Operations Research Project.docx

A Report on OPERATIONS RESEARCH TERM PROJECT

Prepared for: Dr. AKM Kais Bin Zaman Assistant Professor Deptartment of Industrial and Production Engineering

Prepared by: Md. Golam Kibria, Student No.: 0908013 Md. Rassel Sarker, Student No.: 0908014 Level: 03, Term: 01 Dept. of Industrial and Production Engineering Submitted on: 20. 06. 2013

Bangladesh University of Engineering and Technology, Dhaka

ACKNOWLEDGEMENT

Our project report would be incomplete without thanking some people. We would like to acknowledge the people who helped and supported us throughout the project work. First and foremost we would like to convey sincere gratitude to our course teacher, Dr. AKM Kais Bin Zaman, Assistant Professor, Department of IPE, BUET. He has helped us in developing and clarifying our concepts on the course and this project by providing us his valued teaching. While working for this project, his teaching became an integrated part of our thoughts and ideas. We would also like to express our gratitude and sincere thanks to Mr. Ramiz Uddin, owner of the retail shop under study. He provided us most of the required data needed for the project work. Thanks also go to the employees of the shop who helped us by all means while collecting data for the study. Finally, we would like to thank our classmates for their generous support and encouragement.

Contents

List of Illustrations

ABSTRACT Operations Research (OR) is a quantitative analysis of complex management problems. From this analysis, management can make an objective decision.

The main objective of this project is to apply different class concepts to a real life problem. For completing the project work, we have to handle various phases of an OR study. This will help us in defining and solving more complex practical optimization problems in the future.

This project study is irrevocably important. Engineering is not all about reading books and learning techniques. It is valueless if we do not implement it in our day-to-day life. This project report is an outcome of practical implementation of Operations Research that we have learned in the classes during the course. The method we utilized for the purpose of the project study is called „Integer Linear Programming‟.

INTRODUCTION

Operations Research (OR) emerges During World War II. It began as an interdisciplinary activity in the military to solve complex problems. It has grown in the last 50 years to a fully developed engineering discipline. OR is considered as a platform of established mathematical models. OR has achieved characteristics from mathematics, engineering, business, computer science, economics, and statistics. Applications of OR is seen in business, industry, government, and military. The optimization problem we have solved is faced by the owner of „Boimela and Photocopy‟, a retail shop of books, photocopy and other study materials. Mr. Ramiz Uddin is the owner of the shop. It is situated near the Mirpur – 1 bus stand in Dhaka. “Boimela and Photocopy” is a renowned shop established in 1995 as a small retail shop of books and other educational accessories. It has been fulfilling a variety of demand of the general people. The shop owner first installed a photocopier in 2003. The shop was then only

150 square feet. At present, the size of the shop is 500 square feet and there are 4 copiers (two Toshiba 2860 and two Toshiba 3560). There are six workers that include four copiers‟ operators. The shop opens at 9 am and closes at 8 pm daily.

PROBLEM DESCRIPTION

Mr. Ramiz Uddin, the owner of “Boimela & Photocopy”, thinks that he is not currently utilizing all his resources and hence his current yearly profit is not maximized. He has a lot of empty space in his shop. Since the demand for photocopy is very high at present, he thinks he should install more copiers in the shop to maximize his profit. He wants to buy some reconditioned copiers. There are six types of copiers he is considering to buy. Each of the type has its own privilege and expense. It is not clear to him how many of each type of copiers would be most profitable. We begin by having discussions with Mr. Ramiz Uddin to identify the objective, variables, constraints and parameters of the problem. These discussions led to developing the following definition of the problem: Determine how many of each type of copiers should be bought in order to maximize Mr. Ramiz Uddin‟s total yearly profit, subject to the restrictions imposed by the limited space for new machines, owner‟s allocated budget for buying new copiers, current demand, and photocopy quality that the owner intends to serve the customers to make them pleased as well as to keep the fame of his shop.

Methodology Used for Solving the Problem

Since the number of copiers cannot be fractions, the methodology to solve the problem is Integer Programming. To solve the problem we have used Microsoft Word‟s Excel Solver.

Integer Programming (IP):

An integer programming (IP) problem is a mathematical programming problem in which some or all of the variables are restricted to assume only integer values. If all variables take integer values, then the problem is called a pure IP. On the other hand, if both integer and continuous variables coexist, the problem is called a mixed integer program (MIP). Those variables whose values can only be either 0 or 1 are called binary variables. Consequently, IP problems that contain only binary variables sometimes are called binary integer programming (BIP) problems. An IP problem can be either linear or non-linear.

Our project is to solve a pure Integer Linear Programming. With the use of the matrix notation, an IP problem can be represented as below.

minimize Z = CX subject to, AX ≥ b X ≥ 0 and integer

Here,  X = the vector of integer variables,  C = the coefficient vector of the objective function,  A = the coefficient matrix of the constraints,  b = right-hand-side vector of the constraints, All elements of d, B, and b are known constants. Some constraints may be equations and of the type “≤.”.

Numerical Calculation Data Collection:

Table 2: Paper Cost Type of Papers

Price (Tk / 500 pages)

A4 Offset

420

A4 Normal

210

Legal Normal

250

Table 3: Photocopy Price Type of Papers

Single Side (Tk per piece)

Both Sides (Tk per piece)

A4 Offset

2

2.5

A4 Normal

1.5

2.0

Legal Normal

1.5

2.0

Table 4: Rating for image quality compare to original copy (on a scale of 100) Excellent

90 – 100

Very good

80 – 89

Good

75 – 79

Moderate

70 – 74

Bad

60 – 69

Very bad

0 – 59

Table 5: Other Expenses Electricity (Tk per copier in a month)

Rent of Shop (Tk per month)

1000

25000

Table 6: Image quality compare to original copy

Name of copiers

Image quality (on a scale of 100)

Toshiba 2860

80*

Toshiba 3560

85*

Toshiba 2060

75*

Toshiba 4560

85*

Canon 1215

70*

Toshiba eStudio 456

90*

Table 7: No. of Different Type of Copies Copier Name Toshiba 2860 Toshiba 3560 Toshiba 2060 Toshiba 4560 Canon 1215 Toshiba eStudio 456

A4 size Offset

A4 size Normal

Legal Size Normal

Both Sides Single Side Both Sides Single Side Both Sides Single Side 100

100

300

200

175

50

120

120

360

240

210

60

80*

80*

240*

160*

140*

40*

170*

160*

500*

335*

290*

85*

50*

50*

160*

100*

100*

30*

190*

190*

570*

370*

310*

100*

(* Represents assumed data)

Other Information: 1. Available space for copiers: 200 square feet. 2. Available money for buying new copiers: 1000000 Tk. 3. Current demand: 20000 copies per day. 4. Amount of money that the owner is willing to spend for maintenance of the machines: 10000 Tk per month. 5. Maximum expected paper waste or copy waste: A4 size offset: 80 copies per day. A4 size Normal: 150 copies per day. Legal size Normal: 150 copies per day. 6. Image quality the owner wishes to provide: above 80 on a scale of 100.

Calculation of Profit Per Year from Various Copy Machines Copier’s Model – Toshiba 2860: Revenue from Toshiba 2860 Revenue from number of copies: Type of paper

No. of copies per Price per copy,

Total income per day,

day, n

p (Tk.)

n×p (Tk.)

A4 size Offset (Both Sides)

100

2.5

250

A4 size Offset (Single Sides)

100

2

200

A4 size Normal (Both Sides)

300

2

600

A4 size Normal (Single Sides)

200

1.5

300

Legal size Normal (Both Sides)

175

2

350

Legal size Normal (Single Sides)

50

1.5

75

Total

1775

Total income per day = 1775 Tk So, Total income per year = 1775 × 30 × 12 Tk = 639000 Tk Revenue from Salvage Value: Time value of money should be considered here. Buying Cost: 90,000 TK Life: 6 yrs Salvage: 10,000 TK Interest Rate: 5% (Assumed)

Revenue per year from salvage value = 10000 (A|F, 5%, 6) Tk = 10000 × 0.147 Tk = 1470 Tk

Total Revenue from Toshiba 2860 per year = (639000 + 1470) Tk = 640470 Tk Expenses incurred from Toshiba 2860 Machine Cost = 90000 Tk Life: 6 years Machine cost per year = 90000 (A|P, 5%, 6) Tk = 90000 × 0.197 Tk = 17730 Tk Maintenance cost per month = 1000 Tk Maintenance cost per year = 1000 × 12 Tk = 12000 Tk Operator‟s salary per month = 10000 Tk Operator‟s salary per year = 10000 × 12 Tk = 120000 Tk

Ink price per packet (190 gm) = 225 Tk No. of copies can be done using one packet = 6000 Capacities (Copies per day) = 1500 Ink cost per year =

225  1500  30  12 Tk 6000

= 20250 Tk Paper Cost Type of Paper

Cost per 500 Cost per unit

No. of paper

Total cost per

Total cost per

pages (Tk)

page (Tk)

used per day

day (Tk)

year (Tk)

A4 size offset

420

0.84

200

168

60480

A4 size normal

210

0.42

500

210

75600

A4 size legal

250

0.5

225

112.5

40500

Total

176580

Paper wastage

Type of Paper

Cost per 500 Cost per unit

No. of paper

Total

Total wastage

pages (Tk)

wasted per day

wastage

cost per year

page (Tk)

cost per day

(Tk)

(Tk) A4 size offset

420

0.84

8

6.72

2419

A4 size normal

210

0.42

15

6.3

2268

A4 size legal

250

0.5

15

7.5

2700

Total

7387

Electricity Cost: Electricity cost per month = 1000 Tk Electricity cost per year = 12000 Tk

Total Rent for 500 square feet area per month = 25000 Tk Space occupied by Toshiba 2860 = 25 square feet Rent expense for Toshiba 2860 per month =

25000  25 Tk 500

= 1250 Tk Rent expense for Toshiba 2860 per year = 1250 × 12 Tk = 15000 Tk

Total Expense for Toshiba 2860 Type of expense

Expense (Tk)

Machine cost

17730

Maintenance cost

12000

Operator expense

120000

Ink cost

20250

Paper cost

176580

Paper wastage expense

7387

Electricity cost

12000

Rent expense

15000

Total expense

380947

Profit from Toshiba 2860 per year = Total revenue - Total expense = (640470 - 380947) Tk = 259523 Tk

Like Toshiba 2860 we have evaluated total revenue, total expense and profit per year for other copiers‟ model. They are given in the following sections.

Copier’s Model - Toshiba 3560:

Total Revenue from Toshiba 3560 = 769005 Tk Total Expense for Toshiba 3560: Type of expense

Expense (Tk)

Machine cost

21670

Maintenance cost

14400

Operator expense

144000

Ink cost

24300

Paper cost

211896

Paper wastage expense

8143

Electricity cost

12000

Rent expense

15000

Total expense

451409

Copier’s Model - Toshiba 2060: Total Revenue for Toshiba 2060 = 512376 Tk Total Expense for Toshiba 2060 Type of expense

Expense (Tk)

Machine cost

12805

Maintenance cost

12000

Operator expense

120000

Ink cost

19440

Paper cost

141264

Paper wastage expense

6480

Electricity cost

12000

Rent expense

15000

Total expense

338989

Copier’s Model - Toshiba 4560: Total Revenue from Toshiba 4560 = 1066005 Tk Total Expense for Toshiba 4560

Type of expense

Expense (Tk)

Machine cost

23640

Maintenance cost

14400

Operator expense

144000

Ink cost

27000

Paper cost

293544

Paper wastage expense

33869

Electricity cost

12000

Rent expense

15000

Total expense

563453

Copier’s Model – Canon 1215: Total Revenue from Canon 1215 = 339135 Tk Total Expense for Canon 1215 Type of expense

Expense (Tk)

Machine cost

6895

Maintenance cost

13200

Operator expense

96000

Ink cost

14400

Paper cost

92952

Paper wastage expense

4162

Electricity cost

12000

Rent expense

15000

Total expense

254609

Copier’s Model – eStudio 4560: Total Expense for Toshiba eStudio 4560 = 1198140 Tk Total Expense for Toshiba eStudio 4560 Type of expense Machine cost

Expense (Tk) 25610

Maintenance cost

120000

Operator expense

180000

Ink cost

39200

Paper cost

330840

Paper wastage expense

6091

Electricity cost

12000

Rent expense

15000

Total expense

728741

Summary of per year revenue, expense and profit from different copiers calculated using collected data: Type of copiers

Revenue per year

Expense per year

Profit per year (Tk)

(Tk)

(Tk)

Toshiba 2860

640470

380947

259523

Toshiba 3560

769005

451409

317596

Toshiba 2060

512376

338989

173387

Toshiba 4560

1066005

563453

502552

Canon 1215

339135

254609

84526

Toshiba eStudio 456

1198140

728741

469399

Formulation as a Linear Integer Programming Problem:

To formulate the mathematical (linear programming) model for this problem, let X1 = number of copiers of the type Toshiba 2860. X2 = number of copiers of the type Toshiba 3560. X3 = number of copiers of the type Toshiba 2060. X4 = number of copiers of the type Toshiba 4560. X5 = number of copiers of the type Canon 1215. X6 = number of copiers of the type Toshiba eStudio 456. Z = Total profit per year. Thus, X1, X2, X3, X4, X5 and X6 are the decision variable for the model.

Table 5: Calculated Profit for different type of copier Name of Machines

Profit (Tk per year)

Toshiba 2860

259523

Toshiba 3560

317596

Toshiba 2060

173387

Toshiba 4560

502552

Canon 1215

84526

Toshiba eStudio 456

469399

The objective is to choose the values of X1, X2, X3, X4, X5 and X6 so as to maximize Z = 259523 X1 + 317596 X2 + 173387 X3 + 502552 X4 + 84526 X5 + 469399 X6, subject to the restrictions imposed on their values by the limited space in the shop.

(1) Each of the copiers occupies 25 square feet and available space for new copiers is 200 square feet. This restriction is expressed mathematically by the inequality 25 X1 + 25 X2 + 25 X3 + 25 X4 +25 X5 + 25 X6 ≤ 200. (2) From Table 1 we find the market price of different reconditioned copiers. Since the owner allocated 1000000 Tk. for new copiers, another restriction can be mathematically expressed by the inequality 90000 X1 + 110000 X2 + 65000 X3 + 120000 X4 + 35000 X5 + 130000 X6 ≤ 1000000. (3) The capacity of each copier is tabulated in table 1. Since the owner thinks the current demand is 20000 copies per day, this restriction can be mathematically expressed by another inequality 1500 X1 + 1800 X2 + 1200 X3 + 2500 X4 + 800 X5 + 2800 X6 ≤ 20000.

(4) The maintenance cost for each of the copiers is given in table 1. The owner does not wish to spend more than 10000 Tk. per month for the maintenance. This restriction is expressed mathematically by the inequality 1000 X1 + 1200 X2 + 1000 X3 + 1200 X4 +1100 X5 + 1000 X6 ≤ 10000. (5) Paper wastage for each type of paper in each machine is given in table 7. The owner is not willing to waste more than 80 A4 size offset paper, 150 A4 size normal paper and 150 legal size normal paper per day. These restrictions can be mathematically expressed by the inequalities: (i) 8 X1 + 8 X2 + 5 X3 + 12 X4 + 5 X5 + 8 X6 ≤ 80, (ii) 15 X1 + 20 X2 + 15 X3 + 20 X4 + 8 X5 + 10 X6 ≤ 150 (iii) 15 X1 + 15 X2 + 15 X3 + 20 X4 + 8 X5 + 12 X6 ≤ 150 (6) Rating for image quality compare to original copy is given in table 5. The owner wishes to provide a minimum image quality of 80 on a scale of 100. This image quality restriction can be expressed mathematically by the following inequality: 80 X 1  85 X 2  75 X 3  85 X 4  70 X 5  90 X 6  80 X1  X 2  X 3  X 4  X 5  X 6

The final form of this equation yields: 0 X1 + 5 X2 - 5 X3 + 5 X4 - 10 X5 + 10 X6 ≥ 0

To summarize, in a mathematical language of linear integer programming, the problem is to choose values of X1, X2, X3, X4, X5 and X6 so as to Maximize Z = 259523 X1 + 317596 X2 + 173387 X3 + 502552 X4 + 84526 X5 + 469399 X6, subject to the constraints 25 X1 + 25 X2 + 25 X3 + 25 X4 +25 X5 + 25 X6 ≤ 200 90000 X1 + 110000 X2 + 65000 X3 + 120000 X4 + 35000 X5 + 130000 X6 ≤ 1000000 1500 X1 + 1800 X2 + 1200 X3 + 2500 X4 + 800 X5 + 2800 X6 ≤ 20000 1000 X1 + 1200 X2 + 1000 X3 + 1200 X4 +1100 X5 + 1000 X6 ≤ 10000 8 X1 + 8 X2 + 5 X3 + 12 X4 + 5 X5 + 8 X6 ≤ 80 15 X1 + 20 X2 + 15 X3 + 20 X4 + 8 X5 + 10 X6 ≤ 150 15 X1 + 15 X2 + 15 X3 + 20 X4 + 8 X5 + 12 X6 ≤ 150 0 X1 + 5 X2 - 5 X3 + 5 X4 - 10 X5 + 10 X6 ≥ 0 Xj ≥ 0, for j = 1, 2. . . 6. And Xj is integer, for j = 1, 2. . . 6.

Solution: The mathematical model formulated in the previous section has been solved using the Excel Solver. The optimal solution which maximizes the objective function is (X1, X2, X3, X4, X5, X6) = (0, 2, 0, 4, 0, 2) The corresponding value of Z = 3584198 Tk The screenshot of the Excel Solver is attached here.

Figure 1: Solution and optimized objective function value obtained using the Excel Solver

Conclusion During our project work we first defined the problem, and then collected data. To represent the problem, we formulated a mathematical model. Finally, we developed a computer-based procedure, i.e. the Excel Solver, to solve the problem from the developed model. The most challenging and yet most interesting phase of this OR study was the mathematical formulation of the real-life system. After identifying the problem we detected the parameters and the variables which are involved in this problem. To keep the model as simple as possible we selected those variables which seemed most influential. Then we stated verbal relationship among these variables based on collected data. After completing this project work, we have gathered a great deal of knowledge which, we believe, we will be able to implement very efficiently in the future. Since OR has its applications in defense, industry and in all public system, the importance of having a clear knowledge about Operations Research is beyond description.

References