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UBC Theses and Dissertations

Bayesian decision analysis for pavement management Bein, Piotr 1981

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BAYESIAN DECISION ANALYSIS ROR PAVEMENT MANAGEMENT by PIOTR BEIN > P o l y t e k n i s k K a n d i d a t , The T e c h n i c a l U n i v e r s i t y o f Denmark, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (The Depar tmen t o f C i v i l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t he r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA DOCTOR OF PHILOSOPHY i ri May i 1981 P i o t r B e i n , 1981 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l ica t ion o f th is thes is fo r f i n a n c i a l gain s h a l l not be allowed without my wri t ten permission. Department of Oi^^nJ^ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1WS Date ^ l u ^ e - \G , \<38j i i . ABSTRACT Ideally, pavement management is a process of sequential decisions on a network of pavement sections. The network is subjected to uncertainties arising from material variability, random traffic, and fluctuating environ-mental inputs. The pavement manager optimizes the whole system subject to resource constraints, and avoids suboptimization of sections. The optimiza-tion process accounts for the dynamics of the pavement system. In addition to objective data the manager seeks information from a number of experts, and considers selected social-politica1 factors and also potential imple-mentation difficulties. Nine advanced schemes that have been developed for various pavement administrations are compared to the ideal. Although the schemes employ methods capable of handling the pavement system's complexities in isolation, not one can account for all complexities simultaneously. Bayesian decision analysis with recent extensions is useful for attacking the problem at hand. The method prescribes that when a decision maker is faced with a choice in an uncertain situation, he should pick the alternative with the maximum expected utility. To illustrate the potential of Bayesian decision analysis for pavement management, the author develops a Markov decision model for the operation of one pavement section. Consequences in each stage are evaluated by multi-attribute utility. The states are built of multiple pavement variables, such as strength, texture, roughness, etc. Group opinion and network opti-mization are recommended for future research, and decision analysis sug-gested as a promising way to attack these more complex problems. This thesis emphasizes the utility part of decision analysis, while it modifies an existing approach to handle the probability part. A procedure is developed for Bayesian updating of Markov transition matrices^* i i where t h e p r i o r d i s t r i b u t i o n s a r e o f t h e b e t a c l a s s , and a r e based on s u r v e y s o f pavement c o n d i t i o n and on e n g i n e e r i n g judgement. P r e f e r e n c e s o f s i x e n g i n e e r s a r e e l i c i t e d and t e s t e d i n a s i m u l a t e d d e c i s i o n s i t u a t i o n . M u 1 t i a t t r i b u t e u t i l i t y t h e o r y i s a r e a s o n a b l e a p p r o x i -m a t i o n o f the e l i c i t e d v a l u e judgements and p r o v i d e s an e x p e d i e n t a n a l y t i c a l t o o l . The model i s programmed i n PL1 and an example problem i s a n a l y s e d by a computer. C o n c l u s i o n s d i s c u s s t h e pavement maintenance problem from the d e c i s i o n a n a l y t i c a l p e r s p e c t i v e . A r e v i s i o n i s recommended of the w i d e s p r e a d a d d i t i v e e v a l u a t i o n models from the s t a n d p o i n t o f p r i n c i p l e s f o r r a t i o n a l c h o i c e . Those a r e a s o f d e c i s i o n t h e o r y w h i c h may be o f i n t e r e s t t o t h e pavement e n g i n e e r , and t o t h e c i v i l e n g i n e e r i n g e n e r a l , a r e s u g g e s t e d f o r f u r t h e r s t u d y and m o n i t o r i n g . i v. TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENT ; i x 1. INTRODUCTION . .. 1 2. DIFFICULTIES OF PAVEMENT MANAGEMENT 2 2.1 Pavement System and O b j e c t i v e s 2 2.2 The P r o c e s s o f Making Pavement D e c i s i o n s 4 2.3 E x i s t i n g Pavement Management Schemes 6 2.4 Symptoms o f t h e Problem 10 3. PROBLEM DEFINITION 13 3.1 C h a r a c t e r i s t i c s o f t h e Pavement System 13 3.2 C a p a b i l i t i e s o f t h e E x i s t i n g Schemes 16 3.3 The Problem 19 4. PREVIEW OF THE PROPOSED APPROACH 22 4.1 S t r u c t u r e o f O b j e c t i v e s and A t t r i b u t e s 22 4.2 M u l t i a t t r i b u t e Consequences 23 4.3 Markov D e c i s i o n P r o c e s s w i t h Expected U t i 1 i t y C r i t e r i o n 2k 5. THE BASIC MODEL AND LIMITATIONS 27 5.1 The B a s i c B a y e s i a n D e c i s i o n Model 27 5.2 Unidimensiona1 U t i l i t y Theory 30 5-3 R i s k A t t i t u d e s 35 5.4 L i m i t a t i o n s 38 6. MULT I ATTRIBUTE UTILITY THEORY AND CONTENDERS 42 6.1 M u l t i p l e O b j e c t i v e s D e c i s i o n Methods 42 6.2 Approaches t o M u 1 t i a t t r i b u t e U t i l i t y 44 6.3 The D e c o m p o s i t i o n Approach 46 6.4 The M u l t i a t t r i b u t e U t i 1 i t y Model 50 7. MARKOV DECISION PROCESS FORMULATION 53 7.1 The S e q u e n t i a l D e c i s i o n Problem 53 7.2 Markov D e c i s i o n P r o c e s s 53 7.3 Optimal P o l i c i e s and Dynamic Programming 56 7.4 S e p a r a b l e , R i s k A v e r s e U t i l i t y F u n c t i o n s 59 8. APPLICATION 64 8.1 Summary of Method 64 8.2 H y p o t h e t i c a l Example Prob lem 66 8.3 P r o b a b i l i t i e s and U p d a t i n g 68 8.4 Consequences 72 8.5 E l i c i t a t i o n o f P r e f e r e n c e s 75 8.6 B e h a v i o u r o f the Model 83 9. CONCLUSIONS AND SUGGESTIONS FOR FUTURE STUDY 87 9-1 Cone 1 us i ons 87 9.2 The Pavement M a i n t e n a n c e Prob lem 89 9-3 D e c i s i o n A n a l y s i s in C i v i l E n g i n e e r i n g 92 9.4 P r o m i s i n g S tudy A r e a s i n D e c i s i o n A n a l y s i s 9^+ 9-5 F i n a l Comment 96 REFERENCES 97 APPENDIX A : FORMS FOR TESTS ON PREFERENCES 101 APPENDIX B: DERIVATION OF CONSTANTS FOR EXPONENTIAL UTILITY CURVES 104 APPENDIX C: RESULTS OF QUALITATIVE ASSESSMENT . . 107 APPENDIX D: COMPUTER PROGRAM 'BELLMAN' 120 APPENDIX E: COMPUTER PROGRAM LISTING 130 APPENDIX F: SAMPLE OUTPUT 139 LIST OF TABLES Page 1. SUMMARY OF PAVEMENT MANAGEMENT SCHEMES 7 2. CLASSIFICATION OF EXISTING MODELS 20 3. VARIATION OF TRADEOFFS AND CERTAINTY EQUIVALENTS WITH LEVELS OF COMPLEMENTARY ATTRIBUTES 78 4. EXPLANATION OF JOINT STATES 84 5. RESULTS OF FOUR COMPUTATIONAL RUNS 84 v i LIST OF FIGURES Page 1. Pavement System and i t s O b j e c t i v e s R e l a t i v e t o B r o a d e r Systems 3 2. S e l e c t i n g Optimum A l t e r n a t i v e f o r New Pavement and Subsequent O v e r l a y s 5 3. C o m p l e x i t i e s i n t h e Pavement System . 1,4 k. B a s i c Model o f B a y e s i a n D e c i s i o n A n a l y s i s 29 5. C a l c u l u s o f L o t t e r i e s 31 6. F i v e - p o i n t Assessment o f U t i l i t y F u n c t i o n 34 I. Forms o f a M o n o t o n i c a l l y I n c r e a s i n g U t i l i t y F u n c t i o n and D e c i s i o n Maker's R i s k A t t i t u d e s 37 8. G e o m e t r i c I n t e r p r e t a t i o n o f U t i l i t y Independence ^9 9- Summary o f P r o p o s e d Method °5 10. S t r u c t u r e o f O b j e c t i v e s , C o n t r o l l a b l e F a c t o r s and A t t r i b u t e s 67 I I . S k e t c h e s o f the HALF/BEST F u n c t i o n s 10.6 12. T r a d e - o f f User S a f e t y and Economy 109 13- T r a d e - o f f User S a f e t y and N o i s e 109 14. T r a d e - o f f User Economy and Time 110 15. T r a d e - o f f User Economy and Agency Cost 111 16. T r a d e - o f f Agency C o s t and A c c e s s 112 17. T r a d e - o f f User Economy and Jobs ] 13 18. T r a d e - o f f Jobs and G r a v e l 114 19. C e r t a i n t y E q u i v a l e n t s f o r S u b j e c t A 115 20. C e r t a i n t y E q u i v a l e n t s f o r S u b j e c t B 116 21. C e r t a i n t y E q u i v a l e n t s f o r S u b j e c t C 117 22. C e r t a i n t y E q u i v a l e n t s f o r S u b j e c t D 118 23. C e r t a i n t y E q u i v a l e n t s f o r S u b j e c t E 119 24. Main B l o c k s o f the Program BELLMAN 121 v i i 25. S u b p r o g r a m MARKOV 122 26. S u b p r o g r a m VON-NEUMANN 124 2 7 . S u b p r o g r a m R A I F F A 126 28. S u b p r o g r a m KEENEY 127 2 9 . S u b p r o g r a m HOWARD 128 ACKNOWLEDGEMENT I am indebted to several people in connection with t h i s t h e s i s . Dr. Francis P.D. Navin supervised t h i s t h e s i s . He improved the s t y l e of d r a f t v e r s i o n s of chapters. I had valu a b l e d i s c u s s i o n s w i t h Dr. W i l l i a m F. Caselton and Dr. Samuel 0. Russell at various stages of t h i s work. Professor Kenneth R. MacCrimmon and Dr. Shelby L. Brumelle of the Fac u l t y of Commerce and Business A d m i n i s t r a t i o n , and Professor James V. Zidek from the Mathematics Department guided me in t o d e c i s i o n theory. I wish to thank a l l these U.B.C. teachers f o r t h e i r time. Professor Richard F. Meyer and Dr. Gyorgy G. Karady of the Harvard Graduate School of Business A d m i n i s t r a t i o n c l a r i f i e d my i n t e r p r e t a t i o n of temporal u t i l i t y theory. I hope to express my si n c e r e thanks f o r t h i s help. I f e e l g r a t e f u l to the members of the Examination Committee and to the External Examiner f o r t h e i r comments and c o n s t r u c t i v e c r i t i c i s m which r e s u l t e d , h o p e f u l l y , in an improvement of the 1980 ve r s i o n of the t h e s i s . I g i v e the warmest thanks t o my w i f e , Iwona, and c h i l d r e n , Maja and Magnus, f o r moral support and forbearance. I appreciate the f i n a n c i a l support provided f o r t h i s work by the National Research Council and A l l i e d Chemical Canada Ltd. 1 CHAPTER 1 INTRODUCTION Roads and s t r e e t s p l a y a v i t a l r o l e i n a n a t i o n ' s t r a n s p o r t a t i o n s y stem, and pavements a r e an i m p o r t a n t component o f t h e s e n e t w o r k s . Manage-ment of pavements r e q u i r e s a s e t o f p r a c t i c e s t h a t can h e l p i n a c h i e v i n g the be s t v a l u e p o s s i b l e f o r t h e a v a i l a b l e r e s o u r c e s . A r e s e a r c h p i o n e e r e d by Canadian e n g i n e e r s has m a t e r i a l i z e d i n a s e t o f such p r a c t i c e s ( 1 ) . The p r o c e d u r e makes an e f f i c i e n t use o f e x i s t i n g knowledge and a l l o w s the r e -p l a c i n g o f component p a r t s as b e t t e r methods become a v a i l a b l e . ' T h i s t h e s i s i n v e s t i g a t e s whether t h e s t a t e o f t h e a r t can be advanced by a p p l y i n g new methods t h a t have r e c e n t l y emerged f o r d e c i s i o n a n a l y s i s o f complex systems. The s t u d y i s c a r r i e d out i n t h r e e s t e p s . F i r s t , t he e x i s t i n g pavement management p r o c e d u r e s a r e examined and compared w i t h an i d e a l i z e d method t o i d e n t i f y gaps t h a t may r e q u i r e b r i d g i n g . Second, a methodology t h a t i s c a p a b l e o f p r o v i d i n g t h e b r i d g e i s r e v i e w e d . T h i r d , i t i s a p p l i e d t o the o p t i m i z a t i o n o f maintenance of one pavement s e c t i o n . The p r e s e n t a t i o n f o l l o w s the same s t e p s . C h a p t e r 2 o u t l i n e s t h e pavement system and t h e e x i s t i n g management methods. The c o m p l e x i t i e s i n h e r e n t i n t h e system a r e compared t o t h e methods' c a p a b i l i t i e s i n Chapter 3 -The c h a p t e r c o n c l u d e s t h a t more r e a l i s t i c models a r e r e q u i r e d f o r d e c i s i o n making. C h a p t e r k i n t r o d u c e s t h e proposed methodology w h i c h i s p r e s e n t e d in more d e t a i l i n t h e subsequent c h a p t e r s . C h a p t e r 8 p r e s e n t s an a p p l i c a t i o n o f t h e t h e o r y t o a ^ h y p o t h e t i c a l example. The t h e s i s c l o s e s w i t h c o n c l u s i o n s and recommendations f o r f u t u r e r e s e a r c h . 2 " . CHAPTER 2 DIFFICULTIES OF PAVEMENT MANAGEMENT Many road and s t r e e t pavements t h a t were b u i I t a f t e r t h e Second World War r e a c h t h e end o f t h e d e s i g n e d s e r v i c e l i f e and now r e q u i r e u p g r a d i n g . Newer f a c i l i t i e s d e t e r i o r a t e f a s t e r under i n c r e a s i n g t r a f f i c than i n i t i a l l y p r e d i c t e d . Combined w i t h a s h o r t a g e o f p u b l i c f u n d s , t h e s e f a c t s pose a c h a l l e n g e f o r pavement management. B u i l d i n g roads w i l l no doubt c o n t i n u e but g r e a t e r e f f o r t s must be d i v e r t e d t o l o o k i n g a f t e r t h e a l r e a d y b u i l t system. The o b j e c t i v e s t h a t p r e s e n t l y g u i d e pavement management have t o a c c o u n t f o r a g r o w i n g p u b l i c c o n c e r n about the s a f e t y , t h e e n v i r o n m e n t a l , and t h e n a t u r a l r e s o u r c e e f f e c t s o f t r a n s p o r t a t i o n . The e x p e r i e n c e w h i c h a c c u -m u l a t e d from the d e s i g n o f new f a c i l i t i e s can c e r t a i n l y be u t i l i z e d t o s o l v e t h e problem t h a t now f a c e s pavement e n g i n e e r s but t h e d e c i s i o n s i t u a t i o n i s new and a f r e s h approach may be a b l e t o p r o v i d e s o l u t i o n s t h a t b e t t e r respond t o t h e p r e s e n t c i r c u m s t a n c e s . T h i s c h a p t e r p r e s e n t s p r e l i m i n a r i e s o f the s e a r c h f o r a b e t t e r mangement ap p r o a c h . The o b j e c t i v e s o f t h e pavement system a r e reviewed as t hey h e l p t o d i a g n o s e the problems i n t h e e x i s t i n g management p r o c e d u r e s . A sample o f t h e most advanced schemes f o r pavement management i s s t u d i e d from a d e c i s i o n a n a l y s i s p o i n t o f v i e w , and symptoms o f t h e problem a r e d i s c u s s e d . 2.1.Pavement System and O b j e c t i v e s Problem a r e a s i n management p r a c t i c e s can be d i a g n o s e d t h r o u g h an i n v e s t i g a t i o n o f o b j e c t i v e s t h a t g u i d e t h e agency's d e c i s i o n s . T h i s method h e l p s t o q u i c k l y r e a c h a c o n c l u s i o n t h a t o v e r v i e w s a problem. To d e f i n e pavement management o b j e c t i v e s , l e t ' s i s o l a t e a "pavement s y s t e m " - a whole w h i c h c o m p r i s e s a l l f a c t o r t h a t s h o u l d c o n c e r n a pavement manager. The pave-ment system becomes a subsystem when p l a c e d i n a h i e r a r c h y under broader systems. I f i t i s t o f i t i n t o t h e h i e r a r c h y , i t s o b j e c t i v e s must be compa-t i b l e w i t h t h o s e o f t h e h i g h e r - l e v e l systems. SYSTEMS HIERARCHY SYSTEMS OBJECTIVES THE WORLD NATIONAL COMMUNITY T PUBLIC FACILITY SYSTEMS ROADS y ROAD PAVEMENTS Maximize w e l l - b e i n g of humans (p resent and f u t u r e gene ra t i ons ) and w e l l - b e i n g o f na tu r e . Maximize peop l e ' s we l l b e i n g . Economic e f f i c i e n c y , Employment, Regional r e d i s t r i b u t i o n of income, Q u a l i t y of 1 i f e , Na t i ona l u n i t y Other? Sub jec t to r e sou r ce , s o c i a l and p o l i t i c a l c o n s t r a i n t s , o p t im i z e : Access , C a p i t a l , ope ra t i on and maintenance c o s t s , Q u a l i t y of s e r v i c e Other? Subject to r e sou r ce , s o c i a l and p o l i t i c a l c o n s t r a i n t s , o p t im i z e : S a f e t y , comfor t , exped iency , economy f o r the u s e r . C a p i t a l and maintenance cos t s o f the a d m i n i s t r a t i o n . Q u a l i t y of l i f e e f f e c t s on nonusers , Job c r e a t i o n i n unemployment p e r i o d s . Other? FIGURE 1 : PAVEMENT SUBSYSTEM AND ITS OBJECTIVES RELATIVE TO BROADER SYSTEMS F i g u r e 1 i l l u s t r a t e s t h i s i d e a . The pavement system f a l l s i n t o t h e c l a s s o f p u b l i c f a c i l i t y systems ( 2 ) . These systems a r e supposed t o meet a v a r i e t y of p u b l i c demands, economic and noneconomic. The economic o b j e c t i v e s encompass economic e f f i c i e n c y , maximum employment and r e g i o n a l r e d i s t r i b u t i o n of income ( 2 ) . The noneconomic o b j e c t i v e s i n c l u d e q u a l i t y o f w a t e r , a i r , land and b i o r e s o u r c e s , a e s t h e t i c s , p u b l i c h e a l t h , community p r e s e r v a t i o n , r e s o u r c e c o n s e r v a t i o n ( 4 ) , as w e l l as n a t i o n a l u n i t y and s e c u r i t y , and p e o p l e ' s p r e s t i g e needs ( 2 ) . Other o b j e c t i v e s may emerge i n the f u t u r e as the p u b l i c a t t i t u d e s and t h e economic r e a l i t y change. Most o f t h e o b j e c t i v e s a r e a c c o u n t e d f o r i n the r e g i o n a l p l a n n i n g but s t i l l t h e r e remains a m u l t i p l i c i t y o f v a 1 u e s f o r c o n s i d e r a t i o n by t h e pavement manager. The f o l l o w i n g l i s t i s c o m p i l e d from ( 1 ) and ( 3 ) : 1. Economic e f f i c i e n c y f o r t h e agency. 2. Adequate l o a d - c a r r y i n g c a p a c i t y . 3. L i m i t e d p h y s i c a l d e t e r i o r a t i o n due t o t r a f f i c and env i ronment. k. Economy, s a f e t y , s e r v i c e a b i l i t y f o r the u s e r . 5- Good a e s t h e t i c s . 6. L i m i t e d n o i s e and a i r p o l l u t i o n d u r i n g c o n s t r u c t i o n . 7. A c c e s s t o o t h e r major t r a n s p o r t f a c i l i t i e s , o t h e r c o m m u n i t i e s , e t c . 8. L i m i t e d d i s r u p t i o n of a d j o i n i n g land use. 9. Use o f l o c a l m a t e r i a l s and l a b o u r . 10. C o n s e r v a t i o n of a g g r e g a t e s . 11. P r e v e n t a d v e r s e changes i n t r a f f i c volumes as they a f f e c t l o c a l economy. 12. Response t o p u b l i c a t t i t u d e s and c o m p l a i n t s . S e v e r a l p o i n t s a r e e v i d e n t from t h e l i s t . I t i s dynamic as w i t n e s s e d by r e c e n t a d d i t i o n o f t h e e nergy c o n s e r v a t i o n o b j e c t i v e . ! D i f f e r e n t a d m i n i s -t r a t i o n s w i l l put d i f f e r e n t emphasis on v a r i o u s o b j e c t i v e s , a c c o r d i n g t o t h e l o c a l r e a l i t y . There i s c o n t e x t u a l o v e r l a p between some o b j e c t i v e s . I t may be v e r y d i f f i c u l t t o f i n d a common e v a l u a t o r f o r a l l o b j e c t i v e s . Many o b j e c -t i v e s a r e c o n f l i c t i n g and may r e q u i r e d i f f i c u l t t r a d e - o f f s . 2.2.The P r o c e s s o f Making Pavement D e c i s i o n s Many a g e n c i e s have e s t a b l i s h e d a s e t of p r o c e d u r e s , here c a l l e d "pavement management scheme", f o r a i d i n g d e c i s i o n s . The schemes e v o l v e from l o c a l e x p e r i e n c e and, c o n s e q u e n t l y , s i m i l a r problems a r e d e a l t w i t h by d i f f e r e n t methods. The schemes a r e d e v e l o p e d t o v a r i o u s ends: i n d i v i d u a l p r o j e c t s or n e t w o r k s ; new d e s i g n s > o r upg rad i ng o f c u e x j s t i rig f a'c.i ] i t i es ; % • highway pavement or s t r e e t s . R e g a r d l e s s the p u r p o s e - o f a scheme, options--always e x i s t and an optima 1 d e c i s i o n . i s sought. T h i s . d e c i s i o n p r o c e s s c o n s i s t s of " f o u r phases: - . ,; 1. G e n e r a t e a l t e r n a t i v e s , 2. P r e d i c t pavement c o n d i t i o n f o r each a l t e r n a t i v e , 3. E v a l u a t e consequences o f h a v i n g a pavement i n a c o n d i t i o n , k. S e l e c t an a l t e r n a t i v e w h i c h o f f e r s t h e " b e s t " consequence. PHASE OF DECISION PROCESS PROCESSING OUTPUTS Generate a l t e r n a t i v e s I P r e d i c t pavement c o n d i t i o n E v a l u a t e consequences S e l e c t b e s t a l t e r n a t i v e M a t e r i a l s and o v e r l a y t i m i n g Loads Other f a c t o r s D e t e r i o r a t i o n and performance models Judgement Economic optimum and near optima B e s t c o m b i n a t i o n o f new pavement f o l l o w e d by o v e r l a y s FIGURE 2: SELECTING OPTIMUM ALTERNATIVE FOR NEW PAVEMENT AND SUBSEQUENT OVERLAYS As an example t a k e the c h o i c e o f t h e be s t a l t e r n a t i v e among s e v e r a l p r o p o s a l s o f a new pavement and subsequent o v e r l a y s ( F i g u r e 2). Pavement c o n d i t i o n i s p r e d i c t e d by t h e d e t e r i o r a t i o n model from the m a t e r i a l s , c l i m a t e and l o a d i n p u t s . The c o n d i t i o n v a r i a b l e s a r e t r a n s f o r m e d i n t h e performance model i n t o a measure t h a t i s r e l a t e d t o u s e r c o s t s . The p r e s e n t w o r t h o f c o n s t r u c t i o n , m a intenance and s a l v a g e a r e a l s o c a l c u l a t e d . The t o t a l p r e s e n t w o r t h o f agency and u s e r c o s t s p r o v i d e a b a s i s f o r s e l e c t i o n o f t h e optimum. The optimum and the n e a r - o p t i m a a r e e v a l u a t e d by judgement t o a c c o u n t f o r c r i t e r i a o t h e r than economic, and t h e b e s t a l t e r n a t i v e i s i d e n t i f i e d . > 6. 2 . 3 . E x i s t i n g Pavement Management Schemes T a b l e 1 summarizes n i n e pavement management schemes a c c o r d i n g t o t h e f o u r phases o f d e c i s i o n - m a k i n g . I t i s e v i d e n t t h a t t h e f o u r phase p a t t e r n a p p l i e s t o a l l schemes i r r e s p e c t i v e o f the d e c i s i o n s i t u a t i o n . Phase 1 - G e n e r a t i o n o f A l t e r n a t i v e s : T h i s phase v a r i e s w i t h the purpose o f a scheme. For A, B and C ( t h e a l t e r n a t i v e s a r e a number o f p o s s i b l e t h i c k n e s s d e s i g n s combined w i t h sub-sequent o v e r l a y s . Schemes D and E c o n s i d e r maintenance o p t i o n s f o r one pavement s e c t i o n o v e r t i m e , w h i l e F, G, H and I do i t f o r a network. The more complex f o r m u l a t i o n s (D, E, F, G) have th e a l t e r n a t i v e s c o n s t r u c t e d by the o p t i m i z a t i o n model such as l i n e a r programming. Phase 2 - P r e d i c t i o n o f Pavement C o n d i t i o n : A pavement d e t e r i o r a t e s under t r a f f i c and c l i m a t i c l o a d s and i t s c o n d i t i o n i s d e s c r i b e d by unevenness, r u t t i n g , c r a c k i n g , wear, e t c . These v a r i a b l e s a r e c a l c u l a t e d from r e g r e s s i o n e q u a t i o n s (5) , a r e j u d g e d from p e r i o d i c s u r v e y s (10, 13, 1 4 ) , a r e p r e d i c t e d by Markov models (9, 11, 1 2 ) , o r a r e o b t a i n e d by means o f complex m e c h a n i s t i c f o r m u l a e (7, 8 ) . S i g n i f i c a n t p r o g r e s s has been made in t h e development o f a l l approaches but t h e e m p i r i c a l methods a r e s t i l l f a v o u r e d by t h e p r a c t i s i n g e n g i n e e r . S i m p l i c i t y , speed, r e a d i l y a v a i l a b l e i n p u t d a t a a r e the main advantages o f t h e s e methods. They can a l s o accommodate e n g i n e e r i n g judgement and t h e o r e t i c a l r e s u l t s i n ad-d i t i o n t o e x p e r i m e n t a l d a t a (9, 11, 12) - a f e a t u r e not o f f e r e d by the m e c h a n i s t i c methods. Once th e d e t e r i o r a t i o n v a r i a b l e s a r e e s t i m a t e d , o t h e r models p r e d i c t t h e measure o f performance f o r pavements. These models u s u a l l y r e l a t e pavement d e t e r i o r a t i o n v a r i a b l e s t o a s e r v i c e a b i 1 i t y i ndex (8, 9, 11, 12) a l t h o u g h some management schemes (5, 7) proceed d i r e c t l y t o t h i s phase o f a n a l y s i s from the b a s i c d e s i g n i n p u t v a r i a b l e s - pavement t h i c k n e s s o r d e f l e c t i o n , t y p e o f s u b grade, t r a f f i c , age, c l i m a t i c f a c t o r s . The f o r m u l a e a r e e m p i r i c a l , s i g n i f y i n g t h a t p e r f o r m a n c e i s d i f f i c u l t t o Pur-pose Scheme | and | Generation | Prediction of Pavement Condition Evaluation Reference | of Alternatives | Present State | Future State of Consequences A 1 (5) j Single project | construction/ | overlay | "AASHO Interim Guide" regression: PSI = f (thickness, t r a f f i c , climate, subgrade) Stochastic variation of inputs Total cost = administration cost + user cost ION B | (6, 7) J as A ' j E l a s t i c layer analysis and f i e l d deflection. AASHO and Brampton Road Tests regression: RCI = f (deflection, t r a f f i c , climate, subgrade, age) as A 'NE SECT C I (8) j as A | Viscoelastic layer analys is | AASHO Road Test regression: j PSI = f (roughness, rutting, | cracking). Stochastic | variation of inputs as A o D | (9) J Maintenance | options for one j pavement section | Subjective and objective evaluation of pavement | Two-variable state Markov | model. Bayesian updating as A Parametric user costs E 1 (11) 1 as D | as D | One-variable state | Markov Model as A F 1 (10) j Maintenance j options for net- | work scheduled j next year | as D | Monitoring and judgement of j any number of deterioration j variables Total potential gains of pavement rating over long period TORK G | (12) 1 Maintenance options| for urban pavement | network j Subjective rating | as E | Bayesian updating as A EH w 5 3 H | (13) I as F | as D j Monitoring and judgement j of PSI, deflection, cracking j and skid resistance Final Index = weighted four performance variables I | (14) 1 as F | Yearly monitoring ana 2. Maintenance cost, f l e c t i v i t y , 6. Noise judgement of: 1. Skid resistance 3. Roughness, 4. Rutting, 5. Re- Mult id imens ional quality index Abbreviations: AASHO = American Association of State Highway O f f i c i a l s PSI = Present Serviceability Index RCI = Riding Comfort Index Note: "Maintenance" stands for routine maintenance, surface treatments, overlaying and rebuilding with or without recycling. T A B L E 1: S U M M A R Y O F N I N E P A V E M E N T M A N A G E M E N T S C H E M E S -| Selection ot Optimum Alternative 1 1 | Product | User j Scheme | Objective Function | E x p l i c i t Constraints | Supplementary Constraints A j Minimize present worth j of t o t a l cost over | f i n i t e time horizon j Time between overlays, | | capital for construction, | j pavement geometry, service | Accounted for by judgement of decision maker at. higher leve l | L i s t of econo-| mically optimal| j designs on j | project basis | Texas, Florida B | as A j as A | as A | as A j Ontario C | as A | as A | as A | as A Massachusetts D j as A | I n f i n i t e time horizon | as A j as A jOptimal policy = |f (pavement (condition, Iweight of user |costs) C a l i f o r n i a E j as A j as A j as A joptimal policy = jf (pavement |condition) Washington F j Maximize total gains j of pavement rating | over network and time j Available capital, j j supplies, equipment, labour, | overhead, minimum quality | | standards 1 None [Maintenance ( p r i o r i t i e s on |network basis Texas G | Minimize user costs j over network and time j Service + agency budget | as A | as F Cit y of Waterloo H j Maximize Final Index | as A | as A | as F Utah I | Sat i s f y or exceed | quality standards | Budget, minimum pavement | | quality 1 None | as F 1 Denmark T A B L E 1: ( c o n t i n u e d ) 9. model m e c h a n i s t i c a l l y . One index i s used t o measure performance i n most c a s e s . However, t h e more comprehensive schemes employ m u l t i d i m e n s i o n a l i n d i c e s ( 9 , 10, 13, 14) w h i c h can f a c i l i t a t e d e c i s i o n making t h r o u g h " s p e c i a l i z a t i o n " o f c o s t f u n c t i o n s i n a r e a s o f pavement s a f e t y , r i d e a b i l i t y , s t r u c t u r a l adequacy, etc.:. To produce the h i s t o r y o f p e r f o r m a n c e , d e t e r i o r a t i o n v a r i a b l e s a r e g e n e r a t e d f o r each y e a r o f t h e a n a l y s i s p e r i o d and s u b s t i t u t e d i n t o t h e s e r v i c e a b i l i t y r e g r e s s i o n ( 5 , 7, 8 ) . A s i m p l e r approach employs the Markov model t o d e s c r i b e t h e "motion o f p e r f o r m a n c e index t h r o u g h t i m e " ( 9 , J 1', 12). S t i l l o t h e r s p r e f e r t o e s t i m a t e p e r f o r m a n c e h i s t o r i e s from e x i s t i n g d a t a and e n g i n e e r i n g judgement (10, 13). One system has no need f o r p r e d i c t i o n s , as d e c i s i o n s a r e made each year a f t e r c o l l e c t i o n of d e t e r i o r a t i o n d a t a i n the f i e l d ( 1 4 ) . Phase 3 ~ E v a l u a t i o n o f A l t e r n a t i v e s : Most o f t h e r e s e a r c h e r s s t u d i e d measure th e d e s i r a b i l i t y o f a p r o j e c t i n money ( 5 , 6, 8, 9, 11, 12). They m i n i m i z e the t o t a l annual c o s t w h i c h e q u a l s the c o s t t o t h e a d m i n i s t r a t i o n , p l u s the e x t r a c o s t s i n c u r r e d by the pavement u s e r s as a r e s u l t o f pavement d e t e r i o r a t i o n . The agency c o s t s a r e c a l c u l a t e d assuming u n i t m a t e r i a l and l a b o u r p r i c e s and c o n s t a n t d i s c o u n t r a t e f o r t h e a n a l y s i s p e r i o d o f 10 t o kO y e a r s . User c o s t s a r e assumed t o be a f u n c t i o n o f t r a v e l l i n g speed, w h i c h i s assumed t o depend l a r g e l y on t h e roughness o f pavement. P r o p o n e n t s o f t h e non-monetary v a l u a t i o n of a l t e r -n a t i v e s employ a f a i r l y c omprehensive l i s t o f d e t e r i o r a t i o n i n d i c e s . The model o f Lu (10) can accommodate any number o f r a t i n g i n d i c e s o f pavement c o n d i t i o n . The o b j e c t i v e f u n c t i o n i s a sum o f p o t e n t i a l l o n g - t e r m g a i n s o f pavement r a t i n g r e s u l t i n g from one m a intenance a c t i o n . I t i s maximized i n a z e r o - o n e l i n e a r programming model c a p a b l e o f o p t i m i z i n g an e x t e n s i v e road network. P h a s e k - O p t i m i z a t i o n : T h e o p t i m i z a t i o n p r o d u c e s a l i s t o f t h e l e a s t e x p e n s i v e a l t e r n a t i v e s f o r a p r o j e c t , a n d t h e a u t h o r s i n d i c a t e t h a t a d e c i s i o n m a k e r a t a h i g h e r l e v e l s h o u l d s u b j e c t i v e l y r e - e x a m i n e t h e l i s t w i t h r e s p e c t t o t e c h n i c a l , s o c i a l o r p o l i t i c a l f a c t o r s t h a t h a v e n o t b e e n q u a n t i f i e d i n t h e c o r e m o d e l s . F u r t h e r d e c i s i o n a n a l y s i s o f t h e f e a s i b l e d e s i g n s o n t h e p a v e m e n t n e t w o r k b a s i s i s r e q u i r e d t o a s c e r t a i n a t r u e o v e r a l l n e t w o r k o p t i m i z a t i o n r a t h e r t h a n p r o j e c t - b y - p r o j e c t s u b o p t i m i z a t i o n . A r e l a t i v e l y s i m p l e p r o c e d u r e i s e m p l o y e d b y p r a g m a t i c a l l y o r i e n t e d a g e n c i e s . D e t e r i o r a t i o n i n d i c e s f o r e a c h p a v e m e n t s e c t i o n a r e l i s t e d i n d e c r e a s i n g o r d e r o n a p r i o r i t y l i s t ( 1 3 ) . T h e D a n e s ( 1 4 ) h a v e d e v e l o p e d t h e s i m p l e s t m o d e l . F o r e a c h r o a d s e c t i o n t h e i n d i c e s o f s a f e t y , m a i n t e n a n c e e c o n o m y , r i d i n g c o m f o r t , r u t t i n g , r e f l e c t i v i t y a n d n o i s e a r e l i s t e d a s a m u l t i d i m e n s i o n a l i n d e x i n t h e a b o v e o r d e r o f p r i o r i t y a n d c o m p a r e d t o m u l t i -d i m e n s i o n a l q u a l i t y s t a n d a r d s , a t w h i c h m a i n t e n a n c e a c t i o n s s h o u l d b e c a r r i e d o u t . A p r i o r i t y l i s t i s p r o d u c e d , m a i n t e n a n c e c o s t s c a l c u l a t e d a n d a f t e r c o m p a r i n g t o t h e b u d g e t , a n u m b e r o f p r o j e c t s f r o m t h e t o p o f t h e l i s t s l a t e d f o r n e x t i m p l e m e n t a t i o n . 2 . h . S y m p t o m s o f t h e P r o b l e m T h e a p p r o a c h e s r e p r e s e n t e d i n T a b l e 1 h a v e a w i d e r a n g e o f p r o c e -d u r e s , m e t h o d o l o g i e s a n d m a t h e m a t i c a l m o d e l s . T h e d i v e r s i t y o f m e t h o d s m a y d e m o n s t r a t e a d i s s a t i s f a c t i o n w i t h t h e p r e s e n t s t a t e a n d a n e e d f o r a b e t t e r a p p r o a c h . I t i s n o t d i f f i c u l t t o f i n d s y m p t o m s w h i c h i n d i c a t e t h a t a p r o b l e m d o e s i n d e e d e x i s t . C o m p a r i s o n o f h y p o t h e t i c a l o b j e c t i v e s o u t l i n e d i n F i g u r e 1 t o t h e i r p r a c t i c a l c o u n t e r p a r t s r e v e a l s t h a t e c o n o m i c e f f i c i e n c y a n d r i d i n g c o m f o r t a r e o v e r e m p h a s i z e d a t t h e e x p e n s e o f o t h e r i m p o r t a n t o b j e c t i v e s . 11. The nonuser consequences such as excessive n o i s e , exhaust and v i -b r a t i o n s due to rough road surfaces and frequent maintenance operations (15), are neglected. One would hope that these o b j e c t i v e s are contained in con-s t r a i n t s but the formulation t r a n s f e r the e v a l u a t i o n task onto an u n i d e n t i -f i e d d e c i s i o n maker who uses s u b j e c t i v e methods. In the p u b l i c i n t e r e s t a l l o b j e c t i v e s should receive due c o n s i d e r a t i o n , p r e f e r a b l y by a q u a n t i t a t i v e method i f the task i s too complex f o r a human b r a i n to handle. The management schemes reviewed g e n e r a l l y f a i l to meet even incom-p l e t e o b j e c t i v e s . Several authors have recognized that s a f e t y must be included i n t o the management systems (5, 6, 13, 15) but they f a i l to s p e c i f y a t t r i b u t e s s u i t a b l e f o r measurement. Skid r e s i s t a n c e i s a popular measure, but r e f l e c t i v i t y , width of paved shoulders and d r a i n a b i l i t y a l s o have a d e f i n i t e impact on road s a f e t y under dry and wet weather c o n d i t i o n s f o r day and night (16). The m a j o r i t y of management systems use the concept of s e r v i c e a b i l i t y . Although conceived f o r measuring performance of pavements in the broad area of road users' o b j e c t i v e s (17), the o r i g i n a l idea has been d i s t o r t e d . The s e r v i c e a b i l i t y i n d i c e s that have emerged in p r a c t i c e do not include user s a f e t y and account only f o r the comfort of users. Numerous schemes e r r o -neously r e l a t e accident costs e x c l u s i v e l y to the pavement roughness which i s the basis f o r the narrowly defined pavement s e r v i c e a b i l i t y . This i s c l e a r l y a rather inadequate account as theory and experience p i n p o i n t that pavement surface texture and d r a i n a b i l i t y are the d e c i s i v e f a c t o r s f o r t r a f f i c s a f e t y . For a n a l y t i c a l convenience, too many schemes a l s o operate with s c a l a r measures such as the present s e r v i c e a b i l i t y index, where multidimensional i n d i c e s are needed to evaluate pavement performance. To sum up, in s p i t e of successful developments in measurement and e v a l u a t i o n of o b j e c t i v e s other than economic, the management schemes cannot include the f i n d i n g s because of modeling d i f f i c u l t i e s , and they s t i c k by the concepts developed f o r now i n v a l i d d e c i s i o n s i t u a t i o n s . The f o l l o w i n g chapter defines the c h a r a c t e r i s t i c s of an i d e a l i z e d d e c i s i o n a i d f o r pavement management. The ideal i s compared to the sample schemes of Table 1. These steps lead to the formulation of the t h e s i s problem in terms of the requirements f o r a bet t e r method. CHAPTER 3 PROBLEM DEFINITION 3 • 1 . C h a r a c t e r i s t i c s o f the Pavement System T h e o r e t i c a l l y , managing pavements i s a p r o c e s s o f making s e q u e n t i a l d e c i s i o n s on a m u 1 t i o b j e c t i v e network o f s e c t i o n s s u b j e c t e d t o v a r i a b l e m a t e r i a l p r o p e r t i e s , u n c e r t a i n t r a f f i c and f l u c t u a t i n g e n v i r o n m e n t a l i n p u t s . I d e a l l y , the pavement manager o p t i m i z e s the whole network and a v o i d s sub-o p t i m i z a t i o n o f s e c t i o n s . In a d d i t i o n t o o b j e c t i v e d a t a , the manager seeks i n f o r m a t i o n from a number o f e x p e r t s and c o n s i d e r s s e l e c t e d s o c i a 1 - p o l i t i c a 1 f a c t o r s and a l s o p o t e n t i a l i m p l e m e n t a t i o n d i f f i c u l t i e s . F i g u r e 3 summarizes c h a r a c t e r i s t i c s o f the pavement system. Network O p t i m i z a t i o n : O p t i m i z i n g complex systems must o f t e n be approached by d e c o m p o s i t i o n i n t o s i m p l e r subsystems t o f a c i l i t a t e a n a l y s i s . The network o f highways o r s t r e e t s i s broken down i n t o s e c t i o n s . V/hen the i n f o r m a t i o n about i n d i v i d u a l s e c t i o n s i s s y n t h e t i z e d t o a r r i v e a t a s o l u t i o n o p t i m a l f o r the network, i t must d e s c r i b e i n t e r a c t i o n s between s e c t i o n s o r the system w i l l be m i s r e p r e -s e n t e d and s u b o p t i m i z a t i o n may r e s u l t . M u l t i p l e O b j e c t i v e s : The o b j e c t i v e s a r e m u l t i p l e , c o n f l i c t i n g and change i n t i m e . They can be a d e q u a t e l y d e s c r i b e d o n l y by incommensurable a t t r i b u t e s and some a r e i n t a n -g i b l e . I t i s o f t e n d i f f i c u l t f o r the pavement manager t o i d e n t i f y the p e o p l e who g a i n o r l o s e i n a p a r t i c u l a r s i t u a t i o n , t o d e r i v e the r e l e v a n t o b j e c t i v e s and t o d e c i d e how they s h o u l d be w e i g h t e d . A common measure must be d e v i s e d f o r a l l a t t r i b u t e s so t h a t t r a d e - o f f s and e v a l u a t i o n o f outcomes can be made. The e v a l u a t i o n method must produce a s i m p l e i n d e x , f o r even one pavement s e c t i o n r e q u i r e s a l a r g e number o f a l t e r n a t i v e s t o be c o n s i d e r e d f o r d e s i g n MULTIPLE OBJECTIVES 1. Define objectives 2. Weight objectives 3 . Select attributes 4 . Trade-off attributes 5 . Aggregate 1 IMPLEMENTATION 1. Models simple to understand 2 . Established concepts 3. Summary models 4 . Data a v a i l a b i l i t y 5. Time and cost UNCERTAINTY 1. Objective 2 . Subjective 1 I GROUP OPINION 1. On technical questions 2. On s o c i a l questions 3. E l i c i t 4 . Aggregate TIME  DEPENDENCE 1. Physical inputs 2. Socia l inputs 3 . Present decision a f f e c t s future 4 . Periodic Updating I DECOMPOSITION 1. Optimum for network 2 . Preserve interactions i f decomposed F I G U R E 3 : C O M P L E X I T I E S I N T H E P A V E M E N T S Y S T E M ( B L O C K H E A D I N G S ) , I N T E R A C T I O N S ( A R R O W S ) A N D P R O P E R T I E S D E S I R E D F R O M A N I D E A L M A N A G E M E N T M E T H O D O L O G Y ( C O N T E N T S O F B L O C K S ) * a n d m a i n t e n a n c e . U n c e r t a i n t y : P a v e m e n t d e c i s i o n s a r e m a d e i n a n a t m o s p h e r e o f u n c e r t a i n t y . T h a t i s , t h e o u t c o m e r e s u l t i n g f r o m a n y a c t i o n i s s e l d o m k n o w n w i t h c e r t a i n t y d u e t o r a n d o m i n p u t s i n t o t h e p a v e m e n t s y s t e m . T r a f f i c a n d c l i m a t i c l o a d s a r e u n p r e d i c t a b l e . M a t e r i a l p r o p e r t i e s c h a n g e w i t h c o n s t r u c t i o n t e c h n i q u e , p l a c e a n d t i m e . M e c h a n i s t i c s t r u c t u r a l m o d e l s c a n n o t p r o v i d e a c c u r a t e i n f o r m a t i o n a n d c o n d i t i o n s u r v e y s a r e s u b j e c t t o e r r o r s . F u t u r e c o s t s , d i s c o u n t r a t e s a n d b u d g e t s m a y a t b e s t b e g u e s s e d . O b j e c t i v e s a r e d y n a m i c , a n d m a k e t h e a s s e s s m e n t o f f u t u r e d e s i r a b i l i t y o f a c t i o n s e x t r e m e l y d i f f i c u l t . P h i l o s o p h i c c o m p l i c a t i o n s m a y a r i s e i n m o d e l i n g b e c a u s e t h e r e a r e t w o t y p e s o f u n c e r t a i n t i e s . O n e i s a s s o c i a t e d w i t h r a n d o m i n p u t s t h a t o c c u r i n t h e s y s t e m r e p e a t e d l y a n d c a n b e a s s e s s e d b y s t a t i s t i c a l m e t h o d s . T h e o t h e r c a n n o t b e m e a s u r e d " o b j e c t i v e l y " a n d i s e x p r e s s e d b y t h e d e g r e e o f b e l i e f a n a n a l y s t h a s a b o u t i t s o c c u r r e n c e . G r o u p O p i n i o n : T h e p h y s i c a l p a v e m e n t s y s t e m r e q u i r e s t e c h n i c a l e x p e r t i s e t o o p e r a t e a n d e x i s t s i n a c o m p l e x s o c i a l - p o l i t i c a l e n v i r o n m e n t . I t f o l l o w s t h a t t h e d e c i s i o n p r o c e s s s h o u l d i d e a l l y i n v o l v e a n u m b e r o f t e c h n i c a l e x p e r t s a s w e l l a s r e p r e s e n t a t i v e s o f s o c i a l g r o u p s t h a t m a y b e a f f e c t e d b y t h e d e c i s i o n . T h e e x p e r t s h a v e d i f f e r e n t o p i n i o n s a b o u t t h e p r o b a b i l i t i e s o f o u t c o m e s w h e r e a s t h e s o c i a l g r o u p s h a v e d i f f e r e n t p r e f e r e n c e s f o r c o n s e q u e n c e s o f a c t i o n s . I n e a c h c a s e , t h e i n d i v i d u a l j u d g e m e n t s m u s t b e e l i c i t e d b y t h e a n a l y s t a n d c o m b i n e d i n t o a r e p r e s e n t a t i v e f u n c t i o n . I n s i t u a t i o n s r e q u i r i n g i n t e r p e r s o n a l c o m p a r i s o n s , c o n f l i c t s o f t e n o c c u r . E x p e r t s s t i c k b y t h e i r o p i n i o n o f t e n s o l e l y f o r p e r s o n a l r e a s o n s . S o c i a l g r o u p s e x e r t p r e s s u r e o n e a c h o t h e r i n o r d e r t o a c h i e v e t h e i r o w n o b j e c t i v e s . 16. T i m e D e p e n d e n c e : T h e i s s u e o f s e q u e n t i a l d e p e n d e n c e o f a c t i o n s i s t y p i c a l o f r e n e w a l s i t u a t i o n s a n d h a s b e e n r e c o g n i z e d i n p a v e m e n t m a i n t e n a n c e . A n a c t i o n i s e x e c u t e d p e r i o d i c a l l y o n a p a v e m e n t s e c t i o n a n d a f f e c t s t h e s e q u e l . W h e n d e c i s i o n a n a l y s i s e x t e n d s o v e r l o n g e r p e r i o d s , t h e t i m e - d e p e n d e n t f a c t o r s m u s t b e p r e d i c t e d . P r e d i c t i o n i s a s t r a i g h t f o r w a r d e x e r c i s e f o r p r o c e s s e s t h a t a r e m e a s u r a b l e a n d b e h a v e i n a s t e a d y - s t a t e f a s h i o n . T r a f f i c l o a d s , a v a i l a b l e r e s o u r c e s , m a t e r i a l p r i c e s , h o w e v e r , m a y f o l l o w r a n d o m c y c l e s a n d s o c i o - p o l i t i c a l c o m p o n e n t s o f t h e p a v e m e n t s y s t e m u s u a l l y c h a n g e u n p r e d i c t -a b l y . T h e b e s t k n o w n m e t h o d o f a c c o u n t i n g f o r t i m e v a r i a b i l i t y i n p u b l i c s y s t e m s i s t h e p e r i o d i c u p d a t i n g o f m a n a g e m e n t s c h e m e s . I m p l e m e n t a t i o n : T h e p a v e m e n t m a n a g e m e n t f r a m e w o r k s s h o u l d b e u p d a t e d p e r i o d i c a l l y n o t o n l y t o a d a p t t o t h e c h a n g e s i n p h y s i c a l a n d s o c i o - p o l i t i c a l f a c t o r s b u t a l s o t o a c c o m m o d a t e n e w f i n d i n g s i n m o d e l i n g a n d o p e r a t i o n o f t h e s y s t e m . C h a n g e s i n a n o r g a n i z a t i o n l e a d t o i m p l e m e n t a t i o n p r o b l e m s . D e p a r t m e n t s a r e r e o r g a n -i z e d a n d r e s t a f f e d , n o v e l e q u i p m e n t i s a c q u i r e d , n e w p r o c e d u r e s d r a w n . T h e e m p l o y e e s m u s t a d a p t t o t h e s e c h a n g e s b u t i t t a k e s t i m e , m o n e y a n d e f f o r t b e f o r e t h e m a n a g i n g o r g a n i z a t i o n c a n o p e r a t e t h e p a v e m e n t s y s t e m e f f e c t i v e l y . E v e n t h e " b e s t " m e t h o d o l o g y f o r p a v e m e n t m a n a g e m e n t w i l l f a l l s h o r t o f e x p e c t -a t i o n s , i f i t d o e s n o t a c c o u n t f o r t h e p h y s i c a l a n d b e h a v i o u r a l c o n s t r a i n t s i m p o s e d o n t h e o r g a n i z a t i o n . 3 . 2 . C a p a b i 1 i t i e s o f t h e E x i s t i n g S c h e m e s T h e r e i s a l a r g e g a p b e t w e e n t h e p r a c t i c a l m a n a g e m e n t s c h e m e s r e p r e -s e n t e d i n T a b l e 1 a n d t h e i d e a l m e t h o d o u t l i n e d i n F i g u r e 3-M u l t i p l e O b j e c t i v e s : T h e l a r g e s t d i s p a r i t y b e t w e e n t h e m o d e l s r e v i e w e d a n d t h a t p r o p o s e d i s i n t h e a r e a o f s y s t e m o b j e c t i v e s . M o s t s c h e m e s c o n s i d e r r i d i n g c o m f o r t a n d s t r u c t u r a l c o n d i t i o n b e c a u s e p e r f o r m a n c e m e a s u r e s f o r t h e s e o b j e c t i v e s a r e w e l l e s t a b l i s h e d a n d c o n v e r t i b l e t o m o n e y v a l u e s . O t h e r o b j e c t i v e s a r e d e a l t w i t h m a r g i n a l l y b e c a u s e t h e y c a n n o t b e t r e a t e d b y m o n e t a r y c o s t m o d e l s . W h e n e v a l u a t i o n m o d e l s d o a c c o u n t f o r t h e n o n m o n e t a r y f a c t o r s , t h e s e q u e n t i a l d e p e n d e n c e o f c o n s t r u c t i o n a n d m a i n t e n a n c e a c t i v i t i e s i s i g n o r e d a s w e l l a s u n c e r t a i n t i e s ( 1 0 , 1 3 , 1 4 ) . N o - s y s t e m a t i c p r o c e d u r e i s r e p o r t e d w h i c h c o u l d i d e n t i f y t h e s e t o f o b j e c t i v e s i m p o r t a n t f r o m t h e p o i n t s o f v i e w o f s o c i a l g r o u p s a f f e c t e d b y t h e d e c i s i o n s . T h e s t u d i e d s c h e m e s l e a v e a n i m p r e s s i o n t h a t a s e t o f o b j e c t i v e s i s a s s u m e d u n c r i t i c a l l y a s t h e p o i n t o f d e p a r t u r e . A n a l y s t s s p e n d m o s t t i m e o n d e v e l o p i n g t h e p h y s i c a l r e p r e s e n t a t i o n o f t h e s y s t e m . N e t w o r k O p t i m i z a t i o n : W i t h t w o e x c e p t i o n s ( 1 0 , 1 2 ) t h e m a n a g e m e n t s c h e m e s f i r s t o p t i m i z e i n d i v i d u a l p a v e m e n t s e c t i o n s a n d t h e n a r r a n g e t h e m i n a p r i o r i t y l i s t . C r i t e r i a f o r p l a c i n g p r o j e c t s o n t h e l i s t i n c l u d e d e c i s i o n m a k e r ' s j u d g e m e n t o n i n t a n g i b l e s i n a d d i t i o n t o d o l l a r v a l u e a n d r a t i n g o f m e a s u r a b l e a t t r i -b u t e s . I t i s h a r d t o b e l i e v e , h o w e v e r , t h a t t h e m a n a g e r i s a b l e t o p r o c e s s m e n t a l l y a l a r g e n u m b e r o f i n t a n g i b l e f a c t o r s f o r i n t e r r e l a t e d p r o j e c t s . M o r e o v e r , t h e a p p r o a c h o f s u b o p t i m i z i n g s o m e s e c t i o n s t o m a t c h b u d g e t i s l e s s r a t i o n a l t h a n o p t i m i z i n g t h e n e t w o r k , s u b j e c t t o b u d g e t c o n s t r a i n t . U n c e r t a i n t y : U n c e r t a i n t y i s a c c o u n t e d f o r b y r a n d o m i n p u t v a r i a b l e s f o r w h i c h t h e d i s t r i b u t i o n s a r e a s s u m e d t o b e k n o w n ( 5 , 8 ) . T h i s a p p r o a c h i s n o t r e a l i s t i c s i n c e s a m p l e s i z e s a r e u s u a l l y t o o s m a l l t o c o n v e r g e t o t h e s i z e s o f p o p u l a -t i o n s w h o s e d i s t r i b u t i o n s a r e t o b e e s t i m a t e d . N e i t h e r i s a d e t e r m i n i s t i c a p p r o a c h ( 7 , 1 0 , 1 3 , 1 4 ) p r a c t i c a l b e c a u s e v a r i a b l e s w i t h e x t r e m e l y n a r r o w m a r g i n s o f u n c e r t a i n t y a r e r a r e , m e c h a n i s t i c m o d e l s i n a c c u r a t e a n d s a m p l i n g s u b j e c t t o e r r o r . B a y e s i a n f o r m u l a t i o n o f t h e s y s t e m i n p u t s r e f l e c t s t h e f a c t t h a t t h e i n f o r m a t i o n a b o u t t h e s y s t e m i s i m p e r f e c t , a n d i t a l s o a l l o w s f o r a g r a d u a l i m p r o v e m e n t o f k n o w l e d g e a s n e w o b j e c t i v e i n f o r m a t i o n a n d j u d g e m e n t a r e a c q u i r e d . M a n y m a n a g e m e n t m e t h o d s m a k e u s e o f t h i s c o n c e p t (9,11,12) b u t o n l y a p p l y B a y e s i a n i n f e r e n c e p r i n c i p l e s t o t h e p e r f o r m a n c e m o d e l . C o s t s a n d b e n e f i t s , h o w e v e r , m a y b e e v e n l e s s p r e d i c t a b l e t h a n p a v e m e n t p e r f o r m a n c e . G r o u p O p i n i o n : T h e a d v a n t a g e s o f c o m b i n i n g o b j e c t i v e i n f o r m a t i o n w i t h s u b j e c t i v e j u d g e m e n t s a r e r e c o g n i z e d b y s o m e m a n a g e m e n t s c h e m e s b u t n o n e s u g g e s t s h o w t o a g g l o m e r a t e t h e d i f f e r i n g i n d i v i d u a l o p i n i o n s . T h e f a i l u r e o c c u r s n o t o n l y f o r u n c e r t a i n t e c h n i c a l i n p u t s b u t a l s o i n p r e f e r e n c e s f o r o u t c o m e s a n d s o c i a l i n t e r e s t s . T h e d e s i r a b l e m e t h o d e l i c i t s i n f o r m a t i o n i n a n a t m o s p h e r e f r e e o f i n t e r p e r s o n a l p r e s s u r e s a n d o b j e c t i v e l y a g g r e g a t e s i n d i v i d u a l o p i n i o n s . I m p 1 e m e n t a t i o n : P a v e m e n t m a n a g e m e n t r e c o g n i z e s t h e i m p l e m e n t a t i o n a n d a d a p t a t i o n p r o b l e m s . C o m p u t e r s o f t w a r e m a k e s a l l o w a n c e s f o r m o d e l a n d d a t a u p d a t i n g . S o m e f r a m e w o r k s h a v e b e e n r e o r g a n i z e d a n d c o m p u t a t i o n a l l y i m p r o v e d t w o o r t h r e e t i m e s w i t h i n t h e l a s t d e c a d e (5,8,13). T h e o b j e c t i v e s h a v e n o t b e e n r e v i s e d , a n d m a n y s c h e m e s t e n d t o o v e r s i m p l i f y w h i l e o t h e r s r e m a i n e x t r e m e l y c o m p l e x . T h e c h o i c e o f t o o s i m p l e a m o d e l c a n n o t b e d e f e n d e d i n t i m e s w h e n m a n y m e t h o d s a r e a v a i l a b l e . O n t h e o t h e r h a n d , s o m e s c h e m e s (5 • 8) e m p l o y c o m p l e x m e c h a n i s t i c f o r m u l a e t o p r e d i c t p e r f o r m a n c e v a r i a b l e s u s u a l l y f r o m l a b o r a t o r y d a t a . T h e s e m o d e l s a r e t o o i d e a l i z e d f o r a s y s t e m t h a t m o r e a p p r o p r i a t e l y r e q u i r e s s i m u l a t i o n r a t h e r t h a n m o d e l i n g . T i m e D e p e n d e n c e : S e q u e n t i a l a n a l y s i s o r t i m e d e p e n d e n c e h a s b e e n w i d e l y a c c e p t e d i n p a v e m e n t m a n a g e m e n t C5,6,8,9,11,12) b u t i s c o n f i n e d t o a c t i o n s e x p r e s s a b l e i n m o n e t a r y t e r m s . T h e m o n e t a r y v a l u e s a r e t h e n c o n v e r t e d t o a p r e s e n t w o r t h . W h e n o t h e r c o n s e q u e n c e s a r e i n c l u d e d t h e m o d e l s f a i l t o r e f l e c t t h e f a c t t h a t a n a c t i o n i s t a k e n p e r i o d i c a l l y a n d a f f e c t s t h e s e q u e l (10,13,1*0-A n a l g o r i t h m i s n e e d e d t h a t c a n e v a l u a t e m u 1 t i a t t r i b u t e o u t c o m e s o c c u r r i n g s e q u e n t i a l l y . A c r u c i a l p r o p e r t y o f t h e a l g o r i t h m i s t o r e l a t e p r o b a b l e v a l u e s g e n e r a t e d a t d i f f e r e n t d a t e s t o s o m e c o m m o n m e a s u r e . 3 - 3 - T h e P r o b l e m T h e s t u d i e d m a n a g e m e n t s c h e m e s e m p l o y m o d e l s t h a t c a n n o t h a n d l e c o m p l e x i t i e s p o s e d b y t h e p a v e m e n t s y s t e m . T h e g r e a t e s t d i f f i c u l t y a r i s e s b e c a u s e a n a 1 y s t s " c a n n o t c o m b i n e m u l t i a t t r i b u t e e v a l u a t i o n , u n c e r t a i n t y , s e q u e n t i a l d e p e n d e n c e o f a c t i o n s a n d n e t w o r k o p t i m i z a t i o n i n o n e m e t h o d . T a b l e 2 s u g g e s t s t h a t i n c l u s i o n o f a s e t o f c o m p l e x i t i e s e x c l u d e s t h e c o m p l e m e n t a r y s e t f r o m t h e m o d e l a n d m e t h o d s m a y b e g r o u p e d a c c o r d i n g l y . M e t h o d s A , C , D , E a c c o u n t f o r u n c e r t a i n t i e s a n d s e q u e n t i a l d e p e n d e n c e o f a c t i o n s b u t c a n n e i t h e r c o n s i d e r a l l o b j e c t i v e s n o r o p t i m i z e t h e n e t w o r k . M e t h o d B c a n b e a d d e d t o t h i s g r o u p i f t h e p e r f o r m a n c e m o d e l i s m a d e p r o b a -b i l i s t i c a s i n (5). T h e s e m e t h o d s m a y b e m o d i f i e d f o r n e t w o r k o p t i m i z a t i o n u s i n g a t e c h n i q u e e m p l o y e d i n (12) b u t m u l t i p l e o b j e c t i v e s s t i l l c a n n o t b e h a n d l e d . M e t h o d s H a n d I a c c o u n t f o r a l l o b j e c t i v e s b u t u n c e r t a i n t i e s , t i m e d e p e n d e n c e a n d n e t w o r k o p t i m i z a t i o n a r e i g n o r e d . M e t h o d F i s s u p e r i o r t o H a n d I w i t h r e s p e c t t o n e t w o r k o p t i m i z a t i o n b u t r a t e s l o w i n t h e o v e r a l l e v a 1 u a t i o n . A n a l y s t s s i m p l i f y t h e p a v e m e n t s y s t e m f o r a n a l y t i c a l a n d p r a c t i c a l c o n v e n i e n c e . M o d e l i n g c o n c e a l s t h e m o s t r e l e v a n t a s p e c t s a n d h a r d l y r a t i o n a l o p e r a t i o n o f t h e s y s t e m m a y r e s u l t . S i m p l i f i c a t i o n s m u s t b e a c o m p r o m i s e b e t w e e n t h e i m p l e m e n t a t i o n a d v a n t a g e s a n d t h e l o s s o f r e a l i s m t h a t m a y a f f e c t Pavement Management Scheme from Table 1 Property of Ideal Method A,C, D,E B G H,I F A l l objectives included No No No Yes Yes Optimizes a project Yes Yes Yes Yes Yes network No No Yes No Yes Accounts f( or physical state Yes No Yes No No uncertainty of costs, benefits No No No No No Time dependence of actions Yes Yes Yes No No Aggregates group opinion No No No No No C L A S S I F I C A T I O N O F E X I S T I N G M O D E L S F O R P A V E M E N T M A N A G E M E N T A C C O R D I N G T O T H E C O M P L E X I T I E S T H A T C A N B E H A N D L E D . t h e e f f e c t i v e n e s s o f r e s o u r c e a l l o c a t i o n t o p a v e m e n t m a i n t e n a n c e . T h e p r o b l e m r e q u i r e s a s e a r c h f o r a m e t h o d t h a t c a n u n i f y a l l t h e c o m p l e x i t i e s o f p a v e m e n t m a n a g e m e n t - a t a s k b r o a d i n s c o p e . T h e s c o p e o f t h i s t h e s i s , h o w e v e r , i s c o n f i n e d t o t h e c h o i c e o f o p t i m u m a l t e r n a t i v e s f o r t h e m a i n t e n a n c e a n d r e h a b i l i t a t i o n o f o n e p a v e m e n t s e c t i o n w h e n p a v e m e n t c o n d i t i o n d a t a a r e a v a i l a b l e a n d p r e f e r e n c e s o f p a v e m e n t m a n a g e r k n o w n . T h i s t y p e o f c h o i c e o c c u r s a t t h e p r o j e c t l e v e l o f p a v e m e n t m a n a g e m e n t . H o w e v e r , i n d i v i d u a l p r o j e c t s m u s t b e c o m p a r e d a t a h i g h e r l e v e l i n o r d e r t o s a t i s f y t h e o b j e c t i v e s o f t h e w h o l e p a v e m e n t n e t w o r k . N e t w o r k o p t i m i z a t i o n a n d g r o u p o p i n i o n a r e n o t a d d r e s s e d i n t h i s t h e s i s , b e c a u s e i t s o b j e c t i v e i s t o i n v e s t i g a t e t h e a p p l i c a b i l i t y o f d e c i s i o n a n a l y s i s t o t h e s i m p l e r c a s e o f o n l y o n e s e c t i o n a n d o n e d e c i s i o n m a k e r . I t i s h o p e d t h a t u n d e r s t a n d i n g o f t h e p r o b l e m f o r o n e p a v e m e n t s e c t i o n w i l l h e l p a t t a c k t h e m o r e c o m p l e x p r o b l e m b y d e c i s i o n a n a l y t i c a l m e t h o d s . C H A P T E R k P R E V I E W O F T H E P R O P O S E D A P P R O A C H M a n a g i n g p a v e m e n t s m a y b e v i e w e d a s a n o p t i m a l c o n t r o l p r o c e s s . C h o o s i n g a c t i o n s f o r t h e o p t i m a l c o n t r o l o f a s y s t e m m a y b e a c c o m p l i s h e d b y t h e B a y e s i a n d e c i s i o n m e t h o d o l o g y . T h e b a s i c m o d e l i s e x p l a i n e d i n e n g i n e e r i n g t e r m s i n ( 1 8 ) . A l t h o u g h t h i s b a s i c m o d e l s u p p l i e s a c o n v e n i e n t m e a s u r e o f t h e d e c i s i o n m a k e r ' s p r e f e r e n c e s a n d p r e s c r i b e s a c r i t e r i o n f o r c h o o s i n g t h e o p t i m a l a c t i o n i n u n c e r t a i n d e c i s i o n s i t u a t i o n s , i t m u s t b e e x t e n d e d t o h a n d l e t h e c o m p l e x i t i e s o f t h e p a v e -m e n t s y s t e m . T h e f o l l o w i n g c h a p t e r s d e s c r i b e t h e s y n t h e s i s . o f s o m e r e c e n t e x t e n s i o n s i n d e c i s i o n t h e o r y , a n d t h e M a r k o v m o d e l o f p a v e m e n t b e h a v i o u r , i n t o a n e w o p t i m i z a t i o n a p p r o a c h t o p a v e m e n t m a i n t e n a n c e . T o d e m o n s t r a t e t h e f e a s i b i l i t y o f t h e p r o p o s e d m o d e l , i t i s p r o g r a m m e d f o r t h e c o m p u t e r a n d u s e d t o a n a l y z e a h y p o t h e t i c a l p r o b l e m n u m e r i c a l l y . k.1•Structure o f O b j e c t i v e s a n d A t t r i b u t e s T o e v a l u a t e t h e a l t e r n a t i v e a c t i o n s t h e m a n a g e r m u s t f i r s t k n o w w h a t o b j e c t i v e s h e w o u l d l i k e t o a c c o m p l i s h . A n u m e r i c a l e v a l u a t o r , c a l l e d a n a t t r i b u t e , m u s t b e i d e n t i f i e d f o r e a c h o b j e c t i v e i n o r d e r t o m e a s u r e t h e e x t e n t t o w h i c h a n o b j e c t i v e i s a c h i e v e d . T h e a n a l y s t m u s t i n t e r a c t w i t h t h e d e c i s i o n m a k e r t o p r o d u c e a c o m -p r e h e n s i v e l i s t o f o b j e c t i v e s a n d a t t r i b u t e s . A n u m b e r o f s y s t e m a t i c m e t h o d s h a v e b e e n p r o p o s e d t o t h i s e n d . T h e D e l p h i m e t h o d (20) m a y b e u s e d t o e l i c i t o b j e c t i v e s a n d t h e i r i n t e r a c t i o n s f r o m s o c i a l g r o u p s t h a t w i l l b e a f f e c t e d b y a d e c i s i o n . T h e s t r u c t u r e o f o b j e c t i v e s m a y b e i d e n t i f i e d f r o m t h i s i n f o r m a t i o n b y a m e t h o d c a l l e d b y S a g e (19) " a n 23. i n t e r p r e t i v e s t r u c t u r a l m o d e l i n g " . F o r a s i m p l e r s t r u c t u r e , l i t e r a t u r e s u r v e y s a n d i n s i g r i t i n t o t h e s y s t e m s u f f i c e t o p r o d u c e a l l r e l e v a n t o b j e c t i v e s . E x p e r t s a r e a l s o u s e f u l i n g e n e r a t i n g o b j e c t i v e s b u t c a r e m u s t b e e x e r c i s e d t o e l i m i n a t e b i a s . I t i s i m p o r t a n t t o i n c l u d e o n l y t h o s e o b j e c t i v e s t h a t w i l l h a v e a s i g n i f i c a n t b e a r i n g o n t h e f i n a l s e l e c t i o n , b e c a u s e e v a l u a t i o n s b e c o m e t o o e x p e n s i v e a n d t i m e c o n s u m i n g a s t h e i r s c o p e i n c r e a s e s . O n e p r a c t i c a l t e c h n i q u e t h a t c a n h e l p t h e a n a l y s t s c r e e n p o t e n t i a l o b j e c t i v e s , w i t h o u t c o n s u m i n g a g r e a t d e a l o f e f f o r t , h a s b e e n d e s c r i b e d a n d t e s t e d (21). E a c h a t t r i b u t e m u s t c l e a r l y . c o n v e y t h e e x t e n t t h a t t h e a s s o c i a t e d o b j e c t i v e i s a c h i e v e d . A n a t t r i b u t e t o b e u s e f u l m u s t s a t i s f y t w o r e q u i r e -m e n t s . F i r s t , f o r e a c h a l t e r n a t i v e i t m u s t b e p o s s i b l e t o o b t a i n a p r o b a -b i l i t y d i s t r i b u t i o n o v e r t h e p o s s i b l e l e v e l s o f a t t r i b u t e . S e c o n d , t h e d e c i s i o n m a k e r m u s t b e a b l e t o e x p r e s s h i s u t i l i t y f u n c t i o n f o r a l l v a l u e s t h a t t h e a t t r i b u t e c a n a s s u m e . T h e r e s o u r c e s a v a i l a b l e f o r d a t a g a t h e r i n g a n d p r o c e s s i n g , t h e t i m e a n d t h e p o l i t i c a l c o n s t r a i n t s s i g n i f i c a n t l y a f f e c t t h e c h o i c e o f a t t r i b u t e s . T h e f i n a l s e t o f a t t r i b u t e s m u s t q u a n t i f y t h e c o n s e q u e n c e s o f a c t i o n s a n d e n a b l e t h e d e c i s i o n m a k e r t o c o m p a r e a l t e r n a t i v e s . T h e r e i s n o u n i v e r s a l l y a p p l i c a b l e s e t o f o b j e c t i v e s ; i t d e p e n d s o n t h e s o c i o - e c o n o m i c e n v i r o n m e n t o f t h e p a v e m e n t s y s t e m a n d c a n v a r y b e t w e e n c o u n t r i e s a n d b e t w e e n r e g i o n s i n t h e s a m e c o u n t r y . I f t h e o b j e c t i v e s a n d a t t r i b u t e s a r e n o t a d e q u a t e t h e n t h e c h o s e n a c t i o n s m a y b e n o n o p t i m a l a n d h a v e s o m e u n w a n t e d e f f e c t s t h a t w e r e n o t a c c o u n t e d f o r i n t h e e v a l u a t i o n . k.2.Mult i a t t r i b u t e C o n s e q u e n c e s U t i l i t y t h e o r y q u a n t i f i e s a d e c i s i o n m a k e r ' s j u d g e m e n t o f t h e d e s i r -a b i l i t y o f a c t i o n s i n p r o b a b i l i s t i c c h o i c e s i t u a t i o n s . A u t i l i t y f u n c t i o n a s s o c i a t e s a n u m e r i c a l i n d e x w i t h e a c h p o s s i b l e c o n s e q u e n c e . T h e i n d i c e s 2 4 . r e f l e c t t h e p r e f e r e n t i a l r a n k i n g o f t h e c o n s e q u e n c e s . U t i l i t y t h e o r y w a s o r i g i n a l l y d e v e l o p e d t o e v a l u a t e o n e - a t t r i b u t e c o n s e q u e n c e s b u t h a s b e e n r e c e n t l y e x t e n d e d t o m u l t i a t t r i b u t e o u t c o m e s ( 2 2 ) . T h e t h e o r y r e s o l v e s t h e d i f f i c u l t i e s t r a d i t i o n a l l y e n c o u n t e r e d i n t h e e v a l u a -t i o n o f i n t a n g i b l e a n d i n c o m p a r a b l e a t t r i b u t e s . T h e u t i l i t y f u n c t i o n s y n t h e -s i z e s p i e c e s o f i n f o r m a t i o n r e l e v a n t f o r o p t i m a l c h o i c e : t h e c o m p l e t e l i s t o f o b j e c t i v e s , t h e d e c i s i o n m a k e r ' s a t t i t u d e s t o w a r d r i s k a n d t h e t r a d e - o f f s b e t w e e n o b j e c t i v e s . N u m e r o u s a p p l i c a t i o n s o f t h e m u 1 t i a t t r i b u t e u t i l i t y m o d e l h a v e d e m o n -s t r a t e d t h a t i t i s a p p r o p r i a t e f o r m a n y r e a l i s t i c p r o b l e m s a n d c a n b e o p e r a t i o n a l l y v e r i f i e d i n p r a c t i c e . K e e n y a n d R a i f f a ( 2 2 ) r e p o r t o n a p p l i -c a t i o n s t o a i r p o l l u t i o n c o n t r o l , e m e r g e n c y o p e r a t i o n s , n u c l e a r p o w e r p l a n t s i t i n g , s e l e c t i o n o f c o m p u t e r s y s t e m s , c o n s u l t i n g c o m p a n y p o l i c y , a i r p o r t p l a n n i n g . L a t e r a p p l i c a t i o n s i n c l u d e s a l m o n m a n a g e m e n t ( 2 3 ) , w a t e r r e s o u r c e d e v e l o p m e n t ( 2 4 ) , s t a t e e n e r g y p o l i c y ( 2 5 ) . T h e n u m b e r o f a t t r i b u t e s c o n s i d e r e d r a n g e s f r o m f i v e t o t w e l v e . 4 . 3 . M a r k o v D e c i s i o n P r o c e s s w i t h E x p e c t e d U t i l i t y C r i t e r i o n A p a v e m e n t m a y b e v i e w e d a s a d y n a m i c s y s t e m t h a t u n d e r g o e s p r o b a b i -l i s t i c d e t e r i o r a t i o n . A t e v e r y p o i n t i n t i m e i t i s i n a n o b s e r v a b l e c o n d i t i o n t h a t c a n b e d e s c r i b e d b y a n u m b e r o f v a r i a b l e s j o i n t l y d e f i n i n g t h e s t a t e o f t h e s y s t e m . A s t h e p a v e m e n t d e t e r i o r a t e s d u e t o t r a f f i c a n d e n v i r o n m e n t a l f o r c e s , i t m a k e s t r a n s i t i o n s t o s t a t e s o f w o r s e c o n d i t i o n . T h e p a v e m e n t m a n a g e r c a n , t h r o u g h a p e r i o d i c c h o i c e o f a c t i o n , u p g r a d e t h e p a v e m e n t . A l t h o u g h h e c a n n o t t o t a l l y c o n t r o l w h i c h t r a n s i t i o n w i l l o c c u r a f t e r w a r d s , h e c a n a f f e c t i t s . p r o b a b i 1 i t y . W h e n v i e w e d i n a t e m p o r a l s e t t i n g , t h e s y s t e m u n d e r g o e s s t a t e t r a n -s i t i o n s a t e q u a l i n t e r v a l s , t e r m e d s t a g e s . A t e a c h s t a g e a n a c t i o n i s t a k e n a n d a m u 1 t i a t t r i b u t e c o n s e q u e n c e r e s u l t s . C o m p a r e d t o t h e b a s i c B a y e s i a n d e c i s i o n m o d e l t h e r e i s a s e q u e n c e o f a c t i o n s — c a l l e d p o l i c y — a n d a s s o c i a t e d o u t c o m e s r a t h e r t h a n a s i n g l e a c t i o n a n d c o n s e q u e n c e . I n a d d i t i o n t o t h e m u l t i a t t r i b u t e u t i l i t y f u n c t i o n f o r e v e r y s t a g e , t h e d e c i s i o n m a k e r m u s t d e f i n e a t e m p o r a l u t i l i t y f u n c t i o n t o e v a l u a t e e v e r y s e q u e n c e o f o u t -c o m e s w i t h i n t h e p l a n n i n g h o r i z o n . T h e m a n a g e m e n t p r o b l e m i s t o c h o o s e a n o p t i m a l p o l i c y t h a t m a x i m i z e s t h e e x p e c t e d u t i l i t y o f t h e f u t u r e s t r e a m o f o u t c o m e s . T h e s o l u t i o n m a y b e a p p r o a c h e d i n t w o w a y s . T h e f i r s t a p p r o a c h n e g l e c t s t h e t e m p o r a l p r e f e r e n c e s o f t h e d e c i s i o n m a k e r . A l l p o l i c i e s a r e r e p r e s e n t e d i n a m u l t i s t a g e d e c i s i o n t r e e . T h e c o n s e q u e n c e s i n a s t r e a m a s s o c i a t e d w i t h a p o l i c y a r e a d d e d u p a t t h e t i p o f t h e d e c i s i o n p a t h . E v e r y p a t h i n t h e d e c i s i o n t r e e i s t h e n e v a l u a t e d a n d t h e o n e w i t h m a x i m u m e x p e c t e d u t i l i t y i d e n t i f i e s t h e o p t i m a l p o l i c y . T h i s a p p r o a c h d o e s n o t a c c o u n t f o r t h e d e c i s i o n m a k e r ' s a t t i t u d e s t o w a r d r i s k s t h a t a r i s e i n s e q u e n t i a l d e c i s i o n p r o b l e m s . I t i s n o t t h e s u m o f i n d i v i d u a l c o n s e q u e n c e s b u t t h e i r t i m i n g , m a g n i t u d e a n d p r o b a b i l i t y t h a t m a t t e r i n a d y n a m i c d e c i s i o n p r o b l e m . M o r e i m p o r t a n t , t h e s o l u t i o n i s n o t y e t c o m p u t a t i o n a l l y f e a s i b l e . A s m a l l p r o b l e m i n v o l v i n g o n l y 6 s t a g e s , 5 s t a t e s a n d 2 a c t i o n s r e q u i r e s t h e e v a l u a t i o n o f m o r e t h a n o n e m i l l i o n d e c i s i o n p a t h s . T h e a l t e r n a t i v e a p p r o a c h e m p l o y s a M a r k o v d e c i s i o n m o d e l w h i c h i s c o m p u t a t i o n a l l y e f f i c i e n t i f a p p l i e d w i t h d y n a m i c p r o g r a m m i n g (26). T h e m o d e l i s u s e d e x t e n s i v e l y f o r t h e a n a l y s i s o f s e q u e n t i a l d e c i s i o n s . M a n y a u t h o r s h a v e f o u n d i t w e l l s u i t e d f o r t h e o p t i m a l c o n t r o l o f d y n a m i c s y s t e m s t h a t u n d e r g o d e t e r i o r a t i o n (27,28,29). S m i t h ( 3 0 ) i n c o r p o r a t e d t h e m o d e l i n t o a p a v e m e n t m a n a g e m e n t f r a m e w o r k . G e r c h a k a n d W a t e r s ( 3 1 ) p r o p o s e d K l e i n ' s f o r m u l a t i o n (28) f o r t h e e c o n o m i c a n a l y s i s o f r o a d m a i n t e n a n c e . 26. T h e a p p l i c a t i o n p r o p o s e d i n t h i s t h e s i s d i f f e r s f r o m p r e v i o u s w o r k . T h e c o n s e q u e n c e s — ' r e w a r d s ' i n M a r k o v p r o c e s s t e r m i n o l o g y — c o v e r a l l r e l e v a n t o b j e c t i v e s r a t h e r t h a n j u s t t h o s e e x p r e s s a b l e i n m o n e y . T h e y a r e e v a l u a t e d b y t h e u t i l i t y f u n c t i o n w h i c h c a n i n c l u d e t h e d e c i s i o n m a k e r ' s p o s i t i o n t o w a r d r i s k s p o s e d b y t h e i n t e r a c t i o n s b e t w e e n p r o b a b i l i t i e s o f s t a t e s , m a g n i t u d e s o f c o n s e q u e n c e s a n d t i m i n g o f d e c i s i o n s . T h e d y n a m i c s o f p a v e m e n t d e t e r i o r a t i o n i s a s s u m e d t o b e M a r k o v i a n . T h e s e q u e n c e o f s u c c e s s i v e s t a t e s o f t h e s y s t e m f o r m s a d i s c r e t e c h a i n w i t h t r a n s i t i o n p r o b a b i l i t i e s . T h e M a r k o v p r o p e r t y o f t h e c h a i n i m p l i e s t h a t t h e s y s t e m h a s n o m e m o r y i n t h e s e n s e t h a t o n c e i t i s i n a s t a t e , t h e p a s t e v e n t s w h i c h l e d i t t o e n t e r t h e s t a t e w i l l n o t i n f l u e n c e t h e s t a t e s i t e n t e r s i n t h e f u t u r e . H o l b r o o k (32) h a s s h o w n t h a t t h e M a r k o v a s s u m p t i o n i s a p p l i c a b l e t o t h e d e t e r i o r a t i o n o f c o n c r e t e p a v e m e n t j o i n t s . T h r e e i n d e p e n d e n t s t u d i e s h a v e d e m o n s t r a t e d t h a t M a r k o v i a n d e t e r i o r a t i o n i s r e a s o n a b l e f o r f l e x i b l e p a v e m e n t s — t h e t h e o r y , e n g i n e e r i n g j u d g e m e n t a n d e x p e r i e n c e a g r e e (30,33,3^)-D y n a m i c p r o g r a m m i n g c a n s o l v e t h e M a r k o v d e c i s i o n p r o b l e m h a v i n g t h e e x p e c t e d t e m p o r a l u t i l i t y c r i t e r i o n , p r o v i d e d t h e o b j e c t i v e f u n c t i o n h a s a p a r t i c u l a r f o r m . O b j e c t i v e f u n c t i o n s d e c o m p o s e a s r e q u i r e d b y t h e d y n a m i c p r o g r a m m i n g t e c h n i q u e i f t h e y a r e s e p a r a b l e (35). T h e s e p a r a b l e u t i l i t y f u n c t i o n s m u s t b e s c r e e n e d t o e l i m i n a t e t h o s e f o r m s w h i c h d o n o t m e e t b e h a v i o u r a l a s s u m p t i o n s d e s c r i b i n g t h e d e c i s i o n m a k e r ' s a t t i t u d e t o w a r d s r i s k i n s e q u e n t i a l c h o i c e p r o b l e m s (36). I t s e e m s n a t u r a l t h a t d e c i s i o n m a k e r s r e s p o n s i b l e f o r t h e o p e r a t i o n o f p u b l i c f a c i l i t i e s b e t e m p o r a l l y r i s k a v e r s e . O k s m a n (37) h a s d e r i v e d g e n e r a l r e s u l t s f o r t w o a p p e a l i n g f o r m s o f s e p a r a b l e , r i s k a v e r s e u t i l i t y f u n c t i o n s t h a t c a n b e u s e d i n a M a r k o v d e c i s i o n m o d e l . One o f t h e s e f u n c t i o n s i s e m p l o y e d i n t h e e x a m p l e o f C h a p t e r 8. C H A P T E R 5 T H E B A S I C M O D E L A N D L I M I T A T I O N S 5-1 - T h e B a s i c B a y e s i a n D e c i s i o n M o d e l D e c i s i o n a n a l y t i c a l m e t h o d s a r e e i t h e r d e s c r i p t i v e o r n o r m a t i v e . D e s c r i p t i v e m e t h o d s a r e u s e d i n p s y c h o l o g y t o s t u d y h u m a n b e h a v i o u r i n c o m p l e x c h o i c e s i t u a t i o n s . N o r m a t i v e m e t h o d s p r o v i d e r u l e s w h i c h w h e n f o l l o w e d b y t h e d e c i s i o n m a k e r r e m o v e i n c o n s i s t e n c i e s n e c e s s a r i l y p r e s e n t i n h u m a n d e c i s i o n s . A c c o r d i n g t o R a i f f a (38), B a y e s i a n d e c i s i o n a n a l y s i s i s c o n d i t i o n a l l y n o r m a t i v e ; i f t h e d e c i s i o n m a k e r w i s h e s t o f o l l o w c e r t a i n l o g i c a l p r i n c i p l e s o f r a t i o n a l c h o i c e , t h e n t h e p r o c e d u r e p r o v i d e s a m e a n s f o r a s s u r i n g t h a t h i s d e c i s i o n s a r e c o n s i s t e n t w i t h t h e s e p r i n c i p l e s . T h e w o r d B a y e s i a n h a s t w o i m p l i c a t i o n s . B a y e s s u g g e s t e d i n t h e 18 th c e n t u r y t h a t s u b j e c t i v e p r o b a b i l i t y j u d g e m e n t s s h o u l d b e c o m b i n e d w i t h o b j e c t i v e p r o b a b i l i t i e s c a l c u l a t e d f r o m t h e r e l a t i v e f r e q u e n c i e s . T h e s e p r i n c i p l e s a r e m o s t r e l e v a n t f o r e n g i n e e r i n g d e c i s i o n s (18), a n d h a v e b e e n a p p l i e d t o p a v e m e n t s (30). B a y e s ' p h i l o s o p h y h a s b e e n e x t e n d e d t o t h e e v a l u a t i o n p h a s e o f d e c i s i o n s . T h e d e c i s i o n m a k e r m u s t e v e n t u a l l y u s e p r e f e r e n c e s , s u b j e c t i v e b y d e f i n i t i o n , i n o r d e r t o . i d e n t i f y t h e b e s t a c t i o n f o r a s o c i o - t e c h n i c a l s y s t e m . I n p r a c t i c e , h o w e v e r , m a j o r e m p h a s i s i s o n t h e o b j e c t i v e f a c t s t h a t c a n n o t p o s s i b l y c a p t u r e a l l a s p e c t s o f d e c i s i o n s . I t i s n o t u n c o m m o n t h a t t h e d e c i s i o n m a k e r i s s u p p l i e d w i t h r e s u l t s o f t i m e - c o n s u m i n g a n d c o s t l y i n v e s t i g a t i o n s a n d h a s o n l y a l i m i t e d t i m e t o a d d t h e s u b j e c t i v e e l e m e n t s , o f t e n w i t h o u t t h e a i d o f a n y r a t i o n a l t o o l s . F o r e x a m p l e , t h e u s e r s a f e t y a n d e n v i r o n m e n t a l f a c t o r s a r e e x c l u d e d f r o m a n a l y t i c a l l y a d v a n c e d p a v e m e n t m a n a g e m e n t s c h e m e s . T h e i m b a l a n c e m a y b e c o n v e n i e n t f o r t h o s e w h o c o n c e a l v a l u e j u d g e m e n t s . I n t h e o p e r a t i o n o f e n g i n e e r i n g f a c i l i t i e s , h o w e v e r , t h e 28. j u s t i f i c a t i o n o f t e n s e e m s o u t o f p l a c e . B a y e s i a n d e c i s i o n a n a l y s i s h a s m e t a c c e p t a n c e i n a b r o a d r a n g e o f a p p l i c a t i o n s . T h e s u c c e s s c a n u n d o u b t e d l y b e a t t r i b u t e d t o : t h e r e l a t i v e f i d e l i t y w i t h w h i c h t h e r e a l w o r l d i s r e p r e -s e n t e d b y B a y e s i a n d e c i s i o n m o d e l s , f l e x i b l e d a t a r e q u i r e m e n t s a n d c o n c e p -t u a l s i m p l i c i t y o f t h e a p p r o a c h . A d e c i s i o n m o d e l i s r e p r e s e n t e d i n F i g u r e k. I t c o n s i s t s o f a s e t o f p o s s i b l e a c t i o n s , aj,...,a a n d a s e t o f p o s s i b l e s t a t e s , s ^ , . . . , s o f t h e s y s t e m u n d e r a n a l y s i s . T h e s t a t e s a r e u n c e r t a i n a n d d e s c r i b e d b y p r o b a -b i l i t y d i s t r i b u t i o n s . P r o b a b i l i t i e s c a n b e e s t i m a t e d b y e n g i n e e r i n g j u d g e -m e n t , i f n e c e s s a r y , a n d c o m b i n e d w i t h o b j e c t i v e e s t i m a t e s t h r o u g h B a y e s 1 t h e o r e m . W h e n a c t i o n a . i s t a k e n a n d t r u e s t a t e S j o c c u r s , a c o n s e q u e n c e C . . r e s u l t s . T h e d e s i r a b i l i t y o f C . f o r t h e d e c i s i o n m a k e r i s e x p r e s s e d i j y IJ b y a n u m e r i c a l m e a s u r e u „ c a l l e d u t i l i t y . T h e s y s t e m o f a c t i o n f o r k s f o l l o w e d b y t h e c h a n c e f o r k f o r m s a s t r u c t u r e c a l l e d t h e d e c i s i o n t r e e . W h e n t h e s t a t e s a r e u n c e r t a i n t h e t h e o r y p r e s c r i b e s t h e c h o i c e o f a n a c t i o n w i t h t h e h i g h e s t e x p e c t e d u t i l i t y . T h e a p p r o a c h i s s t r a i g h t f o r w a r d . T h e m a j o r t a s k o f t h e d e c i s i o n a n a l y s i s i s t o s p e c i f y p r o b a b i l i t i e s a n d u t i l i t i e s . P r o b a b i l i t y t h e o r y a l l o w s t h e m a n a g e r t o m a k e m a x i m u m u s e o f i n f o r m a t i o n a v a i l a b l e , w h i l e u t i l i t y t h e o r y g u a r a n t e e s t h a t t h e c h o i c e w i l l r e l f e c t t h e d e c i s i o n m a k e r ' s t r u e p r e f e r e n c e . T h e b a s i c m o d e l i s g e n e r a l a n d m a y b e i n t e r p r e t e d a c c o r d i n g t o t h e a n a l y t i c a l n e e d s . N o r e s t r i c t i o n i s p l a c e d o n t h e d i m e n s i o n a l i t y o f a c o n s e q u e n c e w h i c h c a n b e d e s c r i b e d b y a v e c t o r o f a t t r i b u t e s i n m u l t i -o b j e c t i v e d e c i s i o n p r o b l e m s . A c t i o n a . c a n d e n o t e a s e q u e n c e o f d e c i s i o n s o v e r a p e r i o d . S t a t e S j b e c o m e s t h e n a s e q u e n c e o f s t a t e s t h a t c a n o c c u r w i t h j o i n t p r o b a b i l i t y p . . d e c i s i o n c h a n c e E x p e c t e d u t i l i t y o f a c t i o n a . : E ( u ( a . ) ) = . Z . p . u ( C . j ) B e s t a c t i o n : a " -*->- u ( a ) = m a x E ( u ( a « ) ) i F I G U R E k: B A S I C M O D E L O F B A Y E S I A N D E C I S I O N A N A L Y S I S 5 - 2 . U n i d i m e n s i o n a l U t i l i t y T h e o r y B e f o r e d i s c u s s i n g u t i l i t y t h e o r y , t h e n o t i o n o f a l o t t e r y m u s t b e i n t r o d u c e d . A l o t t e r y i s a r i s k y o p t i o n w h i c h r e s u l t s i n r p r i z e s C j ^ , - . -r C r w i t h p r o b a b i l i t i e s p^ ,p2, ••• , P r , r e s p e c t i v e l y , w h e r e E ^ _ ^ p ^ = 1. T h e g e n e r a l l o t t e r y i s d e n o t e d L = ( p ^ C ^ , p 2^2»• • • > P r ^ r ) a n c ' s h o w n i n F i g u r e 5A. B y c o n v e n t i o n t h e p r i z e s a r e i n d e x e d f r o m t h e m o s t p r e f e r r e d ( C ^ = C ) t o t h e l e a s t p r e f e r r e d ( C r = C ° ) . T h e b r a n c h e s e m a n a t i n g f r o m a c h a n c e n o d e o f a c t i o n a . i n F i g u r e k m a y b e t h o u g h t o f a s a l o t t e r y f a c i n g t h e d e c i s i o n m a k e r w h e n a . i s c h o s e n . T h e p r i z e s o f t h i s l o t t e r y , C . j , o b t a i n w i t h p r o b a b i 1 i t i e s P j . T h e u n i d i m e n s i o n a 1 u t i l i t y t h e o r y p r e s u p p o s e s t h a t t h e l o t t e r y p r i z e s c a n b e d e s c r i b e d b y a s i n g l e a t t r i b u t e , s o t h a t n o t r a d e - o f f s a r e i n v o l v e d w i t h i n a p r i z e . T h e t h e o r y i s f o u n d e d o n s i x a s s u m p t i o n s (39). A s s u m p t i o n 1: A n i n d i v i d u a l c a n o r d e r a n y t w o o u t c o m e s a c c o r d i n g t o a p r e f e r e n c e o r i n d i f f e r e n c e r e l a t i o n . T h i s r e l a t i o n i s a l s o t r a n s i t i v e , s o t h a t i f C | > ( w h i c h i s r e a d i s i n d i f f e r e n t o r p r e f e r r e d t o a n d > C ^ , t h e n C , > C v 1 ^ 3 A s s u m p t i o n 2: W h e n d i f f e r e n t c h a n c e m e c h a n i s m s l e a d t o t w o e q u a l o u t c o m e s , t h e d e c i s i o n m a k e r w i l l b e i n d i f f e r e n t b e t w e e n t h e t w o . I t m e a n s t h a t h e f i n d s n o u t i l i t y i n g a m b l i n g . A c o m p o u n d l o t t e r y m a y b e r e d u c e d b y t h e r u l e s o f p r o b a b i l i t y a n d t h e d e c i s i o n m a k e r w i l l f i n d b o t h l o t t e r i e s e q u a l l y a t t r a c t i v e , f o r e x a m p l e , a n d i n F i g u r e 5 B . A s s u m p t i o n J,: E a c h p r i z e C . o f a l o t t e r y i s i n d i f f e r e n t t o s o c a l l e d b a s i c l o t t e r y fe. i n v o l v i n g o n l y t h e m o s t p r e f e r r e d a n d l e a s t p r e f e r r e d t p r i z e s , C 1 a n d C , 3 1 . F I G U R E 5: C A L C U L U S O F L O T T E R I E S A . G e n e r a l l o t t e r y L w i t h r p o s s i b l e p r i z e s B . A v e r a g i n g o u t b y A s s u m p t i o n 2 C . B a s i c l o t t e r y o f A s s u m p t i o n 3 D . S u b s t i t u t i o n p r i n c i p l e o f A s s u m p t i o n 4 w i t h p r o b a b i l i t i e s TT . a n d (1 - TT . ) , r e s p e c t i v e l y ( F i g u r e 5C) . A s s u m p t i o n k: I n a n y l o t t e r y , t h e p r i z e C . m a y b e r e p l a c e d b y i t s e q u i v a l e n t b a s i c l o t t e r y fe. c o n s t r u c t e d a c c o r d i n g t o A s s u m p t i o n 3 w i t h o u t a l t e r i n g t h e p r e f e r e n c e f o r t h a t l o t t e r y . S u b s t i t u t i o n o f fe. f o r C . c r e a t e s a c o m p o u n d l o t t e r y ( F i g u r e 5 D ) , w h i c h m a y b e r e d u c e d t o a s i m p l e l o t t e r y a s d e s c r i b e d i n A s s u m p t i o n 2 . A s s u m p t i o n 5: T h e p r e f e r e n c e a n d i n d i f f e r e n c e r e l a t i o n s a m o n g l o t t e r i e s a r e t r a n s i -t i v e . A f t e r C . i n L a r e r e p l a c e d b y fe. f o r a l l i , t h e l o t t e r y i s e x p r e s s e d i n t e r m s o f o n l y a n d C R - A p p l y i n g A s s u m p t i o n 2 r e d u c e s s u c h l o t t e r y t o a s i m p l e t w o - o u t c o m e l o t t e r y a n d w e g e t b y A s s u m p t i o n 5 t h a t L i s i n d i f f e r -e n t ( i n d i c a t e d b y M t o t h e t w o - o u t c o m e l o t t e r y : L = ( p 1 C 1 , . . . , p . C . , . . . , p R C R ) ^ (p 1C l ,. . .,p.fe. ,. . . ,p rC r) ^ ( p 1 C L , . . . , p . ( T T . C " , (1-Tr.)c°) , . . . , p R C R ) ( i r C * , ( l - i r ) C ° ) , w h e r e TT = p^TT^ + ^>^^2 + • • • + P ^ r ' * o F o r c o n v e n i e n c e , t h e t w o - o u t c o m e l o t t e r y i s d e n o t e d ( C , T T , C ) . A s s u m p t i o n 6: F o r a n y t w o l o t t e r i e s L = ( C " , T T , C ° ) a n d L 1 = (c" , i r 1 , C ° ) , L > L 1 i f a n d o n l y i f TT > TT 1 . W e c a n d e t e r m i n e w h i c h o f t w o l o t t e r i e s i s p r e f e r r e d b y a p p l y i n g A s s u m p t i o n s 1 t h r o u g h 5 a n d c o m p a r i n g t h e r e s u l t a n t TT a n d TT 1 . T h e t a s k o f c o m p a r i n g t w o a c t i o n s , p o s s i b l y d e s c r i b e d b y v e r y c o m p l e x b r a n c h e s i n t h e d e c i s i o n t r e e , i s t h u s r e d u c e d t o c o m p a r i n g t w o n u m b e r s . I f a c t i o n a ^ r e s u l t s i n L = ( C , T T , C ° ) a n d a c t i o n i n L ' = ( C , T r ' , C ° ) , t h e n a ^ i s p r e f e r r e d t o i f a n d o n l y i f i r i s g r e a t e r t h a n i r 1 . I t i s e x a c t l y t h e p r o p e r t y r e q u i r e d f r o m t h e u t i l i t y f u n c t i o n , h e n c e TT i s s u c h a f u n c t i o n . N o t i n g t h a t o u t c o m e s C 1 a n d C a r e e q u i v a l e n t t o (C",1.0,C°) and (c",0.0 ,C°), r e s p e c t i v e l y , i t f o l l o w s t h a t u(c") = 1.0 and u ( C ° ) = 0.0. For the i n t e r m e d i a t e v a l u e s , C . , u t i l i t y i s e l i c i t e d f rom the d e c i s i o n maker . He i s asked what p r o b a b i l i t y TT. would make the b a s i c l o t t e r y ( F i g u r e 5C) i n d i f f e r e n t to C . . T h i s s u b j e c t i v e p r o b a b i l i t y i s e q u a l t o u ( C . ) and C . i s c a l l e d the c e r t a i n t y e q u i v a l e n t o f the l o t t e r y . An a l t e r n a t i v e method might be e a s i e r f o r some d e c i s i o n m a k e r s . I ns tead o f f i x i n g C. and j u d g i n g TT . , he e s t i m a t e s the c e r t a i n t y e q u i v a l e n t o f a l o t t e r y w i t h f i x e d p r o b a b i l i t i e s ( F i g u r e 6 t o p ) . S i n c e the e x p e c t e d u t i l i t y o f t h i s l o t t e r y i s 0.5 u(C ) + 0.5 u ( C ° ) = 0.5?1 + 0.5*0 = 0.5, the e s t i m a t e d c e r t a i n t y e q u i v a l e n t has u t i l i t y 0.5 and i s denoted C ^ . N e x t , we e l i c i t the c e r t a i n t y e q u i v a l e n t s f o r two o t h e r l o t t e r i e s ( F i g u r e 6). S i n c e the e x p e c t e d u t i l i t i e s o f t h e s e l o t t e r i e s e q u a l 0.75 and 0.25 r e s p e c t i v e l y , and c o r r e s p o n d to the c e r t a i n t y e q u i v a l e n t s C ^ and C > we have f i v e p o i n t s t o f a i r the u t i l i t y f u n c t i o n t h r o u g h ( F i g u r e 6 b o t t o m ) . Once the u t i l i t y f u n c t i o n i s d e f i n e d on a l l o u t c o m e s , the e x p e c t e d u t i l i t y o f the whole l o t t e r y r e s u l t i n g f rom a c t i o n a . may be c a l c u l a t e d u ( a . ) = Z .p . IT . = £ . p . u ( C . .) . ' J J J J J U By A s s u m p t i o n 6 the b e s t a c t i o n , a , max imizes the e x p e c t e d u t i l i t y , u (a ) = max u (a . ) . i 1 Because u t i l i t y i s an i n d i c a t o r o f p r e f e r e n c e r a t h e r than an a b s o l u t e m e a s u r e , i t i s u n i q u e up to a p o s i t i v e l i n e a r t r a n s f o r m a t i o n o f the form u 1 ( a . ) = a + b u ( a . ) , b > 0. The two f u n c t i o n s u and u 1 a r e s t r a t e g i c a l l y e q u i v a l e n t and w i l l rank c o n s e q u e n c e s in i d e n t i c a l o r d e r . An a n a l o g y in p h y s i c s i s t e m p e r a t u r e measurement by d i f f e r e n t s c a l e s . V a r i o u s t e m p e r a t u r e s can be o r d e r e d f rom h i g h e s t t o lo w e s t i d e n t i c a l l y by the C e l s i u s , F a h r e n h e i t and K e l v i n s c a l e s . 5.3-Risk A t t ? tudes D e c i s i o n maker's r i s k ' a t t i t u d e s have i m p o r t a n t i m p l i c a t i o n s on the f u n c t i o n a l form o f t h e u t i l i t y f u n c t i o n . C o n s i d e r f i r s t the monotonica11y i n c r e a s i n g f u n c t i o n s . When the c e r t a i n t y e q u i v a l e n t o f any l o t t e r y i s e q u a l t o t he e x p e c t e d outcome o f t h e l o t t e r y , t h e d e c i s i o n maker i s r i s k n e u t r a l . When the c e r t a i n t y e q u i v a l e n t i s l e s s than t h e e x p e c t e d outcome, he i s s a i d t o be r i s k a v e r s e because he w i l l s e t t l e f o r l e s s than the e x p e c t e d outcome o f the l o t t e r y i n o r d e r t o a v o i d r i s k t a k i n g . The d i f f e r e n c e between the e x p e c t e d outcome and the c e r t a i n t y e q u i v -a l e n t i s c a l l e d the r i s k premium, s i n c e t h i s i s what he i s w i l l i n g t o pay i n o r d e r t o a v o i d r i s k . I f the d e c i s i o n maker's c e r t a i n t y e q u i v a l e n t exceeds the e x p e c t e d outcome, he i s r i s k prone ( r i s k s e e k i n g ) . The r i s k premium i s n e g a t i v e f o r a r i s k s e e k i n g p e r s o n , t h a t i s , the person must be p a i d t o g i v e up g a m b l i n g . E q u i v a l e n t l y , such an i n d i v i d u a l i s w i l l i n g t o pay i n o r d e r t o seek r i s k . For m onotonica1ly d e c r e a s i n g f u n c t i o n s , the r i s k premium i s d e f i n e d as the c e r t a i n t y e q u i v a l e n t minus the e x p e c t e d outcome so t h a t the s i g n o f r i s k premium i s the same as f o r i n c r e a s i n g f u n c t i o n s . An a l t e r n a t i v e d e f i n i t i o n o f r i s k a t t i t u d e s does n o t depend on whether the u t i l i t y f u n c t i o n i s monotonica1ly i n c r e a s i n g o r d e c r e a s i n g . R i s k a v e r s e i n d i v i d u a l s p r e f e r the c e r t a i n t y e q u i v a l e n t t o any l o t t e r y t h a t has an e x p e c t e d p r i z e e q u a l t o the s u r e o p t i o n . R i s k prone p e r s o n s p r e f e r the l o t t e r y o v e r the c e r t a i n t y e q u i v a l e n t and r i s k n e u t r a l i n d i v i d u a l s a r e i n d i f f e r e n t between the two o p t i o n s . T h i s d e f i n i t i o n p r o v i d e s means f o r a q u i c k check o f the shape o f u t i l i t y f u n c t i o n . R i s k a v e r s i o n i m p l i e s concave u t i l i t y f u n c t i o n s . C o n s i d e r a r i s k n e u t r a l p i e c e o f a u t i l i t y f u n c t i o n i n F i g u r e 7- The c e r t a i n t y e q u i v a l e n t C c o i n c i d e s w i t h t h e e x p e c t e d outcome, C = O .5C2 + Q . 5 C j , o f the l o t t e r y shown on the l e f t and t h e r i s k premium i s z e r o . When the r i s k premium i n c r e a s e s by moving the c e r t a i n t y e q u i v a l e n t t o p o s i t i o n C^, the f u n c t i o n becomes concave, s i n c e t he u t i l i t y o f i s the same as t h a t o f C^, and the same as t h a t o f the l o t t e r y . By moving the c e r t a i n t y e q u i v a l e n t t o Cp, one can d e m o n s t r a t e t h a t r i s k proneness i m p l i e s convex u t i l i t y f u n c t i o n s . The f a c t t h a t a p a r t i c u l a r r i s k a t t i t u d e d e t e r m i n e s the shape o f a u t i l i t y f u n c t i o n has been e x p l o i t e d t o d e v i s e an a n a l y t i c a l measure o f r i s k b e h a v i o u r . The l o c a l r i s k a v e r s i o n a t C i s d e f i n e d f o r monotonica11y i n c r e a s i n g f u n c t i o n by r(C ) = - u " ( C ) / u ' (C) (5.1) where u" and u 1 a r e the second and the f i r s t d e r i v a t i v e s o f the u t i l i t y f u n c t i o n w i t h r e s p e c t t o C. The shape o f the f u n c t i o n i s measured by u", whic h i s n e g a t i v e , z e r o o r p o s i t i v e when the u t i l i t y f u n c t i o n i s concave, s t r a i g h t l i n e o r convex, r e s p e c t i v e l y . D i v i s i o n by u" en s u r e s t h a t s t r a t e g i -c a l l y e q u i v a l e n t f u n c t i o n s have the same r i s k p r o p e r t i e s . For m o n o t o n i c a 1 l y d e c r e a s i n g f u n c t i o n s t he p o s i t i v e o f u" r a t h e r than the n e g a t i v e i s s u b s t i -t u t e d i n t o t he f o r m u l a . When r(C) i s p o s i t i v e ( n e g a t i v e , z e r o ) f o r a l l C, then u(C) i s concave (convex, s t r a i g h t l i n e ) everywhere and the d e c i s i o n maker i s r i s k a v e r s e ( p r o n e , n e u t r a l ) . When r(C) i s an i n c r e a s i n g ( c o n s t a n t , d e c r e a s i n g ) f u n c t i o n o f C, then the r i s k a t t i t u d e i s a d d i t i o n a l l y q u a l i f i e d . For example, a p o s i t i v e r d e c r e a s i n g i n C i n d i c a t e s a d e c r e a s i n g l y r i s k a v e r s e u t i l i t y f u n c t i o n . The r i s k a v e r s i o n f u n c t i o n i s thus a c o n v e n i e n t a n a l y t i c a l t o o l f o r s e l e c t i n g a u t i l i t y f u n c t i o n t h a t conforms t o the o b s e r v e d r i s k b e h a v i o u r o f the d e c i s i o n maker. u(C) FIGURE 7: FORMS OF A MONOTONICALLY INCREASING UTILITY FUNCTION AND DECISION MAKER'S RISK ATTITUDES 5 . 4 . L i m i t a t i o n s o f t h e B a s i c M o d e l T h e o p p o n e n t s o f B a y e s i a n m e t h o d o l o g y h a v e s c r u t i n i z e d t h e a s s u m p -t i o n s o f u t i l i t y t h e o r y . B e h a v i o u r a l e x p e r i m e n t s d e m o n s t r a t e d t h a t h u m a n s u b j e c t s d o n o t a l w a y s e x h i b i t t r a n s i t i v e p r e f e r e n c e s ( A s s u m p t i o n 1). S i n c e u t i l i t y t h e o r y i s n o r m a t i v e , t h e d e c i s i o n m a k e r m a y r e m o v e h i s i n c o n s i s -t e n c i e s i f h e w i s h e s t o s a t i s f y p r i n c i p l e s o f r a t i o n a l c h o i c e . S u b j e c t s u s u a l l y i m p r o v e w h e n i n t r a n s i t i v i t y i s p o i n t e d o u t t o t h e m (38). T h e w r i t e r h y p o t h e s i z e s t h a t o t h e r a s s u m p t i o n s c a n n o t b e d e f e n d e d b y u s i n g t h e n o r m a t i v e v e r s u s d e s c r i p t i v e r a t i o n a l e . T h e p r o c e d u r e o f a s s i g n i n g u t i l i t y t o a n o u t c o m e ( A s s u m p t i o n 3) i s s u p p o s e d t o c a p t u r e t h e d e c i s i o n m a k e r ' s a t t i t u d e t o w a r d t h e r i s k s t h a t h e f a c e s i n h i s d e c i s i o n s . H o w e v e r , t h e g a m b l i n g s i t u a t i o n o f t h e b a s i c l o t t e r y ( F i g u r e 5 C ) s i m u l a t e s a s i m p l e r i s k s i t u a t i o n i n v o l v i n g t w o f i x e d g r e a t l y d i v e r g e n t o u t c o m e s a n d c a n n o t p o s s i b l y c a p t u r e p r e f e r e n c e s f o r t h e a c t u a l s i t u a t i o n b e i n g a n a l y z e d . W h e n u t i l i t y i s s u b s t i t u t e d f o r o u t c o m e s ( A s s u m p t i o n k) a n d a d e c i s i o n b r a n c h i s r e d u c e d t o a s i m p l e l o t t e r y ( A s s u m p t i o n 5), A s s u m p t i o n 2 c a n n o t h o l d . D e t a i l s a b o u t t h e a c t u a l r i s k s a r e l o s t a n d t h e s i m p l e l o t t e r y c a n n o t b e r e g a r d e d e q u i v a l e n t t o t h e a c t u a l d e c i s i o n p r o b l e m . T h e h y p o t h e s i s i s s u p p o r t e d b y t h e a p p a r e n t i n c o n s i s t e n c y o f b e h a v i o u r i n t h e A l l a i s p a r a d o x d i s c u s s e d b y R a i f f a (38). T h e s u b j e c t s a r e i n c o n s i s t e n t w i t h t h e a x i o m s o f u t i l i t y t h e o r y b e c a u s e t h e y v i e w t h e t o t a l c h o i c e s i t u a t i o n w i t h i n t e r r e l a t i o n s b e t w e e n r i s k s r a t h e r t h a n d e c o m p o s e t h e a n a l y s i s i n t o i n d e p e n d e n t l o t t e r i e s . R a i f f a a p p r o a c h e s r e s o l u t i o n o f t h e A l l a i s p a r a d o x f r o m t h e n o r m a t i v e s t a n d p o i n t . H e d e m a n d s f r o m s u b j e c t s t h a t A s s u m p t i o n 2 h o l d s , w h i c h i s h u m a n l y i m p o s s i b l e . A s a n a l t e r n a t i v e , h e s u g g e s t s i n c l u d i n g t h e u t i l i t y f r o m g a m b l i n g i n t o t h e c o n s e q u e n c e s . W h i l e t h e l a s t t r i c k w i l l w o r k f o r s i m p l e d e c i s i o n t r e e s , i t w i l l n o t b e u s e f u l , i n t h e p r e s e n t w r i t e r ' s o p i n i o n , i n t h e a n a l y s i s o f c o m p l e x p r o b l e m s . I n e x t e n s i v e d e c i s i o n t r e e s i t i s d i f f i c u l t , i f n o t i m p o s s i b l e , t o r e a l i z e t h e s c a l e o f g a m b l i n g i n v o l v e d a n d t o a s s e s s t h e r i s k o f e v e r y i n d i v i d u a l p a t h . S i m i l a r o b j e c t i o n s a r e m a d e b y D r e y f u s a n d D r e y f u s (hO). T h e y c o n c l u d e t h a t d e c i s i o n a n a l y s i s ^ a p p r o p r i a t e f o r s o m e p r o b l e m s . T h e r e a r e , f o r a r i s k - a v e r s e p e r s o n , d i s u t i l i t i e s a s s o c i a t e d w i t h c h o o s i n g t o r i s k a r i s k , a c t u a l l y t a k i n g a r i s k a n d f e e l i n g g u i l t y a f t e r l o o s i n g a g a m b l e i f i t c o u l d h a v e b e e n a v o i d e d . T h e a n a l y s i s i s a p p l i c a b l e w h e r e e a c h p o s s i b l e d e c i s i o n - o u t c o m e p a t h c a n b e i n d i v i d u a l l y v a l u e d w i t h o u t c o n s i d e r a t i o n s a n d e t h i c a l j u d g e m e n t s o f t h i s k i n d . A l s o a m e n a b l e a r e t h o s e c o m p l e x p r o b l e m s w h e r e t h e e x p e r t i s e w h i c h i s b a s e d o n i n t u i t i o n a b o u t t o t a l s i t u a t i o n s r e p r e s e n t e d b y a l a r g e b r a n c h o f t h e d e c i s i o n t r e e c a n n o t b e o b t a i n e d . T h e a u t h o r s (kO) f u r t h e r s u g g e s t t h a t t h e r e m a y b e a c o n n e c t i o n b e t w e e n t h e a p p l i c a t i o n o f d e c i s i o n a n a l y s i s a s t h e m o d e l o f r a t i o n a l i t y a n d t h e t e n d e n c y t o w a r d s a p r o g r e s s i v e l y i m p e r s o n a l a p p r o a c h t o t h e c o m p l e x d e c i s i o n s n o w f a c i n g o u r s o c i e t y . T h e i r c o n c l u s i o n s e e m s p e s s i m i s t i c c o n s i d e r i n g t h a t t h e t o o l h a s n o t y e t b e e n u s e d e x t e n s i v e l y . M o r e o v e r , i t i s n o t c e r t a i n w h e t h e r a l t e r n a t i v e m e t h o d s o f f e r a d v a n t a g e s o v e r d e c i s i o n a n a l y s i s . I t i s d o u b t f u l t h a t t h e y c a n h a n d l e t h e c o m p l e x i t i e s o f t h e e m e r g i n g p r o b l e m s . D e c i s i o n a n a l y s i s a p p e a r s t h e b e s t t o o l a v a i l a b l e a n d w i l l r e m a i n s o u n t i l a b e t t e r a p p r o a c h i s d e v e l o p e d . T h e p r e c e d i n g a r g u m e n t m a y b y i t s e l f d e f e n d t h e a p p l i c a b i l i t y o f B a y e s i a n d e c i s i o n m o d e l t o t h e a n a l y s i s o f t h e p a v e m e n t s y s t e m , b u t a f u r t h e r t w o f o l l o w f r o m t h e c h a r a c t e r i s t i c f e a t u r e s o f t h i s s y s t e m . I n o n e -o f - a - t i m e d e c i s i o n p r o b l e m s — f o r e x a m p l e , w h e t h e r t o b u i l d a d a m o r n o t — t h e d i s u t i l i t y o f t h e w o r s t o u t c o m e , e v e n i f i t w e r e t o o c c u r w i t h v e r y s m a l l p r o b a b i l i t y , a f f e c t s t h e d e c i s i o n m a k e r ' s p e r c e p t i o n o f t h e r i s k s o f a l o t t e r y a n d u p s e t s A s s u m p t i o n 2. B e c a u s e p a v e m e n t d e c i s i o n s a r e r e p e a t a b l e b o t h o v e r t h e n e t w o r k a n d t i m e , m a x i m i z i n g e x p e c t e d u t i l i t y a p p e a r s t o b e a v a l i d d e c i s i o n c r i t e r i o n f o r t h e o p e r a t i o n o f p a v e m e n t s . T h e c o n s e q u e n c e s r e o c c u r a n d t h e d e c i s i o n m a k e r m a y t o l e r a t e t h e l e s s d e s i r a b l e o u t c o m e s a s l o n g a s t h e a c c u m u l a t e d e f f e c t i s n o t w o r s e t h a n t h a t o f a n i n f r e q u e n t b u t m o r e d e s i r a b l e c o n s e q u e n c e . T h e t h i r d a r g u m e n t f o l l o w s f r o m t h e c o n d i t i o n r e g a r d e d b y D r e y f u s a n d D r e y f u s (kO) n e c e s s a r y f o r a p p l i c a t i o n o f d e c i s i o n a n a l y s i s . I t i s u n l i k e l y t h a t t h e p a v e m e n t m a n a g e r c a n d e v e l o p e x p e r t i s e b a s e d o n i n t u i t i o n a b o u t t h e s e q u e n t i a l o p e r a t i o n o f t h e m u 1 t i o b j e c t i v e s y s t e m a n d , b y t h e l a c k o f c o n t e n -d e r s , B a y e s i a n d e c i s i o n a n a l y s i s i s a v a l i d t o o l f o r t h i s a n a l y s i s . T h e u n i d i m e n s i o n a l u t i l i t y m o d e l c a n n o t b e a p p l i e d i n t h e p r e s e n t f o r m t o t h e p a v e m e n t p r o b l e m . W h e n t h e o u t c o m e s a r e r e p r e s e n t e d b y a t t r i -b u t e v e c t o r s r a t h e r t h a n s i n g l e n u m b e r s , i t i s n o l o n g e r p o s s i b l e t o d e f i n e u t i l i t y f u n c t i o n b e c a u s e h u m a n s c a n n o t o r d e r c o m p l e x o u t c o m e s a s r e q u i r e d b y A s s u m p t i o n 1 . M o r e o v e r , m a i n t e n a n c e d e c i s i o n s f o r m a s e q u e n c e o f a c t i o n s t a k e n p e r i o d i c a l l y a n d t h e d e c i s i o n m a k e r i s i n t e r e s t e d i n f i n d i n g t h e " b e s t " s e q u e n c e f o r a g i v e n a n a l y s i s p e r i o d . B e c a u s e t h e t i m e d i m e n s i o n i s a d d e d t o t h e d e c i s i o n t r e e , t h e u t i l i t y f u n c t i o n m u s t e n c o d e p r e f e r e n c e s f o r t h e p r o b a b i l i s t i c o u t c o m e s d i s t r i b u t e d o v e r t h e p e r i o d . T h a t s e q u e n c e o f a c t i o n s i s o p t i m a l w h i c h m a x i m i z e s t h e e x p e c t e d u t i l i t y d e f i n e d o n t h e s e q u e n c e o f o u t c o m e s . F u r t h e r p r a g m a t i c d i f f i c u l t i e s w i t h t h e c o n c e p t o f u t i l i t y a r i s e w h e n t h e d e c i s i o n a f f e c t s s e v e r a l i n t e r e s t g r o u p s a n d t h e d e c i s i o n m a k e r m u s t t r a n s f o r m t h e i r p r e f e r e n c e s i n t o h i s o w n . T h i s p r o b l e m i s d i s c u s s e d b y B e n j a m i n a n d C o r n e l l ( 1 8 ) i n t h e c o n t e x t o f p r o f e s s i o n a l d e c i s i o n s b y c i v i l e n g i n e e r s . I n p u b l i c f a c i l i t y s y s t e m s , s u c h a s t h e p a v e m e n t s y s t e m , s o m e m e c h a n i s m s m u s t b e p r o v i d e d , w h i c h w i l l a s c e r t a i n r e p r e s e n t a t i o n o f u t i l i t y f u n c t i o n s o f a l l i n t e r e s t g r o u p s i n t h e d e c i s i o n m a k i n g p r o c e s s . T h e s e c o m p l i c a t i o n s h a v e b e e n r e s o l v e d b y t h e o r e t i c i a n s . T h e e x t e n s i o n s r e q u i r e a d d i t i o n a l a s s u m p t i o n s a b o u t t h e s y s t e m a n d i t i s m a n d a -t o r y t o c h e c k t h e m b e f o r e t h e e x t e n d e d m o d e l i s a p p l i e d . T h e c h a p t e r s t h a t f o l l o w e x p l o r e t h o s e a p p r o a c h e s t o m u l t l a t t r i b u t e c o n s e q u e n c e s a n d t o s e q u e n t i a l p r o b l e m s w h i c h a r e b o t h c o m p a t i b l e w i t h t h e B a y e s i a n d e c i s i o n s c h e m e . G r o u p o p i n i o n c a n b e a c c o m m o d a t e d b y s u i t a b l e m o d e l s , a n d i s n o t c o n s i d e r e d . CHAPTER 6 MULT I ATTRIBUTE UTILITY THEORY AND CONTENDERS 6 . 1 . 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 Methods Of m u l t i p l e o b j e c t i v e s , u n c e r t a i n t y > and time dependence i n d e c i s i o n a n a l y t i c a l problems the f i r s t c o m p l e x i t y has h i s t o r i c a l l y a b s o r b e d most a t t e n t i o n by t h e o r e t i c i a n s p r o b a b l y because i t i s t h e s i m p l e s t . I n t e r -a c t i o n between th e f i r s t two c o m p l e x i t i e s can be e f f e c t i v e l y h a n d l e d o n l y by the m u 1 t i a t t r i b u t e u t i l i t y t h e o r y , w h i l e some s p e c i a l i z a t i o n s o f t h i s t h e o r y can a l s o a t t a c k problems h a v i n g a l l t h r e e c o m p l e x i t i e s and i n t e r -a c t i o n s . For an a n a l y s t who .wishes to e x p l i c i t l y p r e s e r v e t h e s e complex-i t i e s i n t h e s t r u c t u r e o f a d e c i s i o n a n a l y t i c a l model, m u l t i a t t r i b u t e u t i l i t y a p p l i e d i n the B a y e s i a n d e c i s i o n a n a l y s i s has no c o n t e n d e r s . The o t h e r models a r e w o r t h m e n t i o n i n g , though, f o r i t i s o f t e n p o s s i b l e t o s t r u c t u r e a problem so t h a t i t i s amenable t o the s i m p l e r methods. D e c i s i o n a n a l y t i c a l metholodogy may be c l a s s i f i e d as n o r m a t i v e o r d e s c r i p t i v e . N o r m a t i v e a n a l y s i s e x p l a i n s what r u l e s a d e c i s i o n maker s h o u l d f o l l o w i n o r d e r t o improve h i s d e c i s i o n s . N o r m a t i v e methods, such as the B a y e s i a n d e c i s i o n t h e o r y , h e l p t h e d e c i s i o n maker remove i n c o n s i s t e n c i e s from the d e c i s i o n making p r o c e s s . D e s c r i p t i v e a n a l y s i s e x p l a i n s how a c t u a l l y d e c i s i o n s a r e made by i n d i v i d u a l s who a r e not a i d e d by t h e n o r m a t i v e methods. D e s c r i p t i v e methods a r e o f i n t e r e s t m a i n l y t o the p s y c h o l o g i s t s who a t t e m p t t o p r e d i c t human b e h a v i o u r i n complex c h o i c e s i t u a t i o n s . R e search i n e v a l u a t i o n o f m u 1 t i o b j e c t i v e c h o i c e s has been growing e x p o n e n t i a l l y i n the l a s t twenty y e a r s and a l a r g e number o f methods r e s u l t e d . Many a u t h o r s have a t t e m p t e d t o o v e r v i e w the f i e l d . MacCrimmon's (41) o v e r -v i e w seems most c o m p r e h e n s i v e , a l t h o u g h the m u 1 t i a t t r i b u t e u t i l i t y t h e o r y i s u n d e r r e p r e s e n t e d . Only some o f what he l a b e l s " W e i g h t i n g Methods" a r e c o m p a t i b l e w i t h t h e B a y e s i a n d e c i s i o n a n a l y s i s . " S e q u e n t i a l E l i m i n a t i o n M e t h o d s " s u c h a s e l i m i n a t i o n b y a s p e c t s , d o m i n a n c e a n d c o m p a r i s o n t o s t a n d -a r d s a r e o f t e n u s e d f o r f i l t e r i n g a d m i s s i b l e a l t e r n a t i v e s i n t h e B a y e s i a n a n a l y s i s . " M a t h e m a t i c a l P r o g r a m m i n g M e t h o d s " t h a t r e q u i r e d e c i s i o n m a k e r ' s i n t e r a c t i o n w i t h t h e p r o g r a m o f f e r f e a t u r e s d e e m e d i n t e r e s t i n g f o r t h e n o r m a t i v e a s s e s s m e n t o f c o m p l e x u t i l i t i e s . D e s c r i p t i v e m e t h o d s i n u t i l i t y t h e o r y i n f e r p r e f e r e n c e s o f d e c i s i o n m a k e r s f r o m p a s t d e c i s i o n s i t u a t i o n s o r b y d i r e c t q u e s t i o n i n g . T h e s e m e t h o d s a r e b a s e d o n m u l t i p l e r e g r e s s i o n t e c h n i q u e s a n d r e l a t e i n t u i t i v e j u d g e m e n t s o f m u 1 t i a t t r i b u t e a l t e r n a t i v e s t o v a l u e s o f t h e a t t r i b u t e s . T h e s e m e t h o d s r e q u i r e t h a t t h e d e c i s i o n m a k e r e v a l u a t e c o m p l e x s i t u a t i o n s r e p e t i t i v e l y . R e g r e s s i o n m o d e l s t o b e m e a n i n g f u l s h o u l d b e b a s e d o n a l a r g e n u m b e r o f o b s e r v a t i o n s . P s y c h o l o g i s t s h a v e f o u n d t h a t h u m a n a b i l i t y t o p e r f o r m r e p e t i t i v e m e n t a l t a s k s w i t h o u t e r r o r i s l i m i t e d a n d j u d g e m e n t o f s i t u a t i o n s i n v o l v i n g m o r e t h a n f i v e o r s i x v a r i a b l e s a t a t i m e i s s u b j e c t t o r a n d o m e r r o r s . F o r a r e a s o n a b l e n u m b e r o f a t t r i b u t e s , h o w e v e r , d e s c r i p t i v e u t i l i t y t e c h n i q u e s c o u l d b e u s e d a s a n i n t e r m e d i a t e s t e p i n t h e a s s e s s m e n t o f n o r m a t i v e u t i l i t i e s . I n t e r a c t i v e m u I t i o b j e c t i v e p r o g r a m m i n g m e t h o d s a r e a l s o i n t e r e s t i n g f o r t h e n o r m a t i v e e v a l u a t i o n o f u t i l i t i e s . T h e s e m e t h o d s d o n o t r e q u i r e e x p l i c i t k n o w l e d g e o f t h e d e c i s i o n m a k e r ' s u t i l i t y f u n c t i o n s , b u t u s e s i t o n a n i n t e r a c t i v e b a s i s b y a s k i n g c e r t a i n q u e s t i o n s o f h i m . T h e f a c t t h a t t h e s e m e t h o d s . a c t i v e l y i n v o l v e t h e d e c i s i o n m a k e r m a y f a c i l i t a t e t h e i r i m p l e m e n t a t i o n a n d a c c e p t a n c e o f t h e s o l u t i o n s . T h e p r o c e s s " t e a c h e s " t h e d e c i s i o n m a k e r t o r e c o g n i z e w h a t h e c o n s i d e r s a s g o o d s o l u t i o n s a n d i m p o r t a n t o b j e c t i v e s . T o t e a c h t h e d e c i s i o n m a k e r , a n u m b e r o f c a l c u l a t i o n a n d d e c i s i o n - m a k i n g c y c l e s a r e r e q u i r e d . D u r i n g t h e d e c i s i o n - m a k i n g p h a s e o f e a c h c y c l e , t h e d e c i s i o n m a k e r e x a m i n e s the r e s u l t s o f t h e c a l c u l a t i o n phase and d e v e l o p s new i n s i g h t s and i n f o r m a -t i o n about h i s o b j e c t i v e s . T h i s i n f o r m a t i o n i n t u r n i s used i n the c a l c u l a t i o n phase o f the next c y c l e , t h e r e b y p r o v i d i n g a g u i d e f o r t h e s e a r c h o f the b e s t compromise. The i n t e r a c t i v e methods have t h e s e b e h a v i o u r a l a s s u m p t i o n s i n common: 1 . An i n d i v i d u a l cannot e x p r e s s h i s p r e f e r e n c e s a n a l y t i c a l l y . R a t h e r , they w i l l i m p l i c i t l y g u i d e h i s s e a r c h f o r the " b e s t " s o l u t i o n . 2. The d e c i s i o n maker f i r s t c o n c e n t r a t e s on what he f e e l s a r e the more s i g n i f i c a n t a s p e c t s o f the a l t e r n a t i v e s and then proceeds t o t h e l e s s i m p o r t a n t . 3 . E x p l o r a t i o n o f the f e a s i b l e a l t e r n a t i v e s i s a l e a r n i n g p r o c e s s and the i n f o r m a t i o n so g e n e r a t e d f e e d s back and changes t h e ' d e c i s i o n maker's p r e f e r e n c e s . I f t h e s e a s s u m p t i o n s h o l d , the assessment o f m u 1 t i a t t r i b u t e u t i l i t i e s w i l l p o s s i b l y b e n e f i t from a d o p t i n g t h e f e e d b a c k f e a t u r e o f the i n t e r a c t i v e m u 1 t i o b j e c t i v e programming. 6.2.Approaches t o M u 1 t i a t t r i b u t e U t i l i t y L e t X = X1"X„"...--X be an outcome s p a c e , where X. i s the i th a t t r i b u t e . A s p e c i f i c outcome i s d e s i g n a t e d by x o r (x.. ,x„, . . . ,x ) , I C. p where x. d e s i g n a t e s a s p e c i f i c amount o f X. f o r i = 1 , 2 , . . . , p . The symbol x = (x^ ,X£ ,..•» Xp") d e s i g n a t e s the most d e s i r a b l e outcome and x° = ( x ^ ° , X 2 ° , . . . , X p ° ) d e s i g n a t e s the l e a s t d e s i r a b l e . We a r e i n t e r e s t e d i n a s s e s s i n g the u t i l i t y f u n c t i o n o v e r X, denoted u ( x j , X 2 » . . . , x ) o r u ( x ) . We might have the f o l l o w i n g a t t r i b u t e v e c t o r i n a pavement d e c i s i o n problem r e s u l t i n g from a c t i o n a. and s t a t e s.: xC1 , j ) = ($ c o s t ( i . , j ) , c om f o r t s c o r e o f u s e r s person-days o f maintenance employment ( i , j ) ) E x p e c t a t i o n o f m u l t i a t t r i b u t e u t i l i t y t a ken w i t h r e s p e c t t o p r o b a b i l i t i e s o f s t a t e s E ( u ( x ( i , j ) ) ) = Z j P j u ( x ( i , j ) ) (6.1) p r o v i d e s a n - a p p r o p r i a t e c r i t e r i o n f o r c h o o s i n g between a l t e r n a t i v e a c t i o n s . A c t i o n a. i s a t l e a s t as d e s i r a b l e as a. i f and o n l y i f i k E ( u ( x ( i , j ) ) ) > E ( u ( x ( . k , j ) ) ) (6.2) The u t i l i t y f u n c t i o n i s assumed t o be m o n o t o n i c a 1 l y i n c r e a s i n g i n each Xj and bounded. U t i l i t i e s o f the extreme outcomes a r e c o n v e n t i o n a l l y s e t p r e s e r v e s the p r o p e r t i e s o f u t i l i t y f u n c t i o n . A number o f approaches have been d e v e l o p e d t o c a l c u l a t e m u l t i a t t r i b u t e u t i l i t i e s ( 4 2 ) . I f the s e t o f p o s s i b l e outcomes i s s m a l l and a t t r i b u t e s o n l y a few, i t may be r e a s o n a b l e t o a s s i g n a u t i l i t y t o each o f t h e s e d i r e c t l y . The p r o c e d u r e i s s i m i l a r t o t h a t used i n e l i c i t a t i o n o f u n i d i m e n s i o n a l u t i l i t y ( F i g u r e 5C) . For each x a p r o b a b i l i t y TT i s a s s e s s e d such t h a t the outcome i s i n d i f f e r e n t t o the b a s i c l o t t e r y y i e l d i n g e i t h e r x w i t h p r o b a b i l -i t y IT o r x° w i t h p r o b a b i l i t y (1 - IT). The d i r e c t approach cannot be a p p l i e d t o l a r g e p r o b l e m s . I n d i r e c t approaches reduce the d i m e n s i o n o f t h e a t t r i b u t e v e c t o r t h r o u g h t r a n s f o r m a t i o n . One method f i r s t t r a d e s a l l a t t r i b u t e s o f f i n t o one and then a s c r i b e s a u t i l i t y f u n c t i o n t o t h i s s i n g l e a t t r i b u t e . A p p l i c a t i o n o f t h i s p r o c e d u r e i s l i m i t e d s i n c e i n r e a l - w o r l d systems not a l l a t t r i b u t e s a r e c o n v e r t i b l e i n t o one. u(x°) = 0 and u (x ) = 1 but any p o s i t i v e l i n e a r t r a n s f o r m a t i o n u 1 (x) = a + b u (x) , b > 0 (6.3) A n o t h e r method t r a n s f o r m s r i s k l e s s a d d i t i v e u t i l i t y f u n c t i o n s i n t o r i s k y u t i l i t y f u n c t i o n s . To c a l c u l a t e the r i s k l e s s f u n c t i o n , s c o r e s a r e a s s i g n e d to each a t t r i b u t e and summed up f o r each ou tcome. T h i s type o f u t i l i t y f u n c t i o n i s used in t r a n s p o r t a t i o n f o r e v a l u a t i o n o f a l t e r n a t i v e p l a n s by the a d d i t i v e r a t i n g s c a l e p r o c e d u r e s . To c o n v e r t the r i s k l e s s u t i l i t y f u n c t i o n to a form s u i t a b l e f o r use in p r o b a b i l i s t i c c h o i c e s , s e v e r a l outcomes a r e s e l e c t e d w h i c h c o v e r the f u l l range o f the s c o r e s and "which a r e e a s y to compare w i t h the two ext reme outcomes x and x . U t i l i -t i e s a r e nex t a s s i g n e d to t h e s e s e l e c t e d outcomes by the d i r e c t method and p l o t t e d a g a i n s t the r a t i n g s c o r e s f o r the s e l e c t e d o u t c o m e s . The u t i l i t y f u n c t i o n i s thus d e f i n e d on a s i n g l e a t t r i b u t e " r a t i n g s c o r e " . The main drawback o f t h i s method i s t h a t i t does r e q u i r e the d i r e c t a s s e s s m e n t s o f u t i l i t y o f s e v e r a l m u1 t i a t t r i b u t e o u t c o m e s . When the number o f a t t r i b u t e s i s l a r g e , t h e s e judgements p l a c e a v e r y heavy load on the d e c i s i o n maker . He must s i m u l t a n e o u s l y c o n s i d e r two p r o b a b i l i t i e s and a l l o f the a t t r i b u t e s o f t h r e e o u t c o m e s : x , x and x ° . Sometimes i t may be p o s s i b l e to reduce the d i m e n s i o n o f x by t r a d i n g o f f some o f the a t t r i b u t e s . Imp lementa t ion may be r e l a t i v e l y e a s y because the a d d i t i v e r a t i n g s c a l e t e c h n i q u e i s w e l l e s t a b l i s h e d in t r a n s p o r t a t i o n p l a n n i n g . 6.3-The D e c o m p o s i t i o n A p p r o a c h O t h e r i n d i r e c t a p p r o a c h e s b r e a k down the m u1 t i a t t r i b u t e a s s e s s m e n t p r o c e d u r e i n t o p a r t s . We would l i k e to f i n d a s c a l a r v a l u e d f u n c t i o n o f a s i m p l e form such t h a t u(x) = u ( x 1 , x 2 , . . . , x p ) = f (u.j (x j ) , u 2 ( x 2 ) , . . . . , y p ( x )) where u . i s a u t i l i t y f u n c t i o n o f a t t r i b u t e x . , f o r i = 1,2 p . If t h i s d e c o m p o s i t i o n i s p o s s i b l e the a s s e s s m e n t o f the p - d i m e n s i o n a l u t i l i t y f u n c t i o n then r e s o l v e s i n t o an a s s e s s m e n t o f p un id imens iona1 u t i l i t y f u n c t i o n s . The t h e o r e t i c i a n s have f o c u s s e d on the i d e n t i f i c a t i o n o f a s s u m p t i o n s w h i c h p e r m i t one t o f i n d such a s i m p l e form o f the m u l t i -a t t r i b u t e u t i l i t y f u n c t i o n . The a s s u m p t i o n s c o n c e r n t h e c o n c e p t s o f u t i l i t y independence and p r e f e r e n t i a l independence ( 2 2 ) . T h e i r r o l e i n m u l t i a t t r i b u t e u t i l i t y t h e o r y i s s i m i l a r t o t h a t o f s t a t i s t i c independence i n m u l t i v a r i a t e p r o b a b i l i t y t h e o r y . The more independence c o n d i t i o n s e x i s t among a t t r i b u t e s , the s i m p l e r the f u n c t i o n f and c o n s e q u e n t l y the e a s i e r the assessment. A t t r i b u t e v e c t o r Y, where Y <^  X , i s u t i l i t y independent o f i t s complement Y ( s h o r t l y : Y i s u t i l i t y independent) i f the c o n d i t i o n a l p r e f e r e n c e o r d e r f o r l o t t e r i e s i n v o l v i n g o n l y changes i n t h e l e v e l s o f a t t r i b u t e s i n Y does not depend on the l e v e l s a t w h i c h the a t t r i b u t e s i n Y a r e h e l d f i x e d . L e t X = ( X ^ X ^ . X ^ ) and Y = (X,) and c o n s i d e r the f o l l o w i n g l o t t e r y : where a t t r i b u t e X 1 has l e v e l x' i n one p r i z e and x'J i n the o t h e r , but the Suppose the d e c i s i o n maker f e e l s t h a t h i s c e r t a i n t y e q u i v a l e n t f o r the above l o t t e r y i s Xy I f the l e v e l s o f and X^ a r e n e x t changed and the d e c i s i o n maker s t i l l a s s i g n s x^ t o the new l o t t e r y , h i s p r e f e r e n c e o r d e r f o r the l o t t e r i e s does not depend on the l e v e l s o f and X^. I f t h i s i s v a l i d f o r any p a i r o f l e v e l s o f X^, t h e c o n d i t i o n a l u t i l i t y f u n c t i o n o v e r X p g i v e n t h a t X^ i s f i x e d a t any v a l u e , w i l l be a p o s i t i v e l i n e a r t r a n s f o r m a t i o n of the c o n d i t i o n a l u t i l i t y f u n c t i o n o v e r X 1 , g i v e n t h a t Xj i s f i x e d a t any o t h e r v a l u e . r e m a i n i n g a t t r i b u t e s a r e h e l d a t t h e l o w est l e v e l s x° and x° i n b oth p r i z e s . S i n c e u t i l i t y f u n c t i o n s a r e unique up t o a p o s i t i v e l i n e a r - 48. transformation, u t i l i t y independence implies that u(y,y) = f ( y ) + g ( y ) u ( y , y ' ) , f o r a l l y and y, where f and g correspond to a and b in (6.3) and y' i s a r b i t r a r i l y chosen s p e c i f i c amount of Y. Functions f and g depend, in general, on the s p e c i f i c value of y 1 but not on the v a r i a b l e y. If the conventional s c a l i n g i s chosen and y 1 set equal to the worst value y°, then f(y ) = u(y°,y) and u(y,y) = u(y°,y) + g(y) u(y,y°) (6.4) U t i l i t y independence i s i n t e r p r e t e d in Figure 8 f o r two a t t r i b u t e s , Y and Y = Z. The t h i c k e s t l i n e represents equation (6.4) and the c u r v i l i n -ear surface i s obtained by c a l c u l a t i n g (6.4) f o r a l l y. If other cuts are made through the surface perpendicular to z a x i s , the i n t e r s e c t i o n s w i l l a l l have the same general shape as the t h i c k l i n e . Their e l e v a t i o n above the yz plane w i l l vary with z but they w i l l rank y i d e n t i c a l l y f o r a l l z. Figure 8 demonstrates that u t i l i t y independence i s not r e f l e x i v e . When Y i s u t i l i t y independent of Z the reverse i s not a u t o m a t i c a l l y true. The u t i l i t y curve over Z f o r y f i x e d at y° has a r i s k prone shape. This shape changes to r i s k averse at y . Functions u(y,*) have thus d i f f e r e n t general shape depending on y and are not s t r a t e g i c a l l y equivalent in t h i s case. To make the v e r i f i cat i:6n of u t i l i t y independence assumptions p r a c t i c a l , the set Y should contain only one a t t r i b u t e . However, when X. are u t i l i t y independent of X., i = 1,2,...,p, the f u n c t i o n a l form of f i s rather complex and 2 P - 2 constants must be assessed. Even f o r a problem with f i v e a t t r i b u t e s the number of assessments becomes p r o h i b i t i v e l y high. The simplest form of the m u 1 t i a t t r i b u t e u t i l i t y f u n c t i o n w i t h only p constants to be assessed r e s u l t s when the a t t r i b u t e s are mutually u t i 1 i t y independent, that i s when every subset of (X.j ,X2 ,. . . ,X ) i s u t i l i t y independent of i t s complement. The c o n d i t i o n cannot be used, though, since Y i s u t i l i t y independent o f Z Z i s not u t i l i t y independent o f Y FIGURE 8: GEOMETRIC INTERPRETATION OF UTILITY INDEPENDENCE FOR TWO ATTRIBUTES, WHERE z = y OF EQUATION (6.4). VX5 i t i s n o t p o s s i b l e f o r h u m a n s t o c o n s i s t e n t l y j u d g e l o t t e r i e s h a v i n g v a r i a b l e . l e v e l s i n m o r e t h a n a b o u t t w o a t t r i b u t e s . F o r t u n a t e l y , s e v e r a l w e a k e r c o n d i t i o n s i m p l y m u t u a l u t i l i t y i n d e p e n d -e n c e a n d d r a s t i c a l l y r e d u c e t h e c o m p l e x i t y o f r e q u i r e d v e r i f i c a t i o n s ( T h e o r e m 6.2 i n (22)). O n e c o n d i t i o n e x p l o i t s p r e f e r e n t i a l i n d e p e n d e n c e a s w e l l a s u t i l i t y i n d e p e n d e n c e a n d h a s b e e n f o u n d p a r t i c u l a r l y u s e f u l i n a p p 1 i c a t i o n s . P r e f e r e n t i a l i n d e p e n d e n c e i s d e f i n e d i n a m a n n e r s i m i l a r t o u t i l i t y i n d e p e n d e n c e , e x c e p t t h a t i t c o n c e r n s p r e f e r e n c e s f o r d e t e r m i n i s t i c o u t c o m e s r a t h e r t h a n l o t t e r i e s . A t t r i b u t e v e c t o r Y i s p r e f e r e n t i a l l y i n d e p e n d e n t o f Y ( s h o r t l y : Y i s p r e f e r e n t i a l l y i n d e p e n d e n t ) i f t h e p r e f e r e n c e o r d e r o f o u t c o m e s i n v o l v i n g o n l y c h a n g e s i n t h e l e v e l s i n Y d o e s n o t d e p e n d o n t h e l e v e l s a t w h i c h a t t r i b u t e s i n Y a r e h e l d f i x e d . T h e p r o p e r t y i s r e f l e x i v e . F o r e x a m p l e , i f ( X ^ . X ^ ) i s p r e f e r e n t i a l l y i n d e p e n d e n t o f X ^ , t h e n i f t h e o u t c o m e ( x l J j X ^ . x ^ ) i s p r e f e r r e d t o (x^ ' ,x'2 , x ^ ) f o r o n e v a l u e o f X ^ i t m u s t b e p r e f e r r e d f o r a l l p o s s i b l e x ^ . T h i s i s e q u i v a l e n t t o a s s u m i n g t h a t t r a d e - o f f s u n d e r c e r t a i n t y b e t w e e n v a r i o u s a m o u n t s o f X ^ a n d X^ d o n o t d e p e n d o n X ^ - T h e p r e f e r e n t i a l i n d e p e n d e n c e a s s u m p t i o n i m p l i e s t h a t t h e i n d i f f e r -e n c e c u r v e s o v e r X^zX2 a r e t h e s a m e r e g a r d l e s s o f t h e v a l u e o f X ^ . 6. k . T h e M u l t i a t t r i b u t e U t i l i t y M o d e l A s u i t a b l e c o m b i n a t i o n o f p r e f e r e n t i a l a n d u t i l i t y i n d e p e n d e n c e c o n d i t i o n s t h a t a r e e q u i v a l e n t t o t h e m u t u a l u t i l i t y i n d e p e n d e n c e a r e e x p l o i t e d i n t h e m o s t i m p o r t a n t r e s u l t i n t h e d e c o m p o s i t i o n a p p r o a c h . I f t h e n u m b e r o f a t t r i b u t e s p > 3 a n d , f o r s o m e X . , ( X . , X . ) i s p r e f e r e n t i a l l y i n d e p e n d e n t f o r a l l j ^ i , a n d X . i s u t i l i t y i n d e p e n d e n t , t h e n e i t h e r u ( x ) = Zp. , k . u . ( x . ) , i f Z .k. = 1 (6.5a) i = i i i i i i o r 1 + k u ( x ) = IJP = 1 (1 + k k . u . ( x . ) ) , i f S ; k . t 1 (6.5b) / -'- — 0\ w h e r e u a n d u . a r e u t i l i t y f u n c t i o n s s c a l e d f r o m 0 t o 1; k . = u ( x . " , x . ) a r e s c a l i n g c o n s t a n t s w i t h 0 < k. < 1; 0 < k < » i f E k . < 1; a n d -1 < k < 0 i f E k . > 1. K e e n e y a n d R a i f f a (22) d e s c r i b e i n d e t a i l t h e a s s e s s m e n t a n d t h e c o n s i s t e n c y c h e c k s o f t h e s c a l i n g c o n s t a n t s k . a n d k . T h e r e i s a n i n t e r e s t i n g i n t e r p r e t a t i o n o f t h e i n t e r a c t i o n f a c t o r k i n (6.5b). C o n s i d e r t w o l o t t e r i e s w i t h t w o - a t t r i b u t e p r i z e s , w h e r e y 1 > y " , z 1 i > z " a n d p r e f e r e n c e s a r e i n c r e a s i n g i n b o t h Y a n d Z . T h e m a r g i n a l p r o b a b i l i t y d i s t r i b u t i o n s o f y a n d z a r e i d e n t i c a l i n b o t h l o t t e r i e s b u t t h e i r j o i n t d i s t r i b u t i o n s a r e n o t . o f f e r s a 50-50 c h a n c e a t . e i t h e r t h e b e t t e r ( y ' , z ' ) o r t h e w o r s e p r i z e ( y " , z " ) . i s l e s s d r a m a t i c ; o n e c a n n o t w i n t h e b e t t e r p r i z e b u t c a n n o t l o o s e a s m u c h a s i n L | , e i t h e r . T o p r e f e r o v e r i m p l i e s t h a t i t i s i m p o r t a n t t o d o w e l l i n t e r m s o f a t l e a s t o n e a t t r i b u t e . T h u s Y a n d Z c a n b e t h o u g h t o f a s s u b s t i t u t e s f o r e a c h o t h e r . O n t h e o t h e r h a n d , i f a p e r s o n p r e f e r s i t m e a n s t h a t h e o r s h e n e e d s t o d o w e l l i n b o t h a t t r i b u t e s . T h e f u l l w o r t h o f h a v i n g o n e a t t r i b u t e a t h i g h l e v e l c a n n o t o t h e r w i s e b e e x p l o i t e d a n d Y a n d Z c o m p l e m e n t e a c h o t h e r . F i n a l l y , i f n o i n t e r a c t i o n o f p r e f e r e n c e e x i s t s b e t w e e n Y a n d Z , b o t h l o t t e r i e s a r e j u d g e d e q u i v a l e n t . T h e a p p r o p r i a t e f o r m o f (6.5) c a n b e d e t e r m i n e d p r i o r t o t h e a s s e s s -m e n t o f k . . O n c e t h e i n d e p e n d e n c e c o n d i t i o n s o f (6.5) a r e v e r i f i e d , i t i s s u f f i c i e n t t o p r o b e t h e p r e f e r e n c e f o r t h e a b o v e l o t t e r i e s . I f t h e p r e f e r -e n c e i s c o n s i s t e n t f o r a l l y 1 > y " , z 1 > z " t h e n k ^ 0 a n d (6.5b) i s a p p r o p r i a t e . I f ^ f o r a l l y 1 > y " , z 1 > z " , t h e n Y a n d Z a r e a d d i t i v e i n d e p e n d e n t , k = 0, a n d (6.5a) i s t h e v a l i d m o d e l . T h i s c a n b e s h o w n b y c a l c u l a t i n g e x p e c t e d u t i l i t i e s u s i n g t h e m u l t i l i n e a r . - u t i l i t y f u n c t i o n u ( y , z ) = u ( y , z ) + u ( y »z) + k u ( Y » z 0 ) u ( y o > z ) T h i s f o r m . g e n e r a 1 i z e s (6.5) f o r p = 2 w h e n Y a n d Z a r e m u t u a l l y u t i l i t y i n d e p e n d e n t . T h e p r o o f s c a n b e f o u n d i n (22) f o r p = 2 a n d p £ 3. F o r m (6.6) y i e l d s t h e f o l l o w i n g e x p e c t e d u t i l i t i e s : u ( L ^ = k A u ( y , z ' ) + k B u ( y , z") a n d u ( L ) = k A u ( y ,z") + k B u ( y , z 1 ) w h e r e A = u ( y ' , Z Q ) a n d B = u ( y " , z ) . T h e t e r m s w h i c h o c c u r f o r b o t h a n d l_2 a r e o m i t t e d b y u s i n g a p o s i t i v e l i n e a r t r a n s f o r m a t i o n (6.3). I t f o l l o w s f r o m t h e m o n o t o n i c i t y c o n d i t i o n t h a t A > B . B y s u b s t i t u t i n g u ( y Q , z ' ) = v + d a n d u ( y Q , z " ) = v - d , v > 0, d > 0, a n d u s i n g a p o s i t i v e l i n e a r t r a n s f o r m a t i o n a g a i n , t h e f o l l o w i n g o b t a i n s u(L ) = k d ( A - B ) u ( L 2 ) = k d ( B - A ) W h e n > t h e n u ( L ^ ) > uil^) a n d t h i s i m p l i e s k > 0. < i m p l i e s k < 0, w h e r e a s L , ^ . L _ o n l y i f k = 0. C H A P T E R 7 M A R K O V D E C I S I O N P R O C E S S F O R M U L A T I O N 7.1. T h e S e q u e n t i a l D e c i s i o n P r o b l e m C o n s i d e r a n e n g i n e e r r e s p o n s i b l e f o r m a n a g i n g t h e m a i n t e n a n c e o f a p a v e m e n t s e c t i o n . A t e v e r y p o i n t i n t i m e t h e p a v e m e n t i s i n a p a r t i c u l a r k n o w n c o n d i t i o n d e s c r i b e d b y a n u m b e r o f v a r i a b l e s w h i c h d e f i n e t h e s t a t e o f t h e s y s t e m . T h e s y s t e m m a k e s s t a t e t r a n s i t i o n s a s t h e p a v e m e n t d e t e r i o r a t e s d u e t o t r a f f i c a n d e n v i r o n m e n t a l f o r c e s o r w h e n t h e c o n d i t i o n i s u p g r a d e d b y m a i n t e n a n c e . A l t h o u g h t h e e n g i n e e r c a n n o t t o t a l l y c o n t r o l w h i c h t r a n s i t i o n w i l l o c c u r , h e c a n , b y h i s c h o i c e o f a c t i o n a f f e c t t h e p r o b a b i l i t y o f a n y p a r t i c u l a r t r a n s i t i o n . T h e v a r i o u s d e c i s i o n s e n t a i l r e w a r d s c o n s i s t i n g o f c o s t s a n d b e n e f i t s a n d t h e m a n a g e r h a s a p r e f e r e n c e s t r u c t u r e f o r e v e r y s e q u e n c e o f r e w a r d s . H i s o b j e c t i v e , a t a n y p o i n t i n t i m e , i s t o c h o o s e h i s a c t i o n s s o a s t o m a x i m i z e t h e e x p e c t e d u t i l i t y o f t h e f u t u r e s t r e a m o f r e w a r d s . Art_ ? i m p o r t a n t f e a t u r e o f t h e p r o c e s s i s t h a t d e c i s i o n s a r e m a d e p e r i o d i c a l l y , b u t e a c h d e c i s i o n i n f l u e n c e s a l l o f t h e f o l l o w i n g r e w a r d s . T h i s t y p e o f p r o b l e m c a n b e d i s p l a y e d g r a p h i c a l l y b y a m u l t i p e r i o d d e c i s i o n t r e e b u i l t f r o m t h e b a s i c t r e e o f F i g u r e k. S t a r t i n g i n a p a r t i c u -l a r s t a t e t h e p a v e m e n t m a n a g e r h a s s e v e r a l o p t i o n s . O n c e h e e x e r c i s e s a n o p t i o n , t h e p a v e m e n t e n t e r s a n e w s t a t e w i t h a k n o w n p r o b a b i l i t y a n d t h e p r o c e s s r e p e a t s . T h e d e c i s i o n m a k e r ' s p r e f e r e n c e s f o r d i f f e r e n t s t a t e s o f p a v e m e n t c o n d i t i o n a r e s o l e l y d e t e r m i n e d b y f u t u r e t r a n s i t i o n r e w a r d s h e c a n a c h i e v e f r o m s t a r t i n g i n t h e s e s t a t e s . 7.2. M a r k o v D e c i s i o n P r o c e s s T h e p a v e m e n t d e c i s i o n p r o b l e m c a n b e s t r u c t u r e d b y t h e M a r k o v d e c i s i o n m o d e l (26). T h e m o d e l e m p l o y s a s t a t e s p a c e , a n a c t i o n s p a c e , a l a w o f d e t e r i o r a t i o n , a n d r e w a r d s . W e c o n s i d e r a s y s t e m t h a t c a n b e o b s e r v e d p e r i o d i c a l l y a n d i t s c o n d i t i o n c l a s s i f i e d a s o n e o f t h e c o u n t a b l e s t a t e s i = 1,2 , . . . , N . S t a t e 1 i s t h e i n i t i a l s t a t e o f t h e s y s t e m a n d r e p r e s e n t s t h e p r o c e s s b e f o r e a n y d e t e r i o r a t i o n t a k e s p l a c e . S t a t e N i s t h e t e r m i n a l s t a t e a f t e r w h i c h n o f u r t h e r d e t e r i o r a t i o n c a n t a k e p l a c e . N o o r d e r i n g i s i m p l i e d i n t h e l a b e l l i n g o f t h e i n t e r m e d i a t e s t a t e s . A f t e r d e t e r m i n i n g t h e s t a t e o f t h e s y s t e m , o n e o f a n u m b e r o f a c t i o n s m u s t b e t a k e n . F o r e a c h s t a t e i i n t h e s t a t e s p a c e t h e r e i s a f i n i t e s e t A . o f p o s s i b l e a c t i o n s a v a i l a b l e w h e n t h e s y s t e m i s i n s t a t e 1. W h e n t h e s y s t e m i s i n s t a t e i a n d w e t a k e a c t i o n k e A . , t w o t h i n g s h a p p e n . F i r s t , t h e s y s -t e m m o v e s t o a n e w s t a t e j w i t h p r o b a b i l i t y p . j ( k ) w h i c h i s d e t e r m i n e d b y t h e l a w o f d e t e r i o r a t i o n . S e c o n d , c o n d i t i o n a l o n t h e e v e n t t h a t t h e n e w s t a t e i s j , w e o b t a i n a r e w a r d c . . ( k ) . i j W h e n v i e w e d i n a t e m p o r a l s e t t i n g , t h e s y s t e m u n d e r g o e s s t a t e t r a n s i t i o n s a t e q u a l i n t e r v a l s t e r m e d s t a g e s . A t e a c h s t a g e a n a c t i o n i s t a k e n a n d a r e w a r d r e s u l t s . W h e n t h e t i m e s p a n o f i n t e r e s t c o n s i s t s o f n s t a g e s , w e a d a p t t h e c o n v e n t i o n o f i n d e x i n g t r a n s i t i o n t i m e s s o t h a t t i s t h e n u m b e r o f s t a g e s r e m a i n i n g t o t h e e n d o f t h e a n a l y s i s p e r i o d , i . e . t = n i s t h e f i r s t s t a g e ' s b e g i n n i n g a n d t = 0 i s t h e t e r m i n a l t i m e . We d e f i n e t h e h i s t o r y o f a n n - s t a g e p r o c e s s , h n , a s t h e s e q u e n c e o f s t a t e s a n d a c t i o n s w h i c h o c c u r b e t w e e n t = n a n d t = 0 h n = ( X ,Y ; X ,,Y -X, ,Y, ; X . ) n n n-1 n-1 .1.1 0 w h e r e X ^ d e n o t e s t h e s t a t e o c c u p i e d a t t i m e t a n d Y i s t h e a c t i o n t a k e n a t t i m e t . T h e r e w a r d s a r e i m p l i e d b y h n s i n c e = i , = k a n d X -j = j s p e c i f y c . j ( k ) . T h e r e v e r s e a s s e r t i o n n e e d n o t b e t r u e f o r i f d i f f e r e n t a c t i o n s l e a d t o i d e n t i c a l r e w a r d s , w e c a n n o t r e c o n s t r u c t h n . T h e d e t e r i o r a t i o n l a w o f t h e s y s t e m i s a s s u m e d t o b e M a r k o v i a n . F o r m a l l y , i f t h e p r e s e n t s t a t e a n d a c t i o n a r e i a n d k , r e s p e c t i v e l y , a n d j i s t h e f o l l o w i n g s t a t e t h e M a r k o v p r o p e r t y i m p l i e s t h a t , P ( X t _ ! - J K X n , Y n ; X t + 1 , Y t + 1 ; X t = i , Y t = k ) = P ( X t _ 1 = j | X t = i , Y t = k ) = P i j ( k ) , (7.1) t h a t i s , t h e t r a n s i t i o n p r o b a b i l i t y f r o m i t o j a t s t a g e t d o e s n o t d e p e n d o n t h e h i s t o r y p r i o r t o t . T h e a p p l i c a b i l i t y o f M a r k o v a s s u m p t i o n t o p a v e m e n t d e t e r i o r a t i o n i s d i s c u s s e d i n C h a p t e r k.3-T h e n o t a t i o n f o r t r a n s i t i o n p r o b a b i l i t i e s i m p l i e s t h a t t h e y d e p e n d o n t h e s t a t e s o c c u p i e d , i , j , a n d t h e a c t i o n t a k e n k , b u t a r e i n d e p e n d e n t o f t h e t i m e i n d e x . I t m e a n s t h a t t h e s a m e l a w o f d e t e r i o r a t i o n g o v e r n s t h e p r o c e s s r e g a r d l e s s o f t h e t i m e p o i n t i n t h e a n a l y s i s p e r i o d . A p r o c e s s w i t h t h i s p r o p e r t y i s c a l l e d s t a t i o n a r y o r h a m o g e n e o u s . C o n t r a r y t o t h e M a r k o v a s s u m p -t i o n , t h e s t a t i o n a r i t y a s s u m p t i o n i s n o t c r i t i c a l a n d t h e d e c i s i o n m o d e l m a y b e m o d i f i e d t o a c c o u n t f o r t i m e v a r i a b i l i t y o f t h e d e t e r i o r a t i o n l a w , t h e s t a t e s p a c e , t h e a c t i o n s p a c e , a n d t h e r e w a r d s . T h e s e e l e m e n t s a r e a s s u m e d s t a t i o n a r y i n t h i s t h e s i s i n o r d e r t o s i m p l i f y t h e p r e s e n t a t i o n . F o r e a c h p o s s i b l e s t a r t i n g s t a t e , i , t h e d e c i s i o n m a k e r c a n c h o o s e h i s a c t i o n f r o m t h e s e t A . . . A v e c t o r f w h i c h a s s i g n s a p a r t i c u l a r a c t i o n k e A . t o e a c h p o s s i b l e s t a t e i s c a l l e d a d e c i s i o n r u l e . O n e d e c i s i o n r u l e m i g h t b e f 1 = ( D O N O T H I N G i f s t a t e 1, DO N O T H I N G i f s t a t e 2, R E P A I R i f s t a t e 3, O V E R L A Y i f s t a t e N ) . A n o t h e r d e c i s i o n r u l e m i g h t d i f f e r f r o m f^ b y h a v i n g DO N O T H I N G f o r s t a t e s 1, 2, 3 , a n d o t h e r a c t i o n s s a m e a s i n f ^ . A n y c o l l e c t i o n o f d e c i s i o n r u l e s i s a d e c i s i o n s e t . T h u s ( f ^ ^ ) i s a d e c i s i o n s e t a n d s o i s e a c h d e c i s i o n r u l e t a k e n s e p a r a t e l y . H o w e v e r , i n o r d e r t o a s s u r e t h a t t h e a n a l y s i s c o n s i d e r s a l l p o s s i b l e a l t e r n a t i v e s a s r e q u i r e d b y t h e p r i n c i p l e s o f s y s t e m s a n a l y s i s , t h e d e c i s i o n s e t m u s t b e t h e C a r t e s i a n p r o d u c t o f t h e i n d i v i d u a l a c t i o n s e t s , A , * A , A . . . * A . S u c h a d e c i s i o n s e t c o n t a i n s a l l p o s s i b l e d e c i s i o n r u l e s and i s c a l l e d complete. Since the d e c i s i o n maker . must e x e r c i s e a sequence of d e c i s i o n s over time, we define a p o l i c y . An n-period pol i c y TT^ i s a sequence of n d e c i s i o n r u l e s , TT = (f ,f ,,...,1=,) • (7-2) n n' n-1 1 If a l l d e c i s i o n rules are the same, the p o l i c y i s c a l l e d s t a t i o n a r y and i s denoted ( f n ) . Suppose that at time n the system i s in s t a t e i and the d e c i s i o n maker wants to pursue the n-period p o l i c y T r n = (f , . . . , f ^ ) . Since the de- :' t e r i o r a t i o n law i s p r o b a b i l i s t i c , he cannot p r e d i c t the r e s u l t a n t sequence of events with c e r t a i n t y . The future h i s t o r y i s a random v a r i a b l e n^Ojir ) and has a p r o b a b i l i t y mass f u n c t i o n m ( ? i n ( 0 ) . For a p a r t i c u l a r h i s t o r y h^ given the s t a r t i n g c o n d i t i o n s , the p r o b a b i l i t y m(^ n(v) = h£) = and ^ k m k = ^' Given a h i s t o r y , m^  can be c a l c u l a t e d as the product of one-stage t r a n s i t i o n p r o b a b i l i t i e s since the t r a n s i t i o n s between states are independent events. 7.3-Optimal P o l i c i e s and Dynamic Programming Associated w i t h each h i s t o r y i s a sequence (c ,c ^ , . . . , C q ) of one-stage outcomes. Here, i s a vector reward at stage t . The d e c i s i o n maker has a preference s t r u c t u r e defined on a l l sequences of rewards and encoded in the u t i l i t y f u n c t i o n U(c ,... ,c ). Since the h i s t o r i e s are random v a r i a b l e s ' n' o c o n d i t i o n a l on the s t a r t i n g s t a t e and the p o l i c y used, the associated reward streams are sequences of random rewards. The u t i l i t y of rewards associated with h i s t o r y ri n(i,TT N) i s thus u(i,rr n) = u ( ^ ( V X n , , f n ) , V i ^ n - i - V z - V ^ where ^ denotes random v a r i a b l e , X t i s s t a t e in which stage t begins, C Q ^ 0 ) is reward associated w i t h terminating in s t a t e X and the s t a r t i n g c o n d i t i o n s are X = i and f = k. n n The d e c i s i o n problem i s to choose an optimal sequence of a c t i o n s w i t h which to manage the system over the a n a l y s i s period to maximize expected u t i l i t y . If we s t a r t in s t a t e i and pursue p o l i c y TT , the expected u t i l i t y of the streams of rewards p o s s i b l e in the future i s v. (TT ) = E^n /. \U ( i ,TT ) i n h (i ,TT ) ' n If one t r i e d to f i n d the optimal p o l i c y in a problem i n v o l v i n g N s t a t e s , K acti o n s and T stages, one would have to evaluate about N(K*N)^ paths in the d e c i s i o n tree corresponding to a l l p o s s i b l e h i s t o r i e s . A r e a l i s t i c problem may have N = 10,K = k and T = 10 and w i l l require about k0^ c a l c u l a t i o n s . This i s a p r o h i b i t i v e number even f o r a high speed computer. C l e a r l y , a more powerful technique i s required f o r computational e f f i c i e n c y . The s t r u c t u r e of Markov d e c i s i o n problems i s s u i t a b l e f o r o p t i m i -z a t i o n by the dynamic programming technique ( 2 6 ) which i s based on the p r i n c i p l e of o p t i m a l i t y . The p r i n c i p l e a s s e r t s that regardless of the present s t a t e of the system and the present d e c i s i o n , the remaining d e c i s i o n s in the sequence must c o n s t i t u t e an optimal p o l i c y w i t h regard to the s t a t e r e s u l t i n g from the present d e c i s i o n . For our purpose, " d e c i s i o n " should be int e r p r e t e d as " d e c i s i o n r u l e " . If i r n = (f , f n_ ,. . . , f .j) i s an optimal n-stage p o l i c y , then T r n _ ^ = ( f n _ i , • • • ,f -j) must be an optimal (n-1) stage p o l i c y and i r n = { f ^ , i y ^ _ ^ ) . By using the p r i n c i p l e of o p t i m a l i t y r e c u r s i v e l y , one can s t a r t from the l a s t stage and by backward induction i d e n t i f y the optimal sequence of d e c i s i o n r u l e s . A n a l y s i s of the n-stage problem i s thus s i m p l i f i e d to n problems of f i n d i n g an optimal d e c i s i o n r u l e , f ^ . f o r one stage at a time by the f o l l o w i n g c r i t e r i o n yCfJ . v V > v ( f t , V | ) . f o r a J 1 V (7.3) Here, y (•) denotes a vector w i t h components v.(«), i = 1, N, that i s a v e c t o r o f e x p e c t e d u t i l i t i e s f o r a l l s t a r t i n g s t a t e s . I t i s t e r m e d t h e r e t u r n f u n c t i o n . T h e l e f t h a n d s i d e v e c t o r i n t h e a b o v e c r i t e r i o n i s s a i d t o d o m i n a t e t h e o t h e r , t h a t i s V ; ( f ' t , T T t 2 1 ) > y-Cf t , T r t2 1) f o r a l l i , V . ( f l 7 1 " / . , ) > V . ( f 4. > IT ' . \ c i t , t - T i x t t-1) f o r s o m e i . D o m i n a n c e i s g u a r a n t e e d w h e n t h e d e c i s i o n s e t i s c o m p l e t e . T h e d o m i n a t i n g p o l i c y ( f ^ T T j . ^ ) i s t h e o p t i m a l f j - p e r i o d p o l i c y . I t i s n o t u n i q u e . D y n a m i c p r o g r a m m i n g m a k e s t h e a n a l y s i s c o m p u t a t i o n a l l y f e a s i b l e b u t p u t s r e s t r i c t i o n s o n t h e f o r m o f t h e o b j e c t i v e f u n c t i o n . S e q u e n t i a l d e c i s i o n p r o b l e m s w i t h e x p e c t e d u t i l i t y i n t h e o b j e c t i v e f u n c t i o n a r e d i f f i c u l t t o s o l v e , b u t r e c e n t l y m a t h e m a t i c a l f o r m u l a t i o n s h a v e e m e r g e d f o r t h i s c l a s s o f M a r k o v d e c i s i o n p r o c e s s e s (37). I n g e n e r a l , o b j e c t i v e f u n c t i o n s d e c o m p o s e a s r e q u i r e d f o r t h e a p p l i c a t i o n o f d y n a m i c p r o g r a m m i n g i f t h e y a r e s e p a r a b l e (35). A f u n c t i o n i s s e p a r a b l e i f I f , i n a d d i t i o n , g , a n d g ^ a r e r e a l v a l u e d f u n c t i o n s , a n d g . i s a m o n o - ' t o n i c a l l y n o n d e c r e a s i n g f u n c t i o n o f g 2 f o r e v e r y c ( • ) , t h e n m a x f n , . . . , f l g ( c n ( • ) , . • • , c 1 ( • ) ) = m a x g 1 ( c n ( « ) ; m a x g 2 ( c ^ ( • ) , • • • ,Cj (•)) • (7-4) f n f n - 1 f 1 T h e m a x i m i z a t i o n w i t h r e s p e c t t o ( f .j , . . . , f . ) m o v e s i n s i d e t h e e x p e c t e d u t i l i t y o f t h e n t h s t a g e a n d b a c k w a r d i n d u c t i o n i s n o w p o s s i b l e . T o s h o w t h i s , d e f i n e y ( T T n _ 1 ) = m a x g ^ c ^ ( • ) , • • • , c . j ( • ) ) f n - 1 ' - f 1 a n d w e h a v e y ( i T n ) = m a x g ( c ( * ) , . . . ( • ) ) f n ' * ' * ' f 1 = m a x g i ( c n ( X n , X n _ r f n ) ; y U ^ ) ) . (7-5). n G i v e n t h e o p t i m a l r e t u r n f u n c t i o n V(TT «) f o r s t a g e t-1 , w e c a n f i n d t h e o p t i m a l d e c i s i o n r u l e f o r s t a g e t . I f w e s t a r t i n s t a g e 1 a n d f i n d y ( f ! | ) , w e c a n n e x t i d e n t i f y t h e o p t i m a l 2nd s t a g e d e c i s i o n r u l e a n d t h e o p t i m a l r e t u r n f u n c t i o n yCf^.f.) = V(TT2) . G i v e n t h e s e , w e c a n f i n d f ^ a n d ytf^-rr^) = v(TT^) . T h e : s e q u e n c e o f o p t i m a l d e c i s i o n r u l e s f o r m s a n o p t i m a l p o l i c y . T h e i n f o r m a t i o n a b o u t a l l s t a g e s t-1, t-2, 1 f o l l o w i n g s t a g e t i s s u m m a r i z e d b y t h e o p t i m a l r e t u r n f u n c t i o n y(iT B y v i r t u e o f t h e p r i n c i p l e o f o p t i m a l i t y , i t i s n e v e r n e c e s s a r y t o a n a l y s e t h e e f f e c t o f a c u r r e n t d e c i s i o n r u l e o n t h o s e m a d e i n t h e r e m a i n i n g s t a g e s . F o r t h i s r e a s o n i t i s p o s s i b l e t o o p t i m i z e o v e r o n e s t a g e a t a t i m e . 7 - 4 . S e p a r a b l e , R i s k - A v e r s e T e m p o r a l U t i l i t y F u n c t i o n s C h o o s i n g t h e o p t i m a l p o l i c y T f n b y t h e c r i t e r i o n o f m a x i m u m e x p e c t e d u t i l i t y c a n b e s t a t e d a s V(TT*) = m a x E ^ n U ( c n ( • ) , . . - , c 1 ( • ) , c ( X ) ) n - j. / . \ n i o o f , . . . , r . h ( i ,TT ) n ' 1 n w h e r e c t ( « ) = c ( X , X j , f ) a n d c o ( X Q ) i s t h e t e r m i n a l r e w a r d . I n o r d e r t o u s e d y n a m i c p r o g r a m m i n g , t h e m a x i m i z a t i o n w i t h r e s p e c t t o ( f ^ , . . . , f ^ ) m u s t m o v e i n s i d e t h e n t h s t a g e e x p e c t e d u t i l i t y a n d t h e e x p e c t a t i o n i s c a l c u l a t e d o v e r o n e s t a g e r a t h e r t h a n t h e n - s t a g e h i s t o r y : y ( i r * ) = m a x E% , } U ( c n ( 0 , y U ^ ) ) . (7.6) f n n T h e o p t i m a l r e t u r n f u n c t i o n f o r s t a g e 1 d e p e n d s o n t h e t e r m i n a l r e w a r d C 0 ' X 0 ) , v ( f ' j ) = m a x E £ ^ U ( • ) , C q ( X Q ) ) (7.7) c 1 ; • i • - o o 1 F r o m t h i s , t h e o p t i m a l p o l i c y m a y b e d e t e r m i n e d b y p r o c e e d i n g o n e s t a g e a t a t i m e , u n t i l s t a g e n i s r e a c h e d , p r o v i d e d t h e r e t u r n f u n c t i o n c o n f o r m s t o t h e c o n d i t i o n s o f s e p a r a b i l i t y a n d m o n o t o n i c i t y . T h e s e p a r a b l e u t i l i t y f u n c t i o n s m u s t b e s c r e e n e d t o e l i m i n a t e t h o s e f o r m s w h i c h d o n o t m e e t b e h a v i o u r a l a s s u m p t i o n s d e s c r i b i n g t h e d e c i s i o n m a k e r ' s a t t i t u d e t o w a r d r i s k i n m u l t i s t a g e g a m b l e s . W h e n r e w a r d s c . i n a n n - p e r i o d l o t t e r y a r e t h o u g h t o f a s a t t r i b u t e s x . i n t h e s e n s e o f C h a p t e r 6 .3 , i t i s p o s s i b l e t o u s e a m o d e l l i k e (6.5) i n a t e m p o r a l s e t t i n g , U ( c n , . . . , c o ) = ^ = 1 K t u t ( c t ) , i f E t K t = 1 (7.8a) o r 1 + K U ( c p , . . . , c o ) = n " = 1 ( l + K K t u t ( c t ) ) , i f Z t K t * 1, (7.8b) w h e r e U i s t h e m u l t i p e r i o d u t i l i t y f u n c t i o n ; u ^ a r e t h e u t i l i t y f u n c t i o n s f o r s i n g l e p e r i o d s ; K t a r e t h e s c a l i n g c o n s t a n t s f o r s i n g l e p e r i o d s ; a n d K i s t h e t e m p o r a l i n t e r a c t i o n f a c t o r . M e y e r s h o w s i n C h a p t e r 9 o f K e e n e y a n d R a i f f a ' s b o o k (22) t h a t t h e m u t u a l u t i l i t y i n d e p e n d e n c e c o n d i t i o n s w h i c h a r e n e c e s s a r y f o r (6.5) t o h o l d i n a t e m p o r a l s i t u a t i o n a r e i m p l i e d b y t w o b e h a v i o u r a l f e a t u r e s . F i r s t , i t i s a c c e p t a b l e f o r t h e d e c i s i o n m a k e r t o m a k e h i s d e c i s i o n a b o u t t h e f u t u r e ( i n e a c h p e r i o d ) w i t h o u t r e g a r d t o h i s p a s t s t r e a m o f r e w a r d s . S e c o n d , t h e t e r m i n a l r e w a r d d o e s n o t a f f e c t t h e d e c i s i o n s m a d e p r i o r t o t h e t e r m i n a l t i m e . T e s t i n g t h e a d d i t i v e i n d e p e n d e n c e a s s u m p t i o n w i l l d e t e r m i n e w h i c h o f t h e m o d e l s (7-8a) a n d (7•8b) i s a p p r o p r i a t e , a n d w h a t s i g n t h e i n t e r a c t i o n f a c t o r K a s s u m e s . T h e r i s k b e h a v i o u r i n a t w o - s t a g e p r o b l e m m a y b e t e s t e d u s i n g t h e f o l l o w i n g l o t t e r i e s , w h e r e c ' > c : ( c , c ) L o t t e r y L^ g i v e s t h e d e c i s i o n m a k e r a 50-50 c h a n c e o f r e c e i v i n g e i t h e r t h e s m a l l e r r e w a r d s , c , i n b o t h s t a g e s , o r t h e g r e a t e r r e w a r d , c 1 , i n b o t h s t a g e s . I n , t h e d e c i s i o n m a k e r r e c e i v e s e i t h e r t h e s m a l l e r r e w a r d f o l l o w e d b y t h e g r e a t e r r e w a r d i n t h e s e c o n d s t a g e , o r , w i t h e q u a l p r o b a b i -l i t y , h e r e c e i v e s a s t r e a m o f r e w a r d s i n t h e r e v e r s e d o r d e r . T h e d e c i s i o n m a k e r i s t e m p o r a l l y r i s k a v e r s e f o r t w o - p e r i o d r e w a r d s t r e a m s i f h e d o e s n o t p r e f e r L^ t o L ^ ( L 2 > L ^ ) f o r a l l c a n d c 1 . H e i s r i s k n e u t r a l i f a n d o n l y i f L^ ^ f o r a l l c a n d c ' . F i n a l l y , h e i s r i s k s e e k i n g i f L^ > L ^ f o r a l l c a n d c 1 . T h e f i r s t l o t t e r y m a y b e r e g a r d e d a s r e c e i v i n g " a l l t h e b e s t o r a l l t h e w o r s t " w i t h a 50-50 c h a n c e . T h e s e c o n d l o t t e r y ' s c o n s e q u e n c e s a s s u r e t h a t t h e d e c i s i o n m a k e r w i l l r e c e i v e o n e o f t h e m o r e d e s i r a b l e r e w a r d s a n d o n e o f t h e l e s s . R i s k a v e r s i o n i s a t t r i b u t e d t o p r e f e r r i n g s o m e g o o d a n d s o m e b a d t o a l l o r n o t h i n g . T h e c o n c e p t o f t e m p o r a l r i s k a t t i t u d e s m a y b e e x t e n d e d t o m u l t i s t a g e l o t t e r i e s . I t s e e m s n a t u r a l t h a t d e c i s i o n m a k e r s r e s p o n s i b l e f o r t h e o p e r a t i o n o f p u b l i c s y s t e m s b e t e m p o r a l l y r i s k a v e r s e . A s s u m i n g t h e c o n t r a r y w o u l d i m p l y t h a t t h e d e c i s i o n m a k e r i s i n d i f f e r e n t o r p r e f e r s t h e n - p e r i o d l o t t e r y t o t h e s e q u e n c e ( c , c , . . . c ) w h e r e c a n d c a r e t h e b e s t a n d t h e w o r s t r e w a r d s , r e s p e c t i v e l y , a n d c i s t h e c e r t a i n t y e q u i v a l e n t f o r t h e o n e - p e r i o d l o t t e r y . c o c L o t t e r y L ^ g i v e s h i m a 50% c h a n c e o f t h e w o r s t r e w a r d s f o r n y e a r s . P r e f e r -r i n g t h i s l o t t e r y t o t h e s e q u e n c e o f c e r t a i n t y e q u i v a l e n t s c o u l d p o s s i b l y l e a d t o d e c i s i o n s w i t h c a t a s t r o p h i c o u t c o m e s . I n d i f f e r e n c e w o u l d i m p l y t h a t h e b e h a v e s a s i f h e w e r e o n l y f a c i n g a 50% c h a n c e o f t h e w o r s t r e w a r d f o r o n e s t a g e . O n e c a n c o n c l u d e f o r m o s t o f t h e p u b l i c d e c i s i o n s i t u a t i o n s t h a t t h e a d d i t i v e m o d e l (7-8a) i s n o t v a l i d a n d K i n (7-8b) h a s t o b e n e g a t i v e . F o r m (7.8b) i s k n o w n a s t h e n e g a t i v e m u l t i p l i c a t i v e (36,43). I m p o s i n g a c o n d i t i o n o f s t a t i o n a r i t y o n t h e d e c i s i o n m a k e r s i m p l i f i e s (7-8b), 1 + K U ( c n , . . . , c o ) = n " = 1 (1 + b u ( c t ) ) (7-9) w h e r e b = KK^,; = c o n s t , t = 1 , . . . , n . S t a t i o n a r i t y i m p l i e s t h a t i f f a c e d a t t i m e t w i t h a d e c i s i o n p r o b l e m w h i c h a f f e c t s f u t u r e r e w a r d s t r e a m s , t h e d e c i s i o n m a k e r w i l l m a k e a d e c i s i o n w h i c h i s i n d e p e n d e n t o f t h e a b s o l u t e t i m e . W h e n K i s n e g a t i v e , n o t e t h a t W ( c ) = W ( c , . . . , c ) a -(] + K U ( c , . . . , c ) ) n o n ' ' o a n d w ( c t ) = -(1 + b u ( c ) ) , b < 0 a r e u t i l i t y f u n c t i o n s o v e r C a n d C ^ . , r e s p e c t i v e l y , s o -w ( c ) = n j = 1 ( - w ( c t ) ) = n j = 1 d + b u ( c t ) ) . (i.1 .0) T h e r i s k a v e r s i o n f u n c t i o n c a n b e c a l c u l a t e d f o r t h e s o c a l l e d l e v e l s t r e a m s o f r e w a r d s , t h a t i s w h e n c = e f o r a l l t . T h e n (7-10) b e c o m e s W ( c ) = - 0 + b u ( e ) ) n (7.11) f o r w h i c h t h e r i s k a v e r s i o n f u n c t i o n (5-1) i s - W " / W ' = - u " ( e ) / u ' ( e ) - ( n - 1 ) b u 1 ( e ) /(1 + b u ( e ) ) . F o r n = 1, t h e r i s k a v e r s i o n i s t h a t o f U a s e x p e c t e d . A s t h e n u m b e r o f p e r i o d s i n c r e a s e s , t h e r i s k a v e r s i o n i n c r e a s e s l i n e a r l y w i t h i n n . C a l c u l a t i o n o f t h e r i s k a v e r s i o n f u n c t i o n f o r t h e a d d i t i v e m o d e l (7-8a) u n d e r c o n d i t i o n s o f s t a t i o n a r i t y a n d l e v e l s t r e a m s s h o w s t h a t t h e n - p e r i o d r i s k a v e r s i o n i s t h e s a m e a s t h e s i n g l e p e r i o d r i s k a v e r s i o n : U ( c ) = n a u ( e ) , a = K f o r a l l t a n d - U " ( c ) / U ' ( c ) = - u " ( e ) / u 1 ( e ) . C H A P T E R 8 A P P L I C A T I O N 8 . 1 . S u m m a r y o f M e t h o d F i g u r e 9 o u t l i n e s t h e p r o p o s e d m e t h o d - T h e m a n a g e r d e c i d e s w h i c h o b j e c t i v e s a n d a t t r i b u t e s a r e r e l e v a n t ( C h a p t e r 4 . 1 ) . T h e a n a l y s t e l i c i t s t h e m a n a g e r ' s p r e f e r e n c e s f o r m u l t i a t t r i b u t e c o n s e q u e n c e s a n d s e l e c t s a n a p p r o p r i a t e u t i l i t y m o d e l f o r o n e - s t a g e c o n s e q u e n c e s . ( C h a p t e r 6 . 3 ) . ; T h e p l a n n i n g h o r i z o n i s c h o s e n a n d t h e a n a l y s t d e c i d e s w h i c h t e m p o r a l u t i l i t y f u n c t i o n b e s t r e p r e s e n t s t h e m a n a g e r ' s r i s k a t t i t u d e s ( C h a p t e r 7.^). T h e -f u n c t i o n t r a n s f o r m s t h e s e q u e n c e o f o n e - s t a g e u t i l i t i e s i n t o a c r i t e r i o n i n d e x i n t h e d y n a m i c p r o g r a m m i n g a l g o r i t h m . T h e a v a i l a b l e a c t i o n s , t h e a s s o c i a t e d c o n s e q u e n c e s a n d t h e l a w o f d e t e r i o r a t i o n a r e i n p u t i n t o t h e o p t i m i z a t i o n a l g o r i t h m . A o n e - s t a g e u t i l i t y f u n c t i o n t r a n s f o r m s m u l t i a t t r i b u t e c o n s e q u e n c e s i n t o s i n g l e v a l u e s t o b e u s e d i n e v a l u a t i o n . T h e l a w o f d e t e r i o r a t i o n i s r e p r e s e n t e d b y M a r k o v t r a n s i t i o n m a t r i c e s . U s i n g t h e s e , t h e d y n a m i c p r o g r a m m i n g a l g o r i t h m c a l c u l a t e s t h e e x p e c t e d t e m p o r a l u t i l i t y a n d i d e n t i f i e s a n o p t i m a l p o l i c y ( C h a p t e r 7 .3). T o i l l u s t r a t e t h e m e t h o d i t i s a p p l i e d t o a h y p o t h e t i c a l e x a m p l e w h i c h i s p r o c e s s e d b y a c o m p u t e r p r o g r a m . T h i s t e s t a p p l i c a t i o n e m p h a s i z e s t h e u t i l i t y p a r t o f d e c i s i o n a n a l y s i s . T h e u t i l i t i e s a r e e l i c i t e d f r o m a g r o u p o f e n g i n e e r s , w h i l e t h e s t r u c t u r e o f o b j e c t i v e s a n d t h e M a r k o v t r a n -s i t i o n m a t r i c e s a r e a s s u m e d . T h e e l i c i t a t i o n e x e r c i s e t e s t s t h e b e h a v i o u r o f s u b j e c t s i n a s i m u l a t e d p a v e m e n t d e c i s i o n s i t u a t i o n a n d p r o v i d e s a b a s i s f o r s e l e c t i n g a n a p p r o p r i a t e p r e f e r e n c e m o d e l . T e m p o r a l p r e f e r e n c e s a r e t e s t e d s u p e r f i c i a l l y a s t h e t h e o r y i s s t i l l i n t h e d e v e l o p m e n t s t a g e . C o m p u t a t i o n a l r u n s o n s e v e r a l v e r s i o n s o f t h e p r o b l e m p r o b e w h e t h e r t h e m o d e l b e h a v e s r e a s o n a b l y f o r s i m p l y v a r i e d i n p u t s . T h e s e t e s t s i n v e s t i -g a t e t h e e f f e c t o f v a r i a t i o n i n t h e e l i c i t e d s c a l i n g c o n s t a n t s . T h e e f f e c t Derive s t r u c t u r e of o b j e c t i v e s and a t t r i b u t e s E l i c i t U t i l i t i e s F I G U R E 9- S U M M A R Y O F P R O P O S E D M E T H O D 66. on the optimal p o l i c y of Markov t r a n s i t i o n p r o b a b i l i t i e s and of the temporal u t i l i t y are a l s o t e s t e d . The number of st a t e s i s set at a minimum and the formulas f o r c a l c u -l a t i n g consequences are s i m p l i f i e d to f a c i l i t a t e judgement of the model's behaviour. For t h i s reason, .the program may not be adopted f o r immediate a p p l i c a t i o n by a highway department. However, these s i m p l i f i c a t i o n s can be e a s i l y r e c t i f i e d and the model implemented, provided the user subscribes to the underlying assumptions. 8.2.Hypothetical Example Problem A Department of Highways (DOH) requires a plan f o r the upkeep of a 20 km long s e c t i o n of a four-la n e suburban highway. The o b j e c t i v e s f o r managing the pavement s e c t i o n r e f l e c t DOH's r o l e of a p u b l i c servant (Figure 10). The road user o b j e c t i v e s of s a f e t y , economy, and t r a v e l time concern the average d a i l y t r a f f i c (ADT) of 60,000 vehicles/day that use the se c t i o n at an operating speed of 100 km/hr. The p u b l i x o b j e c t i v e s include DOH's cost of maintaining the s e c t i o n , access to a d j o i n i n g communities pro-vided by the s e c t i o n , job c r e a t i o n in the region, and conservation of g r a v e l . The nonusers are 2000 households in a t r a f f i c noise zone along the s e c t i o n . DOH d e s i r e s to minimize the agency cost f o r the s e c t i o n , because the unused funds can be u t i l i z e d on other l i n k s in the regional network. The region experiences unemployment and a shortage of c o n s t r u c t i o n aggregates. To ame-l i o r a t e the s i t u a t i o n , DOH w i l l maximize employment and minimize the quantity of gravel required f o r highway maintenance. DOH has the f o l l o w i n g a c t i o n s a v a i l a b l e : 1. do nothing, 2. r o u t i n e maintenance ( f i l l cracks and po t h o l e s ) , 3. seal coat and chips (waterstop with rock chips r o l l e d f o r good t i r e f r i c t i o n ) , C O w > H EH C J W PQ O C O o EH U <fj PM w En D H EH EH 00 EH H D USERS PROVIDE SAFETY PROVIDE ECONOMY MINIMIZE DELAY TEXTURE ROUGHNESS ROUGHNESS DRAINABILITY | ACTION SAFE TRAFFIC 10 ADT EXTRA TRAFFIC OPERATING DELAY COST SAVED PREVENTED i o 6 $ /YEAR 10 HOURS / YEAR MINIMIZE COST ACTION COST SAVED i o 6 $ /YEAR PUBLIC MAXIMIZE ACCESS STRENGTH TRAFFIC SECURED ACCESS 10 ADT CREATE JOBS ACTION • JOBS CREATED NO. OF JOBS/ YEAR CONSERVE GRAVEL ACTION VOLUME OF GRAVEL SAVED TONNES /YEAR NONUSERS MINIMIZE NOISE TEXTURE RESIDENTS PROTECTED FROM NOISE 10" HOUSEHOLDS FIGURE 10: STRUCTURE OF OBJECTIVES, CONTROLLABLE FACTORS AND ATTRIBUTES ON ^ 1 k. o v e r l a y ( l a y e r o f a s p h a l t c o n c r e t e l a i d o n e x i s t i n g p a v e m e n t ) . O n e o f t h e s e a c t i o n s w i l l b e i m p l e m e n t e d e v e r y y e a r a n d DOH w i s h e s t o d e t e r m i n e t h a t s e q u e n c e o f a c t i o n s w h i c h w i l l s a t i s f y t h e o b j e c t i v e s i n a b e s t w a y p o s s i b l e w i t h i n a p l a n n i n g h o r i z o n . DOH r e c o g n i z e s t h e p r o b a b i l i s -t i c n a t u r e o f p a v e m e n t b e h a v i o u r a n d w i l l r e p r e s e n t t h e l a w s o f p a v e m e n t d e t e r i o r a t i o n b y M a r k o v t r a n s i t i o n m a t r i c e s . F i g u r e 1 0 o u t l i n e s t h e o b j e c t i v e s , t h e r e l a t e d f a c t o r s c o n t r o l l a b l e b y t h e p a v e m e n t m a n a g e r , a n d t h e m e a s u r e s o f e f f e c t i v e n e s s ( a t t r i b u t e s ) f o r a l l o b j e c t i v e s . T h e a t t r i b u t e s a r e e i t h e r a d o p t e d f r o m h i g h w a y t r a n s p o r t a -t i o n p l a n n i n g o r a s s u m e d a d h o c . T h e y : a r e f u r t h e r e x p l a i n e d i n C h a p t e r 8 . 3 -S a f e t y c a n b e m e a s u r e d b y t h e n u m b e r o f a c c i d e n t s t h a t o c c u r d u e t o i n a d e q u a t e p a v e m e n t s . M i n i m i z i n g t h e n u m b e r o f a c c i d e n t s c a n b e a c c o m p l i s h e d b y c o n t r o l l i n g t h e c u a s a t i v e p a v e m e n t c h a r a c t e r i s t i c s : d r a i n a b i 1 i t y , t e x t u r e a n d c o l o u r . A s s u m e t h a t l i g h t c o l o u r a n d h a r s h m i c r o t e x t u r e a r e a l w a y s p r o v i d e d a s a m a t t e r o f g o o d e n g i n e e r i n g p r a c t i c e . T h e n s a f e t y i s a f u n c t i o n o f d r a i n a b i l i t y m e a s u r e d b y d e p t h o f r u t s a n d m a c r o t e x t u r e m e a s u r e d b y d e p t h o f p a v e m e n t s u r f a c e t e x t u r e . 8 . 3 • P r o b a b i 1 i t i e s a n d U p d a t i n g T h e s t a t e o f t h e p a v e m e n t s y s t e m c a n b e d e s c r i b e d b y a n u m b e r o f v a r i a b l e s . T h e s e v a r i a b l e s a r e e i t h e r u n d e r t h e p a v e m e n t m a n a g e r ' s c o n t r o l ( p a v e m e n t c o n d i t i o n v a r i a b l e s ) o r o u t s i d e h i s c o n t r o l ( t r a f f i c , n u m b e r o f r e s i d e n t s ) . T h e c o n d i t i o n o f p a v e m e n t s c a n b e p r e d i c t e d f r o m t h e i r p a s t p e r f o r -m a n c e b y t h e M a r k o v t r a n s i t i o n m o d e l . A s t a t e o f t h e p a v e m e n t i s j o i n t l y d e f i n e d b y a n u m b e r o f p a v e m e n t v a r i a b l e s s u c h a s r o u g h n e s s , t e x t u r e , r u t t i n g a n d s t r e n g t h . E a c h o f t h e s e f o l l o w s i t s o w n l a w o f d e t e r i o r a t i o n g i v e n b y a M a r k o v t r a n s i t i o n m a t r i x w h i c h d e p e n d s o n t h e a c t i o n a p p l i e d . T h i s l a w m a y b e c o n d i t i o n a l o n t h e v a l u e s o f o t h e r v a r i a b l e s a n d a l s o m a y c h a n g e w i t h t i m e , b u t t h e s e v a r i a t i o n s a r e n o t i n c l u d e d h e r e f o r t h e s a k e o f s i m p l i c i t y o f p r e s e n t a t i o n . M a r k o v t r a n s i t i o n m a t r i c e s f o r p a v e m e n t c o n d i t i o n v a r i a b l e s c a n b e e a s i l y o b t a i n e d f r o m p a v e m e n t c o n d i t i o n s u r v e y s . C o n s i d e r c o n s t r u c t i o n o f t h e m a t r i x f o r a p a v e m e n t v a r i a b l e u n d e r a c t i o n k : J I n o r d e r t o e s t i m a t e p . j ( k ) t a k e a s a m p l e o f s i z e n j ( k ) o n t h o s e p a v e m e n t s w h i c h w e r e a t s t a t e i o n e s t a g e a g o a n d w e r e s u b j e c t e d t o a c t i o n k s i n c e t h a t t i m e . C o u n t t h e n u m b e r o f p a v e m e n t s i n t h e s a m p l e w h i c h a r e i n s t a t e j a t p r e s e n t , r j j ( k ) . T h e b e s t e s t i m a t e o f P j j ( k ) i s s i m p l y t h e r a t i o o f r j j ( k ) a n d n ; ( k ) . . P a v e m e n t n e t w o r k s a r e u s u a l l y s u r v e y e d p e r i o d i c a l l y a n d d a t a s o o b t a i n e d p r o v i d e a b a s i s f o r g r a d u a l u p d a t i n g o f M a r k o v t r a n s i t i o n m a t r i c e s b y u s i n g B a y e s 1 f o r m u l a . S a m p l i n g f o r r . j ^ m a y b e r e g a r d e d B e r n o u l l i t r i a l s a n d t h e n t h e s a m p 1 e - 1 i k e l i h o o d f u n c t i o n i s g i v e n b y t h e b i n o m i a l f o r m u l a . I t i s c o m p u t a t i o n a l l y c o n v e n i e n t t o r e p r e s e n t t h e p r i o r i n f o r m a t i o n a b o u t p . j b y a b e t a d i s t r i b u t i o n , f , w h i c h i s - c o m p a t i b l e w i t h ; o r a . c o n j u g a t e o f , t h e b i n o m i a ! s a m p l e . 1 1 k e l i h o o d ' f u n c t i o n , L . S m i t h (30) h a s s u g g e s t e d t h e u s e o f a n o r m a l c o n j u g a t e b u t t h e m e t h o d p r e s e n t e d h e r e a p p e a r s t o b e m o r e n a t u r a l f o r a t l e a s t t w o r e a s o n s . F i r s t , w h i l e t h e G a u s s i a n a p p r o x i m a t i o n o f t h e p r i o r d i s t r i b u t i o n m a y b e g o o d a r o u n d t h e m e a n , i t c a n n o t b e g o o d i n t h e t a i l s b e c a u s e p . ^ i s a r a n d o m v a r i a b l e f r o m a r e s t r i c t e d i n t e r v a l o f n u m b e r s (0,1) r a t h e r t h a n f r o m a n i n f i n i t e i n t e r v a l ( - 0 0 , 0 0 ) . S e c o n d , t h e r e i s n o r e a s o n t o a s s u m e t h a t f " , L and f are bell-shaped and symmetric as implied by the normal d i s t r i b u t i o n . The p r i o r beta d i s t r i b u t i o n i s given by f ' C p . j ) = B , p . j e x p ( r i j , - l ) ( l - p . j ) e x p ( n l . - r j j ' - l ) (8.1) where B1 i s a normalizing constant, and the primed symbols r e f e r to a sample c o l l e c t e d at previous survey. The mean and variance are: P,j - r . j ' / n , ' (8.2) o1 = P;:(1-P..)/(n f'+!). (8.3) With the present a v a i l a b i l i t y of e l e c t r o n i c c a l c u l a t o r s and computers, the c l a s s of beta d i s t r i b u t i o n s does not seem to be less t r a c t a b l e than the normal d i s t r i b u t i o n . The c l a s s i s very f l e x i b l e f o r use in d e s c r i b i n g e m p i r i c a l data on p r o b a b i l i t i e s . The d i s t r i b u t i o n s can assume a wide v a r i e t y of symmetric and asymmetric shapes, i n c l u d i n g the b e l l , the re c t a n g l e , the t r i a n g l e and the J-shape. When the 1 ike 1ihood f u n c t i o n i s binomial and P beta, then f " i s al s o a beta d i s t r i b u t i o n with parameters r. ." = r. . + r 1 U i j i j i " = n i + n i ' that i s f " ( P i j ) = B" p j j e x p ( r i j " - l ) ( l - p . j ) e x p ( n i j " - - r.y-1) The s i m p l i c i t y of an updating procedure i s obvious. Pavement behaviour i s summarized at any time by r j j " and n.". A f t e r the next c o n d i t i o n survey i s c a r r i e d out, the present p o s t e r i o r information becomes p r i o r and i s updated to a new p o s t e r i o r by Bayes' theorem using the new sample. Once the p r i o r has been updated by the new sample, i t may be discarded, f o r i t s information contents i s now stored i n the new p o s t e r i o r . At any time t , the mean and variance f o r a Markov t r a n s i t i o n p r o b a b i l i t y are t t p n ( t ) = Z r At)/ Z n.-.(*) '.., J t=l J t-1 2 - t a = p . . ( t ) ( l - p . . ( t ) ) / ( l + E n.. ( t ) ) 'J 1 J t = l 1 J 71 • where r j j (t) and n.(t) are sample r e s u l t s obtained by c o n d i t i o n survey at the beginning of stage t . This updating procedure secures the most current information a v a i l a b l e f o r the pavement manager and minimizes record keeping. A consequence of Bayes' theorem i s of p a r t i c u l a r i n t e r e s t from an information value standpoint. In those cases where the p r i o r source of information or the other one i s r e l a t i v e l y weak, the shape of the p o s t e r i o r d i s t r i b u t i o n i s very nearly that of the dominant d i s t r i b u t i o n . When a l t e r -n a t i v e d i s t r i b u t i o n s are a v a i l a b l e , f o r example one based on a c o n d i t i o n survey and one based on p r e d i c t i v e mechanistic formulae, then the analyst can choose that one which i s more "peaked". Moreover, whatever p r i o r pro-b a b i l i t i e s are adopted at time t = l , they w i l l be swamped by the data incoming at subsequent c o n d i t i o n surveys. They can thus be chosen uniformly d i s t r i b u t e d i f no b e t t e r estimate i s a v a i l a b l e at t = l . It i s p o s s i b l e to set up Markov t r a n s i t i o n matrices based on engi-neering judgement, without having the data from pavement c o n d i t i o n surveys a v a i l a b l e . Engineering judgement represents an experience that i s often superior to the r e s u l t s of p r e d i c t i o n s based on l i m i t e d c o n d i t i o n surveys and a n a l y t i c a l models. In order to tap t h i s experience and to s t r u c t u r e i t to a form s u i t a b l e f o r d e c i s i o n a n a l y s i s one can proceed two ways. One, ask the engineer f o r an estimate of r . . ' and n.' and then use these in (8.1). Two, ask f o r estimates of p j j and a in (8.2) and (8.3), solve f o r r . ^ 1 and n j 1 and use these in (8.1). Because other work (30,33,3*0 have demonstrated a good command of the p r o b a b i l i s t i c part of d e c i s i o n a n a l y s i s f o r pavements, t h i s t h e s i s puts less emphasis on p r o b a b i l i t i e s r e l a t i v e to u t i l i t i e s . Although a p r a c t i c a l a p p l i c a t i o n w i l l operate with about three to f i v e s t a t e s to describe each pavement c o n d i t i o n v a r i a b l e , the program uses only two s t a t e s - acceptable and unacceptable, and a l l Markov t r a n s i t i o n matrices are of the order of two. This s i m p l i f i c a t i o n o f f e r s advantages as well as disadvantages. The programming i s kept reasonably simple and program outputs can be evaluated by i n t u i t i o n . Due to an i n t e r a c t i o n between p r o b a b i l i t i e s and u t i l i t i e s in the d e c i s i o n model the r e s u l t s may be d i s t o r t e d , however, even i f the u t i l i t y part is accurate. This f a c t should be remembered when r e s u l t s of the present program are i n t e r p r e t e d . 8.4.Consequences Consequences are vectors which c o n s i s t of a l l a t t r i b u t e s and accrue in each stage. The numerical values of a t t r i b u t e s depend on the s t a t e at stage's beginning, on the a c t i o n chosen, and on the s t a t e at stage's end. It i s assumed in the presented model that a c t i o n s are applied at stage's beginning and the s t a r t i n g s t a t e changes i n s t a n t l y i n t o the end s t a t e . A consequence thus depends on the a c t i o n and on the end s t a t e only. This assumption may d i s t o r t r e s u l t s f o r short planning horizons. If t h i s is a v a l i d concern, then consequences can be made dependent on the s t a r t i n g s t a t e . The refinement i s omitted in t h i s example. To c a l c u l a t e a consequence given end s t a t e , the s t a t e v a r i a b l e s must enter r e s p e c t i v e a t t r i b u t e f u n c t i o n s . The s t a t e i s described by t r a f f i c , number of r e s i d e n t s and by pavement c o n d i t i o n v a r i a b l e s . A p a r t i c u l a r a t t r i b u t e , however, depends on a subset of these v a r i a b l e s , as explained below. The worst values f o r a l l a t t r i b u t e s are chosen at zero and the best values are always greater than zero. This a s c e r t a i n s increasing u t i l i t y funct ions. The t r a f f i c and number of re s i d e n t s are assumed constant in the example. If necessary, a time f u n c t i o n can c a l c u l a t e f u t u r e values of these v a r i a b l e s , while p r o b a b i l i t y d i s t r i b u t i o n s can express u n c e r t a i n t y about these va1ues. T r a f f i c s a f e t y i s measured by the proportion of t r a f f i c that i s prevented from hazards a r i s i n g from inadequate pavement p r o p e r t i e s . The best l e v e l of sa f e t y i s achieved when a l l t r a f f i c i s sa f e , the worst when a l l t r a f f i c i s exposed to hazards. The a t t r i b u t e f o r s a f e t y i s a f u n c t i o n of t r a f f i c , pavement d r a i n a b i l i t y and t e x t u r e . For the simple case of e i t h e r acceptable or unacceptable pavement v a r i a b l e s , X ] = Safe T r a f f i c = F ( D ) * F ( T ) * T r a f f i c where D and T denote d r a i n a b i l i t y (depth of ruts) and texture (depth of macrotexture), r e s p e c t i v e l y ; F(*) = 1 i f pavement v a r i a b l e acceptable and F(•) = 0 otherwi se. Road user economy i s measured by the d o l l a r savings in v e h i c l e operating costs which are achieved r e l a t i v e to a rough pavement c o n d i t i o n . Highest savings o b t a i n on smooth pavements and zero savings - on the roughest pavements, X 2 = Operating Cost Savings = = F(R)*Length*Traffic*Cost where F(R) = 1 i f pavement roughness acceptable, F(R) = 0 otherwise; Length = t o t a l length of lanes on the road s e c t i o n ; Cost = v e h i c l e operating cost per one v e h i c l e kilometre t r a v e l l e d on rough pavement (59). User time delay is an a t t r i b u t e that depends on pavement roughness and a c t i o n taken. Rough surfaces cause the road users to slow down and so do maintenance operations on the road. There e x i s t accurate data which make i t p o s s i b l e to c a l c u l a t e both types of delays, but a simpler formula i s used here. To make the u t i l i t y f u n c t i o n i n c r e a s i n g , the savings in t r a v e l time r e l a t i v e to the worst c o n d i t i o n are chosen f o r the a t t r i b u t e . X-j = Travel Time Savings = = Delay -(F(R)*Length*Cyc1es + + E(A)/365)*Time*Traffic w h e r e D e l a y = m a x i m u m d e l a y p o s s i b l e ; F ( R ) = 1 i f r o u g h n e s s u n a c c e p t a b l e , F ( R ) = 0 o t h e r w i s e ; L e n g t h = t o t a l l e n g t h o f l a n e s o n t h e r o a d s e c t i o n ; C y c l e s = n u m b e r o f v e h i c l e s l o w d o w n c y c l e s p e r k m ; T i m e = a v e r a g e e x c e s s t i m e p e r s l o w d o w n c y c l e ; E ( A ) = n u m b e r o f d a y s t h a t a m a i n t e n a n c e o p e r a t i o n w i l l c a u s e t r a f f i c s l o w d o w n s . B o t h C y c l e s a n d T i m e a r e t y p i c a l v a l u e s c o m -p i l e d f r o m (59), w h i l e E ( A ) i s a s s u m e d f o r e a c h a c t i o n . A g e n c y c o s t i s e q u a l t o t h e s a v i n g s i n e x p e n d i t u r e s o n p a v e m e n t m a i n t e n a n c e r e l a t i v e t o t h e m o s t e x p e n s i v e m a i n t e n a n c e a c t i o n . T h e a c c e s s o b j e c t i v e r e f l e c t s t h e f a c t t h a t o n e o f t h e m o s t i m p o r t a n t t a s k s o f p a v e m e n t e n g i n e e r s i s t h e p r o t e c t i o n o f t h e f a c i l i t y f r o m s t r u c -t u r a l d e t e r i o r a t i o n . T h e s e r i o u s n e s s o f t h i s c o n s e q u e n c e i n c r e a s e s w i t h t h e i m p o r t a n c e w h i c h g i v e n r o a d s e c t i o n h a s i n a t r a n s p o r t a t i o n n e t w o r k . L e t t h e a t t r i b u t e b e , X[j = A c c e s s P r o v i d e d = F ( S ) " T r a f f i c w h e r e F ( S ) =1 i f p a v e m e n t s t r e n g t h m e a s u r e d f . e x . b y t h e B e n k e l m a n b e a m i s a c c e p t a b l e , a n d F ( S ) = 0 o t h e r w i s e . T h e n u m b e r o f j o b s t h a t a r e n e c e s s a r y f o r c a r r y i n g o u t a m a i n t e n a n c e a c t i o n i s a s t r a i g h t f o r w a r d a t t r i b u t e f o r t h e e m p l o y m e n t o b j e c t i v e . " O v e r l a y i n g " i s m o s t l a b o r i o u s a n d " d o n o t h i n g " d o e s n o t c r e a t e a n y e m p l o y m e n t . G r a v e l s a v e d i s m e a s u r e d i n t o n n e s r e l a t i v e t o t h e a c t i o n w h i c h c o n -s u m e s m o s t o f t h e m a t e r i a l . " D o n o t h i n g " a n d " r o u t i n e m a i n t e n a n c e " c a n p r o v i d e l a r g e s t s a v i n g s i n g r a v e l . T h e e f f e c t o f t r a f f i c n o i s e o n n o n u s e r s i s a l s o m e a s u r e d b y a s i m p l e f u n c t i o n . T h e n o i s e i s g e n e r a t e d b y t i r e s o n d e e p l y t e x t u r e d p a v e m e n t s . W h e n p a v e m e n t t e x t u r e . i s s h a l l o w , t h e n o i s e l e v e l d o e s n o t a f f e c t a n y o f t h e r e s i d e n t s i n t h e n o i s e z o n e a l o n g t h e r o a d s e c t i o n . W h e n p a v e m e n t t e x -t u r e i s d e e p , a l l r e s i d e n t s i n t h e z o n e a r e a f f e c t e d : Xg = Noise Prevented = F(T)"Residents where F(T) = 0 i f pavement texture i s deep enough to cause unacceptable noise l e v e l s and F(T) = 1 otherwise; Residents = number of households in the noise zone. 8 . 5 - E 1 i c i t a t ion of Preferences Five cooperative persons were asked to imagine themselves in the r o l e of a pavement manager. They were motivated by the p o s s i b i l i t y of e x p l o r i n g t h e i r own preferences i n a d e c i s i o n a n a l y t i c a l game. A l l of these people are engineers and i t i s l i k e l y that t h e i r value judgements are s i m i l a r to those of a pavement manager. Each subject was introduced to the d e c i s i o n problem i n d i v i d u a l l y . He or she was asked to act as an: i m p a r t i a l d e c i s i o n maker who i s not sub-j e c t e d to any i n s t i t u t i o n a l , personal or p o l i t i c a l pressures. The person f i r s t became f a m i l i a r w i t h the d e c i s i o n scenario and with the o b j e c t i v e s as in Chapters 8.2 and 8.4. If the subject was showing signs of misunder-standing, a d d i t i o n a l explanations were provided. The e l i c i t a t i o n of preferences was c a r r i e d out in two phases. The f i r s t phase assessed preferences of the f i v e subjects q u a l i t a t i v e l y in order to i d e n t i f y p o s s i b l e i n t e r a c t i o n s between o b j e c t i v e s . The second phase e l i c i t e d u t i l i t y f u n c t i o n s from one subject f o r use in the d e c i s i o n model . Q u a l i t a t i v e Assessment The e l i c i t a t i o n was aided by a s e r i e s of forms on which the subjects marked t h e i r answers. The p r e f e r e n t i a l independence was tested using a square graph (Appendix A) on which the o r d i n a t e represents values of one a t t r i b u t e , X., and the absc i s s a represents another a t t r i b u t e , X^. The o r i g i n represents an outcome with both X. and X^ at the worst l e v e l s and the top r i g h t corner represents an outcome with both a t t r i b u t e s at the best 76. l e v e l s . The s q u a r e thus c o n t a i n s a l l p o s s i b l e p a i r s ( X j , X ^ ) . The v a l u e s a t w h i c h t h e complementary a t t r i b u t e s a r e h e l d f i x e d a r e p r e s e n t e d i n an accompanying t a b l e . T r a d e - o f f s s h a l l be i n v e s t i g a t e d i n the whole domain ( X j , X ^ ) . For p r a c t i c a l r e a s o n s , however, o n l y t h r e e p o i n t s were chosen: t h e upper r i g h t c o r n e r , t h e c e n t r e o f t h e s q u a r e , and a p o i n t c l o s e r t o t h e lower l e f t c o r n e r . The s u b j e c t i s shown the square and the t a b l e , and i s asked w h i c h o f the two a t t r i b u t e s would he or she r a t h e r drop from t h e c u r r e n t v a l u e t o t h e w o r s t v a l u e , g i v e n o t h e r a t t r i b u t e s have l e v e l s as i n the t a b l e . I f the answer i s X j , t h e s u b j e c t must e s t i m a t e how much of Xj s h o u l d be g i v e n up i n exchange f o r k e e p i n g X^ a t t h e c u r r e n t v a l u e . T h i s answer i s marked on the s q u a r e . Each of t h e t h r e e ( X j , X^) p a i r s was t e s t e d w i t h the f o l l o w i n g f o u r c a ses o f t h e complementary a t t r i b u t e s , i n o r d e r t o i n v e s t i g a g e whether t r a d e - o f f s change when t h e l e v e l s o f a t t r i b u t e s change: Case 1 = a l l complementary a t t r i b u t e s a t t h e b e s t l e v e l s , Case 2 = a l l complementary a t t r i b u t e s a r e b e s t , e x c e p t " J o b s " , " G r a v e l " and " N o i s e " w h i c h a r e a t the w o r s t l e v e l s , Case 3 = a l l complementary a t t r i b u t e s a t the b e s t l e v e l s , e x c e p t the r o a d - u s e r ones, w h i c h a r e a t the w o r s t l e v e l s , Case 4 = al1 complementary a t t r i b u t e s a t the w o r s t l e v e l s . These f o u r c a s e s a r e c o n t a i n e d on one s h e e t (see Appendix A ) . For n = 8 a t t r i b u t e s , n - 1 = 1 s h e e t s a r e r e q u i r e d t o t e s t t r a d e - o f f s . Thus, 3 x 4 x 7 = 84 q u e s t i o n s (3 q u e s t i o n s p e r s q u a r e , 4 s q u a r e s per s h e e t , 7 s h e e t s ) would have t o be answered by each s u b j e c t . T h i s number appears t o be h i g h , but answers f o l l o w r e l a t i v e l y f a s t a f t e r a "warming-up" p e r i o d o f about f i v e q u e s t i o n s . Moreover, n u m e r i c a l e s t i m a t e s a r e not r e q u i r e d f o r a q u a l i t a t i v e s t u d y and s t a t e m e n t s by t h e s u b j e c t s about t r a d e - o f f s r e l a t i v e t o p r e v i o u s l y answered cases were regarded as s u f f i c i e n t answers. 77. T h e f i v e s u b j e c t s w e r e d i v i d e d i n t o t w o g r o u p s . T h e f i r s t g r o u p ( s u b j e c t s A , D , E ) f i r s t m a d e t r a d e - o f f s o n a l l o f t h e t o p s q u a r e s o f r a n d o m l y o r d e r e d s h e e t s , t h e n o n a l l t h e s e c o n d s q u a r e s o f r e a r r a n g e d s h e e t s , t h e n o n t h e t h i r d s q u a r e s , a n d f i n a l l y o n t h e f o u r t h s q u a r e s . T h i s s c h e m e a s c e r t a i n s t h a t t h e s u b j e c t k e e p s i n m i n d t h e c o m p l e m e n t a r y a t t r i b u t e v a l u e s u n t i l a n e w c a s e o f c o m p l e m e n t a r y a t t r i b u t e s i s c o n s i d e r e d . T h e o r d e r o f s h e e t s w a s r a n d o m a n d d i f f e r e n t f o r e a c h c a s e . T h e s e c o n d g r o u p ( s u b j e c t s B , C ) a n s w e r e d q u e s t i o n s s h e e t b y s h e e t , b u t t h e o r d e r o f s q u a r e s o n e a c h s h e e t w a s k e p t r a n d o m . T h i s s e c o n d s c h e m e . r e q u i r e s t h a t t h e s u b j e c t c o n s t a n t l y c h a n g e t h e m e n t a l p i c t u r e o f t h e c o m p l e m e n t a r y a t t r i b u t e l e v e l s a s t h e q u e s t i o n s m o v e r a n d o m l y b e t w e e n t h e f o u r c a s e s . T h e p r o c e s s i s m o r e e x h a u s t i n g a n d t i m e - c o n s u m i n g t h a n t h e f i r s t s c h e m e . T h e f o r m s f o r t e s t i n g u t i l i t y i n d e p e n d e n c e a l s o c o n t a i n f o u r c a s e s o f c o m p l e m e n t a r y a t t r i b u t e s , b u t h a v e f o u r l o t t e r i e s i n s t e a d o f f o u r t r a d e -o f f s q u a r e s ( s e e A p p e n d i x A ) . T h e s u b j e c t i s a s k e d t o e s t i m a t e t h e c e r t a i n t y e q u i v a l e n t o f a l o t t e r y i n v o l v i n g o n e a t t r i b u t e , c o n d i t i o n a l o n t h e r e m a i n i n g a t t r i b u t e s b e i n g h e l d f i x e d . A t o t a l o f 3 x h = 12 c e r t a i n t y e q u i v a l e n t s (3 p o i n t s f o r o n e c o n d i t i o n a l u t i l i t y c u r v e a s F i g u r e 6, k c a s e s t o c o n d i -t i o n u p o n ) w o u l d h a v e t o b e g i v e n b y e a c h s u b j e c t . H o w e v e r , i t w a s j u d g e d t h a t a s k i n g f o r C Q a n d e i t h e r CQ . 2 5 o r C g i s s u f f i c i e n t i n a q u a l i -t a t i v e s t u d y . C o n s e q u e n t l y , t w o c e r t a i n t y e q u i v a l e n t s w e r e e l i c i t e d f r o m e a c h s u b j e c t f o r o n e a t t r i b u t e . R e s u l t s o f t h e q u a l i t a t i v e a s s e s s m e n t a r e p r e s e n t e d i n A p p e n d i x C a n d a r e s u m m a r i z e d i n T a b l e 3. T h e p r e f e r e n t i a l a n d u t i l i t y i n d e p e n d e n c e a s s u m p t i o n s a p p e a r t o b e a g o o d a p p r o x i m a t i o n f o r t h r e e o f t h e f i v e s u b j e c t s , a n d m a y b e a w o r k a b l e a p p r o x i m a t i o n f o r t h e r e m a i n i n g t w o s u b j e c t s . S u b j e c t V a r i a t i o n i n Trade-Offs V a r i a t i o n i n C e r t a i n t y E q u i v a l e n t s A I n v a r i a n t f o r 7 p a i r s I n v a r i a n t f o r 2 a t t r i b u t e s B I n v a r i a n t f o r 7 p a i r s I n v a r i a n t f o r 2 a t t r i b u t e s C I n v a r i a n t f o r 3 p a i r s . Changed f o r 3 p a i r s by l e s s than 10%. Dropped 1 a t t r i b u t e . I n v a r i a n t f o r 1 a t t r i b u t e . Changed f o r 1 a t t r i b u t e . D I n v a r i a n t f o r 6 p a i r s . Changed f o r 1 p a i r by l e s s than 10%. I n v a r i a n t f o r 2 a t t r i b u t e s E I n v a r i a n t f o r 7 p a i r s I n v a r i a n t f o r 2 a t t r i b u t e s TABLE 3: VARIATION OF TRADEOFFS AND CERTAINTY EQUIVALENTS WITH LEVELS OF COMPLEMENTARY ATTRIBUTES C O Q u a n t i t a t i v e Assessment Preferences were e l i c i t e d from subject E two days a f t e r the f i r s t phase. U t i l i t y f u n c t i o n s over s i n g l e a t t r i b u t e s were assessed by the cer-t a i n t y equivalent method as in Figure 6. Each u t i l i t y curve was smoothed by hand through the f i v e p o i n t s , and the HALF value was estimated from the curve. The HALF values were used as an input i n t o the computer program to approximate u t i l i t y f u n c t i o n s by the exponential form based on a theory developed i n Appendix B. Although these forms do not reproduce the hand-smoothed curves e x a c t l y the d i s c r e p a n c i e s are comparable to the u n c e r t a i n -t i e s inherent i n smoothing. A f t e r the u t i l i t y curves were e s t a b l i s h e d the subject was asked to rank order the a t t r i b u t e s . They were a l l set at t h e i r l e a s t d e s i r a b l e l e v e l s equal zero, and the subject had to decide which one he would most l i k e to have at i t s best l e v e l . The s c a l i n g f a c t o r associated with t h i s a t t r i b u t e i s the l a r g e s t . The a t t r i b u t e he would next p r e f e r to have alone at i t s most d e s i r a b l e l e v e l has the second la r g e s t s c a l i n g f a c t o r , and so on. The f o l l o w i n g ranking was obtained. k 5 > k l > k 2 > k 3 > k 4 - > k 7 > k 8 > k 6 - ( 8 - 6 ) Three i n e q u a l i t i e s were a l s o e l i c i t e d f o r consistency checks, k 6 + k 7 + k 8 < k 2 (8.7a)' k 2 + k 3 < k] (8.7b) k 1 + k 2 + k 3 > k 5 (8.7c) To enumerate the s e a l i n g constants, t r a d e - o f f s were considered on p a i r s of a t t r i b u t e s . The subject f i r s t chooses the a t t r i b u t e , X^, which he would rather push from zero to the best value, x^* and the subject has to f i n d an i n d i f f e r e n t lower value, -x^. The i n d i f f e r e n c e p a i r implies an equation of the form, k i = k k V A ) Seven i n d i f f e r e n c e p a i r s provided seven equations of the above form. 8 0 . T h e s e e q u a t i o n s a r e n o t i n d e p e n d e n t , a n d a n a d d i t i o n a l o n e i s , r e q u i r e d i n o r d e r t o f i n d a u n i q u e s o l u t i o n f o r t h e v e c t o r o f s c a l i n g c o n s t a n t s . T h e e q u a t i o n w a s s o u g h t b y a s k i n g t h e s u b j e c t t h i s q u e s t i o n : H e r e i s a l o t t e r y y i e l d i n g a l l a t t r i b u t e s a t t h e i r b e s t l e v e l s w i t h p r o b a -b i l i t y p o r a l 1 a t t r i b u t e s e q u a l z e r o w i t h p r o b a b i l i t y 1 - p . E s t i m a t e p s o t h a t t h e l o t t e r y b e c o m e s i n d i f f e r e n t t o a c o n s e q u e n c e w i t h S A F E T Y a t i t s b e s t l e v e l a n d a l l o t h e r a t t r i b u t e s e q u a l z e r o . T h e a n s w e r t o t h i s q u e s t i o n c a m e w i t h m u c h m o r e d i f f i c u l t y c o m p a r e d t o t r a d e - o f f q u e s t i o n s . T h e s u b j e c t e v e n t u a l l y e s t i m a t e d a b a n d f o r p r a t h e r t h a n a s i n g l e v a l u e . S i m i l a r q u e s t i o n w a s r e p e a t e d a s a c o n s i s t e n c y c h e c k w i t h U S E R E C O N O M Y a s t h e s i n g l e a t t r i b u t e a t t h e b e s t l e v e l . T h e a n s w e r c a m e a g a i n i n a b a n d e d f o r m , a n d t w o a d d i t i o n a l t r a d e - o f f q u e s t i o n s w e r e p o s e d . T h u s a s e t o f t w e l v e e q u a t i o n s r e s u l t e d , i n c l u d i n g f o u r r e d u n -d a n t e q u a t i o n s f o r c o n s i s t e n c y c h e c k s . T o s o l v e t h e s e t , i t w a s f i r s t n e c e s s a r y t o f i x t h e t w o i m p r e c i s e s c a l i n g f a c t o r s t h a t w e r e g i v e n i n a b a n d e d f o r m : k j = 0.60 ± 0 . 1 0 k ^ = 0 . 2 0 ± 0 . 0 5 T h e l o w , t h e m e d i u m , a n d t h e h i g h v a l u e o f k j w a s s u b s t i t u t e d i n t o t h e s e t a n d t h u s t h r e e s o l u t i o n . . v e c t o r s w e r e o b t a i n e d . T h e s a m e p r o c e d u r e w a s r e p e a t e d w i t h k ^ . T h e e x e r c i s e r e v e a l e d t w o p r o b l e m s . F i r s t , t h e o b t a i n e d v a l u e s f o r k / p k ^ , k y a n d k g d i s a g r e e d w i t h ( 8 . 6 ) . S e c o n d , k^> 0 . 1 8 c a u s e d k r , t o e x c e e d u n i t y . T h e f i r s t p r o b l e m w a s f o u n d t o b e i m p l a n t e d i n i n c o n s i s t e n t t r a d e - o f f s i n v o l v i n g X / j , X ^ , X y a n d X g . T h e s e w e r e r e e x a m i n e d w i t h t h e s u b j e c t , a n d t h e t r a d e - o f f s w e r e c o r r e c t e d . T h e s e c o n d p r o b l e m s i m p l y i n d i -c a t e d t h a t t h e s u b j e c t h a s s p e c i f i e d h i s b a n d t o o w i d e a n d a l l s o l u t i o n s c o n -t a i n i n g k , > 0 . 1 8 m u s t b e n e g l e c t e d a s i n c o n s i s t e n t . The s e t o f e q u a t i o n s was now s o l v e d f o r t h e t h r e e v a l u e s o f k j , and f o r k^ = 0.15 and k^ = 0.17- The s o l u t i o n s c o m p l i e d t o (8.6) and (8.7) but o n l y t h o s e o b t a i n e d by s u b s t i t u t i n g k^ = 0.60 and k^ = 0.15 were i n a r e a s o n a b l e agreement. In f a c t , the agreement was t o l e s s than \% a b s o l u t e e r r o r . T h i s e r r o r was c o r r e c t e d by t h e s u b j e c t t h r o u g h a f i n e - t u n i n g o f two t r a d e - o f f s . The f o l l o w i n g s e t of c o n s i s t e n t e q u a t i o n s r e s u l t e d : k 2 = k, u, ( 3 30 x 10 ADT) = 0.60 k, k 8 = k, u, ( 8 x 1 0 3 ADT) = 0.16 k. k 3 = k 2 u 2 1 6 2.4 x 10. $/year) = 0..67 k 2 k 4 k 2 u 2 6 1.5 x 10 $/year) = 0.42 k 2 k 6 = k 2 u 2 6 0.7 x 10 $/year) = 0.20 k 2 k k = k 5 U 5 ( 3 6 x 10 ADT) = 0.18 k 5 k 6 = k y u 7 ( 16.0 t / y e a r ) = 0.58 k 7 k 6 = k 8 u 8 J 5 0 0 h o u s e h o l d s ) = 0.75 kg k 8 = k^ u/j 6 ,0.9 x 10 $/year) = 0.63 k^ k l = 0.60 k 4 = 0.15. l o w i ng s o l u t i o n o b t a i n s : k l = 0.600 k 2 = O.36O k 3 = 0.240 k k = 0.151 k 5 = 0.839 k 6 = 0.072 ; k7 = 0.124 k 8 = 0.096. The sum o f t h e s e a l i n g c o n s t a n t s i s g r e a t e r than one. Thus, t h e m u l t i -a t t r i b u t e u t i l i t y model i s m u l t i p l i c a t i v e , and t h e i n t e r a c t i o n f a c t o r k must be n e g a t i v e . I t i s computed by subprogram KEENEY (Appendix D). Time Dependence: The c h o i c e o f p l a n n i n g h o r i z o n must be a compromise between a n a l y -t i c a l c o n v e n i e n c e and r e a l i s m . For v e r y l a r g e time h o r i z o n s i t i s p o s s i b l e t o r e g a r d t h e Markov d e c i s i o n p r o c e s s as i f i t were t o c o n t i n u e i n d e f i n i t e l y . The t e r m i n a l r e t u r n s a r e so d i s t a n t t h a t t h e y have on l y a 1 1 i mi ted •- e f f e c t --on the d e c i s i o n maker's b e h a v i o u r . The o p t i m a l p o l i c y may become s t a t i o n a r y and w i l l c o n s t i t u t e a r u l e f o r t h e " b e s t " o p e r a t i o n o f t h e pavement system i n t he long r u n. The l a r g e h o r i z o n approach i s c l e a r l y u n r e a l i s t i c . The u n c e r t a i n t i e s about d i s t a n t f u t u r e become t o o l a r g e t o s e r v e a u s e f u l p u r p o s e . T e c h n o l o -g i c a l i n n o v a t i o n w i l l a f f e c t t r a f f i c and pavement m a t e r i a l s w h i c h w i l l cause the pavement d e t e r i o r a t i o n laws and the s e t o f a v a i l a b l e a c t i o n s t o change. The f i n a n c i a l p i c t u r e and the s o c i a l - p o l i t i c a l r e a l i t y even f o r a f u t u r e as c l o s e as ten y e a r s i s u n p r e d i c t a b l e . These d i s a d v a n t a g e s a r e d i m i n i s h e d f o r s h o r t p l a n n i n g h o r i z o n s . S t a t i o n a r i t y may r e a s o n a b l y be assumed f o r a l l the model's e l e m e n t s : p r o -b a b i l i t i e s , u t i l i t i e s and a c t i o n s . The e f f e c t o f t e r m i n a l r e t u r n s may, however, a f f e c t the r e s u l t s o f o p t i m i z a t i o n more than f o r a l o n g e r h o r i z o n . A s h o r t time h o r i z o n o f f i v e y e a r s i s assumed f o r t h i s example. The Markov t r a n s i t i o n p r o b a b i l i t i e s a r e s t a t i o n a r y w i t h i n t h i s p e r i o d and so a r e the u n i a t t r i b u t e u t i l i t y f u n c t i o n s and t h e s c a l i n g c o n s t a n t s i n the m u l t i -a t t r i b u t e u t i l i t y model. S u b j e c t E was t e s t e d q u a l i t a t i v e l y f o r h i s temporal p r e f e r e n c e s by a p r o c e d u r e t h a t i s o u t l i n e d i n C h a p t e r 7-4. He i n d i c a t e d temporal r i s k a v e r s i o n , but q u a n t i f i c a t i o n o f c o n s t a n t s f o r t h e a p p l i c a b l e model (7.8b) was not a t t e m p t e d because no e l i c i t a t i o n p r o c e d u r e i s y e t a v a i l a b l e . I n s t e a d , t h e temporal s c a l i n g c o n s t a n t s were a l l s e t equal 0 . 3 , which i m p l i e s temporal r i s k a v e r s i o n . 83. 8.6.Behaviour o f the Model The model as o u t l i n e d i n F i g u r e 9 i s programmed f o r computer. The program, named BELLMAN, i s d e s c r i b e d i n Appendix D and l i s t e d i n A p pendix E. The program was t e s t e d on a problem t h a t has been p r e v i o u s l y s o l v e d m a n u a l l y . R e s u l t s o f t h i s t e s t were i d e n t i c a l t o t h o s e o b t a i n e d by manual c o m p u t a t i o n . The example problem i s based p a r t on assumed d a t a and p a r t on r e a l d a t a . Assumed a r e Markov t r a n s i t i o n m a t r i c e s , r e s o u r c e r e q u i r e m e n t s f o r a c t i o n s , and d a t a f o r c a l c u l a t i o n o f consequences. Real d a t a a r e p a r a m e t e r s of p r e f e r e n c e s e l i c i t e d from one o f the s u b j e c t s ( C h a p t e r 8.5). The program p r i n t s out (Appendix F) the i n p u t s , and a l s o t h e com-puted j o i n t s t o c h a s t i c m a t r i c e s , the e x p o n e n t i a l u n i a t t r i b u t e u t i l i t y ; , c u r v e s , t h e m a t r i c e s o f consequences, the t y p e o f m u l t i a t t r i b u t e and temporal u t i 1 i t y mode], and t h e optimum d e c i s i o n s . The l a s t d e c i s i o n r u l e p r i n t e d o u t i s the s t a t i o n a r y r u l e , and i t i s o f g r e a t e s t i n t e r e s t f o r the l o n g - t e r m o p e r a t i o n of a pavement s e c t i o n . For each j o i n t s t a t e the s t a t i o n a r y r u l e s p e c i f i e s an a c t i o n w h i c h a f t e r i m p l e m e n t a t i o n w i l l r e s u l t i n consequences t h a t a r e o p t i m a l r e l a t i v e t o t h e d e c i s i o n maker's p r e f e r e n c e s . J o i n t s t a t e s a r e e x p l a i n e d i n T a b l e 4 i n terms of s t a t e s o f t e x t u r e , d r a i n a b i 1 i t y , roughness and s t r e n g t h . J o i n t s t a t e 1, f o r example, has a l l t h e s e pavement v a r i a b l e s a t a c c e p t a b l e l e v e l s , whereas i n j o i n t s t a t e 16 a l l v a r i a b l e s a r e a t u n a c c e p t a b l e l e v e l s . The s t e a d y - s t a t e d e c i s i o n r u l e f o r t h e o r i g i n a l d a t a i s shown i n T a b l e 5 under Run 1. Subsequent runs t e s t the r e s p o n s e of t h e model t o i n p u t v a r i a t i o n s . Run 2 has t h e s c a l i n g c o n s t a n t f o r AGENCY COST i n c r e a s e d t o 0.30 and f o r ACCESS d e c r e a s e d t o 0.60. The w e i g h t of SAFETY thus i n c r e a s e s r e l a t i v e t o ACCESS. The e f f e c t i s t h a t SEAL and ROUTINE s u p p l a n t OVERLAY in s t a t e s 3, 5 and 6. One can e x p e c t t h i s change, f o r OVERLAY i s t h e most e x p e n s i v e a c t i o n . The s u p p l a n t i n g a c t i o n s a r e n e a r l y as e f f e c t i v e t h r o u g h pavement t e x t u r e on t h e SAFETY o b j e c t i v e , w h i c h r e c e i v e s r e l a t i v e l y STATE OF PAVEMENT VARIABLE l = a c c e p t a b l e ; 2 = not a c c e p t a b l e JOINT STATE TEXTURE DR&INABILITY SOOGHNESS STRENGTH 1 1 1 1 2 2 1 1 1 3 1 2 1 1 4 2 2 1 1 5 1 1 2 1 6 2 1 2 1 7 1 2 2 1 8 2 2 2 1 9 1 1 1 2 10 2 1 1 2 11 1 2 1 2 12 2 2 1 2 13 1 1 2 2 14 2 1 2 2 15 1 2 2 2 16 2 2 2 2 TABLE 4: EXPLANATION OF JOINT STATES JOINT STATE STATIONARY DECISION RULES RUN 1 RUN 2 RUN 3 RUN 4 RUN 5 1 SEAL SEAL SEAL ' S E A L OVERLAY 2 ' SEAL ' SEAL ' SEAL • SEAL OVERLAY 3 OVERLAY ROUTINE OVERLAY ROUTINE OVERLAY OVERLAY OVERLAY OVERLAY ROUTINE OVERLAY 5 OVERLAY ROUTINE OVERLAY ROUTINE OVERLAY 6 OVERLAY SEAL OVERLAY ROUTINE OVERLAY 7 OVERLAY OVERLAY OVERLAY OVERLAY OVERLAY 8 OVERLAY OVERLAY OVERLAY ROUTINE OVERLAY 9 OVERLAY OVERLAY OVERLAY SEAL OVERLAY 10 OVERLAY , OVERLAY OVERLAY ROUTINE OVERLAY 11 OVERLAY • OVERLAY OVERLAY OVERLAY OVERLAY 12 OVERLAY OVERLAY OVERLAY ROUTINE OVERLAY 13 OVERLAY OVERLAY OVERLAY OVERLAY OVERLAY 14 OVERLAY OVERLAY OVERLAY OVERLAY OVERLAY 15 OVERLAY OVERLAY OVERLAY OVERLAY OVERLAY 16 OVERLAY OVERLAY OVERLAY OVERLAY OVERLAY TABLE 5: RESULTS OF FIVE COMPUTATIONAL RUNS h i g h w e i g h t i n Run 2, a f t e r t h e main c o m p e t i t o r , ACCESS, c a r r i e s s m a l l e r w e i g h t . Run 3 t e s t s whether the p o l i c y changes when t h e s c a l i n g c o n s t a n t s a r e n o r m a l i z e d t o sum up t o one. T h i s i m p l i e s an a d d i t i v e m u l t i a t t r i b u t e u t i l i t y model. W i t h g i v e n d a t a , t h e r e i s no change i n t h e o p t i m a l p o l i c y . The r e s u l t s h o u l d not be g e n e r a l i z e d , however, s i n c e the o u t p u t i s a r e s u l t of i n t e r a c t i o n s between the u t i l i t y , the p r o b a b i l i t y and the o t h e r i n p u t s . I t i s c o n c e i v a b l e t h a t w i t h d i f f e r e n t d a t a and more a c c u r a t e model t h e r e would be a change. i:n-the o p t i m a l p o l i c y . Run k has w e i g h t s f o r a l l a t t r i -b u t e s e q u a l . The e f f e c t on the d e c i s i o n r u l e i s s u b s t a n t i a l , but OVERLAY i s s t i l l chosen f o r t h e most d e t e r i o r a t e d s t a t e s . Run 5 d e m o n s t r a t e s t h e e f f e c t o f Markov t r a n s i t i o n p r o b a b i l i t i e s on t h e p o l i c y . OVERLAY i s made more e f f e c t i v e on pavement t e x t u r e r e l a t i v e t o Run 1. The change i s from 0.50 t o 0.90 f o r p j ] and p^^ and c o n s e q u e n t l y SEAL i s e l i m i n a t e d from t h e d e c i s i o n r u l e . The o r i g i n a l s e t of i n p u t s i s not s u i t a b l e f o r t e s t i n g the temporal u t i l i t y s c a l i n g , because the s t e a d y -s t a t e r u l e o b t a i n s i n t h e f i r s t s t a g e i t e r a t i o n . A d i f f e r e n t d a t a s e t was made up w h i c h p roduces a t h r e e - s t a g e o p t i m a l p o l i c y . A change i n the tem-p o r a l s c a l i n g f a c t o r s from 0.3 t o 0.9 s l i g h t l y a f f e c t s the o p t i m a l p o l i c y , but the s t a t i o n a r y d e c i s i o n r u l e remains unchanged. However, f o r a s c a l i n g c o n s t a n t 0.5 f o r t h e most d i s t a n t s t a g e i n t h e f u t u r e and l i n e a r l y d e c r e a s i n g t o 0.1 f o r t h e p r e s e n t , the p o l i c y becomes more p r o t e c t i v e . ROUTINE i s s u p p l a n t e d by OVERLAY - an i n t u i t i v e l y c o r r e c t r e s u l t f o r a s i t u a t i o n w i t h t h e f u t u r e more " i m p o r t a n t " than t h e p r e s e n t . The computer runs have a l s o shown t h a t t h e model i s e c o n o m i c a l l y f e a s i b l e . W i t h f i v e s t a g e s , f o u r a c t i o n s , e i g h t a t t r i b u t e s , f o u r pavement v a r i a b l e s , and two s t a t e s f o r each v a r i a b l e , i t t a k e s f i v e seconds o f CPU t i m e t o c o m p i l e the program. Once c o m p i l e d t h e program r e q u i r e 0.7 CPU seconds t o e x e c u t e a problem. The e x e c u t i o n t i m e i s r o u g h l y p r o p o r t i o n a l t o the number of j o i n t s t a t e s N, N = l p where p = number o f pavement v a r i a b l e s , 1 = number o f s t a t e s f o r each pavement v a r i a b l e . Suppose t h a t a problem has p = k, 1 = 4 . Compared t o the p r e s e n t problem (p = k, 1 = 2 ) , t h e CPU w i l l need 16 t i m e s l o n g e r t i m e t o e x e c u t e , i . e . about 10 seconds. A problem w i t h p = 5 and 1 = 3 r e q u i r e s a p p r o x i m a t e l y t h e same e x e c u t i o n t i m e . At p r e s e n t r a t e s i t amounts t o about two d o l l a r s f o r CPU t i m e . CHAPTER 9 CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY T h i s c h a p t e r s t a r t s w i t h c o n c l u s i o n s r e g a r d i n g the a p p l i c a t i o n o f d e c i s i o n a n a l y s i s t o t h e maintenance o f one s e c t i o n . I t i s f o l l o w e d by a d i s c u s s i o n o f t h e pavement maintenance problem. Because the c o m p l e x i t i e s i n h e r e n t i n t h e pavement problem a r e w i d e s p r e a d i n e n g i n e e r i n g d e c i s i o n s , C h a p t e r 9-3 d i s c u s s e s t h e a p p l i c a b i l i t y o f d e c i s i o n a n a l y s i s t o o t h e r a r e a s of c i v i l e n g i n e e r i n g . C h a p t e r S.k s u g g e s t s s t u d y a r e a s w h i c h may be of i n t e r e s t n ot o n l y t o pavement management but a l s o t o c i v i l e n g i n e e r i n g i n genera 1. 9 . 1 . C o n c l u s i o n s The proposed d e c i s i o n model a p p l i e d t o the o p t i m i z a t i o n o f maintenance f o r one pavement s e c t i o n s a s t i s f i e s t he c h a r a c t e r i s t i c s r e q u i r e d from an i d e a l method ( F i g u r e 3)- The o n l y e x c e p t i o n i s network o p t i m i z a t i o n w h i c h was e x c l u d e d from t h e sco p e o f t h i s t h e s i s as i t r e q u i r e s more r e s e a r c h . M u l t i p l e o b j e c t i v e s a r e handled by t h e m u 1 t i a t t r i b u t e u t i l i t y a p proach. U n c e r t a i n t y o f pavement b e h a v i o u r i s e x p r e s s e d i n t h e form o f Markov t r a n s i t i o n m a t r i c e s t h a t a r e u p d a t e a b l e and can accommodate e n g i n e e r -ing judgement. R i s k s a r i s i n g from t h e s e u n c e r t a i n t i e s a r e encoded i n t h e u t i l i t y f u n c t i o n s f o r s i n g l e a t t r i b u t e s . Group o p i n i o n was not i n c l u d e d i n t o t h i s t h e s i s but can be r o u t i n e l y a c c o u n t e d f o r by any model t h a t has p r o b a b i 1 i t i e s and u t i l i t i e s f o r i n p u t s . Time dependence i s modeled by t h e temporal u t i l i t y a p p l i e d i n t h e Markov d e c i s i o n scheme. A l t h o u g h t h e s k e l e t o n o f t h e model i s c o n c e p t u a l l y s i m p l e , t h e m u 1 t i a t t r i b u t e and t h e temporal u t i l i t y components c r e a t e more c o m p l e x i t y . P o t e n t i a l i m p l e m e n t a t i o n o b s t a c l e s due t o c o m p l e x i t y a r e not i n s u r m o u n t a b l e and o c c u r or do not o c c u r depending on the a t t i t u d e towards i n n o v a t i o n and improvement i n the u s e r ' s o r g a n i z a t i o n . The model's elements have an " o v e r v i e w " e f f e c t , and a v o i d g o i n g i n t o too g r e a t d e t a i l t h a t o f t e n i s u n n e c e s s a r y . The Markov model o f pavement b e h a v i o u r summarizes the u n c e r t a i n e f f e c t s o f t r a f f i c , c l i m a t e and m a t e r i a l s , and f a c i l i t a t e s t h e i n c l u s i o n o f t h e time d i m e n s i o n . The m u l t i a t t r i b u t e u t i l i t y model a g g r e g a t e s incommensurable consequences i n t o one number, and the temporal u t i l i t y model makes i t p o s s i b l e t o reduce t h e number o f dimen-s i o n s by f u r t h e r a g g r e g a t i o n . Data r e q u i r e d by the model a r e commonly a v a i l a b l e . Markov t r a n s i t i o n m a t r i c e s become a s t a n d a r d i t e m o f i n f o r m a t i o n as the a p proach i s g a i n i n g r e c o g n i t i o n i n many departments o f t r a n s p o r t a t i o n and m u n i c i p a l i t i e s ( 9 , 1 1 , 1 2 ) . Markov t r a n s i t i o n m a t r i c e s can be e a s i l y e s t a b l i s h e d from pave-ment c o n d i t i o n s u r v e y s , w h i c h now a r e a common p r a c t i c e . The u t i l i t y d a t a d e r i v e d from the manager's p r e f e r e n c e s i s a l s o o b t a i n a b l e , and the c o s t and time r e q u i r e d a r e d i s p r o p o r t i o n a t e l y low com-pared t o a pavement c o n d i t i o n s u r v e y . I t took about t h r e e hours i n t h i s r e s e a r c h t o f a m i l i a r i z e an e n g i n e e r w i t h t h e o b j e c t i v e s and a t t r i b u t e s , t o check t h e b e h a v i o u r a l a s s u m p t i o n s by a q u a l i t a t i v e e l i c i t a t i o n p r o c e d u r e , and t o e l i c i t the t r a d e - o f f s and the u t i l i t y c u r v e s f o r e i g h t a t t r i b u t e s . Add one hour f o r f i n e t u n i n g o f answers a f t e r c o n s i s t e n c y c h e c k s . Even i f a r e a l s i t u a t i o n w i l l r e q u i r e ten t i m e s as much t i m e f o r e l i c i t a t i o n and r e v i s i o n o f p r e f e r e n c e s a f t e r f e e d b a c k from program r u n s , one week o f work does not seem to o e x c e s s i v e f o r i n f o r m a t i o n t h a t can be reused f o r many pavement s e c t i o n s and f o r a number o f y e a r s . The model behaves w e l l , a t l e a s t as judged from r e s p o n s e s to i n p u t changes t h a t can be p r e d i c t e d by i n t u i t i o n . I t i s r e c o g n i z e d , however, t h a t o n l y an e x t e n s i v e t e s t i n g on r e a l - w o r l d d a t a can d e t e r m i n e t h e u s e f u l n e s s o f the model. The number o f l e v e l s f o r pavement v a r i a b l e s must i n c r e a s e and t h e f o r m u l a s f o r q u a n t i f i c a t i o n o f consequences must improve b e f o r e u s e f u l r e s u l t s can be e x p e c t e d from the model. The computer runs d e m o n s t r a t e t h a t the model i s c o m p u t a t i o n a l l y and e c o n o m i c a l l y f e a s i b l e f o r the number o f v a r i a b l e s t h a t more l i k e l y w i l l o c c u r i n a r e a l a p p l i c a t i o n . There i s c e r t a i n l y room f o r i m p r o v i n g program e f f i c i e n c y by a p r o f e s s i o n a l programmer. A p o s s i b l e a d o p t i o n o f t h e model t o s u i t l o c a l needs w i l l be enhanced by the f l e x i b i l i t y i n s e t t i n g up the s t r u c t u r e o f o b j e c t i v e s , i n d e c i d i n g w h i c h pavement v a r i a b l e s a r e i m p o r t a n t , and i n s e l e c t i n g f o r m u l a s f o r c a l -c u l a t i o n o f consequences. Markov t r a n s i t i o n m a t r i c e s a l s o r e f l e c t l o c a l c o n -d i t i o n s . They a r e l e s s e x p e n s i v e to produce than r e g r e s s i o n e q u a t i o n s , and a r e more a c c u r a t e models o f pavement b e h a v i o u r f o r management purposes than a r e systems o f m e c h a n i s t i c f o r m u l a s . W i t h o u t d o u b t , i n t r o d u c i n g a new model and then p r o v i d i n g d a t a t o o p e r a t e i t w i l l c o s t t i m e and money. However, the u s e r can d e c i d e how many v a r i a b l e s need t o be i n c l u d e d and what compromise s h o u l d be made between a v a i l a b l e r e s o u r c e s and the q u a l i t y o f model's o u t p u t . 9.2.The Pavement Maintenance Problem Approaches c u r r e n t l y used f o r o p t i m i z i n g pavement maintenance cannot a c c o u n t even f o r the most i m p o r t a n t c o m p l e x i t i e s posed by t h e pavement system. T h i s r e s e a r c h u ncovers t h a t d e c i s i o n a n a l y s i s under u n c e r t a i n t y w i t h r e c e n t e x t e n s i o n s can e f f e c t i v e l y a p p roach a l l the c o m p l e x i t i e s i n an u n i f i e d f a s h i o n , and produce a r a t i o n a l answer. Through the i n q u i r y i n t o t h e s t r u c t u r e o f pavement manager's o b j e c -t i v e s , d e c i s i o n a n a l y s i s p r o v i d e s a framework f o r the much needed d e f i n i t i o n o f t h e problem's scope. The p r o c e s s r e v e a l s d i v e r g e n t s o c i a l i n t e r e s t s , i d e n t i f i e s u n c e r t a i n v a r i a b l e s o f t h e pavement system, and h i g h l i g h t s t h e i m p o r t a n c e o f i n c l u d i n g the t i m e d i m e n s i o n i n t o a n a l y s i s . Once a l l r e l e v a n t 90. f a c t o r s a r e s o r t e d o u t , t h e pavement manager knows what i n f o r m a t i o n must be c o l l e c t e d and p r o c e s s e d i n o r d e r t o use the d e c i s i o n model. U n l i k e many management methods s t u d i e d , d e c i s i o n a n a l y s i s c o r r e c t l y p l a c e s the d a t a g a t h e r i n g phase a f t e r the p r o b l e m d e f i n i t i o n , s h i f t s the emphasis from roughness t o o t h e r i m p o r t a n t pavement v a r i a b l e s , and s i g n i f i e s t h o s e o b j e c -t i v e s t h a t c a n n o t be t r a n s l a t e d i n t o money. A l t h o u g h the a d d i t i v e models p r e v a i l i n the e x i s t i n g pavement manage-ment schemes, t h e i r v a 1 i d i t y i s q u e s t i o n a b l e from a r i g o r o u s p o i n t o f v i e w . O b j e c t i v e s o f the pavement sy s t e m i n t e r a c t i n a manner r e s e m b l i n g a s u b s t i -t u t i o n : , i f not a complementary, e f f e c t between a t t r i b u t e s . The time span o f t h e pavement pr o b l e m i s s u f f i c i e n t l y l o n g t o make the temporal r i s k f a c t o r s s i g n f i c a n t . In s h o r t , a r a t i o n a l pavement manager has a p r e f e r e n c e s t r u c -t u r e w h i c h , a c c o r d i n g t o the m u l t i a t t r i b u t e u t i l i t y t h e o r y , e x c l u d e s the a d d i t i v e models from c o n s i d e r a t i o n . A Markov d e c i s i o n p r o c e s s f o r m u l a t i o n i s proposed i n t h i s t h e s i s , and the e x p e c t e d m u l t i a t t r i b u t e u t i l i t y a p p e a rs i n the o b j e c t i v e f u n c t i o n . I n t r o d u c t i o n o f t h i s new c r i t e r i o n does not c r e a t e any s p e c i a l c o m p u t a t i o n a l o r o p e r a t i o n a l problems i n p r e s e n t f o r m u l a t i o n . The model behaves w e l l , and a r e a l - w o r l d problem o f o p t i m i z i n g t h e maintenance o f one pavement s e c t i o n w i l l t a ke o n l y a m a r g i n a l amount o f computer t i m e . I f i n t e r p e r s o n a l c o m p a r i s o n s i s the i s s u e i n pavement management, one w i l l i n v e s t i g a t e what scheme i s most s u i t a b l e f o r a g g r e g a t i n g v a r i o u s o p i n -i o n s . F i r s t o f a l l one may i n v e s t i g a t e whether the pavement manager may be c o n s i d e r e d a b e n e v o l e n t d i c t a t o r o f c h o i c e - a c a s e most e x p e d i e n t to a n a l y s e . O t h e r s i t u a t i o n s can a l s o be h a n d l e d as mentioned i n C h a p t e r 3.2. The r e a l p roblem o f maintenance may be a y e a r l y budget w h i c h i s i n s u f f i c i e n t t o upgrade a l l s e c t i o n s t o the s a t i s f a c t o r y s t a t e o f r e p a i r . When a l l the money i s s p e n t on a few s e c t i o n s i n a g i v e n y e a r , f u t u r e main-tenance o f the n e g l e c t e d s e c t i o n s may p rove more c o s t l y than i f the budget was d i s t r i b u t e d o v e r t h e network more u n i f o r m l y . I f so - and r e s e a r c h i s needed i n t o t h i s p roblem - then an o p t i m i z a t i o n model must be b u i l t t o a c c o u n t f o r network i n t e r a c t i o n s . The c r i t e r i o n may be t o m i n i m i z e a d v e r s e e f f e c t s o f u n d e r i n v e s t m e n t i n m a i n t e n a n c e , or e q u i v a l e n t - t o maximize the e f f e c t i v e n e s s o f m a i n t e n a n c e . D e c i s i o n a n a l y s i s w i l l s t i l l be r e l e v a n t f o r t h e network problem. The d e t a i l s o f one s e c t i o n b e h a v i o u r may need t o be compressed i n t o a more con-c i s e model, and s t a t e s w i l l r e p r e s e n t d i f f e r e n t v a r i a b l e s . S e c t i o n s w i l l p r o b a b l y be c a t e g o r i z e d a c c o r d i n g t o t h e i r i m p o r t a n c e . D e c i s i o n maker's p r e f e r e n c e s w i l l r e f e r t o t h e whole network r e g i o n , r a t h e r than t o t h e microcosmos o f one pavement s e c t i o n . A l t e r n a t i v e s may become f r a c t i o n s o f th e budget a s s i g n e d t o each subnetwork, and be t r a n s l a t e d i n t o p a r t i c u l a r a c t i v i t i e s t o match s u b r e g i o n s ' m a i n t e n a n c e p r a c t i c e and r e s o u r c e s . D e c i s i o n a n a l y s i s w i l l a l l o w one t o i n v e s t i g a t e what s a c r i f i c e s can be imposed on the u s e r s and nonusers i n o r d e r t o f u l f i l l t h e p u b l i c o b j e c t i v e s l i k e c o s t m i n i m i z a t i o n and p r o t e c t i n g the highway i n v e s t m e n t . The t h e o r e t i c i a n s may want t o l o o k i n t o t h e p o s s i b i l i t i e s o f d e v e l o p i n g a m u l t i a t t r i b u t e u t i l i t y model f o r s p a t i a l systems such as n e t w o r k s . I t seems t h a t e x t e n s i o n o f the model t o network c o n s i d e r a t i o n s w i l l s u p p r e s s t h e r e s o l u t i o n on t h e time d i m e n s i o n . I f t h e maintenance budget can be p r e d i c t e d o n l y a few y e a r s i n advance, then i t makes sense t o have an a p p r o x i m a t e s i m u l a t i o n o f t h e network's l o n g - t i m e b e h a v i o u r and a more a c c u -r a t e model t o d e c i d e y e a r l y on t h e d i s t r i b u t i o n o f m a i n t e n a n c e o v e r the network. The model o f Lu and L y t t o n (10) i l l u s t r a t e s t h i s p h i l o s o p h y . U n t i l a B a y e s i a n d e c i s i o n model i s d e v e l o p e d f o r network o p t i m i z a t i o n , the u s e r may want t o f o l l o w t h e p r e s e n t l y p o p u l a r approach of s e c t i o n - b y -s e c t i o n o p t i m i z a t i o n . The i n d i v i d u a l o p t i m a o b t a i n e d by a model such as the one proposed i n t h i s t h e s i s can be matched y e a r l y w i t h a v a i l a b l e r e s o u r c e s by u s i n g a r e s o u r c e a l l o c a t i o n model. 92. 9.3-Dec i s ion A n a l y s i s i n C i v i l E n g i n e e r i n g C i v i l e n g i n e e r s d e s i g n , b u i l d and o p e r a t e s o c i o - t e c h n i c a l systems. A l t h o u g h the emphasis i s on t h e t e c h n i c a l s i d e , the e n g i n e e r must c o n s i d e r the s o c i a l , economic and sometimes t h e p o l i t i c a l a s p e c t s of a problem as w e l l . T h i s f a c t g i v e s r i s e t o m u l t i p l e o b j e c t i v e s and t h e i m p l i c a t i o n s o f group o p i n i o n must i n e v i t a b l y f o l l o w . The e n g i n e e r i n g systems a r e b u i l t t o s e r v e f o r a l e n g t h o f t i m e comparable t o t h e span o f human l i f e . The t i m e d i m e n s i o n cannot be n e g l e c t e d , but i t i n t r o d u c e s u n c e r t a i n t i e s about t h e f u t u r e and makes i t n e c e s s a r y t o c o n s i d e r r i s k s . Other t y p e s o f u n c e r t a i n t y and r i s k e n t e r many problems because c i v i l e n g i n e e r i n g works a r e b u i l t o f v a r i a b l e m a t e r i a l s , a r e s u b j e c t t o random loads and e n v i r o n m e n t a l e f f e c t s , and o f t e n c r u c i a l i n f o r m a t i o n r e q u i r e d by t h e e n g i n e e r i s vague. D e c i s i o n a n a l y s i s appears e x t r e m e l y r e l e v a n t f o r c i v i l e n g i n e e r i n g p r a c t i c e . Numerous a p p l i c a t i o n s i n a wide s e l e c t i o n o f e n g i n e e r i n g a r e a s have a l r e a d y d e m o n s t r a t e d t h a t t h i s method i s a p p r o p r i a t e f o r most r e a 1 i s t i c and complex p r o b l e m s , and can be v e r i f i e d i n p r a c t i c e . I n t a n g i b l e s such as s a f e t y , e n v i r o n m e n t a l impact, a e s t h e t i c s and p o l i t i c a l f e a s i b i l i t y a r e i n c l u d e d b e s i d e o b j e c t i v e s e x p r e s s i b l e i n money. U n c e r t a i n t i e s a r e e f f e c -t i v e l y h a n d l e d by p r o b a b i l i t i e s o f f a c t o r s t h a t a r e under t h e d e c i s i o n maker's c o n t r o l o r o u t s i d e h i s c o n t r o l . Time d e p e n d e n c i e s a r e a c c o u n t e d f o r by d i v i d i n g t h e a n a l y s i s h o r i z o n i n t o a s u i t a b l e number o f p e r i o d s . A p p r o p r i a t e w e i g h t s a r e a s s i g n e d t o p e r i o d s , g i v e n t h e i r r e l a t i v e i m p o r t a n c e and t h e c e r t a i n t y w i t h w h i c h the v a r i a b l e s a r e known f o r each p e r i o d . Temporal r i s k s e n s i t i v i t y i s i n c o r p o -r a t e d i n t o t h e model, as w e l l as the a t t i t u d e s a s s o c i a t e d w i t h t r a d e - o f f s between o b j e c t i v e s . The i n t e r e s t s of many groups can be i n c l u d e d i n the e v a l u a t i o n o f a l t e r n a t i v e s , t h e c o n f l i c t s i l l u m i n a t e d and r e s o l v e d . D e c i s i o n a n a l y s i s can remove many p o t e n t i a l i m p l e m e n t a t i o n d i f f i c u l -t i e s . The method can be u n i v e r s a l l y a p p l i e d t o problems i n f i l t r a t e d by any s u b s e t o f t h e above c o m p l e x i t i e s . I t r e p r e s e n t s t h e r e a l problem r e l a t i v e l y f a i t h f u l l y , i s c o n c e p t u a l l y s i m p l e , and has f a i r l y f l e x i b l e r e q u i r e m e n t s f o r d a t a . D e c i s i o n a n a l y s i s b r e a k s t h e complex d e c i s i o n t a s k i n t o p a r t s t h a t can be i n d e p e n d e n t l y a s s e s s e d by v a r i o u s e x p e r t s , and then combined w i t h due r e g a r d t o i n t e r a c t i o n s . The component p a r t s r e f l e c t the d e c i s i o n making p r o c e s s and a r e l a i d out i n a c l e a r scheme. With o n l y a r e a s o n a b l e e f f o r t t h e u s e r can u n d e r s t a n d t h e model and i t s f u n c t i o n . I f n e c e s s a r y , t h e method can r i g o r o u s l y use i n f o r m a t i o n based on e n g i n e e r i n g judgement i n a d d i t i o n t o the o b j e c t i v e d a t a . The u p d a t i n g of i n f o r m a t i o n can a l s o be r o u t i n e l y h a n d l e d . D e c i s i o n a n a l y s i s can accom-modate component models w h i c h b e s t s u i t the p r a c t i c a l r e q u i r e m e n t s of t h e problem a t hand. These f l e x i b i l i t i e s c o n s t i t u t e a most i m p o r t a n t a s s e t , s i n c e e n g i n e e r i n g d e c i s i o n s o f t e n have t o be made w i t h o n l y l i m i t e d d a t a ava i 1 a b l e . D e c i s i o n a n a l y s i s i s i n f a c t a major p a r t o f the scheme f o r the s c i e n t i f i c method o f i n q u i r y . Once a problem i s i d e n t i f i e d , d e c i s i o n a n a l y s i s p r o v i d e s a s t r u c t u r e d methodology f o r the f o r m u l a t i o n o f the p r oblem, t h e m o d e l i n g o f t h e i d e n t i f i e d system, and t h e g e n e r a t i o n and e v a l u a t i o n o f a l t e r n a t i v e s . I t p r o v i d e s an a i d w h i c h can a s s i s t the d e c i s i o n maker i n d i f f i c u l t problems o f c h o i c e . He can a l s o employ i t as a l e a r n i n g t o o l . The impacts o f d i f f e r e n t a l t e r n a t i v e s may be a s s e s s e d a n a l y t i c a l l y , and t h i s i n f o r m a t i o n used as a f e e d b a c k t o improve the next d e c i s i o n . Because t h e model has a m e a n i n g f u l s t r u c t u r e , the e f f e c t of a change i n i n p u t s can be e v a l u a t e d by s e n s i t i v i t y a n a l y s e s . These w i l l h e l p the u s e r t o g a i n f u r t h e r i n s i g h t i n t o t h e problem. Through s u p p l y i n g t h e model, the a n a l y s t does not usurp the d e c i s i o n making c a p a c i t y . The m o d e l i n g p r o c e s s r e q u i r e s a s t r o n g i n t e r a c t i o n w i t h t h e u s e r i n a l l phases. W i t h t h e a n a l y s t ' s h e l p , t h e d e c i s i o n maker i s f o r c e d t o t h i n k hard about t h e o b j e c t i v e s , the t r a d e - o f f s , and the v a l u e s he a t t a c h e s t o d i f f e r e n t consequences o f d e c i s i o n s . The answers p r o v i d e a f u n c t i o n w h i c h i s u n i q u e f o r a d e c i s i o n maker, i n t h a t i t r e f l e c t s h i s -not t h e a n a l y s t ' s - r e l a t i v e d e s i r a b i l i t y o f each consequence. Some managers may o b j e c t t o t h e time r e q u i r e d f o r t h i s e l i c i t a t i o n p r o c e s s , but no s u b s t i t u t i o n e x i s t s f o r d e c i s i o n maker's judgement. U n t i l an a l t e r n a t i v e method i s i n v e n t e d , d e c i s i o n a n a l y s i s i s s u p e r i o r f o r t h o s e p e o p l e who want t o make, o r who h e l p o t h e r s t o make, r a t i o n a l c h o i c e s i n complex s i t u a t i o n s . 9 - 4 . P r o m i s i n g Study Areas i n D e c i s i o n A n a l y s i s Many r e s e a r c h e r s p o i n t out t h a t the i n a c c e s s i b i l i t y of d e c i s i o n makers i s a main o b s t a c l e i n d e c i s i o n a n a l y s e s o f m a j o r p r o j e c t s . I t may be w o r t h -w h i l e t o i n v e s t i g a t e how f a r t h e a n a l y s i s can be c a r r i e d out w i t h o u t an a c c u r a t e knowledge o f the d e c i s i o n maker's p r e f e r e n c e s t r u c t u r e . S i m p l i f i -c a t i o n s w i l l most l i k e l y be d i s c o v e r e d t h r o u g h s e n s i t i v i t y a n a l y s e s , once a d e c i s i o n model i s s e t up f o r a p a r t i c u l a r problem. In any c a s e , a i d s such as i n t e r a c t i v e computer programs t h a t can q u i c k l y t e a c h a d e c i s i o n maker about h i s p r e f e r e n c e s , may s u b s t a n t i a l l y reduce t h e problem o f u n a v a i l a b l e p r e f e r e n c e d a t a . N e v e r t h e l e s s , the p o p u l a r b e l i e f t h a t s e n i o r managers do not have time t o have t h e i r p r e f e r e n c e s q u a n t i f i e d must be i n v e s t i g a t e d . The time n e c e s s a r y f o r e l i c i t a t i o n i s r e a s o n a b l y s h o r t and i t a p p e a r s t h a t r a t h e r than r e f l e c t a t i m e problem, t h e b e l i e f may have a d i f f e r e n t r e a s o n ; t h e managers may f e a r t o l o s e p a r t of t h e i r d e c i s i o n making a u t h o r i t y i f a u t i l i t y model i s a p p l i e d . I t s h o u l d be c l e a r a t t h i s p o i n t t h a t the f e a r i s not j u s t i f i e d . D e c i s i o n a n a l y s i s s h o u l d be seen by the managers as a p o w e r f u l t o o l t h a t c c a n h e l p them t o a r r i v e a t a b e t t e r d e c i s i o n more e f f i c i e n t l y , w i t h f u l l use o f t h e i r v a l u e judgements. T h i s argument i s p a r t i c u l a r l y v a l i d f o r complex 95. problems where few i f any managers a r e a b l e t o p r o c e s s a l l v a r i a b l e s r a t i o n a l l y w i t h o u t r e s o r t i n g t o a n a l y t i c a l a i d s . The p r e c e d i n g o b j e c t i o n can be s o f t e n e d by e d u c a t i n g the managers on t h e s t r e n g t h s and on t h e n a t u r e o f d e c i s i o n a n a l y s i s . Some managers o b j e c t t o a n a l y s i s because they do not want t o d i s c l o s e t h e p r e f e r e n c e s f o r p o l i t i c a l r e a s o n s . There i s not much one can do t o a b a t e such a s t a n c e w i t h o u t up-s e t t i n g t he d i s t r i b u t i o n o f p o l i t i c a l power. P u b l i c s e r v a n t s , however, may be persuaded by t h e i n c r e a s i n g demands from the p u b l i c f o r b e t t e r a c c o u n t a -b i l i t y o f t h e governments t h e y e l e c t . In t h e l i g h t o f new f i n d i n g s i n n o r m a t i v e t h e o r y o f c h o i c e , i t i s mandatory t o r e v i s e t h e a d d i t i v e e v a l u a t i o n models w h i c h p r a c t i c a l l y dominate th e a n a l y t i c a l f i e l d s o f t r a n s p o r t a t i o n , urban and r e g i o n a l p l a n n i n g , e n v i -ronmental impact s t u d i e s , and many o t h e r c i v i l e n g i n e e r i n g and r e l a t e d e n deavours. T h i s r e s e a r c h has found an e v i d e n c e o f i n t e r a c t i o n s between d e c i s i o n c r i t e r i a , f o r w h i c h a d d i t i v e models cannot a c c o u n t . There appears t o be a f r u i t f u l ground f o r r e s e a r c h i n g new approaches t o o p t i m i z a t i o n . Many p r a c t i t i o n e r s q u e s t i o n the u s e f u l n e s s o f p r e d i c t i n g the " b e s t " c h o i c e , because the p r e s e n t r a t e o f change may make t h e c a l c u l a -t i o n s o b s o l e t e b e f o r e an a l t e r n a t i v e i s implemented. The p h i l o s o p h y i s t o t a k e a c t i o n s w h i c h a l l o w t h e d e c i s i o n maker the g r e a t e s t f l e x i b i l i t y t o adapt t o t h e f u t u r e , r a t h e r i t h a n s e a r c h f o r t h e a b s o l u t e l y b e s t s o l u t i o n s g i v e n p r e s e n t i n f o r m a t i o n . M u l t i p l e c r i t e r i a e v a l u a t i o n models may be adapte d t o s u i t t h i s r e q u i r e m e n t by i n t r o d u c t i o n o f an a d d i t i o n a l o b j e c t i v e w h i c h w i l l measure a c t i o n s ' f l e x i b i l i t y . Developments i n temporal u t i l i t y t h e o r y s h o u l d be s t u d i e d , f o r one may e x p e c t f i n d i n g s u s e f u l f o r problems where t i m e i s a f a c t o r . I n t e r e s -t i n g f o r m u l a t i o n s emerge than can r e a l i s t i c a l l y h a n d l e s e q u e n t i a l a n a l y s e s ( 2 2 ) . By r e l a x i n g the temporal u t i l i t y a s s u m p t i o n s , t h e s o - c a l l e d s e m i -s e p a r a b l e form i s o b t a i n e d . U g e n e r a l i z e s t h e a d d i t i v e and t h e m u l t i p l i -c a t i v e forms a t t h e expense o f a more complex s t r u c t u r e . One may want t o i n v e s t i g a t e whether t h e added c o m p l e x i t y has a s i g n i f i c a n t impact on the r e s u l t s . The a n a l y s t w i l l f i n d t h e s e m i - s e p a r a b l e f u n c t i o n f a i r l y u s e f u l , however, f o r problems where time can be d i v i d e d i n t o an immediate f u t u r e f o r whi c h a l l f a c t o r s a r e r e a s o n a b l y w e l l known, and a d i s t a n t f u t u r e - t h e vague and i 1 1 - p e r c e i v e d y e a r s beyond t h e h o r i z o n . Other u t i l i t y s t r u c t u r e s a r e b e i n g d e v e l o p e d f o r streams o f consequences w i t h an u n c e r t a i n h o r i z o n , and f o r s i t u a t i o n s where p a s t e x p e r i e n c e does a f f e c t p r e f e r e n c e s f o r the f u t u r e . Both c a s e s may be r e l e v a n t f o r s e q u e n t i a l a n a l y s e s i n c i v i l e n g i n e e r i n g . W i t h r e s p e c t t o group d e c i s i o n s , some p h i l o s o p h i c a l q u e s t i o n s about a g g r e g a t i n g i n d i v i d u a l s ' p r o b a b i l i t i e s and u t i l i t i e s remain unanswered, but t h e e n g i n e e r w i l l f i n d many w o r k a b l e approaches - both a x i o m a t i c and i n t u i -t i v e (22,38,56,58). The d e c i s i o n model need not be reshaped t o a c c e p t i n p u t s r e p r e s e n t a t i v e o f a g r o u p , because t h e s e a r e t r a n s f o r m e d i n t o f u n c t i o n s c o m p a t i b l e w i t h t h e model. 9 . 5 . F i n a l Comment The w r i t e r f e e l s t h a t t h e t h e s i s a c c o m p l i s h e s t h e o b j e c t i v e s t a t e d i n Chap t e r 3. The pavement maintenance problem has been s t r u c t u r e d f o r one s e c t i o n w i t h due r e g a r d t o a l l r e l e v a n t c o m p l e x i t i e s . A new o p t i m i z a t i o n model has been s y n t h e t i z e d from powerfu1 developments i n d e c i s i o n a n a l y s i s and pavement t h e o r y . The model a c c o u n t s f o r a l l c o m p l e x i t i e s o f t h e pave-ment system e x c e p t network o p t i m i z a t i o n , w h i c h c a n , h o p e f u l l y , be a t t a c k e d by a d e c i s i o n a n a l y t i c a l a p p roach i n f u t u r e r e s e a r c h . 9,7-REFERENCES 1. Roads and T r a n s p o r t a t i o n A s s o c i a t i o n o f C a n a d a , Pavement Management G u i d e , 1977 2. S h o r t r e e d , J . , " T r a n s p o r t a t i o n Investment D e c i s i o n s " , P r o c e e d i n g s Roads and T r a n s p o r t a t i o n A s s o c i a t i o n o f C a n a d a , 1977-3. H a a s , R. and Hudson , R . W . , Pavement Management S y s t e m s - , M c G r a w - H i l l , New Y o r k , 1978. 4. M o t h e r a l , J . G . , e t a l . , " S y s t e m s C o n s i d e r a t i o n s in Freeway P l a n n i n g , D e s i g n and C o n s t r u c t i o n " , E n g i n e e r i n g I s s u e s , V o l . 103, No. E12, A p r i l 1977-5. L y t t o n , R . L . , M c F a r l a n d , W . F . and S c h a f e r , D . L . , " F l e x i b l e Pavement D e s i g n and Management Systems A p p r o a c h I m p l e m e n t a t i o n " , NCHRP Repor t 160, 1975-6. K h e r , R . , " E c o n o m i c A n a l y s i s o f E l e m e n t s in Pavement D e s i g n " , T r a n s p o r t a t i o n R e s e a r c h Record 572, 1976. 7. J u n g , F . W . , . " S u b s y s t e m f o r P r e d i c t i n g F l e x i b l e Pavement P e r f o r m a n c e " , T r a n s p o r t a t i o n R e s e a r c h Record 572, 1976. 8. 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A . , Bauman, C. and H a a s , R . , "An Inven tory and P r i o r i t y Programming System f o r M u n i c i p a l Pavement Improvements" , P r o c e e d i n g s Roads and T r a n s p o r t a t i o n A s s o c i a t i o n o f C a n a d a , 1977. 13- A n d e r s o n , D . I . , P e t e r s o n , D . E . and S h e p h e r d , L . W . , " R e h a b i l i t a t i o n D e c i s i o n M o d e l " , T r a n s p o r t a t i o n R e s e a r c h Record 633, 1977-14. D a n i s h Roads D i r e c t o r a t e ( V e j d i r e k t o r a t e t ) , Vej r e g e l f o r s l a g v e d r o e r e n d e p r i o r i t e r i n g a f s 1 i d 1 a g s f o r n y e l s e ( P r o p o s a l f o r P r i o r i t y A n a l y s i s o f O v e r l a y s ) , 1977-15- F r i e s z , T . L . and Z w i e b a c k , J . M . , " D i s c u s s i o n " to " G e n e r a l C o n c e p t s o f System A n a l y s i s as A p p l i e d to P a v e m e n t s " , T r a n s p o r t a t ion  R e s e a r c h Record 512, 1974. 98. 16. 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B y e r , P.H. and Sa u n d e r s , D., " C h o o s i n g O b j e c t i v e s i n T r a n s p o r t a t i o n P l a n n i n g " , T r a n s p o r t a t i o n E n g i n e e r i n g J o u r n a l o f ASCE, V o l . 105, No. TE1, J a n u a r y 1979-22. Keeney, R.L. and R a i f f a , H., D e c i s i o n s w i t h M u l t i p l e O b j e c t i v e s , W i l e y , New Y o r k , 1976. 23- Keeney, R.L., "A U t i l i t y F u n c t i o n f o r Examining P o l i c y A f f e c t i n g Salmon on the Skeena R i v e r " , J o u r n a l o f the F i s h e r i e s R esearch  Board o f Canada, V o l . 3k, 1977\ 2k. Keeney, R.L. and Wood, E.F., "An I l l u s t r a t i v e Example o f the Use o f M u l t i a t t r i b u t e U t i l i t y Theory f o r Water Resource P l a n n i n g " , Water Resources R e s e a r c h , V o l . 13, August 1977-25. 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R e i d e l P u b l i s h i n g , D o r d r e c h t , H o l l a n d , 1975-43. M e y e r , R . F . , "On the R e l a t i o n s h i p Among the U t i l i t y o f A s s e t s , the U t i l i t y o f C o n s u m p t i o n , and Investment S t r a t e g y in an U n c e r t a i n , but Time I n v a r i a n t W o r l d " , OR 69: P r o c e e d i n g s o f the F i f t h  I n t e r n a t i o n a l C o n f e r e n c e on O p e r a t i o n a l R e s e a r c h , Lawrence ( e d i t o r ) , T a v i s t o c k P u b l i c a t i o n s , L o n d o n , 1970. 44. O r g a n i z a t i o n f o r Economic C o - o p e r a t i o n and Deve lopment , Road R e s e a r c h G r o u p , Hazardous Road L o c a t i o n s : I d e n t i f i c a t i o n and C o u n t e r M e a s u r e s , P a r i s , T 9 7 ^ 45. O r g a n i z a t i o n f o r Economic C o - o p e r a t i o n and D e v e l o p m e n t , Road R e s e a r c h G r o u p , A d v e r s e W e a t h e r , Reduced V i s i b i l i t y and Road S a f e t y , P a r i s , TsTT. 100. 46. I v e y , D.L. and G a l l a w a y , B.M., "Tire-Pavement F r i c t i o n : A V i t a l D e s i gn O b j e c t i v e " , Highway Re s e a r c h Record 471, 1973-47- Dunlap, D.F., e t a l . , " I n f l u e n c e o f Combined Highway Grade and H o r i z o n t a l A l i g n m e n t on S k i d d i n g " , NCHRP Report 184, 1978. 48. L e e s , G., Katekh d a , J . e . d . , Bond, R. and W i l l i a m s , A.R., "The Design and Perfor m a n c e o f High F r i c t i o n Dense A s p h a l t s " , T r a n s p o r t a t i o n R e s e a r c h Record 624, 1976. 49. 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APPENDIX A FORMS FOR TESTS ON PREFERENCES These forms a r e used t o examine q u a l i t a t i v e l y i f a s u b j e c t ' s p r e -f e r e n c e s change when l e v e l s o f t h e complementary a t t r i b u t e s v a r y . The p r o -c e d u r e i s d e s c r i b e d i n C h a p t e r 8.5- The f i r s t form i s f o r t e s t i n g p r e -f e r e n t i a l independence f o r " u s e r s a f e t y " and "economy". S i m i l a r forms were p r e p a r e d f o r t h e r e m a i n i n g p a i r s o f a t t r i b u t e s but a r e not reproduced h e r e . The second form i s f o r t e s t i n g u t i 1 i t y independence f o r " u s e r s a f e t y " . S i m i l a r forms f o r the o t h e r a t t r i b u t e s a r e not r e p r o d u c e d . TRADE-OFFS BETWEEN SAFETY (lO^ADT) AND ECONOMY ( l O b $ / y r ) S a f e t y 3.6 Economy 3.6 Economy Time Cost A c c e s s Jobs G r a v e l Noi se Time Cost A c c e s s Jobs G r a v e l No i se 0.7 min h r s / y r 1.6 min $ / y r 50,000 ADT 240 j o b s / y r 32 t o n n e s / y r 2,000 hous e h o l d s 0.7 min h r s / y r 1.6 min $ / y r 50,000 ADT 0 0 0 3.6 Economy 3.6 Economy Time Co s t A c c e s s Jobs G r a v e l N o i s e Time Cost A c c e s s Jobs G r a v e l Noi se 1.6 min $/yr 50,000 ADT 240 j o b s / y r 32 t o n n e s / y r 2,000 h o u s e h o l d s 0 0 0 0 0 0 LOTTERIES FOR AGENCY COST (10 $/yr) S a f e t y 50,000 ADT / ^ ' ^ Economy 3.6 min $/yr 0.5/ Time 0.7 min h r s / y r \/ A c c e s s 50,000 ADT Jobs 240 j o b s / y r Q G r a v e l 32 t o n n e s / y r N o i s e 2,000 h o u s e h o l d s S a f e t y 50,000 ADT / 1.6 Economy 3-6 min $/yr 0.5 / / Time 0.7 min h r s / y r '/ A c c e s s 50,000 ADT Q ^ \ Jobs 0 \ 0 G r a v e l 0 N o i s e 0 S a f e t y 0 <1 .6 Economy 0 Time 0 Ac c e s s ' 50,000 ADT Jobs 240 j o b s / y r g G r a v e l 32 t o n n e s / y r N o i s e 2,000 h o u s e h o l d s S a f e t y 0 / 1.6 / Economy 0 0 . 5 / / Time 0 V A c c e s s 0 0 - 5 \ j o b s 0 0 G r a v e l 0 N o i s e 0 104. APPENDIX B DERIVATION OF CONSTANTS FOR EXPONENTIAL UTILITY CURVES A p p l i c a t i o n s o f m u l t i a t t r i b u t e u t i l i t y t h e o r y ( C h a p t e r 4.2) have demonstrated t h a t u t i l i t y c u r v e s w i t h c o n s t a n t r i s k p r o p e r t i e s can a p p r o x i -mate d e c i s i o n maker's b e h a v i o u r a c c u r a t e l y enough f o r a f i r s t c u t a n a l y s i s and sometimes s u f f i c i e n t l y even f o r a more d e t a i l e d a n a l y s i s . The c u r v e s o f Chapte r 8 a r e c o n s t r u c t e d u s i n g t h i s a pproach. I t can be shown g r a p h i c a l l y t h a t an i n c r e a s i n g e x p o n e n t i a l f u n c t i o n t h a t i s bounded between 0 and 1 has t h i s form u(x) = a ( l - e x p ( c x ) ) ; x > 0 where a < 0, c > 0 f o r a r i s k prone b e h a v i o u r and a > 0, c < f o r a r i s k a v e r s e b e h a v i o u r . Two p o i n t s o f t h i s c u r v e a r e d e f i n e d by the r e q u i r e d sea 1i ng. u (WORST) = 0 u(BEST) = 1 where WORST i s the minimum ( l e a s t d e s i r a b l e ) and BEST i s the most d e s i r a b l e x. The WORST v a l u e e q u a l s 0 f o r a l l a t t r i b u t e s i n the example problem. A t h i r d p o i n t w i l l be s p e c i f i e d by t h e d e c i s i o n maker. The s i m p l e s t c a n d i d a t e f o r t h i s p o i n t i s t h e x ( l a b e l l e d HALF) whose u t i l i t y e q u a l s 0.5, u(HALF) = 0.5 When HALF/BEST < 0.5 then t h e u t i l i t y f u n c t i o n i s concave and the d e c i s i o n maker r i s k a v e r s e . When HALF/BEST > 0.5 then t he f u n c t i o n i s convex ( r i s k p r o n e ) . When HALF/BEST = 0.5 the f u n c t i o n i s s t r a i g h t l i n e ( r i s k n e u t r a l ) . R i s k prone case u(x) = a ( e x p ( c x ) - 1 ) ) ; 0 < x < BEST, a > 0, c > 0 (B.1) cx = 1n a - l n ( a + u ( x ) ) I. u(HALF) = 0.5 cHALF = In a - l n ( a + 0.5) (B.2) I I. u(BEST) = 1 cBEST = In a - l n ( a + 1) (B.3) D i v i d i n g (B.2) by (B.3) y i e l d s HALF/BEST = ( i n a - l n ( a + 0 . 5 ) ) / ( l n a - l n ( a + D),a > 0 (B.4) Co n s t a n t a can be i t e r a t e d from t h i s e q u a t i o n and then c can be o b t a i n e d from (B.3)-R i s k a v e r s e case u(x) = - a ( e x p ( - c x ) - 1 ) , 0 < x < BEST, a > 0, c > 0 (B.5) c x = l n a - In (a - u (x)) 105.. u(HALF) 0.5 cHALF 1 n a l n ( a 0.5) (B.6) u(BEST) cBEST 1 n a l n ( a D (B.7) D i v i d i n g (B.6) by (B.7) y i e l d s HALF/BEST = ( I n a - l n ( a - 0 . 5 ) ) / ( l n a - l n ( a - 1 ) ) , a > 1 (B . 8 ) I t e r a t i o n by computer w i l l y i e l d c o n s t a n t a, and c can be c a l c u l a t e d from F i g u r e 11 i s a s k e t c h o f HALF/BEST as g i v e n by (B .4) and ( B . 8 ) . A s u i t a b l e t r a n s f o r m a t i o n o f the r i s k a v e r s e c a s e can make the f u n c t i o n i d e n t i c a l t o the r i s k prone c a s e . To show t h i s , n o t i c e t h a t i f the lower c u r v e i s s h i f t e d t o the o r i g i n i t w i l l form a s y m m e t r i c a l image o f the upper c u r v e p r o v i d e d both have the same shape. 1 - HALF/BEST = = 1 - ( l n ( a + 1) - l n ( a + 1 - 0 . 5 ) ) / ( l n ( a + 1) - l n ( a + 1 - 1 ) = 1 - ( l n ( a + 1) - l n ( a + 0 . 5 ) ) / ( l n ( a + 1) - In a) = ( l n ( a + 1) - In a - l n ( a + 1) + l n ( a + 0 . 5 ) ) / ( l n ( a + 1) - In a) = ( l r i ( a + 0.5) - In a ) / ( l n ( a + 1) - In a ) , a > 0 and the r i g h t hand s i d e o f t h i s e q u a t i o n i s indeed i d e n t i c a l t o the r i s k prone case ( B . 4 ) . I t e r a t i o n f o r both cases can then be han d l e d by one p r o c e d u r e , p r o v i d e d the f o l l o w i n g t r a n s f o r m a t i o n s a r e made i n the r i s k a v e r s e c a s e : HALF t r a n s f o r m e d = BEST - HALF and a = 1 + a c a l c u l a t e d . The program f o r i t e r a t i n g the c o n s t a n t s from HALF/BEST r a t i o i s d e s c r i b e d i n Append i x D. (B.7) HALF/BEST FIGURE U . SKETCHES OF THE HALF /BEST FUNCTIONS 107. APPENDIX C RESULTS OF QUALITATIVE ASSESSMENT The t e s t s were c a r r i e d out on s u b j e c t s A, B, C, D and E u s i n g forms shown i n Appendix A and a c c o r d i n g t o a p r o c e d u r e d e s c r i b e d i n Chapter 8.5-F i g u r e s 12 t o 18 c o r r e s p o n d t o one t r a d e - o f f p a i r o f a t t r i b u t e s e a c h , and F i g u r e s 19 t o 23 c o r r e s p o n d t o c e r t a i n t y e q u i v a l e n t s f o r one a t t r i b u t e each. Cases 1 t o 4 o f complementary a t t r i b u t e s a r e on t h e a b s c i s s a . The u n i t on t h e o r d i n a t e i s p e r c e n t o f t h e BEST (most d e s i r a b l e ) v a l u e o f t h e "more i m p o r t a n t " a t t r i b u t e . P e r c e n t a g e s , as s c a l e d from t h e respond s h e e t s w i t h 5 p e r c e n t p r e c i s i o n , r a t h e r than a b s o l u t e u n i t s a r e used t o make co m p a r i s o n s between t r a d e - o f f p a i r s p o s s i b l e . Note t h a t t h e r e a r e d i f f e r e n c e s o f o p i n i o n as t o w h i c h a t t r i b u t e i s "more i m p o r t a n t " ( F i g u r e s 14, 15, 17, 1 8 ) . L e t t e r s on t h e g r a p h s d e n o t e s u b j e c t s and l i n e s c o n n e c t t h e i r answers s c a l e d from q u e s t i o n n a i r e s h e e t s . Mean and s t a n d a r d d e v i a t i o n a r e g i v e n f o r th o s e s u b j e c t s who s t a t e d t h a t t h e i r t r a d e - o f f s o r r i s k a t t i t u d e s d i d not change w i t h Cases 1 t o 4. One can e x p e c t some v a r i a n c e i n answers from one s u b j e c t even i f t h e p e r s o n v e r b a l l y s t a t e s t h a t c h a n g i n g t h e c a s e s from 1 t o 4 does not a f f e c t t h e t r a d e - o f f s . I t i s i n t e r e s t i n g t o note t h a t s u b j e c t s A and D e x h i b i t e d a s t a n d a r d d e v i a t i o n o f up t o 13 p e r c e n t o f t h e maximum a t t r i b u t e v a l u e , but m a i n t a i n e d t h a t t h e i r p r e f e r e n c e s d i d not change. F i g u r e s 12 t o 18 show t r e n d s f o r t r a d e - o f f s made i n the upper r i g h t hand c o r n e r o f t h e t r a d e - o f f s q u a r e . In t h e m i d d l e r e g i o n o f t h e t r a d e - o f f domain, the p a t t e r n was v e r y s i m i l a r , e x c e p t f o r t h e magnitude o f the t r a d e -o f f . The s t a n d a r d d e v i a t i o n s never exceeded 8 p e r c e n t and s u b j e c t s C and D e x h i b i t e d a change i n t r a d e - o f f s a t c a s e s 3 and 4 f o r one a t t r i b u t e each. In t h e lower l e f t c o r n e r o f t h e square a l l s u b j e c t s had d i f f i c u l t y t o make 108. a c c u r a t e t r a d e - o f f s due t o s m a l l q u a n t i t i e s i n v o l v e d . S u b j e c t s B, D, E s t a t e d t h a t t h e i r t r a d e - o f f s would not be a f f e c t e d by t h e v a r i a t i o n of c a s e s 1 t o k. S u b j e c t s A and C i n d i c a t e d t h a t t h e i r t r a d e - o f f s would be s i m i l a r i n d i r e c t i o n and p r o p o r t i o n a l i n magnitude t o t r a d e - o f f s i n t h e upper r i g h t c o r n e r . The g r a p h s s u g g e s t t h a t f o r s u b j e c t s A, B and E t h e p r e f e r e n t i a l independence a s s u m p t i o n s h o l d w e l l . S u b j e c t C v i o l a t e d p r e f e r e n t i a l i n d e -pendence a s s u m p t i o n s f o r t h r e e out o f seven t r a d e - o f f p a i r s , w h i l e D v i o l a t e d one a s s u m p t i o n . The v i o l a t i o n s o c c u r a t c a s e s 3 and k o f t h e complementary a t t r i b u t e s , but a r e r e l a t i v e l y s m a l l and w i t h i n 5 t o 10 p e r c e n t o f t h e maximum a t t r i b u t e v a l u e r e l a t i v e t o c a s e s 1 and 2. The u t i l i t y independence a s s u m p t i o n i s f u l f i l l e d by s u b j e c t s A, B, D and E. S u b j e c t C was u t i l i t y independent f o r one o f the two a t t r i b u t e s t e s t e d , but s t a t e d a change i n t h e c e r t a i n t y e q u i v a l e n t w i t h c a s e k. In c o n c l u s i o n , t h e p r e f e r e n t i a l and u t i l i t y independence a s s u m p t i o n s appear t o be a good a p p r o x i m a t i on f or t i i r e e of t h e f i v e s u b j e c t s , and may be a w o r k a b l e a p p r o x i m a t i o n f o r t h e r e m a i n i n g two s u b j e c t s . 109. FIGURE 12: TRADE-OFF USER SAFETY AND ECONOMY LU LL < CO CO Lil CD 1 0 0 f BC-8 0 6 0 I 4 0 D-A-E-BC-- D -.A' -E--BC-- D --A-3 -BC -D -A mean =80, s.d. = 5 4 C A S E FIGURE 13: T.RADE-OFF USER SAFETY AND NOISE FIGURE 14: TRADE-OFF USER ECONOMY AND TIME 111 1 0 0 £ >-o 8 0 o LU 6 0 (/> LU CD 4 0 f 4 C A S E 1 0 0 t CO o O 8 0 > o z LU O < CO LU CD 6 0 f 4 0 f B- B- -B-A mean = 8 8 , s.d. = 3 •B 4 C A S E FIGURE 15: TRADE-OFF USER ECONOMY AND AGENCY COST 112. 1 0 0 t 8 0 CO CO LU O 2 6 0 CO LU CO 4 0 t mean = 85, s.d. = 5 mean = 73, s.d. = 6 C A S E FIGURE 16: TRADE-OFF AGENCY COST AND ACCESS 113. 1 0 0 + 8 0 CO 00 o CO LU m 6 0 I A mean = 81, s.d. = 6 D ! mean = 63, s.d. = 12 4 0 4 C A S E A 1 0 0 f o o o LU CO LU CO o. 8 0 t 6 0 I 4 0 E-B-E-•B-E-•B-•E -B C A S E FIGURE. 11: TRADE-OFF USER ECONOMY AND JOBS / FIGURE 18: TRADE-OFF JOBS AND GRAVEL 115. 1 0 0 ^ 8 0 LU LL I-10 LU CD 4 0 t 0.75 x = 68, s.d. = 4 * = 45, s.d. = 6 C 0 .50 4 C A S E 1 0 0 t co o o >• o LU O < lO LU CD 8 0 6 0 [ 4 0 I -A 0.75 0.50 4 C A S E FIGURE 19: CERTAINTY EQUIVALENTS FOR SUBJECT A 1 0 0 i CO o o > o z LU O < CO LU CD 8 0 6 0 4 0 t B-B--B- -B-•B- -B-•B -B 0 . 7 5 0 . 5 0 4 C A S E 1 0 0 ? CO CO LU O O < CO LU CD 8 0 f 6 0 | 4 0 B-- 4 — -B- •B-C 0 . 5 0 •B 0 . 2 5 4 C A S E FIGURE 20: CERTAINTY EQUIVALENTS FOR SUBJECT B 1 0 0 t 8 0 > I— LU LL < t o I-(/> LU CO 6 0 4 0 ^ 0 . 7 5 x = 7 2 , C 0 . 5 0 4 C A S E FIGURE 21: CERTAINTY EQUIVALENTS FOR SUBJECT C 118. 100 f CO o o 80 >-o z LU <5 60 CO LU CD 40 D-D-- 4 — D--D-•D-•D- -D 0 . 7 5 0 . 5 0 C A S E FIGURE 22: CERTAINTY EQUIVALENTS FOR SUBJECT D i o o T FIGURE 23: CERTAINTY EQUIVALENTS FOR SUBJECT E APPENDIX D COMPUTER PROGRAM 'BELLMAN' The program, c a l l e d BELLMAN, i s w r i t t e n i n PL1 w h i c h o f f e r s advan-t a g e s o v e r FORTRAN i n h a n d l i n g m a t r i c e s , d a t a c o n v e r s i o n , i n p u t and o u t p u t f o r m a t s , language e f f i c i e n c y and program s t r u c t u r i n g . The main program i s d i v i d e d i n t o modules c o r r e s p o n d i n g t o d i f f e r e n t p a r t s o f the d e c i s i o n a n a l y s i s . The s t r u c t u r e i s shown i n F i g u r e 24. Appendix E c o n t a i n s l i s t i n g o f t h e program. Input f i l e s : Raw d a t a a r e s t o r e d i n f i l e s f o r easy o v e r v i e w and changes a t a c o n v e r s a t i o n a l t e r m i n a l . The c o n t e n t s o f t h e s e f i l e s a r e read i n a t the b e g i n n i n g of BELLMAN and p r e p r o c e s s e d f o r use i n the subprograms t h a t f o l l o w . Note t h a t each l i n e i n an i n p u t f i l e must end w i t h a comma. F i l e MISCELL c o n t a i n s m i s c e l l a n e o u s d a t a and program o p e r a t i n g v a r i a b l e s . F i l e ACTIONS l i s t s names f o r a v a i l a b l e a c t i o n s , t h e i r c o s t s and o t h e r r e s o u r c e r e q u i r e m e n t s . F i l e PAVARS l i s t s names o f pavement v a r i a b l e s and elementsi of Markov t r a n s i t i o n m a t r i c e s f o r each a c t i o n . F i l e ATTRIBS l i s t s names of a t t r i b u t e s , u n i t s o f measurement, the HALF/BEST r a t i o s and the s c a l i n g c o n s t a n t s f o r t h e m u 1 t i a t t r i b u t e model ( 6 . 5 ) . F i l e TEMPOS c o n t a i n s the temporal s c a l i n g c o n s t a n t s , K T ( T ) , and the t e r m i n a l v a l u e o f temporal u t i l i t y , TVSTAR, f o r t h e model ( 7 - 8 ) . Subprogram MARKOV: MARKOV computes the j o i n t s t o c h a s t i c m a t r i x Q. f o r each a c t i o n as o u t l i n e d i n F i g u r e 25. The subprogram t a k e s the Markov t r a n s i t i o n m a t r i x o f f i r s t pavement v a r i a b l e P(1) and s c a l a r m u l t i p l i e s i t by t h e t r a n s i t i o n m a t r i x o f t h e second pavement v a r i a b l e P ( 2 ) . The r e s u l t i s then s c a l a r m u l t i p l i e d by t h e t h i r d t r a n s i t i o n m a t r i x , and so on u n t i l t h e l a s t pavement 121 c S t a r t BELLMAN Read raw d a t a from i n p u t f i l e s and p r i n t o ut I Compute j o i n t s t o c h a s t i c m a t r i x ' MARKOV 1 > r F i t ut i 1 i t y c u r v e s 1 VON_NEUMAN 1 f Compute m a t r i c e s o f consequences and u n i a t t r i b u t e u t i l i t i e s ' R A I F F A 1 \ f Compute m a t r i x o f m u l t i a t t r i b u t e u t i l i t i e s 'KEENEY' > f Maximi ze: tempora1 e x p e c t e d u t i l i t y 'HOWARD' 1 Stop 'BELLMAN' FIGURE. 24: MAIN BLOCKS OF THE PROGRAM 'BELLMAN 122. ^ S t a r t 'MARKOV' > t Increment OLDQ = = 1 a c t i o n A t > Increment pave-ment v a r i a b l e K IE 0. = P(K)'OLDC) No A l l 1 C ? \ , Y e s A l l / !\ ? > f Yes F i l l i n LABEL L P r i n t o ut LABEL P r i n t o ut Q f o r each A c Return t o 1 BELLMAN No > > OLDQ = Q 3 FIGURE 25. SUBPROGRAM 1 MARKOV 1 v a r i a b l e , N, Q'= P ( N ) * ( P ( N - 1 ) * ••••*(P(2J*(P(D)i ••• ). (D.1) A s u b s e t o f the l o o p s f o r t h i s e q u a t i o n i s then r e p e a t e d io o r d e r t o s t o r e the s t a t e numbers o f pavement v a r i a b l e s t h a t e n t e r a j o i n t s t a t e number. The numbers a r e s t o r e d i n LABEL and make i t p o s s i b l e t o r e f e r back t o i n d i v i d u a l pavement v a r i a b l e s f o r t h e c a l c u l a t i o n of consequences. The e x p l a n a t i o n o f j o i n t s t a t e numbers, and t h e j o i n t t r a n s i t i o n m a t r i c e s a r e p r i n t e d o u t . The l a t t e r i s checked by t h e program and an e r r o r message produced i f a m a t r i x i s not s t o c h a s t i c . Subprogram VON NEUMANN: T h i s subprogram a p p r o x i m a t e s t h e u n i a t t r i b u t e u t i l i t y f u n c t i o n s by e x p o n e n t i a l c u r v e s . I t i s based on t h e t h e o r y i n Appendix B. The program ( F i g u r e 26) s t a r t s from a t e s t on the r i s k a t t i t u d e . For n e u t r a l c a s e s , t h e p r o c e d u r e computes t h e s l o p e of s t r a i g h t - l i n e u t i l i t y c u r v e . For t h e n o n n e u t r a l c a s e s HALF/BEST r a t i o i s t e s t e d and c o n s t a n t a o f (8.1) i t e r a t e d . I f t h e c a s e i s r i s k a v e r s e , i t i s t r a n s f o r d t o r i s k prone. I t e r a t i o n s t a r t s a t A = 0 w i t h an increment DELTA =0.1. The f u n c -t i o n s EXA i s t h e r i g h t hand s i d e o f (B.4) whereas EX i s t h e o t h e r s i d e . EPS i s the v a l u e on w h i c h a check i s made whether the i t e r a t e d A i s l a r g e r than the t r u e A. As soon as t h i s i s t r u e , t h e program checks i f t h e d i f f e r e n c e between b o t h s i d e s of (B.4) i s s m a l l enough t o t e r m i n a t e the i t e r a t i o n . If the d i f f e r e n c e i s t o o l a r g e , A i s reduced by one increment and then i n c r e a s e d by a new increment DELTA/10. T h i s c o n t i n u e s u n t i l EPS i s s u f f i -c i e n t l y s m a l l . The c u r r e n t A i s then t h e c o r r e c t v a l u e o f c o n s t a n t a i n Formula (B.1) i f t h e c a s e i s r i s k prone. For r i s k a v e r s e c a s e , A i s t r a n s -formed by a d d i n g 1 and c o n s t a n t c c a l c u l a t e d . Subprogram RAIFFA: Once t h e j o i n t s t a t e s a r e d e f i n e d i n terms o f s t a t e s o f i n d i v i d u a l pavement v a r i a b l e s and u n i a t t r i b u t e u t i l i t y c u r v e s f i t t e d f o r a l l a t t r i b u t e s , 124. S t a r t 1 VON NEUMANN' Increment a t t r i b u t e A = 0 DELTA = 0.1 n e u t r a l R i s k a t t i t u d e ? a v e r s e T r a n s f o r m t o r i s k prone A = A H - DELTA EPS = No EPS > 0 ? Yes EPS p r e c i s e ? No A = A - DELTA DELTA = DELTA/1C \^ Yes Ri sk prone? No T r a n s f o r m t o r i s k a v e r s e Yes P r i n t o u t uit i 1 i t y c u r v e No A l l a t t r i b u t e s ? Yes Return t o 1 BELLMAN 1FIGURE 26. SUBPROGRAM 'VON NEUMANN' RAIFFA computes t h e m a t r i x o f consequences CONS and c o r r e s p o n d i n g m a t r i x of u n i a t t r i b u t e u t i l i t i e s U. The l a t t e r i s checked f o r p r o p e r s i g n and s c a l i n g . The f o r m u l a s f o r computing a t t r i b u t e s X must be changed i f t h e s t r u c t u r e o f o b j e c t i v e s changes o r t h e number of s t a t e s o f an i n d i v i d u a l pavement v a r i a b l e exceeds 2. See F i g u r e 27 f o r b l o c k d i agram. Subprogram KEENEY: T h i s subprogram ( F i g u r e 28) a g g r e g a t e s a v e c t o r o f u n i a t t r i b u t e u t i l i t i e s i n t o t h e m u l t i a t t r i b u t e u t i l i t y s c a l a r f o r a l l a c t i o n s and a l l j o i n t s t a t e s a t a s t a g e ' s end. It f i r s t t e s t s t he a d d i t i v e independence c o n d i t i o n by c h e c k i n g t he sum o f s c a l i n g f a c t o r s . Complementary c a s e s ( C h a p t e r 6.4) a r e r e j e c t e d by the program and t e r m i n a t e e x e c u t i o n . Supplementary c a s e s cause t h e program t o i t e r a t e t h e i n t e r a c t i o n f a c t o r KAY a c c o r d i n g t o t h e t h e o r y i n S e c t i o n 6.6.5 o f r e f e r e n c e (22). The p r o c e d u r e f o r i t e r a t i n g KAY i s , e x c e p t f o r the t e s t i n g f o r r i s k a t t i t u d e , s i m i l a r t o t h a t used f o r i t e r a t i n g A i n VON NEUMANN. A f t e r KAY i s o b t a i n e d , KEENEY computes the m a t r i x o f m u l t i -a t t r i b u t e u t i l i t i e s , MUTIL, f o r a l l a c t i o n s by t h e m u l t i p l i c a t i v e model. The a d d i t i v e model i s used f o r problems i n w h i c h the s c a l i n g f a c t o r s sum up t o one. Subprogram HOWARD: F i g u r e 29 o u t l i n e s t h i s program. HOWARD s t a r t s w i t h t e s t i n g whether t h e temporal u t i l i t y m o d e l . i s a d d i t i v e o r m u l t i p l i c a t i v e . M u l t i p l i c a t i v e c a s e s c a u s e HOWARD t o i t e r a t e t he temporal i n t e r a c t i o n f a c t o r KAYT. The t e s t i n g and i n t e r a t i o n a r e v e r y s i m i l a r t o t h o s e i n KEENEY, e x c e p t f o r n o t a t i o n . The dynamic programming l o o p s c o n t a i n r e c u r s i v e e q u a t i o n s f o r both t h e a d d i t i v e and the m u l t i p l i c a t i v e model o f temporal u t i 1 i t y . S t a r t 'RAIFFA' 1ncrement j o i n t s t a t e 1 \ r 1 ncrement a c t i on A > f X(*) = ... CONS (A, 1 ,'• ) = x(*) No A l 1 A? No Yes A l l I? Yes P r i n t out CONS y f T r a n s f o r m CONS i n t o m a t r i x o f un i a t t r i bute u t i l i t i e s \ f c Return t o 'BELLMAN' FIGURE 2 ? : SUBPROGRAM 'RAIFFA' No No c c = 1 1ncrement a c t ion A r 1ncrement j o i n t s t a t e 1 > r Add i t i ve model A l 1 I ? Yes A l 1 A ? Yes P r i n t out case S t a r t "KEENEY 1 ZKK 7 I t e r a t e KAY 1ncrement a c t i o n A f 1ncrement j o i n t s t a t e 1 • i f Mu 11 i p mo 1 i c a t i ve del <1 A l 1 I ? No Yes A l l A ? No Yes P r i n t o ut case R e t u r n t o 'BELLMAN Termi nate 'BELLMAN* FIGURE 23:: SUBPROGRAM ' KEENEY ' I t e r a t e KAYT 1 ncrement st a g e T 1ncrement s t a t e 1 1ncrement a c t i o n A ^ f T e r m i n a t e " ^ H ' 1 BELLMAN 1 P r i n t out opt imal p o l i c y c ADD ? > 1 'B \ ,'0'B Mu 11 i p i i ca-t i ve mode 1 \ f Choose maximum Return, t o 'BELLMAN' 3 FIGURE 29: SUBPROGRAM 'HOWARD' CO 129. HOWARD p r i n t s out t h e o p t i m a l d e c i s i o n r u l e s f o r a l l s t a g e s . I f the p o l i c y c o n s i s t s o f fewer d e c i s i o n r u l e s than t h e r e a r e s t a g e s , t h e l a s t r u l e p r i n t e d o u t i s t h e s t e a d y - s t a t e d e c i s i o n r u l e . 130. APPENDIX E COMPUTER PROGRAM LISTING 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 8 19 2 0 21 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 31 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 4 0 41 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 0 51 5 2 5 3 54 5 5 5 6 5 7 5 8 5 9 6 0 B E L L M A N : PROC O P T I O N S ( M A I N ) R E O R D E R ; - D C L ( M I S C E L L , A C T I O N S , A T T R I B S , P A V A R S , T E M P O S ) F I L E S T R E A M ; O D C L E P S I L O N I N I T ( l E - 6 ) ; /* P R E C I S I O N O F C O M P U T A T I O N S */ O D C L P R E C I N I T d E - 2 ) ; /* P R E C I S I O N OF E M P I R I C A L E S T I M A T E S *•/ O D C L ( A, A B E S T , I , J , K, L , M, N, T ) F I X E D B I N C 1 5 , 0 ) ; O D C L ( V B E S T , V H O L D ) F L O A T ; O D C L NAME<2> C H A R ( 2 0 ) V A R Y I N G ; O D C L ( E N D A C T , E N D A T T , E N D P A V , E N D M I S , E N D T E M ) B I T ( 1 > I N I T ( ' O ' B ) ; OON E N D F I L E ( M I S C E L L ) E N D M I S = ' 1 ' B ON E N D F I L E < A C T I O N S > E N D A C T = ' 1 'B ON E N D F I L E < A T T R I B S ) E N D A T T = ' 1 ' B ON E N D F I L E ( P A V A R S > E N D P A V = ' 1 ' B ON E N D F I L E ( T E M P O S ) E N D T E M = ' 1 ' E ; O O P E N F I L E ( A C T I O N S ) I N P U T , F I L E ( A T T R I B S ) I N P U T , F I L E ( P A V A R S ) I N P U T , F I L E ( M I S C E L L ) I N P U T , F I L E ( T E M P O S ) I N P U T ; - / * • # * R E A D IN AND P R I N T OUT M I S C E L L A N E O U S D A T A : I D E N T I F I C A T I O N O F C U R R E N T RUN, A V E R A G E D A I L Y T R A F F I C ( 1 0 0 0 V E H I C L E S / D A Y ) , NUMBER OF R E S I D E N T S ( 1 0 0 0 H O U S E H O L D S >, L E N G T H O F P A V E M E N T S E C T I O N ( K M ) , NUMBER OF L A N E S , E X T R A V E H I C L E O P E R A T I N G C O S T ( 4 / V E H I C L E K M ) , D E L A Y PER V E H I C L E S L O W D O W N - C Y C L E ( H O U R S ) , SLOWDOWN C Y C L E S PER KM(1./KM>, A C T I O N S , A T T R I B U T E S , P A V E M E N T V A R I A B L E S S T A G E S OF O F OF O F O F NUMBER NUMBER NUMBER NUMBER NUMBER T E M P O R A L S C A L I N G C O N S T A N T S O D C L RUNNAME C H A R ( 6 0 ) V A R Y I N G ; O D C L ( T R A F F I C , R E S I D E N T S , L E N G T H , L A N E S , E V 0 C D 3 T , D E L A Y , C Y C L E S ) F L O A T ; O D C L ( N A C T , N A T T , N P A V , N S T G ) F I X E D B I N ( 1 5 , 0 ) ; O G E T F I L E ( M I S C E L L ) D A T A ( R U N N A M E , T R A F F I C , R E S I D E N T S , L E N G T H , L A N E S , E V O C O S T , D E L A Y , C Y C L E S , N A C T , N A T T , N P A V , N S T G ) ; P U T P A G E E D I T ( R U N N A M E , ' T R A F F I C = ' , T R A F F I C , ' * 1 0 0 0 V E H I C L E S / D A Y ' , ' R E S I D E N T S ' ' , R E S I D E N T S , ' * 1 0 0 0 H O U S E H O L D S ' , ' L E N G T H - ' , L E N G T H , ' K M ' , ' L A N E S = ' , L A N E S , ' ' , ' E X T R A V E H I C L E O P E R A T I N G C O S T = ' , E V O C O S T , ' D O L L A R S / V E H I C L E K M ' ' D E L A Y = ' , D E L A Y , ' H O U R S ' , ' C Y C L E S = ' , C Y C L E S , •' / K M ' , 'NUMBER OF A C T I O N S = ' , N A C T , 'NUMBER DF A T T R I B U T E S = ' , N A T T , 'NUMBER OF P A V E M E N T V A R I A B L E S = ' , N P A V , 'NUMBER O F S T A G E S = ' , N S T G ) ( A , S K I P ( 2 ) , ( 4 ) ( A , F ( 4 ) , A , S K I P ( 2 ) ) , ( 3 ) ( A , F ( 6 , 3 ) , A , S K I P ( 2 ) /, ( 4 ) ( A , F ( 3 > , S K I P ( 2 ) ) ) ; 0 N J 0 I N T = 2 # # N P A V ; /* NUMBER OF J O I N T S T A T E S */ - B E G I N ; /» V A R I A B L E S D E C L A R E D IN T H E F O L L O W I N G A R E IN ONE B L O C K W I T H T H E S U B P R O G R A M S AND N E E D NOT B E C T L •»/ O D C L X < N A T T ) F L O A T ; O D C L ( C O S T ( N A C T ) , J O B S ( N A C T ) , G R A V E L ( N A C T ) , T I M E ( N A C T ) > F L O A T ; O D C L P ( N P A V , N A C T , 2, 2 ) F L O A T ; 131 • 61 O D C L Q ( N A C T , N J O I N T , N J O I N T ) F L O A T ; 6 2 O D C L C O N S ( N A C T , N J O I N T , N A T T ) F L O A T , 6 3 O D C L L A B E L ( N J O I N T , N P A V ) F I X E D B I N U 5 , 0 ) ; 6 4 O D C L C ( N A T T , 3 ) F L O A T ; 6 5 O D C L U ( N A C T , N J O I N T , N A T T ) F L O A T ; 6 6 O D C L M U T I L ( N A C T , N J O I N T ) F L O A T ; 6 7 O D C L ( A T T N A M E < N A T T > , U N I T ( N A T T ) ) C H A R ( 2 0 ) V A R Y I N G ; 6 8 O D C L ( A C T N A M E ( N A C T ) , P A V N A M E ( N P A V ) ) C H A R ( 2 0 > V A R Y I N G ; 6 9 - / * * * « R E A D I N AND P R I N T OUT A C T I O N ' S NAME AND R E S O U R C E S 7 0 R E Q U I R E D PER LANE»KM: C O S T ( * > , J O B S ( P E R S O N * D A Y S ) , 71 G R A V E L ( T O N S > , T I M E ( D A Y S / C R E W ) 7 2 »***/ 7 3 O D C L ( B E S T ( N A T T > , H A L F ( N A T T ) , K M N A T T ) ) F L O A T ; 7 4 O D C L TEMP(2»NACT> F L O A T ; 7 5 O P U T L I N E ( 3 1 ) E D I T 7 6 ( ' A C T I O N S W I T H R E S O U R C E S R E Q U I R E D PER L A N E * K M ' ) ( A ) ; 7 7 P U T S K I P ( 2 ) E D I T 7 8 ( ' A C T I O N ' , ' C O S T ' , ' J O B S ' , ' G R A V E L ' , ' T I M E ' ) 7 9 ( A , C O L ( 1 1 ) , A , C O L ( 2 2 ) , A, C O L ( 3 7 ) , A, C O L ( 4 5 ) , A ) ; 8 0 P U T S K I P E D I T ( ' ( D O L L A R S ) ' , ' ( P E R S O N - D A Y S ) ' , ' ( T O N S ) ' , ' ( D A Y S ) ' ) 81 ( C O L ( 1 1 ) , ( 4 ) ( A , X ( 2 ) ) ) ; 8 2 ODO 1=1 T O N A C T ; 8 3 G E T F I L E ( A C T I O N S ) L I S T ( A , N A M E ( 1 ) , ( T E M P ( K ) DO K = l TO 4 ) > ; 8 4 AC T N A M E ( A > = N A M E ( 1 ) ; 8 5 C 0 S T ( A ) = T E M P ( 1 ) ; 8 6 J 0 B S ( A ) = T E M P ( 2 > ; 8 7 G R A V E L ( A ) = T E M P ( 3 ) ; . 8 3 T I M E ( A ) = T E M P ( 4 > ; 8 9 P U T S K I P ( 2 ) E D I T 9 0 ( A C T N A M E ( A > , C O S T ( A ) , J O B S ( A ) , G R A V E L ( A ) , T I M E ( A ) ) 91 ( A ( 1 0 ) , C O L ( 1 1 ) , F ( 7 ) , C O L ( 2 3 ) . F ( 4 , 1 ) , C O L ( 3 8 ) , F ( 4 ) , 9 2 C O L ( 4 5 ) , F ( 4 , 1 ) ) ; 9 3 END; 9 4 - / * * + • C A L C U L A T E ' B E S T ' - V A L U E FOR A L L A T T R I B U T E S 9 5 ###*/ 9 6 O D C L C M A X C O S T , M A X J O B S , M A X G R A V E L , M A X T I M E ) F L O A T I N I T ( O ) ; 9 7 ODO 1=1 TO N A C T ; 9 8 M A X C O S T = M A X ( M A X C O S T , C O S T C I ) ) ; 9 9 M A X J O B S = M A X ( M A X J O B S , J O B S ( I ) ) ; 1 0 0 M A X G R A V E L = M A X ( M A X G R A V E L , G R A V E L ( I ) ) ; 101 M A X T I M E = M A X ( M A X T I M E , T I M E ( I ) ) ; 1 0 2 END; 1 0 3 O B E S T ( 1 ) , B E S T ( 5 ) = T R A F F I C ; 1 0 4 B E S T ( 3 ) = ( C Y C L E S * 3 6 4 + M A X T I M E * L A N E S ) * L E N G T H * D E L A Y * T R A F F I C ; 1 0 5 B E S T ( 4 ) = M A X C 0 S T * L E N G T H * L A N E S / 1 0 0 0 ; 1 0 6 BEST(6)=MAXJOBS»LENGTH*LANES; 1 0 7 B E S T ( 2 ) = L E N G T H * T R A F F I C » E V 0 C 0 S T * 3 6 4 ; 1 0 8 B E S T ( 7 ) = M A X G R A V E L * L E N G T H * L A N E S / 1 0 0 0 ; 1 0 9 E E S T ( 8 > = R E S I D E N T S ; 1 1 0 - / * * * * R E A D IN AND P R I N T OUT P A V E M E N T V A R I A B L E ' S NAME AND 111 T R A N S I T I O N P R O B A B I L I T I E S FOR A L L A C T I O N S 1 1 2 * » » * / 1 1 3 ODO 1=1 T O N P A V ; 1 1 4 P U T L I N E ( 3 0 ) ; 1 1 5 G E T F I L E ( P A V A R S ) L I S T ( K , N A M E ( 1 ) , T E M P ) ; 1 1 6 P A V N A M E ( K ) = N A M E ( 1 ) ; 1 1 7 0 DO A = l TO N A C T ; 1 1 8 P ( K , A , 1 , 1 ) = T E M P ( 2 * A - 1 ) ; 1 1 9 P ( K , A , 2 , 1 ) = T E M P ( 2 * A ) ; 1 2 0 F ( K , A , * , 2 ) = 1 - P ( K , A, #, 1 ) ; 132. 121 P U T S K I P ( 2 ) E D I T 1 2 2 ( ' M A R K O V T R A N S I T I O N M A T R I X FOR ' , P A V N A M E ( K > , 1 2 3 'UNDER A C T I O N = ' , A C T N A M E ( A ) > ( ( 2 ) A , S K I P , ( 2 ) A ) ; 1 2 4 P U T S K I P E D I T ( ' 1 ' , ' 2 ' ) ( C 0 L ( 2 1 ) - A . X ( 5 ) , A ) ; 1 2 5 P U T E D I T 1 2 6 ( '1 ' , P ( K , A , 1, 1 ) , P ( K , A, 1, 2 ) , ' 2 ' , P ( K , A, 2, 1 ) , P ( K , A , 2, 2 ) ) 1 2 7 ( S K I P , C O L ( 1 8 ) , A , ( 2 > ( F ( 6 , 2 ) ) ) ; 1 2 8 END, 1 2 9 E N D ; 1 3 0 - / • * » * R E A D I N A T T R I B U T E ' S NAME, U N I T , H A L F / B E S T R A T I O AND 131 S C A L I N G C O N S T A N T 1 3 2 * # # * / 1 3 3 ODD 1=1 TO N A T T ; 1 3 4 G E T F I L E ( A T T R I B S ) L I S T ( L , N A M E , T E M P ( 1 ) , T E M P ( 2 ) ) ; 1 3 5 A T T N A M E ( L > = N A M E ( 1 ) ; 1 3 6 U N I T ( L ) = N A M E ( 2 ) ; 1 3 7 H A L F ( L ) = T E M P ( 1 > ; 1 3 8 K K ( L > = T E M P < 2 > ; 1 3 9 END; 1 4 0 - / » * * » R E A D I N T E M P O R A L S C A L I N G F A C T O R S 141 AND T E R M I N A L V A L U E O F T E M P O R A L U T I L I T Y 1 4 2 *»*••»/ 1 4 3 O D C L T V S T A R F L O A T ; 1 4 4 O D C L K T ( N S T G ) F L O A T , 1 4 5 O G E T F I L E ( T E M P O S ) L I S T ( ( T , K T ( T ) DO T = l T O N S T G ) , T V S T A R ) ; 1 4 6 O C A L L MARKOV; 1 4 7 O C A L L VON_NEUMANN; 1 4 8 O C A L L R A I F F A ; 1 4 9 O C A L L K E E N E Y ; 1 5 0 O C A L L HOWARD, 151 1/* S C A L A R M U L T I P L Y I N D I V I D U A L MARKOV T R A N S I T I O N M A T R I C E S 1 5 2 TO P R O D U C E T H E J O I N T S T O C H A S T I C M A T R I X Q ( A , 1 1 , J J ) 1 5 3 A = A C T I O N 1 5 4 I I = v F R O M ' J O I N T S T A T E 1 5 5 J J = * T O ' J O I N T S T A T E */ 1 5 6 OMARKOV: P R O C ; 1 5 7 O D C L ( A , 1 1 , 1 2 , I I , J 1 , J 2 , J J , N ) F I X E D B I N ( 1 5 , 0 ) ; 1 5 8 D C L O L D Q ( N A C T , N J O I N T , N J O I N T ) F L O A T ; 1 5 9 D C L C A R R Y ( N P A V ) F I X E D B I N ( I 5 , 0 ) ; 1 6 0 ODO A = l TO N A C T ; /» GO THROUGH A C T I O N S •/ 161 G ( A , * , # > = ! ; 1 6 2 DO K = l T O NPAV; /* GO THROUGH P A V E M E N T V A R I A B L E S */ 1 6 3 N = 2 * * ( K - 1 ) ; 1 6 4 O L D Q ( A , *, * >=Q ( A . *> * ) ; 1 6 5 DO 11=1 TO 2; /* ROWS I I , COLUMNS J I »/ 1 6 6 DO J l = l TO 2; /* OF 2 X 2 T R A N S I T I O N M A T R I X */ 1 6 7 DO 12=1 T O N; /* ROWS 1 2 , C O L U M N S J 2 +/ 1 6 8 I I = ( I l - l ) * N + I 2 i /* OF J O I N T S T O C H A S T I C M A T R I X */ 1 6 9 DO J 2 = l T O N; /* C O M P U T E D SO F A R */ 1 7 0 J J = ( J i - 1 > * N + J 2 ; 171 Q ( A , I I , J J ) = P ( K , A , I I , J I ) • f r O L D Q ( A , 1 2 , J 2 > ; 1 7 2 END; /# J 2 */ 1 7 3 END; /* 12 */ 1 7 4 END; /* J I */ 1 7 5 END; /* I I #/ 1 7 6 END; /* K */ 1 7 7 END; /» A */ 1 7 8 Q/« R E P E A T K, 1 1 , 1 2 L O O P S T O F I L L IN ' L A B E L ' M A T R I X */ 1 7 9 ODO K = i TO N P A V ; 1 8 0 N = 2 » * ( K - 1 ) ; 133 181 DO 11=1 TO 2; 1 8 2 DO 12=1 TO N; 1 8 3 I I = ( I 1 - 1 ) * N + I 2 ; 1 8 4 L A B E L ( I I , K ) = I 1 ; 1 8 5 END; /* 12 */ 1 8 6 E N D ; /* I I */ 1 8 7 /* F I L L I N T H E R E S T OF ' L A B E L ' M A T R I X */ 1 8 8 DO L = l T O N J 0 I N T / ( 2 * N ) - 1; /* NO. O F G R O U P S */ 1 8 9 DO M=l T O 2 * N ; /* NO. I N E A C H GROUP «/ 1 9 0 L A B E L ( L * < 2»N >+M. K ) = L A B E L ( M , K >; 191 END; /» M »/ 1 9 2 END; /* L */ 1 9 3 E N D ; /* K */ 1 9 4 0/» P R I N T OUT E X P L A N A T I O N OF J O I N T S T A T E NUMBERS 1 9 5 IN T E R M S O F S T A T E S P A V E M E N T V A R I A B L E S A R E IN */ 1 9 6 O P U T S K I P ( 6 ) E D I T ( ' L A B E L S FOR J O I N T S T A T E S ' ) ( A ) ; 197 P U T S K I P ( 2 > E D I T 1 9 8 ( ' S T A T E O F P A V E M E N T V A R I A B L E NUMBER: ' ) ( C O L ( 1 7 ) . A ) ; 1 9 9 P U T S K I P E D I T ( ( K DO K = l T O N P A V ! > ( C 0 L ( 7 ) , ( N P A V ) ( X ( 9 ) , F ( 2 > > ) ; 2 0 0 P U T S K I P E D I T ( ' J O I N T S T A T E ' , ( P A V N A M E ( K ) DO K = l TO N P A V ) > 2 0 1 ( A , ( N P A V ) ( X ( 2 ) / A ) ) ; 2 0 2 P U T E D I T ( ( J , L A B E L ( J , *> DO J = l TO N J O I N T ) ) 2 0 3 ( S K I P , C O L ( B > , F ( 2 ) , ( N P A V ) ( X ( 8 ) , F ( 2 > > > ; 2 0 4 0 / * P R I N T OUT J O I N T S T O C H A S T I C M A T R I X FOR E A C H A C T I O N */ 2 0 5 ODO A = l T O N A C T ; 2 0 6 P U T L I N E ( 3 1 ) E D I T 2 0 7 ( ' 1 0 0 * J 0 I N T S T O C H A S T I C M A T R I X FOR A C T I O N = ' , A C T N A M E ( A ) ) ( A , A ) ; 2 0 8 O P U T S K I P ( 2 > E D I T ( ( K DO K = l TO N J O I N T ) > ( X ( 4 ) , ( N J O I N T ) F ( 3 > ) ; 2 0 9 P U T S K I P ; 2 1 0 DO 1=1 TO N J O I N T ; 2 1 1 I F A B S ( S U M ( G ( A , I , # > > - l ) > E P S I L O N T H E N 2 1 2 P U T S K I P E D I T C R O W SUM WRONG I N R O W = ' , I ) (A , F ( 2 > ) ; 2 1 3 I F A N Y ( Q ( A , I , # ) < 0 ) T H E N 2 1 4 P U T S K I P E D I T ( ' P R O B A B I L I T Y - C O I N ROW= ', I ) ( A , F ( 2 ) ) ; 2 1 5 P U T S K I P E D I T ( 1 , 1 0 0 * 0 ( A , I , * ) ) ( F ( 2 ) , X ( 2 ) , ( N J O I N T > F ( 3 ) ) ; 2 1 6 END; /* I #/ 2 1 7 END; /» A */ 2 1 8 O R E T U R N ; 2 1 9 END MARKOV; 2 2 0 ! / • F I T E X P O N E N T I A L C U R V E S TO U T I L I T Y F U N C T I O N S G I V E N BY: 2 2 1 U ( 0 ) = 0 , U ( H A L F ) = 0 . 5, U ( B E S T ) = 1 . 0 */ 2 2 2 0 V 0 N _ N E U M A N N : PROC R E T U R N S ( F L O A T ) ; 2 2 3 O D C L ( A , B , D E L T A , E P S , E X , E X A , U, X ) F L O A T ; 2 2 4 C = 0 ; /• I N I T I A L A L L C O E F F I C I E N T S FOR E X P O N E N T I A L C U R V E S */ 2 2 5 0 / * T E S T E A C H A T T R I B U T E FOR R I S K A T T I T U D E */ 2 2 6 O T A K E : 2 2 7 DO 1=1 T O N A T T ; 2 2 8 I F K K ( I ) > 0 T H E N DO; /* I G N O R E A T T R I B U T E S WITH K K = 0 */ 2 2 9 H A L F ( I ) = H A L F ( I ) » B E S T ( I > ; 2 3 0 E X = H A L F ( I ) / B E S T ( I ) ; 2 3 1 I F A B S ( E X - . 5 X P R E C T H E N GO TO N E U T R A L ; 2 3 2 A = 0 ; D E L T A = 0 . 1; /* I N I T I A L T H E S E A R C H V A R I A B L E S */ 2 3 3 I F E X - . 5 < P R E C T H E N E X = 1 - E X ; /# C O N V E R T A V E R S E TO P R O N E 4/ 2 3 4 GO TO P R O N E ; 2 3 5 O S C A L E : A = A - D E L T A ; /* R E S C A L E T H E S E A R C H V A R I A B L E S */ 2 3 6 D E L T A = D E L T A / 1 0 ; 2 3 7 O P R O N E : E P S = 1 ; 2 3 8 DO W H I L E ( E P S > 0 ) ; 2 3 9 A = A + D E L T A ; 2 4 0 E X A = L O G ( A + . 5 ) - L 0 G ( A ) ; 134. 241 E X A = E X A / ( L 0 G ( A + 1 ) - L O G ( A ) ) ; 242 EPS=(EXA-EX)/EA; 243 END; 244 I F E P S O E P S I L O N THEN GO TO SCALE; 245 B = ( L 0 G ( A + 1 ) - L 0 G ( A ) ) / B E S T ( I ) ; 246 I F H A L F ( I ) / B E S T ( I ) - . 5 C PREC THEN GO TO AVERSE; 247 C ( I . 1 ) = - A ; 248 C ( I , 2 ) = B ; 249 GO TO EXIT; 250 OAVERSE:A-A+l ; 251 C ( I , 1>=A; 252 C ( I , 2 ) = - B ; 2 5 3 GO TO EXIT; 254 ONEUTRAL: C ( I , 3 ) = 1 / B E S T ( I > ; 2 5 5 0/* PRINT OUT INFORMATION ABOUT ATTRIBUTES 256 AND SKETCH U T I L I T Y CURVES F I T T E D »/ 257 OEXIT:. 258 PUT L I N E C 3 1 ) E D I T C I , ATTNAMECI), ' ( ' , U N I T ( I ) , ')'> 259 <F<2>,X ( l ) , (4)A>; 2 6 0 PUT S K I P E D I T ( ' S C A L I N G CONSTANT KK<', I, ')=',KK(I>> 261 ( A , F ( 1 ) , A , F ( 5 , 2 ) i ; 262 PUT S K I P E D I T ( 'BEST=', B E S T < I ) , 'HALF= ' , H A L F ( I ) ) 2 6 3 ( ( 2 ) ( A , F ( 6 , 1>, S K I P ) ) ; 264 I F A B S ( H A L F ( I ) / B E S T ( I ) - . 5 X P R E C THEN 265 PUT S K I P E D I T ( ' R I S K NEUTRAL: U(X) =',C(1,3>, '*X ' > 266 ( A , F ( 6 , 3 ) < A); 267 E L S E I F (HALF( I ) / B E S T C I ) - . 5 X P R E C THEN 268 PUT S K I P E D I T 269 ('RISK AVERSE: U ( X ) = ',C(1, 1), '#(1-EXP(',C(1,2>, '*X ) ) ') 2 7 0 ( A . ( 2 > ( F ( 6 . 3 ) , A > ) ; 271 E L S E PUT S K I P E D I T 272 ('RISK PRONE: U(X> =',C(1, 1>, ' » ( 1 - E X P ( ' , C ( 1 , 2 ) , '*X) ) ') 2 7 3 ( A , ( 2 ) ( F ( 6 , 3 ) , A ) ) ; 274 PUT E D I T 2 7 5 ( '0', '1', ( 2 0 ) ' _'. ' U ( X ) ' ) ( S K I P ( 2 ) , A , X ( 3 7 ) , A - S K I P , A , A ) ; 276 U=0; 2 7 7 DO K= l TO 20; 278 X = K » B E S T ( I ) / 2 0 ; 2 7 9 U = C ( I , 1 > * ( 1 - E X P ( C ( I - 2 > * X ) ) ; 280 U=U+C(I,3)*X; 281 U=40#(U>; 282 I F U<=1 THEN 283 PUT E D I T ( '* ' ) ( C O L U >. A) ; 284 E L S E PUT E D I T ( ' ! ' . ' * ' ) ( C O L ( 1 ) , A, C 0 L ( U + 1 ) , A) ; 285 END; 286 0 END; /* I F K K ( I ) > 0 */ 287 OEND TAKE; 288 ORETURN; 289 END VON_NEUMANN; 290 1/* COMPUTE CONSEQUENCES AND UNIATTRIBUTE U T I L I T I E S */ 291 ORAIFFA: PROC RETURNS(FLOAT); 292 ODCL F(NPAV,2) FLOAT I N I T ( ( 2 » N P A V ) 0 ) ; 2 9 3 F ( » , 1 ) = 1 . 0 ; / * F ( K , J ) = l WHEN PAVEMENT VARIABLE 'K' 294 I S IN ACCEPTABLE STATE ( J = l ) */ 2 9 5 0/* COMPUTE A CONSEQUENCE (= VECTOR OF ATTRIBUTES) 296 FOR EACH ACTION AND J O I N T STATE * / 297 ODO 1=1 TO NJOINT; /* RUN THROUGH J O I N T S T A T E S */ 298 DO A= l TO NACT; /* RUN THROUGH ACTIONS * / 2 9 9 X ( 1 ) = F ( 1 i L A B E L ( I • 1 ) ) * F ( 2 , L A B E L ( 1 , 2 ) ) * T R A F F I C ; 300 X ( 3 ) = ( l - F ( 3 , L A B E L ( I , 3 ) ) ) » C Y C L E S # 3 6 4 + T I M E ( A ) * L A N E S ; 135. 3 0 0 . 5 X ( 3 > = B E S T ( 3 >-X < 3 ) * L £NGTH*DELAY*TRAFFIC< 3 0 2 X ( 4 ) = B E S T ( 4 ) - C O S T < A)»LENGTH#LANES/1000; 3 0 3 X ( 5 ) = F ( 4 , L A B E L d , 4 ) ) * T R A F F I C , 3 0 4 X ( 6 ) = J 0 B S ( A > * L E N G T H * L A N E S ; 3 0 5 X ( 2 ) = F ( 3 . L A B E L ( I , 3 ) > * L E N G T H * T R A F F I C * E V 0 C 0 S T « 3 6 4 ; 3 0 6 X ( 7 ) = B E S T ( 7 ) - G R A V E L ( A ) * L E N G T H * L A N E S / 1 0 0 0 ; 3 0 7 X ( B ) = ( 1 - F ( 1 , L A B E L ( I , 1 ) ) P R E S I D E N T S ; 3 0 8 0 C O N S ( A . I , * > = X ; /* A = A C T I 0 N , I = J 0 I N T S T A T E , * = J = A T T R I B U T E */ 3 0 9 0 END; /* A «/ 3 1 0 E N D ; /* I */ 3 1 1 0 / * P R I N T OUT M A T R I X OF C O N S E Q U E N C E S FOR E A C H A C T I O N */ 3 1 2 ODO A = l TO N A C T ; 3 1 3 P U T L I N E C 2 5 ) E D I T 3 1 4 ( ' M A T R I X O F C O N S E Q U E N C E S FOR A C T I O N = ' , A C T N A M E ( A ) ) ( A , A ) ; 3 1 5 P U T S K I P ( 2 ) E D I T 3 1 6 ( ' A T T R I B U T E ' , (K DO K = l TO N A T T ) ) ( A , C O L ( 1 3 ) , ( N A T T > ( F ( 2 ) , X ( 5 ) > > ; 3 1 7 DO J = l T O N J O I N T ; 3 1 8 P U T S K I P E D I T ( ' S T A T E ' , J , ( C O N S ( A , J , K ) DO K = l TO N A T T ) ) 3 1 9 ( A , F ( 3 ) , ( N A T T ) ( X ( l > , F ( 6 , 2 ) ) > ; 3 2 0 END; /* J */ 3 2 1 E N D ; /* A */ 3 2 2 0 / * T R A N S F O R M M A T R I X O F C O N S E Q U E N C E S I N T O M A T R I X OF 3 2 3 U N I A T T R I B U T E U T I L I T I E S 3 2 4 U ( A , I , J ) : A = A C T I O N , I = J O I N T S T A T E , J = A T T R I B U T E */ 3 2 5 ODO A = l T O N A C T ; 3 2 6 DO 1=1 TO N J O I N T ; 3 2 7 DO J = l TO N A T T ; 3 2 S U ( A , I , J ) = C ( J , 1 > + ( 1 - E X P ( C ( J , 2 > * C 0 N S ( A , I , J ) ) ) 3 2 9 + C ( J , 3 > * C 0 N S ( A , I , J ) ; 3 2 0 I F U ( A , I , J X O i ( 1 - U ( A , I , J ) K - E P S I L O N T H E N DO; 3 3 1 P U T S K I P ( 5 ) E D I T 3 3 2 ( 'WRONG U N I A T T R I B U T E U T I L I T Y = ' , U ( A , I , J ) , 3 3 3 ' FOR A C T I O N = ' , A C T N A M E ( A ) , ' A T T R I B U T E = ' , A T T N A M E < J ) , 3 3 4 ' J O I N T S T A T E = ' , J ) ( A , F ( 7 , 2 ) , ( 5 > A, F ( 2 ) >; 3 3 5 END; 3 3 6 E N D ; /* J */ 3 3 7 E N D ; /* I */ 3 3 8 E N D ; /* A »/ 3 3 9 O R E T U R N ; 3 4 0 OEND R A I F F A ; 3 4 1 1/* C O M P U T E M U L T I A T T R I B U T E U T I L I T I E S BY A P P R O P R I A T E MODEL */ 3 4 2 O K E E N E Y : PROC R E T U R N S ( F L O A T ) ; 3 4 3 O D C L ( D E L T A , E P S , K A Y , L E F T , R I G H T ) F L O A T ; 3 4 4 0 / * T E S T T H E A D D I T I V E I N D E P E N D E N C E C O N D I T I O N */ 3 4 5 O I F ( l - S U M ( K K ) > > P R E C T H E N DO; 3 4 6 P U T P A G E E D I T ( ( 1 0 ) ' * ' , ' SUM O F KK < l ' X A , A ) ; 3 4 7 GO TO A D I E U ; /* T E R M I N A T E E X E C U T I O N */ 3 4 8 E N D ; 3 4 9 E L S E .IF A B S ( l - S U M ( K K ) > O P R E C T H E N GO TO ADD I T ; 3 5 0 0/« I T E R A T E T H E I N T E R A C T I O N F A C T O R KAY */ 3 5 1 O K A Y = 0 ; /* I N I T I A L T H E S E A R C H V A R I A B L E S */ 3 5 2 D E L T A = . 1 ; 3 5 3 GO TO I T E R A T E ; 3 5 4 O S C A L E : K A Y = K A Y + D E L T A ; /* R E S C A L E T H E S E A R C H V A R I A B L E S •*/ 3 5 5 D E L T A = D E L T A / 1 0 ; 3 5 6 0 I T E R A T E : E P S = 1 ; 3 5 7 DO W H I L E ( E P S > E P S I L O N ) ; 3 5 8 K A Y = K A Y - D E L T A ; 3 5 9 L E F T = 1 + K A Y ; 3 6 0 R I G H T = i ; 136. 3 6 1 DO 1=1 TO N A T T ; 3 6 2 RIGHT=RIGHT«(1+KAY*KK(I>>; 3 6 3 END; 3 6 4 E P S = L E F T - R I G H T ; 3 6 5 END ; 3 6 6 O I F E P S O E P S I L O N T H E N 0 0 T O S C A L E ; 3 6 7 C M U L T I P : / * M U L T I P L I C A T I V E M U L T I A T T R I B U T E U T I L I T Y MODEL */ 3 6 8 DO A = l TO N A C T ; 3 6 9 DO 1=1 TO N J O I N T ; 3 7 0 M U T I L ( A , I>=PR0D(1+KAY»KK»U(A, I , * ) ) ; 3 7 1 M U T I L ( A , I ) = ( M U T I L ( A , D - D / K A Y ; 3 7 2 E N D ; /* I */ 3 7 3 END; /» A »/ 3 7 4 P U T S K I P ( 5 > E D I T ( ' M U L T I A T T R I B U T E U T I L I T Y I S ' X A ) ; 3 7 5 P U T S K I P E D I T ( ' N E G A T I V E M U L T I P L I C A T I V E , S U M ( K K ( I ) ) = ' , 3 7 6 S U M ( K K ) X A , F ( 6 , 3 ) >; 3 7 7 P U T S K I P E D I T < ' I N T E R A C T I O N F A C T O R K A Y = ' . K A Y > ( A . F ( 6 , 3 ) ) ; 3 7 8 0 GO TO E X I T ; 3 7 9 O A D D I T : /* A D D I T I V E M U L T I A T T R I B U T E U T I L I T Y MODEL »/ 3 8 0 DO A = l T O N A C T ; 3 8 1 DO 1=1 TO N J O I N T ; 3 8 2 M U T I L < A . I > = S U M ( K K * U ( A , I , * ) ) ; 3 8 3 E N D ; /* I */ 3 8 4 E N D ; /* A •*/ 3 8 5 P U T S K I P ( 5 ) E D I T ( ' M U L T I A T T R I B U T E U T I L I T Y I S A D D I T I V E ' X A ) ; 3 8 6 O E X I T : 3 8 7 O R E T U R N ; 3 8 8 OEND K E E N E Y ; 3 8 9 1/* T E S T FOR T H E T E M P O R A L U T I L I T Y MODEL 3 9 0 AND C O M P U T E T H E O P T I M U M P O L I C Y */ 3 9 1 OHOWARD: PROC R E T U R N S ( F L O A T ) ; 3 9 2 O D C L V S T A R ( N J O I N T , 0 : N S T G ) F L O A T ; 3 9 3 V S T A R ( * , 0 ) = T V S T A R , /* S E T T H E T E R M I N A L V A L U E S */ 3 9 4 D C L A S T A R ( N J O I N T , N S T G ) F I X E D B I N M 5 , 0 ) ; 3 9 5 D C L ( R U L E ( N J O I N T ) , O L D R U L E ( N J O I N T ) > F l X E D B I N ( 1 5 , 0 ) ; 3 9 6 R U L E = - 1 ; 3 9 7 D C L ( D E L T A , E P S , K A Y T , L E F T , R I G H T ) F L O A T ; 3 9 8 D C L ADD B I T ( l ) I N I T C O ' B ) ; 3 9 9 O P U T S K I P ( 5 ) E D I T 4 0 0 ( ' T E R M I N A L V A L U E S OF T E M P O R A L U T I L I T Y = ' , T V S T A R ) ( A , F ( 5 , 2 ) ) ; 4 0 1 P U T S K I P E D I T ( ' T E M P O R A L S C A L I N G F A C T O R S : ' X A ) ; 4 0 2 DO T = l T O N S T G ; 4 0 3 P U T S K I P E D I T ( ' K T ( ' , T , ' ) = ' , K T ( T > ) ( A , F ( 2 > , A , F ( 5 , 2 ) ) ; 4 0 4 E N D ; 4 0 5 0/# T E S T T H E A D D I T I V E I N D E P E N D E N C E C O N D I T I O N */ 4 0 6 O I F ( i - S U M ( K T ) ) > P R E C T H E N DO; 4 0 7 P U T P A G E E D I T ( ( 1 0 ) ' * ' , 'SUM OF K T < l ' X A , A ) ; 4 0 8 GO TO A D I E U ; /* T E R M I N A T E E X E C U T I O N */ 4 0 9 E N D ; 4 1 0 E L S E I F A B S ( 1 - S U M ( K T ) ) < = P R E C T H E N DO; 4 1 1 P U T S K I P E D I T ( ' A D D I T I V E T E M P O R A L U T I L I T Y ' ) ( A ) ; 4 1 2 V S T A R ( * . 0 > = T V S T A R + 1 ; /* S C A L I N G P O S I T I V E */ 4 1 3 A D D = ' 1 ' B ; 4 1 4 GO TO D Y N A M I C ; 4 1 5 E N D ; 4 1 6 0/« I T E R A T E T H E T E M P O R A L I N T E R A C T I O N F A C T O R K A Y T */ 4 1 7 0 K A Y T = 0 ; /* I N I T I A L T H E S E A R C H V A R I A B L E S */ 4 1 8 DELTA-*. 1; 4 1 9 GO TO I T E R A T E ; 4 2 0 O S C A L E : K A Y T = K A Y T + D E L T A ; /* R E S C A L E T H E S E A R C H V A R I A B L E S */ 137. 4 2 1 D E L T A = D E L T A / 1 0 ; 4 2 2 0 I T E R A T E : E P S = 1 ; 4 2 3 DO W H I L E ( E P S > E P S I L O N ) ; 4 2 4 K A Y T = K A Y T - D E L T A ; 4 2 5 L E F T = l + K A Y T i 4 2 6 R I G H T = 1 ; 4 2 7 DO 1=1 TO N S T G ; 4 2 8 R I G H T = R I G H T * ( 1 + K A Y T * K T ( I ) ) ; 4 2 9 END; 4 3 0 E P S = L E F T - R I G H T ; 4 3 1 END ; 4 3 2 O I F E P S O E P S I L D N T H E N GO T O S C A L E ; 4 3 3 O P U T S K I P ED I T ( ' T E M P O R A L U T I L I T Y I S ' X A ) ; 4 3 4 P U T S K I P E D I T ( ' N E G A T I V E M U L T I P L I C A T I V E . S U M < K T ( I ) > = ' , 4 3 5 SUM<KT) X A , F ( 6 , 3 ) ) ; 4 3 6 P U T S K I P E D I T ( ' T E M P O R A L I N T E R A C T I O N F A C T O R K A Y T = ' , K A Y T ) 4 3 7 < A , F ( 6 , 3 ) ) ; 4 3 8 0 / * D Y N A M I C PROGRAMMING A L G O R I T H M */ 4 3 9 /* T = S T A G E ; C O U N T S FROM T H E F U T U R E BACKWARDS 4 4 0 I = ' F R O M ' S T A T E 4 4 1 * = J = ' T O ' S T A T E 4 4 2 A = A C T I O N 4 4 3 */ 4 4 4 D Y N A M I C : 4 4 5 ODO T = l TO N S T G ; /* GO THROUGH S T A G E S */ 4 4 6 O L D R U L E = R U L E ; 4 4 7 DO 1=1 T O N J O I N T ; /* GO T H R O U G H J O I N T S T A T E S »/ 4 4 8 V B E S T = - 1 ; 4 4 9 A B E S T = 1 ; 4 5 0 DO A = l TO N A C T ; /* GO T H R O U G H A C T I O N S */ 4 5 1 0 /* C O M P U T E E X P E C T A T I O N BY R E C U R S I V E E Q U A T I O N •*/ 4 5 2 0 I F ADD T H E N DO; /* A D D I T I V E T E M P O R A L U T I L I T Y */ 4 5 3 V H O L D = S U M ( Q ( A , I , * > * ( K T ( T ) » M U T I L ( A , * > + V S T A R ( * , T - 1 ) ) ) ; 4 5 4 END; /* A D D I T I V E */ 4 5 5 0 E L S E DO; /* M U L T I P L I C A T I V E T E M P O R A L U T I L I T Y */ 4 5 6 V H O L D = S U M ( Q ( A , I , * > * < 1 + K A Y T * K T ( T ) * M U T I L ( A , * ) ) * 4 5 7 V S T A R ( « , T - 1 ) ) ; 4 5 8 END; /» M U L T I P L I C A T I V E »/ 4 5 9 0 I F V H O L D > V B E S T T H E N DO; /» C H O O S E MAXIMUM U T I L I T Y */ 4 6 0 V B E S T = V H O L D ; 4 6 1 A B E S T = A ; 4 6 2 END; /* C H O O S E */ 4 6 3 0 END; /» A */ 4 6 4 V S T A R ( I , T ) = V B E S T ; 4 6 5 A S T A R ( I , T ) = A B E S T ; 4 6 6 R U L E ( I ) = A S T A R ( I , T ) ; 4 6 7 END; /* I */ 4 6 8 I F A L L ( O L D R U L E = R U L E ) T H E N GO TO S T A T I O N A R Y ; 4 6 9 END; /* T */ 4 7 0 O S T A T I O N A R Y : T H 0 R = T - 1 ; 4 7 1 0 / * D I S P L A Y O P T I M U M D E C I S I O N R U L E S «/ 4 7 2 O P U T S K I P ( 5 ) E D I T C O P T I M U M D E C I S I O N S ' X A ) ; 4 7 3 P U T S K I P ( 2 ) E D I T ( ' S T A G E I N D E X C O U N T S FROM T H E P R E S E N T ' X A ) ; 4 7 4 P U T S K I P ( 2 ) E D I T 4 7 5 ( ' S T A G E ' , ' S T A T E ' , ' A C T I O N ' , ' U T I L I T Y ' ) ( ( 4 ) ( A , X ( 4 ) ) ) ; 4 7 6 DO T=THOR T O 1 B Y - 1 ; 4 7 7 P U T S K I P E D I T ( ( 3 6 ) ' * ' > ( A ) ; 4 7 8 DO 1=1 T O N J O I N T ; 4 7 9 P U T S K I P E D I T ( ( T H O R - T + 1 ) , I , A C T N A M E ( A S T A R ( I , T ) ) , V S T A R ( I , T ) ) 4 8 0 ( X ( 3 ) , F ( 2 ) - X ( 6 ) , F ( 2 ) , X ( 4 ) , A ( 7 ) , X ( 4 ) , F ( 6 , 3 ) ) ; 4 8 1 E N D ; /* I »/ 4 8 2 E N D ; /* T «/ 4 8 3 I F THOR < N S T G T H E N P U T S K I P ( 2 ) E D I T 4 8 4 ( ' P O L I C Y B E C O M E S S T A T I O N A R Y I N S T A G E I T E R A T I O N = ' , T H O R ) 4 8 5 < A , F < 2 > ) ; 4 8 6 P U T S K I P ( 3 ) ; 4 8 7 O R E T U R N ; 4 8 8 OEND HOWARD; 4 8 9 O E N D . /* E N D O F R E A D - I N B L O C K */ 4 9 0 O R E T U R N ; 4 9 1 O A D I E U : END B E L L M A N ; d o f F i l e APPENDIX F SAMPLE OUTPUT I L L U S T R A T I V E E X A M P L E T R A F F I C = 5 0 -»1000 V E H I C L E S / D A Y R E S I D E N T S = 2 »1G00 H O U S E H O L D S L E N G T H = 2 0 KM L A N E S = 4 E X T R A V E H I C L E O P E R A T I N G C O S T = 0 . 0 1 0 D O L L A R S / V E H I C L E KM D E L A Y = 0 . 0 1 7 HOURS C Y C L E S = 0. 1 0 0 /KM NUMBER O F A C T I O N S = 4 NUMBER O F A T T R I B U T E S = 8 NUMBER O F P A V E M E N T V A R I A B L E S ' 4 NUMBER O F S T A G E S = 5 A C T I O N S W I T H R E S O U R C E S R E Q U I R E D PER L A N E * K M A C T I O N C O S T J O B S G R A V E L T I M E ( D O L L A R S ) ( P E R S O N - D A Y S ) ( T O N S ) ( D A Y S ) NONE 0 0 . 0 0 • G. 0 R O U T I N E 1 0 0 0. 5 0 0. 0 S E A L 1 0 0 0 1 . 0 5 0 0 . 2 O V E R L A Y 2 0 0 0 0 2 . 5 4 0 0 1 . 0 MARKOV T R A N S I T I O N MATKIX FOR T E X T U R E UNDER A C T I O N = N O N E 1 2 1 0 . 8 5 0 . 15 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T H I X FOR T E X T U R E UNDER AC T I Q N = R G U T I N E 1 2 1 0 . 9 0 0 . 10 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T R I X FOR T E X T U R E UNDER A C T I Q N = S E A l _ 1 2 1 1 . 0 0 0 . 0 0 2 1 . 0 0 0 . 0 0 MARKOV T R A N S I T I O N M A T R I X FOR T E X T U R E UNDER ACTION=OV£RLAY 1 2 1 0 . 9 0 0 . 1 0 2 0. 9 0 0 . 10 MARKOV TRANSITION MATRIX FOR RUTTING UNDER ACTION=NONE 1 2 1 0. 8 0 0. 2 0 2 0 . 0 0 1 . 0 0 MARKOV TRANSITION MATRIX FOR RUTTING UNDER AC TIQN=RGUTIHE 1 2 1 0. 8 0 0. 2 0 2 0 . 0 0 1 . 0 0 MARKOV TRANSITION MATKIX FOR RUTTING UNDER ACTION=SEAL 1 2 1 0 . 8 0 0. 2 0 2 0 . 0 0 1 . 0 0 MARKOV TRANSITION MATRIX FOR RUTTING UNDER AC TIQN=OVERLAY 1 2 1 1. 0 0 • 0 . 0 0 2 i . 0 0 0. 0 0 MARKOV T R A N S I T I O N M A T R I X FOR R O U G H N E S S UKPiR ACTION ~ WOM£ j 2 1 0. 8 0 0. 2 0 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T K l X FOR R O U G H N E S S UNDER A C T I O N = R O U T I N t 1 2 '1 0 . 9 0 0. 10 2 0. 9 0 0. 1 0 MARKOV T R A N S I T I O N M A T R I X FOR R O U G H N E S S UNDER A C T I O N = S E A L 1 2 1 0. 9 0 0. 10 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T R I X FOR R O U G H N E S S UNDER A C T I O N = 0 V E R L A Y 1 2 1 1 . 0 0 0 . 0 0 2 1 . 0 0 0 . 0 0 MARKOV T R A N S I T I O N MATKIX FOR S T R E N G T H UNDER AC T I O N = N O N E 1 2 1 0. 9 5 0. 0 5 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N MATh'IX FOR S T R E N G T H UNDER A C T I O N = R C U T I N c 1 2 1 0. 9 5 0. 0 5 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T R I X FOR S T R E N G T H UNDER A C T I C N = S E A L 1 2 1 0. 9 5 0. 0 5 2 0 . 0 0 1 . 0 0 MARKOV T R A N S I T I O N M A T R I X FOR S T R E N G T H UNDER A C T I O N s O V E R L A Y 1 2 1 i . 0 0 0 . 0 0 2 1. 0 0 0. 0 0 L A B E L S FOR J O I N T S T A T t S S T A T E OF P A V E M E N T V A R I A E L E NUMBER: 1 2 3 4 J O I N T S T A T E T E X T U R F R U T T I N G R O U G H N E S S S T R E N G T H 1 1 l 1 2 2 i 1 3 1 2 * 4 2 2 1 5 1 1 2 6 2 1 2 7 1 2 2 S 2 2 2 9 1 1 i 1 0 2 1 i 11 1 2 1 1 2 2 2 i 13 1 1 2 14 2 1 2 1 5 1 2 d 1 6 2 2 2 2 2 2 2 2 2 1 0 0 * J 0 I N T S T O C H A S T I C M A T R I X FOR A C T I C N = N O N E 1 2 3 4 6 7 8 9 10 11 1 2 13 14 15 16 1 5 2 9 13 2 1 3 2 3 1 3 0 1 0 1 0 0 0 2 0 61 0 15 0 15 0 4 0 3 0 1 0 1 0 0 3 0 0 6 5 i i 0 0 16 3 0 0 3 1 0 0 i 0 4 0 0 0 7 6 0 0 0 19 0 0 0 4 0 0 0 1 5 0 0 0 0 6 5 11 16 3 0 0 0 0 3 1 i 0 6 0 0 0 0 0 7 6 0 1 9 0 0 0 0 0 4 0 1 7 0 0 0 0 0 0 81 14 0 0 0 C 0 0 4 1 8 0 0 0 G 0 0 0 9 5 0 0 0 0 0 0 0 5 9 ' 0 0 0 0 0 0 0 0 54 10 14 2 14 2 3 1 10 0 0 0 0 0 0 0 0 0 6 4 0 16 0 16 0 4 11 0 0 0 0 0 0 0 0 0 0 6 8 12 0 0 17 3 12 0 0 0 0 0 0 0 0 0 0 0 S C 0 0 0 2 0 13 0 0 0 0 0 0 0 0 0 0 0 0 6 8 12 17 3 14 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 2 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 5 15 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 * J 0 I N T S T O C H A S T I C M A T R I X FOR ACTION=ROUTIIME 1 2 3 4 1 6 2 7 15 2 2 0 6 8 0 17 3 0 0 7 7 9 4 0 0 0 3 5 5 6 2 7 15 2 6 0 6 8 0 1 7 7 0 0 7 7 9 8 0 0 0 3 5 9 O O O O 10 0 0 0 0 11 0 0 0 0 12 0 0 0 0 13 0 0 0 0 14 0 0 0 0 15 0 0 0 0 1 6 0 0 0 0 5 & 7 8 9 10 7 1 2 0 3 0 0 8 0 a. 0 4 0 0 9 l 0 0 0 0 0 9 0 0 7 1 2 0 3 0 0 8 0 2 0 4 0 0 9 1 0 0 0 0 0 9 0 0 0 0 0 0 6 5 7 0 0 0 0 0 7 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 7 0 0 0 0 0 7 2 0 0 0 0 0 0 0 0 0 0 0 0 11 1 2 13 14 15 16 1 0 0 0 0 0 0 1 0 0 0 0 4 n 0 0 0 0 0 5 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 5 0 0 0 0 1 6 2 7 1 2 0 0 18 0 8 0 2 81 9 0 0 9 1 0 9 0 0 0 0 10 1 6 2 7 1 2 0 0 1 8 0 8 0 2 8 1 9 0 0 9 1 0 9 0 0 0 0 10 1 0 0 * J 0 I N T S T O C H A S T I C M A T R I X FOR A C T I 0 N = S E A L i 2 3 4 5 6 7 8 9 10 11 1 2 13 14 15 16 1 6 8 0 17 0 s 0 2 0 4 0 1 0 0 0 0 0 cL 6 8 0 17 0 e 0 2 0 4 0 1 0 0 0 0 0 3 0 0 8 5 0 0 0 9 0 0 0 5 0 0 0 0 0 4 0 0 8 5 0 0 0 9 0 0 0 5 0 0 0 0 0 5 0 0 0 0 7 6 0 19 0 0 0 0 0 4 0 1 0 6 0 0 0 0 7 6 0 1 9 0 0 0 0 0 4 0 1 0 7 0 0 0 0 0 0 9 5 0 0 0 0 0 0 0 5 0 8 0 0 0 0 0 0 9 5 0 0 0 0 0 0 0 5 0 9 0 0 0 0 0 0 0 0 7 2 0 1 8 0 S 0 2 0 10 0 0 0 0 0 0 0 0 7 2 0 1 8 0 8 0 2 0 11 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 10 0 12 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 10 0 13 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 2 0 0 14 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 2 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 73 X — > tn r X TI II z m c t o H Ul 73 • > O r c x II O o IU o * x tn tn m o t n > H r cn II « > z n o m O! -H o o < • o o z TJ 0) 73 H O > C z - o m II o o o o 0- c o m .X m O*Ul*>£JIU'-*O-O<IlvJ0-CJl-t»t«)IU>-i o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c o o o I •=> in H 3 a 73 > o a II o < ni 73 r o OJ *» tn m •fl o M -E-145. 2 U S E R C O S T S A V E D ( 1 0 0 0 D O L L A R S / Y E A R ) S C A L I N G C 0 N S T A N 1 K K ( 2 > = 0 . 3 6 B E S T = 3 6 4 0 . 0 H A L F = 1 8 2 0 . 0 R I S K N E U T R A L : U<X> = 0 . 0 0 0 * X U ( X > 3 D E L A Y P R E V E N T E D ( 1 0 0 0 H O U R S / Y E A R ) S C A L I N G CONSTANT K K ( 3 ) = 0 . 2 4 B E S T = 6 7 3 . 1 H A L F = 3 3 6 . 5 R I S K N E U T R A L : U ( X > = 0. 0 0 1 * X U ( X ) 4 A G E N C Y C O S T SAVfcD ( 1 0 0 0 D O L L A R S / Y E A R > S C A L I N G CONSTAN 1 K K ( 4 > = 0. 15 B E S T = 1 6 0 0 . 0 H A L F = 7 2 0 . 0 R I S K A V E R S E : U(X>=- 3. 0 1 7 * ( 1 - E X P ( - 0 . C 0 0 * X ) ) U ( X ) 5 A C C E S S P R O V I D E D ( 1 0 0 0 V E H I C L E S / D A Y ) S C A L I N G C O N S T A N T K K ( 5 ) = 0 . 8 4 B E S T = 5 0 . 0 H A L F = 1 7 . 0 R I S K A V E R S E : U ( X > = 1 . 3 3 8 * ( 1 - E X P ( - 0 . 0 2 8 * X ) ) U ( X i 147. b J O B S C R E A T E D < P E R S O N * D A Y S / Y E A R > S C A L I N G C O N S T A N i K M 6 > = 0. 0 7 B E S T = 2 0 0 . 0 H A L F = 1 0 0 . 0 R I S K N E U T R A L : U(X>=- 0. 0 0 5 * X U<X> •* 7 G R A V E L S A V E D < 1 0 C 0 T O N S / Y E A R ) S C A L I N G CONSTANT K K C ? > = 0 . 1 2 B E S T = 3 2 . 0 H A L F = 1 2 . S R I S K A V E R S E : U<X>=- 1. 7 B 4 * < l - E X P < - 0 . 0 2 6 * X > i U ( X ) 8 N O I S E P R E V E N T E D C l O O O H O U S E H O L D S ) S C A L I N G C O N S T A N I K K ( 8 > = 0 . 1 0 B E S T = 2. 0 H A L F = 1 . 0 R I S K N E U T R A L : U ( X > - 0. 5 0 0 * X 149. M A T R I X O F C O N S F Q U E N C E S FOR A C T I O N = N Q M E A T T R I B U T E 1 2 3 4 5 6 7 8 S T A T E 1 5 0 . 0 0 6 4 0 . 00 6 7 3 . 0 6 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 0. 0 0 S T A T E 2 0. 0 0 6 4 0 . 00 6 7 3 . 0 6 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 2. 0 0 S T A T E 3 0 . 0 0 6 4 0 . 00 6 7 3 . 0 6 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 0. 0 0 S T A T E 4 0. 0 0 6 4 0 . 00 6 7 3 . 0 6 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 2. 0 0 S T A T E 5 5 0 . 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 0 . 0 0 S T A T E 6 0. 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 2. 0 0 S T A T E 7 0. 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 0. 0 0 S T A T E 8 0. 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 5 0 . 0 0 0. 0 0 3 2 . 0 0 •n c 0 0 S T A T E 9 5 0 . 0 0 6 4 0 . 0 0 6 7 3 . 0 6 6 0 0 . 0 0 0. 0 0 0. 0 0 3 2 . 0 0 0. 0 0 S T A T E 10 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 6 0 0 . 0 0 0 . 0 0 0. 0 0 3 2 . 0 0 00 S T A T E 11 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 6 0 0 . 0 0 0 . 0 0 0. CO 3 2 . 0 0 0. 0 0 S T A T E 1 2 0. 0 0 6 4 0 . CO 6 7 3 . 0 6 6 0 0 . 0 0 0. 0 0 0. 0 0 3 2 . 0 0 2. 00 S T A T E 1 3 5 0 . 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 0 . 0 0 0. CO 3 2 . 0 0 0. 0 0 S T A T E 14 0. o o - 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 0 . 0 0 0. 0 0 3 2 . 0 0 2. 0 0 S T A T E 1 5 0. 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 0. 0 0 0. 0 0 3 2 . 0 0 0. 0 0 S T A T E 16 0. 0 0 0. 0 0 6 6 . 6 4 6 0 0 . 0 0 0. 0 0 0. CO 3 2 . 0 0 2. 0 0 M A T R I X O F C O N S F G U E N C E S FOR A C T I O N = R O U T I N E A T T R I B U T E 1 2 3 4 5 6 7 8 S T A T E i 5 0 . 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 o. 0 0 S T A T E 2 0. 0 0 6 4 0 . CO 6 7 3 . 0 6 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0 0 S T A T E 3 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E A 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 5 0 . 0 0 4 0 . CO 3 2 . 0 0 •—\ <=_. 0 0 S T A T E 5 5 0 . 0 0 0. CO 6 6 . 6 4 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E 6 0. 0 0 0. CO 6 4 5 9 2 . 0 0 5 0 . 0 0 4 0 . CO 3 2 . 0 0 0 0 S T A T E 7 0. 0 0 0. CO 6 6 . 6 4 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E S 0 . 0 0 0. 0 0 6 6 . 6 4 5 9 2 . 0 0 5 0 . 0 0 4 0 . 0 0 3 2 . 0 0 2. 0 0 S T A T E 9 5 0 . 0 0 6 4 0 . CO 6 7 3 . 0 6 5 9 2 . 0 0 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E 10 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 0. 0 0 4 0 . 0 0 3 2 . 0 0 2. 0 0 S T A T E 11 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E 12 . 0. 0 0 6 4 0 . 0 0 6 7 3 . 0 6 5 9 2 . 0 0 0. 0 0 4 0 . 0 0 3 2 . 0 0 2. 0 0 S T A T E 13 5 0 . 0 0 0. 00 6 6 . 6 4 5 9 2 . 0 0 0 . 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E 11 0. 0 0 0. 0 0 6 6 . 6 4 5 9 2 . 0 0 0. 0 0 4 0 . 0 0 3 2 . 0 0 2. 0 0 S T A T E 15 0. 0 0 0. CO 6 6 . 6 4 5 9 2 . 0 0 0. 0 0 4 0 . 0 0 3 2 . 0 0 0. 0 0 S T A T E 16 0. 0 0 0. CO 6 6 . 6 4 5 9 2 . 0 0 0 . 0 0 4 0 . 0 0 3 2 . 0 0 2. 0 0 150. M A T R I X O F C O N S E Q U E N C E S FOR A C T I O N - S E A L . A T T R I B U T E 1 e.' 3 4 5 6 7 8 S T A T E i 5 0 . 0 0 6 4 0 . 0 0 6 5 9 . 7 4 5 2 0 . 0 0 5 0 . 0 0 8 0 . CO 2 8 . 0 0 0. 00 S T A T E 2 0. 0 0 6 4 0 . CO 6 5 9 . 7 4 5 2 0 . 0 0 5 0 . 0 0 8 0 . CO 2 8 . 0 0 2. 00 S T A T E 3 0. 0 0 6 4 0 . 0 0 6 5 9 . 7 4 5 2 0 . 0 0 5 0 . 0 0 8 0 . 0 0 2 8 . 0 0 0. 0 0 S T A T E A 0. 00 6 4 0 . 0 0 6 5 9 . 7 4 5 2 0 . CO 5 0 . 0 0 8 0 . CO 2 8 . 0 0 2. 0 0 S T A T E 5 5 0 . 0 0 0. CO 5 2 . 3 1 5 2 0 . 0 0 5 0 . 0 0 e o . CO 2 3 . 0 0 0. 0 0 S T A T E 6 0. 0 0 0. 0 0 5 3 . 3 1 5 2 0 . 0 0 5 0 . 0 0 8 0 . 0 0 2 8 . 0 0 2. 0 0 S T A T E 7 0. 0 0 0. CO 5 2 . 31 5 2 0 . 0 0 5 0 . 0 0 8 0 . 0 0 2 3 . 0 0 0. 0 0 S T A T E 3 0. 0 0 0. CO 5 3 . 3 1 5 2 0 . 0 0 5 0 . 0 0 8 0 . CO 2 8 . 0 0 00 S T A T E 9 5 0 . 0 0 6 4 0 . 0 0 6 5 9 . 7 4 5 2 0 . 0 0 0 . 0 0 8 0 . CO 2 8 . 0 0 0. 00 S T A T E 10 0. 00 6 4 0 . CO 6 5 9 . 7 4 5 2 0 . 0 0 0 . 0 0 8 0 . CO 2 8 . 0 0 2. 00 S T A T E 11 0. 0 0 6 4 0 . 0 0 6 5 9 . 7 4 5 2 0 . 0 0 0 . 0 0 8 0 . CO 2 8 . 0 0 0. 00 S T A T E 12 0. 0 0 6 4 0 . CO 6 5 9 . 7 4 5 2 0 . 0 0 0 . 0 0 8 0 . CO 2 8 . 0 0 00 S T A T E 13 5 0 . 0 0 0. CO 5 3 . 3 1 5 2 0 . 0 0 0 . 0 0 8 0 . 0 0 2 8 . 0 0 0. 00 S T A T E 14 0. 0 0 0. CO 5 3 . 3 1 5 2 0 . 0 0 0 . 0 0 8 0 . 0 0 2 3 . 0 0 . 2. 00 S T A T E 15 0. 00 0. CO 5 3 . 3 1 5 2 0 . 0 0 0 . 0 0 8 0 . 0 0 2 3 . 0 0 0. 00 S T A T E 16 0. 00 0. CO 5 3 . 3 1 5 2 0 . 0 0 0 . 0 0 8 0 . 0 0 2 3 . 0 0 2. 00 M A T R I X O F C O N S E Q U E N C E S FOR A C T I O N - O V E R L A Y A T T R I B U T E 1 2 3 4 5 6 7 8 S T A T E 1 5 0 . 0 0 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 5 0 . 0 0 2 0 0 . CO 0. 0 0 0. 00 S T A T E 2 0 . 0 0 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 5 0 . 0 0 2 0 0 . 0 0 0. 0 0 2. 00 S T A T E 3 0. 0 0 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 5 0 . 0 0 2 0 0 . 0 0 0. 0 0 0. 00 S T A T E 4 0. 0 0 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 5 0 . 0 0 2 0 0 . CO 0. 0 0 00 S T A T E 5 5 0 . 0 0 0. CO 0. 0 0 0. 0 0 5 0 . 0 0 2 0 0 . 0 0 0. 0 0 0. 00 S T A T E 6 0 . 0 0 0. CO 0. 0 0 0. 0 0 5 0 . 0 0 2 0 0 . 0 0 0. 0 0 2. 00 S T A T E 7 0 . 0 0 0. CO 0. 0 0 0. 0 0 5 0 . 0 0 2 0 0 . CO 0 . OG 0. 00 S T A T E 8 0. 0 0 0. 0 0 0. 0 0 0. 0 0 5 0 . 0 0 2 0 0 . CO 0. 0 0 2. 00 S T A T E o 5 0 . 0 0 6 4 0 . 0 0 6 0 6 . 4 2 0. 0 0 0. 0 0 2 0 0 . 0 0 0 . 0 0 0. 00 S T A T E 10 0. 00 6 4 0 . 0 0 6 0 6 . 4 2 0. 0 0 C. 0 0 2 0 0 . 0 0 0. 0 0 00 S T A T E 11 0. 00 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 0 . 0 0 2 0 0 . 0 0 0. 0 0 0. 00 S T A T E 12 0. OG 6 4 0 . CO 6 0 6 . 4 2 0. 0 0 C. 0 0 2 0 0 . 0 0 0. 0 0 2. 00 S T A T E 13 5 0 . 0 0 0. CO 0. 0 0 0. 0 0 0. 0 0 2 0 0 . 0 0 0. 0 0 0. 00 S T A T E 14 0. 0 0 0. 00 0. 0 0 0. 0 0 0 . 0 0 2 0 0 . 0 0 0. 0 0 2. 00 S T A T E 15 0. 0 0 0. 00 0. 0 0 0. 0 0 0 . 0 0 2 0 0 . CO 0. 0 0 0. 00 S T A T E 1 6 0. 0 0 0. CO 0. 0 0 0. 0 0 0. 0 0 2 0 0 . 0 0 0. 0 0 2. 00 M U L T I A T T R I B U T E U T I L I T Y I S N E G A T I V E M U L T I P L I C A T I V E . SUM<KK(I>>=- 2 . 4 8 2 I N T E R A C T I O N F A C T O R K A Y = - 0 . 9 7 7 T E R M I N A L V A L U E S OF T E M P O R A L U T I L I T Y = - 1 . 0 0 T E M P O R A L S C A L I N G F A C T O R S : KT< 1) = 0. 3 0 KT< 2 ) - 0 . 3 0 K T ( 3 ) = 0 . 3 0 K T ( 4 ) = 0 . 3 0 K T ( 5 ) = 0. 3 0 T E M P O R A L U T I L I T Y I S N E G A T I V E M U L T I P L I C A T I V E , S U M ( K T ( I > > = 1 . 5 0 0 T E M P O R A L I N T E R A C T I O N F A C T O R K A Y T - - 0 . 6 S 1 O P T I M U M D E C I S I O N S S T A G E I N D E X C O U N T S FROM Trie P R E S F N T S T A G E S T A T E A C T I O N U T I L I T Y 1 O V E R L A Y - 0 . 7 9 ? 2 O V F R L A Y - 0 . 7 9 9 3 O V E R L A Y - 0 . 7 9 ? a. O V E R L A Y - 0 . 7 9 9 5 O V F R L A Y - 0 . 7 9 9 6 O V E R L A Y - 0 . 7 9 9 7 O V E R L A Y - 0 . 7 9 9 6 O V E R L A Y - 0 . 7 9 ? 9 Qv'FRV A Y - 0 . 7 9 9 10 O V E R L A Y - 0 . 7 9 9 U O V F R L A Y - 0 . 7 9 ? 12 O V F R L A Y - 0 . 7 9 9 13 O V E R L A Y - 0 . 7 9 9 11 O V F R L A Y - 0 . 7 9 9 15 O V F R l A Y - 0 . 7 9 9 16 O V E R L A Y - 0 . 7 9 9 P O L I C Y B E C O M E S S T A T I O N A R Y IN S T A G E I T E R A T I O N 

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