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].

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Arrow, K. J . ,

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John Wiley and

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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 , "

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

3.

Bergson, A.,

~Management

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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.

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

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Making,

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9.

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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 _

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15. 16.

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18. 19.

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--

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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"

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U. o f

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

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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.

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

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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

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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.

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and Leitmann, G . ,

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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

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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

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

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(

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

=

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