Ang a. H-S, Probability Concepts in Engineering Planning and Design, 1984

Probability Concepts in Eng·ineering· Planning· and Desig·n VOLU1l1E II

DECISI01V, RISK, AND RELIABILITY

ALFREDO H-S. ANG Professor of CiL·il Engineering Unh·ersity of Illinois at Urbana-Champaign

WILSON H. TANG Professpr of Civil Engineering University oflllinois at Urbana-Champaign.

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JOHN WILEY & SONS New York

•

Chichester

Brisbane

Toronto

Singapore

:,

Dedicated to Myrtle and Bernadette

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Copyright l!:l 1984 by John Wiley & Sons. Inc. All righis reserved. Published simultaneousi)· in Canada. Reproduction or translation of any pan "of this worl; beyond that permined by Sectic>ns 107 and 108 of the 1976 United States Copyrij!ht Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department. John Wiley & Sons.

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Librtzry of onxrtss

cllttzloxinx in Publictztion DIIUI:

!Revised for volume 2) Ans. Alfredo

Hua-Sin~;.

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Probability concepts in engineering planninl! and design. Includes bibliographical references and in~exes. Contentl: , .. I. Basic principles-v. ::!. Decision. risl.:. and reliability. I. Engineering-Statistical methods. !!. Probabilities. I. Tang. Wilson H. TA340.AS 620 .. 00422"01519::! 75-589.:::

Printed in the Republic of Sinppore

10 9 8 7 6

s4

3 2

preface. _

Like Volume I, this volume emphasizes the applications of probability and statistics in engineering. It is built on the basic principles contained in Volume I but additional more advanced tools are developed. including statistical decision analysis. Markov and queueing models. the statistics of extreme values, Monte Carlo simulation. and system reliability. Again, the necessary concepts are introduced and illustrated within the context of engineering problems. Indeed, the developrr.ent of the logical concepts and the illustration of established principles in engineering applications are · the main objectives of the volume. The tools provided should form the quantitative bases for risk evaluation, control, and management, and intelligent decision making under uncertainty. In some instances. problems in several diverse areas are used to illustrate the same or similar principles. This was purposely done to demonstrate the universal application of the pertinent concepts and also to provide readers with wider choices of illustrative appii~ations. Much of the mathematica( material is, of course, available in the literature and some of it is in text form: for example, Gumbel (1958) on extreme value statistics; Rubinstein ( 1981) on Monte Carlo simulation; Raiffa (1970) and Schlaifer ( 1969) on statistical decision analysis: and Parzen (1962) and Saaty (1961) on Markov and queueing processes. However, the concepts that are especially useful for engineering planning and design are emphasized: moreover, these are presented purposely in the context of engineering significance and in terms that are more easily comprehensible to engineers (supplemented with numerous illustrative examples). Much of the material in Chapters 6 and 7 on basic and systems reliability is. for the most part, only currently available in scientific and technical journals-in this regard. only typical references to the available literature are cited. . as there is no intention to be exhaustive. The material is suitable for two advanced undergraduate or graduate lrvel courses. One would cover engineering decision and risk analysis, based on Chapters 2. 3, and 5 of the present volume supplemented by Chapter 8 of Volume I; the other would focus on system reliability and design and may be developed· using the material contained in Chapters 4, 6. and 7. Both courses would require a prerequisite .background in introductory applied probability and statistics, such as the material in Volume I. During the preparation a:nd development of the material for this volume, the authors are indebted in many ways to colleagues and students-to former students who had· to endure imperfect and incomplete versions of the material; to our colleague Professor Y. K. Wen for numerous discussions and suggestions over the years; and to Professors A. der Kiureghian, M. Shinozuka, E. Vanmarcke, and J. T. P. Yao for constructive suggestions and incisive reviews of the manuscript. A number of former graduate students also contributed to the development ·or some of the material including numerical calculations of the examples; in particular. the contributions and assis.tance of R. M. Bennett, C. T. Chu, H . Pearce, v

vi

PREFACE

J. Pires, I. Sidi, M. Yamamoto, and A. Zerva are greatly appreciated. Finally, our thanks to Claudia Cook for her patience and expert typing of several versions of the manuscript material, and to R. Winburn for his professional artwork of the figures.

A. H-S. Ang AND W. H. Tang

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contents l.

Introduction

1.1 Aims and Scope of Volume II 1.2

Essence and Emphasis 1.2.1 Decision Analysis 1.2.2 Markov, Queueing, and Availability Models 1.2.3 Statistical Theory of Extremes 1.2.4 Monte Carlo Simulation 1.2.5 Reliability and Reliability-Based Design 1.2.6 Systems Reliability

I I 2 3 3 3 4

2.

Decision AJIO:lysis

5

2.1

Introduction 2.1.1 Simple Risk-Decision Problems 2.1.2 Characteristics of a General Decision Problem

5 9

.......

2.2 The Decision Model 2.2.1 Decision Tree 2.2.2 Decision Criteria 2.2.3 Decision Based on Exis~ing Information-Prior Analysis 2.2.4 Decision with Additio'nal Information-Terminal Analysis 2,.2.5 Preposterior Analysis 2~.6

2.2.7 2.2.8 2.3

2.4

Value of Information Sensitivity of Decision to Error in Probability Estimation More Examples

5

10 10 13 16 25 28 30 32 33

2.3.1 2.3.2

Applications in Sampling and Estimation Bayes Point Estimator Optimal Sample Siz~

47 47 50

Elementary Concepts of Utility Theory 2.4.1 Axioms of Utility Theory 2.4.2 Utility Function 2.4.3 Determination of Utility Values 2.4.4 Utility Function of Monetary Value 2.4.5 Utility Function of Other Variables 2.4.6 Maximum Expected Utility 2.4.7 Common Types of Utility Functions 2.4.8 _ ~nsitivity of Expected Utility to Form of Utility Function

56 56 57 60 63 66. 67 70 73

viii 2.5

2.6

CONTENTS

2.5.1 2.5.2 '2.5.3

Assessment of Probability Values Bases of Probability Estimation Subjective Probability ~ubjective Distribution of a Random Variable

74 74 75 76

2.6.1 2.6.2

Decisions with Multiple Objectives Weighted Objective Decision Analysis Multiattribute Utility Approach

77 78 84

Problems for Chapter 2

96

3. . Markov, Queueing, ~nd Availability Models

112

The Markm· Chain 3.1.1 Introduction 3.1.2 The Basic Model 3.1.3 State Probabilities 3.1.4 Steady State Probabilities 3.1.5 First Passage Probabilities 3.1.6 Recurrent, Tran~ient, and Absorbing States 3.1.7 Random Time Between Stages 3.1.8 Continuous Parameter Markov Chain

112 112 113 118 li2 125 136 138

3.2

Queueing Models 3.2.1 Introduction 3.2.2 Poisson Process Arrivals and Depanures 3.2.3 Poisson Arrivals and Arbitrarily Distributed Service Time 3.2.4 Distribution of Waiting Time

140 140 140 149 154

3.3

Availability Problems 3.3.1 Introduction 3.3.2 Continuous Inspection-Repair Process 3.3.3 Inspection and Repairs at Regular Intervals 3.3.4. Deteriorating Systems

156 156 156 166

3.1

\

...

....

112

176

Problems for Chapter 3

. 178

4. Statistics of Extremes

186

4.1

Introduction 4.1.1 Engineering Significance of Extreme Values 4.1.2 Objective and Coverage

186 186 187

4.2

Probability Distribution of Extremes 4.2.1 Exact Distributions

187 187'

COiHENTS

194

Asymptotic Distributions The Symmetry Principle

.:!04

4.3

The Three Asymptotic Forms 4.3.1 Gumbel's Classification 4.3.2 The Type I Asymptotic Form 4.3.3 The Type II Asymptotic Form 4.3.4 The Type III Asymptotic Form 4.3.5 Convergence Criteria

206 206 207 216 222 228

4.4

Extremal Probability Papers 4.4.1 The Gumbel Extremal Probability Paper 4.4.2 The Logarithmic Extremal Probability Paper

235 235 238

4.5

Exceedance Probability 4.5.1 Distribution-Free Probability 4.5.2 Significance of Probability Distribution

246 246 249

4.6

Estimation of Extremal Parameters 4.6.1 Remarks on Extreme Value Estimation .. 4.6.2 The Method of Moments 4.6.3 The Method of Order Statistics 4.6•.£ Parameters of Other Asymptotic Distributions 4.6.5 Graphical Method

255 255 256 256 261 266

Problems for Chapter 4

268

5.

Mont~

274

5.1

Introduction 5.1.1 An Example of Monte Carlo Simulation

5.2

Generation of Random Numbers 5.2.1 Random Numbers with Standard Uniform Distribution · 5.2.2 Continuous Random Variables 5.2.3 Discrete Random Variables 5.2.4 Generation of Jointly Distributed Random Numbers 5.2.5 Error Associated with Sample Size

281 288 290 291

5.3

Variance Reduction Techniques 5.3.1 Antithetic Variates 5.3.2 Correlated Sampling 5.3.3 Control Variates

292 292 296 298

5.4

Application Examples

' 301

Problems for Chapter 5

326

4.2..2 4.2.3

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_;~

Carlo Simulation

274 275 278 279

X

CONTENTS

6. Reliability and Reliability-Based Design

333

6.1 Reliability of Engineered Systems

333

6.2 Analysis and Assessment of Reliability 6.2.1 Basic Problem 6.2.2 Second-Moment Formulation 6.2.3 Linear Perforniance Functions 6.2.4 Nonlinear Performance "Functions

334 334 340

6.3

Modeling and Analysis:ofUncertainty Sources and Types of Uncertainty Quantification ofUncertainty Measures Analysis of Uncertainty · An Alternative Representation of Uncenaimy

6.3.1 6.3.2 6.3.3 6.3.4

6.4 hobability-Based Design· Criteria 6.4.1 On Design and Design Criteria. 6.4.2 Second-Mc;>ment Criteria

347 359 383 384 387 390 406 412 412 414

Prob_lems for Chapter 6 ·< .

434

7.

Systems Reliability

448

7.1

Introduction

448

7.2

Multiple Failure Modes 7.2.1 Probability Bounds

448 450

7.3

Redundant and Nonredundant Systems 7.3.1 Series Systems 7.3.2 Parallel Systems · 7.3.3 Combined Series-Parallel Systems

471 473

7.4 Fault 7.4.1 7.4.2 7.4.3 7.5

Tree, Event Tree Analyses The Fault Tree Diagram Probability Evaluation Event Tree Analysis

474

478 485 486 495 498

Approximate Methods 7.5.1 The PNET Method

504

Problems for Chapter 1

517

505

CONTElVTS

xi"

Appendix A-Tables

531

Table A."I Values of the Gamma Function

532

Table A.2 CDF of the Standard Extremal VariateS

533 '

Table A.3 The Three Types of Asymptotic Extremal Distributions

537

Table A.4. Weights a1 and b1 for Leiblein's Order Statistics Estimator

538

Table A.S Relations Among Parameters of the T)i:,e III Asymptotic Distribution

539

Appendix B-TransforTTUJtion of Nonir.orrnal Variates to Independent Normal Variates

540

The Rosenb_latt Transformation Determination·of Safety Index

544

B.l B.2

540

References

549

Inde;c

559

-·

.

1. Introduction 1.1

AIMS AND SCOPE OF VOLUME II

The principal aims of the present volume parallel those of Volume I; namely, the modeling of engineering problems containing uncertainties and the analysis of their effects on system performance, and the development of bases for design and decision making under conditions of uncertainty. The basic principles and elementary tools introduced in Volume I are supplemented here with additional advanced tools and concepts, including statistical decision theory, Markov and queueing processes, and the statistical theory of extreme values. The discussion of numerical tools for probability calculations would not be complete without mention ofthe Monte Carlo simulation methods; one chapter summarizes the elements of this calculathnal procedure. One of the most important applications ·of probability concepts in engineering is in the evaluation of the safety and reliability of engineering systems, and the formulation of associated design criteria. Recent developments for these purposes are the topics of the final two chapters, with the last chapter devoted specifically to the analysis of the reliability of technological systems. The role of judgment is emphasized when dealing with practical problems; engineering judgments are often necessary irrespective of theoretical sophistication. However, the proper place and role for judgments are delineated within the overall analysis of uncertainty and of its effects on decision, risk, and reliability. Moreover, consistent with a probabilistic approach,judgments have to be expressed in probability or statistical terms; wherever expert judgments are expressed in conventional or det'erministic terms (which is often the case), they have to be translated into appropriate probabilistic terms. Methods for these purposes are suggested. · 1.2 ESSENCE AND EMPHASIS

The main thrust and emphasis in each of the ensuing chapters may be summarized as follows. 1.2.1. Decision Analysis (Chapter 2)

The goal of most engineering analysis is to provide information or the basis for decision making. Engineering decision making could range from simply selecting the size of a column in a structure, to selecting the site for a major dam, to deciding whether nuclear power is a viable energy s~urce. Unfortunately, uncertainties are invariably present in practically all facets of engineering decision making. In this I

2

INTRODUCTION

light, some measure of risk is unavoidable in any decision made (i.e., alternative selected) during the planning and design of an engineering system. A systematic framework for decision analysis under uncertainty, in which the feasible alternatives . are identified and the respective consequences evaluated, could be very useful. The tools for such decision analysis are the subject of Chapter 2 · · The decision tree model is introduced to identify the necessary components of a decision problem, consisting of the feasible alternatives, the possible outcomes associated with each alternative and respective probabilities, and the potential consequences associated with each alternative. In short, the decision tree provides an organized outline of all the information relevant to a systematic decision analysis. .Various engineering examples are used to illustrate the concepts involved in formal decision analysis. The concept of value of informarion is discussed with respect to whether or not additional i~onnation should be gathered before making a final decision. The probiem assOciated with sampling, riamely, determining the parameter estimates ·for design, and the development of optimal sampling plans may also be formulated as a decision problem. , Elementary concepts of utility theory are introduCed a~ a generalizi:d measure of value, on the basis of which the relative significance of various potential consequences of a decision may be evaluated. Specific c:ase studies of complex engineering decision problems are discussed, including those itivolving multiple objectives. l.l.2_i- Markov, Queueing, and Availability Models (Chapter 3)

1be ,performance characteristics of an engineering system may frequently be classified into different discrete states. For example, the potential effects of an eartbquake on a structure may be classified under several states corresponding to di:>tinct damage levels; the financial status (e.g., _in terms of cash flow) of an engineering contractor may represent the various states; including bankruptcy ; . and the number of vehicles waiting (queu,e length) at a toll booth could represent a state of a toll system. The conditions of these systems may change or move from one state to another in accordance with some probability law. The probability that the system will be in a particular state after a given number of moves may then be of interest. The Markov chain is a probability model specifically designed to analyze a system with multiple states. The basic concepts of the discrete parameter homogeneous Markov chain model are first introduced. Queueing models are then shown to be examples of a continuous parameter Markov chain. Steady state queue length probabilities are derived for specific queueing systems with Poisson arrivals. The performance of an engineering system may often be divided into two states; for example, safe and unsafe states, or operating and nonoperating states. The study of the transient behavior of this two-state system is known as the availability problem. Problems of system availability (system in a safe state) ·may include a maintenance program, that is, inspection and repair at regular intervals, in which case the renewal theory is a useful model. Availability problems of interest in engineering would include the availability at a given time for a system with or without maintenance. ·

ESSENCE AND ntPHASIS

3

A number of engineering applications of the Markov process, including the queueing and availability models. are illustrated in Chapter 3. 1.2.3 Statistical Theory of Extremes (Chapter 4)

Extreme values or extremal conditions of physical phenomena are of special interest in many engineering problems. especially in those concerned with the safety and reliability of engineering systems, and/or involving natural hazards. The statistical theory of extreme values. therefore; is of special significance in engineering problems concerned with risk and reliability. Much of the theory of statistical extremes is available in the literature; however, the available material is primarily of a mathematical nature and is hardly accessible to many engineers. One .Jf the purposes of Chapter 4is to summarize and h,ighlight the essential concepts of statistical extremes, and to emphasize the practical significance of these concepts in engineering problems. In particular, the asymptotic property of the statistics of extremes enhances the significance of the extremal theory; understanding this concept is essential for many applications. The asymptotic property and its associated concepts are, therefore, stressed throughout the chapter, and illustrated with numerous examples of engineering problems. 1.2.4 Monte Carlo Simularloo~Cbapter 5) As the complexity of an engineering system increases, the required analytical model may become extremely difficult to formulate mathematically unless gross idealization and simplifications are invoked; moreover, in some cases even if a formulation is possible, the required solution may be analytically intractable. In these'instances, a probabilistic solution may be obtained through Monte Carlo simulations. Monte Carlo simulation is simply a repeated process of generating deterministic solutions to a given problem; each solution corresponds to a set of deterministic values of the underlying random variables. The· main element of a Monte Carlo simulation procedure is the generation .of random numbers from a specified distribution; systematic and efficient methods for generating such 'random numbers from several common probability distributions are summarized and illustrated. · Because a Monte Carlo solution generally requires a large number of repetitions, particularly for problems i.tivolving very rare events, its application to complex · problems could be costly. There is, therefore, good reason to use the Monte Carlo

approach with some. caution;_generally, it should be used only as a last re5ort, that is, when analytical or approximate methods are unavailable or inadequate. Often, Monte Carlo solutions may be_the only means for checking or validating an. approximate method of probability calculations. ·· A number of case study problems are illustrated, to demonstrate the simulation procedure as well the type of information· that may be derived from a Monte Carlo calculation. ,

as

1.2.5 Reliability_aiid Reliability-Based De5ign (Chapter 6)

.

.

The safety and/or performance of an engineering system is invariably the principal technical objective of an engineering design. In order to achieve some desired level of reliability, proper methods fo_r its evaluation are, of course, required. As this

4

INTRODUCTION

must invariably be done in the presence of uncertainty, the proper measure of reliability or safety may only be stated in the context of probability. Indeed, consistent levels of safety and reliability may be achieved only if the criteria for desigri are based on such probabilistic measures of reliability. · Engineering reliability and its .significance in engineering design is a -rapidly growing field; the most recent and practically useful developments are presented in this chapter. Applications in the various fields of engineering are illustrated with emphasis on civil engineering. It goes without saying that problems of safety and reliability arise because of uncertainty in design. The quantification and analysis of uncertainty are, therefore, central .issues in the evaluation of reliability and the development of associated reliability-based design. The statistical bases and methods for these purposes are widely illustrated in this chapter.

1.2.6 Systems Reliability ((:bapter 7) ·..

.

.

Although the reliability of the total system is of principal concern, system reliability is nonetheless a function of.the reliabilities of it!' constituent components ithat are covered in Chapter 6); that is, the determination of system reliability must invariably be based on reliability information for the components. Moreov~. engineering designs are invariably performed at the component level, from which the reliability of the system . is evaluated through analysis; techniques for this purpose are presented in Chapter 7. From a reliability standpoint, a system is characterized by multiple modes of failure, in which each of the potential failure modes may be composed of component failure events that are in series or in·parallel, or combinations thereof(especially for a redilndant system). In the case of a general redundant system, the redundancy may be of;the standby or active type. Depending on whether the redundancies are active • ·. ~.nriby, the reliability of the system as well.~!:~ its analysis will be different. In the case of complex systems, the identification of the potential modes of failure may be quite involved, requiring a systematic procedure for identification, such as the fault tree model. Moreover, the potential consequences of a failure (or initiating event) of a system may vary, depending on the subsequent events that follow the particular initiating event. The event rree model may be used to facilitate the systematic identification of all potential consequencr-~. The applications of the fault tree and event tree models are illustrated with a variety o( engineering examples in this final chapter.

···'

....

2. Decision Analysis· 2.1 INTRODUCTION Making technical decisions is a .necessary part of ~~gineering planning and design : in fact, the primary responsibility of an engineer is to make decisions. Often, such decisions have to be based on predictions and information that invariably contain uncertainty. Under such conditions, risk is virtually unavoidable. Through. probabilistic modeling and analysis, uncertainties may be modeled and assessed properly, and their effects on a given decision accounted for systematically. In this manner, the risk associated with each decision alternative may be delineated and, if desired or necessary, measures taken to control or minimize the corresponding possible consequences. Decision problems in engineering planning and design often also require the consideration of nontechriidi.l:factors, such as social preference or acceptance. environmental impact, and sometimes even political implications. In these latter cases, the selection of the .. best" decision alternative ~annot be governed solely by technical considerations. A systematic framework that will permit the consideration of all facets of a decision problem is the decision 11UJdel. The elements of the decision model and the analyses involved in a decision problem are developed in this chapter. Both single- and multiple-objective engineering decision problems are discussed and illustrated.

2.1.1 Simple Risk-Decision Problems

\

In any decision analysis, the set of decision (or design) variables should first be identified and defined. For example, in the design of storm sewers, the engineer may select a pipe diameter to provide a desired Bow capacity. Because of variability

in the future storm runoff; there is a probability that the ftow capacity of this pipe may not be sufficient. Since this probability is related to the pipe size, the probability level may also be the decision variable instead of the pipe diameter. In order to rank various designs, an objective function is usually defined in terms of the decision variables. This function is frequently-expressed in monetary units representing total benefit or total cost. It is obvious that the optimal design will be determined by the values of the decision _variables that will maximize the benefit or minimize the loss function. In some cases, in which the objective function is a continuous function of the decision variables, the calculus of maximization and minimization provides a convenient tool for optimization. Formally, if X 1, X 2 , ••• , XN denote the set of decision variables, the optimal design will be

5

6

DECISION ANALYSIS

given by values of the decision variables satisfying the following set of equations: oF(Xl, ... ,X.)=O·

ax,

i = 1, 2, ... , n

•

(2.1)

where F(X ~o ••• , X.) is the objective function. The second partial derivatives may be examined to determine whether Eq. 2.1 yields the maximum or minimum objective function. The procedure is illustrated in the following examples.

EXAMPLE 2J

(excerpted from Shuler,1967)

A contractor is preparing a bid for a construction project. Based on experience, it is his judgment that the probability of winning a bid depends on his bid ratio. R, as follows: .

-

.

..

0.6 ~ R ~ ·t.6

p = f.6 - R;

.

.

'

in which R is the ratio of lris !;lid. price to the total estimated cost (Fig. E2.1 ). O_bviously. th~ decision ,·arlable here is the bid price B that the contract~r will submit The corresponding objective function may be the expected profh from the project;

\ X = (B -

C)p

+ 0 · (I

- p)

B-C

=-c-.pC = (R -._l)pC = (-Rz

+ 26R-

1.6)C;

for0.6

~

R S 1.6

'where C is the estimated cost o( construction. The optimal decision requires the maximization :or.x with respect to R; thus, setting · dX dR = C(-2R + 2.6) 0 ~·

..

=

we obtain R = 1.3. Since dl X fdR 2 < 0 at R = i.3,' ihe bid 'f'atio of 1.3 will maximize X. Therefore, the optimal bid for the contractor should be 1.3 times the cstiinated cost. The above solution assumes that there is no monetary gain or loss if the contractor does not win the job. Suppose that the construction crew would be idle and the contractor will be operating at a loss L if he fails to win ~is job. ln such a Case, the objective function should include an expected loss; suppose L = O.lC. Then, the expected overall gain would be X = (R - l)pC- (l - p)L

= [(R - 1)(1.6- R)- (R - 0.6) x O.l)C = [-R' + 2SR- 1.54)C p

1.0

p

= l6- R

0.2 07-~~~--~~--~----~

0.6

Figur~ £2.1

QB

• lO

1.2

1.6

,:

R, • .

Probability of winning versus bid 'ratio.

INTRODUCTION

- .7

Then,

dX

-

dR

= -2R + 2.5

=0

yields R = 1.25. Hence, the optimal bid becomes 1.25 times the estimated cost. A lower bid is required here so that the contractor will increase his chance of winning the bid, thus reducing the chance of idling his construction crew.

EXAMPLE 2.2 The construction of a bridge pier requires the installation of a cofferdam in a ·river. Suppose the occurrence of floods follows a Poisson distribution with a mean occurrence rate of 1.5 times per year, and the elevation of each flood is exponentially distributed with a mean of 5 feet above normal water leveL Each time the cofferdam is ovcrtoppcd,"the expected loss resulting from a possible delay in construction and pumping cost is estimated to be $25,000. Since the flood elevation can be predicted sufficiently ahead of time to· effect the evacuation of personnel working inside the cofferdam, the possibility of loss due !O workmen being trapPed may be neglected. Suppose that the construction cost of the cofferdam is given by

c. = c. + 3000h where c. includes the cost of a colfcrdam foundation and ncccssary construction to dam the river to nonnal water level, arid "h is the height of the cofferdam (in feet) above normal water level. Determine the optimal height of the cofferdam if it is expected to be used over a two-year period. A convenient choice of the objective function here is the expected monetary loss Cr. which would consist of the construction cost and the expected total Joss from the flooding of the cofferdam during the two years of service. The expected loss in each flood is C = E(loss Iovertopping)P(overtopping)

Hence, the expected total loss from all floods during the two-year period {the time value of money neglected) is

..

C1 = l:E(lossli floods)P(i floods in 2years) 1•0

"' 31e- 3 = L:i·(25,000e- 1115) - 1•0 I! ., Jle-J = i - .,-

zs;oroe-111' 1•0 L:

I.

= 75,000e- 1115 The total expected monetary loss becomes

Cr= C.+ C1 = c.+ 3000h + 75,000e~lll' Clearly, his the only deCiSion variable; whereas c. is a constant independent of h. Differentiating

Cr with respect to h, we obtain the optimal cofferdam heigh~ as foUows:-

d~r =

3000

+ 75,000e- 1115( -!)

=0

8

DECISION ANALYSIS

-

Construct ion Cost, Cc

E

Pumpino + Oetoy Cost, Ct 0~~~~--~~~--~~~--~~--~ 0 2 3 'I 5 6 7 B 9 10 II

h (in feet) Figur~

£2.1 Costs as functions of cofferdam elevation above normal water level.

or Thus,

h.,.. = 5 In (5) = 8.05 ft Therefore. the optimal height of the cofferdam should be approximately 8 feet above the normal wat~ level of. the river. Figure E22 shows the various cost components as functions of the decision variable h. It may be observed that as h increases, the construction cost increases; whctcas the expected flood Joss decreases. The total expected loss takes on a minimum when the slopes of these two functions arc equal but of opposite signs. At this point, the marginal increase in Cc is balanced by the marginal decrease inC1 ,)be value of c., which was assumed to be S40,000 in Fig. E2.2; simply adds a constant value to the ordinate at each value of h, and does not affect the determination of the optimal value of h. The probability that a cofferdam with h = 8.05 It will be overtopped in a flood is given by . .. 1 . . P(flood level > 8.05') = Se-;tJ' dx = 0.2

J.

1 .0$

Alternatively, the objective function may be formulated using the probability of overtopping, p, as the decision variable. Since the construction cost is a function of h, we need to express h in terms of the overtopping probability p. Based on the exponential distnbution assumed for the: flood level it can be shown that 1

p=

.. J.• 5

-e-~"

dx

or

· h ·= -5 Inp Hence, the total expected cost becomes

CT =

c. + 3000(- 5 In p) + 3(25.000)p

=c.- 15,000 In p +·15,000p Differentiating CT with respect to p, we obtain the optimal probability as follows :

.

dC 1 . -d T = -15,000 '- + 75,000 = 0 I p

p

.

I .

INTRODC/CT/Ol'i

9

from which

p.;_. ·= 0.2 which is the same as obtained earlier. Based on this value of p... . the corresponding optimal elevation of the cofferdam (above normal water level) is determined as

h.,.= -5 In p.,.. = - 5 In (0.2) , · = 8.05 ft The first approach described above directly gives the optimal value of the design variable, whereas the second formulation directly yields the optimal probability level In cases involving several cofferdams, each subject to a different distribution of floods, expressing the . overall objective function iD. terms of a common rislt level could simplify the optimization process. In any case, the two methods should lead to identical optimal designs, provided there is a correspondence between the design variables and the probability of overtopping.

2.1.2

Characteristics of a General Decision Problem

The applicability of the optimization procedure presented above, based on calculus, is limited; first' of all; the p~jective function must be expressed as a continuous function of the decision variables. Unfortunately, this may not be the case for many engineering decision problems. Consider a dam that has been proposed as a: possible solution for flood control in a certain drainage basin. Its elevation may be a decision variable; the darn site may also need to be determined. Moreover, other forms of flood control strategies such as diversion channels, levees, multiple reservoirs may be possible alternatives to be considered in the decision analysis. Thus, it will be difficult, if not impossible, to obtain a continuous objective function · in terms of all the decision variables. Sometimes, a decision may not be based solely on the available information. If time permits, additional information may be collected prior to the final selection among the feasible design alternatives. In engineering problems, additional data could take the form oflaboratory or field tests, or further research. Since there could be a variety of such schemes for collecting additional information, this would further expand the spectrum of alternatives. In short, in more general decision problems, a framework for systematic analysis

is required. Specifically, the decision analysis should at least include the-following components: (1) A list of all feasible alternatives, including the acquisition of additional information, if appropriate. (2) A list of all possible outcomes associated with each alternative.

(3) An estimation of the probability associated with each possible outcome. (4) An evaluation of the .consequences associated with each combination of alternative and outcome. (5) The criterion for decision. (6) A systematic evaluation of all alternatives. A decision model that considers all these basic components is presented in Section42

10

DECISION ANAL YS/S

2.2 THE DECISION MODEL l.l.l Decision Tree· The various components of a decision problem may be integrated into a formal Ia yout in the form of a decision tree, consisting of the sequence of decisions-namely, a list of feasible alternatives ;the possible outcomes associated with each alternative; the corresponding probability assignments; monetary consequences and utility evaluations (see Section 2.4).1n other words, the decision tree integrates the relevant systematic manner suitable for an components of the decision analysis in analytical evaluation of the optimal alternative. Probability models of engineering analysis and design may be used to estimate the relative likelihoods of the possible outcomes, and appropriate value or utility models evaluate the relative desirability of each consequence. . . Figure 2.1 shows a generic example of a decision tree with three ahernatives, in which the third alternative involves performing an experiment to gather additional information prior to any final decision. The word "experiment" should be" interpreted in a broad sense; covering any method of gathering additional data. The . following notations are used in Fig. 21 :.

a

a1 = Alternative i.

. ··

Outcomej. · """' · e1 = Experiment k designed to gather additional information. z, = Experimental outcome l. u(-ri'1, 8) = Utility value corresponding tci alternative a1 and outcome 81; if the ·:: utility depends pn experiment e1 and the corresponding experimental ~outcome z, it will be denoted as u(e1 , z" a;. 81). · (Ji =

_,;..:..:...;...:=..::.:..!..::.:..!~:....

u(e 1 ,11,o2

'-"-- R)

= «>[100 -

w)J

(100 + j1fi:X:J + 1600

The decision tree for the test alternative is shown in Fi~ E219c. The distribution of W at the chance node A is N(50, 30). The decision analysis starts with node B, where the expected monetary values of the two alternatives are · EMV(no

~~:ti~n) = -s~(~

and EMV(cxcavation) = -0.8

40

DECISION ANALYSIS

No Excavation

0

Test A

:

Excovaticn

..:: .

:-:-.:.• ' ·•·•

No Liquefaction

-0.8

-- -.· . -. · •. (1.0)

Figuu £2.19~ Decision tree for test alternative.

·Therefore, whether or not to excavate depends "on the ·value of,... obtained from the test. The critical value of w occurs at

(~J - S41\.to,jl--5J = -0.8 . or ... =

-40j2~-·e~8)

\

"'.56.29 .

Ifw is less than .56.29, EMV(excavation) > EMV(no excavation), and the optimal alternative is to excavate, with a corresponding expected monetary value of -0.8. On the other hand, if w exceeds .56.29, EMV(excavation) < EMV(no excavation), the preferred alternative is not to excavate, with a corresponding expected monetary value of- 5C!l( -w;40j2). At node A, the expected monetary value of the tes_t alternative is thus

J'

6 19

EMV(test) =

-•

-0.8 f,.{w) dv.·

+

f... -Sill(40.,.;'21 "~\

fw{w) dw

U.l9

N(.50, 30). After simplification, .56.29- 50) - 5 f"" ~ ('--~:-)··-· 1 ---0.8~ -- ~ exp[ (

where f,.{w) is the PDF of EMV(test) =

30

=

sb.lt

4!:J.../2 .J1n 3(1

.50)1] dw

_1 ;("'-2 30

-0.47 - 0.16

= -0.63

where !he integral has been evaluated numerically. When we compare this with the expected monetary value of the optimal alternative without the test. from part (b). the value of the test in monetary terms is

VI

= -0.63 -

( -0.8)

= $0.17 million EXAMPLE 2.·10 (excerpted from Chamberlain, 1970) The water quality in the Great Lakes is gradually being degraded because of industrial pollution. Three alternatives have been proposed by the Great Lakes Management to improve the situation; namely: a1 : a1:

Build a new efficient treatment plant for SJO million. Conduct research to develop a sccndary pr~cc;ssor: tht;