as a virtual press (N þ 1) which obviously will not produce any juice. With the exception of press 1, which processes fresh pulp and hence will not be hydrated, the remaining 2–N units can be hydrated in any proportion by juice provided by one or more of the following presses in the set including press (N þ 1). In summary, press j (2  j  N) can receive juice from any press between (j þ 1) and (N þ 1) as can be observed in Figure 25.4. In what follows all equations are referred to 1 kg of fresh pulp.

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W0

Press 1

W1

Hydration tank 2

W1+V2h

W2

Press 2

WN⫺1

Hydration tank N

WN⫺1+VNL

JN

V2h = V2,N+1 + V2N + V2,N−1+ ... +V23

VN,N+1

V2,N+1

BN = V2N + V3N +

V 2,N

WN

Press N

BN+1 +VN−1,N

Water (Press N+1)

V2,3 J 2 = C2

J1 = C1

CN

FIGURE 25.4 General configuration of a train of N presses in series.

Let Vjk be the juice produced by press k which is sent to hydrate the pomace in press j, and Vjh the sum of all wetting streams flowing onto press j from all presses located downstream of it, as given by Equation 25.14: Vjh ¼

N þ1 X

Vjk

(25:14)

k¼jþ1

An elemental material balance around press j gives: Wj1 þ Vjh ¼ Wj þ Jj

(25:15)

A similar balance on soluble solids leads to Wj1 Xj1 þ

N þ1 X

Vjk Yk ¼ Wj Xj þ Jj Yj

(25:16)

k¼jþ1

Taking into account the definition of yield hj given above, it is hj ¼ 1  Wj, and calculation of the difference of pomace mass delivered by two consecutive presses is straightforward: Wj1  Wj ¼ hj  hj1

(25:17)

Equation 25.15 can be written: Jj ¼ hj  hj1 þ Vjh

(25:18)

Mass balance is completed with the following equations (Figure 25.4): Jj ¼ Cj þ Bj where Cj is the amount of juice sent to process Bj is the fraction used for hydration of the preceding presses

ß 2008 by Taylor & Francis Group, LLC.

(25:19)

Hence: Bj ¼

j1 X

(25:20)

Vkj

k¼2

If W0 is the mass flow rate of fresh pulp, assumed to be 1 kg, the yield of the first press is equal to the juice mass flow rate produced in that unit: J1 ¼ h 1 ¼ W 0  W 1

(25:21)

Overall mass balance: W0 þ BNþ1 ¼ WN þ

N X

Cj

(25:22)

Cj Yj

(25:23)

j¼1

Overall soluble solids mass balance: W0 X0 ¼ WN XN þ

N X j¼1

Now, Equation 25.5, which was written for the second press of a set of two, can be generalized to N presses, plus fresh water (‘‘press’’ N þ 1): Yteor,j ¼

Wj1 Xj1 þ Vjh Yh Wj1 þ Vjh

(25:24)

Yjh is the average concentration of Vjh, as given by Equation 25.25 and is equivalent to YH in Equation 25.5, while Vjh is equivalent to H. Nþ1 P

Yjh ¼

Vjk Yk

k¼jþ1

Vjh

(25:25)

As in the simpler case of two presses, Yteor,j cannot be attained in practical operation, but a lower equilibrium value Yj* < Yteor,j will be the actual concentration of the juice, while the concentration of the liquid originally wetting the pomace, Xj1, is reduced to a value Xj* > Yteor,j , in equilibrium with Yj* . The diffusion factor defined by Equation 25.6, takes the form: Zj ¼

Yj*  Yjh Yteor, j  Yjh

(25:26)

As before, Zj < 1 and in a perfect mixer Yj* tends to Yteor,j, and Zj tends to 1. Xj* is given by the amount of soluble solids contained in Wj, and can be calculated by difference: Xj* ¼ (Wj1 Xj1 þ Vjh Yjh  Yj* Vjh )=Wj1

ß 2008 by Taylor & Francis Group, LLC.

(25:27)

However, actual values Xj and Yj leaving the press are equal to equilibrium ones in the particular case of (hj  hj1) ¼ 0 only. Should it be the case, press j is expected to produce the same amount of pomace than press (j  1), independent of the degree of hydration, so if no water is added, no juice is produced. Otherwise, any amount of water added should leave the press as juice Jj and the final concentrations in the juice and the pomace liquid would be Yj ¼ Yj* and Xj ¼ Xj* respectively. If (hj  hj1) is not equal to zero, concentrations are to be calculated in a different manner, as has been explained for the simpler case of two presses. Let (hj  hj1) be > 0. This means that press j will extract all the liquid added plus some of the liquid entering with the pomace. In this case: Xj ¼ Xj*

for (hj  hj1 )  0

Yj ¼ (Wj1 Xj1 þ Vjh Yjh  Wj Xj )=Jj

for (hj  hj1 ) > 0

(25:28) (25:29)

Symmetrically, for (hj  hj1) < 0, press j will not extract all the wetting liquid added, but some of it will remain diluting the pomace. Hence: Yj ¼ Yj*

for (hj  hj1 )  0

Xj ¼ (Wj1 Xj1 þ Vjh Yh  Jj Yj )=Wj

for (hj  hj1 ) < 0

(25:30) (25:31)

Equations 25.28 through 25.31 are valid for presses 2–N; for press 1 results Y1 ¼ X0 and X1 ¼ X0.

25.4 ECONOMICAL BALANCES As it was already said, it is not advisable, from an economical point of view, to hydrate as much as physically possible, because in an ACJ plant water has to be evaporated to reach the desired concentration of the final product. A simple economical balance will give us the tool to find the optimum point of operation. Profit, in terms of money units per unit time, is calculated by means of Equation 25.32: G ¼ Jc (Pv  CT )  CF

(25:32)

in which fixed costs CF and selling price Pv are data, and concentrate flow rate Jc and variable costs CT are calculated as follows: 1. Juice concentrate flow rate, Jc: Jc ¼ JF BF =Bc

(25:33)

Calculation of juice mass flow rate fed to the evaporator, JF, depends on the situation considered, which in turn changes along the year:

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a. Saturated evaporation (SE), corresponding to a situation of raw material surplus, usually encountered during harvest season, which makes the evaporator to work at full capacity. b. Evaporation idle capacity (IC), corresponding to a shortage of raw material, usually encountered during postharvest season, making the evaporator to work ‘‘on demand.’’ In the first case, JF is a function of the actual evaporation capacity and is calculated as follows: JFSE ¼

Ev (1  BF =Bc )

(25:34)

In the second case, JF is calculated as a function of the mass flow rate of processed pulp, by means of Equation 25.35: JFIC ¼ Rc P(W0 X0  WN XN )

(25:35)

When JF is calculated by Equation 25.34, the pulp mass flow rate that can be processed is obtained by rearranging Equation 25.33 and using the obtained value of JF: P¼

JFSE Rc (W0 X0  WN XN )

(25:36)

If, on the other hand, JF is calculated by means of Equation 25.35, the dependent parameter is the water to be evaporated, and it is calculated as the difference between the juice flow rate fed to the evaporator and the concentrate: Ev ¼ JFIC  Jc ¼ JFIC (1  BF =BC )

(25:37)

2. Variable costs, CT: Factor CT in Equation 25.32 is the sum of four terms: CT ¼ CM þ Cp þ Cz þ CE

(25:38)

CM ¼ PM Rj

(25:39)

Where

being parameter Rj the ratio between pressed pulp and the amount of concentrate produced: Rj ¼ P=Jc

(25:40)

Cp is a value that should be entered by the user, and Cz is the cost of enzyme treatment that could be done to the pulp before pressing in order to increase the yield,

ß 2008 by Taylor & Francis Group, LLC.

and depends on the enzymes unit price, Pz, and the applied doses, Dz, and it is calculated by Equation 25.41: Cz ¼ Pz Dz Rj

(25:41)

Regarding the evaporation cost CE, when raw material is the process bottleneck (situation b) as mentioned above, it is calculated by means of Equation 25.42: CE ¼ Ev Cv =JC

(25:42)

where Cv is the cost of evaporation of 1 kg of water. Conversely, when the evaporator is saturated, evaporation cost is a constant included in CP, and CE ¼ 0.

25.5 OPTIMIZATION Among all possible arrangements between presses, optimization seeks the best hydration flowsheet, having as objective function either the maximum profit or the maximum sugar recovery. With reference to Figure 25.4, Vij is a fraction of Jj, and can be expressed as Vij ¼ hij Jj

(25:43)

where hij is a factor between 0 and 1. Considering Equations 25.19 and 25.20, inequality 25.44 must be true: j1 X

hij  1

(25:44)

i¼2

Coefficients hij are the elements of a matrix, named connectivity matrix because its elements define the amount of juice from press k that goes to wet press j. This matrix has (N  1) rows, representing the hydrated presses from 2 to N, and (N  1) columns, identifying the source presses, from 3 to (N þ 1). We define a variable named pitch, noted p, a natural number equal or larger than 1, to determine the minimum amount of juice allowed to flow from one press to another. The pitch is related to hij by a simple expression, valid for i < j: hij ¼ kij =p

(25:45)

Fractions of juice going to clarification are defined as cj ¼ Cj =Jj

(25:46)

The cj’s can be added to the right hand side of inequality 25.45 to make it an equation: j1 X i¼2

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(hij þ cj ) ¼ 1

j ¼ 3, . . . , N þ 1

(25:47)

Coefficients cj can be related to pitch p by means of factors kjj with a relationship similar to Equation 25.45: cjj ¼ kij =p

(25:48)

Then Equation 25.47 becomes: j X

kij ¼ p

(25:49)

i ¼2

kij is also a natural number such that 0  kij  p and Equation 25.49 must be satisfied N  1 times, namely for j ¼ 3, . . . , (N þ 1). Calculations to be performed are all material and economical balances for all possible combinations of kij. It must be noted that for a given p, there are (j  2) degrees of freedom in each press, except the fresh water supply, virtual press (N þ 1), for which element kNþ1,Nþ1 is always zero, making (j  3) degrees of freedom. As an example let us consider a set of 4 presses, therefore Equation 25.49 can be posed 3 times. Let p ¼ 2, which means that the juice delivered by any press is to be divided in two equal streams, one going to another press and the remaining half going either to clarification or to a third press. The resulting equations are k23 þ k33 ¼ 2 k24 þ k34 þ k44 ¼ 2

(25:50)

k25 þ k35 þ k45 ¼ 2 The possible values of the kij are 0, 1, and 2 respectively, while the values k33, k44, and k45 will be calculated as a result from those assigned to k23, k24, k34, k25, and k35. The complete set of cases to be studied is represented by matrix A shown in Table 25.1. In this example 108 alternatives will be evaluated. For instance, if we assume k23 ¼ 1, k33 ¼ 1, k24 ¼ 1, k34 ¼ 1, k44 ¼ 0, k25 ¼ 0, k35 ¼ 2, k45 ¼ 0, the hij’s and cj’s will be those shown in matrix B at Table 25.2. Elements of matrix B show that fresh water will wet press 3 only, juice from press 4 will wet presses 2 and 3 in equal parts, while 50% of juice from press 3 will go to press 2 and the remaining 50% will go to clarification. If p ¼ 4, the number of cases to be evaluated rises up to 1125, and for p ¼ 10, the number of calculations will be 47,916. If the train has 5 presses, for p ¼ 1, cases evaluated will be 96, for p ¼ 2 the number will be 1,800, for p ¼ 4 the number of cases will grow to 91,875, and for p ¼ 5 they will be 395,136. This means an important growth of the number of calculations as p increases. Optimization can be performed with either the net profit or the maximum sugar recovery as objective function. One constraint to be taken into account is the amount of fresh water added to the last press: It cannot be unlimited so a certain maximum hydration must be defined. A practical way is to take it as a fraction of the total amount of pulp to be processed, selected with some heuristic criterion.

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TABLE 25.1 Matrix A: Connectivity Matrix for Four Presses in Series and p ¼ 2 k23

k33

k24

k34

k44

k25

k35

k45

Submatrix-2 Submatrix-1 0

2

0

0

2

0

0

2

0

2

0

0

2

0

1

1

0

2

0

0

2

0

2

0

0

2

0

0

2

1

0

1

0

2

0

0

2

1

1

0

0

2

0

0

2

2

0

0

0

2

0

1

1

0

2

0

2

0

0

2

1

0

1

0

2

1

1

0

0

2

2

0

0

1

1

2

0

Submatrix-1 is repeated

Submatrix-2 is repeated

The concept of ‘‘critical hydration’’ (Elustondo and Urbicain, 1992) which is the minimum amount of water required for all the presses to produce some juice should be taken into account to place a lower bound to the hydration. Hence, the algorithm evaluates all possible configurations starting with the critical hydration as the minimum value. The best configuration is selected for this hydration level, and then the hydration level is increased in a certain arbitrary step, and the procedure is repeated until the maximum selected hydration is reached.

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TABLE 25.2 Matrix B: hij’s and cj’s for Shaded Cells in the Connectivity Matrix hij

3

4

5

cj

2 3 4

0, 5 0 0

0, 5 0, 5 0

0 1 0

0, 5 0 0

It is worth to mention that not all calculated configurations are physically or practically feasible, so some heuristics are introduced to discard them immediately. For instance, the total hydration could be larger than the critical, but could be distributed in such a way that for some particular press it is not. In that case the configuration is eliminated. Another case is when the amount of liquid entering to a given press is such that the unit would be flooded; it is also discarded. One heuristic adopted is that Vjh in press j must be lower than 1.5 times the mass flow rate of pomace fed to the press. If at a certain level of hydration all configurations are discarded because that condition is not satisfied, it means that no further hydration is possible, and the iteration stops. In any case, once all best configurations have been selected at each level of hydration, the algorithm selects the best among them, with its corresponding hydration flow rate. The algorithm described was programmed in FORTRAN language and tested with different examples of both saturated evaporator and raw material shortage seasons (González, 1990; González et al., 2000).

25.6 EXAMPLE As an example, let us consider a set of five presses in series in the situation of saturated evaporation. The values considered are shown in Table 25.3, and the results are displayed in Table 25.4. A value of p ¼ 2 was considered. The optimal flowsheet for the selected example with the optimum degree of hydration found is shown in Figure 25.5. The raw material feed is taken as 100% of P. The flowrate of water to press 5 is 30% of P. The exhausted pomace at a rate of 30% P leaves the system from press 5 while a flow rate of juice equal to P is collected from presses 1 (60%) and 2 (40%) and goes to clarification. Juice from presses 3, 4, and 5 is used to hydrate upstream presses at the rates indicated in the figure. Finally it should be remarked that most of the examples showed an optimum for a purely countercurrent flow arrangement, as could be expected. The result is a complex function of many economical and technical variables involved, so each case must be analyzed considering its cost structure and processing capacity at each particular season.

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TABLE 25.3 Data for the Example Press #

1

2

3

4

5

Yield (%) Z

60

80 0.7

70 0.7

70 0.7

70 0.7

Hydration step (%) Evaporation capacity (kg=h) Clarification yield Enzymes addition (kg=kg of raw material) Brix raw material Brix concentrate Raw material price ($=kg) Concentrate selling price ($=kg) Enzymes price ($=kg) Processing cost ($=kg of concentrate) Fixed cost ($=h)

10 25,000 0.92 40  106 12 70.5 50  103 0.98 40 0.49 300

TABLE 25.4 Results for the Example Hydration (%) 10 20 30 40 50 60

Concentrate Production kg=h

Fruit Consumption kg=kg conc.

Profit $=h

Water kg=h

Raw Material kg=h

Brix Presses

Recovery %

5,128 4,898 4,492 4,079 3,731 3,426

7.98 7.37 7.14 7.04 6.97 6.94

86.9 223.8 235.0 205.3 175.5 143.4

4,093 7,222 9,617 11,494 13,012 14,261

40,935 36,109 32,057 28,735 26,025 23,768

12.00 11.55 10.74 9.89 9.16 8.50

83.33 90.24 93.22 94.44 95.37 95.90

Pure water 30 P =100 Press 1 40 Tank 2 h = 0.6

Press 2 20 h = 0.8

Tank 3 22.5

20

Press 3 30 h = 0.7 15

40

30

15

30

15

0

0

FIGURE 25.5 Results for a train of five presses, 30% hydration.

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Press 5 30 h = 0.7

12.5

12.5

60

Press 4 30 Tank 5 h = 0.7

15

7.5

7.5

Tank 4

0

C = 100

NOMENCLATURE Bc BF Bj CE CV Cj CM Cp CT Cz Dz Ev G GF hij Jc JF Jj kij p P PM Pv Pz Rc Vjk Wj Xj Xj* Yj Zj

Brix degrees of concentrate Brix of juice fed to evaporator Juice flow rate from press j Concentrate evaporation cost Water evaporation cost Juice flow rate to clarification from press j Raw material cost Processing cost Total cost Enzymes treatment cost Enzymes addition Water evaporated Net profit Fixed costs Fraction of juice from press j wetting press i Concentrate mass flow rate Juice fed to the evaporator Juice from press j Factor in Equation 25.45 Pitch, variable in Equation 25.45 Fresh pulp mass flow rate Raw material price Concentrate selling price Enzyme price Clarification yield Juice flow rate wetting press j from press k Pomace from press j Sugar concentration in pomace from press j Sugar equilibrium concentration in pomace from press j Sugar concentration in juice from press j Diffusion factor

GREEK LETTER hj

Press yield

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kg soluble solids=kg concentrate kg soluble solids=kg concentrate (kg=kg of raw material) $=kg concentrate $=kg vapor produced kg=kg of raw material $=kg concentrate $=kg concentrate $=kg concentrate $=kg concentrate kg=kg of raw material kg=h $=h $=h

kg=h kg=h kg=kg of raw material

kg=h $=kg of raw material $=kg concentrate $=kg enzyme kg of raw material=kg conc. kg=kg of raw material kg=kg of raw material kg=kg pomace kg =kg pomace kg=kg juice

REFERENCES Al-Mashat, S.H.I. and Zuritz, C.A., Stress relaxation behavior of apple pomace and effect of temperature, processing aid and compaction rate on juice yield, J. Food Eng., 20, 247, 1993. Bazhal, M.I., Lebovka, N.I., and Vorobiev, E., Pulsed electric field treatment of apple tissue during compression for juice extraction, J. Food Eng., 50, 129, 2001. Demir, N. et al., The use of commercial pectinase in fruit juice industry. Part 3: Immobilized pectinase for mash treatment, J. Food Eng., 47, 275, 2001. Elustondo, M.P. and Urbicain, M.J., Critical hydration in a series of presses for apple juice manufacture, J. Food Eng., 17, 217, 1992. González, M.T., Programa Hidrat, PLAPIQUI, 1990. González, M.T., Elustondo, M.P., and Urbicain, M.J., Optimizing apple juice extraction in multiple presses, J. Food Eng., 17, 217, 224, 2000. Lanoisellé, J.L. et al., Modeling of solid=liquid expression for cellular materials, AIChE J., 42, 2057, 1996. Lebovka, N.I., Praporscic, I., and Vorobiev, E., Combined treatment of apples by pulsed electric fields and by heating at moderate temperature, J. Food Eng., 65, 211, 2004. Urbicain, M.J., Elustondo M.P., and Ramos M.A., Computer simulation of an ACJ plant, Symposium of the International Federation of Fruit Juice Producers, Paris, France, 1990.

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26

Optimization of Canned Food Processing Ricardo Simpson and Arthur A. Teixeira

CONTENTS 26.1 26.2 26.3

26.4

26.5

Introduction .............................................................................................. 562 Canning Fundamentals ............................................................................ 562 Batch Retort Processing in Commercial Food Canneries ....................... 564 26.3.1 Problem Structure for Batch Processing in Canned Food Plants.................................................................................. 564 26.3.2 Batch Retort Operation in Canned Food Plants.......................... 566 26.3.3 Hierarchical Approach................................................................. 566 26.3.4 Retort Scheduling ....................................................................... 567 Criteria for Optimal Design and Operation of Batch Retorts Food Canning Plants................................................................................ 571 26.4.1 Maximizing Net Present Value of Capital Investment ............... 571 26.4.2 Energy Consumption .................................................................. 577 26.4.2.1 Model Development .................................................... 577 26.4.2.2 Mathematical Model for Food Material ...................... 578 26.4.2.3 Mass and Energy Balances during Venting ................ 578 26.4.2.4 Mass and Energy Consumption between Venting and Holding Time (to Reach Process Temperature) ... 580 26.4.2.5 Mass and Energy Balance during Holding Time ........ 582 26.4.3 Simultaneous Processing of Different Product Lots in the Same Retort ...................................................................... 583 26.4.3.1 Simultaneous Sterilization Characterization ............... 584 26.4.3.2 Mathematical Formulation for Simultaneous Sterilization ................................................................. 585 26.4.3.3 Computational Procedure ............................................ 586 26.4.3.4 Mathematical Formulation of Mixed Integer Linear Programming Model ................................................... 586 26.4.3.5 Expected Advantages on the Implementation of Simultaneous Sterilization ...................................... 588 26.4.4 New Package Systems and Their Impact on Energy Consumption ............................................................................... 590 Conclusion ............................................................................................... 591

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Nomenclature ........................................................................................................ 591 Sub Indices............................................................................................... 593 Greek Letters............................................................................................ 593 References ............................................................................................................. 594

26.1 INTRODUCTION The process of ‘‘canning’’ foods first began with the use of glass jars and was invented and developed on a small commercial scale in France in 1795 by Nicholas Appert. He was a French chef who was determined to win the prize of 12,000 francs offered by Napoleon for a way to prevent military food supplies from spoiling. Appert canned meats and vegetables in jars sealed with pitch and by 1804 opened his first vacuum-packing plant. It was a French military secret that soon leaked across the English Channel (Holdsworth, 1997). In 1810, an Englishman, Peter Durand, took the process one step further and developed a method of sealing food into unbreakable tin containers which was perfected by Bryan Dorkin and John Hall who set up the first commercial canning factory in England in 1813. It was not until 1860 that Louis Pasteur provided the explanation for canning’s effectiveness when he was able to demonstrate that the growth of microorganisms is the cause of food spoilage. A number of inventions and improvements followed, and by the 1860s, the time it took to process food in a can had been reduced from 6 h to 30 min. Canned foods were soon commonplace. Tin-coated steel, semirigid plastic containers, and flexible retortable pouches are used today. During the 1920s, Bigelow, the first director of the National Canners’ Association in Washington DC, and colleagues developed a method of determining thermal processes based on heat penetration measurements in cans of food (Bigelow et al., 1920). One of these colleagues, Charles Olin Ball, subsequently developed mathematical methods for the calculation of process times at specified temperatures (Ball, 1923, 1928). These became known as the Ball Formula methods and brought him worldwide recognition as an expert in this subject (Ball and Olson, 1957). Most of the subsequent developments in the subject have been based on these workers’ early concepts. The basic principles of canning have not changed dramatically since Nicholas Appert and Peter Durand. Sufficient heat to destroy microorganisms is applied to foods packed into sealed or ‘‘airtight’’ containers capable of withstanding the rigors of exposure to saturated steam at high pressure and temperature. In most food canneries around the world, these processes are carried out in steam retorts or autoclaves. A typical cannery would consist of a battery of retorts with each retort operating in a batch mode. The purpose of this chapter is to explore a number of criteria for optimal design and operation of batch retort processes.

26.2 CANNING FUNDAMENTALS Sterilization of canned foods by thermal processing has a long successful tradition and will most likely continue to be an important method of food preservation far into the future. Thermally processed canned foods offer significant convenience to the consumer. They are shelf stable with extended shelf lives in the range of 1–4 years or more at

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ambient temperature and very economic with respect to cost for value to the consumer. Commercial food sterilization by thermal processing in batch retorts has been the most commonly used method of manufacture in food canneries worldwide during the last 75 years. The operation of retorts is carried out in three stages (venting, heating, and cooling). The target of the thermal process in the retort is the heat inactivation of bacterial spores that might be present in the product. Heating is accomplished with saturated steam under pressure to achieve sterilizing temperatures within the retort in the range (1058C–1308C) for a specified time sufficient to guarantee commercial sterility (Simpson et al., 2006). Cooling is normally carried out by introducing water at ambient temperature into the retort under overriding air pressure to avoid sudden pressure drop in the retort and associated pressure stress in the containers that would otherwise occur. It is important to point out that the degree of microbial inactivation or level of sterilization is referred to as the lethality of the process (Fo). This is the target value that must be reached when calculating the process time needed at a specified retort temperature in establishing a thermal process for any product. Calculation of thermal processes is carried out by the general method of Bigelow and his collaborators (Bigelow et al., 1920) or the Ball formula method (Ball, 1923, 1928) mentioned previously. For totally historical reasons (prior to high-speed digital computers), the general method was rarely used because it required very time-consuming tedious stepwise integration carried out by hand. For this reason, the Ball formula method was the method of choice in spite of its relative lack of precision compared with the general method. Over the years, the Ball formula method has experienced a series of modifications which enhance the calculations (Ball et al., 1928; Hayakawa, 1971; Stumbo, 1973; Pham, 1987, 1990). In spite of all these modifications, it continues to perform with less precision than the general method. Therefore, process authorities routinely incorporate a host of ‘‘safety factors’’ when using the Ball formula method to compensate for this lack of precision. One of the reasons for continued popular use of the Ball formula method is its considerable versatility when compared with the general method. The formula method allows an easy recalculation of the process for different operating conditions (retort temperature-TRT, initial temperature-IT, etc.). However, whenever precision is important to minimize unnecessary overprocessing, process authorities turn to the General method. Use of this method typically results in process times approximately 15%–25% shorter than those calculated by the Ball formula method (Spinak and Wiley, 1982; Simpson et al., 2000, 2003b). In addition to improved product quality, this has important economic implication with respect to production capacity and energy utilization in the manufacturing process. Throughout the food canning industry, thermal processing in retorts has normally been carried out under carefully controlled constant retort temperature (CRT). However, recent work reported in the literature has raised interest in the use of variable retort temperature profiles (VRT) to potentially optimize product quality by maximizing nutrient retention in surface regions of the product or most importantly minimizing process time and energy consumption (Banga et al., 1991; Almonacid et al., 1993; Noronha et al., 1993; Durance, 1997; Simpson et al., 2004). Practical utilization of these variable temperature profiles is currently an area of study and research with a view of having these variable temperature profiles connected to automation and control of the commercial sterilization process.

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26.3 BATCH RETORT PROCESSING IN COMMERCIAL FOOD CANNERIES As stated earlier, batch retort processing has been widely practiced in commercial food canneries throughout the canned food industry. Although high speed processing with continuous rotary or hydrostatic retort systems can be found in very large canning factories, such systems are not economically feasible in the majority of small- to medium-sized canneries (Simpson et al., 2003a). In this chapter, batch retort processing will be analyzed with a retrospective and a prospective view. The industrial engineering approach to the batch processing problem structure will be defined in relation to canned food plants. This will be used in optimizing the design and operation of a food canning plant with specific procedures such as retort scheduling in a cook room with a battery of retorts as an example. Finally, we will try to look to the future to speculate on new advances that might be made in this large and diverse field.

26.3.1 PROBLEM STRUCTURE

FOR

BATCH PROCESSING

IN

CANNED FOOD PLANTS

Batch processing with a battery of individual retorts is a common mode of operation in many food-canning plants (canneries). Although high-speed processing with continuous rotary or hydrostatic retort systems can be found in very large canning factories (where they are cost justified by high volume throughput), such systems are not economically feasible in the majority of small- to medium-sized canneries (Norback and Rattunde, 1991). In such smaller canneries, retort operations are carried out as batch processes in a cook room in which the battery of retorts is located. Although the unloading and reloading operations for each retort are labor intensive, a well-designed and managed cook room can operate with surprising efficiency if it has the optimum number of retorts and the optimum schedule of retort operation. This type of optimization in the use of scheduling to maximize efficiency of batch processing plants has become well known and is commonly practiced in many process industries. Several models, methods, and implementation issues related to this topic have been published in the process engineering literature (Rippin, 1993; Barbosa and Macchietto, 1993; Kondili et al., 1993; Lee and Reklaitis, 1995a,b; Reklaitis, 1996). However, specific application to retort batteries in food-canning plants has not been addressed in the food process engineering literature. Food canneries with batch retort operations are somewhat unique in that the process line as a whole is usually a continuous process in that unit operations both upstream and downstream from the retort cook room are normally continuous (product preparation, filling, closing, labeling, case packing, etc.). Although retorting is carried out as a batch process within the cook room, unprocessed cans enter and processed cans exit the cook room continuously at the same rate (Figure 26.1). Since the entire process line operates continuously, food canneries are often overlooked as batch process industries. The focus of this chapter is to apply these batch process optimization techniques only to the retort operations within the cook room, and not the entire process line of the cannery.

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Continuous operated

Batch operated

Continuous operated

1 Containers 2 Container filling

Raw material preparation

Exhausting

Containers sealing

Covering liquid

.

Container packaging

. . . . NA Sterilization stage

FIGURE 26.1 General simplified flow diagram for a canning plant.

Labeling

Secondary packaging

Warehousing and packing

Food processing, and thermal processing in particular, is an industry confronted with strong global competitiveness. Continuous innovation and improvement of processing procedures and facilities is needed. Although the literature in food science and thermal processing is very extensive, most of the references deal with the microbiological and biochemical aspects of the process or with engineering analysis of a single unit process operation and rarely analyze the processing operations in the context of manufacturing efficiency. The early stages of a project usually involve studies of alternative processes, plant configurations, and type of equipment. Among problems confronted by canned food, plants with batch retort operations are peak energy=labor demand, underutilization of plant capacity, and underutilization of individual retorts.

26.3.2 BATCH RETORT OPERATION

IN

CANNED FOOD PLANTS

In batch retort operations, maximum energy demand occurs only during the first few minutes of the process cycle to accommodate the venting step while very little is needed thereafter in maintaining process temperature. Likewise, peak labor demand occurs only during loading and unloading operations and is not required during the holding time at processing temperature. In order to minimize peak energy demand, it is customary to operate the retorts in a staggered schedule so that no more than one retort is venting at any one time. Similar rationale applies to labor demand so that no more than one retort is being loaded or unloaded at any one time. Too few retorts in a battery can leave labor unutilized while too many will leave retorts unutilized. The optimum number will maximize utilization of labor and equipment thus minimizing ongoing processing costs. Alternatively, the optimum number of retorts may be based upon maximizing the economic rate of return on the capital investment in the project measured in terms of net present value which takes many additional factors beyond processing costs into account. In the case of maximizing output from a fixed number of retorts for different products and container sizes, isolethal processes can be identified for each of the various products (alternative combinations of retort temperature and process time that deliver the same lethality) from which a common set of process conditions can be chosen for simultaneous processing of different product lots in the same retort.

26.3.3 HIERARCHICAL APPROACH The hierarchical approach consists of successive refinements, and the design procedure is similar to the hierarchical planning strategy discussed in the artificial intelligence (AI) literature (Douglas, 1988). In contrast to normal true batch processes, canned food plants are operated with just one stage functioning in a batch mode. During normal operation of the sterilization stage (Figure 26.2), the various retort units are filled with cans, perform the retorting process for a specified period, and then they shut down and the cycle is repeated. As previously mentioned, in canned food plants, all units, with the exception of retorts, operate continuously. The distinction between batch and continuous processes are sometimes somewhat ‘‘fuzzy’’ (Douglas, 1988). According to the literature, when a plant has one or two

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Bleeder vent

Cooling water

Steam

Condensate

FIGURE 26.2 Still vertical retort (cross-sectional view of vertical retort used for study).

batch operations with large production rates that otherwise operate continuously, they are normally referred to as a continuous process. Although most of the food science and food engineering literature refers to a canning plant as a batch plant, when the sterilization stage is operated in batch mode, and the hierarchical approach is applied, it is assumed that it is better to classify it as a continuous process. The design effort will be to decide whether a concept is sufficiently promising from an economic point of view that a more detailed study could be justified. As shown earlier, in our specific case, the flow scheme of the process is presented in Figure 26.1. Although some exceptions to this flow scheme could be justified, the following analysis will consider it as a general flow scheme for canned food plants. The main target in the following sections will be to decide the optimum number of retorts that can be allocated in a canned food plant. The approach will be to identify a general procedure that can be applied to canned food plants. Finally, to decide and optimize canned food plant design and operation the Net Present Value (or Net Present Worth) profitability evaluation method should be utilized.

26.3.4 RETORT SCHEDULING Batch processing in food canneries consists of loading and unloading individual batch retorts with baskets or crates of food containers that have been filled and sealed just prior to the retorting operation. Each retort process cycle begins with purging of all the atmospheric air from the retort (venting) with inflow of steam at maximum flow rate, and then bringing the retort up to operating pressure=temperature, at which time the flow rate of steam falls off dramatically to the relatively low level required to maintain process temperature. The retort is then held at the process temperature for

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the length of time calculated to achieve the target lethality (Fo value) specified for the product. At the end of this process time, steam to the retort is shut off and cooling water is introduced to accomplish the cool down process after which the retort can be opened and unloaded. One of the factors that should be considered to decide retort scheduling is the energy demand profile during sterilization processing (Almonacid et al., 1993). In batch retort operations, maximum energy demand occurs only during the first few minutes of the process cycle to accomplish the high steam flow venting step. Very little steam is needed thereafter to compensate for the bleeder (and convection and radiation losses) in maintaining process temperature (Barreiro et al., 1984; Bhowmik et al., 1985). A typical representation of the energy demand profile during one cycle of a retort sterilization process is shown in Figure 26.3. As shown, at the initial stage of the process a high peak of energy consumption occurs (venting before reaching the retort temperature), later decreasing dramatically and finally reaching a low and constant value (convection, radiation, and bleeder). Thus, the energy demand for the whole plant will be conditioned upon this acute venting demand in the sterilization process of each retort operating cycle. To minimize the boiler capacity and maximize energy utilization, it is necessary to determine adequate scheduling for each individual retort. Likewise, peak labor demand occurs only during loading and unloading operations and is not required during the holding time at processing temperature. Therefore, a labor demand profile would have a similar pattern to the energy demand profile. In order to minimize this peak energy and labor demands, the retort must operate in a staggered schedule so that no more than one retort is venting at any one time, nor being loaded or unloaded at any one time. When a battery consists of the optimum number of retorts for one labor crew, the workers will be constantly loading and unloading a retort throughout the workday, and each retort will be venting in-turn one at a time. 4

200 180

Temperature

160

Energy

3.5

2.5

120

2

100 80

1.5

60 1 40 0.5

20

0

0 0

20

40 Time (min)

FIGURE 26.3 Energy consumption in a batch cycle.

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60

80

Energy (MJ)

Temperature (⬚C)

3 140

1

2 . Q Can/min

. .

Q Can/min

. . NA

Sterilization step

FIGURE 26.4 Diagram for operation of a battery with optimum number (NA) of retorts such that the cook room system operates with continuous inflow and outflow of product.

Under these optimum circumstances, unprocessed product will flow into and processed product will flow out of the retort battery system as though it were a continuous system as shown in Figure 26.4 while the energy profile will appear as in Figure 26.5. The optimum number of retorts in the battery will maximize utilization of labor and equipment, thus minimizing unit-processing costs. Too few retorts in a battery can leave labor unutilized while too many will leave retorts unutilized. A Gantt 6

Energy (MJ)

5 4 3 2 1 0 0

10

20

30

40 50 Time (min)

60

70

80

FIGURE 26.5 Energy demand profile from retort battery operating with optimum number of retorts and venting scheduling.

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No. of autoclaves

NA

tc

.

tp

td

. . . . . . .

tc

2 1

tc

tp

tp

td

td

tc

tp

td

Time

FIGURE 26.6 Gantt chart showing temporal programming schedule of the battery retort system operation.

chart showing the temporal programming schedule of the battery retort system (Figure 26.6) can be used as a first step in determining the optimum number of retorts. Optimum operation of the retort battery can be achieved if the loading step of the last retort starts at the same time as the first retort finishes its cycle and is ready for unloading. This means that the loading time multiplied by the number of retorts must fit within the total time to load, process, and unload one retort. This relationship can be expressed mathematically: tc þ tp þ td ¼ tc NA where NA is number of retorts tc, tp, and td are loading, process, and unloading times, respectively Considering that loading and unloading times are equal (tc ¼ td), we get: NA ¼ 2 þ

tp tc

(26:1)

Therefore 3  NA  1, and the minimum number of retorts for optimum operation under this criterion is 3. The number of retorts for any given situation will depend upon the ratio of process time to loading=unloading time. Moreover, according to the operation scheme presented in Figure 26.4, the following mathematical relationships can relate the plant production capacity (Q) to loading time and retort size: Qtc ¼ KV

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(26:2)

Rearranging Equation 26.2 and replacing tc from Equation 26.1, it is possible to obtain an expression for production capacity (Q) as a function of processing time (tp) and retort number (NA) as follows: Q¼

KV (NA  2) tp

(26:3)

From Equation 26.3, it is possible to infer that production capacity is directly influenced by process temperature because the higher the process temperature the shorter the process time, and the higher the production capacity (more batches per day).

26.4 CRITERIA FOR OPTIMAL DESIGN AND OPERATION OF BATCH RETORTS FOOD CANNING PLANTS 26.4.1 MAXIMIZING NET PRESENT VALUE

OF

CAPITAL INVESTMENT

A criterion to optimize plant design and operation is to determine the number of retorts that will maximize the net present value (NPV) of the invested capital for the new process line. This can be approached on the basis of microeconomics. Equation 26.4 is the expression for NPV: NPV ¼ I þ

n X j ¼1

bj (1 þ i) j

(26:4)

where two main terms can be distinguished, total investment (I) and annual benefits (bj). The total investment for the project will be expressed as the cost requirement in retorts, fittings, boiler, general equipment, construction and engineering, working capital, etc. Expressing the total investment mathematically: I ¼ IN þ IF þ IB þ IX

(26:5)

According to Guthrie (1969), the cost of equipment (retorts, etc.) could be expressed as being in proportion to its capacity. Therefore, in the specific case of retorts, a mathematical expression is C R ¼ kR V a ,

0 zlb then update zlb, the maximum number of boxes obtained so far. In this case, if zlb ¼ zub then exit to step 3. Step 3: Return the best found solution zlb. If zlb ¼ zub then the pattern to (L, W) is optimal. Figure 29.6 presents the pattern generated by algorithm 1 for the example (L, W, l, w) ¼ (20, 20, 4, 3) compared to the pattern generated by Smith and De Cani’s

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(a)

(b)

FIGURE 29.6 Patterns with 32 and 33 boxes obtained for (L, W, l, w) ¼ (20, 20, 4, 3) by Smith and De Cani’s four-block heuristic and Bichoff and Dowsland’s five-block heuristic (algorithm 1).

four-block heuristic. Since the pallet is squared, P ¼ Q ¼ {(0, 6), (1, 5), (2, 4), (3, 2), (4, 1), (5, 0)}. Note that this pattern is optimal given that the remaining area is less than the area of a box face. It is obtained utilizing the efficient partition (2, 4) at the four borders of the pallet, which yields (L1, W1) ¼ (8, 12), (L2, W2) ¼ (12, 8), (L3, W3) ¼ (4, 4), (L4, W4) ¼ (12, 8), and (L5, W5) ¼ (8, 12), according to Equations 29.11 through 29.15. Other block heuristics of 7 and 9 blocks and the diagonal solution are described in Nelissen (1994). Figure 29.7 illustrates patterns with more than 5 blocks. Note that none of them can be generated by algorithm 1. In the next section, we present a refinement of algorithm 1, which is able to produce first-order patterns of any number of blocks.

29.4.2 M&M HEURISTIC The B&D heuristic can be seen as an implicit enumeration method of a number of a guillotine and first-order nonguillotine cuts to the rectangle (L, W). As shown in Morabito and Morales (1998), this heuristic can be improved as follows: During the enumeration of algorithm 1 to (L, W), for each cut examined, apply recursively

(a)

(b)

FIGURE 29.7 Example (L, W, l, w) ¼ (42, 39, 9, 4): (a) Pattern with 44 boxes obtained by algorithm 2 and (b) an optimal pattern with 45 boxes obtained by algorithm 3.

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algorithm 1 to each block (Li, Wi) generated by this cut, and so on, until all blocks generated by all cuts are so small that the algorithm can not be applied. Note that this is a simple recursive procedure based on the nonguillotine B&D heuristic (recursive procedures for guillotine problems were explored by Herz, 1972). The procedure can be represented as a tree search; each node of the tree corresponds to a block, and each set of a maximum of 5 arcs emerging from a node corresponds to a set of guillotine cuts or a first-order cut. Each time algorithm 1 is applied to a block, the search goes down one level in the tree. If the algorithm is always applied to the most recent generated node, the tree is traversed according to a depth-first search strategy (in fact, the recursive procedure utilizes a backtracking strategy which becomes more evident with the description of algorithm 2 below). The depth of the tree depends on the sizes of the boxes and the pallet. Let n be the level of a node in the tree. In problems where the depth of the tree is large, memory limitations may require imposing an upper bound N on the level of the traversed nodes. In this way, the B&D heuristic can be seen as the particular case of the recursive procedure when N ¼ 1. Let us redefine zi in (Equation 29.8) as zi ¼ max {bLi =lcbWi =wc, bLi =wcbWi =lc},

if i ¼ 1, . . . , 5

(29:21)

In the sequel, we present a procedure (algorithm 2) which can be easily coded in a computer language supporting recursion, such as Pascal and C. Note that the routine B&D (L0, W0, n) is recursive (see step 2.2.2.1 of the algorithm) and has the parameters: length L0  L, width W0  W, and level n of the block or node in the tree. Unlike algorithm 1, the best solution produced by algorithm 2 corresponds to a firstorder pattern not limited to 5 blocks. Algorithm 2.

The M&M recursive procedure

Initialization: Determine the sets P in (Equation 29.9) and Q in (Equation 29.10) to (L, W). Make z ¼ B & D(L, W, 1), i.e., call the recursive routine B&D with the pallet size at level 1. Routine B&D(L0, W0, n): Returns the maximum number of faces in the block (L0, W0) at level n: Step 1: Determine the subsets of P and Q to (L0, W0), say P(L0) and Q(W0). Compute zlb in Equation 29.18 and zub minimum of Equation 29.20 and Barnes’ bound to (L0, W0). If zlb ¼ 0 or zlb ¼ zub then go to step 3, otherwise go to step 2. Step 2: For each (r1, s1) 2 P(L0) and (r2, s2) 2 P(L0) so that r2l þ s2w  r1l þ s1w, and for each (t1, u1) 2 Q(W0) and (t2, u2) 2 Q(W0) so that t2l þ u2w  t1l þ u1w, do: Step 2.1: Determine the sizes (Li, Wi), i ¼ 1, . . . , 5 according to Equations 29.11 through 29.15. Step 2.2: If L1, L4, W4, and W5 are not null and there is no overlap (i.e., if Equations 29.16 through 29.17 are false), then:

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Step 2.2.1: If the depth limit of the treePis not reached (i.e., if n < N): Step 2.2.2.1: Then, determine z0 ¼ 5i¼1 B&D(Li , Wi , n þ 1). P Step 2.2.2.2: Else, determine z0 ¼ 5i¼1 zi using Equation 29.21. Step 2.2.2: If z0 > zlb then update zlb, the maximum number of boxes obtained so far to (L0,W0). In this case, if zlb ¼ zub then exit to step 3. Step 3: Return the best found solution zlb. If zlb ¼ zub, then the pattern to (L0, W0) is optimal. It should be remarked that even for N ¼ 1, algorithm 2 is heuristic, i.e., it does not ensure that the optimal nonguillotine pattern of the MPL can be found. Moreover, it does not ensure the best first-order nonguillotine pattern of the MPL can be found. The last remark can be illustrated by the following example. Consider the instance (L, W, l, w) ¼ (42, 39, 9, 4) previously considered in Figure 29.2, and for which P ¼ {(0, 10), (1, 8), (2, 6), (3, 3), (4, 1)} and Q ¼ {(0, 9), (1, 7), (2, 5), (3, 3), (4, 0)}. Figure 29.7a and b respectively illustrate the best pattern obtained by algorithm 2 without limiting the tree depth (i.e., N ¼ 1) and an optimal pattern for the instance (note its similarity with the pattern of Figure 29.2b). Both patterns are first-order type, however, the optimal pattern lays out one more box. To be obtained, the latter pattern involves examining a partition (r, 4) in the upper border of the pallet (Figure 29.7b), which does not exist in P. Morabito and Morales (1998) modify algorithm 2 in order to ensure that it does find the best first-order nonguillotine pattern of the MPL at the cost of a higher computational effort. This can be done as follows: in step 2, instead of examining just the efficient partitions (sets P and Q in Equations 29.9 and 29.10), the algorithm should examine all points of the sets X and Y in Equations 29.1 and 29.2 (or, similarly, the raster points of sets X0 and Y0 in Equations 29.3 and 29.4). Taking this into account, in the example of Figure 29.7, we obtain X and Y as illustrated in Section 29.3 (with jXj ¼ 27 and jYj ¼ 24 elements, respectively); since the partitions of the upper border of the pallet now belong to X, the optimal pattern can be obtained. In order to implement this refinement, expressions 29.11 through 29.15 are substituted as follows. For each x1 2 X, x2 2 X, y1 2 Y, and y2 2 Y, the following five blocks are examined: (L1 , W1 ) ¼ (x1 , W  W4 )

(29:22)

(L2 , W2 ) ¼ (L  L1 , W  W5 )

(29:23)

(L3 , W3 ) ¼ (L  L1  L5 , W  W2  W4 )

(29:24)

(L4 , W4 ) ¼ (x2 , y1 )

(29:25)

(L5 , W5 ) ¼ (L  L4 , y2 )

(29:26)

This set of expressions allow algorithm 2 to find the optimal pattern of Figure 29.7b for N  3. Before presenting this improved version of algorithm 2, let us discuss how step 2.2 can be modified to reduce the search by discarding symmetric patterns (note that symmetric patterns can be discardable since they lay out the same number of

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x1 1 y1

3

5 y2

4 x2

(a)

1 (b)

x1 2

4

y1 x2

y2

2 5

x1 1 y1 = 0 (c)

5

(d)

y2

x2 x1 1

y1 = 0

2

x2

2

2

y2 = 0

y1

5

y2

x1 = 0 x2 = 0

FIGURE 29.8 (a) 5-block, (b) 4-block, (c) 3-block, and (d) 2-block symmetries.

boxes). Figure 29.8 depicts four types of symmetry: 5-block, 4-block, 3-block, and 2-block symmetries. They are avoided if x1, x2, y1, and y2 in Equations 29.22 through 29.26 are such that one of the four (mutually exclusive) rules is satisfied: 5-blocks: x1 > 0, y1 > 0, x2 > x1, y2 > y1 and either x1 þ x2 < L or (x1 þ x2 ¼ L and y1 þ y2  W). 4-blocks: x1 > 0, y1 > 0 and either (x2 ¼ x1, y2 > y1 and x1  bL=2c) or (x2 > x1, y2 ¼ y1 and y1  bW=2c) or (x2 ¼ x1, y2 ¼ y1, x1  bL=2c and y1  bW=2c). 3-blocks: either (x1 > 0, y1 ¼ 0, x2 ¼ x1 and 0 < y2  bW=2c) or (x1 ¼ 0, y1 > 0, 0 < x2  bL=2c and y2 ¼ y1). 2-blocks: either (0 < x1  bL=2c, y1 ¼ 0, x2 ¼ x1 and y2 ¼ 0) or (x1 ¼ 0, 0 < y1  bW=2c, x2 ¼ 0 and y2 ¼ y1). These rules also prevent overlap. Step 2.2.2.1 in algorithm 2 can be also modified to avoid unnecessary recursions. Before computing all B&D(Li, Wi, n þ 1), i ¼ 1, . . . , 5, we can check if the best current solution to (L0, W0), zlb, is attainable by z0. While there exists a chance to find a solution better than zlb, we continue evaluating each B&D(Li, Wi, n þ 1); otherwise, we abandon the current examination. The modified recursive procedure is presented below as algorithm 3. Similarly to algorithm 2, the routine B&D(L0, W0, n) is recursive (step 2.1.2.1 of the algorithm) and has the parameters length L0  L, width W0  W, and level n of the block or node in

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the tree. The solution produced by algorithm 3 with N ¼ 1 corresponds to the best first-order nonguillotine pattern to (L, W). Algorithm 3.

The M&M modified recursive procedure

Initialization: Determine the sets X in Equation 29.1 and Y in Equation 29.2 to (L, W). Make z ¼ B&D(L, W, 1), i.e., call the recursive routine B&D with the pallet size at level 1. Routine B&D(L0, W0, n): Returns the maximum number of faces in the block (L0, W0) at level n: Step 1: Determine the subsets of X and Y to (L0, W0), say X(L0) and Y(W0). Compute zlb in Equation 29.18 and zub (minimum of Equation 29.20 and Barnes’ bound) to (L0, W0), where L* and W* in Equation 29.20 are defined accordingly. If zlb ¼ 0 or zlb ¼ zub then go to step 3, otherwise go to step 2. Step 2: For each x1 2 X(L0) and x2 2 X(L0) so that x2  x1, and for each y1 2 Y(W0) and y2 2 Y(W0) so that y2  y1, do: Step 2.1: If one of the four rules (5-blocks, 4-blocks, 3-blocks, or 2-blocks) holds, then: Step 2.1.1: Determine the sizes (Li, Wi), i ¼ 1, . . . , 5 according to Equations 29.22 through 29.26. Step 2.1.2: If the depth limit of the tree is not reached (i.e., if n < N): Step 2.1.2.1: Then make z0 ¼ 0 and compute the upper bound zub for i each (Li,W Pi), say zub . If zlb  5i¼1 ziub then go to step 2.1.3, else determine z1 ¼ B&D (L1, WP 1, n þ 1). If zlb  5i¼2 ziub then go to step 2.1.3, else determine z2 ¼ B&D (L2, WP 2, n þ 1). P If zlb  2i¼1 zi þ 5i¼3 ziub then go to step 2.1.3, else determine z3 ¼ B&D(L 3, WP 3, n þ 1). P If zlb  3i¼1 zi þ 5i¼4 ziub then go to step 2.1.3, else determine z4 ¼ B&D(L P 4, W4, n þ 1). If zlb P 4i¼1 zi þ z5ub then go to step 2.1.3, else determine n þ 1). z0 ¼ 4i¼1 zi þ B&D(L5 ,W5 , P Step 2.1.2.2: Else determine z0 ¼ 5i¼1 zi using Equation 29.21. Step 2.1.3: If z0 > zlb then update zlb, the maximum number of boxes obtained so far to (L0, W0). In this case, if zlb ¼ zub then exit to step 3. Step 3: Return the best found solution zlb. If zlb ¼ zub then the pattern to (L0, W0) is optimal. Morabito and Morales (1998) considers two additional modifications in algorithm 3 in order to improve its computational performance, motivated by the discussion in Herz (1972): 1. In step 1, there is a tradeoff between the undesirable cost of obtaining high quality lower bounds zlb to (L0, W0) and the benefits of their savings on the search process. Computational experiments showed that it is very effective

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to start with a good lower bound. Thus, for (L0, W0) ¼ (L, W),zlb is obtained by calling algorithm 1 instead of computing it from Equation 29.18. 2. Tree search may still involve multiple B&D callings for the same node (L0, W0), despite being drastically reduced by step 2.1. After the first time B&D(L0, W0) is computed, we can update the lower bound zlb of (L0, W0), store its value, and simply retrieve it each time B&D(L0, W0) is re-called. Instead of O(LW) numbers, the memory requirements are O(jXjjYj) since only the values with respect to the elements in X [ {L*} and Y [ {W*} must be stored (these requirements can be even diminished if we use the raster points). These considerations are similarly applied for computing the upper bound zub of each (L0, W0). The fact of saving information of the already solved subproblems (of the current problem being solved) is related to the idea of creating a pool of solutions as described in Brunetta and Gregoire (2005). Note that algorithms 2 and 3 perform a backtracking search that traverse the tree without using additional information which could indicate the most promising paths to be first followed and would enable the search to be (eventually) more efficient. Although not considered here, the use of informed search strategies such as best-first search is an appealing line of research. Further refinements of the recursive five-block heuristic to improve its computational performance are discussed in Birgin et al. (2007). The extension of the recursive five-block heuristic to the three-dimensional case of packing boxes of sizes (l, w, h) into a container (L, W, H) is studied in Lins et al. (2002). In this case, the procedure becomes a recursive nine-block heuristic.

29.5 COMPUTATIONAL EXPERIMENTS AND RESULTS Morabito et al. (2000) analyze the application of algorithm 3 to solve problems of arranging products (packed in boxes) on pallets, and arranging loaded pallets on trucks. Algorithm 3 was coded in Pascal language and the solutions of the experiments were obtained in a few seconds using a Pentium microcomputer. We initially discuss the outcomes obtained by applying the algorithm to solve thousands of randomly generated realistic examples. Then, in order to assess the effectiveness of the solutions in practice, we analyze the results of the approach applied to two Brazilian case studies: a food company distribution center and a wholesale distribution center. The use of the approach to optimize the sizes of packages, pallets, and trucks is also discussed. In the following tables, the symbol U denotes the area utilization (as a percentage) and BL the number of boxes per layer for each specified set of examples. Associated averages and standard deviations are respectively referred to as U, BL, su and sB.

29.5.1 RANDOM EXPERIMENTS In addition to optimizing the loading patterns, the MPL solutions can be used for either the optimal pallet sizing or the selection of the best among a set of standard pallets. In Morabito et al.’s (2000) study, 30 independent samples of 1000 random

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TABLE 29.1 Computational Performance of Algorithm 3 for 30 Samples of 1000 Random Examples (Pallets P1–P4) Pallet P1 P2 P3 P4

L (mm)

W (mm)

L=W

Area (m2)

Average of U (%)

Standard Deviations of U, s U

95% Confidence Interval

1200 1100 1200 1200

1000 1100 800 1200

1.2 1.0 1.5 1.0

1.20 1.21 0.96 1.44

84.3 80.9 78.9 83.5

0.2 0.3 0.4 0.3

[84.2, 84.4] [80.8, 81.1] [78.6, 79.2] [83.3, 83.7]

examples were generated (a total of 30,000 examples), and for each example, algorithm 3 (with depth limit N ¼ 3) was applied to obtain the utilization of the pallet area. The dimensions (l, w) of each example were uniformly sorted from the intervals 200  l  600 and 150  w  450, as suggested by Wright (1984). These ranges of box dimensions are typical in the United States as well as in Brazil. Indeed, the data of the two companies studied below also belong to these intervals. Table 29.1 presents the average results of the 30 samples for four different pallets (P1–P4). Pallet P1, of size 1200
1000 mm, is the PBR and the ISO series 2 pallet recommended since 1980 (Section 29.2); pallet P2, of size 1100
1100 mm, is the ISO series 1 pallet; pallet P3, of size 1200
800 mm, is the Europallet adopted by UIC (Union International des Chemins de Fer) since 1961, and pallet P4 is a hypothetical pallet of size 1200
1200 mm. Results comprise for each pallet, the average of the 30 U values, and their corresponding standard deviation (s U ) and 95% confidence interval. Pallet P1 yielded the highest average of mean utilization (84.3%), followed by pallets P4, P2, and P3 (note in Table 29.1 that the 95% confidence intervals do not overlap). Moreover, pallet P1 had the smallest standard deviation of mean utilization, which means that the variability of the box lengths and widths had less impact on the mean utilization of pallet P1, in comparison with the other pallets. Table 29.2 presents the average results of one of the 30 samples in Table 29.1. The numbers in the penultimate column correspond to U and the associated sU (not to be confused with the standard deviation of the average mean utilization of Table 29.1). As expected, for all pallets P1–P4, the mean utilization is near the average of 30 mean utilizations while the standard deviation (of utilization) is much higher than the standard deviation of the average mean utilization in Table 29.1. Besides pallets P1–P4, Table 29.2 also includes pallets P5–P8 (pallets P9–P12 are discussed later) with the same area as the square pallet P2 (i.e., 1.21 m2), but different ratios L=W varying from 1.25 to 2. Pallets P5–P8 can be seen as ‘‘rectangularizations’’ of pallet P2. The last column in Table 29.2 presents BL and their corresponding standard deviation sB (in brackets). Observe that the standard deviation is large in comparison with the mean, since the sample contains different box sizes from (200, 150) to (600, 450) mm. Note that pallet P5(L=W ¼ 1.25) had the best performance (84.1%) among the pallets of the same area (P2 and P5–P8), almost as good as pallet P1(L=W ¼ 1.2).

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TABLE 29.2 Computational Performance of Algorithm 3 for a Sample of 1000 Random Examples with [l, w, sl, sw] ¼ [400, 300, 118, 86] (Pallets P1–P12) Pallet P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12

L (mm)

W (mm)

L=W

Area (m2)

U [sU] (%)

BL [sB] (Units=Layer)

1200 1100 1200 1200 1230 1347 1455 1556 1154 1180 1205 1255

1000 1100 800 1200 984 898 831 778 1049 1026 1004 965

1.2 1.0 1.5 1.0 1.25 1.5 1.75 2.0 1.10 1.15 1.20 1.30

1.20 1.21 0.96 1.44 1.21 1.21 1.21 1.21 1.21 1.21 1.21 1.21

84.4 [8.7] 81.0 [11.5] 78.4 [13.2] 83.3 [10.6] 84.1 [8.6] 83.0 [8.9] 82.6 [10.9] 80.9 [13.0] 83.2 [9.7] 83.8 [9.0] 84.3 [8.7] 83.6 [9.1]

10.2 [5.3] 10.0 [5.5] 7.7 [4.3] 12.2 [6.5] 10.3 [5.4] 10.1 [5.5] 10.1 [5.5] 10.0 [5.5] 10.3 [5.6] 10.3 [5.4] 10.3 [5.3] 10.3 [5.5]

Mean area utilization (%)

To investigate if the optimal performance for this sample would be obtained by a pallet with a ratio close to 1.2, four other pallets, P9–P12, were defined with the same area as pallet P2 (1.21 m2), but different ratios L=W varying from 1.1 to 1.3, as shown in Table 29.2. Note that pallet P11, with L=W ¼ 1.2, has the highest U (84.3%) among pallets P5–P12, which is very close to pallet P1’s utilization (84.4%). Figure 29.9 depicts the performance of the pallets of equal area, suggesting that the best pallet for the present sample data (l ¼ 400, w ¼ 300) is very close to the PBR ratio, L=W ¼ 1.2. Observe that the optimal ratio does not need to be equal to the expected box length=width ratio of the data l=w ¼ 1.33. Morabito et al. (2000) also generated an additional set of 1000 examples with both l and w uniformly sorted from the interval [150, 600] (yielding l=w ¼ 1.0) to show that the result of Figure 29.9 is not strongly related to the fact that the box ratio used in the experiments averaged close to 1.2. The mean area utilization for pallets

85 84 83 82 81 80 79 78 1

1.1

1.15

1.2

1.25

1.3

1.5

1.75

2

L /W

FIGURE 29.9 Performance of algorithm 3 with pallets P2 and P5–P12 (Area ¼ 1.21 m2).

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P2 (L=W ¼ 1.0), P9 (L=W ¼ 1.1), P10 (L=W ¼ 1.15), P11 (L=W ¼ 1.2), P5 (L=W ¼ 1.25), and P12 (L=W ¼ 1.3) are 79.7%, 82.2%, 82.4%, 82.5%, 81.7%, and 80.9%, respectively, indicating that the optimal ratio for this data set (l ¼ 375, w ¼ 375) is also very close to L=W ¼ 1.2. It should be remarked that this simple approach could be useful for supporting decisions in either the design of pallets as a function of the product mix of the company, or the selection of the most appropriate pallet (among the set of candidate pallets) for that product mix.

29.5.2 CASE STUDIES

OF

LOADING PACKED PRODUCTS

ON

PALLETS

The effectiveness of the approach in practice is also reported in Morabito et al. (2000), Morabito and Farago (2002) and Oliveira and Morabito (2006), where algorithm 3 was applied to the loading of packed products on pallets in different case studies. In this section, we analyze the results obtained for two of them. The first case study was performed in one of the distribution centers of a food company (company A), and the second refers to a wholesale distribution center (company B). A sample of 148 and 78 products, together with their corresponding loading patterns, was randomly collected in each company, respectively. Table 29.3 presents U and BL (and corresponding standard deviations in brackets) obtained by the algorithm for pallets P1–P4 with the data of companies A and B. It is worth noting that values of U in Table 29.3 are higher than those of Table 29.2, suggesting that the random samples of Section 29.2 represent more unfavorable instances than those found in practice. However, the relative performance between pallets P1–P4 in Table 29.2 was approximately maintained in Table 29.3. Observe that pallet P1 had again the best performance (i.e., 86.0% and 89.5%, respectively), followed by pallets P2 or P4, and pallet P3. In fact, both companies were utilizing pallet P1 before this study, but with loading patterns worse than those of Table 29.3.

TABLE 29.3 Computational Performance of Algorithm 3 for a Sample of 148 Products of Company A and 78 Products of Company B (Pallets P1–P4) Sample Source [l, w, sl, sw] Company A [436, 279, 85, 61]

Company B [330, 270, 107, 86]

Pallet

L (mm)

W (mm)

L=W

Area (m2)

U [sU] (%)

BL [sB] (Units=Layer)

P1 P2 P3 P4 P1 P2 P3 P4

1200 1100 1200 1200 1200 1100 1200 1200

1000 1100 800 1200 1000 1100 800 1200

1.2 1.0 1.5 1.0 1.2 1.0 1.5 1.0

1.20 1.21 0.96 1.44 1.20 1.21 0.96 1.44

86.0 [7.3] 84.7 [9.6] 80.1 [10.1] 84.4 [8.1] 89.5 [6.2] 86.6 [8.7] 86.0 [8.5] 88.3 [7.3]

9.9 [5.7] 9.8 [5.9] 7.4 [4.7] 11.7 [6.9] 15.1 [8.2] 14.9 [8.3] 11.7 [6.5] 17.9 [9.8]

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The results were further analyzed by separating the samples in two disjoint sets: 1. Set of products whose loading patterns have sizes less than or equal to the pallet size 1200
1000 mm. 2. Set of products whose loading patterns have sizes greater than 1200
1000 mm. In this case, the usual tolerance of both companies is 25 mm in each border of the pallet. Therefore, let us define pallet P13 as pallet P1 with a border tolerance of 25 mm, i.e., a pallet of size 1250
1050 mm. Table 29.4 compares the average results of the two companies with the average results obtained by the algorithm for both pallets P1 and P13. The loading patterns were produced with sets of products of each company with mean values l and w, and corresponding standard deviations sl and sw. The algorithm produced a mean utilization ranging from 88.1% to 91% of the total area, corresponding to gains relative to the company’s solution of 0.6% and 3.6%. The smallest gain corresponds to only 7 products (out of 71). Figure 29.10 illustrates the corresponding patterns of one of these products. In the same way, the largest difference corresponds to 22 products (out of 56) for which the algorithm found better loading patterns. The difference between the algorithm (for pallet P13) and company A results was 1.4% (87.3%–85.9%), corresponding to 22 out of 77 products. For company B, that difference was 2.6% (90.5%–87.9%), corresponding to 9 out of 22 products. It is interesting to note that there are patterns found by the algorithm for pallet P1 (i.e., without border tolerance) which load the same number of boxes per layer as patterns used by the companies, requiring border tolerances. Figure 29.11 illustrates an example found in Table 29.4 for the 22 product sample of company B—observe that both patterns arrange the same number of boxes (i.e., 21), however, the pattern used by the company has a larger size (1200
1035 mm) than the one found by the algorithm. It is worth remarking that the sizes of the boxes sampled in companies A and B belong to the intervals of Section 29.5.1 suggested by Wright (1984).

29.5.3 CASE STUDIES

OF

LOADING PALLETS

ON

TRUCKS

Morabito et al. (2000) also analyze the application of algorithm 3 to load standard pallets on standard trucks. The input data now becomes the pallet size (L,W) and the truck size (L, V), where L and V are, respectively, the effective internal length and width of the trucks. As before, the problem consists of arranging the maximum number of rectangles (L, W) and (W, L) into the larger rectangle (L, V), i.e., a quadruple (L, V, L, W). In those experiments, the trucks are three semi-trailers (here denoted by T1–T3) commonly used in the Brazilian trucking industry. Table 29.5 presents, for each pallet P1–P4, the trucks’ area utilization U (in percentage) and the number of pallets per layer BL obtained by the algorithm for T1–T3. Results indicate that for all vehicles, pallet P1 resulted in a utilization of the truck area better than pallet P2. The difference between their utilization can be substantial; reaching as much as 6%.

ß 2008 by Taylor & Francis Group, LLC.

ß 2008 by Taylor & Francis Group, LLC.

TABLE 29.4 Mean Results Obtained by the Algorithm and Companies A and B with Different Samples of Products Sample Source  sl, sw] [l, w,

Pallet

L (mm)

W (mm)

L=W

Area (m2)

Company A 71 Products [433, 272, 85, 65] Company B 56 Products [361, 233, 101, 62] Company A 77 Products [437, 285, 84, 57]

P1

1200

1000

1.2

1.20

P1

1200

1000

1.2

1.20

P1 P13

1200 1250

1000 1050

1.2 1.19

1.20 1.31

Company B 22 Products [320, 292, 104, 80]

P1 P13

1200 1250

1000 1050

1.20 1.19

1.20 1.31

Solution Source

U [sU] (%)

BL [sB] (Units=Layer)

Algorithm Company A Algorithm Company B Algorithm Algorithm Company A Algorithm Algorithm Company B

88.1 [7.3] 87.5 [7.1] 91.0 [4.8] 87.4 [6.2] 84.0 [6.8] 87.3 [6.0] 85.9 [4.6] 85.8 [7.2] 90.5 [3.6] 87.9 [5.9]

10.6 [6.7] 10.5 [6.5] 15.9 [8.5] 15.1 [7.9] 9.2 [4.5] 10.5 [5.3] 10.6 [4.9] 13.4 [7.5] 15.2 [7.7] 14.8 [7.6]

(a)

(b)

FIGURE 29.10 Loading patterns for (L, W, l, w) ¼ (1200, 1000, 348, 208): (a) algorithm’s solution (15 boxes per layer) and (b) company A’s solution (14 boxes per layer).

Examining not only the loading of products on pallets but also the loading of pallets on trucks, allows us to obtain global utilization indices which are useful for evaluating the economic performance of unit load systems in the logistics chain of a company. Based on the results in Table 29.5, we can calculate these indices for the palletized cargo of each company. They are computed by simply multiplying the mean utilization indices of the pallet area (Table 29.4) by the utilization indices of the truck area (Table 29.5). Table 29.6 presents these results for both companies A and B. Table 29.6 shows that pallet P1 dominates the other pallets in terms of global performance. The difference in total area utilization varies from 5% to 9%, i.e., it can be rather substantial. From the point of view of loading pallets onto trucks, note in Table 29.5 that pallet P3 has the best utilization indices. However, as we consider the area utilization with respect to box loading, pallet P1 outperforms pallet P3. It is worth noting that, besides optimizing the loading of pallets on trucks, this simple approach can also be useful in the selection of the most appropriate trucks as a function of the adopted pallets or, vice versa, in the selection of the most adequate pallets as a function of the available trucks. Similarly, the MPL solutions can be applied to help the packaging design as a function of the pallets utilized along the logistics chain of the company. In this case, the short computer runtimes of the

(a)

(b)

FIGURE 29.11 Loading patterns for (a) (L, W, l, w) ¼ (1200, 1000, 345, 160)–algorithm’s solution (21 boxes per layer) and (b) (L, W, l, w) ¼ (1200, 1035, 345, 160)–company B’s solution (21 boxes per layer).

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TABLE 29.5 Area Utilization and Number of Pallets Per Layer for Trucks T1–T3 Pallet Truck

T1 T2 T3

P2

P1

P3

P4

L (mm)

V (mm)

U (%)

BL (Units= Layer)

U (%)

BL (Units= Layer)

U (%)

BL (Units= Layer)

U (%)

BL (Units= Layer)

14,370 13,370 12,470

2,480 2,480 2,480

93.9 93.9 92.8

28 26 24

87.9 87.4 85.7

26 24 22

93.9 95.3 93.9

35 33 31

88.5 95.3 92.8

22 22 20

algorithm allow the evaluation of many alternative packaging sizes (l, w) by means of an explicit enumeration of all possible technically feasible combinations of the values of l and w.

29.6 CONCLUDING REMARKS This chapter has addressed the optimization of palletized cargo loading particularly the case in which the products are packed in identical boxes. We have reviewed optimization models and solution approaches and described recursive heuristic procedures that aim to maximize the pallet’s space utilization. Experiments with random experiments and two case studies in food company and wholesale distribution centers illustrated how a relatively simple approach can be useful not only for the effective loading of products on pallets but also for supporting decisions in the loading of pallets on trucks, and in the design or selection of packages, pallets and trucks. While the state-of-the-art points out the manufacturer’s pallet problem as successfully solved in practice, the benefits that an analysis based on global performance measures can bring to more strategic decisions should be emphasized. We have shown that by extending the scope of the reported solution approaches from

TABLE 29.6 Global Performance of the Palletized Cargo for Companies A and B Truck Sample Source Company A

Company B

T1 T2 T3 T1 T2 T3

Pallet

L (mm)

V (mm)

P1 (%)

P2 (%)

P3 (%)

P4 (%)

14,370 13,370 12,470 14,370 13,370 12,470

2,480 2,480 2,480 2,480 2,480 2,480

80.7 80.7 79.8 84.0 84.0 83.0

74.5 74.0 72.6 76.0 75.7 74.3

75.2 76.4 75.2 80.7 81.9 80.7

74.7 80.5 78.3 78.1 84.1 81.9

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the loading of products on pallets to the loading of pallets on trucks, global utilization indices can be obtained and used for evaluating the economic performance of different unit load systems in the logistics chain.

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Dyckhoff, H., Scheithauer, G., and Terno, J., Cutting and packing, in Annotated Bibliographies in Combinatorial Optimisation, Amico, M., Maffioli, F., and Martello, S., Eds., John Wiley and Sons, New York, 1997, p. 393. ESICUP—Euro Special Interest Group on Cutting and Packing. Available at: < http:==www. apdio.pt=esicup= > (accessed in 2007). Young-Gun, G. and Kang, M., A fast algorithm for two-dimensional pallet loading problems of large size, Eur. J. Oper. Res., 134, 193, 2001. Ghiani, G., Laporte, G., and Musmanno, R., Introduction to Logistics Systems Planning and Control, John Wiley and Sons, New York, 2004. Hadjiconstantinou, E. and Christofides, N., An exact algorithm for general, orthogonal, twodimensional knapsack problems, Eur. J. Oper. Res., 83, 39, 1995. Herbert, A. and Dowsland, K., A family of genetic algorithms for the pallet loading problem, Ann. Oper. Res., 63, 415, 1996. Herz, J.C., Recursive computational procedure for two-dimensional cutting, IBM J. Res. Dev., 16, 462, 1972. Hifi, M., Special issue: Cutting and packing problems, Studia Informatica Universalis, 2, 1, 2002. Hodgson, T., A combined approach to the pallet loading problem, IIE Trans., 14, 176, 1982. Letchford, A. and Amaral, A., Analysis of upper bounds for the pallet loading problem, Eur. J. Oper. Res., 132, 582, 2001. Lins, L., Lins, S., and Morabito, R., An n-tet graph approach for non-guillotine packings of n-dimensional boxes into an n-container, Eur. J. Oper. Res., 141, 421, 2002. Lins, L., Lins, S., and Morabito, R., An L-approach for packing (l,w)-rectangles into rectangular and L-shaped pieces, J. Oper. Res. Soc., 54, 777, 2003. Lodi, A., Martello, S., and Monaci, M., Two-dimensional packing problems: a survey, Eur. J. Oper. Res., 141, 241, 2002. Martello, S., Special issue: Knapsack, packing and cutting, Part II: Multidimensional knapsack and cutting stock problems, INFOR, 32, 4, 1994. Martins, G.H., Packing in two and three dimensions, PhD Dissertation, Naval Postgraduate School, CA, 2002. Martins, G. and Dell, R., The minimum size instance of a pallet loading problem equivalence class, Eur. J. Oper. Res., 179, 17, 2007. Morabito, R. and Arenales, M., An and=or-graph approach to the container loading problem, Int. Trans. Oper. Res., 1, 59, 1994. Morabito, R. and Farago, R., A tight Lagrangean relaxation bound for the manufacturer’s pallet loading problem, Studia Informatica Universalis, 2, 57, 2002. Morabito, R. and Morales, S., A simple and effective recursive procedure for the manufacturer’s pallet loading problem, J. Oper. Res. Soc., 49, 819, 1998. Morabito, R. and Morales, S., Errata: A simple and effective recursive procedure for the manufacturers’ pallet loading problem, J. Oper. Res. Soc., 50, 876, 1999. Morabito, R., Morales, S., and Widmer, J.A., Loading optimization of palletized products on trucks, Transpor. Res. Part E, 36, 285, 2000. Nelissen, J., Solving the pallet loading problem more efficiently, Working Paper, Graduiertenkolleg Informatik und Technik, Aachen, 1994. Nelissen, J., How to use structural constraints to compute an upper bound for the pallet loading problem, Euro. J. Oper. Res., 84, 662, 1995. Oliveira, L.K. and Morabito, R., Métodos exatos baseados em relaxações Lagrangiana e surrogate para o problema de carregamento de paletes do produtor, Pesqui. Operacional, 26, 403, 2006. Pureza, V. and Morabito, R., Some experiments with a simple tabu search algorithm for the manufacturer’s pallet loading problem, Comp. Oper. Res., 33, 804, 2006.

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Ribeiro, G.M. and Lorena, L.A.N., Optimizing the woodpulp stowage using Lagrangean relaxation with clusters, J. Oper. Res. Soc., 59, 600, 2008. Scheithauer, G. and Terno, J., The G4-heuristic for the pallet loading problem, J. Oper. Res. Soc., 47, 511, 1996. Scheithauer, G. and Sommerweiss, G., 4-block heuristic for the rectangle packing problem, Euro. J. Oper. Res., 108, 509, 1998. Smith, A. and De Cani, P., An algorithm to optimize the layout of boxes in pallets, J. Oper. Res. Soc., 31, 573, 1980. Steudel, H., Generating pallet loading patterns: A special case of the two-dimensional cutting stock problem, Manage. Sci., 10, 997, 1979. Pallet, Associação Brasileira de Supermercados, Superhiper, 16, 7, 1990. Sweeney, P. and Paternoster, E., Cutting and packing problems: a categorized, applicationoriented research bibliography, J. Oper. Res. Soc., 43, 691, 1992. Tarnowski, A., Terno, J., and Scheithauer, G., A polynomial-time algorithm for the guillotine pallet loading problem, Infor, 32, 275, 1994. Tecnologística, A logística do big mac, 1, 8, 1996. Tecnologística, Custos logísticos na economia brasileira (Logistic costs in the Brazilian economy), January, 64, 2006. Tsai, R., Malstrom, E., and Kuo, W., Three dimensional palletization of mixed box sizes, IEEE Trans., 25, 64, 1993. Wascher, G., Haussner, H., and Schumann, H., An improved typology of cutting and packing problems, Euro. J. Oper. Res., 183, 1109, 2007. Wascher, G. and Oliveira, J.F., Special issue on cutting and packing, Eur. J. Oper. Res., 183, 1106, 2007. Wang, P. and Waescher, G., Special issue on cutting and packing, Eur. J. Oper. Res., 141, 239, 2002. Wilson, R., 17th Annual State of Logistics Report, CSCPM, June, 2006. Wright, P., Pallet loading configurations for optimal storage and shipping, Paperboard and Packing, December, 46–4, 1984.

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30

Optimization of the Arrays of Impinging Jets Muhiddin Can and A. Burak Etemo
glu

CONTENTS 30.1 30.2

Introduction .............................................................................................. 685 Introduction to Impinging Jet Theory...................................................... 686 30.2.1 General ........................................................................................ 686 30.2.2 Description of the Principal Flow Regimes ................................ 686 30.3 Heat Transfer under Impinging Air Jets .................................................. 687 30.4 Mass Transfer under Impinging Air Jets ................................................. 689 30.5 Optimum Combination of Design Parameters......................................... 690 30.6 Applications of Impinging Air Jets in Food Processing ......................... 694 30.7 Economical Analysis of Food Drying Process for Energy Demand....... 695 30.7.1 Air Power .................................................................................... 696 30.7.2 Power for Air Heating................................................................. 697 30.7.3 Energy Costs ............................................................................... 698 30.8 Conclusion ............................................................................................... 699 Nomenclature ........................................................................................................ 700 Greek Letters.............................................................................................. 700 References ............................................................................................................. 700

30.1 INTRODUCTION Almost any problem in the design, operation, and analysis of industrial processes can be reduced to the problem of determining either largest or smallest value of a function of several variables in the final analysis. Since optimization is the collective process of finding the set of conditions required to achieve the best result from a given situation, optimization techniques should be brought to bear on every task of practical importance. In most aspects of industry, continual improvement is an important feature. Thus, the largest production from given raw materials is desired with the greatest profit from a fixed investment and so on. Optimization is a definitely formal presentation of these ideas. Improvement can be regarded from two view points: economical and technical. Economic improvement provides an overall framework in which a given design should be examined since all problems should be considered within a financial structure. Some aspects, however, may not be directly related to company finances,

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and for this reason, improvements are carried out on a technical basis while focusing on the economical criteria. Any problem investigated in an optimization analysis should have its objective as the system improvement. It should be obvious that, obtaining at least one solution to improve any system is essential. In other words, resulting output can be found by defining the system inputs. If this is not possible, the system can be neither designed nor controlled or optimized.

30.2 INTRODUCTION TO IMPINGING JET THEORY 30.2.1 GENERAL Industrially significant flow-related problems are in fact turbulent which are difficult to treat analytically. As in all problems of forced convective heat transfer, fluid flow aspects of impinging air jets cannot be separated from the heat transfer aspects. Heat transfer coefficients in the impingement region are significantly influenced by the turbulence. Because of the fundamental role it plays for heat transfer, the impingement jet structure is briefly described in this chapter. The boundary layer at solid surface in moving streams of fluid constitutes a resistance to convective heat transfer to or from the surface. Principle of impinging jets is basically to reduce this thickness of the boundary layer to augment the convection.

30.2.2 DESCRIPTION

OF THE

PRINCIPAL FLOW REGIMES

Complex flow regime between two-dimensional (slots) or circular (axi-symmetric) jets is divided into four subregions each containing a basic flow (Figure 30.1). A free jet region occurs when the fluid is discharged from a nozzle or orifice into a fluid at rest. As the jet boundary mixes with and entrains the surrounding fluid (increasing the mass flow), the jet spreads out, and its velocity decreases although total momentum remains constant. The velocity profiles across the two types of jets are similar and spreading of the turbulent boundaries is virtually identical (Martin, 1977).

Xn B

B Nozzle

Potential Core

Nozzle

Free jet region

Z

Primary stagnation point

Wall jet region

Surface

Secondary stagnation region Surface

Impingement region Surface

FIGURE 30.1 Main flow regimes in impinging jets. (Adapted from Can, M., Heat Mass Transfer, 39, 509, 2003.)

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The flow in the impingement region is essentially inviscid and has no predominant direction. The fluid leaving the impingement region then flows parallel to the surface, and a wall jet forms away from the stagnation point (i.e., center of the jet). It consists of a boundary layer region with what is effectively half a free jet superimposed on it. When two adjacent jets impinge on a surface, a secondary stagnation point is set up between them where two wall jets meet (Figure 30.1).

30.3 HEAT TRANSFER UNDER IMPINGING AIR JETS As in all problems of impingement heat transfer, fluid flow aspects of impinging air jets is investigated with heat transfer aspects. A clear understanding of fluid flow patterns associated with impinging jets provides the required basis for a satisfactory understanding of associated heat and mass transfer effects. Heat transfer coefficients in the impingement region are significantly influenced by turbulence, and determining the variation of heat transfer coefficient becomes a significant step in analysis of impingement heat transfer processes. An experimental heat transfer rig used for this purpose is shown in Figure 30.2a where the impinging air jet at ambient temperature is directed onto the horizontal aluminum plate. Temperature of the plate is maintained at 1008C using the steam jacket fitted under its surface. The heat flux data to determine the heat transfer coefficient can be obtained by, for example, using a small ‘‘Gardon’’ type flux-meter (Figure 30.2b). Hardisty and Can (1983) tested different nozzle shapes for their effects on local (ho) and average (h) heat transfer coefficient at stagnation point using the above described system. The data obtained were within approximately
5% of the results obtained from the following nondimensional empirical equation:

Temperature

1=3 Nuo ¼ 0:481Re0:53 E Pr

(30:1)

ΔTs

Surface temperature q

Air temperature Air supply

Plenum chamber

Slot nozzle Sensor flush with heated surface Steam in Leads to XY plotter

Distance Constantan foil

Heat flux meter

Cooling air Heated aluminum plate

Condensate out Traversing screw

(a)

Cylinderical copper body

(b)

Copper wires to voltmeter

FIGURE 30.2 (a) Heat transfer research test rig and (b) Gardon type heat flux meter. (Adapted from Can, M., Heat Mass Transfer, 39, 509, 2003.)

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where Z=B0 ¼ 8, 3000 < Re < 11000 Nuo ¼ hoB0=k ReE ¼ VEB=y For tested nozzle shapes, heat transfer coefficients determined in this study also showed a good agreement with the data given by Gardon and Akfirat (1966). Data from the single slot-nozzle test rig were also used to develop a correlation to determine the average heat transfer coefficient for design purposes. Tests were carried out on nozzles of different slot widths for different nozzle shapes. The results showed that heat transfer data obtained at Z=B0 ¼ 8 was satisfactorily correlated when the effective slot width, B0 (BCD) was used as the characteristic dimension. The heat transfer coefficients, h, were within
5% of the following empirical equation: Nu ¼ 0:224Re0:52 Pr 1=3

(30:2)

A multinozzle test rig was also used to provide data on arrays of nozzles (Figure 30.3). In this rig, horizontal impingement surface (calorimeter) consisted of an aluminum plate heated from below by hot water flowing in a number of narrow rectangular channels, and a large constant head tank maintained the steady-flow conditions. As the hot water passes through the calorimeter, cooling effect of the air jet arrays causes the water temperature to decrease. Energy lost by the water, Q_ w, is expressed by Q_ w ¼ m_ w cpw (Twi  Two )

(30:3)

Overflow Airflow Stirrer

Plenum chamber

3 ⫻ 4 kW 1⫻3 kW

Stirrer

Filter

Ca

lor

im

Slot nozzle arrays

ete

r

Drain

Circulation pump

Hole arrays

Slu

ice

Adaptor box

(a)

(b)

FIGURE 30.3 (a) Multinozzle test rig and (b) Assembly of hole or slot nozzle arrays to plenum chamber. (Adapted from Can, M., Heat Mass Transfer, 39, 509, 2003.)

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The heat transferred from the calorimeter surface to the air jet, Q_ s, is then given for convective heat losses: Q_ s ¼ hAs (Ts  TA ) þ losses

(30:4)

Then, assuming Q_ w is equal to Q_ s gives:   h ¼ m_ w cpw (Twi  Two ) =[As (Ts  TA )]

(30:5)

For a given value of the temperature difference (Ts  TA), average heat transfer coefficient is dependent upon both effective thickness and thermophysical properties of the stagnant film (h ¼ f (V, L, r, m, cp, k, shape of surface)). The heat losses are generally assumed to be less than 1% and negligible.

30.4 MASS TRANSFER UNDER IMPINGING AIR JETS Fluid dynamic processes causing heat transfer across a temperature difference also results in mass transfer across a concentration difference. Heat and mass transfer are analogous (Sherwood and Pigford, 1952). For example, in evaporative drying, vapor concentration is highest at the surface and lowest in the drying air. Due to this concentration difference, the vapor is transferred across the boundary layer. The steady-state flow of vapor away from the surface may be expressed by the mass transfer rate equation: m_ D ¼ hm As (rs  rA )

(30:6)

With the stagnant film hypothesis for a given value of the concentration difference (rs  rA), hm depends upon both effective thickness of the boundary layer and mass transfer properties of the stagnant film: hm ¼ f (V, L, D, m, r, shape of surface). By heat and mass transfer analogy, dimensionless mass transfer relation can be written Sh ¼ C Rea Scb

(30:7)

In a similar manner to Equation 30.1, mass transfer relation for an impinging jet is also given by Sho ¼ 0:481ReE0:53 Sc1=3

(30:8)

During drying, heat is transferred towards the surface while the vapor is also simultaneously transferred. The heat supplied serves for evaporation and, if necessary, raising the system temperature. For a small interval of time Dt, dynamic energy balance becomes: hAs (TA  Ts ) ¼ hm As (rs  rA )hfg þ mcp DT=Dt

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(30:9)

At equilibrium conditions the surface becomes at its wet-bulb temperature satisfying the following energy balance equation: hAs (TA  Ts ) ¼ hm As (rs  rA )hfg

(30:10)

TA  Ts ¼ (hm =h)(rs  rA )hfg

(30:11)

hm =h ¼ (1=rcp )(Pr=Sc)2=3 ¼ Le2=3 =rcp

(30:12)

From Equation 30.10:

where Pr=Sc is Lewis number (Le). Substituting Equation 30.12 in Equation 30.11 then gives: TA  Ts ¼ (1=rcp )Le2=3 (rs  rA )hfg

(30:13)

From Equation 30.8, assuming fixed value of airflow rate and surface geometry in Sh a Sc1=3

(30:14)

and assuming small vapor concentrations, properties of the air=vapor mixture approximate those of pure air, then: hm a D2=3

(30:15)

When the vapor pressure is low, the vapor concentration may be obtained from the ideal gas law. Applying this relation to the vapor at the interface: rs ¼ Ps =RTs

(30:16)

where Ps is the saturated vapor pressure corresponding to Ts. Finally, for a surface at a given temperature: m_ D a D2=3 Ps A

(30:17)

Equation 30.17 is consistent with the evaporative index of Gardner (1960) which was based on the stagnant film hypothesis.

30.5 OPTIMUM COMBINATION OF DESIGN PARAMETERS The optimum combination of design parameters is generally used to obtain the lowest costs in an impingement system. The optimization might be based upon: . .

Nozzle box capital costs: fabrication of nozzles, plenum chamber, materials, maintenance, etc. Nozzle box running costs: pumping power, fuel, etc.

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. .

Total system capital costs: air systems (fan), cooling or heating section, energy for drying, etc. Total system costs: capital, interest, running operation, etc.

These economic factors are in broad terms since actual conditions such as size of operation, type of material, etc., affect total costs and, to some degree, cost ratios. Increasing heat transfer coefficient usually requires a corresponding nonproportional increase in fan power. On the other hand, similar heat transfer coefficients can sometimes be achieved with lower fan power, and hence lower capital and running costs. Such considerations lead to the requirement of optimization studies. In practice, each nozzle shape or configuration has its advantages and disadvantages. For example, Can et al. (2002) showed that slightly better heat transfer performance was achieved under the holes arrays proving a cheaper cost to manufacture. However, choice of such an arrangement obviously places constraints on the design. The importance of such constraints depends largely upon the plant size and allocated funds. To balance these aspects, a slot nozzle array should be smaller and more compact than the ones using holes or round nozzles to obtain a similar heat transfer effect. In addition, two-dimensional (slot) jets are apparently preferred in industrial uses for uniformity or heat transfer across the surface. Figure 30.4, for example,

Z

B(D)

(a)

Orifice

(b)

Contoured nozzle

Z

D n (or X n)

(c)

Semiconfined

(d)

No exhaust ports

Z

Dn (or X n)

(e)

With exhaust ports

FIGURE 30.4 Flow geometries and arrangements. (Adapted from Can, M., Heat Mass Transfer, 39, 509, 2003.)

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shows different nozzle geometries and arrangements used in practice. Selection of nozzle configuration should be based on cost of fabrication, ease of maintenance, and a selection of a suitable fan. The following remarks are largely obtained from the authors’ research on slot nozzles. Nozzle height, Z=B: At the nozzle outlet, the turbulence level becomes relatively low. Due to intense mixing with surrounding air, the turbulence level rises and reaches a maximum at approximately 8 effective slot widths from the nozzle outlet. This requires knowledge of the discharge coefficient. Experiments show that the stagnation point heat transfer coefficients, ho, exhibit a maximum when the impingement surface is positioned at Z=B0 ¼ 8. Similarly, the average heat transfer coefficient h also becomes at its maximum when Z=B0 ¼ 8 for any shape of nozzle configurations (Hardisty and Can, 1983). They also found that the effective width B0 (D0 ) of nozzles used in this study was related to the actual nozzle size B(D) by B0 (D0 )¼ B(D)CD where CD is the discharge coefficient. Nozzle shape: Data on the effect of nozzle shape on heat transfer are presented here obtaining optimum nozzle systems. Although experimental results show that nozzle shape itself does not directly affect the heat transfer rate, cost of nozzle fabrication should be taken into account. Thus, heat transfer data from different nozzles can be correlated using the effective nozzle width B0 ( ¼ B  CD). The experimental data by Hardisty and Can (1983) to determine discharge coefficient is CD ¼ 0:64 VE0:10

(30:18)

These results demonstrated the discharge coefficients only varying with air jet velocity and nozzle shape but not with nozzle width. However, in industrial applications, selection of jet configuration is governed not only by relative magnitude of average heat transfer coefficient but also by desired heating pattern and economics of the fan system. Studies involving arrays of jets were also conducted with sharpedged orifices preferred in particular industrial installations due to their ease of fabrication. Although, simple sharp-edged orifices are easy to fabricate from flat strips, thin materials cannot stand to high plenum pressures, and such simple slot nozzles would bend in practice. Making simple circular holes are advantageous but a tradeoff should be made between strength and cost of material required and corresponding gains from high air velocities because of bending at high plenum pressures. Nozzle width (or diameter), B(D): Because of its influence on air mass flow rate and consequently on capital and running costs, slot width or diameter is also an important design parameter. To select the optimum nozzle width for a multinozzle installation, influence of slot width on heat transfer, blower power, and number of nozzles required with their associated fabrication costs should be assessed. Air velocity, VE: Electrical power is required to provide the pressure rise to attain the intake velocity, to overcome the frictional and flow losses in the ducts and bends and to accelerate air through nozzles. Heat transfer coefficients are proportional to air jet velocity, and an increase in velocity requires a rise in fan power. This may result in an unacceptable rise in noise level.

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TABLE 30.1 Comparison between Slot Nozzles and Circular Holes

P (kW) 2.2 3.0 3.6 4.0 4.2 4.3 4.5

Nozzle Width B ¼ 3 mm, Re ¼ 10000

Nozzle Diameter D ¼ 5 mm, Re ¼ 10000

Number of Slot Nozzles

h (W=m2-K)

Af (%)

Number of Circular Holes

h (W=m2-K)

Af (%)

3 6 12 18 24 30 40

245 265 310 315 300 285 270

0.5 1.0 2.0 3.0 3.9 5.0 6.7

6 11 18 24 30 36 41

180 235 265 252 240 218 205

1.0 1.8 3.0 3.9 5.0 6.0 6.8

Air temperature, TA: Effect of air temperature is briefly discussed and given in Avci and Can (1999). Increasing operating conditions also demand an increase in heating power. In addition to the heat content of the air, there should be a further input to overcome the system losses. In some cases, when the air flow rate is high, air temperature, TA, remains approximately constant. Were air jet temperature not equal to the surrounding temperature, a diluting effect of entrainment would result in dramatic changes in the heat transfer coefficient (Goldstein and Seol, 1991; Polat, 1993). Nozzle pitch, Xn: Nozzle pitch (or center to center spacing of slot nozzles) is directly related to optimum free area Af ( ¼ B=Xn  100), and it directly becomes an optimization parameter. For example, as seen in the results of this chapter, ‘‘free or open area,’’ ratio of the total nozzle outlet area to the heat transfer surface area, was a fundamental parameter. By expressing the results in terms of free area, for example, it might be possible to optimize the ratio of heat transfer to blower power (Table 30.1) when the objective is to maximize the heat transfer coefficient. Table 30.1 shows fan power characteristics for different width and hole diameters. Comparisons were made between the optimum values of free area determined by experiments. These comparisons and percentage discrepancy between them are shown in Tables 30.2 and 30.3. Table 30.1 could also be used to select a nozzle system to suit a specified fan power. The results for circular holes were in good agreement with the various data of Friedman and Mueller (1951) with some differences with the data given by Martin (1977). TABLE 30.2 Comparison for Slot Jets Width of Slot Nozzle B (mm) 2.0 2.5

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Optimum Value of Free Area Af (%) Experimental

Theoretical

Agreement
(%)

4.5 4.5

4.69 4.67

4.0 3.7

TABLE 30.3 Comparison for Circular Holes Diameter of Circular Nozzle D (mm) 5.0 10.0

Optimum Value of Free Area, Af (%) Experimental

Theoretical

Agreement
(%)

3.0 3.0

3.21 3.28

6.5 8.5

Secondary stagnation point region: Despite its significance, little research (Huber and Viskanta, 1994a,b) has been carried out into the secondary stagnation point which occurs midway between two slot jets where two wall jets meet, interact, and flow away from the surface. Because of turbulent mixing, heat transfer is enhanced at the secondary stagnation point where two jets meet. The original statement of design parameters can be written in the form: h ¼ f (VE, B(D), Z=B(Z=D), Xn(Dn)). However, considering only heat transfer, selection of nozzle configuration is entirely dependent upon the following three different parameters: . . .

Nozzle height, Z=B0 ¼ 8 (or Z=D0 ¼ 8) and B0 ¼ B  CD, D0 ¼ D  CD Nozzle width B (or diameter D), the narrowest slot width, B ¼ 2.0 mm and the smallest hole diameter, D ¼ 5.0 mm Nozzle pitch, Xn (or Dn) related to optimum free area Af

where Af ¼ 2.5% for slots (B ¼ 2.0 mm and B ¼ 2.5 mm) and Af ¼ 3.0% for holes (D ¼ 5.0 mm and D ¼ 10.0 mm). Both experimental and theoretical investigations (Can et al., 2002) carried out in this research show that the above three conditions are indeed the most relevant in the design of nozzle systems. The authors also concluded that relating heat transfer data to capital and running costs is a significant process for optimization purposes.

30.6 APPLICATIONS OF IMPINGING AIR JETS IN FOOD PROCESSING A single jet or arrays of air jets, impinging normally on a surface, is used to achieve enhanced heat and mass transfer coefficients for convective heating=cooling processes in food processing. Food processing systems use higher air velocities (10–100 m=s) exiting from nozzles and impinging on the food product. Impingement drying, for example, is an old technology that has been recently applied to food products (Moreira, 2001). Tortilla and potato chips, pizza, crust, pretzel, and crackers have been successfully cooked and baked in air impingement ovens in the food industry. Granular products (coffee beans, cocoa beans, rice, and nuts) are reported

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to dry faster and to heat more uniform due to the pseudofluidized bed effect created by high velocity air from the nozzles during impingement. Physical characteristics of impinging jets such as turbulent mixing in the free jet region, stagnation, boundary-layer formation, recirculation, and their interactions with food products in terms of heat and mass transfer have been reviewed by Sarkar et al. (2004). The discussion included experimental methods used for measurement of heat and mass transfer for single and multiple slot and circular jets. Numerical modeling of air impingement systems was also presented with special consideration of problems arising in food-processing systems. Millsap and Marks (2005) compared the heat transfer coefficient values obtained from a discrete model for food products processed in a commercial moist air impingement oven with the values from previously developed equations for flat plates located under an array of slot nozzles. Attempts of Sarkar and Singh (2004a,b) were to couple flow and heat transfer equations for external boundary layer flow over a frozen food product where the heat transfer equations for fluid flow was solved as an unsteady-state problem in conjunction with internal heat transfer with the model validation for both heat transfer and flow field. When air impingement technology is applied for thawing of frozen foods, resulting effective heat transfer coefficients becomes quite complicated. The airflow from impingement jets result in spatially variable heat transfer coefficient. In addition, transient surface temperature changes during thawing result in effective heat transfer coefficient varying with time. Anderson and Singh (2006a) determined the spatially variable effective heat transfer coefficients as a function of surface temperatures using an inverse method for thawing under a single impinging jet. Anderson and Singh (2006b) developed a two-dimensional model for air impingement thawing of frozen foods and verified the model experimentally. Dirita et al. (2007) focused on air impingement cooling of cylindrical foods focusing on numerical analysis of the initial stage of cooling=chilling operations. Results included the temperature distributions within the food product and at its surface with the associated flow field due to the jet–food interaction. The local, timedependent Nusselt number distribution was shown to be strongly dependent upon the conjugate effect, i.e., the heat transfer rate is altered by conduction in the food, which was neglected in the associated studies available so far.

30.7 ECONOMICAL ANALYSIS OF FOOD DRYING PROCESS FOR ENERGY DEMAND The air flow is generated by fans with electrical power required to produce the pressure rise to attain the intake velocity, to overcome the frictional losses in the ducts and bends, and to accelerate the air through the nozzles. Heat energy should be supplied to the air flow either indirectly in heat exchangers or directly by burning gas in the air stream. Therefore, pressure rise to produce a certain air velocity and amount of supplied heat to raise the air temperature to a certain value can be estimated to establish some comparative basis for optimizing the energy demand and to secure the minimum cost for the process.

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30.7.1 AIR POWER The air accelerates from a low velocity in the plenum chamber or hood to a jet velocity of VE (m=s) in the free air at pressure PA. The velocity in the hood should not exceed 1=6 VE. Bernoulli’s equation for ideal flow can be applied to determine the pressure PH by correcting the air density at the raised temperature of T: PH  PA ¼ DP ¼ 171:6

VE2 (Pa) T

(30:19)

These pressures and heads plotted against air jet velocity VE for different air jet temperatures are shown in Figure 30.5. For n nozzles of width w and slot width B: Jet exit area ¼ n w B (m2 ) Volumetric flow-rate of air per second ¼ Area  Velocity ¼ n w B VE (m3 =s) This air volume should have its pressure raised from PA to PH every second demanding an energy input rate given by Air power N ¼ (n w B VE )171:6

VE2 (W) T

(30:20)

For simplicity, the analysis can be confined to one slot nozzle of width 1 m and slot width 3 mm: Air power per slot nozzle ¼ N ¼ 0:513

VE3 (W) T

(30:21)

Pressure difference, ⌬p, (kPa)

7 6 5

(20⬚C) (40⬚C) (60⬚C) (80⬚C) (100⬚C) (120⬚C)

4 3 2 1 0 0

20

40

60

80

100

Jet velocity, VE (m/s)

FIGURE 30.5 Variation of pressure difference with air jet velocity.

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0.002

Slot nozzle

Air power, N (kW)

Air power, N (kW)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.001 0.005 0

0

20

40

60

80

0

100

Jet velocity, VE (m/s)

(a)

Circular nozzle

0.015

20

40

60

80

100

Jet velocity, VE (m/s)

(20⬚C)

(40⬚C)

N(20⬚C)

N(40⬚C)

(60⬚C)

(80⬚C)

N(60⬚C)

N(80⬚C)

(100⬚C)

(120⬚C)

N(100⬚C)

N(120⬚C)

(b)

FIGURE 30.6 (a) Air power curves for slot nozzle and (b) air power curves for circular hole.

Similarly, for a circular hole of the diameter D ¼ 5.8 mm, Air power per circular hole: N ¼ 0:00453

VE3 (W) T

(30:22)

These power curves for jets at temperatures of 208C–1208C are plotted in Figure 30.6a and b; the velocity is the main parameter with only a slight effect from temperature. Thus, an increase from 40 to 100 m=s leads to almost 15 times the power demanded. To estimate the fan power requirement, required energy to accelerate air from rest at the intake and to overcome frictional losses through the system (with many installations, this might result in an extra 30% power requirement) should be included in the analysis.

30.7.2 POWER

FOR

AIR HEATING

Heat energy for the air from one nozzle can be estimated from the volume flow and the rise in temperature above the ambient air temperature, TA where nozzle area is w B; pD2=4; volumetric flow rate is w B VE; pD2 VE=4 for slot and circular holes; air density at temperature T is rATA=T; and heat flow rate (Q) is m Cp(T  TA). Thus, For ambient temperature TA ¼ 15 C

and rA ¼ 1:225 kg=m3 ;   288:15 VE (kW) (30:23) Heat flow rate per slot nozzle ¼ Q ¼ 1:066 1  T   288:15 VE (kW) (30:24) Heat flow rate per circular hole ¼ Q ¼ 0:00939 1  T

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Q(40⬚C)

Q(60⬚C)

Q(80⬚C)

Q(100⬚C)

Q(120⬚C)

Cost ($/h) Natural gas Fuel-oil 4 Electricity

Heating power, Q (kW)

30

Slot nozzle

0.7672

1.6566

2.8294

0.6905

1.3253

2.2635

20

0.5288

1.1417

1.9499

15

0.4229

0.9133

1.5599

10

0.2712

0.5708

0.9749

5

0.1376

0.2971

0.5075

25

0

0 0

20

40

60

80

100

0.24

0.30

Jet velocity, VE (m/s) 0.06

0

Q(40⬚C)

Heating power, Q (kW)

0.12

0.18

Volumetric flow rate, V (m3/ h)

(a)

Q(60⬚C)

Q(80⬚C)

Q(100⬚C)

Q(120⬚C)

Cost ($/h) Natural gas Fuel-oil 4 Electricity

0.3

Circular nozzle

0.25

0.0068

0.0146

0.0249

0.2

0.0054

0.0117

0.0199

0.15

0.0042

0.0090

0.0155

0.1

0.0028

0.0060

0.0103

0.05

0.0014

0.0031

0.0052 0

0 0

20

40

60

80

100

2.1136

2.6420

Jet velocity, V E (m/s) 0

(b)

0.5284

1.0568

1.5852

Volumetric flow rate (x le-3), V (m3/ h)

FIGURE 30.7 (a) Heating power curves for slot nozzle and (b) heating power curves for circular nozzle.

The effect of air jet velocity and temperature can be seen in Figure 30.7a and b. In addition to air heat content, there needs to be a further input to overcome the system losses, e.g., from burner or heat exchanger, along the ducts and from the hood. This extra demand depends especially on particular design features, amount of lagging, etc. These losses could amount to around 20%.

30.7.3 ENERGY COSTS In order to calculate the energy costs, two different cases are given for slot and circular jets. Working conditions, geometrical properties, and energy cost values can be found in Table 30.4.

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TABLE 30.4 Case Studies for Slot Nozzles and Circular Holes Air jet velocity (m=s) Air jet temperature (8C) Number of slots Air jet power (kW) Fan power assuming 30% extra losses (kW) Heating power (kW) Assuming system losses of 20% (kW)

Case 1. Slot Nozzles

Case 2. Circular Holes

60 80 72 (1 m  3 mm) 72  0.314 ¼ 22.61 (from Figure 30.6a) 29.39

60 80 8175 (5.8 mm) 8175  0.00277 ¼ 0.8311 (from Figure 30.6b) 22.645

72  11.695 ¼ 842.04 (from Figure 30.7a) 1010.45

8175  0.1030 ¼ 842.025 (from Figure 30.7b) 1010.43

Cost per Hour (Electricity Cost: 0.099042 $=kWh) Fan power ($=h)

2.91

2.24

Natural Gas Fuel-Oil Electricity (From Figure 30.7a) Heating power ($=h) Total ($=h)

0.3172 22.84 25.75

0.685 49.32 52.23

1.1699 84.23 87.14

Natural Gas

Fuel-Oil

Electricity

(From Figure 30.7b) 0.00279 22.81 25.05

0.00603 49.30 51.54

0.01030 84.20 86.44

30.8 CONCLUSION 1. One of objective of the heat transfer research presented in this chapter was to investigate the effect of nozzle shape. It was found that knowledge of the discharge coefficient alone can be sufficient to correlate test data from nozzles of different shapes. 2. Proper use of the results presented might lead to either more effective heat and mass transfer process or reduction in costs. 3. Optimization of nozzle arrays under impinging air jets was given for practical purposes. Optimum combination of design parameters presented are simple and have wide applicability. A study of capital and running costs including nozzle design and fan power consumption would be most helpful in the practical aspect. 4. Economical analysis of drying equipments is given for both slot and circular nozzles. Considering only heat transfer, selection of nozzle configuration is entirely dependent upon the following three important parameters: nozzle height, Z=B0 ¼ 8, nozzle width B, the narrowest slot width and nozzle pitch, xn, related to optimum free area, Af. It is also shown that heat transfer coefficients exhibit a maximum at a dimensionless nozzle-to-plate spacing Z=B0 ¼ 8 independent of nozzle shape.

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5. Research given in this chapter provides particularly a rational basis for understanding air jet designers of industrial equipment for food processing operations.

NOMENCLATURE As Af B B0 cp CD D h hm h hfg k m m_ D m_ w Re Nu Nu Pr Twi Two Ts TA VE V_ w

Surface area Free area Nozzle width Effective nozzle width Specific heat Discharge coefficient Diffusion coefficient

m2 % m m J=kg-K

Surface mass transfer coefficient Average heat transfer coefficient Latent heat Thermal conductivity Mass Surface mass transfer rate Water mass flow rate Reynolds number Nusselt number Average Nusselt number Prandtl number Temperature at calorimeter inlet Temperature at calorimeter outlet Surface temperature Free stream temperature Air jet velocity at nozzle exit Volumetric flow rate Nozzle width

m=s W=m2-K J=kg W=m-K kg kg=s kg=s

m2=s

K K K K m=s m3=s m

GREEK LETTERS DT rs rA y

Temperature change Saturated vapor concentration Partial concentration of vapor in drying air Kinematic viscosity

K kg=m3 kg=m3 m2=s

REFERENCES Anderson, B.A. and Singh, R.P., Effective heat transfer coefficient measurement during air impingement thawing using an inverse method, Int. J. Refrigeration, 29, 281, 2006a. Anderson, B.A. and Singh, R.P., Modeling the thawing of frozen foods using air impingement technology, Int. J. Refrigeration, 29, 294, 2006b.

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Avci, A. and Can, M., The analysis of the drying process on unsteady forced convection in thin films of ink, App. Thermal Eng., 19, 641, 1999. Can, M., Experimental optimization of air jets impinging on a continuously moving flat plate, Heat Mass Transfer, 39, 509, 2003. Can, M., Etemoglu, A.B., and Avci, A., Experimental study of convective heat transfer under arrays of impinging air jets from slots and circular holes, Heat Mass Transfer, 38, 251, 2002. Dirita, C., De Bonis, M.V., and Ruocco, G., Analysis of food cooling by jet impingement including inherent conduction, J. Food Eng., 81, 12, 2007. Friedman, S.J. and Mueller, A.C., Heat transfer to flat surfaces, I. Mech. Eng. Proc. General Discus. Heat Transfer, 1951, p. 138. Gardner, T.A., A theory of drying with air, TAPPI, 43, 796, 1960. Gardon, R. and Akfirat, J.C., Heat transfer characteristics of impinging two-dimensional air jet, Trans. ASME J. Heat Transfer, 88, 101, 1966. Goldstein, R.J. and Seol, S., Heat transfer to a row of impinging circular air jets including the effect of entrainment, Int. J. Heat Mass Transfer, 34, 2133, 1991. Hardisty, H. and Can, M., An experimental investigation into the effect of changes in the geometry of a slot nozzle on the heat transfer characteristics of an impinging air jet, Proc. Inst. Mech. Eng., 197C, 7, 1983. Huber, A.M. and Viskanta, R., Convective heat transfer to a confined impinging array of air jets with spent air exits, J. Heat Transfer, 116, 570, 1994a. Huber, A.M. and Viskanta, R., Effect of jet-jet spacing on convective heat transfer to confined impinging arrays of axisymmetric air jets, Int. J. Heat Mass Transfer, 37, 2859, 1994b. Martin, H., Heat and mass transfer between impinging gas jets and solid surfaces, Adv. Heat Transfer, 13, 1, 1977. Millsap, S.C. and Marks, B.P., Condensing-convective boundary conditions in moist air impingement ovens, J. Food Eng., 70, 101, 2005. Moreira, R.G., Impingement drying of foods using hot air and superheated steam, J. Food Eng., 49, 291, 2001. Polat, S., Heat and mass transfer in impingement drying, Drying Tech., 11, 1147, 1993. Sarkar, A. et al., Fluid flow and heat transfer in air jet impingement in food processing, J. Food Sci., 69, CRH113, 2004. Sarkar, A. and Singh, R.P., Modeling flow and heat transfer during freezing of foods in forced airstreams, J. Food Sci., 69, E496, 2004a. Sarkar, A. and Singh, R.P., Air impingement technology for food processing: Visualization studies, Lebensm.-Wiss. u.-Tech., 37, 873, 2004b. Sherwood, T.K. and Pigford, R.L., Absorption and Extraction, McGraw Hill Book Co., New York, 1952.

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31

Optimal Operational Planning in the Fruit Industry Supply Chain Guillermo L. Masini, Aníbal M. Blanco, Noemí C. Petracci, and J. Alberto Bandoni

CONTENTS 31.1 31.2 31.3

Introduction .............................................................................................. 703 Literature Review..................................................................................... 706 FISC Description ..................................................................................... 707 31.3.1 FISC Operations ......................................................................... 709 31.3.2 FISC Model ................................................................................. 709 31.4 Operational Planning Framework ............................................................ 711 31.5 Results and Discussion ............................................................................ 714 31.6 Conclusion and Future Work .................................................................. 723 Appendix A: FISC Optimization Model ............................................................. 726 Appendix B: Optimization Model Parameters...................................................... 736 Nomenclature ........................................................................................................ 741 Abbreviations by initials ............................................................................ 741 Indexes ....................................................................................................... 741 Sets ............................................................................................................. 741 Variables .................................................................................................... 742 Parameters .................................................................................................. 743 References ............................................................................................................. 745

31.1 INTRODUCTION The fruit industry is a major economic activity in many countries. Descriptions and models of fruit industry supply chains (FISC) have recently appeared for the Argentinean and the South African cases (Masini et al., 2003; Masini et al., 2007; Ortmann, 2005). A typical FISC comprehends farms where fruit of different varieties are grown, packaging plants where fresh fruit is packed after cleaning and classification and cold storage facilities to store processed and nonprocessed fruit to prevent quality losses. Depending on the available fruit varieties, peripheral activities such as concentrated

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juice, cider, or oil production are usually present. There exist two major markets for a typical large company of a FISC: regional and overseas, which demand different types of packed fruit and derivate products. In such an interconnected network, a huge amount of possibilities regarding when, where, and how much to purchase and allocate of different goods naturally arise making the FISC a challenging application for Operations Research tools such as mathematical programming planning and scheduling models. Supply chain modeling for optimization can be classified in strategic, tactic, and operational (Shapiro, 2001). At the strategic level, long-term decisions (2–5 years) such as product development and new infrastructure investments are addressed. At a tactical level, midterm decisions are considered (from several months to two years), typically related to sales and inventory policies. Operational optimization deals with daily or weekly decisions such as goods processing, allocation, and transportation within the system facilities. At the three levels, model parameters (costs, raw material availability, demands, etc.) require forecasting for each period of the considered horizons. Some sort of strategy to deal with uncertainty should be also adopted. The resulting optimization models are multiperiod and large-scale in general. Binary variables and nonlinearities may be present depending on the particular system. At the operational level, the planning models also receive current data in the present period (exact amounts of raw material and exact costs) and should be able to deal with unexpected events (out of operation of plants, on spot raw material purchase opportunities, unfavorable climatic episodes, etc.) some of which might cause system disruption. Evidently, operational optimization models radically differ from tactical=strategic ones, at least at the implementation level. In fact, not surprisingly, many approaches for operational planning of such highly disturbed systems consider the supply chain as a dynamic entity under feedback and anticipative control. From a management point of view, supply chains can be centralized or decentralized systems. In centralized supply chains, a decision-making entity concentrates the information and coordinates the operations of the whole system. A large company, which operates nodes in several instances of a sector or industry, constitutes a centralized supply chain. In the fruit case for example, a typical large company usually own farms, packaging plants, concentrated juice plants and cold storage facilities. In this sense the supply chain constitutes a link of own resources. Centralized operation of resources of different players of an industrial activity, although desirable for market reasons, is a complex task and a rare example. In decentralized supply chains each node pursue the optimization of it own operations and share limited information with the other components of the chain. Although supply chain optimization is a mature field, very few contributions on FISC modeling with management purposes have appeared in the literature. According to the authors’ knowledge, only three contributions on the whole FISC system has been presented so far: Masini et al. (2003), Ortmann (2005), and Masini et al. (2007). In Masini et al. (2003), a preliminar planning model of a typical large company operating in the Argentinean pip fruit (apples and pears) supply chain was described.

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Recently, that model was extended in Masini et al. (2007) to address the company’s tactical planning problem. The resulting LP multiperiod model is based on a realistic description of the network topology and spans a 1 year horizon with a weekly resolution in order to comprehend the whole fruit season with the required detail. The model is fed with estimations of fruit production and quality per variety and seeks to maximize the total profit of the business for the given infrastructure. It returns the amounts of packed fruit and concentrated juice that should be allocated in the different markets together with the requirements of third party raw material, storage, and transportation. The model was conceived as a decision support tool to aid in the negotiation instance of the FISC business when sales commitments and third party resources purchase have to be decided. In Ortmann (2005), the South African fresh fruit export supply chain was described in detail and models to optimize the fruit business with emphasis in foreign markets were proposed. The resulting single period LP models, based on graph theoretic descriptions of the system, were designed to maximize the fruit flow and to minimize transportation costs. They do not consider for example local markets, derivative industries such as juice production or peripheral costs such as fruit production and packaging. Since the study was mainly aimed to analyze the export fruit supply chain infrastructure, the models were designed to identify bottlenecks within the network rather than to provide a planning tool for the system’s managers. The above described contributions address the tactical=strategic levels of the decision making in centralized FISCs. Such models are not appropriate as operational decision support tools, which should be able to capture the real dynamic behavior of the system (Perea Lopez et al., 2003). In fact, the FISC possesses unique features regarding classic supply chains from an operative point of view. No operational optimization models for FISC systems has been presented so far, and the topic addressed in this chapter have been proposed for the first time according to the authors’ knowledge. Classic chains are ‘‘pull systems’’ (Perea Lopez et al., 2003) driven by the orders that the customers place on retailers, which propagates backwards through distribution centers, warehouses, and plants to finally reach raw material suppliers. In such systems, product availability (adequate inventory) is the key factor to achieve good customer satisfaction. Unlike classic chains, FISCs are a sort of ‘‘push systems,’’ driven by the fresh fruit produced in farms each season, which determines the spectrum of final products. Customer demand does not disturb the system in the classical sense since most of the (estimated) final production (packed fruit and juice) is allocated before the season through negotiation between clients and producers (Masini et al., 2007). Therefore the operational planning of the FISC should seek to satisfy as close as possible the already established product delivery commitments rather than to respond to ‘‘online’’ customer demand. In the present contribution, an operational planning framework for a typical FISC is proposed, which explicitly considers the unique operational features of FISC systems, in particular the meaningful uncertainty in the quality and quantity of the raw material (fresh fruit from farms) and the possibility of supply chain

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disruption episodes. The proposed methodology implements a Model Predictive Control philosophy. The chapter is structured as follows. In the following section the existing operational optimization approaches for supply chain systems are reviewed. In Section 31.3, the chosen study case, a large company that operates several nodes within the Argentinean FISC, is described and the corresponding model is presented. In Section 31.4, the proposed operational planning scheme is introduced. Results are presented and discussed in Section 31.5. The chapter closes with a conclusion and future work (Section 31.6).

31.2 LITERATURE REVIEW Operational planning of batch processes is usually known as scheduling. In the face of uncertainty and disruption events re-scheduling is required to keep on feasible and optimal operation. There exists a large body of literature on batch plant re-scheduling. See for example Aytug et al. (2005) for a comprehensive review on the topic. However, as pointed out by Adhitya et al. (2007), none of those re-scheduling methods can be directly applied to continuous systems, such as supply chains, due to inherent differences in scheduling decisions and objectives. Alternatively, a sound approach for operational optimization of supply chains is to conceptualize them as dynamic systems and to apply ‘‘classic’’ knowledge of control theory to operate them. In Perea-Lopez et al. (2001), after the development of a dynamic model to capture the real behavior of a decentralized supply chain, different proportional type ‘‘control laws’’ were analyzed in the face of typical disturbances. The controller gains tuning was addressed as an optimal control problem aimed to minimize total costs. Model predictive control (MPC) frameworks have been also proposed for supply chain operational management. Bose and Pekny (2000) proposed an MPC approach to lump the major elements of supply chain operational optimization: the forecasting model and the optimization model. For given demand information in the future periods (on a weekly basis), a detailed production schedule (MILP) is solved and implemented only for the present period. The objective was to minimize inventory costs while ensuring a target customer service level. Perea Lopez et al. (2003) devised an MPC implementation based on a MILP model of a typical centralized ‘‘pull’’ supply chain. The focus is on the development of a responsive analysis tool to quickly update the decision-making process in the face of changes in the forecasts of demand, while maximizing the total profit of the network. With a similar philosophy, Mestan et al. (2006) proposed an MPC framework to address the operations management of multiproduct supply chain networks for centralized, decentralized, and semidecentralized information systems. As expected, it is demonstrated that centralized operations (total information sharing) renders superior performance regarding inventory management, production scheduling, total profit, and customer satisfaction than decentralized and semidecentralized schemes.

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FISCs possess particular operational features regarding typical chains, making therefore necessary to devise tailored tools for their management. As already commented for example, opposite to typical supply chains, which are pull systems (demand driven), FISCs are push systems (seasonal raw material availability driven). The major source of uncertainty is present in the amount of fruit entering the system each period rather than in final product demand, which is preallocated before the season. In the present contribution an MPC scheme is proposed for the operational management of a FISC, which naturally permits to deal with input uncertainty, as well as with unexpected events, which can disrupt the system operations.

31.3 FISC DESCRIPTION A typical FISC (Figure 31.1) comprises own farms (OFs), third party fruit suppliers (TPSs), storage facilities, packaging plant facilities (PPFs), concentrated juice plant facilities (CJPFs), and clients, among other minor processing nodes. A very detailed description of the FISC infrastructure and business can be found in Masini et al. (2007).

PPFS X142 PPF PFS

NPFS X1

OF

X5 NPFS

X2

PP

X8

X15

RPFC

X13

OCJC

RCPFS X141

WFS

RCPFS

CJPF

X3 FRS

TPS

OPFC

X10 X 14

PFS

X7 X 7F

X12 PPFS

X4

CJS X6

CJP

X11

X9

FRS

PCJS

PCJS

CJS X111

OF

Own farms

TPS Third party supplier PPF Packaging plant facility

CJPF Concentrated juice plant facility FRS

Fruit reception sites

RPFC Regional packed fruit client CJP

Concentrated juice plants

CJS

Concentrated juice storage

OCJC Overseas concentrate juice client

NPFS Not processed fruit storage PP

Packaging plants

PFS Processed fruit storage WFS Waste fruit storage

FIGURE 31.1

PPFS Port processed fruit storage RCPFS Regional client packed fruit storage PCJS Port concentrated juice storage

FISC flowsheet.

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OPFC Overseas packed fruit client

In farms, fruit of different varieties are grown and harvested according to particular harvest periods. The Argentinean pip FISC mostly operates with several varieties of apples and several varieties of pears. Every apple variety is used for ‘‘apple juice’’ production and every pear variety is used for ‘‘pear juice’’ production. Each day of the harvesting period, each variety of fruit is produced in the OFs. Tradable fruit is processed in own PPFs (X1) and waste fruit (damaged or too imperfect nontradable fruit) in CJPFs (X2). A production cost per variety can be associated to each farm, which comprises growing and labor expenses. In PPFs (Blanco et al., 2005) fresh fruit is received, washed, and stored in NPFSs for later classification and packing at PPs. Each PPFs can also receive fresh fruit from TPSs (X3). PPFs produce waste fruit devoted to juice production CJPs (X7F), and packed fruit (X8), which is cold stored (PFSs) previous the delivery. PPs possess a maximum processing capacity and a related operating cost, which considers labor, energy, and maintenance expenses. In general, only one fruit variety, apples or pears, can be processed at a time. PPFs also have some waste fruit storage capacity (WFSs). In CJPFs, juice of the different types of fruit is produced. CJPFs possess fruit reception sites (FRSs) where waste fruit is stored before processing. FRSs receive waste fruit from OFs (X2), TPSs (X4), and PPs (X7). The produced juice is stored in adequate storage facilities in the same plant (CJS) until delivery. CJPs have a maximum processing capacity and a certain operating cost. As PPs, CJPs are also multiproduct plants in the sense that only one fruit variety, apples or pears, can be processed at a time. For any FISC there exist two major markets for packed fruit: regional (RPFC) and overseas (OPFC). Concentrated juice is fully assigned to the overseas market (OCJC). Selling of the different products constitutes the main income of the system. Each product has a particular selling price in each market depending on the type of demand. Two types of ‘‘demands’’ are considered: .

.

‘‘Fixed’’ demand is given by the amounts of packed fruit and juice agreed with the different clients before the fruit season. For such compromises, there is a specific time delivery schedule. ‘‘Eventual’’ demand represents the possibility of allocating product at a certain price in any period (packed fruit regional: X141, packed fruit overseas: X142 and concentrated juice X111).

Products to the overseas market can be stored in port storage facilities (PPFSs and PCJSs). It is also considered that the Regional Client also posses some product storage capacity (RCPFS). Storage Facilities (NPFSs, PFSs, FRSs, PPFSs, and PCJSs) have maximum storage capacities and an associated cost related to the energy required to refrigerate the fruit. There also exists significant transportation within the system. Products are delivered by refrigerated trucks to the regional market and by ship to the overseas market. Fruit is transported by nonrefrigerated trucks among the different nodes of the chain. Therefore, the flows within the system are discrete since only a finite integer number of travels per day can be done with the available truck fleet.

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Transportation costs depend on the distances between the facilities and the type of transportation: refrigerated or not.

31.3.1 FISC OPERATIONS For the purposes of the present operational planning model, it is considered that the company operates as a centralized supply chain since it has complete control of the several instances that constitute its structure. The company also possesses a limited knowledge of third party operation policies and resources (fruit and storage), represented by bounds on the maximum availabilities in certain periods of the season. Once the season begins, the system has to be operated within the current business and productive scenario. This is the operation instance of the business. The current fresh fruit availability determines the resulting product profile and therefore the potential degree of customer satisfaction. In this sense current daily fruit production in farms (amounts and quality per variety) as well as estimations for the future, have to be provided to the model. The current fruit availability is strongly determined by the current fruit harvest, which is highly dependent on climatic conditions (weather temperature, rain, etc.) and pest occurrence. Unfavorable climatic scenarios (late frosts) or pest episodes may render poor harvests in fruit amount and quality. On spot purchase from third party suppliers (X3, X4) can complement own fresh fruit production and reserves. Fluctuations may also occur on costs due to offer=demand tradeoffs, as well as unexpected events such as out of operation of plants and transportation delays, which can cause operations disruption. On the final products extreme of the chain, the system operation is required to meet the ‘‘fixed’’ and ‘‘eventual’’ demands. As previously commented the fixed demand is given by certain amounts of fruit committed in specific delivery dates. Since there exists a maximum amount of product that can be delivered in a particular period due to manipulation and transportation capacity, it is in general not possible to send the whole agreed amount ‘‘exactly’’ in the delivery date. Rather, the deliveries will be necessarily distributed along some time window before the delivery date. This is achieved by distributing the goods between own, client, and port warehouses (PFSs, CJSs, RCPFSs, PPFSs, and PCJSs). Since own storage capacity is cheaper than client and port storage, product is expected to be kept in own facilities as long as possible before delivering. The company can also decide to satisfy some ‘‘on spot (eventual) demand’’ of packed fruit (X141, X142) and concentrated juice (X111). Such product has a higher selling price than that of the fixed demand and it might turn convenient to devote production to this item in detriment of fixed demand satisfaction.

31.3.2 FISC MODEL A very complete mass balance model for the FISC network can be found in Masini et al. (2007). In order to illustrate the operational planning issues of the system, a reduced version, which captures the essential complexity, is adopted in this work. Several features had to be added however to capture enough detail for meaningful decision making.

ß 2008 by Taylor & Francis Group, LLC.

Since the model in Masini et al. (2007) had ‘‘tactical’’ purposes, an LP approach with a weekly time discretization was adopted in that work. For the ‘‘operational’’ purposes of the present model a daily discretization of the timeline is considered in order to take account of short-term uncertainty and potential chain disruption episodes. A major difference regarding the model in Masini et al. (2007) is the consideration of the ‘‘multiproduct nature’’ of PPs and CJPs meaning that either apples or pears can be processed in a particular period or that the plant is completely shutdown. Moreover, the flows within the system are discrete since the transportation of goods between the nodes is done by trucks, and only a finite integer number of travels can be performed per day. Binary variables are introduced to model such integer decisions leading to a MILP formulation. Overall purpose of the operational planning model is to maximize the net profit of the system. The total profit is defined as the total income per product sales (fixed and eventual demands) minus the sum of fruit production and purchase costs, storage costs, transportation and operation costs, and ‘‘client dissatisfaction’’ costs. This last term is a penalization for delivering less amount of product than agreed in the ‘‘fixed demand’’ schedule. For each delivery date it is evaluated by multiplying the difference between the agreed amount of product and the current accumulated delivered product times a specific customer dissatisfaction cost. This specific dissatisfaction cost is a rather ‘‘subjective’’ weight factor, which depends on the client and the type of product, and it can be used to tune the relative importance of the different commitments. Notice that the nondelivery of product is penalized in both, the sales income term and in the client dissatisfaction term. Finally, since significant ‘‘start-up’’ costs exist for both PP and CJP, an additional term is included to account for putting plants back in operation after shutdowns. For the objectives of the present contribution, production and quality distribution per variety in farms is described by average and standard deviation values based on historic data. If available, forecasts should be adopted. Recent technologies allow, for example, the prediction of fruit size based on growing fruit measurements and weather data (Mayorano et al., 2006). The ‘‘eventual demand term’’ is modeled as a maximum amount of product requested by ‘‘on-spot’’ customers at each period generated as a random value from an average and a standard deviation. The complete mathematical model developed on the above description is provided in Appendix A, and the data for the particular case study addressed in this article can be found in Appendix B. For meaningful decision making, the operational planning should be able to somehow account for short-term uncertainty in fruit production, third party fruit availability and costs and prices as well as potential chain disruption events such as unexpected out-of-service of processing plants and storage chambers, reduction in transportation capacity, etc. In order to effectively consider such scenarios, the described model is proposed to be operated within an MPC framework. The proposed strategy is described in detail in the following section.

ß 2008 by Taylor & Francis Group, LLC.

31.4 OPERATIONAL PLANNING FRAMEWORK In order to address the above described operating scenario, an MPC-based scheme is proposed. MPC is a model based control strategy that shows attractive features for multivariable, highly interacting, nonlinear, uncertain systems. It was originally devised for chemical process control and later extended to many other areas due to intrinsic advantages regarding more classic control strategies. In the following, the basics of MPC are presented focusing in the particular implementation for the FISC system. For a comprehensive overview of the topic, the interested reader is referred to Ogunnaike and Ray (1994). MPC implementations consist essentially of three basic elements (Figure 31.2): .

.

.

Reference Trajectory (w(t)): The ‘‘reference trajectory’’ is usually referred as the set-point trajectory representing the desired target for the system controlled outputs. System Output Prediction (y(t)): The ‘‘system output prediction’’ is provided by a convenient dynamic model of the system, such as a multiperiod first principles representation, which allows the estimation of the evolution of the states from the current situation in the face of disturbances. Control Action Sequence (u(t)): The ‘‘control action sequence’’ is given by the manipulated variable profiles, which are the solution of an optimization problem aimed to optimize some specified objective function subject to the system model. Typically it is minimized some distance of the output prediction from the reference trajectory.

The MPC control implementation philosophy can be summarized as follows: for a given state and input variables (disturbances) in the present time, compute the profile of the manipulated variables along the considered time horizon in order to optimize the specified objective function while satisfying the system constraints. In order to accomplish this objective, some estimation of the future values of the disturbances should be assumed. The simplest solution is to consider that the values

w (t )

y (t ) u(t )

t −1

FIGURE 31.2 MPC elements.

ß 2008 by Taylor & Francis Group, LLC.

t

t +1 t + 2

of the disturbances remain constant in the present value within the whole control horizon. Since there exists a large degree of uncertainty in such estimations, it is not sound to implement the whole control profile. Rather, only the first element is actually implemented, and the whole control action is re-evaluated in the next period with updated data. Such a ‘‘rolling horizon’’ control strategy permits to ‘‘walk through the timeline’’ (Bose and Pekny, 2000) to predict how the system should evolve with the current and estimated future inputs and manipulations in order to achieve the desired objective. In order to implement an MPC strategy for supply chain planning, the system has to be conceptualized as a dynamic entity in terms of states, input and outputs (Perea Lopez et al., 2003). Some inputs will constitute ‘‘disturbances’’ to the model and some others ‘‘manipulated variables’’ for control purposes. A subset of the output variables will conform the ‘‘controlled outputs’’ whose values will be desired to follow some predefined trajectory or assume particular values in certain periods of the control horizon. For the FISC system, the following elements are identified to conceptualize the MPC strategy: .

.

.

.

Controlled output: The controlled output is the total profit of the business, which is maximized along the whole planning horizon. The total profit is defined as sales income minus costs. Costs comprehend production, third party purchase, transportation, cold storage costs and customer dissatisfaction. Each term is properly defined in Appendix A. State variables: The state variables are the inventories of the different goods in the storage facilities: fresh fruit (NPFSs, FRSs, WFSs), packed fruit (PFSs, RCPFSs, PPFSs), and concentrated juice (CJSs, PCJSs). Manipulated variables: The manipulated variables are the flows of all the streams of the system (Figure 31.1), which can be adjusted to achieve the optimal plan. Disturbances: The disturbances can occur in almost any parameter and variable of the system. For example, the amount and quality of the entering fruit is a continuous source of uncertainty as well as the costs and prices of the different goods. Operating capacity of the processing plants, storage capacity of the storage facilities, transportation capacity, etc., can also suffer temporary reductions due to unexpected events. Some estimation of all these in the future is required for the planning model, and forecasts could be used if available.

The implementation scheme for the MPC strategy for the FISC operational planning is as follows: Given: a. Delivery schedule given by the amount of packed fruit and concentrated juice to be sent to the different clients in specific periods of the timeline b. Current value of all the inventories of the system (states)

ß 2008 by Taylor & Francis Group, LLC.

c. Current data (in the present period) and estimations (for the next periods) of . Fresh fruit production in own farms (quantity and quality) . Raw material maximum availability from third party suppliers . Eventual product demand . Costs (raw fruit, transportation and operative) . Final product prices . Capacities of processing plants, storage facilities, and transportation between nodes Apply the following algorithm (Figure 31.3): 1. Set t ¼ 1. 2. Solve the MILP planning problem for a time horizon of H periods. The total profit of the system is considered as the objective function. 3. Implement only the solution corresponding to the first period of the considered horizon, this is, effectively transfer the calculated amounts of the different goods between the nodes and process the suggested amounts of the specific varieties in the processing plants to elaborate final products. 4. Store the new ‘‘state’’ of the system (inventories of the different products in the different storage facilities of the chain). 5. Update all the elements of the current data (item c). 6. Set t ¼ t þ 1, Go to 2. It should be noted that since the system is moving in the timeline, the delivery dates become progressively closer and are finally ‘‘left behind.’’ Such ‘‘relativity’’ between the timeline and the ‘‘moving’’ planning horizon is a characteristic feature of the FISC operative planning scheme. This issue is schematically illustrated in

t =1

Update forecasts

Solve MILP

Implement the solution for the first period

t =t +1

FIGURE 31.3 MPC algorithm.

ß 2008 by Taylor & Francis Group, LLC.

t =6

t =7

T

H

t =8

T

H

H

T

FIGURE 31.4 Moving horizon strategy.

Figure 31.4. The vertical dashed bars represent the amounts of a certain product committed in delivery dates 7 and 10. In the next section, the performance of the proposed algorithm is illustrated by means of two operating scenarios. A minimum version of the FISC model described in the previous section and detailed in the Appendices was used in the experiences. The considered chain infrastructure (one PP, one CJP) and the adopted planning horizon (100 days) generate a MILP model that solves in reasonable computation time, while capturing an adequate level of detail of the system.

31.5 RESULTS AND DISCUSSION In order to apply the proposed strategy, the following policy to update the information in step 5 of the algorithm was adopted: . .

Estimations of all uncertain parameters are randomly generated from average and standard deviation values as detailed in Appendix B. In order to simulate ‘‘disturbed’’ scenarios, the values of some certain and uncertain parameters (average and standard deviation) are modified during specific windows of the planning horizon.

Since there are a large number of degrees of freedom in the model, disturbances are compensated by modifying many variables simultaneously along the whole planning horizon. Therefore, it is involved to isolate the effect of some particular disturbance especially if it is of low magnitude.

ß 2008 by Taylor & Francis Group, LLC.

In order to illustrate the approach and extract some meaningful conclusions, a ‘‘nondisturbed’’ base case and a ‘‘dramatically’’ disturbed scenario are analyzed in the following. A 100 days planning horizon (H) was adopted and results are reported for the first 50 days of operation. Scenario 1 Parameters known with certainty are assumed constant in their nominal values, and uncertain parameters are randomly generated from nominal averages and standard deviations along the whole planning horizon. No disturbances or disruption episodes are simulated in this case, meaning that nominal values do not change in any period. This might be considered as a ‘‘base case’’ for comparison with disturbed scenarios. In Figure 31.5, the raw material income is shown for both NPFS (Figure 31.5a) and FRS (Figure 31.5b). It can be seen that fruit from OFs (X1, X2) and TPSs (X3, X4) is available from period 14, which coincides with the beginning of the pear harvest (apple harvest begins in day 77). There is some waste fruit production from the PP before period 14 as a result of the processing of the fruit stored in NPFS (X7F in Figure 31.5b). Fresh fruit from farms (X1 in Figure 31.5b) shows a saw-type profile as result of the adopted stochastic production policy. However, since the number of truck travels per day is an integer number, such behavior is not observed in variable ntX1 (Figure 31.6), which fluctuates between two and three truck travels per day. It should be noted that not completely full trucks are allowed for the fruit transportation. Similar behaviors can be observed for variables X2 and X7F, and ntX2 and ntX7 in Figures 31.5 and 31.6, respectively. TPS fresh fruit purchase is at the maximum possible during most of the reported period (X3 in Figure 31.5a). On the other hand, no waste fruit is demanded at all from TPSs (X4 in Figure 31.5b) since evidently PP waste fruit production (X7F in

35

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Ton

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0 10

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X2

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0

50

10

(b)

50

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FIGURE 31.5 Fresh fruit income (a) NPFS income (fruit from farms (X1) and from TPS (X3)), (b) FRS income (waste fruit from farms (X2), from PP (X7F), and from TPS (X4) ).

ß 2008 by Taylor & Francis Group, LLC.

Trip number

3 ntX 1 ntX 2 ntX 7 2

1

0 10

20

30

40

50

Days

FIGURE 31.6

Truck travels (fruit from farms (ntX1), from TPS (ntX2), and from PP (ntX7) ).

Figure 31.5b) sufficiently complements OFs waste fruit income for juice production (X2 in Figure 31.5b). Waste fruit production in farms and in the PP (X2 and X7F in Figure 31.5b, respectively) is similar in average. This results from the fact that although the waste fruit fraction is larger in farms than in the PP (see Appendix B) the amount of processed fruit at the PP (X5 in Figure 31.8) is larger than that harvested in farms (X1 in Figure 31.5a). Regarding storage, NPFS (Figure 31.7) results from the balance between the incoming fruit (Figure 31.5a) and the processed fruit in the PP (X5 in Figure 31.8).

14 1200

NPFS FRS

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1000 10 8

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0

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FIGURE 31.7 Nonprocessed material inventories.

ß 2008 by Taylor & Francis Group, LLC.

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FIGURE 31.8 Production in PP (X5) and CJP (X6).

The steep slope in the first periods of NPFS in Figure 31.7 indicates that the existing fruit is being processed but no new fruit is entering the system during that time. The same trend is observed later on but with a lower slope since fresh fruit enters the system from period 14 on. The inventory in the FRS (Figure 31.7) is the result of the balance between the incoming waste fruit (Figure 31.5b) and the fruit being processed at the CJP (X6 in Figure 31.8). In the first 14 days, only waste fruit from the PP feeds the FRS (X7F in Figure 31.5b). That fruit is immediately processed at the CJP. As pear starts entering the system, the FRS shows a more or less erratic behavior depending on the production level at the CJP. It can be seen from Figure 31.8 that the PP presents continuous operation at different production levels (X5) during the reported period (not shut-downs). The CJP on the other hand, verifies three shutdowns (X6) during that time. In the current practice, plant shutdowns are not usually implemented when fruit is available for processing due to the significant related start-up costs. However, the global economic balance considered in the model can activate such option to optimize the operations. Figure 31.9 shows the evolution of the inventories of final products in own (a) and third party storage (b) for both packed fruit and concentrated juice. Such inventories result from the balance of the produced in the PP and the CJP and the delivered to meet the fixed and eventual demands (not shown). It can be observed that, as expected, own storage is preferred as long as possible since it has a lower cost (Figure 31.9a). Third party storage (Figure 31.9b) is required however in the proximity of the delivery dates. For example, it can be observed from Figure 31.10a that there are regional client commitments of packed fruit in days 20 and 40 (among others). Important withdraws from PFS are observed those days in Figure 31.9a which coincide with high peaks in RPFCS in Figure 31.9b.

ß 2008 by Taylor & Francis Group, LLC.

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1800 1600 1400 1200

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6 400 PPFS RPFCS PCJS

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2

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100 1 50 0

0 10 (b)

20

30

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50

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FIGURE 31.9 Product inventories (a) own storage, (b) port and client storage.

However, some amounts of fruit are also sent to RPFCS the previous periods to ensure adequate satisfaction. Figure 31.10 shows the amount of dissatisfaction in packed fruit (a) and concentrated juice (b) in the different dates of the fixed demand schedule. Different levels of dissatisfaction are allowed regarding the particular weighting factors for the different commitments (Appendix B). The degree of dissatisfaction is also influenced by the ‘‘eventual demand’’ component (Figure 31.11). Since eventual product is allocated at a higher price than the corresponding to the fixed demand, it can be observed that rather large amounts of product are devoted to satisfy on spot sales. For example, significant volumes of packed pear (although not at as much as requested, see Appendix B) are assigned to this item from periods 14 on (Figure 31.11a), while

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450 Overseas-apple Overseas-pear Regional-apple Regional-pear

400 350

Ton

300 250 200 150 100 50 0 20

(a)

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16 14 Apple Pear

⫻1000 gallon

12 10 8 6 4 2 0 15 (b)

34 Days

FIGURE 31.10 Client dissatisfaction (fixed demand) (a) packed fruit, (b) concentrated juice.

allowing some dissatisfaction for the pre-established deliveries (Figure 31.10a). For packed apple on the other hand, since there are not fixed deliveries in the reported period all the existing apple in NPFS before the beginning of the apple harvest is assigned to the eventual demand at a requested average of 1000 kg=day (Figure 31.11a). For the juice case, apple juice for example is not available at all for the eventual sales from periods 35–49 (Figure 31.11b) since complete satisfaction of such product is preferred for the 34 period juice delivery date (Figure 31.10b). Scenario 2 The second analyzed scenario considers the shutdown of the PP from periods 20–30. This was modeled by setting variable X5 to zero during those periods (Figure 31.14).

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FIGURE 31.11 On spot sales (eventual demand) (a) packed fruit, (b) concentrated juice.

Such a situation can simulate a programmed maintenance stop, an employee strike, or some unexpected shutdown of the PP. From Figure 31.12a it can be seen that the fruit income to NPFS (X1) remains (stochastically) unchanged regarding the base case. On the other hand, no waste enters the FRS from PP in periods 20–30 as expected, since no fruit is processed in the PP during that time (X7F in Figure 31.12b). Due to this shortage in waste fruit, the CJP operates at low levels and is even forced to stop four times during days 20–30 (X6 in Figure 31.14). The FRS shows a decreasing trend during that time with peaks in the days that the CJP stops processing (Figure 31.13). An accumulation of non-processed fruit is observed in the NPFS (Figure 31.13) during periods 20–30, as well as a sustained reduction in the PFS during that time (Figure 31.15a). Large deficits in both, packed fruit and concentrated juice, are

ß 2008 by Taylor & Francis Group, LLC.

30 25

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FIGURE 31.12 Fresh fruit income (a) NPFS income (fruit from farms (X1) and from TPS (X3) ), (b) FRS income (waste fruit from farms (X2), from PPs (X7F), and from TPS (X4)).

observed in the delivery dates after the shutdown (Figure 31.16a and b) as a consequence of the stop in the fruit processing. Opposite to the base case where a large deficit in pear juice is allowed in period 15 (Figure 31.10b), in this second scenario pear juice deficit is greatly reduced for such delivery date (Figure 31.16b). This behavior can be also followed from the CJS profiles. For the base case, no juice is withdrawn to attend the 15 period delivery date commitment, while a large amount is removed for period 34 (Figure 31.9a). In this second scenario an important delivery is done in period 15 but the 34 period delivery date is left unattended (Figure 31.15a). In order to process the accumulated fruit, the PP operates at it largest capacity the days following the stop (periods 31–40) (X5 in Figure 31.14). As well, a significant increase is observed in CJP activity in those periods (X6 in Figure 31.14) regarding the base case (X6 in Figure 31.8).

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FIGURE 31.13

Nonprocessed material inventories.

In order to better accomplish fixed demand deliveries, the amount of packed pear assigned to satisfy the eventual demand is greatly reduced (Figure 31.17a) regarding the base case (Figure 31.11a). For the juice case, opposite to the first scenario, a large dissatisfaction of apple juice is allowed for the 34 delivery date (Figure 31.16b), while assigning such production to attend the eventual demand (Figure 31.17b).

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FIGURE 31.14

Production in PP and CJP.

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FIGURE 31.15

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Product inventories (a) own storage, (b) port and client storage.

31.6 CONCLUSION AND FUTURE WORK In this chapter an operational planning framework for the FISC management has been proposed, and its performance was illustrated through a couple of scenarios. The proposed methodology relies on a model predictive strategy, which naturally allows the handling of short-term uncertainty and potential chain disruption episodes. The system is designed to support current decisions based on forecasts of the main data but in a dynamic fashion, meaning that the forecasts are updated each period for the whole horizon including in that way new information as soon as available.

ß 2008 by Taylor & Francis Group, LLC.

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FIGURE 31.16 Client dissatisfaction (fixed demand) (a) packed fruit, (b) concentrated juice.

The studied scenarios were a ‘‘base case,’’ which reflects the normal operation of the chain, and a disturbed scenario, which simulated a disruption episode in the system. In order to cope with disturbances, ‘‘control actions’’ are simultaneously distributed among the many available ‘‘manipulations’’ along the whole planning horizon. Therefore, in order to draw some significant conclusions about the system behavior, the disturbed scenario was a rather dramatic disruption episode, namely the out of service of the main processing facility of the system. Obvious trends could be observed in mass balance related variables, for example the inventories of the different goods in the system. Other variables, such as customer dissatisfaction levels, eventual demand satisfaction, and operative levels in processing plants, present less evident responses since they result from an economic balance, which include the rather subjective ‘‘customer dissatisfaction weights.’’

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FIGURE 31.17

20 Days

On spot sales (eventual demand) (a) packed fruit, (b) concentrated juice.

An important issue to be considered for practical implementation of the proposed scheme is the computational burden of the MILP model to be solved in step 2 of the algorithm. The number of integer variables in the model depends on the number of nodes of the chain under study as well as on the length of the adopted planning horizon (H). It is expected that even for a modest number of nodes in the system (OFs, PPs, and CJPs) and a medium-size planning horizon, the model would not solve in practical time. Therefore, some strategy should be adopted to address large FISC systems in at least whole-fruit-season length time horizons.

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APPENDIX A: FISC OPTIMIZATION MODEL This appendix presents the complete mathematical model for the optimization of the FISC used in this chapter. See Figure 31.1 for a schematic reference to the main variables. Fruit production in own farms The most important source of raw material of the FISC is the fruit harvested in own farms (OFs). Historic records were used to estimate the average and standard deviation values for the daily production of each variety (AOFPv and SDOFPv, respectively). The current daily production in farms OFPt,v is calculated from those via a Monte Carlo simulation, Equation A.31.1. OFPt,v ¼ MCS{AOFPv , SDOFPv }

(A:31:1)

8v, t2HPv Waste fruit production in farms (fraction), woft,v, is calculated in Equation A.31.2 from a given average (Awofv) and standard deviation (SDwofv) also estimated from historical data: wof t,v ¼ MCS{Awofv , SDwofv }

(A:31:2)

8v, t2HPv The amounts of tradable fruit sent to packaging plants facilities, PPFs, (X1ft,v) and the waste fruit sent to concentrated juice plants facilities, CJPs, (X2ft,v) are therefore calculated as follows: X2f t,v ¼ wof t,v OFPt,v

(A:31:3)

X1f t,v þ X2f t,v ¼ OFPt,v

(A:31:4)

8v, t2HPv Delivery of fresh fruit from own farms Farm production is transported by trucks to PPFs (fresh fruit X1) and CJPFs (waste fruit X2). Equations A.31.5 and A.31.6 set bounds on the maximum volumes that can be transported with the available truck fleet. X1t,v  X1f t,v

(A:31:5)

X2t,v  X2f t,v

(A:31:6)

8v, t2HPv Fruit reception at PPFs and stock balance at NPFSs Fresh fruit reaching PPFs is stored in nonprocessed fruit refrigerated chambers, NPFSs. This stock consists of the fruit from own farms (X1) and from third party suppliers (X3) received during harvest and postharvest periods (EHPv).

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From NPFSs, fruit is fed to the packaging plant (PPs) to be processed (X5). The corresponding material balance at NPFSs is as follows: NPFSt,v ¼ NPFSt1,v þ X1t00 ,v þ X3t0 ,v  X5t,v

(A:31:7)

NPFS0,v ¼ NPFS0v

(A:31:8)

8v, 8t, t0 2 EHPv, t00 2 HPv where NPFS0v is the initial stock of fruit. The availability of third party fruit (X3) has a bound which is modeled using an average maximum (AMaxAvTPFv) and a standard deviation (SDMaxAvTPFv) obtained from historical records or forecasts, as follows: X3t,v  MaxAvTPFt,v MaxAvTPFt,v ¼ MCS{AMaxAvTPFv , SDMaxAvTPFv }

(A:31:9) (A:31:10)

8v, t 2 EHPv Fruit processed at PPs The fruit admitted at PPs is classified according to the quality and size required by each client. The fruit that does not meet quality standards is discarded as waste for juice production. Based on historical records (represented by an average Awppv, and a standard deviation, SDwppv) waste fractions corresponding to each variety can be estimated. The fruit discarded in the classification process (X7) is represented with fraction wppt,v as shown in Equations A.31.11 and A.31.12: X7t,v ¼ wppt,v X5t,v

(A:31:11)

wppt,v ¼ MCS{Awppv , SDwppv }

(A:31:12)

8v, 8t The overall balance at PPs is as follows: X7t,v þ X8t,v ¼ X5t,v

(A:31:13)

8v, 8t The fruit produced in the PPs is devoted to the overseas clients, c ¼ o, and to the regional clients, c ¼ r, which demand different types of quality and size. The estimation of the fraction for each client is based on historical records of production (Equations A.31.14, A.31.15, and A.31.16). X8t,v ¼ fct,v,c¼o X8t,v þ fct,v,c¼r X8t,v

(A:31:14)

fct,v,c¼o ¼ MCS{Afcv , SDfcv }

(A:31:15)

fct,v,c¼r ¼ 1  fct,v,c¼o

(A:31:16)

8v, 8t

ß 2008 by Taylor & Francis Group, LLC.

Waste fruit from PPs and stock balances at WFSs The waste fruit (X7) produced by the classification process at the PPs is stored (WFSs) before shipment to CJPFs (X7F). The corresponding balance is represented by Equations A.31.17 and A.31.18 WFSt,v ¼ WFSt1,v þ X7t,v  X7Ft,v

(A:31:17)

WFS0,v ¼ WFS0v

(A:31:18)

8v, 8t where WFS0v is the initial stock. Fruit balance at PFSs The fruit stored at PFSs is delivered to overseas (X10) and regional (X14) customers. There also exists the possibility of eventual sales of fruit of regional (X141) and overseas (X142) qualities to potential on-spot customers. The corresponding balances at PFSs are as follows: PFSt,v,c¼0 ¼ PFSt1,v,c¼0 þ fct,v,c¼0 X8t,v  X10t,v  X142t,v

(A:31:19)

PFSt,v,c¼r ¼ PFSt1,v,c¼r þ fct,v,c¼r X8t,v  X14t,v  X141t,v

(A:31:20)

PFS0,v,c ¼ PFS0v,c

(A:31:21)

8v, 8t, 8c where PFS0v,c is the initial stock. Fruit balance at the port refrigeration chambers PPFSs Fruit for overseas clients (X10) is stored in the port refrigerated chambers (PPFS) awaiting for the scheduled date (DSPFOEPFO,v) for the shipment (X12). The amount of refrigerated fruit at the port is calculated through the corresponding mass balances, Equations A.31.22 and A.31.23 PPFSt,v ¼ PPFSt1,v þ X10t,v þ X12t0 ,v

(A:31:22)

PPFS0,v ¼ PPFS0v

(A:31:23)

8v, 8t, t0 2 DSPFOEPFO,v where PPFS0v is the initial stock. Fruit balance at refrigerated chambers of regional clients RPFCSs Fruit for regional clients (X14) is stored in the refrigerated facilities at the client’s place (RCPFS) waiting for the reception (X15) at the scheduled date (DSPFREPFR,v). This amount of fruit is calculated through Equations A.31.24 and A.31.25. RCPFSt,v ¼ RCPFSt1,v þ X14t,v  X15t0 ,v

(A:31:24)

RCPFS0,v ¼ RCPFS0v

(A:31:25)

8v, 8t, t0 2 DSPFREPFR,v where RCPFS0v is the initial stock.

ß 2008 by Taylor & Francis Group, LLC.

Fruit reception at CJPFs and balance at the FRSs The waste fruit reaching CJPFs is stored in the reception site (FRSs). Such fruit comes from own farms (X2) and third party suppliers (X4) during the harvest and postharvest (EHPv) periods and from PPFs (X7F). FRSs provide the raw material (X6) for the concentrated juice plants (CJPs). The corresponding balances are FRSt,v ¼ FRSt1,v þ X2t00 ,v þ X4t0 ,v þ X7Ft,v  X6t,v FRS0,v ¼ FRS0v

(A:31:26) (A:31:27)

8v, 8t, t0 2 EHPv, t00 2 HPv where FRS0v is the initial stock. Waste fruit form third party suppliers (X4) is bounded (MaxAvTPWt,v). Such bound is generated by an average (AMaxAvTPWv) and a standard deviation (SDMaxAvTPWv) estimated from historical records or forecasts on third party waste fruit availability: X4t,v  MaxAvTPWt,v MaxAvTPWt,v ¼ MCS{AMaxAvTPWv , SDMaxAvTPWv }

(A:31:28) (A:31:29)

8v, t 2 EHPv Fruit processing at CJPs and juice storage at CJSs The concentrated juice plant yield is estimated from a given conversion factor (cftojv) for each variety: X9t,v ¼ cftojv X6t,v

(A:31:30)

8v, 8t The produced juice is stored in the refrigerated facility (CJSs) waiting for shipment to the overseas customers (X11) or for eventual sales (X111). The balances at CJSs are CJSt,v ¼ CJSt1,v þ X9t,v  X11t,v  X111t,v CJS0,v ¼ CJS0v

(A:31:31) (A:31:32)

8v, 8t where CJS0v is the initial stock. Juice balances at port storage facilities PCJSs Concentrated juice for overseas clients (X11) is stored in the port storage facilities (PCJS) pending for the shipments (X13) at the scheduled dates (DSCJOECJO,v). The corresponding balances are given by Equations A.31.33 and A.31.34: PCJSt,v ¼ PCJSt1,v þ X11t,v  X13t0 ,v

ß 2008 by Taylor & Francis Group, LLC.

(A:31:33)

PCJS0,v ¼ PCJS0v

(A:31:34)

8v, 8t, t0 2 DSCJOECJO,v where PCJS0v is the initial stock. Maximum inventories for fruit and juice Equations A.31.35 through A.31.40 set maximum capacities for product storage facilities: PPFs (MaxCapFS), ports (MaxCapPPFS and MaxCapPCJS for fruit and juice, respectively), regional clients (MaxCapRCPFS), CJPFs (MaxCapFRS and MaxCapCJS for waste fruit and juice, respectively): Sv (NPFSt,v þ WFSt,v þ Sc PFSt,v,c )  MaxCapFS

(A:31:35)

Sv PPFSt,v  MaxCapPPFS

(A:31:36)

Sv PCJSt,v  MaxCapPCJS

(A:31:37)

Sv RCPFSt,v  MaxCapRCPFS

(A:31:38)

Sv FRSt,v  MaxCapFRS

(A:31:39)

Sv CJSt,v  MaxCapCJS

(A:31:40)

8t Bounds and constraints for fruit processing at PPs Packaging plants can either operate within a certain production range (MinProcCapPP–MaxProcCapPP) or be completely shutdown. Additionally, they can processes only one fruit variety each day. These conditions are modeled in Equations A.31.41 and A.31.42 with the aid of binary variables (yPPt,v): yPPt,v MinProcCapPP  X5t,v  yPPt,v MaxProcCapPP Sv yPPt,v  1

(A:31:41) (A:31:42)

8v, 8t Bounds and constraints on fruit processing at CJPs Similar operating conditions as for the PPs hold for CJPs. Equations A.31.43 and A.31.44 model this situation using binary variables yCJPt,v: yCJPt,v MinProcCapCJP  X6t,v  yCJPt,v MaxProcCapCJP Sv yCJPt,v  1

(A:31:43) (A:31:44)

8v, 8t Customer dissatisfaction The company has trade commitments with customers made in the previous season (volume of products and delivery dates). Failure to fulfill these commitments is

ß 2008 by Taylor & Francis Group, LLC.

penalized with an additional cost depending on the degree of shortfall. For the overseas client, OPFC, the deficit (deltaPFOEPFO,v) is evaluated in terms of the agreed (AAPFOEPFO,v) and effectively shipped volumes (X12): deltaPFOEPFO,v ¼ AAPFOEPFO,v  X12t,v

(A:31:45)

X12t,v  AAPFOEPFO,v

(A:31:46)

8v, t 2 DSPFOEPFO,v In a similar fashion, for regional fruit (RPFC) and juice (OCJC) clients: deltaPFREPFR,v ¼ AAPFREPFR,v  X15t,v

(A:31:47)

X15t,v  AAPFREPFR,v

(A:31:48)

deltaCJOECJO,v ¼ AACJOECJO,v  X13t,v

(A:31:49)

X13t,v  AACJOECJO,v

(A:31:50)

8v, t 2 DSPFREPFR,v

8v, t 2 DSCJOECJO,v Bounds on eventual clients sales Based on historical records the volume purchased by eventual customers is characterized by an average and a standard deviation. These bounds are modeled with Equations A.31.51 through A.31.56: For fresh fruit regional quality (X141): X141t,v  MaxAvECRt,v MaxAvECRt,v ¼ MCS{AMaxAvECRv , SDMaxAvECRv }

(A:31:51) (A:31:52)

8v, 8t For fresh fruit overseas quality (X142): X142t,v  MaxAvECOt,v MaxAvECOt,v ¼ MCS{AMaxAvECOv , SDMaxAvECOv }

(A:31:53) (A:31:54)

8v, 8t For concentrated juice (X111): X111t,v  MaxAvECJt,v MaxAvECJt,v ¼ MCS{AMaxAvECJv , SDMaxAvECJv } 8v, 8t

ß 2008 by Taylor & Francis Group, LLC.

(A:31:55) (A:31:56)

Fruit transportation The transport of fruit between the different nodes of the system is performed by a truck fleet. The required number of daily trips (integer variable) ntX1t, ntX2t and ntX7t, for the nonrefrigerated transport of raw fruit X1, X2, and X7F is estimated depending on the truck load capacity per trip, npfttc: Sv X1t,v ¼ ntX1t npfttc

(A:31:57)

Sv X2t,v ¼ ntX2t npfttc

(A:31:58)

Sv X7Ft,v ¼ ntX7t npfttc

(A:31:59)

t 2 HPv

8t The number of daily trips, ntX10t and ntX14t, for the refrigerated transport of packed fruit, X10 and X14, is estimated as a function of the truck load capacity per trip, pfttc: Sv X10t,v ¼ ntX10t pfttc

(A:31:60)

Sv X14t,v ¼ ntX14t pfttc

(A:31:61)

8t The number of daily trips, ntX11t, for the refrigerated transport of concentrated juice, X11, is estimated as a function of the truck load capacity per trip, cjttc: Sv X11t,v ¼ ntX11t cjttc

(A:31:62)

8t Fruit admission in PPs and CJPs TPSs provide full loaded trucks of raw fruit: Sv X3t,v ¼ ntX3t npfttc

(A:31:63)

Sv X4t,v ¼ ntX4t npfttc

(A:31:64)

t 2 EHPv Due to handling capacity, processing plants can admit a limited number of trucks per day (MaxAUIPP for PPFs and MaxAUICJP for CJPFs): ntX1t þ ntX3t0  MaxAUIPP t 2 HPv, t0 2 EHPv

ß 2008 by Taylor & Francis Group, LLC.

(A:31:65)

ntX2t0 þ ntX4t00 þ ntX7t  MaxAUICJP

(A:31:66)

8t, t0 2 HPv, t00 2 EHPv Constraint on the dispatching of trucks with finished products to customers Due to operational reasons, there is a maximum number of trucks that can be dispatched daily from plants to clients (MaxDUIPP for PPFs and MaxDUICJP for CJPFs): ntX10t þ ntX14t  MaxDUIPP

(A:31:67)

ntX11t  MaxDUICJP

(A:31:68)

8t

8t Costs: The production cost at OFs (FrPrCt) is evaluated as a production cost per kilogram, fpct,v, times the volume of produced fruit (OFPt,v): FrPrCt ¼ Sv fpct,v OFPt,v

(A:31:69)

t 2 HPv The raw material purchase cost from TPS (FrPuCt) is calculated as the specific purchase price times the purchased volume for both tradable and waste fruit: FrPuCt ¼ Sv (fpuct,v X3t,v þ wpuct,v X4t,v )

(A:31:70)

t 2 EHPv The inventory costs (SCt) are functions of the stored volumes in the different storage facilities: SCt ¼ Sv (fsct {NPFSt,v þ WFSt,v þ Sc PFSt,v,c }) þ Sv (pfsct {PPFSt,v þ RCPFSt,v }) þ Sv frsct FRSt,v þ Sv cjsct CJSt,v þ Sv pcjsct PCJSt,v

(A:31:71)

8t where fsct is the specific fruit storage cost in PPFs pfsct is the specific fruit storage cost in ports and regional customers places frsct is the specific waste fruit storage cost in reception sites of CJPFs cjsct and pcjsct are the specific juice storage costs in CJPFs, and ports, respectively The transportation cost (TCt) is a function of the type of cargo (raw fruit, packed fruit, waste fruit and juice), the type of cost (fixed and variable), the number of trips and the distances between facilities. The fixed cost depends on the type of conveyor

ß 2008 by Taylor & Francis Group, LLC.

unit: tfcnpf ($=travel) for nonrefrigerated transportation (nonprocessed fruit and juice) and tfcpf ($=travel) for refrigerated transportation (packed fruit). The variable cost depends on the load capacity of the trucks (npfttc (kg=unit) for nonprocessed fruit, pfttc (kg=unit) for packed fruit and cjttc (gallons=unit) for juice), on the specific cost per volume and distance (fftct ($=kg=km) for nonprocessed fruit, pftct ($=kg=km) for processed fruit and cjtct ($=gallon-km) for juice) and on the distances between facilities (km) (dofnpfs between OF and PPFs, doffrs between OF and CJPFs, dppfrs between PPFs and CJPFs, dpfspfc between PPFs and RCPFS, dpfsppfs between PPFs and PPFS and dcjspcjs between CJPFs and PCJS). TCt ¼ tfcnpf (ntX1t þ ntX2t þ ntX7t þ ntX11t ) þ tfcpf (ntX10t þ ntX14t ) þ fftct (npfttc {dofnpfs ntX1t þ doffrs ntX2t þ dppfrs ntX7Ft }) þ pftct (pfttc {dpfsppfs ntX10t þ dpfspfc ntX14t }) þ cjtct cjttc dcjspcjs ntX11t

(A:31:72)

8t The operating cost (OCt) is evaluated as a function of the processed fruit in PPs (X5) and of the produced juice in CJPs (X9): OCt ¼ ppoct Sv X5t,v þ cjpoct Sv X9t,v

(A:31:73)

8t where the specific production costs are ppoct ($=kg) for PPs and cjpoct ($=gallon) for CJPs. The penalty cost due to client dissatisfaction (CICt) is evaluated as a function of the deficit in the delivered product times a weighting factor: CICt ¼ SEPFR,v cicpfr deltaPFREPFR,v þ SEPFO,v cicpfo deltaPFOEPFO,v þ SECJO,v ciccjo deltaCJOECJO,v

(A:31:74)

t 2 DSPFREPFR,v, t 2 DSPFOEPFO,v, t 2 DSCJOECJO,v where cicpfr ($=kg), cicpfo ($=kg), and ciccjo ($=gallon) are the penalization factors. There are switching costs in PPs and CJPs (CCVt) when the variety being processed changes (from pears to apples and from apples to pears) due to minor adjustments required in the process equipment. To evaluate this cost, it is computed if a variety change is produced in some period using variables ZFt for PPs and ZJt for CJPs:

ß 2008 by Taylor & Francis Group, LLC.

ZFt  yPPt,v¼p  yPPt1,v¼p

(A:31:75)

ZFt  yPPt1,v¼p  yPPt,v¼p

(A:31:76)

0  ZFt  1

(A:31:77)

ZJt  yCJPt,v¼p  yCJPt1,v¼p

(A:31:78)

ZJt  yCJPt1,v¼p  yCJPt,v¼p

(A:31:79)

0  ZJt  1

(A:31:80)

8t Continuous variables ZFt and ZJt are forced to take the value 0 when there is no variety change from one day to the following, and 1 otherwise. The switching cost is calculated as follows: CCVt ¼ ZFt CCVF þ ZJt CCVJ

(A:31:81)

8t where CCVF is the cost per change in PPs and CCVJ in CJPs. There are also start-up costs (CPMt) in PPs and CJPs when plants are put back into operation after shutdowns. In order to evaluate this cost, the number of times that the plants are started up is calculated with the aid of continuous variables ZPPt for PPs and ZCJPt for CJPs, which are forced to take values 0 or 1: ZPPt  Sv (yPPt,v  yPPt1,v )

(A:31:82)

0  ZPPt  1

(A:31:83)

ZCJPt  Sv (yCJPt,v  yCJPt1,v )

(A:31:84)

0  ZCJPt  1

(A:31:85)

8t The start-up cost is calculated as CPMt ¼ ZPPt CPMPP þ ZCJPt CPMCJP

(A:31:86)

8t where CPMPP and CPMCJP are the start-up costs for PPs and CJPs, respectively Sales income The sales income (INCOMt) depends on the price of the final products and the corresponding delivered volumes: INCOMt ¼ Sv (pfpov {X12t,v þ X141t,v þ X142t,v }) þ Sv (pfprv X15t,v ) þ Sv (cjpov {X13t,v þ X111t,v })

(A:31:87)

8t where pfpov and pfprv are the prices of packed fruit of overseas and regional qualities respectively cjpov is the price of concentrated juice

ß 2008 by Taylor & Francis Group, LLC.

Objective function The objective function (BEN) is the gross profit calculated as the difference between incomes and costs for the considered planning horizon: BEN ¼ St (INCOMt  {FrPrCt þ FrPuCt þ SCt þ TCt þ OCt þ CICt þ CCVt þ CPMt })

(A:31:88)

8t Fruit industry supply chain optimization problem The optimization problem consists in the maximization of the gross profit, subject to the described model: Maximize BEN ¼ Gross Profit (Equation A.31.88) s.t. {SC operating model: Equations A:31:1 and A:31:88}

(A:31:89)

Problem (A.31.89) is a MILP model whose solution provides the optimal time profiles of the inventories and the flow-rates of each stream, along with the levels of production in the processing plants, subject to the constraints on raw material availability and processing, storage and transportation capacities.

APPENDIX B: OPTIMIZATION MODEL PARAMETERS

TABLE 31.1 Harvest Seasons: Production (OFPt,v) and Waste Fraction (woft,v) at OFs v

Variety

HPv

—

—

Days

p a

pear apple

14 to 100 77 to 100

AOFPv

SDOFPv

Awofv

kg=day 27,000 38,000

% 3,300 4,000

18 20

TABLE 31.2 Production at PP Fraction of Tradable Quality (fct,v,c) and Fraction of Waste (wppt,v) v — p a

Afcv 30 40

ß 2008 by Taylor & Francis Group, LLC.

SDfcv 3 4

%

SDwofv

Awppv

SDwppv

10 10

1 1

1.7 1.9

TABLE 31.3 Maximum Availability at TPSs: Fresh Fruit (MaxAvTPFt,v) and Waste Fruit (MaxAvTPWt,v) v AMaxAvTPFv SDMaxAvTPFv AMaxAvTPWv SDMaxAvTPWv

—

p

a

ton=day

12 1 5 0.5

30 4 8 0.9

TABLE 31.4 Maximum Availability for Eventual Clients: Fruit of Regional Quality (MaxAvECRt,v), Fruit of Overseas Quality (MaxAvECOt,v), Concentrated Juice (MaxAvECJt,v) v

— kg=day

AMaxAvECRv SDMaxAvECRv AMaxAvECOv SDMaxAvECOv AMaxAvECJv SDMaxAvECJv

gallon=day

p

a

500 50 500 50 50 5

500 50 500 50 50 5

TABLE 31.5 Commercial Compromises Packed fruit (ton=shipment) Regional Client, AAPFREPFR,v v

EPFR1

EPFR2

EPFR3

EPFR4

EPFR5

p a

400 230

400 230

400 230

400 230

400 230

Overseas Client, AAPFOEPFO,v v

EPFO1

EPFO2

EPFO3

EPFO4

EPFO5

p a

170 100

170 100

170 100

170 100

170 100

Concentrated juice (gallon=shipment) Overseas Clients, AACJOECJO,v v

ECJO1

ECJO2

ECJO3

ECJO4

ECJO5

p a

8500 6000

8500 6000

8500 6000

8500 6000

8500 6000

ß 2008 by Taylor & Francis Group, LLC.

TABLE 31.6 Shipments Timetable (Day) Regional Client, DSPFREPFR,v v

EPFR1

EPFR2

EPFR3

EPFR4

EPFR5

p a

20 20

40 40

60 60

80 80

100 100

v

EPFO1

EPFO2

EPFO3

EPFO4

EPFO5

p a

20 80

35 85

42 90

56 95

70 100

Overseas Client, DSPFOEPFO,v

Overseas Client, DSCJOECJO,v v

ECJO1

ECJO2

ECJO3

ECJO4

ECJO5

p a

15 15

34 34

73 73

87 87

100 100

TABLE 31.7 Initial Stocks v NPFS0v PFS0v,c PPFS0v FRS0v CJS0v PCJS0v WFS0v RCPFS0v

—

p

Ton

0 r 0 0 0 0 0 0 0

Gallon Ton

a o 0

1200 r 700 0 0 5120 0 0 0

o 510

TABLE 31.8 Maximum Stock Capacity MaxCapFS MaxCapPPFS MaxCapRCPFS MaxCapFRS MaxCapCJS MaxCapPCJS

ß 2008 by Taylor & Francis Group, LLC.

Ton

Gallon 1000 gallons

2,600 2,500 2,000 100 15,000 5,000

TABLE 31.9 Production Capacity PP (kg=day) MinProcCapPP MaxProcCapPP

20,000 60,000

CJP (gallon=day) MinProcCapCJP MaxProcCapCJP

5,000 17,000

TABLE 31.10 Transportation Distances (km) dofnpfs doffrs dppfrs dpfspfc dpfsppfs dcjspcjs

55 78 22 450 500 500

TABLE 31.11 Production Costs in OFs and Raw Material Price from TPSs ($=kg) v fpct,v fpuct,v wpuct,v

—

p

a

t 2 HPv t 2 EHPv

0.1179 0.45 0.20

0.0984 0.38 0.18

TABLE 31.12 Fruit to Juice Conversion v

cftojv (gallon=kg)

p a

0.0323 0.0313

ß 2008 by Taylor & Francis Group, LLC.

TABLE 31.13 Storage Costs 8t (Day) fsct frsct pfsct cjsct pcjsct

$=ton

$=1000 gallons

1.20 0.10 1.80 1.95 3.95

TABLE 31.14 Transportation Costs Fixed tfcnpf tfcpf

$=trip

200 500

$=ton  km

0.50 0.70 5.00

Variable fftct pftct cjtct

$=1000 gallons  km

TABLE 31.15 Transportation Capacity npfttc pfttc cjttc

kg=trip gallon=trip

10000 30000 5280

TABLE 31.16 Operating Costs 8t Production ppoct cjpoct

$=kg $=gallon

0.55 2.10

$

7000 20000

$

200 500

Starting-up CPMPP CPMCJP Variety change CCVF CCVJ

ß 2008 by Taylor & Francis Group, LLC.

TABLE 31.17 Penalization Factors for Product Dissatisfaction cicpfr cicpfo ciccjo

$=kg

1.20 2.10 5.00

$=gallon

TABLE 31.18 Sale Price of Final Products v pfpov pfprv cjpov

— $=kg $=gallon

p

a

2.50 1.60 7.50

2.30 1.50 6.95

NOMENCLATURE ABBREVIATIONS CJP CJPF OCJC OF OPFC PP PPF RPFC TPS

BY INITIALS

Concentrated juice plant Concentrated juice plant facility Overseas concentrated juice client Own farms Overseas packed fruit client Packaging and classification plant Packaging plant facility Regional packaged fruit client Third party suppliers

INDEXES c ECJO EPFO EPFR t v

Packed fruit client Shipping of concentrated juice to overseas market Shipping of packed fruit to overseas market Shipping of packed fruit to regional market Time period (days) Fruit variety

SETS DSCJOECJO,v DSPFOEPFO,v

Timetable for shipping of concentrated juice to overseas clients Timetable for shipping of packed fruit to overseas client

ß 2008 by Taylor & Francis Group, LLC.

DSPFREPFR,v EHPv HPv

Timetable for dispatching of packed fruit to regional client Extended harvest period Harvest period

VARIABLES Mass Flow Variables X1t,v X1Ft,v X2t,v X2Ft,v X3t,v X4t,v X5t,v X6t,v X7t,v X7Ft,v X8t,v X9t,v X10t,v X11t,v X12t,v X13t,v X14t,v X15t,v X111t,v X141t,v X142t,v

Fresh fruit from OFs to PPFs Own fresh fruit for PPFs Waste fruit from OFs to CJPFs Own waste fruit for CJPFs Fresh fruit from TPSs to PPFs Waste fruit from TPSs to CJPFs Fresh fruit feed to PPs Waste fruit feed to CJPs Waste fruit produced at PPs Waste fruit from PPFs to CJPFs Packed fruit produced at PPs Concentrated juice produced at CJPs Packed fruit of overseas quality from PPFs to PPFS Concentrated juice from CJPFs to PCJS Packed fruit to overseas client Concentrated juice to overseas client Packed fruit of regional quality from PPFs to RCPFS Packed fruit to regional client Concentrated juice to eventual client Packed fruit of regional quality to eventual client Packed fruit of overseas quality to eventual client

Inventory Variables CJSt,v FRSt,v NPFSt,v PCJSt,v PFSt,v,c PPFSt,v RCPFSt,v WFSt,v

Concentrated juice stock in CJPFs Waste fruit stock in CJPFs Nonprocessed fresh fruit stock in PPFs Port concentrated juice stock Packed fruit stock in PPs Port packed fruit stock Regional client packed fruit stock Waste fruit stock in PPFs

Economic Variables BEN CCVt CICt CPMt

Gross profit Change of variety cost Client dissatisfaction cost Plant start-up cost

ß 2008 by Taylor & Francis Group, LLC.

FrPrCt FrPuCt INCOMt OCt SCt TCt

Own farm production cost TPS raw material cost Sales income Operative cost Stock cost Transportation cost

Operation Variables Binaries yCJPt,v yPPt,v

Operation=No-operation of CJPs Operation=No-operation of PPs

Continues ZCJPt ZFt ZJt ZPPt

Counter Counter Counter Counter

of of of of

starting-ups of CJPs variety changes in PPs variety changes in CJPs starting-ups in PPs

Transportation Variables (Integer) ntX1t ntX2t ntX3t ntX4t ntX7t ntX10t ntX11t ntX14t

Number Number Number Number Number Number Number Number

of trips for X1 of trips for X2 of trucks for X3 of trucks for X4 of trips for X7F of trips for X10 of trips for X11 of trips for X14

PARAMETERS AACJOECJO,v AAPFOEPFO,v AAPFREPFR,v Afcv AMaxAvECJv AMaxAvECOv AMaxAvECRv AMaxAvTPFv AMaxAvTPWv AOFPv Awofv Awppv

Commercial compromise with OCJCs Commercial compromise with OPFCs Commercial compromise with RPFCs Average fraction of overseas quality fruit produced at PPs Average maximum availability of juice for eventual clients Average maximum availability of fresh fruit of overseas quality for eventual clients Average maximum availability of fresh fruit of regional quality for eventual clients Average maximum availability of fresh fruit at TPSs Average maximum availability of waste fruit at TPSs Average production in OFs Average fraction of waste fruit production in OFs Average fraction of waste fruit produced at PPs

ß 2008 by Taylor & Francis Group, LLC.

CCVF CCVJ cftojv Ciccjo Cicpfo Cicpfr cjpoct cjpov CJS0v cjsct cjtct Cjttc CPMCJP CPMPP Dcjspcjs Doffrs Dofnpfs Dpfspfc Dpfsppfs Dppfrs fct,v,c fftct fpct,v fpuct,v FRS0v frsct fsct MaxAvECJt,v MaxAvECOt,v MaxAvECRt,v MaxAvTPFt,v MaxAvTPWt,v MaxCapCJS MaxCapFRS MaxCapFS MaxCapPCJS MaxCapPPFS MaxCapRCPFS MaxProcCapCJP MaxProcCapPP MinProcCapCJP MinProcCapPP NPFS0v

Cost of switching of fruit variety at PPs Cost of switching of fruit variety at CJPs Fruit to juice conversion in CJPs Penalization for dissatisfaction to OCJCs Penalization for dissatisfaction to OPFCs Penalization for dissatisfaction to RPFCs Operating cost of CJPs Concentrated juice selling price Initial juice stock in CJSs Cost of cold production in CJPFs Cost of concentrated juice transportation Truck capacity for concentrated juice transportation Starting-up cost of CJPs Starting-up cost of PPs Distance between CJPFs and PCJSs Distance between OFs and CJPFs Distance between OF and PPFs Distance between PPFs and RCPFS Distance between PPFs and PPFS Distance between PPFs and CJPFs Fraction of fruit produced at PPs of quality c Cost of nonprocessed fruit transportation Production cost in OFs Purchase cost of fresh fruit from TPSs Initial waste fruit stock in FRSs Storage cost in CJPFs Storage cost in PPFs Maximum juice availability for eventual client Maximum fresh fruit availability of overseas quality for eventual client Maximum fresh fruit availability of regional quality for eventual client Maximum fresh fruit availability from TPSs Maximum waste fruit availability from TPSs Maximum stock capacity of CJPFs Maximum stock capacity of FRSs Maximum stock capacity of PPFs Maximum stock capacity of PCJSs Maximum stock capacity of PPFSs Maximum stock capacity of RCPFSs Maximum processing of CJPs Maximum processing capacity of PPs Minimum production capacity of CJPs Minimum production capacity of PPs Initial stock of nonprocessed fruit in NPFSs

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Npfttc OFPt,v PCJS0v pcjsct pfpov pfprv PFS0v pfsct pftct Pfttc PPFS0v ppoct RCPFS0v SDfcv SDMaxAvECJv SDMaxAvECOv SDMaxAvECRv SDMaxAvTPFv SDMaxAvTPWv SDOFPv SDwofv SDwppv Tfcnpf Tfcpf WFS0v woft,v wppt,v wpuct,v

Truck capacity for nonprocessed fruit Fruit production in OFs Initial stock of juice in PCJSs Storage cost of PCJSs Price of packed fruit of overseas quality Price of packed fruit of regional quality Initial stock of packed fruit in PFSs Storage cost in PPFSs and RCPFSs Transportation cost of packed fruit Truck capacity for packed fruit Initial stock of processed fruit in PPFSs Operating cost of PPs Initial stock of processed fruit in RCPFSs Standard deviation of the fraction of fruit of overseas quality produced in PPs Standard deviation of maximum availability of juice for eventual clients Standard deviation of maximum availability of packed fruit of overseas quality for eventual clients Standard deviation of maximum availability of packed fruit of regional quality for eventual clients Standard deviation of maximum availability of fresh fruit from TPSs Standard deviation of maximum availability of waste fruit from TPSs Standard deviation of production in OFs Standard deviation of waste fruit in OFs Standard deviation of fraction of waste fruit produced in PPs Fixed cost for non refrigerated transportation Fixed cost for refrigerated transportation Initial stock of waste fruit in WFSs Fraction of waste fruit in the production of OFs Fraction of waste fruit produced in PPs Purchase price of waste fruit from TPSs

REFERENCES Adhitya, A., Srinivasan, R., and Karimi, I.A., A model based rescheduling framework for managing abnormal supply chain events, Comp. Chem. Eng., 31, 496, 2007. Aytug, H. et al., Executing production schedules in the face of uncertainties: A review and some future directions, Eur. J. Oper. Res., 161, 86, 2005. Blanco, A.M. et al., Operations management of a packaging plant in the fruit industry, J. Food Eng., 70, 299, 2005. Bose, S. and Pekny, J.F., A model predictive framework for planning and scheduling problems: A case study for consumers goods supply chain, Comp. Chem. Eng., 24, 329, 2000.

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Masini, G., Petracci, N., and Bandoni, A., Supply chain planning in the fruit industry, Proceedings of FOCAPO 2003, Coral Springs-Miami, FL. Masini, G.L. et al., Supply chain tactical optimization in the fruit industry, in Supply Chain Optimization, Papageorgiou, L. and Georgiadis, M. (Eds.), John Wiley and Sons, New York, Volume 4, Part II, pp. 121–172, 2007. Available online at Mayorano, F.J. et al., A computational model for forecasting of fruit size, RIA, 35, 143, 2006. Mestan, E., Turkay, M., and Arkun, Y., Optimization of operations in supply chain systems using hybrid systems approach and model predictive control, Ind. Eng. Chem. Res., 45, 6493, 2006. Ogunnaike, B.A. and Ray, W.H., Process Dynamics, Modeling and Control, Oxford University Press, London, UK, 1994. Ortmann, F.G., Modeling the South African fresh fruit export supply chain, MSc Thesis, 2005. Perea-Lopez, E. et al., Dynamic modeling and decentralized control of supply chains, Ind. Eng. Chem. Res., 40, 3369, 2001. Perea-Lopez, E., Ydstie, B.E., and Grossmann, I.E., A model predictive control strategy for supply chain optimization, Comp. Chem. Eng., 27, 1201, 2003. Shapiro, J.F., Modeling the Supply Chain, Duxbury-Thompson Learning, Pacific Grove, California, 2001.

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32

Optimizing the Management of Curing Chambers Jose Bon and Antonio Mulet

CONTENTS 32.1 32.2

Introduction .............................................................................................. 747 Materials and Methods ............................................................................ 749 32.2.1 Raw Material ............................................................................... 749 32.2.2 Curing Model .............................................................................. 749 32.2.3 Experimental Procedure .............................................................. 749 32.2.4 Management System ................................................................... 751 32.2.4.1 Modeling the Curing Process ...................................... 751 32.2.4.2 Curing Process Optimization Model (Decision Making) ....................................................... 752 32.2.4.3 Control System ............................................................ 753 32.3 Results and Discussion ............................................................................ 754 32.3.1 Curing Model .............................................................................. 754 32.3.2 Management System ................................................................... 756 32.3.3 Optimization Tool ....................................................................... 757 32.3.4 Control......................................................................................... 759 32.3.5 Communication ........................................................................... 760 32.4 Conclusion ............................................................................................... 760 Acknowledgments................................................................................................. 760 Nomenclature ........................................................................................................ 760 Greek Letters............................................................................................ 761 References ............................................................................................................. 761

32.1 INTRODUCTION Curing processes are common in meat and milk industries. Especially in small firms, products from different batches are often cured in the same chamber bringing an added intricacy to the management of the curing process (Tscheuschner, 2001; Fellows, 2007). The curing process requires maintaining relatively high levels of humidity in the chamber and avoiding unnecessary local changes in order to minimize problems of

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either case hardening or mould proliferation. This needs careful attention of temperature and humidity control (Kramer, 2000; Moreira, 2001; Young et al., 2006). On the other hand, changes in the controlled variables (air temperature and relative humidity) will affect the product batches present in the curing chamber in a different way, which is something that would be necessary to take into account when dealing with curing chamber management. All these reasons point to why it is desirable to avoid or minimize variations around the set points of moisture and temperature in the chamber and to obtain and quantify the information needed about the behavior of the product. Consequently, a suitable management of the curing chamber would need a wide range of considerations, such as knowing how dependent the evolution of the product’s curing state is on the environmental conditions, how the environmental conditions affect the different product batches, and how the batch load influences the chamber dynamics. As the main objective of the curing management is to improve product quality (Mittal, 1997; Tscheuschner, 2001; Fellows, 2007), it is desirable to (1) evaluate how changes in the controlled variables will affect the different batches in the chamber; (2) quantify the variation of the batches in the chamber with time; (3) quantify and evaluate the effect of new curing protocols used by producers to accelerate processing; (4) establish how to attain a desired curing state in a given time for a batch; and (5) optimize the quantity of product in the chamber undergoing a curing state in a given period. Therefore, the curing processes in the food processing industry need global management=optimization in order to achieve the quality levels set by producers, and the desired improvement in costs. Such management means carrying out objective production studies in order to set up optimization problems. To attain this objective, mathematical models that simulate the curing processes and also model the decisions of the operator are needed (Trystram and Courtois, 1997; Irudayaraj, 2001; Datta and Rattray, 2005; Sablani et al., 2006). To approach the global management of the curing process, both tasks (process control and decision making) should be linked in order to enable the decisions to be made in a relatively rapid and objective way. Therefore, an integrated tool is necessary for the management, control, and optimization of curing processes (Mulet, 1994; Mulet et al., 1998; Irudayaraj, 2001; Seider et al., 2003; Bequette, 2003; Bon, 2005; Bon et al., 2005; Foo et al., 2005). As a consequence, a tool with three basic components could be considered: (1) a component devoted to managing the curing chamber that can store and manage the information about the inputs and outputs, update batch information, manage the communication with the producer, estimate the curing state of the batches, and assess the necessary links with other components; (2) an optimization component which determines the optimal values of the decision variables (T and RH) by considering the objectives and restrictions fixed by the producer; (3) a chamber control component, considering a description of a model of the effect of the actuators (cooling equipment, dehumidifier, and humidifier) on the controlled variables (T and RH). The tasks carried out by those components could be grouped in two systems, one directed towards management optimization and the other towards chamber control. In order to show the development of a system for the optimal management of curing chambers, an example based on a cheese curing process will be addressed.

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32.2 MATERIALS AND METHODS 32.2.1 RAW MATERIAL To develop the management system, the elaboration process of Mahón cheese (Minorca, Spain) (Denomination of Origin) was used, and several controlled batches were obtained from a collaborating company. The pieces were manufactured using pasteurized milk according to the usual guidelines of the company.

32.2.2 CURING MODEL Simal et al. (2001) showed that a diffusion model would adequately describe the weight losses of Mahón Cheese. Evidently, curing is a complex process which includes not only water loss but also complex biochemical and textural transformations. Nevertheless, according to literature, as water loss is a kinetic phenomenon, it is related to the cheese characteristics, and therefore seemingly, as a first approach in the modeling of the process, it is possible to use the moisture content as an indicator of the curing state. Considering only the first term of the diffusion model for the intersection of three infinite slabs of different thicknesses (Figure 32.1), an approximation of the average dimensionless moisture content can be described according to the following equation:
  12 þ 22 p2 De t t  te ¼ e 4L1 4L2 c¼ to  te

(32:1)

The evolution of moisture content over time was studied by Tarrazó et al. (1995), using controlled production of Mahón cheese, at three temperatures, 58C, 108C, and 158C in chambers with controlled relative humidity at 85%. The effect of ambient humidity was also addressed considering three different relative humidity values of 80%, 85%, and 90%, keeping the temperature constant at 128C. The data obtained from these production processes were used to fit to the mathematical model.

32.2.3 EXPERIMENTAL PROCEDURE To obtain curing=drying data while avoiding the system inertia and the interaction between the relative humidity and the temperature in the curing chamber, a

2L2

2L1 2L2

FIGURE 32.1 Shape of Mahón cheese.

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FIGURE 32.2 Microchamber.

microchamber was set up where low levels of air velocity were maintained similar to what happens in the industrial chambers. The microchamber with controlled relative humidity measured 40  35  22 cm (Figure 32.2) and was placed inside a larger one where the temperature was controlled. To control the air humidity in the microchamber, an air pump of 5 W and 200 L=h maximum air flow was used, and by means of an appropriate net of ducts, the ambient air (outside the microchamber) was circulated through a glass column with silica-gel (Figure 32.3). To obtain a dumping effect inside the chamber, different salts were introduced according to the required humidity level. The salts were chosen according to the equilibrium humidity of the corresponding saturated dissolutions (Greenspan, 1977). Cheese pieces were introduced into the microchamber according to a preestablished time table; the piece dimensions were measured and weighed at preset times with a Mettler PM 4000 balance (0.1 g accuracy) during the curing process. Controlled temperature chamber

Timer

Air distribution duct

Fan Air exit Controller

Controlled humidity chamber Silica-gel column

Net Cheese

Cheese

Pump

FIGURE 32.3 Scheme assembly microchamber.

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Humidity probe

Cheese

Saturated salt solution

PC

Cooling equipment

Hood Shelves

Heater

Condensate

Control box T RH

Air

Control system

Management system Humidifier H

FIGURE 32.4 Pilot chamber.

A pilot curing chamber of 9.73 m3 (Figure 32.4) was used to study and to develop the control system of the environmental conditions. For setting up the control system and in order to eliminate the effect of air streams in the chamber, air was distributed using hoods made of fabrics with a wide enough mesh for good air circulation. To control the humidity in the pilot chamber, two actuators were used: A heater placed at the exit from the evaporator and a Defensor 505 humidifier, which mechanically atomized water (around 500 g=h) in the order of micrometers. A device from Galltec (Polyga mod. FG 120 PT 100) was used as a relative humidity probe. The range of measurement covers from 0% to 100% relative humidity with 2.5% accuracy for humidity values greater than 40%. Pt100 probes were used for temperature measurement with a range of between 2008C and 8508C with a precision of 0.18C and an accuracy of 0.1%. A model TTM-109-1-N-A, TOHO controller was used for noncomputer controlled experiments to receive the signals from the probes and convert them in order to assure the link between the probes and the computer. The global management and control of the pilot curing chamber were carried out by means of two different networked PCs. One computer carried out the tasks linked to chamber control and the other one dealt with process management.

32.2.4 MANAGEMENT SYSTEM 32.2.4.1

Modeling the Curing Process

To fit the model, experiments were carried out both in the microchamber and in the pilot chamber. As main initial data, the weight and the dimensions of each piece were determined, and these measurements were periodically repeated. The pieces were turned over daily in order to facilitate the airing on both sides. As a general rule, in order to characterize the cheeses adequately, they were grouped into batches, each batch including cheeses of similar characteristics

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produced at the same time. To be able to assess the curing state of a cheese, concept of ‘‘curing state’’ of a given batch was defined as ‘‘the time needed for the same type (size, producer) of cheese in standard environmental conditions to reach the curing level of the batch.’’ Obviously the same ‘‘Curing State’’ does not mean the same characteristics for different sized cheeses. In order to quantify this property (curing state), a unit called the ‘‘standard day’’ has been defined as ‘‘the evolution during a day of a cheese submitted to standard environmental conditions.’’ A temperature of 128C and humidity of 85% were taken as standard environmental conditions. Consequently, the curing state could be addressed as ‘‘Number of days that the standard cheese would need, submitted to the conditions standard of curing, to reach the characteristics of the cheese evaluated.’’ The drying model Equation 32.1 was considered to compute the curing state. Evidently, to estimate the parameters of the mathematical model Equation 32.1, experimental information was needed (time, moisture content, environmental conditions). To identify the values of the parameters, a tool in Excel 2003 was developed by means of the creation of tables (experimental values, values calculated with the model), the definition of an appropriate objective function (average of the sum of the squared differences between the calculated and experimental values), and the use of the optimization tool ‘‘Solver’’ of Excel, in this case defining the corresponding parameter in the mathematical model (effective diffusivity, De) as decision variable. 32.2.4.2

Curing Process Optimization Model (Decision Making)

The procedure to be established should satisfy the requirements defined by the producer (operator); such requirements are directly related to the quality standards of the product and indirectly related to costs or benefits. In a general framework, the producer will define his trade policy, influenced to a greater or lesser degree by the process and from these considerations he will establish the optimization sought. In other words, he will satisfy his requirements with the resources available. Considering the general methodology of formulation of an optimization problem (Edgar and Himmelblau, 2001), the following stages could be set up: 1. Identification of the decision variables: temperature and relative humidity of the chamber (T ^ RH). 2. Establishing of the functional restrictions. The user can determine the intervals in which the decision variables can vary. 3. Objective function (mathematical expression of what the producer wants to optimize). The objective will depend on the producer’s expectations on demand of a type of cheese (curing days). By changing the environmental conditions, it is possible to satisfy these expectations in different ways. The problem is not obvious due to the existence of different batches. As an example, the objective to be attained could read as follows: Quantity of cheese that would show a state or range of maturity (interval of standard days) at a given elapsed time.

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It can happen that the inputs in the chamber present a strong seasonal variation. To avoid and detect these circumstances that would influence the solution of the optimization and would interfere with the aims of the producer, two models have been defined on what is aimed to be optimized. The first model for the objective function considers only the batches that would be included within the range of curing state considered. The second model would look for the closest possible fit to the specified range taking all the batches into consideration. The use of both models to examine the results will accurately show the effects of the distribution of the chamber load. If the load is relatively uniform along the time, the objective function model used will be of no importance. Nevertheless, for slightly uniform loads, the results differ and the producer must analyze the results and decide according to his needs. Since the existence of different batches led to more complicated optimization, a direct optimization method (SIMPLEX) was considered to solve this problem (Edgar and Himmelblau, 2001; García et al., 2006). The optimum search is always carried out inside the set of possible solutions, a set limited by the constraints to which the decision variables (temperature and relative humidity) are restricted. 32.2.4.3

Control System

Due to the fact that a curing chamber usually has a ‘‘large capacity,’’ and the fact that a model with sufficient precision to allow the application of a ‘‘feed-forward’’ control is usually not available (Whitman et al., 2004), an on–off type feedback control was taken as a suitable and useful control scheme for curing chambers. To minimize the amplitude of the oscillatory variations of the temperature and relative humidity of the chamber, a mathematical model for the behavior of the chambers and the corresponding sensors and actuators are required. The ‘‘input’’ control variables were the working times of the compressor, the dehumidifier, (evaporator and resistance) and the humidifier. To control functioning of the curing chambers, it is necessary to know the effect of the input control on the variables which are considered (temperature and relative humidity) to have an influence on the curing process (Trystram and Courtois, 1997). Then, there is a need to obtain the models which describe the following effects on the chamber: . . . .

Effect of the cooling equipment Effect of the dehumidifier (evaporator and resistance on) Effect of the humidifier Expected variation of temperature and humidity in the chamber, without control

Every effect can be mathematically described from the application of energy and mass balances in the chamber. To carry out the mass and energy balances, some relationships like, for example, the equation that describes the speed of transfer of water between cheeses and the chamber environment, are also needed (Equation 32.1). For every effect, a system of differential equations that links the evolution of the environmental conditions (temperature and relative humidity) depending on the working time of the actuator (cooling equipment, humidifier, and dehumidifier) can be obtained.

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Since an on–off controller was chosen to control the working time of the actuators, the mathematical model was discretized applying in this case the Z transform (Lipták, 2005). The equations describing the advance of variables (T, X) as a result of the actuators were obtained by inverting the Z transform equations with this general form: Tk ¼ aTkk1 þ ebDt Tk1

Xk ¼ aXkk2 þ ebDt Xk1

(32:2)

That is to say, the predicted value of temperature and relative humidity for an instant k is described depending on the measured values of the above mentioned variables in previous samplings. The k constant indicates the time of reading of the environmental conditions (temperature and relative humidity), and the parameters k1 and k2 quantify the dead times for temperature and moisture content. The values of the parameters a, b, a, and b change according to the effect considered.

32.3 RESULTS AND DISCUSSION The problem approach is based on two systems that act independently, but are interconnected. The control system deals with the procedures relative to the chamber, and the management system deals with the procedures linked with the chamber management according to the instructions of the operator. To develop the content of both systems, different actions were carried out so as to establish the necessary models and program different tools required to set up the procedures.

32.3.1 CURING MODEL The curing process was, as a first approach, linked to water losses. An example of moisture content losses is shown in Figure 32.5. As can be observed in this figure, Time (days) 0 0 Log dimensionless moisture

−0.1 −0.2

20

40 60 80 100 Ln y = −0.0072 t + 0.0008 R 2 = 0.998 Standard deviation = 4  10 −6 %

120

−0.3 −0.4 −0.5 −0.6

Ln y = −0.0036 t + 0.0831 R 2 = 0.998 Standard deviation = 810−5 %

−0.7 Period 1

Period 2

FIGURE 32.5 Fitting of the water loss model, RH 85%.

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140

160

TABLE 32.1 Effective Diffusion Coefficient (m2=s), T ¼ 128C De (m2=s) RH (%) 80 85 90

te (kg=kg) 0.193 0.206 0.293

Period 1 11

5.868  10 4.431  1011 4.050  1011

Period 2 1.976  1011 2.215  1011 3.603  1011

the first section of the curve (up to approximately 20 days) does not fit the same straight line as the other points. This could be due to a change in product characteristics during the curing time, and as a consequence two periods must be considered in the model. It could also be due to the fact that, at 20 days after introducing the cheeses into the chambers for curing, they were surface treated with an antifungal paint in order to avoid later fungal infections, this being a particular characteristic of the producer. For these reasons, two experimental straight-line curves could be fitted, one before and another after the first 20 days, thus defining two curing periods. The fitting of the curing model (Equation 32.1) to the experimental data allows identification of the effective diffusion coefficient De. Some results obtained according to this procedure are shown in Table 32.1. To obtain the curing model, the external resistance to mass transfer was neglected. As a consequence the effective diffusion coefficient can change with the relative humidity as shown in Table 32.1. The influence of the temperature on the diffusion coefficient was analyzed. It can be observed that the temperature influence follows the Arrhenius equation, as Equation 32.3 shows at 85% of relative humidity: Ln(De ) ¼ 0:4187  7108:4

1 T þ 273:16

R2 ¼ 0:95

(32:3)

In order to test the accuracy of the model, the explained variance for each of the periods considered was calculated. In the first period, fitting of the model to the experimental results was good in all the cases considered, the explained variance being 99% and the standard deviation 0.00004. In the second period, the fit was equally good, the explained variance being 99.7% and the standard deviation also 0.00004. Therefore, the proposed model shows enough precision to achieve the desired objective. It should be indicated that the value of the constants of the model may differ among the producers (technological differences), and an adaptation may be necessary. The results of the model showed, as a first approach, that it is possible to use it to define the curing of the Mahón Cheese in standard days for a particular producer.

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32.3.2 MANAGEMENT SYSTEM The managing procedure looks to centralize the information and the joint treatment of the available information in order to reach the objectives sought. According to the methodology previously considered, a system was created thus allowing the management of the curing process of the existing batches in the chamber. The development of the management system was carried out using the programming language Visual Basic version 6.0 for Windows XP, 32 bits. Figure 32.6 shows the screen of main form of the management system program. The system was built addressing different managing areas according to their function; there is a main program and several subroutines, thus not only allowing memory to be saved but also improving the flexibility in the programming task and facilitating further improvements. The management system includes the following basic tools: 1. Database with information about the batches (history and quantity) currently in the chamber: . Batch characterization including a code for every batch, its date of entry, and its curing state on that date. . Residence time (batch history). . Record of environmental conditions. To estimate the curing state of a batch, it is necessary to know the record of environmental conditions to which the batch has been submitted. The control computer periodically

FIGURE 32.6 Main screen.

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2.

3.

4.

5.

stores the information of temperature and relative humidity in the chamber. At the end of every day, the computer control calculates the average value of the above mentioned variables (temperature and relative humidity) and stores the statistically significant information in a daily file. The management system computer can communicate with the control computer, and the managing program updates the field environmental conditions of the database on a daily basis. . Current curing state of every batch (at the last calculated time). Tool for entry and exit batches. When information about entry and exit of pieces into the chamber is communicated to the management tool, the managing program immediately updates the database with the file of stocks. Tool for application of the model to estimate the state of the batches according to their history (past) and the expected actions in the chamber (future). Updating tool that allows the transmission to the managing program of real time information (measurements) on properties of a certain batch in a given instant. From this data, the management program can give a more precise estimation of the current curing state of the batch. Tool for optimization of the curing process. As this tool is one of the main features of the system, it will be examined in detail.

It is necessary to indicate that the managing program has an interactive character, thus the user can inquire about the content of the data files at any moment by selecting the right buttons.

32.3.3 OPTIMIZATION TOOL The tool developed for the optimization of the curing process includes the following procedures: 1. Definition of the objective to be attained. This procedure allows the producer to define the meaning of ‘‘optimal result’’ to the managing program considering four main aspects: . Controllable environmental conditions in the curing chamber (temperature and relative humidity) . Type of products in the chamber and their actual curing state . Elapsed time from the current instant . Preferred curing state for the selected product when the above mentioned lapsed time goes on The scene would be a curing chamber in which it is possible to control the temperature and the relative humidity where there are different batches. In these circumstances, and also assuming there will be a future input in the chamber, the producer decides that, at a future time, he wants to have the maximum quantity of one type (curing state) of Mahón cheese. In other words, the producer fixes a time horizon and a curing degree.

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2. From the objective selected, the program will define the objective function. The objective function mathematically describes what the producer wants to optimize. Two objective functions were defined: . First objective function (Equations 32.4 through 32.8) is an addition with the same number of terms as the number of batches considered (existing batches, NE, and forecast batches, NF). In every addend for the corresponding batch, it is estimated whether the curing state (standard days, dt) belongs to the established range of maturity (dt1, dt2, standard days). The function is weighed by the number of cheeses in the batch if the estimated curing time belongs to the range considered, or by zero if not. OF1 ¼ OF1i þ OF1j NE X

OF1i ¼

(32:4)

OF1I(dt (i))

(32:5)

OF1J(dt ( j))

(32:6)

i¼1

OF1j ¼

NF X j¼1

 OF1i(dt (i)) ¼  OF1j(dt (j)) ¼ .

NL(i, t) $ 2 [dt1 , dt2 ] = [dt1 , dt2 ] 0 $ dt (i) 2

(32:7)

NLP( j, t) $ 2 [dt1 , dt2 ] = [dt1 , dt2 ] 0 $ dt (j) 2

(32:8)

The second objective function (Equation 32.9) is also an addition with the same number of terms as the number of batches considered. In every addend for the corresponding batch, the absolute value of the difference of its curing state (standard days, dt) compared to the average value of the established curing range (dt1, dt2, standard days) weighed for the number of cheeses is estimated.

OF2 ¼

X i

      dt1 þ dt2  X dt1 þ dt2    NL(i, t)dt (i)  NLP(j, t) dt (j)  þ 2  2  j

(32:9)

The first function would be used when the aim was to obtain the maximum number of cheeses inside an interval (to maximize OF1), and the second when the aim was to obtain the maximum number of cheeses whose curing state was as near as possible to the average value of the interval (to minimize OF2). 3. Once the criterion is defined, the optimization program is executed and the optimal values for the decision variables (temperature and relative humidity) are obtained. The management system will ask the user if the optimal

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results obtained for the decision variables are going to be the chamber set points from the current instant onwards. If the user accepts, the managing computer sends the new set points to the control computer. At the end of the process, the producer is provided with information in order to make optimal decisions, according to the definition of the problem.

32.3.4 CONTROL In order to control the temperature of the curing chamber, an on–off control with differential actuation (0.28C) was considered. To apply the developed mathematical model Equation 32.2 that relates the evolution of the environmental conditions (temperature and relative humidity) with the working time of the actuators, it was necessary to estimate the values of the model parameters (a, b, a, b), the dead times, and the sampling interval. By means of the application of steps (on–off) of the actuators in the pilot chamber (Figure 32.4), the corresponding dead times (response times of the controllable variables of the chamber; temperature and relative humidity), the times when the actuators are activated, and the appropriate intervals of sampling, were obtained (Lipták, 2005). From experimental data, the parameters of the control model (a, b, a, b) were identified. For that purpose, the arithmetic mean of the squared differences between the experimental values and the corresponding calculated values was taken as the objective function to minimize. A close fit was observed, the explained variance being above 99%, the parameters of the model corresponding to each of the considered effects (effect of the cooling equipment, the dehumidifier, the humidifier, as well as the evolution of the temperature and humidity of the air in the chamber, without control, in an unsteady state) were fitted. In order to apply the control procedure, a program was set up to control the functioning of a large curing chamber, allowing the control computer to communicate with the management computer through a network. The control environment includes several tools: 1. Optimization of temperature=moisture cycles in order to minimize the extent of the chamber oscillations 2. Management of the records of the environmental conditions, records that will be used by the management program 3. Link between the control computer and the management computer Visual Basic version 6 was used as the programming language, for use in Windows XP, 32 bits. Considering the set point at 128C, it was observed that the system had a dead time of approximately 64 s. The temperature control had a standard deviation of 0.228C, and an average value of 11.98C was obtained. In the microchamber, the relative humidity was also controlled satisfactorily, maintained in an interval of 0.9 (at 80%), 0.5 (85%), and 0.6 (90%).

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32.3.5 COMMUNICATION In order to reach the goal defined by the user, the management program needs information about the batches present in the chamber and about the environmental conditions to which they have been exposed, and therefore, subprograms were developed to allow for communication with the control program which is executed in the control computer. The management and control computer use the DDE (dynamic data exchange) to communicate.

32.4 CONCLUSION From an analysis of the literature, it can be concluded that a global management of the production process in the food processing industry is generally seen to be of great interest. This is especially true when there is a curing stage, such as the curing stage of Mahón cheese. Global management could be adequately developed from two independent but interconnected systems. A drying model was developed as a first approach describing the curing of Mahón cheese, which allowed the curing state to be expressed in standard days. The model accurately reproduces the evolution of the average moisture in cheeses. The management system was found to be quite useful as, on screen, the producer can choose required objectives among different choices. Other objective functions, such as costs, can also be included. The management tool allows the operator to explore the effect of his potential decisions on the production process and to evaluate the effects of the changes of the temperature and the relative humidity of the chamber on the existing batches. In a large chamber (34.8 m3), the application of the models developed for the control reduces the amplitude of the oscillation of the temperature around the set point by around 31%. This leads to a shorter working time of the cooling equipment (a reduction of about 66.9%), which has the disadvantage of, in turn, leading to an increase in the on–off frequency of the cooling equipment (it increases by approximately 66%). The cheese producers who had the system installed considered the global results to be highly satisfactory.

ACKNOWLEDGMENTS This study was carried out in collaboration with the company ‘‘Hort de Sant Patrici,’’ manufacturer of Mahón cheese D.O. and financed by the Conselleria d’Agricultura of Illes Balears.

NOMENCLATURE a b De dt

Parameter of the model for the temperature control Parameter of the model for the temperature control Effective diffusivity Standard days

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m2=s days

dt1

Lower value of the closed interval of standard days Higher value of the closed interval dt2 of standard days k Indicator of the sampling time Relationship between dead time for temperature k1 and sampling interval Relationship between dead time for absolute k2 humidity and sampling interval Half thickness of a Mahón cheese D.O. L1 L2 Half width of a Mahón cheese D.O. NE Existing batches NF Forecast batches NL(i, t) Number of cheeses belonging to batch i at instant t NLP(j, t) Number of cheeses that will be introduced into the chamber belonging to batch j at instant t OF1 First objective function OF2 Second objective function RH Relative humidity t Time T Temperature X Absolute moisture content (dry basis)

days days

m m

% s 8C kg water=kg dry air

GREEK LETTERS Dt c a b t te to

Sampling interval Dimensionless moisture content Parameter of the model for the temperature control Parameter of the model for the humidity control Average moisture content (dry basis) Equilibrium moisture content (dry basis) Initial moisture content (dry basis)

s

kg water=kg dry solid kg water=kg dry solid kg water=kg dry solid

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