Multiobjective Optimization in Water Resources Systems Surrogate Worth Trade-off Method

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MULTIOBJECTIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS

The Surrogate Worth Trade-off Method

DEVELOPMENTS I N WATER SCIENCE. 3 advisory editor VEN TE CHOW Professor o f Hydraulic Engineering Hydrosystems Laboratory Civil Engineering Building University of Illinois Urbana, Ill., U.S.A.

MULTI0BJECTIVE 0PTIMIZATI0N IN

WATER RESOURCES SYSTEMS The Surrogate Worth Trade-off Method YACOV Y. HAIMES Systems Engineering Department Case Institute of Technology Case Western Reserve University Cleveland, O h i o

WARREN A. HALL Civil Engineering Department Colorado State University F o r t Collins, Colorado

HERBERT T. FREEDMAN Systems Engineering Department University of Pennsylvania Philadelphia, Pennsylvania

ELSEVIER SCIENTIFIC PUBLISHING COMPANY AMSTERDAM

- O X E O R D - NEW YORK

1975

ELSEVIER SCIENTIFIC PUBLISHING COMPANY 335 Jan van Galenstraat P.O. Box 21 1, Amsterdam, The Netherlands

AMERICAN ELSEVIER PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue New York, New York 10017

ISBN: 0-444-4131 3-8 Copyright 0 1975 b y Elsevier Scientific Publishing Company, Amsterdam All rights reserved. No part of this publication may b e reproduced, stored i n a retrieval system, or transmitted in any f o r m or b y any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Scientific Publishing Company, Jan van Galenstraat 335, Amsterdam Printed in Thp Netherlands

Pheljuce 1.arge s c a l e systems, and i n p a r t i c u l a r w a t e r a t e s p e c i a l problems w h i c h make g i e s qu'ite d i f f i c u l t and meaningless i f n o t

each o f

treated with

a c t u a l l y misleading.

from three important l a r g e number

the application o f

unless

w h i c h may make

First,

decision-makers and/or

decisions o r influence

t h e r e i s a l a r g e number o f

stem

there i s a

,

constituencies

decisions according

to his

Second, even f o r any one

non-commensurable o b j e c t i v e s t o

T h i r d , t h e r e i s a v e r y l a r g e elemen-t o f u n c e r t a i n t v T h i s element i s due t o tiis: i i i o k

i n v i r t u a l l y a l l water resources decisions. deoree o f

insight quite

H o s t o f -these d i f f i c u l t i e s

o m , but d i f f e r e n t , versions o f the desired goals. be o p t i r i i z e d .

o p t i m i z a t i o n methodolo-

considerable

c h a r a c t e r i s t i c s o f t h e s e systems.

o f quasi-independent

d e c i s i o n maker,

r e s o u r c e s systems, c r e -

irreversibility of

these decisions coupled

w i t h both hydrologic

u n c e r t a i n t y and i n a b i l i t y t o p r e d i c t t h e f u t u r e w i t h r e a s o n a b l e a c c u r a c y . The r e c e n t t r e n d

i n water resources

however, has been t o e l e v a t e

many o f t h e

and

other

real

non-commensurable

p a r i t y w i t h economic e f f i c i e n c y as c r i t e r i a f o r e x c e l l e n c e . o f a vector

world

problms,

objectives

to

This inclusion

o f o b j e c t i v e f u n c t i o n s i n t r o d u c e s a new dimension i n t h e f i e l d s

o f m o d e l i n g , m a t h e m a t i c a l programming and o p t i m a l c o n t r o l . sents a comprehensive s u r v e y o f t h e r n u l t i p l e o b j e c t i v e problems,

T h i s book r e p r e -

methods p r e s e n t l y a v a i l a b l e f o r s o l v i n g

and c o n c e n t r a t e s on a

new p o w e r f u l and o p e r a -

t i o n a l method, namely t h e S u r r o g a t e Worth T r a d e - o f f (SWT) Method.

T h i s met-

hod s i m p l i f i e s t h e i n t e r a c t i o n between d e c i s i o n - m a k e r s and systems a n a l y s t s , and enables them t o d e t e r m i n e a b e s t p o l i c y v i a a v e r y moderate i n t e r a c t i o n . The method

i s applicable t o s t a t i c

( o p t i m a l c o n t r o l ) systems. tional algorithms are poration o f

( m a t h e m a t i c a l programming)

and dynamic

T h e o r e t i c a l bases as w e l l as d e t a i l e d

d i s c u s s e d w i t h s e v e r a l example problems.

special properties o f

w a t e r r e s o u r c e s systems

computaThe i n c o r -

(e.g. r i s k

and

u n c e r t a i n t y ) i n t o a m u l t i o b j e c t i v e framework i s a n a l y z e d and a p p l i c a t i o n s o f t h e SWT

method a r e p r e s e n t e d .

I n particular, sensitivity, irreversibility

and o p t i m a l i t y a r e s t u d i e d i n d e t a i l

as m u l t i p l e o b j e c t i v e f u n c t i o n s .

The

a v a i l a b i l i t y of aqes a n d

o p er at i o n al methodologies such a s

enhances t h e systems modeling and

the SLIT method,

encour-

p a t t e r n of thinking in multiob-

j e c t i v e f u n c t io n terms.

Thus, more r e a l i s t i c analyses may r e s u l t by e lim inating t h e needs f o r a s i n g l e o b j e c t i v e formulation.

This book should s er v e c o l l e g e s t u de nts and profe ssors , p r a c t i c a l engineers, and managers involved i n t h e decision-making process of re a l world problems whether i n

water resources systems or o t h e r l a r g e s c a l e sys-

tems.

a p p l i c a t i o n include

P o s s i bl e a r e a s of

delivery,

conirnunication, urban and

transportation,

health c a r e

housing development, environmental

and

energy problems and mar,y o t h e r a r e a s , where noncommensurable o b j e c t i v e funct i o n s dominate t h e s e oroblems.

Yacov Y . Hairnes

lloveniber 1974

Warren A . Hall Herbert T . Freedman

Vi

Acknow Zedgments

The authors wish t o tnank

who have contributed t o t h i s book by t h e i r comments, c r i t i q u e , and suggestions. Amonq these individuals a r e Professor L . S. Lasdon (Case lJestern Reserve U n i v e r s i t y ) , Professor David Marks (MIT), Professor Jared Cohon (Johns Hopkins Univ e r s i t y ) , Professor Charles Howe (University of Colorado), and Dr. W. S c o t t Nainis (Arthur 0. L i t t l e , I n c . ) . Special thanks a r e due t o Prasanta Das who has d i l i q e n t l y proofread t h e f i n a l manuscript and t o Sue Reeves who typed the manuscript. Last and not l e a s t , we thank a l l t h e qraduate students i n t h e Hater Resources Proqram, Systems Enqineerinq Department, a t Case llestern Reserve University, who have o f f e r e d many comments and suqqestions durinq t h e preparation o f t h i s book. The cooperation of t h e American Geophysical Union, and t h e American Society of Civil Enqineers, by releasinq copyriqhted material t o be included i n t h e book, i s appreciated. a l l individuals

The preparation of the material i n t h i s book was

p a r t i a l l y supported

by t h e National Science Foundation, Research Applied t o National Proqram, under research p r o j e c t : "Mu1 t i level Approach f o r Deoional Resource P1 annino and Flanaqement", a r e due t o Dr. Richard Kolf project officers.

Grant Number

61-34026.

Needs Water

Special thanks

and t o Dr. Lawrence Tombauqh who

served

as

TABLE OF CONTENTS __ Page

Preface

V

v ii

Ac know1 edgements Table o f Contents

viii

L i s t o f Figures

xiii xiv

L i s t o f Tables

1.

2.

3.

Fundamentals i n M u l t i p l e O b j e c t i v e Problems

1

1.1

Introduction

1

1.2

H u l t i o b j e c t i v e s i n Water Resources Systems

2

1.3

Problem D e f i n i t i o n

4

1.4

T e r m i n o l o g y and Concept o f N o n - i n f e r i o r S o l u t i o n s

1.5

Overview o f Book

11

Footnotes

12

References

13

6

S o l u t i o n Methodologies f o r M u l t i p l e O b j e c t i v e Problems

15

2.1

Introduction

15

2.2

U t i l i t y Functions

15

2.3

I n d i f f e r e n c e Functions

16

2.4

L e x i c o g r a p h i c Approach

16

2.5

P a r a m e t r i c Approach

17

2.6

The F - C o n s t r a i n t Approach

19

2.7

Goal Programming

22

2.8

The Goal A t t a i n m e n t Method

25

2.9

A d a p t i v e Search Approach

27

2.10 I n t e r a c t i v e Approaches

28

2.11 O t h e r Approaches

28

Footnotes

29

References

30

The S u r r o g a t e Worth T r a d e - o f f Method

34

3.1

General Approach

34

3.2

The D e r i v a t i o n o f t h e T r a d e - o f f Rate F u n c t i o n

37

3.3

Computational Procedure f o r C o n s t r u c t i n g t h e

3.4

Trade-off Function

41

The S u r r o g a t e Worth F u n c t i o n

43

viii

CONTENTS

3.5 Computational Procedure for Finding the Preferred Solutions 3.5.1 Decision Space Surrogate Worth Function 3.5.2 A-Space Surrogate Worth Function 3.5.3 Objective Function Space Surrogate Worth Function 3.6 Geometric Interpretation of the SWT Method 3.7 Summary Footnotes References

ix

46 46 47 50 53 55 57 57

4. The SlJT Method for Static Two-Objective Problems 4.1 Computa tional Efficiencies 4.1.1 Limits on E~ 4.1.2 Trade-off and Worth Relationships 4.1.3 Reversion to the Decision Space 4.1.4 Regressions 4.1.5 Finding the Indifference Band 4.2 The Static Two-Objective E-Constraint (STE) Algorithm 4.2.1 The Algorithm 4.2.2 Sample Problem 4.3 The Multiplier Approach 4.3.1 Limitations of the Multiplier Approach 4.4 The Static Two-Objective Combined (STC) Algorithm 4.4.1 The Algorithm 4.4.2 Sample Problem 4.5 The Static Two-Objective Multiplier (STM) Algorithm 4.5.1 The Algorithm 4.5.2 Sample Problem 4.6 Summary Footnotes References

58 58 58 59 61 62 63 63 64 66 69 70 70 71 73 74 76 76 78 79 79

5. The SWT Method for Dynamic Two-Objective Problems 5.1 Introductory Analysis 5.2 Dynamic Problems in c-Constraint Form 5.3 Dynamic Two-Objective E-Constraint (DTE) Algorithm 5.3.1 The Algorithm 5.3.2 Sampl e Problem 5.4 Dynamic Two-Objective Combined (DTC) Algorithm 5.4.1 The Algorithm

81 82 82 84 a4 89

94

94

X

CONTENTS

__ Page

5.4.2 Sample Problem 5.5 Dynamic Two-Objective Multiplier (DTM) Algorithm 5.5.1 The Algorithm 5.5.2 Sample Problem 5.6 Summary Footnotes References

97 100 100 102 103 104 104

6. The SWT Method for Static n-Objective Problems 6.1 Surrogate Worth Functions 6.2 Preferred Solutions and Consistency 6.3 Computational Efficiencies 6.3.1 Limits on E . J 6.3.2 Reversion to the Decision Space 6.3.3 Multiple Regression 6.3.4 Finding the Indifference Band 6.4 The Static n-Objective &-Constraint (SNE) Algorithm 6.5 The Static n-Objective Multiplier (SNM) Algorithm 6.6 Summary Footnotes References

105 105 106 109 109 110 111 111 112 115 117 117 118

7. The 7.1 7.2 7.3 7.4

119 119 120 123 125 126 126

SLIT Method for Dynamic n-Objective Problems Introductory Analysis The Dynamic n-Objective &-Constraint (DNE) Algorithm The Dynamic n-Objective Multiplier (DNM) Algorithm Summary Footnotes References

8. Applications of the SWT Method t o Water Resources Problems 8.1 The Reid-Vemuri Example Problem

8.2 8.3 8.4 8.5 8.6 8.7 8.8

Solution to the Reid-Vemuri Problem Discussion of Results Stream Resource Allocation Problem Solution of Stream Resource Allocation Problem Discussion of Results Northern California Water System Solution of California Water System Mu1 tiobjective Problem

127 127 128 132 132 135

141 142 148

xi

CONTENTS

Footnotes References 9. Multiobjective Water Quality Models 9.1 Introduction 9.2 Water Quality Goals and Objectives 9.2.1 Primary Objectives 9.2.2 Secondary Objectives 9.3 General Problem Formulation 9.4 Formulation of an Example Problem 9.5 Application of the SWT Method to the Three Water Quality Objective Problem 9.6 Summary and Conclusions Footnotes References 10. Sensitivity, Stability, Risk and Irreversibility as Multiple Objectives 10.1 Introduction 10.2 System Characteristics Related to the Evaluation of Risk 10.2.1 Sensitivity 10.2.2 Responsivity 10.2.3 Sta bi 1 i t y 10.2.4 Irreversibil i ty 10.3 Sources of Uncertainties and Errors in Modeling 10.3.1 Model Topology (g1) 10.3.2 Model Parameters (ci2) 10.3.3 Model Scope (g3) 10.3.4 Data (q) 10.3.5 Optimization Techniques (%) 10.3.6 Human Subjectivity (%) 10.4 Formulation of Risk Objectives for Water Resources Systems 10.5 Measurement of Risk-Related Characteristics 10.6 Summary and Conclusions Footnotes References 11. Epilogue 11.1 Advantages of the SWT Method 11.2 Further Development of the SWT Method

154 154 156 156 156 156 158 158 161 165 167 168 168 170 170 171 172 174 175 175 176 177 178 178 179 179 181 185 186 188 188 189 189 190

CONTENTS

Author Index S u b j e c t Index

xii

192 194

LIST OF FIGURES

1-1

D e c i s i o n F u n c t i o n Spaces f o r Example 1

1-2

N o n - i n f e r i o r P o i n t s f o r Convex B i c r i t e r i o n Problem

10

2-1

P a r a m e t r i c Approach

20

2-2

D u a l i t y Gaps i n t h e P a r a m e t r i c Approach

20

2-3

€ - C o n s t r a i n t Approach

23

2-4

Goal A t t a i n m e n t Method

26

3-1

43

3-2

Regression F i t t i n g o f h . . v s . f . 1J J D e t e r m i n a t i o n o f t h e I n d i f f e r e n c e Band

48

3-3

h-Space Worth F u n c t i o n s f o r Non-Convex Problems

49

7

3-4

A-Space Worth F u n c t i o n s f o r L i n e a r Problems

51

3-5

Geometric I n t e r p r e t a t i o n o f Worth F u n c t i o n

54

3-6

R o l e o f SWT Method

56

4-1

Non-Continuous T r a d e - o f f Curve

60

4-2

Flowchart f o r S t a t i c Two-Objective €-Constraint Algorithm

65

4-3

F a i l u r e o f t h e M u l t i p l i e r Approach

71

4-4

F l o w c h a r t f o r S t a t i c T w o - O b j e c t i v e Combined A l g o r i t h m

72

4-5

Flowchart f o r S t a t i c Two-Objective M u l t i p l i e r Algorithm

77

5-1

F l o w c h a r t f o r Dynamic T w o - O b j e c t i v e c - C o n s t r a i n t A l g o r i t h m

a5

5-2

F l o w c h a r t f o r Dynamic T w o - O b j e c t i v e Combined A l g o r i t h m

95

5-3

F l o w c h a r t f o r Dynamic T w o - O b j e c t i v e M u l t i p l i e r A l g o r i t h m

101

6-1

Flowchart f o r S t a t i c n-Objective €-Constraint Algorithm

113

6-2

Flowchart f o r S t a t i c n-Objective M u l t i p l i e r Algorithm

116

7-1

F l o w c h a r t f o r Dynamic n - O b j e c t i v e € - C o n s t r a i n t A l g o r i t h m

121

7-2

F l o w c h a r t f o r Dynamic n - O b j e c t i v e M u l t i p l i e r A l g o r i t h m

124

8-1

A Two R e s e r v o i r System

143

9-1

ith Subsystem R e p r e s e n t a t i o n

158

10-1

S e n s i t i v i t y Band

173

V a r i a b l e Span o f C o n t r o l

1a7

10-2

xiii

LIST OF TABLES

4-1 4-2

Results of S t a t i c Two-Objective E-Constraint Problem, s t e p 5 OM Responses f o r S t a t i c Two-Objective &-Constraint Problem

4-3 4-4

DM Responses f o r S t a t i c Two-Objective Combined Problem

68 75 75

4-5

DM Responses f o r S t a t i c Two-Objective M u l t i p l i e r Problem

78

5-1

Results of t h e Dynamic Two-Objective €-Constraint Problem,

Results of S t a t i c Two-Objective Combined Problem, s t e p 4

68

92 93 99

step 5 DM Responses f o r Dynamic Two-Objective ELConstraint Problem Results o f Dynamic Two-Objective Combined Problem, s t e p 3 DM Responses f o r Dynamic Two-Objective Combined Problem OM ResDonses f o r DTM Problem

103

8-1

Non-Inferior Points and Decision-Maker Responses

130

8-2

136

8-3

Physical Constants f o r Stream Allocation Problem Non-Inferior Points and DM Responses f o r Stream

8-4 8-5 8-6

A 1 1 oca t i on Pro bl em Data f o r T r i n i t y River Subsystem Example Results f o r T r i n i t y River Subsystem Example Preferred Solutions f o r California Water Project Example

140 149 150

5-2 5-3

5-4 5-5

xiv

99

153

Chapter 1

FUNDAMENTALS IN MULTIPLE OBJECTIVE PROBLEMS

1.1

INTRODUCTION A recent

problems which

trend in

systems a n a l y s i s has

have more than

been the

one o b j e c t i v e func tion.

important i n t h e study of l a r g e s c a l e systems, where e r a l c o n f l i c t i n g and non-commensurable

c onside ra tion of This i s e s p e c i a l l y

t h e r e tend t o be sev-

objectives t h a t

the system modeler

can i d e n t i f y . For example, i n water resources planning, one wants t o maximize b o t h economic e f f i c i e n c y , which i s measured in monetary u n i t s , and environmental q u a l i t y ,

tion. ered,

which i s measured in

u n i t s of p o l l u t a n t concentra-

T r a d i t i o n a l l y , only one o b j e c t i v e (economic e f f i c i e n c y ) was consid-

or somehow

with t he o t h e r o b j e c t i v e s being included a s c o n s t r a i n t s ,

However, s o c i e t y i s placing

made cominensurate with t h e primary o b j e c t i v e .

an i n c r e a si n g importance on non-pecuniary o b j e c t i v e s which a r e d i f f i c u l t t o q u a n t i f y monetarily.

Multiple

objective analysis

has been

a pplie d t o a

wide v a r i e t y o f problems including t r a n s p o r t a t i o n , p r o j e c t s e l e c t i o n f o r research a c t i v i t i e s , economic production, the q u a l i t y of l i f e , managing an academic department, game theory, and many o t h e r s . A fundamental c h a r a c t e r i s t i c of decision

processes i s

the develop-

ment o f l o g i c a l bases f o r el i mi n at i n g from f u r t h e r c onside ra tion l a r g e numbers

of otherwise p o s s i b l e d e c i s i o n s ,

most d e s i r a b l e d eci s i o n i s

with reasonable

n o t i n ad v er t ently l o s t .

assurance t h a t the

The reduced

field o f

p o s s i b i l i t i e s can then be more e a s i l y analyzed by a de c ision maker i n orde r t o a r r i v e a t a f i n a l d eci s i o n . I f two o r more

o b j ect i v es a r e n o t commensurable, then t h e r e i s gen-

e r a l l y no s i n g l e optimum d eci s i o n .

Despite t h i s a s s e r t i o n , de c isions which

involve a p p a r en t l y non-commensurable o b j e c t i v e s a r e reached every day by A s u b s t a n t i a l percentage of the se individua ls a r e m i l l i o n s o f people. quite sure could be

they made t h e b e s t

d eci s i o n - - b e s t i n the sense t h a t no o t h e r

demonstrated as s u p e r i o r .

i s one of f i n di n g t h e

Thus i t would appear t h a t the problem

means o f reducing non-commensurable o b j e c t i v e s t o a n

a p p r o p r i a t e common denominator. Much of p r i ci n g theory

i n economics

is

devot-ed t o

t h i s que stion.

Physically non-commensurable q u a n t i t i e s which a r e traded in l a r g e numbers i n a " f r e e , " non-coercive market, appear t o have been r a t h e r well commens u r a t e d i n monetary u n i t s . This has encouraged development of s t r a t e g i e s I

2

MULl"'lOBJECT1 VE OPPIMIZATIOW IF1 WN'ER RMOUNCES SYSTEM5

t o c r e a t e , by law, t h e i n s t i t u t i o n a l equivalent of a market f o r t h e remaining non-commensurable objectives of water resources. "Pollution c e r t i f i c a t e s , e f f l u e n t charges, s c a r c i t y based p r i c i n g , " e t c . a r e examples of t h i s approach.' While very a t t r a c t i v e in some r e s p e c t s , i t i s c l e a r from other i n s t i t u t i o n a l l y managed markets t h a t these economic a r t i f i c e s in many ways

may be

f a r from

adequate t o

would in f a c t represent even

c r e a t e pseudo-market

the important o b j e c t i v e s

conditions which

t o any s a t i s f a c t o r y

degree. The reluctance of the p o l i t i c a l system t o adopt such pseudo-market i n s t i t u t i o n s , and t h e acceptance of d i r e c t p o l i t i c a l a l l o c a t i o n suggests t h a t f o r t h e immediate f u t u r e , a t l e a s t , i t will be necessary t o seek o t h e r a l t e r n a t i v e s f o r t r e a t i n g the non-commensurable o b j e c t i v e problem. Thus t h e development of mathematical techniques f o r the s o l u t i o n of multiple o b j e c t i v e problems i s q u i t e important. The purpose of t h i s book i s t o i n v e s t i g a t e computational procedures f o r the s o l u t i o n of multiple obj e c t i v e problems concentrating on t h e surrogate worth tradeoff (SWT) method. This chapter w i l l present the mathematical formulation of the genera l multiple o b j e c t i v e problem, and discuss the concepts and terminology i n herent t o such problems. 1.2

MULTIOBJECTIVES IN WATER RESOURCES SYSTEMS

Water resources systems c r e a t e special problems which make t h e app l i c a t i o n of c l a s s i c a l optimization methodologies q u i t e d i f f i c u l t a n d , u n l e s s t r e a t e d with considerable i n s i g h t , q u i t e meaningless i f not a c t u a l l y misleading. Most of these d i f f i c u l t i e s stem from t h r e e important characteri s t i c s of these systems. F i r s t , t h e r e i s a l a r g e number o f quasi-independent decision makers and/or c o n s t i t u e n c i e s , each of which may make o r influence decisions according t o t h e i r own, possibly d i f f e r e n t , versions of the desired goals. Second, even f o r any one decision maker, t h e r e i s a l a r g e number of non-commensurable o b j e c t i v e s t o be optimized. Third, t h e r e i s a very l a r g e element o f uncertainty and r i s k i n v i r t u a l l y a l l water resources d e c i s i o n s . This element i s due t o the high degree of i r r e v e r s i b i l i t y of these decisions coupled with both hydrologic uncertainty and i n a b i l i t y t o p r e d i c t the f u t u r e with reasonable accuracy. So long a s one o b j e c t i v e (e.g. economic e f f i c i e n c y ) dominates over a l l o t h e r s and a s i n g l e point of view ( e . g . n a t i o n a l ) can be a s s e r t e d as primary, t h e optimization can proceed along c l a s s i c a l l i n e s using e i t h e r

judgment o r mathematical d eci s i o n models a s d e s i r e d , where secondary obje c ti v e s a n d points of view can be taken i n t o account t h r o u g h judgment-based constraints. To a l i mi t ed e x t e n t t h e judgement-based c o n s t r a i n t s can be parameterized and/or s u b j ect ed t o a s e n s i t i v i t y a n a l y s i s . The r ecen t

trend i n water

r es o u rc e s, however,

has been t o e l e v a t e

many of t h e non-commensurable o b j e c t i v e s t o p a r i t y with economic e f f i c i e n c y a s c r i t e r i a f o r ex cel l en ce. Water resource p r o j e c t s a r e g en er ally jectives.

This f a c t

constructed t o se rve multiob-

i s i n h er en t i n the na ture

of almost

any l a r g e - s c a l e

project,

e . g . , r e s e r v o i r s , dams, aqueducts, t h e development of groundwater

systems,

and so o n .

water f o r i r r i g a t i o n ,

A large reservior

c re a te d by a high

municipal a n d i n d u s t r i a l needs,

and recreation f a c i l i t i e s ,

dam may supply

provide f o r f i s h i n g

improve navigation and flood control c a p a b i l i t -

i e s , generate h y d r o e l e c t r i c power, maintain s u i t a b l e water q u a l i t y f o r both

ground

and s u r f a c e

water recharge,

wat er ,

provide a b uffe r f o r d r o u g h t years and ground-

improve r e l a t e d land use

and prevent damages from runoff,

a n d enhance t h e regional development i n terms of a b e t t e r economy and qual-

i t y of l i f e .

I n regional planning of water a n d r e l a t e d land re sourc e s, t h e

simultaneous co n s i d er at i o n of more t h a n one p r o j e c t i s o f t e n e s s e n t i a l d u e t o t h e i n t e r a c t i o n s a n d coupling t h a t e x i s t among them.3 C l e a r l y , t h e p r o blem of mu1 t i p r o j e c t s - - mu l t i o b j ect i v es planning becomes t r u l y l a r g e s c a l e and complex. Probably one of t h e major reasons f o r the r e l a t i v e s c a r c i t y of mu1 t i o b j e c t i v e

formulations

and co n side ra tions

i n the 1 i t e r a t u r e , n o t

n e c e s s a r i l y l imi t ed t o water resources systems, i s t h a t u n t i l r e c e n t l y , a l most a l l t h e s o l u t i o n s t r a t e g i e s developed involved a s i n g l e o b j e c t i v e function.

Optimization techniques

and methodologies a r e viewed here as s o l -

t o t h e mathematical

ution s t r a t e g i e s t h a t a r e applied

o b j e c t i v e f u n ct i o n and a s e t of c o n s t r a i n t s . objective

f u n ct i o n s introduces

a new dimension in the f i e l d s of modeling,

mathematical programming, and optimal c o n t r o l , i c a l notion

model defined by a n

The inc lusion of a vector of

of a n optimal s o l u t i o n

e s p e c i a l l y sinc e t h e numer-

g en era lly w ill

not e x i s t f o r a ve c tor

optimization problem, as wi l l be discussed l a t e r . 4 I t i s important t o note, however,

t h a t both judgment

have one common

and mathematical models of de c ision processes

f e a t u r e i n t h a t they u t i l i z e a logic a l argument t o elimin-

a t e l a r g e numbers o f p o s s i b l e d eci s i o n s e t s from f u r t h e r contention f o r t h e " bes t " deci s i on. Some procedures which can accomplish c o n t e x t w i l l be explored i n t h i s book.

t h i s in t h e m ultiple -obje c tive

A number of s t u d i e s have been conducted which i n c l u d e m u l t i p l e obj e c t i v e s i n water resources planning. The Corps of E n g i n e e r s 5 used t h r e e o b j e c t i v e s ( n a t i o n a l income, r e g i o n a l development, and environmental q u a l -

i t y ) in their

study of

t h e North

s t u d i e d the

M i l l e r and Byers6

income f o r an a g r i c u l t u r a l a r e a . between net

Atlantic region

tradeoff

o f the

United S t a t e s .

between environmental q u a l i t y and

Cohon and Marks7 e v a l u a t e d

the t r a d e o f f

n a t i o n a l income and e q u i t y of r e g i o n a l income d i s t r i b u t i o n f o r

a developing c o u n t r y .

Major* took r e g i o n a l development i n t o a c c o u n t i n t r a -

ditional cost-benefit analysis.

O'Riordanq used the o b j e c t i v e s o f economic

growth, environmental q u a l i t y and s o c i a l w e l l - b e i n g f o r r i v e r b a s i n p l a n ning i n Canada. Monarchi e t a l l o p r e s e n t e d a s e q u e n t i a l t e c h n i q u e which should

e n a b l e t h e d e c i s i o n maker t o d e t e r m i n e a s a t i s f a c t o r y s o l u t i o n from

non-inferior points.

An a n a l y s i s of t h e a p p l i c a b i l i t y o f v a r i o u s m u l t i p l e r e s o u r c e s problems has been c a r r i e d o u t by

o b j e c t i v e techniques t o water Cohon.ll 1.3

PROBLEM DEFINITION For n o t a t i o n a l

convenience,

define t h e general vector optimization

t o be: Problem _ _ _ _ 1-1:

...,

m i n i f , ( Z ) , f2(xJ,

fn(xJl

X -

Subject t o

,

gk(L) 5 3 Where

xis

m

...,

n, are n

g,(x), k = 1,2,

...,

m, a r e m constraint functions

=

For s i m p l i c i t y i n n o t a t i o n , and :: 0 ) .

objective functions

I t can

equality constraints are not present.

t h a t each e q u a l i t y (3 0

...,

1,2,

1 -

constraints

1, 2,

=

an N - dimensional v e c t o r o f d e c i s i o n v a r i a b l e s . f.(x), i

be assumed

k

c o n s t r a i n t was r e p l a c e d

by two i n e q u a l i t y

T h u s , t h e r e i s no l o s s i n g e n e r a l i t y

by

s i d e r i n g t h e compact n o t a t i o n o f a system o f i n e q u a l i t y c o n s t r a i n t s . assumed

t h a t a l l functions

may be

nonlinear in

L.

Convexity o r

conIt is other

p r o p e r t i e s may be assumed when needed. lems N

,','

x

where

E

n.

Note t h a t in most re a l world prob-

This can be w r i t t e n more compactly in vector notation a s :

RN

R N i s t h e d eci s i o n v ect o r , f :

+

R n i s t h e o b j e c t i v e func tion

R" i s t h e c o n s t r a i n t v e c t o r , and 0 E Rm i s a vector whose v e c t o r , 9: R N elements a r e a l l zero. The meaning of minimizing a ve c tor w ill be disc uss-

ed in the next s e c t i o n .

The d e f i n i t i o n of

5

"

must a l s o be c l a r i f i e d f o r

"

v e c t o r s: Definition 1 : _____

...

all i = 1 , 2,

.

Rk where t h e s u b s c r i p t

For any two v e c t o r s ,

E

k,

R k and

z

E

s

and only i f yi r: z i f o r

if

i denotes t h e i t h element o f the

vector. the

The c o n s t r a i n t s decision vector

unique value

S

=

If(&)

[JET

~

f(x);

g(x) r: 0 determine a f e a s i b l e s e t T of values f o r = {x/g(&) 50). Each vector x E T determines a

x; T

thus t h e r e e x i s t s a s e t S of f e a s i b l e values f o r

T(x);

The mu l t i p l e o b j e c t i v e problem can be considered a s:

1.

Min

f(x)

MIN

f(x)

o r as s . t . f(&) 5 This d u a l i t y wi l l be presented l a t e r . these i d e a s :

EXAMPLE 1 : I

_

~

s

s.t. 5

E

7

useful f o r understanding the various solution methods

The following examples w ill serve t o i l l u s t r a t e some

of

6

0 I x2 I 5 ( o r g 3

=

x2 - 5

0 , g4

=

- x2 I 0)

For t h i s problem T ( t h e feasib.ie s e t f o r x_) and S ( t h e f e a s i b l e s e t f o r f(&)) a r e shown i n f i g u r e s 1-1-a and 1-1-b r e s p e c t i v e l y . 1.4

TERMINOLOGY A N D CONCEPT OF NON-INFERIOR SOLUTIONS

I t i s important here t o d ef i n e some of the terms used in multi-objective analysis. F i r s t , t h e d e f i n i t i o n of optimal i s d i f f e r e n t than f o r t h e case of a s i n g l e o b j e c t i v e f u n ct i o n : Definition 1-2:

An optimal s o l u t i o n i s one which a t t a i n s the minimum value of a l l of t h e o b j e c t i v e s simultaneously;

_f (-x )

s.t.

x

E

T i f and only i f

L* L*

i s an optimal solution t o the problem MIN T and

E

f(~*)I

f(x) f o r

all

Optimal s o l u t i o n s a r e a l s o known a s supe rior s o l u t i o n s . t h e r e i s no optimal the

minimum value

x

T. I n general

E

s o l u t i o n t o a mu l t i - obje c tive problem. of f l i s 0 which

I n example 1 , t h e minimum thus the se two minima cannot

occurs a t x1 = 0 ,

value of f 2 i s 0 which occurs a t x1 = x2 = 5; be a t t a i n e d simultaneously.

while

Let Ti be t h e s o l u t i o n t o min

fi(x)

X -

Subject t o

for all i = 1 , 2,

..., n

i . e . , f . i s t he global minimum o f t h e i t h o b j e c t i v e func tion while ignoring 1

a l l t h e o t h e r ( n - 1 ) o b j e c t i v e s . Looking a t t h i s in the func tiona l space, = i,,f ) where 7 . i s a s defined above, then f S means t h a t if

(r,,

.. .

n

1

-

no optimal s o l u t i o n e x i s t s . F o r example 1 , f=(O,O) which i s not in S a s can I n t h e following example t h e r e i s an optimal be seen from f i g u r e 1-1-b. solution :

7

T

x2

Figure 1-1-a.

Decision Space

f2

Figure 1-1-b. Figure 1 - 1 .

Functional Space

Decision and Functional Spaces f o r Example 1

8

EXAMPLE 2:

g2

=

- x2

s 0

The mininium value of each objective is zero and these can be attained simultaneously when x1 = x2 = 0. The optimal solution is x* = (0,C); f(x*)= (0,O). There has been some work done12 on determining when optimal solutions to a multiple objective problem will exist. Since, however, optimal solutions generally do not exist, one must be satisfied with obtaining noninferior solutions. The concept of non-inferior solutions, also known as Pareto optimum or efficient solutions, is basic to economics in general and particularly for competitive equilibrium. K ~ o p m a n s ' ~defined an efficient point for mu1 tiobjective functions in economics as follows: "A possible point in the commodity space is called efficient whenever an increase in one of its coordinates (the net output of one good) can be achieved only at the cost of a decrease in some other coordinate (the net output of another good)." Kuhn and Tucker'" extended the theory of nonlinear programming for one objective function to a vector minimization problem and introduced necessary and sufficient conditions for non-inferior solutions. A formal definition of a non-inferior solution is given below: Definition 1-3:

A non-inferior solution is one in which no decrease can be obtained in any of the objectives without causing a simultaneous increase in at least one of the other objectives; x* is a non-inferior solution to the problem MIN f(&) s.t. x E T if and only if there does not exist any 5 t T such that : f(&*) and f.(x) f.(x*) for some i = 1 , 2, ...,n. This 1 1 solution is obviously not unique. We define the non-inferior set as NI = {f(~)ls is a non-inferior solution]. In example 1, the solutions x2 = 5, 0 s x1 5 5 are all non-inferior

f(x)

solutions. I n t h e functional space ( f i g u r e 1 - 1 - b ) t h i s corresponds t o t h e l i n e f , = 5 - f,. I t has been shown15 t h a t a l l f ( l i ) ~ N I must l i e on t h e boundary of S .

This i s

obvious,

reduction could be achieved

s i n ce f o r any point

i n one o b j e c t i v e

in the i n t e r i o r , a

without changing

the others

by moving in a negative d i r e c t i o n p a r a l l e l t o t h a t a x i s a s f a r a s p o s s i b l e , t h a t i s , u n t i l a boundary i s reached. be in

the n o n - i n f er i o r s e t

s c r i b e d subsequently,

Necessary c onditions f o r a point t o

have been developed16;

however,

a r e in general

the

methods t o be de-

simpler t o use than d i r e c t

a p p l i c a t i o n of t h es e co n d i t i o n s i n determining the non-infe rior s e t .

I t i s easily verified

t h a t f o r any two convex f u n c t i o n s , t h e non-every point in the i n t e r v a l between t h e minima [ x l * , x2*] of t h e b i c r i t e r i o n problem depicted in Figure 1 - 2 , i s a n o n - i n f er i o r p o i n t . For nonconvex func tions, however, t h e noni n f e r i o r s e t may be non-connected. Since the n o n - i n f er i o r s e t i s on the boundary of S , i t forms a surf a c e in R n which can be described by T * ( f l , f 2 , . . . . f n ) = 0. This can be solved t o g e t f i * ( f l , ..., f i - l , f i + l ,...,f n ) c a l l e d the tra de -off func tions s i n c e they show how much t h e value of f i must change t o s t a y in the non-inf e r i o r s e t when t h e values of t h e o t h e r o b j e c t i v e s change. The r a t e of change of t h e t r ad eo f f f u n ct i o n with r e s p e c t t o f i a l s o forms useful funJ c t i o n s which a r e c a l l e d t h e t r ad eo f f r a t e func tions 1..; T i j ( f l , . . . , f i - l , i n f e r i o r s e t i s continuous.

fit,

,.., f n )

For example,

1J

a f 1. * / a f l.. Some a u t h o r s have modified t h e d e f i n i t i o n of a non-infe rior s o l u t i o n

t o exclude

=

those p o i n t s

where a f i r s t orde r improvement

in one o b j e c t i v e

can be made a t t h e expense of only a second orde r degradation i n another17; t h a t i s , t h e p o i n t s where any of t h e t r ade off r a t e func tions a r e e i t h e r zero or i n f i n i t e a r e n o t properly n o n - i n fe rior s o l u t i o n s . I n most c a s e s , t h e determination of t h e non-infe rior s e t i s n o t suff i c i e n t ; t h e systems a n a l y s t must choose one de c ision which i s by some definition "best".

Thus,

ad d i t i o n al c r i t e r i a

must be introduced t o d i s t i n -

guish t h e " b e s t " of t h e n o n - i n f er i o r s o l u t i o n s .

Although some authors re -

t a i n t h e term optimal f o r t h i s " b e s t " s o l u t i o n , the word

" pre fe rre d"

will

be used i n t h i s book t o avoid ambiguity. D e f i n i t i o n 1-% A p r e f e r r ed

s o l u t i o n i s a n o n - i n f erior

s o l u t i o n which i s chosen a s

the f i n a l d e c i s i o n t h r o u g h some ad d i t i o n al c r i t e r i a .

10

Figure 1 - 2 .

Non-Inferior Points for Convex Bicriterion Problem.

f ( x-)

For each value

s o c i e t y from f i ( x ) u n i t s

E

Rn t h e r e i s some b e n e f i t t h a t would accrue t o

of each o b j e c t i v e i = 1 , 2 ,

..., n .

This b e n e f i t

Since each o b j e c t i v e i s bei s c a l l e d t h e u t i l i t y f u n ct i o n ( u : Rn -f R 1 ) . ing minimized, i t must be t r u e t h a t i t i s p r e f e r a b l e t o have l e s s of each; thus t h e u t i l i t y objective.

f u n ct i o n i s monotonically decreasing with re spe c t t o each

U t i l i t y f u n ct i o n s a r e a l s o known a s soc ia l preference o r s o c i a l

welfare f u n c t i o n s . Consider t h e s u r f aces of equal u t i l i t y in R n ( u ( f ) e t y i s i n d i f f e r e n t between any p o i n t s on the se s u r f a c e s , c a l l e d s o c i a l i n d i f f e r e n c e or i s o - p r ef er enc e s u r f a c e s .

=

c o n s t a n t ) . Sociand thus they a r e

These w i l l be use-

f u l l a t e r i n f i n d i n g p r ef er r ed s o l u t i o n s . The p o i n t s where t h e s o c i a l i n d i f f e r e n c e surfa c e s a r e tangent t o t h e n o n - i n f e r i o r s e t a r e known as t h e i n d i f f e r e n c e b a n d . Definition 1 - 5 :

ior set

The i n d i f f e r e n c e band i s defined t o be t h e subse t o f t h e non-inferwhere t h e improvement of one o b j e c t i v e func tion i s e quiva le nt in

t h e mind of t h e d eci s i o n maker t o t h e necessary degradation of t h e o t h e r s . 1.5

OVERVIEW OF BOOK The next ch ap t er attempts t o give a comprehensive survey of t h e m u l -

t i p l e o b j e c t i v e problem and t h e various approaches and techniques a v a i l a b l e for the solution

o f such problems.

Chapter 3 w i l l

pre se nt s t i l l another

approach t o solving mu l t i p l e o b j e c t i v e problems - the surroga te worth tra de o f f (SWT) method18. which

Chapter 4 d es cr i b es

can be implemented

Three algorithms f o r

some computational

f o r two-objective

problems

efficiencies

in t h e SWT method.

implementing t h e SWT approach in s t a t i c two-objective

problems a r e presented;

one employing t h e € - c o n s t r a i n t approach in solving

m u l t i p l e o b j e c t i v e problems;

one using Lagrange m u l t i p l i e r s in a v a r i a t i o n

of t h e parametric approach,and one using a combination of the se two. Chapt e r 5 modifies t h e r e s u l t s of ch ap t er 4 i n orde r t o apply t h e SWT t o dynami c o p t i m i z a t i on problems ; included a r e t h r e e analogous a lgorithm s. Chapters 6 and 7 modify

t h e r e s u l t s of t h e previous two c ha pte rs t o encompass p r o b -

two o b j e c t i v e s ( f o r s t a t i c and dynamic c a se s, re spe c tt h e r e l a t i o n s h i p s between t h e various worth func1 , 2 , ... , n , j = 1 , 2 , ... , n , i # j . Chapter 8 a p p l i e s the

lems with more than ively), tions Wij,

and i n v e s t i g a t e i =

SWT method t o

t h r e e problems i n water resources. Chapter 9 a p p l i e s the

SWT

: ;Y

12

method

to

water qtiality

problems.

Chapter

10

discusses

the

incor-

p o r a t i o n of s e n s i t i v i t y , i r r e v e r s i b i l i t y and r i s k a s m u l t i p l e o b j e c t i v e s i n water resources systems. t h e book,

The f i n a l c h a p t e r summarizes the major themes o f

and i n d i c a t e s a r e a s where i t s implementation may prove f r u i t f u l .

FOOTNOTES 1.

D i s c u s s i o n s and examples of t h e s e s t r a t e g i e s can be found

i n Kneese

and Bower [1968], Hainies e t a1 [1972] and o t h e r s .

2.

The o r i g i n a l development can be found i n Hainies and Hall r19741.

3.

Multiproject analysis

i s d i s c u s s e d by many

e t a1 [1962], Howe arid E a s t e r 119711,

a u t h o r s i n c l u d i n g Maass

Hall and Dracup r19701, I s a r d

e t a1 [1972] and H a i i e s and Hall [1974].

4.

The f i r s t d i s c u s s i o n of a v e c t o r of o b j e c t i v e s i s i n K u h n and Tucker [ 19501 .

5.

See Carps of E n g i n e e r s 119721.

6.

See M i l l e r and Byers [1973].

7.

See Cohon and Marks [1973].

8.

See Major [1969].

9.

See O'Riordan [1973].

in.

See Monarchi e t a1 r19731.

11.

Cohon [1973] p r o v i d e s

c l a s s i f i c a t i o n s f o r the v a r i o u s t e c h n i q u e s a s

well a s s t u d y i n g t h e i r a p p l i c a b i l i t y t o w a t e r r e s o u r c e s problems.

12.

Athans and Geering [1973] p r o v i d e

necessary

and s u f f i c i e n t

condi-

t i o n s f o r t h e e x i s t e n c e of s u p e r i o r s o l u t i o n s . 13.

See Koopnians 119511.

14.

Again s e e Kuhn and Tucker [1950].

15.

A formal proof i s g i v e n i n Reid and C i t r o n r19711.

16.

D i f f e r e n t forms can be found i n and

17.

K u h n and Tucker [1950], C h u

[1970]

DaCunha and Polak r19671.

T h e most g e n e r a l d e f i n i t i o n

of p r o p e r

n o n - i n f e r i o r i t y i s given

B e o f f r i o n [1968]. Other d e f i n i t i o n s can be found i n K u h n and [1950] and K1 i n g e r [1964].

by

Tucker

13 18

The o r i g i n a l development can be found

in

Haimes and Hall 119741.

R E FEKEPJCES 1.

Athans, M . ,

and G e e r i n g , H .

P., "Necessary and S u f f i c i e n t C o n d i t i o n s

f o r D i f f e r e n t i a b l e Nonscalar-Valued F u n c t i o n s t o A t t a i n Extrema," IEEE T r a n s a c t i o n s , v o l . AC

-

18, no. 2 , 1973.

2.

C h u , K . C . , "On t h e N o n - i n f e r i o r S e t f o r Systems with Vector-Valued O b j e c t i v e F u n c t i o n s , " IEEE T r a n s a c t i o n s , v o l . AC-15, no. 5 , 1970.

3.

Cohon, J . L . ,

"An Assessment

of M u l t i o b j e c t i v e

Solution techniques

f o r River Basin P l a n n i n g P r o b l e m s , " Ph.D. D i s s e r t a t i o n , M . I . T . , 1973.

4.

Cohon, J . L . , and Marks, D . H . ,

" M u l t i o b j e c t i v e S c r e e n i n g Models and

Clater Resource I n v e s t m e n t " Water Resources R e s e a r c h , vol . 9 ,

no.

4 , 1973. 5.

Corps of E n g i n e e r s , "N.A.R. Water Resources S t u d y , " Appendix T,1972.

6.

DaCunha, N . O . , and P o l a k , E . ,

"Constrained

Minimization under Vec-

t o r Valued C r i t e r i a i n F i n i t e Dimensional

Spaces,"

Journal

of-

Math. A n a l . and Appl . , v o l . 1 9 , no. 1 , 1967. 7

I .

G e o f f r i o n , A . M., " P r o p e r E f f i c i e n c y and t h e Theory of Vector

Maxi-

m i z a t i o n , " Journal of Math. Anal. and Rppl-., v o l . 2 2 , no. 3 , 1 9 6 8 .

8.

Haimes, Y . Y . ,

9.

Haiines, Y . Y . ,

" H i e r a r c h i c a l Modeling f o r t h e Planning and Management of a T o t a l Regional Wdter Resources S y s t e m " P r e s e n t e d a t the IFAC Syniposiuni on Control of Water Resources Systems, I s r a e l , September 17 - 21, 1 9 7 3 . Also Automatica, J a n . 1975. and H a l l , W . A . ,

" M u l t i o b j e c t i v e s i n Water Resources

Systems A n a l y s i s : The S u r r o g a t e Worth T r a d e o f f Method," Water Resources Research.., v o l . 10, no. 4, 1974. 10.

Haimes, Y . Y . ,

Kaplan, 11. A . ,

and H u s a r , M . A . ,

"A

Multilevel

proach t o Determine Optimal T a x a t i o n f o r the Abatement

of

ApWater

P o l l u t i o n , " Water Resources R e s e a r c h , v o l . 8 , no. 4 , 1972. 11.

H a l l , W . A . , and J . A .

Dracup, Water Resources Systems E n g i n e e r i n g ,

PlcGraw-Hill Book Company,

12.

Howe, C . W . ,

N. Y . , 1970.

and K . Id. E a s t e r , I n t e r b a s i n T r a n s f e r s of Water,

Econ-

omics I s s t i e s and I m p a c t s , T h e John Hopkins Press, B a l t i m o r e , l 9 7 1 .

14

to Reqional Science, The M. 1.1. P r e ss, Cambridge, Massachusetts,l972.

13.

I s a r d , W . e t a l , Methods

of Reqional Anal.ysis;

an Introduc tion

14.

Klinger, A., "Vector Valued Performance C r i t e r i a , " I E E E Transactions,

v o l . AC - 9 , no. 1 , 1964. 15.

Kneese, A. V., and B . 1 . Bower,

m i n g Water Qua1it.y: E c z -

rjjcsLTechnoloqy, I n s t i t u t i o n s , The Johns Hopkins Pre ss, n o r e , Maryland, 1968. 16.

Kocpmans, T . C . , "Analysis of Production a s a n E f f i c i e n t Combination of A c t i v i t i e s , " A c t i v i t y Analysis of Production,

Cowles

s i o n Monograph 13, Edited by T . C . Koopmans, Wiley, N . Y . , 17.

Balti-

Commis1951.

A . W . Tucker, Nonlinear Programming, Proceedings on Mathematical S t a t i s t i c s and Proba b i l i-t y , pp. 481-492, University o f Ca lifornia P r e s s , Berkeley, Cal i f o r n i a , 1950.

Kuhn, H . W . ,

dnd

Second Berkeley Svm-m

18.

Maass, A . e t a l , Desiqn of Water-Resource Systems, Harvard Univers i t y P r e s s , Cambridge, Massachusetts, 1962.

19.

Major, D.C., "Benefit-Cost Ratios f o r Proje c ts in Multiple Objective Investment Programs," Water Resources Research, vol. 5 , no.6, 1969.

20.

M i l l e r , W . L., and D. M . Byers, "Development and Display of Multiple Objective P r o j ect Impacts," Water Resources Research, v o l . 9 , no. 1 . 1973.

21.

Monarchi, 0. G . ,

C.C.

K i s i e l , and L . Duckstein. " I n t e r a c t i v e Flulti-

o b j e c t i v e Programming i n Water Resources," LeG-, 22.

Water

Resources

Re-

vol . 8 , Nov. 4 , 1973.

O'Riordan, J . , " A n Approach t o Evaluation i n Multiple Objective River Basin P l an n i n g , " Canada Department o f Environment,Vancouver, B. C . , 1973.

23.

"On Non-inferior Performance Index Reild, R. W., and S . J . C i t r o n , ' Je c t o r s , " Journal o f Optimization Theory and A pplic a tions, vol. 7 , n o . 1 , 1971.

Chapter 2

SOLUTION METHODOLOGIES FOR MULTIPLE OBJECTIVE PROBLEMS 2.1

INTRODUCTION

two approaches t o t h e s o l u t i o n of problems with

There a r e b a s i c a l l y multiple objectives.

One can e i t h e r attempt t o find the pre fe rre d s o l u t i o n

d i r e c t l y , o r f i r s t g en er at e t h e n o n - i n f er ior s e t a n d then f i n d the

pre fe r-

A t h i r d school of thought i s t h a t the sys-

red s o l u t i o n from among t h e s e .

tems a n a l y s t should be concerned only with developing t h e non-infe rior s o l -

(DM)

u t i o n s; t h e d e ci s i o n maker

can then choose on his own which of the se

s o l u t i o n s t o implement'. I t seems l o g i c a l , however, t h a t t h e OM w i l l d e s i r e some s o r t of f u r t h e r a n a l y s i s t o f i n d t h e pre fe rre d solution: i f t h i s a n a l y s i s can be q u a n t i f i e d and systemized t o reduce t h e s u b j e c t i v i t y , then a more a c c u r at e (according t o the c r i t e r i a introduced) s o l u t i o n w i l l be f o u n d a n d s o c i e t y w i l l presumably be b e t t e r o f f . 2.2

UTILITY FUNCTIONS

The f i r s t type of d i r e c t approach i s t h e u t i l i t y func tion approach; t h i s assumes t h a t t h e u t i l i t y f u n c t i o n , u ( + ) , which can be used t o commens u r a t e t h e various o b j e c t i v e s with adequate accuracy, i s known.The r e d s o l u t i o n i s defined a s t h e one which maximizes s o c i e t y ' s can be found d i r e c t l y by s o l v i n g : MAX u(f(x)) s u b j e c t t o

x

Since t h e u t i l i t y i s o b j e c t i v e , t h e p r ef er r ed set2.

monotonically decreasing

s o l u t i o n wi l l be an

pre fe r-

utility; E T.

this

with re spe c t t o each

element of t h e

non-infe rior

The major drawback t o t h i s approach i s t h a t , in ge ne ra l, the u t i l i t y

function cannot be determined. Much work has been done in decision theory on how individual a n d s o c i e t a l u t i l i t y f u nc tions may be approximated3. Theo r e t i c a l l y , a d eci s i o n maker r e f l e c t s t h e d e s i r e s of his c o n s t i t u e n t s by some method

of aggregating

individual

preferences.

Many s t u d i e s assume

n additive u t i l i t i e s

(g(f) =

1

i=l

u i ( f i ) ) ; t h e im plic a tions of t h i s assumption

have been analyzed e x t e n s i v e l y 4 ; one imp1 i c a t i o n

i s t h a t the indifference

between f i and f . i s independent of t h e values of the o t h e r o b j e c t i v e s f o r J a l l i a n d j which i s g en er al l y u n r e a l i s t i c . The d e f i n i t i o n of pre fe rre d s o l u t i o n used

in t h i s u t i l i t y f u n ct i o n approach w ill be used again in some 15

o f t h e approaches t o be d e s c r i b e d subsequently.

2.3

INDIFFCRENCE FUNCTIONS t h a t indifference functions are easier t o

I t i s c j e r l e r a l l y conceded5

deteriiiine than

the actual u t i l i t y

d i n a l comparisons. prefers

to

I(%)t h a n

f t i n c t i o n s i n c e t h e y c a n be f o u n d b y o r -

f o r a d e c i s i o n maker

i t is easier

t o d e t e r m i n e i f he

i t i s t o d e t e r m i n e how much a d d i t i o n a l u t i l i t y

i s derived from I ( x l )

(as i m p l i e d i n t h e u t i l i t y approach).

w o r k has been done

using

ution.

Indifference functions

surate since they r e l a t e terms o f another. was known

Thus,

recent

i n d i f f e r e n c e curves t o f i n d t h e p r e f e r r e d solc a n be u s e d t o make t h e o b j e c t i v e s commen-

how much i n c r e m e n t s i n one o b j e c t i v e a r e w o r t h i n

Briskin6

assumed t h e f o r m o f t h e i n d i f f e r e n c e f u n c t i o n

( e x p o n e n t i a l ) f o r a t w o o b j e c t i v e ( m i n i m i z e t i m e and c o s t ) p r o b -

lem.

The c o n s t a n t s i n t h e e q u a t i o n a r e d e t e r m i n e d b y q u e s t i o n i n g t h e

find

several p o i n t s on t h e curve;

t h e n s i i b s t i t u t i n g f o r f,

DM t o

i n t e r m s o f f,

frorv t h e i n d i f f e r e n c e e q u a t i o n t h e p r e f e r r e d s o l u t i o n c a n b e f o u n d b y s o l v ing:

The p r o b l e m w i t h t h i s a p p r o a c h i s t h a t i t assumes t h e i n d i f f e r e n c e e q u a t i o n w i l l b e t h e same e v e r y w h e r e i n t h e t h e case.

f e a s i b l e space.

Another approach u s i n g i n d i f f e r e n c e

This i s generally

not

f u n c t i o n s i s t o f i n d where

drie o f t h e i n d i f f e r e n c e s u r f a c e s i s t a n g e n t t o t h e t r a d e o f f f u n c t i o n ;

this

w i l l g i b e t h e p r e f e r r e d s o l u t i o n i n t h e maximum u t i l i t y s e n s e .

2.4

L E X I C O G R A P H I C APPROACH The l e x i c o g r a p h i c a p p r o a c h 7 r e q u i r e s

i s d e f i n e d t o be

that the objectives

A preferred

i n o r d e r o f i m p o r t a n c e b y t h e DM.

be ranked

s o l u t i o n f o r t h i s approach

one w h i c h s i m u l t a n e o u s l y m i n i m i z e s as many o f t h e o b j e c t -

i v e s as p o s s i b l e , s t a r t i n g w i t h t h e m o s t i m p o r t a n t and g o i n g down t h e h i e r archy.

L e t fl(x)b e t h e

fl(x) s.t. 5

E

most i i n p o r t a n t

T i s solved f o r a l l subject t o

5

then t h e problem

possible solutions;

a l l s o l u t i o n s t o t h i s p r o b l e m yl.

f7(K) i s minimized

objective;

Then t h e n e x t E

y1

MIN

we c a l l t h e s e t o f

tnost i m p o r t a n t o b j e c t i v e

t o f i n d t h e s o l u t i o n s e t y2.

This

process i s repeated u n t i l a l l n o b j e c t i v e s have been considered. I f the s o l u t i o n s e t yi a t t h e i t h i t e r a t i o n has only one element, then t h i s w i l l be the s o l u t i o n t o t h e e n t i r e problem; t h e obje c tive s ranked l e s s important f i a r e ignored by t h i s method. The r a t i o n a l e f o r t h i s approach i s t h a t i n d i v i d u al s tend t o make d eci s i o n s i n t h i s manner5. This lexicograph-

than

i c d e f i n i t i o n of p r ef er r ed was used by McGrew and Haimes f o r t h e problem o f Note t h a t the solution w ill

j o i n t system i d e n t i f i c a t i o n and o p t i mi zat iong.

be very s e n s i t i v e t o t h e ranking by t h e DM, a n d thus t h e a n a l y s t should exe r c i s e caution

i n applying t h i s method

when two obje c tive s

a r e of nearly

equal importance. To i l l u s t r a t e

consider again example 1 from Chapter

t h i s approach,

1 , a n d assume t h a t t h e DM has decided t h a t f , i s the most important obje c ti v e . F i r s t , minimize f l = x1 s . t . 0 s x i :: 5 , 0 s x, :: 5 and ge t the s o l -

ution s e t y1 = i ( x l , x 2 ) / x 1 = 0 , 0 s x2 s 5;. Then, minimize f , = 10 - x I - x 2 , s . t . x1 = 0 , 0 L x2 L 5 . The s o l u t i o n i s x1 = 0 , x 2 = 5 , f , = 0 , f , = 5 ; t h i s i s t h e p r ef er r ed s o l u t i o n a n d i s shown a s point A in f i g u r e s 1 - 1 a and 1-1-b. 19

A v a r i a t i o n of t h i s method was proposed by blaltz ary o b j e c t i v e i s minimized, t h e second keeping

the f i r s t

o b j e c t i v e within

objective i s a certain DM).

( t h i s percentage i s determined by t h e

; a f t e r the prim-

minimized s u b j e c t

percentage

The t h i r d

to

of i t s optimum

objective

i s then

minimized keeping t h e f i r s t two within a c e r t a i n percentage o f the values found in t h e previous s t e p . This process i s repeated u n t i l a l l the o b j e c t ives have been considered.

This approach reduces t h e sens t i v i t y somewhat,

b u t t h e same caveat i s necessary. 2.5

PARAMETRIC APPROJCJ Assume t h a t t h e r e l a t i v e importance o f the n o b j e c t ves i s known a n d

c o n st a n t . ities.

From a u t i l i t y viewpoint

t h i s implies a d d i t i v e and l i n e a r u t i l -

Then t h e p r ef er r ed s o l u t i o n i s found by solving:

n

MIN

1

oi f i ( x )

i=l

where oi > 0 a r e

t h e weighting

importance of t h e

objectives

c o e f f i c i e n t s which (usually

t h e oi

are

determine the r e l a t i v e normalized

so t h a t

18

T o 1. = ] ) . For s i m p l i c i t y i n n o t a t i o n , l e t 0 re pre se nt t h e row ve c tor ( o l , 02, . . . , o ) , where t h e s u p e r s c r i p t T denotes t h e transpose ope ra tion. n I1

i=l

One drawback of t h i s approach, weight

t o be given t o

however, i s t h a t t h e proper r e l a t i v e

any o b j e c t i v e on t h i s s c a l e

i s usua lly a func tion

n o t only of t h e q u a n t i t y of t h a t o b j e c t i v e produced, b u t a l s o of t h e q u a n t i t i e s of a l l o t h e r o b j e c t i v e s t o be produced. Even monetary value i s s u b j e c t t o t h i s c r i t i c i s m : To most i n d i v i dua ls a second $1,000 i s not equa l l y a s important or valuable as t h e f i r s t , and t h e values of each usua lly depend heavily on t h e i n d i v i d u a l ' s at t ai n m e nt l e v e l s of o t h e r physical a n d social objectives.

A t least

i n t h eo r y ,

t h i s means t h a t

t h e method should r e q u i r e t h e

consensus of t h e ex p er t s on t h e " p r i c e " of each o b j e c t i v e f o r a l l possible combinations of l e v e l s

of o b j e c t i v e s a t t a i n e d .

This complexity

can

be

eliminated i f t h e d eci s i o n only adds o r s u b t r a c t s a ne gligibly small increment t o t h e t o t a l of any o b j e c t i v e . as a l l

the e x p er t s were f u l l y aware

I t could then be argued t h a t , so long of t h e s t a t u s quo,

t h e e r r o r due t o

considering t h e values of each investment of each o b j e c t i v e a s independent

quo should not be s e r i o u s .

Llhen dealing with national policy i s s u e s , how-

e v e r , e i t h e r t h e proposed o r expected changes in o b j e c t i v e l e v e l s a r e t r i -

o r t h e de c isions a r e intended t o produce major increments i n t h e various o b j e c t i v e s . I n t h e l a t t e r c a s e t h e problem of commensuration reappears with g r e a t s i g n i f i c a n c e . The parametric approach was used by Major1' in modifying cost-benef i t a n a l y s i s f o r water r es o u r ces planning. Although t h e assumption of known a n d c o n s t an t values f o r t h e r e l a t i v e importance of t h e obje c tive s i s g e n e r a l l y n o t accep t ab l e1 2 , t h i s method can be used t o ge ne ra te points i n t h e n o n - i n f e r io r s e t by u t i l i z i n g various values of 0;t h i s was f i r s t s u g gested by E v e r et t 1 3 f o r resource a l l o c a t i o n problems. For convex problems ( t h a t i s , when t h e t r ad eo f f f u n ct i o n i s convex), t h e parametric approach g e n e r a t e s the e n t i r e n o n - i n f er i o r s e t 1 4 . Reid and V e m ~ r i ' ~found t h e t r a d e o f f f u n c ti o n f; a s a f u n ct i o n of 0 when t h e o b j e c t i v e func tions a r e posynomials, b u t Reid and Citron16 found t h a t even simple systems ge ne ra ll y g i v e i n c r e d i b l y complex f u n ct i o n s f o r f;(g). Geoffrion" developed a parametric procedure f o r f i n d i n g t h e e n t i r e non-infe rior s e t f o r t h e c a se The paraof maximizing two concave f u n ct i o n s with concave c o n s t r a i n t s . metric approach has been used iii l i n e a r problems in developing a modified simplex method t o determine t h e e n t i r e non-infe rior s e t l s . Geoffrionl' * used t h e parametric method t o f i n d t h e n o n-infe rior de c ision ve c tor 5 a s a f u n c t i o n of ~1 f o r t h e two o b j e c t i v e cas e where CY = o1 a n d 1-a=02 . The v i a l ( i n which cas e t h e a n a l y s i s i s unimportant)

utility

function

(assumed k n o w n )

could then be found a s a func tion o f

a n d then maximized t o f i n d t h e p r ef er r ed s o l u t i o n .

McGrew

01

Haimes20

and

used t h e parametric approach i t e r a t i v e l y t o converge t o t h e lexicographic p r e f e r r e d s o l u t i o n f o r t h e j o i n t i d e n t i f i c a t i o n and optimization problems. To i l l u s t r a t e t h e d i r e c t form of t h e parametric approach, consider again example 1 ,

and suppose t h e DM has decided t h a t o b j e c t i v e

times as important as o b j e c t i v e f , .

MIN

The s o l u t i o n t o t h i s

XI

+ 3(10 -

i s three

The problem then becomes: -

XI

problem i s x1

f2

=

30 -

~ 2 =)

5 , x2

2x1

5, f ,

=

=

-

3x2

5 , f,

0;

=

t h i s pre-

f e r r e d s o l u t i on i s shown as p o i n t B i n f i g u r e s 1-1-a a n d 1 - 1 - b . The parametric approach can be i n t e r p r e t e d geometrically a s follow s. The s e t L = c(x)igT = c } (where c i s a c o n s t a n t ) , de fine s a hyperplane2l .in R n with outward normal 0. The minimization o f 0T .f($ can be

-f(x)

-

t h i s hyperplane L with fixe d

viewed a s moving

2

in a negative d i r e c t i o n a s

f a r a s p o s s i b le keeping L f 7 S non-null.

This minimum w ill ge ne ra lly occur

where L i s tangent t o S ( b u t n o t always;

t h i s w ill

This i s depicted f o r t h e two o b j e c t i v e case

cause problems l a t e r ) . L

in f i g u r e 2 - 1 ;

is a

line

with slope - 02/Oi; t h e minimum f o r t h i s value of 0 occurs a t point A . For non-convex problems, when one t r i e s t o find t h e e n t i r e non-inferior set,

t h e problem of d u a l i t y gaps a r i s e s ; some points in t h e non-in-

f e r i o r s e t cannot be f i g u r e 2 - 2.

found f o r

any value of

The l i n e L which i s

tangent

0.

Consider

a t A with slope

t h e point A in

-

02/01

can be

moved f a r t h e r i n a negative d i r e c t i o n y n t i l i t i s tangent a t point B. t h e parametric

Thus

approach with t h i s value of o w i l l find points B and C , b u t

not p o i n t A. Geometrically speaking, p o i n t s i n the non-infe rior s e t which do not have a supporting hyperplane cannot be found by t h e parametric method. The

-

term d u a l i t y g a p comes from t h e f a c t t h a t t h e parametric problem MIN gT i s r e l a t e d t o t h e dual of t h e problem MIN f l ( x ) s . t . f . ( x ) $ y . j = 2 ,

f(x) 3,

..., n

J -

where t h e y . a r e c o n s t a n t s . J

J

These gaps can be explained in terms

of d u a l i t y t h e o r y 2 2 . 2.6

THE €-CONSTRAINT APPROACH The d i r e c t

form o f t h e € - c o n s t r a i n t

approach23 re quire s the DM

to

MULTIOBJECTIVE OPTIMIZATION IN WATER RESOlJRCES SYSTEMS

20

Figure 2-1

fl

Parametric Approach

I

0

f2

Figure 2-2

Duality Gaps i n t h e Parametric Approach

s p e c i f y t h e maximum a l l o w a b l e l e v e l s (c2, ives (f2,

...,f n ) ;

f3,

the preferred

c3,

...

E ~ f )o r t h e n - 1 o b j e c t -

s o l u t i o n i s t h e one which s o l v e s t h e

f o l l o w i n g problem:

s.t.

f .(x) 5 E. j = J J

... ,n

2, 3 ,

From a u t i l i t y v i e w p o i n t ,

any o b j e c t i v e c o u l d be chosen as f,.

O f course,

t h i s approach

says t h a t t h e b e n e f i t t o s o c i e t y

s t a n t as l o n g as t h e l e v e l f u l above t h i s l e v e l .

E.

f r o m o b j e c t i v e f . i s conJ i s n o t exceeded, b u t becomes i n f i n i t e l y harm-

J I n o t h e r words t h e u t i l i t y f u n c t i o n i s a d d i t i v e w i t h

.

I

- m ; f . > E . J J

T h i s approach can a l s o be roach i f

E

j

i n t e r p r e t e d i n terms o f t h e l e x i c o g r a p h i c appi s i n t e r p r e t e d as t h e s a t i s f a c t i o n l e v e l o f t h e jth o b j e c t i v e ,

and fl a s t h e l e a s t i m p o r t a n t o b j e c t i v e . Then t h e s e t yn-l o f s o l u t i o n s t o t h e n - l s t s t a g e would be i x l f . ( x ) s E ~ j, = 2 , 3 , ..., n, and E T;} J t h u s t h e E - c o n s t r a i n t approach c a n be i n t e r p r e t e d as t h e nth i t e r a t i o n i n t h e l e x i c o g r a p h i c approach.

x

The d e t e r m i n a t i o n

o f t h e maximum

l e v e l s as w e l l as t h e

assumption

a b o u t t h i s f o r m o f p r e f e r e n c e a r e o f t e n q u e s t i o n a b l e i n r e a l problems; howe v e r , t h i s method

can be used

t y p e s o f problems

by v a r y i n g t h e

t o generate

the non-inferior

values o f t h e

E.

J

,j = 2, 3,

set

for all

..., n .

Spe-

c i f i c methods f o r a c h i e v i n g t h i s w i l l be d e s c r i b e d i n l a t e r c h a p t e r s . approach has been used f o r t h e j o i n t

This

i d e n t i f i c a t i o n and o p t i m i z a t i o n p r o b -

l e m ~ ,and ~ ~ f o r w a t e r r e s o u r c e s problems w i t h l i n e a r o b j e c t i v e s and c o n s t r a i n t ~ ~ Haimes ~ . e t a126

prove t h a t t h i s

s o l u t i o n s f o r t h e two o b j e c t i v e cases.

approach does

give non-inferior

Pasternak and PassyZ7 used a com-

b i n a t i o n o f t h e p a r a m e t r i c and € - c o n s t r a i n t approaches t o f i n d t h e r e d (maAjmum programming

s o l u t i o n t o a non-convex, problem.

two o b j e c t i v e ,

Due t o t h e n o n - c o n v e x i t y ,

Prefer-

0-1 i n t e g e r

t h e parametric

approach

c o u l d n o t be used a l o n e , and t h i s mixed approach was f o u n d t o be more e f f i c i e n t t h a n u s i n g a s t r a i g h t & - c o n s t r a i n t approach.

22

MULTIOBJECTIVE OPPIMIZA TIOPJ Ifl WATER RESOURCES SYSTEMS

To i l l u s t r a t e t h e d i r e c t f o r m o f t h e € - c o n s t r a i n t approach, c o n s i d e r a g a i n example 1, and suppose t h a t t h e DM has d e c i d e d t h a t t h e maximum l e v e l s o c i e t y can t o l e r a t e o f f 2 i s 3 u n i t s . Min

The problem i s t h e n :

x1

The s o l u t i o n t o t h i s problem i s X I = 2, x2 = 5, f,

2, f, = 3.

=

This pre-

f e r r e d s o l u t i o n i s shown as p o i n t C i n f i g u r e s 1-1-a and 1-1-b. G e o m e t r i c a l l y , t h i s approach

adds a d d i t i o n a l

c o n s t r a i n t s which r e -

duce t h e f e a s i b l e d e c i s i o n space T, o r e q u i v a l e n t l y t h e f e a s i b l e f u n c t i o n a l

space S,

D e f i n e T.' J

1 5 ~ T . lf o r j = 2, 3, now

J

=

tzlf.(x) 5 J -

...,n

E

1 for j n S; ?I

=

and S' = S

2, 3,

..., n;

Sin...

then S '

nSi

j

. The

=

{f(x)

problem i s

M I N fl(x) s.t.

L(&)

&

S'

Each c o n s t r a i n t f . ( x ) E . d e f i n e s t h e h a l f - s p a c e i n Rnon t h e n e g a t i v e s i d e J J The i n t e r s e c t i o n o f a hyperplane perpendicular t o t h e f . a x i s a t f = J j 'j* o f a l l o f t h e s e h a l f - s p a c e s w i t h S g i v e s t h e new f e a s i b l e space S ' . This i s d e p i c t e d f o r t h e two o b j e c t i v e case the half-plane t o the l e f t o f the l i n e

i n f i g u r e 2-3: f2=E2.

s'

I

=

SAS2 where s ' i s

Note t h a t t h i s approach can

d e t e r m i n e t h e e n t i r e n u n - i n f e r i o r s e t even f o r non-convex problems.

2.7

GOAL PROGRAMMING The

goal programming method r e q u i r e s t h e DM t o s e t g o a l s

would l i k e each o b j e c t i v e t o a t t a i n . as t h e one w h i c h m i n i m i z e s vector o f

A

t h e d e v i a t i o n s from t h e s e t goals.

g o a l s s e t b y t h e DM f o r t h e

i c a l f o r m u l a t i o n o f the problem i s

t h a t he

p r e f e r r e d s o l u t i o n is then defined o b j e c t i v e s by

2;t h e n

Denote t h e t h e mathemat-

23

I

fi

*

f,

(€2)

0

Figure 2-3.

€2

€-Constraint Approach.

f2

24

Ml/LTTOEdECTl

s.t. 5 where 1 1 .

1I

denotes any

E

t'r

UPTlMrZATION I!/ WATER RFSOURCES SYSTEMS

T

.

Note that the goal vector f does not have

to be in the feasible set S; for example, in example 1 the goal vector could be f^ T= (0, 0). In fact, if the goal vector is in S, then this method may yield an inferior solution. This approach was developed by Charnes and Cooper29 for linear problems. Using the sum of the absolute values of the deviations as the norm, + they keep the problem lipear by defining vectors of slack variables y 3 0 + - + + and 1- 2 0 such that f(5)- f = Y -1 ; Yi (the ith component of y ) is then the over-attainment of the 'ti objective, and yi- (the i t h component o f y - ) is the under-attainment of the ith objective. The problem then becomes: MIN

n

1

+ yi + yi-

i=l s.t.

+ f(L) - f = Lv_

-

-

y

^T To illustrate this approach, consider again example 1 and assume _fl= + + + (0, 0). The problem is then MIN y1 + y 2 + Y, s.t. XI = Y1 10 - x1 + x 2 = y2 - y2 , 0 < x1 s 5, 0 s x2 6 5. Note that y1 was not necessary since f, = x 1 cannot go below 0 (fl cannot be under-attained). The soluution to this problem is that any point on the line fl + f, = 5 minimizes this objective and i s d preferred solution. A special case of this method is known as the mean square approach. This assumes that the ith component of the goal vector will be Ti, the minimum value of f.(x) s.t. 5 E T, and uses a least square norm. Salukva1 dze30 applied this method to optimal control problems; it can be shown3I that the solution to this problem is automatically in the non-inferior set. This approach eliminates the decision maker entirely and thus this definition of preferred solution will generally not maximize the benefit to society. In general, the problem with goal programming is that an equal importance is placed on each objective. If weighting factors are introduced to counteract this, then the problem of determining the weights for any I

25

SOLlJTlON ME'I'HODOL,OCIES

real problem arises. Just as in the parametric case, the non-inferior set can be generated by varying the weighting factors, but it can be shown32 that this method suffers from the same duality gap problem as the parametric approach when the problems are non-convex. 2.8

THE GOAL ATTAINMENT METHOD

A variant of the goal programming method is the goal attainment method33. In this approach, a vector o f weights g relating the relative under or over attainment of the desired goals must be determined by the DM in addition to the goal vector f. The preferred solution solves the prob1 em: MIN z

where z i s a scalar variable unrestricted in sign. This approach can also be used to generate the non-inferior set; using f a s in the mean square aps.t. x- E proach (the ith component of is T i , the minimum value o f fi(5) T), the entire non-inferior set can be found by varying g, even for nonn

1 wi = 1. This i=l approach has been successfully applied to static problems(economic dispatch in Power system control) and dynamic problems (the load-frequency control problem in regulator design). This approach is depicted for the two objective case in figure 2-4. g and f fix the direction of the vector f. + g z, and the minimum value of z occurs where this vector first intersects S. To illustrate the use of this method consider again example 1. Assume that the goal vector set by the DM is (0,0)T and that he decides the relative over-attainment o f f2 should be 1/3 the over-attainment o f f l . Then gT = ( . 7 5 , . 2 5 ) and the goal attainment problem becomes: convex problems.

Again g is generally normalized so that

Min z s.t.

XI

- .75 2 (0

26

S

n f2

F i g u r e 2-4.

Goal A t t a i n m e n t Method.

f2

27

The solution to this problem is z = 5, x 2 = 5, x1 = 15/4, f: 5/4. This is depicted as point D in figures 1-1-a and 1-1-b. 2.9

=

15/4, f2

=

ADAPTIVE SEARCH APPROACH

This approach34 is useful if one is only interested in determining non-inferior values. One starts with an initial non-inferior vector -0 x in the decision space; the ith component of % is found by solving:

MIN fi(x)

Then new solutions are generated with the following recursive formula:

where a. controls the step size, 2 is the Jacobian matrix of partial der1 illatives of the objectives with respect to the decision variables, ci controls the direction and +c controls the feasibility of the solution. Each new solution is then checkec for non-inferiority as follows; if any two of the gradients are o f opposite sign, or the point i s on the boundary of T, then it is a candidate for non-inferiority. After a large number of steps e . g . 1000) are completed, the candidates for non-inferiority are compared; those that are toc close together or inferior are e1iminated;the remaining ones should be a good approximation to the non-inferior set. At this point, a regression or interpolation can be used to determine an analytic equation; it is generally easier to accomplish this in the functional space. The drawbacks of this approach are that the computationaleffort can become immense when there are many decisions,and that no means of choosing a preferred solution from the non-inferior ones is presented.

28 2.10

MULTIOBeJECYfIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS

INTERACTIVE APPROACHES

Much work has been

done r e c e n t l y i n i n t e r a c t i v e approaches t o f i n d -

ing preferred solutions;

i n these

f i n d the preferred solution step o f the

w i t h questions

search i n order

An i n t e r a c t i v e

methods a search

t o determine

approach developed

procedure

i s used t o

b e i n g asked o f t h e

DM a t each

a new e s t i m a t e o f t h e s o l u t i o n .

by G e o f f r i o n

and examined by

Feinberg

uses i n d i f f e r e n c e f u n c t i o n s t o d e t e r m i n e t h e maximum u t i l i t y p r e f e r r e d s o l -

tio on^^.

Since the s o c i a l

utility,

t h e normal t o t h i s s u r f a c e i s t h e d i r e c t i o n o f g r e a t e s t

indifference function

o f the u t i l i t y function. by

q u e s t i o n i n g t h e DM,

i s a surface o f constant

Thus one can p i c k some i n i t i a l p o i n t

x

increase E

T, and

t h e s o c i a l i n d i f f e r e n c e f u n c t i o n around t h i s p o i n t

can be found; t h e normal t o t h i s f u n c t i o n i s t h e d i r e c t i o n t o move i n p i c k f o r &.

i n g the n e x t approximation

The s t e p s i z e i s f o u n d

by c a l c u l a t i n g

f ( x ) f o r d i f f e r e n t s t e p s i z e s ( k e e p i n g & E T) and by q u e s t i o n i n g t h e DM t o f i n d t h e one he p r e f e r s most. T e r m i n a t i o n o c c u r s when t h e improvement between s t e p s i s l e s s t h a n some s p e c i f i e d v a l u e .

The a t t r i b u t e s o f t h i s app-

o f the u t i l i t y

r o a c h a r e t h a t no assumptions a b o u t t h e f o m

function

are

necessary, and t h a t t h e DM need o n l y c o n s i d e r r e l a t i v e preference^^^. 2.11

OTHER APPROACHES

There a r e a number o f lems which

can o n l y

those i s t o model

o t h e r approaches t o m u l t i p l e

be mentioned h e r e

t h e decision-making

o b j e c t i v e prob-

due t o space l i m i t a t i o n s . process i t s e l f 3 7 ,

One o f

using techniques

such as game t h e o r y t o d e t e r m i n e t h e p r e f e r r e d s o l u t i o n ; t h e s e models, however, t e n d t o bc c o m p u t a t i o n a l l y p r o h i b i t i v e . ered the m u l t i - o b j e c t i v e

O t h e r a u t h o r s 3 8 have c o n s i d -

problem f o r t h e case

where t h e r e a r e o n l y two o r

t h r e e n o n - i n f e r i o r p o i n t s f r o m w h i c h t h e p r e f e r r e d s o l u t i o n must be chosen, developing s p e c i a l i z e d techniques f o r these s i t u a t i o n s . i s t o generate a stronger

A n o t h e r approach39

partial ordering o f the non-inferior

solutions;

t h i s method a l l o w s f o r u n c e r t a i n t i e s on t h e P a r t o f t h e DM, b u t as a r e s u l t

w i l l o n l y e l i m i n a t e some o f t h e n o n - i n f e r i o r p o i n t s . lem i s d e c i s i o n

making under u n c e r t a i n t y .

Another t y p e o f p r o b -

Much work has been done i n t h e

area40, b u t t h i s goes beyond t h e scope o f t h i s t e x t .

29 FOOTNOTES

1.

T h i s was suggested by Zadeh [1963].

2.

T h i s was shown by G e o f f r i o n [1967];

3.

Many s t u d i e s i n economic t h e o r y

a l s o see d e f i n i t i o n 4.

have been d e v o t e d t o t h i s q u e s t i o n ;

see, f o r example, Arrow [1963] o r Bergson [1954]. 4.

See F i s h b u r n [1967],

5.

A good d i s c u s s i o n of t h i s q u e s t i o n can be f o u n d i n Arrow [1963].

6.

See B r i s k i n [1966].

o r Keeney [1972] among o t h e r s .

7.

T h i s was f i r s t i n t r o d u c e d by Georgescu-Roegen [1954].

8.

There i s a r e l a t i o n s h i p between t h e l e x i c o g r a p h i c approach and u t i l -

9.

See McGrew and Haimes [1974].

10.

See W a l t z [1967].

11.

M a j o r [1969] was

i t y t h e o r y ; see Robinson and Day [1972].

the f i r s t t o

use m u l t i p l e o b j e c t i v e

analysis f o r

problems i n w a t e r r e s o u r c e s p l a n n i n g . 12.

See t h e c r i t i c i s m o f Freeman and Haveman [1970] f o r example.

13.

See E v e r e t t [1963].

14.

T h i s i s proven b y G e o f f r i o n [1968].

15.

The problem t o w h i c h R e i d

and Vemuri [1971] a p p l y

their results i s

d i s c u s s e d f u r t h e r i n Chapter 8. 16.

See R e i d and C i t r o n [1971].

17.

See G e o f f r i o n [1966].

18.

Various authors

have proposed ways o f a c c o m p l i s h i n g t h i s .

and Zeleny [1973],

Sengupta [1972],

See Yu

and Evans and S t e u e r [1972].

19.

See G e o f f r i o n [1967].

20.

See McGrew and Haimes [1974].

21.

A h y p e r p l a n e i s t h e g e n e r a l i z a t i o n o f a p l a n e i n t o n dimensions;

a

one d i m e n s i o n a l h y p e r p l a n e i s a l i n e . 22.

F o r an e x p l a n a t i o n o f

d u a l i t y gaps

see Lasdon [1968]

o r Gembicki

23.

A good d e s c r i p t i o n o f t h i s approach can be found i n Hairnes [1973b].

24.

Applications t o the

[ 19731. j o i n t i d e n t i f i c a t i o n and o p t i m i z a t i o n problems

can be f o u n d i n Hairnes and

Wismer [1972] and Olagundoye and Hairnes

[ 19731, 25.

This technique

was used by

Byers [1973]. 26.

See Haimes e t a1 [1971].

Cohon and Marks

[1973] and M i l l e r and

30 27.

See P a s t e r n a k and Passy [1972].

28.

The g e n e r a l u s e o f

29.

The o r i g i n a l development o f goal programming i s g i v e n i n Charnes and

norms i s d i s c u s s e d by

Yu

Salukvadze [1974] and

and Lietniann [1974]. Cooper [1961]. 30.

See Salukvadze [1971].

31.

T h i s i s proven by Huang [1972].

32.

The proof can be found i n Gembicki [1973].

33.

The o r i g i n a l development

o f t h e goal a t t a i n m e n t method i s g i v e n

by

Genibi c k i [ 1 9731. approach i s g i v e n i n

34.

The o r i g i n a l development of the a d a p t i v e s e a r c h

35.

Beeson and Meisel [1971]. T h i s i n t e r a c t i v e approach i s d e s c r i b e d i n d e t a i l by G e o f f r i o n [1970] and F e i n b e r g [1972].

36.

Descriptions of

37.

A nuniber of t h e s e models a r e d e s c r i b e d i n Cohon [1973] and Cochrane

other interactive

[1971], F e i n b e r g [1972],

a p p r o a c h e s can be found

in

Roy

and Cohon [1973].

and Zeleny [1973]. some o t h e r s

38.

A good example i s by Maier-Rothe and S t a n k a r d [1970]; a r e d e s c r i b e d i n MacCrimmon [1972].

39.

T h i s approach i s d e s c r i b e d i n Roy [1971] and Cohon [1973].

40.

The g e n e r a l problem i s d e s c r i b e d i n R a i f f a [1968]; o t h e r models can be found i n Cochrane and Zeleny [1973].

REFERENCES 1.

Arrow, K. J . ,

S o c i a l Choice and I n d i v i d u a l V a l u e s ,

John Wiley and

2.

Beeson, R . M . and M e i s e l , W.S., "The O p t i m i z a t i o n o f Complex Systems

S o n s , N.Y., 1963.

w i t h Respect t o M u l t i p l e C r i t e r i a , " -__ and C y b e r n e t i c s

Proceedings:

Systems,

Man

Conference, Anaheim, C a l . , 1971.

4.

"On t h e Concept o f S o c i a l W e l f a r e , " Q u a r t e r l y J o u r n a l of Economics, vol . 6 8 , 1954. B r i s k i n , L . E . , "A Method of Unifying M u l t i p l e O b j e c t i v e F u n c t i o n s , "

5.

Charnes, A . , and Cooper, W . W . ,

3.

Bergson, A.,

~Management

S c i e n c e , v o l . 12B, no. 1 0 , 1966. Management Models and I n d u s t r i a l Ap-

P J i c a t i o n o f L i n e a r Programming, v o l . 1 ,

John Wiley

and Sons,

N.Y., 1961.

6.

Cochrane, J . L., and Z e l e n y , M . , W

l

e C r i t e r i a Decision

Making,

31 of South Carolina Press, Columbia, S.C., 1973. Cohon, J. L., and 0. H . Marks, "Multiobjective Screeninq Models U.

7.

8.

9.

10.

and Water Resource Investment," Water Resources Research, vol . 9, no. 4, 1973. Cohon, J. L., "An Assessment of Multiobjective Solution Techniques for River Basin Planning Problems," Ph.D. Dissertation, M. I. T. 1973. Evans, J.R. and Steuer, R.E., "Generating Efficient Extreme Points in Linear Multiple Objective Programming: Two Algorithms and Computing Experience," Presented at Seminar on Multiple Criteria Decision Making, U. of South Carolina, 1972. Everett, H., 1 1 1 , "Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources," Operations Research, vol. 1 1 , 1963. Feinberg, A., "An Experimental Investigation of an Interactive Approach for Multi-criterion Optimization with An Application to Academic Resource Allocation," Western Management Science Institute, Working paper no. 186, 1972. Fishburn, P. C., "Methods of Estimating Additive Utilities: Manaae_- Science, vol. 13, no. 7, 1967. ment Freeman, A. M., 111, and Haveman, R. H., "Benefit-Cost Analysis and Multiple Objectives: Current Issues in Water Resources Planning," Water Resource Research, vol. 6, no. 6 , 1970. Gembicki, F., "Vector Optimization for Control with Performance and Parameter Sensitivity Indices," Ph.D. Dissertation, Case Western Reserve University, 1973. Geoffrion, A . M., "Strictly Concave Parametric Programming," Management Science, vol. 13, no. 3, 1966. Geof f rion , A.M. "Sol vi ng Bi cri terion Mathemat ical Programs," tions Research, vol. 15, no. 1 , 1967. Geoffrion, A.M., "Proper Efficiency and the Theory of Vector Maximization," Journal of Math. Anal. and Appl., vol. 22, no. 3, 1 968. Geoffrion, A.M., "Vector Maximal Decomposition Programming," Western Management Science Institute, Working paper no. 164, 1970. Georgescu-Roegen, N., "Choice, Expectations, and Measurability," Quarterly Journal of Economics, vol. 64, 1954. Haimes, Y. Y., "Integrated System Identification and Optimization," _ _ I _

11.

12.

13.

14.

15. 16.

17.

18. 19.

20.

--

32

MULTIOBJECTIVE OPTIMTZAT TON TN WATER RESOURCES SYSTEMS i n C o n t r o l and Dynamic Systems: Advances i n Theory and A p p l i c a t i o n s , v o l . 10, Academic Press, I n c . , 1973 .

21.

Haimes, Y . Y . ,

Lasdon, L.S. and Wisner, D.A.,

"On a B i c r i t e r i o n F o r -

m u l a t i o n o f t h e Problems o f I n t e g r a t e d System I d e n t i f i c a t i o n and System O p t i m i z a t i o n , " 22.

Haimes, Y . Y.,

I E E E T r a n s a c t i o n s , v o l . SMC-1, 1971.

and Wismer, D. A.,

Combined Problem o f

" A Computational Approach t o t h e

O p t i m i z a t i o n and Parameter

Identification,"

Automatica, v o l . 8, 1973. 23.

Huang, S.C.,

"Note

on t h e

O b j e c t i v e Functions,"

Mean-Square S t r a t e g y f o r V e c t o r Valued

J o u r n a l of O p t i m i z a t i o n Theory and A p p l i c a -

t i o n s , v o l . 9, no. 5, 1972. 24.

Keeney, R. L.,

25.

Lasdon,L.S.

26.

MacCrirnmon,K.R.,

" M u l t i p l i c a t i v e U t i l i t y Functions," M1.T. Operation

Research Center, T e c h n i c a l R e p o r t no. 70, 1972. " D u a l i t y and Decomposition i n M a t h e m a t i c a l Programming"

I E E E T r a n s a c t i o n s , v o l . SSC-4, no. 2, 1968. "An Overview o f M u l t i p l e O b j e c t i v e D e c i s i o n Making"

Presented a t Seminar on M u l t i p l e C r i t e r i a D e c i s i o n Making,

U. o f

South C a r o l i n a , 1972. 27.

Maier-Rothe, C. and Stankard, M.F. J r . , "A L i n e a r Programming Approach t o Choosing Between M u l t i - o b j e c t i v e A l t e r n a t i v e s , "

presented

a t t h e 7 t h M a t h e m a t i c a l Programming Symposium, t h e Hague, 1970. 28.

M a j o r , D.C.

"Benefit-Cost Ratios f o r P r o j e c t s i n M u l t i p l e Objective

I n v e s t m e n t Programs," Water Resources Research,

v o l . 5,

no. 6,

1969. 29.

McGrew, D.R.,

"A P a r a m e t r i c S o l u t i o n t o t h e J o i n t

and Haimes, Y. Y . ,

System I d e n t i f i c a t i o n and O p t i m i z a t i o n Problem," -___ m i z a t i o q Theory and A p p l i c a t i o n s ,

30.

M i l l e r , W. L.,

and Byers, D.M.,

Journal o f Opti-

v o l . 13, no. 5, 1974.

"Development

and D i s p l a y o f M u l t i -

p l e O b j e c t i v e P r o j e c t Impacts," Water Resources Research, v o l . 9, no. 1, 1973. 31.

Olagundoye,

O., and Haimes, Y . Y.,

"The E p s i l o n - C o n s t r a i n t Approach

f o r S o l v i n g B i c r i t e r i o n Programs,"

SRC T e c h n i c a l Report.

CWRU,

1973. 32.

Pasternak, H.,

and Passy, Y . ,

t e r i o n Functions," 33.

R a i f f a , H.,

34.

Reid, R. W . ,

"Annual A c t i v i t y P l a n n i n g w i t h B i c r i -

Technion mimeograph s e r i e s no. 110, 1972.

D e c i s i o n A n a l y s i s , Addison-Wesley, and C i t r o n , S. J.,

Reading, Mass, 1968.

"On N o n - I n f e r i o r Performance Index

V e c t o r s , " J o u r n a l o f O p t i m i z a t i o n The0r.y and A p p l i c a t i o n s , v o l . 7 , no. 1, 1971.

35.

Reid, R.

W., and Vemuri, V . ,

"On t h e N o n - I n f e r i o r Index Approach t o

Large S c a l e Mu1 t i - C r i t e r i a Systems," J o u r n a l o f t h e F r a n k l i n I n s t i t u t e , v o l . 291, no. 4 , 1971. 36.

Robinson, S.M.,

and Day, R.H.,

"Economic D e c i s i o n s w i t h L

**

Utility"

S o c i a l Systems Research I n s t i t u t e , U . o f Wisconsin, Paper no.7227 1972. 37.

Roy, B.,

38.

Salukvadze, M . E .

"Problems and

Methods w i t h M u l t i p l e O b j e c t i v e F u n c t i o n s , "

Mathematical Programming, v o l . 1, no. 2, 1971. "Optimization o f Vector Functionals:

ming o f Optimal T r a j e c t o r i e s , "

Tbilisi,

The Program-

t r a n s l a t e d from

Avto-

m a t i k a i Telemekhanika, no. 8, 1971. 39.

Salukvadze, M . E .

"On t h e E x i s t e n c e o f S o l u t i o n s i n Problems o f Op-

t i m i z a t i o n under V e c t o r Valued C r i t e r i a , "

Journal

o f Optimiza-

t i o n Theory and A p p l i c a t i o n , v o l . 13, no. 2, 1974. 40.

Sengupta, S. S., Programmes,"

" P r o b a b i l i t i e s o f Optima i n M u l t i - O b j e c t i v e L i n e a r p r e s e n t e d a t Seminar on M u l t i p l e C r i t e r i a

Decision

Making a t U. o f South C a r o l i n a , 1972. 41.

Waltz, F.

M.,

Criteria," 42.

Yu, P.L.

"An E n g i n e e r i n g Approach: H i e r a r c h i c a l O p t i m i z a t i o n I E E E T r a n s a c t i o n s , v o l . AC-12, 1967.

and Leitmann, G . ,

"Compromise S o l u t i o n s , Domination S t r u c -

t u r e s , and S a l u k v a d z e ' s S o l u t i o n , " J o u r n a l o f O p t i m i z a t i o n Theory and A p p l i c a t i o n s , v o l . 13, no. 3, 1974. 43.

Yu, P. L . , and Zeleny, M., t h e L i n e a r Cases and a

"The S e t o f a l l Non-Dominated S o l u t i o n s i n Multi-Criteria

Simplex Method,"

U.

of

Rochester, Center f o r System Science, 1973. 44.

Zadeh, L. A . ,

" O p t i m a l i t y and Non-Scalar Valued Performance C r i t e r i a "

I E E E T r a n s a c t i o n s , v o l . AC-8,

no. 1, 1963.

Chaoter 3

THE SURROGATE WORTH TRADE-OFF METHOD TIi? S[i~?‘lOgCLtP k h f h i/rude-066 hje,tbd

recognizes t h a t

optimization

theory i s u s u al l y much more concerned with the r e l a t i v e value of a dditiona l increments o f t h e various

non-commensurable o b j e c t i v e s ,

o f each o b j e c t i v e f u n c t i o n ,

a t a given

value

Fur-

then i t i s w j t h t h e i r a bsolute va lue s.’

thermore, given any c u r r e n t s e t o f o b j e c t i v e

levels attained,

e a s i e r f o r t h e d eci s i o n makers t o a s s e s s the r e l a t i v e value of

it

i s much

the

tra de -

between any two

objectives t h a n I n a d d i t i o n , the optimiz a t i o n procedure can be developed so t h a t i t r e q u i r e s no more t h a n a n a s s essment of whether a n ad d i t i o n al q u a n t i t y o f one o b j e c t i v e i s worth more or l e s s than t h a t which may be l o s t from anothe r, given t h e l e v e l s o f each. The o r d i n a l approach can then be u t i l i z e d with much l e s s concern f o r the p o t e n t i a l d i s t o r t i o n s i n r e l a t i v e ev al u ations introduced by attempting t o com:nensurate t h e t o t a l value o f a l l o b j e c t i v e s concerned. Since t h e dimension of t h e d eci sion space N f o r most re a l world problenis i s g en er al l y higher than t h a t o f the functional space n ( N dec i s i o n s and n o b j e c t i v e s M >>n), a f u r t h e r s i m p l i f i c a t i o n i s t o e s t a b l i s h d e c i si o n s i n t h e functional sDace and then l a t e r transform t h i s information i n t o t h e d e c i si o n space. o f f o f marginal

i n cr eas es a n d

i t i s f o r them t o a s s e s s

3.1

decreases

t h e i r ab s o l u t e va lue s.

G E N E R A L APPROACH

The bas i c concept of t h e Surrogate Worth Trade-off method f o r

non-

commensurate m u l t i - o b j e c t i v e o p t i mi zat i o n w ill be explained through a simp l i f i e d exa:nple of commensurate mu l t i - o b j e c tive optim iz a tion. unconstrained, two o b j e c t i v e , one d eci s i o n v a r i a b l e

optimization

in which b o t h o b j e c t i v e s a r e measurable i n t h e same u n i t s , val ue. min f l ( x ) + f 2 ( x ) Applying t h e c l a s s i c a l c a l c u l u s o ptim iz a tion

34

Consider a n problem

e . g . , monetary

Thus d f l / d f 2

=

- 1

defines o p t i m a l i t y

s a r y and s u f f i c i e n t c o n d i t i o n s and t e s t s .

s u b j e c t t o t h e usual neces-

I t w i l l be

noted

i s the trade-off

ratio

trade-off r a t i o

a t o p t i m a l i t y m u s t e q u a l m i n u s one when

f u l l y coinmensurate u n i t s . small increment i n i s significant.

fl

that

between o b j e c t i v e one and o b j e c t i v e two, Note t h a t i t i s t h i s

dfl/df2 hence t h e

fl and f 2 a r e i n

r a t i o o f the value o f the

f2 t h a t

t o t h e value o f t h e r e s u l t i n g increment i n

E x c e p t t o t h e e x t e n t t h a t t h e s e i n c r e m e n t a l v a l u e s depend

fl and

upon t h e a t t a i n e d l e v e l o f b o t h o b j e c t i v e s , a b s o l u t e v a l u e s o f do n o t a p p e a r i n t h e o p t i m a l i t y e q u a t i o n .

The c o n c e r n i s f o r t h e

f2 relative

v a l u e o f t h e increments, g i v e n an a t t a i n e d l e v e l o f achievement o f b o t h obj e c t i v e s ( w h e t h e r o r n o t t h e y have t h e same u n i t s o f m e a s u r e m e n t ) .

fl and f 2 b e measured i n d i f f e r e n t u n i t s

Next l e t e.g.,

from a r i v e r - r e s e r v o i r

f i r m w a t e r and f i r m e n e r g y

t h i s case dfl/df2

o r dimensions,

operations.

In

d e f i n e s t h e t r a d e - o f f r a t i o T12. A t o p t i m a l i t y , t h e com-

m e n s u r a t e d v a l u e o f T12 m u s t e q u a l m i n u s one.

tion o f objectives

f, and f 2

of

f

2

are i n d i f -

the true

a t t h e known a t t a i n e d l e v e l s o f

.

(but

t o t h e t r u e ( b u t unknown) p e r

unknown) p e r u n i t w o r t h o f a n y i n c r e m e n t A f l u n i t worth o f any increment A f 2

S i n c e fl a n d

!di2 , t h e r a t i o

f e r e n t u n i t s l e t T12 be m u l t i p l i e d by

satisfac-

I f i t c o u l d be d e t e r m i n e d , t h i s r a t i o

Wi2

w o u l d b e t h e w o r t h c o e f f i c i e n t f o r t h e t r a d e - o f f T12. By d e f i n i t i o n , i n a non-commensurate p r o b l e m mined f o r a l l values o f e i t h e r space); otherwise t h e o b j e c t i v e s

x

W!.

c a n n o t be d e t e r -

1J

( d e c i s i o n space) o r fl and f 2 ( o b j e c t i v e c o u l d b e commensurated

and s t a n d a r d

op-

t i m i z a t i o n techniques applied. However, c o n s i d e r a " s u r r o g a t e w o r t h f u n c t i o n " W12, the following properties.

F i r s t , i t has a p o s i t i v e v a l u e

maker c o n s i d e r s t h a t t h e t r u e w o r t h of a f l of

Af2

.

i s greater

which r e s u l t s i n i n d i f f e r e n c e ;

assign t h e value o f zero

a b l e i n f o r m a t i o n , cannot determine whether t h e

o r d i n a l sense.

WI2

Afl

Using o r d i n a r y

In

i n c r e m e n t a l g a i n i n one obl o s s i n another.

and i m p l e m e n t a t i o n

Finally in

an

+5 r e p r e s e n t s a s t r o n g e r f e e l i n g t h a t

i s greater than

The s u r r o g a t e w o r t h f u n c t i o n i t s construction

i s true.

t o any d e c i s i o n

i s t h a t i t i s monotonically consistent

That i s , a value o f

the true worth o f

worth

t h a t i s , t h e d e c i s i o n maker, w i t h t h e a v a i l -

j e c t i v e i s o r i s n o t p r e f e r a b l e t o t h e necessary the t h i r d property o f

i f the decision

than t h e t r u e

Second, i t has a n e g a t i v e v a l u e i f t h e o p p o s i t e

combination these two p r o p e r t i e s

which possesses

Af2

t h a n does a v a l u e o f 1-3.

now has a l l t h e p r o p e r t i e s needed f o r i n finding the

preferred

slope i n t e r c e p t o r curve f i t t i n g procedures

solution

.

f o r successive

approximation, t h e zero of t h i s p a r t i c u l a r surroga te worth quickly found.

By d e f i n i t i o n of t h e

v a l e n t t o marginal l o s s equal

function can be

zero va lue , such a solution i s equi-

t o marginal

ga in, hence the followipg d e f i -

n i t i o n of a p r ef er r ed s o l u t i o n . Definition 3 - l : A p r ef er r ed s o l u t i o n i s defined t o be any non-infe rior f e a s i b l p s o l u t i o n which belongs t o t h e i n d i f f e r e n c e band. To sunimarize t h e concept behind t h e SWT method, function i s s u b s t i t u t e d f o r t h e t r u e

( b u t unknown)

surroga te

worth

worth function

which

J.

( i f k n o w n ) would commensurate cal t r a d e o f f r a t i o s

T..

1J

.

t h e numerator and denominator of t h e physiThe s u r r o g at e has the property of monotonicity

a n d a value of zero ( o r o t h e r

a r b i t r a r y number) whenever the value of

numerator of t h e t r a d e o f f r a t i o s

equals

t h e value

of t h e

the

denominator.

Thus when the s u r r o g at e worth f u n ct i o n has a value of z e ro, the corresponding s o l u t i o n i s within t h e band of i n d i f f e r e n c e and no o t h e r solution

can

be judged s u p er i o r t o i t . I n p r a c t i c e a l l t h a t i s required of t h e de c ision maker i s t o d e t e r mine whether o r not a n incremental gain i s worth the corresponding increment a l l o s s f o r any t r ad eo f f T . . , a n d i f n o t , which i s greater.Computationa1 1J e f f i c i e n c y i s gained i f he a l s o es t i mat es how f a r from equal (or i n d i s t i n g u i sh a b l e ) t h e worth of t h e proposed r e s u l t s a r e . Worth need be evaluated only in t h e r e l a t i v e s en s e, e . g . , whether

Af.

1

A f . in worth J

and

the

improvement between s u cces s i v e t r i a l s i s " l a r g e " o r " sm a ll" . The computational procedure may be executed in de c ision space,

in

o b j e c t i v e space o r in t r ad e- o f f r a t i o space. Since o b j e c t i v e s a r e normally f a r fewer i n number than d eci s i o n v a r i a b l e s , i t w ill usua lly be p r e f e r a b l e

t o work i n o b j e c t i v e or t r ad e- o f f r a t i o space.

Usually t h e number of

j e c t i v e s t o be considered simultaneously a r e of the orde r of

10 or

ob-

less,

while t h e number of d eci s i o n v a r i a b l e s may be of the orde r of a thousand. k i s r e s u l t s i n a decided computational advantage f o r the .use o f o b j e c t i v e space.

I t i s a l s o more r e d l i s t i c f o r i n t e r a c t i o n s with the de c ision maker.

He makes h i s judgement on t h e b as i s of one tra de -off r a t i o a t a time, given the corresponding l e v e l s of attainment of a l l of the o b j e c t i v e s . When t h e d eci s i o n v ect o r a r e t h r e e o r more non-commensurate s t i l l apply.

x

has a g r e a t many

components and t h e r e

o b j e c t i v e s t h e concepts described above

Methods wi l l be developed i n c ha pte rs s i x and seven f o r com-

puting t h e t r a d e- o f f r a t i o s s i g n i f i c a n t computational c o s t .

t o the a n a l y s i s with a

minimum of

The approach t o t h e computations in t h e general problem in subsequent s e c t i o n s

begins by f i n d i ng t h e

t i v e function s u b j e c t t o t h e ith

described

minimum value of each objec-

system of c o n s t r a i n t s .

The minimum of

the

o b j e c t i v e f u n ct i o n , ignoring a l l o t h e r ( n - 1 ) o b j e c t i v e s , i s determined

a s i n se c t i o n 1 . 4 and denoted by

F i . The next s t e p i s t o formulate the mul-

t i o b j e c t i v e problem i n t h e € - c o n s t r a i n t form ( a s discussed in se c tion 2 . 6 ) ; the rnaxinium t o l e r a b l e l e v e l s E . wi l l be r e l a t e d t o 7 . i n the next sec3

J

tion. By considering one o b j e c t i v e f u nc tion a s primary a n d a l l o t h e r s

as

c o n s t r a i n t s a t minimum s a t i s f i c i n g l e v e l s , t h e Lagrange m u l t i p l i e r s r e l a t e d

t o the o t h e r ( n - 1 ) o b j e c t i v e s wi l l be zero o r non-zero. If non-zero, p a r t i c u l a r c o n s t r a i n t does l i m i t t h e optimum. I t w ill be shown t h a t

that non-

zero Lagrange m u l t i p l i e r s correspond t o the non-infe rior s e t of s o l u t i o n s , while t h e zero Lagrange m u l t i p l i e r s correspond t o t h e i n f e r i o r s e t of solutions.

Furthermore, the s e t of non-zero Lagrange m u l t i p l i e r s re pre se nt t h e

s e t o f t r a d e -o f f r a t i o s between t h e p r i n cipa l o b j e c t i v e a n d each of thecons t r a i n i n q o b j e c t i v e s r e s p e c t i v e l y . C l ea rly, the se Lagrange m u 1 t i p 1 i e r s a r e f u n c t i o n s o f t h e optimal l ev el a t t a i n e d by t h e princ ipa l o b j e c t i v e function a s well a s of t h e l e v e l s of a l l o b j e c t i v e s s a t i s f i e d a s e q u a l i t y (binding) constraints.

Consequently, t h e s e Lagrange mu1 t i p l i e r s form

a

matrix

of

t r a d e - o f f r a t e f u n ct i o n s . 3.2

THE ~ ~ DERIVATION _ _ _ OF _ THE TRADE-OFF RATE FUhlCTIOJ

Given t h e mu l t i o b j ect i v e problem posed in problem 1-1, t h e t r a d e off r a t e f u n ct i o n between t h e i t h a n d j t h func tions denoted by T . . ( x ) i s 1J

defined a s follows:

where

or equivalently, T..

1J

The f u n ct i o n s T. . ( x ) have t h e property t h a t 1J -

-

38

Tij(x)

=

1

,

for i = j

and

Plore will be s a i d on t h e p r o p e r t i e s of t h e tra de -off r a t e func tions in sut)seqjent s e c t i o n s .

T..(x) 1J -

The d e r i v a t i o n and determination of t h e func tions

T . . ( x ) i s of primary importance i n t h e SWT method. The d i r e c t u t i l i z a t i o n 1J of eqiiation (1 ) however, i s c l e a r l y impractical and computationally prohibitive.

Thus, an a l t e r n a t i v e approach must be sought.

The

d u a l i t y awl Lagrange m u l t i p l i e r s as well as t h e t - c o n s t r a i n t

concept approach

of are

u t i l i z e d i n subsequent s e c t i o n s of t h i s book t o provide both the inform a tion needed and a b a s i s f o r co n s t r u ct i n g t h e t r ade -off r a t e m a trix.2 The following development shows t h a t t h e tra de -off r a t e func tions can be found from t h e values of t h e dual v a r i a b l e s a ssoc ia te d with t h e cons t r a i n t s i n a reformulated problem. Reformulate t h e system in problem 1-1 a s follow s: s u bje c t t o

min f l ( x ) X -

Problem 3-1:

f.(x) J --

, j = 2 , 3 , ..., n

tj

and gk(&

5 0

;

k = 1,2,

..., m

f . , the

minimum value

where

of

the

j t h o b j e c t i v e when a l l o t h e r o b j e c t i v e s a r e ignored ( s e e s e c t i o n 1 . 4 ) ,

Note t h a t

and

F .

J

i s defined i n terms of

J

t h a t t h e F . a r e t h e d e v i a t i o n s from t h i s minimum value. J

t h e value

t.

J

Thus i . re pre se nts J

i n t h e o b j e c t i v e space whose f . a x i s i s s h i f t e d t o i J

vdlues of F . wi l l be varied p ar amet r i cal l y in t h e process J t h e trade-off f u n c t i o n s . Form t he g en er al i zed Lagrangian, L , t o problem 3-1:

j '

of

The

c onstruc ting

where

k

tlk,

=

rn

n

1 , 2 , . . . , m , and

llj, j = 2,3,

...,

n

are

generalized

The s u b s c r i p t l j i n X denotes t h a t h i s t h e Lagrange

Lagrange r i u l t i p l i e r s .

m u l t i p l i e r associated ( i n t h e c - c o n s t r a i n t vector o p t i m i z a t i o n prob1em)with t h e f i r s t o b j e c t i v e and t h e jth c o n s t r a i n t . i

Ij

The

Lagrange

multiplier

w i l l be s u b s e q u e n t l y g e n e r a l i z e d t o be h . . a s s o c i a t e d w i t h t h e ithob1J

j e c t i v e f u n c t i o n and t h e jth c o n s t r a i n t . I n order t o l i m i t the derivation t h i s book, d e n o t e by

x

of

...,

N, w h i c h s a t i s f y t h e

Similarly, l e t

n be t h e s e t o f a l l

t h e s e t o f a l l xi,

Kuhn-Tucker c o n d i t i o n s i n problem 3-1.

t h e Kuhn-Tucker c o n d i t i o n s i n

i

=

1,2,

Lagrange m u l t i p l i e r s w h i c h s a t i s f y t h e Kuhn-Tucker c o n d i t i o n s . For s t a t i o n -

x,

A . ( k = 1,2, ..., m; j = 2,3, i J k , and 15 Tucker3 c o n d i t i o n s o f i n t e r e s t t o o u r a n a l y s i s a r e :

a r y values o f

h

1 j ( f J. ( -x ) -

lj C l e a r l y , equation

-

E.)

J

0 ,

0 , j

=

...,

2,3,

j =

(3a) holds o n l y i f

h

=

.

...,

...,

2,3,

n ) , t h e Kuhn

n

(3a)

n

(3b)

f . ( x ) - F . = 0, o r b o t h . J J j = 2,3, ..., n then t h e

= 0, o r

15 f o r any

Note, however t h a t i f f . ( x ) - E . < 0 J J = 0 . For t h e case where t h e jth c o n s t r a i n t i s i n a c t i v e corresponding h 1j ( n o t b i n d i n g ) , t h e c o r r e s p o n d i n g Lagrange M u l t i p l i e r ( d u a l v a r i a b l e o r shadow p r i c e ) i s i d e n t i c a l l y z e r o .

The s e t o f i n a c t i v e

s t r a i n t s associated w i t h a s p e c i f i c value o f {j : 5

=

E

x

Denote t h e s e t o f a c t i v e ( b i n d i n g value o f

by

r. J

A(

‘j

)

=

‘j

(non-binding)

w i l l be denoted by

I(

...,

n)

; f .J ( x-) - ‘j < 0 ; j = 2,3,

c o n s t r a i n t s associated w i t h

a

con-

) “j

specific

A(tj):

ij

: x ~ x ; f X- .) J

-

€ .

J

=

0 ;j

=

2,3,

...,

From e q u a t i o n ( 3 a ) , i t i s c l e a r t h a t a l l A . c o r r e s p o n d i n g t o 13 I n a d d i t i o n , a l l x . corresponding t o

nl

for jcI(c.) J f o r jcA(e.) 1J J J a r e n o n - n e g a t i v e and n o t n e c e s s a r i l y z e r o . Denote t h e s e h . by A . ( A ( F ~ ) ) . 1J 15

a r e i d e n t i c a l l y zero.

E.

J

E.

. ( A ( E . ) ) , j = 2 , 3 . . . , n i s of spe c ia l 1J J s i n c e i t i n d i c a t e s t h e marginal b e n e f i t ( c o s t ) of the o b j e c t i v e The value of

A

due t o a n ad d i t i o n al u n i t of i s derived'+:

fl($

Note, however, t h a t f o r y

.

E.

J

t

func tion

From equation ( 2 ) , t h e following

xl j

x and

interest

E

n, u k €

ci

, for all

j

and

k,

fl(x)

=

L

Also note t h a t f o r a l l

A l j ( A ( ~ j ) )f,. ( x ) = J these c o n s t r a i n t s a r e a c t i v e ) . Therefore:

cj

, j

= 2,3,

..., n

(since

C l e a r l y , t h i s equation can be generalized where the index of performance i s t h e i t h o b j e c t i v e function of problem 1 - 1 r a t h e r than the f i r s t one. t h i s case t h e index i should r ep l ace t h e index 1 in x yie lding X... I j 1J Accordingly: Aij(A(~j))

i

#

,j

-

=

afi ( 5 ) (4)

i,j = 1,2,

;

In

..., n

af. ( x ) Thus

T.. 1J

or

1 - can be found by c a l c u l a t i n g - A . . which i s af.(x)

1J

J -

obtainable from t h e o v er al l system Lagrangian a s w ill be discussed quently.

I t is

important

to

note

that

the tra de -off

A ~ ~ ( A ( L i~s ) ap ) p l i cab l e t o any noncommensurable func tions. l e t t h e u n i t s of f i ( x ) be $ , and t h e u n i t s of ( d i ss o l v e d oxygen).

f.(x) J -

be

subse-

rate

func tion For example ,

pounds

of

DO

Then the u n i t s o f A.. are $/DO.

Equation ( 4 ) i s v al i d

1J

for all

r a t i o i s v a l i d only when t h e j t h

A . .(A(€.)); i.e.,

the tra de -off 1J J c o n s t r a i n t i s a c t i v e ( b i n d i n g ) . I t can be

41 shown t h a t a d i r e c t correspondence e x i s t s between x . . A ( € . ) ) (A.. a s s o c i 1J J 1J a t e d with t h e a c t i v e c o n s t r a i n t s ) and t h e non-infe rior s e t t o problem 1 - 1 , A i j ( I ( ~ j ) ) ( A , . as s o ci at ed with t h e i n a c t ve c o n s t r a i n t s ) a n d 1J Consider t h e cas e where f o r some t . t h e corresponding J A . . i s zero, i . e . , h . . = ~ . . ( I ( E . ) ) Except . f o r t h e degenerate c a s e , t h i s 1J 1J 1J J means t h a t t h e r e i s no improvement in t h e o b j e c t i v e func tion f i ( x ) even a t

a n d between

the inferior s e t .

t h e expense of f u r t h e r degradation of t h e o b j e c t i v e f . ( x ) . This s o l u t i o n J c l e a r l y belongs t o t h e i n f e r i o r s e t . The degenerate c a se where theLagrange riiultiplier corresponding t o an a c t i v e c o n s t r a i n t i s zero has been s t u d i e d 5 . i s defined here The s o l u t i o n corresponding t o such ~ . . ( A ( E . )= ) 0 1J J t o be a s s o c i a ted with t h e i n f e r i o r s e t ; i . e . , degenerate s o l u t i o n s a r e considered a s i nf er i o r s o l u t i o n s . Consider next t h e cas e where f o r some

the corresponding A . . = J 1J 0 , i . e . , t h e r e i s a degradation i n t h e

This means t h a t A . . 1J j t h o b j e c t i v e f u n ct i o n f o r a n improvement in A ~ ~ ( Aj)). ( I

since

A,.

ior s e t .

1J

=

C .

the

ith

o b j e c t i v e func tion

df.(X) - --l-a f ,(iT . Thus. t h i s s o l u t i o n corresponds t o t h e non-inferJ -

I n sumrnary, s i n c e only t h e n o n - i n fe rior s o l u t i o n s a r e of i n t e r e s t , only > , . . ( A ( JJ

' j

))

;'

0

need be considered.

For s i m p l i c i t y in n o t a t i o n ,

l e t t e r A, i n d icat i n g a c t i v e c o n s t r a i n t , will be ).

dropped

and t h e

the

symbol

. . ( 'j ) w i l l be used h e r e a f t e r .

i j

The p o s s i b l e ex i s t en ce

of a d u a l i t y gap6

and i t s e f f e c t on

the

SNT method i s discussed i n ch ap t er 4. Note t h a t even i f a d u a l i t y g a p does e x i s t , t h e E- co n s t r ai n t method s t i l l ge ne ra te s a l l non-infe rior solu-

tions.

However, a given value of t h e t r ade -off r a t e func tion

X..

1J

, may

corresoond t o more than one n o n - i n f er i o r s o l u t i o n . 3.3

COMPUTATIONAL PROCEDURE FOR CONSTRUCTING THE TRADE-OFF FUNCTION I n t h i s s e c t i o n , a p o s s i b l e approach i s presented f o r ge ne ra ting

t h e t r a d e - o f f r a t i o s T . . = d f i / a f . b y c a l c u l a t i n g the Lagrange m u l t i p l i 1J J e r s X . . t o problem 3-1. F i r s t A 1 2 wi l l be found a s a func tion of E~ . 1J The system given by problem 3-1 i s solved f o r k values of r 2 , 1 2 K say 1 2 , t 2 , ..., t 2 , where a l l o t h e r E . , j = 3 , 4 , . . . , n a r e held J

f i x e d a t some l ev el k = 1,3,

...,

K,

~0

J

.

Of co u r s e,

only

the

positive

k

h12(t2)

,

a r e of i n t e r e s t (corresponding t o t h e non-infe rior solu-

K t i o n s ) . S i n c e t h e s e X12(c2)

are positive,

...,

( a c t i v e c c l n s t r a i n t s ) , f o r k = 1,2,

(k

=

...,

1,2,

tional value o f

K)

k

i t must be t h a t

K.

f2(x)

k

=

c2

k

each v a l u e o f f 2 ( $

Thus, f o r

where t h e c o n s t r a i n t s a r e a c t i v e , a c o r r e s p o n d i n g f u n c -

n12(fi(x)) ( k

=

...,

1,2,

K) i s generated. A t t h i s

a r e n r e s s i o n a n a l y s i s may be p e r f o r m e d i f d e s i r e d , t o y i e l d a

least

a r e s a p p r o x i m a t i o n t o t h e f u n c t i o n h 1 2 ( f 2 ( x ) ) ( f i g u r e 3-1 d e p i c t s a r a t i c f u n c t i o n f i t by r e g r e s s i o n ) . a functio,. o f the values o f s i t i v e t o these levels o f

,j

t.

J

,

t.

J

sququad-

N o t e , however, t h a t h 1 2 ( f 2 ( x ) ) i s a l s o =

3,4, ..., n.

I f t h e f u n c t i o n i s sen-

..., n , then a m u l t i p l e r e g r e s s i o n

j = 3,4,

I t w i l l b e shown i n C h a p t e r 4

a n a l . v s i s
stage

that often

the

t r a d e - o f f r a t e f u n c t i o n h . . need be c o n s t r u c t e d o n l y i n t h e v i c i n i t y o f t h e 1J i n d i - f f e r e : i c e band. I n t e r p o l a t i o n a n d c u r v e f i t t i n g p r o c e d u r e s c a n a l s o be u s e d t o a : p r o x i m a t e A . . n e a r t h e i n d i f f e r e n c e band. This 1J computati?nal e f f o r t involved i n t h e regression a n a l y s i s . Similarly, the trade-off r a t e function agairi t h e prime o b j e c t i v e f u n c t i o n i s f l ( x ) , K d i f f e r e n t values o f i ,

0

f 2 ,t 4 , . . . ,

=

2,3,

t.3\

c-; . This

, k

. . . , n,

=

and p r o b l e m 3 - 1 i s s o l v e d f o r with

i s repeated

a

fixed

1,2,

level

of

t o generate the trade-off

4,5, . . . , n. Once a l l t r a d e - o f f r a t e f u n c t i o n s

have k e n g e n e r a t e d , t h e p r i n c e o b j e c t i v e

and t h u s a l l t r a d e - o f f r a t e f u n c t i o n s X . . , i # j , j = 1J c a n be g e n e r a t e d . t o the

the

can be generated, where

1,2, ..., K ,

technique

for j =

r a t e f u n c t i o n s Xlj, Ilj, j

I.

minimizes

it11

s changed

,2,

. . . ,n

When a l l o f t h e f u n c t i o n s X . . h a v e been d e t e r m i n e d f o r a l l i, j = 1J into

. . . , n, i # j , i t i s sometimes d e s i r a b l e t o t r a n s f o r m t h e h . .

func':ions o f t h e d e c i s i o n v a r i a b l e s

x

1.l E

X.

T h i s t r a n s f o r m a t i o n e x i s t s and

c a n be shown t o b e u n i q u e w i t h c e r t a i n assumed p r o p e r t i e s o n t h e f . ( x ) and J , j = 1,2, ..., n . \..('.(x)) 1J J Exarnpl e : Assume t h a t f ( x )

i s q i v e n as l i n e a r f u n c t i o n o f

2-

f2(x)

and t h a t

h12(f2(x))

=

f

a 2 x 2 + a3

i s given as a qlradratic function o f f2(X):

X12(f2(x)) Then A 1 2

alx,

x:

=

blf2(x)

+

can be o b t a i n e d as a f u n c t i o n o f

b2(f2(x))2

x as

+

b3

follows:

43

Regression F i t t i n g of A . . ( f . ( x ) ) a s a function of f . ( X )

Figure 3-1.

1J

x 1 2( x )

+ c2x2 + c 3x 1 x 2

= C,Xl

~

(xl

+

J

'4'1

J -

-

2 +

'5'2

2 + C

6

x

T h e o r e t i c a l l y t h e following r e l a t i o n s should hold:

For computational advantage t h i s property can be used

t o check t h e consis-

tenc,y of t h e regression functions and t o ensure t h a t no e x c e s s i v e r e g r e s s i o n functiondl deviations occur. 3.4

THE SURROGATE WORTH FUNCTION Assume t h a t a p a r t i c u l a r value of t h e trade-off r a t i o

between two o b j e c t i v e s

fi(x)

and f . ( x ) J

--

A , . , i f j, ' J has been computed. If t h i s p a r t i -

44

i”

CP

1

- -

x

c u l a r value Nere by chance t h e p r ef er r ed s o l u t i o n , then t o t h e de c ision maker,

> . . u n i t s of f . ( x ) would be ex act ly equal in 1J

value

1 -

t o one u n i t of

The term p r ef er r ed s o l u t i o n was defined t o be any non-infe rior f e a s i b l e s o l u t i o n belonging t o a subset of the non-infe rior s e t - t h e f.(x-). J

-

indiffereiice band

where t h e worth of an improvement in one o b j e c t i v e i s

equivalent t o t h e corresponding degradation in a nothe r.

Since the a t t a i n e d

l e v e l s of a l l o b j e c t i v e s a r e known, i t should be r e l a t i v e l y simple t o

ans-

I s t h e given marginal change ( A . . ) in t h e i t h o b j e c t i v e 1J function ( f i ( x ) ) worth more or l e s s t h a n a one u n i t change in t h e j t h obj e c t i v e f u n c t io n ( f . ( x ) ) ? I f t h e f i r s t a l t e r n a t i v e i s believed t o be t r u e , J -then o b j e c t i v e f i ( x ) should be decreased a t t h e expense of f . ( x ) , a n d vic e J versa. I n e i t h e r cas e t h e d e s i r e i s t o l o c a t e the p a r t i c u l a r s e t of tra de off r a t i o s h . . which a r e simultaneously n eutra l t o t h i s que stion. 1J A s u r r o g at e worth f u n ct i o n W . . i # j ; j = 1 , 2 , . . . , n, can be TJ’ h.. e stim a ting t h e d e s i r a b i l i t y of defined a s a n y monotonic f u n ct i o n of 1J For example, t h e s c a l e could range from - 10 t o + l o , the t r a d e - o f f h . .. TJ with a -10 i n di cat i n g t h a t h . . marginal u n i t s of o b j e c t i v e i a r e muchless wer t h i s q u e s t i o n :

1J

d e s i r a b l e t h a n an ad d i t i o n al u n i t of j , a +10 i n d i c a t i n g t h e o p p o s i t e , a n d

a zero s i g n i f y i n g a n even t r a d e ( i . e . , belongs t o t h e i n d i f f e r e n c e band) A s i m i l a r s u r r o g at e worth f u n ct i o n can be defined f o r each tra de -off r a t i o . The preferred s o l u t i o n i s where t h e s e l e c t e d tra de -off r a t i o s make t h e s u r r o g a t e w o r t h fu n ct i o n s simultaneously equal t o 0 ( o r such othe r number a s may be designated a s t h e measure of an even t r a d e ) . In p r a c t i c e t h e r e wi l l u s u al l y be a band of i n d i f f e r e n c e o f a t t a i n e d o b j e c t i v e s near any p r ef er r ed s o l u t i o n thus i d e n t i f i e d . That i s , t h e r e w i l l be a range of values of

f i and

f. J

over which t h e

de c ision maker

would

i s worth more o r l e s s than Af. . BY 1 J approaching t h e zero using only t h e p o s i t i v e limb of t h e surroga te worth function one bound of t h e band of i n d i f f e r e n c e i s e s t a b l i s h e d . By approach-

n o t be prepared t o a s s e r t t h a t Af.

ing zero using only t h e negative limb, t h e opposite

bound

is

determined,

thus e s t a b l i s h i n g t h e band of i n d i f f e r e n c e . Any s u r r o g at e wcrth f u n ct i o n can be used so l o n g a s i t has a spe c if i e d value a t i n d i f f e r e n c e ( e . g . , zero) and i s monotonic. For example, function could a r b i t r a r i l y be assigned a value o f

+10

the

for the f i r s t t r i a l

non-inferior s o l u t i o n i f t h e worth of Afi

worth of

Afi

c’

Af. J

.

> Af., a n d a value of -10 i f t h e J By maintaining monotonic c onsiste nc y, the form of a

s a t i s f a c t o r y su r r o g at e worth f u n ct i o n can be determined b,y successive t r i a l s of non-inferior s o l u t i o n s .

I n particular,

e a r l i e r t r i a l s may be used

e s t i m a t e where t h e zero might be t o a c c e l e r a t e movement towards t h e

to

indif-

f e r e n c e band. I f t h e process were r e p e a t e d u s i n g d i f f e r e n t s t a r t i n g p o i n t s , o r i f

d i f f e r e n t p o s i t i v e ( o r n e g a t i v e ) v a l u e s were a s s i g n e d a t i n t e r m e d i a t e steps, d i f f e r e n t s u r r o g a t e w o r t h f u n c t i o n s would be g e n e r a t e d .

However, b y d e f i n i -

t i o n o f t h e z e r o o f t h e s e f u n c t i o n s , t h e same s o l u t i o n ( o r band o f i n d i f f e r e n c e ) would r e s u l t . The uniqueness o f t h e r e s u l t i n g band o f i n d i f f e r e n c e p e r i i i i t s a p p l i c a t i o n o f t h e s u r r o g a t e w o r t h method t o s i t u a t i o n s where t h e r e i s more t h a n I f a l l have t h e same s t a n d a r d s o f v a l u e , t h e samedeci-

one d e c i s i o n maker.

s i o n ( o r band o f i n d i f f e r e n c e ) g a t e f u n c t i o n s developed.

w i l l be d e t e r m i n e d r e g a r d l e s s o f t h e s u r r o -

I f the plural

d e c i s i o n makers

have

different

s t a n d a r d s o f v a l u e , as woiild n o r m a l l y be expected, t h e s u r r o g a t e w o r t h m e t h od has t h e p r o p e r t y o f d e f i n i n g t h e i n d i f f e r e n c e band a t any and a l l l e v e l s o f unanimity o r l a c k t h e r e o f .

The band o f i n d i f f e r e n c e f o r u n a n i m i t y would

be t h a t s e t o f d e c i s i o n s i n c l u d i n g a l l d e c i s i o n m a k e r s ‘ bands o f rence.

By d e f i n i n g n e s t e d i n t e r v a l s , each c o n t a i n i n g

one

less

indiffedecision

m a k e r ’ s band o f i n d i f f e r e n c e , bands o f i n d i f f e r e n c e f o r l e s s t h a n u n a m i n i o u s m a j o r i t i e s can be a p p r o x i m a t e d .

A p p r o x i n a t i o n stems f r o m t h e f a c t t h a t some

r u l e w i l l have t o be d e v i s e d t o d e t e r m i n e w h i c h band o f i n d i f f e r e n c e s h o u l d be d e l e t e d n e x t .

An o b v i o u s r u l e m i g h t be t o s p e c i f y t h a t c a n d i d a t e

which

would n a r r o w t h e g r o u p band o f i n d i f f e r e n c e by t h e g r e a t e s t amount. A l t e r n a t i v e l y , a s p e c i f i c d e c i s i o n c o u l d be “ d e f i n e d ” as p r e f e r r e d i f i t i s i n c l u ded i n t h e maximum number o f i n d i v i d u a l i n d i f f e r e n c e bands.

O f course, a l l

such r u l e s a r e somewhat a r b i t r a r y even when t h e y seem i n t u i t i v e l y r e a s o n a b l e o r appeal t o some l o g i c o f s o c i a l j u s t i c e .

1

I n summary, t h e s u r r o g a t e w o r t h f u n c t i o n

W..

1J

associated w i t h

the

ith and j t h o b j e c t i v e s can be d e f i n e d as any monotonic f u n c t i o n s a t i s f y i n g : ,’

=

w.. 1J

‘:

0 when X . . m a r g i n a l u n i t s o f fi(&) 1J

a r e p r e f e r r e d o v e r one

m a r g i n a l u n i t o f f . ( x ) , g i v e n t h e l e v e l o f achievement

J -

o f a1 1 t h e o b j e c t i v e s .

0 when 1.. m a r g i n a l u n i t s o f f i ( x ) 1J

a r e e q u i v a l e n t t o one

m a r g i n a l u n i t o f f . ( x ) , g i v e n t h e l e v e l o f achievement J -o f a1 1 t h e o b j e c t i v e s . 0

when A . . m a r g i n a l u n i t s o f f i ( $ 1J

a r e n o t p r e f e r r e d over

one m a r g i n a l u n i t o f f . ( x ) , g i v e n t h e l e v e l o f a c h i e v e -

.!-

inerit o f a l l t h e o b j e c t i v e s . I t i s important t o note here t h a t t h e d e c i s i o n maker(s) i s provided

with the t r a d e- o f f value ( v i a t h e t r ad e- o ff r a t e f u n c t i o n ) j e c t i v ? functions a t functions.

a

of any two

ob-

s t a t e d l ev el of attainment of a l l of the o b j e c t i v e

Furthermore, a l l t r ad e- o f f values generated from t h e

tra de -off

f u n c t i o n a r e as s o ci at ed with t h e n o n - i n f e rior s e t . Hence, i t i s e vide nt t h a t i t i s always p o s s i b l e t o g en er at e a s u r r o g a t e w o r t h func tion whichwill in t u r n determine t h e band of i n d i f f e r e n c e of

A..

(ifj, i,j

=

...,

1,2,

1J

n)

a n d thus t h e p r ef er r ed s o l u t i o n s t o t h e m ulti-obje c tive problem ( t h e compu-

t a t i o n a l procedure f o r f i n d i n g t h e o v er al l solution t o problem 3-1 from t h e su r r o g a t e worth f u n ct i o n s i s discussed i n t h e next s e c t i o n ) . The s p e c i f i c values of t h e s u r r o g ate w o r t h W . . assigned by

the

1J

DM

f o r each value of x . . a r e unimportant, needing only t o be monoton.ic in h . . 1J 1J and t o p a s s through zero a t i n d i f f e r e n c e . T h u s W . . a r e ordina l measures, 1J

compared t o t h e car d i n al measures required by t h e u t i l i t y func tion approach. F i n a l l y , i t i s important t o note t h a t t h e above a na lyse s areconducted in t h e f u nct i o n al space, f l ( x ) ,

, f n ( x ) , and not i n

the

de c ision

space, x l , . . . , x N . This i s of co u r s e, a c l e a r advantage s i n c e t y p i c a l l y n (one may have a hundred o r a thousand de c isions even with j u s t t h r e e

N

~

t o ten o b j e c t i v e f u n c t i o n s ) . 3.5

CXJMJUTATIONAL PROCEDURE FOR FINDING THE P R E F E R R E D SOLUTIONS The s u r r o g at e worth f u n ct i o n described i n t h e previous section

a ss i g n s a s c a l a r value (on a n o r d i n al s c a l e ) t o any non-infe rior

t o t h e m u l t i p l e o b j e c t i v e problem 1 - 1 .

solution

However, t h e r e a r e t h r e e

different

ways of s p e c i fy i n g a n o n - i n f er i o r s o l u t i o n - by i t s de c ision space coordina t e s , by i t s o b j e c t i v e f u n ct i o n space co o rdina te s, o r by i t s tra de -off r a t e values X. . a t t h e n o n - i n f er i o r p o i n t . 1J

Thus,

there a r e three possibilities

f o r t h e su r r o g at e worth f u n ct i o n W . . ( x . ) , b J . . ( f . ) o r W . . ( A . . ) . J 1J J 1J 1J 1J 3.5.1

Decision Space Surrogate Worth Function I n t h i s approach, t h e s u r r o g at e

worth

func tion i s

developed a s a

function of t h e d eci s i o n s x l . . . x N ; however, t h e r e a r e several reasons why t h i s i s g e n e r al l y n o t f e a s i b l e . F i r s t , t h e values of t h e de c isions a r e not DM , s i n c e he i s more

p a r t i c u l a r l y rel ev an t o r meaningful t o t h e

familiar

(and more d i r e c t l y concerned) with t h e o b j e c t i v e s and t h e i r tra de -off values. Second, t h e d e ci s i o n space i s g en er al l y more

complicated than

the

others

s i n c e t h e r e a r e u s u al l y many more d eci s i o ns than o b j e c t i v e s . Third, onemust r e s t r i c t t h e values of t h e d eci s i o n s t o n on-infe rior va lue s.

In t h e

func-

t i o n a l space t h i s can be e a s i l y done s i n c e any n-1 of t h e o b j e c t i v e s s p e c i f y a n o n - i n f e r i o r point v i a t h e t r ad e- o f f f u n c t i o n . d e c i si c n space, however

(see figure 1-1).

This i s impossible in t h e T h u s t h e problem of finding a

point where both s i ra bl e.

W..(x) = 1J

0 and x i s non-inferior makes this approach unde-

3.5.2 ?--Space Surrogate Worth Function In this approach, the surrogate worth function W . . is developed as 1J a function of A... In general, values of h . . for n-1 different j are re1J

1J

quired to specify a non-inferior point, but for the present, this approximation will be maintained. For several distinct values of 1 . . , the decision 1J tnaker is asked whether x . . units of fi(x) are more or less or equally p r e 1J A linear combination of the two answers ferred to one unit o f f.(x). J W . . ( A . . ) nearest zero can be made (see figure 3-2). Then the value of ~~

1J

1J

i . is . chosen such that W . . ( i . . ) = 0 on the line segment fitting the 1J 1J 1J two values of h . . . With this estimate X.., the corresponding W..(A..) is 1J 1J 1J 1J * requested and the process repeated until indifference is found at X . . . 1J Then the indifference band is assumed to exist within a neighborhood of A*... 1J Additional questions to the decision maker can be asked in the neighborhood * ofA.. to determine the approximate limits of the band of indifference. 1J * Having determine all x . . for i , j = 1,2, . . . , n , the following system 1J o f relations should be solved simultaneously in the decision space x: X..

=

1J

~

Problem 3-2:

~-

Solve:

hij(x)

=

A?. 1J

. i,j '

=

1,2, ..., n

such that This problem represents n2 -n equations with the constraints derived from the Kuhn-Tucker conditions that should be imposed on 5 ~ . Note that since normally N ':, n, the number of equations in problem 3-2, (n2 -n) may still be much smaller than N. The constraints on 5~ introduce additional inequalities which in turn limit the solution space to only feasible non-inferior solutions. A non-unique preferred (indifference) solution can be expected and is of course, accepted, as in any optimization problem with a single objective function. For the case where A . .(f.(x)) is constant over a given 1J J variation in f.(x), as may happen with linear objective functions, an elasJ The tradeticity tertn can be augmented to A.. for determining W . . ( A . . ) . 1J 1J 1J off rate funct.ion x . . can be replaced by the elasticity trade-off rate 1J function. ~

48

w 10

W -J

a

0 v)

sz- o n

a

0

- 10

F i g u r e 3-2.

A,. = -

'J

D e t e r m i n a t i o n o f t h e i n d i f f e r e n c e band

f.(x)

afi(x)

L ~

fi(x)

at-Jo

A l t e r n a t i v e e l a s t i c i t y f u n c t i o n s may b e c o n s t r u c t e d t o c i r c u m v e n t t h i s p r o b lem.

Other techniques a r e described i n t h e n e x t s e c t i o n . Although t h i s approach appears

suring that

x is

to

s u f f e r from t h e problem o f

in-

n o n - i n f e r i o r when r e v e r t i n g t o t h e d e c i s i o n space, a means

o f a v o i d i n g t h i s w i l l b e d i s c u s s e d i n s e c t i o n s 4 . 1 . 3 and 6 . 3 . 2

.

A

more

s e v e r e p r o b l e m , however, i s t h a t t h i s a p p r o a c h i s g u a r a n t e e d o n l y when t h e t r a d e - o f f f u n c t i o n i s c o n v e x and n o n - l i n e a r .

ii

! I

49

I /

f vs.f 2 1 I n such a case, i t i s o b v i o u s t h a t

C o n s i d e r t h e t w o o b j e c t i v e s t a t i c c a s e where t h e t r a d e - o f f i s non-convex as shown i n f i g u r e 3 - 3 - a . '12(f2)= -df,/df

2

i s n o t Q unique f u n c t i o n over t h e range o f values o f f2 i n

t h e n c n - i n f e r i o r r e g i o n (shown i n f i g u r e 3 - 3 - b ) .

(Note t h a t t h e t o t a l det-i-

v a t i v e i s p r e s e n t s i n c e t h e r e a r e o n l y two o b j e c t i v e s . ) longer a well defined function o f

x , ~ s, i n c e

Hence, w o r t h i s n o

f o r some v a l u e s o f

A12

there

w i l l be m o r e t h a n one c o r r e s p o n d i n g p o i n t ( f ,f ) on t h e t r a d e - o f f c u r v e , 1 2 and i n g e n e r a l , a d i f f e r e n t w o r t h w i l l b e a s s i g n e d t o e a c h p o i n t . Thus the w o r t h may a p p e a r as i n f i g u r e 3 - 3 - c .

O f c o u r s e , i n some c a s e s W12

i n d e p e n d e n t of fl a n d f 2 , so t h a t W12(X12) w i l l b e a w e l l d e f i n e d

may

( a s shown i n f i g u r e 3 - 3 - d ) .

f

,

XI2

0

f2

F i q u r e 3-3-a.

Tradeoff Function

0

f

F i g u r e 3-3-b.

*

T r a d e o f f Rate Function

WI 2

0

F i g u r e 3-3-c.

F i g u r e 3-3.

P o s s i b l e Worth Curve

F i g u r e 3-3-d.

P o s s i b l e Worth Curve

>,-Space W o r t h F u n c t i o n s f o r Non-Convex Problems

be

function

When w o r t h i s not a well defined f unc tion of A . . however, t h i s tends 1J t o cause two types of problems. F i r s t , i f t h e w o r t h i s found only a t c e r t a i n values o f x12 a n d a polynomial f i t i s attempted, t h e s o l u t i o n t o W 1 2 ( A 1 2 ) *= 0 w i l l probably be q u i t e d i f f e r e n t from t h e t r u e pre fe rre d tra de -off r a t e x12. Second,

assume t h a t enough information i s obtained in t h e f i r s t attempt

f i n d t h e a c t u a l shape of t h e

be coniputationally i n f e a s i b l e ) ; can be determined.

then t h e t r u e pre fe rre d tra de -off

However, t h e r e may be several points

curve which have t h e same s l o p e of t h e system model t o f i n d

to

multi-valued worth curve ( i n general t h i s w ill

*

on t h e

r a t e A* 12 tra de -off

Thus, when h 1 2 i s r e l a t e d back t o t h e p r ef er r ed d e c ision v e c t o r , the model w ill not -

be a b l e t o d i scer n which of t h es e p o i n t s i s the pre fe rre d s o l u t i o n . The o t h e r pathological cas e i s where X 12(f2) i s c onsta nt over a c e r t a i n i n t e r v a l ( s e e f i g u r e s 3-4-a and 3 - 4 - b). This always occurs i n l i n e a r problerns a n d sometimes in non-linear ones. Since X12 t h e r e will be several values of

i s again not one-to-one

t h e w o r t h corresponding

to

some values of

Furthermore, in t h e l i n e a r problem,

s i n c e h I 2 ( f 2 ) i s a l s o disc ontinux12. ous t h e r e a r e no values of t h e w o r t h f o r o t h e r values o f x (see figure *I2 3-4-c). Thus, i n c o r r e c t o r even i n f e a s i b l e r e s u l t s f o r A,2 may be found. E a r l i e r i n t h i s s e c t i o n , i t was suggested t o use a transformation 5 2 = f2 A12/fl t o circumvent t h e l i n e a r i t y o f h 1 2 ( f 2 ) . The r a t i o n a l e i s t o take t h e e l a s t i c i t y i n t o account; t h e r e a r e s t i l l two problems present w i t h t h i s approach, however: ( 1 ) A i 2 ( f 2 ) may s t i l l be discontinuous and thus the worth func tion w i l l a l s o be d i s co n t i n u o u s , aga'in with p o ssibly i n f e a s i b l e r e s u l t s ; note however t h a t i t may be p o s s i b l e t o circumvent t h i s problem b y some s o r t of re g r e s s i o n over t h e s e i n t e r v a l s s i n ce t h e d i s c o n t i n u i t i e s a r e bounded. I ( 2 ) The geometric i n t e r p r e t a t i o n i s now l o s t s i n c e A 1 2 i s no longer t h e negative o f t h e s l o p e of t h e t r ad e- o f f curve - and the d e f i n i t i o n of the p r e f e r r e d s o l u t i o n must be changed. Although t h e h-space approach appears t o have many drawbacks, 'it i s t h e s i m p l e s t f o r some problems. Algorithms using t h i s approach a r e included i n subsequent ch ap t er s . 3.5.3

Objective Function Space Surrogate Worth Function A simple means of surmounting t h es e d i f f i c u l t i e s i s t o

use

the ob-

j e c t i v e function space s u r r o g at e worth f u n c t i o n . This w ill be developed here f o r t h e two-objective cas e; t h e g e n e r a l i z a t i o n f o r n o b j e c t i v e s i s in c h a p t e r 6.

Note t h a t

provided

i n general t h e values of n-1 o f t h e o b j e c t i v e

are

51

f

,

0

0

f2

Figure 3-4-a.

Tradeoff Function

f2

Figure 3-4-b.

Tradeoff Rate Function

WI 2

0 -

Figure 3-4-c Figure 3-4.

P o s s i b l e Worth Curve

A-Space Worth Functions f o r Linear Problems

needed t o s p e c i f y a n o n - i n f er i o r p o i n t , so t h a t W . . w ill be

n-1 o f t h e f

..

1.I

a

func tion

of

J

For t he two-objective problem, t h e f i r s t segment of t h e SWT method provides t h e f u n ct i o n s f ; ( f 2 ) and t h e corresponding

X12(f2)

=

- df;/df2.

Using t h e same d e f i n i t i o n of worth a s p r eviously, t h e surroga te w o r t h t i o n W12 can be developed a s a f u n ct i o n o f f 2 d i r e c t l y , and W12(f2)= 0 The pre fe rre d tra de -off ved t o f i n d f f ( t h e p r ef er r ed value o f f 2 ) .

func-

solrate

52

j ; 2

c

J

is

12

( f * ) and t h e p r e f e r r e d v a l u e o f fl i s f;(f;) 2

For any value

f2

.

, t h e d e c i s i o n maker i s s t i l l a s k e d how

much

A ~ ~ ~ ( a: d ~d i )t i o n a l u n i t s o f f 1 a r e w o r t h r e l a t i v e t o one a d d i t i o n a l u n i t o f

* -

f 2 , g i v e n f 2 u n i t s o f f 2 and f l ( f 2 ) say, on t h e s c a l e f r o m

-

u n i t s o f fl

.

H i s r e l a t i v e assessment,

1 0 t o + 10, i s t h e v a l u e W 1 2 ( f 2 ) . By a s k i n g enough

q u e s t i o n s a t v a r i o u s v a l u e s o f f 2 , p o i n t s c a n b e g e n e r a t e d so t h a t t h e f u n c t i o n s Id

12

( f ) can be approximated ( e . g . ,

by r e g r e s s i o n ) .

2

Since f o r each v a l u e o f i n the non-inferior region (i.e.,

f2 Xl2

t h e r e i s o n l y one v a l u e o f X 1 2 and and fl a r e b o t h f u n c t i o n s o f f

t h o i i y h f 2 may n o t b e a f u n c t i o n o f h 1 2 ) ,

W12(f2)w i l l b e a u n i q u e

function

o f f 2 , even f o r t h e non-convex a n d l i n e a r c a s e s c a u s i n g t h e p r e v i o u s

culties. val.

fl

even diffi-

C o n s i d e r t h e c a s e where h 1 2 ( f 2 ) was c o n s t a n t o v e r a c e r t a i n i n t e r -

T h i s new a p p r o a c h

i m p l i c i t l y takes e l a s t i c i t y i n t o account since

can f i n d t h e w o r t h a t d i f f e r e n t v a l u e s o f i s constant.

one

i n t h e i n t e r v a l where h , 2 ( f 2 )

f2 A l s o n o t e t h a t t h i s a p p r o a c h a l l o w s t h e d e c i s i o n maker

terniine t h e actual e l a s t i c i t y r a t h e r than

t o de-

u s i n g a n y p r e c o n c e i v e d v a l u e such

as f2/fl. I n many p r o b l e m s , t h e r e may be more t h a n one s o l u t i o n t o W ( f )=O; 12 2 each o f t h e s e i s a p r e f e r r e d s o - l u t i o n ( t h e s o c i a l i n d i f f e r e n c e c u r v e i s t a n g e n t t o t h e t r a d e - o f f c u r v e a t m o r e t h a n o n e p o i n t ) and some o t h e r c r i t e r i o n i s n e c e s s a r y t o c h o o s e among them.

Furthermore, w i t h t h i s approach t h e t r a n s f o r m a t i o n t o

the decision

space c a n b e a c c o m p l i s h e d w i t h o u t w o r r y i n g a b o u t D o n - i n f e r i o r i t y . t i o n s t o W12(f2)

*

=

0 are t h e p r e f e r r e d values f2 .

r e d v e c t o r d e c i s i o n s &*, c a n b e o b t a i n e d

The s o l u -

Accordingly, the prefer-

by simply

solving

the

following

o p t i m i z a t i o n problem :

probl ?!!L3-3:

’-

inin f ( x ) ~x

This i s

j u s t a comnion p r o b l e m i n

same as p r o b l e m 3-1 w i t h x*

single objective optimization;

*

it i s

the

replaced by f.. I t s solution y i e l d s the desired 3 J Mote t h a t t h i s a l t e r n a f o r t h e t o t a l v e c t o r o p t i m i z a t i o n p r o b l e m 1-1. t

t i v e method a v o i d s t h e need o f r e p r e s e n t i n g

A.. 1.l

as a f u n c t i o n o f

x(

e.g.,

A . . ( x ) ) ,and consequently el i mi n at es t h e need of solving problem 3-2. 1J Thus i t can be seen t h a t t h e o b j e c t i v e function space i s ge ne ra lly the best domain f o r t h e s u r r o g at e w o r t h f unc tion. Algorithms u t i l i z i n g t h i s approach a r e presented in t h e following cha pte rs. 3.6

GEOMETRIC INTERPRETATION OF THE SWT METHOD The case of two

o b j e c t i v e f u n ct i o ns i s considered f i r s t ;

n

be l a t e r generalized t o

objectives.

t h i s w ill

The f i r s t segment of the SWTmethod

*

develops p o i n t s on t h e t r ad e- o f f curve f l ( f 2 ) and tra de -off r a t e curve X , 2 ( f 2 ) a t various values of f 2 . The d ecision maker i s then questioned t o determine t h e w o r t h corresponding t o various non-infe rior values of

f ; he

2

i s asked t o s p eci f y t h e r e l a t i v e worth ( o n a s c a l e of ( s a y ) t10 t o -10) of a d d i ti o n al u n i t s of f compared t o one a dditiona l u n i t of f2,give n 1 A12(f2) f T ( f 2 ) a n d 7, u n i t s of t h e two o b j e c t i v e s . His assessment of t h i s r e l a t i v e

(or

w o r t h i n d i c a t e s t h e divergence between t h e negative slope of the i n d i f f e r e n c e curve space.

(in)

and

A12,

Consider f i g u r e 3-5.

a t t h e p oint (f;(;2),;2)

The Social I ndiffe re nc e

his)

in t h e functional

(SI)

curves

re pre -

s e n t i n g equal preference a r e i n r e a l i t y u na va ila ble ; however, i t i s assumed t h a t t h e DM i s basing h i s d eci s i o n s on h i s s u b j e c t i v e ide a s of t h e i r form. The t r a d e - o f f curve T i s a l s o g en er al l y impossible t o determine completely; u s u a l l y only a f i n i t e number o f p o i n t s o n the curve w ill be known. A t point A, f o r example, t h e DM i s w i l l i n g t o accept only mA a d d i t i o n a l u n i t s of However h 1A2 > mA so t h a t when asked i f he t o g e t one u n i t l e s s of f

2'

w i l l i n g t o take

fl is

if2 ad d i t i o n al u n i t s of f l f o r one l e s s of f 2 h i s response

A will be "no", thus W 1 2 ( A 12 ) i f he i s w i l l i n g t o take

<

0.

A t p o i n t C , mc

>

Af2

so t h a t

when

asked

C ad d i t i o n al u n i t s of f l f o r one l e s s of f 2 , h i s

C response w i l l be " y e s " , o r Id 1 2 ( A 1 2 )

>

0.

A t point 6 , mB

=

B

A12

SO

t h a t when

asked i f he i s w i l l i n g t o t ak e on h B ad d itiona l u n i t s of f l f o r one l e s s o f 12 B f 2 , h i s response wi l l be i n d i f f e r e n c e , or W 1 2 ( h 1 2 ) = 0. Note t h a t t h e actual numerical value of t h e worth f u n ct i o n s r epre se nts only a r e l a t i v e s c a l i n g : 0 1 i f W ( A ) = +8 and 14 ( A ) = +7, a l l t h a t can be s a i d i s t h a t t h e d i f f e 12 12 12 12 rence between m a n d A 1 2 a t t h e point on t h e tra de -off curve c orre spondingto ho i s g r e a t e r than t h e d i f f e r e n c e between them a t t h e point corresponding 12 t o A ; ~ . These numerical values can be used, however, a s a f i r s t * e stim a te 1 1 . 0 e . g . , 17, i s seven times f a r t h e r from X12 than A,2 i s from X 1 2 . of

~ 7 ~ ;

MULTIOBJECTIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS

54

A t point A, W,*

>

0

A t point 6, W12

=

0

A t point C , W12

i

0

S i s t h e f e a s i b l e s e t in t h e functional space

The thick boundary of S i s t h e trade-off function T S I a r e t h e social i n d i f f e r e n c e curves

a t A, t h e slope of T i s -

X 1A2 ,

t h e slope o f SI i s -m A

a t 6, t h e slope of T i s -

hB 12,

t h e slope of S I i s -m B

a t C , the slope of T i s -

hC 12,

the slope o f S I i s -m C

Figure 3-5.

Geometric I n t e r p r e t a t i o n o f Worth Function.

3.7

SUMMARY

The s u r r o g a t e w o r t h

trade-off

the p r e f e r r e d s o l u t i o n ( i . e . ,

method p r o v i d e s a means o f

finding

maximum u t i l i t y ) by d e t e r m i n i n g t h e p o i n t

tangency between t h e t r a d e - o f f f u n c t i o n and t h e s o c i a l i n d i f f e r e n c e

of

curve.

The t r a d e - o f f r a t e f u n c t i o n s e n a b l e t h e d e c i s i o n maker t o compare t h e s l o p e s o f t h e t r a d e - o f f f u n c t i o n and s o c i a l i n d i f f e r e n c e c u r v e a t v a r i o u s p o i n t s i n t h e f u n c t i o n a l space; t h e s e t r a d e - o f f r a t e f u n c t i o n s were f o u n d Lagrange m u l t i p l i e r s o f t h e

t o be

the

m u l t i p l e o b j e c t i v e problem i n € - c o n s t r a i n t form.

The s u r r o g a t e w o r t h f u n c t i o n a l l o w s i n t e r p o l a t i o n

o f the

DM's responses t o

f i n d the preferred solution. The SWT method can be viewed as an i n t e r m e d i a r y between t h e DM and t h e system response ( f i g u r e 3 - 6 ) , teraction.

The SWT

b o t h s i m p l i f y i n g and q u a n t i f y i n g t h e i r i n -

method i n i t i a l l y i n t e r a c t s w i t h t h e system t o d e t e r m i n e

t h e t r a d e - o f f and t r a d e - o f f r a t e f u n c t i o n s interacts assessment. found.

with the

among t h e o b j e c t i v e s .

DM by d e v e l o p i n g q u e s t i o n s o f r e l a t i v e

By a n a l y z i n g h i s o r d i n a l

It

worth

responses t h e p r e f e r r e d

for

then his

solution i s

T h i s s o l u t i o n ( v i a t h e SWT method) can t h e n be expressed i n terms o f

t h e d e c i s i o n v a r i a b l e s o f t h e system. i n t e r a c t i o n between t h e DM

There i s s t i l l c o n s i d e r a b l e p e r s o n a l s i n c e h i s response w i l l n o r m a l l y

and t h e system

r e f l e c t h i s knowledge o f t h e system and h i s e v a l u a t i o n o f t h e p r e f e r e n c e s o f his constituents. because many o f t h e

The SWT method, however, g r e a t l y s i m p l i f i e s t h e DM's t a s k n o n - i n f e r i o r s o l u t i o n s can be s y s t e m a t i c a l l y e l i m i n a t e d

based on (and f u l l y c o m p a t i b l e w i t h ) h i s knowledge and expressed p r e f e r e n c e s The advantages o f t h e SWT method a r e numerous.

The d e c i s i o n s r e q u i -

r e d by t h e DM a r e m i n i m a l ; he d e a l s o n l y w i t h t h e f u n c t i o n a l space ( w h i c h i s generally

much s m a l l e r ,

direct significance

e a s i e r t o work w i t h , and o f d i r e c t r a t h e r t h a n i n -

when compared t o t h e d e c i s i o n space),

and w i t h o r d i n a l

r e l a t i o n s h i p s between h i s v a l u e s ( r a t h e r t h a n a c t u a l v a l u e s ) . i s made c o n c e r n i n g t h e f o r m o f t h e u t i l i t y f u n c t i o n ; by t h e DM t o a p a r t i c u l a r t r a d e - o f f p r o p e r l y part o f his constituents.

O f course h i s

only that indifference

r e p r e s e n t s i n d i f f e r e n c e on t h e

v a l u e judgements a r e

s u b j e c t i v e , b u t he i s p r o v i d e d w i t h adequate

No assumption

information,

necessarily

and a

logical

framework, t o r a t i o n a l l y and s i m p l y assess and e v a l u a t e h i s p r e f e r e n c e s .

56

PHYSICAL SYSTEM

I

F i g u r e 3-6.

Role o f t h e SWT Method.

I 1

*'

I

57

I

FOOTNOTES 1.

The o r i g i n a l

development of t h i s method can be found

i n Haimes

and

Hall C19741.

2.

The c o n c e p t s of d u a l i t y and

Lagrange m u l t i p l i e r s a r e

discussed i n

E v e r e t t [1963], lasc'ori [1968] and o t h e r s , w h i l e the t - c o n s t r a i n t proach was d i s c u s s e d i n s e c t i o n 2 . 6 ( a l s o see Haimes [ I 9 7 3 3.

These a r e t h e n e c e s s a r y c o n d i t i o n s

ap-

I).

f o r s t a t i o n a r i t y and can be found

i n Kuhn and Tucker [1950] a s well a s i n most o p t i m i z a t i o n t e x t s .

4.

See l u e n b e r g e r [ 19731.

5.

For o t h e r r e s u l t s on t h e d e g e n e r a t e c a s e s e e Olagundoye [1971].

6.

Again s e e E v e r e t t [1963], l a s d o n [1968], o r Gembicki [1973].

REF E R EN C E S

1.

E v e r e t t , H.,

"Generalized lagrange M u l t i p l i e r s

Problerris of Optimum A l l o c a t i o n of R e s o u r c e s , "

Method

for

Solving

v,

v o l . 1 1 , p p . 399-417, 1963.

2.

Getiibicki, F.,

" V e c t o r O p t i m i z a t i o n f o r Control w i t h Performance

Pdrameter S e n s i t i v i t y I n d i c e s , " P h . D . D i s s e r t a t i o n ,

and

Case Western

Reserve U n i v e r s i t y , 1973. 3.

Hairiies, Y . Y . ,

"The I n t e g r a t e d System I d e n t i f i c a t i o n and O p t i m i z a t i o n , "

i n Advances i n Control Systems Theory and A p p l i c a t i o n s , Leondes, E d i t o r , Volume X , Academic P r e s s , N . Y .

4.

Hairiies, Y . Y . ,

and H a l l , W . A . ,

C.

T.

pp.435-518, 1973

" M u l t i o b j e c t i v e s i n Water

.

Resources

Systems A n a l y s i s : The S u r r o g a t e Worth T r a d e o f f Method," Water

Re-

s o u r c e s Research, v o l . 1 0 , no. 4 , pp. 615-624, 1974. 5.

6.

7.

K u h n , H . W . , and Tucker, A. \!. , " N o n l i n e a r Programming," i n Second B e r k e l e y Symposium on Mathematical S t a t i s t i c s and P r o b a b i l i t y , Uni v e r s i t y of C a l i f o r n i a P r e s s , B e r k e l e y , C a l i f o r n i a , 1950. l a s d o n , L.S., " D u a l i t y and Decomposition i n Mathematical Programming,' I E E E T r a n s a c t i o n s , v o l . SSC-4, no. 2 , 1968. Luenberger, D . G . ,

Introduction

t o l i n e a r and N o n l i n e a r Programming,

Addision-Wesley P u b l i s h i n g Company, I n c . , 1973.

8.

Olagundoye, 0. B . ,

" E f f i c i e n c y and t h e t - C o n s t r a i n t Approach f o r Mul-

t i - C r i t e r i o n System," M.S. T h e s i s , Systems E n g i n e e r i n g Department, Case Western Reserve U n i v e r s i t y , C l e v e l a n d , Ohio, 1971.

Chapter 4

THE SWT METHOD FOR STATIC TWO-OBJECTIVE PROBLEMS The o v e r a l l procedure f o r s o l v i n g m u l t i p l e o b j e c t i v e problems with t h e su r r o g a t e worth t r ad e- o f f method can be divided i n t o two segments. The f i r s t p a r t involves t h e development of function

i n o r d er t o

the information provided tion.

information

e d i f y t h e d eci s i o n maker. by t h e

DM's

about

the

choices t o f i n d t h e p r e f e r r e d s o l u -

algorithms presented i n t h i s book can be

The so l u t i o n

tra de -off

The second segment uses classified

according t o t h e approach used f o r each segment - E-c onstra int ( E ) , m ultip l i e r ( M ) , o r combined (C) appropriate - s t a t i c

- a n d t h e types of problems f o r which they a r e

( S ) o r dynamic ( D ) , two-objective ( T ) o r n-obje c tive

(N). c a se .

This c h ap t er wi l l p r es en t algorithms f o r t h e two-objective s t a t i c Innovations t o improve computational e f f i c i e n c y o f t h e SWT method

a r e d e t a i l e d . The a p p l i c a b i l i t y of each algorithm i s disc usse d, and sample problems a r e included t o i l l u s t r a t e t h e i r use. 4.1

COMPUTAT IOr.IAL EFFICIENCIES There a r e several improvements t h a t can be incorporated in t h e SWT

method f o r t h e two-objective cas e. n o n - i n f e r i o r region can be found i n f e r i o r p o i n ts

(non-binding

First, to

c2).

l es se n

maximum values f o r

t2

in

the

t h e e f f o r t wasted by finding

Secondly, i t w ill be shown t h a t once

W12(f2) has been found, W21 i s redundant, b u t i t can be used a s a c onsistency check. T h i r d , a simple method of re ve rting t o t h e de c ision space w i l l be presented.

F i n a l l y , i t wi l l be shown t h a t by using search te c hni-

ques and i n t e r p o l a t i o n s t h e e n t i r e SWT procedure can be accomplished without r e s o r t i n g t o r e g r e s s i o n s . 4.1.1

Limits on

c2

The maximum v al u e,

f21qAX, f o r

€2

i n t h e non-infe rior region i s

found by solving t h e following problem: Problem 4-1 : MIN

s.t. Theorem: value f o r

58

fl(x) E T

x

I f t h e s o l u t i o n v ect o r t o problem 4-1 i s i s f2(X*).

x*,

then t h e maximum

59

STA>''J(,' 'I'WO-OBdEC'I'IVE PROBLEMS

Proof: x

T

-~

E

must

x*

Since

T, f l ( x l )

3

be i n t h e

s o l v e s problem 4-1,

t h e n f o r any o t h e r f e a s i b l e v e c t o r

T h e r e f o r e any &l which a l s o g i v e s f 2 ( q ) > f2(&*)

f,(L*).

i n f e r i o r r e g i o n s i n c e a r e d u c t i o n can be o b t a i n e d i n b o t h

o b j e c t i v e s by u s i n g

x* .

Thus t h e l a r g e s t v a l u e t h a t f 2 can a t t a i n i n t h e

non-inferior region i s f ( x * ) . F o r t h e case where t h e s o l u t i o n v e c t o r x* 2 ~* t o problem 4-1 i s n o t u n i q u e , t h e minimum o f t h e v a l u e s f 2 ( K ) i s used as t h e maximum v a l u e f o r

F

2'

The minimum v a l u e f o r a l l the other objectives

.

c o n s t r a i n t w i l l always

c2

f2MIpJ

i t2

i s f o u n d as i n s e c t i o n

c2

When t h e n o n - i n f e r i o r s e t i s be

b i n d i n g f o r values o f

1 . 4 by i g n o r i n g continuous,

the

i n the interval

c2

< fZMAX . As an example o f t h e case where t h e c o n s t r a i n t i s n o t

b i n d i n g , consider. t h e f o l l o w i n g problem: MIN

fl

=

x

MIN

f2 =

x

s.t.

-3

<

x

<

3 x /3

3

The o b j e c t i v e s a r e shown i n f i g u r e

4-1-a;

t i o n a l space i s d e p i c t e d i n f i g u r e 4-1-b.

t h e f e a s i b l e s e t S i n t h e funcWhen t h e v a l u e

c2 =

0 (which

= -3 and fZlvlAX = 3 ) i s used, t h e s o l u t i o n w i l l be a t x = between fZMIN

f

2

=

every

-1, fl t2

=

- 2 / 3 so t h a t t h e c o n s t r a i n t i s n o t b i n d i n g .

i n t h e i n t e r v a l -1

E~

i

2

S

-

I n t h i s examp e

w i l l be n o n - b i n d i n g .

I n summary, when u s i n g t h e € - c o n s t r a i n t approach, l a r g e numbers o f i n f e r i o r solutions

can be a u t o m a t i c a l l y e l i m i n a t e d by f i n d i n g minimum and

maximum v a l u e s f o r

E ~ .

proach, t h e

I n a d d i t i o n , f o r each s o l u t i o n f o u n d by t h i s ap-

r 2 c o n s t r a i n t s h o u l d be checked t o i n s u r e t h a t i t i s b i n d i n g ;

i f n o t binding, then t h e s o l u t i o n i s i n f e r i o r .

4.1.2

T r a d e - o f f and Worth R e l a t i o n s h i p s The f i r s t segment o f t h e SWT method p r o v i d e s p o i n t s

i n t h e n o n - i n f e r i o r s e t , and t h e t r a d e - o f f r a t e

h12

-fT

= (fl,f2)

=-afl/af2If.

Simi-

l a r l y there i s a value o f t h e t r a d e - o f f r a t e h2, = - af2/afl

If .

Although

-

t h e p r e v i o u s c h a p t e r c o n s i d e r e d h 2 1 as a f u n c t i o n o f fl and h 1 2 a s a f u n c t i o n o f f2, i t i s o b v i o u s t h a t b o t h can be c o n s i d e r e d as a f u n c t i o n o f f 2 s i n c e i n t h e n o n - i n f e r i o r s e t fl i s a known f u n c t i o n o f f 2 ( t h a t i s f;(f2)

).

1

Then f o r any v a l u e o f f 2

,

iZ1(f2) = l/h12(f2)

.

Since

the

60

Figure 4-1-a. Objectives in Decision Space

Figure 4-1-b. Functional Space. Figure 4-1.

Non-Continuous Trade-off Curve.

Note: The feasible set S is the entire curve the non-inferior set is the thick portion.

( f ) are available, Azl(f2) can be f o u n d without re solving t h e "21 2 I t can now be seen t h a t t h e problern with f 2 a s t h e primary o b j e c t i v e . worth f u n c t i o n WZ1 can a l s o be considered a function. o f f 2 ; i t s values a r e

values

determined by asking t h e DM how much

hZ1(T2) a d d i t i o n a l u n i t s of f l , given f

worih in r e l a t i o n t o one a d d i t i o n a l u n i t of

* ^

f l ( f 2 ) u n i t s of f l .

2

f 2 are

u n i t s of f 2 a n d

I t w i l l be assumed t h a t the tra de -off r a t e X 1 2

is

a

good approximation t o t h e change which occurs in t h e non-infe rior value of f , when f 2 i s changed by one u n i t ; t h a t i s

I t w i l l be shown i n ch ap t er s i x t h a t i f s i n c e i n d i f f e r e n c e t o t r ad i n g

If2]

1

>>

W12(f2) =

0

and I f l /

>> \ A l 2 / .

then W Zl(f2) = 0,

a d d i t i o n a l u n i t s o f f l f o r one a d d i t i o n a l

u n i t of f 2 i s t h e same a s i n d i f f e r e n c e t o tra ding one u n i t of f l f o r l/A:2 u n i t s of f 2 .

Thus f i n d i n g WZl(f2) i s redundant.

a r e generally n o t

able to assess t h e i r

W2,(f2) can be found a t each value

However, de c ision makers

preferences a c c u r a t e l y .

f 2 a t which

W12

Since,

i s found simply

by

asking t h e DM one a d d i t i o n a l q u e s t i o n , i t may be useful t o use an averaged w o r t h f u n c t i o n W12(f2) =

1/2 (W12(f2) +

W

21

( f ) ) t o f i n d t h e pre fe rre d

2

solution. Reversion t o t h e D e c i s i o n a c e

4.1.3

The s o l u t i o n of W 1 2 ( f 2 ) = 0 i s t h e pre fe rre d value f; of o b j e c t i v e f2.

The p r e f er r ed d eci s i o n v ect o r

x*

can be found a s in s e c t i o n 3 . 5 . 3 by

solving Problem 4-2-: MIN

f,

(x)

f;

s . t . f 2 (x) s X E T

x*

t o problem 4-2 i s t h e same preI t w i l l now be shown t h a t t h e s o l u t i o n f e r r e d s o l u t i o n t h a t would be found by t h e method described in s e c t i o n 3 . 5 . 2 i f i t i s modified t o remove redundancy.

That

method i s t o

solve

simultaneously:

s.t.

x satisfies

X12(f2(X))

=

A21(fl(x))

=

t h e Kuhn-Tucker co n d i t i ons

However, x , ~ ( ~ ~ =( x1 /)h)Z l ( f , ( x ) ) whenever

*

= 1/hZl

i f t h e DM i s c o n s i s t e n t .

(x i s a non-infe rior p o i n t ) . x i s a non-infe rior p o i n t , a n d

T h u s t h e second equation i s

redun-

62

MULT I OBJECT W E OPTIMIZATION IPJ WATER RESOURCES SYSTEMS

d a n t and t h i s approach becomes: _______ Problem 4 - 3 :

Solve

h12(f2(x))

*

=

such t h a t & m e e t s t h e Kuhn-Tucker

A12

c o n d i t i o n s f o r problem 3-1. I f *.

Theorem: Proof:

s o l v e s problem 4-2,

Note t h a t t h e r e i s a v a l u e

slope o f t h e trade-off f u n c t i o n a t known; t h e n A* 12

A12(f;).

=

t h e n i t s o l v e s problem 4-3.

*

Alp

0 which i s t h e n e g a t i v e o f t h e

>

.

f;

Since t h e

Assume t h e f u n c t i o n constraint

f2($

*

*

A 1 2 ( f 2 ( x ) ) = x12 so

*

x

s o l v e s problem 4-3.

~

~i s (

f

~

)

s f i i s binding

*

f2(x )

( i t s n u l t i p l i e r i s h12 which i s g r e a t e r t h a n z e r o ) , t h e n

*

~

fi.

=

Thus

I n a d d i t i o n , %* a u t o m a t i c a l l y

s a t i s f i e s t h e Kuhn-Tucker c o n d i t i o n s s i n c e i t s o l v e s t h e m i n i m i z a t i o n p r o b lem 4 - 2 , which i s t h e same as problem 3-1 w i t h

*

r e p l a c e d by

c2

f p . Thus

t h i s r e v e r s i o n method which does n o t r e q u i r e knowing h 1 2 ( f 2 ) i n f u n c t i o n a l f o r m i s much s i m p l e r t o use. 4.1.4

Regressions The f i r s t segment o f t h e SWT method p r o v i d e s p o i n t s

i n t h e n o n - i n f e r i o r s e t , and t h e t r a d e - o f f r a t e at f .

h12 =

fT

(fl,f2)

=

- a f l / a f 2 evaluated

For each o f t h e s e p o i n t s , one v a l u e o f t h e w o r t h f u n c t i o n W12(f2)

can be found.

The method d e s c r i b e d i n s e c t i o n 3 . 5 . 2 needed

f u n c t i o n a l form i n order t o g e t

X12(x)which

check and i n r e v e r s i o n

d e c i s i o n space.

t o the

t h e r e i s no need f o r a c o n s i s t e n c y check on

~

~ i n(

However, i n many problems

A12

s i n c e t h e s e v a l u e s can be

found a c c u r a t e l y by s o l v i n g m i n i m i z a t i o n problem 3 t h e approach d e s c r i b e d i n t h e p r e v i o u s

~

i s used b o t h i n a c o n s i s t e n c y

-

1.

One can t h e n u s e

section f o r reverting t o the deci-

Thus one can a v o i d b o t h t h e X12(x) . A 1 2 ( f 2 ) and t h e problem o f t r a n s f o r m i n g t h a t i n t o

s i o n space w i t h o u t needing t o know

x12(x)

regression f o r

which f o r some problems

( s u c h as dynamic ones)

can g i v e a c o m p l i c a t e d o r

unusable r e s u l t . However, r e g r e s s i o n s t o f i n d h 1 2 as an a n a l y t i c f u n c t i o n o f f 2 can s t i l l be employed, i f d e s i r e d by t h e a n a l y s t . be s i m p l e r t o u s e t h e s e

F o r c e r t a i n problems i t may

f u n c t i o n s t o determine a d d i t i o n a l values a t which

t o q u e s t i o n t h e DM r a t h e r t h a n r e s o l v i n g t h e € - c o n s t r a i n t problem f o r each a d d i t i o n a l value. mate

A12(f2)

I n g e n e r a l , one would n o t e x p e c t t o be a b l e t o a p p r o x i -

by a s i m p l e p o l y n o m i a l

w i t h any r e a s o n a b l e degree o f accu-

r a c y b u t i n t e r p o l a t i o n s o r c u r v e f i t t i n g procedures o v e r

small

intervals

between known v a l u e s may be u s e f u l i n d e t e r m i n i n g a d d i t i o n a l v a l u e s ofAl2.

f

~

)

4.1.5

F i n d i n g t h e I n d i f f e r e n c e Band Once t h e n o n - i n f e r i o r s o l u t i o n s have been found,

t i o n e d t o f i n d o u t h i s assessments o f w o r t h .

t e r e s t i s where W12(f2) = 0, t h e f u n c t i o n a l f o r m o f found.

the

DFI i s ques-

Since t h e o n l y value o f W12(f2)

in-

need n o t

be

One approach t o f i n d i n g t h e v a l u e s o f f 2 f o r which 1d12(f2) = 0 i s I n t h i s approach, t h e DM i s asked

a type o f exhaustive search technique.

e q u a l l y spaced n o n - i n f e r i o r v a l u e s o f

t o assess t h e w o r t h a t k . . .f 2 , u n t i l bIl2 changes s i g n .

fp

,

f e r r e d v a l u e o f f 2 i s known t o be between t h e l a s t two t e s t e d v a l u e s . example, i f bIl2(f!)

,

e.g.

As soon a s W12 changes s i g n , t h e p r e -

f;

has a d i f f e r e n t s i g n t h a n kJ12(f;+l)

m v a l u e o f f 2 must be between f 2 and " ;f

.

For

then t h e p r e f e r r e d

The search p r o c e d u r e

can t h e n

be r e s t a r t e d w i t h a s m a l l e r i n c r e m e n t o v e r t h e i n t e r v a l (f!,f$i).

Due t o

t h e m o n o t o n i c i t y o f t h e s u r r o g a t e w o r t h f u n c t i o n , i f two d i f f e r e n t

values

f a r e p r e f e r r e d s o l u t i o n s , t h e n any v a l u e between them w i l l a l s o be a 2 p r e f e r r e d s o l u t i o n . Note t h a t even w i t h t h i s approach, n o t a l l o f t h e non-

of

i n f e r i o r v a l u e s need be t e s t e d . Another approach i s a t y p e o f g r a d i e n t approach o r Newton a p p r o x i m a t i o n method. Wl2(f;).

Two v a l u e s o f t h e w o r t h a r e found, e. g . ,

The n e x t v a l u e o f

t h r o u g h t h e two known v a l u e s h i t s

W12

Once a v a l u e o f f 2 i s f o u n d f o r w h i c h f e r e n c e band

i s determined

ferred solution.

=

0 .

and

W12(fg)

fp t r i e d i s t h e one where a s t r a i g h t

line

Mathematically

W12(f2) = 0, t h e r e s t o f t h e i n d i f -

by f i n d i n g t h e w o r t h a t v a l u e s n e a r t h i s p r e -

T h i s l a t t e r approach may r e q u i r e f e w e r q u e s t i o n s t o t h e

d e c i s i o n maker, b u t more c a l c u l a t i o n i s needed t o d e t e r m i n e w h i c h q u e s t i o n A l s o n o t e t h a t f o r t h e v a l u e o f f 2 a t which t h e DM s h o u l d b e q u e s -

t o ask.

t i o n e d ( f 2 ) , t h e t r a d e - o f f r a t i o may n o t have been f o u n d i n t h e f i r s t segment. f

2

One c o u l d t h e n e i t h e r r e s o l v e t h e o p t i m i z a t i o n problem 3-1 f o r

~1

2

o r regression

t e c h n i q u e s on known

trade-off ratio. when W12

Also

n o n - i n f e r i o r v a l u e s t o approximate t h e

note t h a t these

algorithms are equally applicable

i s c o n s i d e r e d as a f u n c t i o n o f A12;

t h e same e q u a t i o n s h o l d i f f 2

i s r e p l a c e d by h 1 2 .

4.2

=

t o f i n d t h e t r a d e - o f f r a t i o e x a c t l y , o r use c u r v e - f i t t i n g , i n t e r p o l a t i o n

T X L T A T I C TWO-OBJFCTIVE E-CONSTRAINT (STE) ALGORITHM A l g o r i t h m s w i l l now be p r e s e n t e d d e s c r i b i n g t h e c o m p u t a t i o n a l p r o -

cedures f o r s o l v i n g

All of

t w o - o b j e c t i v e p r o b l e m s w i t h t h e SWT method.

t h e s e a l g o r i t h m s assume t h a t t h e fererences; they could e a s i l y

DM i s a b l e t o a c c u r a t e l y a s s e s s h i s p r e -

be m o d i f i e d a s d e s c r i b e d i n

section 4.1.2

The first algorithm u s e s the F-con-

if this is an g n t e n a b l e assumption.

s t r a i n t approach b o t h f o r f i n d i n g t h e n o n - i n f e r i o r p o i n t s and f o r t i n g t o t h e d e c i s i o n space t o f i n d t h e p r e f e r r e d s o l u t i o n .

rever-

A flowchart o f

t h i s a l g o r i t h m i s p r o v i d e d i n f i g u r e 4-2. 4.2.1

Ih-e A l g o r i t h m Step 1:

F i n d t h e minimum v a l u e , fZMIN,f o r f

MIN

f2(5)

s.t.

x

E

2

by solving:

T

The s o l u t i o n t o t h i s p r o b l e m i f fZMIN. S t e p 2:

F i n d t h e maximum v a l u e , fZMAX, f o r f2 by s o l v i n g MIN

fl

s.t. 5

(x) i

T

*

*

I f t h e s o l u t i o n v e c t o r t o t h i s p r o b l e m i s 5 t h e n fZMAX i s f2(5 * i s iiiore t h a n one s o l u t i o n v e c t o r t h e n t h e minimum v a l u e o f

x

f2MAX

; if there

f2(x*)

is

'

Step 3:

Set t h e i n i t i a l value f o r

S t e p 4:

MIN

f,(&

s.t.

f (x) 5 2 -

c2 =

fZMAX

-

A where A > 0.

E2

x i T

-

G e n e r a l l y t h e Kuhn-Tucker c o n d i t i o n s c a n be u s e d t o s o l v e t h i s p r o b l e m . O f c o u r s e , a n y o p t i m i z a t i o n t e c h n i q u e w h i c h i s a p p r o p r i a t e c a n be u t i l i z e d . *

x*

Let f;(t2)

be t h e d e c i s i o n v e c t o r w h i c h s o l v e s t h i s p r o b l e m . =

f,(x*);

p l i e r f o r the

The s o l u t i o n i s

e a c h s o l u t i o n s h o u l d a l s o c o n t a i n h12, t h e L a g r a n g e m u l t i c2

constraint.

is

A l s o a c h e c k t o see i f t h e c o n s t r a i n t

b i n d i n g iiiust b e made a s p a r t o f t h i s s t e p . I f t h e c o n s t r a i n t i s b i n d i n g t h e n f 2

=

f 2 so t h a t t h e o u t p u t s o f t h i s s t e p a r e

value f2

and

*

fl(f2)

If the

F~

c o n s t r a i n t was b i n d i n g i n s t e p 4, t h e n s e t

- A ; otherwise set

E~

= f

If

r2

a t the

I f the constraint i s n o t binding then ignore these values.

= c2.

S t e p 5: t 2

A12(f2)

i s greater than

F*

=

(X*). 2 -~

fZMINt h e n r e t u r n t o s t e p 3; o t h e r w i s e

pro-

65

ue stion DeLi-i,

IQ-I fimin

5.t..

E

1,

M.,kcr

I T

' 7 1-1

21 iax

t

iiqure 4-2.

Flowchart f o r S t a t i c Two-Objective t-Constra int Algorithm.

66

MULTTOBJECTIVE O P T ~ ~ Z A T I OIW N WATER RESOURCES SYSTEMS

ceed t o s t e p 6. Step 6 : For each v a l u e

Develop t h e s u r r o g a t e w o r t h f u n c t i o n W12(f2) f 2 a t which t h e worth i s desired,

f o r h i s assessment o f how much

A12(i2)

ask t h e d e c i s i o n maker

additional units of objective

a r e w o r t h i n r e l a t i o n t o one a d d i t i o n a l u n i t o f o b j e c t i v e

* ^

u n i t s o f f 2 and f l ( f 2 )

units of f l .

as f o l l o w s :

f2

fl

, given

f2 H i s assessment can be made on a r e l a -

t i v e s c a l e o f say -10 t o + l o w i t h z e r o s i g n i f y i n g equal w o r t h ; t h i s a s s e s s ment i s t h e v a l u e o f bIl2

a t f 2 . The s e a r c h t e c h n i q u e s d e s c r i b e d i n s e c t i o n

4.3.5 a r e used t o d e t e r m i n e a t w h i c h v a l u e s o f f 2 t h e w o r t h s h o u l d b e f o u n d . Step 7 .

The p r e c e e d i n g

step i s repeated u n t i l t h e e n t i r e

indif-

f e r e n c e band i s f o u n d . Step 8.

Find t h e p r e f e r r e d d e c i s i o n vector

x^ by

solving:

MIN f l ( x )

< f;

s.t. f2(x)

X E T I n many problems t h e r e w i l l be more t h a n one s o l u t i o n

*

f 2 t o step 7.

In

t h a t case, t h e r e w i l l be more t h a n one p r e f e r r e d s o l u t i o n , and s t e p 8 must

*

be r e p e a t e d f o r each f 2 i n o r d e r t o f i n d a l l o f t h e

preferred

decisions

v e c t o r s ; some o t h e r c r i t e r i a must be i n t r o d u c e d t o d e c i d e among them. Step 9:

A sensitivity

analysis

c o u l d be p e r f o r m e d t o d e t e r m i n e

the p o s s i b l e e f f e c t s o f implementing t h e p r e f e r r e d s o l u t i o n . Step 10: 4.2.2

Stop - t h e s o l u t i o n i s r e a d y t o be implemented.

Sampl e P r o b l em The use o f t h i s a l g o r i t h m w i l l be i l l u s t r a t e d by s o l v i n g example 1

from chapter 1.

The problem i s : MIN

x1

MIN

10 -

s.t.

0

X1

-

x2

< x1 < 5

0 < x 2 < 5 Step 1 :

We f i n d fZMINby s o l v i n g : M I N 1 0 - x1 - X2 s.t.

0

X,

5

The s o l u t i o n t o t h i s

x1

problem i s a t

=

0,

0

< x2

5

5.

Since i t i s not

unique, take the one which g i v es t h e minimum value of f ( x ) 2 0 , x 2 = 5 with f 2 ( x ) = 5 . T h u s f 2 MA X= 5 . Step 3:

Let t i n g

Step 4:

Solve

A =

.5, s e t

which i s x1 =

4.5.

c2 =

MIN x1

s . t . 10 -

-

X1

X2

5

E2

0 < X l < 5 0<X2'5 The simDlex method can be used t o s o l v e t h i s f o r a l l of t h e values of

c2

Reformulating t h e s e equations in simplex format

mri S.t.

x1 X1

+ x 2 - S1 + R1

x1 + x2 + where For

s2 s3

5

=

5

S1,

S2,

S 3 a r e s l ack v a r i a b l e s and R1 i s

=

4.5

t h e f i n a l tableau i s

x l '

;

Z

1;

x2

s1

0

0

-1

0

-1

0

s2 x2

0

0

0

0

Thus t h e s o l u t i o n i s . 5 so the

=

10 -

=

1

1

f;(4.5)

=

l2 0

'2

he a r t i f i c i a l v a r i a b l e .

s3 -1

.5

-1

.5

: :I: :

. 5 ; s i n c e S1 and R1 both equal z e ro,

6 - c o n s t r a i n t i s binding a n d t h i s value i s a c c e pta ble .

The value

of

t h e Lagranqe m u l t i p l i e r i s t h e negative of t h e c o e f f i c i e n t under t h e sla c k v a r i a b l e corresponding t o t h e c 2 c o n s t r a i n t ; thus h12(4.5) S t e p 5:

Since t h e

t2

c o n s t r a i n t i s binding, s e t

= E~

1 .O. =

4.0 and go

back t o s t e p 3 . This i s repeated u n t i l t 2 = 0 i s r e a c h e d ; t h e r e s u l t s a r e summarized in t a b l e 4-1. Step 6 : The d eci s i o n maker i s questioned and assume h i s responses

a r e a s given i n t a b l e 4-2. The exhaustive

search type

algorithm

was used t o determine t h e

values of f 2 a t which t h e worth was found. Linear i n t e r p o l a t i o n was f o r those n o n -i n f er i o r values not found i n t a b l e 4 - 1 . Step 7 :

*

The i n d i f f e r e n c e band i s found t o be 2 . 0 s f 2

5

used

2.3.

TABLE 4 - 1 Results of S t a t i c Two-Objective € - C onstra int Problem (Ste p 5 )

52 ( f 2 )

f2

f;(f*)

4.5

0.5

1 .o

4.0

1.o

1 .o

3.5

1.5

1 .o

3.0

2.0

1.o

2.5

2.5

1 .o

2.0

3.0

1.o

1.5

3.5

1.o

1.o

4.0

1.o

0.5

4.5

1.o

TABLE 4 - 2

DM Responses f o r S t a t i c Two-objective F-Constraint Problem

*

f2

1'11 ( f 2 )

4.5 4.0

+ 10 + 8

3.5

+

7.5

3.0

+

4

2.5

+

I

+

0 0.5

-

1

2.0 2.4 2.3

0

1.9

Step 8: t 2

Find t h e p r ef er r ed d eci s i on ve c tor by solving s t e p 4 with

replaced by each

2.0, the result i s

*

f2

in the indifference band.

x;

=

3.0,

For example, f o r

*

f2

=

The preferred decision vector may not be unique. Also note that there tmay be more than one indifference solution or indifference bands, consequently, the associated preferred decision vectors will likely be different. Step 9: At this point a sensitivity analysis should be performed but such work is beyond the scope of this text. Step 10: :top. Notice that even though the trade-off function was linear, this alqorithm had no problems in finding preferred solutions. THE MULTIPLIER APPROACH One of the major problems with the i-constraint algorithm is that the minimizations required may be complicated to solve. One way tocircumvent this difficulty is to use the multiplier approach, a variation of the parametric method, to find the non-inferior points. Consider the following problem: Problem -4-4. ~. __ FlIN fl + A 1 2 f2 4.3

(x)

s.t.

x

E

(x)

T

This is equivalent to the parametric approach where h 1 2 = e2/el. It can be shown? that the solution vector to problem 4-4 solves the following problem: FlIN fl(x)

x*

s.t. f2(X)5 f2(&*) where A 1 2 is the Lagrange multiplier for the f2 constraint. Thus one can 0 and solve problem 4-4, finding the corresponding set the value of h 1 2 f*1 and f;(non-inferior values of the objectives) to be used in determining the worth function. This method avoids the necessity of finding the mini> 0 assures being in the mum and maximum values for c2 since setting non-inferior region; also the minimizations are simpler since there is one less constraint. It is possible for convex two-objective problems to determine a maximum value for in the non-inferior region. Since the trade-off function f;(f2) i s convex,

will be a monotonically decreasing function

of f2, so that the largest value of x12 will occur at the minimum value of f2 . Thus the following problem can be solved:

70

,VlJI,TIOB~JRC2'iVEOPTiMIZATION IiSi bJAYE3 RESOURCES SYSTEMS

s . t . f ( x ) :: 2 -

x

- t-

fzr.iIrj

T

where f2MI,4i s found as i n s e c t i o n 4 . 1 . 1 . ponding t o t h e f 2 c o n s t r a i n t i s 4.3.1

x

~

The Lagrange m u l t i p l i e r c o r r e s -

~

~

~

~

~

.

L i m i t a t i o n s of t h e M u l t i p l i e r Approach There a r e several

proach.

Although

inferior points,

problems t h a t can a r i s e with t h e m u l t i p l i e r ap-

a l l of t h e s o l u t i o n s not a l l of t h e

d u a l i t y g a p problem4

exists.

found by t h i s

approach a r e non-

n o n - i n f e rior points can be found whenthe Hopefully,

one would be a b l e

t o ge ne ra te

enougii p o i n t s t o a c c u r a t e l y determine t h e worth f u n c t i o n s ; any given p r o b lem would have t o be judged on i t s own merits a s t o whether or n o t t h e information generated by t h e m u l t i p l i e r approach i s adequate. A more s ev er e problem i s t h a t

in t h e m u l t i p l i e r approach does

not always correspond t o t h e ncgative

of t h e slope of t h e tra de -off func-

tion.

Consider t h e f e a s i b l e f u n ct i o n al space

S

shown i n f i g u r e 4-3.

If

t h e s l o p e of t h e t r ad e- o f f curve a t p o i n t A i s - h 102 then any l i n e L with slope - A l 2 where h l z 2 A 102 w i l l a l s o f i n d t h e same point A a t t h e minimum. I n t h i s c a s e , t h e values of

h12 >

0

do not correspond t o - d f l / d f 2 .

In

terms of d u a l i t y t h eo r y , any s p e c i f i c value h 102 may not n e c e s s a r i l y c o r r e s pond t o a s t a t i o n a r y p o i n t of t h e Lagrangian t o problem 3-1.

This occurs whenever t h e r e a r e d i s c o n t i n u i t i e s in t h e slope of the t r a d e - o f f f u n c t i o n a n d i s e s p e c i a l l y prominent i n l i n e a r problems. Thus i t i s a d v i s a b l e t o use such a n approach only f o r non-linear problems. In convex non-linear problems, t h i s e f f e c t appears only a t t h e end points of t h e n o n - i n f e r i or s e t ; f o r non-convex t r ad e -off f u n c t i o n s , i t can a l s o o c c u r where t h e d u a l i t y gaps s t a r t and end.

Since t h e s e ina c c ura c ie s a r e pre se nt

a t only a few p o i n t s i n non-linear problems, t h e accuracy of t h e s u r r o g a t e t o o g r e a t l y i f enough values were generated.

worth f u n c t i o n would not be a f f e c t e d 4.4

THE STATIC TWO-OBJECTIVE COMBINED (STC) ALGORITHM An algorithm wi l l now be presented using t h e m u l t i p l i e r approach

f o r t h e f i r s t segment of t h e s o l u t i o n procedure t o lower t h e computational e f f o r t s r e q u i r ed , while r e t a i n i n g t h e e - c o n s t r a i n t approach f o r t h e re ve rsion t o t h e d e ci s i o n space. f i g u r e 4-4.

A flowchart f o r t h i s algorithm i s provided in

71

S

0' Note:

f2

x12

> h

0 12

Both l i n e s f i n d p o i n t A a t t h e minimum. Figure 4-3.

4.4.1

F a i l u r e o f M u l t i p l i e r Approach.

The Algorjthm Step 1 : Find t h e maximum v al u e f o r h 1 2 by solving:

fl(x)

MIN

s . t . f2(x) < f2MIN

X E T

-

where f Z M I Ni s t h e s o l u t i o n t o

MIN f 2 ( x ) s . t .

p l i e r f o r the f 2 c o n s t r a i n t i s X12MAX. d i f f i c u l t one could s e t

XlZMAX

-

m

x

E

T.

The Lagrange m ulti-

I f t h i s s t e p i s computationally too

.

Step 2: S e l e c t an i n i t i a l value X l 2 such t h a t 0

< X12

< XlZMAX.

72

x* ,f I*

(i^, 12)

3

f 2*

IQuestion Decision Maker Does W,,(f,)

2

1

-1

= O?

s . t . f2(X) 5 f*

w S e n s i t i v i t y Analysis

Figure 4-4.

Flowchart f o r S t a t i c Two-Objective Combined Algorithm.

73

Step 3:

Solve s.t.

MIN f l ( $ + X 1 2

x

t

f2(x

T

* ^

x ) and f2(L) t o find fl(A12) The so l u t i o n vector x- i s s u b s t i t u t e d i n t o f ( ~-

*

dnd f 2 ( h 1 2 ) . These values a r e n o t n e c e s s a r i l y unique.

Step 4 :

I f enough information has been generated, g o on t o s t e p 5 ,

i f n o t , choose a new value of Step 5:

h12

>

0 and go back t o s t e p 3 .

Develop t h e s u r r o g at e w o r t h func tion W 1 2 a s follow s:

For each s e t o f values A 1 2 ,

* ^

*

A

f l ( h 1 2 ) , f 2 ( A 1 2 ) a t which t h e w o r t h i s d e s i r e d ,

ask t h e DM f o r h i s assessment of how much f

1

a dditiona l u n i t s o f o b j e c t i v e

* -

a r e worth i n r e l a t i o n t o one ad d i t i o n al u n i t of f 2 , given f2(X12) u n i t s

* ^

o f f 2 a n d f 1 ( A 1 2 ) u n i t s of f l .

* ^

f2(A12).

The search

His assessment i s then t h e value of W 1 2 a t

techniques described in s e c t i o n

4 . 3 . 5 a r e used

to

deterriiine a t which values of f 2 t h e w o r t h should be found. Step 6 :

The preceeding

s t e p i s repeated u n t i l t h e

ference band i s found. Step 7 :

Find t h e p r ef er r ed d eci s i on ve c tor MIrj

x*

entire indif-

by solving:

fl($

These l a s t two s t e p s a r e t h e same a s i n t h e € - c o n s t r a i n t approach. i f t h e r e i s more than one s o l u t i o n t o s t e p 6 , s t e p 7

*

Again,

must be repeated f o r

each of t h e se s o l u t i o n s f 2 i n o r d er t o f i n d a l l of t h e pre fe rre d solutions. Step 8:

A

sensitivity

a n a l y s i s could be performed t o

determine

t h e p o s s i b l e e f f e c t s of implementing t h e pre fe rre d s o l u t i o n . Step 9 : 4.4.2

Stop!

h_mple ProblemThe use of t h i s algorithm wi l l be i l l u s t r a t e d by applying i t t o t h e

following non-linear problem:

2

MIN

fl

=

x:

t

2 x2

MIN

f2

=

xl

t

3 x

2

where x1 a n d x 2 a r e unbounded. Step 1 :

Since f 2 i s unbounded

( f 2 M I N= -

m),

12MAX

cannot

be

determined.

.

Thus, s e t h12MAX = Step 2:

S e t h12

Step 3:

Solve

.

.5

=

2 2 M I N f ' = x1 + 2 x2

+ h12(x1 + 3 x2) . of' = 0

The n e c e s s a r y c o n d i t i o n f o r a minimum i s

(it

i s also sufficient

since f ' i s convex); thus 2 x1 + h 1 2 and Thus f o r h 1 2

=

.5,

Step 4:

+

x2

fi

fy(.5)

=

=

0

3 h12

=

or

x1

f (.5) 2

D i f f e r e n t values o f

-

5 212

o r x2 = - 3 h12/4

0

*

11/32,

=

h12

=

.

- 11/8.

a r e used i n s t e p

2,

and

the

r e s u l t s obtained a r e g i v e n i n t a b l e 4-3. Step 5:

The s u r r o g a t e

worth

function i s

developed,

and i t i s

assumed t h e DM's a s s e s s m e n t s a r e as g i v e n i n t a b l e 4 - 4 . Using t h e g r a d i e n t

f2

s h o u l d be made a t

=

algorithm

- 2 1/8

.

,

i t i s obvious t h a t t h e t h i r d

*

,Il=2 . 7 5 and f,

l i n e a r i n t e r p o l a t i o n f r o m t a b l e 4-3 t o be that for

f o r t h i s problem

f

*

. 1

neighboring

, a quadratic

f2 = - 2 1/8

Once

values

trial

The c o r r e s p o n d i n g t r a d e - o f f i s f o u n d b y =

55/64.

Note

i n t e r p o l a t i o n w o u l d b e more a c c u r a t e

i s found t o be a p r e f e r r e d s o l u t i o n

, other

are t r i e d t o determine t h e e x t e n t o f t h e i n d i f f e r e n c e

band S t e p 6:

The i n d i f f e r e n c e band i s f o u n d t o b e - 2 1 / 8

Step 7:

We f i n d

f o r each f

K'' b y

.

solving:

MIN

2 2 x1 + 2 x 2

8.t.

x

*

:: f 2 5 - 2

*

+ 3 x 2 :: f 2

1

i n t h e i n d i f f e r e n c e band. S i n c e t h e c o n s t r a i n t m u s t be b i n d i n g 2 * t h i s c a n b e s o l v e d b y s u b s t i t u t i n g x = f 2 - 3 x i n t o t h e o b j e c t i v e equa2 * 1 t i o n and u s i n g v f l = 0. F o r example, t h e p r e f e r r e d s o l u t i o n f o r f 2 = -2.0

*

i s f o u n d t o be x,

4.5

=

-4/ll,

*

x2

=

- 6/11 .

S t e p 8:

S e n s i t i v i t y a n a l y s i s c o u l d be performed here.

S t e p 9:

Stop!

THE STATIC TWO-OBJECTIVE MULTIPLIER (STM) ALGORITHM I n order t o

u s e t h e m u l t i p l i e r a p p r o a c h i n t h e second segment a l s o

( t o revert t o the decision

space i n f i n d i n g t h e p r e f e r r e d d e c i s i o n v e c t o r

75

R e s u l t s o f S t a t i c Two-Objective Combined Problem ( S t e p 4 )

0.5

11132

1.o

1 31%

-

2.0

5 112

- 5 112

3.0

12 3 / 8

- 8 1/4

4.0

22

-

5.0

35 518

- 13 314

137 112

- 27 112

10.0

Table

1 3/8

2 3/4

11

4 - 4

DM Responses f o r S t a t i c Two-Objective Combined Problem

- 1 3/8

+ 1.5

- 2 3/4

- 1.5

- 2 118

0

- 2

0

- 2 1/4

- 1

) , i t i s n e c e s s a r y t o know t h e p r e f e r r e d t r a d e - o f f r a t e hY2.

Since t h e

*

s o l u t i o n o f t h e surrogate worth f u n c t i o n gives t h e p r e f e r r e d value f2 , i t as a f u n c t i o n o f f 2 i n o r d e r t o f i n d AY2This

would be necessary t o know

c o u l d be done by p e r f o r m i n g a r e g r e s s i o n on t h e v a l u e s o f f p and h12 f o u n d i n step 3 o f t h e previous algorithm, b u t t h i s could introduce e r r o r s which may be s i z a b l e .

I n some cases, t h e v a l u e o f f

may be known as a f u n c t i o n o f

h12

2 i n the non-inferior region

d i r e c t l y f r o m t h e necessary c o n d i t i o n s

used i n s o l v i n g s t e p 3. F o r t h e s e problems one c o u l d t h e n f i n d t h e i n v e r s e multiplier

= h12(fl). F o r non-convex problems, u s i n g t h e 12 ( f 2 ) ; t h u s approach t o r e v e r t t o t h e d e c i s i o n space may cause some o f t h e

preferred

s o l u t i o n s t o be

function

h

missed due t o t h e p o s s i b i l i t y o f d u a l i t y gaps.

However, i f t h e o t h e r

a l g o r i t h m s p r o v e i m p o s s i b l e t o s o l v e , t h i s approach

can be used anyway t o

g e t one s o l u t i o n .

s t e p s 5 and 6 below.

Finding

Xl2(f2)

would r e p l a c e

76

'

This algorithm's

vc

.

5

SY

accuracy i s guaranteed o n l y f o r problems i n which f2 i s a one-to-one

t h e t r a d e - o f f f u n c t i o n i s n o n - l i n e a r and convex ( w h e r e function o f

h12

Lases, t h e d e v e l o p n i e n t o f t h e w o r t h f u n c t i o n guaranteed t o be v a l i d . trade-off rate

For these

over t h e range o f f2 i n t h e n o n - i n f e r i o r s e t ) .

*

x12

The s o l u t i o n o f

W12

WI2(xl2)

as a f u n c t i o n o f A12 =

is

0 gives the preferred

A

w h i c h c a n t h e n be u s e d i n a p a r a m e t r i c p r o c e d u r e .

f l o w c h a r t f o r t h i s algorithrri i s given i n f i g u r e 4-5. 4.5.1

The Algorith!! S t e p s 1 t h r o u g h 4 a r e t h e same as i n t h e S t a t i c

T w o - o b j e c t i v e Com-

b i t i e d a l g o r i t h m and w i l l n o t b e r e p e a t e d h e r e . S t e p 5:

D e v e l o p t h e s u r r o g a t e w o r t h f u n c t i o n W12(h12) a s

follows:

* -

* ^

F o r each s e t o f v a l u e s >,12, f ( > ) , f (A ) a t w h i c h t h e w o r t h i s d e s i r e d , 2 12 1 12 ask t h e DM f o r h i s assessment o f how much x12 a d d i t i o n a l u n i t s o f o b j e c t i v e f1 a r e w o r t h

i n r e l a t i o n t o one a d d i t i o n a l

unit o f objective

* ^

f 1 ( A 1 2 ) u n i t s o f fl.

u n i t s o f f 2 and

f2 g i v e n

H i s assessiiient i s t h e n t h e

The s e a r c h t e c h n i q u e s d e s c r i b e d i n s e c t i o n 4 . 3 . 5

v a l u e o f W12 a t h 1 2 .

are

u s e d t o d e t e r m i n e a t w h i c h v a l u e s o f x 1 2 t h e w o r t h s h o u l d be f o u n d . Step 6:

The

preceeding

sIep

is

repeated

until

the

entire

i n d i f f e r e n c e band i s f o u n d . Step 7 :

Find the preferred decision vector

x

by solving:

* MIN

fl(x)

s.t.

5

tl

+ h12

f2(&)

T

T h i s i s t h e same p r o b l e m a s s t e p 3 w i t h

replaced by

* A12;

thus l i t t l e

a d d i t i o n a l c a l c u l a t i o n i s needed. Step 8:

A sensitivity

a n a l y s i s c o u l d be performed here t o d e t e r -

iiiine t h e p o s s i b l e e f f e c t s o f i m p l e m e n t i n g t h e p r e f e r r e d s o l u t i o n . Step 9:

Stop!

4.5.2 % n p l e P r o b l e m C o n s i d e r t h e same p r o b l e m a s i n s e c t i o n 4 . 4 . 2 f 2 i s a one t o one f u n c t i o n o f

developed -

~

- 11

i

x1 = =

of

'12

i n the f i r s t

segment o f t h a t example

~ and~ x 2 / = 32 x 1 2 / 4 .

12

as f o l l o w s .

/4 f o r a n y v a l u e o f

( i n step 3)

S u b s t i t u t i n g back i n t o

x12

0.

I t c a n be shown t h a t

The n e c e s s a r y c o n d i t i o n s showed t h a t

f2(x) g i v e s

f2(A12)

Thus f 2 i s a o n e - t o - o n e f u n c t i o n

and t h e S t a t i c T w o - o b j e c t i v e M u l t i p l i e r a l g o r i t h m i s a p p r o p r i a t e .

S i n c e s t e p s 1 t h r o u g h 4 a r e t h e same a s i n s e c t i o n 4 . 4 . 2 ,

we c o n t i n u e w i t h

77

-

Yes Solve Min

s.t.

fl(x) + E

XT2f2(&)

T

. I

1

I S e n s i t i v i t y Analysis

F i g u r e 4-5.

Flowchart f o r S t a t i c Two-Objective M u l t i p l i e r Algorithm.

step 5. Step 5:

The s u r r o g at e worth f u n ct ion i s developed; assume t h a t t h e These a r e l i s t e d a s a

DM's assessments a r e t h e same as in s e c t i o n 4 . 4 . 2 . function cf A 1 2 i n t a b l e 4-5. Table 4 - 5

DM Responses f o r S t a t i c Two-Objective Mu1 t i p 1 i e r Problem A l2

W12% 2 )

0.5

+ 1.5

1.o

- 1.5

0.75

0

0.7

0

0.6

t l

0.8

- 1

The g r a d i en t search type algorithm was used t o determine t h e values of

Xl2

a t which t h e w o r t h was found.

Linear i n t e r p o l a t i o n was used f o r

those n o n - i n f e r i o r values n o t found i n t a b l e 4-3. * Step 6 : The i n d i f f e r e n c e band i s found t o be . 7 i X 1 2 Step 7 :

Solve

2

2

*

MIN x1 + 2 x 2 t X12(x1

t

3 x2).

*

Taking t h e g r a d i en t and s e t t i n g i t equal t o zero fo; values of t h e d eci s i o n v a r i a b l e s a r e found t o be

*

.

.75

$

x1

X12 =

=

-3/8,

. 7 5 , pre fe rre d

*

x2

=

-

9/16.

This can be repeated f o r o t h e r values of x 1 2 in t h e i n d i f f e r e n c e band. Thus approximately t h e same r e s u l t i s obtained with t h i s algorithm a s with t h e s t a t i c two-objective combined algorithm. 4.6

SUMMARY This c hap t er has presented some modification in computational method

from t h e version of t h e s u r r o g at e worth t r a de -off method described inc ha pt e r 3 t o make i t more g en er al l y a p p l i c a b l e and e a s i e r t o use. Three algorithms f o r implementing t h e SWT method

for the

static

two-objective c a se

were presented along with d i s cu s s i o n s of t h e i r a p p l i c a b i l i t y and simple examples of t h e i r use. The € - c o n s t r a i n t ( S T E ) algorithm i s a p p l i c a b l e t o The combined a l l problems b u t may be d i f f i c u l t computationally t o solve . (STC) algorithm can be applied t o non-linear problems with a probable savings i n t h e computation and only a s l i g h t l o s s of accuracy. The m ulti-

p l i e r (STM)

algorithm i s guaranteed only when t h e tra de -off

func tion

convex a n d n o n- l i n ear , b u t i s g e n e r a l l y t h e most e f f i c i e n t procedure.

is

O f course

, one should

keep i n mind t h a t i t i s d i f f i c u l t t o s t a t e

which algorithm i s b e s t f o r any given problem ; i t i s u p t o t h e a n a l y s t t o use h i s d i s c r e t i o n i n determining which one i s a ppropria te f o r h i s s p e c i f i c situation. The question of how well t h e mathematical equations in t h e o b j e c t i v e f u n c t i o n s r e p r es en t

society's true

objectives

is a

common

problem

in

modeling. For example, t o what e x t e n t does t h e number o f man-hours of usage adequately measure r e c r e a t i o n a l o b j e c t i v e s f o r a r e s e i v o i r ? These problems a r e not e x p l i c i t l y taken i n t o account in t h e SWT method; however, they can be somewhat ameliorated d u r i n g t h e process o f i n t e r a c t i n g w i t h t h e decision maker by i n s u r i n g t h a t he understands e x a c t l y what he i s deciding on.

I n general , one should e x e r c i s e

caution when using t h e SWT method

with o b j e c t i v e s f o r which measures a r e e i t h e r poorly defined o r mathematical equations u n av ai l ab l e. As i n a l l approaches t h e r e wi l l probably e x i s t c e r t a i n pathological For example, a n probleiiis in which none o f t h e algorithms a r e a p p l i c a b l e . i n a b i l i t y t o solve t h e Kuhn-Tucker co n d itions could lead t o t h e i n a b i l i t y

t o find

n o n -i n f er i o r

points.

Thus, one must be c a utious when

t h e s e algorithms t o perverse f u n ct i o n s .

The next

applying

chapter w ill apply t h e

SWT method t o dynamic problems. FOOTNOTES 1.

Note t h a t t h e

"

*

"

p r e f e r r ed values;

i s used t o denote both non-infe rior values a n d it

should

be c l e a r

from t h e

contex which i s

intended. 2.

See Wagner

[1969],

H i l l i e r a n d Lieberman

[1967],

Intriligator

3.

This was f i r s t proven by Ev er et t t-19631. Again see Ev er et t [1963], Lasdon [1968], or Gembicki [1973].

[1971], Taha [1971], o r any t e x t on optimization techniques. 4.

REFERENCES

~_I__.

1.

E v e r e t t , H . 111, "Generalized Lagrange M u l t i p l i e r Method f o r Solving Problems of O p t i m u m Allocation of Resources," Operations Research v o l . 1 1 , 1963.

2.

Gembicki, F . , "Vector Optimization f o r Control with Performance and Parameter S e n s i t i v i t y I n d i c e s , " P h . D . D i s s e r t a t i o n , Case Western Reserve Un i v er s i t y , 1963.

3.

H i l l i e r , F . S . and Lieberman, G . J . , Introduc tion

to

Operations Re-

4. 5. 6.

7.

search, Holden-Day, San Francisco, 1967. Intriligator, M.D., Mathematical Optimization and Economic Theory, Prentice-Hall Inc., Englewood Cliffs, N.J., 1971. Lasdon, L.S. ,"Duality and Decomposition in Mathematical Programming" _________ IEEE Transactions, vol. SSC-4, no. 2, 1968. Taha, H.A., Operations Research; An Introduction, The Macrnillan Company, N . Y . , 1971. blagner, H.M., Prjnciples of Operations Research with Applications to Managerial Decisions, Prentice Hall Inc., Englewood Cliffs, N. J., 1969.

Chapter 5 THE SWT METHOD FOR DYNAMIC TWO-OBJECTIVE PROBLEMS

The preceeding ch ap t er s have attempted t o solve problems where t h e o b j e c t i v e s a r e s t a t i c f u n ct i o n s of a v ec tor of de c ision v a r i a b l e s . This chapter will consider t h e a p p l i c a t i o n o f t h e SWT method t o dynamic systems ?/here t h e o b j e c t i v e s a r e

of the system a s well a s

f u n ct i o n s of t h e s t a t e

of t h e d e c i s i on ( c o n t r o l ) v a r i a b l e s , with both s t a t e s and de c isions time I t i s assumed t h a t t h e s t a t e of t h e systeni a t any time i s a dependent. known f u n c t i o n of time and t h e previous s t a t e s and d e c i s i o n s . The following notation w i 1 1 b

used:

Cn[O,tf]

i s the

s e t of a l l continuous func tions

from t h e closed i n t e r v a l [ o , t f ] i n t o Rn; x ( t ) i s t h e f i r s t d e r i v a t i v e of x with r e sp e c t t o t h e independent v a r i a b l e t .

I t wil

be assumed t h a t t h e

ith

o b j e c t i v e can be formulated a s

follows: +

where

x(t)t C P I O , t f ]

iIf

ai(x(t),u(t),t) dt,

=

1,2,

..., n

i s a v ect o r of s t a t e v a r i a b l e s , u ( t ) t - C r [ O , t f ]

vector of c o n t r o l v a r a b l e s , and dent v a r i a b l e

i

is a

tf i s t h e terminal value of t h e indepen-

which wi l l be considered

fixe d and the saine f o r a l l objec-

The i n t e g r a l term can be viewed a s summing the c o n t r i b u t i o n s ( a . )

tives.

t o t h e o b j e c t i v e a l l along t h e t r a j e c t o r y , while

(li

i s t h e c ontribution

t3

t h e o b j e c t i v e of t h e f i n a l s t a t e of t h e system. Note. however, t h a t form with

f i s c a l a r valued.

may be non-commensurable.

not a l l dynamic Droblems can be p u t i n t o t h i s

For example, t h e values of a i a t d i f f e r e n t t such a s t h e i n t e r e s t r a t e

Plultiplying f a c t o r s

a r e coniinonly used t o commensurate t h e values of g e n e r a l , however,

ai

a t different t

.

In

a i a t each t w o u l d be a d i f f e r e n t non-commensurable ob-

j e c t i v e a n d t he problem would become a n i n f i n i t e - o b j e c t i v e problem. There problems.

a r e various types of c o n s t r a i n t s which may a r i s e in dynamic

End p o i n t c o n s t r a i n t s of t h e form g ( x ( t f ) , t f ) : 0 w ill be inc lu-

ded i n t h i s formulation. be

P a t h c o n s t r a i n t s , such a s N(x(t),u(t),t)s 0 can

a i s o included in the following

s t r a i n t s o f t h i s form

development1 , b u t problems withcon-

tend t o be r a t h e r d i f f i c u l t t o a c ttia lly s o l v e ; f o r

t h e sake of c l a r i t y , they wi l l be avoided in t h i s formulation.

Also note

t h a t problems where tf i s a co n t r o l v a r i a b l e ( e . g . minimum time problems) 81

can be handled by modifying t h e necessary c onditions f o r a minimum. This ch ap t er wi l l show t h e r e l a t i o n s h i p between t h e s t a t i c and dynamic problems.

The dynamic problem wi l l be p u t i n t o

a n d t h e Lagrange m u l t i p l i e r s f o r t h es e c o n s t r a i n t s

r e p r e s e n t t h e elements of t h e t r ad e- o f f r a t e m a trix.

& - c o n s t r a i n t form

w ill be shown t o again Algorithms t h a t a r e

analogous t o t h e s t a t i c cas e wi l l be presented f o r the s o l u t i o n of two objective

dynamic problems,

including sample problems

to i l l u s t r a t e their

use.

5.1

INTRODUCTORY A P I A L Y S E The general m u l t i p l e

o b j e c t i v e dynamic

problem can be w r i t t e n in

vector n o t a t i o n a s :

where

E, a,

and 9 a r e t h e n-vectors whose elements

are f . , a i ,

The f e a s i b l e d eci s i o n space T

and

1

respectively .

$.,

wi ll be a subse t of

w i l l thus be complicated ( o r impossible)

t o work w ith.

again look a t t h e f e a s i b l e f u n ct i o n space

S

=

Cr[O,tf] and However, one can

{-f i g ( t ) =

v(x(t),u(t),t)

with ~ ( 0 a)s given and g ( x ( t f ) , t f ) s 0). Note t h a t S i s a subse t of R n . Since t h e s u r r o g at e worth t r ad e- o f f method ope ra te s prim a rily in t h e funct i o n a l space S , t h i s method wi l l be e a s i l y adaptable t o dynamic

problems.

The n o n - i n f e r i o r s e t wi l l again be on t h e boundary of S, and can be re pre sented by t h e t r ad e- o f f f u n c t i o n , with i t s slope represented by t h e t r a d e off r a t e functions. time and can t h e r e f o r e case.

These wi l l a l s o be s c a l a r valued a n d independent be determined

Thus t h e only d i f f e r e n c e

occurs i n t h e d eci s i o n space;

and u t i l i z e d just a s i n t h e

between t h e s t a t i c

t h e f u n ct i o na l space,

of

static

and dynamic problems S, i s i d e n t i c a l in

both c a s e s . 5.2

DYNAMIC PROBLEMS IN €-CONSTRAINT FORM _____

The two o b j e c t i v e dynamic form d i r e c t l y a s f o l l o ws :

problem

can be p u t i n t o

€-constraint

83

However, since the constraint in this form ing substitution is made. Define a new state variable y such y(Q) = 0 .

The problem then becomes:

Thus the constraint is now included as an end point constraint and can be solved by any o f the available methods.2 It will now be shown that the Lagrange multiplier for the E~ constraint (when it is binding) is the value o f the trade-off rate function at f, = C , . First form the Lagrangian:

where A is a scalar Lagrange multiplier for the E~ constraint, is an mvector o f multipliers for the end point constraints, q ( t ) i s a p-vector o f Lagrange multipliers (which are functions o f time) for the system equa-

v 2 ( t ) i s a s c a l a r Lagrange m u l t i p l i e r ( a l s o a func-

tion constraints, and

t i o n o f time) f o r t h e system equation able y .

I t can now be seen t h a t

c o n s t r a i n t s f o r t h e new s t a t e v a r i -

aL/aa2 =

- X .

As in t h e s t a t i c c a s e , only those values of

F2

points i n t h e n o n - i n f er i o r region wi l l be considered. s t r a i n t must be binding. ftf

a,(K(t),!(t),t)

dt

@z(K(tf)) =

f2

which correspond t o Thus, the

Again, when the optimum i s found,L =

f;.

=

t: con2 $ l ( ~ ( t f ) +)

Since t h e t - c o n s t r a i n t i s binding,

E~

=

JO

y(tf)

f

.

Thus A

=

- af*/af

f o r tlie c 2 c o n s t r a i n t i s r e a l l v X 1 2 ( c 2 ) , evaluated a t f 2 5.3

=

1

and t h e Lagrange m u l t i p l i e r 2 i . e . , t h e tra de -off r a t e func tion

'2 .

DVPlAPlIC TWO-OBJECTIVE c-CONSTRAINT (DTE) ALGORITHM This s e c t i o n wi l l p r es en t

algorithms de sc ribing t h e computational

procedures f o r solving m u l t i p l e o b j e c t i v e dynamic problems which a r e a n a l -

ogous t o t h e s t a t i c al g o r i t h ms . The f i r s t of the se uses t h e E-c onstra int approach f o r both f i n d i n g t h e n o n - i n f er i o r points a n d f o r r e v e r t i n g t o t h e d e c i si o n space t o f i n d t h e p r ef er r ed d eci sion v e c t o r . Note t h a t a l l o f t h e computational e f f i c i e n c i e s developed i n t h e func tiona l space in c ha pte r 4 will be a p p l i c a b l e t o dynamic problems. I n p a r t i c u l a r t h e use of 1d12 a s a f u n c t i o n of f z , t h e methods f o r f i n d i n g l i m i t s on f 2 , t h e avoidance of A r e g r e s s i o n s a n d t h e assumption of a c o n s i s t e n t DM w ill be u t i l i z e d . flowchart f o r t h i s algorithm i s presented i n f i g u r e 5-1. 5.3.1

-Algorithm Step 1 :

Find t h e minimum value f o r f 2 by solving:

The s o l u t i o n t o t h i s problem i s f Z M I N. St e p 2:

Find t h e maxinium v al u e f o r f 2 by solving:

85

Figure 5-1.

Flowchart f o r Dynamic Two-Objective & -Constra int Algorithm.

Continued next page.

MULTIOBJEC7’IVE OPTIMIZATION I N WATER RESOURCES SYSTEMS

No

Solve Min

I -

b l ( x (t,))

F i g u r e 5-1.

+

i n tf

Continued

al(x(t),E(t),t)dtl

87

O Y N A M f C TWO-OBJECTI VE PROBLEMS

If the state vector &*(t) then f2,qAX

$,(x*(t,))

=

0

::

!3(L(tf)'tf)

-

and control vector g*(t) a2(<(t),c(t),t)dt.

+

solve this problem, I f there is more than

one state vector x*(t) and control vector i*(t) solving this problem, then f Z M A Xis the minimum value o f

$,(x*(tf))

Step 3: Set the ini-tial value

E~

+

':j

=

fZMAX - a where A > 0 .

a2(c(t),<(t)

,t)dt.

Step 4: Solve the problem $l(x(tf))

MIN

j.:f

+

al(x(t),u(t),t)dt

s.t. i(t)

=

x(x(t),u(t),t);

~ ( 0 )given

dt)

=

a2(x(t),u(t),t);

Y(O)

s(x(tf)tf) .Y(tf)

+

5

=

o

o_

02(X(tf))

:: €2

The general approach to solving this problem is to define the Hamiltonian: H(x(t),u(t),t) = a,(x(t),u(t),t) +d(t)-x(&(t),!L(t),t) + v2(t) . The necessary conditions for a minimum then become functions are omitted for clarity): 1)

aH/au

=

0

(the arguments o f the

The

f i r s t six

find u(t),i(t) to find

conditions are solved

as i n any optimal c o n t r o l problem t o

and

x12.

y ( t ) i n terms o f

Assume t h a t t h e

12'

Then s u b s t i t u t i n g

t2. tions,

Condition 7

can then be used

c o n s t r a i n t i s b i n d i n g , o r y(t,)+$,(x(t,))=

E~

y ( t f ) and

i-(tf)

t h s e q u a t i o n can be s o l v e d f o r

found from t h e f i r s t s i x

i12. If the

t h a n z e r o , t h e n t h e second p a r t o f t h e c o n d i t i o n (Al2

value

>

0)

condi-

i12 i: l e s s i s v i o l a t e d and

t h e assump i o n o f a b i n d i n g c o n s t r a i n t was t h e r e f o r e i n c o r r e c t ; t h u s 0

and t h e c o n s t r a i n t i s n o n - b i n d i n g .

A12

t h e s o l u t i o n i s i n t h e i n f e r i o r r e g i o n , t h i s s o l u t i o n w i l l be i g n o r e d .

i,,i s

t 2 . The v a l u e

f:(f2),

then the value o f

Xl2

back i n t o

f2

=

i ( t ) and & ( t ) .

I f t h e c2 c o n s t r a i n t was b i n d i n g i n s t e p 4, t h e n

S t e p 5:

.

at

which corresponds t o t h e s o l u t i o n o f t h e minimiza-

t i o n problem, can be found by s u b s t i t u t i n g

A

A,2(f2)

If

solution

i s g r e a t e r t h a n z e r o , t h e n t h e a s s u m p t i o n was c o r r e c t and t h e i s i n the non-inferior region.

=

S i n c e a n o n - b i n d i n g c o n s t r a i n t means

E~

=

E~

-

O t h e r w i s e l o w e r r2 t o t h e v a l u e a t t a i n e d b y f 2 a t t h e s o l u t i o n t o s t e p

4 - set

f 2

=

y ( t f ) + d2(X(tf)),

step 4 w i t h the value changes c o n d i t i o n

7

=

where y ( t ) and i ( t ) a r e t h e v a l u e s f o u n d i n 0

substituted.

Note t h a t changing

r e t u r n t o s t e p 3; o t h e r w i s e c o n t i n u e

If

E~

i s g r e a t e r t h a n f2MIN

o n t o s t e p 6.

D e v e l o p t h e s u r r o g a t e w o r t h f u n c t i o n WI2(f2)

S t e p 6:

only

so l i t t l e e x t r a work i s

o f t h e necessary conditions,

r e q u i r e d t o o b t a i n more n o n - i n f e r i o r p o i n t s .

E~

e x a c t l y as i n

t h e s t a t i c case; r e c a l l t h a t t h i s can b e a c h i e v e d s i n c e f2 i s s c a l a r valued.

?,

a t w h i c h t h e w o r t h i s d e s i r e d , a s k t h e DM f o r h i s a s s e s s -

ment o f how much

A 1 2 ( f 2 ) a d d i t i o n a l u n i t s o f o b j e c t i v e fl a r e w o r t h i n r e -

F o r each v a l u e

l a t i o n t o one a d d i t i o n a l u n i t o f o b j e c t i v e *

f2, g i v e n

f2 u n i t s o f

f 2 and

A

fl(f2)

u n i t s o f fl.

-10 t o

+10

H i s a s s e s s m e n t i s made o n a n o r d i n a l s c a l e , s a y f r o m

w i t h z e r o s i g n i f y i n g e q u i v a l e n t w o r t h ; t h i s assessment i s

v a l u e o f W12 a t f p .

The s e a r c h t e c h n i q u e s d e s c r i b e d i n s e c t i o n 4 . 3 . 5

used t o determine a t which v a l u e s o f f2 t h e w o r t h should be found.

the are

Step 7: The preceeding step is repeated until the entire indifference band is found. * (t) and control Vectors Step 8: Find the preferred state vectors * * * u (t) by solving step 4 with E~ rep aced by f2 for each f2 in the indifference band:

x

Since the necessary conditions 1 through 6 are the same as in step 4, they * do not have to be resolved. A l s o the constraint must be binding since f2 must correspond to a non-inferior point. Thus, little extra computation i s * necessary to revert to the decision space. If there i s more than one f2 in step 7, then there is more than one preferred solution, and step 8 must be * repeated for each f2 in order to find all o f the preferred solutions; some other criteria must then be introduced to decide among them. Step 9: A sensitivity analysis could be performed to determine the possible effects of implementing the preferred solution. Step 10: Stop! In most real problems, it will be impossible to analytically solve the siniul taneous boundary value differential equations in the necessary conditions of steps 1 , 2, 4 and 8 . However, there are numerical approximation techniques available wh ch can often be used, e.g , quasilinearization, gradient methods, and ne ghboring extremal methods 3 5.3.2

Sample Problem The use of this algor thm will be illustrated w th the following ex-

ample: Example 5-1:

1

M I N x(1)

+

u2(t) dt 0

MIN

x(1)

1

(u(t) - 5)2 dt

t

0

s.t. ;c(t)

=

u(t)

;

x(0)

=

10

90 where

MULTIOBJECTIVE OPI'IMIZATION IN WATER RESOURCES SYSTEMS and

x

are

u

C1 [0,1].

F

F i n d f2MINby s o l v i n g :

Step 1 :

i'

x(1) +

MIN

-

5)

x(0)

=

(U

2

dt

JO

s . t . ic

u

=

;

10

H = (u - 5)

The H a m i l t o n i a n f o r t h i s problem i s

2

+

v u

The necessary c o n d i t i o n s a r e : 1)

aH

2)

c , = - - aH-

2 ( u - 5 ) + v =

-

au

-

0

ax

i

3)

x(0)

;

=

tion 1 giving

u(t)

1

=

10

v(t) = 1

.

9/2

=

v(1)

;

=

The second c o n d i t i o n i m p l i e s

0

w h i c h can be s u b s t i t u t e d i n t o c o n d i u ( t ) i s a c o n s t a n t , c o n d i t i o n 3 can be

Since

x ( t ) = 10 + 9 t / 2 .

easily integrated t o give

Thus t h e s o l u t i o n

to

this

problem i s fZMIN = 14.75. Step 2:

Find

MIN

fZMAX by s o l v i n g :

j1

+

x(1)

u2 d t

0

s.t.

i

=

; x(0)

u

The H a m i l t o n i a n f o r t h i s problem i s

H

10

=

= u

2

+ v u

.

The necessary c o n d i t i o n s a r e :

l )

aH au

2)

;

3)

i

2 u

=

= =

- aH ax

-

u

+ =

; x(0)

The second c o n d i t i o n i m p l i e s tion 1 giving 10 - t / 2 .

v

u ( t ) = - 1/2.

0

=

0

v(1)

; =

10

v(t)

=

1

=

1

w h i c h can be s u b s t i t u t e d i n t o c o n d i -

Condition 3 i s then i n t e g r a t e d t o g i v e x ( t )

These v a l u e s o f x ( t ) and u ( t ) a r e s u b s t i t u t e d i n t o f 2 =

x(1)

=

+

91

D Y N A M I C TWO-OBJECTIVE PHUBLEMS

Step 3:

S et t h e i n i t i a l value f o r

=

E~

39.0; f o r o t h e r i t e r a t i o n s

A will be 1 . 0 .

Step 4 :

Reformu a t e t h e problem in & - c o n s t r a i n t form:

MIN

fl

I

+

x(l

u2 d t

' 0

s.t.

i

=

u

;

y

=

(u

-

+

Y(1)

x(1)

x(0) 5f

=

10

y(0)

;

0

=

s €2 H

The Hamiltonian f o r t h i s problem i s

u2 + v

=

1

u +

v2(u - 5)

2

The necessary co n d i t i o n s a r e : 1)

aH

-

2u +

v1

2)

.jl

=

- -

-

3)

.j2

=

0

4)

i

=

u

5)

j

=

(u - 5)2

ax

0

2v2(u-5)

$1)

;

$1)

;

;

+

1

+

h12

=

x(0)

=

0

=

10

=

; y (0)

=

0

v l ( t ) = 1 + h 1 2 and v 2 ( t ) = h 1 2

Conditions 2 and 3 g i v e

.

S u b s t i t u t i o n i n t o co n d i t i o n 1 g i v es :

u(t)

=

-

(9 X l 2

1 ) / ( 2 + 2 h12)

I n t e g r a t i n g c o n d i t i o n s 4 and 5 g i v e s :

x(t) Y(t) Assume t h a t t h e

E~

= =

10

+ (9

(11

+

h12

Xl2)

2

- 1) t

/ ( 2 + 2 h12)

t /(2 + 2

h,2)2

c o n s t r a i n t i s binding s o t h a t :

Y(1)

+

x(1)

= E2

S u b s t i t u t i n g equations (2) and ( 3 ) i n t o ( 4 ) g i v e s : + (118 - 8 E ~ h) 1 2 + (159 - 4 (59-4 E ~ )

E

~

=)

0

For

t2

=

39.0 t h e s o l u t i o n i s

x~~

.0153; t h e n e g a t i v e r o o t f o r X 1 2 i s i g -

=

nored because t h i s would n o t meet t h e

X12

*

> 0 requirement.

f 1 ( 3 9 . 0 ) i s now

x 12 = .0153 i n t o t h e e x p r e s s i o n s f o r x ( t ) ( e q u a t i o n s ( 1 ) and ( 2 ) ) and s o l v i n g

found by s u b s t j t u t i n g

fl

=

x(1)

+

u2 dt;

the result i s fy(39.0)

and

u(t)

9.7557

=

JO

S t e p 5:

S i n c e t h e c o n s t r a i n t was b i n d i n g , s e t

E~

=

3 8 . 0 and go back

c2 = 1 4 . 0 ; t h e r e s u l t s

a r e summarized As a check xZ1 was found a t v a l u e s of f l c o r r e s p o n d i n g t o the v a l u e s i n column 2 of t a b l e 5-1; t h e e r r o r between A 1 2 and ' / A l 2 was l e s s t h a n .0001 i n a l l c a s e s . S t e p 6: The d e c i s i o n maker i s q u e s t i o n e d and i t i s assumed t h a t h i s t o s t e p 3.

This i s repeated u n t i l

i n t a b l e 5-1.

r e s p o n s e s a r e a s given i n t a b l e 5-2. The e x h a u s t i v e s e a r c h t y p e a l g o r i t h m was used t o d e t e r m i n e the v a l u e s of

f2

a t which t h e worth was found.

L i n e a r i n t e r p o l a t i o n was

used

t h o s e n o n - i n f e r i o r v a l u e s n o t found i n t a b l e 5-1. Table 5 - 1 R e s u l t s of Dynamic Two-Objective E - C o n s t r a i n t Problem ( S t e p 5 )

*

52 ( f 2 )

f2

fl (f2)

15.0

30.00

9.00

16.0

24.82

3.47

17.0

22.00

2.33

18.0

19.97

1.77

19.0

18.38

1.43

20.0

17.09

1.18

21 . o

16.00

1 .oo 0.86 0.74

22.0

15.07

23.0

14.28

24.0

13.59

0.64

25.0

12.98

0.56

26.0

12.46

0.49

27.0

12.00

0.43

28.0

11.60

0.37

29.0

11.25

0.32

30.0

10.95

0.28

for

93 Table 5-1 ( C o n t ’ d )

X12(f2)

f2

31 .O 32.0

10.69

0.24

10.47

0.20

33.0

10.28

0.17

34.0

10.13

0.14

35.0

10.00

0.11

36.0

9.90

0.08

37.0 38.0

9.83 9.78

39.0

9.76

0.06 0.04 0.02

Table 5

-

2

UM Responses f o r Dynamic Two-Objective € - C o n s t r a i n t Problem

S t e p 7: from t a b l e 5-2. c i s i o n maker. S t e p 8:

f2 15.0

W12(f2) - 7

16.0

- 6

17.0

- 5

18.0

- 4

19.0

- 2

20.0

- 0.5

21 . o

0

22.0

+ 0.5

20.5

- 0.5

21.5

f

0.5

=

21.0 a s can be seen

One p r e f e r r e d v a l u e i s

f

* 2

Note t h a t t h e i n d i f f e r e n c e band i s v e r y small f o r t h i s

hY2 =

s u b s t i t u t e d i n t o e q u a t i o n s ( 1 ) and ( 2 ) t o g e t the =

10

f

2t

de-

*

The v a l u e f 2 = 2 1 . 0 can be used d i r e c t l y i n p l a c e o f

e q u a t i o n ( 5 ) t o g e t the p r e f e r r e d t r a d e - o f f r a t e

x*(t)

directly

and p r e f e r r e d c o n t r o l v a r i a b l e

E~

in

t h i s i s then

1.O;

preferred s t a t e variable

u*(t)

v a l u e of o b j e c t i v e f l can a l s o be found t o be 1 6 . 0 .

=

2

.

The p r e f e r r e d

94

MUL7IOBJECTIVE OPTilvlIZATION IW WATER RESOURCES SYSTEMS

5.4

DYNAMIC TWO-OBJECTIVE COMBINED (DTC) ALGORITHFI

One o f t h e

m a j o r problems w i t h t h e

t-constraint algorithm i s

To l o w e r

t h e minimizations r e q u i r e d a r e o f t e n q u i t e complicated t o solve. the computational

e f f o r t s required,

m u l t i p l i e r approach f o r t h e f i r s t [-constraint space.

f '

If

t)

a l g o r i t h m i s developed

segment o f t h e

s o l u t i o n procedure;

the

approach v i l l s t i l l be used f o r t h e r e v e r s i o n t o t h e d e c i s i o n

x12

a,(x(t),u(t),t)i

dt

i s m i n i m i z e d s u b j e c t t o t h e same system c o n s t r a i n t s ,

=

x(x(t),u(t),t),x(fl)

g i v e n , and g ( x ( t f ) , t f )

Maximum v a l u e s f o r

setting

h12

>

0

t h e s t e p s o f f i n d i n g l i m i t s f o r c2 can be

guarantees a n o n - i n f e r i o r p o i n t , avoided.

s 0,t h e n t h i s i s e q u i v a -

= O ~ / A ~ .S i n c e

l e n t t o t h e p a r a m e t r i c approach w i t h

can be found, as i n t h e s t a t i c case, by

A12

p u t t i n g t h e problem i n c - c o n s t r a i n t f o r m and s o l v i n g i t w i t h f21,,I,.l

using the

J u s t as i n t h e s t a t i c case one can d e f i n e a new o b j e c t i v e :

+

-~ x(

an

that

E~

r e p l a c e d by

where fZMI,{i s f o u n d as i n t h e p r e v i o u s a l g o r i t h m ; t h e m u l t i p l i e r c o r -

r e s p o n d i n g t o t h e c2 c o n s t r a i n t i s t h e n x ~ ~ ~ , ,However, ~ ~ . it i s f e l t t h e need f o r

XlZHAX

computational

burden.

i s n o t l a r g e enough t o j u s t i f y t h e g r e a t l y If

A

~

i s n~ o t known, ~ , one ~c o u l d ~ keep

that

increased increasing

A , ~u n t i l two c o n s e c u t i v e v a l u e s g i v e t h e same n o n - i n f e r i o r p o i n t . S i n c e i t i n v o l v e s one l e s s c o n s t r a i n t , t h i s method s h o u l d t o s o l v e t h a n u s i n g t h e E - c o n s t r a i n t approach. proach a r e t h e same

as f o r t h e

be s i m p l e r

The l i m i t a t i o n s o f t h i s ap-

s t a t i c case; i t i s s u b j e c t t o i n a c c u r a c i e s

f o r problems where t h e t r a d e - o f f f u n c t i o n i s non-convex o r l i n e a r . same r e a s o n i n g as i n t h e s t a t i c case

(see s e c t i o n 4.3.1),

By t h e

t h e use o f t h i s

method i s n o t recommended f o r l i n e a r problems.

However, i t may be used i n

o t h e r cases w i t h h o p e f u l l y

A flowchart o f t h i s

small inaccuracies.

r i t h m i s p r o v i d e d i n f i g u r e 5-2. 5.4.1

The A l g o r i t h m

x12

Step 1 :

Set

Step 2:

S o l v e t h e f o l l o w i n g problem: PIIN

bl(x(tf))

=

h12

+

' O

i12+2(x(tf))

+ ~,,a,(x(t),u(t),t)i

dt

+

11'

ia,(x(t),u(t),t))

algo-

95

11 YN/IiMI I' 'I'WO-0RJKC''T T'JE PFOBLEMS

,

I-

x*(t), ~

-,

,*(t),fl*,

fZ*

Sensitivity Analysis

Figure 5-2. Flowchart for Dynamic Two-objective Combined Algorithm.

MULTIOHJECI'I V E OPT I N I Z A T I O N IPJ MAI'ER RESOURCES SYSTEMS

96

s.t. i(t)

=

x(x(t),u(t),t)

$l(X(tf).tf)

s

; X(O)

given

0

The general approach to solving this problem is to define the Hamiltonian: H(x(t),u(t),t)

=

+

al(r(t),u(t),t) vT -1

*

+

h12a2(x(t),g(t),t)

Y(x(t),u(t),t)

-

Again the arguments of the functions will be dropped for simplicity. necessary conditions for a minimum are:

3)

=

4)

$9

Y

= o

The

; ~ ( 0 )given ;

u

2

0

These are solved as in any optimal control problem to find i(t) and i(t) which can then be substitdted into the original objectives to find

For any value of A 1 2 there may be more than one solution x(t) and y(t); in * * that case there is more than one fl and f2 corresponding to the trade-off rate value Step 3: If it is felt that enough information has already been generated, then proceed to step 4. If not, chose a new value of h 1 2 0 and return to step 2. Step 4: Develop the surrogate worth function W12(f2). For each set * * * of values h I 2 , f1(h12), f2(h12! at which the worth i s desired, ask the DM for his assessment of how much h I 2 additional units of objective f, are * ^ worth in relation to one additional unit of f , given f2(h12) units of f2 2 * * and f 1 (A 12) units of fl. His assessment is then the value of W12 at f;(:12). The search techniques described in section 4.3.5 are used to determine at which values of f2 the worth should be found.

Step 5:

The p r e c e e d i n g

step i s repeated u n t i l t h e e n t i r e i n d i f f e -

rence band i s f o u n d . S t e p 6:

Find t h e preferred s t a t e vectors

x* ( t )

*

and c o n t r o l v e c t o r s

owing problem f o r each f 2 i n t h e i n d i f f e r e n c e band

ven

The c o n s t r a i n t w i l l be b i n d i n g s i n c e

*

f 2 must be i n t h e n o n - i n f e r i o r

s e t , so t h e i n e q u a l i t y c o n s t r a i n t can be r e p l a c e d by an e q u a l i t y .

T h i s can

be s o l v e d j u s t as i n t h e t - c o n s t r a i n t method by a p p l i c a t i o n o f t h e n e c e s s a r y

*

I f t h e r e i s more t h a n one f 2 i n s t e p 5, t h i s s t e p must be r e -

conditions.

peated f o r each

one i n o r d e r t o d e t e r m i n e a l l o f t h e p r e f e r r e d s o l u t i o n s .

T h i s r e v e r s i o n t o t h e d e c i s i o n space does add a g r e a t d e a l o f c o m p u t a t i o n a l complexity,

but t h e e n t i r e a l g o r i t h m i s s t i l l simpler computationally than

the [-constraint algorithm. Step 7 :

A sensitivity

a n a l y s i s can be p e r f o r m e d t o d e t e r m i n e

the

possible e f f e c t s o f implementing t h e p r e f e r r e d s o l u t i o n . Step 6: 5.4.2

Stop!

Sample Problem The use o f t h i s a l g o r i t h m w i l l be

i l l u s t r a t e d w i t h t h e same problem

as was used f o r t h e Dynamic Two O b j e c t i v e r - c o n s t r a i n t a l g o r i t h m i n s e c t i o n 5.3.2. Example 5 - 2 : I

MIN

f 2 = x(1)

+

(u(t)

-

5)’

dt

0

s.t.

i(t) x,u

Step 1 :

Set

x12

=

u(t)

c 1 [0,1]

E

=

1.0

; x(0)

=

10

MULT TOBcJECTIVE OPT iWMIZATION IN WAY'ER RESOIJRCES SYSTEMS

98 Step 2:

HIN ( 1 + h12) x ( 1 ) +

Solve

x

s.t.

u

=

; x(0)

1:

u2 + x12(u-5) =

2

dt

10

The H a m i l t o n i a n f o r t h i s problem i s H = u2 + h 1 2 ( u - 5 ) 2 + v1 u The necessary c o n d i t i o n s f o r a minimum a r e :

1)

aH

2)

+

=

-

3)

x

=

u

+

2u

-

au

aH ax

-

-

=

0

x(0)

;

Condition 2 implies u (t)

+ v1

2 Xl2(U-5) ;

Vl(l)

1 +

=

substituting into condition 1

hI2;

gives

Integrating condition 3 yields:

- 1 ) / ( 2 + 2A12).

( 9 AI2

x ( t ) = 10 + ( 9 A 1 2

x 12

1 +

=

10

=

vl(t)

0

=

- l ) t / ( 2 + 2 X12).

S u b s t i t u t i n g i n t o fl and f 2 y i e l d s : f;(A12)

=

(9 A12-1)2/(2+2 A12)2

f;(X12)

=

(11

For

= 1.0,

A12

-t

+

10

(gX12 - 1)/(2+2 X 1 2 )

+ A12) 2 / ( 2 + 2 Xl2)2 + 1 0 + fy(l.0)

16.0,

=

( 9 x 1 2 - 1 ) / ( 2 + 2 X12)

fi(l.0)

=

21.0

.

The problem i s s o l v e d f o r o t h e r v a l u e s o f

Step 3 :

A12

> 0 and t h e

r e s u l t s a r e summarized i n t a b l e 5 - 3 . Step 4:

-

The DM i s q u e s t i o n e d

i n t a b l e 5-4. Step 5: The r e s t value o f f

*

One p r e f e r r e d v a l u e i s f 2 = 21.0 as can be

o f the indifference 2

assume h i s responses a r e as

band

seen

S o l v e t h e problem:

mi

~ ( i )+

jl

s.t.

x

=

u

: x(0)

j

=

(U-5y

y(1)

+ x(1)

u2

The H a m i l t o n i a n f o r t h i s problem i s The necessary c o n d i t i o n s a r e :

dt 10

=

; y(0) $

directly.

c o u l d be f o u n d by q u e s t i o n i n g t h e D M a t

near 21.0.

Step 6:

given

=

0

21.0

H

=

u

2

+ v 1 u + v 2( u - 5 )

2

.

5 - 3

TABLE

R e s u l t s of Dynamic T w o - O b j e c t i v e Combined Problem ( S t e p 3 )

.25

10.75

30.75

.5

12.53

25.86

.75

14.34

22.91

1.o

16.00

21 .oo

2.5

22.50

16.79

5.0

27.11

15.44

7.5

29.21

15.10

10.0

30.41

14.96

TABLE 5 - 4

DM Responses f o r Dynamic T w o - O b j e c t i v e Combined Problem f2

bJl 2 ( f 2 )

14.96

- 10

15.10

- 6.5

15.44

- 6

16.79

- 5.5

21 .oo

0

22.91

t l

1)

;n* 3H

-

2 u + v1 + 2 y ( u - 5 )

2)

\i,

=

aH

3)

c2

=

0

4)

x

=

u

5)

j

=

(U-5)(

6)

y(1) + x(1)

0

-

ax

v2(l)

; ;

$1)

;

x(0)

0 1 + XI2

x12 10

; y(0) =

=

= =

=

=

0

21.0

These e q u a t i o n s can be s o l v e d as i n t h e example i n s e c t i o n 5.3.2. t i o n i s found t o be x * ( t )

=

10

+

2 t , u * ( t ) = 2, f *1

=

16.0, f;

=

The s o l u 21.0

.

T h i s s o l u t i o n method was much s i m p l e r t h a n t h e dynamic t w o - o b j e c t i v e s - c o n s t r a i n t method.

The l a t t e r

s o l u t i o n of a

required t h e

n o n - i n f e r i o r p o i n t found,

while t h i s

qua dra tic equation f o r each

approach required on1.y a d i r e c t sub-

s t i t u t i o n . The d i f f e r e n c e i n computational complexity i s even more pronounced f o r

more complicated problems.

f u n c t i o n i s convex and n o n - l i n ear , method.

Since in t h i s there

problem t h e

tra de -off

i s no l o s s of accuracy with t h i s

The r ev er s i o n t o t h e d eci s i o n space r e q u i r e s more e f f o r t with t h e

dynamic two-objective

combined algorithm,

b u t i t does not o f f s e t t h e gain

in t h e f i r s t segment. 5'5

DYFIAFIIC TWO-OBJECTIVE MULTIPLIER (DTII) ALGORITHA ~

I t is the decision

obvious t h a t t o use space would

t h e m u l t i p l i e r approach f o r re ve rsion t o

make t h e computation even e a s i e r .

*

In

orde r t o

accomplish t h i s , t h e p r ef er r ed t r ad e- o f f r a t e X 1 2 corresponding t o t h e pre-

*

I n some problems t h e non-infe rior value of f 2 may be known as an a n a l y t i c f u n ct i o n of x 1 2 from the necessary c onditions One could then in s t e p 2 of t h e dynamic two-objective combined a lgorithm . * * f i n d t h e i n v e rs e f u n ct i o n X1 2 ( f 2 ) a n d thus know X 1 2 = X 1 2 ( f 2 ) . I n o t h e r c a s e s , a r e g r es s i o n could be performed on t h e values of X 1 2 and f 2 found in s t e p 2, although t h i s may introduce l a r g e e r r o r s . For non-convex problems, using t h e m u l t i p l i e r method f o r r e v e r t i n g t o t h e de c ision space may a l s o cause some of t h e p r ef er r ed s o l u t i o n s t o be missed due t o t h e p o s s i b i l i t y of d u a l i t y gaps. However, i f t h e o t h e r methods prove t o be impossible t o s o l v e , t h i s approach may be used anyway, re pla c ing s t e p s 4 and 5 below. Thus, t h i s a l g o r i t h m ' s accuracy i s guaranteed only f o r problems with nonl i n e a r , convex t r ad e- o f f functions(when f 2 i s a one-to-one func tion of A 1 2 over t h e range of f 2 in t h e n o n - i n f er i o r s e t ) . For the se c a s e s , the be st * way of f i n d i n g X 1 2 i s t o use t h e approach of developing the worth W12 a s a f u n c t i o n of A 1 2 . The s o l u t i o n of W 1 2 ( h 1 2 ) = 0 gives t h e pre fe rre d tra de -off * rate X12. A flowchart of t h i s algorithm i s given in f i g u r e 5-3. f e r r e d value f 2 must be known.

5.5.1

The Algorithm Steps 1 through 3

a r e t h e same a s in t h e dynamic two-objective com-

bined algorithm a n d wi l l n o t be repeated he re . Step 4: of values

Develop t h e s u r r o g at e w o r t h func tion W 1 2 ( ? t , 2 ) .

* ^

A12, f l ( x 1 2 ) , f 2 ( X I 2 ) a t which the worth i s d e s i r e d , ask t h e DM

f o r h i s assessment of how much

h12

a d d i t i o n a l u n i t s of o b j e c t i v e *

w o r t h i n r e l a t i o n t o one a d d i t i o n a l u n i t of and

* ^

f l ( X 1 2 ) u n i t s of f l .

The search

For e a c h s e t

* -

f 2 given

fl

are

A

f2(X 12) u n i t s of f 2

His assessment i s then t h e value of W 1 2 a t X 1 2 .

techniques described

i n s e c t i o n 4.3.5 a r e used t o determine a t

v Sensitivit

Figure 5-3.

Anal s i s

Flowchart f o r Dynamic Two-Objective M u l t i p l i e r Algorithm.

102

MULTIOB,TECT TVE OPTIMIZATION TN WATER RESOlJRCES SYSTEMS

which values of Step 5:

x 1 2 t h e worth should be found. The preceeding

step i s

repeated u n t i l t h e e n t i r e i n d i f f e -

rence band i s found. Find t h e p r ef er r ed s t a t e ve c tors 5- t ) and c ontrol ve c tors solving t h e following problem f o r each h Y 2 in the indifference

Step 6:

u*(t) by band.

+1 ( x ( t f ) )+ t

+ 2 ( x ( t f1)

a,(x(t)

s.t. i(t)

+

11'

{a l ( x ( t

,u(t), t )1 d t

x ( L ( t ) , L ( t ) , t ) ; ~ ( 0 )given

=

*

thus t h e Note t h a t t h i s i s i d e n t i c a l t o s t e p 2 with h 1 2 replaced by necessary c o n di t i o n s need not be resolved a n d l i t t l e e x t r a computation i s necessary . Step 7 :

a n a l y s i s could be performed t o determine t h e

A sensitivity

p o s s i b l e e f f e c t s of implementing t h e p r ef erre d s o l u t i o n . Step G :

Stop!

This procedure i s by f a r t h e e a s i e s t t o s olve . 5.5.2

Sample Problem This algorithm wi l l be i l l u s t r a t e d with the same problem a s the pre-

vious two a l g o r i t h ms . 5-2 ( s e c t i o n

Since s t e p s

1 through 3 a r e t h e same a s in example

5 . 4 . 2 ) they wi l l not be repeated here.

Step 4: The s u r r o g a t e worth f u n ct i o n W 1 2 ( ~ , 2 ) i s developed a n d i t i s assumed t h a t t h e DM's assessments a r e t h e same a s given in example 5-2. These a r e given a s a f u n ct i o n of h 1 2 i n t a b l e 5-5.

*

One p r ef er r ed value h 1 2 i s 1 . 0 a s can be seen d i r e c t l y . The

Step 5:

r e s t of t h e i n d i f f e r e n c e band could be found by questioning t h e DM a t values of

x12

near 1 . 0 . Step 6: Solve t h e problem: 2 x(1 ) +

FlIN

s.t.

i

=

1;

{ u 2 + (u-5)')

u ; x(0)

=

dt

10

Since t h e necessary co n d i t i o n s a r e t h e same a s in s t e p 2 , t o give

x*(t)

=

10 + 2 t ,

u*(t)

=

2,

*

they can be used

f ; = 21.0, f l = 16.0.

I t i s obvious

103 TABLE

5 - 5

DM Responses f o r Dynamic T w o - O b j e c t i v e M u l t i p l i e r Problem

52 - 10

10.0

7.5

-

6.5

5.0 2.5

-

6

-

5.5

+

1

1 .o

0

.75

f r o m t h i s s i m p l e example t h a t t h e dynamic t w o - o b j e c t i v e m u l t i p l i e r a l g o r i t h m provides a g r e a t saving i n computational e f f o r t . 5.6

SUMMARY

This chapter

has demonstrated t h e a p p l i c a t i o n o f t h e s u r r o g a t e w o r t h

t r a d e - o f f method t o dynamic problems. the

SLIT

method f o r t h e dynamic

w i t h discussions The

o f t h e i r a p p l i c a b i l i t y and

t - c o n s t r a i n t (DTE)

d i f f i c u l t t o solve

a great

6-constraint

s i m p l e examples

but the reversion t o the decision

savings i n computational e f f o r t .

space in

i n t h e f i r s t segment may i n -

The m u l t i p l i e r a l g o r i t h m

(DTrl)

i s guaranteed but effects a

only great

Thus, i f t h e & - c o n s t r a i n t a l g o r i t h m p r o v e s

o r impossible t o solve,

despite i t s inaccuracies,

prob-

f o r t h e f i r s t seg-

t e c h n i o u e may s t i l l p r e s e n t d i f f i c u l t i e s ;

t h e t r a d e - o f f f u n c t i o n i s convex and n o n - l i n e a r , the multiplier

algorithm

may be

t o a t l e a s t g e t an a p p r o x i m a t e answer.

c h a p t e r w i l l c o n s i d e r t h e use o f t h e two o b j e c t i v e s .

o f t h e i r use.

t o a l l problems b u t may be

saving i n computation

o f t h e m u l t i p l i e r technique

troduce inaccuracies.

difficult

implementing

presented, along

The combined (DTC) a l g o r i t h m can be a p p l i e d t o

ment ( f i n d i n q n o n - i n f e r i o r p o i n t s )

when

for

case were

( t h i s d i f f i c u l t y i s even more s a l i e n t i n dynamic

n o n - l i n e a r problems w i t h

a d d i t i o n , t h e use

algorithms

algorithm i s applicable

lems t h a n i n s t a t i c ones ) .

which uses t h e

Three

two o b j e c t i v e

SNT

The

used, next

method i n problems w i t h more t h a n

104 FOOTNOTES 1.

The m o d i f i c a t i o n s n e c e s s a r y f o r s u c h c o n s t r a i n t s c a n be f o u n d i n a n y optimal control

t e x t such as

B r y s o n and Ho

[1969],

o r A t h a n s and

F a l b [1966].

2.

A g a i n see B r y s o n a n d Ho [1969] o r A t h a n s and F a l b [1966!.

3.

Ibid.

___ REFERENCES

1.

Athans,

M.

and F a l b , P.

L., O p t i m a l C o n t r o l ;

_______ T h e o r y and I t s A p p l i c a t i o n s , M c G r a w - H i l l ,

2.

B r y s o n , A . E.,

a n d Ho, Y . C . ,

Waltham, Mass.,

1969.

An I n t r o d u c t i o n t o t h e

N.

Y., 1966.

A p p l i e d O p t i m a l C o n t r o l , G i n n and Co.,

Chapter

6

THE SWT METHOD FOR STATIC n-OBJECTIVE PROBLEMS The previous ch ap t er s have considered t h e a p p l i c a t i o n of t h e SWT meof m u l t i p l e o b j e c t i v e problems with only two

t h o d t o t h e s p eci al c l a s s

jectives.

These approaches wi l l be generalized in t h i s a n d t h e next

t e r t o problems with function

W.. 1J

n

objectives.

The d e f i n i t i o n of t h e surroga te

and p r ef er r ed s o l u t i o n s wi ll be modified,

of a n a c c u r a t e d eci s i o n maker wi l l be s t u d i e d . c i e s found f o r two o b j e c t i v e

problems wi ll be

obchapworth

a n d t h e assumption

The computational e f f i c i e n extended t o the

n objective

c a s e , and algorithms f o r t h e s o l u t i o n of the se problems w ill be presented. LURROGATE WORTH FUNCTIONS

6.1

Consider t h e w o b j e c t i v e problem in c - c o n s t r a i n t form: ~

Probl ern 6-1.: ~

~

MIN

fl(&)

s.t.

f(r)

$

5

X L T

.-

where f t Rn-'

i s t h e v ect o r 1 ( f 2 , f 3 , . . . , f n ) T and

t

Rn-'

i s t h e ve c tor

( I ? ,f 3 , ..., t n )T . I f a l l t h e c o n s t r a i n t s a r e binding, then t h e solution * n-1 t o t h i s problem produces t h e values of /i and f l a t f = 5 , where $ E R i s t h e v e c t o r of Lagrange m u l t i p l i e r s

-1 (h12,

x13,

. . . , A ~ a~n d) ~ f; i s

the

a t f i x ed values of t h e othe r o b j e c t i v e s . It is * . and f l wi l l depend on a l l of the values f k , k = 15 T 2 , 3 , . . . , n and n o t j u s t f . ; thus each A . i s a func tion of 2 = ( f 2 , f 3 ,.., f n ) , J * 1J and t h e n o n - i n f er i o r value of f l , f l , i s a l s o a func tion of f. n o n - i n f e r i o r value of

fl

obvious t h a t f o r any j,

A

A t t h i s p o i n t , a review function i s a p p r o p r i a t e .

of t h e

d e f i n i t i o n of

t h e surroga te worth

The value of t h e s u r r o g ate worth func tion W . . i s t h e de c i1J sion maker's assessment of how much ( s ay on a s c a l e from - 10 t o f 10 with

D e f i n i t i o n 6-1:

-.

zero s i g n i f y i n g i n d i f f e r e n c e )

he p r e f e r s

f o r one marginal u n i t of

given t h e

,..., f corresponding t o

f., J

tra ding x . . marginal u n i t s of f i 1J values of a l l of t h e objectives

. . Note t h a t W . . i 0 means t h a t the DM does 1J 1J W.. < 0 means t h a t t h e DM p r e f e r s not t o make 1J such a trade, and W . . = 0 implies i n d i f f e r e n c e . 1J Since t h e remainder of t h i s book d e a l s only with finding t h e surro-

fl

n

A.

p r e f e r making such a t r a d e ,

105

g a t e worth, t h e word " w o r t h " wi l l a l s o be used t o mean surroga te worth. I n order t o a s s e s s t h e worth W . a t a given value f . f o r t h e j t h o b 1J * J j e c t i v e , t h e DM will need t o know t h e values h . and f l corresponding t o 1J f . . However, f o r any given f . t h e r e wi l l be many d i f f e r e n t values of ), . J * J 1J a n d f l , depending on t h e values of f k , k = 2 , 3 , ..., n , k f j , a n d thus t h e r e w i l l be many d i f f e r e n t values of t h e worth. The re fore , t h e w o r t h cannot be considered a s a f u n ct i o n of

f . al o n e; W . must a l s o be a func tion of f. The J 1J a t f i s t h e DM's assessment of how much A l j ( f ) a dditiona l u n i t s

value of W lj of f l a r e w o r t h in r e l a t i o n t o one ad d i t iona l u n i t of f ., given t h e i - l s t J * component of f u n i t s of o b j e c t i v e f i ( f o r i = 2 , 3 , . . . , n ) and f , ( f ) u n i t s of One could a l s o extend t h e approach o f s e c t i o n 3 . 5 . 2 of considering W . fl. 1J a s a f u n c t i o n oc h l j ; i n t h i s extension W . w ill be a func tion of A Just 1J -1 . as i n t h e two o b j e c t i v e c a s e , t h i s can only be done when A i s a one-to-1 one f u n c t i o n of f or e l s e t h e w o r t h a t some values o f Al w ill not be unique.

I t i s p o s s i b l e a s a f i r s t approximation t o consider W . . a s a func tion of f . 1J J alone i n o r d e r t o narrow down t h e range of t h e pre fe rre d s o l u t i o n . 5.2

P R_ E F_E R_R E_D_SOLUTIONS ~ _

AND CONSISTENCY

x.. of t h e t r ad e- o f f r a t e matrix ( t h e matrix of a l l x.. 1J 1J j = 1 , 2 , . . . , n , i # j ) can be considered a func tion of f; t h e f i r s t

Any element for i ,

segment of t h e SWT method would produce

X.. 1J

a s a func tion of ( f l , f 2 , . . , f i - l ,

f i + l , . . . , f n ) b u t s i n ce f l i s a f u n ct i o n of

x.. 1J

region, - afi/;>:. J

If

f 2 , f 3 , . . , f n i n t h e non-infe rior can be considered a f u n ct i o n o f ( f 2 , f 3 ,.., f n ) = f. Since i .=. 1J

, E u l e r ' s chain r u l e f o r p a r t i a l d e r i v a t i v e s

;iz / - dy

(ax = ay

az

-

ax

)

-.

can be used t o g et t h e following r e l a t i o n s h i p between t h e x . . : 1J

S i m i l a r l y , using t h e f a c t t h a t ax/ay X1J ..(f) ~

=

=

l / s , i t follows t h a t :

l / AJ1 . . (f)

(2)

The g e n er al i zat i o n of t h e previous s e c t i o n shows t h a t W . . i s a funct i o n of

( f l , f 2 ,... , f i - l ,

of f 2 , f 3 , . . . ,f n ,

1J

fitl,

. . . ,f n ) . However, sinc e f 1 i s a func tion

Ld.. can be considered a func tion of

E.

When t h e r e a r e

n

o b j e c t i v e s t h e r e are1:* - n worth f u n c t i o n s W . . (f),i = 1 , 2 , ... ,n, j = 1J 1 , 2 , . . . , n , i # j . Each value W . . ( f ) i n d i c a t e s t h e D M ' s assessment of "how 1J f a r " A . . ( f ) i s from t h e negative of t h e r a t e of change ( i n t h e d i r e c t i o n of 1J -

f . ) o f t h e s o c i a l i n d i f f e r e n c e surface, a t t h e p o i n t i n t h e f u n c t i o n a l J . T h i s i s i m p o s s i b l e t o d e p i c t g r a p h i c a l l y , b u t can be space (f;(z);E) viewed as t h e g e n e r a l i z a t i o n o f f i g u r e 3-3. The a c t u a l

numerical value o f

r e l a t i v e assessments,

b u t when

equal z e r o , t h e s o c i a l

the worth functions a r e again o n l y

a l l o f t h e worth

i n d i f f e r e n c e and

f u n c t i o n s simultaneously

trade-off

f u n c t i o n s a r e tangent.

Thus t h e f o l l o w i n g d e f i n i t i o n s can be made.

___-__ D e f i n i t i o n 6-2:

i s d e f i n e d as any n o n - i n f e r i o r p o i n t

A preferred solution

* *

. . .n,

( f ;f ) i n t h e f u n c t i o n a l space such t h a t a l l W . .(-f*) = 0 f o r i = l , Z , 1 -1J

-D~e _f i _n i_t i_o n_ 6 - 3 :

A

preferred decision vector

d e c i s i o n v e c t o r &* such t h a t f,(x*) The r e l a t i o n s h i p s

=

f:

and

is

defined

x(&*)

=

as any f e a s i b l e

?*.

f u n c t i o n s w i l l now be d e r i v e d .

between t h e w o r t h

There a r e two assumptions t h a t must be made.

The f i r s t i s t h a t t h e t r a d e -

o f f r a t e x . . i s a good d p p r o x i m a t i o n t o t h e change which would o c c u r i n t h e 1J n o n - i n f e r i o r v a l u e o f f . when f . i s changed by one u n i t and a l l t h e o t h e r 1 J o b j e c t i v e s are h e l d constant. I n g e n e r a l t h i s w i l l o n l y be e x a c t l y t r u e i n t h e l i m i t as t h e i n c r e m e n t s go t o zero, t i o n as l o n g as valid,

Ifi

and

1

b u t w i l l be t r u e as an

>> l A i j l .

approxima-

I f t h i s assumption

i s not

t h e n t h e q u e s t i o n s asked o f t h e DM s h o u l d be m o d i f i e d so t h a t he i s

comparing c A . .

1J

that

I f j / >> 1

CA..

u n i t s o f f . w i t h c u n i t s o f f . where c > o i s a number such J' t h e change w h i c h would o c c u r i n t h e

i s a good a p p r o x i m a t i o n t o

1J n o n - i n f e r i o r value o f

o b j e c t i v e s a r e held constant.

f . when

f . i s changed b y c u n i t s and a l l t h e o t h e r J Note t h a t i f c i s t o o s m a l l , t h e n c A . . w i l l

he i n d i s t i n g u i s h a b l e f r o m z e r o

t o t h e DM,

1

1J

and he w i l l be u n a b l e t o answer

the questions. The

second assumption

r a t e l y assess h i s p r e f e r e n c e s

i s t h a t t h e d e c i s i o n maker i s a b l e t o accui n t h e sense o f always

v a l u e s f o r t h e i n d i f f e r e n c e band. to

determine

ences.

how

well

i d e n t i f y i n g t h e same

An e x p e r i m e n t was performed by F e i n b e r g ?

decision

makers c o u l d assess

their prefer-

He found t h a t i n a t h r e e o b j e c t i v e problem (power, c o s t and economy

i n a u t o m o b i l e s e l e c t i o n ) , d e c i s i o n makers would t r a d e power f o r c o s t , f o r economy and economy f o r power is,

a t some p o i n t s .

cost

T h i s means t h a t t h e DM

i n r e a l i t y , i n d i f f e r e n t a t t h e s e p o i n t s and y e t i s u n a b l e t o r e c o g n i z e

this fact.

That i s ,

he i s a s s i g n i n g

w o r t h when he has no b a s i s f o r i t . must be c a r e f u l i n

positive or

A conclusion

negative values t o t h e i s t h a t i n p r a c t i c e , one

e x c l u d i n g f r o m t h e i n d i f f e r e n c e band v a l u e s whose w o r t h

functions a r e near zer o . If t h e DM i s unable t o accu r at el y a s s e s s h i s pre fe re nc e s, then t h e r e will g e n e r a l l y be no p o i n t a t which a l l of t h e W . .(f)= 0. Therefore, i t 1J will be assumed t h a t t h e DM i s reasonably a c c u r a t e . A n a l t e r n a t i v e i s d i s cussed a t t h e end of t h i s s e c t i o n . Theorem 6-1: __ Proof:

W . .(f) = 0 1J

ginal u n i t s of

fi

If W i j ( f )

0 then l d . . ( f )

=

0.

=

J1

means t h a t t h e r e i s a n equal preference f o r

A . (.):

mar-

f. J of t r ad i ng any c onsta nt m ultiple c

point

1J a t t h e non-infe rior

and one marginal u n i t of

Note t h a t t h e worth A. .(f) (f;(F);f-). 1J marginal u n i t s of f . f o r c marginal u n i t s of f . a t (fy(2);L)i s t h e same 1 J f o r any value of c f o r which t h e f i r s t assumption holds. Specifically, l e t c

then t h e r e i s an equal preference f o r one marginal u n i t of f i

1/Aij(f);

=

and l / h . .(f)marginal u n i t s of f . .

Using

J

1J

equation

(2)

i t i s found t h a t

t h e r e i s a n equal preference f o r one marginal u n i t of f i

and h . . ( f ) margiJ1 Then by t h e d e f i n i t i o n of W . . ( f ) , i t follows J1 -

*

-

nal u n i t s of f . a t ( f , ( f ) ; z ) . J t h a t Wji(+) = 0. Q.E.D. Unfortunately,

t h e r e i s no

simple r e l a t i o n s h i p

between the values

b I . . ( f ) , W . ( f ) and W . . ( f ) because of t h e f a c t t h a t the s c a l e s on which the 1J kJ 1J worth i s assessed a r e o r d i n al and thus two d i f f e r e n t values of t h e worth -

~~

cannot be added o r mu l t i p l i ed with any meaning.

However,

i t can s t i l l

be

shown t h a t n - 1 of t h e w o r t h f u n ct i o n s ( W . .(f), j = 1 , 2 , . . .n , j # i ) a r e 1J s u f f i c i e n t t o determine t h e p r ef er r ed s o l u t i o n s . The following relationships w i l l be used. Theorem 6-2: j f k

, then

W.

Jk

(f*)

Given =

f"

0.

*

such t h a t W . .(f) 1J

=

0 and

blik(f*)

=

0 where

*

-

W . .(f*)= 0 implies t h a t W..(f ) = 0 from theorem 6-1, so t h a t 1J J1 x t h e r e i s a n equal preference f o r h . . ( f ) marginal u n i t s of f . and one rnar-

Proof:

* *

*

ginal u n i t of f i a t ( f , ( f );f

?',-

; similarly

wik(?*)

=

o

J

implies

w k 1. (-f * )

=

0

so t h a t t h e r e i s an ?qua1 preference f o r

A . ( f * ) marginal u n i t s of f k and

one marginal u n i t of f i a t ( f l ( f ) ; f ) .

A c o n s i s t e n t DM y i e l d s

* *

*

kl -

an

equal

preference f o r h . . ( f * ) marginal u n i t s of f . and h k i ( f ' k ) marginal u n i t s o f f k . J1 J Thus, ~ . . ( f * ) / h ( f * ) car g i n al u n i t s o f f j and one marqinal u n i t of f k w ill be equaj\ypref!irred. Using equation ( 1 ) shows t h 3 t A . marginal u n i t s Jk of f i and one marginal u n i t of f k a r e e qua lly pre fe rre d a t the point (f;(f*);f*).

A t this

W . ( f * ) = 0. Q.E.D. Jk p o i n t t h e d e f i n i t i o n o f p re fe rre d s o l u t i o n can be modified.

Thus

Definition .. 6-4: A preferred solution is defined as any non-inferior point -* ) = 0 for j = 2 , 3 , (f;;f~*) in the functional space such that Idlj (f Note that the difference between definitions 6-2 and 6-4 is that in the forrner the property \&I. .(f*) = 0 must hold for all i , while in the latter it 1J need hold only for i = 1 . This new definition will be siloem to be equivalent to the previous one by the following theorem: ~~Theorem 6-3: Solving b l l j ( f ) = 0 simultaneously for j = 2,3, ..., n is equivalent to solving bl..(f) = 0 for i = 1,2, . . . , n, j = 1,2,..., n, j # i. 1J = 0 for all i and j, then it solves Proof: Obviously if f-* solves Hi;(!) the subset W,j(f) = 0 for j = 2,3: . . . , n. If f" solves W .(f) = 0 for j = 13 2 , 3 , ..., n, then by theorem 6-2, I,lij(f-*) = 0 for any i = 1,2 ,..., n , j= 2,3 , . . . , n, i # j, since Wli(f*) = c) and W . ( f * ) = 0. Also Wil(f*) = 0 for 13 any i = 2,3, n from the theorem 6-1. Therefore, all o f the ldij(f) = 9 for i = 1,2 ,...,n,j = 1,2, ...,n, i # j and the two problems are equivalent. ~

~

~

...,

Q.E.D.

If the DM is unable to accurately assess his preference then there will generally be no point at which all 0.i t i l e Hij(f) = 0. In this case, one could solve the sets o f n-1 simultaneous equations id. .(f)= 0 ; j = 1J 1 : 2 , . . . ,n, j f i separately for each i to get n different solutions and then dcfine the preferred solution f* as the average:

c;

f

*

~-

=

n ),

*

f./n. However, for the remainder of this book it will be assumed

-1

i =1 that tbe inaccuracies of the DH are negligible. 6.3 C OMP UTAT IONA L E F F I C I ENC I E S This section will attempt to extend the computational efficiencies devolored for the two objective case to n objective problem. Limits on the values for t . reversion to the decision space to find the preferred J' decisitn vector, and the use of regressions and search techniques will be studiec'. 5.3.1 Limits on t . J Unfortunately, when there are more than two objectives, it i s impossible to use the same approach as in the two-objective case to determine a rnaximuni value in the non-inferior region for the objectives f ., j=2,3,. . ,n. J Recall that when there were only two objectives, the maximum value of f2 was found by finding the solution &* to the problem mi fl(XI

s.t.

xc

T

110

~ ~ L ~ ’ I ~ B , J ~ OPTIMIZATTON T T v E I N WATER RESOURCES SYSTEMS

fZMAX = f 2 ( & * ) . Any value of y which gave f 2 ( K )

and setting

i nferi or since jectives,

f l ( x ) :: f l ( K * )

however, a value of

by d e f i n i t i o n of y*.

x which

gives

f2(y)

>

>

f2(& *) was

For t h r e e o r more obf2(K*) i s n o t neces-

s a r i l y i n f e r i o r s i n c e even though f l ( x ) :: f,(K*) i s s t i l l t r u e , f . ( x ) < f . ( x * ) J J,maximum may hold f o r some j g r e a t e r than 2 . T h u s f 2 ( 5 * ) i s no longer the

.

value f o r f 2 . problems.

However, t h e r e a r e ways of finding maximum values f o r c e r t a i n

I n many problems, tile c o n s t r a i n t s w ill determine a maximum

feas-

i b l e value f o r each o b j e c t i v e f . which can be viewed a s an upper bound f o r J I n o t h e r words, t h e f e a s i b l e s e t S i n t h e func tiona l space i s bounded in “j. 211 d i r e c t i o n s . I n o t h e r problems, one can determine a maximum value of any objective f .

a t f i x ed values of t h e o t h e r o b j e c t i v e s

J

fk, k

=

2,3,

...n ,

Fixing t h e values of t h es e n - 2 o b j e c t i v e s reduces t h e problem t o a

k # j.

two-di-

mensional one, so t h a t t h e maximum value of f . can be found by solving: J MIN

fl(y)

s . t . f k ( x ) 5 Ek X

k

=

2,3,

. . . ,n,

k # j

E T

x*

If t h e so l u t i on t o t h i s problem i s , and a l l of t h e E~ c o n s t r a i n t s a r e bindina, then t h e maximum value of f . a t t he se fixe d values of o t h e r objectives i s

*

J

f.(x ).

Note t h a t f o r d i f f e r e n t values of the o t h e r o b j e c t i v e s ,

J -

t h e r e i s no guarantee t h a t t h i s wi l l s t i l l be t h e maximum value of f . t h i s j’ problem would have t o be resolved w i t h t h e new E~ in orde r t o find the new maximum value f o r f . . I n a d d i t i o n , t h i s approach may prove q u i t e d i f f i c u l t J

t o imp1 ement. The minimum value f o r each o b j e c t i v e f . can s t i l l be determined. I t will be t h e so l u t i o n t o t h e problem

MIN

fj(x)

5.t.

x

E

J

T

Reversion t o t h e Decision Space

6.3.2

The reversion t o t h e d eci s i o n space can be performed analogously t o

*

y and pre fe rre d value f l of o b j e c t i v e f l can be found a s described in s e c t i o n 3 . 5 . 3 by s o l ving t h e following problem: Problem 6 - 2 : -_____ t h e two o b j e c t i v e cas e.

*

MIN

fl($

The p r ef er r ed d ec ision vector

111

f(x) ,< f"

s.t.

X

c T

The a l t e r n a t i v e approach i s as d e s c r i b e d i n s e c t i o n 3 . 5 . 2 , Problem 6-3:

namely:

Solve t h e n-1 s i m u l t a n e o u s e q u a t i o n s :

_ _ _ _ I _

*

I I ~ ( ~ ( =L &l ) ) such t h a t x m e e t s t h e Kuhn-Tucker c o n d i t i o n s f o r problem 6-1. Theorem 6-4: Any s o l u t i o n

x*

t o t h e problem 6-2 a l s o s o l v e s problem

6-3. P r o o f : Assume -~ f(x

) ::

*

Ll(L)

i s known. Then

/ly

= /Ll(f*).

Since a l l t h e

constraints

must be b i n d i n g ( t h e i r m u l t i p l i e r s a r e t h e components o f

a r e a l l g r e a t e r than zero),

then

f(x*) = ?*.

Thus

/11 (f(&*))

=

*

A1 which

$

so &*

would be found by problem 6-3. Again

x*

a u t o m a t i c a l l y s a t i s f i e s t h e Kuhn-Tucker c o n d i t i o n s s i n c e i t

s o l v e s t h e m i n i m i z a t i o n problem 6-2, w h i c h i s i d e n t i c a l t o problem 6-1 the s p e c i f i c value

g

*

=

.

for

Q.E.D.

Thus i t i s n o t necessary t o know Al(f)

i n f u n c t i o n a l form. Note t h a t

t h e approach o f s o l v i n g problem 6 - 2 i s g e n e r a l l y s i m p l e r . 6.3.3

M a t i p 1 e Regressions

*

and fl a r e now f u n c t i o n s o f f2,f 3,...,fn, i t i s generally 1j even more i m p e r a t i v e t h a n i n t h e two o b j e c t i v e case t o a v o i d t h e m u l t i p l e Since h

r e g r e s s i o n s w h i c h would be necessary t o f i n d t h e i r f u n c t i o n a l forms. t u n a t e l y by s o l v i n g problem 6-1 f o r any g i v e n v a l u e o f the values

&,(f)

and

worth f u n c t i o n s W s o l v i n g problem

(f), j

d i f f e r e n t v a l u e s o f each t h e f u n c t i o n a l forms. t o determine

=

for

5 which i s binding,

g a r e found and t h u s t h e v a l u e o f t h e 2 , 3 , ... n, a t f = 5 can be developed. Thus by

ff(L) a t

lj -

6-1

For-

q

=

d i f f e r e n t values o f

g which a r e b i n d i n g , q

j = 2,3,..., n, can be f o u n d w i t h o u t knowing

Wlj,

The o p t i o n o f f i n d i n g t h e f u n c t i o n a l forms i n o r d e r

more v a l u e s a t w h i c h t o ask

t h e DM

a d d i t i o n a l questions

is

s t i l l a v a i l a b l e , b u t t h e number o f p o i n t s n e c e s s a r y t o g e t an a c c u r a t e m u l t i p l e regression w i l l generally

be i n o r d i n a t e l y l a r g e and w i l l t h u s n o t be

included i n these algorithms. A l t e r n a t i v e l y ,

i n t e r p o l a t i o n o r curve f i t t i n g

t e c h n i q u e s can be used i n p l a c e of r e g r e s s i o n s . 6.3.4

r i n d i n g t h e I n d i f f e r e n c e Band The search t e c h n i q u e s f o r f i n d i n g t h e i n d i f f e r e n c e band d e s c r i b e d i n

c h a p t e r f o u r can be extended

t o t h e n - o b j e c t i v e case,

a l t h o u g h now

there

a r e n-1 w o r t h f u n c t i o n s which must s i m u l t a n e o u s l y equal z e r o . For t h e f i n d values

exhaustive search technique,

f o r the

n-1

one would q u e s t i o n t h e

w o r t h f u n c t i o n s a t e q u a l l y spaced p o i n t s

DM t o i n the

I f none of t h es e h a d a l l n - 1 worth func tions equal t o zero,

function space. then

c l o s e s t t o zero would be

o t h e r values near t h e ones

tried until the

i n d i f f e r e n c e b a n d i s found. Some e f f o r t can be saved by n o t t r y i n g a l l of t h e e q u a l l y spaced p o i n t s on t h e f i r s t pass. Rather, a s soon a s a point i s f o u n d where a l l of t h e worth f u n ct i o n s a r e near z e ro,

the search procedure

should be r e s t a r t e d with smaller increments from t h a t point. Again, i t may c u r v e - f i t t i n g o r re gre ssion t o known

be necessary t o apply i n t e r p o l a t i o n ,

n o n - i n f e r i o r values in o r d er t o have t h e information t o question t h e DM. A g r a d i en t approach

can a l s o be

used t o determine which func tion

space value t o t r y n ex t .

The g r ad i en t approach r e q u i r e s the e quiva le nt of information a b o u t d e r i v a t i v e s . Therefore, n-1 values of each of t h e n - 1 w o r t h f u n c t i o n s a r e required t o determine t h i s information f o r an

t o t r y next a f t e r f o 0

w (f; f1 = fo ~~

A2, f 3 ) ,

+

13

-

n-objec-

For example, in a t h r e e o b j e c t i v e problem, t o determine w h a t

t i v e problem.

=

(f;.f;.)', f;,

W12

f!

one must find + A3) a n d W13(f;.

(f: +

W12

fi +

A3).

A2, f i ) ,

Then

J--1(W,7(To),

This approach may a l s o r e q u i r e i n t e r p o l a t i o n s , c u r v e - f i t t i n g , o r r e gression when t h e

DM must be questioned a t non-infe rior values

the trade-off r a t i o s a r e n o t known. e n t approach

Both

a r e a l s o a p p l i c a b l e when t h e

for

which

t h e exhaustive search and gra diA-space surroga te w o r t h

func-

t i o n s a r e used by r ep l aci n g f . with A .. J 15 6.4

THE STATIC n-OBJECTIVE €-CONSTRAINT (SNE) ALGORITHM

Algorithms d es cr i b i n g t h e use of t he SWT method f o r n-obje c tive p r o b lems will now be presented.

The

f i r s t of the se uses t h e

6-c onstra int

ap-

proach b o t h f o r f i n d i n g t h e n o n - i n f er i o r points and f o r r e v e r t i n g t o t h e d e c i si o n space. The water resources problems in c ha pte r 8 w ill be solved by t h i s method a n d thus no example i s presented here. i n f i g u r e 6-1. Step 1 :

A flowchart i s Drovided

Find t h e minimum value of f . by solving: 3

MIN

fj(x)

113

T Set j

2

Question "cision s.t.

X-

E

Maker

T

j - l s tComponent o f f+,lin

1Lax Yes

find

i f possible

t Choose

c,,,

.>

g

>

fmin

I Sens i t i v i ty Ana1,ys i s

Yes

F i g u r e 6-1.

Flowchart f o r S t a t i c n-Objective c-Constraint Algorithm.

s.t.

x

E

T

The s o l u t i o n t o t h i s problem i s t h e repeated f o r a l l j

=

2,3,

j-lst

component of

fMIN;

this stepis

...,n.

I f p o s s i b l e t h e maximum v al u es , fqAx should be found here. Step 2: S e t t h e i n i t i a l values f o r Solve

Step 3:

MIN

f >

fMIN

fl(x)

s . t . f($ 5 X E T

*

x

Let fr(L)

=

be t h e d eci s i o n vector which s o l v es t h i s problem. The s o l u t i o n i s f , ( K * ) ; each s o l u t i o n should a l s o c onta in .,(.), t h e vector of La-

grange m u l t i p l i e r s f o r t h e c o n s t r a i n t s . binding then

If a l l of t h e

c constraints *

f so t h a t t h e outputs of t h i s s t e p a r e f l ( f ) a n d

=

I f any of t h e g c o n s t r a i n t s a r e not binding then

f

-. = i.

_n,(f)

ignore

are at

the se

values. Step 4 :

I f enough information

s t e p 5 ; otherwise s e l e c t new values of

has been generated then proceed t o

c>

hMIN and return

One

t o s t e p 3.

method of s e l e c t i n g new values i s t o s t a r t with very l a r g e values f o r r a n d decrease each f.(x) ~~

:

i.

J

by some number

A . i 0 each i t e r a t i o n i f t h e c o n s t r a i n t J i f t h i s c o n s t r a i n t i s not binding then s e t

'j i s binding;

"j

-

f.(X*). J --

DM i s p o s s i b l e on a re a l time b a s i s , then t h e

I f i n t e r a c t i o n with t h e

search techniques described i n

section 6.3.4

can be used in choosing new

values of i_ . Step 5 :

Develop t h e s u r r o g at e w o r t h func tions W 1 2 ( f )

follows: For each s e t of values d e s i r e d , ask t h e

c, k 1 ( f ) ,

and f;(f)

,. . . ,Wln(f)

a t which t h e worth

as is

DM f o r h i s assessment of how much h l j ( f ) a dditiona l u n i t s

o f objective f l a r e w o r t h

i n r e l a t i o n t o one a dditiona l u n i t of o b j e c t i v e

u n i t s of f l and t h e j - l s t component o f u n i t s of f . . H i s f . given f;(f) J J assessment on a s c a l e of -10 t o +10 with zero signifying equal preference i s the value W l j ( f ) . This i s repeated f o r a l l j = 2 , 3 , ..., n . Step 6:

Repeat s t e p 5 u n t i l a value

*

f i s found such t h a t a l l of

the worth f u n c t i o n s W . ( f * ) , j = 2 , 3 , . . . , n equal z e r o . Other values near * 1J f can be t r i e d t o determine t h e ex t en t of the i n d i f f e r e n c e band. Step 7 : The p r ef er r ed d eci s i o n v ector E* i s found by solving:

~

MIN

f,(?)

s.t.

f(x) I f*

c r T

*

If t h e r e i s more than one s o l u t i o n f t o s t e p 6 , then t h i s s t e p must be r e peated f o r each one i n order t o f i n d a l l o f t h e pre fe rre d s o l u t i o n s . Step 8 :

A s e n s i t i v i t y a n a l y s i s should be performed t o determine t h e

p o s s i b l e e f f e c t s of implementing t h e p r ef erre d s o l u t i o n . Step 9 : 6.5

Stop!

THE STATIC n-OBJECTIVE MULTIPLIER (SNM) ALGORITHM I n order t o determine t h e zeroes of t h e worth func tions and thus t h e

.____

preferred s o l u t i o n s , values f o r t h e

i t i s g en er al l y necessary

worth f u n c t i o n s ;

t o have a l a r g e number

thus a l a r g e number

minimizations i n t h e f i r s t segment a r e r equire d. proach g e n e r a l l y i s much more e f f i c i e n t than t h e generating n o n - i n f er i o r p o i n t s , problems d e s p i t e t h e p o s s i b l e

i t is

of s o l u t i o n s

Since t h e m u l t i p l i e r apE-c onstra int approach in

useful in

n-obje c tive

non-linear

i n accu r aci es caused by non-convexities

s e c t i o n 4 . 3 . 1 ) . For l i n e a r problems, t h e E-c onstra int approach w ill not too d i f f i c u l t due

t o t h e a v a i l a b i l i t y of t h e simplex method.

algorithm w i l l not be presented here; f i r s t t h r e e s t e p s of

it

t h i s algorithm

of

t o the

can be generated

by

(see be

The mixed using

the

followed by s t e p s 5 through 9 of the

- c o n s t r a i n t al g o r i t h m. Ilhen t h e r e a r e more than two o b j e c t i v e s , t h e r e i s no simple means of f i ndi ng ,iljMAX. For t h i s cas e i t i s assumed t h a t XljMAX = for j = t

2,3, ... n.

A flowchart of t h i s algorithm i s given in f i g u r e 6-2.

Step 1 :

Choose i n i t i a l values f o r

Step 2 :

Solve

MIN f l ( x ) + s.t.

x

E

-1

Al

>

0

.

f(x) -

1

The s o l u t i o n v ect o r &* i s s u b s t i t u t e d i n t o f l ( x ) and

f(x) t o

find f;(Al)and

E*(L, 1. Step 3 :

I f enough information has been generated, go on t o s t e p

i f n o t , choose a new value f o r Step 4:

A1

>

4;

0 and go back t o s t e p 2 .

Develop t h e s u r r o g a t e w o r t h func tions

as follows: For each s e t of values A1, f;(Al),f*(Al)

Wl2(A1) ,. . . , Wln(Ll) a t which t h e w o r t h

is

st

A ., which i s t h e j - 1 1J a d d i t i o n a l u n i t s of o b j e c t i v e f l a r e w o r t h in r e l a t i o n t o st one a d d i t i o n a l u n i t of o b j e c t i v e f . , given f;(Ll) u n i t s of f l and t h e j - 1 J component of f*(/i,) u n i t s of f . His assessment on a s c a l e of -10 t o +10 i s J' This s t e p i s repeated f o r a l l j = 2 , 3 , . . . , n . t h e value of W l j ( A l ) .

d e s i r e d , ask t h e DM f o r h i s assessment of how much

component of

116

t Solve ::in j f l ( x )

I

s.t.

.+

4;

- f(x)/ -

4

x

E

T

X*>fl*(AJ1 ,f*(h 1

Has enough information

"'O

k t i o o s e new

4

]

been aenerated?

u Sensitivity Analysis

Figure 6 - 2 .

Flowchart f o r S t a t i c n-Objective Multiplier Algorithm.

117

I 1’7’

rj-

*

j =

Step 5: Repeat s t e p 4 u n t i l Ly i s found such t h a t W . . ( A ) = 0 f o r * 1 J -1 2 , 3 , . . . n. Additional values near L, may be t r i e d t o determine t h e

e x t e n t of t h e i n d i f f e r e n c e band.

*

Find t h e p r ef er r ed d eci s i on ve c tor 5 by solving

Step 6 :

MIN

fl(x) + L1T* f(x) *

s.t. 5

t

T

Since t h i s i s t h e same problem a s s t e p 2 with

L1

replaced by

4; ,

little

a d d i t i o n a l computation wi l l be necessary. A s e n s i t i v i t y a n a l y s i s could be performed t o determine t h e Step 7 : p o s s i b l e e f f e c t s of implementing t h e p r ef e rre d s o l u t i o n . Step 8 : Stop! 6.6

SUMMARY This chapter has extended t h e algorithms of

chapter four f o r s t a t i c

n o b j e c t i v e problems. The W . a r e found t o be func tions of f 2 , f 3 , . . . , f n (or 15 A12,X,3, . . . , A l n ) and t h e p r ef er r ed s o l u tion i s defined t o be t h a t f o r which a l l of t h e addition,

Wlj,

j = 2,3,

...,

n

simultaneously equal

t h e use of t h e o t h er worth f u n c tions

W.

.,

1J

i

=

z e ro.

2,3, . . . ,n , j

In =

1 , 2 , . . . ,n , j # i , i s shown t o be redundant under c e r t a i n assumptions. The computational e f f i c i e n c i e s of avoiding r e g r e s s i o n s , finding maximum values using search techniques, and r ev e rsion t o the de c ision space devefor ‘j’ loped f o r two o b j e c t i v e problems were modified f o r the n-obje c tive c a se .

Two algorithms were presented; t h e c - c o n s t r a i n t approach i s useful f o r l i n e a r (and l i n e ar i zed ) p r o b l ems , b u t i n o t h e r problems may r e q u i r e too much coiiiputational e f f o r t in o r d er t o g en er at e enough values t o have v a l i d approximations t o t h e worth f u n ct i o n s . The m u l t i p l i e r approach i s subje c t t o inaccuracies

due t o non-convexities

in t h e trade-off f u n c t i o n s , b u t i s t o solve a s i t has n-1 fewer c o n s t r a i n t s . Mixed algorithms can a l s o be used a s a compromise between e f f i c i e n c y a n d accuracy. The next ch ap t er wi l l d i s cu s s t h e modifications necessary f o r dynamic n - o b j e c t i ve problems. g e n e r a l l y much simpler

FOOTNOTES 1.

This new

d e f i n i t i o n of

book . 2.

See Feinberg [1972].

f wi l l be used throughout t h e r e s t of t h e

118

MUL,TIOBdECTTVE OPTIMIZATION IN WATER HESOURCES SYSTEMS

REFERENCES

1.

Feinberg, A . , "An Experimental I n ve stiga tion o f an I n t e r a c t i v e Approach f o r M u l t i - c r i t e r i o n Optimization with An Application t o Academic Resource Al l o cat i o n , "

Western Management Science I n s t i -

t u t e , Working paper no. 186, 1972.

Chapter 7 THE SWT METHOD FOR DYNAMIC n-OBJECTIVE PROBLEMS This chapter i s both a m o d i f i c a t i o n e x t e n s i o n o f c h a p t e r 5.

o f t h e p r e v i o u s c h a p t e r and an

A n a l y s i s o f dynamic problems w i t h

more t h a n

two

...,@n)T

and

o b j e c t i v e s and a l g o r i t h m s f o r t h e i r s o l u t i o n s a r e p r e s e n t e d . INTRODUCTORY ANALYSIS _____

7.1

For n o t a t i o n a l convenience d e f i n e

a

t

R’-’

=

. . . , an)T.

(a2,a3,

d e f i n e d such t h a t $ ( t ) =

& ER”’

(@,,@,,

=

A v e c t o r o f new s t a t e v a r i a b l e s

a ( t~) , u ( t ) , t )

and y(0) =

f o l l o w i n g n o t a t i o n i s as d e f ned i n c h a p t e r 5.

0 ; the

y

F

is

R”’

remainder o f t h e

?he problem i n € - c o n s t r a i n t

f o r m t h e n becomes: _ Problem _ 7-1 _ : ~M I N

$, ( x ( t f )

s.t. i ( t ) A

t )

x(x(t),u(t),t)

=

a(x(t),u(t),t)

=

M t f ) ’ t f ) d t f )

+

; ~ ( 0 )g i v e n ;

y(0)

0

=

< 0s

p(x(t,))

g

Note t h a t a g a i n t h e o b j e c t i v e f u n c t i o n s a r e s c a l a r v a l u e d s i n c e t h e y

Also

a r e i n t e g r a l s over time.

n o t e t h a t problems where

tf

i s a control

v a r i a b l e ( e . g . minimum t i m e p r o b l e m s ) , as w e l l as t h o s e w i t h p a t h c o n s t r a i -

s

nts g(x(t)u(t),t)

0 can

a l s o be handled w i t h t h e a l g o r i t h m s d e s c r i b e d

in

t h i s c h a p t e r by m o d i f y i n g t h e n e c e s s a r y c o n d i t i o n s f o r a minimum.l Since t h e

f u n c t i o n a l space f o r

t h e f u n c t i o n a l space f o r c h a p t e r w h i c h was t h i s chapter.

...,

s t a t i c ones

conducted i n t h e

t h e dynamic problem i s t h e same

f u n c t i o n a l space w i l l be a p p l i c a b l e t o

I n review, t h e worth f u n c t i o n s M..,

n, i f j a r e f u n c t i o n s o f f2,f,,...,

preferred solution simultaneously.

* *

(fl;f

)

as

(Rn), a11 a n a l y s i s i n t h e p r e c e e d i n g

fn o

i s where a l l

i = 1,2,

1J r h 12,h13,

W .(f) =

1J

-

0

..., n,

...,

>ln,

j = 1,2,

and

f o r j = 2,3,

the

...,n

The two assumptions o f c h a p t e r 6 w i l l s t i l l be m a i n t a i n e d .

They a r e t h a t t h e t r a d e - o f f

r a t e i s a good a p p r o x i m a t i o n t o t h e change which

would o c c u r i n t h e n o n - i n f e r i o r v a l u e o f fi when f . i s changed by one u n i t , J and t h a t t h e DM i s a b l e t o a c c u r a t e l y assess h i s p r e f e r e n c e s . Then, t h e o rems 6-1 t h r o u g h 6-3 w i l l s t i l l h o l d .

The f o l l o w i n g

definition w i l l also

119

be made. Definitipn 7-1:

A p r ef er r ed control v ect or a n d a pre fe rre d s t a t e v e c t o r a r e

*

defined a s any f e a s i b l e control v ect o r u- ( t ) and

any f e a s i b l e s t a t e ve c tor

?* ( t ) such t h a t

The computational dynamic problems. f o r each

f. J

e f f i c i e n c i e s in c ha pte r 6

I n particular,

minimuin

a r e also applicable t o

( a n d sometimes maximum) values

can be found, a n d search techniques can be used t o f i n d where

or i n t e r p o l a t i o n s t o f i n d 1: a n d f l a t o t h e r values of f m a y be used t o avoid -1 I n a d d i t i o n , t h e pre fe rre d control and s t a t e vecresolving problem 7-1. t o r s can be found andlogously t o s e c t i o n 6 . 3 . 2 .

a l l of

7.2

t h e worth f u n ct i o n s

?HE DYNAMIC An

equal zer o .

n-OEJECTIVE

algorithm

*

c

Also,

multiple

-CONSTRAINT ( D N~E ) ALGORITHM _ _ _ -~

wi l l now be presented which uses t h e € - c o n s t r a i n t ap-

proach f o r b o t h

f i n d i n g t h e n o n - i n f er i o r

decision space.

A flowchart i s provided in f i g u r e 7-1.

Step 1 :

reverting t o the

J

repeated f o r a l l j =

points a n d f o r

Find t h e minimum value of f . by solving:

The s o l u t i o n t o t h i s problem i s t h e j - l s t j

re gre ssions

=

2,3, ..., n.

component of

t h i s step i s

I f p o s sible t h e maximum value of each f . , J

2 , 3 , . . . , n should be found a s described in s e c t i o n 6.3.1. Step 2:

Choose i n i t i a l values f o r 5

St e p 3 :

Solve problem

7-1.

> f

-#IN ' The s o l u t i o n i s

should a l s o c o n t ai n A 1 ( c ) , t h e v ect o r of Lagrange constraints.

I f a l l of t h e s e c o n t r a i n t s a r e binding,

f * ( t ) ; each s o l u t i o n 1 multipliers f o r the then

=

f so

that

= 5. I f any of t h e outputs of t h i s s t e p a r e f;(f) a n d L,(f) a t t h e value the c o n s t r a i n t s a r e n o t binding then ignore the se va lue s.

121

Choose

fmax

>

5

> f

.

Figure 7-1. Flowchart f o r Dynamic n-Objective €-Constraint Algorithm Continued next page.

i,2

t

1-

Are a l l : - C o n s t r a i n t s I Yes

Binding?

I

Q u e s t i o n D e c i s i o n Maker I

F i g u r e 7-1 Continued.

'lo

I

Step 4:

I f enough i n f o r m a t i o n has

been g e n e r a t e d ,

t h e n proceed

and r e t u r n t o s t e p 3 . t o s t e p 5; o t h e r w i s e s e l e c t new v a l u e s f o r 5 > f +II N Step 5: Develop t h e s u r r o g a t e w o r t h f u n c t i o n s V12(?), . . .,Ldln(f) as follows.

For each s e t o f v a l u e s

f. A l ( f ) ,

f;(f)

a t which t h e worth i s

s i r e d , a s k t h e DFI f o r h i s assessment o f how much hlj(f)

de-

additional units o f

o b j e c t i v e fl a r e w o r t h i n r e l a t i o n t o one a d d i t i o n a l

u n i t o f objective f . J' g i v e n f;(f) u n i t s o f fl and t h e j - l s tcomponent o f f u n i t s o f f . . H i s a s s J essment on a s c a l e o f -10 t o + l o i s t h e v a l u e W . ( f ) . T h i s i s r e p e a t e d f o r 1J a l l j = 2,3, ..., n . Repeat s t e p 5 u n t i l

Step 6: j = 2,3,

...,

n.

z* i s *f o u n d such t h a t W 15. ( f* )

A d d i t i o n a l v a l u e s near

e x t e n t o f t h e i n d i f f e r e n c e band. Step 7: u*(t)

0 for

=

-

f may be t r i e d t o d e t e r m i n e

Find t h e p r e f e r r e d s t a t e v e c t o r

by s o l v i n g problem 7-1 w i t h 5 r e p l a c e d by f*.

*

*

x

the

( t ) and c o n t r o l v e c t o r I f t h e r e i s more

than

f t o s t e p 6, t h e n t h i s s t e p must be r e p e a t e d f o r each one i n

one s o l u t i o n

order t o f i n d a l l o f t h e p r e f e r r e d s o l u t i o n s . S t e p 8: A s e n s i t i v i t y a n a l y s i s c o u l d be p e r f o r m e d h e r e t o d e t e r m i n e the p o s s i b l e e f f e c t s o f implementing t h e p r e f e r r e d s o l u t i o n . Step 9: 7.3

Stop!

THE DYNAMIC n-OBJECTIVE FIULTIPLIER (DNM) ALGORITHM J u s t as i n t h e s t a t i c case,

t i o n s induces

i n t h e f i r s t segment. computation

f i n d i n g t h e zeroes

a need f o r a l a r g e number o f

U s u a l l y t h e r i u l t i p l i e r approach

per s o l u t i o n than the

r a c i e s s h o u l d be s m a l l compared terpolation;

thus i t i s

o f t h e worth func-

solutions t o the minimizations r e q u i r e s much l e s s

& - c o n s t r a i n t apnroach,

w i t h the inaccuracies i n regression o r i n -

o f t e n best

f o r n o n - l i n e a r problems.

a l g o r i t h m f o r dynamic problems w i l l n o t be p r e s e n t e d here; r a t e d by u s i n g t h e f i r s t t h r e e

and i t s i n a c c u The

mixed

i t can be gene-

s t e p s o f t h i s a l g o r i t h n i f o l l o \ r e d by s t e p s 5

through 9 o f t h e previous a l g o r i t h m . A f l o w c h a r t o f t h i s a l g o r i t h m i s given i n f i g u r e 7-2.

Al

> 0

Step 1 :

Set i n i t i a l values f o r

Step 2:

S o l v e t h e f o l l o w i n g problem:

-

.

F i g u r e 7-2.

F l o w c h a r t f o r Dynamic n - O b j e c t i v e M u l t i p l i e r A l g o r i t h m

dE(tf)'tf) The so l u t i o n s t a t e v ect o r into

fl =

t

4

x*(t) a n d control ve c tor i * ( t ) a r e s u b s t i t u t e d

+ , ( 5 ( t f )+ ) [tf a , ( x ( t ) , u ( t ) , t ) d t

t o get f y ( L l ) and into

'0

Step 3 : I f enough information has been ge ne ra te d, proceed t o s t e p 4; otherwise choose a new value Step 4:

,

0

and r e t u r n t o s t e p 2 .

Develop t h e s u r r o g at e worth func tions

W

1 2 (1p' L ) , . . . , W l n ( L l )

For each s e t of values A f ;(Ll),E*(L1) a t which t h e worth i s -1 ' d e s i r e d , ask t h e DY f o r h i s assessment of how much A a d d i t i o n a l u n i t s of I j o b j e c t i v e f l a r e w o r t h i n r e l a t i o n t o one a dditiona l u n i t of f . given J f:(Jhl) u n i t s of f l and t h e j - l s t component of IT.hl) u n i t s of f . . His a s s J essnient on a s c a l e of -10 t o +I(? i s t h e value W l j ( A l ) . This s t e p i s repeated f o r a l l j = 2,3, . . . , n. * * Step 5 : Repeat s t e p 4 u n t i l A1 i s found such t h a t \dlj(!il) = 0 f o r * j = 2 , 3 , . . . , n. Additional values near L~ may be t r i e d t o determine the e x t e n t of t h e i n d i f f e r e n c e band. * Step 6: Find t h e p r ef er r ed s t a t e ve c tor 5 ( t ) and pre fe rre d control * * vector u ( t ) by solving t h e problem i n s t e p 2 with A, replaced by If * t h e r e was more than one p r ef er r ed s o l u t i o n in s t e p 5 , then t h i s s t e p i s repeated f o r each one i n o r d e r t o f i n d a l l of t h e pre fe rre d s o l u t i o n s . Step 7 : A s e n s i t i v i t y a n a l y s i s could be performed t o determine the p o s s i b l e e f f e c t s of implementing t h e p r ef erre d s o l u t i o n . Step 0: S t o p ! as follows.

7.4

S!IM/ARy This chapter has extended t h e algorithms of chapter 5 , and modified

t h e r e s u l t s of ch ap t er 6 f o r dynamic n - o bje c tive problems. The L-c onstra int approach i s useful f o r l i n e a r (and l i n e a r i z e d ) probleins, b u t in o t h e r problems may r e q u i r e too much computation i n orde r t o ge ne ra te enough values t o have v a l i d approximations t o t h e worth f u n c t i o n s . The m u 1 t i p l i e r approach i s subject

t o i naccu r aci es due t o non-convexities in t h e tra de -off f u n c t i o n s ,

b u t i s g e n e r a ll y

much simpler

Hixed algorithms

can a l s o be used

accuracy.

t o s o l v e a s i t has a s a compromise

n-1

less

constraints.

between e f f i c i e n c y and

126

MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS The n e x t c h a p t e r s w i l l p r o v i d e examples o f t h e use o f t h e SWT method

i n problems i n w a t e r r e s o u r c e p l a n n i n g .

FOOTNOTES 1.

A derivation problems

o f t h e necessary c o n d i t i o n s

can be f o u n d

i n Bryson and Ho

f o r a minimum i n dynamic [1969]

o r o t h e r books on

optimal c o n t r o l . RE F E R E N C E S

1.

Bryson, A . E . ,

and Ho, Y . C . ,

Waltham, Mass.,

1969.

A p p l i e d O p t i m a l C o n t r o l . Ginn and

Co.,

Chapter 8 APPLICATIONS OF THE SWT METHOD TO WATER RESOURCES PROBLEMS E R E I D - V E M U R I EXAMPLE PROBLEM

8.1

R e i d and V e m u r i l

introduced the

following

multiobjective function

problem i n w a t e r r e s o u r c e s p l a n n i n g : "

...

A dam o f f i n i t e h e i g h t impounds w a t e r i n t h e r e s e r v o i r

and t h a t w a t e r i s r e q u i r e d t o be r e l e a s e d f o r v a r i o u s p u r poses such as f l o o d c o n t r o l , i r r i g a t i o n , i n d u s t r i a l and u r b a n use, and power g e n e r a t i o n .

The r e s e r v o i r may a l s o be used

f o r f i s h and w i l d l i f e enhancement, r e c r e a t i o n , s a l i n i t y and p o l l u t i o n c o n t r o l , mandatory r e l e a s e s t o s a t i s f y r i p a r i a n r i g h t s o f downstream u s e r s and so f o r t h .

T h e problem i s

e s s e n t i a l l y one o f d e t e r m i n i n g t h e s t o r a g e c a p a c i t y o f t h e r e s e r v o i r so as t o maximize t h e n e t b e n e f i t s a c c r u e d . . . " There a r e two d e c i s i o n v a r i a b l e s : x1

=

x2

=

T o t a l man Iiours d e v o t e d t o b u i l d i n g t h e dam. Mean r a d i u s o f t h e l a k e impounded i n so:ie

fashion.

There a r e t h r e e o b j e c t i v e f u n c t i o n s : f.,(x1,x2)

=

Capital cost o f the project.

f2(x2)

=

The w a t e r l o s s ( v o l u m e l y e a r ) due t o e v a p o r a t i o n .

f3(x1.x2)

=

The t o t a l volume c a p a c i t y o f t h e r e s e r v o i r .

I n order t o

change t h e volume o b j e c t i v e t o a m i n i m i z a t i o n problem, t h e r e -

c i p r o c a l f u n c t i o n f ( x ,x ) was formed: 3 1 2

where

A l l d e c i s i o n s and o b j e c t i v e s a r e c o n s t r a i n e d t o be n o n - n e y a t i v e . t h i s problem

i s f a r from

representing a

Although

r e a l i s t i c w a t e r r e s o u r c e problem

( t h e r e a r e o n l y two d e c i s i o n v a r i a b l e s ) , i t was chosen because o f t h e gener a l i n t e r e s t t h a t R e i d and Vemuri had g e n e r a t e d b y t h e i r paper. 127

Reid a n d Veniuri were s a t i s f i e d with determining t i o n s via t h e i r proposed approach. non-inferior

solutions

non-infe rior

solu-

With t h e SldT method, not only t h e same

( i n a d d i t i o n t o o t h e r s ) can be ge ne ra te d, b u t a l s o

the t r a d e - o f f r a t e f u n c t i o n s and t h e

ultimate

pre fe rre d

solution

t o the

whole problem can be determined assuming t h e e xiste nc e of a de c ision maker. SOLUTIOFI T O T t l E R E I D - V E M U R I PROBLEM This s o l u t i o n procedure wi l l i l l u s t r a t e t h e approach of considering

8.2

t h e su r r o g a t e worth f u n ct i o n s a s f u n ct i o n s of the

.

=

1J

described in s e c t i o n

The f i r s t s t e p of t h e SWT method i s t o f i n d t h e minimum values

3.5.2.

each o b j e c t i v e f u n c t i o n .

x1

A..

f,

C l ear l y

=

0,

f2 =

0 a t x2

=

0 , and f 3

=

for 0 at

t - c o n s t r a i n t formulation i s now adopted t o ge ne ra te ,Il2 and

The

'13: Probleri 8-1 :

.0lXl

.02

(x,)

MIN

e

subject t o

1/2 x2

2

e

c

2

(x2) c2

- 0 . 0 0 5 ~ ~ -0.01 -2 (x,) x2 : r3

x1

,

0 ,

x2

0

.

+ ll3

(e

>,

Then t h e Lagrangian i s formed:

.01 x L

=

e

.02 l(xl)

2

x2 + x12 ( I / Z x2 -

-, 0 0 5 ~ ~ t2)

(x,)

- . 01

-2

x 2 -

t3)

The Kuhn-Tucker necessary co n d i t i o n s f o r a minimum a r e :

The above c o n d i t i o n s were solved f o r various values of c 2 and E~ (inc luding some of t h e values from Table 1 , Reid and Vemuri)' via the Newton-Raphson Method.

The r e s u l t s a r e presented i n Table 8-1. The f i r s t two columns of Table 8-1 a r e the s e l e c t e d values of

t 3

( o r e q u i v a l en t l y f 2 a n d f 3 ) .

c2

and

Columns t h r e e and four a r e t h e non-infer-

i o r values of t h e d eci s i o n v a r i a b l e s corresponding t o t h e chosen values f o r and C o l u m n f i v e i s t h e correspondinq non-infe rior value of objec$3. Columns s i x and seven a r e t h e t ra de -off r a t i o s . Note t h a t column tive f l .

/

129

::

h = -af / a f 13 s i n c e the d e c i s i o n maker

.

seven i s t h e r a t i o .-,ifl/2f3

i S

This i s required in

place of

x13

=

f a m i l i a r w i t h t h e volume Of t h e r e s e r -

v o i r , f ( x x ) , r a t h e r t h a n i t s r e c i p r o c a l f ( x x ) This t r a d e - o f f r a t i o 3 1' 2 3 1 ' 2 ' can be found a s f o l l o w s :

An a t t e m p t t o u s e m u l t i p l e r e g r e s s i o n a n a l y s i s f o r t h e c o n s t r u c t i o n of X 1 2 and ) 1 3 a s a n a l y t i c f u n c t i o n s of f 2 and f 3 u s i n g t h e w i d e band of n o n - i n f e r i o r p o i n t s ( s e e Table 8-1) r e s u l t e d i n c o r r e l a t i o n c o e f f i c i e n t o f o n l y .80. This i s

attributed to the

e x p o n e n t i a l n a t u r e of

the o b j e c t i v e f u n c t i o n s .

Consequently, t h e a l t e r n a t i v e approach o f a v o i d i n g r e g r e s s i o n s was a d o p t e d , where t h e d e c i s i o n maker o r o v i d e d t h e s u r r o g a t e worth v a l u e s I d l 2 and \)Il3, f o r t h o s e v a l u e s of

A 1 2 and h 1 3 , g i v e n i n Table 8 - 1 . The c o r r e s p o n d i n g f , , and f 3 can a l s o be found i n t h e t a b l e . I f i t i s n e c e s s a r y t o d e t e r m i n e t h e worth a t v a l u e s o f X 1 2 and A 1 3 n o t i n Table 8 - 1 , then i n t e r p o l a t i o n o r

f,,

m u l t i p l e r e q r e s s i o n n e a r t h e d e s i r e d v a l u e s can be u s e d . The v a l u e s of t h e s u r r o g a t e worth f u n c t i o n s g e n e r a t e d w i t h a " d e c i s i o n maker" a r e t a b u l a t e d i n columns 8 and 9 of T a b l e 8-1. than one s e t of

1J

c o r r e s p o n d i n g v a l u e s of I l 2 , Table 8-1,

Note t h a t more

t r a d e - o f f s r e s u l t e d i n an i n d i f f e r e n c e band, W . .

=

0.

The

x13, f l , f 2 , and f 3 can be r e a d d i r e c t l y from

rows 9 , 2 5 , 30, and 32.

All s o l u t i o n s

corresponding t o t h e s e

rows a r e p r e f e r r e d ; t h e y a r e n o n - i n f e r i o r s o l u t i o n s which belong t o t h e i n d i f f e r e n c e band. The d e c i s i o n v a r i a b l e s c o r r e s p o n d i n g t o the above p r e f e r r e d s o l u t i o n s

The s i m p l e s t way i n t h i s example i s t o T h u s , f o r example, row 9 p r o v i d e s the f o l l o w i n g optimal dec i s i o n s and v a l u e s o f the o b j e c t i v e f u n c t i o n s : x1 = 1 7 2 . 9 5 ; x2 = 3 8 . 7 3 ; f l = 9374.98 ; can be o b t a i n e d i n s e v e r a l ways. use T a b l e 8 - 1 .

f2

=

750.00;

f3

=

3750.00

I n o t h e r problems, the methods f o r r e v e r t i n g t o t h e d e c i s i o n bed i n s e c t i o n 6 . 3 . 2 may be r e q u i r e d .

space descri-

TABLE

8-1

N O N - I N F E R I O R P O I N T S AND D E C I S I O ? I MAKER RESPONSES

f2

f3

x1

X

2

fl

x 12

'1 3

-

i"l 2

3

2.00

+ 8

+ 6

4.00

t 2

+ 2

~~

1

250.00

500.00

0.70

22.36

499.95

2.00

2

250.00

1 000. 00

128.91

22.36

2000.00

8.00

3

250.00

1750.00

239.59

22.36

61 24.45

24.50

- 7.30

- 2

- 2

4

250.00

2503.00

310.41

22.36

12499.99

50.00

-10.00

- 5

- 5

5

250.00

3750.00

391.04

22.36

281 24.09

112.49

-1 5 . 0 0

-1 0

6 7

250.00

5000.30

448.28

22.36

49984.46

199.88

-19.99

-1 0

750.00

1750.00

24.43

38.73

2041 .46

2.72

- 2.33

+ 7

8

750.00

2500.00

93.09

38.73

41 66.41

5.55

- 3.33

+ 4

-1 -1 t +

9

750.00

3750.00

172.95

38.73

9374.98

12.50

- 5.00

0

0

10

750.00

5000.00

229.91

38.73

16665.71

22.22

- 6.67

- 2

- 2

11

100.00

1750.00

421 .71

14.14

1531 0.72

153.09

-1 7 . 5 0

-1 0

-1 0

12

500.00

1750.00

102.65

31.62

3062.14

6.12

- 3.50

+ 4

+ 3

13

100.00

3750.00

573.53

14.14

70310.77

703.09

-37.50

-10

-1 0

14

500.00

3750.00

253.27

31.62

14060.19

28.12

- 7.50

- 3

- 3

15

1000.00

3750.00

116.19

44.72

7029.45

7.03

3.75

+ 3

+ 2

16

106.00

473.00

150.47

14.56

1055.33

9.96

-

17

33.40

150.00

151.74

8.17

336.83

10.08

0

0 5 3

4.46

0

+ 1

- 4.49

0

18

334.00

1500.00

151.74

25.85

3368.26

10.08

- 4.49

0

19

1060.00

4730.00

150.47

46.04

10553.25

9.96

- 4.46

0

20

31.60

316.00

310.41

7.95

1580.00

50.00

-10.00

- 5

+ + + -

1 l 1 5

21

3.34

150.00

6C9.47

2.58

3367.91

1008.25

-44.90

-1 0

-? 0

Non-Inferior P o i n t s and Cecision Maker Responses

T a b l e 8-1 ( C o n t ' d )

+

x13

f2

f3

22

59.50

841.00

379.22

10.91

5943.42

99.89

-14.13

-1 0

- 9

23

88.90

562.00

21 9 . 3 8

13.37

1776.34

19.9s

- 6.32

- 2

- 1

24

33.40

1500.00

609.47

8.17

33679.12

1008.25

-44.90

-1 0

-1 0

25

100.00

500.00

172.95

14.14

1250.00

12.50

-

0

26

100.00

1000.00

31C.41

14.14

5000.00

50.00

-10.00

- 5

27

100.00

5000.00

630.86

14.14

124971 . 6 5

1249.43

-49.98

-1 0

28

500.00

1000.00

0.70

31 .G2

999.89

2.00

- 2.00

0 - 5 -1 0 + 6 - 5

X

li

X

2

fl

2 ___

5.00

29

500.00

5000.00

310.41

31.62

24999.99

50.00

-10.00

+ 8 - 5

30

1000.00

5000.00

172.95

44.72

12499.97

12.50

- 5.00

0

0

31

100.00

2500.00

492.62

14.14

31 209.63

311.69

-24.95

-1 0

-1 0

32

500.00

2500.00

172.95

31.62

6249.99

12.50

-

0 + 7

+ 5

33

1000.00

2500.00

37.39

44.72

31 25.00

3.12

5.00

- 2.50

0

DISCUSSION OF RESULTS

8.3

I n g e n e r al , one may need ad d i t i o n al a n a l y s i s in the c a se where and

i s no row with both W 1 2

equal t o z e ro.

W13

there

I n t h i s case, a multiple

r e g r e s s i o n or i n t e r p o l a t i o n s can be conducted t o obta in ld12 and W 1 3 each a s a f u n c t i o n of

t o zero.

Xl2

and X 1 3

Then one

near values where t h e worth func tions a r e

would have t o

W 1 2 ( X 1 2 , ~ 1 3) = 0 and W13(A12,A13)

close t h e equations f o r

s o l v e simultaneously =

0 t o obta in e stim a te s

for

A;~

and

^* x13.

a m u l t i p l e r eg r es s i o n or i n t e r p o l a t i o n s can be conducted f o r f 2

Similarly,

and f 3 a s f u n ct i o n s of A 1 2 a n d h l 3 i n o r d e r t o provide t h e necessary i n f o r -

*

maiion t o t h e DM. F i n a l l y , one would s o l v e problem 8-1 f o r

*

and

t 3

=

* * - *

f3(A12,A13) a s described i n s e c t i o n 6 . 3 . 2 .

were avoided

i n t h i s example o f t h e cas e where

* ^*

f2(h12,x13)

These complications

s i n c e f o u r values were found

s u r r o g a t e worth f u n ct i o n s were zer o . lustration

E~ =

f o r which both

The next example w ill provide an i l -

no p r ef er r e d s o l u t i o n i s found in t h e t a b l e .

Note t h a t t h e worth f u n ct i o n s W 2 3 , W 2 1 , W 3 1 and b132 were not found s i n c e i t was assumed t h a t t h e DM i s a c c u r a t e l y as s essing his pre fe re nc e s. STREAM RESOURCE A L L O C A T I O N PROBLEM

8.4

This s e c t i o n wi l l demonstrate t h e use of the surroga te worth tra de off method i n another water resources problem. streani resources viewpoint3.

The problem

has been s t u d i ed ex t ensive ly from

The s t a t i c

n-objective

t h e a l l o c a t i o n of

-

the s i n g l e o b j e c t i v e

E - constra int algorithm described

in

Chapter 6 w i l l be used t o s o l v e t h i s a s a m u l t i p l e o b j e c t i v e problem.

For i l l u s t r a t i v e purposes, co n s i d er a physical system c o n s i s t i n g of a r e s e r v o i r upstream of a s e r i e s o f n municipal and i n d u s t r i a l use rs d i s charging i n t o t h e r i v e r . The r e s e r v o i r is used f o r water supply can a l s o be used f o r low flow augmentatidn ( r e l e a s i n g of water down-

and

stream t o d i l u t e w a s t e s ) . which

has t h e a b i l i t y

demand)

t o co n t r o l t h e amount of B . O . D .

discharged d a i l y

limitations);

W e assume t h e e x i s t e n c e of a regional a u t h o r i t y i n t o t h e stream

the regional a u t h o r i t y

( b i o l o g i c a l oxygen

by each user ( e . g . via e f f l u e n t

i s a l s o re sponsible

for

re gula ting

water re1 e a s e from t h e r e s e r v o i r . Thus t h e d eci s i o n maker f o r o u r problem i s t h e regiondl a uthorityw hic h determines t h e e f f l u e n t l i m i t a t i o n s ( X of tre a tm e nt) f o r each u s e r , and t h e amount of water t o be r el eas ed from t h e r e s e r v o i r f o r low flow augmentation. Define t h e d e ci s i o n v a r i a b l e s as f o l l o ws : c e n t treatment t o be used a t t h e mal).,

xi, i

=

1,2, ..., n , i s t h e per-

i t h treatment p l a n t (expressed a s a de c i-

and y i s t h e amount of water

( i n u n i t s of Fo where Fo i s t h e unaug-

mented i n i t i a l f l o w

i n t h e stream)

t o be r e l e a s e d f r o m

the reservoir for

f l o w augmentation. I t i s assumed t h a t t h e o b j e c t i v e s o f t h e DM a r e t h r e e f o l d .

1:

M i n i m i z e t h e t o t a l c o s t o f waste t r e a t m e n t i n t h e r e g i o n ( i t i s assumed

that the cost o f

r e l e a s i n g w a t e r f o r f l o w augmentation i s n e g l i g i b l e s i n c e

t h e dam i s a l r e a d y p r e s e n t ) .

The c o s t

f u n c t i o n s f o r each i n d i v i d u a l u s e r

as g i v e n by Hass w i l l be used f o r t h e Miami R i v e r . removed:

t h e p e r c e n t B.O.D. Costi

160.8 + 26.7qi

=

These a r e q u a d r a t i c i n

+

+ 255.7qi)

(640.7

(xi - .45)

2

where qi i s t h e t o t a l waste w a t e r l o a d g e n e r a t e d by t h e ith user i n m i l l i o n g a l l o n s p e r day.

Thus t h e t o t a l c o s t t o

t h e r e g i o n i s t h e sum o f t h e i n -

dividual costs: n

fl

=

)

1

i=l where 2:

= 160.8

txi

+

(X.

+ 26.7qi

Bi

xi

and

ai

Maximize t h e w a t e r i n

- .45) =

2

640.7 + 255.7qi

the reservoir available f o r

on e v a p o r a t i o n , r a i n f a l l , c a p a c i t y , e t c .

depends

This

S , t h e amount o f w a t e r a v a i l a b l e t o be r e l e a s e d

lumped i n t o one parameter ( i n u n i t s o f Fo)

water supply.

All o f these f a c t o r s a r e

f o r t h e p e r i o d being studied.

Then t h i s second o b j e c t i v e

becomes : Maximize 3:

Minimize the

o f the water

f2

=

S - y

p o l l u t i o n i n t h e stream.

i m m e d i a t e l y downstream

I t i s assumed t h a t t h e q u a l i t y

o f t h e f i n a l user

(point A i n figure

8 - 1 ) i s i m p o r t a n t t o t h e DM. For example, t h i s area may be used f o r r e c r e a t i o n a l purposes

so t h a t t h e q u a l i t y must

so t h a t t o a v o i d l e g a l problems t h e q u a l i t y

o f another j u r i s d i c t i o n a l area must be m a i n t a i n e d .

The measure o f q u a l i t y t o be used i s t h e c o n c e n t r a t i o n

o f d i s s o l v e d oxygen, D.O., The B.O.D.

basic

be m a i n t a i n e d , o r i t may be p a r t

i n milligrams per l i t e r .

Streeter-Phelps

discharged i n t o t h e r i v e r

e q u a t i o n 4 i s used t o r e l a t e t h e a m o u n t o f to

the

D.O.

l e v e l a t any

point i n the

stream. F o l l o w i n g t h e approach o f Hass, t h e r i v e r i s d i v i d e d i n t o n reaches ( t h e p o i n t where each u s e r d i s c h a r g e s h i s wastes d e f i n e s a new r e a c h ) . Then the

D.O. c o n c e n t r a t i o n a t p o i n t A i s : n D.O.

=

[

z i =1

aixi

+

b,y

+ (b,-cl)l/[(c2-b2)

-

b2y1

134

MUI,TIOBJEC”IVE OPTIMIZATION I N WATER RESOURCES SYSTEMS

-

c2

=

(Fn

w.

=

u n t r e a t e d B.O.D.

1

i R. = J

F0)q l o a d ( l b s / d a y ) o f t h e ith user

J

exp ( -

1

m= i

rmtm)

=

j c kmtm) m=i unaugmented r i v e r f l o w a t b e g i n n i n g o f r e a c h i ( c f / d a y )

=

i n i t i a l c o n c e n t r a t i o n o f D.O.

Si

=

saturation level f o r

r. J t. J

=

r e a e r a t i o n c o e f f i c i e n t i n reach j

=

time o f t r a v e l f o r reach j

k. J

=

deoxygenation c o e f f i c i e n t f o r reach j

q

=

conversion f a c t o r from l b s / c f t o m g / l i t e r

cT j

=

D.O.

K’j

=

Fi c

0

This i s b a s i c a l l y

exp ( -

i n stream ( m g / l i t e r )

O2 i n ith reach ( m g / l i t e r )

c o n t e n t ( l b s l d a y ) o f added f l o w i n r e a c h j

a restatement o f

e q u a t i o n A-14 o f t h e appendix o f Hass.

Thus c b j e c t i v e t h r e e becomes: n Maximize

[

c

aixi

+ bly

+

(b,-cl)]/[(c2-b2)

-

b2y]

i=l There a r e a l s o c o n s t r a i n t s on t h i s system. t h a t each u s e r employ

a t l e a s t primary treatment

Legal c o n s t r a i n t s r e q u i r e (assumed t o be 4 5 % ) , and

135

APPLTCAY'IONS TO WATER RESOURCES PROBLEMS

Thus

o v e r 99% t r e a t m e n t i s p r e s e n t l y p h y s i c a l l y i m p o s s i b l e .

s x 1.

.45

.99

5

for

...,

i = 1,2,

.

n

S i m i l a r l y , t h e amount o f f l o w augmentation must be between z e r o and t h e amount a v a i l a b l e (S

3.47) so t h a t 0

=

<

< 3.47

y

Introducing the transformation x i

=

xi

- .45 f o r each i, t h e o v e r a l l

p r o b l em becomes : n MIN

Problem 8-2:

fl

I "

' L

ai +

7:

=

i=1

6.X. 1 1

[(C2-b2) - b2Y1 s.t.

8-5.

SOLUTI0r.I

OF STREAM

0

< x 1.

o

s

y

.54

i

i

s

RESOURCE ALLOCATION PROBLEM

The f i r s t t a s k i s t o p u t problem 8-2 i n t o s t a n d a r d f o r m .

The second

and t h i r d o b j e c t i v e s a r e t r a n s f e r r e d i n t o m i n i m i z a t i o n s by u s i n g t h e f a c t that M I N f . = - MIN(- f . ) . Thus d e f i n e f; = - f 2 and f; = - f 3 ; the J J a l g o r i t h m can now be i n i t i a t e d . S t e p 1: The minimum v a l u e o f f; i s f o u n d by s o l v i n g MIN

y - S

s.t. 0

y 5

$

s

Using t h e v a l u e S = 3.47 f r o m Hass, fZMIN must equal -3.47 s i n c e t h e can a l s o be f o u n d .

This obviously occurs a t y

=

miniI

F o r t h i s problem

mum o b v i o u s l y o c c u r s a t y = 0.

t h e maximum v a l u e f o r f 2 3.47 and i s fZMAX = 0.

The minimum v a l u e o f f i i s f o u n d by s o l v i n g MIN

-

n

[

n

c a 1. x :1 + .45

i=l

c ai + bly + (bl-c1)]/[(c2-b2) i=l I

s.t. It i s

O s y s S ;

obvious t h a t t h e

level (f3)

O s x .1 s . 5 4

i=2,3

minimum v a l u e o f

,...,

This g i v e s t h e value

f3F,Ip4 = - 7.97.

n

f3 i s t h e p o i n t where t h e

i s maximum, w h i c h must o c c u r where each I

- b2y]

0.0.

x.' = .54 and y = 3.47. 1

o c c u r s where y = 0 S i m i l a r l y , f3MAX

I

and each x i

=

0 ; thus i t i s found t h a t f3MAX= -4.78.

Note t h a t the

values

a r e developed from t h e d a t a i n Hass; these

of tlie c o n s t a n t s a i , b l , b 2 , c 1 , c 2

Rewrite tlie problem -in t - c o n s t r a i n t form:

S t e p 2:

n MIN

( 9

tr.+(3.x.L

j:

1

i=l

s.t.

s

y -

1

1

:: c 2

n - [

I

n a.x.

1 1

i =1

f

c a i + bly + (bl-c,)l/[(c2-b2)-b2yl s

.45

i=l

o

. x i : .54 f o r

0

s

y

i

=

1,2,

..., n

< 3.47

TABLE 8-2 P h y s i c a l C o n s t a n t s f o r Stream A l l o c a t i o n Problem i

a. 1

qi

1

3331.84

45.2

2

342.96

4.7

3

1539.69

4.2

4

886.02

3.6

5

73 . 8 3 172.06

0.5 1.2

6

7

189.40

0.8

8

433.07

0.6

9

199.94

0.5

10

1913.91

3.2

11

1741.59

8.4

12

722.94

2.7

13

238.59

0.6

14

3633.82

12.1

5

2266.56

8.4

-1

bl b2

= =

7650.31 -971.27

c 1 c2

= =

1740.36 1930.14

t3

137 B r i n g i n g a l l o f t h e c o n s t a n t s o v e r t o t h e r i g h t hand s i d e g i v e s :

s.t.

.

s

o .

xl

y

i

1

: y

0

+

t2

for i

.54

I,Z,

=

...,

n

3.47

$

T h i s p r o b l e i i i c a n be e a s i l y s o l v e d b y q u a d r a t i c p r o g r a m m i n g ( f l P ) . ues of t h e c o n s t a n t s qi

8-2.

used i n f i n d i n g

A s t h e r i g h t hand s i d e o f t h e

f3

The v a l -

a n d B~ a r e a l s o g i v e n i n T a b l e

ai

inequality i s negative, i n order t o

u s e q u c d r a t i c p r o g r a m i n g t h e c o n s t r a i n t m u s t be changed t o n

n a.x i=l

+

( b l - b z F 3 ) ~ :: ( c l - b l )

- ( C 2 - b 2 ) c 3 - . 4 5 i=l 7 a 1.

Let

Q1

=

bl

- b

4,

=

s

+ f2

2'3 n

Q3

c1 - bl - (c2 - b 2 ) c 3

=

-

c a 1.

.45

i=l

The p r o b l e m t h e n becomes : P r o_ b l_ e m~ 8_- 3_: ~-

-

n

PIIN

'2

+

ai

B.X. 1 1

i=l

n ):

a.x. 1

i=1

1

f l i x

O

S t e p s 3 ,?, 4 :

<

Y

+

QIY ? Q3

s

.54

i 3.47

A q u a d r a t i c programming

upper bounding technique5

t o eliminate the

xi

s o l u t i o n procedure, u s i n g an

< .54 and

y:'

3.47

constr-

w L w m ~ E c m m OPTIMIZATION IW WATER RESOURCES SYSTEMS

138

a i n t s , was w r i t t e n a n d implemented on t h e GE 4060 computer. ved f o r 33 d i f f e r e n t values of

of which 30 were binding.

This was s o l The s o l u t i o n t o

t h e q u a d r a t i c programming problem includes t h e Lagrange m u l t i p l i e r s A3

f o r t h e c o n s t r a i n t s i n problem 8-3. Notice t h a t

+ ,ifl/"Q3

( t h e plus s i g n i s

"*

To f i n d

straint).

x12

X2

- afl/aQ2 and

corresponds t o a

h3

and =

2

con-

note t h a t -

=

Since Q, = S +

p r es en t because

=

E~

and

E~

=

f

2

2

=

- a f l . dQ2 . aQ2 d ~ 2 d f 2

=

- f 2 then h 1 2 = -

afl/af2

~

~

2'

~

i s negative ( i . e . , - a f l / a f 2 i s p o s i t i v e ) s i n c e

I t i s a p p r o p r i at e t h a t

i f t h e amount a v a i l a b l e f o r supply i n c r e a s e s , then t h e c o s t t o achieve t h e same q u a l i t y wi l l a l s o i n cr eas e ( s i n c e t h e flow augmentation w ill be l e s s , more treatment wi l l be r eq u i r ed

.

To f i n d

note t h a t

h13,

2f1 - afl/af3 = -

=

-

aQ1 di,

plow,

r3

= f

3

=

- f

so 2 = - 1; 3' df3

a l s o dQ3 = b2 - c 2 and d Q l / d s 3 dE3 ~

The only problem i s f i n d i n g afl/aQ1.

=

- b2.

Note t h a t t h e Lagrangian f o r problem

8-3 i s :

n 1= fl

+

six;

A2(y-Q2) - X 3 1=1

Since I= f Thus

h13

1

a t t h e optimum,

afl/aQ1

2 3. - X3(b2 - c 2 )

b X

=

=

+

4 1 Y - Q3

aL/aQl

=

- x3

y.

.

The f i r s t f i v e columns of Table 8-3 show t h e r e s u l t s o f t h e s e s t e p s . The u n i t s f o r

f 2 were changed from u n i t s of

Fo t o m i l l i o n cubic f e e t per

day a n d f o r X 1 2 from $ / u n i t s of Fo t o $ / million cubic f e e t per day. Step 5: The DM i s questioned and h i s responses a r e given i n t h e l a s t two columns o f

Table 8-3.

i n t h e water r es o u r ces group.

The DFI f o r t h i s problem was a graduate s t u d e n t In o r d er t o

impart an understanding of t h e

system f o r which he was making d e c i s i o n s , t h e o b j e c t i v e s were discussed and t h e upper Figure 8-1.

and lower

bounds f o r each o b j e c t i v e

were presented

along with

The d e s c r i p t i o n given was a s follow s:

"A p o ss i b l e o p er at i n g p o i n t f o r t h e system i s

f 2 m i l l i o n cubic f e e t

139

APPL ICATl ONS TO WATER RESOURCES PROBLEMS of water a v a i l a b l e

a D.O.

f o r supply,

l e ve l of f 3 mg/l a t point A in

the

A t t h i s p o i n t , would you be w i l l i n g t o pay in o r d er t o have 1 m i l l i o n more cubic f e e t of

stream, z n d a c o s t of f; $/day. an a d d i t i o n a l

$/day

-

water a v z i l a b l e ?

Rate your

wi l l i n g n es s on a s c a l e from -10

(totally un-

w i l l i n g ) t o t 1 0 ( t o t a l l y w i l l i n g ) with zero s i g n i f y i n g i n d i f f e r e n c e . l a r l y , r a t e your wi l l i n g n es s t o pay

Simi-

d o l l a r s per

day

l ev el 1 m g / l i t e r . "

t o r a i s e t h e D.O. The u n i t

- A13

an additiona l

increments f o r

f2

( 1 m i l l i o n cubic f e e t of w a te r) and f 3

( 1 m g / l i t e r ) were chosen t o be l a r g e enough so t h a t t h e DN would be a b l e t o perceive t h e d i f f e r e n c e . For example, t h e DM would probably c onside r t h e d i f f e r e n c e between 5 . 0 a n d 5.1 m g / l i t e r t o be n e g l i g i b l e and t h u s que stions about r a i s i n g t h e D . O .

l ev el by . 1 mg/l would have no s i g n i f i c a n c e .

same time, t h e u n i t i n c r m e n t s must

A t the

be small compared t o t h e a bsolute v a l -

ues of t h e o b j e c t i v e s , s i n c e A . = - Afl/Af. i s only t h e c a se i n t h e l i m i t , 15 J a s t h e increments go t o zer o . Thus, i f t h e u n i t increments a r e too l a r g e , X 1 2 i s n o t a good approximation t o t h e change - Afl/Af2 . I t i s f e l t that t h e values chosen f i t both c r i t e r i a . Step 6:

Since none of t h e n o n - i n f e rior values i n Table 8-3 had both

worth f u n c t i o n s equal t o z e r o , m u l t i p l e re gre ssions were performed t o approximate N 1 2 ( f 2 , f 3 ) and W1 3 ( f 2 , f 3 ) . Due t o t h e r e l a t i v e and s u b j e c t i v e n a t u r e of t h e D M ' s responses, good a s any.

The c o r r e l a t i o n These two

i t i s f e l t t h a t a l i n e a r approximation i s a s

The r e s u l t s of t h e r eg r es s i on were: W12(f2,f3)

2

47.55 - .34 f 2 - 4.94 f 3

W,3(f2,f3)

2

67.66 - .15 f 2

c o e f f i c i e n t s were

2

-

.721

9.88 f 3 and

R 2 = .868 r e s p e c t i v e l y .

equations were solved simultaneously t o

g i v e an e stim a te o f t h e

*

=

*

= 6.06. The corresponding tra de off r a 3 = -61.2a nd A;?= -1670 and t h e t i o s were found by i n t e r p o l a t i o n t o be corresponding n o n - i n f er i o r value of f l found by i n t e r p o l a t i o n i s f; = 5987.

p r e f e r r e d values

When t h e s e

f2

=

51.77 and f

R

values were given t o t h e

DM,

Note t h a t i n s t ead of i n t e r p o l a t i o n s ,

he assigned W12

=

0 and W13 = 0.

t h e problem i n s t e p 3 could have been

*

resolved t o f i n d t h e t r a d e - o f f r a t e s and n on-infe rior value of f l f o r f 2 and f;

. Step 7 :

The r ev er s i o n t o t h e d eci sion space i s performed by solving

t h e q u a d r a t i c programming problem i n s t e p 3 with and

f;.

*

The r e s u l t s g i v e

*

'*

xi

, i

=

E~

and

E~

replaced by f;

1,2, . . . ,n which must be transformed

back t o x i , and y . The p r ef er r ed s o l u t i o n i s :

TABLE

8-3

P i o n - i n f e r i o r P o i n t s and DFI Responses f o r Stream A1 l o c a t i o n Problem ~

f2

;

f 3____ _ _____ f

92

'13

+ 10

+ 10

+ 10 9

+ +

8

- 904

+

+

6

"1 3 A l2 ~-~

50.54

5.0

4994

-

0.67

-

12

40.43

5.5

4994

- 1.89

-

48

46.66

5.5

5073

-23.53

- 532

+

52.88

5.5

5287

-45.16

7

9

24.88

6.0

4995

- 2.43

-

+ 10

+ 10

31.10

6.0

5052

-1 5 . 9 2

- 58%

+

+

+

98

38.88

6.0

5242

-32.78

-1 075

46.66

6.0

5562

-49.64

-1 422

52.88

6.0

591 2

-63.13

-1 599

-

9

5

3

i - 4

0

+

3

5

-

3

+

3.89

6.5

5031

- 9.30

- 652

+ 10

15.55

6.5

521 9

-22.91

-1 41 2

+

7

3 0

23.33

6.5

5432

-31 . 9 8

-1791

+

5

-

1

31 . 1 0

6.5

571 6

-41.05

-2067

+

3

-

4

38.88

6.5

6070

-50.13

-2241

0

-

5

46.66

6.5 6.5

6496 6887

-59.20

52.88

-66.46

-231 2 -2295

-

5 9

- 10

3.89

7.0

6074

-31 . 9 7

-3521

-

1

-

4

-

5

- 10

15.55

7.0

6479

-37,49

-3630

+ 10 + 3

23.33

7.0

6785

-41 . 1 7

-3621

+

31.10

7.0

7120

-44.85

-3547

+

2

-

7

38.88

7.0

7483

-48.53

-3407

-

1

-

7

-3 202

-

4

- 10

4

46.66

7.0

7874

-52.21

52.88

7.0

8208

-55.15

-2991

- 10

- 10

3.89

7.5

8552

-24.93

-6390

+ 10

- 10

15.55

7.5

8849

-25.95

-5848

+

9

- 10

23.33

7.5

9053

-26.63

-5451

+

7

- 10

31.10

7.5

9263

-27.30

-5026

+

5

- 10

38.88

7.5

9478

-27.98

-4573

+ 1

- 10

46.66

7.5

9698

-28.66

-4092

-

3

- 10

52.88

7.5

9878

-29.21

-3687

-

7

- 10

141

Step 8 :

A t t h i s point

some s o r t of

s e n s i t i v i t y a n a l y s i s could be

however, such work i s beyond the scope of t h i s b o o k .

performed;

The only

check made was t o inform t h e DM of t h e r e s u l t s , and he f e l t t h a t i t did express h i s preferences f a i r l y a c c u r a t e l y .

8.6

DISCUSSION OF RESllLTS The s u r r o g a t e worth t r ad e- o f f method was discovered t o be e a s i l y im-

plementable f o r t h i s problem. a l r e a d y a v a i l a b l e from Hassl, abling t h e value of

The mathematical model f o r t h i s problem was t h e only change necessary being t h a t of

of t h a t model.

t o change. The f a c t t h a t t h e

imitting t h e use

o f q u ad r at i c programming

c.3

en-

This re quire d only s l i g h t modification problem was formulated i n a manner perg r e a t l y reduced

t h e programming

and debugging time. Quadratic programming uses t h e simplex method f o r which there a r e

many packages a v a i l a b l e ;

must be made a r e f o r

t h e only s p e c i f i c

m odific a tions t h a t

i n t e r f a c i n g t h e data, which did not r e q u i r e

e f f o r t t o implement in t h i s s i t u a t i o n .

too much

I n a d d i t i o n , t h e simplex technique

i s a very e f f i c i e n t procedure. T h i r t y d i f f e r e n t n o n - i n f er i o r p o i n t s were found using l e s s t h a n twent y minutes of computer time of t h e GE 4060 d i g i t a l computer (which i s equiv a l e n t t o approximately two o r t h r e e minutes on t h e Univac each n o n - i n f e r io r p o i n t took approximately t h i r t y a s did t h e t h r e e i n f e r i o r

(non-binding)

seconds of computer time

solutions.

t h e d e c i si o n maker r eq u i r ed l e s s than t h i r t y minutes, appeared t o be more than s u f f i c i e n t t o allow preferences.

1108 computer);

The i n t e r a c t i o n with b u t t h i s time period

him t o adequately express his

These p r ef er en ces were then used in a m u l t i p l e l i n e a r re gre s-

sion t o f i n d a n e s t i m a t e of t h e p r ef er r ed s o l u t i o n ;

again t h e r e a r e numer-

ous computer packages a v a i l a b l e t o perform re gre ssions so l i t t l e e f f o r t i s necessary. I n t e r p o l a t i o n s were used t o f i n d t h e corresponding tra de -off r a t e s a n d t h e DM was questioned a g a i n . The re ve rsion t o t h e de c ision space required only one more i t e r a t i o n of t h e q u a dra tic programming procedure. I n summary, t h e o v er al l

implecientation and

s o l u t i o n via t h e

SLIT method

was

142

M U I TIOBJECTIVE OPTTMIZATION IN CJATER IiESOURCES SYSTEMS

e a s i l y accomplished.

Despite t h e s i m p l i c i t y of t h e problem in t h i s example

i t i s a n approximation t o r eal

situations,

and thus

t h e surroga te

worth

t r a d e - o f f method appears t o have g r e a t p o t e n t i a l f o r use in re a l problems. NORTHERN C A L I F O R N J A WATER SYSTEM Another problem f o r which m u l t i p l e o b j e c t i v e a n a l y s i s i s a ppropria te

8.7

i s that

of r e s e r v o i r o p er at i o n s

b a si s.

A l a r g e s c a l e example

, j e c t i v e problem

i s the analysis

f o r water and energy

of t h e Northern C a l i f o r n i a

t h e monthly o p er at i o n s

economic r e t u r n s

from a

t o produce

multi r e s e r v o i r system

I n r e a l i t y there a r e

a s a sinGle ob-

of t h i s problem considered

which considered horizon.

on a month t o m o n t h

a t l e a s t three

firm

over a

water system7 water and power

50 ye a r

planning

o b j e c t i v e s which should be

considered i n t h e scheduling of r e s e r v o i r r e l e a s e s .

These a r e t h e

made a v a i l a b l e , water made a v a i l a b l e , and c o s t of ope ra tion.

energy

The C a l i f o r -

nia water systems a n a l y s i s attempted t o commensurate t h e s e o b j e c t i v e s , augmenting energy and water i n t o monetary terms by using t h e p r i c e f o r which However, in g e n e r a l , they could be sold a s a commensurating c o e f f i c i e n t . t h i s p r i c e would not be a co n s t an t b u t would depend on t h e amount of energy a n d water a v a i l a b l e as well as on time.

Rather than attempt a p r i c e evalu-

a t i o n o f a l l p o s s i b l e combinations of water and energy a t each of 600 time p e r i o d s, rnultiobjective a n a l y s i s can be used a s a much simpler approach. The system modeled here i s a s i m p l i f i c a t i o n of t h e T r i n i t y subsystem in t h e Northern C al i f o r n i a water system. system.

There a r e two r e s e r v o i r s in t h i s

The f i r s t r e l e a s e s water through a power p l a n t ; t h e second can r e -

l e a s e water e i t h e r f o r supply o r downstream t h r o u g h another power p l a n t and

t o t h e ocean. The s t o r a g e cap aci t y of t h e second r e s e r v o i r i s n e g l i g i b l e t o t h e f i r s t so t h a t i t s l ev el can be considered c onsta nt o n a I n o t h er words, a l l water coming i n t o t h e second (lower) monthly b a s i s . r e s e r v o i r in any period i s e i t h e r r el eas ed t o t h e r i v e r o r used f o r supply a n d a d d i t i o n a l energy ( s e e Figure 8 - 1 ) . For t h i s system, t h e r e a r e two d eci sions which must be made f o r each compared

planning period - namely l e a se s a r e denoted

t h e r e l e a s e s from t h e two r e s e r v o i r s .

xi and r i r e s p e c t i v e l y .

Note t h a t in r e a l i t y t h e s e v a l -

ues a r e t h e average r e l e a s e s f o r t h e i t h period. a r e considered,

These r e -

If te n

planning periods

a s in t h e following approximation, t h e r e a r e then 20 d e c i -

s i o n s. I t i s a l s o assumed t h a t t h e two r e s e r v o i r ope ra tions

are

not inde-

pendent.

I n ot h er words, t h e r e i s some agency re sponsible f o r t h e ove ra ll

operation

of t h e system

and t h i s agency

w ill determine

the releases

to

APl 'L1 CAT i O N S TO WA!l'l?l< KESOURCBS

PROBLEMS

5

143

optii,ially o p e r at e t h e system. There a r e two ways

energy o b j e c t i v e . The f i r s t t o t h e region i s t h e amount of energy which can be guaranteed during ap l an n i n g pe riod. Since i n d u s t r i a l concerns recognizes t h a t

of formulating t h e

what i s important

or p r i v a t e u t i l i t i e s u t i l i z i n g t h e energy w ill usua lly need t o recover l a r g e amounts of c a p i t a l over s u b s t a n t i a l periods of time t o make use of t h e energy, t h e r e must be some reasonable assurance t h a t the a n t i c i p a t e d level of energy wi l l be a v a i l a b l e a s planned. This "guaranteed" energy i s known a s "firin" energy. Additional energy above t h e guaranteed amount ( " d u m p " energy) i s useful i n t h a t i t can re pla c e o t h e r sources of energy f o r t h e i n d u s t r i e s when i t i s a v a i l a b l e , b u t i s somewhat l e s s valuable t h a n the guaranteed a m o u n t . I f E i i s t h e amount of energy generated by t h e system i n period i , then t h e firm energy can be w r i t t e n a s Firm Energy = Idin i E i i , o r M i n i a . E f ' t where t h e f r a c t i o n of t h e firm le ve l Ef i s a i and i s i i 1 possibly d i f f e r e n t f o r each i . Note t h a t in g e n e r a l , t h e u t i l i t y i s concerned with t h e amount of energy guaranteed per day r a t h e r t h a n per month, b u t f o r t h i s p a r t i c u l a r system energy c o n t r a c t s a r e in f a c t based on mont h l y demands during 'on peak' p er i o d s . Another approach i s t o ignore t h e d i f f e r e n c e between firm a n d dump enerLiy a n d assume t h a t t h e t o t a l amount of energy produced i s what i s of value t o t h e region r eg ar d l es s of when produced. This i s a s i m p l i f i c a t i o n which may o r niay not be j u s t i f i e d f o r any p a r t i c u l a r re gion. If i t i s j u s t i f i e d , t h e energy o u t p u t can be w r i t t e n a s Total Energy = 7: E. . i ' The energy E i wi l l now be r e l a t e d t o t h e d e c i s i o n s . Consider f i r s t t h e energy produced by a r e l e a s e of water from t h e f i r s t r e s e r v o i r through t h e power p l a n t . This energy would be proportional t o the amount of water r e l e a s e d , x i , and a l s o t o t h e l ev el of water in t h e r e s e r v o i r . The le ve l of t h e water in t h e r e s e r v o i r , h , i s a f u nc tion o f t h e volume of water s t o r e d , q . where g e n e r a l l y i t can be approximated by a func tion of the form h

=

q1'3. Other more p r e c i s e forms o r even a t a b u l a t i o n can be used. Note t h a t t h e s t o r a g e volume changes

over

time

and

i s dependent

on t h e inflows y i a n d t h e amount of water re le a se d each pe riod. Define qi t o be t h e s t o r a g e volume a t t h e beginning of period i . Note t h a t t h e amount of st o r a g e a t t h e end of period i i s equal t o t h e amount of s t o r a g e a t t h e The following equation i s used t o r e l a t e t h e beginning of period i + 1 . s t o r a g e volumes i n s u cces s i v e p er i o d s .

qi+1

where

=

q . - x. + y. 1

1

1

y i i s t h e inflow t o and

period i . A l t l i o u g h t h e inflow

-

e.

1

e i i s t h e evaporation from the r e s e r v o i r in

y i i s a s t o c h a s t i c parameter, t h e values of

yi will be assumed d e t e r m i n i s t i c and known, and w ill be based on c r i t i c a l period hydroloqic a n a l y s i s .

t h e worst sequence values f o r y i found

That is,

in a 50-year sample of hvdrologic records w ill be used.

Since t h i s " c r i t i -

c a l " period w i l l control t h e maximum f i r m o u t p u t f o r t h a t p a r t i c u l a r hydrog r a p h t h e r e s u l t s can be shown t o be i d e n t i c a l with the 10 period sequence.

The evaporation tional t o the

from t h e r e s e r v o i r in period s u r f ace a r e a .

This

i w ill in general be propor-

can usua lly be

approximated

by

e.

=

Since q changes between i and i + l , e i w ill vary over time. The average value

e 1. ,

over one period i can be approximated by

I n g e n e r a l , t h e evaporation r a t e , p . wi l l vary with t h e season . 1

Replacing

t h i s i n t h e s t a t e equation f o r q . gives 1

I t w i l l a l s o be assumed

'chat t h e i n i t i a l s t o r a g e in t h e r e s e r v o i r , q l ,

is

known Returning t o the average value o f t h e head energy.

c a l c u l a t i o n o f the head,

over t h e period must be Thus t h e average head, h , i s :

i t i s apparent

t h a t the

used in determining the

The energy o u t p u t from t h e f i r s t r e s e r v o i r can thus be w ritte n a s :

The energy o u t p u t from t h e second r e s e r v o i r i s much simpler t o c a lc u-

146

MULT IOHJECTIVE OPT [MIZATION I N WATER RESOURCES SYSTEMS

l a t e s i n c e t h e head i s co n s t an t due t o a t i o n i t w i l l be assumed

o t h e r than x i i s n e g l i g i b l e . culated as

E.

1

1

two energy o u t p u t s .

the maximum c a p a b i l i t y

I n addi-

The energy from t h i s r e s e r v o i r i s e a s i l y c a l -

Therefore t h e t o t a l energy

= yr..

the sum of t h e s e

c onsta nt s t o r a g e l e v e l .

t h a t evaporation i s n e g l i g i b l e and t h a t t h e inflow output of the system i s

I n r e a l i t y , t h e r e may be l i m i t s on of the power p l a n t s , b u t

o f energy output f o r each

t h e se a r e simple c o n s t r a i n t s which wi l l be ignored in t h i s a n a l y s i s . The o b j e c t i v e of water supply a l s o can be considered in two ways. The guaranteed "f i r m" water can be a s e p a r a t e o b j e c t i v e , o r the t o t a l water output over t h e t en period span can be t h e measure chosen.FromFigure 8-1 i t can be seen t h a t t h e amount a v a i l a b l e f o r water supply, iod i s W i

=

Wi,

in any per-

x. - r . . 1

1

The f i n a l o b j e c t i v e i s a c o s t f u n ct i on. I t w ill be assumed t h a t t h e c o s t of power generation a t each p l an t i s proportional t o t h e amount of water passing through t h e power p l a n t , a n d

t h a t t h e v a r i a b l e c o s t of water

suppiy ( l a r g e l y pumping energy f o r d i s t r i b u t i o n ) i s proportional t o the amThen t h e v a r i a b l e c o s t f o r t h e i t h period can be o u n t of water s u p p l i ed . written as C i = a i x i + bi ri + c . ( x . - r . ) = ( a . + c . ) x . + ( b . - c . ) r . 1 1 1 1 1 1 1 1 1 I n g e n e r a l , non-linear c o s t f u n ct i o n s may be more a p p r o p r i a t e b u t l i n e a r i t y

in t h i s i n st a nce

i s reasonable and wi l l be assumed f o r s i m p l i c i t y

without

l o s s of g e n e r a l i t y i n s o f a r a s m u l t i p l e o b j e c t i v e a n a l y s i s i s concerned. which must be considered f o r t h i s system i s

A final constraint

the m o u n t of s t o r a g e a t t h e end of any period cannot of t h e r e s e r v o i r and must be above t h e minimum voir operation,

(Qmin

I

qi+l

exceed

required f o r

t h e c a pa c ity proper r e s e r -

Qmax).

I

The t h r e e o b j e c t i v e o p t i mi zat i o n problem can be summarized a s Maximize firm energy:

Max [Min

1

{

l/3

7 (qi

+ q i + l ) x i + y r .1? ]

1

Maximize firm water:

Max [Min { x i - r i l l

Minimize c o s t :

Min [ z( ai + c i ) x i + ( b1- - c1. ) r .1] i

i

Subject t o

'13

q i +l

+6 2 qi+l

=

q. 1

D 2

2/3

q. - x. 1

1

+

y

i

i = 1,2,

...,

10.

that

147 The alternative formulation replaces the first two objectives by Maximize total energy

=

MAX

[c i

1

v3

Ib

(qi + qi+l) xi + yr.] . 1

xi - ri] . i Note that the cost objective does not change. To make this formulation more realistic, constraints requiring a minimum output of water and energy in each period could be included. Both of these problems are highly non-linear and are difficult to solve. When put into &-constraint form, the first formulation becomes: Maximize total water

=

MAX [X

4) x.1 - r.1 :: t 3 5) ri 3 0 i = 1,2,..., 10. (Note that xi 3 0 is not necessary as a constraint since this condition must be met for constraints 4 and 5 to hold.) With this formulation, there are 20 decision variables, one objective function, 10 equality constraints and 50 inequality constraints. The problem may be reformulated by using the equality constraints.

and substituting for x. The formulation becomes: Problem 8-4:

+

(bi - ci) ri 1 n

143

+ yi - r i > c3

...

(i

=

1,2,

(i

=

1 , 2 , . . . , 10)

)

10)

with bounds on t h e d eci s i o n v a r i a b l e s of 1)

Qmin

2)

ri

s qi+1 s Qma, >

0

The d e c i si o n v a r i a b l e s a r e r 1. and q 1. +1 f o r i q 1 i s a known c o n s t a n t . ) With t h i s formulation

t h e r e a r e 20

=

1,2,

...,

10. (Remember t h a t

bounded de c ision v a r i a b l e s , one

o b j e c t i v e f u n c t i o n , a n d 20 i n e q u a l i t y c o n s t r a i n t s . SOLUTION OF C A L I F O R N I A WATER SYSTEM MULTIOBJEEIVE PROBLEM

8.8

The formulation involving c o s t , t h r e e o b j e c t i v e s was System.

firm w a te r, solved on t h e

implemented and

and firm energy a s t h e Univac

1108 Computer

The values used f o r t h e co n s t an t s a r e l i s t e d in Table 8-4. Finding

t h e minimuin

a n d maximum values f o r

non-linear programming problem programming time,

c2

and

E~

r e q u i r e s t h e solution of a

w i t h minimax o b j e c t i v e s .

an approximate lower l i m i t of zero

In o r d e r t o save

i s used f o r b o t h ob-

The maximurn f o r f 2 i s taken as 300 MWH s i n c e no f e a s i b l e solut i o n could be found f o r any higher values. S i m i l a r l y , t h e maximurn f o r f i s 3 7500 AF. This approximation may lead t o some non-binding s o l u t i o n s (recogn i z a b l e by h = 0 ) which must then be d i s carde d. jectives.

The minimization i n problem 8-4 was solved f o r d i f f e r e n t values of and

t 3

c2

in t h e ranges defined i n t h e preceeding paragraph. These values were

chosen t o be approximately e q u i d i s t a n t

i n t h e f e a s i b l e region and a r e l i s -

ted i n t h e f i r s t two columns of Table 8-5.

Note t h a t t h e r e

rent constraints for

I n some c a s e s , only one w ill be

c2

in problem 8-4.

are

10 d i f f e -

binding, and t h e corresponding non-zero Lagrange m u l t i p l i e r i s the tra de -off ratio A12.

Should t h e r e be more than one non-zero m u l t i p l i e r , t h e one with value corresponds t o t h e most binding c o n s t r a i n t a n d

t h e l a r g e s t a b so l u t e

thus r e p r e s e n t s t h e t r ad e- o f f r a t i o .

I t i s important t o remember t h a t X 1 2

d e sc r i b e s t h e change i n t h e optimal value of f l d u e t o a n incremental change i n f only f o r small increments. This i s e s p e c i a l l y r e l e v a n t here sinc e a

2

l a r g e increment may cause

a d i f f e r e n t one of t h e ten c o n s t r a i n t s t o become

149 TABLE

8-4

D a t a f o r T r i n i t y R i v e r Subsystem Example

Y ( KAF 1

Period

a($/AF)

b(S/AF)

c($/AF)

1

18.0

3.00

2.00

0.60

2

22.5

3.00

2.00

0.70

3

25.0

5.00

2.50

0.60

4

30.0

3.00

2.00

0.50

5

27.5

2.50

1.50

0.50

6

15.0

2.00

1 .oo

0.40

7

10.0

1 .oo

0.50

0.30

8

10.0

1.50

1 .oo

0.40

9

15.0

2.50

2.00

0.50

10

17.0

4.00

2.50

0.60

(Y

=

200 FlWH/AF/Ft

p

=

.04 A F / A c r e

y

=

1500 MblH/AF

91

=

180 KAF

Qmin

=

100 KAF

=

400 KAF

Qmax

MU1,'T'IOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS

150

TABLE

___

8-5

R e s u l t s f o r T r i n i t y River Subsystem Example Computational Model

F i r s t run

f2

A l2

'1 3

'dl 2

1' 3 +lo

100

0

22

-.036

-3.6

+ 8

100

2.5

82

-.028

-5.6

+ 9

- 2

150

0

32

-.036

-3.6

+ 4

150

2.5

85

-.036

-4.6

+ 7

150

5.0

163

-.028

-5.6

+ 2

+lo - 1 - 5

-3.6

+ 2

+lO

200

0

43

-. 036

200

2.5

95

-.037

-3.6

0

0

-.028

-5.6

0 - 4

- 6 +10 - 7

200

Second run

f3

Decision Maker

250

5.0 0

165 54

-.036

-3.6

250

2.5

106

-.037

-3.6

250

5.0

166

-.028

-5.6

- 1 - 4

250

7.5

246

-.028

-5.6

- 3

-10

300

0

65

-.036

-3.5

-

+10

300

2.5

117

-.037

-3.6

- 7

- 1

6

0

300

5.0

170

-.036

-4.6

- 5

- 8

300

7.5

247

-.028

-5.6

- 4

-10

220

2.5

99

-.037

-3.6

0

0

180

2.5

91

200

3.0

106

-.037 -.037

-3.6 -3.6

+ 2 0

0 - 1

200

2.0

85

-.037

-3.6

0

+ 6

f;

are

$1000

f2

are

FIWH/period

f3

are

KAF/period

Note:

the units of

Preferred Solution

Preferred Solution

APl)l,/ I'AT7ON.7

151

7 ' 0 WATER HFSOUHCES PROBLEMS

-

af / af will be ne ga tive , sinc e a n i n c r e a s e 1 2 in c o s t ( f l ) w i l l y i e l d a n i n cr eas e i n firm energy ( f 2 ) . All of t h e above

binding. Note a l s o t h a t h 1 2 =

analysis also applies t o

x13

and

E~

.

The s o l u t i o n procedure u t i l i z e d

f o r problem 8-4

i s t h e generalized

A computer reduced g r a d i e n t ( G R G ) algorithm f o r non-1 ine a r optim iz a tion8. package i s a v a i l a b l e f o r t h e Univac 1108 Computer System. Seventeen values of

c2

and

were used as i n p u t , however one of them did not converge t o a

E~

so l u t i o n within solutions.

t h e time l i m i t of

seconds of computer time.

and four were i n f e r i o r

The r e s u l t s of t h i s computation provide t h e non-

f l a n d t h e trade-off r a t i o s h 1 2 and

i n f e r i o r value of each p a i r

twenty-seconds,

The s o l u t i o n s which did converge r e q u i r e d , on t h e average, four

( L ~ , L ~ ) .

x13

corresponding t o

Those r e s u l t s a r e l i s t e d in t h e middle t h r e e columns of

Table 8-5. The decision-maker ( D M ) was i n t er r oga te d with que stions analogous t o those used f o r t h e stream resource a l l o c a t i o n problem. For example, t h e question corresponding t o row 2 i n Table 8-5 "Given an ope ra ting polic y c o s t i n g $ 82,000 f o r t h e ten planning periods which guarantees 2500 a c re f e e t of firm water per period and 100 megawatt hours of firm power per peri o d , how w i l l i n g would you be t o spend an a dditiona l $2800 t o inc re a se t h e firm energy by 100 MWH/period? A t t h e same p o i n t , w h a t would your w i l l i n g ness be t o spending an ad d i t i o n al a c r e f e e t per period?"

$5600 t o inc re a se t h e firm water by 1000

The responses of t h e DM a r e l i s t e d in t h e l a s t

two

columns of t a b l e 8-5. One p r ef er r ed s o l u t i o n i s found i n t h i s t a b l e - namel y , f; = $95,000, f2f = 200 MWH/period, f i = 2500AF/period. Four o t h e r values of

t2

and

*

E~

*

near f 2 and f 3 were used in t h e GRG algorithm t o t r y t o

determine t h e e x t e n t of t h e i n d i f f e r e n c e band.

These computations a n d r e -

s u l t s a r e l i s t e d i n t h e l a s t f o u r rows of Table 8-5. I t can be seen t h a t t h e i n d i f f e r e n c e band extends approximately from 200 s f; s 220, f i 2.5, thus including values o t h e r

*

f 2 = 210, f

* 3

=

2.5

than those found from Table 8-5;

f o r example,

i s a l s o a p r ef er r ed s o l u t i o n .

Reversion t o t h e d eci s i o n space i s performed f o r values in t h e i n d i f f e r e n c e band, using t h e GRG program. Results f o r t h r e e e q u i d i s t a n t values i n t h e i n d i f f e r e n c e band a r e presented i n Table 8-5. Again t h e s u r r o g at e worth t r ad e- o f f method i s a b l e t o develop a solut i o n even though t h e mu l t i o b j ect i v e problem i s highly non-linear with 20 d e c i si o n s and 30 c o n s t r a i n t s a n d t h r e e o b j e c t i v e s (of which two a r e of t h e minimax f o r m a t ) . Although a r eal d eci s i o n maker was not u t i l i z e d , t h e SWT

inethod could have s o l u t i o n of t h e

found t h e i n d i f f e r e n c e band mu l t i o b j ect i v e problem

d i f f i c u l t p a r t of t h e procedure,

f o r solving

f o r any DM responses.

b u t the availability

non-linear probleiiis rendered

the Lagrange m u l t i p l i e r s f o r t h e

The

in E-c onstra int format i s the most of t h e GRG

t h i s s t e p simple.

package

I n addition,

€ - c o n s t r a i n t s a r e a utom a tic a lly output b y

the GRG. The a c t u al decision-maker f o r t h i s model of t h e T r i n i t y River subsystem would

of course

select a different

s e t of values

of water power and

c o s t t h a n was s el ect ed by t h e s u b s t i t u t e decision-maker used f o r t h i s example.

However, the number o f i t e r a t i o n s a n d t h e form of the c a l c u l a t i o n i s

e s s e n t i a l l y t he same. I t i s , of co u r s e, p o s s i b l e t o as s ign p r i c e s t o the energy a n d water produced a s was done by Hall and Shephardg. This procedure in e f f e c t , presumes t h a t a l l u n i t s of water a n d a l l u n i t s of energy a r e i n d i s t i n g u i s h a b l e i n value r e g a rd l es s of c o s t of production or l e v e l s of production.

cases t h i s i s approximately c o r r e c t , c o n t r a c t s s p e ci f y f i x ed p r i c e s iiiaker i s t h e monetary

I n some

p a r t i c u l a r l y when the power atid water

only i n t e r e s t of t h e de c ision of c o s t . I n most c i v i l systems,

a n d when the

r e t u r n i n excess

however, t h e l a t t e r i s n o t s t r i c t l y t r u e ,

even when

revenue a r e f i x ed i n advance of a n a l y s i s .

The s i m p l i c i t y of t h e SWT method

i s immediately apparent i f t h e

the prices

producing

"v al u e" of water and energy i s not based on

an a r b i t r a r y p r i c e b u t r a t h e r on t i i e i r impact on soc ia l goa ls.

Evaluating

a " p r i c e f o r a l l conbinations of wat er , energy and c o s t l e v e l s f o r the 600 m o n t h planning horizon would be a d i f f i c u l t t a s k . I n t h i s p a r t i c u l a r example i t i s p o s sible t o e stim a te t h e "cash value" of t h e marginal u n i t s of water and energy f o r t h e pre fe rre d s o l u t i o n s . Note

t h a t t h e product of t h e true-worth r a t i o times t h e tra de -off r a t i o must equ-1 a t o p t i m a l i t y . Since t h e DM co n s ide rs t h a t t h e pre fe rre d s o l u t i o n s he a t t a i n e d a r e o p t i mal , i s t h e marginal monetary p r i c e of M W H , i . e . , $37 per megawatt-hour a t t h e margin i n our example,and $3.60 per a c - f t . i s

al

t h e moneta$yvalue of water a t themargin f o r o u r DM.These a r e t h e average u n i t values and nothing can be deduced regarding the t o t a l vcllue of t h e water a n d power produced except

tion.

t h a t t h e i r sum exceeds the t o t a l c o s t of produc-

c, 0 W

a i

0

.->

in S

c,

.-0 3 7

0

Lo

i

m

.

h

*

N

.

N

N

r .

N

.

c r 7

m

w

r.r. N

m

m r . N

eh

h

m .

m

i

N

N N

N N

m N

m N

N d

h

N

m

N

r

m

m r.

L

.r

x

.r

m

.

N

m

.

a,

i

N

r.

N

r. .

t n . N

e

m

N

r

'-

r.

h

Lo N

W

N

r.

0 co

m

7

m M

r.

N

W N

W

m

r. r.

0) N

i

N

01 N

r.

a

co N

Lo

N

N co

W N

W N

r. N

N

.r

a, N

0

0 5J

0 05 N

5J

m

r. N

i

r.

m N

03

i

N

N

N

03

N

m

M N

M

m

c9

N W

m N

m

N N 00

Lo

co

m r.

co r. N

0 N

r .

m

m

N

m

m

N

N

r. N

m

i

r.

N

0

m

N

0

m

m

r

+

7

.r

cr

N

I1

m m

X

.r

m

m

m

N 03 N

m

N Lo

Lo

N

r.

N

N

e-

a

0,

r.

m

m

m

e-

co

N

r. N

N

W

e-

N Lo

cu

co

% co

N

c3 N

W

m 0 N

N

m

0

m

L

.r-

X

N

I1

Lo

I1

r

0

I1

m

h

.r

m

m 7

+

.r

i

5

N

Lo

N

I1

Lo

I1

0 0

m /I

*

0

m

F-

m

N

m

N

m

m m

d

m

Lo

m

m m

7-

e-

e-

co

.r

L

W r. 0

m

N h

m N

m

m 00 N

dLo

N 5J

0 N

co N

m m h

N

m N Lo N

Lo

co

N N

W

m

0 N

N

m m

7-

+

i

.r

U

I1

N

m I1

0 N N

N

153

154

MULTTOBJECTIVE' OP7'IMIZATJON TN WATER RESOiIRCES SYSTEMS

FOOTNOTES

1

This m u l t i o b j e c t i v e

problem

and Vemuri

t h e i r s o l u t i o n was

[1974];

i s described

i n R e i d and Vemuri [1971]

t o f i n d an a n a l y t i c f u n c t i o n

f o r the non-inferior set. 2.

Rows

16-24

Vemuri

i n T a b l e 8-1

[1971].

correspond t o t h e values

*

Note t h a t t h e i r

t a k e n f r o m Reid-

*

J1 corresponds t o f 2 , and J2 t o f

* 1

due t o t h e way t h e o b j e c t i v e s were d e f i n e d . 3.

Some o f t h e a u t h o r s who

4.

The

5.

A description

Haimes e t a1 [1972],

have s t u d i e d t h i s problem

are

Hass [1970],

and Liebman and Lynn [1966].

original formulation

o f t h i s equation

i s f o u n d i n S t r e e t e r and

Phelps [1925]. of

upper bounding

techniques

can be found

in

Taha

[1971]. 6. 7.

B o t h t h e d a t a and t h e b a s i c model a r e t a k e n f r o m Hass [1970]. See H a l l and Shephard [1967] o r H a l l and Dracup [1970] f o r a d e s c r i p t i o n o f t h e C a l i f o r n i a water

s u p p l y system and i t s

analysis i n

the

s i n g l e o b j e c t i v e format 8.

T h i s a l g o r i t h m i s d e s c r i b e d b y Lasdon e t a1 [1973]

9.

Again see H a l l and Shephard [1967].

REFERENCES

1.

Y., Kaplan,

Haimes, Y .

M. A . ,

and Husar, M.A.,

" A M u l t i l e v e l Approach

t o D e t e r m i n i n g Optimal T a x a t i o n f o r t h e Abatement o f Water tion,"

2.

~

H a l l , W. A . and Dracup, J . A.,

Water Resources Systems

McGraw-Hill Book Company, N.W., 3.

H a l l , W.A. and R.W. sources C e n t e r . Hass, J. E . ,

Engineering,

1970.

Shephard, "Optimum O p e r a t i o n s f o r P l a n n i n g o f a Com-

p l e x Water Resources System,"

4.

Pollu-

Water Resources Research, v o l . 8, no. 4, 1972.

U n i v e r s i t y o f C a l i f o r n i a Water

Re-

C o n t r i b u t i o n # l 2 2 , Los Angeles, 1967.

"Optimal T a x i n g

f o r t h e Abatement o f Water P o l l u t i o n , "

Water Resources Research, v o l . 6, no. 2, 1970.

5.

Lasdon, L. S . ,

Fox, R. L . , and Ratner, M W . ,

"Nonlinear

Using t h e G e n e r a l i z e d Reduced G r a d i e n t Method," serve U n i v e r s i t y ,

O p e r a t i o n s Research Department

Optimization

Case Western

Re-

T e c h n i c a l Memo-

randum no. 325, 1973. 6.

Liebman, J.C.,

an3 Lynn, W.R.,

"The Optimal A l l o c a t i o n o f Stream D i s -

s o l v e d Oxygen," Water Resources Research, v o l . 2, n o . 3, 1966.

APPL TI;11'IONS TO WATER RESOURCES PROBLEMS

7.

"On the N o n - i n f e r i o r Index Approach t o J o u r n a l o f the F r a n k l i n Ins t i t u t e , v o l . 291, no. 4 , 1971. S t r e e t e r , H.W., and P h e l p s , E . B . , "Study o f the P o l l u t i o n and Natural P u r i f i c a t i o n o f the Ohio R i v e r , " P u b l i c H e a l t h B u l l e t i n no. 146, 1925. Taha, H. A., O p e r a t i o n s Research: An I n t r o d u c t i o n , The Macmillan Company, N. Y., 1971. Vemuri, V . , " M u l t i p l e O j b e c t i v e O p t i m i z a t i o n i n Water Resource Syst e m s , " N a t e r Resources Research, v o l . 1 0 , no. 1 , 1974. Reid, R . W.,

and Vemuri, V . ,

Large S c a l e M u l t i - C r i t e r i a Systems,"

~~

8.

9. 10.

155

Chapter 9 MULTIOBJEC IVE WATER QUALITY MODELS

INLROJKCJIIbJ

9.1

I n a world where new c r s es continue t o overshadow previous ones a n d t h e attempted s o l u t i o n of one c r i s i s i s c e r t a i n t o a f f e c t previous c r i s e s a n d c r e a t e new ones,

where m u l t i p l e and

ofte n

noncommensurable goals a n d

o b j e c t i v e s ( o f t e n i n c o n f l i c t a n d competition with each o t h e r ) c h a r a c t e r i z e our society,

needed.

t o systems modeling

a cau t i o u s approach

and optim iz a tion i s

This sober and r e a l i s t i c approach should recognize t h e mutual in-

t,eractions among t h e various g o al s sion rilaker(s)

in analyzing t h e

and o b j e c t i v e s a n d should a i d t h e de c i-

trade-offs

among various

o b j e c t i v e s in a

q u a n t i t a t i v e way. A fundaiiiental and

almost axiomatic p r e r e q u i s i t e f o r the se models t o

be r e a l i s t i c , a n d thus be considered f o r a n ultim a te u t i l i z a t i o n by the dec i s i o n r!iaker(s), i s t h a t they be s u s c e p t i b l e t o m u l t i p l e o b j e c t i v e func tions in t h e i r noncoiTliliensurab~e forms and u n i t s . The second fundamental p r e r e q u i s i t e i s t h a t t h e r e should e x i s t solut i o n niethodologies which

a r e capable of

analyzing

and optimizing ( i n the

mind o f t h e decision maker(s) ) t h es e m u l t i p l e o b j e c t i v e s . The present lack of mathematical models with m u l t i p l e o b j e c t i v e funct i o n s can be a t t r i b u t e d p r i mar i l y t o t h e past lack of ope ra tiona l methodolog i e s capable of t i v e functions.

analyzing and optimizing The

m u l t i p l e noncommensurable objec-

Surrogate Worth Trade-off

methodology t h a t f u l f i l l s t h e

(SWT)

second p r e r e q u i s i t e .

Method i s one such

Systems engineers and

systems modelers can now move t o f u l f i l l t h e f i r s t p r e r e q u i s i t e f o r r e a l i s t i c models by co n s t r u ct i n g t h e proper m u l t i p l e o b j e c t i v e func tions.

9.2

!.YTER

QUALITY GOALS AND OBJECTIVES Two major c l a s s e s of water q u a l i t y o b j e c t i v e s may he i d e n t i f i e d . The

f i r s t c l a s s , primary o b j e c t i v e s , corresponds d i r e c t l y t o water q u a l i t y s t a n ddrds a n d thus depends upon t h e p a r t i c u l a r water r e s o u r c e ' s physic a l, chemic a l , a n d biological c h a r a c t e r i s t i c s . Secondary o b j e c t i v e s ,

t h e second c l a s s i f i c a t i o n , correspond b o t h t o

t h e impact of water q u a l i t y l ev el

on t h e u t i l i z a t i o n o f t h e water resource

a n d t o t h e impact o f water resource use on q u a l i t y .

9.2.1

Primary ObjectivesThe Great Lakes llater Quality Agreement

between t h e U . S . and Canada

provides a l i s t i n g of primary o b j e c t i v e s and goals in water q u a l i t y 1 . These

156

o b j e c t i v e s a n d g o al s a r e presented and s p e c i f i e d in the Act e i t h e r i n ternis o f upper

o r lower c o n s t r a i n t s ( l e v e l s of achievement) o r in terms o f objec-

tives t o

be minimized

o r maximized.

The following i s a sample of

water

q u a l i t y goals and o b j ect i v es :

( Phenols and o t h er o b j ect i ona ble t a s t e and odor ni i n i iii i ze

mini''1ize

I

I I

producing substances Temperature change t h a t would adversely a f f e c t any

I

l o cal o r general use of t he water Mercury and o t h e r t o x i c heavy metals

iiinimize

\

minimize

I

I I

( P e r s i s t e n t pest control products a n d o t h e r

1

p e r s i s t e n t organic contaminants t h a t a r e toxic o r harmful t o h u m a n , animal, or a qua tic l i f e

1 I

{ 5 e t t l e a b l e and suspended m a t e r i a l s 1

rriiniiiiizc miniill-ire

{

minimize

{ R ad i o act i v i t y 1

O i l , petrochemicals, and immiscible substances

1

a n d goals i n terms o f upper and lower c o n s t r a i n t s : t o t a l col iforms : 1000 per 100 m i l l i l i t e r s

f e c a l c o l i f o r ms

,< 200 per 100 m i l l i l i t e r s

dissolved oxygen

6 . 0 mi l l grams per l i t e r

t o t a l dissolved s o l i d s

,<

Iron

: 0. 3 mi l l grains per l i t e r s 8.5

6.7

<

pH

200 mill grams per l i t e r

ower achievement a c t u a l l y corresponds I n most c a s e s , t h e upper o r t o u l t i m a t e des i r ed goals r ep r es en t i n g s h o r t , inte rm e dia te , o r l o n g term perspectives. Qu an t i t at i v e co n s i d er at i o n of the se goa ls 1s c o n s t r a i n t s within a formal model may, t h e r e f o r e , introduce a se ve re i n t r a c t a b i l i t y in t h e i r f u r t h e r a n a l y s i s , a s well a s a misre pre se nta tion of the re a l world t h a t i s being modeled. Two conclusions t h a t may be drawn from the above g o a l s a n d o b j e c t i v e s : ( i ) There e x i s t mu l t i p l e noncommensurable o b j e c t i v e s in water qua1 i t y co n t r o l and management. ( i i ) Objectives and g o al s included in a mathematical model a s c o n s t r a i n t s (such a s upper l i m i t o n coliforms o r lower l i m i t o n dissolved oxygen) will g e n e r a l l y ( a n d most l i k e l y ) lead t o s o l u t i o n s on t h e boundaries of those c o n s t r a i n t s . T h u s important information concerning t h e e f f e c t of re la xing one c o n s t r a i n t on t h e improvement o f another i s not r e a d i l y ava i 1able.

MUr,,TIOHJE(‘TI V E 0PTIMIZATIO.V IIL’ CU’ATER RESOURCES SYSTEMS

158

9.2.2 Secondary O b j e c t i v e s I n g e n e r a l , t h e secondary o b j e c t i v e s a r e n o t d e f i n e d as p r e c i s e l y o r q u a n t i t a t i v e l y as t h e p r i m a r y o b j e c t i v e s .

The

following

is a

sample o f

secondary o b j e c t i v e s : (i) (ii) (iii)

Reduction i n the l e v e l o f a l g a l growth Restoration o f year-round aerobic c o n d i t i o n s R e s t o r a t i o n o f t h e w a t e r body f o r t h e purpose o f swimming, f i s h i n g , and r e c r e a t i o n

(iv)

M i n i m i z a t i o n o f any h e a l t h hazards

C l e a r l y , a s y s t e m a t i c and q u a n t i t a t i v e methodology c a p a b l e o f a n a l y z i n g t h e t r a d e - o f f s among

a l l o b j e c t i v e s i s needed.

The S u r r o g a t e Worth T r a d e - o f f

(SWT) Method f i l l s t h i s need as w i l l be d i s c u s s e d s u b s e q u e n t l y . 9.3

GENERAL PROBLEM FORMULATION Most e x i s t i n g s i n g l e o b j e c t i v e models f o r

w a t e r q u a l i t y c o n t r o l and

management can be extended t o i n c l u d e m u l t i p l e o b j e c t i v e f u n c t i o n s .

As an

a s i n g l e o b j e c t i v e f u n c t i o n w a t e r q u a l i t y model2 w i l l be ex-

illustration,

tended t o t h e case o f m u l t i p l e o b j e c t i v e s . Given a system i n t o

w a t e r r e s o u r c e s system,

N subsystems

it i s

c o n v e n i e n t t o decompose t h e

(N reaches i n t h e case o f a r i v e r ) .

This allows

modeling and a n a l y s i s o f a l l o f t h e system i n p u t s and responses.

yi

Let the vector

l,2,,

. . ,N,

where U. = -1

be t h e i n p u t ( p o l l u t i o n ) t o t h e ith subsystem,i

1. iM

[U.11, Ui2,...,U.

=

The f i r s t element o f t h e v e c t o r

Ui, Uil, may r e p r e s e n t w a t e r q u a n t i t y and t h e Ui2,....U. may r e p r e s e n t iM d i f f e r e n t w a t e r q u a l i t y c h a r a c t e r i s t i c s ( e . g . , BOD, pH, t e m p e r a t u r e , t o t a l dissolved solids, e t c . ) .

4 S.

-1

F i g u r e 9.1

ithsubsystem r d p r e s e n t a t i o n

MlJL7'i OBcJh'C"71V E

159

WA TF'R QUAL i T Y MODELS

Let be t h e i n p u t v e c t o r coming i n t o t h e ith subsystem f r o m

W.

-1

o t h e r subsystems, subsystem g o i n g t o o t h e r be t h e o u t p u t v e c t o r o f t h e ith

V.

--1

subsystems, and subsystem, be t h e d e c i s i o n v e c t o r o f t h e ith

S.

-1

where

Wi,

vi,

Figure 1 ) .

and S . a r e o f t h e same dimension as -1

Clearly,

equal V . + S . . -1

Wi

t h e v e c t o r sum

vi

The subsystem O u t p u t s

-1

+

U., i . e . , M-dimension ( s e e

-1

lJi i s meaningless and Si a r e assumed t o be

and

does

not

represen-

t e d by t h e f o l l o w i n g f u n c t i o n s .

where -1 y.

=

bil,

Yi2,

...

-1 0.

=

[Oil,

oi2,

...

i

= 1 , 2,

)

YiM]

)

OiM]

..., N

A t p r e s e n t , n o t a l l components o f t h e f u n c t i o n s Y . and a . a r e known t o w a t e r -1

q u a l i t y experts.

-1

T h i s however c o n s t i t u t e s no l i m i t a t i o n t o t h e model, be-

cause whenever t h e f u n c t i o n a l r e l a t i o n s h i p o f any component o f LJi i s known, t h a t component can be i n s e r t e d and t h u s b e c o n s i d e r e d i n t h e a n a l y s i s . Natur a l l y , t h e presence o r absence o f a component o f LJi has a s t r o n g t h e model.

e f f e c t on

Many c u r r e n t m a t h e m a t i c a l models c o n s i d e r as components o f

o n l y BOD o r DO and assume t h e S t r e e t e r - P h e l p s f u n c t i o n a l r e l a t i o n s h i p 3 . i s evident,

however, t h a t t h e g r e a t e r

t h e number o f components o f

ci

U.

-1

It

that

a r e c o n s i d e r e d and a n a l y z e d , t h e m o r e a c c u r a t e and r e p r e s e n t a t i v e t h e mathem a t i c a l model becomes. Finally, l e t

G(U,

-

W, 5) 5

0

be k - d i m e n s i o n a l v e c t o r o f c o n s t r a i n t s .

The i n e q u a l i t y c o n s t r a i n t s i n c l u d e

e q u a l i t y c o n s t r a i n t s and r e p r e s e n t t h e p h y s i c a l , l e g a l , economic, and o t h e r system c o n s t r a i n t s . Note t h a t

u

=

[U -1

:. -2: u '

-

s

=

[S1

1

w

=

[W -1

:. w-2: '

__

-

:. $1

z2; ;

a r e auqirrented v e c t o r s . The s i n g l e c r i t e r i a optimization problem, e . g . , t h e minimization of 1 of treatinent c o s t f u n ct i o n s of each subsystem, F i ( U i , S . ) :

the sui:i F

-1

can be extended t o include o t h e r o b j e c t i v e s .

For example, l e t

be t h e j t k l o b j e c t i v e f u n ct i o n f o r t h e water resources system, e . g . : for j

=

1,

F1 i s t h e above c o s t f u n c t i o n , ( E q . 1 ) ;

for j

=

2,

F

for j

=

3,

F 3 may be t o t a l mercury a n d othe r toxic heavy metals;

for j

=

4, F4 may be t o t a l s e t t l e a b l e and suspended m a t e r i a l s ;

for j

=

5,

F

2 may be t o t a l phenols;

5 may be t o t a l o i l , petrochemicals, a n d immiscible

s u b s t an ces . C l e a r l y , t h e o v er al l o b j e c t i v e f u n ct i o n f o r the water re sourc e s system w ill g e n e r a l l y n o t be a simple summation of t h e va rious o b j e c t i v e s .

Also, note t l i a t t h e r e i s no discrepancy i n t h e f a c t t h a t t h e mth component of U., U i m , .-1 riia,y be m c x u r y co n cen t r at i o n and t h a t t h e j t h o b j e c t i v e func tion FJ i s a l s o r:ercury concentratioii. The o v er al l model f o r water q u a l i t y c ontrol and management can be written a s : S;isteni 9-1 : ninimize

{F

1

(LJ,S ) , . . . , ~ ~511 (1,

UlS b jec t t o t h e con s t r a i n t s

s 1 . 1

G(U-,W,S)

:'

0

Ceconposition and h l t i l e v e l Approach" can be applied f o r solving t h e overa l l problem

where t h e Surrogate ldorth

higher level c oo r d i n at o r .

Trade-off Method i s

The d i s cu s s i o n

of t h e

utilized a s

a

hie ra rc hic a l-m ultile ve l

approach i s beyond t h e scope of t h i s book. I n the

following

wi 1 1 be d i scu s sed .

section,

examples f o r

multiobjective

systems

9.4

FORMULATION PROBLEM .~~ ._ ~ .-~ O_F A N EXANPLE __.. ..____~ The following formulation i s an i n t e g r a t i o n and extension of several

s i n g l e o b j e c t i v e f u n ct i o n models.s I n t h e model discussed here, t h e ve c tors lJi,5 ,LIi, and l i (forthe a r e t h r e e dimensional v e c t o r s ; t h e elements of which a r e

i t h subsystem)

a s s o c i a t e d with b i o l o g i cal oxygen demand loa d,

thermal loa d, and a lga e con-

centration respectively. Models which a r e concerned with t h e e f f e c t o f biologic a l oxygen demand (c(03)load on t h e dissolved oxygen ( D O ) in t h e stream a r e s i m i l a r t o t h e The s i n g l e objec-

stream resource a l l o c a t i o n model described in Chapter 8 .

t i v e f u n c t i o n i s t h e t o t a l c o s t of wastewater treatment f o r a l l use rs along t h e stream. The streaiii i s segmented i n t o Pi r ea c he s, each of which i s a ssoc ia te d with u s e r s ( p o l l u t e r s ) who niay d i s ch ar g e organic wastes i n t o t h e stream. The S t r eet er - P h el p s r e l a t i o n 6 i s u t i l i z e d t o transform minimum d i s solved oxygen

standards f o r each

reach i n t o a s e t of

linear inequalities

r e l a t i n g upstream treatment l e v e l s t o downstream d e c i s i o n s . The c o n s t r a i n t s r e q u i r e t h a t the supply of a v a i l a b l e oxygen f o r the organic decomposition process i n each reach ( t h a t a v a i l a b l e above the quali t y standard requirement)

must be equal t o o r exceed t h e demand imposed by

BOD loads discharged i n t o t h a t reach and a l l reaches preceding i t .

Thus, f o r reach i i t i s r eq u i r ed t h a t d i l u l l ( l - s l l ) + d 1. 2 u 21 ( 1 - s Z 1 ) +

+ d.. u. (1 - s i l ) 11 11

where

u.

51

=

d..

=

e.

1

i

=

1,2,

...,

N

(2)

g r o s s b i o l o g i cal oxygen demand ( B O D ) load introduced in p u t ( 1 bs/day)

=

1J

s e.

a t t h e beginning of t h e j t h reach t h a t has a p o l l u t i n g

8 .

51

...

.

th f r a c t i o n of u . removed through treatment by t h e j 51 pol 1 u t e r . pounds of oxygen demanded by t h e decomposition of a pound o f BOD discharged by t h e j t h p o l l u t e r in reach i.

=

a m o u n t of d i s s o l v ed oxygen a v a i l a b l e f o r t h e decomposition process ( t o t a l a v a i l a b l e l e s s standard requirement) i n reach i pcr u n i t of flow.

I n a d d i t i o n , o t h e r r e s t r i c t i o n s on s . ' s r e q u i r e a t l e a s t primary treatment, 51

thus

0.45 : s .

51

s 0.99

j

=

1,2, ..., H

162

MUL7’IOU,JBC7 TVE OP?I?4IZATl ON IN WATER RESOURCES SYSTEMS

An o b j e c t i v e f u n ct i o n as s o ci at ed with t h i s model i s t o minimize --1 F (S1):

the

t o t a l c o s t of wastewater t r eat men t . -F1

Ilin

(S1)

‘jl where

s

=

-1 and

--1 f (sjl)

[Sll

I41

1 A

160.8 + 26.7 ’1. + €40.7 ( s . - 0.45) J 51

=

f

255.7 q . ( s . J 51 0.99

-

2

0.45) 2

:: s . s 51 q . i s t he p l an t s i z e i n mi l l i o n g a l l o n s per day.

\ / h e r e 0.45 and,

,szl ) . . . ) s

J

(3)

Hass derived equa-

t i o n ( 3 ) from F r an k el ’ s d a t a f o r t h e !!iami River in Ohio.7 The niodel presented by Foley, and Foley a n d tiaimes8 concerned i t s e l f with o t h e r water q u a l i t y s t an d ar d s i n a d d i t i o n t o DO. These were thermal pollution and algae. Let:

u.

52

raw load of energy introduced a t t h e beginning of t h e

=

j t h reach

sj2

=

v. 57

=

percentage of waste heat ( u . ) removed by cooling towers 52 temperature of water leaving t h e j t h reach t o t h e j + l S t reach

A s e t o i c o n s t r a i n t s on t h e thermal p o l l u t i o n can be introduced.

vj2

:: v j 2 (Max)

vj2

=

Vj2(Uj2’

Sj-1 , 2 ’ s j - 2 , 2 . .

. . ’S12’V02’t)

where v . (Max)

=

maximum temperature allowed

V

=

i n i t i a l temperature of flow e nte ring t h e f i r s t reach

=

time

52

02

t

Equation ( 4 ) can be r e w r i t t e n a s :

vj2

=

v 5. 2 ( w 5. 2 ’ u j 2 3 t ,

wj2

=

temperature of t h e water ente ring t h e j t h reach

where from t h e ( j - 1 ) reach a t time t .

MULT 1OHJECTIVE WATEII QUALITY MODELS

163

C r i t i c a l f l o w c o n d i t i o n s i m p l y d r o p p i n g t h e t i m e dependence i n t h e exponent i a l decay as g i v e n by Lesbosquetg. 2,

the decisions

I n a development s i m i l a r t o

equation

a r e r e l a t e d t o t h e t e m p e r a t u r e v . by t h e f o l l o w i n g 52

s.

52

equation :

where

'j2

=

b . kJ

=

j k c= l bkj u k 2 ( 1 - 5 k 2 ) ~ ~~

. Jv2 (Max),

j = 1,2,

...,

N

(5)

c o n s t a n t s r e l a t i n g t h e t r e a t m e n t o f t h e raw l o a d u

k2

t o a decrease i n t e m p e r a t u r e i n t h e jth r e a c h .

Note t h a t t h e u . a r e g i v e n i n u n i t s commensurable w i t h v . byassump52 52 t i o n , and t h a t t h e e n t i r e f l o w o f t h e r i v e r i s u t i l i z e d by t h e stream power p l a n t s ; o t h e r w i s e e q u a t i o n ( 5 ) must be m o d i f i e d . The c o s t f u n c t i o n must be m o d i f i e d i n t h e p r e s e n t c o n t e x t t o i n c l u d e the additional

where

f!(s. ) J 52

z2

=

cost associated

=

s? + c . J J 2 J

01.

rs12, s 2 2 ,

with

t h e removal

of

thermal

pollution,

'

... ,

'N2'

and 0 1 . and c . a r e c o s t c o e f f i c i e n t s o f t h e system. Since i t i s desired t o J J -1 m i n i m i z e t h e c o s t o f t r e a t m e n t o f t h e r m a l p o l l u t i o n , F (5,) can be added t o the cost function.

T h i s y i e l d s a new c o s t f u n c t i o n

F (513 -2 s )

=

I F1(Z1)

+

F1(5S)l

T h e r e f o r e , t h e f i r s t o b j e c t i v e f u n c t i o n ( i n commensurable d o l l a r s ) i s : Min

s s

1 F (Z1,

5,)

-1 '-2 subject t o the for

c o n s t r a i n t s discussed

thermal p o l l u t i o n

expresses

previously.

the desire t o

The o b j e c t i v e f u n c t i o n minimize t h e

temperature

change t h a t would a d v e r s e l y a f f e c t any l o c a l o r g e n e r a l u s e o f t h e w a t e r .

S i n c e 'J2 i s g i v e n as a f u n c t i o n o f

5,

( i n e q u a t i o n 5 ) t h e second ob-

j e c t i v e f u n c t i o n c a n be r e w r i t t e n as a f u n c t i o n o f

z2 o n l y :

..

164

.

The o b j e c t i v e f o r a l g a e c o n c e n t r a t i o n i s d e r i v e d a s f o l l o w s : Foil owi n q e a i 1 e y l

vj3

=

exp

(u,

a2 (

+1 0~. 1 6 v . ) 52

~

j

=

1,2, ..., N

(6)

where = a l g a e ( p h y t o p l a n k t o n ) c o n c e n t r a t i o n a t the end of t h e j t h r e a c h 53 and ' u ~ ,k = 1 , 2 , 3 a r e c o n s t a n t s c h a r a c t e r i s t i c t o t h e s t r e a m . In a p r e -

v.

liminary study,

these constants

v a l u e s of w a t e r d e p t h ,

are

determined

s o l a r i n t e n s i t y , and

t h a t a l g a e growth depends on

b.y the

assumed c r i t i c a l

nutrient concentration.

thermal l o a d removal

z2 v i a

tion ( 6 ) .

Note

the V . i n equa52

The o b j e c t i v e f u n c t i o n r e p r e s e n t s t h e d e s i r e t o minimize t h e maximum algae concentration for a l l reaches.

Thus

Max v . j = 1,2, ..., N j 53' To summarize, the o v e r a l l mathematical model i n c l u d e s t h r e e

F3(5,)

=

noncommensur-

a b l e o b j e c t i v e f u n c t i o n s , a s well a s s e v e r a l c o n s t r a i n t s : MIN

n

F1(S1,Z2) =

2 160.8 + 26.7 q . + 640.7 ( s . -.45) J 51

1

j=l

+ 255.7 q . ( s . - .45) ' J

MIN Subject t o

F

3

(z2) =

.45

0 s

i

31

Max v . 53 j

c

s. 51

'j2

2

+ a.s. + c. J 52 J

'

.99

1 .oo

j

=

1,2,

...,

N

j

=

1,2,

...,

N

9.5

A P P L I C A T I O N OF THE SWT METHOD TO THE T H R E E WATER QUALITY OBJECTIVE _PROBLEM _ __

The three o b j e c t i v e s defined in t h e previous s e c t i o n a r e t o be o p t i iiiized a s symbolically expressed by system 9-1, subject t o t h e e x i s t i n g physical constraints. I n order t o generate t h e trade-off f u n c t i o n s ,

A , . , t h e vector o p t i 1J mization problem i s r e w r i t t e n in the c - c o n s t r a i n t form a s follows:

s -2s -1' subject t o the constraints

where the v a r i a b l e s t . a r e r e l a t e d t o f . (the minimum of t h e j t h o b j e c t i v e J J function, while ignoring a l l o t h e r ( n - 1 ) o b j e c t i v e s ) a s follows: -

f. J

"j

where

6.

J

= = i'

min F~ -

f.

J 0

+ 6. J

The system's Lagrangian, L , i s :

where

i s a vector of Lagrange m u l t i p l i e r s .

The trade-off f u n c t i o n s A 1 2 and A13 a r e determined by solving equation ( 7 ) . A d e t a i l e d computational discussion on t h e construction of t h e trade-off functions was given in Chapter 3. Note t h a t the value of

i s the r a t i o of the

incremental gain in

o b j e c t i v e 1 ( c o s t minimization) t o t h e incremental l o s s i n o b j e c t i v e 2 (temperature) and a value of A 1 3 i s t h e trade-off r a t i o between c o s t and algae production. The A . . corresponding t o t h e binding c o n s t r a i n t s a r e 1J associated with t h e noninferior s o l u t i o n and thus a r e of i n t e r e s t , t h e s e 2 2 i . a. r e also positive. The term ( F - E ~ ) r e p r e s e n t s t h e amount by which F 1J

exceeds t h e t a r g e t a t t a i n e d level of temperature, E ~ . T h e Lagrange multi2- E( ~F) = 0 when equation ( 7 ) i s minimized i s t h e plier which makes A ~ ~ "shadow price" o r marginal trade-off value, expressed in d o l l a r c o s t per

166

MULTTOBJECTIVE OPTIMIZATION IN WATEE RESOURCES SYSTEMS

u n i t temperature increase. The L a g r a n g i a n i s s o l v e d f o r d i f f e r e n t v a l u e s o f ponding t o

each s o l u t i o n a r e

a minimum o f

c 2 , E ~ , h 1 2 , and

values o f

range, t h e t r a d e - o f f

By c h a n g i n g c 2 and

r a t e f u n c t i o n s X12(F

E~

and

E ~ .

Corres-

o b j e c t i v e f u n c t i o n 1, F1*,

1*

, E ~ , E ~ )and

E~

and

over a reasonable

X1:](F

1*

, E ~ , E ~ )

can

be computed. R e w r i t i n g t h e v e c t o r m i n i m i z a t i o n problem

posed i n t h e € - c o n s t r a i n t

f o r m where F2 i s t h e p r i m a r y o b j e c t i v e y i e l d s t h e same s o l u t i o n . The t r a d e o f f r a t e f u n c t i o n s xZ1 and xZ3 c o r r e s p o n d i n g t o t h e L a g r a n g i a n , L 2 , (where

L2

F

=

2

+ AZ1(F 1 and x13

are related t o

The above s t e p s c a n be an i n t e r a c t i o n

with the

t a k e n on a s t r i c t l y a n a l y t i c a l b a s i s w i t h o u t

decision-maker.

The f o l l o w i n g

steps i n v o l v e t h e

decision-maker. S e l e c t any s e t ( F temperature r i s e ,

1*

,

E ~ , c3)

as " o p t i m i z e d " a t t a i n e d l e v e l s o f c o s t ,

and a l g a e p r o d u c t i o n r e s u l t i n g

from t h e "optimal" d e c i -

s i o n determined by s o l v i n g e q u a t i o n (7). For t h i s s e t X 1 2 and A13 Begin w i t h

a r e known.

The d e c i s i o n - m a k e r i s asked whether o r n o t he would g i v e

A12.

up one u n i t o f t e m p e r a t u r e i n o r d e r t o g a i n

h12

units o f cost.

yes, he i s asked t o a s s i g n a n u m e r i c a l v a l u e between s t r o n g l y he would f e e l a b o u t t h a t t r a d e , senging a s t r o n g d r i v e t o

I f he says

0 and +10 t o show how

z e r o b e i n g i n d i f f e r e n t , 10 r e p r e -

g a i n c o s t a t t h e expense

o f temperature.

I f he

says no, t h e n he a s s i g n s a v a l u e between 0 and -10 t o r e p r e s e n t how s t r o n g l y he f e e l s toward t h e o p p o s i t e d i r e c t i o n o f t r a d i n g .

value corresponding t o e2and

E~

The s u r r o g a t e w o r t h

( a s s i g n e d by t h e d e c i s i o n - m a k e r ) i s denoted

by W 1 2 ( ~ 2 3c 3 ) . A t t h e same t i m e t h e d e c i s i o n - m a k e r i s asked h i s p r e f e r e n c e w i t h r e s -

He i s , o f c o u r s e , i n f o r m e d what t h e 1* a t t a i n e d l e v e l s o f c o s t , t e m p e r a t u r e , and a l g a e p r o d u c t i o n would be ( F , 1* c2, E ~ ) . Note t h a t F1* = F ( E ~ , F ~ hence ) t h e r e a r e r e a l l y o n l y two i n d e pect t o

i n e x a c t l y t h e same way.

A13

pendent v a r i a b l e s i n t h e o b j e c t i v e space. L e t us presume, f o r t h e sake o f i l l u s t r a t i o n , t h a t t h e d e c i s i o n - m a k e r gave a v a l u e o f +8 t o t h e surrogate worth o f f2

and

t3

are required.

<

E2'

E3

surrogate worth o f

a t other c2, s3.

and a v a l u e o f +3 t o t h e

T h i s i n d i c a t e s a decrease i n b o t h

The d e c i s i o n - m a k e r i s t h e n asked f o r t h e s u r r o g a t e 1* I I W ' ( E ~ , E ~ ) c,o r r e s p o n d i n g t o F ( E ~ , E ~ ) , and where I

worth value, E2

h12

<

F3.

1

Xi2,

Xi3,

The d e c i s i o n - m a k e r i s asked t o make a " c o n s i s t e n t "

esti-

167

!4UI,T IOBJEC'i'IVE WATER QUALITY MODELS

mate o f

W;,(E;,

I

and

E);

Since h i s past value o f

W13(€,,

,

I

E ~ ) .

That i s ,

W,;

and

W12

<

Wi3

< W13.

W12 was t8 he m i g h t say +4 i f he f e l t you had made

a c o n s i d e r a b l e improvement b u t he would s t i l l t r a d e t e m p e r a t u r e f o r c o s t r e He m i g h t say +1 i f he w a s n ' t q u i t e so empha-

duction rather emphatically.

I

tic.

S i m i l a r a n a l y s i s can be performed f o r W13. o f values o f t h e

W i t h t h e s e two s e t s and W13

s u r r o g a t e w o r t h f u n c t i o n s W12

i t i s n o w p o s s i b l e t o make a l i n e a r i n t e r p o l a t i o n ( o r e x t r a p o l a t i o n )

to f i n d the point

a t w h i c h b o t h W12 and W13 would equal z e r o i f t h e

(E~,E;)

s u r r o g a t e w o r t h f u n c t i o n s were l i n e a r .

This point,

0

0

i s t h e n used

(E,,E~),

as a t h i r d t r i a l v a l u e and t h e process r e p e a t e d u n t i l t h e d e c i s i o n - m a k e r i s u n a b l e t o say w i t h c e r t a i n t y whether he would t r a d e f u r t h e r o r n o t .

Such a

s i t u a t i o n corresponds t o

a z e r o o f t h e s u r r o g a t e w o r t h f u n c t i o n and t o t h e

value o f

f u n c t i o n which

t h e r e a l worth

c o s t r e d u c t i o n t o t h e worth

equates t h e w o r t h

o f the gain i n

o f t h e l o s s i n t e m p e r a t u r e c o n t r o l and t h a t o f

c o s t r e d u c t i o n t o algae c o n t r o l .

T h i s i s a p r e f e r r e d s o l u t i o n i n t h e sense

t h a t no knowledge e x i s t s by w h i c h t h e d e c i s i o n - m a k e r c o u l d a s s e r t a " b e t t e r " solution. By c a r e f u l l y a p p r o a c h i n g t h e z e r o o f t h e s u r r o g a t e w o r t h f u n c t i o n f r o m

the p o s i t i v e side only

(i.e.,

i n d i f f e r e n c e can be found. a

" r i g h t hand"

value o f

(c,,

t

bound can be d e t e r m i n e d . E

~

)

f o r both

band o f i n d i f f e r e n c e and any

W13,

W12 and

I f not, then t h e range

found.

v a l u e s ) a " l e f t hand" bound on t h e band o f

By r e p e a t i n g t h e process f r o m t h e n e g a t i v e s i d e

E~~

(E,,E~)

5

E~

I f b o t h bounds

a r e a t t h e same

t h e n a u n i q u e s o l u t i o n has been 5

E , ~

and

E~~

5

E~

:E~~ i s t h e

i n t h i s r a n g e i s as good as any o t h e r .

A l l d e c i s i o n s , S1 and S, p e r t a i n i n g t o any p r e f e r r e d s o l u t i o n a r e a l s o i m p l i e d f o r t h e band of i n d i f f e r e n c e and can be d i r e c t l y c a l c u l a t e d as d e s c r i b e d i n Chapter 6. 9.6

SUMMARY AND CONCLUSIONS The S u r r o g a t e Worth T r a d e - o f f Method i s p a r t i c u l a r l y u s e f u l f o r w a t e r

q u a l i t y problems

where c o s t ,

d i s s o l v e d oxygen, t e m p e r a t u r e , BOD, e t c . a r e

t h e measures o f g o a l s ,

b u t f o r w h i c h no r a t i o n a l p r o c e d u r e f o r commensura-

tion

i s available

i n common

"worths"

units

or likely

t o become

available.

The

o f t h e s e l e v e l s of g o a l a t t a i n m e n t a r e n o t u n i v e r s a l b u t a r e v e r y

s i t e sensitive.

T h i s p r o c e d u r e a v o i d s t h e problem o f a t t e m p t i n g t o d e t e r -

mine common u t i l i t y so

t h a t o p t i m i z a t i o n can be accomplished and, i n s t e a d ,

accomplishes a f u n c t i o n a l o p t i m i z a t i o n i n m u l t i p l e o b j e c t i v e space and t h e n e v a l u a t e s o n l y p a r i t y o f t r a d e - o f f i n t h i s o p t i m i z e d o b j e c t i v e space. For t h e p r o c e d u r e t o be s t r i c t l y c o r r e c t t h e measures o f t h e o b j e c t i v e s (DO, pH, e t c . )

must be e i t h e r t r u e measures o f t h e a c t u a l o b j e c t i v e s

o r be m o n o t o n i c a l l y r e l a t e d t o them. A good example o f a meabure which does n o t meet t h i s c r i t e r i a i s t h e use o f " v i s i t o r - d a y s ' ' a s a measure o f a n objective t o

maximize a r e c r e a t i o n a l o b j e c t i v e .

i n c r e a s e s the o b j e c t i v e i s enhanced,

O b v i o u s l y , a s t h i s index

b u t o n l y up t o some unknown l i m i t

s a t u r a t i o n p o i n t beyond which a d d i t i o n a l v i s i t o r recreational experience i n t o a nightmare.

or

d a y s can t u r n an e x p e c t e d

However, such "improper" i n d i c e s

by a n o m a l i e s i n d e c i s i o n - m a k e r r e s p o n s e s so t h a t no

will reveal themselves p r a c t i c a l hartii i s done.

3.

FOOTNOTES The e n t i r e l i s t can be found i n G r e a t Lakes I l a t e r Q u a l i t y [1972]. T h i s model i s d e s c r i b e d i n d e t a i l by Haimes [1971]. The d e r i v a t i o n o f t h i s r e l a t i o n s h i p can be found i n S t r e e t e r and

4.

T h e m u l t i l e v e l a p p r o a c h i s p r e s e n t e d i n Haimes 119731.

5.

The f o l l o w i n g m u l t i o b j e c t i v e model was p r e s e n t e d by Hairnes and Hall [1975]. Among the s i n g l e o b j e c t i v e w a t e r q u a l i t y models a r e t h o s e o f

1. 2.

Phelps 113251.

Haas 119701, HaimPs, Foley and Yu r19721, Haimes, Kaplan

and Husar

7.

r19721, Foley 119711 and Foley and Haimes [1973]. Again, s e e S t r e e t e r and P h e l p s [1925]. S e e Haas 119701 f o r t h e d e r i v a t i o n and Frankel 119651 f o r t h e d a t a .

8.

See Foley [1971! and Foley and Haimes 119731.

9.

T h i s r e l a t i o n s h i p was o r i g i n a l l y d e r i v e d by Lesbosquet [1946].

10.

The d e r i v a t i o n of t h i s e q u a t i o n i s p r e s e n t e d by B a i l e y [197O].

6.

References ______ 1. Bailey, T.E.,

" E s t u a r i n e Oxygen Resources-Phytosynthesis and Reaera-

t i o n " , J o u r n a l of t h e S a n i t a r y E n g i n e e r i n g D i v i s i o n , ASCE, v o l .

36,

no. SA2, P r o c . Paper 7215, A p r i l 1970, pp. 279-296. 2.

F o l e y , J . b l . , ~ l i r l t i l e v e lC o n t r o l of H a t e r O u a l i t y , H.S. T h e s i s , Case

3.

Western Reserve U n i v e r s i t y , C l e v e l a n d , Ohio, June 1971. F o l e y , J . I!.,and Y . Y . Haimes, " P l u l t i - l e v e l C o n t r o l of M u l t i - p o l l u t a n t System", ASCE J o u r n a l of the Environmental E n g i n e e r i n g Division v o l . 99, no. EE3, p p . 253-268, J u n e 1973.

4.

F r a n k e l , R . J . , "Llater Q u a l i t y Model f o r

Planagenient:

An Engineering-Economic

Domestic H a s t e ? i s n o s a l , P h . D . D i s s e r t a t i o n , U n i v e r s i t y

of C a l i f o r n i a , a t B e r k e l e y , C a l i f . , 1965. 5.

Great Lakes H a t e r Q u a l i t y ,

agreement

between the

United S t a t e s o f

America and Canada, s i g n e d a t Ottawa, A p r i l 1 5 , 1972.

6.

Hairnes, Y . Y . , and W . A . H a l l , " M u l t i o b j e c t i v e s i n Water Resources __ Water Systems A n a l y s i s : The S u r r o g a t e Worth T r a d e - o f f Method", Resources . . Research, v o l . 10, no. 4, August 1974, pp. 615-624.

~

7.

Haiiiies, Y . Y . ,

" M o d e l i n g and C o n t r o l o f t h e P o l l u t i o n o f Water Resour-

ces Systems Via M u l t i l e v e l Approach", Water Resources B u l l e t i n , v o l .

8.

7, no. 1, Feb. 1971, pp. 104-112. ~~i,,,,,,, y . y . , and H a l l , W . A., " A n a l y s i s o f

Multiple

Objectives

in

Water Q u a l i t y " , Presented a t t h e s p e c i a l ASCE Conference a t C o r n e l l U n i v e r s i t y , I t h a c a , N.Y.,

June 26-28,

1974. To

appear i n t h e J o u r -

n a l o f ASCE H y d r a u l i c D i v i s i o n , 1975.

9.

Haiiiies, Y . Y . ,

"Decomposition and M u l t i l e v e l Approach i n Modeling

Management o f Water Resources System", pp. 348-368, o f Large ~S c a l e Problems, ~

P u b l i s h i n g L o . , Amsterdam, 10.

Haiiiies, Y . Y . ,

Foley, J . W . ,

D . M. Hiiiinielblau,

and

Decomposition

E d i t o r , North Holland

1973. and

YU,

W.,"Computational

Results

for

Water P o l l u t i o n T a x a t i o n Using M u l t i l e v e l Approach," Water Resources ~

11.

G u l l e t i n , v o l . 8, no. 4, Aug. 1972, pp. 761-772.

Haimes, Y . Y . ,

Kaplan, M. A . ,

t o D e t e r m i n i n g Optimal

and Husar, M. A . ,

"A M u l t i l e v e l Approach

T a x a t i o n f o r t h e Abatement o f

Water P o l l u -

t i o n " , Water Resources Research, v o l . 8, no. 4, Aug. 1972, pp. 851860. 12.

Haas, J . E . ,

"Optimal T a x i n g f o r t h e Abatement o f Water P o l l u t i o n , "

w a t e r Resources Research, v o l . 6, no. 2, A p r i l 1970, pp. 353-365. 13.

Lesbosquet, M.,

"Cooling-Water B e n e f i t s f r o m I n c r e a s e d R i v e r Flows,"

Journal o f t h e

New England Water Works A s s o c i a t i o n ,

v o l . 60, June

1946, pp. 111-116. 14.

S t r e e t e r , H.W.,

and Phelps,

E.

B.,

"Study o f t h e P o l l u t i o n and

Natu-

r a l P u r i f i c a t i o n o f t h e Ohio R i v e r " , P u b l i c H e a l t h B u l l e t i n No. F e b r u a r y 1925.

Chapter 10 SENSITIVITY, STABILITY, RISK AND IRREVERSIBILITY AS MULTIPLE OBJECTIVES

1 0.1

I NTRODUCTI ON _ _ ~ _____ Water resources p r o i p c t s a r e planned,

ted and modified under While in

numerous

risks

general t h e tcrms r i s k

designed, c onstruc te d, opera-

and unc ontrolla ble

uncertainties.

and u n ce rta inty can denote t h e same t h i n a

( i t i s r i s k y because i t i s u n c e r t a i n )

i t i s useful f o r a n a l y t i c a l purposes

t o d e f i n e t h e se s e p a r a t e l y a s two d i s t i n c t concepts. Risk i s c ha ra c te riz e d of events following reasonably well known o r measurable p r o b a b i l i t i e s , even though t h e s p e c i f i c time o r s p a t i a l sequence o f occurence of events cannot be determined. I n water resources problems by a frequency d i s t r i b u t i o n

f o r example,

a common cause of

risk i s the

a ssoc ia te d

Past hydrological records

a r e u s u al l y a v a i l a b l e

probability distribution,

b u t any s p e c i f i c

hydrologic input.

t o reasonably

sequence of

define the

events i s l a r g e l y

c o n t r o l l e d by chance. I n c o n s t r a s t t o r i s k , unc e rta inty i s c h a r a c t e r i z e d b y t h e absence of any known reasonably v a l i d p r o b a b i l i t y d i s t r i b u t i o n of events.

The term r i s k i s assigned t o measurable chance c o n t r o l l e d f a c t o r s ,

while u n c e r t a i n t y a p p l i e s t o a l l o t h e r s .

In

water resources f o r example,

t h e r e a r e u n c e r t a i n t i e s as s o ci at ed with:

-

The growth of p o p u l at i o n , i n d u s t r y , a g r i c u l t u r e and

-

The projected c o s t of l a b o r , m a t e r i a l , and i n f l a t i o n . The assessment of f u t u r e advancement in e ngine e ring,

-

The p r o j ect ed b e n e f i t s as s o ci at ed with the p r o j e c t s .

urban a r e a s .

s c i e n ce, and technology.

In a d d i t i o n , t h e r e a r e important u n c e r t a i n t i e s and r i s k s introduced by both t h e system and man's attempts t o model i t . There a r e

many types of

r i s k and

unc e rta inty in

water re sourc e s,

most of them well known t o t h e p r a c t i t i o n e r s in t h e f i e l d . O f these, the one most important f o r t h e d eci s i o n process modeler i s t h d t r e l a t e d t o t h e An p r e c i si o n with which t h e co n t r o l v a r i a b l e s can in f a c t be c o n t r o l l e d . "optimal" s o l ut i o n i n t h e sense of minimizing t h e d i s t a n c e , time o r c o s t involved i n c r os s i n g a deep gorge might be t o c ross hand a n d hand on any convenient c a b l e such a s a high v o l t ag e e l e c t r i c a l conductor. For most of us, however, such a d eci s i o n would be unthinkable because we know t h a t we lack t h e necessary p r eci s i o n o f co n t r o l with r e s p e c t t o t h i s p a r t i c u l a r de170

:MJS 1 :i7 1 ViT Y, STA H I CITY, RXSK

A IVD I RREVERSIBIZITY

171

cision variable. While an extreme example, it does serve to illustrate that the classical concept of optimum (signifying best) is by itself inadequate - - the degree of control of the significant system responses must be also considered as a decision parameter for a great many water resource systems . To the extent that the effect of the lack of control can be characterized by a probability distribution, the corresponding uncertainty is reduced to risk, and in a few cases in water resources systems, but by no means all (or even most), risk can to some extent be allowed for by optimizing the mathematical expectation of the results. In fact, where mathematical expectation is a valid criteria for optimality, it can be shown that it is not always necessary to know the probability distribution in any great detail. If, on the other hand, the risk problem is characterized by infrequent decision, irreversibility, or both, optimization o f mathematical expectation may lead to very serious errors. Since these two characteristics tend to dominate the risk and uncertainty situations involved in water resources management, their proper consideration as separate noncommensurable objectives is essential in most water resources decision models. Other systems characteristics which may increase or mitigate the effects of risk and uncertainty are sensitivity, responsivity and stability. Modeling inaccuracies can be important also. The discussion in this chapter of these system and modeling characteristics is intended to stimulate the considerable amount of further research needed to evaluate their impact on risk and uncertainty as objective functions. Some proposals on how this might be done for the sensitivity characteristic are suggested. A better knowledge and understanding of these characteristics and their relationship to risk and uncertainty will permit these crucial factors to be properly incorporated into multiobjective analysis. 10.2 SYSTEM CHARACTERISTICS RELATED TO THE EVALUATION OF RISK Water resource systems have a number of characteristics associated with the stochastic nature of the system inputs, outputs, and states. Four of these characteristics are discussed here: Sensitivity. Responsivity Stability. Irreversi bil i ty. These are the major elements of prototype systems involved in the definition of risks as indices of performance. As an additional characteristic,

-

.

I72

the p r e c i si o n of t h e modpl rdtely.

Although i t i s

r ep r es en t i n g t h e system w ill be discussed separecognized t h a t t h e

current

state-of-the-art i n

systelris a n a l y s i s i s n o t y e t f u l l y capable of q u a n t i t a t i v e l y t r e a t i n g a l l of these c h a r a c t e r i s t i c s , i t i s e s s e n t i a l t h a t they be considered a s thoroughl y a s p o s s i b l e.

They a r e d e s c r i p t i v e l y defined a s follows:

S e n s j t j v j J i i s t h e systern c h a r a c t e r i s t i c r e l a t i n g t h e changes in t h e

system's index of performance ( o r o u t p u t ) t o expected v a r i a t i o n s in t h e dec i s i o n v a r i a b l e s , uncontrolled parameters, c o n s t r a i n t l e v e l s or t h e m ode l' s coefficients. Fe2ponsivi~Ly i s t h e system c h a r a c t e r i s t i c of being dynamically r e s ponsive t o changes ( i n cl u d i n g random v a r i a t i o n s ) in t h e de c isions over time. This measures

t h e a b i l i t y of t h e

s i g n i f i c a n t responses

of the

system t o

follow t h e changes i n a v a r i a b l e d eci s i o n in time and/or space. S t a b i l i t y - i s a system c h a r a c t e r i s t i c r e l a t e d t o t h e degree of v a r i a -

~ . .

tiori of t h e

mean

systern response

y i e l d s a n i n v a r i a n t inean response

t o fixed

decisions.

A

s t a b l e systeni

t o t h e mean value of a de c ision

set.

A

systerii may be s t a b l e a n d s t i l l have a n important random component. I r r - ~ y - e r - ~ j ~ b ~i sJ ya~ systeni c h a r a c t e r i s t i c r e l a t e d t o t h e degree of d i f f i c u l t y involved i n r e s t o r i n g previous s t a t e s o r c onditions once the systeiii has been a l t e r e d bya d eci s i o n ( i n cl u d ing t h e " de c ision" t o d o nothing). Some examples

of each of

t h e above

four c h a r a c t e r i s t i c s

w ill

be

given i n o r d e r t o c l a r i f y t h e concepts. 19.2.1 Sensi t i v i t y I t i s p o s s i b l e t o c o n s t r u c t hypothetical s i t u a t i o n s in which t h e det e r m i n i s t i c mathematical

optimum d eci s i o n would

be t h e worst possible u n -

l e s s t h e decision v a r i a b l e can be very p r e c i s e l y c o n t r o l l e d . i l l u s t r a t e s such a s i t u a t i o n in which i t i s presumed t h a t v a r i a b l e can be any value w i t h

Figure 10.1 t h e de c ision

x a n d t h a t x ma,y take on C' t he se l i m i t s . The d e t e r m i n i s t i c

c o n t r o l l e d only within l i m i t s , equal l i k e l i h o o d within

imathematical maximum i s obviously f a r from being the p r a c t i c a l optimum

de* I n t h i s contrived example, x 2 i s c l e a r l y a " b e t t e r " de c ision t h a n * x1 unless t h e d eci s i o n maker i s more i n t e r e s t e d in gambling than r i s k avoi-

cision. dance.

Even i f t h e example i s t r e a t e d by maximizing t h e mathematical expect a t i o n of f ( x ) , i t does n o t follow t h a t a r e s u l t i n g "optimum" a t x1 i s sup e r i o r t o x 2 . For t h i s t o be t r u e t h e ap propria te o b j e c t i v e must indeed be

o r minimization of t h e expected value of f ( x ) . This i s a major co n s i de ra tion. The "gamblers' r u i n " problem i s t h e c l a s s i c a l example where t h i s i s c l e a r l y not the 0 b j e c t i v e . l t h e rriaxiniization

seldom t r u e

where r i s k i s

173

A(x)

Figure 10-1 . Sensitivity Band

174

MlJI,l'TOHcJBC2'IVE OPTIMIZATION I!$ WATER RESOURCES SYSTEMS

As i n many p r a c t i c a l problems i n w a t e r r e s o u r c e s ,

t h e d e c i s i o n which maxi-

mizes t h e expected v a l u e o f t h e r e t u r n i n t h i s problem w i l l a l s o c o r r e s p o n d t o a maximization o f

the r i s k o f g e t t i n g

l i t t l e or

t h e r e a r e two noncommensurable o b j e c t i v e s i n t h i s

nothing.

In reality

case, avoidance o f r i s k

and g a i n i n g economic r e t u r n . The presumption t h a t

the objective

i s t h e m a x i m i z a t i o n o f expected

v a l u e i ?s i m p l y one p a r t i c u l a r method o f commensurating r i s k and r e t u r n . I t would appear t o be v a l i d i n t h o s e i n s t a n c e s where t h e number o f d e c i s i o n s t o be niade o v e r t i m e

and space i s l a r g e

enough t o i n s u r e

a high probability

t h a t such a r e t u r n w i l l i n f a c t be r e a l i z e d i n t h e l o n g r u n , d e s p i t e i n t e r m e d i a t e ups and downs.

The expected v a l u e as an i n d e x o f performance (ob-

j e c t i v e f u n c t i o n ) c l e a r l y does n o t a p p l y t o s i t u a t i o n s i n w h i c h o n l y a v e r y (and t h e i r consequences) w i l l be i n v o l v e d .

few d e c i s i o n s

i n any s i t u a t i o n where

the objective function i t s e l f

discontinuously a l t e r e d i n t o a

c o m p l e t e l y new f o r m f o r a l l v a l u e s o f x, i n

t h e e v e n t t h a t x chanced t o f a l l i n a p a r t i c u l a r sub-range. numbers o f s i t u a t i o n s i n tions are

Nor i s i t v a l i d

i s a u t o m a t i c a l l y and There a r e l a r g e

w a t e r r e s o u r c e s where one o r b o t h o f t h e s e excep-

a p p l i c a b l e and,

hence,

where t h e expected v a l u e

criterion for

r i s k and r e t u r n w i l l n o t r e p r e s e n t t h e o b j e c t i v e s o f t h e

commensuration o f r e a l system. 1 0 . 2 . 2 Respons iv it y

R e s p o n s i v i t y i s t h e c a p a b i l i t y o f t h e system t o respond i n a reasona b l e t i m e frame t o a v a r i a b l e ( c h a n q i n g ) d e c i s i o n .

I t i t generally related

t o " f r i c t i o n s " i n t h e system and d e l a y e d response.

One o f t h e most impor-

tant responsivity characteristics long lead time means o f

o f w a t e r and o t h e r

u s u a l l y r e q u i r e d t o observe a need,

m e e t i n g t h a t need,

t o develop a

c i v i l systems i s t h e t o conceive a p o s s i b l e

p r e l i m i n a r y plan,

t o obtain

a

basic p o l i c i t a l approval o f t h e p l a n d u r i n g a " p o l i t i c a l hassle period", t o complete t h e f i n a l d e s i g n , cision.

This i s

f o r t y o r f i f t y years. i s seldom

and t o c o n s t r u c t o r o t h e r w i s e implement t h e de-

o f t e n i n excess o f

Even f o r s m a l l , a l m o s t i n c o n s e q u e n t i a l problems

l e s s t h a n two y e a r s .

much more r a p i d r a t e ,

t w e n t y - y e a r s , and sometimes more t h a n

Since o b j e c t i v e s

can and do

it

change a t a

t h i s f o r m o f r e s p o n s i v i t y has become e x c e e d i n g l y i m -

p o r t a n t i n w a t e r r e s o u r c e s planning, There sources.

a r e many o t h e r f o r m s

A c l a s s i c example

channel aqueduct system,

o f responsivity occurring

i s time delay i n

down an open

Another i s t h e r e l a t e d p r o b l e m o f f l o o d r o u t i n g .

Yet a n o t h e r i s t h e a b i l i t y o f a "move" to

r o u t i n g water

i n water r e -

t y p e supplemental i r r i g a t i o n system

cover the e n t i r e f i e l d i n t h e f a c e o f drought.

The response of h y d r o -

175 e l e c t r i c systems t o

rapid f l u c t u a t i o n s in demanci i s an economica l y useful

responsivity of these systems. The response of water use t o p r i c e i s another very importan

el ement water resources systems. I n many instances involving the use of water, c o s t s which vary with t h e amount of water used a r e q u i t e small r e l a t i v e t o c o s t s which a r e i n s e n s i t i v e t o t h e volume of use ( l a r g e l y i r r e v e r s i b l e c a p i t a l investment). This may r e s u l t in a response delay of the order of magnitude of t h e economic l i f e of t h e investments involved.

of responsivity in

10.2.3 Stability S t a b i l i t y i s a measure of t h e r e s i s t a n c e t o non-decision o f t h e mean response of t h e system.

ponse of t h e system

mod f i c a t i o n

Frequently in water resources t h e r e s -

will vary appreciably even f o r

a fixed d e c i s on. If the e f f e c t of t h e v a r i a t i o n i s t o r e t u r n t h e system automatical y t o t h e "output" or o b j e c t i v e value represented by t h e d e c i s i o n , t h e decision system i s s t a b l e . I f , on t h e other hand, a u t o - c a t a l y t i c e f f e c t s cause t h e r e sponse t o move away from t h a t intended by t h e d e c i s i o n s , t h e decision system i s unstable. Many water resources and o t h e r c i v i l systems have hiqhly unstable decision systems.

One obvious example i s t h e flood control deci-

I t has been a s s e r t e d t h a t providing p a r t i a l flood c o n t r o l , sion system. commensurate with one s e t of predicted f u t u r e c o n d i t i o n s , has r e s u l t s in a t t r a c t i n g more economic a c t i v i t y i n t o t h e "protected" area - - making t h e o r i g i n a l decision f o r p a r t i a l control q u i t e improper f o r t h e a u t o c a t a l y t i c a l l y changed s i t u a t i o n . Transportation routing i s another c l a s s i c example. I n most instances water based r e c r e a t i o n has a l s o responded i n an unstable way. On the o t h e r hand, many O F the past e s t i m a t e s of f u t u r e water needs, made u p t o f i f t y years in advancc ( e . g . Mulholland's almost p e r f e c t timeframe estimate o f need f o r have proved

1600 c f s f o r Southern C a l i f o r n i a coastal a r e a ) 2

t o be remarkably uncanny,

suggesting t h a t highly s t a b l e s e l f -

f u l f i l l i n g e f f e c t s may be involved in these c a s e s . I r r e v e r s i bi 1 i t y I r r e v e r s i b i l i t y i s a measure of the d i f f i c u l t y in returning a system t o i t s o r i g i n a l s t a t e once a decision change has been made. Suicide i s an extreme example of an i r r e v e r s i b l e decision. I n o t h e r c a s e s , the decision 10.2.4

can be i s the

Humpty-Dumpty reversed b u t only a t l a r g e social o r economic c o s t . l i t e r a r y p e r s o n i f i c a t i o n o f t h i s very important o b j e c t i v e o f water

resource and many o t h e r c i v i l systems. Some decisions a r e completely i r r e v e r s i b l e b u t in a continuous sense. That i s ,

t h e s t a t e of t h e system,

S

, can be changed by a r b i t r a r i l y small

increlrlents over tiiire, t , or space in one d i r e c t i o n b u t i t can n o t be reversed.

blatheiiiatically,

t h i s form can

burn f o s s i l fuel b u t we cannot "unburn" i t . i r r e v e r s i b l e in

a S / a t t 0. We can Other de c isions a r e completely

be represented b y

I n some cases t h e i r r e v e r s i b i l i t y i s a

either direction.

!;latter of degree ( i . e . s o ci al a n d economic c o s t ) , e i t h e r continuous o r d i s A highway i s a n

continuous. b i l i t y " since

e x c e l l e n t example

i t can be removed or

of a v a r i a b l e " i r r e v e r s i -

expanded only a t

considerably g r e a t e r

c o s t t h a n i f t h e proper d eci s i o n had been made o r i g i n a l l y . 1 0 . 3 SOURCES- OL- U ~ J ~ - R ~ I J J ~AND _ ERRORS I rd MODELJH~A Not a l l of t h e u n c e r t a i n t i e s or r i s k s involved i n systems a n a l y s i s a r e as s o ci at ed with t h e system i t s e l f . certainty,

a l l too o f t en ignored in t h e

water

resources

A significant un-

quest f o r q u a n t i t a t i v e p r e d i c t i v e

i s t h a t r e l a t e d t o t h e degree t o which the various models used ac-

iiiodels,

t u a l l y r e p r e sen t t h e s i g n i f i c a n t behavior of the re a l system being modeled.

These

u n c e r t a i n t i e s can be

parameters,

introduced through

t h e model's topology,

a n d t h e d a t a c o l l e c t i o n and processing techniques.

its

Model u n -

c e r t a i n t i e s wi l l o f t e n be introduced through human e r r o r of both commission a n d oriiission.

An

irlatherriatical model

"optimized"

t h e s i g n i f i c a n t behavior f a c t t h a t some of t h e able o f

d eci s i o n s e t i s

t r u l y optimal only i f t h e t h a t de c ision s e t c l o s e l y re pre se nts The of t h e act u al system over time a n d space.

used t o g en er at e

socio-economic elements

r e a c ti n g co mp et i t i v el y o r

choice of d e c is i o n s e t only

of t h e re a l system a r e cap-

complenientarily t o t h e decision-maker's

emphasizes t h i s shortcoming of most mathemati-

cal models. I n f a c t , t h e r e a r e a c t u a l l y no c i v i l systems involving a s i n g l e decision-maker, d e s p i t e t h i s customary assumption in ootimal de c ision modeling. The necessary obtained

condition f o r reasonable

t h r o u g h o p t i mi zat i o n i s

u t i l i t y o f a n y de c ision

set

t h a t t h e important responses of t h e re a l

systern t o t h o s e d eci s i o n s a r e t h e same as those produced b y t h e mathematical !!lode1 w i t h i n a t o l e r a b l e e r r o r .

Since water resources

de c isions a r e very

i t niay be d i f f i c u l t t o e va lua te modeling e r r o r s , l e t

often made only once,

This s i o n i f i c a n t alone reduce them t o q u a n t i t a t i v e p r o b a b i l i t y measures. source of u n c er t ai n t y i s probably one of t h e major reasons f o r t h e slow and cautious adoption i n c i v i l systems of t h e products of re se a rc h, p a r t i c u l a r l y systems a n a l y s i s niodeling. any nraxiniization o r

The v a l i d i t y of

minimization problem depends

t h e accuracy with which

t h e optimal s o l u t i o n

x* t o

(among o t h e r t h i n g s ) on

t h e mathematical model re pre se nts the re a l system,

I n p a r t i c u l a r , t h i s accuracy depends on t he c l o s e n e s s t o t h e re a l system

of

177

the iiiodrl ' s i n p u t - o u t p u t crrors

relationships.

cdn be as s o ci at ed with t h e

The sources of u n c e r t a i n t i e s and

following s i x major c a t e g o r i e s of model

chardcteristics: ( i ) Model Topclogy - ( t x , ) ( i i ) Model Parameters ( i i i ) Model Scope o r Focus - ( y 3 ) (iv)

(v)

Data

(n

-

4

)

Optimization Technique

-

(F~)

( v i ) Human S u b j e c t i v i t y The above s i x c a t e g o r i e s a r e discussed h e r e a f t e r in some d e t a i l . 1 0.3 . 1

Mods! ._Topp_l ogy (9 ) Model t o p o l o g y r e f e r s t o t h e o r d e r ,

equdtions

degree a n d forin of the system of

For example, a dynamic system

which r ep r es en t t h e r eal system.

represented by a system o f d i f f e r e n t i a l equations (ordina ry or p a r t i a l ) ; d s t a t i c system might be represented by s e t s of a l g e b r a i c equa-

iriight be

t i o n s such a s polynomials, e t c . Consider f o r example, a groundwater

system o f both confined a n d u n -

I n o r d er t o model t h e dynamic response of t h e hydraulic

confined a q u i f e r s .

t o any f u t u r e demands (withdrawals or recharge) on t h e

head i n t h e a qu i f er

groundwater system, one may use a system of d i f f e r e n t i a l e qua tions. second o r d e r p a r t i a l

d i f f e r e n t i a l equations may

t h e confined a q u i f e r ,

be adequate

whereas nonlinear second orde r

Linear

f o r modeling

partial differential

equntions ( P D E ) might be needed f o r the unconfined a q u i f e r . Furthermore, a hoiiiogeneous a q u i f e r may be adequately modeled by a two-dimensional system, but a

s t r a t i f i e d a n d non-homogeneous one

dimensional

PDE,

etc.

Clearly,

topology over another i n t r o d u ces

o u g h t t o be modeled by a t h r e e a s e l e c t i o n o f one model

in each c a s e ,

a source of

uncertainties

and e r r o r s in

t h e accuracy o f t h e model's r e p r e s e n t a t i o n . Model topology i s tiiriization.

p a r t i c u l a r l y important in

de c ision m a k i n g f o r op-

Almost any f u n ct i o n al form can be used t o approximate t h e ab-

s o l u t e value of any cau s e- ef f ect r e l a t i o n s h i p .

However, optimal de c isions

concerned with the magnitude of the se func tions as with t h e i r d e r i v a t i v e s (or incremental r a t i o s ) . Thus a l i n e a r l e a s t squares r e -

are usually n o t as

gression model o f a

b a s i c a l l y n o n - l i n ear response,

because of t h e charac-

t e r i s t i c s o f l i n e a r system o p t i mi zat i o n , i s ver,y a p t t o s e l e c t " de c isions" a t point,s which i n f a c t have t h e g r e a t e s t e r r o r in t h e re pre se nta tion of

the true derivative. 10.3.2 Model Parameters

(q)

Once the model topology has been selected, the choice of model parameters (often called parameter identification, parameter estimation, system identification, model calibration, etc.) determines the accuracy with which the system model represents the real system. Consider the groundwater system discussed earlier. Once the customary system of parabolic partial differential equations is selected, the proper values of the coefficients need to be determined (e.g. storativity and transmissivity as functions of the spatial coordinates). This parameter estimation (identification) process introduces a source of uncertainties and errors in the accuracy of the calculated values of the parameters and in turn in the model itself. 10.3.3 Model Scope

(ECX)

Model scope refers to the type and level of resolution used in the model for the description o f the real system. Four major descriptions are identified in water resources systems. These are: (i) Temporal description. ( i i ) Physical-Hydrological description. ( i i i ) Political-Geographical description. (iv) Goal or Functional description. The above descriptions are discussed in some detail hereafter. The characteristic parameters of uncertainties and errors associated with the selection of the model scope is denoted by the set EX. In referring again to the ground water system, one may wish to study the behavior (response) of the system under planned development for short, intermediate and long-term planning horizons (temporal description). The groundwater system itself, which may consist of several aquifers, may be decomposed on the basis of the physical-hydrological characteristics or political-geographical boundaries. Finally, if the groundwater system is to be managed as part of a larger water resources system with concern for water quality, storage, recharge, and so on, then different decompositions may be more advantageous, such as goal description. Clearly, while these four descriptions have individual merits, each describes the system from a narrowed point of view. The system in totality may never be well-represented by any one description, and thus the selection of model's scope introduces yet another source of uncertainties and errors in the system's repre-

sentation.

Scope i s a

c o n t r o l l e d by many

p a r t i c u l a r l y important f a c t o r Even so,

what different objectives. a single

though

where t h e system i s

r e l a t i v e l y independent decision-makers, each with some"rational"

such systems

are

ofte n modeled

as

decision-maker was a t t h e helm, i . e . a s i f

a

s i n g l e point of view can be a s s e r t e d . 10.3.4

Ea&

(q)

Access t o

sufficient representative

da ta f o r

c a l i b r a t i o n , i d e n t i f i c a t i o n , t e s t i n g , Val i d a t i o n

model

constructions,

and hopefully implementa-

t i o n , i s obviously a very important element in systems a n a l y s i s . lack of e i t h e r accu r at e o r

Cle a rly a

s u f f i c i e n t data due t o t h e c o l l e c t i o n , a c q u i s i -

t i o n , processing, a n a l y s i s ,

etc.,

may cause s u b s t a n t i a l e r r o r s in t h e re-

sul t s . Consider again t h e

above groundwater system: t h e value of t h e

model

parameters determined through t h e i d e n t i f i c a t i o n procedure i s l i k e l y t o depend on t h e a v a i l a b l e d a t a .

An i n s u f f i c i e n t number of sampling s i t e s , the

number of samples, and sampling accuracy (w ithin each s p a t i a l l o c a t i o n ) may introduce s i q n i f i c a n t sources of u n c e r t a i n t i e s and e r r o r s in t h e system model. 1 0 . 3 . 5 Optimization Techniques

(%)

Once t h e mathematical model

has been c onstruc te d and i t s

parameters

i d e n t i f i e d , t he s e l e c t i o n and a p p l i c a t i o n of s u i t a b l e optimization methodol o g i e s ( s o l u t i o n s t r a t e g i e s ) introduces anothe r source of u n c e r t a i n t i e s and e r r o r s i n t h e s o l u t i o n derived from t h e system

I n t h e groundwater

model.

system discussed e a r l i e r , t h e s e l e c t i o n of t h e method of numerical i n t e g r a t i o n of t h e system of

PDE with t h e as s o ci a te d grid s i z e , boundary and i n i -

t i a l conditions,

computer s t o r a g e c a p a c i t y and accuracy,

duce sources of

u n c e r t a i n t i e s and e r r o r s in t h e s o l u t i o n .

ample, c o n s i d e r a

nonlinear o b j e c t i v e f u nc tion

A s another ex-

with a nonlinear system of

i n e q u a l i t y c o n s t r a i n t s r ep r es en t i n g t h e behavior of ply system.

e t c . , a l l intro-

a power and water sup-

I f t h e o p t i mi zat i o n method a pplie d f o r solving t h i s system i s

t h e simplex method racy of t h e s o l u t i o n

( v i a l i n e a r i z a t i o n of t h e system model), then t h e accuobtained may be q u estiona ble .

This

particularly i s

t r u e f o r highly nonlinear systems.

I t i s important t o note t h a t t h e s e l e c t i o n of the optim iz a tion nique g e n e r a l l y co i n ci d es (or should) sequently, t h e model

with t h e model's c o n s t r u c t i o n .

any exchange between t h e s o p h i s t i c a t i o n and t h e accuracy

(or approximation)

tech-

Con-

( o r simp1 i f i c a t i o n ) of

o f t h e solution

should be

made a t an e a r l i e r s t a g e o f t h e s y s t e m ' s a n a l y s i s . 10.3.6

Human S u b j e c t i v i t y

(%)

Human s u b j e c t i v i t y s t r o n g l y i n f l u e n c e s t h e outcome o f ses i n w a t e r r e s o u r c e s ( a s w e l l as i n o t h e r a r e a s ) .

systems a n a l y -

Human s u b j e c t i v i t y may

include: (i) (ii) ( i ii)

The background, t r a i n i n g and e x p e r i e n c e o f t h e a n a l y s t ( s ) , Personal p r e f e r e n c e and s e l f - i n t e r e s t ,

and

Proficiency.

C l e a r l y , human s u b j e c t i v i t y can i n f l u e n c e a l l o f t h e o t h e r f i v e m a j o r c a t e g o r i e s o f model c h a r a c t e r i s t i c s .

A c i v i l engineer, a l l addressing

a h y d r o l o g i s t o r a systems e n g i n e e r ,

t h e problem o f p l a n n i n g

development and

p r e d i c t i n g t h e waterhead

f o r example,

f o r t h e above

ground w a t e r system

response t o

w i t h d r a w a l s and r e -

charges, may each c o n c e i v e a d i f f e r e n t approach o r methodology. W h i l e human s u b j e c t i v i t y plays a 'cry important r o l e categories o f

: ! i u d ~ lc h a r a c t e r i s t i c s ,

u n c e r t a i n t i e s and e r r o r s available i n t h i s respect. e f f e c t r e l a t i o n s h i p s here,

i n t h e s e l e c t i o n o f a l l t h e major

each o f w h i c h

i n t h e system model,

i n t r o d u c e s sources o f

no q u a n t i t a t i v e a n a l y s i s i s

R a t h e r t h a n a t t e m p t t o q u a n t i f y such cause and t h e importance o f

each c h a r a c t e r i s t i c i s i n d i -

c a t e d and a framework f o r t h e i r a n a l y s i s i s suggested. I n a n a l y z i n g t h e sources

o f u n c e r t a i n t i e s and e r r o r s

as t h e y a f f e c t

s e n s i t i v i t y , s t a b i l i t y , i r r e v e r s i b i l i t y and u l t i m a t e l y o p t i m a l i t y , t h e system's analyst

may have a v a i l a b l e

the following

knowledge

a b o u t t h e aug-

mented v e c t o r ~ ' 1: (a)

A complete knowledge o f g i s a v a i l a b l e namely,

(b)

i s a deterministic variable.

The v e c t o r g i s a s t o c h a s t i c v a r i a b l e b u t an e s t i m a t e o f i t s probability distribution function i s available.

(c)

The v e c t o r 5 i s a s t o c h a s t i c v a r i a b l e where no knowledge i s a v a i l a b l e on t h e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n .

I t i s assumed t h a t

can be 01.

constructed r e l a t i n g s e n s i t i v i t y ,

Furthermore,

t h e knowledge of any e v e n t ,

some a n a l y t i c a l

functions

s t a b i l i t y and i r r e v e r s i b i l i t y t o

depending on w h i c h element o f

i s under c o n s i d e r a t i o n ,

i t s mean and v a r i a n c e can v a r y between

noncommensurable o b j e c t i v e f u n c t i o n s

( a ) and

(c).

In

w i l l r e s u l t regardless o f

E.

The S u r r o g a t e M o r t h T r a d e - o f f (SWT) Method

u t i l i z e d t o solve t h i s

problem o f n o n c o m e n s u r a b i l i t y among m u l t i -

t h e degree of knowledge o f can be

f o r any g i v e n system

objective functions.

:;6".Si 7 ' 1 I/! T'Y, S'l7lL?rl,l!!Y, RISK

10.4

181

AND .L?REVERSIBILITY

-____ FORMULATION OF R I S K OBJECTIVES FOR WATER RESOURCE SYSTEMS

There p r o b a b l y i s o b j e c t i v e s i n general,

no s t a n d a r d approach given t h e almost

t o the specification o f r i s k

i n f i n i t e possible

number o f com-

b i n a t i o n s o f system-modeling c h a r a c t e r i s t i c s d i s c u s s e d i n s e c t i o n 10.3.

To

a l a r g e e x t e n t , each case may have t o be t r e a t e d de novo t o a s s u r e t h a t t h e m o d e l i n g e r r o r s i n t r o d u c e d by s t a n d a r d i z e d approaches do n o t i n t r o d u c e more u n c e r t a i n t y t h a n t h e r i s k element b e i n g a n a l y z e d . The b a s i c q u e s t i o n i s : v o i r going dry. of failing to

R i s k o f what?

I t may be t h e r i s k o f a r e s e r -

I n e x a c t l y t h e same p h y s i c a l s i t u a t i o n , meet a minimum

water"or " f i r m energy'.

i t may be t h e r i s k

prescribed level o f service,

such as

"firm

T h i r d l y , i t may be t h e r i s k o f d i v e r g e n c e f r o m t h e

prescribed level

o f service.

simultaneously.

The f i r s t two would c o n s t i t u t e d i s c r e t e u n i t s o f t h e r i s k

ob.jective v e c t o r .

I n fact,

The l a t t e r c o n s t i t u t e s an

f i n i t e s e t o f components between For example,

a l l t h r e e r i s k elements may e x i s t o b j e c t i v e v e c t o r w i t h an i n -

z e r o and t h e p r e s c r i b e d l e v e l o f s e r v i c e .

i t may be d e s i r a b l e t o

know t h e r i s k o f f a i l i n g by 500 c u b i c

and b y 1500 c f s , and i t may be j u s t as d e s i r a b l e t o

f e e t p e r second

(cfs) know t h e r i s k a s s o c i a t e d w i t h 1000 c f s o r any o t h e r p o i n t .

however,

t h i s continuous vector

number o f d i s c r e t e p o i n t s p o i n t s e s t i m a t e d by

o f objectives

(e.9.

0, 500, 1000, 1500)

interpolation or

I n most

can be modeled

curve f i t t i n g i f

cases,

a t a small

and t h e i n t e r m e d i a t e the

corresponding

r i s k and o p t i m a l p o l i c i e s a r e r e l a t i v e l y i n s e n s i t i v e . For example, l e t

F ( x ) be t h e e x p e c t e d n e t economic b e n e f i t o f s e l e c -

t i n g a l e v e l o f s e r v i c e s o f x. ble while

I t i s d e s i r e d t o s e t F ( x ) as h i g h as p o s s i -

r e s e r v o i r ( o f t o t a l c a p a c i t y f) ) max w i l l n o t be s u f f i c i e n t t o p r o v i d e minimum

minimizing the r i s k t h a t the

and t h e s t o c h a s t i c i n f l o w

y(t)

l e v e l o f s e r v i c e a t a l l times w i t h i n t h e next n time periods. haps t h e s i m p l e s t f o r m

o f t h e r i s k problem,

This i s per-

b u t i t serves t o i l l u s t r a t e a

number o f i m p o r t a n t c h a r a c t e r i s t i c s w h i c h must be c a r e f u l l y c o n s i d e r e d . Proceeding w i t h t h i s

simple formulation,

a knowledge o f t h e s t a t i s -

t i c a l c h a r a c t e r i s t i c s o f t h e h y d r o l o g y a l l o w s development o f a l a r g e number o f " e q u a l l y l i k e l y " h y d r o g r a p h i c sequences o f n t i m e p e r i o d s each. d a y ' s r e s e r v o i r l e v e l be

qo u n i t s , and qi

Let to-

be t h e s t o r a g e a t t i m e p e r i o d i;

t h e w a t e r r e l e a s e a t t i m e p e r i o d i, ri, w i l l be l i m i t e d by t h e w a t e r i n f l o w a t p e r i o d i, yi,

as w e l l as t h e maximum c a p a c i t y o f t h e r e s e r v o i r ,

(Imax.

Any s p e c i f i e d f e a s i b l e r e l e a s e p o l i c y x, s h o u l d s a t i s f y t h e c o n s t r a i n t : i n f l o w - a v a i q a b l e s t o r a g e space ,< x

. For

time

p e r i o d i:

The actual r e s e r v o i r r e l e a s e s

r.

1

might then be e s t a b l i s h e d

f o r the

policy x . r.

=

x

r.

=

v. -

1

y 1. - (pmax -

if 1

(Omax

- oii)

s x

Oil

if

Yi + q i

$

yi + q i

xi

>

y. + q < x i i ( o r any o t h e r ) f i x e d decision r u l e ,

r . = y. + qi 1

1

Using t h i s

if

t h e s e t of equally l i k e l y

x a t least hydrooraphs can be used t o determine t h e p r o b a b i l i t y t h a t r i n period time horizon. I n t h i s way t h e p r o b a b i l i t y o f f a i l u r e

once i n any

t o meet minimum s e r v i c e l e v e l s a t l e a s t once i n n time periods i s calcul a t e d as a function of s e r v i c e l e v e l . If t h i s q u a n t i t y i s desiqnated as

P,(x),

then the vector optimization problem i s

max

[1 - P n ( x ) , F ( x ) ]

X

Subject t o :

c o n s t r a i n t s on input hydrology, c o n s t r a i n t on r e s e r v o i r c a p a c i t y , and non-negative c o n s t r a i n t s on i n i t i a l

Since

P n ( x ) and

r e s e r v o i r conditions. F ( x ) a r e fundamentally d i f f e r e n t

quantities,

this

i s a vector optimizationof noncommensurable functions and i t can be t r e a t e d using t h e Surrogate Worth Trade-of-F Nethod. Note t h a t t h e optimum policy a n d acceptable r i s k l e v e l s will depend on t h e i n i t i a l storage level chosen, hence t h i s represents a family of optimizations. There a r e several o t h e r r e p r e s e n t a t i o n s of t h e r i s k element of t h i s problem. For example, the r i s k o b j e c t i v e can be defined a s t h e p r o b a b i l i t y t h a t t,he decision level x will not r e s u l t in a f a i l u r e w i t h i n n time p e r i ods ( n = 1,2,3,

..., N ) . I n t h i s case a p r o b a b i l i t y d i s t r i b u t i o n can be gene-

rated f o r P n ( x ) f o r each level of x considered. optimization problems of the form max [ I - P n ( x ) , f ( x ) ]

The r e s u l t i s a family of

n

=

1,2,

...,

N

Subject t o c o n s t r a i n t s a s before. Once again Method.

t h e problem can

be t r e a t e d using t h e Surrogate Worth Trade-off

A t t h i s point most readers will have probably wondered why we did n o t simply determine t h e p r o b a b i l i t y of f a i l u r e , a s s e s s a n approqriate economic penalty function a n d proceed t o maximize t h e mathematical expectation of t h e r e s u l t i n g s i n g l e economic o b j e c t i v e . This i s a very v a l i d question

a n d i n c e r t a i n circumstances

logy t o follow.

i t would

be t h e c o r r e c t de c ision model topo-

The v a l i d i t y of t h i s approach,

s k i l l a n d accuracy of

however, depends upon the

co n s t r u ct i n g t h e penalty func tion f o r dropping below

a d e l i v e r y of x i n any time period. I n f a c t , t h i s i s what t h e Surrogate Worth Trade-off Method does, except t h a t inste a d of attempting t o e va lua te t h e penalty ( a very s u b j e c t i v e mat t er i n r i s k c a s e s ) , a t t e n t i o n i s focused on t h e simpler question whether t h e decision-maker i s w i l l i n g to a c c e pt a s p e c i f i c (computable) i n cr eas e i n r i s k i n orde r t o obta in a s p e c i f i c (computable)

i n c r eas e i n h i s b e n e f i t .

r e a l l y necessary t o

As was shown in

know t h e answer t o t h e

Chapter 3 ,

i t is not

l a t t e r question i n an a b s o l u t e

q u a n t i t a t i v e sense, b u t r a t h e r only i n t h e ordina l (rank o r d e r ) o r q u a l i t a t i v e sense of one being of g r e a t e r v al u e t h a n t h e o t h e r . A penalty func tion on t h e o t h e r hand must be numerically a c c u r a t e over a l l possible values of x , otherwise t h e d e r i v a t i v e s on which optim iz a tion u s u a l l y r e s t s may be badly i n e r r o r . If t h e proper p en al t y f unc tion can be a c c u r a t e l y determined, both methods should lead t o i d e n t i c a l r e s u l t s . The v a l i d i t y of using a penalty f u nc tion and optimizing mathematical e x p e c t a t i o n i s open t o s e r i o u s question in a number of counts in problems involving water resource systems. To be v a l i d t h e process must in f a c t e l i m i n a t e t h e b a s i s f o r a r es i d u al r i s k o b j e c t i v e , and t h i s w ill be Two major types of problems p o s s i b l e only i f c e r t a i n co n d i t i o n s a r e met. a r e discussed below: i n t h e f i r s t t y p e, t h e decision-maker must expect t o have a l a r g e number of a p p l i c a t i o n s of t h e d e c i s i o n , l a r g e enough so t h a t h i s a c t u a l experience with t h e d eci s i o n can reasonably be expected t o be an adequate unbiased sample of t h e corresponding p r o b a b i l i t y d i s t r i b u t i o n s . A decision-maker who o n l y g e t s one t r i a l with i t s corresponding r e s u l t , i s n o t r e a d i l y consoled by t h e mathematical expe c ta tion. tle s t i l l m u s t consider s e p a r a t e l y wh et h er t h e r i s k i s worth t h e gain. .Anyone would be w i l l i n g t o make a s e r i e s of 1,000,000 b et s of $1 . O O each on t h e black numbers on t h e r o u l e t t e t a b l e i f he were paid even money f o r t h e two green house numbers a s well a s t h e black. However, very few would bet $1,000,000 with one and only one bet allowable u n d e r e x a c t l y t h e same circumstances. The mathematical expectancy i s e x a c t l y t h e same, b u t t h e r e l a t i v e d e s i r a b i l i t y i s obviously q u i t e d i f f e r e n t . I n a d d i ti o n t o t h e requirement t h a t an adequate number o f experiences a r e p o s s i b l e , t h e mathematical ex pe c ta tion must a l s o be r e l a t e d t o o t h e r r i s k producing components of t h e system such a s i r r e v e r s i b i l i t y , s t a For example, firm power c o n t r a c t s in t h e Central b i l i t y and r e s p o n s i v i t y . Valley p r o j e c t of C al i f o r n i a r e q u i r e t h e le ve l of firm power c ontra c te d t o

MULTiOBJECTiVE OP'I'iMi ZA TIOW i N WA!l%'R HFSOURCES SYSTEMS

104 drop t o the

l o w e s t power o u t p u t a c t u a l l y d e l i v e r e d ,

f a l l s below t h e " f i r m " c o n t r a c t l e v e l . t h e economic

whenever power o u t p u t

This constitutes a d i s c o n t i n u i t y i n

objective function i t s e l f ,

and

mathematical expectation

as

n o r m a l l y d e f i n e d i s t h u s n o t adequate f o r t h e r i s k element concerned. The t y p e o f

risk-return

problem d e s c r i b e d above i s e s s e n t i a l l y

one

o f r i s k produced by t h e s e n s i t i v i t y o f t h e parameters d e s c r i b i n g t h e system.

A

second t y p e o f r i s k problem i s t h a t produced by s e n s i t i v i t y t o t h e

decision.

T h i s t y p e o c c u r s whenever

t h e d e c i s i o n v a r i a b l e c a n n o t be p r e -

c i s e l y c o n t r o l l e d b u t v a r i e s about t h e d e c i s i o n p o i n t , responding variance i n output.

r e s u l t i n g i m a cor-

i f t h e proper

Once a g a i n ,

conditions are

met and an a p p r o p r i a t e p e n a l t y f u n c t i o n c a n be d e t e r m i n e d o v e r t h e r a n g e o f variance, i t i s appropriate

t o optimize

settinq the decision variable t o i t s

t h e mathematical

expectation

of

I f these c o n d i t i o n s a r e

mean v a l u e .

n o t met, t h e p r o b l e m a g a i n becomes a m u l t i o b j e c t i v e o p t i m i z a t i o n . On o c c a s i o n , i t may be v e r y d i f f i c u l t t o assess t h e r i s k o t h e r t h a n q u a l i t a t i v e o r judgmental

terms.

p e r se

in

However, knowing t h a t r i s k i s

r e l a t e d t o s e n s i t i v i t y w i l l p e r m i t a m u l t i o b j e c t i v e a n a l y s i s i n v o l v i n g sens i t i v i t y and r e t u r n ,

where s e n s i t i v i t y s u b s t i t u t e s

To do

for risk.

so,

t h i s r e q u i r e s u s i n q a d e f i n i t i o n o f s e n s i t i v i t y which r e f l e c t s these q u a l i t a t i v e and judgmental f a c t o r s . the variation u f

t h e d e c i s i o n a b o u t i t s s e l e c t e d v a l u e x m a y n o t be

( o r r e a d i l y determinable), control " that x

y e t i t may

be p o s s i b l e t o

known

e s t i m a t e a "Span o f

i n c l u d i n g t h e most l i k e l y s i g n i f i c a n t v a r i a t i o n i n x .

xC

may be a f u n c t i o n o f

C

F o r example, a p r o b a b i l i t y d i s t r i b u t i o n o f

x since the a b i l i t y t o control

Note

x may depend on

i t s magnitude. An i n t e r e s t i n g problem

i n sensitivity

s t r u c t i o n sequencing problem4.

arises

i n t h e optimal

The methodology developed

con-

t h e r e determines

t h e o p t i m a l o r d e r o f c o n s t r u c t i o n f o r N w a t e r s u p p l y p r o j e c t s , each

having

a s p e c i f i c f i x e d c a p a c i t y o f Qi.

demand

Any g i v e n e s t i m a t e o f

the

future

f o r s e r v i c e r e s u l t s i n some s p e c i f i c o r d e r o f c o n s t r u c t i o n , depending on t h e t i m e p a t t e r n o f t h i s demand.

However, t h e l a t t e r i s an u n c e r t a i n t y u s u a l l y

o b t a i n e d by e x t r a p o l a t i o n processes which, w h i l e sometimes v a l i d f o r periods i n t o t h e f u t u r e , creases. quirement)

become much more

I f i t i s presumed t h a t

an e q u a l l y l i k e l y

f u n c t i o n can be e s t i m a t e d f o r a

o f past trends

f u t u r e demand ( o r r e -

s h o r t t i m e p e r i o d on t h e b a s i s

( e s t i m a t i n g t h e mean and s t a n d a r d

from past variances),

short

u n r e l i a b l e as t h e t i m e span i n -

deviation

of

the error

i t i s p o s s i b l e t o g e n e r a t e a number o f p o s s i b l e f u t -

u r e demand f u n c t i o n s by methods s i m i l a r t o t h o s e used t o q e n e r a t e s i m u l a t e d h y d r o l o g i c sequences.

I n t h e absence o f any o t h e r knowledge we can a s s e r t

185 t h a t t h e s e a r e b e s t p o s s i b l e e s t i m a t e s o f a number o f "equa 1Y 1 k e l y " f u t u r e demands. Using such a s e t o f e q u a l l y l i k e l y f u t u r e demand f u n c i o n s ma1 sequence o f p r o j e c t c o n s t r u c t i o n f o r puted.

the opti-

each s i m u l a t e d f u g r e can be com-

I f they a r e e q u a l l y l i k e l y , i t i s p o s s i b l e t o determine t h e frequ-

ency w i t h

which any p a r t i c u l a r

p r o j e c t would

be

constructed f i r s t .

The

rank o r d e r o f t h i s f r e q u e n c y r e p r e s e n t s a n u m e r i c a l measure o f t h e s e n s i t i v i t y o f t h e supply-use

system t o t h e

i n i t i a l project

decision

under t h e

c o n d i t i o n s o f demand u n c e r t a i n t y .

10.5

MEASUREMENT OF RISK-RELATED CHARACTERISTICS I n s e c t i o n 2,

four risk-related

systems were i d e n t i f i e d

characteristics

and d e f i n e d i n

o f water resources

a d e s c r i p t i v e sense.

However, i n

order t o incorporate these c h a r a c t e r i s t i c s i n t o d e c i s i o n analysis, q u a n t i t a t i v e measures w i l l

be r e q u i r e d f o r each,

and none i s p r e s e n t l y a v a i l a b l e .

The purpose o f t h i s s e c t i o n i s t o e x p l o r e i n a t e n t a t i v e way some p o t e n t i a l measures o f

sensitivity leaving

q u a n t i t a t i v e expression o f r e s p o n s i v i t y

,

s t a b i l i t y and i r r e v e r s i b i l i t y f o r f u t u r e d i s c u s s i o n . I n t h i s discussion

t h e measurement

of sensitivity

i s approached

from t h e p o i n t o f v i e w o f r i s k and u n c e r t a i n t y r a t h e r t h a n f r o m t h a t o f i t s more a b s t r a c t m a t h e m a t i c a l c o n n o t a t i o n . However, i t w i l l be u s e f u l t o b e g i n with the latter. Let the various causative factors

o f r i s k and u n c e r t a i n t y

i n t o t h e m o d e l i n g a n a l y s i s be i d e n t i f i e d by a1 , ral definition,

one measure o f

systems o u t p u t ,

objective,

sensitivity

a2,.

. . ,an.

i s the r a t e o f

o r decision w i t h respect t o

w e l l as t h e r a t e a t w h i c h t h i s r a t e i t s e l f i s c h a n g i n g .

entering

From i t s genechange o f any

t h e f a c t o r s a . as J Thus a v e c t o r mea-

s u r e o f t h e s e n s i t i v i t y o f systems o u t p u t s yi a t a p o i n t c o u l d be expressed as :

sy

=

where I n any

~

practica

-

11

denotes a norm ( t h e l e a s t squares as an example).

application only

ew terms o f t h i s v e c t o r o f

the f i r s t

d e r i v a t i v e s would be s i g n i f i c a n t . S i m i l a r expressions fi(x,s),

and f o r t h e

can be

written

for

*

" o p t i m a l " p o l i c y v e c t o r xi.

the

system

There a r e a l s o s e n s i t i -

v i t i e s o f each o f t h e above w i t h r e s p e c t t o each o t h e r , i . e . , t i v e may be s e n s i t i v e t o

objectives system o b j e c -

r i s k s o r u n c e r t a i n t i e s , t o t h e system o u t p u t

yi,

186

M U I ~ ~ ~ ~ D B ~ J E CGY TP VTEI M ~ Z ~ ~ TIII T ~ MWER O~J

o r t o t h e d e c i s i o n p o l i cy

*

A m m u ~ cSYSTEMS ~s

A s i m i l a r d e f i n i t i o n can be made with r e s p ect t o any o r a l l components o f the c o n s t r a i n t vector g(x,~). Once a s e n s i t i v i t y norm i s d ef i n ed , t h e systems a n a l y s t may seek t o minimize t h i s norm, along with minimizing t h e ove ra ll system o b j e c t i v e s f i ( L , g ) . This c l e 3 r l y lends i t s e l f t o noncommensurable ve c tor minimization problems where t h ? s u r r o g at e worth t r ad e- o f f method can be applied f o r i t s s o l u t i o n . The "span of c o n t r o l " ma,y have an important influe nc e on t h e s e l e c t i o n of t h e proper norm f o r t h e model's s e n s i t i v i t y . Consider Figure 10.2 where t h ? graph i s Given of t h r e e f u n ct i o ns denoted b y Case I , 11, and 111, * a l l of which possess t h e same maximum a t a . C l e a r l y , each of t h e t h r e e f u n c t i o n s has a d i f f e r e n t s e n s i t i v i t y t o a. Accordingly, a d i f f e r e n t norm based on af, '?f, aa

or

aaZ

111 r e s p e c t i v e l y .

a3f

-aa3

x.

1

and v i c e v e rsa .

advantageous in Cases I , 11, and

may prove t o be

Note t h a t t h e span of control of Case I , f o r example, i s

much w i d x t h a n t h a t of Case I 1 or Case 111. Unfortunately t h e use of v ar i o u s o rde rs of d e r i v a t i v e s a s a

measure

o f s e n s i t i v i t y has t h e s er i o u s f a i l i n g of beinq v a l i d only within t h e imme-

d i a t e neighborhood of t h e decisioned p o i n t and objective f i .

x r and i t s a ssoc ia te d o u t p u t y 1.

However t h e r i s k or u nc e rta inty f a c t o r a i may cause t h e

actual x i , yi and/or f . t o d ev i at e s u b s t a n t i a l l y from t h e decisioned values 1 x y , yi and/or f ? . A n i r r i g a t o r may "decide" t o apply 3 inches of water a t 1 a n i r r i g a t i o n b u t h i s a b i l i t y t o control t h a t de c ision a t e x a c t l y 3 inches leaves much t o be d e s i r e d .

His act u al i r r i g a t i o n may be anywhere from two t o f o u r inches with corresponding v a r i a t i o n s in o u t p u t ( s o i 1 m o i s t u r e a v a i l -

a b i l i t y ) or o b j e c t i v e s ( p r o f i t on t h e c r o p ) . 10.6

S U I U A R Y AND CONCLUSIONS

In t h i s

chapter a number o f

c e r t a i n t y have

been t e n t a t i v e l y

q u es t i ons a ssoc ia te d with r i s k explored f o r t h e

purpose of

and u n -

stim ula tinq

f u r t h e r a n a l y s i s and research i n t o t h e q u a n t i f i c a t i o n s of the se f a c t o r s f o r use i n m u l t i - o b j ect i v e o p t i mi zat i o n a n a l y s i s .

A g r e a t many problems e x i s t

i n water resources s,ystems and o t h e r c i v i l s,ystems involving resources in which avoidance of objective. formulated

r i s k and u n cer t ai n t y

If suitable

a r e o f t e n in f a c t

q u a n t i t a t i v e measures of

t h e dominating

the se o b j e c t i v e s can be

then t h e s u r r o g at e worth t r ad e-off method can be used t o d e t e r -

mine t h e optimal

or a t l e a s t

s u p er i o r

combinations of r i s k

and

various

forms of r e t u r n . Direct situations.

measures of Example of

r i s k t o be

avoided

can be defined

hydrologic r i s k q u a n t i f i c a t i o n

in certain

( t o be minimized)

187

I ai

F i g u r e 10.2

a

V a r i a b l e Span o f C o n t r o l

were developed as an example o f t r e a t i n g r i s k due t o chance c o n t r o l l e d nond e c i s i o n e d i n p u t s o r system parameters. More complex r i s k and u n c e r t a i n t y s i t u a t i o n s d e v e l o p when cannot be made w i t h

decisions

p r e c i s i o n c o n t r o l b u t r a t h e r w i l l vary about t h e d e c i -

s i o n v a l u e s i n some random o r q u a s i random manner. When t h e number o f r e p e titions of may be

that decision i s

meaningless

t o the

a l s o s m a l l so t h a t decision-maker,

mathematical expectations

an i m p o r t a n t

formof r i s k i s

introduced. Such i m p r e c i s i o n

o f control

may

introduce

t h r o u g h s e v e r a l systems c h a r a c t e r i s t i c s . were i d e n t i f i e d and

d e s c r i p t i v e l y defined:

risk

or

uncertainty

Four such system c h a r a c t e r i s t i c s sensitivity,

responsivity,

s t a b i l i t y and i r r e v e r s i b i l i t y . I n a d d i t i o n s e v e r a l t y p e s o f m o d e l i n g e r r o r s were i d e n t i f i e d w h i c h can l e a d t o i m p r e c i s e contro1,imprecisepredictionsof

188

MULTTOAJECTTVE OP7'IMIZATTON IIJ WLTlL? RESOURCES SYSTEMS

t h e r e a l response o r b o t h , hence h a v i n g a c c e n t u a t e r i s k and u n c e r t a i n t y . Because t h e y

equivalent a b i l i t y t o

a r e somewhat s i n g u l a r l y r e l a t e d

t o the

create

or

s p e c i f i c sys-

tems concerned, g e n e r a l i z a t i o n s on r e s p o n s i v i t y , s t a b i l i t y and i r r e v e r s i b i l i t y a r e n o t d i s c u s s e d i n t h i s more g e n e r a l d i s c u s s i o n .

o t h e r hand would appear t o

be amenable t o more g e n e r a l i z e d q u a n t i f i c a t i o n s Each o f t h e above measures i s u s e f u l

as d i s c u s s e d i n t h e p r e v i o u s s e c t i o n . under

successively

S e n s i t i v i t y on t h e

more g e n e r a l c i r c u m s t a n c e s

ranging

f r o m systems con-

t r o l l a b l e w i t h i n c l o s e l i m i t s t o t h o s e w h i c h can be o n l y a p p r o x i m a t e l y cont r o l l e d w i t h i n broad l i m i t s . T h i s somewhat

preliminary analysis

q u a n t i t a t i v e measures o f r i s k can be be o p t i m i z e d i n a

and d i s c u s s i o n

indicates

that

d e f i n e d and u t i l i z e d as o b j e c t i v e s t o

multi-objective control.

I n some i n s t a n c e s even u n c e r -

t a i n t y (no p r o b a b i l i t y d i s t r i b u t i o n d a t a ) can be t r e a t e d a d e q u a t e l y . An i n d i c a t i o n , however, i s n o t an accomplished f a c t and much i n s i g h t and a n a l y s i s

\Jill

be r e q u i r e d t o q u a n t i f y

i n common w a t e r r e s o u r c e s systems multi-objective decision o v e r r i d i n g importance

the major r i s k f a c t o r s i n v o l v e d

adequately t o

analysis.

allow t h e i r inclusions i n

Because o f t h e s i n g u T a r and sometimes

o f t h i s issue

i t i s hoped t h a t t h i s d i s c u s s i o n w i l l

s t i m u l a t e t h a t i n s i g h t and a n a l y s i s .

FOOTNOTES 1.

The

g a m b l e r ' s r u i n problem

i s d e s c r i b e d b y H a l l and Dracup, [1970].

2.

These e s t i m a t e s a r e d e s c r i b e d by Nadeau, [1950].

3.

The development o f

4.

The o p t i m a l

t h e d e s c r i p t i o n s f o r water resources

systems can

be found i n Haimes and Macko, [1973]. c o n s t r u c t i o n sequencing

problem

i s solved

by B u t c h e r ,

and W. A. H a l l ,

"Dynamic

Proqramming

Haimes and H a l l [1969]. References 1.

B u t c h e r , W. S . ,

Y . Y . Haimes,

f o r t h e Optimal Sequencing o f Water Supply P r o j e c t , " Water ces Research, v o l . 5, no. 6, p. 1196,

J . A. Dracup,

2.

H a l l , W. A. and

3.

Nadeau, R. A.,

4.

Haimes Y . Y . and D. Macko,

McGraw-Hill Book Co.,

Resour-

1969.

Water Resources Systems E n g i n e e r i n g ,

New York, 1970.

The Water Seekers,

Doubleday and Company, I n c . ,

Flew

York, 1950. ces Management",

" H i e r a r c h i c a l S t r u c t u r e s i n Water Resour-

IEEE-Systems, Man, and C y b e r n e t i c s , v o l . SMC-3,

no. 4, pp. 396-402, 1973.

Chapter 11 EPILOGUE T h i s c h a p t e r summarizes t h e c h a r a c t e r i s t i c s

and

advantages o f

the

SWT method and i n d i c a t e s f u r t h e r e x t e n s i o n s o f t h e method. M u l t i p l e o b j e c t i v e p l a n n i n g and d e c i s i o n making i s a n i m p o r t a n t p r o b lem f o r most c i v i l systems.

The v a r i o u s o b j e c t i v e s i n v o l v e d g e n e r a l l y can-

n o t be r e p r e s e n t e d i n common u n i t s , h e n c e t o f i n d t h e b e s t p o l i c y , t h e d e c i s i o n maker must make a m e n t a l a n a l y s i s o f t h e t r a d e - o f f s

that

might

be

achieved. The SWT method combines t h e s e same mental processes w i t h mathemat i c a l a n a l y s i s i n a p r o c e d u r e w h i c h p r o v i d e s a c o n v e r g i n g s y s t e m a t i c approach.

The m a t h e m a t i c a l a n a l y s i s a p p l i e s o n l y t o t h e q u a n t i t a t i v e f u n c t i o n s .

High p r e c i s i o n f u n c t i o n s

a r e n o t d i l u t e d by

low p r e c i s i o n f u n c t i o n s since

each i s a s e p a r a t e and d i s t i n c t v e c t o r t h r o u g h o u t . respond o n l y

t o h i s sense o f

The d e c i s i o n - m a k e r need

satisfaction o f levels o f objective attained

"AA" v s . " A B " t r a d e - o f f p o s s i b i l i -

and h i s sense o f d e s i r a b i l i t y o f s i m p l e

T h i s i s e x a c t l y what a d e c i s i o n - m a k e r always does or a t l e a s t t r i e s

ties. t o do.

The a n a l y s i s assumes

t h a t those

are the best trades

possible a t

t h a t l e v e l o f a t t a i n m e n t , where s u b - o p t i m a l c o m b i n a t i o n s a r e n o t p r e s e n t e d . Thus t h e SWT method models a process v e r y s i m i l a r t o t h e r e a l d e c i s i o n p r o cess where a s i n g l e d e c i s i o n - m a k e r i s i n v o l v e d , y e t

it substantially

re-

duces t h e number o f c o m b i n a t i o n s o f l e v e l s o f o b j e c t i v e s a t t a i n e d t h a t need be c o n s i d e r e d . 11.1

ADVANTAGES OF THE SWT METHOD The m a j o r

c h a r a c t e r i s t i c s and

advantages

o f the

surrogate worth

t r a d e - o f f method a r e : (i)

Non-commensurable o b j e c t i v e f u n c t i o n s can be handled qua n t it a t iv e l y .

(ii)

The s u r r o g a t e w o r t h f u n c t i o n s , which r e l a t e t h e d e c i s i o n maker's preferences t o t h e n o n - i n f e r i o r s o l u t i o n s through t h e t r a d e - o f f f u n c t i o n s , can be c o n s t r u c t e d i n t h e f u n c t i o n a l space and o n l y l a t e r be t r a n s f o r m e d i n t o t h e d e c i s i o n space.

(iii)

The d e c i s i o n - m a k e r i n t e r a c t s w i t h t h e m a t h e m a t i c a l model a t a g e n e r a l and a v e r y moderate l e v e l .

He makes d e c i -

s i o n s on h i s s u b j e c t i v e p r e f e r e n c e i n t h e f u n c t i o n a l space (more f a m i l i a r and m e a n i n g f u l t o him) r a t h e r t h a n i n t h e d e c i s i o n space.

This i s p a r t i c u l a r l y important since the

d i m e n s i o n a l i t y o f t h e d e c i s i o n space i s g e n e r a l l y much

189

190

MULTIOBJECTIVE OPTIPIZATIOW I!] WATER RESOURCES SYSTEMS

l a r g e r t h a n t h e d i m e n s i o n a l i t y o f t h e f u n c t i o n a l space. (iv)

The SWT method p r o v i d e s t h e d e c i s i o n maker w i t h a d d i t i o n a l q u a n t i t a t i v e i n f o r m a t i o n on t h e n o n - i n f e r i o r ( P a r e t o optimum) space.

I n particular, the trade-off functions

afi

(A,. = ij

~

af,

,i#

j , i , j = 1,2

,..., n ) ,

which a r e t h e

J

s l o p e s o f t h e n o n - i n f e r i o r c u r v e s i n t h e f u n c t i o n a l space, a r e o f s i g n i f i c a n t i m p o r t a n c e t o t h e d e c i s i o n - m a k e r by p r o v i d i n g t h e r e l a t i v e t r a d e - o f f s a t anv l e v e l o f o b j e c t i v e achievement between a n y two o b j e c t i v e f u n c t i o n s . (v)

Computational f e a s i b i l i t y and t r a c t a b i l i t y have been demonstrated t h r o u g h t h e s o l u t i o n o f s e v e r a l example problems.

(vi)

The a p p l i c a b i l i t y o f m u l t i o b j e c t i v e a n a l y s i s v i a t h e s u r r o g a t e w o r t h t r a d e - o f f method t o s e v e r a l problems in

water

resources planning--water q u a l i t y mainten-

ance, r e s e r v o i r o p e r a t i o n and c o n s t r u c t i o n , e t c . - -

has

been demonstrated. (vii)

The a v a i l a b i l i t y o f o p e r a t i o n a l m e t h o d o l o g i e s , such as t h e SWT method, encourages and enhances t h e systems m o d e l i n g and p a t t e r n o f t h i n k i n g i n m u l t i o b j e c t i v e f u n c t i o n a l terms.

Thus more r e a l i s t i c a n a l y s e s may r e s u l t

by e l i m i n a t i n g t h e need f o r a s i n g l e o b j e c t i v e f u n c t i o n f o r mu 1a t ion .

71.2

FURTHER DEVELOPMENT OF SWT METHOD W h i l e t h e SWT methodcan be u t i l i z e d t o advantage i n most m u l t i o b j e c -

t i v e o p t i m i z a t i o n problems

t h e r e i s s t i l l a number

o f a r e a s i n w h i c h sub-

s t a n t i a l improvement i n e f f e c t i v e n e s s s h o u l d be p o s s i b l e . One o f t h e i m p l i c i t assumptions i n t h e development o f t h e SWT method i s that there i s

a s i n g l e q u a n t i t a t i v e value

DM's assessment o f p r e f e r e n c e . tems,

w h i c h can be a s s i g n e d as t h e

I n most r e a l problems i n v o l v i n g c i v i l s y s -

t h e r e i s a number o f d e c i s i o n - m a k e r s who w i l l have d i v e r s e o p i n i o n s ,

hence t h e r e can be s e v e r a l such q u a n t i t a t i v e v a l u e s . cussed b r i e f l y i n s e c t i o n 3 . 4 .

T h i s problem was d i s -

With f u r t h e r refinements,

the

application

t o m u l t i o b j e c t i v e o p t i m i z a t i o n i n v o l v i n g m u l t i p l e d e c i s i o n - m a k e r s s h o u l d be feasible. I n addition,

r e f i n e m e n t s may be u s e f u l f o r d e t e r m i n i n g t h e t y p e s o f

q u e s t i o n s t o ask d e c i s i o n - m a k e r s .

The r e q u i r e m e n t

that

t h e increment

of

191

each o b j e c t i v e be smal , must be c o u n t e r - b a l a n c e d i t must s t i l l be l a r g e enough f o r

the

by t h e r e q u i r e m e n t t h a t

DM t o be a b l e t o p e r c e i v e t h e d i f f e -

rences. Thus f a r , a me hod f o r d e t e r m i n i n g t h e optimum s i z e o f t h e s e i n c r e ments f o r a l l problems has n o t been found.

G u i d e l i n e s t o improve t h e con-

d e c i s i o n - m a k e r s w i l l a l s o a i d i n a p p l y i n g t h e SWT

s i s t e n c y and a c c u r a c y of method t o r e a l problems.

The s e n s i t i v i t y a n a l y s e s

rithms o f chapters t h r e e through s i x should

suggested i n t h e a l g o -

a l s o be u t i l i z e d when a p p l i c a -

t i o n s t o r e a l problems a r e c o n s i d e r e d . One o f t h e advantages o f t h e SWT method i s t h a t a l m o s t any a l g o r i t h m f o r d e t e r m i n i n g n o n - i n f e r i o r p o i n t s can be i n c o r p o r a t e d i n t o t h e procedure,

so t h a t as

new and improved a l g o r i t h m s a r e developed, t h e SWT approach can

be c o n s t a n t l y updated. h.. = 1J

is a

- afi/8f.

The o n l y

requirement i s

that the trade-off rates

must be d e t e r m i n a b l e as p a r t o f t h e s o l u t i o n .

J T h i s book has c o n s i d e r e d o n l y d e t e r m i n i s t i c problems;

l a r g e amount o f work on

however, t h e r e

d e c i s i o n t h e o r y f o r problems w i t h s t o c h a s t i c

d e c i s i o n s , o b j e c t i v e s and c o n s t r a i n t s .

A d a p t i n g t h e SWT approach t o t h e s e

problems would i n c r e a s e i t s u s e f u l n e s s .

I n p a r t i c u l a r , t h e SWT method can

t r e a t s i g n i f i c a n t f a c t o r s o f r i s k and u n c e r t a i n t y as o b j e c t i v e s . The dynamic

problems i n t h i s

book c o n s i d e r e d o n l y o b j e c t i v e s which

were presumed n o t t o change w i t h t i m e , ( e . g . method c o u l d be

pendent-- where t h e v a l u e tant (fi(t)

i n t e g r a l s over t i m e ) .

The SWT

extended t o dynamic problems whose o b j e c t i v e s a r e t i m e deo f some f u n c t i o n a t each p o i n t i n t i m e i s impor-

= Li(x(t),u(t),t)

f o r i = 1,2,

..., n ) o r w h e r e

t h e i n t e g r a l up t o t

t h a t p o i n t i s i m p o r t a n t o v e r t h e e n t i r e t r a j e c t o r y ( f i ( t ) = i L . ( x ( s ) , u ( s ) , s ) ds 0 ' ..., n ) . For example such problems may a r i s e i n l o n g - t e r m p l a n -

f o r i = i,2,

n i n g problems where t h e c a p i t a l c o s t as w e l l as t h e o p e r a t i o n s , maintenance and replacement c o s t i s i m p o r t a n t . Problems o f t h i s f o r m w i l l r e q u i r e g r e a t m o d i f i c a t i o n s i n c e t h e t r a d e - o f f r a t e s x.. and t h e w o r t h f u n c t i o n s W . . w i l l 1J

1J

a l s o be f u n c t i o n s o f t i m e . a p p l i c a t i o n i s t o use t h e s u r r o g a t e w o r t h t r a d e - o f f

Another a r e a of method as a c o o r d i n a t o r models i n m u l t i l e v e l

i n m u l t i l e v e l h i e r a r c h i c a l models.

models o f t e n have

S i n c e t h e sub-

noncommensurable and competing ob-

j e c t i v e s , a c o o r d i n a t i n g a n a l y s i s w h i c h can h a n d l e m u l t i p l e o b j e c t i v e p r o b lems i s u s e f u l . Finally,

the

most

i m p o r t a n t a p p l i c a t i o n i s t o implement t h e

method i n problems w i t h r e a l d e c i s i o n - m a k e r s .

SWT

AUTHOR I N D E X

Arrow, K . J .

28, 107

F e i n b e r g , A.

15, 16

Athans, PI. 8, 81, 83, 89

Fishburn, P.C.

B a i l e y , T.E.

164

Foley, J.W.

Beeson, R.M.

27

Fox, R.L.

15

161, 162 150

Bergson, A . 15

F r a n k e l , R.J.

Bower, B.T.

Freeman, A.M.

I11

Geering, H.P.

8

2

B r i s k i n , L.E.

16 81, 83, 89, 119

Bryson, A.E. B u t c h e r , W.S. Bvers, D.M. Charnes,

A.

Chu, K.C.

162

19, 25, 41, 70

Gembicki, F. G e o f f r i o n , A.M.

184

18

9, 15, 18, 28

4, 21

Georgescu-Roegen,

24

Haimes, Y.Y.

N.

16

2, 3, 11, 17, 19, 21,

34, 38, 131, 158, 160, 161, 162,

9

178, 184 C i t r o n , S.J.

9, 18

Cochrane, J . L .

Haas, J.E.

131, 140, 161, 162

Hall,, W.A.

2, 11, 34, 141, 151, 172,

28

4, 21, 28

Cohon, J . L .

184 Cooner, W . W .

24

Corps o f Engineers Day, R.H.

Havernan, R.H.

18

H i l l i e r , F.S.

64

4

17

Ho, Y . C . DaCunha, N.D.

81, 33, 89, 119

9 Howe, C.W.

Dracup, J.A. D u c k s t e i n , L. E a s t e r , K.W.

3

3, 141, 172 Huang, S.C.

24

Husar, H . A .

2, 161

4 3 I n t r i l i g a t o r , M.D.

Evans, J . R .

64

18 I s a r d , W.

3

E v e r e t t , H. 111 18, 38, 41, 69, 70 F a l b , P.L.

192

Kaplan, M.A.

2, 161

Keeney, R.L.

15, 131

81, 83, 89

193

AUTHOR INDEX Phelps, E.B.

4

K i s i e l , C.C. K l i n g e r , A.

Polak, E.

9

Kneese, A.V. Koopmans, T . C .

Kuhn, H.W.

6 2 , 6 4 , 79, 111 1 9 , 21, 3 8 , 41, 7 0 ,

163

Lieberman, G.J.

Lietmann, G .

64

Robinson, S.M. Rov, B.

28

40

131

28

Taha, H.A.

Maier-Rothe, C.

Veinuri, V .

1 3 2 , 1 5 9 , 161 18

1 8 , 1 2 7 , 128

28

W a l t z , F.M.

Major, D.C.

4

Idismer, D.A.

Marks, D . H .

4 , 21

Yu, P . L .

17, 19

Yu, W.

P l e i s e l , W.S.

27

Zadeh, L . A .

M i l l e r , W.L.

4 , 21

Zeleny, M .

Monarchi, D.G.

4 175

01agundo,ye, 0.

21, 41 4

21 21

64 17 21

1 8 , 24

McGrew, D . R .

P a s t e r n a k , H.

28

6 4 , 136

Wagner, H . M .

178

O'Riordan, J.

18

T u c k e r , A.W. 3 , 8 , 9 , 3 9 , 47, 6 1 , 6 2 , 6 4 , 7 9 , 111

3

Nadeau, R.A.

24

S t a n k a r d , M.F., J r .

Steuer, R.E.

MacCrimmon, K . R .

P a s s y , U.

17

S t r e e t e r , H.W.

24

Luenberger, D.G.

Placko, D.

9, 18, 127, 128

S e n g u p t a , S.S.

131

Liebman, C . C .

Maass, A .

R e i d , R.W.

150

S a l u k v a d z e , M.E.

Lebosquet, M .

Lynn, W.R.

28

R a t n e r , M.W.

8

3, 8, 9 , 3 9 , 47, 6 1 ,

Lasdon, L.S. 150

9

Raiffa, H.

2

1 5 9 , 161

161 15 1 8 , 28

SUBJECT INDEX

Adaptive Search

Cost-Benefit Analysis

27

Allocation of resources Aqueducts Aquifer

2 , 18

Curve f i t t i n g

181

3, 174 Dams

177, 1 7 8

Auto-catalytic e f f e c t s

175

3

Decision maker

( e n t i r e book)

Decision making B i c r i t e r i o n Problems

BOD

4 , 18

9, 10, 11

161

-

( e n t i r e book)

i n groups models of

45 28

under u n c e r t a i n t y

28

Causative f a c t o r s o f r i s k and uncertainty 185

Decision space 5-7, 2 2 , 27, 34-36, 46-48, 58-62, 74, 8 2 , 84, 89, 94, 97, 100, 107, 110, 112, 117, 120

Central Val 1 ey p r o j e c t

Decomposition

183

Chain Rule for p a r t i a l derivat i v e s 106 Combined algorithms 94-100, 104

70-74,

Commensuration; value c r i t e r i o n of Consistency

174

Control v a r i a b l e s 81, 87, 89, 93, 97, 102, 119-125 125

139 194

184

Di f f e r e n t i a1 equations

-

system of

177

D i f f e r e n t i a l equations ( p a r a b o l i c )

-

129,

system of

178

Dissolved oxygen (DO) 40, 161 D i s t r i b u t i o n of income ( r e g i o n a l ) Drought

Convexity 4 , 9-10, 18-21, 25, 48-50, 70, 75-76, 78, 94, 100, 103, 115, 125 Correlation c o e f f i c i e n t

Demand functions

D i f f e r e n t i a l equations ( p a r t i a l ) - system of 177

43, 61-62, 84

Control vector

100

4

174

Duality gaps

Dump energy

19, 25, 41, 70, 75, 100 144

Dy nam i c n- o b j ec t iv e €-cons t r a i n t (DNE) a1 gor i thm 120-1 23

195

SIJBJECT INDEX

Dynamic n - o b j e c t i v e mu1 t i p 1 i e r (DNM) a1 g o r i thm 123-1 25

E f f l u e n t charges

Dynamic 2 - o b j e c t i v e combined

End p o i n t c o n s t r a i n t s

81, 83

Environmental q u a l i t y

1, 3-4, 12

(DTC) a l g o r i t h m 94-97 Example

2

E l a s t i c i t y Functions

47-48,

50, 52

97-1 00 E r r o r sources i n modeling

176-177

Dynamic 2 - o b j e c t i v e E - c o n s t r a i n t (DTE) a l g o r i t h m Example

84-89

89-94

Exhaustive search

Dynamic 2 - o b j e c t i v e mu1 t i p l i e r (DTM) a1 g o r i thm 100-1 02 Exarnpl e

Euler's chain r u l e

102-1 03

63-69

84-94, 99, 103, 112-1159 117, 120-123

63, 68, 92, 111-

112 E x p e c t a t i o n (mathematical)

171, 183

184

E x t r a pol a t i on

E-constraint algorithms

106

F e a s i b l e d e c i s i o n space 34-36, 46-48,

5-7, 22, 27,

58-62, 74, 82, 84,

89, 94, 97, 100, 107, 110, 112,

E - c o n s t r a i n t method 11, 19, 21 -23, 37-42, 58-74, 82-94, 105, 112-1159

117, 120 F i r m energy 35, 144, 181 - water 35, 181

117, 120-123, 125 Flood c o n t r o l E - c o n s t r a i n t form 128, 147

3, 175

82, 105, F u n c t i o n space

5-7,

19-20, 22-27,

34, 36, 46, 49-55, 59-60,

cj:

l i m i t s on

109

70-71, 82, 84, 107, 109-110, 119

Economic c o m p e t i t i v e equilibrium 8 cost

efficiency growth

Game t h e o r y

1, 28

175 Goal a t t a i n m e n t program

1-3

Goal programming

4

p r i c i n g theory production

1, 2, 18

G r a d i e n t search

22, 24-25 63, 78, 112

1 Ground water systems

E f f i c i e n t solutions

6-9,

94, 100, 105-

109, 112, 119-120, 129, 165

3, 178

18,

21-22, 25, 27, 41, 46-49, 52, 58-63, 69-70, 78, 82, 84, 88-89,

25-27

128,

Guaranteed energy

Hamiltonian

144

87, 90-91, 96, 98

Human S u b j e c t i v i t y i n Water resources modeling

180

MULTIOBJECTIVE OP!TIMIZATION I N WATER RESOURCES SYSTEMS

196 H y d r o e l e c t r i c systems

174, 179

Hydrographic s e q u e n c e s Hydrologic sequences tlydrology

180 183

1 3 8 , 165

19, 22

Identification

Lagrangian 166

(see systems

I n d i c e s of performance

171

I n d i f f e r e n c e band 1 1 , 36, 44-48, 63, 6 6 , 68, 73-74, 7 8 , 8 9 , 97, 1 0 2 , 111-114, 1 1 7 , 1 2 3 , 125, 1 2 9 , 151

11, 16

1

rlarginal u n i t

Mean

1 8 0 , 183

Mean s q u a r e approach 24

I n t e r a c t i v e methods

28

Miami R i v e r

I n t e g e r programming

21

M i n i m u m Time problems

I n t e r p o l a t i o n 27, 47-48, 58, 6 2 , 6 8 , 74, 78, 92, 111112, 139, 131 I r r e v e r s i b i l i t y 170, 172, 175176, 185, 187, 188 Irrigation

3 , 1 7 4 , 186

1 , 3 , 55-56,

79

3 7 , 41, 88,

81

165

1 0 5 , 106, 1 0 7 , 152

Mathematical modeling

81

16-17, 1 9 ,

L i n e a r problems 1 8 , 2 4 , 4 7 , 50-52, 66-70, 9 4 , 115, 125

Marginal t r a d e - o f f v a l u e

11

I n f i n i t e o b j e c t i v e problem

Jntwest rate

1-3

Lexicographic approaches 21

Ma nag eme n t

Indifference functions 28, 53-55, 107 Inferior solutions 110

39-40, 7 0 , 8 3 , 128, 1 6 5 ,

Large s c a l e problems

identification)

I n d i f f e r e n c e hand d e f i n i t i o n of

3 9 , 4 7 , 61-62,

Lagranqe m u l t i p l i e r s 1 1 , 37-43, 55, 64, 69-71, 82-84, 105, 1 1 4 , i z n ,

2

Hyperplanes

Muhn Tucker c o n d i t i o n s 64, 7 9 , 1 1 1 , 1 2 8

162

Model Topol ogy

81-82

176-1 77

calibration

178

data collection

1 7 6 , 179

d a t a p r o c e s s i n g 1 7 6 , 179 parameters 178 s c o p e 178 M u l t i l e v e l approach

160

F l u l t i o b j e c t i v e w a t e r q u a l i t y models ,lacobian m a t r i x

27, 712

156 g e n e r a l problem f o r m u l a t i o n 158-1 60

SUBJECT I N D E X

197 N o r t h e r n C a l i f o r n i a w a t e r system

M u l t i o b j e c t i v e w a t e r qua1 i t y models example Multiple objectives regressions

142-1 48

161-164 ( e n t i r e book)

solution

148-154

42, 111, 139 O b j e c t i v e f u n c t i o n space

M u l t i p l i e r algorithms

74-78,

(see

f u n c t i o n s space)

100-1 03, 115-1 17, 123-1 25, Optimal c o n s t r u c t i o n sequence problem approach

69-70, 94, 100,

184

103, 115-117, 123-125 Optimal Control M u n i c i p a l and i n d u s t r i a l w a t e r use

( s e e dynamic o p t i -

m ization)

3

solutions

1, 3, 6-8

definition N-objective algorithms

Optimal p o l i c y v e c t o r

11 9-1 25 Naviqation

fi

105-1 17, Optimal s o l u t i o n

3

validity N e i g h b o r i n g e x t e r n a l methods

185

170, 171

176

89 Optimization technique

Newton a p p r o x i m a t i o n method

Optimum d e c i s i o n

78, 112 Newton Raphson method

128 (all

over t e x t )

6-9,

18,

109, 112, 119-120, 128, 129, 165

81, 119

Penalty function

183

Pollution certificates Power q e n e r a t i o n

2

3

Preferred control vector

definition of

120

8

necessary c o n d i t i o n s

Non-linearity

Path c o n s t r a i n t s

69-70, 78, 82,

34, 88-89, 94, 100, 105-

proper

( s e e non-

i n f e r i o r solutions)

25, 27, 41, 46-49,

52, 58-63,

11, 17-20, 25,

69, 94 Pareto-optimal s o l u t i o n s

82, 97, 106

Non-inferior solutions 21-22,

172, 176

P a r a m e t r i c approach

Noncommensurable o b j e c t i v e s

Non-inferior set

178

63,

8-9

Preferred decision vector

107, 114,

117

9 4, 50, 70-78,

103, 115, 123

100,

Preferred solutions 44-45,

9, 15-28, 35-36,

66, 68, 73-78, 89, 93,

97-98, 100-102, 107-110, 115, Norms

24, 185, 186

117, 120-125 definition

9

MULTIOBJECTIVE OPTIMIZATION IN WATEr7 RESOURCES SYSTEMS

198

Preferred s t a t e vector

Probabi 1 it y d i s t r i b u t i o n s

74, 76, 89, 97, 102, 115, 117, 125, 141, 184, 185

183

P r o j e c t s e l e c t i o n f o r research

1

Proper non-i n f e r i o r s o l u t i o n s (see n o n - i n f e r i o r s o l u t i o n s ) P r o t o t y p e systems

Simplex method Slack v a r i a b l e s

18, 67 24, 67

Social i n d i f f e r e n c e functions

171

-cost functions

137

(see u t i l i t y

functions)

1, 3

-preference functions -welfare functions

Social indifference surface

2, 18

Recreational use o f water resources projects

3, 79, 168, 174 3, 4

Reqional development Reqressions

27, 42-43,

Soil moisture

Span of c o n t r o l

50, 58,

mization 187

Reservoirs

127

128-132

3, 35, 79

171, 185

187, 188

172, 173-1 74,

187, 188

170, 172, 175, 185,

Ql, 83, 87, 89, 93,

97, inz, 119-125

algorithm

112-115, 132

S t a t i c n - o b j e c t i v e m u l t i p l i e r (SNM) 11 5-1 17

St a t i c two-o b j e c t i ve combi ned (STC) alqorithm

70-74

S t a t i c t w o - o b j e c t i v e € - C o n s t r a i n t (STE)

R i sk-re1 a t e d c h a r a c t e r i s t i c s

measurement

State variables

a1 g o r i thm

R e s p o n s i v i t y i n mu1 t i o b j e c t i v e optimization

186

S t a t i c n - o b j e c t i v e e - c o n s t r a i n t (SNE)

R e s p o n s i b i l i t y as a system characteristic

125

S t a b i l i t y i n mu1 t i o b j e c t i v e o p t i -

117, 120, 123 Reid-Vemuri example problem

107

186

Solution state vector

62, 75, 84, 100, 109, 112,

solution

(see u t i l i t y

functions)

172

Resource a l l o c a t i o n

(see

u t i l i t y functions)

Q u a s i l i n e a r i z a t i o n 89 Random v a r i a t i o n s

11, 16

28, 53-55, 107

Q u a d r a t i c proqramming Ouality o f l i f e

3, 12, 66, 72-

S e n s i t i v i t y analysis

123, 125

185

Risks i n water resource systems 2, 12, 170, 181-185, 187

algorithm

63-69

S t a t i c t w o - o b j e c t i v e M u l t i p l i e r (STM) algorithm

74-79

Stochastic i n f l o w

181

199

SUBJECT INDEX

S t o r a t i v i ty

Three w a t e r qua1 it y o b j e c t i v e

178

p r o b l em

Stream r e s o u r c e a1 l o c a t i o n problem

58-63,

135-142

S t r e e t e r - P h e l ps e q u a t i o n

133,

(see o p t i m a l

S u r r o g a t e Worth F u n c t i o n s 43-55,

58-63,

88-91,

inn,

24,

66-69, 73-78

114, 115, 123,

125 S u r r o g a t e Worth T r a d e - o f f Method 132, 165, 182, 186

- f o r dynamic n - o b j e c t i v e p r o b l ems

75, 82-84,

81-103

9, 35-55,

93, 96, 100,

106-107, 119

Trade-off r a t i o s

36, 46-51,

112

(see t r a d e - o f f

r a t e functions) Transmi ss i v it y

1 78

Transportation

1, 175

Treatment c o s t f u n c t i o n T r i n i t y subsystem

160

142, 149, 150

Two-objective algorithms

119-1 26

- f o r dynamic t w o - o b j e c t i v e problems

Trade-off r a t e f u n c t i o n s

T r a d e - o f f r a t i o space

solutions)

34-62,

70-71, 82, 107, 117

59-63,

159, 161 Superior s o l u t i o n s

9, 16, 18, 49-55,

Trade-off functions

132-135

solution

165-1 68

63-79, 81-

103 Two r e s e r v o i r system

143

-for s t a t i c n-objective problems

105-117

U n c e r t a i n t y and e r r o r i n s o l u t i o n

- f o r s t a t i c two-objective problems

63-69

179, 180, 186 U n c e r t a i n t y i n w a t e r r e s o u r c e systems

System d e s c r i p t i o n i n w a t e r

2, 12, 170

r e s o u r c e s (see Model scope) -goal o r f u n c t i o n a l des c r i p t i o n ( s e e Model scope)

U n s t a b l e d e c i s i o n system Utility functions

175

11, 15-19, 21, 28,

46

-pol i t i c a l - g e o g r a p h i c a l d e s c r i p t i o n ( s e e Model scope)

- p hy s ica 1 -hydro 1og ica 1 d escription

(see Model scope)

-temporal d e s c r i p t i o n

(see

Variance

180

Vector o p t i m i z a t i o n

Model scope) Systems I d e n t i f i c a t i o n

17, 19,

21, 177 Thermal p o l l u t i o n

Water need

175

Water q u a l i t y 162, 163

(see m u l t i p l e

objective optimization)

Water r e c h a r g e

178 178

200

MULTIOBJECTDE OPTIMIZATION I N WATER RESOURCES SYSTEMS

Water r e l e a s e policy Water resource systems Water s t o r a q e Worth functions

181 1-4, 79

178 (see Surroqate

Worth f u n c t i o n s )