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Optimal long-term development and operation of irrigation systems with storage under hydrological uncertainty Igwe, Okay Cyril 1977

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OPTIMAL LONG-TERM DEVELOPMENT AND OPERATION OF IRRIGATION SYSTEMS WITH STORAGE UNDER HYDROLOGICAL UNCERTAINTY BY OKAY CYRIL IGWE, P.Eng. B.Sc. (Cum Laude) Eng., Technion, I s r a e l I n s t i t u t e of Technology, H a i f a , 1969 M.Sc. (Engineering Hydrology), U n i v e r s i t y o f Guelph 1971. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the F a c u l t y o f Graduate S t u d i e s I n t e r d i s c i p l i n a r y Hydrology and Water Resource Systems Major i n C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1976. Okay Cyril Igwe, 1977 © In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the regu i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f Interdisciplinary Hydrology The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date December 20, 1976 ABSTRACT The scarcity of water resources is increasing and mcLH.Qi.noJL quantities of water are becoming more important. The need for sound and conservative management of water supply is imperative. Intensified agfU.cuZtun.at pro-duction, to {dtd the. ever-increasing world population, is fmquJjU.ng matte, irrigation, which is the. heaviest consumptive use of water. It is definable, therefore, to seek new techniques that can modify the iAAigation water Supply regime. Such a modification Implies the uAgent need for the development of an iAAigation water supply regime dictated by seasonal. hydAologic consider-ations and agricultural production that is technologically fully controlled on the. basis of long-fiange stochastic considerations. The fact that most observed historical hydAologic data, are usually short and may constitute poor representation of the possibilities for long-term planning in irrigation systems management, reinforces the postulatlon that any meaningful, approach to the. optimal development and operation of irrigation systems must take, full cognizance of hydrological uncertainty. To achieve optimum competence in irrigation systems management undeA . the predominating constraint of hydrological uncertainty, a methodology that first considers the systems operational policies as well as several levels of water consumption is necessary. To be realistic with operating rules one has to consider the stochastic variability in irrigation planning and thus has to consider uncertainty and risk relating to the major decision input information-[hydrologic information),; and other considerations other than strictly maximizing expected economic monetary value must be brought into the model formulation. Kny meaningful planning in agricultural water utilization has to be man-centered in approach and must provide objective i i analysis of Subjective considerations. The above. rationale, led to the de.veJLopme.nt of a stochastic Bayesian decision The.oh.y optimization model, which specifies expe.aA.ed uXiZity as the. criterial obje.ctA.ve. fu.nctA.on to maximize, and which could be. realistically employed to identify the. best decision-criXerion and ade.qa.ate. policies {on. optimal long-term planning, de.velopme.nt and operation of irrigation systems with storage (index hydrological uncertainty. The. model, which is behavioral In approach, is applied to the. Nicola Valley lHAA.gatl.on district located In the. dry, semi-arid interior of Brutish Columbia. Two AjiAA.gatA.on opeAatlonal procedures, two de.cibi.on criXeria, and diffeAcnt cAop response function* are employed in the. analysis to identify the. best planning policy {on, astute, irrigation systems management in the. region. The. results obtained from the. model indicate, that the. optimal areas to irrigate undeA hydAological uncertainty are de.pe.nde.nt on the degree of hydro -logical uncertainty, the systems operating ph.oce.dure, the. crops irrigated and their responses to water, and, on the. decision criterion and utility function* employed. Post-optimal analyses indicate, that optimal policies obtained are. very sensitive to discretized probability distribution of, thz uncertain states of nature., crop response function, uiiliXy function and decision criterion, and system operating ph.oce.dure employed. Tor Hicola Valley Irrigation District the model shows that the practice of irrigating more alfalfa hectarage at a water consumption level that ii, below the designated maximum water requirement of alfalfa, - Procedure 11, ii, superior to the practice of irrigating less hectarage to maximum consumptive use of crop and maximum water holding capacity of the soil, - Procedure I. It is also shown that the criterion of maximizing total expected utility, EU, is superior to the criterion of maximizing total expected monetary value, EMI/, under uncer-tainty and risk. The model also shows that iX is desirable to have some hydrologlcal forecasting device. In the Nicola region for improved output from the model. Thus, the model has considerable promise as a valid tool for optimal long-term Irrigation systems management decision-making under hydro logical uncertainty. i v TABLE OF CONTENTS ABSTRACT . . LIST OF TABLES LIST OF FIGURES AND ILLUSTRATIONS ACKNOWLEDGEMENTS . . . CHAPTER 1. INTRODUCTION . . . . . 2. REVIEW OF LITERATURE 9 3. CONTEXT AND OBJECTIVES OF STUDY . . . . . 22 3.1 Context 2 2 3.2 O b j e c t i v e s of Study . . . . • . . . 2 4 4. DEVELOPMENT OF DECISION THEORY STOCHASTIC OPTIMIZATION MODEL . . 2 6 4.1 Major Approaches to Programming Under Risk and U n c e r t a i n t y 2 6 4.2 O p t i m i z a t i o n of Expected Values . . . 30 4.3 D e c i s i o n Theory Approach to S t o c h a s t i c O p t i m i z a t i o n Problems . . . . . . 34 4.4 D e c i s i o n Theory S t o c h a s t i c O p t i m i z a t i o n Model Development 36 4.5 O b j e c t i v e F u n c t i o n 3 7 4.6 Assumptions 3 9 4.7 C o n s t r a i n t s 4Q 4.8 D e r i v a t i o n of U t i l i t y F u n c t i o n . . . 40 4.9 H y d r o l o g i c a l U n c e r t a i n t y Element . . . 44 4.10 Optimum Choice C r i t e r i o n : Maximum Expected U t i l i t y 4 5 4.11 Crop Response F u n c t i o n s . . . . . . 45 (Chapter 4 continued ) v Page i i i x CHAPTER Page 4 (continued) 4.11.1 Procedure I 4 6 4.11.2 Procedure I I . . . . . . . 47 4.12 S e n s i t i v i t y A n a l y s i s . . . . . . 4 8 4.13 Bayesian D e c i s i o n S t r a t e g y . . . . 50 4.14 Optimal D e c i s i o n S t r a t e g i e s Over Time 55 5. APPLICATION OF MODEL TO MEDIUM-SCALE IRRIGATION SYSTEM 5 4 5.1 I n t r o d u c t i o n . . 5 4 5.2 Model Inputs . 5 7 5.2.1 Inflow a n a l y s i s . . . . . 58 5.2.2 H y d r o l o g i c a l S t a t e s of nature and e s t i m a t i o n o f p r i o r p r o b a b i l i t i e s . . . . . . 60 5.2.3 A c t i o n or d e c i s i o n a l t e r n a t i v e s 6 2 5.2.4 Consumptive uses o f a l f a l f a and i r r i g a t i o n water requirement . 6 2 5.2.5 Crop net value . . . . 6 6 5.2.6 Crop response f u n c t i o n s . . . 69 5.3 Computational Procedures 6 9 5.3.1 D e c i s i o n c r i t e r i o n based on maximum t o t a l expected monetary value 69 5.3.2 D e c i s i o n c r i t e r i o n based on maximum t o t a l expected u t i l i t y . 7 2 5.3.3 E s t a b l i s h i n g the u t i l i t y f u n c t i o n 7 2 5.4 Model Output 8 6 5.4.1 Output from Procedure I . . . 8 6 5.4.2 Output from Procedure I I . . . 8 6 (Chapter 5 continued) v i CHAPTER Page 5 (continued) 5.5 S e n s i t i v i t y A n a l y s i s I l l 5.5.1 S e n s i t i v i t y of o p t i m a l p o l i c y to p r i o r p r o b a b i l i t i e s . . . I l l 5.5.2 Minimum p r o b a b i l i t i e s , P(9^) t h a t l e a v e the p r e f e r r e d a c t i o n optimal 123 5.5.3 S e n s i t i v i t y of o p t i m a l p o l i c y to crop response f u n c t i o n . . . 127 5.5.4 S e n s i t i v i t y of o p t i m a l p o l i c y to u t i l i t y f u n c t i o n 133 5.5.5 S e n s i t i v i t y o f o p t i m a l p o l i c y to assumptions 134 a) U n c e r t a i n t y i n y i e l d s , f a c t o r p r i c e s , and incomes . . . . 134 b) U n c e r t a i n t y i n i r r i g a t i o n water requirement . . . . 146 5.6 Computation of Bayesian S t r a t e g i e s . 161 5.6.1 C a l c u l a t i o n of c o n d i t i o n a l p r o b a b i l i t i e s , P(Z/0) . . . 162 5.6.2 C a l c u l a t i o n of j o i n t p r o b a b i l i t i e s of 0 and Z,P(Z/0) P (0) . . . 162 5.6.3 C a l c u l a t i o n of p o s t e r i o r p r o b a b i l i t i e s , P(0i/Z) . . . 164 5.6.4 Computation of Bayesian s t r a t e g i e s 164 5.6.5 Value o f experiment . . . . 1 7 l 5.7 Optimal D e c i s i o n S t r a t e g i e s Over Time 173 6 APPLICATION OF MODEL TO REGIONAL PLANNING IN LARGE-SCALE IRRIGATION"SYSTEMS . . . 177 6.1 I n t r o d u c t i o n 17 7 6.2 A p p l i c a t i o n to N i c o l a V a l l e y I r r i g a t i o n 179 (Chapter 6 continued) v i i CHAPTER Page 6 (continued) 6.2.1 Inflow a n a l y s i s I 8 7 6.2.2 H y d r o l o g i c a l s t a t e s of nature and e s t i m a t i o n o f p r i o r p r o b a b i l i t i e s . . . . . . 189 6.2.3 D e c i s i o n a l t e r n a t i v e s . . . . 190 6.3 Model Output: Second I r r i g a t i o n P e r i o d 19 7 6.3.1 Output from Procedure I 19 7 6.3.2 Output from Procedure I I . . . 19 7 6.4 Model Output: F i r s t I r r i g a t i o n P e r i o d 202 6.4.1 Output from Procedure I . 20 2 6.4.2 Output from Procedure I I . . . 20 2 7. DISCUSSION. OF RESULTS . . . . . . . 217 7.1 D i s c u s s i o n . . . . . . . . . 217 7.2 Probable E r r o r . 2 3 0 8. CONCLUSIONS AND RECOMMENDATIONS . . . . 23 4 8.1 C o n c l u s i o n s . . . . . . . . . 234 8.2 Recommendations 238 8.3 A d d i t i o n a l Research Requirements . . 240 REFERENCES . . . . . . 242 APPENDICES 26 0 Appendix A. LISTING OF COMPUTER PROGRAMS . . 26 0 PROGRAM A-1 261 PROGRAM A-2 . 26 7 Appendix B. AVAILABLE HYDROLOGIC AND HYDRO-METEOROLOGICAL DATA FOR NICOLA VALLEY 274 v i i i LIST OF TABLES Table • Page 5.1 Computation of P r i o r P r o b a b i l i t i e s of Inflow, Quilchena Creek 61 5.2 Computed Monthly Consumptive Uses of A l f a l f a f o r Quilchena Area 67 5.3 Monetary Pay-Off T a b l e : E M V - C r i t e r i o r , Procedure I 73 5.4 U t i l i t y Pay-Off T a b l e : E U - C r i t e r i o n , Procedure I 8 0 5.5 T o t a l Expected Monetary Value, EMV, f o r T=2, Procedure I 8 7 5.6 T o t a l Expected U t i l i t y , EU, f o r T=2, Procedure I 8 9 5.7 Optimal Areas Under I r r i g a t i o n : Procedure I 9 3 5.8 T o t a l Expected Monetary Value, EMV, Using S t i m u l a t o r y Crop Response F r a c t i o n f o r T=2, Procedure I I 9 4 5.9 T o t a l Expected U t i l i t y , EU, Using S t i m u l a t o r y Crop Response F u n c t i o n f o r T=2, Procedure II 9 6 5.10 T o t a l Expected Monetary Value, EMV, Using Non-S t i m u l a t o r y Crop Response F u n c t i o n f o r T=2, Procedure II 98 5.11 T o t a l Expected U t i l i t y , EU, Using Non-Stimulatory Crop Response F u n c t i o n f o r T=2, Procedure II 10 0 5.12 Optimal Areas Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n , Procedure I I 10 7 5.13 Optimal Areas Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n , Procedure I I 108 i x Table Page 5.14 P r i o r P r o b a b i l i t i e s of Quilchena Creek Flow Assuming the P r i n c i p l e of I n s u f f i c i e n t Reason 112 5.15 Optimal Areas Under I r r i g a t i o n Using S u b j e c t i v e P r o b a b i l i t i e s , Procedure I 120 5.16 Optimal Areas Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s , Procedure I I 121 5.17 Optimal Areas Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s , Procedure I I 12 2 5.18-1 Computation of Minimum P r o b a b i l i t i e s That Leave P r e f e r r e d A c t i o n Optimal, Procedure I 125 5.18-2 Computation of Minimum P r o b a b i l i t i e s That Leave P r e f e r r e d A c t i o n Optimal, Procedure I I 129 5.19 Optimal Areas Under I r r i g a t i o n f o r Changed Crop Net Value, Procedure I 14 3 5.20 Optimal Areas Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n and Changed Crop Net Value, Procedure I I 14 4 5.21 Optimal Areas Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n and Changed Crop Net Value, Procedure I I 14 5 5.22 Computation of P r i o r P r o b a b i l i t i e s of I r r i g a t i o n Water Requirement f o r A l f a l f a , Quilchena Area 148 5.23 Optimal Areas Under I r r i g a t i o n C o n s i d e r i n g Uncer-t a i n t y i n I r r i g a t i o n Water Requirement, Procedure I 158 x Table 5.24 Optimal Areas Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n and C o n s i d e r i n g U n c e r t a i n t y i n I r r i g a t i o n Water Requirement, Procedure I I 5.25 Optimal Areas Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n and C o n s i d e r i n g U n c e r t a i n t y i n I r r i g a t i o n Water Requirement, Procedure I I 5.26(A) Number of Years of Occurrence of V a r i o u s Combinations of H y d r o l o g i c a l C o n d i t i o n s i n A p r i l and the Subse-quent Summer I r r i g a t i o n P e r i o d , and C a l c u l a t i o n of C o n d i t i o n a l P r o b a b i l i t i e s , P(Z/0) 5.26(B) C a l c u l a t i o n of J o i n t P r o b a b i l i t i e s of 9 and Z, P(Z/0) P(0) 5.26(C) C a l c u l a t i o n of P o s t e r i o r P r o b a b i l i t i e s , P(9i/Z) 5.27 Computation of Bayesian S t r a t e g y Using P o s t e r i o r P r o b a b i l i t i e s , Procedure I 5.28 Computation of Bayesian S t r a t e g y Using P o s t e r i o r P r o b a b i l i t i e s and S t i m u l a t o r y Crop Response Func-t i o n , Procedure II 5.2 9 Computation of Bayesian S t r a t e g y Using P o s t e r i o r P r o b a b i l i t i e s and Non-Stimulatory Crop Response F u n c t i o n , Procedure I I 5.30 Computation of Net Worth of A l t e r n a t i v e I r r i g a t i o n Operation S t r a t e g i e s 6.1 Mean Monthly Flows f o r Major Inflow Sources f o r N i c o l a V a l l e y Region During the I r r i g a t i o n Season x i Table Page 6.2 T o t a l Flows of Major Inflow Sources f o r N i c o l a V a l l e y Region During the I r r i g a t i o n Season 179 6.3 Computation of P r i o r P r o b a b i l i t i e s of Inflow from N i c o l a R i v e r , Quilchena Creek and Moose Creek During the I r r i g a t i o n Season 191 6.4 Frequency A n a l y s i s of Inflow from Major Surface Water Sources i n N i c o l a V a l l e y Region During the I r r i g a t i o n Season 19 2 6.5 Optimal Areas Under I r r i g a t i o n f o r Second I r r i g a t i o n P e r i o d , Procedure I 198 6.6 Optimal Areas Under I r r i g a t i o n f o r Second I r r i g a t i o n P e r i o d Using S t i m u l a t o r y Crop Response F u n c t i o n , Procedure I I 201 6.7 Optimal Areas Under I r r i g a t i o n f o r Second I r r i g a t i o n P e r i o d Using Non-Stimulatory Crop Response Func-t i o n , Procedure I I 20 5 6.8 Optimal Areas f o r Regional Planning i n Large-Scale I r r i g a t i o n Using Two I r r i g a t i o n P e r i o d s 215 x i i LIST OF FIGURES AND ILLUSTRATIONS F i g u r e Page 5.1 C e l l Frequencies of Quilchena Creek Flow During the I r r i g a t i o n Season 63 5.2 Step Crop Response F u n c t i o n , Procedure I 70 5.3(a) S t i m u l a t o r y , E x p o n e n t i a l Crop Response F u n c t i o n , Procedure I I 71 5.3(b) Non-Stimulatory Second-Order Polynomial Crop Response F u n c t i o n , Procedure II 71 5.4(a) L i n e a r U t i l i t y F u n c t i o n 78 5.4(b) Non-Linear U t i l i t y F u n c t i o n Derived f o r Quilchena Ranch I r r i g a t i o n System Management D e c i s i o n -Making 7 9 5.5 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n f o r Procedure I 9 1 5.6 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n f o r Procedure I 9 2 5.7 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I 103 5.8 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure II 10 4 5.9 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 10 5 x i i i F i g u r e Page 5.10 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure II • 10 6 5.11 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using S u b j e c t i v e P r o b a b i l i t i e s f o r Procedure I 114 5.12 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using S u b j e c t i v e P r o b a b i l i t i e s f o r Procedure I 115 5.13 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s f o r Procedure I I 116 5.14 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s f o r Procedure II 117 5.15 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s f o r Pro-cedure I I 118 5.16 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n and S u b j e c t i v e P r o b a b i l i t i e s f o r Procedure II 119 5.17 Minimum P r o b a b i l i t i e s , P(9ft), That Leave the P r e f e r r e d A c t i o n Optimal 13 2 5.18 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n With Changed Crop Net Value, Pro-cedure I 13.7 x i v F i g u r e Page 5.19 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n w ith Changed Crop Net Value, Procedure I 13 8 5.2 0 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n w i t h Changed Crop Net Value and Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I 139 5.21 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n w ith Changed Crop Net Value and Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I 140 5.22 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n with.Changed Crop Net Value and Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure II 141 5.23 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n with Changed Crop Net Value and Using Non-S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I 142 5.24 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n with U n c e r t a i n t y i n Consumptive Use and Inflow I n t e g r a t e d f o r Procedure I 15 2 5.25 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n w ith U n c e r t a i n t y i n Consumptive Use and Inflow I n t e g r a t e d f o r Procedure I 15 3 5.26 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n with U n c e r t a i n t y i n Consumptive Use and Inflow I n t e g r a t e d and Using S t i m u l a t o r y Crop xv F i g u r e Page Response F u n c t i o n f o r Procedure I I 154 5.27 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n w ith U n c e r t a i n t y i n Consumptive Use and i n f l o w I n t e g r a t e d and Using. S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure II 15 5 5.28 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n with U n c e r t a i n t y i n Consumptive Use and Inflow I n t e g r a t e d and Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure II 156 5.29 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n w ith U n c e r t a i n t y i n Consumptive Use and Inflow I n t e g r a t e d and Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 157 5.30 Cumulative P o s t e r i o r P r o b a b i l i t y D i s t r i b u t i o n of I r r i g a t i o n Season Inflow 167 6.1 Major Sources of Surface Inflow i n t o N i c o l a Lake 181 6.2 Monthly I r r i g a t i o n Water Requirement f o r A l f a l f a i n N i c o l a V a l l e y Region 182 6.3 Mean Monthly T o t a l Inflow i n t o N i c o l a R e s e r v o i r During the I r r i g a t i o n Season 184 6.4 Cumulative Frequency A n a l y s i s of Inflow During the I r r i g a t i o n Season f o r N i c o l a V a l l e y Regional System 193 6.5 P r i o r P r o b a b i l i t y D i s t r i b u t i o n of Inflow During. the I r r i g a t i o n Season 194 x v i F i g u r e Page 6.6 Schematic R e p r e s e n t a t i o n of the 2-Period I r r i g a t i o n O peration 196 6.7 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n , Procedure I 199 6.8 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n , Procedure I 2 00 6.9 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure II 20 3 6.10 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I 204 6.11 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 206 6.12 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 2 07 6.13 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s , Pro-cedure I 2 08 6.14 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s , Procedure I 2 09 6.15 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s Using x v i i F i g u r e Page S t i m u l a t o r y Crop Response F u n c t i o n f o r Pro-cedure II 21i 6.16 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s Using S t i m u l a t o r y Crop Response F u n c t i o n f o r Procedure I I " 212 6.17 T o t a l Expected Monetary Value Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 213 6.18 T o t a l Expected U t i l i t y Versus Area Under I r r i g a t i o n f o r the 2 - I r r i g a t i o n P e r i o d s Using Non-Stimulatory Crop Response F u n c t i o n f o r Procedure I I 214 x v i i i ACKNOWLEDGEMENTS I wish to express my appreciation to many people for t h e i r assistance, cooperation and encouragement. F i r s t , I wish to express my gratitude to the Canadian International Develop-ment Agency for the scholarship award, without which support t h i s study would not have been possible. Sincere thanks are due to Professor (Emeritus) T.L. Coulthard and Professor S.O. Russell for t h e i r guidance and for serving as major Professors. Apprecia-tion i s also extended to the other members of my. Committee, Professor I.K. Fox, Dr. 0. Slaymaker, Dr. I.E. Podmore, and Dr. J.W. Zahradnik for t h e i r cooperation. Due thanks are extended to Dr. J.W. Zahradnik and to the Department of Bio-Resource Engineer-ing for the f a c i l i t i e s provided for t h i s study. I am e s p e c i a l l y grateful to Professor and Mrs. T.L.. Coulthard for t h e i r much appreciated encouragement, kindness, and h o s p i t a l i t y . . A special note of thanks i s extended to Mrs. Enid Stewart for her kind advices and encouragement, and for typing t h i s manuscript. I would, l i k e to thank my wife, U zoamaka, and son Obiorah, for t h e i r r e l e n t l e s s patience, understanding, and encouragement. I wish to express appreciation to my Mother, Madam A.U. Igwe, and my parents-in-law, Chief and Mrs. F.C. Okoro, for t h e i r thoughtfulness, encouragement, and support. F i n a l l y , I wish to express my indebtedness to my l a t e Father, Chief M.N. Igwe, for a l l his s a c r i f i c e s and guidance; to him I proudly dedicate t h i s d i s s e r t a t i o n . xix • 1 CHAPTER 1 INTRODUCTION The continuous growth of demands placed on fresh water resources makes i t indispensable to "seek" new methods and unconventional techniques of water management. While the abundance of water, in quantity as well as quality, has been taken for granted by many, with the rapid increase i n water demands i n recent years there i s a growing appreciation and respect for the value of water and the need for sound and conservative management of the supply. Today the dimensions of water problems throughout the world are enormous. With the expansion i n world population, and the complementary ser-vices and industries t h i s population requires, and with the increase i n personal incomes, the pressures on water resources have mounted. Consequently, water resources are becoming l i m i t e d i n some regions of the world and competition for water among d i f f e r e n t water users has become more acute. A l l water uses - both consumptive and non-consumptive -have shared i n the enormous increase i n demands. Domestic use has increased as populations have grown, as urbanization has taken place and personal incomes have r i s e n . The increase in i n d u s t r i a l uses has p a r a l e l l e d the spectacular growth i n industry the world has seen i n recent decades. The production of food for the soaring population has required a tremendous augmentation of a g r i c u l t u r a l water supplies because an 2 enormous i n c r e a s e i n i r r i g a t i o n as w e l l as f e r t i l i z e r s , new lands and improved p l a n t v a r i e t i e s , have been necessary to produce the a g r i c u l t u r a l commodities r e q u i r e d . Water i s a b s o l u t e l y v i t a l f o r a g r i c u l t u r e , and e q u a l l y important, a g r i c u l t u r e i s the l a r g e s t consumer of water i n many r e g i o n s o f the world. As the l a r g e s t consumptive user of water, a g r i c u l t u r e ' s importance to the water economy i s obvious. The f r u i t s o f past r e s e a r c h i n a g r i c u l t u r a l water management, i r r i g a t i o n , and drainage, have had, and continue to have, an important impact on the management of a n a t i o n ' s t o t a l water resources and, thereby, on the w e l f a r e of a l l c i t i z e n s , the necessary p r o d u c t i o n of food, and the n a t i o n ' s economy. A d d i t i o n a l , expanded e f f o r t i s r e q u i r e d to provide the framework f o r s o l u t i o n s r e s p o n s i b l e to the g r e a t e r needs r e s u l t i n g from i n c r e a s e d p o p u l a t i o n p r e s s u r e s , i n c r e a s e d wealth, p o p u l a t i o n s h i f t s , and i n c r e a s e d awareness of e c o l -o g i c a l and environmental c o n s i d e r a t i o n s . Thus, i n most cases one of the important demands on a multi-purpose water management system would be f o r i r r i g a t i o n water. As the need f o r h i g h l y c e n t r a l i z e d water resources systems i n c r e a s e s , so a l s o does the need f o r improved and b e t t e r p l a n n i n g techniques and r e s e a r c h f o r the op t i m a l a l l o -c a t i o n o f water to the v a r i o u s water uses. The volume of s t o r e d water a v a i l a b l e i n comparison to the area t h a t i s or might be i r r i g a t e d i s a c r i t i c a l r e l a t i o n s h i p and t h e r e f o r e of paramount importance. T h i s r e l a t i o n s h i p , t o a c o n s i d e r a b l e .3 extent, determines the e f f e c t of the i r r i g a t i o n sub-system upon the t o t a l water management system, i n which the a r t i -f i c i a l water storage reservoir i s the basic element. This implies that the d i f f e r e n t sub-systems i n the superior complex water supply system have to be optimized, both in space and time, to arr i v e at an optimal design and operation of the superior system. The use of water resources as a production input i n agriculture has resulted i n a meaningful process of transfor-mation i n agriculture in semi-arid and a r i d regions. The need for sound and conservative management of water supply has been learned. Inherent i n the transformation i s the objective to reduce dependency on the seasonal a v a i l a b i l i t y of water i n order to extend the period of a g r i c u l t u r a l production over the entire year or a major part of i t and to reduce the impact of droughts. This actually r e f l e c t s the transformation of rain-fed agriculture (where natural supply conditions determine production patterns) to i r r i g a t e d farming, where complete or near complete control of water application makes i t possible to almost disregard the natural supply conditions and provide a water supply program adjusted to the preferred demand schedule. In recent years, numerous studies have been made which indicate that the need for water exceeds the r e l i a b l e y i e l d of available resources. Since the sc a r c i t y of water i s increasing, a question arises as to whether instead of 4 t r y i n g to i n c r e a s e the r e l i a b l e y i e l d , i t i s b e t t e r to adapt use to the seasonal v a r i a t i o n s i n supply which today r e s u l t i n l i t t l e or no use of a major p a r t of the water flow. To adapt use to the v a r i a t i o n s i n supply, i m p l i e s mod-i f i c a t i o n i n the supply regime which i s d i c t a t e d by seasonal h y d r o l o g i c a l c o n d i t i o n s . Under the envisaged m o d i f i e d water supply p o l i c y , a g r i c u l t u r a l p r o d u c t i o n w i l l be dependent on seasonal v a r i a t i o n s , but w i l l d i f f e r from r a i n - f e d a g r i c u l t u r e i n being t e c h n o l o g i c a l l y f u l l y c o n t r o l l e d on the b a s i s of long-range s t o c h a s t i c c o n s i d e r a t i o n s w i t h r e s p e c t t o both s u r f a c e and ground water r e s o u r c e s . T h i s approach would r e p l a c e the present p r a c t i c e i n which a f i x e d predetermined amount of water i s a l l o c a t e d to the a g r i c u l t u r a l s e c t o r through a h i g h l y i n s t i t u t i o n a l i z e d system of water r i g h t s , without r e g a r d f o r the v a r i a b l e h y d r o l o g i c a l c o n d i t i o n s . In the past t h i s has r e s u l t e d i n under-utLlization of the storage c a p a c i t y i n a drought year, or tremendous wastes i f e x c e s s i v e flows occur i n a given year. U n f o r t u n a t e l y , observed h y d r o l o g i c a l records are u s u a l l y s h o r t . The water r e s o u r c e s systems planner i s o f t e n handi-capped i n the o p t i m a l p l a n n i n g of the system elements because of inadequate h y d r o l o g i c a l data. Most h i s t o r i c a l h y d r o l o g i c a l data have s h o r t r e c o r d s and may c o n s t i t u t e poor r e p r e s e n t a t i o n of the p o s s i b i l i t i e s . Thus, the r e l i a b i l i t y of most h i s t o r -i c a l h y d r o l o g i c a l data i s q u e s t i o n a b l e . Again, a l l hydro-l o g i c a l processes are more or l e s s s t o c h a s t i c . In most cases 5 they have been assumed d e t e r m i n i s t i c or p r o b a b i l i s t i c o n l y to s i m p l i f y t h e i r a n a l y s i s . A c c o r d i n g l y , any meaningful approach to the optimal development and o p e r a t i o n of water reso u r c e s systems must take f u l l cognizance of h y d r o l o g i c a l u n c e r t a i n t y , both s u r -face and subsurface. More r e c e n t l y , s p e c i a l a t t e n t i o n i s being focussed on the systems approach to the s o l u t i o n of l a r g e water management systems as the e s t a b l i s h m e n t of such systems, i n c l u d i n g l a r g e - s c a l e i r r i g a t i o n schemes, has a d i r e c t e f f e c t on the g e o g r a p h i c a l environment i n c l u d i n g i t s b i o p h y s i c a l f a c t o r s and e x e r t s s t r o n g pressure on the ecology o f landscape and p o p u l a t i o n . The theme of t h i s d i s s e r t a t i o n i s to develop an o p t i m i z a t i o n model, which w i l l be s u i t e d to the development and o p e r a t i o n o f i r r i g a t i o n systems under h y d r o l o g i c a l uncer-t a i n t y . I t i s intended t h a t the model be f l e x i b l e , man-o r i e n t e d , and be as c l o s e as p o s s i b l e to r e a l - l i f e s i t u a t i o n s , as f a r as i r r i g a t i o n systems management i s concerned. For any i r r i g a t i o n systems p l a n n i n g model to be r e a l i s t i c and o p t i m a l , one has to know about the -operation of the system. There i s a h i e r a r c h y of s o l u t i o n s t h a t i s sought i n the management of i r r i g a t i o n systems when l a r g e storage volumes are r e q u i r e d , Roefs [94]. In c h r o n o l o g i c a l order the d e c i s i o n s t h a t a water r e s o u r c e s manager must make a r e : 1. the s p e c i f i c water r e s o u r c e s p r o j e c t to be b u i l t ; 6 2 . the proper time at which the p r o j e c t should be b u i l t ; 3. the s c a l e of the p r o j e c t ; 4 . the t a r g e t output which should be s e t i n a s p e c i f i e d time of o p e r a t i o n ; 5 . the o p e r a t i o n r u l e s f o r the p r o j e c t s p e c i f i e d ; 6. the r e a l time o p e r a t i o n a l c o n t r o l d e c i s i o n s . The f i r s t t h ree d e c i s i o n s can be c h a r a c t e r i z e d as pl a n n i n g d e c i s i o n s . The f o u r t h d e c i s i o n can be regarded as a r i s k a l l o c a t i o n d e c i s i o n . The f i f t h d e c i s i o n i s the oper-a t i o n p l a n . The s i x t h d e c i s i o n can be c h a r a c t e r i z e d as a " r e a l - t i m e " management d e c i s i o n , which i s the adjustment of d e c i s i o n s w i t h i n a f i n a l time s c a l e to f i t the o p e r a t i o n r u l e s . The sequence of d e c i s i o n s to be made e x h i b i t s some p r o p e r t i e s which complicate the a n a l y s i s . The f i r s t t hree p l a n n i n g d e c i s i o n s are interdependent. Worse s t i l l , these are dependent on the f o u r t h ( r i s k a l l o c a t i o n d e c i s i o n ) and f i f t h ( o p e r a t i o n a l plan) d e c i s i o n s . In other words, one cannot determine the op t i m a l s c a l e of the p r o j e c t without f i r s t , or a t l e a s t c o n c u r r e n t l y , c o n s i d e r i n g the d e f i n i t i o n of an o p t i m a l t a r g e t output f o r the p r o j e c t and the d e f i n i t i o n of an o p t i m a l s e t of o p e r a t i n g r u l e s f o r the p r o j e c t and t a r g e t . Thus, whether or not the p r o j e c t e x i s t s and the t a r g e t outputs have been s e t , the r e a l i s t i c approach to f o l l o w i s to a c t as i f they have and s o l v e the s p e c i f i c problem of o p e r a t i o n 7 r u l e d e f i n i t i o n f i r s t , s i n c e adequate estimates of o p e r a t i n g r u l e s and the e f f e c t s t h e r e o f are t o o l s t h a t the d e c i s i o n maker who i s ' s e t t i n g c o n t r a c t l e v e l s on p l a n n i n g a water r e -sources system must have. The aforementioned h i e r a r c h y of d e c i s i o n s a p p l i e s whether d e t e r m i n i s t i c o p t i m i z a t i o n or s t o c h a s t i c o p t i m i z a t i o n are used. In e i t h e r case an e f f i c i e n t procedure f o r d e t e r -mining o p t i m a l o p e r a t i n g r u l e s i s a key to a l l the other s o l u t i o n s r e q u i r e d . Thus, t h i s study s t a r t s with o p e r a t i o n of i r r i g a t i o n systems under u n c e r t a i n t y as a b a s i s f o r the development of a s t o c h a s t i c d e c i s i o n theory o p t i m i z a t i o n model which c o u l d be m e a n i n g f u l l y and r e a l i s t i c a l l y employed f o r p l a n n i n g purposes and d e s i g n i n i r r i g a t i o n systems under h y d r o l o g i c a l u n c e r t a i n t y . A comprehensive review of l i t e r a t u r e r e l a t i n g to the a p p l i c a t i o n s of the d i f f e r e n t systems techniques to the s o l u t i o n of problems i n i r r i g a t i o n systems management i s presented i n Chapter 2. Chapter 3 d e f i n e s the o b j e c t i v e s of study, namely, to develop a simple model f o r making o p e r a t i n g d e c i s i o n s f o r an i r r i g a t i o n p r o j e c t , t e s t i t s s e n s i t i v i t y and then g e n e r a l i z e i t f o r i r r i g a t i o n management pl a n n i n g d e c i s i o n s . Chapter 4 examines a l t e r n a t i v e approaches such as Dynamic programming, L i n e a r programming, and other forms of q u a d r a t i c programming techniques and h i g h l i g h t s the f e a t u r e s of Bayesian D e c i s i o n Theory which make i t the most a p p r o p r i a t e approach 8 for i r r i g a t i o n systems management decision-making under un-certainty and r i s k . The p r i n c i p l e of the Bayesian Decision Theory i s examined and the theory applied to develop a simple stochastic optimization model that can be s p e c i f i c a l l y employed for i r r i g a t i o n systems management decisions under hydrological uncertainty. In Chapter 5 the actual study done employing the model developed i n Chapter 4 i s described, on the assumption that a l l decisions regarding water use are made at the farm l e v e l for small to medium i r r i g a t i o n establishments; t h i s chapter gives the results i n terms of d i f f e r e n t operating pro-cedures, u t i l i t y functions, and crop response functions. Results of s e n s i t i v i t y analysis are also presented. Chapter 6 looks at the application of the model for regional planning in a large-scale i r r i g a t i o n system on the assumption that the regional water system i s highly centralized and that decisions are made by a central authority; r e s u l t s are, again, given i n terms of d i f f e r e n t operating procedures, u t i l i t y functions, crop response functions, and for the d i f f e r e n t i r r i g a t i o n periods. The reader i s directed to Table 6.8. In Chapter 7 the results from Chapters 5 and 6 are discussed, and the sources of probable errors i n the outputs are examined. Chapter 8 presents the conclusions from the study, and recommendations for further improvement and future d i r e c t i o n s . The computational procedures adopted i n the study are summarized i n the computer programs, A - l and A-2, for procedures I and II, respectively in Appendix A. •9 CHAPTER 2 REVIEW OF LITERATURE The most r e c e n t t r e n d i n the a n a l y s i s and design of water resources systems i s the a p p l i c a t i o n of systems approach employing the concepts of systems e n g i n e e r i n g . Planners and engineers i n the water resource f i e l d are c o n t i n u a l l y and r e l e n t l e s s l y attempting to s o l v e water resources problems by the use of e i t h e r the s i m u l a t i o n techniques or the o p e r a t i o n s r e s e a r c h technique. D i f f e r e n t o p t i m i z a t i o n models have been employed by d i f f e r e n t R i v e r B a s i n A u t h o r i t i e s , p a r t i c u l a r l y i n the United S t a t e s , to improve t h e i r systems o p e r a t i n g schedules and outputs. The i n c r e a s i n g use of o p t i m i z a t i o n models as a handy t o o l i n water resources systems s t u d i e s and r e s e a r c h has captured the i n t e r e s t s of many h y d r o l o g i s t s , engineers, planners and those i n v o l v e d i n water resources systems management. The p o p u l a r i t y and importance being a t t a c h e d to s i m u l a t i o n and o p e r a t i o n r e s e a r c h techniques have soared r e c e n t l y because of the wide a v a i l a b i l i t y of high-speed d i g i t a l computers and easy access to these computers by those who need them. The s i m u l a t i o n a n a l y s i s i s intended to reproduce the behavior or performance of a r i v e r b a s i n system on the computer i n every important aspect of the systems v a r i a b l e s , such as system u n i t s , i n p u t s , outputs, and ranges of s c a l e , Maas e t a l . [71]. S i m u l a t i o n r e p r e s e n t s a l l the i n h e r e n t c h a r a c t e r i s -t i c s and probable responses of the system c o n t r o l by a model t h a t i s l a r g e l y a r i t h m e t i c and a l g e b r a i c i n nature but 10 i n c l u d e s a l s o some mathematical, l o g i c a l p r o c e s s e s . The Harvard Water Program has been simulated i n the computer, Maas et a l [71]. The g r e a t e s t l i m i t a t i o n of the s i m u l a t i o n a n a l y s i s i s t h a t the technique r e q u i r e s data of good r e p r e s e n t a t i o n of the p o s s i b l e occurrences of the h y d r o l o g i c events. The use of o p e r a t i o n s r e s e a r c h techniques i n water r e s o u r c e s p l a n n i n g and development was developed l a r g e l y d u r i n g the l a s t two decades. S p e c i a l c o n t r i b u t i o n s t o the knowledge are p a r t i c u l a r l y due to a team of r e s e a r c h workers p a r t i c i p a t i n g i n the Harvard Water Program, Chow [27], Maas and Hufschmidt [69,70] and Maas et a l [71]. The systems approach has made remarkable progress i n the a n a l y s i s of components and subsystems from which the s y n t h e s i s of the complete system may be p o s s i b l e , Buras [19]. T h i s e n t i r e 'Operation i s aimed a t producing a whole s e r i e s of a l -t e r n a t i v e s which can be ranked i n accordance with a gi v e n c r i t e r i o n , and i n which the b e n e f i c i a l and/or d e t r i m e n t a l e f f e c t s of each i s c l e a r l y d e f i n e d . The systems e n g i n e e r i n g approach attempts to shorten the time l a g between the appearance of needs (e.g. i n c r e a s e d demand f o r water) and the p r o d u c t i o n of new hardware and/or software* (o p e r a t i n g procedures) t h a t s a t i s f y these needs, Buras [19]. In t h i s approach v a r i a b l e s d e s c r i b i n g components or s t a t e s of a system can be d e f i n e d , and r e l a t i o n s h i p s between them repre s e n t e d , through equations i n a mathematical model. These r e l a t i o n s h i p s , whether l i n e a r or n o n - l i n e a r , can be p r o p e r l y e v a l u a t e d by a v a r i e t y of techniques, some of which have been made p o s s i b l e 11 by the advances i n computer s c i e n c e . However, i f the problems were r e d u c i b l e t o a s e t of mathematical e x p r e s s i o n s , there would have been no need t o i n v e n t systems e n g i n e e r i n g : perhaps numerical a n a l y s i s would have been s u f f i c i e n t i n many cases. But i n water resources problems there are a l s o many s o c i a l and p o l i t i c a l f a c t o r s t h a t must be given due c o n s i d e r a t i o n . When there i s no r e a l i s t i c way to a s s i g n values t o these f a c t o r s , t h e i r e f f e c t upon the system as a whole can be eva l u a t e d by h a n d l i n g them as c o n s t r a i n t s . These c o n s t r a i n t s are the requirements o f the design c r i t e r i a to which the o p t i m i z a t i o n i s s u b j e c t e d . The c o n s t r a i n t s may be t e c h n i c a l , budgetary, s o c i a l o r p o l i t i c a l , and the b e n e f i t s may be r e a l or i m p l i e d . Hence the optimal design i s s u b j e c t to t e c h n i c a l as w e l l as economic and s o c i o - p o l i t i c a l l i m i t a t i o n s . The mathematical or a n a l y t i c a l models employed i n the o p e r a t i o n s r e s e a r c h techniques f o r making optimal d e c i s i o n s are g e n e r a l l y c a t e g o r i z e d a c c o r d i n g to the nature of the hydrological data. When the p r o b a b i l i t y of h y d r o l o g i c a l data i s ignored , the model i s known as a " D e t e r m i n i s t i c Model". When the hydrological u n c e r t a i n t y and/or ot h e r u n c e r t a i n t i e s are con s i d e r e d , the model i s c a l l e d a " S t o c h a s t i c Model". The use of models are advantageous i n the sense t h a t i t give s the a n a l y s t an i n s i g h t i n t o the behavior o f the system under v a r i o u s c o n d i t i o n s of p l a n n i n g and o p e r a t i o n , and t h e r e f o r e w i l l enable him to base h i s d e c i s i o n on the expected behavior of the system and thus to reduce h i s dependence on experience 12 and i n t u i t i o n . O p t i m i z a t i o n models most f r e q u e n t l y used i n c l u d e d e t e r m i n i s t i c l i n e a r programming, d e t e r m i n i s t i c dynamic programming, s t o c h a s t i c l i n e a r programming, s t o c h a s t i c dynamic programming and v a r i o u s o t h e r forms. I t i s becoming accepted p r a c t i c e f o r many government agencies i n the water res o u r c e s f i e l d t o c o n s t r u c t models of complex r i v e r b a s i n systems as a i d s i n the comparison of a l t e r n a t i v e investments. Since i r r i g a t i o n i s the g r e a t e s t consumptive use of water i n most semi-arid, r e g i o n s , i t i s to be expected t h a t these p l a n n i n g models have, i n one way or the o t h e r , i n c o r p o r a t e d the i r r i g a t i o n subsystem i n t o the e n t i r e s u p e r i o r system. In r e g i o n s where the water supply systems are not h i g h l y c e n t r a l i z e d the p l a n n i n g models have tended to concen-t r a t e on e f f i c i e n t i r r i g a t i o n p l a n n i n g and development as the s o l e o b j e c t i v e . S e v e r a l s t u d i e s have a p p l i e d system a n a l y s i s techniques to aspects o f the i r r i g a t i o n p l a n n i n g problem. A l l the approaches have con c e n t r a t e d mainly on two major aspects of i r r i g a t i o n v i z : shortage o f water supply as the main cause l i m i t i n g the expansion of the p r o d u c t i o n acreage; and, inadequate water u t i l i z a t i o n e f f i c i e n c y r e s u l t i n g i n high water l o s s e s . The former may be due to the f a c t t h a t the r e s e r v o i r i s too s m a l l t o be e f f e c t i v e i n modifying the h i g h l y f l u c t u a t i n g water supply and peak demand, thus r e s u l t i n g i n frequent occurrence of drought and o c c a s i o n a l water overflow ( s p i l l a g e ) from the r e s e r v o i r . The l a t t e r may be a r e s u l t o f poor 13 cropping p a t t e r n s and schedules (or l a n d preparation) which can r e s u l t i n i n e f f i c i e n t u t i l i z a t i o n of a given water supply. D i f f e r e n t c a t e g o r i e s o f systems a n a l y s i s techniques have been a p p l i e d to i r r i g a t i o n system p l a n n i n g . • S e v e r a l c a t e g o r i e s of s t u d i e s c o u l d be i d e n t i f i e d . The f i r s t category of s t u d i e s p r i m a r i l y i n v e s t i g a t e s the f a c t o r s a f f e c t i n g the optimal c r o p p i n g p a t t e r n s of an a g r i c u l t u r a l p r o j e c t . Hutton [ 5 9 ] reviews the a p p l i c a t i o n of o p e r a t i o n s r e s e a r c h to farm problems. Agrawal and Heady [ 2 ] d i s c u s s r e c e n t trends i n o p e r a t i o n s r e s e a r c h method f o r a g r i c u l t u r a l d e c i s i o n s . M a r t i n , Burdak, and Young [75] use a cropping p a t t e r n model i n tandem with an analog computer model of the a q u i f e r to examine i n t e r t e m p o r a l e f f e c t s of a d e c l i n i n g water t a b l e . Mann, Moore and J o h l [ 7 4 ] examine s h i f t s i n cropping p a t t e r n s f o r Punjabi farmers under v a r y i n g t e c h n o l o g i c a l c o n d i t i o n s . Strong [ 1 0 6 ] s t u d i e s the t r a d e - o f f between the v a l u e of water saved through i n c r e a s e d e f f i c i e n c y and i n c r e a s e d leakage c o s t s . Gotsch [ 4 2 ] i s con-cerned with the e f f e c t s on farmers' supply curves of the i n s t a l l a t i o n of i r r i g a t i o n w e l l s . Huang, Liang and Wu [ 5 8 ] have, more r e c e n t l y , attempted o p t i m i z i n g water u t i l i z a t i o n of the L a l a m i l o i r r i g a t e d area i n Hawaii through m u l t i p l e crop s c h e d u l i n g . A l s o Buras, N i r and A l p e r o v i t s [ 2 0 ] adopted the system approach i n an attempt to s o l v e the s c h e d u l i n g of i r r i g a t i o n a c t i v i t i e s , f o r a p r e d e t e r -mined, f i x e d , q u a n t i t y of i r r i g a t i o n water, s o l i d - s e t s p r i n k l i n g , f o r an i r r i g a t i o n season of 2 4 0 days, f o r mixed cropping area 14 of 124 hecta r e s i n Western G a l i l e e i n I s r a e l . A second s e t of model s t u d i e s attempts to optimize the c o n j u g a t i v e o p e r a t i o n of ground and s u r f a c e water systems i n an i r r i g a t i o n c o n t e x t . Aron [ 5 ] a p p l i e d dynamic pro-gramming to p a r t of the Santa C l a r a V a l l e y i n C a l i f o r n i a t o optimize s u r f a c e water-groundwater a l l o c a t i o n s over time. Bear and L e v i n [ 1 0 , 1 1 ] used dynamic programming to determine o p e r a t i n g s t r a t e g i e s f o r an a q u i f e r with f i x e d i n s t a l l a t i o n s , i s o l a t e d from the remainder of the eco n o m i c - h y d r o l o g i c a l system. Buras [ 1 7 ] c o n s i d e r e d the op t i m a l o p e r a t i o n of a combined s u r f a c e r e s e r v o i r - a q u i f e r system. Burt [ 2 1 ] examined ground-water u t i l i z a t i o n over time through the use of two models -one with underground storage only and the other with a s u r f a c e r e s e r v o i r and an underground r e s e r v o i r . Duchstein and K i s i e l [ 3 5 ] reviewed these models and provide a comprehensive b i b l i o g r a p h y . A t h i r d category i s concerned w i t h s t u d i e s t h a t employ l i n e a r programming models to examine s p a t i a l water a l l o c a t i o n s i n a s t a t i c - d e t e r m i n i s t i c m i l i e u . Dracup [ 3 4 ] c o n s i d e r e d f i v e sources of water and three water u s e r s . C a s t l e and Lindborg [ 2 4 ] a l l o c a t e d water between two a g r i c u l t u r a l areas. M i l l i g a n ' s study [ 8 0 ] based i t s water a l l o c a t i o n i n p a r t on water balance i n t e r a c t i o n s . McConnen and Menon [ 7 6 ] analyzed the i n t e g r a t e d use of ground and s u r f a c e water, f o r i r r i g a t i o n . S p o f f o r d [ 101 ] employed n o n - l i n e a r programming techniques to analyze the a l l o c a t i o n of s u r f a c e and groundwater 15 s u p p l i e s of v a r y i n g q u a l i t i e s between i r r i g a t e d areas i n West P a k i s t a n . Champion and G l a s e r [ 2 5 ] used a l i n e a r program to a l l o c a t e i r r i g a t i o n water i n a c a p i t a l budgeting study. Smith [ 1 0 0 ] used a parametric l i n e a r programming model t o determine the i r r i g a t i o n systems investment f o r an i r r i g a t i o n p r o j e c t i n Bangladesh. The i r r i g a t i o n p r o j e c t l i n e a r program u t i l i z e d the c o s t s of a l t e r n a t i v e means of p r o v i d i n g i r r i g a t i o n water and subsurface drainage, a water balance of the r e g i o n based on parameters estimated i n a separate i n v e s t i g a t i o n of ground-water resources and i r r i g a t i o n e f f i c i e n c i e s , and crop b e n e f i t s , t o determine the p r o j e c t c o n f i g u r a t i o n t h a t maximizes net b e n e f i t s . Another c l a s s of o p t i m i z a t i o n models which have been used i n i r r i g a t i o n systems p l a n n i n g d e a l s with p r e l i m i n a r y s c r e e n i n g models. Dorfman [ 3 3 ] i s the base r e f e r e n c e f o r any of these s t u d i e s . Blanchard [ 1 6 ] was one of the f i r s t to use l i n e a r programming to screen a l t e r n a t i v e s i n a complex r i v e r system comprising s u r f a c e and groundwater s u p p l i e s and s e v e r a l competing water uses. Thomas and R e v e l l e [ 1 1 1 ] examined optimal u t i l i z a t i o n of the water impounded by the Aswan High Dam f o r both power and i r r i g a t i o n . O n i g k e i t , Kim and Schmid [ 8 6 ] employed a s t o c h a s t i c model to determine optimal s u r f a c e r e s e r v o i r s i z e s , a dynamic programming model to a s c e r t a i n r e s e r v o i r i r r i g a t e d area combinations, and a l i n e a r programming model to optimize la n d and water a l l o c a t i o n s w i t h i n an i r r i g a t e d area i n the P l a i n of Thessaly. 16 The most r e c e n t development i n the f i e l d o f i r r i g a -t i o n systems o p t i m i z a t i o n i n c l u d e s a model c l a s s which concen-t r a t e s on the optimal o p e r a t i o n of a g i v e n i r r i g a t i o n system f o r improved water use e f f i c i e n c y , improved e l e c t r i c a l power c o n s e r v a t i o n f o r i r r i g a t i o n w e l l s , and improved crop y i e l d . T h i s c l a s s of s t u d i e s i s r e l a t i v e l y more r e a l i s t i c i n i r r i g a -t i o n i n t h a t the models i n t h i s c l a s s s t a r t with g i v e n crop p r o d u c t i o n f u n c t i o n s and some have attempted to base t h e i r water a l l o c a t i o n s i n p a r t on water balance i n t e r a c t i o n s . F l i n n and Musgrave [40] used dynamic programming f o r d e t e r m i n i s t i c hydrology. Others t h a t have employed dynamic programming i n t h e i r s t u d i e s i n c l u d e H a l l and Buras [45] when seasonal d i s t r i -b u t i o n of water a p p l i c a t i o n i s known but not the a l l o c a t i o n t o d i f f e r e n t crops; H a l l and Butcher [48 ], Dudley, Howell and Musgrave [36] and Aron [5] f o r a l l o c a t i n g a g i v e n water supply to a s i n g l e crop; Burt and Stauber [22] f o r a l l o c a t i n g p r i c e d water to a s i n g l e crop; and deLucia [31] f o r a one crop i r r i g a t i o n system under s t o c h a s t i c regimes. A l p e r o v i t s [3] Buras, N i r and A l p e r o v i t s [20] developed a model which e s p e c i a l l y adopts the systems approach and uses a s i m u l a t i o n a l g o r i t h m . H a l l and Dracup [50] i s a source of good r e f e r e n c e m a t e r i a l f o r t h i s c l a s s of s t u d i e s . Windsor and Chow [123] have proposed m u l t i l e v e l o p t i m i z a t i o n schemes comprising one crop, i n t e r t e m p o r a l water a p p l i c a t i o n o p t i m i z a t i o n i n l e v e l one and a l l o c a t i o n of water to crops and s o i l s i n l e v e l two. 17 Other r e l e v a n t s t u d i e s i n c l u d e those of Bargur and Gablinger [ 7 , 8 ] ; R u s s e l l [ 9 5 ] who developed a method based on d e c i s i o n theory f o r the o p e r a t i o n of the Okanagan Lake i n B r i t i s h Columbia, a lake r e g u l a t e d f o r both f l o o d c o n t r o l and storage of water f o r i r r i g a t i o n ; a l s o Stewart, M i s r a , P r u i t t and Hagan [ 1 0 5 ] and,Stetson, Watts, Corey and N e l s o n [ 1 0 3 ] . As mentioned elsewhere In t h i s l i t e r a t u r e review, i r r i g a t i o n subsystem i s u s u a l l y an i n t e g r a l p a r t of any m u l t i -purpose or complex water resources systems investment and o p e r a t i o n . In most s e m i - a r i d r e g i o n s , where i r r i g a t i o n i s v i t a l l y needed, and on a l a r g e - s c a l e , water resources develop-ment normally s t a r t s with i r r i g a t i o n as the i n i t i a l or top p r i o r i t y o b j e c t i v e . In such cases the i n i t i a l p l a n n i n g and design c o n s i d e r a t i o n s f o r the water resource elements u s u a l l y have the o b j e c t i v e of maximizing outputs f o r i r r i g a t i o n water. I f i r r i g a t i o n system p l a n n i n g does not s t a r t o f f i n the r i g h t d i r e c t i o n , and i f a l l the necessary aspects i n the design and o p e r a t i o n o f systems are not c o n c u r r e n t l y c o n s i d e r e d , the whole i r r i g a t i o n p r o j e c t i s bound t o f a i l , as has been r e p o r t e d i n the pas t . Although many d e t e r m i n i s t i c and s t o c h a s t i c o p e r a t i o n s r e s e a r c h techniques have been a p p l i e d to solve, the problem of i r r i g a t i o n systems management, many of these approaches have, i n g e n e r a l , p r o v i d e d o n l y p a r t i a l s o l u t i o n s to t h i s complex problem. Most s t u d i e s , a v a i l a b l e i n l i t e r a t u r e , have concen-t r a t e d mainly on the e f f i c i e n t o p e r a t i o n o f a s i n g l e r e s e r v o i r f o r i r r i g a t i o n . Some r e s e a r c h e r s have used the d e t e r m i n i s t i c 18 approach while many others have more r e c e n t l y come to r e a l i z e the i m p l i c a t i o n s of s t o c h a s t i c v a r i a b i l i t y i n i r r i g a t i o n p l a n n i n g . These have c o n s i d e r e d the i r r i g a t i o n o p e r a t i n g process to be s t o c h a s t i c , s i n c e the i r r i g a t i o n water r e q u i r e -ments, groundwater recharge, q u a n t i t i e s of s u r f a c e r u n o f f , c o s t of i r r i g a t i o n , crop y i e l d s , and v a l u e s are a l l a s s o c i a t e d with high u n c e r t a i n t i e s and r i s k s . Many methodologies have been p u b l i s h e d on the search of an optimum o p e r a t i n g p o l i c y f o r a s i n g l e r e s e r v o i r f o r e f f i c i e n t use of water. H a l l and h i s co-workers [ 4 5 , 4 7 , 4 4 , 4 9 ] , s t u d i e d e x t e n s i v e l y the optimum o p e r a t i o n p o l i c y of a r e s e r v o i r with d e t e r m i n i s t i c stream flows. Young [ 1 2 7 ] , f o r m u l a t e d the o p t i m a l p o l i c y by u s i n g the Monte C a r l o and r e g r e s s i o n methods.. S e v e r a l others have used s t o c h a s t i c m o d e l l i n g approach. Prominent among these are Buras [ 1 8 ] and Butcher [ 2 3 ] who used a s t o c h a s t i c dynamic programming technique; Falkson [ 3 7 ] a p p l i e d Howard's p o l i c y i t e r a t i o n method; Loucks [ 6 5 , 66 ] f a n ( j Thomas [ 1 1 0 ] employed s t o c h a s t i c l i n e a r programming techniques i n d e r i v i n g the optimum o p e r a t i n g p o l i c y . Smith. [ 1 0 0 ] used a s t o c h a s t i c , chance-constrained pro-gramming to a r r i v e at o p e r a t i n g d e c i s i o n s f o r an i r r i g a t i o n p r o j e c t i n Bangladesh. R u s s e l l [ 9 5 ] p r o v i d e s the only example of a study, t h a t the w r i t e r i s aware of t h a t uses a s t o c h a s t i c approach based on d e c i s i o n theory to develop opera-t i n g r u l e s f o r the Okanagan Lake which i s r e g u l a t e d f o r both 19 c o n t r o l of f l o o d s and storage of water f o r i r r i g a t i o n purposes. Huang, L i a n g , and Wu [ 5 8 ] adopted a s t o c h a s t i c l i n e a r pro-gramming wi t h recourse i n a study to determine optimal c r o p p i n g p a t t e r n f o r the p r o d u c t i o n of d i f f e r e n t types of. vegetables having d i f f e r e n t water consumption r a t e s d u r i n g d i f f e r e n t growth p e r i o d s , i n the L a l a m i l o area of Hawaii. They used a d i f f e r e n t approach and attempted the p o s s i b i l i t y of modifying crop water demand p a t t e r n s i n order t o i n c r e a s e an i r r i g a t i o n r e s e r v o i r water u t i l i z a t i o n e f f i c i e n c y . Their model attempts to modify crop water demand p a t t e r n s by the proper s e l e c t i o n of crop type, l i m i t a t i o n o f acreage and d e t e r m i n a t i o n of p l a n t i n g time. T h i s approach allowed them to design optimal cropping p a t t e r n s t h a t w i l l e f f i c i e n t l y u t i l i z e a given water supply (that i s , r e s e r v o i r c a p a c i t y i s predetermined and f i x e d ) . An e x c e l l e n t comparison of v a r i o u s s t o c h a s t i c models can be found i n the paper by Loucks and Falkson [ 6 7 ] -In t h a t paper, i t i s shown t h a t a unique optimal p o l i c y can be obtained i r r e s p e c t i v e o f s t o c h a s t i c model used. In the l i t e r a t u r e c i t e d , c e r t a i n d e c i s i o n v a r i a b l e s such as cr o p p i n g p a t t e r n s , water demand p a t t e r n s f o r i r r i g a -t i o n , and r e s e r v o i r c a p a c i t i e s , have been c o n s i d e r e d t o be predetermined (exogenous v a r i a b l e s ) . Smith [ 1 0 0 ] attempted to avoid t h i s by u s i n g chance-constrained programming approach i n which the i r r i g a t i o n water requirements, groundwater recharge, s u r f a c e r u n o f f , c o s t s of i r r i g a t i o n and crop values 20 are stochastic - r a i n f a l l and prices are random. However, the acreage under i r r i g a t i o n for the f i v e crops considered were predetermined and fixed at one m i l l i o n acres for each crop considered i n any decision period. Another, l i m i t a t i o n i s that a l l the models considered are e s s e n t i a l l y s t a t i c . For well defined model uses t h i s i s not a serious limitation, Smith [99] since the aim of the systems approach i s to seek and find optimum solutions within a certain time period. However, since s t r u c t u r a l changes w i l l occur i n the socio-economics of a l l countries i n the developing world, such as South Asia, and A f r i c a , s t a t i c assumptions regarding socio-economic factors, power and c o n f l i c t within the society (or i r r i g a t i o n project) must lead to error regarding project impact. Thus, such s t a t i c models, with t h e i r inherent assumptions, must be modified for successful applications i n the developing countries. In most of the l i t e r a t u r e c i t e d the optimization proceeded with the maximization of one type of value c r i t e r i o n the net benefits s o l e l y expressed i n monetary or expected monetary values. Russell [9 5] provides the only example of a study that deviates from th i s general approach. In the development of his stochastic model for the operation of the Okanagan Lake system, he attempts to maximize "expected u t i l i t y values". It should be noted that an economically optimal system need not be a system that i s optimal for 21 s o c i e t y . I t i s e s s e n t i a l to take cognizance of the i n t e r -p l a y of p h y s i c a l f a c t o r s , t e c h n o l o g i c a l p r ocesses, a g r i -c u l t u r a l p r a c t i c e s , and s o c i a l o r g a n i z a t i o n s i n the management of water r e s o u r c e s . For any i r r i g a t i o n systems'planning study to be meaningful, the study should s t a r t w i t h o p e r a t i o n a l r u l e s . To be r e a l i s t i c w i t h o p e r a t i n g r u l e s one has to c o n s i d e r the s t o c h a s t i c v a r i a b i l i t y i n i r r i g a t i o n p l a n n i n g and thus has to c o n s i d e r u n c e r t a i n t y and r i s k r e l a t i n g t o the major uncer-t a i n t y element - inadequate data i n hydrology; and other c o n s i d e r a t i o n s other than s t r i c t l y maximizing expected monetary v a l u e s have to be brought i n t o the model f o r m u l a t i o n . 22 CHAPTER 3 CONTEXT AND OBJECTIVES OF STUDY 3.1 Context Since i r r i g a t e d a g r i c u l t u r e i s the l a r g e s t con-sumptive user of water, h y d r o l o g i c a l i n f o r m a t i o n i s a key i n p u t i n the management of i r r i g a t i o n p r o j e c t s . In almost a l l circumstances t h i s key i n p u t i s v a r i a b l e and to some extent u n p r e d i c t a b l e , and thus i r r i g a t o r s need to be a b l e to d e a l w i t h i t . Although a system based on water r i g h t s , which has been i n use i n western developed c o u n t r i e s works w e l l , r i g i d i t i e s are i n t r o d u c e d i n t o the system which may not a l l o w f l e x i b i l e p l a n n i n g . C o n s t r a i n t s i n t r o d u c e d by water r i g h t s do not apply at the l e v e l of i n d i v i d u a l farms and i n many de v e l o p i n g c o u n t r i e s . Because of expected s t r u c t u r a l changes and consequent p o l i c y changes w i t h regards to power and con-f l i c t w i t h i n d e v e l o p i n g s o c i e t i e s , most models t h a t have worked s a t i s f a c t o r i l y i n developed c o u n t r i e s may not be very s u c c e s s f u l i n poor, d e v e l o p i n g c o u n t r i e s . I t i s necessary, t h e r e f o r e , to develop a technique which c o u l d be u n i v e r s a l l y adapted f o r o p t i m a l i r r i g a t i o n p r o j e c t e v a l u a t i o n and decision-making i n both the developed and d e v e l o p i n g c o u n t r i e s . A p r a c t i c a l o p t i m i z a t i o n a l g o r i t h m f o r competent i r r i g a t i o n systems management decision-making should be developed w i t h the f o l l o w i n g c r i t e r i a i n mind: i ) Concepts of model should be g e n e r a l l y a p p l i c a b l e t o s i t u a t i o n s 23 in the r i c h as well as the poor countries, and to . d i f f e r e n t cultures; the model approach should be such that technological bias i s minimized and the model could be re a d i l y applicable to ind i v i d u a l farm l e v e l i r r i g a t i o n decisions as well as regional and a high degree of centralized economic and water management control. i i ) For the model to be successfully employed for planning purposes the analysis should st a r t with operational rules and should be e x p l i c i t and comprehensive i n output. i i i ) A behavioral approach which stresses current human needs in the optimization period, both economic and s o c i a l welfare, rather than economic p r o f i t a b i l i t y , i s desirable. iv) The approach should enhance the processes of communication with the public, within the planning organization, and with other organizations. vj The model should foster the transmission of highly tech-n i c a l information to laymen and decision-making echelons i n an i n t e l l i g i b l e form that includes information concerning value judgements of the technicans and decision analysts. vi) The model can be employed when hydrological data are scarce, can assure that available data are used e f f i c i e n t l y and e f f e c t i v e l y , and can provide objective analysis of sub-j e c t i v e considerations. v i i ) The model should be s u f f i c i e n t l y adaptable so that modi-f i c a t i o n s can be made to r e f l e c t changing agency requirements 24 or to focus more s h a r p l y on an unusual aspect of an i r r i g -a t i o n p r o j e c t . v i i i ) The model must achieve a s a t i s f a c t o r y balance between r e a l i s m and c o m p u t a b i l i t y . A model meeting the above s t a t e d c r i t e r i a i s not p r e s e n t l y a v a i l a b l e . 3..2 O b j e c t i v e s of Study Th e r e f o r e , the purpose of t h i s d i s s e r t a t i o n i s to develop a methodology f o r decision-making i n r e l a t i o n to long-term and o p e r a t i o n of i r r i g a t i o n systems with storage, under h y d r o l o g i c a l u n c e r t a i n t y . The s p e c i f i c o b j e c t i v e s are: 1. To develop a simple s t o c h a s t i c o p t i m i z a t i o n model which can p r o v i d e o b j e c t i v e a n a l y s i s of s u b j e c t i v e c o n s i d e r -a t i o n s f o r o p e r a t i o n decision-making - s p e c i f i c a l l y how to decide on the area of l a n d to prepare f o r i r r i -g a t i o n at the beginning of the growing season, under v a r y i n g degrees of h y d r o l o g i c a l u n c e r t a i n t y , d i f f e r e n t systems o p e r a t i n g procedures, d i f f e r e n t crop response f u n c t i o n s , d i f f e r e n t d e c i s i o n c r i t e r i a , and d i f f e r e n t i r r i g a t i o n p e r i o d s . 2. To make the model as r e a l i s t i c as p o s s i b l e f o r c o n s i d -e r a t i o n s t h a t apply i n a c t u a l o p e r a t i o n of i r r i g a t i o n systems. 3. To check the s e n s i t i v i t y of the o p t i m a l p o l i c y to changes 25 i n h y d r o l o g i c a l u n c e r t a i n t y , crop response f u n c t i o n s , d i f f e r e n t o p e r a t i o n a l r u l e s , and d i f f e r e n t d e c i s i o n c r i t e r i a , employing data from the N i c o l a V a l l e y I r r i -g a t i o n D i s t r i c t i n B r i t i s h Columbia. 4 . To i d e n t i f y the f l e x i b i l i t y , p r a c t i c a b i l i t y , and limi^-t a t i o n s of the approach to the management of resources i n the de v e l o p i n g c o u n t r i e s . Once such a technique f o r determining the optimal i r r i g a b l e areas under h y d r o l o g i c a l u n c e r t a i n t y i s developed, i t co u l d serve as a key to the s o l u t i o n of many problems r e l a t i n g to i r r i g a t i o n systems management decision-making, and c o u l d be i n s t r u m e n t a l t o the improvement of a l r e a d y e x i s t i n g o p e r a t i o n a l models. CHAPTER 4 DEVELOPMENT OF DECISION THEORY" STOCHASTIC OPTIMIZATION MODEL 4.1 Major Approaches to Programming Under Risk and U n c e r t a i n t y . I r r i g a t i o n systems plann i n g , development and opera-t i o n have to be done wit h adequate c o n s i d e r a t i o n s f o r long-term s t o c h a s t i c h y d r o l o g i c a l i n p u t r a t h e r than the c o n s e r v a t i v e approach of a d e t e r m i n i s t i c f i x e d q u a n t i t y of f i r m water supply f o r i r r i g a t i o n throughout the l i f e of the p r o j e c t . The problem of s t o c h a s t i c v a r i a b i l i t y i n h y d r o l o g i c a l phenomena has been before man s i n c e the dawn of c i v i l i z a t i o n and i t has been s o l v e d i n s t i t u t i o n a l l y by the c r e a t i o n of the system of water r i g h t s , w i t h a t t e n t i o n focused on p r i o r i t y , q u a n t i t y , and more r e c e n t l y , q u a l i t y of use. I n s t i t u t i o n a l s o l u t i o n s are seldom o p t i m a l a t any one time. Though s p e c i f i c system of water r i g h t s may be q u i t e v a r i a b l e , f o r example the r o t a t i o n system i n use among i r r i g a t o r s p r a c t i s i n g the r i g h t s of p r i o r a p p r o p r i a t i o n i n the western U n i t e d S t a t e s , i n each case where the water supply v a r i e s randomly about an expected mean, the systems of r i g h t s i n c o r p o r a t e s some " c l a u s e " t h a t i n c l u d e s some form of p r i o r i t y . More r e c e n t l y w i t h the advent of the concept of systems e n g i n e e r i n g , d i f f e r e n t programming techniques have been a p p l i e d to d i f f e r e n t r i v e r - r e s e r v o i r systems. Major approaches to programming under r i s k i n c l u d e the Dynamic Pro-gramming, the L i n e a r Programming, and v a r i o u s other forms of 27 q u a d r a t i c programming techniques. In the dynamic programming approach the o b j e c t i v e , i n the face of h y d r o l o g i c a l u n c e r t a i n t y , i s c o n c e i v a b l y the maximization of expected monetary val u e s of the t o t a l r e t u r n . T h i s approach presents a d i f f i c u l t y i n the fundamental i s s u e of the s u i t a b i l i t y of expected monetary value as the o b j e c t i v e i n dynamic programming. The e s s e n t i a l d e f i c i e n c y of the a n a l y s i s i s t h a t the value of the consequent s t a t e under an optimal p o l i c y , as computed from the formulated r e c u r s i v e equation and c o n s t r a i n t s , i s r e p l a c e d by the optimum expected monetary value of the consequent s t a t e under a p o l i c y which at each stage maximizes the expected monetary value of the remain-i n g s t a g e s . Another shortcoming i s the f a i l u r e t o s i m u l t a n e o u s l y take account of the non-commensurate o b j e c t i v e of a v o i d i n g r i s k . I t i s p r e c i s e l y the same f a i l u r e which causes most of the d i f f i c u l t i e s i n any o t h e r s t o c h a s t i c maximization problem which does not r e c o g n i z e r i s k , H a l l and Dracup [50]. S e v e r a l forms of l i n e a r programming f o r m u l a t i o n s have been proposed f o r s o l v i n g i r r i g a t i o n systems o p t i m i z a t i o n pro-blems under r i s k . F i r s t o f these i s the s t o c h a s t i c l i n e a r programming with r e c o u r s e . In t h i s approach the problem i s p a r t i t i o n e d i n t o two or more stages. In the f i r s t stage, before any s t o c h a s t i c events have oc c u r r e d , c e r t a i n d e c i s i o n v a r i a b l e s are s e l e c t e d , such as i r r i g a t i o n storage c a p a c i t i e s and cropping p a t t e r n s , (exogeneous v a r i a b l e s ) . In the second 28 stage the s t o c h a s t i c events occur and some c o n s t r a i n t s may be v i o l a t e d . The d e c i s i o n maker can meet the c o n s t r a i n t s , however, by making a s e r i e s of second stage d e c i s i o n s t h a t may i n c u r a l o s s due t o shortage or s u r p l u s . The l i n e a r f o r m u l a t i o n o f t h i s approach may be s t a t e d mathematically as f o l l o w s : Max Z(x) = c'x - E [Min q'y] ) su b j e c t to Ax = b ) ( 4 - 1 ) Tx + .Wy = x > 0, y > 0 ) where, c = a v e c t o r of p r i c e s ; b = a v e c t o r of resource a v a i l a b i l i t i e s ; A = a matrix of t e c h n o l o g i c a l c o e f f i c i e n t s ; x = a v e c t o r of d e c i s i o n v a r i a b l e s ; q = a v e c t o r of p e n a l t i e s i n c u r r e d due to d e v i a t i o n s from the t a r g e t ; T and W = ma t r i c e s o f t e c h n o l o g i c a l c o e f f i c i e n t s ; X = a v e c t o r of s t o c h a s t i c elements; y = a v e c t o r of second stage d e c i s i o n v a r i a b l e s ; and c, q, T, W, and X may c o n t a i n s t o c h a s t i c elements. The d e c i s i o n maker chooses x (to s a t i s f y Ax = b and x ^ 0) such t h a t when the s t o c h a s t i c v a r i a b l e s are observed, y can be chosen to s a t i s f y y >. 0 and Wy = X - Tx, and to maximize the d i f f e r e n c e between the expected f i r s t stage b e n e f i t s , and second stage c o s t s . An obvious advantage of t h i s approach i s th a t i t pr o v i d e s a framework f o r the i n c o r p o r a t i o n o f crop response f u n c t i o n s and 29 other technical data into a programming model, either i n the objective function or as constraints. However, the most obvious drawback to the use of stochastic programming with recourse Is the d i f f i c u l t y i n obtaining the requisite loss functions for water d e f i c i t s e s p e c i a l l y for crops where l i t t l e of the needed research has been performed. These data have not been employed i n the past for decision making and l i t t l e experience has been accumulated to aid i n 'their estimation. Another form of stochastic l i n e a r programming i s the chance-constrained programming technique. In t h i s approach, constraint v i o l a t i o n s are allowed a certain proportion of the time and the e x p l i c i t costs of constraint v i o l a t i o n s are not required. A mathematical formulation of th i s approach may be written as follows: Max f (c 1 x) ) subject to ) ( 4 - 2 ) Pr {Ax _< b} >^  a; x >^  0 ) where, Pr = the p r o b a b i l i t y operator; a = a vector of pro b a b i l i t y measures. In general, the decision variables, x, are selected by a rule depending on the stochastic variables> A, b, and c; thus x = D(A, b, c ) . The form of D i s often s p e c i f i e d in advance, for example, to be l i n e a r . The functional, f ( c ' x), i s usually i n the form of an expected monetary value, thus 30 Max E (c' x), where E symbolizes expected value of a function. Although t h i s technique appears to be simple, i t s t i l l remains a question of fact to j u s t i f y a l l oversimplifying assumptions and procedures i m p l i c i t in the approach. The exclusion of basic technological functions, such as crop response functions and loss functions indicates that the v a l i d i t y of t h i s approach may not stand the test of time, as computational techniques improve, as agronomists provide the necessary technological functions, and as water control and hydrologic engineering become a more sophisticated process. Another quadratic programming approach to stochastic optimization problems i s concerned with deriving the d i s t r i b u -tion function of the objective function. Most research done in t h i s aspect has been more on the empirical estimation of this d i s t r i b u t i o n function and hence provides l i t t l e contribu-t i o n in p r a c t i c a l i r r i g a t i o n decision-making. 4.2 Optimization of Expected Values The basic d i f f i c u l t y of optimizing expected values i s seen- to rest i n the role of r i s k . Despite various attempts to account for r i s k through increased interest rates, large penalty for going broke, or similar a r b i t r a r y methods, r i s k and return are two non-commensurate components of the objective function. Where r i s k i s either quite small or the consequences of r i s k are n e g l i g i b l e , i t can be neglected and the expected 31 value can be o p t i m i z e d without s e r i o u s problems. Where r i s k i s not n e g l i g i b l e , c o n s i d e r a b l e c a u t i o n must be e x e r c i s e d and the t r u e r i s k and i t s consequences e v a l u a t e d independently f o r the supposed o p t i m a l p o l i c y . One consequence of r i s k o f t e n overlooked i n a n a l y s i s i s t h a t i n many cases once a f a i l u r e o c c u r s , a l l o f the formu-l a t i o n s f o r the o b j e c t i v e f u n c t i o n f o r a l l f u t u r e time are i n v a l i d a t e d . For example, a drought which r e s u l t s i n the l o s s of p e r e n n i a l crops, such as deciduous or c i t r u s f r u i t s and nut t r e e s , w i l l o f t e n cause the l o s s o f the investment and s e v e r a l y e a r s ' growth as w e l l . In some aspects of water resources design, f a i l u r e i s tantamount to t e r m i n a t i o n o f the system. A l l o f these events, should they a c t u a l l y occur, i n v a l i d a t e the o b j e c t i v e f u n c t i o n s used i n the c o n v e n t i o n a l programming a n a l y s i s employed to s o l v e s t o c h a s t i c o p t i m i z a t i o n problems. For these reasons, water resources systems i n g e n e r a l , and most i r r i g a t i o n systems i n p a r t i c u l a r , embody s u b s t a n t i a l elements of gambler's r u i n , hence r i s k should not be ign o r e d i n the i l l u s i o n t h a t i t w i l l disappear, nor should i t be conce a l e d i n an a r b i t r a r i l y chosen i n t e r e s t r a t e o r l a r g e p e n a l t y , however l a r g e . The e n t i r e o p e r a t i o n of systems a n a l y s i s i s aimed at producing a whole s e r i e s o f a l t e r n a t i v e s which can be ranked i n accordance with a g i v e n c r i t e r i o n , and i n which the bene-f i c i a l and/or d e t r i m e n t a l e f f e c t s of each i s c l e a r l y d e f i n e d . 32 In most of the l i t e r a t u r e c i t e d the c r i t e r i a l function for the alternatives being considered i s usually the present value of the monetary net benefits for deterministic optimization models, or the expected monetary value for stochastic models. When confronted with the task of making a decision, that i s choosing between d i f f e r e n t alternative actions i n an uncertain environ-ment, the decision maker resorts to gambling (theory of games) with respect to the d i f f e r e n t l i k e l y states of nature that could occur within the decision period. The l i k e l i h o o d of occurrence, P (9i) , of these d i f f e r e n t states of nature, 9 i , may be equal or d i f f e r e n t . In these exi s t i n g situations of uncertainty the concepts of pr o b a b i l i t y are v i t a l i n a r r i v i n g at a reasonable decision. It becomes a question of simple r a t i o n a l thinking to argue without much j u s t i f i c a t i o n that choosing the action or strategy with the maximum expected value appears to be a reasonable c r i t e r i o n of choice. The major problem that might arise i s to i d e n t i f y whether maximizing expected monetary value,[EMV],or maximizing some other c r i t e r i a l function, such as expected u t i l i t y , [ E U ] , gives a better choice c r i t e r i o n . In as much as there are classes of cases i n which maximizing expected monetary value i s i d e n t i c a l with maximizing expected u t i l i t y there are other classes of cases in which monetary values do not measure accurately the r e a l consequences of the gains or losses involved i n the decision problem. A t y p i c a l case i s one where 33 t h e r e are two a l t e r n a t i v e a c t i o n s , A i , and two s t a t e s of nature, 0 i , w i t h e q u a l l y l i k e l y p o s s i b i l i t i e s of occurrence, t h a t i s , each has a p r o b a b i l i t y of occurrence of h a l f d u r i n g the d e c i s i o n p e r i o d . I f the a l t e r n a t i v e a c t i o n ,A^ ., has monetary val u e s of 0(0^) and OO^) and a l t e r n a t i v e a c t i o n , A 2 has monetary v a l u e s Afe^) and -A(Q^) then the expected monetary value would be EMV (A^) = 0; EMV (A 2) = 0, and a d e c i s i o n maker would be i n d i f f e r e n t between a c t i o n s A^ and A^ when expected monetary ga i n (or l o s s ) i s used as a choice c r i t e r i o n . C l e a r l y most d e c i s i o n makers would not be i n d i f f e r e n t p a r t i c u l a r l y i f A's g a i n or l o s s were l a r g e i n r e l a t i o n to t h e i r c u r r e n t wealth and i f t h i s choice were a non-repeated d e c i s i o n . Assuming t h a t i n general, management d e c i s i o n makers have a r e a l a v e r s i o n t o u n c e r t a i n t y , a s u p e r i o r c r i t e r i o n f o r d e c i s i o n making under u n c e r t a i n t y u s i n g expected va l u e s has to be e s t a b l i s h e d . I t i s necessary, t h e r e f o r e , to d e r i v e a f u n c t i o n which can adequately t r a n s l a t e monetary values to another s c a l e which a c c u r a t e l y measures the s u b j e c t i v e value t h a t a d e c i s i o n maker attaches to d i f f e r e n t l e v e l s of wealth. T h i s s u b j e c t i v e s c a l e i s d e f i n e d as a u t i l i t y s c a l e and the f u n c t i o n i s a u t i l i t y f u n c t i o n . S o l v i n g a d e c i s i o n - u n d e r - r i s k problem by u s i n g the c r i t e r i o n of maximizing expected u t i l i t y has been shown to p rovide d e c i s i o n s which are c o n s i s t e n t w i t h the d e c i s i o n 34 maker's p r e f e r e n c e s among r i s k y a l t e r n a t i v e a c t i o n s . Because of t h i s c o n s i s t e n c y , u t i l i t y a n a l y s i s seems to be a s u p e r i o r way of accounting f o r r i s k as a parameter i n a d e c i s i o n problem, O f f i c e r and H a l t e r [84]. The u t i l i t y f u n c t i o n reduces the dimensions of p r e f e r e n c e s f o r r i s k y s i t u a t i o n s t o a s i n g l e dimension, thus l e s s e n i n g the ambiguity and s i m p l i f y i n g the a n a l y s i s o f the d e c i s i o n problem. The use of u t i l i t y a n a l y s i s as a d e s c r i p t i v e a i d to d e c i s i o n making i s g a i n i n g wide acceptance by d i f f e r e n t experts i n v o l v e d i n d e c i s i o n making r e s e a r c h . O f f i c e r and H a l t e r [84] present a comprehensive review of d i f f e r e n t t h e o r e t i c a l and p r a c t i c a l models of u t i l i t y e s t i m a t i o n . 4.3 D e c i s i o n Theory Approach to S t o c h a s t i c  O p t i m i z a t i o n Problems The theory of t h i s s t o c h a s t i c o p t i m i z a t i o n model r e s t s on the premises t h a t the problem of i r r i g a t i o n systems manage-ment i s to be viewed as one of d e c i s i o n making under u n c e r t a i n t y , and t h a t the c r i t e r i a l f u n c t i o n t o maximize i s the t o t a l expected u t i l i t y of the o b j e c t i v e f u n c t i o n . The major sources of u n c e r t a i n t y , and hence r i s k i n the management of i r r i g a t i o n systems emanate from the f o l l o w i n g : a) V a r i a b i l i t y i n h y d r o l o g i c events, both i n time and space, d u r i n g the growing p e r i o d . b) Crops v a r y i n g widely i n the e x p e c t a t i o n and v a r i a n c e of net income. The major v a r i a t i o n s i n net income stem 3 5 from variations in crop y i e l d s , and in product and factor p r i c e s . c) The unpredictable impact of i r r i g a t i o n return flow on the water q u a l i t y of the receiving stream.' Under the e x i s t i n g uncertainties, Bayesian decision making theory i s a handy tool which could form the basis of a meaningful optimization model. This model would then guide the choice between strategic options which w i l l be d i c -tated by the d i f f e r e n t states of nature. S t a t i s t i c a l decision theory stipulates that the decision maker choose that action which maximizes his expected u t i l i t y , which i s defined as the sum of the u t i l i t y of each possible outcome mult i p l i e d by i t s p r o b a b i l i t y of occurrence. Bayesian decision theory has the advantage of per-mitting the use of the decision maker's subjective p r o b a b i l i t i e s . Also-Bayesian approach incorporates some elements of f l e x i b i l i t y in that Bayes' theorem can be used to update the p r o b a b i l i t y d i s t r i b u t i o n as further information becomes available. Thus, the analyst can always improve his estimates in the future. This i s p r e c i s e l y the s i t u a t i o n encountered i n the development and operation of i r r i g a t i o n systems, where with time additional information about climatic conditions, increased pressures on limited water supply as a r e s u l t of increased and competing demand/or change i n policy i s continually coming to l i g h t . 36 4.4 Decision Theory Stochastic Optimization  Model Development In' the model the area under i r r i g a t i o n i s not con-sidered as an exogeneous variable. The a n a l y t i c a l solution requires that the in d i v i d u a l factors be expressed as functions of one variable, namely the area of the i r r i g a t i o n system whose optimum acreage we are seeking. The net value of a given crop, j , i s expressed by the formula: where by: n. = TR. - TVC. (4-3) 3 3 3 IIj = net value of unit area of crop j ; TRj = t o t a l revenue; TVC. = t o t a l variable cost, attributable to ^ crop j ; For any action, A , the monetary value i s given M(A..) = T RA. " T V C A . ( 4 " 4 ) Hi 3 3 and the expected monetary value [EMV] i s given by: n where [E(M)], = EMV =• E M(A..) . p(0.) (4-5) A. . , T l 1 3 1=1 J M(A..) = monetary value of action A. when -,1 state of nature, 0. i s -1 being considered; A. = area of crop, j , (hectares) under -1 i r r i g a t i o n ; p(0.) = pr o b a b i l i t y of occurrence of state 1 of nature, 9^; 37 n = number of hydrological states of nature being considered i n the decision problem, i . e . number of p r o b a b i l i t y i n t e r v a l s . [E(M)]A. m EMV = the expected monetary value of -1 action A. summed over a l l the states of nature, n. 4.5 Objective Function The objective of maximizing aggregate consumption benefits i s rarely unchallenged i n i r r i g a t i o n project planning and development although i t i s normally employed because of i t s r e l a t i v e convenience. I t makes no mathematical difference whether other objectives are included i n the objective function with appropriate weights or as constraints. A c o n f l i c t i n g objective i s s e l f - s u f f i c i e n c y i n food grain production which may lead to production of fewer high value crops for sale on world markets but which may also r e l i e v e a dependency on food aid subject to serious p o l i t i c a l s trings. Another objective i s income r e d i s t r i b u t i o n , either by regions or by classes. Regional income d i s t r i b u t i o n must be analyzed on a scale larger than an i r r i g a t i o n project, but class d i s t r i b u t i o n of income i s a phenomenon profoundly affected by water control schemes. ' In situations where there are c o n f l i c t s between these d i f f e r e n t objectives, the decision theory analysis becomes a more desirable approach under uncertainty, and the objective function i s stipulated as maximization of expected u t i l i t y of 38 a c t i o n A.., j = 1 to N, over a l l the s t a t e s of nature, 9^, i = 1 to n. The maximization of expected u t i l i t y i s dependent p r i m a r i l y on f i n d i n g a s u i t a b l e u t i l i t y f u n c t i o n . U s u a l l y u t i l i t i e s can be expressed as f u n c t i o n s of monetary g a i n s . Thus, i f U(A..) i s the u t i l i t y of a c t i o n A. when s t a t e of nature 0^ i s being c o n s i d e r e d then: U ( A j ± ) =• f[M(A j ;.)] (4-6) The expected u t i l i t y , EU, i s given by: n [E(U)] A. = EU = E U(A..) . p(0.) (4-7 3 i = l • 3 1 1 The o b j e c t i v e f u n c t i o n i s expressed thus: F = MAX [E(U) ] A. (4-8) J N n i . e . F = MAX E E U(A..) . p(9.) (4-9) j = l i = l D 1 1 where, j = 1 to N, a c t i o n s or d e c i s i o n a l t e r n a t i v e s i = 1 t o n, h y d r o l o g i c a l s t a t e s of nature. The procedure of a n a l y s i s l i e s i n comparing the expected u t i l i t i e s [E(U)]A_., f o r d i f f e r e n t v a r i a b l e areas and choosing t h a t which i s most f a v o r a b l e . The most f a v o r a b l e v a r i a n t (area) i s the extreme value of [E(U)]A... The r e s u l t of the a n a l y t i c a l s o l u t i o n l i e s i n dete r m i n i n g the extreme or maxima of the equation by s e t t i n g the d e r i v a t i v e of equation (4 - 7) equal to zero: [E(U) ]A_. = f (A..) 5 i . e . d{[E(U)]A.} d A . 7 D from which the optimum value of A., may be determined. 4.6 Assumptions The d e r i v a t i o n of the u t i l i t y f u n c t i o n and the subsequent a n a l y s i s w i l l be based on the f o l l o w i n g assumptions: 1) M o n o c u l t u r a l i r r i g a t i o n f o r a given crop, j , w i t h a given crop response f u n c t i o n . One crop harvested per i r r i g a t i o n season. 2) Consumptive use, C_. , of crop j i s known, and crop uses water a t a constant r a t e throughout the growing p e r i o d of l e n g t h , T_. . 3) I r r i g a t i o n water i s used as e f f i c i e n t l y as i s t e c h n o l o g i c 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 . 4) A uniform p r i c e f o r water i s envisaged and the r e c e i p t s f o r the t o t a l amount of water s u p p l i e d f o r i r r i g a t i o n s are assigned to the water management system. 5) The storage r e s e r v o i r c a p a c i t y i s c o n s i d e r e d exogenous and s u f f i c i e n t to impound any amount r e q u i r e d to s a t i s f y crop needs. 6) R e s e r v o i r i s a s i n g l e purpose type, being used to impound water f o r i r r i g a t i o n purposes o n l y . 7) Only n a t u r a l r u n o f f s u p p l i e s water to the r e s e r v o i r . (4-10) (4-11) 40 8) Water which has not been a p p l i e d to land and i s s p i l l e d has no v a l u e . 9) I r r i g a b l e land i s not a l i m i t i n g f a c t o r . 10) The i r r i g a t i o n system management d e c i s i o n s are s o l e l y a t the i r r i g a t i o n e n t e r p r i s e manager's d i s c r e t i o n . 4.7 C o n s t r a i n t s (1) Water consumption c o n s t r a i n t The t o t a l amount of water used to i r r i g a t e an area, A j , should not exceed the t o t a l amount of i n f l o w d u r i n g the growing p e r i o d , T.. . (a) I n s t i t u t i o n a l c o n s t r a i n t The q u a n t i t y and q u a l i t y of water r e l e a s e d f o r down-stream purposes (e.g. f i s h e r i e s ) should be enough to meet r e g u l a t o r y agency's s p e c i f i e d standards. 4.8 D e r i v a t i o n of U t i l i t y F u n c t i o n The maximization of expected u t i l i t y i s dependent p r i m a r i l y on f i n d i n g a s u i t a b l e u t i l i t y f u n c t i o n . H a l t e r [ 5 1 ] has presented the b a s i c t h e o r e t i c a l framework f o r Bayesian d e c i s i o n making p a t t e r n e d a f t e r Chernoff and Moses [26]. Dean [30] has p r o v i d e d a number of e m p i r i c a l a p p l i c a t i o n s of the framework to problems o f optimum s t o c k i n g r a t e s on C a l i f o r n i a range lan d , and a l s o p o i n t e d out some o f the problems encountered i n a p p l y i n g the theory. The d e r i v a t i o n of u t i l i t y f u n c t i o n s 41 seems to be one of the greatest hinderances to decision making research. These functions are of paramount importance i f the suggestions of Tedford [ 1 0 7 ] for solving decision problems under uncertainty and r i s k are to be noticed by actual decision makers. Developments i n Bayesian decision making have recently given new impetus to u t i l i t y measurement, Hildreth [54] • The successful derivation of u t i l i t y functions in a p r a c t i c a l setting permits Bayesian decision-making concepts to be applied i n t h e i r e n t i r e t y to a l l decision problems under risk, O f f i c e r and Halter [84]. It has been stated e a r l i e r that u t i l i t i e s can be adequately expressed as functions of monetary gains or losses. If i t i s to be assumed that monetary values adequately r e f l e c t payoffs, then a l i n e a r u t i l i t y function for money i s obtained. n In t h i s case the term E U (A..) i n equation (4-9) can be n i=l D 1 replaced by E M (A..), and thus equation (4-7) becomes the i=l J X same as equation (4-5) and the objective function simply becomes : N n F = MAX E E M (A..) . p(9.) (4-12) j=l i =l 3 1 1 If the assumption that monetary values adequately r e f l e c t payoffs i s relaxed, then the objective function to maximize i s as given i n equation (4-9) and the form of the function, U(A j ±) = f [M(A j d)] has to be established. Thus, the function to be derived has 42 to be able to r e l a t e monetary values to another s c a l e which p r o v i d e s o b j e c t i v e a n a l y s i s of s u b j e c t i v e c o n s i d e r a t i o n s , t h a t i s a s c a l e which measures the s u b j e c t i v e value t h a t a d e c i s i o n maker attaches to d i f f e r e n t l e v e l s of wealth. There-f o r e , the u t i l i t y s c a l e i s intended to express the i r r i g a t i o n manager's s a t i s f a c t i o n ( u t i l i t y ) , o f which monetary val u e s are j u s t an important component. I t i s assumed t h a t the i r r i g a t i o n manager has the r i g h t to make a l l d e c i s i o n s concerning the i r r i g a t i o n system management. The attempt, t h e r e f o r e , i s to convert the d e c i s i o n maker's p r e f e r e n c e s among r i s k y p r o spects i n t o u t i l i t y numbers and thus a u t i l i t y f u n c t i o n f o r monetary gains or l o s s e s can be d e r i v e d . Given the i r r i g a t i o n manager's u t i l i t y f u n c t i o n , h i s d e c i s i o n problem can be s o l v e d by maxi-mi z i n g expected u t i l i t y . Thus, expected u t i l i t y i s an index t h a t any d e c i s i o n maker w i l l maximize i f he i s to remain c o n s i s t e n t with h i s p r e f e r e n c e s among r i s k y p r o s p e c t s . Having d e r i v e d the u t i l i t y f u n c t i o n f o r a d e c i s i o n maker, he can r e t a i n h i s p r e f e r e n c e s f o r r i s k y p r o spects i n more complex d e c i s i o n problems by u s i n g h i s u t i l i t y f u n c t i o n to s o l v e them, t h a t i s , by maximizing expected u t i l i t y . O f f i c e r and H a l t e r [84] p r o v i d e a comprehensive a n a l y s i s of u t i l i t y f u n c t i o n d e r i v a t i o n f o r a d e c i s i o n maker. The von Neumann-Morgenstern (N-M) model has been adopted i n t h i s study [84,114]. To account f o r s u b j e c t i v e f e e l i n g s toward r i s k 43 i n a d e c i s i o n problem the monetary outcomes are converted to u t i l i t y numbers. From the computed monetary pay o f f t a b l e the most f a v o r a b l e outcome to the d e c i s i o n maker i s s e l e c t e d . A l s o the l e a s t f a v o r a b l e outcome i s s e l e c t e d . I t i s e s s e n t i a l f o r the maximum and minimum u t i l i t y values to be u n i t y and zero on t h i s s c a l e (or 100 and z e r o ) ^ r e s p e c t i v e l y . By so doing an o r i g i n and a s c a l e f o r measuring u t i l i t i e s have been a r b i t r a r i l y s e l e c t e d . For example U[M(A..)] = 100 ) J X ) (4-13) U[-M(Aj i)] 0 ) I t i s then p o s s i b l e t o c a l c u l a t e the weighted average u t i l i t y f o r any r e f e r e n c e c o n t r a c t having p r o b a b i l i t y n of winning M(A^) [and 1 - n o f l o s i n g M(A_.^) ] . The u t i l i t y i s equal t o the u t i l i t y of the " c e r t a i n cash" amount because the " c e r t a i n cash" amount i s e q u i v a l e n t t o t h a t r e f e r e n c e c o n t r a c t . For any monetary value , say +$50,000 between +M (A..^) and -M t n e u t i l i t y i s computed from the r e l a t i o n s h i p : U($50,000) = U[+$M(A j : L)] n + U[-$M(A j i)] (l-Il) (4-14) where -$M(Aj i) < $50,000 < +$M(A^i) , and n is. o b tained from a choice t a b l e f o r f i n d i n g i n d i f f e r e n c e p o i n t s between c e r t a i n cash and v a r i o u s r e f e r e n c e c o n t r a c t s . H a l t e r and Dean [5 3], p r ovide an example of such a choice t a b l e f o r a given d e c i s i o n maker. The u t i l i t y f u n c t i o n i s then obtained by p l o t t i n g these u t i l i t y numbers as a f u n c t i o n of monetary gains or l o s s e s . Since the monetary value i s d i r e c t l y r e l a t e d to the 44 area, A.., the u t i l i t i e s can be expressed d i r e c t l y as f u n c t i o n s of the areas (actions) under i r r i g a t i o n , v i z : U ( A j ± ) = f (A..) (4-15) 4.9 H y d r o l o g i c a l U n c e r t a i n t y Element The emphasis, i n the course of the development of the s t o c h a s t i c o p t i m i z a t i o n model w i l l be on the h y d r o l o g i c a l u n c e r t a i n t y . Since i t i s assumed t h a t the i n f l o w i n t o the i r r i g a t i o n storage r e s e r v o i r i s from n a t u r a l r u n o f f o n l y , the v a r i a b i l i t y i n h y d r o l o g i c event w i l l be the dominant f a c t o r c o n t r o l l i n g the p r o d u c t i o n i n i r r i g a t e d a g r i c u l t u r e . Since optimum ch o i c e under h y d r o l o g i c a l u n c e r t a i n t y i s d e s i r e d , the u t i l i t y v a l u e s have to be t r a n s l a t e d i n t o expected values o f u t i l i t y c o n s i s t e n t w i t h the designated c h o i c e c r i t e r i o n . Thus a p r o b a b i l i t y d i s t r i b u t i o n or p r o b a b i l i t y d e n s i t y f u n c t i o n of the predominant i n f l o w u n c e r t a i n t y element i s r e q u i r e d . The a n a l y s i s r e q u i r e s t h a t a p r o b a b i l i t y d i s t r i b u t i o n of i n f l o w s i n t o the i r r i g a t i o n storage r e s e r v o i r be estimated. I t i s , t h e r e f o r e , necessary to d e f i n e v a r i o u s s t a t e s of nature with r e s p e c t to hydrology. Seven h y d r o l o g i c a l s t a t e s of nature are d e s i g n a t e d as f o l l o w s : 0^ r e p r e s e n t s very poor h y d r o l o g i c a l s t a t e © 2 r e p r e s e n t s poor h y d r o l o g i c a l s t a t e 0^ r e p r e s e n t s f a i r h y d r o l o g i c a l s t a t e 0 d r e s p r e s e n t s normal h y d r o l o g i c a l s t a t e 0^ represents good hydrological state represents very good hydrological state 0^  represents excellent hydrological state Estimates of p r i o r p r o b a b i l i t i e s , p(0^), can be obtained from available h i s t o r i c a l hydrological records by carrying out a frequency analysis of long-term recorded flows. From t h i s analysis the p r o b a b i l i t y of d i f f e r e n t magnitudes of inflow can be derived. 4.10 Optimum Choice C r i t e r i o n : Maximum Expected U t i l i t y From equation (4-7) the t o t a l expected u t i l i t i e s are computed for the d i f f e r e n t decision alternatives (areas under i r r i g a t i o n ) by multiplying the i n d i v i d u a l u t i l i t i e s U(A_.^) by the p r o b a b i l i t i e s p(9^) and summing over the seven hydrological states of nature (uncertainty element). The optimum area to i r r i g a t e at any given time w i l l be that area which has the highest expected u t i l i t y , that i s that area for which d { [E (U) ] A. } J = o d A. U-3 4.11 Crop Response Functions Different crops exhibit d i f f e r e n t responses when supplied with varying amounts of water, a l l other production inputs being constant. It i s not an easy task to establish 46 these crop response f u n c t i o n s f o r the i n d i v i d u a l p r o d u c t i o n i n p u t s s i n c e the e f f e c t s of each i n p u t on crop y i e l d s are i n t e r r e l a t e d and interdependent. Thus the p r e c i s e e stablishment o f these t e c h n o l o g i c a l f u n c t i o n s r e q u i r e s exhaustive techniques and i s a l s o time-consuming. This e x p l a i n s the reason why. any a p p r e c i -able e f f o r t has not been made by agronomists and i r r i g a t i o n engineers i n t h i s r e s p e c t and consequently, why these f u n c t i o n s are not e a s i l y a v a i l a b l e i n l i t e r a t u r e . In t h i s study the a n a l y s i s has proceeded with assumptions about crop response f u n c t i o n s , b e a r i n g i n mind the g e n e r a l l y accepted trends by experts [115, 116, 125, 129]. Because of l a c k of adequate i n f o r m a t i o n on the p r e c i s e r e l a t i o n s h i p s between water consumption and crop y i e l d two broad procedures were adopted and assumptions made as f o l l o w s : 4.11.1 Procedure I In t h i s procedure i t i s assumed t h a t no crop can be produced without adequate i r r i g a t i o n water supply. The crop, j , uses water a t a constant r a t e (average weekly consumptive use rate) throughout the growing season o f le n g t h , T^. Crop d i e s i f not watered a l l season a t t h i s r a t e . T h i s i m p l i e s t h a t the area under i r r i g a t i o n must be kept at such a l e v e l t h a t any t o t a l seasonal flow, Q T , w i l l adequately i r r i g a t e the area at j constant r a t e . I f Qj i s the q u a n t i t y of water needed to adequately i r r i g a t e an area, A\ , throughout the growing season T_. , t h a t 47 i s Q_. i s the q u a n t i t y of water r e q u i r e d to keep the moisture l e v e l i n the p l a n t - s o i l regime a t , o r near f i e l d c a p a c i t y , then Q. = A. C. T. (4-16) 3 3 3 3 n=m Q T = E q(9 i) (4-17) m=l where q(©-) = monthly i n f l o w i n t o the storage r e s e r v o i r . m = number of months i n the growing season. I f Q T . > Q . f U(Q T ) = U(A Y ), Y = max. j j 3 3 3 where U = symbolizes u t i l i t y , and Yj = y i e l d of crop, j . I f Q T <"Q., U(Q ) < U(A Y ), and Y = 0. j j 3 3 3 Thus i n t h i s procedure the crop response f u n c t i o n i s assumed to be a s t e p - f u n c t i o n . A v a r i a b l e area schedule depending on the h y d r o l o g i c a l s t a t e of nature obtained a f t e r p l a n t i n g an i n i t i a l area, A , i s proposed. That i s , the area A i s to be o o cut back to A. i n such a manner t h a t : 3 Q T =• Q- = A. C. T,. (4-18) 4.11.2 Procedure II In t h i s procedure the o r i g i n a l a r ea,-A , p l a n t e d under u n c e r t a i n t y , i s not to be cut back to optimal area, A_. , i n the event t h a t < . Instead the seeded area, A q , i s to be u n i f o r m l y i r r i g a t e d r e g a r d l e s s of the supply, Q T . In j 48 t h i s case the s t o c h a s t i c maximization procedure r e c o g n i z e s r i s k by a s s i g n i n g a p e n a l t y f u n c t i o n to the s i t u a t i o n , Q T < Q. by s a c r i f i c i n g o p t i m a l y i e l d f o r m a i n t a i n i n g the seeded area, A Q , a t a moisture l e v e l below t h a t r e q u i r e d f o r op t i m a l output, t h a t i s below the f i e l d c a p a c i t y or the designated consumptive use l e v e l . In t h i s case t y p i c a l crop response f u n c t i o n s are assumed. E s t a b l i s h e d p r i n c i p l e s of resource use have been borrowed to develop both s t i m u l a t o r y and non-s t i m u l a t o r y r e l a t i o n s between resource a v a i l a b i l i t y and resource u t i l i z a t i o n [115, 116]. Since inadequate water does not reduce p r o d u c t i o n l i n e a r l y , but r a t h e r reduces i t n o n - l i n e a r l y , non-convexly, and at times d i s c o n t i n u o u s l y and, s i n c e expected values are used l i n e a r p o s t u l a t e s would be inadequate [104, 125, 129']. I f l e s s water i s a v a i l a b l e , t h a t i s < Q., the y i e l d i s presumed to f a l l o f f , e v e n t u a l l y r e a c h i n g zero. I f more water i s a p p l i e d than the consumptive use requirement the y i e l d i s assumed to h o l d at the maximum y i e l d . Since the crop response f u n c t i o n i s not p r e c i s e l y known, i t i s not p o s s i b l e to know the consumptive use t h a t g i v e s the maximum yield,. Thus a r a t i o , R, of water a p p l i e d to water r e q u i r e d has been adopted, and a y i e l d c o e f f i c i e n t i n s t e a d o f r e a l y i e l d i s a l s o used i n the a n a l y s i s , and computer programming. 4.12 S e n s i t i v i t y A n a l y s i s The obvious advantage of p r e s e n t i n g i r r i g a t i o n 49 management d e c i s i o n problems i n the Bayes' d e c i s i o n theory framework i s t h a t i t f o r c e s the i r r i g a t i o n o p e r a t o r to focus on the aspects of r i s k i n h e r e n t i n the problem. In the past recommendations were based on s i n g l e valued estimates which do not a l l o w the i r r i g a t i o n d e c i s i o n maker any f l e x i b i l i t y to a d j u s t to the r i s k i n the s i t u a t i o n . From the d e c i s i o n theory s t o c h a s t i c maximization model presented i t should be noted t h a t a p a r t i c u l a r choice or s e t of c h o i c e s w i l l be optimum f o r some p r o b a b i l i t y d i s t r i b u t i o n over the s t a t e of nature. This p a r t l y e x p l a i n s the reason why d i f f e r e n t i r r i g a t i o n o p e r a t o r s employ d i f f e r e n t s t r a t e g i e s i n a c t u a l r e a l — l i f e s i t u a t i o n s w i t h r e s p e c t to t h e i r i r r i g a t i o n systems expansion and o p e r a t i n g p o l i c i e s under the same p r e v a i l -i n g c o n d i t i o n s of u n c e r t a i n t y - d i f f e r e n t s u b j e c t i v e judgements l e a d i n g to d i f f e r e n t p r i o r p r o b a b i l i t y d i s t r i b u t i o n s over the same h y d r o l o g i c a l s t a t e s of nature ( u n c e r t a i n t y ) . A l s o t h i s approach to i r r i g a t i o n d e c i s i o n problems i s s e n s i t i v e to each i r r i g a t i o n o p e r a t o r ' s u t i l i t y f u n c t i o n , and the crop response f u n c t i o n assumed. Thus, a s e n s i t i v i t y a n a l y s i s w i t h r e s p e c t to the above mentioned s u b j e c t i v e estimates of p r o b a b i l i t i e s , u t i l i t y f u n c t i o n s , and crop response f u n c t i o n s w i l l e n r i c h t h i s type of s u b j e c t i v e i r r i g a t i o n systems management decision-making. A l s o , the economic and p h y s i c a l impacts of i m p l i c i t assumptions made i n the procedures, as w e l l as the s e n s i t i v i t y of the o b j e c t i v e 50 f u n c t i o n to changed assumptions and p o l i c y w i l l be r e f l e c t e d i n the s e n s i t i v i t y a n a l y s i s . T h i s a n a l y s i s w i l l i n d i c a t e the range of valu e s of the d i f f e r e n t f a c t o r s w i t h i n which the optimum choice pr s t r a t e g y s t i l l remains optimal.- Furthermore, s e n s i t i v i t y a n a l y s i s tends to smooth out i n c i d e n t a l e r r o r s i n s u b j e c t i v e assessments made by the i r r i g a t i o n management d e c i s i o n makers. 4.13 Bayesian D e c i s i o n S t r a t e g y The aim of the systems approach to water resource systems problems i s to f i n d s o l u t i o n s which are o p t i m a l l y competent w i t h i n a c e r t a i n time p e r i o d f o r a s p e c i f i c r e g i o n of i n t e r e s t . Most c o n v e n t i o n a l systems techniques f o r h a n d l i n g s t o c h a s t i c o p t i m i z a t i o n problems are v a l i d only f o r the s p e c i f i e d time p e r i o d s . Under u n c e r t a i n t y d e c i s i o n theory i s most s u i t e d f o r s o l v i n g s t o c h a s t i c o p t i m i z a t i o n problems because i t draws a formal d i s t i n c t i o n between d e c i s i o n s and outcomes by c o n s i d e r i n g the u n c e r t a i n t i e s i n h e r e n t i n the d e c i s i o n problem. The whole process takes cognizance of the value of a d d i t i o n a l i n f o r m a t i o n , making be s t use of l i m i t e d data while i n d i c a t i n g the areas of g r e a t e s t u n c e r t a i n t y , areas i n which the need f o r a d d i t i o n a l i n f o r m a t i o n i s most p r e s s i n g . A l s o the a s s o c i a t e d s e n s i t i v i t y a n a l y s i s w i l l improve the r e l i a b i l i t y of the outcomes. Moving backward i n time, the d e c i s i o n maker can e v a l u a t e the outcome of c e r t a i n o b s e r v a t i o n s , and i n so doing can update h i s informa-51 t i o n and modify h i s a c t i o n s so t h a t the choice which he a c t u a l l y makes i s most l i k e l y to be i n harmony with the r e a l i z e d event, or to be an improved d e c i s i o n . Thus f o r long-term o p t i m a l development and o p e r a t i o n of i r r i g a t i o n systems under u n c e r t a i n t y the above f e a t u r e of d e c i s i o n theory o p t i m i z a t i o n technique h i g h l i g h t s i t as the most v a l i d approach as t h i s would i n d i c a t e the o p t i m a l d e c i s i o n s t r a t e g i e s over time. In many cases, i n the optimum a l l o c a t i o n of a s t o c h a s t i c water supply f o r i r r i g a t i o n purposes, the q u a n t i t y of water which might be a v a i l a b l e from the s t o c h a s t i c supply can be estimated with a f a i r degree of accuracy a t or near the b e g i n n i n g of the growing season from snow surveys, r e s e r v o i r c o n d i t i o n s , or other h y d r o m e t e r e o l o g i c a l o b s e r v a t i o n s . Where t h i s i s t r u e , the s t o c h a s t i c (non-firm) supply can be u t i l i z e d w ith c o n s i d e r a b l e confidence f o r c e r t a i n crops. I f the h y d r o l o g i c a l s t a t e of nature c o u l d be pre-d i c t e d p r i o r to the d e c i s i o n with a h i g h degree of accuracy, d e c i s i o n s c o u l d then be a d j u s t e d to the c o n d i t i o n s expected and, over a long time p e r i o d , the value of gains ( u t i l i t y ) i n c r e a s e d . I f the i r r i g a t i o n management system c o u l d "spy" on the h y d r o l o g i c a l s t a t e of nature through some o b s e r v a t i o n , experiment, or other f o r e c a s t i n g d e v ice i t would be p o s s i b l e to estimate p o s t e r i o r p r o b a b i l i t i e s P(0^/Z) using Bayes 1 formula: 52 p(e i ) p(z/e±) p ( e i / z ) = p(e.) P ( z / e j + + p(e ) p ( z / e ) ( 4 _ 1 9 ) l l n n where • P(0 i / Z ) i s the posterior p r o b a b i l i t y d i s t r i b u t i o n of outcome 9., which i s a weighting of the p r i o r p r o b a b i l i t i e s P(9.) by the conditional p r o b a b i l i t i e s 1 P ( Z / 9 . ) ; that i s the likelihoods of acquiring the s p e c i f i c information , Z,given the possible values of the random variable , 0. . I Z represents a set of predictors, Z 9^ represents hydrological states of nature. n i s the number of hydrological states of nature being considered. The posterior probabilities, can then be used to derive the action that maximizes expected value, given the Z observed prior to K a p a r t i c u l a r decision. An optimum action can thus be derived for any observed value of Z . This set of actions are c a l l e d the "Bayesian Strategy" - a complete set of rules that would t e l l the i r r i g a t i o n management decision maker how to act i n response to any observed hydrological condition , 1V. The worth of the additional information or the measure of the usefulness of the hydrological predicting device employed i s the increase i n expected value from employing posterior (data problem) rather than p r i o r (no data problem) p r o b a b i l i t i e s . This gain i s c a l l e d the "value of the experiment". 4.14 Optimal Decision Strategies Over Time If the i r r i g a t i o n manager's agribusiness growth 53 c o u l d be simulated over a p e r i o d o f time comparisons among the two b a s i c i r r i g a t i o n development and o p e r a t i o n procedures, v i z : the schedule of c u t t i n g back an o r i g i n a l seeded area, A. under DO' u n c e r t a i n t y to an area, A. , such t h a t Q >_ Q . , t h a t i s D T T j 3 i r r i g a t i n g A., to maximum water-holding c a p a c i t y ( f i e l d c a p a c i t y ) of the s o i l and maximum consumptive use of the crop, t h a t i s maximum y i e l d ; and the schedule of m a i n t a i n i n g the area, A^, and i r r i g a t i n g the whole area below the consumptive use l e v e l of the crops, t h a t i s reduced y i e l d s , c o u l d be made from the r e s u l t s o f a s e r i e s of such s i m u l a t i o n s . As a b a s i s f o r com-pa r i n g the a l t e r n a t i v e s t r a t e g i e s growth i n net worth of the business a t the end of the p e r i o d would o r d i n a r i l y be s e l e c t e d , i f i t i s assumed t h a t f i n a n c i a l growth i s a g o a l . However s i n c e the d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model presen-ted has maximization of expected u t i l i t y as an o b j e c t i v e , the net worth must be converted t o u t i l i t y , u s i n g the u t i l i t y f u n c t i o n , f o r c o n s i s t e n c y . 54 CHAPTER 5 APPLICATION OF MODEL TO MEDIUM-SCALE IRRIGATION SYSTEM 5 .1 I n t r o d u c t i o n In many a r i d , and s e m i - a r i d regions o f the world i r r i g a t i o n i s necessary f o r year-round crop p r o d u c t i o n . Many farmers take the r e s p o n s i b i l i t y o f developing t h e i r own i r r i -g a t i o n systems and f i n a n c i n g these systems. These farmers u s u a l l y do not have the necessary funds to c o n t r a c t and guarantee t h e i r i r r i g a t i o n water s u p p l i e s with the v a r i o u s and h i g h l y o r g a n i z e d water companies. The water systems are not c e n t r a l i z e d e i t h e r , and thus each i r r i g a t o r designs and operates h i s own c o l l e c t i o n , s t o r a ge, and water d i s t r i b u t i o n system. The i r r i g a t o r normally goes i n f o r a low-cost system and g e n e r a l l y d i v e r t s h i s water from a nearby creek. Because of the high v a r i a b i l i t y i n the flow o f t h i s supply source he normally attempts to have storage f a c i l i t i e s f o r impounding water d u r i n g p e r i o d s o f high creek flow f o r use d u r i n g d r i e r and low-flow p e r i o d s when he i r r i g a t e s . Because of the f l u c t u a t i n g nature of h i s source o f supply the farmer; i s never c e r t a i n of the amount of water t h a t would be a v a i l a b l e f o r h i s o p e r a t i o n from year t o year, and consequently he takes some r i s k , each year, i n d e c i d i n g what hectarage to seed f o r i r r i g a t i o n at the beginning of the i r r i g a t i o n season. In re g i o n s where a l l water has been d e c l a r e d as the p r o p e r t y of the S t a t e , i n s t i t u t i o n a l c o n s t r a i n t s demand that^each i r r i g a t o r 55 o b t a i n s a permit t o draw water f o r a p p l i c a t i o n on h i s l a n d . In cases where there i s no w r i t t e n system of water r i g h t s law, the i r r i g a t o r u s u a l l y assumes t h a t he can use the water f l o w i n g i n h i s immediate v i c i n i t y as he wishes. However, i n both cases, the common problem i s how to decide on the s c a l e of i r r i g a t i o n p r o j e c t a t the s t a r t o f the scheme and, on an adequate development or expansion, and o p e r a t i o n a l p o l i c y f o r h i s system, b e a r i n g i n mind the heavy c o n s t r a i n i n g f a c t o r of h y d r o l o g i c a l u n c e r t a i n t y and the r i s k s i n v o l v e d i n t h i s type of a g r i b u s i n e s s . U s u a l l y i r r i g a t o r s c o n f r o n t e d with t h i s type of s i t u a t i o n tend t o minimize the r i s k s i n v o l v e d by assuming t h a t the worst c o n d i t i o n w i l l occur each year. So, they normally determine the maximum i r r i g a b l e hectarage e i t h e r u s i n g the "worst" observed h y d r o l o g i c a l s t a t e o r , the s o - c a l l e d normal or average c o n d i t i o n s . T h i s approach ensures them of a c e r t a i n guaranteed crop h a r v e s t each year. This s i t u a t i o n becomes more complicated where the farmer engages i n i r r i g a t i o n of pasture f o r l i v e s t o c k o p e r a t i o n . T h i s type of o r g a n i z a t i o n i s r a t h e r i n f l e x i b l e i n t h a t cow numbers cannot be changed e a s i l y from year to year i n response to h y d r o l o g i c a l c o n d i t i o n s and feed supply. The rancher's dilemma i s to decide on the s i z e of herd t o m a i n t a i n : i f the herd i s too s m a l l , feed w i l l be wasted i n many years; i f the herd i s too l a r g e , the feed supply i n many years w i l l be inadequate, and expensive supple-mental feeds must be purchased. r>6 Such i s the s i t u a t i o n t h a t a t y p i c a l rancher i n the Quilchena Creek area of N i c o l a V a l l e y i n the s e m i - a r i d 5 d r y i n t e r i o r o f B r i t i s h Columbia has to cope with i n the o p e r a t i o n of h i s ran c h i n g . The success o f the whole beef cow herd l i v e s t o c k o p e r a t i o n (cow-calf operation) i n t h i s area i s e n t i r e l y dependent on the a s t u t e management of the rancher's h a y - a l f a l f a i r r i g a t i o n system. The area i s c h a r a c t e r i z e d by high e v a p o t r a n s p i r a t i o n and lack of r a i n f a l l d u r i n g the summer growing season. A summary of the c l i m a t i c f e a t u r e s of the whole N i c o l a V a l l e y r e g i o n w i l l be given i n Chapter 6, under r e g i o n a l a p p l i c a t i o n of the model. The rancher's l i v e s t o c k o p e r a t i o n and expansion p o l i c y i s h e a v i l y c o n s t r a i n e d by u n c e r t a i n t y i n h y d r o l o g i c a l input f o r . h i s i r r i g a t i o n system which i s very v i t a l f o r the success o f the whole e n t e r p r i s e . At present v a s t areas of f o o t h i l l l a n d i s a v a i l a b l e f o r i r r i g a t i o n development and thus la n d i s not a l i m i t i n g f a c t o r i n expansion p o l i c y i n the area. The Quilchena ranch management i s t y p i c a l of the standard p r a c t i c e adopted by a l l ranchers i n the area. A t y p i c a l p a t t e r n of ranch o p e r a t i o n i s to feed the c a t t l e d u r i n g w i n t e r and t u r n them out t o graze on range be l o n g i n g to the rancher i n s p r i n g , on g r a z i n g land on the p l a t e a u l e a s e d from the government i n summer, and again to the rancher's own range i n f a l l . The main forage crop i s a l f a l f a with c l o v e r and oats s e r v i n g as cover crops, and the a l f a l f a needs to be seeded 57 every 4 t o 5 years; with adequate water supply 3 " c u t s " can be harvested per year. At present most ranchers h a r v e s t a maximum of 1 1 /2 -2 crops per year. Current y i e l d s are of the order of 2 . 0 - 4 . 0 tons per acre per c u t of a l f a l f a , with 2.5 tons per acre i n most y e a r s . To achieve optimal competence i n i t s cow-calf opera-t i o n the Quilchena ranch management has to improve i t s d e c i s i o n -making process. T h i s i m p l i e s t h a t the output i n i t s hay-a l f a l f a i r r i g a t i o n system has to become more e f f i c i e n t and more e f f e c t i v e . The d e c i s i o n model developed i n Chapter 4 i s hereby a p p l i e d t o the Quilchena ranch i r r i g a t i o n system management problem under h y d r o l o g i c a l u n c e r t a i n t y . 5.2 Model Inputs B a s i c a l l y the d e c i s i o n model i n p u t s i n f o r m a t i o n on i n f l o w d e r i v e d from a n a l y s i s of h i s t o r i c a l flow d a t a . T h i s g i v e s some i d e a of the v a r i a b i l i t y o f the creek's flow from year to year d u r i n g the growing p e r i o d . S t a t i s t i c a l a n a l y s i s was c a r r i e d out on the a v a i l a b l e h i s t o r i c a l flow r e c o r d t o estimate p r i o r p r o b a b i l i t i e s of d i f f e r e n t flows which d e f i n e the d i f f e r -ent h y d r o l o g i c a l s t a t e s of nature. The a c t i o n s , or the d e c i s i o n a l t e r n a t i v e s , are d e f i n e d as the d i f f e r e n t h e c t a r e s of range land t h a t can be i r r i g a t e d . The consumptive uses of the a l f a l f a crop f o r the a l t e r n a t i v e areas are computed, assuming c o n d i t i o n s o f c e r t a i n t y . A l s o assuming t h a t p r i c e s and y i e l d s are c e r t a i n and s t a b l e , and assuming a crop-response f u n c t i o n of a l f a l f a i n the area, the crop net valu e s 58 a s s o c i a t e d with each d e c i s i o n a l t e r n a t i v e are computed. The model then transforms these i n p u t s s e q u e n t i a l l y through a d e c i s i o n theory framework i n t o monetary p a y - o f f e n t r y f o r each s t a t e - a c t i o n combination. A s u i t a b l e u t i l i t y f u n c t i o n i s d e r i v e d ; the monetary p a y - o f f e n t r i e s are converted i n t o u t i l i t i e s , and expected u t i l i t i e s are then computed f o r each s t a t e - a c t i o n combination and entered i n a t a b l e i n the matrix f a s h i o n . These expected u t i l i t i e s are summed i n v e r t i c a l columns f o r each a c t i o n over a l l the seven rows d e f i n i n g the h y d r o l o g i c a l s t a t e s of nature and the a c t i o n with the maximum t o t a l expected u t i l i t y i s s e l e c t e d as the optimal p o l i c y . 5.2.1 Inflow a n a l y s i s In the Quilchena area, the source of water f o r i r r i g a t i n g the a l f a l f a crop on the Quilchena ranch i s the Quilchena creek. T h i s creek does not have any a p p r e c i a b l e l e n g t h o f continuous flow measurement. The a v a i l a b l e stream-flow r e c o r d f o r t h i s creek i s "spotty" The drainage area above the s t a t i o n where the creek's d i s c h a r g e measurement i s made i s 30 4 square m i l e s . Most o f the creek's flow d u r i n g the growing season (May to September i n c l u s i v e ) occurs d u r i n g the f r e s h e t p e r i o d , May and June. The creek's flow d u r i n g J u l y , August and September i s i n r e c e s s i o n stages. T h i s t r e n d i s e a s i l y n o t i c e a b l e from Table B . l i n Appendix B. Since there was no a p p r e c i a b l e l e n g t h of streamflow h i s t o r i c a l r e c o r d f o r s t a t i s t i c a l a n a l y s i s purposes, the a v a i l a b l e "spotty" data 59 has to be i n c r e a s e d . Therefore the 88 years of a v a i l a b l e r a i n f a l l data was used f o r t h i s purpose. Table B.2 g i v e s the summary of a v a i l a b l e r a i n f a l l data f o r the growing p e r i o d . To estimate a r a i n f a l l - r u n o f f r e l a t i o n s h i p f o r Quilchena creek sub-basin a number of days i n August and September 19 74 was s e l e c t e d , and the d i s c h a r g e hydrograph p l o t t e d as shown i n F i g u r e B . l . The t o t a l r a i n f a l l producing the r u n o f f f o r t h i s dry p e r i o d was obtained from Table B.2. Table B.3 g i v e s the creek's measured d a i l y d i s c h a r g e s f o r J u l y , August and September 1974. From the r a i n f a l l - r u n o f f a n a l y s i s i t was e s t a b l i s h e d t h a t the r u n o f f can be assumed to be zero, s i n c e the r u n o f f c o e f f i c i e n t i s 0.05. However to convert the r a i n f a l l i n t o flow data t h i s c o e f f i c i e n t was used to convert a l l the monthly r a i n f a l l data i n t o the corresponding flow data f o r the 88 years of r e c o r d as shown i n Table B.4. As mentioned e a r l i e r most of the Quilchena creek's streamflow, and hence the i r r i g a t i o n water supply comes from snowmelt r u n o f f d u r i n g the s p r i n g f r e s h e t p e r i o d , May-June. From Tables B.4 and B . l i t was roughly e s t a b l i s h e d t h a t the t o t a l Quilchena creek's flow f o r the growing p e r i o d , May to September, was about 4 times t h a t of the p o s s i b l e flow estimated from r a i n f a l l only f o r the same p e r i o d . Therefore a l l the t o t a l i n f l o w and average monthly i n f l o w estimated i n Table B.4 were m u l t i p l i e d by a f a c t o r of 4 to get the 88-year r e c o r d of i n f l o w shown i n Table B.5, and used f o r subsequent s t a t i s t i c a l a n a l y s i s . 60 5.2.2 H y d r o l o g i c a l s t a t e s o f nature and e s t i m a t i o n of  p r i o r p r o b a b i l i t i e s Based on the 88 years of flow data given i n Appendix B, Table B.5, a s t a t i s t i c a l a n a l y s i s was c a r r i e d out. The 3 3 t o t a l monthly flow data ranges from a minimum of 6,32 0 x 10 m 3 3 i n 1929 to a maximum of 40,000 x 10 m i n 1903. An average 3 3 t o t a l i n f l o w of 16,900 x 10 m was computed f o r the growing p e r i o d , May 1 to September 30, and the standard d e v i a t i o n was 3 3 ± 3,000 x 10 m f o r the growing p e r i o d . On the b a s i s of the range of i n f l o w data, the average t o t a l i n f l o w , and the standard d e v i a t i o n , the h y d r o l o g i c a l s t a t e s of nature were d e f i n e d as f o l l o w s : 3 3 0-^  : Very poor i n f l o w c o n d i t i o n ; i n f l o w l e s s than 9,000 x 10 m . 9 2 : Poor i n f l o w c o n d i t i o n ; 9,000 x 10 3 - 12,000 x 10 3 m3. 3 3 3 9 3 : F a i r i n f l o w c o n d i t i o n ; 12,000 x 10 - 15,000 x 10 m . 3 ' 3 3 e4 : Normal i n f l o w c o n d i t i o n ; 15,000 x 10 - 18,000 x 10 m . 0 C : Good i n f l o w c o n d i t i o n ; 18,000 x 10 3 - 21,000 x 10 3 m3. D 3 3 3 0 & : Very good i n f l o w c o n d i t i o n ; 21,000 x 10 - 24,000 x 10 m . 3 3 0_ : E x c e l l e n t i n f l o w c o n d i t i o n ; 24,000 x 10 m or over. The d i s t r i b u t i o n of i n f l o w c o n d i t i o n s was approximated by simply d i v i d i n g the o b s e r v a t i o n s i n t o c l a s s i n t e r v a l s , ( c e l l s ) as shown i n F i g u r e 5.1 and c a l c u l a t i n g the frequency, r e l a t i v e frequency, and cumulative frequency of each c e l l , as shown i n Table 5.1. The r e l a t i v e f r e q u e n c i e s are the e m p i r i c a l p r i o r p r o b a b i l i t i e s of average i n f l o w c o n d i t i o n s f o r Quilchena TABLE 5 . 1 COMPUTATION OF PRIOR PROBABILITIES OF INFLOW, QUILCHENA CREEK Inflow Index P, , c .T , Interval i.e States of Nature ~ V) Frequency . No. of Years Observed n (0i) Relative Frequency i.e. Prior Probability n (a)/N = P(9.) Cumulative Relative Frequency < 9,000X103 9 1 : Very Poor: - 3a (6,000X103-9,000X103 12 0.140 0.140 Poor : - 2a 9,000X103-12,000Xl03 14 0.160 0.300 Fair : - l a 12,000X103-15,000X103 14 0.160 0.460 Normal : a=0 15,000X103-18,000X103 17 0.190 0.650 V Good : +la 18,000X103-21,000X103 3 0.030 0.680 V Very Good: +2a 21,000X103-24,000X103 16 0.180 0.860 (24,000X103-27,000X103) V Excellent: +3a 24,000X103 or over 12 0.140 1.000 7 N= E n(0.)=88 i=l 1 7 E P(0.) = 1.000 i=i 1 62 Creek. F i g u r e s 5.1(a), 5.1(b), and 5.1(c) are the p l o t s of c e l l frequency, r e l a t i v e frequency, and cumulative frequency f o r t o t a l growing p e r i o d i n f l o w s of Quilchena Creek. The t o t a l monthly i n f l o w s f o r the growing p e r i o d are ranked i n descending order of magnitude and the p r o b a b i l i t y of exceedance of flows c a l c u l a t e d as shown i n Table B.6. F i g u r e B.2 i s the cumulative p r o b a b i l i t y d i s t r i b u t i o n of streamflow f o r Quilchena Creek, t h a t i s the p r o b a b i l i t y t h a t a given supply w i l l be e q u a l l e d or exceeded i n the f i v e month growing p e r i o d . 5.2.3 A c t i o n or d e c i s i o n a l t e r n a t i v e s The Quilchena range management has ample land a v a i l a b l e f o r expanding i t s i r r i g a t i o n system but i t i s l i m i t e d by the l a c k of water supply from the Quilchena Creek. Up t o 5,000 h e c t a r e s of f o o t h i l l i r r i g a b l e land i s a v a i l a b l e . The a c t i o n s or d e c i s i o n a l t e r n a t i v e s are the d i f f e r e n t hectarages which the ranching management can i r r i g a t e , and t h i s ranges from 100 h e c t a r e s (250 acres) to 5,000 h e c t a r e s (12,500 acres) i n i n t e r v a l s of 100 h e c t a r e s . Thus there are 50 a c t i o n s under c o n s i d e r a t i o n . 5.2.4 Consumptive uses of a l f a l f a and i r r i g a t i o n water  requirement The monthly consumptive uses of a l f a l f a f o r the growing p e r i o d were computed u s i n g the B l a n e y - C r i d d l e e q u a t i o n : 63 20 Z 15 c CD 3 10 cr cu £ 5 o u tT o.20 c ~ <D . O ^ 5 0.15 CD 2 - a 0. 10 2 o S a 0.05 0.00 1 1 1 1 1 — (a) -1 1 1 1 i ( b ) 5000 10000 15000 20000 25000 30000 35000 Inflow ( I0 3 m 3 ) Fig. 5. I CELL FREQUENCIES OF QUILCHENA CREEK FLOW DURING THE IRRIGATION SEASON . c = I § o x 2 5 - 4 ( 5 * 1 where, C = monthly consumptive use (mm) K = consumptive use c o e f f i c i e n t t = mean d a i l y maximum temperature (°F) p = percentage of d a y l i g h t hours at 50°N l a t i t u d e . The monthly l e a c h i n g requirements were computed using the f o l l o w i n g e q u a t i o n : D, L.R. = ~ (5.2 i where, L.R. = l e a c h i n g monthly requirement = depth of drainage water (mm)/month = depth of i r r i g a t i o n water (mm)/month. In Quilchena area a maximum l e a c h i n g requirement of 8 percent has been estimated,, Wilcocks [121]. D, ^ . = • 0 . 0 8 (5.3 i i . e . - D , = 0 . 0 8 D . - (5.4 d l At Quilchena area f o r J u l y , K = 0.85, f o r a l f a l f a t = 7 7.6 °F p = 11% 0.85 X 77.6 X 11 v , c = Too ^ 65 = 7.25 X.25.4 = 18 4.15 mm/July month. Average J u l y d a i l y consumptive use = "*"^4' ='5.94 mm/day. D d = 0.08 X 184.15 mm/month = 14.7 3 mm/month 14 73 Average d a i l y , = — ~ ~ = 0.4 75 mm/day .". Monthly consumptive use of a l f a l f a = 184.15 + 14.73 = 19 8.8 8 mm/July month (7.83 in) Average d a i l y consumptive use = 5.94 + 0.475 = 6 . 415 mm/day (0.2525 in) 0.25 i n = 6.35 mm/day In Quilchena area the predominant s o i l type i s s i l t loam. The i n f i l t r a t i o n c a p a c i t y i s about 5.08 mm/hr (0.20 i n / h r ) , and the s o i l water h o l d i n g c a p a c i t y i s 190.5 mm/m (2.5 i n / f t ) . The a v a i l a b i l i t y c o e f f i c i e n t i s assumed as 0.60 and the s o i l water d e f i c i t , i s t h e r e f o r e , 0.60 X 190.5 or 114.3 mm/m (1.50 i n / f t ) . For 0.67 m (2 feet) r o o t i n g depth, the s o i l water d e f i c i t i s 114.3 X 0.67 or 76.58- mm (1.50 X 2 =3.0 in/2 f t ) . For e f f i c i e n c y of a p p l i c a t i o n = 55%, depth of water to be a p p l i e d = 76.58 mm = 139.24 mm (3.0 in/0.55 = 5.45 i n 0.55 or o.4 5 f t ) . Therefore assume depth o f water to be a p p l i e d = 152.40 mm (6 i n or 0.50 f t ) . 66 With the above i n f o r m a t i o n i t i s estimated t h a t f o r a growing p e r i o d of 1 5 3 days (May to September i n c l u s i v e ) and an average d a i l y consumptive use of 6 . 3 5 mm/day ( 0 . 2 5 in/day) f o r 3 3 a l f a l f a , the Net I r r i g a t i o n Requirement w i l l be 1 7 3 7 X 10 m ( 1 4 0 0 a c r e - f e e t ) f o r a 1 0 0 hectare ( 2 5 0 acre) a l f a l f a p l o t . 3 3 Assume Net I r r i g a t i o n Water Requirement = 1 7 5 0 X 10 m  f o r 1 0 0 h e c t a r e , Table 5 . 2 g i v e s the monthly values of con-sumptive uses f o r a l f a l f a a t Quilchena, f o r the growing p e r i o d . 5 . 2 . 5 Crop net value Before the "pay-off" f o r each s t a t e - a c t i o n combination can be computed data on a l f a l f a y i e l d s and r e t u r n s on a l f a l f a , , as w e l l as v a r i a b l e c o s t s must be obtained. In Quilchena area y i e l d of a l f a l f a . v a r i e s between 2 . 0 - 4 . 0 tons/acre with average of 2 . 5 tons/acre i n most years. A y i e l d of 2 . 0 t o n s / a c r e , t h a t i s 5 . 0 0 tons/hectare w i l l be assumed. Return from h a r v e s t e d a l f a l f a crop ranges from $ 6 0 - $ 1 2 0 / t o n depending on q u a l i t y of product and year, which i n t u r n depends on the amount of sunshine (heat) and water a p p l i e d . T o t a l revenue i s assumed as $ 6 1 2 . 5 / h e c t a r e . The t o t a l v a r i a b l e c o s t s are made up of the p l a n t i n g c o s t s and, h a r v e s t i n g p l u s swathing and b a l i n g c o s t s . P l a n t i n g c o s t i s c a l c u l a t e d as $ 1 7 5 . 0 / h e c t a r e and h a r v e s t i n g (plus swathing and bali n g ) c o s t i s $ 1 2 . 5 / h e c t a r e . These c a l c u l a t i o n s were made assuming the type of i r r i g a t i o n system to be a s p r i n k l e r system, a movable tractor-end-tow type, w i t h e l e c t r i c power as source of power f o r pumps. I t 67 TABLE 5-2 COMPUTED MONTHLY CONSUMPTIVE USES OF ALFALFA FOR OJJILCHENA AREA* MONTH 'KTCAL (ins) MAY JUNE JULY AUGUST SEPTEMBER ° r, (mm) Monthly Consumptive Use (ins) 6.30 7.00 7.83 1, .20 6 .20 34.53 Daily Consumptive Use (ins) 0.20 0.23 0.25 0 .23 0 .21 Monthly Consumptive Use (mm) 160.02 177.80 198.88 182 .88 157 .48 877.06 Daily Consumptive Use (mm)' 5.08 5.84 6.35 5 .84 5 .33 *Based on B l a n e y - C r i d d l e Equation 68 should be s t a t e d t h a t i n c a l c u l a t i n g v a r i a b l e c o s t s , d e p r e c i a -t i o n (a d e f e r r a b l e cost) and unpaid i n t e r e s t on investment ( o p p o r t u n i t y cost) have not been i n c l u d e d as c o s t s because these are assumed to be the same f o r a l l a c t i o n s under c o n s i d e r a t i o n . Thus the net crop value which i s the net farm income t h a t would accrue to the rancher f o r the s p e c i f i c a c t i o n and h y d r o l o g i c a l s t a t e of nature i s c a l c u l a t e d , f o r a u n i t area ( h e c t a r e ) , from equation (4-3): IT. = TR. - TVC . 3 3 3 TR. = $612.5/hectare 3 TVC . = PC.' + HC . (5.5) 3 3 3 = ($175 + $12.5)/hectare = $187.5/hectare i . e . IK' = $612.5 - $187.5 IL = $425.0/hectare For any given d e c i s i o n a l t e r n a t i v e ( a c t i o n ) , A_., and f o r s t a t e of nature, 9^, the t o t a l crop net value i s computed from equation (4-4) : M / 7 1 v '= TR, - TVC„ . (A . . ) A . A . 31 3 3 For s i m p l i c i t y i n computational procedure, the two cuts of a l f a l f a with y i e l d of 2.0 tons/acre each have been assumed as one cut with y i e l d of 4.0 t o n s / a c r e , t h a t i s 10.0 tons/hectare f o r season and r e t u r n i s assumed at $60/ton approximately. 69 5.2.6 Crop response f u n c t i o n s In S e c t i o n 4.11.1 and 4.11.2 the two procedures employed i n t h i s study have been s t a t e d . C o n s i s t e n t with the u n d e r l y i n g assumptions of Procedure I, the crop-response f u n c t i o n i s rep r e s e n t e d as a s t e p - f u n c t i o n i n F i g u r e 5.2. In Procedure II the approach d i c t a t e s t h a t more r e a l i s t i c assump-t i o n s be made concerning the r e l a t i o n s h i p between water a p p l i e d and crop y i e l d . F i g u r e 5.3(a) i s the crop-response f u n c t i o n obtained by assuming a s t i m u l a t o r y r e l a t i o n between resource a v a i l a b i l i t y and output i n resource u t i l i z a t i o n . I t i s an ex p o n e n t i a l crop-response f u n c t i o n . F i g u r e 5.3(b) i s a non-s t i m u l a t o r y type of crop-response f u n c t i o n . I t i s a second order polynomial e q u a t i o n . 5.3 Computational Procedure 5.3.1 D e c i s i o n c r i t e r i o n based on maximum t o t a l expected  monetary value The computional procedures are the same f o r the two broad approaches adopted i n the study. The d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model s e q u e n t i a l l y i n p u t s a l l the data d e s c r i b e d i n the pre c e d i n g s e c t i o n s . With the h y d r o l o g i c a l s t a t e s of nature, (0^), and the a c t i o n s , or d e c i s i o n a l t e r n a -t i v e s , (A_.) d e f i n e d , the monetary p a y - o f f f o r each s t a t e -a c t i o n combination i s computed and entered i n a pay-off t a b l e i n a matrix format. I f i t i s assumed t h a t monetary gains and l o s s e s adequately r e f l e c t p a y - o f f s then t h i s i m p l i e s a l i n e a r u t i l i t y f u n c t i o n f o r the d e c i s i o n maker. Under t h i s assumption 0.20 0.40 0.60 0.80 R(I,J)= Ratio of Water applied Max.water requirement of crop for max.yield Fig,5.2 STEP CROP - RESPONSE FUNCTION • PROCEDURE I 71 . 0 0 0 . 8 0 0 . 6 0 0 . 4 0 0 . 2 0 0 . 0 0 f± = 1 . 0 0 CT = 0 . 2 5 Y L D ( I , J ) = / o - ° - , 8 7 5 ( R l I ' J ) - ^ ) 2 / a ' 2 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0 0 . 7 0 0 . 8 0 0 . 9 0 1 . 0 0 _, T D , Water applied R(I,J)=Ratio of — ;  Max.water requirement of crop for max. yield F i g . 5 . 3 ( a ) STI ML)LATORY,EXPONENTIAL CROP-RESPONSE FUNCTION : PROCEDURE I I . . 0 0 0 . 8 0 0 . 6 0 0 . 4 0 0 . 2 0 0 . 0 0 For 0 . 2 5 < R ( I ,J )< l .00 Y L D ( I , J ) = - I . 0 3 + 4 . 8 0 ( I , J ) - 2 . 8 0 ( R ( I , J ) ) 5 0 0 . 1 0 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0 0 . 6 0 0 . 7 0 0 . 8 0 0 . 9 0 1 . 0 0 ,.' Water applied R(I,J) = Ratio of TT : 7 ^ — —— Max.water requirement of crop for max. yield Fig. 5.3(b) NON-STIMULATORY SECOND-ORDER, POLYNOMIAL CROP-RESPONSE FUNCTION « PROCEDURE II. 72 the p r o b a b i l i t i e s a s s o c i a t e d with the d i f f e r e n t h y d r o l o g i c a l s t a t e s of nature are used t o transform the computed monetary gains i n the pa y - o f f t a b l e to the corresponding t o t a l expected monetary value (EMV) u s i n g equation (4-5) of Chapter 4. Then us i n g equation (4-12) of the same Chapter 4, the t o t a l maximum expected monetary v a l u e , (EMMAXI), i s computed, and the d e c i s i o n a l t e r n a t i v e corresponding to t h i s v a l u e , A.. , i s s e l e c t e d as the o p t i m a l area to i r r i g a t e . 5.3.2 D e c i s i o n c r i t e r i o n based on maximum t o t a l  expected u t i l i t y I f the monetary gains or l o s s e s do not adequately r e f l e c t p a y - o f f s , then the d e c i s i o n i s not made on the b a s i s of e q uation (4-12). The monetary gains or l o s s e s i n the pay-o f f t a b l e have to be transformed i n t o e q u i v a l e n t u t i l i t i e s u s i n g an e s t a b l i s h e d u t i l i t y f u n c t i o n . 5.3.3 E s t a b l i s h i n g the u t i l i t y f u n c t i o n In Chapter 4, S e c t i o n 4-8, the procedure f o r d e r i v i n g a d e c i s i o n maker's u t i l i t y f u n c t i o n was d i s c u s s e d . The von Neuman-Morgenstern (N-M)model P-14,84] -was used to transform the monetary pa y - o f f t a b l e i n t o u t i l i t i e s ( U ^ ) . Table 5.3 i s a sample monetary p a y - o f f t a b l e . F i g u r e 5.4(a) i s the case when the u t i l i t y f u n c t i o n i s l i n e a r . F igure 5.4(b) i s the n o n - l i n e a r u t i l i t y f u n c t i o n d e r i v e d f o r Quilchena Ranch management:. Table 5.4 i s the u t i l i t y p a y - o f f t a b l e d e r i v e d from Table 5.4 and Fi g u r e 5.4(b). TABLE 5.3 MONETARY PAY-OFF TABLE: EVM-CRITERION: PROCEDURE I D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 1 2 3 4 5. 6 7 8 9 10 H y d r o l o g i c a l Status 4 of Nature ($ x 10 ) 0 i e r Very Poor 4.25 8.50 12.75 17. 00 14.11 8.36 2.61 -3.14 -8.89 -14.64 Poor 4.25 8 r 50 12.75 17.00 21.25 25.50 19.75 14.00 8.25 2.50 e 3 : F a i r 4.25 8.50 12.75 17 . 00 21.25 25. 50 29.75 31.14 25.39 19.64 V Normal 4.25 8.50 12.75 17.00 21.25 25.50 29.75 34 .00 38.25 36.79 9 5 : Good 4.25 8. 50 12.75 17. 00 21.25 25.50 29.75 34.00 38. 25 42.50 V Very Good 4.25 8.50 12 . 75 17.00 21.25 25.50 29.75 34 .00 38.25 42.50 e 7 : E x c e l l e n t 4.25 8.50 12.75 17. 00 21.25 25 . 50 29.75 34.00 38.25 42.50 Continued Table 5.3 Continued. D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 11 12 13 14 15 16 17 18 19 20 H y d r o l o g i c a l Status of Nature 9 i Very Poor -20. 39 -26. 14 -31. 89 -37. 64 -43.39 -49. 14 -54. 89 -60. 64 -66. 39 -72. 14 9 2 : Poor -3. 50 -9. 00 -14. 75 -20. 50 -26.25 -32 . 00 -37. 75 -43. 50 -4 9. 25 -55. 00 e 3 : F a i r 13. 89 8. 14 2. 3 9 -3. 36 -9.11 -14 . 86 -20. 61 -26. 36 -32. 11 -37 . 86 Normal 31. 04 25. 29 19. 53 13. 79 8.04 2. 29 -3. 46 -9. 21 -14. 96 -20. 71 e 5 : Good 46. 75 42. 43 36. 68 30. 93 25.18 19. 43 13. 68 7. 93 2. 18 -3. 57 9 6 : Very Good 46. 75 51. 00 53. 82 48. 07 42.32 36. 57 30. 82 2.5. 07 1.9. 32 13. 57 8 v E x c e l l e n t 46. 75 51. 00 55. 25 59. 50 59.46 53. 71 47. 96 42 . 21 36. 46 30. 71 Continued Table 5.3 Continued D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 21 22 23 24 25 26 27 28 29 30 H y d r o l o g i c a l Status ($ x 10 4 of Nature 0 i V Very Poor -77. 89 -83. 64 -89. 39 -95. 14 -100 .89 -106. 64 -112. 39 -118 . 14 -123. 8 9 -129. 64 Poor -60. 75 -66. 50 -72 . 2 5 -78. 00 -83 .75 -89 . 50 -95. 2 5 -101. 00 -106. 75 -112 . 50 e 3 : F a i r -43. 61 -49. 36 -55. 11 -60. 86 -66 .61 -71. 36 -78. 11 -83. 86 -89. 61 -95. 36 9 4 : Normal -26. 46 -32. 21 -37'. 96 -43. 71 -49 .46 -55. 21 -60. 96 -66. 71 -72. 46 -78. 21 e 5 : Good -9. 32 -15. 07 -2 0. 82 -26 . 57 -32 .32 -38. 07 -4 3. 82 -4 9. 57 -55. 32 -61. 07 9 6 : Very.Good 7. 82 2. 07 -3. 68 -9. 43 -15 .18 -20 . 93 -26. 68 -32. 43 -38. 18 -43. 93 e 7 : E x c e l l e n t 24. 96 19. 21 13. 46 7. 71 1 .96 -3. 79 -9. 54 -15. 29 -21. 04 -26. 79 Continued .... Table 5 . 3 Continued D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 31 32 33 34 35 36 37 38 39 40 H y d r o l o g i c a l Status Very Poor - 1 3 5 . 39 - 1 4 1 . 14 - 1 4 6 . 89 - 1 5 2 . 64 - 1 5 8 . 39 - 1 6 4 .14 - 1 6 9 . 89 - 1 7 5 . 64 - 1 8 1 . 39 - 1 8 7 . 14 e 2 : Poor - 1 1 8 . 25 - 1 2 4 . 00 - 1 2 9 . 75 - 1 3 5 . 50 - 1 4 1 . 25 - 1 4 7 . 0 0 - 1 5 2 . 75 - 1 5 8 . 50 - 1 6 4 . 25 - 1 7 0 . 00 e 3 : F a i r - 1 0 1 . 11 - 1 0 6 . 8 6 - 1 1 2 . 61 - 1 1 8 . 36 - 1 2 4 . 11 - 1 2 9 . 8 6 - 1 3 5 . 61 - 1 4 1 . 36 - 1 4 7 . 11 - 1 5 2 . 86 e 4 : Normal - 8 3 . 96 - 8 9 . 71 - 9 5 . 46 - 1 0 1 . 21 - 1 0 6 . 96 - 1 1 2 .71 - 1 1 8 . 46 - 1 2 4 . 21 - 1 2 9 . 96 - 1 3 5 . 71 9 5 : Good - 6 6 . 82 - 7 2 . 57 - 7 8 . 32 - 8 4 . 07 - 8 9 . 82 - 9 5 . 5 7 - 1 0 1 . 32 - 1 0 7 . 07 -112 . 82 - 1 1 8 . 57 9 6 : Very Good - 4 9 . 68 - 5 5 . 43 - 6 1 . 18 - 6 6 . 93 - 7 2 . 68 - 7 8 . 4 3 - 8 4 . 18 - 8 9 . 93 - 9 5 . 68 - 1 0 1 . 43 e 7 : E x c e l l e n t - 3 2 . 54 - 3 8 . 29 - 4 4 . 04 - 4 9 . 79 - 5 5 . 54 - 6 1 . 2 9 - 6 7 . 04 - 7 2 . 7 9 - 7 8 . 54 - 8 4 . 29 Continued .... OS Table 5 . 3 Continued D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 41 • 42 43 44 45 46 47 48 49 50 H y d r o l o g i c a l Status of Nature Very Poor - 1 9 2 . 89 - 1 9 8 . 64 - 2 1 0 . 14 - 2 1 5 . 89 - 2 0 4 . 39 - 2 2 1 . 64 - 2 2 7 . 39 - 2 3 3 .14 - 2 3 8 . 89 - 2 4 4 . 64 e 2 : Poor - 1 7 5 . 75 - 1 8 1 . 50 - 1 9 3 . 00 - 1 9 8 . 75 - 1 8 7 . 25 - 2 0 4 . 50 - 2 1 0 . 25 - 2 1 6 . 00 - 2 2 1 . 75 - 2 2 7 . 50 e 3 : F a i r - 1 5 8 . 61 - 1 6 4 . 36 - 1 7 5 . 86 - 1 8 1 . 61 - 1 7 0 . 11 - 1 8 7 . 36 - 1 9 3 . 11 - 1 9 8 . 8 6 - 2 0 4 . 61 - 2 1 0 . 36 9 4 : Normal - 1 4 1 . 46 - 1 4 7 . 21 - 1 5 8 , 71 - 1 6 4 . 46 - 1 5 2 . 96 - 1 7 0 . 21 - 1 7 5 . 96 - 1 8 1 .71 - 1 8 7 . 46 - 1 9 3 . 21 e 5 : Good - 1 2 4 . 32 - 1 3 0 . 07 - 1 4 1 . 57 - 1 4 7 . 32 - 1 3 5 . 82 - 1 5 3 . 07 - 1 5 8 . 82 - 1 6 4 . 5 7 - 1 7 0 . 32 - 1 7 6 . 07 V Very Good - 1 0 7 . 18 - 1 1 2 . 93 - 1 2 4 . 43 - 1 3 0 . 18 - 1 1 8 . 6 8 - 1 3 5 . 93 - 1 4 1 . 68 - 1 4 7 . 4 3 - 1 5 3 . 18 - 1 5 8 . 93 E x c e l l e n t - 9 0 . 04 - 9 5 . 79 - 1 0 7 . 29 - 1 1 3 . 04 - 1 0 1 . 54 - 1 1 8 . 79 - 1 2 4 . 54 - 1 3 0 . 2 9 - 1 3 6 . 04 - 1 4 1 . 79 F i g . 5 . 4 ( a ) LINEAR UTILITY FUNCTION 20 I 0 0 I I 1 1 1 1 I . I 1 \ ^ . ^ - F o r -4XI0 5<M< + 6 X | 0 5 _ — f U = 72.0 + 7.64M-0.49M 2 _ A— For-6X I0 5<M<-4X |0 5 — / U = 100.80 + I6.80M / 1 1 1 1 1 1 1 1 I . I 5 4 3 2 Monetary losses I 0 Monetary gains(Mx8IO ) Fig.5.4(b) NON-LINEAR UTILITY FUNCTION DERIVED FOR QUILCHENA RANCH IRRIGATION SYSTEM MANAGEMENT DECISION - MAKI NG TABLE 5.4 UTILITY PAY-OFF TABLE: EU CRITERION: PROCEDURE I. D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 10 H y d r o l o g i c a l Status of Nature 0 i UTILITY e l : Very Poor 75. 16 7 8.14 80. 94 83. 57 81. 80 78 . 04 73. 96 . 69. 55 64 .82 59 .76 e 2 : Poor 75. 16 78.14 80. 94 83. 57 86. 02 88. 30 85. 18 81. 74 77 . 97 73. 88 e 3 : F a i r 75. 16 78.14 80. 94 83. 57 36. 02 88. 30 90. 39 91. 04 8 8 .24 85.12 9 4 : Normal 75. 16 78.14 80. 94 33. 57 86. 02 88. 30 90. 39 92. 31 94 . 05 93.47 0 5: Good 75. 16 78.14 80. 94 83. 57 86 . 02 88 . 30 90. 39 92. 31 94 .05 95. 62 V Very Good 75. 16 78.14 80. 94 83. 57 86. 02 88. 30 90. 39 92. 31 94 . 05 95.62 0 7: E x c e l l e n t 75. 16 78.14 80. 94 83. 57 86. 02 38. 30 90. 39 92. 31 94 .05 95.62 Continued CO o Table 5.4 Continued D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 11 12 13 14 15 16 17 18 19 20 H y d r o l o g i c a l Status of Nature UTILITY 6 i Very Poor 54. 38 48. 68 42. 65 36 . 30 27.90 18. 24 8. 58 o. 00 0. 00 0. 00 e 2 : Poor 69. 47 64. 73 59. 66 54. 28 48.57 42. 53 36. 18 27. 72 18. 06 8. 40 e 3 : F a i r 81. 67 77. 90 73. 80 69. 38 64 .64 59. 57 54. 18 48. 46 42. 42 36. 05 9 4 : Normal 90. 99 88. 19 85 . 06 81. 60 77.82 73. 72 69. 29 64. 54 59. 47 54. 07 9 5 : Good 97. 01 95. 59 93. 4 3 90. 94 88.13 84 . 99 81. 53 77. 75 73. 64 69. 21 V Very Good 97. 01 98. 22 98. 93 97. 40 95.56 93. 39 90. 89 88. 07 84. 93 81. 47 9 - 7 : E x c e l l e n t 97. 01 98. 22 99. 25 100. 00 100.00 98. 90 97. 37 95. 52 93. 34 90. 84 CO Continued .... Table 5 . 4 Continued .. D e c i s i o n A l t e r n a t i v e s (hectares x 10 ) , Aj 21 22 23 24 25 26 27 28 29 30 H y d r o l o g i c a l Status of Nature 9 i UTILITY e r Very Poor 0 . 0 0 0 . 00 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 00 0 . 00 e 2 : Poor 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 0 . 00 0 . 0 0 0 . 0 0 0 . 00 0 . 0 0 e 3 : F a i r 2 7 . 5 4 17. 88 8 . 2 2 0 . 0 0 0 . 0 0 0 . 00 0 . 0 0 0 . 0 0 0 . 00 0 . 0 0 Normal 4 8 . 3 5 4 2 . 30 3 5 . 9 3 2 7 . 3 6 1 7 . 7 0 8 . 0 4 0 . 0 0 0 . 0 0 0 . 0 0 0 . 0 0 9 5 : Good 6 4 . 45 5 9 . 37 5 3 . 9 7 4 8 . 2 4 4 2 , 1 9 3 5 . 8 1 2 7 . 1 8 1 7 . 5 2 7 . 8 6 0 . 0 0 V Very Good 7 7 . 6 8 7 3 . 5 6 6 9 . 1 2 64 . 3 6 5 9 . 2 7 5 3 . 86 4 8 . 1 3 4 2 . 07 3 5 . 6 9 27 . 00 67 = E x c e l l e n t 8 8 . 0 2 8 4 . 87 8 1 . 4 0 7 7 . 6 0 7 3 . 4 8 6 9 . 0 4 6 4 . 2 7 5 9 . 1 8 5 3 . 76 4 8 . 0 2 CO Continued .... Table 5.4 Continued D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 31 32 33 34 35 36 37 38 39 40 H y d r o l o g i c a l S tatus of Nature UTILITY ©i 9 1 : Very Poor 0. 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0 . 00 e 2 : Poor 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0. 00 0.00 0.00 e 3 : F a i r 0.00 0. 00 0. 00 0.00 0.00 0. 00 0.00 0.00 0.00 0.00 V Normal 0. 00 0.00 0.00 0.00 0.00 0. 00 0.00 0.00 0.00 0. 00 Good 0.00 0. 00 0. 00 0.00 0.00 0, 00 0.00 0.00 0.00 0 .00 V Very Good 17.34 7. 68 0. 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 V E x c e l l e n t 41.96 35.57 2 6.82 17.16 7.50 0.00 0.00 0.00 0.00 0. 00 Continued Table 5.4 Continued .. D e c i s i o n A l t e r n a t i v e s (hectares x 10 ), Aj 41 42 43 44 45 46 4 7 48 49 50 H y d r o l o g i c a l S t a t u s of Nature UTILITY e i Very Poor 0.00 0. 00 0.00 0. 0 0 0.00 0 .00 0.00 0.00 0.00 0.00 e 2 : Poor 0. 00 0.00 0. 00 0.00 0.00 0. 00 0.00 0.00 0. 00 0.00 9 3 : F a i r 0. 00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 9 4 : Normal 0.00 0.00 0.0 0 0.00 0.00 0 . 00 0. 00 0 . 00 0.00 0. 00 9 5 : Good 0. 00 0.00 0.00 0.00 0.00 0.00 0. 00 0 . 00 0.00 0. 00 9 6 : Very Good 0.0.0 0.0 0 0. 00 0.00 0.00 0.00 0. 00 0.00 0.00 0.00 9_: E x c e l l e n t 0. 00 0.00 0.00 0.00 0.00 0.00 0. 00 0.00 0.00 0.00 co 8 5 Using equation (4-7) the u t i l i t i e s are converted i n t o expected u t i l i t i e s f o r each s t a t e - a c t i o n combination and the t o t a l expected u t i l i t y v a l u e , (EU), i s obtained f o r each a c t i o n , A_. , by summing over a l l the seven s t a t e s of nature. Equation (4-9) i s then employed to compute the t o t a l maximum expected u t i l i t y (UVMAXI) _. , and the d e c i s i o n a l t e r n a t i v e c o rresponding to t h i s t o t a l maximum expected u t i l i t y , A_. , i s chosen as the optimal area to i r r i g a t e under the e x i s t i n g s i t u a t i o n of h y d r o l o g i c a l u n c e r t a i n t y . In both Procedures I and I I the c r i t e r i a l f u n c t i o n t h a t one attempts t o maximize i s the area under i r r i g a t i o n . The v a r i a b l e i r r i g a b l e areas are chosen under u n c e r t a i n t y . Whereas Procedure I attempts to cut back the o r i g i n a l seeded area, A., to an optimal area, A., a t time, T, Procedure I I adopts a d i f f e r e n t approach and attempts t o maintain the o r i g i n a l seeded area, A. , and a l l o c a t e s whatever water i s -*o a v a i l a b l e f o r i r r i g a t i o n evenly to the area, A. , from time, -*o T, to the end of the i r r i g a t i o n p e r i o d , where T i s the l e n g t h of time i n months a f t e r seeding. Computer Programs A - l and A-2, used i n computations f o r Procedure I and I I r e s p e c t i v e l y , are given i n Appendix A. In the a n a l y s i s f i v e d i f f e r e n t values of T, T=0 to 4, were assumed. This was done, i n s t e a d of us i n g j u s t one T-value, so t h a t the e f f e c t of time, when a check i s made on the water a v a i l a b l e f o r i r r i g a t i o n , as i t a f f e c t s the optimal p o l i c y , c o u l d be assessed. 86 5.4 Model Output 5.4.1 Output from Procedure I The outputs from the model u s i n g Procedure I are e i t h e r the t o t a l expected monetary values or the t o t a l expected u t i l i t y v alues f o r the d i f f e r e n t d e c i s i o n a l t e r n a t i v e s (areas under i r r i g a t i o n i n hectares) f o r any chosen T-value. Table 5.5 g i v e s the t o t a l expected monetary values c o r r e s -ponding to the d e c i s i o n a l t e r n a t i v e s f o r T = 2; Table 5.6 g i v e s the t o t a l expected u t i l i t y v a l u e s c o r r e s p o n d i n g t o the d e c i s i o n a l t e r n a t i v e s f o r T=2. The outputs f o r a l l the values of T, T = 0 to 4, are represented g r a p h i c a l l y f o r EMV-c r i t e r i o n and E U - c r i t e r i o n i n F i g u r e s 5.5 and 5.6, r e s p e c t i v e l y . From F i g u r e s 5.5, and 5.6, the optimal areas to i r r i g a t e under h y d r o l o g i c a l u n c e r t a i n t y adopting the assumptions of Procedure I f o r Quilchena ranch are summarized i n Table 5.7. 5.4.2 Output from Procedure I I S i m i l a r l y Table 5.8 g i v e s the t o t a l expected monetary valu e s corresponding to the d e c i s i o n a l t e r n a t i v e s , and Table 5.9 g i v e s the t o t a l expected u t i l i t i e s c orresponding to the d e c i s i o n a l t e r n a t i v e s u s i n g the s t i m u l a t o r y crop-response f u n c t i o n , f o r T = 2, r e s p e c t i v e l y . Table 5.10 and Table 5.11 are the corresponding t a b l e s , u sing the non-stimulatory crop-response f u n c t i o n f o r T = 2, r e s p e c t i v e l y . The outputs f o r a l l the values of T, T = 0 to 4, are shown g r a p h i c a l l y f o r 87 TABLE 5.5 TOTAL EXPECTED MONETARY VALUE, EMV FOR T = 2 : PROCEDURE I D e c i s i o n A l t e r n a t i v e , (Hectares x 10 ) , A.. T o t a l Expected Monetary Value, ( $ 1 0 4 ) , EMV.. 1 4. 25 2 8. 50 3 12. 75 4 17. 00 5 20. 25 6 23. 10 7 24. 35 8 25 14 9 24 79 10 23. 36 11 21 11 12 18 60 13 15 79 14 11 44 15 6. 50 16 0. 74 17 -5 01 18 -10 76 19 -16 51 20 -22 26 21 -28 .01 22 -33 . 76 (continued. . .) TABLE 5.5 (continued) D e c i s i o n A l t e r n a t i v e , (Hectares x 10 2) , A.. T o t a l Expected Monetary Value, ( $ 1 0 4 ) , EMV.. 23 -39.51 24 -45.26 25 -51.01 26 -56.76 27 -62.51 28 -68.26 29 -74.01 30 -79.76 31 -85.51 32 -91.26 33 -97.01 34 -102.76 35 -108.51 36 -114.26 37 -120.01 38 -125.76 .39 -131.51 40 -137.26 41 -143.01 42 -148.76 43 -154.51 44 -160.26 45 -166.01 46 -171.76 47 -177.51 48 -183.26 49 -189.01 50 -194.76 89 TABLE 5.6 TOTAL EXPECTED UTILITY, EU, FOR 1 = 2 : PROCEDURE I D e c i s i o n A l t e r n a t i v e (Hectares x 10 2) , A_. T o t a l Expected U t i l i t y EU . 3 1 . 75.16 2 78.14 3 80.94 4 83.57 5 85.43 6 86.86 7 87.26 8 87.23 9 86.46 10 85.03 11 83.04 12 80.69 13 77.99 14 74.6 5 15 70.66 16 66.10 17 61.26 18 56.01 19 51.54 20 46.84 21 41.83 22 37.80 23 33.60 24 29.09 25 25.59 (continued) TABLE 5.6 (continued) D e c i s i o n A l t e r n a t i v e (Hectares x 10^), A.. T o t a l Expected U t i l i t y EU . D 26 21.96 27 18.48 28 16.38 29 14.19 30 11.58 31 9.00 32 6.36 33 3.75 34 2.40 35 1.05 36 0.00 37 0.00 38 0.00 39 0.00 40 0.00 41 0.00 42 0.00 43 0.00 44 0.00 45 0.00 46 0.00 47 0.00 48 0.00 49 0 .00 50 0.00 Area under irrigation (hectares) F i g . 5 . 5 TOTAL EXPECTED MONETARY VALUE VERSUS AREA UNDER IRRIGATION FOR PROCEDURE I. 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) .6 TOTAL EXPECTED UTILITY VERSUS AREA UNDER IRRIGATION FOR PROCEDURE I . 93 TABLE 5.7 OPTIMAL AREAS UNDER IRRIGATION: PROCEDURE I Total Expected Monetary Value, EMV, Criterion Total Expected U t i l i t y , EU, Criterion Time Index (Months) Optimal Area Under Irrigation (Hectares) Total Maximum Expected Monetary Value ($ 1,000) Optimal Area Under Irrigation (Hectares) Total Maximum Expected U t i l i t y T = 0 1,300 318.00 1,000 90.21 • T = 1 900 282.00 900 88.80 T = 2 800 252.00 700 87.26 T =' 3 700 220.00 600 85.99 T = 4 600 184.00 500 84.04 94 TABLE 5.8 TOTAL EXPECTED MONETARY VALUE, EMV, USING STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE II : T = 2 D e c i s i o n A l t e r n a t i v e (Hectares x 1 0 2 ) , A. D T o t a l Expected Monetary Value ( $ 1 0 4 ) , EMV.. 1 4.25 2 8.50 3 12.75 4 17.00 5 20.78 6 23.74 7 26.03 8 27.76 9 28.79 10 29.32 11 29.04 12 28.46 13 27.87 14 2 6.92 15 25.55 16 23.59 17 21. 31 18 18.96 19 16.73 20 14.68 21 12.86 22 11.28 23 9.90 24 8.72 (continued..) TABLE 5.8 (continued) D e c i s i o n A l t e r n a t i v e (Hectares x 10 ), A.. T o t a l Expected Monetary Value ($10 4), EMV\ 25 7.71 26 6.85 27 6.10 28 5.47 29 4.92 30 4.45 31 4. 04 32 3.68 33 3. 37 34 3.10 35 2.86 36 2.65 37 2.46 38 2.29 . 39 2.14 40 2.01 41 1.89. 42 1.79 43 1.69 44 1.60 45 1.52 46 1.45 47 1. 38 48 1. 32 49 1.27 50 1.22 TABLE 5.9 TOTAL EXPECTED UTILITY, EU, USING STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE II : T = 2 D e c i s i o n A l t e r n a t i v e (Hectares x 10 2) , A.. T o t a l Expected U t i l i t y , EU . 3 1 75 .16 2 78 .14 3 80 .94 4 83 .57 5 85 . 76 ; 6 8 7 .28 7 88 . 31 8 88 .92 9 89 .11 10 89 .01 11 88 .55 12 87 . 91 13 87 .21 14 86 .42 15 85 .47 16 8 4 54 17 83 65 18 82. 75 19 81. 80 20 80. 89 21 80. 02 22 79. 22 23 78. 49 24 77. 83 25 77. 24 (continued. . TABLE 5.9 (continued) D e c i s i o n A l t e r n a t i v e (Hectares x 10 ), A.. T o t a l Expected U t i l i t y EU . J 26 76.72 27 76.26 28 75.86 29 75.50 . 30 31 75.19 74.91 32 74.67 33 74.46 34 74.27 35 74.10 36 .73.95 37 73.82 38 73,70 39 73.59 40 73.50 41 73.41 42 73.33 43 73.26 44 73.20 45 73.14 46 73.09 47 73.04 48 73.00 49 72.95 50 72.92 98 TABLE 5.10 TOTAL EXPECTED MONETARY VALUE, EMV, USING NON-STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE I I : T=2. D e c i s i o n A l t e r n a t i v e T o t a l Expected Monetary Value (Hectares x 1 0 2 ) , Aj ($10 4), EMVj 1 4.25 2 8.50 3 12.75 4 17. 00 5 21.26 6 24.49 7 26.92 8 26.33 9 27. 95 10 28.33 11 24.53 12 24.00 13 22.22 14 20.14 15 14.64 16 12.30 17 8.67 18 0.27 19 - 2.90 20 - 6.69 21 . -11.77 22 -16.15 23 -20.90 24 -31.93 25 -35.05 26 -38.31 27 -47.25 28 -49.00 29 -50.75 30 -52.50 31 -54.25 32 -56.00 33 -57.75 34 -59.50 35 -61.25 36 -63.00 37 -64.75 38 -66.50 39 -68.25 40 -70.00 Continued .... 99 Table 5.10 Continued .... D e c i s i o n A l t e r n a t i v e T o t a l Expected Monetary Value (Hectares x 102), Aj ($10 4), EMVj 41 -71.75 42 -73.50 43 -75.25 44 -77.00 45 -78.75 46 -80.50 47 -82.25 48 -84.00 49 -85.75 50 -87.50 100 TABLE 5.11 TOTAL EXPECTED UTILITY, EU, USING NON-STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE I I : T=2. D e c i s i o n A l t e r n a t i v e T o t a l Expected U t i l i t y (Hectares x 1 0 2 ) , Aj EUj 1 75.16 2 78.14 3 80.94 4 83.57 5 86.03 6 87.74 7 88.77 8 87 .40 9 87.81 10 87.42 11 83.63 12 82.83 13 81. 35 14 79.18 15 74.01 16 71.93 17 69.37 18 62 . 51 19 60.75 20 58.70 21 55.49 22 52.76 23 49.40 24 38.67 25 35.29 26 31.64 27 21.42 28 18 .48 29 15.54 30 12.60 31 9.66 32 6.72 33 3.78 34 0.84 35 0.00 36 0.00 37 0. 00 38 0. 00 39 t 0.00 40 0. 00 Continued ... . 101 Table 5.11 Continued .... D e c i s i o n A l t e r n a t i v e T o t a l Expected U t i l i t y (Hectares x 1 0 2 ) , Aj EUj 0. 00 0. 00 0.00 0.00 0. 00 0. 00 0.00 0.00 0. 00 0.00 41 42 43 44 45 46 47 48 49 50 1 0 2 E M V - c r i t e r i o n and E U - c r i t e r i o n , f o r s t i m u l a t o r y crop response f u n c t i o n i n F i g u r e s 5 . 7 and 5 . 8 , r e s p e c t i v e l y . The c o r r e s -ponding p l o t s f o r the n o n - s t i m u l a t o r y crop response f u n c t i o n , f o r E M V - c r i t e r i o n and E U - c r i t e r i o n are shown i n F i g u r e s 5 . 9 and 5 . 1 0 , r e s p e c t i v e l y . From F i g u r e s 5 . 7 and 5 . 8 the optimal areas to i r r i g a t e under u n c e r t a i n t y a r i s i n g from d i f f e r e n t h y d r o l o g i c a l s t a t e s of nature f o r the s t i m u l a t o r y crop response f u n c t i o n f o r Quilchena ranch was summarized i n Table 5 . 1 2 . The corresponding summary t a b l e f o r n o n - s t i m u l a t o r y crop-response f u n c t i o n i s Table 5 . 1 3 , . from F i g u r e s 5 . 9 and 5 . 1 0 . From Table 5 . 7 f o r Procedure I and, Tables 5 . 1 2 and 5 . 1 3 f o r Procedure I I , i t i s obvious t h a t the optimal area to develop f o r i r r i g a t i o n i n any given year w i l l depend on the time index, T, t h a t i s the time a f t e r seeding when some i n f o r m a t i o n on the volume of i n f l o w i n t o the i r r i g a t i o n storage r e s e r v o i r becomes a v a i l a b l e . In the Quilchena area the major source of water supply f o r Quilchena creek flow comes from snowmelt r u n o f f d u r i n g the May and June f r e s h e t p e r i o d . Thus f o r Quilchena ranch management the b e s t time index to use i n a r r i v i n g at a meaningful d e c i s i o n , u s i n g the technique of t h i s study, i s T = 1 or T = 2 where T = 4 corresponds to b e g i n n i n g of June and T = 2 corresponds to b e g i n n i n g of J u l y . I f i t i s assumed t h a t the d e c i s i o n i s made at the middle of June, say, then the optimal. policy 400 4 0 0 800 1200 4400 4800 1600 2000 2400 2800 3200 3600 4000 Area under irrigation (hectares) Fig. 5.7 TOTAL EXPECTED MONETARY VALUE VERSUS AREA UNCER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE II 5200 b 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation ( hectares) Fig. 5.8 TOTAL EXPECTED UTILITY VERSUS AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE II. 400 360r -T = 0 1200 F i g 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) 5.9 TOTAL EXPECTED MONETARY VALUE VERSUS AREA UNDER ^ , Axnov r D H P - R F < ^ P O N S E FUNCTION FOR PROCEDURE II. I I I I 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 M Area under irrigation (hectares) o> Fig. 5.10 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION USING NON -STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE II. TABLE 5.12 OPTIMAL AREAS UNDER IRRIGATION USING THE STIMULATORY CROP RESPONSE FUNCTION: PROCEDURE I I . TOTAL EXPECTED MONETARY VALUE, EMV, CRITERION TOTAL EXPECTED UTILITY,, EU, CRITERION TIME INDEX (months) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED MONETARY VALUE ($1,000) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED UTILITY T = 0 1,600 397.00 1,400 93.25 T = 1 1,300 338.00 1,100 91.00 T = 2 1,000 293.00 900 89.50 T = 3 900 256.00 800 87.50 T = 4 800 234.00 700 86.50 o TABLE 5.13 OPTIMAL AREAS UNDER IRRIGATION USING THE NON-STIMULATORY CROP RESPONSE FUNCTION: PROCEDURE I I . TOTAL EXPECTED MONETARY VALUE, EMV, CRITERION TOTAL EXPECTED UTILITY, EU, CRITERION TIME INDEX (months) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED MONETARY VALUE ($1,000) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED UTILITY T. = 0 1,600 463.00 1,300 95.00 T = 1 1,000 348.00 1,000 92.00 T = 2 1,000 283.00 800 89.00 T = 3 800 240.00 700 86.50 T = 4 600 205.00 600 85.00 o co 109 would be determined by u s i n g the average of T = 1 and T = 2. For procedure I, and adopting the above r a t i o n a l e , the optimal area to develop w i l l be 850 h e c t a r e s (EMV = $267,000) or 800 hectares (EU = 88.03). For procedure I I , the optimal area to develop f o r i r r i g a t i o n w i l l be 1,150 h e c t a r e s (EMV = $315,000), or 1,000 h e c t a r e s (EU = 90.25) f o r the s t i m u l a t o r y crop response f u n c t i o n . For the n o n - s t i m u l a t o r y crop response f u n c t i o n the corresponding optimal areas are 1,000 h e c t a r e s (EMV = $315,500) or 900 h e c t a r e s (EU = 90.50). Thus the m u l t i - v a l u e d estimates presented would allow the d e c i s i o n maker some f l e x i b i l i t y to a d j u s t to the r i s k i n the s i t u a t i o n . Under the assumptions o u t l i n e d e a r l i e r , a l i n e a r u t i l i t y f u n c t i o n given i n F i g u r e 5.4(a) and s u b j e c t i v e p r o b a b i l i t i e s equal to the e m p i r i c a l l y - e s t i m a t e d long-run ( p r i o r ) p r o b a b i l i t i e s g iven i n Table 5.1, i t can be u n e q u i v o c a l l y s a i d t h a t the optimum area to i r r i g a t e under h y d r o l o g i c a l u n c e r t a i n t y i s 850 hectares s i n c e t h i s area p r o v i d e s the maximum t o t a l expected monetary value (EMV) of $267,000. Thus under the approach of procedure I, i t would pay the Quilchena ranch management to i r r i g a t e an area corresponding to " f a i r " h y d r o l o g i c a l c o n d i t i o n s , knowing t h a t c o n d i t i o n s t h i s f a v o r a b l e w i l l probably occur about 16 percent of the time, d u r i n g the d e c i s i o n p e r i o d . When the assumption of a l i n e a r u t i l i t y f u n c t i o n i s r e l a x e d and the q u a d r a t i c polynomial d e r i v e d and shown i n F i g u r e 5.4(b) i s assumed as r e p r e s e n t i n g the Quilchena ranch manager's u t i l i t y f u n c t i o n then, 110 under the approach o f Procedure I, the optimum area to i r r i g a t e becomes 800 h e c t a r e s , s i n c e t h i s area p r o v i d e s the maximum t o t a l expected u t i l i t y value (EU) of 88.03. T h i s area again corresponds t o " f a i r " h y d r o l o g i c a l c o n d i t i o n s . Under the approach of Procedure II and us i n g the s t i m u l a t o r y crop response f u n c t i o n and, assuming a l i n e a r u t i l i t y f u n c t i o n the optimum area to i r r i g a t e i s 1,150 h e c t a r e s , the area which p r o v i d e s the maximum t o t a l expected monetary value (EMV) o f $315,500, and which corresponds to "good" h y d r o l o g i c a l c o n d i t i o n s . However when the q u a d r a t i c u t i l i t y f u n c t i o n i s used the optimum area to i r r i g a t e drops t o 1,000 h e c t a r e s , with t o t a l maximum expected u t i l i t y (EU) of 90.25. Th i s area corresponds t o "normal" h y d r o l o g i c a l c o n d i t i o n s . Under the approach o f Procedure II and employing the non-st i m u l a t o r y crop response f u n c t i o n and, assuming a l i n e a r u t i l i t y f u n c t i o n the optimum area to i r r i g a t e i s 1,000 hectares t h i s p r o v i d i n g the maximum t o t a l expected monetary value (EMV) of $315,500, and t h i s area corresponds t o "normal" h y d r o l o g i c a l c o n d i t i o n s . I f the q u a d r a t i c u t i l i t y f u n c t i o n i s used i n pla c e o f the l i n e a r f u n c t i o n the optimum area t o i r r i g a t e drops t o 900 h e c t a r e s , with t o t a l maximum expected u t i l i t y (EU) o f 90.50, and t h i s area corresponds to "normal" h y d r o l o g i c a l c o n d i t i o n s . C o n s i s t e n t l y the output from the s t o c h a s t i c o p t i m i -z a t i o n model presented shows t h a t , i r r e s p e c t i v e of the I l l procedure adopted and the crop response function used, i n the sit u a t i o n where the u t i l i t y function i s li n e a r , larger hectarages are obtained than when the quadratic u t i l i t y function i s employed. Procedure II gives better results than Procedure I and stimulatory crop response function gives more hectarage than non-stimulatory crop response function for Procedure I I . ^•^ S e n s i t i v i t y Analysis It was pointed out in Chapter 4 that a form of s e n s i t i v i t y analysis would enrich the decision theory stochastic optimization technique, which has been subjectively developed for i r r i g a t i o n systems management decision-making. Thus a s e n s i t i v i t y t e s t i n g with respect to the subjective estimates of p r o b a b i l i t i e s , u t i l i t y functions, and crop response functions would t e s t i f y as to the f l e x i b i l i t y of the model. Also the economic impacts of i m p l i c i t assumptions made i n the procedures with respect to yiel d s and factor prices on the objective function w i l l be r e f l e c t e d i n the s e n s i t i v i t y analysis. 5.5.1 S e n s i t i v i t y of optimal policy to p r i o r p r o b a b i l i t i e s To evaluate how sensitive the optimal p o l i c i e s , arrived at i n Section 5.4, are to the empirically-computed long-run (prior) p r o b a b i l i t i e s , the p r i n c i p l e of i n s u f f i c i e n t reason was employed and a l l the hydrological ntntoa of nature shown in Table 5.1 are now assumed to be equally l i k e l y , that is»,each of the seven categories of flows now have equal p r o b a b i l i t i e s of 1/7 or 0.143, as shown in Table 5.14. This 112 TABLE 5.14 PRIOR PROBABILITIES OF QUILCHENA CREEK FLOW ASSUMING THE PRINCIPLE OF INSUFFICIENT REASON INFLOW INDEX STATES OF NATURE INTERVAL PRIOR PROBABILITY (©i) (m3) P ( e i ) 0,: Very poor: - 3a <9,000xl0 3 (6,000xl0 3-9,OOOxlO 3) 0.143 0 2: Poor: - 2o 9,000xl0 3-12,OOOxlO 3 0.143 0 3: F a i r : - l a 12,O00xl0 3-15,OOOxlO 3 0.143 0 4: Normal: a = 0 15,000x103-18,OOOxlO 3 0.143 0 5: Good: + l a 18,000xl0 3-21,OOOxlO 3 0.143 0_: Very good: + 2a 21,000xl0 3-24,OOOxlO 3 0.143 b 0 : E x c e l l e n t : + 3a (24,000xl0 3-27,OOOxlO 3) 0.143 24,OOOxlO 3 or over 7 £ P(0i) = 1.000 • i = l 113 assumption i s c o n s i s t a n t with the p r e s e n t l y adopted n o t i o n of the ranchers i n the Q u i l c h e n a area. Thus, the p r o b a b i l i t i e s p resented i n Table 5.14 can be assumed to r e p r e s e n t the Q u i l c h e n a ranch i r r i g a t i o n manager's own s u b j e c t i v e p r o b a b i l i t i e s . Employing the computational procedure of S e c t i o n 5.3, the same step-crop response f u n c t i o n f o r Procedure I and, s t i m u l a t o r y and n o n - s t i m u l a t o r y crop response f u n c t i o n s f o r Procedure I I , and the same computer programs A-1 and A-2, the a n a l y s i s was repeated with the new s e t of s u b j e c t i v e p r i o r p r o b a b i l i t i e s . The computer outputs are p l o t t e d i n F i g u r e s 5.11 and 5.12; 5.13 and 5.14, and, 5.15 and 5.16, r e s p e c t i v e l y . The o p t i m a l areas from Procedures I and I I are summarized i n Tables 5.15; 5.16 and 5.17, r e s p e c t i v e l y . From Table 5.15 i t i s observed t h a t , under the assumptions of Procedure I, a d i f f e r e n t o p t i m a l area, v i z : 900 h e c t a r e s (EMV = $273,000), or 850 hectares (EU> 88.25) i s o b t a i n e d when the new s e t of p r i o r p r o b a b i l i t i e s i s s u b s t i t u t e d f o r the e m p i r i c a l p r i o r p r o b a b i l i t i e s with c o r r e s -ponding areas: 850 h e c t a r e s (EMV = $267,000) or 800 h e c t a r e s (EU = 88.03), from Table 5.7 f o r T=l and T=2. From Table 5.16 new o p t i m a l areas are o b t a i n e d , 1,200 hectares (EMV = $320,000) and 1050 liect.aron (i:W 90.44) as compared to 1150 hectares (EMV = $415,500) and 1,000 h e c t a r e s (EU = 90.25) from Table 5.12. From Table 5.17 the new o p t i m a l areas are 1,000 h e c t a r e s (EMV = $310,000) and 850 h e c t a r e s (EU =.90.40) as compared to 1,000 h e c t a r e s 360 T T I Fig.5. II TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION FOR PROCEDURE I. USING SUBJECTIVE PROBABILITIES. 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig. 5.12 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION FOR PROCEDURE I, USING SUBJECTIVE PROBABILITIES. 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig.5.13 TOTAL EXPECTED MONETARY VALUE VS.AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION AND SUBJECTIVE PROBABILITIES FOR PROCEDURE II . 50 1 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig.5.14 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION AND SUBJECTIVE PROBABILITIES FOR PROCEDURE II. Area under irrigation (hectares) Fig.5.15 TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION USING NON-STIMULATORY CROP-RESPONSE FUNCTION AND SUB JECTIVE PROBABILITIES FOR PROCEDURE II. 0 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig.5.16 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION USING NON-STIMULATORY CROP-RESPONSE FUNCTION AND SUBJECTIVE PROBABILITIES FOR PROCEDURE II. TABLE 5.15 OPTIMAL AREAS UNDER IRRIGATION USING SUBJECTIVE PROBABILITIES: PROCEDURE I. TOTAL EXPECTED MONETARY VALUE [EMV] CRITERION TOTAL EXPECTED UTILITY [EU] CRITERION TIME. INDEX (months) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED MONETARY VALUE ($1,000) „ OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED UTILITY T = 0 1,200 327.00 1,100 90.70 T = 1 ' 1,000 291.00 900 89.00 T = 2 800 255.00 800 87.50 . T = 3 600 218.00 600 86.00 T = 4 500 183.00 500 84.20 to o TABLE 5.16 OPTIMAL AREAS UNDER IRRIGATION USING THE STIMULATORY CROP RESPONSE FUNCTION AND SUBJECTIVE PROBABILITIES: PROCEDURE I I . TOTAL EXPECTED MONETARY VALUE, EMV, CRITERION TOTAL EXPECTED UTILITY, EU, CRITERION TIME INDEX (months). OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED MONETARY VALUE ($1,000) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED UTILITY T = 0 1, 600 407.0.0 1,400 93.59 T = 1 1,300 352.00 1,100 91.44 T = 2 1,100 304.00 1,000 89.44 T = 3 1,000 266.00 8 00 87.68 T = 4 800 239.00 800 86.43 TABLE 5.17 OPTIMAL AREAS UNDER IRRIGATION USING THE NON-STIMULATORY CROP RESPONSE FUNCTION AND SUBJECTIVE PROBABILITIES: PROCEDURE I I . TOTAL EXPECTED MONETARY VALUE, EMV, CRITERION TOTAL EXPECTED UTILITY, EU, CRITERION TIME INDEX (months] OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED MONETARY VALUE ($1,000) OPTIMAL AREA UNDER IRRIGATION (hectares) TOTAL MAXIMUM EXPECTED UTILITY T = 0 T = 1 T = 2 T = 3 T = 4 1,600 1,000 1, 000 800 800 475.00 350.00 2 8 8.00 242.00 204.00 1, 300 1,000 700 600 600 95.35 91.97 88.83 86.36 84.83 123 (EMV = $315,500) and 900 h e c t a r e s (EU = 90.50) from Table 5.13. 5.5.2 Minijnum p r o b a b i l i t i e s , P (6 h) t h a t leave the p r e f e r r e d a c t i o n o p t i m a l I t i s noted from the p r e c e d i n g s e c t i o n s t h a t f o r the e m p i r i c a l l y - e s t i m a t e d p r i o r p r o b a b i l i t i e s of Table 5.1 the Quilchena ranch i r r i g a t i o n o p e r a t o r would choose the o p t i m a l areas as given i n summary Tables 5.7; 5.12 and 5.13, using Procedure I or I I , r e s p e c t i v e l y . For Procedure I and f o r time index, T = 2, he would choose A g (800 hectares) i f he employed the EMV d e c i s i o n c r i t e r i o n , or (700 hectares) i f he adopts the EU d e c i s i o n c r i t e r i o n . S i m i l a r l y f o r Procedure II T = 2, he would choose A 1 Q (1,000 hectares) using the EMV-c r i t e r i o n , or A g (900 hectares) u s i n g the E U - c r i t e r i o n f o r the s t i m u l a t o r y crop response f u n c t i o n ; f o r the n o n - s t i m u l a t o r y crop response f u n c t i o n the o p t i m a l a c t i o n s are A 1 Q (1,000 hectares) f o r E M V - c r i t e r i o n , or A g (800 hectares) f o r EU-c r i t e r i o n . Since the e m p i r i c a l l y estimated p r i o r frequency d i s -t r i b u t i o n of the s t a t e s of nature may not e x a c t l y r e f l e c t the Quilchena ranch l o c a l c o n d i t i o n s , i t may be h e l p f u l t o the ranch manager to know how s m a l l the p r o b a b i l i t y of a s p e c i f i c s t a t e would have to be i n order t h a t the p r e f e r r e d a c t i o n a s s o c i a t e d with the s t a t e would s t i l l remain the optimum c h o i c e . For Procedure I, E M V - c r i t e r i o n , A g (800 hectares) corresponds to " f a i r h y d r o l o g i c a l s t a t e " , (©3)• For Procedure I I , EMV-c r i t e r i o n , A 1 0 (1,000 hectares) corresponds t o "normal 124 h y d r o l o g i c a l s t a t e " , ( 9 ^ ) . To compute how small the p r o b a b i l i t y of 9^ would have to be before the manager would p r e f e r another a c t i o n other than A f o r Procedure I, some other d e c i s i o n o -a l t e r n a t i v e s have to be c o n s i d e r e d t ogether with the optimal a c t i o n (A ). Let these other a c t i o n s be A_ , and A n. The a c t i o n s , and the monetary gains or l o s s e s a s s o c i a t e d with each s t a t e - a c t i o n combination are as t a b u l a t e d i n Table 5 . 1 8 ( a ) - l . From the t a b l e i t seems t h a t action. A^ would be optimal i f the d e c i s i o n maker knew f o r c e r t a i n t h a t e i t h e r 9^ or 9^ would occur; and t h a t a c t i o n Ag would be o p t i m a l , i f he knew t h a t 9 would occur with c e r t a i n t y . A c t i o n A j y would be o p t i m a l i f he knew f o r c e r t a i n t h a t e i g h e r 9^, 9^, 0,, or 9_ would occur. Table 5 . 1 8 ( a ) - l shows the p r e f e r r e d a c t i o n s i n i t a l i c s , given the s t a t e s of nature. To compute the minimum p r o b a b i l i t i e s when any a c t i o n A_. i s p r e f e r r e d , the d e v i a t i o n s , d.., of each element of the other a c t i o n s are c a l c u l a t e d from Table 5 . 1 8 ( a ) - l and shown i n Table 5 . 1 8 ( b ) - l . For each a c t i o n the minimum d.. i s s e l e c t e d , and iD c a l l e d d .. as shown i n Table 5.18(b)-1. The minimum proba-mj b i l i t y i s c a l c u l a t e d f o r each s t a t e of nature by the f o l l o w i n g formula: _ - m^k  P k d.. - d , (5.6) hk mk 125 TABLE 5.18(a)-1 EMV - VALUES FOR ACTIONS A-,, A g AND A PROCEDURE I, T = 2 S t a t e s of Nature A c t i o n s 1 2 3 A 7 = 700 Hect. Ag = 800 Hect. A 9 = 900 Hect. E n t r i e s i n ($1,000) 8^ : Very Poor 3.65 -4.40 -12. 45 8 2 : Poor 31.60 22.40 13. 20 8 3 : F a i r 47.60 49. 80 40. 60 8^ : Normal 56. 50 64. 60 72. 70 6 5 : Good 8.90 10.20 11. 50 6 6 : Very Good 53.60 61.20 68. 85 0^ : E x c e l l e n t 41.60 e 47.60 ,5 3. 5 5 TABLE 5 . 1 8 ( b ) - ! DEVIATIONS, d.., WHEN DIFFERENT ACTIONS ARE PREFERRED, ID' EMV-CRITERION, PROCEDURE I, T = 2 d. . When A ID 7 Is P r e f e r r e d d. . When A g Is P r e f e r r e d d. . When 1 13 .^g i s P r e f e r r e d S t a t e s of A c t i o n s , A. State s of A c t i o n s , A. Sta t e s o f A c t i o n s , A. Nature A 8 A J  A 9 Nature A 7 . A J  A 9 Nature A ? A 8 J 9 . 1 2 3 9 . l 1 3 9 . I 1 2 9 1 + 8 . 05 + 1 6 . 1 0 e i - 8 . 05 + 8 . 0 5 6 1 - 1 6 . 1 0 - 8 . 05 6 2 + 9 . 2 0 + 1 8 . 4 0 6 2 -9. 20 + 9 . 2 0 9 2 -18. 40 - 9 . 2 0 9 3 - 2 . 2 0 + 6 . 0 0 9 3 + 2 . 2 0 + 9 . 2 0 6 3 - 6 . 0 0 -9. 20 • 6 4 -8. 10 -1 6.2 0 9 4 + 8 . 1 0 -8.10 84 + 1 6 . 2 0 + 8 . 1 0 . 9 5 - 1 . 3 0 - 2 . 60 9 5 + 1 . 3 0 - 1 . 3 0 6 5 + 2 . 60 + 1 . 3 0 9 6 - 7 . 60 - 1 5 . 2 5 - 9 6 . + 7 . 6 0 - 7 . 6 5 0 6 + 1 5 . 2 5 +7 . 65 9 7. . - 6 . 00 - 1 1 . 9 5 9 7 + 6 . 00 - 5 . 95 6 7 + 1 1 . 9 5 + 5 . 9 5 Minimum of d i j = d m j - 8 . 1 0 - 1 6 . 2 0 - 9 . 2 0 - 8 . 1 0 - 1 8 . 4 0 - 9 . 2 0 .12 7 where i s the maximum p r o b a b i l i t y f o r each s t a t e , 0^, i n a c t i o n , A ; k takes on the same numbers as j except the number o f the s t a t e of nature f o r which the p r o b a b i l i t y i s being c a l c u l a t e d . Table 5.18(c)-1 shows the P - v a l u e s . From these P -K K. values the maximum P = P (0, ) i s s e l e c t e d and t a b u l a t e d i n K h Table 5 .18 (c ) - 1 , as w e l l . Any p r i o r p r o b a b i l i t y on 0^ t h a t i s equal to or g r e a t e r than t h i s P (0^) w i l l make the p r e f e r r e d a c t i o n o p t i m a l . Thus P (0^) v a l u e s , shown i n Table 5.18 ( c ) - l are the minimum p r o b a b i l i t i e s t h a t w i l l change the p r e f e r r e d a c t i o n i n comparison with a t l e a s t one other s t a t e of nature, and not n e c e s s a r i l y a g a i n s t a l l o t h e r s t a t e s of nature. S i m i l a r l y , to compute how s m a l l the p r o b a b i l i t y of 8^ would have to be before another a c t i o n o t h e r than becomes op t i m a l f o r Procedure I I , the same procedure i s used. Tables 5.18(b)-2 and 5.18(c)-2 are computed from Table 5.18(a)-2, showing the EMV v a l u e s f o r a c t i o n s A , A, n, and y i u A l l ' us-'-n9 t n e s t i m u l a t o r y crop response f u n c t i o n f o r Procedure I I , T = 2. F i g u r e 5.17 shows the minimum p r o b a b i l i t i e s i n r e l a t i o n to p r i o r p r o b a b i l i t i e s . 5 . 5 . 3 S e n s i t i v i t y of o p t i m a l p o l i c y to crop response f u n c t i o n The s e n s i t i v i t y of o p t i m a l p o l i c y to crop response f u n c t i o n i s c l e a r l y observed i n the use of s t i m u l a t o r y and n o n - s t i m u l a t o r y crop response f u n c t i o n s employed i n the a n a l y s i s f o r Procedure I I . From F i g u r e s 5.7, 5.8, f o r the s t i m u l a t o r y crop response, and, F i g u r e s 5 . 9 , 5.10, f o r the non-stimulatory TABLE 5.18(c)-! P AND P(9. ) VALUES, EMV-CRITERION, PROCEDURE I, T = 2 A c t i o n s, Aj 1 2 •3 S t a t e s of Nature A 7 A 8 A 9 Maximum 8 . 1 (700 Hectares) (800 Hectares) (900 Hectares) Over k = P(6 h) 6 1 : Very Poor P r e f e r r e d A c t i o n f o r 8]_ 0.501548 0.486443 0.501548 62 : Poor P r e f e r r e d A c t i o n f o r 02 0.468208 0.453195 0.468208 63 F a i r . 0.807018 P r e f e r r e d A c t i o n • f o r 8 3 0.468208 0.807018 P "4 : Normal 0.531792 0.531792 P r e f e r r e d f o r 84 A c t i o n 0.531792 6 5 : Good 0.876190 0.876190 P r e f e r r e d f o r 85 A c t i o n 0.876190 66 : Very Good 0.546805 0.545994 P r e f e r r e d f o r e 6 A c t i o n 0.546805 a "7 . E x c e l l e n t 0.606260 0.607261 P r e f e r r e d f o r 07 A c t i o n 0.607261 129 TABLE 5.18(a)-2 EMV - VALUES FOR ACTIONS A g, A 1 Q AND A PROCEDURE I I , T = 2 A c t i o n s 1 2 3 S t a t e s of Nature A g = 900 Hect. A10 " 1000 Hect. A l l =1100 Hect. E n t r i e s i n ($1,000) 8^ : Very Poor 5.40 3.90 2.90 8 2 : Poor 24.25 18.00 13.40 8 3 : F a i r 51.60 4.4.00 35.60 8^ : Normal 72.70 78.60 74.90 0' 5 : Good 11. 50 12.75 14.00 8 6 : Very Good 68.85 76.50 84.15 8^ : E x c e l l e n t 53.55 59.50 65. 45 TABLE 5 . 1 8 ( b ) - 2 DEVIATIONS, d.., WHEN DIFFERENT ACTIONS ARE PREFERRED, ' l j ' ' EMV-CRITERION, PROCEDURE I I , T = 2 d.. When A g Is P r e f e r r e d d.. When A 13 Is P r e f e r r e d d.. When A,, Is P r e f e r r e d ID 1 1 S t a t e s of A c t i o n s , A- Sta t e s of A c t i o n s , A. S t a t e s o f A c t i o n s , A. Nature j A Nature a J a Nature a J a . A i o A 9 1 1 A 9 A i o 6 . l 2 3 9 . l 1 3 8 . l 1 2 0 1 + 1 . 5 0 + 2 . 5 0 6 1 - 1 . 5 0 + 1 . 00 6 1 - 2 . 5 0 - 1 . 0 0 9 2 + 6 . 2 5 + 1 0 . 8 5 9 2 - 6 . 25 + 4 . 60 82 - 1 0 . 8 5 - 4 . 6 0 9 3 + 7 . 60 + 1 6 . 0 0 9 3 -7. 60 + 8 . 4 0 9 3 -16.00 -8. 40 9 4 - 5 . 9 0 - 2 . 2 0 6 4 • + 5 . 90 + 3 . 7 0 9 4 + 2 . 2 0 - 3 . 7 0 . ' 9 5 - 1 . 25 - 2 . 5 0 6 5 -+ 1 . 2 5 - 1 . 2 5 9 5 + 2 . 5 0 + 1 . 2 5 ' . 6 6 . - 7. 65 -15. 30 9 6 + 7 . 6 5 -7. 65 9 6 + 1 5 . 3 0 + 7 . 6 5 9 7 - 5 . 95 - 5 . 95 6 7 + 5 . 95 - 5 . 9 5 6 7 + 5 . 95 + 5 . 9 5 Minimum of d. . = d . xj m: - 7 . 65 - 1 5 . 3 0 - 7 . 6 0 - 7 . 6 5 - 1 6 . 0 0 - 8 . 4 0 TABLE 5.18(c)-2 Pv AND P(8, ) VALUES, EMV-CRITERION, PROCEDURE I I , T = 2 A c t i o n s -Aj 1 2 3 Maximum St a t e s o f Nature A 7 A 8 A 9 9 . l (900 Hectares) (1,000 Hectares) (1,100 Hectares) Over k = P ( 9 h ) 9 ^ : Very Poor P r e f e r r e d A c t i o n f o r 0 1 0.836066 0.859551 0.859551 8 2 : Poor • P r e f e r r e d A c t i o n f o r 9 2 0.550360 0.585086 0.585086 9 3 : F a i r P r e f e r r e d A c t i o n f o r 9 3 . 0.501639 0.488818 0.501639 9 ^ : Normal 0.566667 P r e f e r r e d A c t i o n f o r 0 4 0.674009 0.674009 9 5 : Good 0.864865 0.870466 P r e f e r r e d A c t i o n f o r 9 5 0.870466 9 g : Very Good 0.511182 0.520124 P r e f e r r e d A c t i o n f o r e 6 0.520124 9_, : E x c e l l e n t 0.728929 0.585366 P r e f e r r e d A c t i o n f o r 9 ? 0. 728929 CO I—1 132 OOi 0.80 0.60 0.40 0.20 0 (a ) PROCEDURE I - - - - PRIOR PROBABILITY I.OOi 0.80 0.60 0.40 0.20 (b) PROCEDURE II — --PRIOR PROBABILITY o1 0 5 10 15 20 25 30 6 3 Total irrigation season inflowUO m ) Fig . 5.17 MINIMUM PROBABILITIES P(en), THAT LEAVE THE PREFERRED ACTION OPTIMAL 133 crop response and, the corresponding summary Tables 5.12 and 5.13, i t i s shown that the proposed decision theory stochastic optimization model i s very sensitive to the crop response functions. For the stimulatory crop response function the optimal areas range from 800 hectares for T = 4 to 1,300 hectares for T = 1, and 600 hectares for T = 4 to 1,000 hectares for T = 1, for the non-stimulatory function, using the EMV-c r i t e r i o n . For the EU-criterion the corresponding ranges are 700 hectares to 1,000 hectares for the stimulatory function and 600 hectares to 1,000 hectares for the non-stimulatory function. Thus, not only are d i f f e r e n t decisions arrived at with d i f f e r e n t p r i o r p r o b a b i l i t i e s but also with d i f f e r e n t crop response functions. 5.5.4 S e n s i t i v i t y of optimal p o l i c y to u t i l i t y function Given sp e c i f i e d p r i o r p r o b a b i l i t i e s and sp e c i f i e d crop response function, and varying the u t i l i t y function of the decision maker the optimal policy i s changed. In both Procedures I and II this aspect of the model i s obvious. From Table 5.7 for Procedure I the optimal policy using the l i n e a r u t i l i t y function seems to be better than the corresponding output using a quadratic u t i l i t y function. From Tables 5.12 and 5.13 for Procedure II the same'trends are observed. The Quilchena ranch manager's u t i l i t y function shown in Figure 5.4(b), which i s quadratic, depicts diminishing marginal u t i l i t y for gains and increasing marginal d i s u t i l i t y for losses. This manager i s 134 "conservative" in the sense that i n a risky s i t u a t i o n , such as that of operating his i r r i g a t i o n system under hydrological uncertainty, he would prefer the action with lower v a r i a b i l i t y , even i f both actions were to have the same expected monetary value. His aversion to r i s k i s high and he does not l i k e "gambling". Thus, this manager i s an expected u t i l i t y maxi-mizer and not an expected monetary value maximizer. Therefore his decision behavior (risk taking) i s better represented by his u t i l i t y function of Figure 5.4(b). If the manager's u t i l i t y function had turned out to be a cubic function a d i f f e r e n t set of optimal areas would r e s u l t . Thus, in general, the model i s very sensitive to u t i l i t y functions. 5.5.5 S e n s i t i v i t y of optimal p o l i c y to assumptions a) Uncertainty i n yield's, factor prices (costs) and incomes (revenues) In computing the crop net value under Section 5.2.5, the variable cost component, TVC\, of equation (4-3) was computed under the assumption that depreciation and, unpaid interest on investment were the same for a l l the decision alternatives under consideration. Thus planting costs and harvesting costs are assumed the same throughout this study, at $175/hectare and $12.5/hectare, respectively. In r e a l i t y ^ f a c t o r prices such as labour costs and costs of other inputs vary from year to year. But the above assumption i s j u s t i f i e d by the fact that, i n Quilchena area, most of the controllable variable cost inputs are provided by the ranchers themselves. Human labor i s not used much since these ranchers have planting and 135 h a r v e s t i n g equipment, and only d e p r e c i a t i o n and unpaid i n t e r e s t are the most s i g n i f i c a n t p a r t of the t o t a l v a r i a b l e c o s t s . In computing the t o t a l revenue component, TR_. , of equation (4-3) the revenue was assumed to be $60/ton, a p p r o x i -mately, and the y i e l d was assumed as 2.0 tons/acre or 5.0 tons/hectare per c u t of a l f a l f a . In the Quilchena area the revenue from a l f a l f a v a r i e s between $60 - $120/ton, and y i e l d v a r i e s from 2 - 4 t o n s / a c r e , t h a t i s , 5 - 1 0 t o n s / h e c t a r e . For s i m p l i c i t y i n the s e n s i t i v i t y a n a l y s i s , with r e s p e c t to u n c e r t a i n t y i n crop net valu e , the p l a n t i n g and h a r v e s t i n g c o s t s are assumed c o n s t a n t . The revenue i s a l s o assumed con s t a n t a t $60/ton d u r i n g the o p t i m i z a t i o n p e r i o d . The y i e l d w i l l be v a r i e d from 2.0 tons/acre or 5.0 tons/hectare to 3.0 tons/acre or 7.5 t o n s / h e c t a r e / c u t of a l f a l f a . For 2 cuts of a l f a l f a t h i s would be e q u i v a l e n t to 15.0 tons/hectare. From equation (5-5), TVC. PC . + HC . 3 3 • 3 = $175 + $12.5 = $187.5/hectare From equation (4-3), n . = TR. - TVC. 3 3 3 = $912.5 - $187.5 = $725/hectare. To t e s t the s e n s i t i v i t y o f optimal p o l i c y to changes i n crop net v a l u e , which i s r e f l e c t e d only i n changes of crop y i e l d , the computer program was rerun u s i n g a net value of 136 $725/hectare (instead of $425/hectare), with the states of nature, t h e i r empirical p r i o r p r o b a b i l i t i e s , and the decision alternatives the same. The output from t h i s analysis i s presented in Figures 5.18, 5.19; and 5;20 and 5.21, 5.22 and 5 - 2 3 f for Procedures I and I I , respectively. The corresponding summary tables are shown i n Tables 5.19, 5.20 and 5.21 for Procedures I and I I , respectively. From Table 5.19 for T = 1 and T = 2, average optimal areas are 950 hectares (EMV = $480,500), and 750 hectares (EU = 95.48), as compared to 850 hectares (EMV = $267,000) and 800 hectares (EU = 88.03) from Table 5.7. The EMV-criterion . shows increase i n optimal hectarage and a swelling EMV-value. The EU-criterion, however, shows a drop in both the hectarage and u t i l i t y , i n d i c a t i n g the theory of diminishing marginal u t i l i t y for gains as manifested i n the u t i l i t y function of Figure 5.4(b). From Table 5.20 optimal values are 1150 hectares (EMV = $538,500) and 850 hectares (EU = 96.48) as compared with 1150 hectares (EMV = $315,500) and 1000 hectares (EU = 90.25) from Table 5.12. Table 5.20 gives 1100 hectares (EMV = $549,500) and 750 hectares (EU = 97.18) as compared with 1000 hectares (EMV = $315,500) and 900 hectares (EU = 90.50) from Table 5.13. The greatest e f f e c t of increasing the crop net value i s i n the EMV-value and not so much i n the EU-criterion. I t should be noted that although the s e n s i t i v i t y of optimal policy to changes i n crop net value (yield) has been 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig.5.18 TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION FOR PROCEDURE I , YIELD = 7.5 TONS PER HECTARE. 5 Fig.5.19 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION FOR PROCEDURE I, YIELD = 7.5 TONS PER HECTARE. 700 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hectares) Fig. 5. 20 TOTA L EXPECTED MONETARY VALUE -VS. AREA UNDER IRRIGATION USING STIMULATORY CROP- RESPONSE FUNCTION FOR PROCEDURE II, YIELD = 7.5 TONS PER HECTARE 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irriga tion ( hectares) Fig.5.21 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE I I , YIELD = 7.5 TONS PER HECTARE. r -Area under irrigation (hectares) Fig. 5.22T0TAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION USING i NON-STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE I I , YIELD = 7.5 TONS PER HECTARE .i 0 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irrigation (hec ta res ) Fig.5.23 TOTAL EXPECTED UTILITY VS.AREA UNDER IRRIGATION USING NON - STI M UL ATOR Y CROP-RESPONSE FUNCTION FOR PROCEDURE II, YIELD = 7.5 TONS PER HECTARE. TABLE 5.19 OPTIMAL AREAS UNDER IRRIGATION FOR CHANGED CROP NET VALUE: PROCEDURE I T o t a l Expected Monetary Value, [EMV], C r i t e r i o n T o t a l Expected U t i l i t y , [EU], C r i t e r i o n Time Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) • Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,300 587.00 900 97.24 T = 1 1,000 510.00 800 96.28 T = 2 900 451.00 700 94.68 T = 3 700 386.00 600 93.12 T = 4 600 327.00 500 90.71 TABLE 5.20 OPTIMAL AREAS UNDER IRRIGATION USING THE STIMULATORY CROP RESPONSE FUNCTION AND CHANGED CROP NET VALUE: PROCEDURE I I T o t a l Expected Monetary Value, [EMV] C r i t e r i o n T o t a l Expected U t i l i t y , [EU], C r i t e r i o n Time Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,600 677.00 1,000 98.15 T = 1 1,300 577.00 900 97. 24 T = 2 1,000 500.00 800 95. 71 T = 3 900 437.00 700 93.92 T = 4 800 398.00 600 92.62 TABLE 5.21 OPTIMAL AREAS UNDER IRRIGATION USING THE NON-SIMULATORY CROP RESPONSE FUNCTION AND CHANGED CROP NET VALUE: PROCEDURE I I T o t a l Expected Monetary Value, [EMV] C r i t e r i o n T o t a l Expected U t i l i t y , [EU] C r i t e r i o n Time Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,600 790.00 900 99.29 T = 1 1,200 598.00 800 98. 32 T = 2 1,000 501.00 7 00 96. 04 T •= 3 800 423.00 600 93.12 T = 4 8 00 366.00 600 91.41 146 performed using only two y i e l d values, 5.0 tons/hectare and 7.5 tons/hectare, the uncertainty associated with y i e l d would have been better represented by taking the range of y i e l d values, from 5-10 tons/hectare and attaching a p r o b a b i l i t y to each value. This was not done because of lack of long-term y i e l d data which could have been used to estimate the empirical p r i o r p r o b a b i l i -t i e s of y i e l d . Also i n this study the emphasis i s on hydrologi-c a l uncertainty and other uncertainties are t r i v i a l . 5.5.5 b) Uncertainty i n i r r i g a t i o n water requirement In the computational procedure i t was assumed that the i r r i g a t i o n water requirement was pr e c i s e l y known and that the . a l f a l f a crop consumed water at a constant d a i l y rate during the entire i r r i g a t i o n season. In pr a c t i c e , t h i s i s not correct. I r r i g a t i o n water requirement varies from year to year, and, even in any given year, from month to month, week to week, and day to day, depending on c l i m a t i c conditions. Typical monthly variations i n the consumptive uses of a l f a l f a i n the Quilchena area during the i r r i g a t i o n season are shown in Table 5.2. An average value of 6.35 mm/day was assumed for the entire i n v e s t i -gation season, May to September, of 15 3 days. For optimal long-term development and operation of i r r i g a t i o n systems in the Quilchena area the long-run (prior) p r o b a b i l i t i e s of t o t a l i r r i g a t i o n seasonal consumptive uses of a l f a l f a ought to be derived from h i s t o r i c a l records, and these employed to improve the p r a c t i c a l a p p l i c a b i l i t y of the proposed decision theory stochastic optimization model. The uncertainties associated 147 with the t o t a l i r r i g a t i o n season inflows Into the i r r i g a t i o n storage reservoir should be combined with the uncertainties associated with the t o t a l growing season i r r i g a t i o n water requirement for a more complete stochastic analysis and output. The estimation of empirical p r i o r p r o b a b i l i t i e s with respect to t o t a l growing season i r r i g a t i o n water requirement has not been done since the data of Tables B.7, B.8, and B.9 of Appendix B were not long enough and not r e l i a b l e . There-fore the range of t o t a l seasonal i r r i g a t i o n water requirements and the p r o b a b i l i t i e s shown i n Table 5.22 have been derived subjectively, using the l i t t l e information derived from the B r i t i s h Columbia I r r i g a t i o n Guide (14). This source indicates that the t o t a l seasonal evapotranspiration for pasture based on a d a i l y water budget for coastal regions of B r i t i s h Columbia varies from year to year. Table B.10, Appendix B, gives 11 years data, 1955-1965, for Agassiz Experimental Station (Canada Department of Agric u l t u r e ) . The range of peak values shown i n Table 5.22 was derived by taking into consideration the t o t a l evapotranspiration of a l f a l f a , summer r a i n f a l l contribution, winter moisture contribution, as well as the leaching requir-ment and other design considerations for a l f a l f a i n the Quilchena area, that i s the hot dry i n t e r i o r of B r i t i s h Columbia. Information obtained from the ranchers in the Quilchena area has also been incorporated i n the subjective assessment of the p r i o r p r o b a b i l i t i e s . For consistency i t was deemed desirable 14 8 TABLE 5.22 COMPUTATION OF PRIOR PROBABILITIES OF IRRIGATION WATER REQUIREMENT FOR ALFALFA, QUILCHENA AREA Sta t e s of Nature (6 k) I r r i g a t i o n Water Requirement Index I n t e r v a l (m 3/Hectare/Season) P r i o r P r o b a b i l i t i e s P ( 0 k ) 6]_ : Very Poor (Extremely hot day) 22,875-24,705 + 3a 0.042 0 2 : Poor (Very, very hot day) 21,045-22,875 +2a 0.095 0 3 : F a i r (Very hot day) 19,215-21,045 + 10 0.155 8 4 : Normal (Hot/warm day) 17,385-19,215 a=0 0.300 0 5 : Good (Cool day) 15,555-17,385 - l a 0.240 0g : Very Good (Very Cool day) 13,725-15,555 -2a 0.135 07 : E x c e l l e n t (Cool/wet day) 11,895-13,725 -3a 0.033 T o t a l 1. 000 149 to define seven states of nature with respect to i r r i g a t i o n water requirement, as shown i n Table 5.22. To assess the e f f e c t of uncertainty i n i r r i g a t i o n water requirement on the optimal p o l i c y , the inflow p r o b a b i l i -t i e s were combined with the i r r i g a t i o n water requirement p r o b a b i l i t i e s for each state of nature. Since the inflows and i r r i g a t i o n water requirement are considered inversely i n t e r -dependent and i n t e r r e l a t e d , both being dir e c t functions of the hydrometereological and cl i m a t i c factors, the two p r o b a b i l i t i e s are combined thus: where, P ( e i e k > = p ( e i ) p ( e k ) / e i ) (5.7) {9. 0, } i s the event that "both 9. and 0, " occur, that i s , {9^ 9^1 i s the compound event; P {9^ 9K> i s the pr o b a b i l i t y that both events occur; P (9^) i s the p r o b a b i l i t y that 9^ occur P (9^/9^) i s the conditional p r o b a b i l i t y of 9^ given that 9^ has occurred. If P (9^/©^) i s replaced simply by P (9^), consistent with Table 5 . 22 , then. P { 9 i 9 k } = P ( 6 i } P ( e k } * ( 5 ' 8 ) When the uncertainties i n inflow and i r r i g a t i o n water requirement are combined, assuming that u t i l i t y function i s non-linear, as shown i n Figure 5.4(b), then equation (4-9) for computing the t o t a l expected u t i l i t y of the decision alternatives i s m o d i f i e d as f o l l o w s : N n n MAX ( E ) E p (0, ) . E p (0. ). • U (A..) i . e . j = l k=l k i = l 1 3 N n n F = MAX ( Z . E E p (0. ) . p (©. ). U (A..) (5 j = l J k=l i = l k 3 S i m i l a r l y , assuming a l i n e a r u t i l i t y f u n c t i o n of Fi g u r e 5.4(a) equation (4-12) i s m o d i f i e d as f o l l o w s : N n n F = MAX ( E ) E p (0 ). E p ( 0 . ) . M (A..) i . e . 'j = l k=l K i = l • 1 3 1 N n n F = MAX ( E . E E p (0,). p (© ) . M (A..) (5 j = l ) k=l i = l •* 1 3 151 Equations 5.9 and 5.10 are used to compute the t o t a l expected u t i l i t i e s , EU, and t o t a l expected monetary v a l u e s , EMV, f o r a l l the a c t i o n s and the outputs are p l o t t e d i n F i g u r e s 5.24 and 5.25; 5.26 and 5.27, and 5.28 and 5.2.9 f o r Procedures I and I I , r e s p e c t i v e l y ; and Tables 5.23, 5.24 and 5.25, r e s p e c t i v e l y . From Tables .5.2 3, and 5.7, .5.25. and 5.12, and Tables 5.25 and 5.13, i t i s r e a d i l y observed t h a t u n c e r t a i n t y i n i r r i g a t i o n water requirement tends to decrease both the expected monetary values EMV and the expected u t i l i t y EU. I t i s a l s o observed t h a t the s t o c h a s t i c o p t i m i z a t i o n model c o n s i s t e n t l y assesses the r i s k i n the s i t u a t i o n as shown by the new o p t i m a l areas v i z : from Table 5.23, 950 hectares (EMV = $44,000), and 950 h e c t a r e s (EU = 12.56) as compared t o 850 hectares (EMV = $267,000) and 800 h e c t a r e s (EU = 88.03), from Table 5.7; from Table 5.24, 1050 h e c t a r e s , (EMV = $50,000), and 1050 h e c t a r e s (EU = 12.85) , as compared t o 1150 h e c t a r e s (EMV = $315,500), and 1000 h e c t a r e s (EU = 90.25), from Table 5.12; from Table 5.25, 1110 h e c t a r e s (EMV .= $54,000) and 1100 h e c t a r e s (EU = 12.94), as compared t o 1000 h e c t a r e s (EMV = $315,500) and 900 h e c t a r e s (EU = 90.50), from Table 5.13. C o n s i s t e n t l y , a l s o , an important e f f e c t of combined u n c e r t a i n t i e s i n i n f l o w and i r r i g a t i o n water requirement i s t h a t of a r r i v i n g at the same opt i m a l area i r r e s p e c t i v e of c r i t e r i o n adopted (EMV or EU), but c o n s i s t e n t l y again, Procedure II g i v e s l a r g e r hectarages than Procedure I. 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Area unde r irrigation (hectares) Fig. 5.24 TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION FOR PROCEDURE 1.WITH UNCE RTAIN T Y IN CONSUMPTIVE USE AND INFLOW INTEGRATED. ^ tn 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irr igation (hecta re s ) Fig.5.25 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION FOR PROCEDURE I, " WITH UNCERTAINTY IN CONSUMPTIVE USE AND INFLOW INTEGRATED. 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area under irriga tion ( hectares ) Fig.5.26 TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE I I , WITH UNCERTAINTY IN CONSUMPTIVE USE AND INFLOW INTEGRATED. 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 4400 4800 5200 Area' under irrigation (hectares). Fig.5.27 TOTAL EXPECTED UTILITY VS.AREA UNDER IRRIGATION USING STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE II, WITH UNCERTAINTY IN CONSUMPTIVE USE AND INFLOW INTEGRATED. 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Area under irrigation (hectares) .5.23 TOTAL EXPECTED MONETARY VALUE VS. AREA UNDER IRRIGATION USING NO^-STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE II, WITH UNCERTAINTY IN CONSUMPTIVE USE AND INFLOW INTEGRATED. Fig.5.29 TOTAL EXPECTED UTILITY VS. AREA UNDER IRRIGATION USING NON-STIMULATORY CROP-RESPONSE FUNCTION FOR PROCEDURE I I , WITH UNCERTAINTY IN CONSUMPTIVE USE AND INFLOW INTEGRATED. TABLE 5.23 OPTIMAL AREAS UNDER IRRIGATION CONSIDERING THE UNCERTAINTY IN IRRIGATION WATER REQUIREMENT: PROCEDURE I T o t a l Expected Monetary Value, [EMV] C r i t e r i o n T o t a l Expected U t i l i t y , [EU], C r i t e r i o n Time Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,100 48.00 1,100 12. 81 T = 1 1,000 45.00 1,000 12. 68 T = 2 900 43.00 900 12. 51 T = 3 800 38.00 800 12.29 T = 4 600 32.00 600 12. 0.4 TABLE 5.24 OPTIMAL AREAS UNDER IRRIGATION USING THE STIMULATORY CROP RESPONSE FUNCTION AND CONSIDER-ING THE UNCERTAINTY IN IRRIGATION WATER REQUIREMENT: PROCEDURE II T o t a l Expected Monetary Value, [EMV] C r i t e r i o n T o t a l Expected U t i l i t y , [EU] C r i t e r i o n Time . Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,400 58.00 1,300 13.15 T = 1 1,100 52.00 1,100 12.94 T = 2 1,000 48. 00 1,000 12.75 T = 3 900 43. 00 900 12.53 T = 4 800 39.00 800 12. 37 TABLE 5.25 OPTIMAL AREAS UNDER IRRIGATION USING THE NON-STIMULATORY CROP RESPONSE FUNCTION AND CONSIDERING UNCERTAINTY IN IRRIGATION WATER REQUIREMENT: PROCEDURE I I T o t a l Expected Monetary Value, [EMV], C r i t e r i o n T o t a l Expected U t i l i t y , [EU], C r i t e r i o n Time Index (Months) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected Monetary Value ($1,000) Optimal Area Under I r r i g a t i o n (Hectares) T o t a l Maximum Expected U t i l i t y T = 0 1,600 72.00 1,400 13. 52 T = 1 1,200 58. 00 1,200 13.08 T = 2 1,000 50.00 1,000 12.79 T = 3 1, 000 42.00 800 12.49 T = 4 800 37.00 800 12.17 161 5.6 Computation of Bayesian S t r a t e g i e s Equation (4-19) showed how the p r i o r p r o b a b i l i t i e s d i s t r i b u t i o n can be r e v i s e d u s i n g c o n d i t i o n a l p r o b a b i l i t i e s i n Baye's formula. Many improved d e c i s i o n s i n i r r i g a t e d a g r i c u l t u r e are made a f t e r we have searched f o r some a d d i t i o n a l i n f o r m a t i o n about the l i k e l i h o o d o f the h y d r o l o g i c a l s t a t e s of nature. Since the h y d r o l o g i c a l i n f o r m a t i o n obtained i s not to be c o n s i d e r e d p e r f e c t i n regard t o p r e d i c t i n g which h y d r o l o g i -c a l s t a t e o f nature w i l l occur, the ou t s t a n d i n g c h a r a c t e r i s t i c of a d d i t i o n a l i n f o r m a t i o n or data from the p r e d i c t i n g device i s t h a t i t comes from some p r o b a b i l i t y d i s t r i b u t i o n , and t h i s . p r o b a b i l i t y d i s t r i b u t i o n c o u l d be i n t e r p r e t e d as p r o v i d i n g the p r o b a b i l i t y of an o b s e r v a t i o n "Z" given the h y d r o l o g i c a l s t a t e of nature, 9^. This p r o b a b i l i t y d i s t r i b u t i o n i s c a l l e d the p o s t e r i o r p r o b a b i l i t y d i s t r i b u t i o n . To analyze the i r r i g a t i o n management d e c i s i o n problem of the t y p i c a l rancher i n the Quilcehan area i n a framework t h a t employs p o s t e r i o r d i s t r i b u t i o n , a t y p i c a l o p e r a t i o n of a seasonal nature has to be co n s i d e r e d . In t h i s o p e r a t i o n , i n s t e a d of a l l year round cow-calf o p e r a t i o n , i t i s c o n s i d e r e d t h a t the c a t t l e are purchased i n e a r l y s p r i n g (during A p r i l ) and s o l d at the end of the summer (around October 1). Thus the i r r i g a t e d hectarage f o r feed p r o d u c t i o n can be v a r i e d e a s i l y from year to year depending on the outlook o f the hydro-l o g i c a l c o n d i t i o n s . T h i s , a l s o , i m p l i e s that the s t o c k i n g r a t e s can be v a r i e d e a s i l y from year t o year depending on the expected l r a y - a l f a l f a t h a t can be p o s s i b l y produced each year. 162 The i r r i g a t i o n o perator might use the observed snow c o n d i t i o n s and temperatures i n e a r l y s p r i n g as a p r e d i c a t o r or i n d i c a t o r of the tr u e i n f l o w c o n d i t i o n d u r i n g the subsequent summer i r r i g a t i o n p e r i o d . The c a s u a l long-term o b s e r v a t i o n of the Quilchena ranch manager i n d i c a t e s t h a t the growing season i n f l o w i s a f u n c t i o n of the amount of w i n t e r s n o w f a l l and e a r l y s p r i n g maximum d a i l y temperatures. 5.6.1 C a l c u l a t i o n s of c o n d i t i o n a l p r o b a b i l i t i e s , P(Z/9) C o n d i t i o n a l p r o b a b i l i t i e s , P(Z/9), are c a l c u a l t e d as shown i n Table 5.26(A). The a v a i l a b l e h i s t o r i c a l h y d r o l o g i c a l (inflow) data are summarized on the l e f t s i d e of the t a b l e . The number of years of occurrence of v a r i o u s combinations of h y d r o l o g i c a l c o n d i t i o n s i n e a r l y s p r i n g ( A p r i l ) and the sub-sequent summer i r r i g a t i o n p e r i o d are shown and these are converted to the c o n d i t i o n a l p r o b a b i l i t i e s , P(Z/9), on the r i g h t s i d e . These c o n d i t i o n a l p r o b a b i l i t i e s sum to 1.000 by rows. 5.6.2 C a l c u l a t i o n of j o i n t p r o b a b i l i t i e s of 9 and Z, P (Z/9) P (9) The c a l c u l a t e d c o n d i t i o n a l p r o b a b i l i t i e s , P (Z/9) i n each row are m u l t i p l i e d by the p r i o r p r o b a b i l i t i e s of the d i f f e r e n t h y d r o l o g i c a l . s t a t e s of nature, P (9^) i n t h a t row to d e r i v e the j o i n t p r o b a b i l i t i e s of the v a r i o u s combinations of 9 and Z. Summing the p r o b a b i l i t i e s i n each column of TABLE 5.26(A) NUMBER OF YEARS OF OCCURRENCE OF VARIOUS COMBINATIONS OF HYDROLOGICAL CONDITIONS IN APRIL AND THE SUBSEQUENT SUMMER IRRIGATION PERIOD, AND CALCULATION OF CONDITIONAL PROBABILITIES, P(Z/6) Summer Hydro-log i c a l Observed Hydrological (Snowpack) Conditions Early Spring (April) (Z^ ) Conditional Probabilities, P(Z/6) Prior Prob-(Inflow) Conditions ( 6 ± ) Z l Z2 Z3 Z4 Z5 Z6 Z7 Very Poor • Poor Fair Nor-mal Good Very Good Excel-ent Total P(Z1/ a) p(z 2/ e) P(Z3/9) p(z4/e) P(z 5/ e) p(z6/e) P(Z7/6) Sum a b i l i t y P(9) 9 1 : Very Poor 8 2 2 - - - 12 0.670 0.170 0.170 - - - - 1.000 0.140 6 2 : Poor 1 9 2 1 1 - - 14 0.070 0.640 0.140 0.070 0. 070 . - - 1.000 0.160 6 3 : Fair - - 7 4 3 - - 14 - - 0.500 0.290 0.210 - - 1.000 0.160 6. : Normal 4 — — 4 9 3 1 - 17 - 0.240 0.530 0.180 0.060 - 1.000 0.190 6g : Good - - 1 2 - .- 03 - - - 0.330 0.670 - - 1.000 0. 030 6 g : Very Good - 1 4 4 6 1 16 - - 0.060 0.250 0.250 0.380 0. 060 1.000 0.180 . 8^ : Excellent - - 3 2 4 3 12 - - - 0.250 0.170 0.330 0.250 1.000 0.140 Total 9 11 16 22 15 11 •4 88 CO 164 Table 5.26(B) g i v e s the marginal p r o b a b i l i t i e s , P (Z), v i z p(z) = P(z/e ]) P i e ^ ' + P f ^ ) p(e 2) + ... + (p(z/e ?) p(e ?) i .e 7 p(z) = z p(z/e.) p(e.) ( 5 . i i ) e=i 1 1 5.6.3 C a l c u l a t i o n of p o s t e r i o r p r o b a b i l i t i e s , P(9^/Z) Table 5.26(C) shows the c a l c u l a t i o n of the p o s t e r i o r d i s t r i b u t i o n P(9/Z) from the components of Baye's formula given i n Table 5.26(B) , v i z : P(9/Z) = P(9) P(Z/9) v P(Z) (5.12) F i g u r e 30 i s the p l o t of the cumulative p o s t e r i o r p r o b a b i l i t y d i s t r i b u t i o n of growing season i n f l o w f o r the v a r i o u s combina-t i o n s of 9 and Z. 5.6.4 Computation of the Bayesian s t r a t e g i e s Tables 5.27, 5.28, and 5.29 are the c a l c u l a t i o n s i n v o l v e d i n d e r i v i n g the Bayesian s t r a t e g i e s u s i n g the p o s t e r i o r p r o b a b i l i t i e s shown i n Table 5.26(C) , f o r Procedures I, and I I , r e s p e c t i v e l y . The expected value of each s t a t e - a c t i o n combination i s c a l c u l a t e d f o r each Z, o b s e r v a t i o n and summari-k zed i n the t a b l e s shown. For any given Z , the expected net value i s obtained f o r a c t i o n , A., and s t a t e of nature, 9., TABLE 5.26(B) CALCULATION OF JOINT PROBABILITIES OF 0 AND Z, P(Z/0) P(0) Observed H y d r o l o g i c a l (Snowpack) C o n d i t i o n s , E a r l y S p r i n g ( A p r i l ) Summer H y d r o l o g i c a l ( I n f l o w ) C o n d i t i o n s <e.) Z l Z2 Z4 Z 5 Z6 Z7 Sum 0^  : Very Poor 0. 090 0. 020 0. 020 - - - -0 2 : Poor 0. 010 0.100 0. 020 0. 010 0. 010 - -03 : F a i r - - 0.080 0.050 0.030 - -• 04 : Normal - - ' 0.050 0.100 0.030 0. 010 05 : Good - - - 0. 010 0. 020 - -06 : Very Good - - 0. 010 0.050 0. 050 0. 070 0.010 ! 0^  : E x c e l l e n t - - - 0. 040 0. 020 0.050 i i 0.040 i P(Z) 0.100 0.120 0.180 0.260 0.160 0.130 0. 050 1. 000 TABLE 5.2 6 (C) CALCULATION OF POSTERIOR PROBABILITIES, P(Q±/Z) Summer Observed H y d r o l o g i c a l (Snowpack) C o n d i t i o n s , E a r l y S p r i n g ( A p r i l ) (Z^) (Inflow) C o n d i t i o n s (0.) Z l Z2 Z 3 Z4 Z 5 Z6 Z7 P ( 6/Z 1) P ( 9/Z 2) P (9/Z 3) P(0/Z 4) P ( 6/Z 5) P ( 6/Z 6) P(9/Z 7) 7,500 x 1 0 3 m 3 0-^:Very Poor 0.900 0.170 0.110 10,500 x 10 3m 3 0 2: Poor 0.100 0.830 0.110 0. 040 0.060 - -13,500 x 10 3m 3 0^: F a i r • _ 0.440. 0.190 0.19 0 _ _ 16,500 x 10 3m 3 0^: Normal 0.280 0.380 0.190 0.08 0 — 19, 500 x 10 3m 3 0 5: Good 0.040 0.130 22,500 x 1 0 3 m 3 0 g: Very Good _ _ 0. 060 0.190 0.300 0. 540 0.200 25,200 x 1 0 3 m 3 0y: E x c e l l e n t _ _ _ 0.16 0 0.130 0. 380 0. 800 Sum 1.000 1. 000 1.000 1. 000 1.000 1. 000 1. 000 167 Fig.5.30 CUMULATIVE POSTERIOR PROBABILITY DISTRIBUTION OF IRRIGATION SEASON INFLOW. T A B L E 5.27 COMPUTATION OF BAYESIAN STRATEGY USING POSTERIOR PROBABILITIES: PROCEDURE I, EMV-CRITERION 168 Hydrological States of Nature: I r r i g a t i o n Season Inflows ( 6 ^ e e e V e r y P o o r P o o r F a i r N o r m a l G o o d V e r y G o o d E x c e l l e n t " 1 0 1 0 0 0 H e c t . A c t i o n s : I r r i g a b l e A r e a s ( H e c t a r e s ) , ( A ^ ) ' " 1 1 1 1 0 0 H e c t . 12 1 2 0 0 H e c t . " 1 3 1 3 0 0 H e c t . "14 1 4 0 0 H e c t . " 1 5 1 5 0 0 H e c t . ( $ 1 , 0 0 0 ) " 1 6 1 6 0 0 H e c t P O ) 82.140 I B S . 000 287. 860 290. 710 4 2 5 . 0 0 0 4 2 5 . 0 0 0 4 2 5 . 0 0 0 6 4 . 6 4 0 1 6 7 . 5 0 0 2 7 0 . 3 6 0 3 7 3 . 2 1 0 407. 500 4 6 7 . 5 0 0 4 6 7 . 5 0 0 4 7 . 1 4 0 1 5 0 . 0 0 0 2 5 2 . 8 6 0 3 5 5 . 7 1 0 4 5 8 . 5 7 0 5 1 0 . 0 0 0 5 1 0 . 0 0 0 I I ! I I E x p e c t e d New I n c o m e , [ E M V ] , f r o m a P e r f e c t P r e d i c t o r = [ ( 8 2 . 1 4 x 0.14) + ( 1 8 5 . 0 x 0.16) + ( 2 8 7 . 8 6 x 0.1 + ( 3 9 0 . 7 1 X 0 . 1 9 ) + ( 4 6 7 . 5 x 0 . 0 3 ) + ( 5 4 3 . 9 3 x O . + ( 6 1 1 . 7 9 x 0 . 1 4 ) ) x 1 , 0 0 0 = $ 3 5 8 , 9 8 0 2 9 . 6 4 0 1 3 2 . 5 0 0 2 3 5 . 3 6 0 3 3 8 . 2 1 0 4 4 1 . 0 7 0 543. 930 5 5 2 . 5 0 0 1 2 . 1 4 0 1 1 5 . 0 0 0 2 1 7 . 8 6 0 3 2 0 . 7 1 0 4 2 3 . 5 7 0 5 2 6 . 4 3 0 5 9 5 . 0 0 0 - 5 . 3 6 0 9 7 . 5 0 0 2 0 0 . 3 6 0 3 0 3 . 2 1 0 4 0 6 . 0 7 0 5 0 8 . 9 3 0 611. 790 - 2 2 . 8 6 0 8 0 . 0 0 0 1 8 2 . 8 6 0 2 8 5 . 7 1 0 3 8 8 . 5 7 0 4 9 1 . 4 3 0 5 9 4 . 2 9 0 0.140 0.160 0.1G0 0.190 0 .030 0.180 0.140 Z O b s e r v a t i o n I I x p e c t e d N e t I n c o m e s [EMV] G i v e n Z O b s e r v a t i o n A 10 A l l A 12 A 13 A 14 A 15 A 16 P ( Z ) Z l 92 .43 74 .93 57 .43 39 .93 22 .43 4 .93 - 1 2 . 57 0 .100 Z2 167 . SI 1 5 0 . 02 13 2 .51 1 1 5 .02 97 51 80 . 02 62 .51 0 .120 Z 3 290 85 2 7 7 .45 2 6 3 .15 2 4 8 .74 2 3 1 24 2 1 3 .74 1 9 6 .24 0 .180 Z 4 376 .31 3 8 2 .20 386 .05 387 93 3 8 0 02 3 6 8 .02 3 5 0 5 1 0 . 2 6 0 Z 5 3 7 8 .02 ' 394 .14 4 0 3 .53 409 28 399 56 3 8 6 53 369 02 0 1 6 0 Z 6 4 2 2 26 459 96 497 66 5 3 0 73 536. 03 5 3 1 56 514 06 0 1 3 0 Z 7 4 2 5 00 4 6 7 50 5 1 0 00 5 5 0 79 5 8 1 29 591 5 7 3 72 0 05 0 .', When Z x i s o b s e r v e d , f o l l o w A 1 Q When Z 2 i s o b s e r v e d , f o l l o w A 1 Q When Z 3 i s o b s e r v e d , f o l l o w A 1 Q When Z 4 i s o b s e r v e d , f o l l o w A 1 3 E x p e c t e d N e t I n c o m e , [ E M V ] , f r o m Z P r e d i c t o r When Z 5 i s o b s e r v e d , f o l l o w A 1 3 When Z f i i s o b s e r v e d , f o l l o w A 1 4 When Z ? i s o b s e r v e d , f o l l o w A 1 5 [ ( 9 2 . 4 3 x 0 . 1 0 ) + ( 1 6 7 . 5 1 x 0.12) + ( 2 9 0 . 8 5 x 0.18) + ( 3 8 7 . 9 3 x 0. 26) + ( 4 0 9 . 28 x 0.16) + ( 5 3 6 . 03 x 0.13) + ( 5 9 1 . 2 2 x 0 . 0 5 ) ] x 1 0 0 0 = $ 3 4 7 , 2 7 0 V a l u e o f a P e r f e c t P r e d i c t o r = V a l u e o f t h e Z P r e d i c t o r $ 3 5 8 , 9 8 0 - $ 3 1 5 , 7 5 7 $ 3 4 7 , 2 7 0 - $ 3 1 5 , 7 5 7 $ 4 3 , 2 2 3 . 0 0 $ 3 1 , 5 1 3 . 0 0 .169 T A B L E 5.2 8 COMPUTATION OF B A Y E S I A N STRATEGY U S I N G P O S T E R I O R P R O B A B I L I T I E S AND STI M U L A T O R Y CROP RESPONSE F U N C T I O N : PROCEDURE I I , E M V - C R I T E R I O N H y d r o l o g i c a l s t a t e s o f N a t u r e : I r r i g a t i o n S e a s o n I n f l o w s ( 6 ^ A c t i o n s : I r r i g a b l e A r e a s ( H e c t a r e s ) A 1 3 1 3 0 0 H e c t . A 1 4 1 4 0 0 H e c t . A 1 5 1 5 0 0 H e c t . A 1 6 1 6 0 0 H e c t . A 1 7 1 7 0 0 H e c t . A 1 8 1 8 0 0 H e c t . A 1 9 1 9 0 0 H e c t . P ( 6 ) ( $ 1 , 0 0 0 ) 8 1 : V e r y P o o r 143. S10 1 4 0 . 3 5 0 1 3 7 . 9 6 0 1 3 6 . 1 8 0 1 3 4 . 9 2 0 1 3 4 . 0 7 0 1 3 3 . 5 8 0 0.140 8 2 : P o o r 231.510 2 2 3 . 4 0 0 2 1 6 . 4 9 0 2 1 0 . 6 5 0 2 0 5 . 7 5 0 2 0 1 . 6 5 0 1 9 8 . 2 5 0 0.160 8 3 : F a i r 336.470 3 2 5 . 0 0 0 3 1 4 . 1 3 0 3 0 4 . 1 6 0 2 9 5 . 2 0 0 2 8 7 . 2 3 0 2 8 0 . 2 0 0 0.160 8 4 : N o r m a l • 440.550 4 3 2 . 1 2 0 4 2 1 . 4 4 0 4 0 9 . 9 5 0 3 9 8 . 4 7 0 3 8 7 . 4 6 0 3 7 7 . 1 4 0 0.190 8 5 : G o o d 5 1 9 . 6 9 0 £ 2 5 . 1 1 0 5 2 2 . 7 9 0 5 1 5 . 7 5 0 5 0 6 . 0 3 0 4 9 4 . 9 7 0 4 8 3 . 4 3 0 0. 0 3 0 6g : V e r y G o o d 5 5 2 . 3 0 0 5 8 3 . 2 2 0 5 9 9 . 6 4 0 6 0 5 . 6 7 0 6 0 4 . 5 9 0 5 9 8 . 8 3 0 5 9 0 . 1 4 0 0 . 1 8 0 6 7 : E x c e l l e n t 5 5 2 . 5 0 0 5 9 5 . 0 0 0 6 3 5 . 9 4 0 6 6 3 . 9 3 0 6 7 9 . 5 9 0 686.110 6 8 6 . 0 6 0 0.140 E x p e c t e d N e t I n c o m e , [ E M V ] , f r o m a P e r f e c t P r e d i c t o r = [ ( 1 4 3 . 5 1 x 0. 14) + ( 2 3 1 . 51 x 0.16) + ( 4 4 0 . 55 x 0 + ( 5 2 5 . 1 1 x 0 . 0 3 ) + ( 6 0 5 . 6 7 x 0 . 1 8 ) + ( 6 8 6 . 1 1 x 0 X 1 0 0 0 = $ 4 1 5 , 5 0 0 E x p e c t e d N e t I n c o m e s , [ E M V ] , G i v e n Z O b s e r v a t i o n Z O b s e r v a t i o n A A 1 3 A 1 4 A 1 5 A 1 6 A 1 7 A 1 8 . A 1 9 . P ( Z ) Z l 1 5 2 . 3 1 1 4 8 . 6 6 1 4 5 . 8 1 1 4 3 . 6 3 1 4 2 . 0 1 1 4 0 . 8 3 1 4 0 . 0 5 0.100 Z 2 216. 55 2 0 9 . 2 0 1 9 8 . 2 9 1 9 7 . 9 9 1 9 3 . 7 1 1 9 0 . 1 6 1 8 7 . 2 6 0.120 . Z 3 345. 80 3 3 8 . 9 9 3 3 1 . 1 9 3 2 3 . 1 1 3 2 2 . 9 7 3 1 5 . 1 2 3 0 0 . 8 0 .0.180 Z 4 4 5 4 . 7 3 4 6 1 . 9 1 465.07 4 6 3 . 9 4 4 5 9 . 5 8 4 5 3 . 2 3 4 4 5 . 7 2 0. 2 6 0 Z 5 4 6 6 . 6 1 4 7 7 . 8 4 4 8 3 . 2 7 483. 33 4 7 9 . 6 6 4 7 3 . 4 8 4 6 5 . 8 8 0.160 Z 6 5 4 3 . 4 3 5 7 5 . 6 1 5 9 9 . 1 9 6 1 2 . 1 5 616.60 6 1 5 . 0 9 6 0 9 . 5 5 0.130 Z 7 5 5 2 . 4 6 5 9 2 . 6 4 6 2 8 . 6 8 6 5 2 . 2 7 6 6 4 . 5 9 668.66 6 6 6 . 8 8 0. 0 5 0 B a y e s i a n S t r a t e g y : When Z 1 i s o b s e r v e d , f o i l o w A 1 3 When Z 2 i s o b s e r v e d , f o l l o w A 1 3 When Z 3 i s o b s e r v e d , f o l l o w A 1 3 When Z 4 i s o b s e r v e d , f o l l o w A 1 5 When Z i s o b s e r v e d , f o l l o w A , , When Z g i s o b s e r v e d , f o l l o w Ay When Z 7 i s o b s e r v e d , f o l l o w A. _ ' 18 E x p e c t e d N e t I n c o m e , [ E M V ] , f r o m Z P r e d i c t o r = [ ( 1 5 2 . 31 x 0. 10) + ( 2 1 6 . 55 x 0. 12) + ( 3 4 5 . 8 0 x 0. 18) + <465.07 x 0.26) + ( 4 8 3 . 33 x 0. 16) + ( 6 1 6 . 60 x 0.13) + ( 6 6 8 . 6 6 x 0 . 0 5 ) ] x 1,000 = $ 4 1 5 , 3 0 0 V a l u e o f a P e r f e c t P r e d i c t o r = $ 4 1 5 , 5 0 0 _ $ 3 9 6 , 9 3 7 = $ 1 8 , 5 6 3 . 0 0 V a l u e o f t h e Z P r e d i c t o r - $ 4 1 5 , 300 - $ 3 9 6 , 937 5 1 0 , 3 6 3 . 0 0 170 TAM.K 5:29 COMPUTATION OF B A Y E S I A N STRATEGY U S I N G P O S T E R I O R P R O B A B I L I T I E S AND NON-STI M U L A T O R Y CROP RESPONSE F U N C T I O N : PROCEDURE I I , E M V - C R I T E R I O N H y d r o l o g i c a l S t a t e s o f N a t u r e : I r r i g a t i o n S e a s o n I n f l o w s , ( 6 i ) A c t i o n s : " 1 3 1 3 0 0 H e c t . 14 1 4 0 0 H e c t . I r r i g a b l e A r e a s ( H e c t a r e s ) , (A..) " 1 5 1 5 0 0 H e c t . " 1 6 1 6 0 0 H e c t . " 1 7 1 7 0 0 H e c t . 8^ : V e r y P o o r 127.080 1 0 5 . 3 1 0 8 2 : P o o r 325.390 3 0 5 . 1 5 0 6 3 : F a i r 469.890 4 5 5 . 0 3 0 8 4 : N o r m a l 5 4 0 . 6 0 0 5 5 4 . 9 5 0 8 5 : G o o d 5 6 7 . 5 0 0 6 0 4 . 9 1 0 6 6 : V e r y G o o d 5 4 0 . 6 0 0 6 0 4 . 9 1 0 6 ? : E x c e l l e n t 5 5 2 . 5 0 0 5 9 5 . 0 0 0 ( $ 1 , 0 0 0 ) 10 1 8 0 0 H e c t . " 1 9 1 9 0 0 H e c t . P ( 8 ) 7 1 . 9 5 0 2 8 1 . 7 7 0 444 . 9 7 0 5 6 1 . 5 5 0 6 3 1 . 4 9 0 6 5 4 . 8 0 0 6 3 1 . 4 9 0 3 7 . 2 8 0 2 5 5 . 8 5 0 4 3 0 . 7 1 0 5 6 7 . 850 6 4 9 . 2 8 0 6 9 2 . 9 9 0 6 9 2 . 9 9 0 1 . 5 4 0 2 2 7 . 8 2 0 4 1 2 . 9 7 0 5 5 6 . 9 7 0 6 5 9 . 8 3 0 7 2 1 . 5 4 0 7 4 2 . 1 1 0 - 3 1 5 . 0 0 0 1 9 8 . 0 5 0 3 9 2 . 3 4 0 5 4 7 . 7 6 0 664. 340 7 4 2 . 0 5 0 7 8 0 . 9 1 0 E x p e c t e d N e t I n c o m e , ( E M V ] , f r o m a P e r f e c t P r e d i c t o r - 3 3 2 . 5 0 0 1 6 6 . 8 0 0 3 6 9 . 2 7 0 5 3 4 . 9 2 0 6 6 3 . 7 6 0 7 5 5 . 7 9 0 811. 010 0.140 0.160 0.160 0.190 0 .030 0.180 0 . 1 4 0 [ ( 1 3 7 . 0 8 x 0. 1 4 ) + ( 3 2 5 . 39 x 0,16) + ( 4 5 9 . 89 x 0.16) + ( 5 C 1 . 85 x 0.19) + ( 6 6 4 . 34 x 0. 03) + ( 7 5 5 . 7 9 x 0 . 1 8 ) + ( 8 1 1 . 0 1 x 0 . 1 4 ) ] x 1,000 = $ 5 2 1 , 0 9 0 Z O b s e r v a t i o n E x p e c t e d N e t I n c o m e s , [ E M V ] , G i v e n Z O b s e r v a t i o n "13 155.91 293. 33 4 3 7 . 0 3 5 1 9 . 6 4 5 1 7 . 4 1 5 4 5 . 1 2 5 5 0 . 1 2 "14 1 2 5 . 3 0 2 7 1 . 1 7 437. 04 5 4 3 . 8 8 5 4 7 . 6 7 5 9 7 . 1 5 5 9 6 . 9 8 "15 9 2 . 9 4 2 4 6 . 1 0 4 3 1 . 2 1 5 5 9 . 9 1 5 6 8 . 7 6 6 3 8 . 4 8 6 3 6 . 1 5 "16 5 9 . 1 4 2 1 8 . 7 0 4 2 0 . 6 4 5 7 4 . 0 8 5 8 6 . 3 3 6 8 2 . 5 0 6 9 2 . 9 9 *17 2 4 . 1 7 1 8 9 . 3 5 4 0 6 . 1 8 5 8 1 . 4 4 5 9 6 . 6 7 7 1 6 . 1 9 7 3 8 . 0 0 B a y e s i a n S t r a t e g y : When i s o b s e r v e d , f o l l o w A ^ When Z^ i s o b s e r v e d , f o l l o w A ^^ When Z j i s o b s e r v e d , f o l l o w A ^ 4 When Z^ i s o b s e r v e d , f o l l o w A ^ g E x p e c t e d N e t I n c o m e , [ E M V ] , f r o m Z P r e d i c t o r "18 - 2 6 3 . 6 9 1 1 0 . 8 3 3 5 7 . 6 6 583.12 601. 00 7 4 1 . 2 8 7 7 3 . 1 4 "19 P ( Z ) 2 8 2 . 5 7 8 1 . 9 1 3 3 9 . 3 8 5 8 0 . 0 1 6 0 0 . 2 7 759.10 799. 97 0.100 0.120 0.180 0 .260 0.160 0.130 0. 0 5 0 When Z 5 i s o b s e r v e d , f o l l o w A 1 8 When Z g i s o b s e r v e d , f o l l o w A l g When Z 7 i s o b s e r v e d , f o l l o w A l g ( ( 1 5 5 . 91 x 0 1 0 1 ) + ( 2 9 3 . 33 x 0 . 1 2 ) + ( 4 3 7 . 04 x 0.18) + ( 5 8 3 . 1 2 x 0.26) + ( 6 0 1 . 0 0 x 0.16) + ( 7 5 9 . 1 0 x 0.13) + ( 7 9 9 . 9 + ( 7 9 9 . 9 7 x 0 . 0 5 ) ] x 1 , 0 0 0 . = $ 5 1 5 9 1 0 V a l u e o f a P e r f e c t P r e d i c t o r = $ 5 2 1 , 0 9 0 = 5 4 6 3 , 0 : 6 = $ 5 8 , 0 3 4 . 0 0 V a l u e o f a Z P r e d i c t o r = ? 5 1 5 , 9 1 0 - $ 4 6 3 , 0 5 6 = $ 5 2 , 8 5 4 . 0 0 171 by the equation: i=7 EMV = I M (A..) X P (9/Z. ) . (5.13) i = l 31 k i The Bayesian s t r a t e g y i s then given f o r each Z o b s e r v a t i o n , Z, , by l o c a t i n g the maximum EMV i n the row 9. f o r A., j = 1 to N, v i z : Bayesian s t r a t e g y o r the optimum s t r a t e g y i s giv e n by: j=N i=7 MAX ( I ) Z M ( A . . ) X P (9/Z,). (5.14) . . . j = l i = l 1 ) 1 K 1 The o p e r a t i o n i s done K times t o d e f i n e the Bayesian s t r a t e -g i e s f o r a l l the o b s e r v a t i o n s , and the r e s u l t s are summarized i n the t a b l e s as shown. 5.6.5 Value of experiment The p a y o f f e n t r i e s i n the upper s e c t i o n o f each t a b l e are the values when the s t a t e s of nature, 9^, are pre-d i c t e d , w i t h c e r t a i n t y , t h a t i s the t r u e s t a t e s are p e r f e c t l y p r e d i c t e d by the Z v a l u e s . When any Z, i s observed 9. i s observed with c e r t a i n t y , t h a t a c t i o n , A^,'which maximizes the payof f f o r that s t a t e i s taken as the optimum s t r a t e g y . These optimum s t r a t e g i e s , given a p e r f e c t p r e d i c t o r , t h a t i s p e r f e c t knowledge about the h y d r o l o g i c a l s t a t e of nature or t o t a l growing season i n f l o w at the beginning o f the i r r i g a t i o n season, T = 0, are obtained f o r a l l the s t a t e s , 9^, i = l t o 7, 172 and m u l t i p l i e d by the p r i o r p r o b a b i l i t i e s f o r each 9^, and summed over a l l the s t a t e s t o o b t a i n the expected net value from the p e r f e c t p r e d i c t o r , v i z : 9.=7 x E M (A . . ) X P (9.) (5.15) 0 = 1 Dx max x i The value o f the p e r f e c t p r e d i c t o r i s given by the d i f f e r e n c e between equation (5.15) and the optimum expected value of the no-data problem. S i m i l a r l y , the expected net value from Z p r e d i c t o r i s c a l c u l a t e d , v i z : V 7 7 \ M ( V m a x X P <V < 5- 1 6 )  Z k _ 1 and the value of the Z p r e d i c t o r , t h a t i s the "value of the Z - p r e d i c t i n g experiment", i s given by the d i f f e r e n c e between equation (5.16) and the optimum expected value of the no-data problem. The expected net value from a p e r f e c t p r e d i c t o r , the value of a p e r f e c t p r e d i c t o r , the expected net value from the Z p r e d i c t o r , and the value of the Z p r e d i c t o r are a l l given i n Tables 5.27 and, 5.28 and 5.29 f o r Procedures I and II f o r the E M V - c r i t e r i o n , r e s p e c t i v e l y . 173 5.7 Optimal D e c i s i o n S t r a t e g i e s Over Time At present, under the e x i s t i n g i n s t i t u t i o n a l con--s t r a i n t s and water permit system, the h i g h u n c e r t a i n t y and r i s k a s s o c i a t e d with r a n c h i n g o p e r a t i o n s , absence of hydro-l o g i c a l f o r e c a s t i n g d e v i c e , and the economic c o n s t r a i n s of the s c a l e of o p e r a t i o n , a t y p i c a l rancher i n the Quilchena area i r r i g a t e s a minimum acreage of rangeland, 1,000 - 1,250 a c r e s , t h a t i s , 500 h e c t a r e s from year t o year. T h i s hectarage corresponds to Procedure I optimum hectarage f o r T = 4, t h a t i s s t a t e of absolute u n c e r t a i n t y or complete ignorance with r e s p e c t t o t o t a l growing season i n f l o w . Assuming an average y i e l d f o r a l f a l f a i r r i g a t e d to be 2.0 t o n s / a c r e / c u t , t h a t i s 5.0 t o n s / h e c t a r e / c u t or 10.0 t o n s / h e c t a r e f o r 2 cuts of a l f a l f a d u r i n g the i r r i g a t i o n season, and a net value of $42.50 per ton, the t o t a l revenue, TR.. , per year = 10.0 X 500 X 42.50 = $212 ,500.0/year. I f i t i s assumed t h a t the Quilchena ranching system operates i n a 5-year c y c l e and t h a t the Quilchena ranch management records a t o t a l revenue of $212,500/year f o r the 5 years, then a t e s t of the a l t e r n a -t i v e d e c i s i o n s t r a t e g i e s can be conducted. Table 5.30 i s the summary of the t o t a l expected (EMV) values f o r the Bayesian s t r a t e g i e s from Z p r e d i c t o r f o r Procedures I and I I , and the no-data type of o p e r a t i o n with constant stream of t o t a l revenue from year to year. I t has TABLE 5.30 COMPUTATION OF NET WORTH OF ALTERNATIVE IRRIGATION OPERATION STRATEGIES -Procedure I Procedure I I Procedure I I I -Step-Crop Response (i) S t i m u l a t o r y Crop Response ( i i ) Non-Stimulatory Crop Response No Data O p e r a t i o n Y e a r ($1,000) 1 : Poor 167.-51 216.55 293.33 212.50 2 : F a i r 290.85 345.80 437.04 212.50 3 : Normal 387.93 465.07 583.12 212.50 4 : Good 409.28 483.33 601.00 212.50 5 : Very Good 536.03 616.60 759.10 212.50 175 been assumed t h a t s t a t e s of nature, 9^, i = 2 to 6 occur i n the 5-year p e r i o d under c o n s i d e r a t i o n . At r percent i n t e r e s t r a t e , the net worth of each a l t e r n a t i v e at the end of t years i s given by t=n. . £ B (1 + r) (5.17) t=0 Z where B , B_, B„ B are stream of annual revenues; o 1 2 n t i s the time p e r i o d i n years ; r i s the r a t e of i n t e r e s t . For Procedure I : [(536.03) + (409.28)(1 + 0.06) + (387.93) -(1 + 0.06) 2 + (290.85) (1 + 0.06) 3 + (167.51) (1 + 0.06) 4] X 1,000 = [536.030 +433.837 + 435.878 + 346.407 + 211.478] X 1,000 = 1963.630 X 1,000 = $1,963,630.00 For Procedure I I ( i ) [ (616 .60) + (483.33) (1.06) + (465.07) (1.1235) +.(345.80) (1.1909) + (216.55) (1.2624)] X 1,000 = [616.600 + 512.330 + 522.506 + 411.813 + 273.373] X 1,000 = $2 , 336 ,622 X 1,000 = $2 , 3 3G , 6 2 2 . 0.0 176 For Procedure I I ( i i ) : [(759.10) + (601.00) X (1.06) + (583.12) (1.1236) + (437.04) (1.1910) + (293.33) (1.2625)] X 1,000 = [759.100 + 637.060 + 655.194 + 520.515 + 370.329] X 1,000 = $2942.198 X 1,000 = $2,942,198.00 For No-Data Operation [(212.5) + (212.5) (1.06) + (212.5) (1.1236) + (212.5) (1.1910) + (212.5) (1.2625)] X 1,000 = 212.5 +225.25 + 238.765 + 253.088 +• 268.281 = $1,197 ,884.00 Thus the o r d e r i n g of s t r a t e g i e s i s Procedure I I , Procedure I, and the no^data o p e r a t i o n with expected values of $2,336,622.00, and $2,942,198.00; $1,963,630.00, and $1,197,884.00, r e s p e c t i v e l y . T h i s o r d e r i n g i s obtained from the E M V - c r i t e r i o n , assuming a l i n e a r u t i l i t y f u n c t i o n with r e s p e c t to the net worth a t the end of f i v e years. I f the u t i l i t y f u n c t i o n i s not l i n e a r , then another u t i l i t y f u n c t i o n would have to be d e r i v e d with r e s p e c t t o f u t u r e net worth, and the expected u t i l i t i e s computed f o r each s t r a t e g y . I t ought to be s t a t e d t h a t the c h o i c e of i n t e r e s t r a t e of 6 percent was a r b i t r a r y . Another i n t e r e s t r a t e can change the above o r d e r i n g of s t r a t e g i e s . 177 CHAPTER 6 APPLICATION OF MODEL TO REGIONAL PLANNING IN LARGE-SCALE IRRIGATION SYSTEMS. 6-1 I n t r o d u c t i o n In the preceding chapter the d e c i s i o n theory o p t i m i z a -t i o n model was a p p l i e d to Quilchena Ranch i r r i g a t i o n system management, under the assumption t h a t d e c i s i o n s r e g a r d i n g water use are made at the farm l e v e l . In t h i s chapter i t i s intended to show t h a t the f e a t u r e s of the d e c i s i o n theory o p t i m i z a t i o n technique are a l s o a p p l i c a b l e to p r o j e c t models. Since the establishment o f h i g h l y c e n t r a l i z e d r e g i o n a l water management systems and i n t e r — b a s i n t r a n s f e r s may become the most popular form of water u t i l i z a t i o n i n the a r i d and s e m i - a r i d areas i n the n o t - t o o - d i s t a n t f u t u r e , i t i s necessary to employ t h i s type of decision-making technique under u n c e r t a i n t y to c l a r i f y the d e c i s i o n s r e q u i r e d o f the r e g i o n a l water management system a u t h o r i t y , to provide meaningful g u i d e l i n e s i n a r r i v i n g at informed d e c i s i o n s , and consequently, to o p t i m a l l y a l l o c a t e the l i m i t e d water resources among the d i f f e r e n t water d e r i v a -t i v e s . Normally the c e n t r a l a u t h o r i t y charged with the r e s p o n s i b i l i t y of proper management of a region's water resources i s concerned p r i m a r i l y with the optimal a l l o c a t i o n and u t i l i z a t i o n of water resources i n the re g i o n or d i s t r i c t . I t i s common p r a c t i c e i n the United S t a t e s f o r the c e n t r a l body to delegate a u t h o r i t y to a water company or an i r r i g a t i o n 17 n d i s t r i c t whose duty i t i s to supply i r r i g a t i o n water to i r r i g a t o r s w i t h i n the r e g i o n . The most common type of water management i n such a s e t t i n g i s f o r the c e n t r a l body to i s s u e permits t o the i r r i g a t o r s a u t h o r i z i n g them t o withdraw s p e c i f i c amounts o f water from a watercourse f o r i r r i g a t i o n purposes, or the d i s t r i c t c o n s t r u c t s a c e n t r a l , r e g i o n a l water supply system i n c l u d i n g a s i z a b l e storage r e s e r v o i r from which i t s u p p l i e s water t o the d i f f e r e n t users i n the r e g i o n . The s p e c i f i c problem here i s f o r the c e n t r a l a u t h o r i t y to determine how many hec t a r e s of land to l i c e n s e f o r i r r i g a t i o n , and to avoid o v e r - l i c e n s i n g . In the dev e l o p i n g c o u n t r i e s a l l l a r g e - s c a l e i r r i g a -t i o n undertakings are administered by F e d e r a l o r State agencies or j o i n t l y by F e d e r a l and St a t e agencies. The d e c i s i o n s made by the p r o j e c t a d m i n i s t r a t o r s may encompass the amount of land to be brought under i r r i g a t i o n and, the mix and sequencing o f crops t o be produced. A l s o because the a v a i l a b l e h y d r o l o g i c a l records are g e n e r a l l y scanty the c e n t r a l a u t h o r i t y ' s foremost problem i s to decide on the hectarage to put under i r r i g a t i o n i n each i r r i g a t i o n p e r i o d . In these c o u n t r i e s the problem i s f u r t h e r complicated by the f a c t t h a t there are o f t e n no c o n s t i t u t e d system of water r i g h t s law i n use. The i r r i g a t o r s , thus, are o f t e n c o n f r o n t e d with developing and o p e r a t i n g t h e i r i r r i g a t i o n systems under the heavy c o n s t r a i n t of h y d r o l o g i c a l u n c e r t a i n t y . I t i s very important, t h e r e f o r e , t h a t a technique, 179 such as the decision theory optimization model developed herein, be applied to provide reasonable guidelines which could be used by the central authorities administering large-scale i r r i g a t i o n enterprises i n a r r i v i n g at improved strategies or p o l i c i e s for the development and operation of large-sclae regional i r r i g a t i o n systems. 6.2 Application to Nicola Valley I r r i g a t i o n The Nicola Valley area i n south central B r i t i s h Columbia i s located i n the dry, semi-arid i n t e r i o r with low p r e c i p i t a t i o n and high evaporation. I t i s an a g r i c u l t u r a l l y -a t t r a c t i v e area, being c l i m a t i c a l l y desirable for r a i s i n g livestock e s p e c i a l l y c a t t l e for beef production. Ranching, therefore, i s the main economic enterprise in agriculture of the Nicola Valley. Since t h i s area i s characterized by semi-a r i d hydrologic phenomena i r r i g a t i o n i s a necessity. Intensive, high l e v e l of alfalfa-hay i r r i g a t i o n i s practised to produce winter feed for c a t t l e . Agriculture i s the largest consumer of water in t h i s region and as a r e s u l t the water requirements of agriculture are the main water needs of the area. The outlook for agriculture i n this area i s very c r i t i c a l for two main reasons: the area i s a dry area and there i s shortage of water r e s u l t i n g from low p r e c i p i t a t i o n . Also the area appears to suffer from an even more decisive shortage of good a g r i c u l t u r a l land, since the amount of land available at the lower elevations i s very limited. Thus the 180 major p h y s i c a l c o n s t r a i n t s to a g r i c u l t u r a l p r o d u c t i o n i n the N i c o l a V a l l e y are lack of adequate lan d and inadequate water supply. The main forage crop i s a l f a l f a which on l y needs to be seeded every 4 to 5 years; with adequate water supply 2-3 "c u t s " of a l f a l f a can be ha r v e s t e d d u r i n g the i r r i g a t i o n season. Current y i e l d s are of the order of 2.0 - 4.0 tons per acre per cut of a l f a l f a i n the N i c o l a V a l l e y r e g i o n . The main s u r f a c e water sources o f the N i c o l a V a l l e y r e g i o n are shown i n F i g u r e 6.1. P r e c i p i t a t i o n i n the growing season i s scanty and r u n o f f i s zero. The seasonal p a t t e r n o f r u n o f f i s f a i r l y con-s t a n t throughout the area. The p a t t e r n f o l l o w s the normal B r i t i s h Columbia seasonal p a t t e r n of low flows i n f a l l and win t e r , h i g h flows, due to snowmelt, i n s p r i n g ( f r e s h e t p e r i o d ) , and d e c l i n i n g flows i n summer. With hi g h e v a p o t r a n s p i r a t i o n r a t e s and low r a i n f a l l , s a l i n i t y i s a dominant problem. Thus the computation of i r r i g a t i o n water requirement should i n c o r p o r a t e requirements f o r adequate l e a c h i n g i n order to maintain adequate s a l t balance i n the i r r i g a t e d area. F i g u r e 6.2 shows the computed monthly consumptive uses plu s l e a c h i n g requirements f o r the i r r i g a t i o n p e r i o d , as w e l l as the growing season r a i n f a l l histogram. Table 6.1 and F i g u r e 6.3 show the mean monthly flow o f the N i c o l a R i v e r , Quilchena Creek, and Moore Creek for. p e r i o d o f r e c o r d , d u r i n g the i r r i g a t i o n season. 182 250 L (a) 200 L 150 J r 100 L 50 J_ May June July August I r r i g a t i o n water equirement I r r i g a t i o n Season Averag R a i n f a l l A September T A B L E 6 . 1 MEAN MONTHLY FLOWS O F MAJOR INFLOW SOURCES FOR NTCO IA V A L L E Y REG ION L U R I N G THE I R R I G A T I O N SEASON ( X 1 0 3 rn 3 ) Y E A R H O N T H May J u n e J u l y A u g S e p t 1 9 1 5 2 3 8 6 9 . 3 3 0 8 9 9 . 9 2 1 9 5 9 . 8 7 9 8 5 . 4 1 9 9 6 . 3 16 7 0 6 9 5 . 5 7 9 7 4 8 . 0 4 9 5 7 3 . 1 1 2 7 1 6 . 6 3 0 1 7 . 5 17 4 6 1 6 7 . 7 . 1 1 7 9 1 7 . 4 3 2 5 7 7 . 2 2 5 9 8 . 2 5 9 9 . 6 1 8 1 9 3 6 0 . 0 1 3 7 6 0 . 0 3 8 8 0 . 0 - 2 3 6 0 . 0 6 8 0 . 0 19 6 3 7 1 0 . 0 1 4 1 7 8 . 0 1 8 2 3 1 . 2 3 3 6 8 . 8 5 9 4 . 5 20 1 3 4 2 3 . 3 4 2 9 1 5 . 2 3 4 0 9 7 . 0 6 8 5 8 . 6 6 8 5 . 9 2 1 2 3 2 3 4 . 3 7 7 0 3 0 . 5 2 6 8 0 8 . 8 . 2 6 0 9 3 . 9 2 5 3 7 9 . 0 1 9 3 3 6 8 8 6 1 . 0 7 9 0 7 5 . 6 2 2 8 9 5 . 0 4 0 5 0 . 6 1 4 0 8 . 9 35 . 1 1 3 2 1 . 3 5 4 9 8 9 . 2 5 4 5 4 8 . 1 1 9 8 4 9 . 1 6 3 2 2 . 3 36 5 1 1 4 5 . 6 4 8 4 2 7 . 7 2 0 5 0 7 . 6 2 7 1 7 . 9 7 4 1 . 2 1 9 4 7 2 1 6 2 7 . 0 1 3 2 2 8 . 6 6 6 1 4 . 3 1 3 9 2 . 5 6 9 6 . 2 4 8 5 2 3 6 9 . 2 1 1 0 1 6 4 . 1 3 7 5 0 7 . 7 2 1 7 0 2 . 6 1 4 3 8 9 . 7 49- 6 8 2 2 7 . 7 4 8 2 7 8 . 1 1 3 0 3 3 . 8 2 7 9 2 . 9 6 6 5 . 0 50 3 4 0 9 2 . 6 8 8 0 0 9 . 9 2 4 1 8 0 . 2 3 0 0 3 . 8 9 0 1 . 1 5 1 7 5 8 8 1 . 4 6 4 7 3 1 . 5 7 8 9 7 . 9 4 6 4 5 . 8 1 5 4 8 . 6 1 9 6 5 8 4 2 9 2 . 3 6 7 4 0 0 . 1 9 9 6 6 . 4 4 0 5 4 . 1 3 2 0 9 . 5 66 4 1 8 6 3 . 4 2 8 6 6 6 . 4 1 1 2 7 4 . 3 4 3 6 9 . 9 1 3 1 1 . 0 67 3 1 5 3 6 . 6 7 6 0 3 0 . 2 1 3 2 Q 1 . 4 1 2 2 2 . 4 2 4 4 . 5 6 8 8 3 8 3 3 . 4 1 0 0 8 5 5 . 4 2 0 4 2 6 . 4 • 4 4 6 8 . 3 3 1 9 1 . 6 69 1 3 9 9 4 0 . 3 3 1 0 3 4 . 4 1 7 5 1 6 . 3 1 1 4 2 . 4 7 6 1 . 6 70 2 5 3 6 8 . 2 3 1 6 7 8 . 7 5 2 3 7 . 7 6 3 1 . 1 1 8 9 . 3 71 7 5 1 5 3 . 6 8 3 6 2 6 . 8 1 8 2 3 5 . 8 4 0 5 2 . 4 3 3 1 5 . 6 72 1 2 7 3 0 9 . 1 1 9 1 5 1 4 . 0 3 4 1 2 0 . 3 . 8 8 0 5 . 2 5 1 3 6 . 4 73 3 7 0 0 7 . 0 2 8 4 9 6 , 3 - 9 3 4 6 . 8 . 9 1 1 . 9 2 2 8 . 0 74 1 0 0 5 7 5 . 0 1 3 9 6 8 7 . 5 3 3 5 2 5 . 0 5 5 8 7 . 5 1 3 9 6 . 9 T O T A L 1 3 9 0 8 6 4 . 8 1 6 6 1 3 4 3 . 5 5 4 7 1 6 2 . 1 1 5 7 3 8 1 . 9 7 8 6 1 0 . 2 3 3 X 10 m 3 3 X 10 d in 3 3 X 10 ni X 1 0 3 m 3 X 1 0 3 m 3 A V E R A G E 5 5 6 3 4 . 6 6 6 4 . 5 3 . 7 2 1 8 8 6 . 5 6 2 9 5 . 3 3 1 4 4 . 4 3 3 X 10 ni X 1 0 3 m 3 X 1 0 3 m 3 X 1 0 3 m 3 X 1 0 3 m 3 4H ,534-SH U , ^1 >-< I i n «o| May June J u l y August Sept. Month 185 The N i c o l a area i s w i t h i n the j u r i s d i c t i o n o f the Kamloops Water D i s t r i c t which i s one of the s e v e r a l admini-s t r a t i v e r e g i ons of the c e n t r a l a u t h o r i t y -- the B r i t i s h Columbia Water Resources S e r v i c e . The Water Rights Branch of the s e r v i c e adopts the d o c t r i n e of p r i o r a p p r o p r i a t i o n and i s s u e s permits and water l i c e n c e s to the d i f f e r e n t water users i n the r e g i o n . Experts from the Water Resources S e r v i c e as w e l l as the ranchers i n the V a l l e y have suddenly come to the r e a l i z a t i o n t h a t with i n c r e a s e d p i t c h f o r development i n the N i c o l a V a l l e y area the region might be o v e r l i c e n s e d with r e s p e c t t o the scanty a v a i l a b l e water r e s o u r c e s . I f t h i s happens the major economic a c t i v i t y , beef p r o d u c t i o n a g r i c u l t u r e , may be j e o p a r d i z e d . The a v a i l a b l e h y d r o l g o i c a l r e c o r d f o r the V a l l e y i s not long enough to be c o n s i d e r e d r e l i a b l e f o r p l a n n i n g purposes and t h i s f u r t h e r complicates the s i t u a t i o n . Recently there has been an i n c r e a s i n g concern about the water s i t u a t i o n i n r e l a t i o n to a g r i c u l t u r e , and a l s o about the expansion of land to be i r r i g a t e d i n the area. One p o s s i b l e way t o a v o i d the s i t u a t i o n i s f o r the ranchers to p r a c t i s e c o n s e r v a t i o n , and t h i s i s a c h i e v a b l e through the use o f a combination of adequate i r r i g a t i o n water a p p l i c a t i o n t e c h n o l o g i e s a v a i l a b l e i n the area. I t i s b e l i e v e d t h a t both c o n t r o l l e d s u r f a c e i r r i g a t i o n and s p r i n k l e r methods w i l l be needed i n the area and both c o u l d be made more e f f i c i e n t . Thus the present method o f i r r i g a t i o n water a p p l i c a t i o n by w i l d 186 f l o o d i n g has to be r e p l a c e d by more e f f i c i e n t c o n t r o l l e d sur-face i r r i g a t i o n systems and s p r i n k l i n g systems f o r adequate o p t i m i z a t i o n of resources and energy use. Other c o n s i d e r a t i o n s l i k e i n t e r b a s i n t r a n s f e r s through a h i g h l y concentrated r e g i o n a l water management system, l i m i t i n g the number of l i c e n c e s i s s u e d i n the area, or u t i l i z a t i o n o f ground water res o u r c e s r e p r e s e n t the range of l i k e l y a l t e r n a t i v e s . However, no matter what p o l i c y i s f i n a l l y adopted the most important l i n e of approach i s to f i r s t determine the optimum hectarage t h a t c o u l d be i r r i g a t e d under the c o n s t r a i n t of h y d r o l o g i c a l u n c e r t a i n t y . T h i s w i l l a i d i n s e t t i n g the g u i d e l i n e s f o r f u t u r e p o l i c i e s f o r water resources u t i l i z a t i o n of the V a l l e y , i r r e s p e c t i v e of the i r r i g a t i o n water a p p l i c a t i o n system used. High water a p p l i c a t i o n e f f i c i e n c y i n c r e a s e s the p r o b a b i l i t y t h a t water w i l l be used econ o m i c a l l y , although i t does not i n s u r e economical use. The economical use o f water i s a f u n c t i o n of the water c o s t s , the crop y i e l d s and the crop va l u e s ; whereas water a p p l i c a t i o n e f f i c i e n c i e s i n v o l v e d i r e c t l y n e i t h e r water c o s t s nor crop y i e l d s . Thus d e c i s i o n s based s o l e l y on water a p p l i c a t i o n e f f i c i e n c y of an i r r i g a t i o n system are not adequate f o r forming p o l i c i e s r e l a t i n g to c o n s e r v a t i o n and o p t i m i z a t i o n o f resources use. The d e c i s i o n theory o p t i m i z a t i o n model developed, i s a p p l i e d to the r e g i o n a l water resources problem of the N i c o l a V a l l e y area. The r e s u l t s from the model w i l l i n d i c a t e to the c e n t r a l water management a u t h o r i t y the o p t i m a l p o l i c i e s t o 187 adopt with r e s p e c t to proper plan n i n g and expansion of i r r i g a t i o n schemes i n the N i c o l a V a l l e y . I t i s assumed t h a t the N i c o l a V a l l e y ' s water management w i l l become h i g h l y c e n t r a l i z e d i n the f u t u r e . A l s o i t i s envisaged that i n the f u t u r e N i c o l a Lake (a n a t u r a l V a l l e y lake) w i l l become the n a t u r a l storage r e s e r v o i r from which water w i l l be drawn f o r i r r i g a t i o n purposes as w e l l as other purposes s i c h as r e c r e a -t i o n and f i s h e r i e s . Thus, the d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model i s a p p l i e d to the p l a n n i n g of r e g i o n a l i r r i g a t i o n i n the N i c o l a V a l l e y d i s t r i c t on the premises of the f o r e g o i n g assumption. The r e s e r v o i r would be r e g u l a t e d and permits r e q u i r e d f o r farmers to draw water at v a r i o u s p o i n t s along the banks of the n a t u r a l r e s e r v o i r system, f o r i r r i g a t i o n purposes. 6.2.1 Inflow a n a l y s i s The t o t a l flows of the major i n f l o w sources d u r i n g the i r r i g a t i o n p e r i o d are shown i n Table 6.2. Only 2 5 years of flow r e c o r d was a v a i l a b l e . As can be observed from the t a b l e the flow r e c o r d i s s p o t t y . The p e r i o d s of r e c o r d are 1915-1921; 1933-1936; 1947-1951 and 1965-1974. There was no snow survey r e c o r d a v a i l a b l e to c a r r y out a r e g r e s s i o n a n a l y s i s which c o u l d have been used to f i l l i n the flows f o r the years of no r e c o r d . I t was assumed t h a t the recorded flows c o u l d have occurred i n any chosen p e r i o d of 25 years an thus the flows were used f o r the s t a t i s t i c a l a n a l y s i s and the TABLE 6.2 TOTAL FLOWS OF MAJOR INFLOW SOURCES FOR NICOLA VALLEY REGION DURING THE IRRIGATION SEASON YEAR Total Growing Season Inflow (10 m ) 1915 86797.5 16 215535.0 17 199860.0 18 40000.0 19 99082.5 20 97980.0 21 178725.0 1933 176115.0 35 147030.0 36 123540.0 1947 43515.0 48 235897.5 49 132997.5 50 150187.5 51 154860.0 1965 168922.5 66 .87397.5 67 122235.0 68 .212775.0 69 . 190395.0 70 63105.0 71 184200.0 72 366885.0 73. VS'.i'jn.O 74 279375.0 189 d e f i n i t i o n of the h d y r o l o g i c a l s t a t e s of nature. 6.2.2 H y d r o l o g i c a l s t a t e s of nature and e s t i m a t i o n of  p r i o r p r o b a b i l i t i e s From Table 6.1 the range i s from a minimum t o t a l 3 3 i r r i g a t i o n season flow of 40,000 X 10 m i n 1918 to a maximum 3 3 . . . of 366,885 X 10 m i n 1972. The computed average i r r i g a t i o n 3 3 . • season i n f l o w i s 19 3,335 X 10 m and the standard d e v i a t i o n 3 3 i s + 45,000 X 10 m . Seven h y d r o l o g i c a l s t a t e s o f nature were thus d e f i n e d : 0^ : Very poor h y d r o l o g i c a l c o n d i t i o n ; i n f l o w l e s s than 85,000 X 10 3 m3. . . 3 02 : Poor h y d r o l g o c i a l c o n d i t i o n s ; 85,000 X 10 -130,000 X 10 3 m3. . . 3 0^ : F a i r h y d r o l o g i c a l c o n d i t i o n ; 130,000 X 10 -175,000 X 10 3 m3. 3 0 4 : Normal h y d r o l o g i c a l c o n d i t i o n s ; 175,000 X 10 -220,000 X 10 3 m3. 3 0 5 : Good h y d r o l o g i c a l c o n d i t i o n s ; 220,000 X 10 265,000 X 10 3 m3. 3 9^ : Very good h y d r o l o g i c a l c o n d i t i o n s ; 265,000 X 10 -.310,000 X 10 3 m3. 3 9 7 : E x c e l l e n t h y d r o l o g i c a l c o n d i t i o n ; 310,000 X 10 or over. The d i s t r i b u t i o n of i n f l o w c o n d i t i o n s was est!mated by simply d i v i d i n g the flow o b s e r v a t i o n s i n t o c l a s s i n t e r v a l s and c a l c u l a t i n g the frequency, r e l a t i v e frequency and 190 cumulative frequency of each c e l l as shown i n Table 6.3. The recorded flows are a l s o ranked i n descending order of magni-tude and frequency a n a l y s i s c a r r i e d out as shown i n Table 6.4. Fi g u r e 6.4 i s the cumulative frequency a n a l y s i s of i n f l o w d u r i n g the i r r i g a t i o n season. The p r i o r p r o b a b i l i t y ( r e l a t i v e frequency) o f the d e f i n e d s t a t e s of nature i s obta i n e d by using the lower and upper l i m i t of the i n f l o w c e l l s i n each c l a s s i n t e r v a l and i n t e r p o l a t i n g from F i g u r e 6.4. The values o b t a i n e d are P.(0 1) = 0.160, P(© 2) = 0.210, P(© 3) = 0.240, p(© 4) = 0 .270, .P(6 ) = 0.080, P(0 g). = 0.030, P(0 ?) = 0.020, and p l o t t e d i n F i g u r e 6.5(a) and 6.5(b). 6.2.3. D e c i s i o n a l t e r n a t i v e s The d e c i s i o n a l t e r n a t i v e s are the p o s s i b l e hectarages which can be i r r i g a t e d d u r i n g the i r r i g a t i o n season i n the N i c o l a V a l l e y r e g i o n u s i n g o n l y the water supply from the major s u r f a c e water s o u r c e s — t h e N i c o l a R i v e r , the Quilchena Creek and the Moore Creek. The t o t a l a v a i l a b l e f o o t h i l l l a n d which can be eco n o m i c a l l y i r r i g a t e d i s about 25,000 hectares (62,500 a c r e s ) . Therefore the c e n t r a l a u t h o r i t y ' s o b j e c t i v e i s to determine how many he c t a r e s of land to l i c e n s e f o r i r r i g a t i o n under the e x i s t i n g h y d r o l o g i c a l u n c e r t a i n t y . The a c t i o n s range from a minimum of 500 hecta r e s to a maximum of 25,000 h e c t a r e s . Thus i f 500 hectare i n t e r v a l s are co n s i d e r e d there w i l l be 50 d e c i s i o n a l t e r n a t i v e s . The crop i s a l f a l f a with consumptive uses and 191 TABLE 6.3 COMPUTATION OF PRIOR PROBABILITIES OF INFLOW FROM NICOLA RIVER, QUILCHENA CREEK AND MOORE CREEK DURING THE IRRIGATION SEASON State of Nature e. Inflow Index Interval (m ) Frequency n(e±)-Relative Frequency n(0) i)/N = P(6) i) Cumulative Relative Frequency 7 E P (9 ) i=l e l 40,000X103--85,000X103:-3a 4 0. 160 0 .160 92 85,000X103--130,000Xl03:-2a 6 0. 240 0 .400 93 130,000X103--175,000X103:-la 5 0. 200 0 .600 e4 175,000X103--220,000Xl03:a=0 7 o. 280 0 .880 95 220,000X103--265,00OXlO3:+la 1 0. 040 0 .920 96 265,000X103--310,000Xl03:+2a 1 • o. 040 0 .960 97 310,000X103--355,000X103:+3a 1 p. 040 1 .000 N= Z n(9.)=25 ZP(9.)-1.000 i=l 1 i=l 1 192 TABLE 6.4 FREQUENCY ANALYSIS OF INFLOW FROM MAJOR SURFACE WATER SOURCES IN NICOLA VALLEY REGION DURING THE IRRIGATION SEASON Year Inflow Rank Return % Flow (10 3 m3) m Period T _ N+l • R m ' Equalled or Exceeded P = 100/TR% 1972 366,885.0 1 26.0.0 3.85 1974 279,375.0 2 13.00 7.69 1948 235,897.5 3 8.67 11.53 1916 215,535.0 4 6.50 • 15.38 1968 212,775.0 ' 5 5.20 19.23 • 1917 199,860.0 6 4.33 23.09 1969 190,395.0 7 •3.71 26.95 1971 . 184,200.0 8 .3.25 30.77 1921 178,725.0 9 2.89 34.60 1933 176,115.0 10 2.60 .38.46 1965 168,922.5 11 2.36 42.37 1951 154,860.0 12 2.17 46.08 1950 150,187.5 13 2.00 50.00 1935 147,030.0 14 1.86 53.76 1949 132,997.5 15 • 1.73 57.80 1936 123,540.0 16 1.63 61.35 1967 122,235.0 17 1.53 ' 65.36 1919 99,082.5 18 1.44 69.44 1920 97,980.0 19 1.37 72.99 1966 87,397.5 20 . 1.30. • 76.92 1915 ' 86,797.5 21 1.24 80.65 1973 75,990.0 22 1.18 84.75 1970 6 3,105.(1 23 1.13 88.50 1947 • .43,515.0 24 1.08 92.59 .1918 40,000.0 25 1.04 ' 96.15 C u m u l a t i v e R e l a t i v e Frequency R e l a t i v e Frequency o 19 5 i r r i g a t i o n water requirement as computed i n S e c t i o n 5.2.4 and shown i n Table 5.2 and F i g u r e 6.2. A l l assumptions made w i t h r e s p e c t t o crop net value i n S e c t i o n 5.2.5, and crop response f u n c t i o n s i n 5.2.6 are a p p l i c a b l e . The computation procedure i s as o u t l i n e d i n 5.3.1 and 5.3.2. The same q u a d r a t i c u t i l i t y f u n c t i o n t h a t was e s t a b l i s h e d i n S e c t i o n 5.3.3 i s assumed f o r c e n t r a l i r r i g a t i o n a d m i n i s t r a t i v e agency's boss. In the N i c o l a V a l l e y r e g i o n , two crops of a l f a l f a are normally cut d u r i n g the i r r i g a t i o n season. Therefore any meaningful r e g i o n a l p l a n n i n g model f o r i r r i g a t i o n water use i n the area has to i n c o r p o r a t e t h i s f a c t as a major f e a t u r e . Thus the e n t i r e i r r i g a t i o n season i s s u b - d i v i d e d i n t o two p e r i o d s : the f i r s t i r r i g a t i o n p e r i o d and the second i r r i g a t i o n p e r i o d , as shown s c h e m a t i c a l l y i n Fig u r e 6.6. The f i r s t a l f a l f a crop i s c u t about h a l f the season and i r r i g a t i o n f o r the second crop s t a r t e d immediately f o r the remaining h a l f of the season. For s i m p l i c i t y the consumptive uses f o r the two p e r i o d s are assumed i d e n t i c a l . In the a n a l y s i s both o p e r a t i n g procedures adopted i n the pr e v i o u s a n a l y s i s : Procedure I and Procedure I I , are used. D i f f e r e n t time index parameters, T = 0 to T = 4 are used to determine the optimal areas to i r r i g a t e under the assumptions of Procedure I and Procedure I I . For N i c o l a r e g i o n , under the schedule of 2 cuts of a l f a l f a i n the i r r i g a t i o n season the a p p r o p r i a t e time index to use i s T = 2. I 1st Period I r r i g a t i o n --Length of I r r i g a t i o n Season--2nd Period I r r i g a t i o n -T=T 1 Benefits from 1st Period I r r i g a t i o n Total Annual Bene from I r r i g a t i o n 1 Benefits from 2nd Period I r r i g a t i o n Optimal Area Irrigated \ Optimal Area Irrigated_ in Second Period i n r i r s t Period Optimum Area Irrigated Annually Figure 6.6 Schematic Representation of the 2-Period.Irrigation Operation 197 This index i s employed to determine the optimal areas to i r r i g a t e i n the f i r s t and second i r r i g a t i o n p e r i o d s . Employing the s t o c h a s t i c d e c i s i o n theory o p t i m i z a t i o n model the optimal hectarage to i r r i g a t e i n the second i r r i g a t i o n p e r i d , A^ i s determined f i r s t , under the c o n s t r a i n t of h y d r o l o g i c a l u n c e r t a i n t y . With determined the optimal area to i r r i g a t e i n the f i r s t i r r i g a t i o n p e r i o d , A^, i s determined by s e a r c h i n g f o r the complementary hectarage which maximizes the t o t a l expected u t i l i t y i n the whole range of the d e c i s i o n a l t e r n a -t i v e s under i n f l o w u n c e r t a i n t y . 6.3 Model Output: Second I r r i g a t i o n P e r i o d 6.3.1 Output from Procedure I The optimal areas to i r r i g a t e under h y d r o l o g i c a l u n c e r t a i n t y are shown i n Table 6.5 f o r the second i r r i g a t i o n p e r i o d and f o r a l l the f i v e values of the time index para-meters, T = 0 to T = 4. The graphs of t o t a l expected monetary value , and t o t a l expected u t i l i t y as a f u n c t i o n of area under i r r i g a t i o n are shown i n F i g u r e s 6.7 and 6.8, r e s p e c t i v e l y . For T = 2, and adopting the E M V - c r i t e r i o n , the optimal area i s 8,500 hecta r e s with t o t a l expected monetary v a l u e , EMV, of $2.33 X 10 6; adopting the E U - c r i t e r i o n the optimal area drops to 6,500 hectares with t o t a l expected u t i l i t y , EU, of 86.0 f o r the second i r r i g a t i o n p e r i o d . 6.3.2 Output from Procedure II For o p e r a t i n g Procedure I I , and u s i n g the s t i m u l a t o r y crop response f u n c t i o n f o r a l f a l f a , the o p t i m a l areas f o r 198 TABLE 6.5 OPTIMAL AREAS UNDER IRRIGATION FOR SECOND IRRIGATION PERIOD: PROCEDURE I EMV-CRITERION EU-CRITERION Time Index Optimal Area Total Expected Optimal Area Total Expected (months) (Hectares) Monetary Value. (Hectares) U t i l i t y ($ioV T =0 11,500 3.02 9,500 90.50 T = 1 9.000 2.68 8,500 87.50 T = 2 8,500 2.33 6,500 86.00 T = 3 6,000 2.00 6_j000 85.50 T = 4 4,000 1.50 . 3^500 82.25 l O Area under I r r i g a t i o n (1CH Hectares) Figure 6.7 T o t a l Expected Monetary Value vs. Area under I r r i g a t i o n : Procedure I IO0-, F Figure 6.8 T o t a l Expected U t i l i t y versus A rea under I r r i g a t i o n : Procedure I. 201 TABLE 6.6 OPTIMAL AREAS UNDER IRRIGATION FOR SECOND IRRIGATION PERIOD USING STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE II EMV-CRITERION EU-CRITERION Time Index Optimal Area Total Expected- Optimal Area Total Expected (Month s) (Hectares) Monetary Value . (Hectares) U t i l i t y ( $ i o 6 ) T = 0 14,500 3.75 13,000 92.40 T = 1 12,000 3.26 11,000 90.40 T = 2 10,000 2.83 9 ,500 08.50 T = 3 9,000 2.72 9,000 86.75 T = 4 6,500 2.20 6^500 86.00 2 0 2 T = 0 to T .= 4 are given i n Table 6.6, and F i g u r e s 6.9, and 6.10, f o r E M V - c r i t e r i o n and E U - c r i t e r i o n , r e s p e c t i v e l y . For T = 2, u s i n g E M V - c r i t e r i o n the optimal area t o be i r r i g a t e d i s 10,000 he c t a r e s with EMV of $2.83 X 10 6; u s i n g EU-c r i t e r i o n the optimal area i s 9,500 hecta r e s with EU of 88.4, f o r the second p e r i o d of i r r i g a t i o n . For the same Procedure I I but employing the non-s t i m u l a t o r y crop response f u n c t i o n the optimal areas computed from the model are given i n Table 6.7 f o r T = 0 to T = 4, and F i g u r e s 6.11 and 6.12 f o r E M V - c r i t e r i o n and E U - c r i t e r i o n , r e s p e c t i v e l y . For T = 2, E M V - c r i t e r i o n i n d i c a t e s optimal area to be 9,500 hecta r e s with EMV of $2.70 X 10 , and E U - c r i t e r i o n gives 9,000 he c t a r e s with EU of 87.0, f o r the second p e r i o d i r r i g a t i o n . 6.4 Model Output: F i r s t I r r i g a t i o n P e r i o d 6.4.1 Output from Procedure I F i g u r e s 6.13 and 6.14 give the model output u s i n g the E M V - c r i t e r i o n and E U - c r i t e r i o n , r e s p e c t i v e l y ^ f o r Procedure I. While the E M V - c r i t e r i o n g i v e s an optimal area of 15,000 hec t a r e s with EMV of $2.50 x 10^ f o r f i r s t p e r i o d i r r i g a t i o n , the E U - c r i t e r i o n g i v e s a lower value of 10,500 hecta r e s with EU of (17.40, Tor tho 171 ml; p e r i o d i r r 1 <|.'i I:. I <>n . 6.4.2 Output from Procedure II Under the o p e r a t i n g schedule of Procedure I I , and employing the s t i m u l a t o r y crop response f u n c t i o n , the EMV-o 6 T o t a l Expected Monetary Value, EMV ($10 ) uo o . <s O 1 — i : r -TO . C ro (II II II 4> u> II H II O r~ r\ -r 205 TABLE 6.7 OPTIMAL AREAS UNDER IRRIGATION FOR SECOND IRRIGATION PERIOD USING NON-STIMULATORY CROP RESPONSE FUNCTION FOR PROCEDURE II EMV-CRITERION EU-CRITERION Time Index Optimal Area Total Expected Optimal Area Total Expected (months) (Hectares) Monetary Value (Hectares) U t i l i t y ($io6) T = 0 14,000 4.27 12,500 94.00 T = 1 12,500 3.26 10,500 90.00 T = 2 9,500 2.70 9,000 87.00 T = 3 7,500 2.31 7,500 86.00 T = 4 6,500 2.12 6,500. 85.00 Figure 6.11 T o t a l Expected Monetary Value Versus Area under I r r i g a t i o n Using Non-Stimulatory Crop Response Function Total Expected Monetary Value, EMV ($10 ) I CO 9o-f Figure 6.14 T o t a l i^r-Dected U t i l i t y vs. Area under I r r i g a t i o n for Procedure I. 210 c r i t e r i o n g i v e s an optimal area of 22,500 he c t a r e s with EMV 6 of $3.33 x 10 , and E U - c r i t e r i o n g i v e s an area of 14,500 he c t a r e s , with EU of 93.5, f o r the f i r s t p e r i o d of i r r i g a t i o n , as i n d i c a t e d i n F i g u r e s 6.15 and 6.16, r e s p e c t i v e l y . Adopting the non-s t i m u l a t o r y crop response f u n c t i o n f o r the same Procedure II an optimal area of 18,000 he c t a r e s with EMV of $2.75 X 10^ i s obtained employing the E M V - c r i t e r i o n E U - c r i t e r i o n g i v e s a value of optimal area of 12,500 hectares with EU of 89.5, f o r the f i r s t p e r i o d of i r r i g a t i o n , as shown i n F i g u r e s 6.17 and 6.18, r e s p e c t i v e l y . Table 6.8 i s the summary t a b l e which gi v e s a l l the v a r i o u s o p t i m a l areas under the v a r i o u s o p e r a t i n g procedures, and u s i n g the two d i f f e r e n t d e c i s i o n c r i t e r i a f o r the f i r s t i r r i g a t i o n and second i r r i g a t i o n p e r i o d s . From the t a b l e i t i s observed t h a t the p l a n n i n g investment d e c i s i o n f o r r e g i o n a l l a r g e - s c a l e i r r i g a t i o n systems depends p r i m a r i l y on the o p e r a t i n g schedule f o r the system, on the crops t o be i r r i g a t e d and t h e i r responses to water, and on the d e c i s i o n c r i t e r i o n adopted. C o n s i s t e n t l y , Procedure I giv e s lower optimal hectarages than Procedure I I . Even f o r a given Procedure, the optimal hectarage i s a f u n c t i o n of the type of crop response f u n c t i o n used. For example f o r Procedure I I , the s t i m u l a t o r y crop response f o r a l f a l f a c o n s i s t e n t l y gave h i g h e r i r r i g a b l e hectarages than the non-s t i m u l a t o r y f u n c t i o n f o r the two d e c i s i o n c r i t e r i a . Again g i v e n an o p e r a t i n g procedure and a 4--C Figure 6.ID Iota I ixpected Monetary Value versus Area under I r r i g a t i o n using Stimulatory Crop Response Function Figure 6.16 T o t a l Expected U t i l i t y versus Area under I r r i g a t i o n Using Stimulatory Crop Response Function for Procedure II. TABLE 6.8 SUMMARY: OPTIMAL AREAS FOR REGIONAL PLANNING IN.LARGE-SCALE IRRIGATION USING TWO IRRIGATION PERIODS Operating Procedure Criterion of Maximizing Total Expected Monetary Value, EMV  1st Irrigation Period 2nd Irrigation Period Optimal Area Total Optimal Area Total (Hectares) Expected (Hectares) Expected Monetary Monetary Value Value Criterion of Maximizing Total Expected U t i l i t y , EU ($106) ($1Q6) 1st Irrigation Period 2nd Irrigation Period Optimal Area (Hectares) Total Expected U t i l i t y Optimal Area (Hectares) Total Expected U t i l i t y Procedure I 15,000 2.50 8,500 2.33 10,500 87.50 6.500 86.00 Procedure II (i) 22,500 3.33 10,000 2.83 14,500 93.50 9,500 88.50 Procedure II (ii) 18,000 2.75 9,500 2.70 12,500 89.50 9,000 87.00 CO 216 s p e c i f i c crop response f u n c t i o n the op t i m a l p l a n n i n g d e c i s i o n i s i n f l u e n c e d by the e v a l u a t i o n d e c i s i o n c r i t e r i o n used. From Table 6.8 Procedures I and I I show high e r optimal i r r i g a b l e areas when t o t a l expected monetary v a l u e s , EMV, i s the o b j e c t i v e f u n c t i o n t h a t one seeks t o maximize, and lower o p t i m a l hectarages when t o t a l expected u t i l i t y v a l u e , EU, i s the c r i t e r i a l f u n c t i o n t h a t i s maximized. In oth e r words, u s i n g a l i n e a r u t i l i t y f u n c t i o n f o r monetary gains or l o s s e s g i v e s higher i r r i g a b l e areas than using a n o n - l i n e a r , q u a d r a t i c u t i l i t y f u n c t i o n . 217 CHAPTER 7 DISCUSSION OF RESULTS 7.1 D i s c u s s i o n The r e s u l t s from the d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model u s i n g d i f f e r e n t o p e r a t i n g procedures, d i f f e r e n t crop response f u n c t i o n s and d i f f e r e n t d e c i s i o n c r i t e r i a are shown i n Chapters 5 and 6, both i n t a b u l a r form and g r a p h i c a l l y . From these r e s u l t s i t i s e v i d e n t t h a t the optimum area to develop f o r i r r i g a t i o n i s h i g h l y dependent on the a v a i l a b l e h y d r o l o g i c a l i n f o r m a t i o n and time of the i r r i g a t i o n season when t h i s i n f o r m a t i o n i s a v a i l a b l e . I t i s obvious from the t a b l e s and graphs t h a t the time index parameters, T, i s the main f a c t o r c o n t r o l l i n g the d e c i s i o n . As T i n c r e a s e s from T = 0 ( p e r f e c t h y d r o l o g i c a l information) to T = 4 (absolute h y d r o l o g i c a l u n c e r t a i n t y ) the o p t i m a l area decreases, i r r e s p e c t i v e of opera-t i n g procedure, crop response function,, and u t i l i t y f u n c t i o n employed. From Tables 5.7, 5.12, and 5.13 the percentage decreases i n hectarage from p e r f e c t i n f o r m a t i o n to absolute u n c e r t a i n t y are 54%, 50%, and 62.5% f o r step crop response f u n c t i o n , s t i m u l a t o r y and n o n - s t i m u l a t o r y crop response f u n c t i o n s , r e s p e c t i v e l y and adopting the t o t a l expected monetary v a l u e , E M V - c r i t e r i o n ; the corresponding percentage decreases adopting the t o t a l expected u t i l i t y , E U - c r i t e r i o n are 50%, 50%, and 54% f o r the three crop response f u n c t i o n s , r e s p e c t i v e l y . A l s o , g e n e r a l l y , f o r a l l the T-values the E M V - c r i t e r i o n c o n s i s t e n t l y 218 gave h i g h e r i r r i g a b l e hectarages than the E U - c r i t e r i o h . Procedure I I r e s u l t e d i n more hectarages than Procedure I; and s t i m u l a t o r y crop response f u n c t i o n gave more hectarage than no n - s t i m u l a t o r y crop response f u n c t i o n f o r Procedure I I . Thus, the optimal d e c i s i o n c h o i c e was more s e n s i t i v e t o h y d r o l o g i c a l u n c e r t a i n t y when a n o n - l i n e a r u t i l i t y f u n c t i o n was employed as the d e c i s i o n c r i t e r i o n . When the u t i l i t y f u n c t i o n was l i n e a r , as shown i n Fi g u r e 5.4(a), maximizing the t o t a l expected monetary v a l u e , EMV, w i l L a l s o maximize t o t a l expected u t i l i t y . For the d i f f e r e n t l e v e l s of h y d r o l o g i c a l u n c e r t a i n t y , i n d i c a t e d i n the time i n d i c e s , T, the E M V - c r i t e r i o n i n d i c a t e d a decrease i n optimal p o l i c y of 30% to 54%, 19% to 50%, and 37.5% to 62.5% f o r the three crop response f u n c t i o n s , r e s p e c t i v e l y ; with the E U - c r i t e r i o n the c o r r e s p o n d i n g decreases are 10% to 50%, 21% to 50%> and 30% to 54%, r e s p e c t i v e l y . Thus, the s m a l l e r ranges i n r e d u c t i o n o f optimal i r r i g a b l e areas i n d i c a t e d f o r the EMV-c r i t e r i o n as the l e v e l of u n c e r t a i n t y i n c r e a s e d show t h a t the c r i t e r i o n o f maximizing t o t a l expected monetary value i s not s e n s i t i v e t o the r i s k s i n v o l v e d i n the s i t u a t i o n . T h i s i s again r e i n f o r c e d by the f a c t t h a t the optimal i r r i g a b l e hec-tarages chosen using the E M V - c r i t e r i o n are b i g g e r than tho corresponding areas using the E U - c r i t e r i o n . The f a c t t h a t the e f f e c t of u n c e r t a i n t y i n h y d r o l o g i c data was more n o t i c e a b l e when the n o n - l i n e a r u t i l i t y f u n c t i o n 219 was used i s very obvious from the graphs of model outputs presented i n Chapters 5 and 6. From the sharper peaks and s m a l l e r l i m i t i n g v a l u e s of i r r i g a b l e hectarages under the d i f f e r e n t l e v e l s of u n c e r t a i n t y f o r the E M V - c r i t e r i o n i t i s obvious t h a t t h i s d e c i s i o n c r i t e r i o n does not d i s c r i m i n a t e between the high l o s s e s which the d e c i s i o n maker cannot absorb, and very h i g h b e n e f i t s which are o b v i o u s l y d e s i r a b l e and can be e a s i l y absorbed by the d e c i s i o n maker. The l i m i t i n g values i n i r r i g a b l e l a n d are much h i g h e r u s i n g the E U - c r i t e r i o n than u s i n g the E M V - c r i t e r i o n . Thus the EU choice c r i t e r i o n g i v e s a wider range of a l t e r n a t i v e s w i t h p o s i t i v e u t i l i t y v a l u e s . The r e s u l t s of the s e n s i t i v i t y a n a l y s i s i n d i c a t e d t h a t the d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model i s s e n s i t i v e to the e m p i r i c a l l y computed p r i o r p r o b a b i l i t i e s of i n f l o w . T h i s i s observable from Tables 5.15, 5.16, and 5.17 f o r Procedures I and I I , r e s p e c t i v e l y . The optimal p o l i c y i s changed when a d i f f e r e n t s e t of p r i o r p r o b a b i l i t i e s i s used. Thus i t was deemed necessary to determine the minimum p r o b a b i l i -t i e s t h a t leave the p r e f e r r e d a c t i o n s o p t i m a l , t h a t i s to estimate how s m a l l the p r o b a b i l i t y of a s p e c i f i c s t a t e of nature would have to be i n order t h a t the p r e f e r r e d a c t i o n , a s s o c i a t e d with t h a t s t a t e would s t i l l remain the p r e f e r r e d a c t i o n . These minimum p r o b a b i l i t i e s are g i v e n i n Tables 5 . 1 8 ( c ) - l and 5.18(c)-2. From the r e s u l t s presented i n Chapters 5 and 6 i t 220 i s obvious t h a t the model i s h i g h l y s e n s i t i v e to crop response f u n c t i o n s . Both the shape and slope of the crop response f u n c t i o n assumed i n f l u e n c e the optimal d e c i s i o n . For a l l l e v e l s of h y d r o l o g i c a l u n c e r t a i n t y c o n s i d e r e d i n t h i s study the step crop response f u n c t i o n gave lower optimal i r r i g a b l e hectarages than e i t h e r the s t i m u l a t o r y , or the no n - s t i m u l a t o r y crop response f u n c t i o n s . T h i s i s t o be e x p l a i n e d by the f a c t t h a t i r r i g a t i n g l e s s area to maximum water requirement of the crop i s l e s s economical than i r r i g a t i n g more area with an amount l e s s than the maximum crop water requirement under the e x i s t i n g assumptions employed i n t h i s study: land i s not as l i m i t i n g as water f o r i r r i g a t i o n . Thus t h i s study has shown t h a t f o r the c o n d i t i o n s e x i s t i n g i n the N i c o l a V a l l e y r e g i o n of B r i t i s h Columbia i t i s b e t t e r to i r r i g a t e more land to a l e v e l l e s s than the maximum water h o l d i n g c a p a c i t y of the s o i l than i r r i g a t i n g s m a l l e r areas t o maximum consumptive use requirements of the crop. T h i s would be i n l i n e w i t h the c o n s e r v a t i o n approach i n u t i l i z a t i o n of i r r i g a t i o n water which has been recommended f o r the area. A l s o t h i s approach.would reduce the amount of i r r i g a t i o n r e t u r n flow and avoid the p o s s i b l e l o n g -term " u n c e r t a i n " e f f e c t s on the a q u a t i c environment of the regi o n . The procedure advocated w i l l bo advantage aim only wlmn astu t e i r r i g a t i o n management p r a c t i c e s geared to a v o i d i n g " o v e r - i r r i g a t i o n " i n the area are e s t a b l i s h e d , and an optimal balance i s sought between the area to put under i r r i g a t i o n and 221 the amount of water to apply per hectare per i r r i g a t i o n . However t h i s r e q u i r e s a p r e c i s e determination of; the f u n c t i o n ro l a t i n y crop water consumption to the y i e l d of: the crop, f o r a l f a l f a and a l l the other p o s s i b l e crops t h a t might be i n t r o d u c e d i n the area i n the f u t u r e . I f t h i s f u n c t i o n i s not w e l l e s t a b l i s h e d i n t h i s area, the l o s s i n y i e l d of crop r e s u l t i n g from s m a l l e r amount of water a p p l i e d might be so g r e a t as to o f f s e t the advantage of the e x t r a hectarage t h a t has been brought under i r r i g a t i o n . To improve the outputs from the model the u n c e r t a i n t y a s s o c i a t e d with crop water requirement was s u b j e c t i v e l y assessed and i n t e g r a t e d with the u n c e r t a i n t y i n i n f l o w . The e f f e c t of i n t e g r a t i n g the u n c e r t a i n t y i n crop water requirement i s the r e d u c t i o n i n expected u t i l i t i e s and expected monetary v a l u e s . These are shown i n Tables 5.23, 5.2 4 and 5.25 f o r Procedures I and I I , r e s p e c t i v e l y ; and i n F i g u r e s 5.23 through 5.28. Again the c r i t e r i o n of maximizing t o t a l expected u t i l i t y i s more s e n s i t i v e to the u n c e r t a i n t y elements than the c r i t e r i o n of maximizing t o t a l expected monetary v a l u e . S e n s i t i v i t y a n a l y s i s was done with r e s p e c t to crop net v a l u e . The g r e a t e s t e f f e c t of i n c r e a s i n g the crop net value i s seen on the c r i t e r i o n of maximizing t o t a l expected monetary v a l u e . Tables 5.19, 5.20 and 5.21 summarize the r e s u l t s of t h i s a n a l y s i s . For the same range of optimal areas the t o t a l expected monetary values are i n c r e a s e d a l o t more than the u n i t i n c r e a s e i n the crop net v a l u e . T h i s aspect of 222 the model i s a good g u i d e l i n e i n d e c i s i o n s i n v o l v i n g the optimal mix of crops to grow i n a r e g i o n of mixed cropping systems, high - v a l u e d cash crops, r o o t crops, v e g e t a b l e s , and p a s t u r e . Under h y d r o l o g i c a l u n c e r t a i n t y and under t o t a l i r r i g a b l e land c o n s t r a i n t s t h i s model can be r e a d i l y a p p l i e d t o determine the o p t i m a l areas of each crop to develop and operate i n the r e g i o n . Another important, i f not the most important, aspect of the d e c i s i o n theory s t o c h a s t i c o p t i m i z a t i o n model developed i n t h i s study i s the u t i l i t y f u n c t i o n of the d e c i s i o n maker. The model outcome i s very s e n s i t i v e t o , and h i g h l y dependent on the form of the u t i l i t y f u n c t i o n employed i n the d e c i s i o n a n a l y s i s . For any given o p e r a t i n g procedure, s e t of p r i o r p r o b a b i l i t i e s , and g i v e n crop response f u n c t i o n , the output from the model w i l l vary e r r a t i c a l l y i f d i f f e r e n t u t i l i t y f u n c t i o n s are used i n the a n a l y s i s . The t a b l e s and graphs of Chapter 5 c l e a r l y d e p i c t s the e f f e c t of u s i n g both the l i n e a r u t i l i t y f u n c t i o n , F i g u r e 5 . 4 ( a ) , and the q u a d r a t i c polynomial u t i l i t y f u n c t i o n , F i g u r e 5 . 4 ( b ) , f o r the Quilchena ranch i r r i g a t i o n system management d e c i s i o n "making. The importance of the u t i l i t y f u n c t i o n as w e l l as i t s r o l e i n the model i s made more e x p l i c i t by the f a c t t h a t even when a s p e c i f i c u t i l i t y f u n c t i o n i s adopted a l t e r i n g the shape or the slope of t h i s f u n c t i o n would r e s u l t i n a d i f f e r e n t model outcome. For example, i f a c u b i c u t i l i t y f u n c t i o n i s s u b s t i t u t e d f o r the q u a d r a t i c f u n c t i o n of F i g u r e 5 . 4 ( b ) , the o r d e r i n g of a c t i o n s may be completely 223 reversed, even though both the cubic and quadratic polynomials may have si m i l a r properties: increasing marginal d i s u t i l i t y for losses, and increasing marginal u t i l i t y for gains. The reversal i n ordering of actions obviously would r e s u l t only from the fact that the i n d i v i d u a l whose u t i l i t y function i s a cubic polynomial has an increasing marginal d i s u t i l i t y for losses which i s less d r a s t i c than that for the i n d i v i d u a l with a second order polynomial u t i l i t y function. Even i f the two u t i l i t y functions were both cubic polynomials but with d i f f e r e n t c o e f f i c i e n t s , the ordering of actions w i l l be d i f f e r e n t for the two cubic functions, merely because of d i f f e r e n t slopes and d i f f e r e n t points of i n f l e c t i o n . In general, the i n d i v i d u a l decision maker's u t i l i t y function can be viewed as a r e f l e c t i o n of his attitude towards r i s k . The steeper the u t i l i t y function of an i n d i v i d u a l i s , the more conservative his attitude towards r i s k . Closely a l l i e d to the u t i l i t y function i s the question of the better c r i t e r i o n to employ i n ordering feasible alternatives when applying the stochastic decision theory optimization model. This aspect of the model presents a r e a l problem for the decision analyst since an action which i s optimal using the c r i t e r i o n of maximizing t o t a l expected monetary value may become the least desirable action when the c r i t e r i o n of maximizing t o t a l expected u t i l i t y value i s employed. It has been shown that the outcomes from the EMV-criterion and the EU-criterion w i l l be the same only when the decision maker's 22 4 u t i l i t y f u n c t i o n i s l i n e a r . In other, cases where monetary values do not measure the r e a l consequences of the gains or l o s s e s i n h e r e n t i n the d e c i s i o n problem the assumption of a l i n e a r u t i l i t y f u n c t i o n would not be v a l i d . Thus, i t becomes necessary to d e r i v e the a p p r o p r i a t e u t i l i t y f u n c t i o n which c o u l d , not o n l y r e f l e c t the d e c i s i o n maker's c u r r e n t s t a t e of wealth but a l s o p r o v i d e d e c i s i o n s t h a t would be c o n s i s t e n t w i t h the d e c i s i o n maker's p r e f e r e n c e s among r i s k y d e c i s i o n a l t e r n a -t i v e s . The f a c t t h a t the r e s u l t s from the model employed i n t h i s study show t h a t , i n most cases, the o p t i m a l i r r i g a b l e areas f o r the d i f f e r e n t l e v e l s o f u n c e r t a i n t y are s m a l l e r f o r the E U - c r i t e r i o n than f o r the E M V - c r i t e r i o n i n d i c a t e s t h a t the r o l e of r i s k i n i r r i g a t i o n systems management i s b e t t e r accounted f o r u s i n g the E U - c r i t e r i o n . Thus, i t can be s a i d t h a t the E U - c r i t e r i o n i s a s u p e r i o r c r i t e r i o n f o r a g r i c u l t u r a l management d e c i s i o n s . In the p a r t i c u l a r case of Quilchena ranch management, i t i s c l e a r l y obvious, from the output t h a t the d e c i s i o n maker i s an expected u t i l i t y maximizer because the optimal i r r i g a b l e areas o b t a i n e d u s i n g the E U - c r i t e r i o n are c l o s e r to the management's p r a c t i c e than the values from the E M V - c r i t e r i o n . For example from Table 5.7, f o r T = 4, the optimal i r r i g a b l e areas are 600 h e c t a r e s (1,500 acres) using the E M V - c r i t e r i o n , and 500 h e c t a r e s (1,250 acres) u s i n g the E U - c r i t e r i o n . The present mode of d e c i s i o n i n Quilchena ranch i s based on long-term accumulated experience and the management 225 on t h i s b a s i s , i r r i g a t e s 1,000 - 1,250 acres (400 - 500 h e c t a r e s ) . Assuming t h a t the E U - c r i t e r i o n i s a s u p e r i o r c r i t e r i o n to be employed i n a r r i v i n g a t b e t t e r d e c i s i o n s with r e s p e c t to i r r i g a t i o n systems management under r i s k and u n c e r t a i n t y , and t h a t i r r i g a t o r s or those i n v o l v e d i n the business of s u p p l y i n g i r r i g a t i o n water are u t i l i t y maximizers, the g r e a t e s t hinderance o f the approach i s l i k e l y to be the accurate d e r i v a t i o n of the i r r i g a t i o n d e c i s i o n maker's u t i l i t y f u n c t i o n . I t i s obvious t h a t farmers i n g e n e r a l have a high a v e r s i o n to r i s k but farmers adopt d i f f e r e n t management p r a c t i c e s because they have d i f f e r e n t a t t i t u d e s toward r i s k . Therefore i t w i l l be erroneous to use one u t i l i t y f u n c t i o n to r e p r e s e n t the a t t i t u d e of farmers i n any given r e g i o n . Thus to be able to compare the d i f f e r e n t a t t i t u d e s of farmers toward r i s k t h e i r i n d i v i d u a l u t i l i t y f u n c t i o n s have to be d e r i v e d . Although e x t e n s i v e f i e l d survey would be r e q u i r e d , i t i s r e l a t i v e l y e a s i e r to d e r i v e the i r r i g a t o r ' s u t i l i t y f u n c t i o n where d e c i s i o n s r e g a r d i n g water use are made at the farm l e v e l , as i n the Quilchena ranch. In a r e g i o n where the water system i s h i g h l y c e n t r a -l i z e d and a c e n t r a l a u t h o r i t y maker, a l l the d e c i n i o n a r e g a r d i n g water use, the task of d e r i v i n g u t i l i t y f u n c t i o n becomes more complicated. T h i s i s so because the a n a l y s t attempts to maximize the expected u t i l i t y of the farmers but the c h i e f 226 d e c i s i o n maker may be an e x e c u t i v e of the c e n t r a l a u t h o r i t y who i s not an i r r i g a t o r h i m s e l f . One p o s s i b l e way to avoid the s i t u a t i o n i s to assume t h a t the c h i e f d e c i s i o n maker d e r i v e s h i s u t i l i t y value from knowledge of the u t i l i t y v alues of the group of i r r i g a t o r s or s e t t l e r s f o r whom the p r o j e c t i s planned. However i n the case of a group recommendation the a n a l y s t or c o n s u l t a n t should use a group u t i l i t y f u n c t i o n as an a i d i n a r r i v i n g at h i s recommendations. He may r e s o r t to m u l t i a t t r i b u t e u t i l i t y theory [ 38, 6 3] and use i t to d e r i v e the r e q u i r e d 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 s ; or the technique adopted by O f f i c e r , H a l t e r , and D i l l o n [85, 73 ] i n t h e i r study i n A u s t r a l i a . An obvious method of d e r i v i n g a group u t i l i t y f u n c t i o n i s to take the average of the i n d i v i d u a l ' s u t i l i t y f u n c t i o n s , t h a t i s , average the c o e f f i c i e n t s of the i n d i v i d u a l f u n c t i o n s . Another method i s t o take the median of the i n d i v i d u a l u t i l i t y f u n c t i o n as the group f u n c t i o n . The group u t i l i t y f u n c t i o n s d e r i v e d u s i n g one of the above techniques may not be p e r f e c t but r i s k - o r i e n t e d u t i l i t y f u n c t i o n approach can provide b e t t e r recommendations than a more t r a d i t i o n a l approach such as expected c o s t m i n i m i z a t i o n which makes no allowance f o r r i s k . A l s o the maximization o f expected u t i l i t y approach i s e s s e n t i a l l y a b e h a v i o r a l approach which i s g e n e r a l l y s u p e r i o r to a l t e r n a t i v e approaches of maximizing expected monetary outcomes. The whole r a t i o n a l e f o r attempting to develop a 227 decision theory stochastic optimization model which can be employed for i r r i g a t i o n system's management decision-making i s that the major uncertainty elements, v i z : uncertainty i n inflow and uncertainty i n crop water requirements could be better-incorporated i n the decision framework under uncertainty. Also •it i s believed that Bayes' approach, which incorporates some element of f l e x i b i l i t y could be employed to update the pro-b a b i l i t y d i s t r i b u t i o n s as further information becomes available. Tables 5.27, 5.28 and 5.29 summarize the computations of Bayesian strategies using posterior p r o b a b i l i t y d i s t r i b u t i o n of inflow. The analysis assumes that a forecasting device i s available and that forecasts of the d i f f e r e n t states of nature are available at the beginning of the i r r i g a t i o n season. The analysis indicates that the establishment of some type of inflow forecasting device may be worthwhile since the values of the Z-predictor for the two Procedures adopted i n t h i s study are positive benefits. This would enhance better i r r i g a t i o n planning and operation decisions for the Quilchena area. In Section 5.7 i t was shown - that over a long-term period (5 year operation period) the net worth of yearly revenues accruing from Procedure II would be larger than that from Procedure I. This comparison was made assuming a 6 percent inte r e s t rate. Thus the strategy of adopting the conservation approach while enhancing the output i n i r r i g a t e d agriculture and preservation of the aquatic environment by minimizing i r r i g a t i o n return flows i s not only a better approach environmentally, 228 but also on economical grounds. Over i r r i g a t i o n i s uneconomi-c a l and long-term cumulative effects of i r r i g a t i o n return flows might be detrimental. Project planners and i r r i g a t o r s have to become sensitive to t h i s . Chapter 6 gives the results of employing the model for regional planning i n large-scale i r r i g a t i o n systems. The Nicola Valley's regional i r r i g a t i o n development and operation are concurrently considered i n the model, instead of the conservative model approach of treating i r r i g a t i o n planning p r i o r to under-standing the mode, and i n i s o l a t i o n , of the operation. It was demonstrated, through the use of the stochastic decision theory optimization model that, under uncertainty and r i s k , the appropriate planning strategy would be f i r s t to analyze the d i f f e r e n t possible modes of operation of the system. Tables 6.5, 6.6, and 6.8, and Figures 6.7 and 6.8, 6.9 and 6.10, and 6.11 and 6.12, give the results of the analyses for operation. These results represent a set of possible preferred actions under varying degrees of uncertainty and r i s k . With the operating decisions sorted out the analysis was extended for planning or policy decision. In the Nicola Valley region the most appropriate time index parameter to employ i n operation and planni of i r r i g a t i o n systems under uncertainty i s determined a3 T=2. For two cuts of a l f a l f a harvested i n the i r r i g a t i o n season, the analysis proceeded by determining the optimal areas for the two i r r i g a t i o n periods. The results for the two periods of i r r i g a t i o n are shown i n Figures 6.13 and 6.14; 6.15 229 and 6.16; 6.17 and 6.18, f o r the d i f f e r e n t o p e r a t i n g p o l i c i e s , f o r the two a l t e r n a t i v e d e c i s i o n c r i t e r i a , EMV and EU, and f o r the d i f f e r e n t crop response f u n c t i o n s , r e s p e c t i v e l y . Taiile 6.8 i s a summary of the model output f o r r e g i o n a l p l a n n i n g i n l a r g e - s c a l e i r r i g a t i o n , f o r a two crop i r r i g a t i o n p e r i o d o p e r a t i o n f o r the N i c o l a V a l l e y r e g i o n of B r i t i s h Columbia. The r e s u l t s are c o n s i s t e n t with the r e s u l t s obtained when the model was a p p l i e d t o a s i t u a t i o n where d e c i s i o n s r e g a r d i n g water use were made a t the farm l e v e l , as was i l l u s t r a t e d i n Chapter 5 f o r the Quilchena ranch i r r i g a t i o n system management. From Table 6.8 i t i s seen t h a t the optimal i r r i g a b l e areas f o r p l a n n i n g purposes depend on the o p e r a t i n g procedure f o r the i r r i g a t i o n system, the crops i r r i g a t e d and t h e i r responses to water, and on the d e c i s i o n c r i t e r i o n employed. I t i s observed t h a t i r r e s p e c t i v e of the o p e r a t i n g procedure, the crop response f u n c t i o n and the d e c i s i o n c r i t e r i a , the optimal areas i n the f i r s t p e r i o d i r r i g a t i o n are h i g h e r than the corresponding optimal areas f o r the second p e r i o d i r r i g a t i o n . T h i s i s t o be expected i n the N i c o l a V a l l e y r e g i o n i n accordance with the p a t t e r n o f consumptive uses and i n f l o w s f o r the two i r r i g a t i o n p e r i o d s as shown i n F i g u r e s 6.2 and 6.3, r e s p e c t i v e l y . Thus f o r improved i r r i g a t i o n system decision-making f o r t h i s r e g i o n , the c e n t r a l a d m i n i s t r a t i v e agency, t h a t i s , the Water Rights Branch of the B r i t i s h Columbia Water Resources S e r v i c e should i n c o r p o r a t e t h i s f a c t when determining the amount of water l i c e n s e s to i s s u e i n the r e g i o n f o r i r r i g a t i o n purposes. Permits 230 that do not take the mode of i r r i g a t i o n operation and hydrologica v a r i a b i l i t y during the i r r i g a t i o n season into account are not adequate i f the p o l i c y of conservation i s to be achieved i n t h i s region. Also the c r i t e r i o n of maximizing t o t a l expected u t i l i t y appears to be a better choice c r i t e r i o n for regional i r r i g a t i o n system planning and management under uncertainty and r i s k . This i s evident from Table 6.8. The EMV-criterion gives optimal i r r i g a b l e areas which are closer to the upper l i m i t of the available, economically i r r i g a b l e land but EU-criterion gives lower optimal areas which indicate that the r i s k s i n the s i t u a t i o n are better accounted for i n t h i s choice c r i t e r i o n . For example, for operating Procedure II and using the stimulatory crop response function, EMV-criterion gives optimal i r r i g a b l e areas as: 22, 500 hectares for the f i r s t i r r i g a t i o n period and 10,000 hectares for the second i r r i g a t i o n period; but EU-i r r i g a t i o n gives optimal hectarages as: 14,500 hectares, and 9,500 hectares for the f i r s t and second i r r i g a t i o n periods, respectively. 7.2 Probable Error The a n a l y t i c a l sequence of the stochastic decision theory optimization model employed in this study embodien aomo assumptions and in lorntudia Le computations in the d i f f e r e n t facets of the transformation of the input data into the output. Thus, i t i s possible that some errors might have been introduced 231 i n the transformation process and the accumulted e f f e c t might have shown i n the model output. Th probable errors i n the model can emmanate from one of the following sources: 1. Assumptions made in the development of the mode1; 2. The degree of r e l i a b i l i t y of the various input data, such as inflows and the accompanying frequency analysis and consequent d e f i n i t i o n of states of nature and the p r i o r p r o b a b i l i t i e s and consumptive use; 3. The estimation of crop response functions; 4. The derivation of the u t i l i t y functions. It was assumed, i n the development of the model that the crop uses water at a constant peak consumptive use rate, 6.35 mm/day throughout the i r r i g a t i o n season. The e f f e c t of t h i s assumption would be r e f l e c t e d i n the model output giving lower optimal i r r i g a b l e areas. The consumptive use rate i s not a constant but varies from day to day and with d i f f e r e n t stages of development of the crop. I f accurate information becomes available on the d a i l y evaporation rates.and on the consumptive uses of the crop for the d i f f e r e n t stages of crop development, such as the vegetative stage and maturity stage, then the model could be modified to y i e l d precise optimal areas to i r r i g a t e in the i r r i g a t i o n season. In the 2-crop i r r i g a t i o n consideration, the optimal i r r i g a b l e areas estimated for the second i r r i g a t i o n 232 period may be s l i g h t l y smaller than the output since the seasonal consumptive use for the second period was assumed the same as the f i r s t period while i n actual practice i t i s s l i g h t l y higher. Because of lack of hydrological data for the Nicola Valley area an i n d i r e c t means was adopted in estimating the long-term flows of the Quilchena Creek. The 88-year flow record was estimated by developing a relationship between r a i n f a l l and runoff, and this was adjusted to account for snowmelt runoff. This analysis was not very r e l i a b l e and might have introduced errors i n the frequency analysis and the consequent estimation of p r i o r p r o b a b i l i t i e s of the states of nature. The quantitative e f f e c t of t h i s error on the model output could not be evaluated. In the regional planning application of the model for the Nicola Valley region only 25 years of flow data were available for frequency analysis and d e f i n i t i o n of p r i o r p r o b a b i l i t i e s of the states of nature. This length of record i s not very adequate and very r e l i a b l e for hydrological s t a t i s t i c a l analysis for planning purposes. The crop response functions employed i n the study were not actual f i e l d determined functions. The functions were assumed and they may not r i g i d l y represent the actual relationship between water consumption and y i e l d of a l f a l f a . Real functions, i f they become available, should be substituted for the assumed functions employed in this study. Also, i f 233 d i f f e r e n t crops are planned for i r r i g a t i o n i n a region these functions must be established for each crop. In the regional application of the model i t was assumed that the s o i l i s homogeneous i n the entire region. This i s not l i k e l y to be true. In such cases composite crop response functions may be necessary to improve the outputs from the model. The quadratic u t i l i t y function of Figure 5.4 (b) which was employed for the EU-criterion may not s t r i c t l y represent the Quilchena creek management's u t i l i t y curve. The curve i s that derived and used for a n a l y t i c a l purposes. If the r e a l u t i l i t y function could be derived this should be used for p r a c t i c a l decision-making i n i r r i g a t i o n system management. The use of an i n d i v i d u a l decision-making u t i l i t y function, even when accurately estimated, for decisions involving a group i s not adequate. Thus, i n the regional planning application of the model the greatest shortcoming might l i e i n the derivation of an adequate u t i l i t y function for the group of i r r i g a t o r s i n the region. The output from Chapter 6 could be improved considerably i f a straightforward technique for deriving group u t i l i t y functions were readily available. 234 CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions From the results of t h i s study, the stochastic decision theory optimization model developed appears to be an appropriate methodology for obtaining optimal p o l i c i e s for planning the development and operation of i r r i g a t i o n systems under uncertainty. The model has demonstrated that i t i s b e n e f i c i a l to develop a water supply regime that i s dictated by long-range stochastic considerations. Thus the technique of this study would be a better substitute for the present practice i n which a fixed, predetermined amount of water i s allocated to i r r i g a t i o n s through a permit system. This i s a highly i n s t i t u t i o n a l i z e d system of water rights, with disregard to the variable hydrological conditions. The results of thi s study have also demonstrated that any meaningful planning policy i s achievable only when studies on the mode of operation of the system have been c a r r i e d out. An optimal competence i n i r r i g a t i o n systems management can only be achieved by concurrently considering a l l the feasible planning alternatives and a l l the feasible operating alternatives in the same time period, and applying d i f f e r e n t decision c r i t e r i a which may be relevant to the decision problem. In t h i s study the model developed was applied to the Nicola Valley i r r i g a t i o n systems management, both at the farm l e v e l and on a regional planning l e v e l . Using the two d i f f e r e n t operating p o l i c i e s and the two decision c r i t e r i a employed i n the study i t was demonstrated that i t i s better to choose optimal operating and investment p o l i c i e s on the basis of maximizing t o t a l expected u t i l i t y since, under uncertainty and r i s k , t h i s c r i t e r i o n can force the decision maker to focus on the aspects of r i s k inherent i n the problem, and also allows him some f l e x i b i l i t y margin to adjust to the r i s k i n the s i t u a t i o n . The u t i l i t y analysis technique i s advantageous i n that i t provides an objective analysis of subjective considerations. In complicated decision problems, such as regional i r r i g a t i o n developmental and operational problems, i t allows the decision makers to break the decision problem into many smaller decision problems which can be quantified by subjective u t i l i t y assessment. The whole process takes cognizance of the value of information, making best use of l i m i t e d data while i n d i c a t i n g the areas of greatest uncertainty, areas i n which the need for additional information i s most pressing. Moving backward in time, the decision-maker can evaluate the outcome of certain observations, aid by so doing can update his information and modify his actions so that the choice he actually makes i s most l i k e l y to be in harmony with the r e a l i z e d event, or to be an improved decision. Thus, under the e x i s t i n g constraint of lack of hydrological information for e f f i c i e n t planning of i r r i g a t i o n systems 236 management, the decision problem can be best structured i n the stochastic decision theory framework developed i n this study. The stochastic decision theory optimization problem can then be solved using the analyst's a b i l i t y to provide an objective analysis.of subjective considerations as well as mathematically sound and e f f i c i e n t programming methods to obtain optimal p o l i c i e s for planning subject to s p e c i f i e d optimal predetermined operational procedure and established u t i l i t y function. By conceptualizing the problems of uncertainty and information seeking with respect to i r r i g a t i o n systems management in the decision-making framework of thi s study, i t i s believed that future research on i r r i g a t i o n systems managerial processes and related aspects of motivation, communication, and organization can proceed more meaningfully. The technique i s simple, man-oriented, and represents an approach that could be conveniently employed to solve r e a l - l i f e , p r a c t i c a l management decision problems i n i r r i g a t i o n planning. The technique s o l i c i t s the opinion and cooperation of the c l i e n t and thus i t offers a behavioral approach to i r r i g a t i o n system management. The u t i l i t y theory focuses on human needs and attempts to maximize these s p e c i f i c needs and preferences ( u t i l i t y ) i n the optimization period. The theory improves communication between the i r r i g a t o r s and the po l i c y makers, and thus i t motivates a l l the parties involved to attempt and achieve t h e i r objectives and goals. The approach gives optimal p o l i c i e s that are consistent 237 with respect to the s p e c i f i e d hydrological states of nature, the operating procedure stipulated, the crop response function in use, and the decision maker's u t i l i t y function. Thus i t i s consistent with the decision maker's value system and ar t i c u l a t e d preferences. The optimal operating and planning decisions obtained for the Nicola Valley region are maximum u t i l i t y for long-term development and operation of the region's i r r i g a t i o n system assuming storage and considering uncertainties pertaining to inflows. Based on the results and considering conservation of water resources i n the region- avoiding o v e r i r r i g a t i o n and preservation of the aquatic environment while enhancing the output i n i r r i g a t e d agriculture,-as the sole objective and a task that ought to be achieved, optimum competence i n the region's i r r i g a t i o n practices can be achieved by abandoning the present practice of "over-application of i r r i g a t i o n water" i n an e f f o r t to apply the maximum consumptive use of the crop, and adopt the practice of i r r i g a t i n g more hectarage at a water consumption l e v e l that i s s l i g h t l y below the maximum water requirement of the crop. That i s , i n the context of thi s study, Procedure II i s more b e n e f i c i a l to the region than Procedure I. HoWever, t h i s requires that an optimal balance be sought between the variable i r r i g a t i o n water supply and the associated uncertainties, and the maximum hectarage to i r r i g a t e in any i r r i g a t i o n period, expressed i n terms of the amount of consumptive use that would 2 3 8 achieve a desirable y i e l d . This policy should be subjected to the needs of the region and modified through s e n s i t i v i t y analysis to obtain optimal decision strategies over substantial time periods. Therefore, i t can be unequivocally concluded that the stochastic decision theory optimization model developed and used i n t h i s study o f f e r s considerable promise for optimal long-term development and operation of i r r i g a t i o n systems with storage under hydrological uncertainty. The model i s stochastic and gives optimal planning p o l i c i e s based on predetermined optimal operating p o l i c i e s . The model uses t o t a l expected u t i l i t y as the c r i t e r i a l function to maximize, and thus i s not s t r i c t l y based on economic c r i t e r i a . The model i s simple, f l e x i b l e , behavioral and man-oriented i n approach, and thus very applicable to r e a l - l i f e situations. Although the model has been applied to the Nicola Valley region i r r i g a t i o n systems management, the features of the model can be extended to many d i f f e r e n t types and scales of i r r i g a t i o n development and operation i n other d i f f e r e n t l o c a l i t i e s . Thus, the results obtained from the stochastic decision theory optimization model developed and used in t h i s study f u l f i l the stated objectives of the study. 8.2 Recommendations As concluded above, the res u l t s obtained from the model f u l f i l the objectives of the study, and i t s features can be 239 extended to other l o c a l i t i e s and f o r d i f f e r e n t system management p r a c t i c e s . The e v a l u a t i o n c r i t e r i o n of maximizing t o t a l expected u t i l i t y as a b a s i s f o r s e l e c t i n g the o p e r a t i n g and developmental p o l i c i e s i n i r r i g a t i o n p r a c t i c e s w i l l be p a r t i c u l a r l y u s e f u l i n the d e v e l o p i n g n a t i o n s of the world, although i t s f e a t u r e s are widely a p p l i c a b l e i n any l o c a l i t y , i r r e s p e c t i v e of the nation's e x i s t i n g stage of development and wealth. The d i f f e r i n g o b j e c t i v e s of development with r e s p e c t to the d i f f e r e n t world c u l t u r e s and l e v e l s of wealth can be best t r a n s l a t e d i n terms of u t i l i t i e s and u t i l i t y s c a l e developed f o r the s p e c i f i c l o c a l i t y . The c o n f l i c t s i n o b j e c t i v e s of i r r i g a t i o n development, s e l f - s u f f i c i e n c y i n f o o d - g r a i n p r o d u c t i o n f o r l o c a l consumption and long-term f o r e i g n trade and f o r e i g n exchange as observed i n such c o u n t r i e s as P a k i s t a n , I n d i a and N i g e r i a , [41] , can be r e s o l v e d by r e s o r t i n g to u t i l i t y a n a l y s i s . In N i g e r i a the s h o r t term o b j e c t i v e of the n a t i o n ' s r e c e n t l a r g e - s c a l e i r r i g a t i o n e s tablishments i s "to produce enough food to feed the n a t i o n and thus o b v i a t e the need f o r r e l i a n c e on c o s t l y imported foods. The long-term o b j e c t i v e i s to r e v e r s e the adverse movement of c a p i t a l by producing s u r p l u s food f o r export, thus c o n t r i b u t i n g to f o r e i g n exchange e a r n i n g s . " [82] . A l s o , the most recent World Bank survey of N i g e r i a noted t h a t 'the- major problem i n expanding i r r i g a t i o n was the absence of b a s i c data. " H y d r o l o g i c a l data are meager or n o n - e x i s t e n t . " [61, .83]. Therefore, under the e x i s t i n g time-240 dependent objectives and lack of adequate hydrological information, the stochastic decision theory optimization model becomes the most applicable management tool since the s i t u a t i o n has to be subjectively assessed and improved i n the l i g h t of more hydrological information becoming available. Thus, i t i s recommended that the features of this model be employed for i r r i g a t i o n systems management decision-making i n Nigeria and the other developing countries i n s i m i l a r situations. For the ex i s t i n g socio-economic and s o c i o - c u l t u r a l situations i n these countries the model could be employed to determine the optimal mix of crops to be i r r i g a t e d as well as the time-dependent rotation sequence that best suites the p a r t i c u l a r needs and preferences of the nation. It i s l i k e l y that Procedure II w i l l be a better procedure to adopt i n these countries but p r a c t i c a l l y i t may not be feasible since the technological functions required to adopt t h i s procedure may be lacking. In general, the model outputs can be improved by carrying out more research on u t i l i t y functions. This i s p a r t i c u l a r l y important i n the case where the i r r i g a t i o n enterprise i s planned for a group of farmers and a group u t i l i t y function would be required. 8.3 Additional research requirements The r e l i a b i l i t y of the results from the model could be improved by obtaining more information on the precise 241 relationships between water consumption and y i e l d of crops. These crop response functions and other necessary technological production functions are very e s s e n t i a l for meaningful application of the model for r e a l i s t i c decision-making i n i r r i g a t i o n systems' management under hydrological uncertainty. More research should be encouraged i n t h i s area. Accurate input data are required i n the model formu-l a t i o n i f r e a l i s t i c and useful results are to be obtained from i t . Also i t may be advantageous to eliminate the less desirable alternatives through a 'pruning' technique when the number of decision alternatives becomes very large. Although Fortran MTS/370 routine i s very e f f i c i e n t for the solution, the output may be very voluminous and may, i n addition, require an increased computer time and consequently increased financing. 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" D i g i t a l Computer Simu-l a t i o n for Solving Management Problems of Conjunctive Groundwater and Surface Water Systems". Water Resources  Research, Vol. 8, pp. 533-556, 1972. 129. Young, R.A., and W.E. Martin. "Modelling Production Response Relations for I r r i g a t i o n Water: Review and Implications". Report 16, Water Resources and Economic Development of the West, Dec. 1967, pp. 1-24. 2 6 0 APPENDIX A 261 L I S T I N G OF +MSOURCE* AT 1 7 : 0 1 : 07 4 C P R O G R A M A - l V 5 6 c c 7 c > 8 c A P P L I C A T I O N OF D E C I S I O N THEORY IN MANAGEMENT OF WATFR RESOURCE- S Y S T E M S 9 c O P T I M I Z A T I O N OF I R R I G A T I O N S Y S T E M S UNDER H Y D R O L O G I C A L U N C E R T A I N T Y 10 c U N C F R T A I N T Y IN INFLOW IS E X A M I N E D W.R.T. I R R I G A T I O N S Y S T E M S DE SI ON-MAX y 11 c ING : D E V E L O P M E N T AND O P E R A T I O N S > 12 c D E C I S I O N C R I T F R I A : MA XI M I Z A T I ON OF E X P E C T E D V A L U E S 13 c A ( I ) I S MATRIX OF V A R I A B L E A R E A S ( H E C T A R E S ) UNDER I R R I G A T I O N - A C T I O N S > 14 c C U ( I ) I S MATRIX OF CROP GROSS C O N S U M P T I V E U S E S ( C U B I C M E T E R S ) CORR^SPON > 15 c DING TO A ( I ) MATRIX 16 C S ( J ) IS MATRIX OF V A R I A B L E I N F L O W S ( C U B I C M E T E R S ) - S T A T C S OF NATURE > 17 C X ( I , J ) I S D I F F E R E N C E B T . S ( J ) AND WATER USED UP IN H ( L ) MONTHS > 18 c H ( L ) IS TIME I N D E X (MONTHS) T H AT I R R I G A T I O N WATER HAS BERN A P P L I E D 19 c Y ( L ) IS AMT.OF WATER REQD.TO I R R I G A T E FOR R E M A I N I N G MNTHS-IN S E A S O N > 20 c A O P T ( I , J ) I S O P T I M A L A R E A THAT CAN BE I R R I G A T E D IN R E M A I N I N G SEASON > 21 c D F L A I S D I F F E R E N C E BT- O R I G I N A L S E E D E D AREA AND AOPT 22 c P ( J ) I S MATRIX OF INFLOW P R O B A B I L I T I E S C O R R E S P O N D I N G TO S ( J ) MATRIX > 23 c C I S U N I T C O N S U M P T I V E U S E OF C R O P l A L F A L F A ) IN C U B I C M E T E R S / H E C T A R E > 2 4 N c T R . I S T O T A L R E V E N U E ( G R O S S INCOME) PER H E C T A R E OF A L F A L F A I N $ 25 \ c PC I S P L A N T I N G COST OF A L F A L F A IN $ PER H E C T A R E > " 26 c HC I S H A R V E S T I N G C O S T OF A L F A L F A I N $ PER H E C T A R E 27 c W I S NET $ V A L U E OF A L F A L F A IN $ PER H E C T A R E 28 c V ( I ) IS MATRIX OF T O T A L NET $ V A L U E C O R R E S P O N D I N G TO A ( I ) MATRIX > 29 c M ( I , J ) I S MATRIX OF NE T MONETARY V A L U E FOR E A C H A C T I O N - S T A T E COMB!NAT I > 30 c ON 31 c F M ( I . J ) IS M A T R I X OF E X P E C T E D MONETARY V A L U E FOR E A C H A C T I O N - S T A T F ' C O M 32 c B I NAT I ON 262 L I S T I N G OF *MSOURCE* AT 1 7 : 0 1 : 0 7 3 3 C M V U ) I S MATRIX DF T O T A L E X P E C T E D MONETARY V A L U E C O R R E S P O N D I N G T O A l l ) •» 34 C MATRIX 35 C EMMAX I S MAX.TOTAL E X P E C T E D MONFTARY V A L U E IN H V I I I MATRIX 36 C U ( I f J ) I S MATRIX OF U T I L I T Y FOR E A C H A C T I O N - S T A T E C O M B I N A T I O N » 3 7 . C E I I U i.l.) 1°. M A T R I X DF E X P E C T E D U T I L I T Y FOR EACH ACT ION -ST A T E CUMHINAT TO 3 0 C U V ( I ) IS MATRIX OF T O T A L E X P E C T E D U T I L I T Y V A L U E C O R R E S P O N D I N G TO A ( I ) 39 C MATRIX t 4 0 C UVMAX I S MAX. T O T A L E X P E C T E D U T I L I T Y V A L U E I N U V ( I ) MATRIX 41 R E A L M 42 R E A L MV » 4 3 D I M E N S I O N A ( 5 0 ) , S ( 7 ) , P ( 7 ) , C U ( 5 0 ) , M ( 5 0 , 7 ) , V ( 5 0 ) , E M ( 5 0 , 7 ) , M V ( 5 0 ) , 4 4 4 U ( 5 0 , 7 ) , E U ( 5 0 , 7 ) , U V ( 5 0 ) , X < 5 0 , 7 ) , A O P T ( 5 0 , 7 ) , D E L A ( 5 0 , 7 > , H ( 5 ) , Y ( 5 ) 4 5 N=50 J 4 6 R E A D ( 5 , 1 ) { A ( I ) , 1 = 1 , 5 0 ) 4 7 1 F O R M A T ( 7 ( 8 F 1 0 . 1 , / I J 4 8 K=7 • 1 4 9 R E A D ( 5 , 4 ) ( S ( J ) , J = 1 , 7 ) , 5 0 4 F 0 R M A T I 7 F 1 1 . 1 ) 51 R E A D t 5 , 7 ) ( P ( J ) , J = l , 7 ) . J 52 7 F 0 R M A T ( 7 F 8 . 3 ) 53 R F A D ( 5 , 2 ) C , W , T R , H C , P C 54 2 F 0 R M A T < F 9 . 1 , F 9 . 1 , F 9 . 1 , F 9 . 1 , F 9 . 1 ) J 55 R F A D ( 5 , 6 l ) ( H ( L ) , L = l , 5 ) .J 5 6 61 FORM A T ( 5 F 6 . 1 ) 5 7 DO 1 1 1 L = l , 5 J 5 8 P R I N T 2 4 • J 5 9 24 FORM A T ( ' 1 ' , ' C OMPUTED E X P E C T E O $ VAL 6 0 4 U E S FOR E A C H A C T I O N - S T A T E C O M B I N A T I O N ' ) -l 61 P R I N T 1 0 2 , H ( L ) 2 6 3 L I S T I N G OP *MSOURCF* AT 1 7 : 0 1 : 0 7 62 102 F O R M A T ( • - • , 12X,'H = » , F 6 . 1 ) . ' 63 C PROGRAM C A L C U L A T F S C O N S U M P T I V E U S E S CU.AND T O T A L CROP NET V A L U E » V 6 4 PR IN T 26 65 26 F O R M A T ( ' - • » ' A ( ! J C U ( I ) V I I ) S( * 6 6 4 J ) P U ) M ( I , J ) E M ( I , J ) ' j 6 7 00 9 0 1=1,50 6 8 APE A =1 » 6 9 C U ( I >-=A'( I ) * C 70 V ( I ) = A ( I ) * W 71 C PROGRAM C A L C U L A T E S T H E P A Y - O F F T AB LE-M ONST ARY GAINS AND L O S S E S AND THE »> 72 C E X P E C T E D INCOME 7 3 DO 66 J = l , 7 74 X l l , j ; - S ( J ) - ( ( C / 5 . ) * H ( L ) * A ( I ) ) » 7 5 Y ( L I = ( C / 5 . ) + A ( I ) * ( 5 « 0 - H ( L ) ) . . . I 7 6 ! F ( X { I , J ) . L T . Y ( L ) ) G 0 T O 18 7 7 M ( I i J ) = A ( I ) M T R - H C - P C ) / 1 0 0 0 0 0 . 0 • 7 8 GO TO 19 7 9 18 A O P T ( I , J ) = X ( I , J ) / { ( C / 5 . ) * 1 5 . 0 - H ( L ) ) ) 8 0 D E L A ( I , J ) = A { I ) - A O P T ( I , J ) \ 81 M( I , J ) = { ( A O P T ( It J ) M T R - H C - P C ) ) - ( D E L A ( I , J )*PC ) J / 1 0 0 0 0 0 . 0 i 82 19 F M ( I , J ) = M ( I , J ) * 1 0 0 0 0 0 - 0 * P ( J ) 8 3 W R I T E 1 6 , 8 ) A ( I ) , C U ( I > t V ( I ) , S ( J ) t P ( J ) t M ( I , J ) , E M ( I , J ) * 84 8 FORM A T ( ' • , 4 X , F 1 0 . 1 t 4 X , F 1 2 . 1 , 4 X , F l 2 . 1 , 4 X , F 1 2 . 1 » 4 X , F 8 . 3 , 4 X , F \ 4 . 4 t 85 4 8 X . F 1 4 . 1 ) 86 66 C O N T I N U E ' 87 9 0 C O N T I N U E 88 C PROGRAM C A L C U L A T E S E X P E C T E D MONETARY VALUE , M V , F O R EACH A C T I O N 8 9 P R I N T 3 6 9 0 36 F O R M A T ( • - • , • M V(I) A ( I ) « ) 264 L I S T I N G OF *MSOURCE* 91 DO 55 1 = 1 , 5 0 92 SUM=0.0 93 DO 7 7 J = l , 7 9 4 SUM = S U M + E M ( I , J ) 95 77 CONTINIJF 96 M V ( I ) = SUM 9 7 W R I T F ( 6 , 9 ) M V ( I ) , A ( I ) 98 9 F O R M A T C O ' , 4 X , F 1 4 . 1 , 6 X , F 1 0 . 1 ) 9 9 55 C O N T I N U E 100 C PROGRAM C A L C U L A T E S MAXIMUM E X P E C T E D MONETARY VALUE,EMMAX « 101 P R I N T 4 6 • • » 1 0 2 4 6 F O R M A T ( • , • EMMAX A l l ) ' ) 103 EMMAX=0.0 < 1 0 4 DO 88 1 = 1 , 5 0 1 0 5 I F ( M V ( I ) . L T . E M M A X ) GO TO 88 106 EMMAX=MV(I) > 1 0 7 E M M A X I = A ( I ) 1 0 8 88 C O N T I N U E 1 0 9 WRITF(6,10)EMMAX,EMMAXI. ) 1 1 0 1 0 F O R M A T ! ' 0 ' , 4 X , F 1 4 . 1 , 6 X , F 1 0 . 1 ) 1 1 1 C PROGRAM C A L C U L A T E S U T I L I T I E S AND E X P E C T E D U T I L I T I E S U S I N G D E R I V E D 1 1 2 C U T I L I T Y F U N C T I O N S AND P R O B A B I L I T I E S OF T H E 7 H Y D R O L O G I C A L S T A T E S O F > 1 1 3 C N A T U R E 1 1 4 P R I N T 3 S 1 1 5 38 F O R M A T ( ' 1 ' , ' COMPUTED E X P E C T E D U T I L >' 1 1 6 4 I T Y FOR E A C H A C T I O N - S T A T E C O M B I N A T I O N ' ) 117 P R I N T 3 4 1 1 8 34 F O R M A T ( « - « , « A ( I ) C U ( I ) S ( J ) •» 1 1 9 4 P ( J ) U ( I , J ) E U ( I , J ) * ) 265 L I S T I N G OF *MSOURCE* AT 1 7 : 0 1 : 0 1 2 0 00 9 9 1 = 1 , 5 0 1 2 1 AREA =1 v . 122 DO 44 J= 1,7 12 3 C U ( I ) = A ( H * C 124 X U , J < > S < J ) - ( ( C / 5 . ) + H < L ) * A ( I ) ) 1 2 5 Y ( L ) = ( C / 5 . ) * A ( I ) * ( 5 . 0 - H ( L ) ) 1 2 6 I F ( X ( I » J ) . L T • Y ( L ) ) G O T O 29 1 2 7 M ( I , J ) = A ( I ) * ( T R - H C - P C ) / 1 0 0 0 0 0 . 0 1 2 8 GO TO 4 9 1 2 9 2 9 A O P T ( I , J ) = X ( I , J ) / ( ( C / 5 . ) * ( 5 . 0 - H ( L ) ) ) 1 3 0 D E L A ( I , J ) = A ( I ) - A O P T ( I , J ) 1 3 1 M ( I , J ) = ( ( A O P T l I , J J * { T R - H C - P C ) ) - ( D E L A d , J ) * P C ) J / 1 0 0 0 0 0 . 0 132 4 9 I F ( M ( I , J ) . L T . - 6 . 0 ) G 0 TO 23 133 I F ( M < I , J ) . G T . - 4 . 0 ) GO TO 22 1 3 4 U ( I , J ) = 1 0 0 . 8 0 + 1 6 . 8 0 * M ( I , J ) 1 3 5 GO TO 59 1.36 23 U U , J ) = 0 . 0 1 3 7 GO TO 59 138 22 I F ( M ( I , J ) . G T . 6 . 0 ) GO TO 2 2 2 139 U ( I , J ) = 7 2 . 0 + 7 . 6 4 * M ( I , J ) - 0 . 4 9 * M ( I , J ) * * 2 1 4 0 GO TO 59 1 4 1 2 2 2 U ( I , J > = 1 0 0 . 0 142 GO TO 59 1 4 3 5 9 E U ( I , J ) = U ( I , J ) * P ( J ) 1 4 4 W R I T E ( 6 , U ) A ( I ) , C U ( I ) , S ( J ) , P ( J ) , U ( I , J ) , E U ( I , J ) 1 4 5 11 F O R M A T ( ' • , 4 X , F 1 0 . I , 4 X , F 1 2 . 1 , 4 X , F l 2 . I , 4 X , F 8 . 3 , 4 X , F 8 . 2 , 4 X , F 8 . 2 ) 1 4 6 4 4 C O N T I N U E 1 4 7 9 9 C O N T I N U E 1 4 8 C PROGRAM SUMS UP T H E E X P E C T E D U T I L I T I E S FOR E A C H A C T I O N TO O B T A I N T H E 4 L I S T I N G OF *MSOURCE* 1 4 9 C T O T A L E X P E C T E D U T I L I T Y , U V 1 5 0 P R I N T 5 6 151 56 FORMAT (•-',' UV ( I ) A ( . I J ' ) 1 5 2 DO 7 9 1 = 1 , 5 0 1 5 3 T O T A L * 0 . 0 ' 1 5 4 DO 3 3 J = l , 7 155 T O T A L = T O T A L + E U ( I , J ) 1 5 6 33 C O N T I N U E 1 5 7 U V ( I ) = T O T A L 158 W R I T E 1 6 , 1 5 ) U V ( I ) , A ( I ) 1 5 9 15 FORM AT C O * » 4 X » F 3 . 2 » 8 X , F 1 0 . 1 ) 1 6 0 7 9 C O N T I N U E 161 C PROGRAM C M C U L A T E S MAXIMUM E X P FC T E D U T I L I T Y , U V M A X 162 P R I N T 9 6 1 6 3 96 F O R M A T f - 1 , ' UVMAX A ( I ) « ) 1 6 4 UVMAX=0.0 1 6 5 DO 3 9 1 = 1 , 5 0 166 I F ( U V ( I ) . L T . U V M A X ) GO TO 3 9 1 6 7 UVMAX=UV(I) 1 6 8 U V M A X I = A ( T ) 169 3 9 C O N T I N U E 1 7 0 W R I T E ( 6 , 1 7 ) U V M A X , U V M A X I 1 7 1 1 7 FORMAT! «0« , 4 X , F 8 . 2 , 6 X , F 1 . 0 . 1 ) 1 7 2 1 1 1 C O N T I N U E 1 7 3 S T O P 1 7 4 END E X E C U T I O N T E R M I N A T E t S I G N O F F 26 7 . LISTING OF *MSOURCE* AT 1707-54 4 C P R O G R A M A-2 • 5 6 c c. 7 c > 8 c APPLICATION OF DECISION THEORY IN MANAGEMENT OF WATER RESOURCE SYSTEMS 9 . c OPTIMIZATION OF IRRIGATION SYSTEMS UNDER HYDROLOGICAL UNCERTAINTY > 10 c UNCERTAINTY IN INFLOW IS EXAMINED W-R.T. IRRIGATION SYSTEMS DESION-MAK > 11 c ING :DEVELOPMENT AND OPERATIONS 12 c DECISION CRITERIA:MAXIMIZATION OF EXPECTED VALUES 13 c A d ) IS MATRIX OF VARIABLE AREAS(HECTARFS) UN DER I RR IG AT 10 N -ACTIONS > 14 c CU(I) IS MATRIX CF CROP GROSS CONSUMPTIVE USES(CUBIC METERS) CORRESPON 15 c DING TO A( I ) MATRIX > 16 CS(J ) IS MATRIX OF VARIABLE I NFL OWS(CU BIC METERS)-STATES OF NATURE > 17 C X ( I , J ) IS DIFFERENCE B T . S ( J ) AND WATER USED UP IN H(L) MONTHS 18 C H(L) IS TIME INDEX (MONTHS) THAT IRRIGATION WATER HAS BEEN APPLIED 19 C Y(L) IS AMT.OF WATER REOO.TO IRRIGATE FOR REMAINING MNTHS.IN SEASON > 20 C R ( I , J ) IS RATIO OF APPLIED WATER TO WATER REOD. FOR OPTIMAL YIELD X 21 c Y L D ( I . J ) IS YIELD COEFFICIENT CORRESPONDING TO R ( I , J ) 22 c P(J ) IS MATRIX OF INFLOW PROBABILITIES CORRESPONDING TO S ( J ) MATRIX > 23 c C IS UNIT CONSUMPTIVE USE OF CROP(ALFALFA) IN CUBIC METERS/HECTARE 24 c TR IS TOTAL REVENUE(GROSS INCOME) PER HECTARE OF ALFALFA IN $ 9 25 c PC IS PLANTING COST OF ALFALFA IN $ PER HECTARE i» 26 c HC IS HARVESTING COST OF ALF -LFA IN $ PER HECTARE 27 c W I S NET $ VALUE OF ALFALFA IN $ PER HECTARE 28 c V( I ) IS MATRIX OF TOTAL NET * VALUE CORRESPONDING TO A( I ) MATRIX •*> 29 c M ( I t J ) IS MATRIX OF NET MONETARY VALUE FOR EACH ACTION-STATE COMBINAT I 30 c ON > 31 c ^MII.J) IS MATRIX OF EXPECTED MONETARY VALUE FOR EACH ACTION-STATE COM 32 c B I NAT I ON 268 L I S T I N G OF *MSOURCE* AT 1 7 : 3 7 : 5 4 33 C'.MV(I) I S MATRIX OF T O T A L E X P E C T E D MONETARY V A L U E C O R R E S P O N D I N G TO A ( I ) • 3 4 C MATRIX 3 5 C EMMAX I S MAX.TOTAL E X P E C T E D MONETARY V A L U E IN M V ( I ) MATRIX 3 6 C - U ( I . J ) I S MATRIX OF U T I L I T Y FOR EACH A C T I O N - S T A T E C O M B I N A T I O N «. 3 7 . C »-;i.UIfJJ IS M A K U X OF H XP EC TED U T I L I T Y FOR E A C H A C T I O N - S T A T E COMBI MAT 10 38 C U V I I ) I S MATRIX OF T O T A L E X P E C T E D U T I L I T Y V A L U E C O R R E S P O N D I N G TO A ( I') ' 3 9 C MATRIX «• 4 0 C UVMAX I S MAX. T O T A L E X P E C T E D U T I L I T Y VALU= IN U V ( I ) MATRIX 41 R E A L M 42 REAL MV ^ 4 3 R E A L MU 4 4 D I M E N S I O N A ( 5 0 ) , S ( 7 ) , P ( 7 ) , C U ( 5 0 ) , M ( 5 0 , 7 ) , V ( 5 0 ) , E M ( 5 0 , 7 ) , M V ( 5 0 ) , 45 4 U ( 5 0 , 7 ) , E U ( 5 0 , 7 ) , U V ( 5 0 ) , X ( 5 0 , 7 ) , A O P T ( 5 0 , 7 ) , 0 E L A ( 5 0 , 7 ) , H ( 5 ) , Y ( 5 ) > 4 6 4 R ( 5 0 , 7 ) , Y L D ( 5 0 , 7 ) 4 7 N=50 48 R E A D ( 5 , 1 J ( A ( I ) , 1 = 1 , 5 0 ) j 4 9 1 F 0 R M A T ( 7 ( 8 F 1 0 . 1 , / ) ) 50 K = 7 51 R E A D 1 5 , 4 ) ( S ( J ) , J = 1 , 7 ) 3 52 4 F 0 R M A T ( 7 F 1 1 . 1 ) 53 R E A D ( 5 , 7 ) ( P ( J ) , J = 1 , 7 ) 54 7 FORMA T ( 7 F 8.3) <j 55 R E A D ( 5 , 2 ) C , W , T R , H C , P C 56 2 F 0 R M A T ( F 9 . 1 , F 9 . 1 , F 9 . 1 , F 9 . 1 , F 9 . 1 ) 5 7 R E A D ( 5 , 6 1 ) ( H ( L ) , L = l , 5) « 58 61 F O R M A T ( 5 F 6 . 1 ) Of 59 R E A D ( 5 , 7 2 ) M U , S I G M A 6 0 72 F 0 R M A T ( F 6 . 1 , F 6 . 2 ) « 61 DO 1 1 1 L = l ,5 -4 269 L I S T I N G OF *MSOURC c* AT 1 7 : 3 7 6 2 P R I N T 2 4 6 3 24 F O R M A T ( • 1 • ,' COMPUTED E X P E C T E D $ VAL 64 4 U E S FOR E A C H A C T I O N - S T A T E C O M B I N A T I O N ' ) 6 5 PR I NT 1 0 2 i H ( L ) 66 102 FORMAT!'-' , 1 2 X » ' H» • , F 6 • 1 ) 6 7 C PROGRAM C A L C U L A T E S C O N S U M P T I V E U S E S CU,AND T O T A L CROP NET V A L U E , V . 68 P R I N T 2 6 6 9 26 F O R M A T ( « - « , i A l l ) CU ( I ) V ( I ) S( 7 0 4 J ) P ( J ) M ( I , J ) E M ( I , J ) « ) 71 DO 9 0 1=1,5 0 72 AREA =1 73 C U ( I ) = A ( I ) * C 74 V ( I ) -<' ( I )*W 75 C PROGRAM C A L C U L A T E S T H E P A Y - O F F T A B L E - M O N E T A R Y G A I N S AND L O S S E S AND THE 76 C FXPrCTF.D INCOME 77 DO 66 J = l ,7 78 X( I > J ) = S ( J ) - ( t C / 5 . ) * H ( L ) * A ( I ) ) 79 Y I L ) = ( C / 5 . ) * A ( I ) M 5 . 0 - H I L ) ) 80 R ( I , J1--XI.I , J ) / Y ( L ) 81 I F ( R ( I , J ) . G T . 1 . 0 ) G 0 TO 18 82 B = ( R U , J ) - M U ) * * 2 83 D=SIGMA**2 8 4 F=B/D 85 T = F * ( - 0 . 1 8 7 5 ) 86 Y L D U , J l = E X P ( T ) ' 87 M ( I , J ) = A ( I ) * ? L D ( I , J ) * ( T R - H C - P C ) / 1 0 0 0 0 0 . 0 88 GO TO 19 89 18 M( I , J ) = A ( I ) * 1 . 0 * ( T R - H C - P C ) / 1 0 0 0 0 0 . 0 9 0 19 EM( I , J ) = M ( I , J ) * 1 0 0 0 0 0 . 0 * P ( J ) 270 Si L I S T I N G OF +MSOURCE* AT 1 7 : 3 7 : 5 91 W R I T E ( 6 , 8 ) A ( I ) , C U ( I ) , V ( I ) , S U ) , P ( J ) , M l I , J ) , E M ( I , J ) 9 2 8 F O R M A T ( ' • , 4 X , F l 0 . 1 , 4 X , F 1 2 . 1 , 4 X , F 1 2 . 1 , 4 X , F 1 2 . 1 , 4 X , F 8 . 3 , 4 X , F 1 4 . 4 , 93 4 0 X , F 1 4 . 1 ) •. 94 ' . 66 C O N T I N U E ^ 95 90 C O N T I N U F 96 C PROGRAM C A L C U L A T E S t r X P E C T E D MONETARY VALU!:, MV , FOR EACH A C T I O N 9 7 P R I N T 3 6 9 8 3 6 F O R M A T ( ' - ' , ' M V ( I ) A ( I ) « ) . 9 9 Of) 5 5 1 * 1 , 5 0 100 SUM = 0.0 •>> 101 DO 77 J = l , 7 102 SUM= SUM+EM(I , J ) 1 0 3 77 C O N T I N U E ^ 104 . M V ( I ) = S U M 1 0 5 W P I T E 1 6 , 9 ) M V ( I ) , A l I ) 1 0 6 9 F O R M A T l < 0 ' , 4 X . F 1 4 . 1 , 6 X , F 1 0 . 1 1 j 1 0 7 55 C O N T I N U E 1 0 8 C PROGRAM C A L C U L A T E S MAXIMUM E X P E C T E O MONETARY VALUE,EMMAX 109 P R I N T 4 6 j 1 1 0 4 6 F O R M A T l ' - ' , ' EMMAX A ( I ) ' ) 111 EMMAX=0.0 112 DO 88 1 = 1, 5 0 CS 1 1 3 I F ( M V ( I ) . L T . E M M A X ) GO TO 8 8 114 EMMAX = M V ( I ) 1 1 5 E M M A X I = A ( I ) . ' -j, 1 1 6 88 C O N T I N U E 117 W R I T E 1 6 , 10)EMMAX,EMMAXI 1 1 8 10 F O R M A T C 0' , 4 X , F l 4 . 1, 6X , F 1 0 .1 > J 1 1 9 C PROGRAM C A L C U L A T E S U T I L I T I E S AND E X P E C T E D U T I L I T I E S U S I N G D E R I V E D 271 L I S T I N G OF *MSOURCE* AT 1 7 : 3 7 : 5 4 1 2 0 C U T I L I T Y F U N C T I O N S AND P R C B A B I L I T I E S CF T H E 7 HY D R O L O G I C A L S T A T E S OF 3 121 C NATURF 122 PR I N T 3 11 123 30 F O R M A T ( » i « , « COMPUTED E X P E C T E D U T I L > J.24 4 I T Y FUR E A C H ACT 1 U N - S T A T E C O M B I N A T I O N ' ) 125 P R I N T 3 4 ' 126 34 F O R M A T ( ' - < t i A l l ) C U ( I ) S I J ) ' 1 2 7 4 P 1 J ) U( I , J ) E U l I , J ) • ) 128 DO 99 1= 1 , 5 0 1 2 9 AREA =1 > 1 3 0 DO 44 J = l , 7 131 C U l I ) = A ( I) *C 132 X I I , J ) = S 1 J ) - l 1 C / 5 . )*H1 L ) * A ( I ) ) * 133 Y ( L J = ( C / 5 . ) * A ( I } *( 5 . 0 - H 1 L ) ) • J» 134 R U , J ) = X 1 I , J ) / Y ( L ) 1 3 5 I F 1 R ( I , J ) . G T . 1 . 0 ) G 0 TO 29 • ' * 136 B=1R1 I , J l - M U ) * * 2 1 3 7 D=SIGMA**2 1 3 8 F=B/D > 139 T = F * l - 0 . 1 8 7 5 ) 1 4 0 Y L O U , J ) = E X P 1 T ) 141 Ml I , J )=A( I ) * Y L D l I , J ) * I T R - H C - P C J / 1 0 0 0 0 0 . 0 . " <* 142 GO TO 4 9 143 2 9 Ml I , J ) = A( I ) * 1 . 0 * 1 T R - H C - P C ) / 1 0 0 0 0 0 . 0 144 4 9 I F I M 1 I , J ) . L T . - 6 . 0 ) G 0 TO 23 <* 145 I F I M 1 I , J ) . G T . - 4 . 0 ) GO TO 22 146 U l I , J 1 = 1 0 0 . 8 0 + 1 6 . 8 0 * M ( I , J ) 1 4 7 GO TO 59 * 148 23 U 1 I , J ) = 0 . 0 /- II L I S T I N G OF *MSOUF.CE* AT 1 7 : 3 7 : 5 ' 1 4 9 GO TO 59 150 22 I F ( M ( I , J ) . G T . 6 - 0 ) GO TO 222 151 U( I , J ) = 7 2 . 0 « - 7 . 6 4 * M ( I , J ) - 0 . 4 9 * M ( I , J ) * * 2 152 GO TO 59 153 222 U ( ! , J ) = 1 0 0 . 0 154 GO TO 5 9 155 59 E U ! I , J ) = U ( I , J ) * P { J ) 1 5 6 K R I T E 1 6 , 1 1 ) A ! I ) , C U ( I ) , S ! J ) , P ! J ) , U l I , J ) , E U ( I , J ) 1 5 7 11 FORMAT{• ' , 4 X , F 1 0 . 1 , 4 X , F 1 2 . 1 , 4 X , F 1 2 . 1,4X, F 8 . 3,4-X,F8 1 5 8 44 C O N T T N U c 1 5 9 99 C O N T I N U E 1 6 0 C PROGRAM SUMS UP THE E X P E C T E D U T I L I T I E S FOR E A C H A C T I O N 161 C T O T A L E X P E C T E D U T I L I T Y , U V 162 P R I N T 5 6 163 56 FORMAT!' -' , ' U V I I ) A l l ) ' } 1 6 4 DO 7 9 1 = 1 , 5 0 1 6 5 T O T A L = 0 . 0 166 DO 3 3 J = 1 , 7 16 7 T O T A L = T O T A L * E U l I , J J 168 33 C O N T I N U E 1 6 9 U V l I ) = T O T A L 1 7 0 WR I T E ( 6 , 1 5 ) U V ( I ) , A ( I ) 1 7 1 15 F O R M A T ( ' 0 ' , 4 X , F 8 . 2 , 8 X , F 1 0 . 1 ) 172 79 C O N T I N U F 1 7 3 C PROGRAM C A L C U L A T E S MAXIMUM E X P E C T E D U T I L I T Y , U V M A X 174 P R I M T 9 6 175 96 F O R M A T ( ' - ' , • UVMAX' A! I ) • ) 1 7 6 UVMAX=0.0 177 DO 39 1 = 1 , 5 0 L I S T I N G OF *MSOURC~* 170 I F ( ' J V ( I ) .LT.UVMAX) GO TO 39 179 UVMAX=UV(T ) 180 U V M A X I = M I ) 181 39 C O N T I N U C 182 . W R I T E ( 6 , 1 7) UVMAX.UVMAXI 133 17 FORM AT (• 0» , 4 X , F 8 . 2 . 6 X , F I 0.1) 104 111 CONTIN'jr 1 8 5 STOP 186 FMO E X E C U T I O N TERMINATE') t S I G N O F F 274 APPENDIX B 275 TABLE B.l (a) NICOLA JUVIR A30Ji£_iLIC_Oi i>_-AKE_ - STATION S j . 08LG049 MONTHLY »NT DI CCHARGES IN CUBIC F EET PER SE " N D FOR Tn t PERIOD OF RECCS3 EAR JA_M FEB .1AR APR MAY JUN J U L A'J^ J E P OCT NOV "EC ME AS 915 9 16 9 17 ::: 71 . 5 27 .5 F e i i 28 1 724 1070 • 5 3 -36 : 37 70.2 1: ' 2 2 . ; 18 26 . 3 ::: 933 — - — — - - - - 7 1 8 2 0 1 35 . ; 12 . 1 1.2 5 5 . 2 -7. 1 965 966 967 — ::: 7 5.2 20 . 3 7 4 1 368 771 6 1 2 260 69 1 87.3 98 . 2 • •. c 3 5 . S 29 . 2 . 4 1 0.74 5.6 1 . 9 I " 968' 969 15.6 20.8 3D. j 17.3 28 .8 1 5.9 23.5 3".9 ' '6 5 2 '230 823 2 = 2 • 62 1 5 1 25 7 • 0 22.0 ' 1.3 28 . 9 3C . 3 :i . 5 2 . 5 158 1 5 3 970' 13.3 26 .7 21.8 5 1 . 7 223 2S3 16.2 5 . 1 1 . 5 2 . 2 2 . 9 1 . S 57 .C 97 3- 23 .5 25.1. 32.6 ? 5 : 325 25 3 8 2.3 3 . 1 2 . 0 1.3 4 . 1 3 . 7 67 .5 EAN 18.3 26 . 9 21 .8 •0 .5 5 39 5 1 6 • 69 3 1 . 5 1 3 3 . 8 2 1.3 • - . 1 110 LOCATION 50 120 11 3 2 r 21 23 H DRAINAGE A REGULATED SO MILES NICOLA RIVER ABOVE NICOLA ANNUAL EXTREMES OF DISCHARGE IN CFS AND ANNUAL -AKE - STA" TOTAL DiSt L CN NO. 08LG049 --ARGE IN AC-FT -Z THE P E R I ; EAR MAXIMUM INSTANTANEOUS DISCHARGE MAXIMUM DA IL J I S C L:ARGE MINIMUM DAI DISCHARGE TOTAL CJ15£ 915 665 CFS ; ; . MAY 2 1 1 3 0 CFS CN . SEP 1 7 9 16 800 c:-s ; s .'UN 1. 23 0 CFS IN SEP 22 9 17 1250 CFS c s JUN 1 1 0 CFS CN SEP 17 9)3 . . . 98 1 C F ? CN JUN 8 8 5 CFS ON SE? 28 965 1 120 CFS CN JUN 4 28 0 CFS ON SEP 1 4 966 - 582 CFS CN MAY 1 4 6 7 CFS ON SEP 30 967 1050 CFS CN JUN 5 0 50 CFS CN OCT 1 3 • 968 1900 CFS C N MAY 25 1 7 CFS CN JAN 1 0 • 1 '* ZCO A C - F T 969 2000 CFS CN MAY 1 7 • 0 80 CFS CN DEC 1 3 " 1: JO AC-FT 970 — 50 1 CFS CN JUN 6 1 2 CFS CN AUG 31 ' • 1 7 0 0 A C - F T 973 600 CFS ON MAY 27 0 70 CFS CN SEP 26 • > 5 3C0 A C - F T • - EXTREME RECORDE: FOR THE PERIOD OF RECORD 7 j = ;o A C - r i 276 TABLE B.l (b) IjOORE CREEK NEAR QUILCHENA - STATION NO. 08LG01I MONTHLY AND ANNUAL MS,\N DISCHARGES IN CUBIC FEET PER SECOND FOR THE PERIOD OF RECORD FEB MAR LAT 50 18 25 N LONG 120 26 35 w APR MAY JUN J U L AUG S E ? OCT NOV DEC KEA! 19. 2 5 , . 6 3 . 6 23.9 47 . 2 11. 5 5. , 7 5. . 2 4.4 02 55 . 5 1 3 . 2 3 . . 9 3 . . 0 5 . 6 20.5 1 8 . 8 1 . 6 1 , . 3 0 . . 97 . -11.5 83 .5 12. 6 3. 3 3, , 3 2. . 6 3.1 17.7 1 0 . 9 ii . 8 1 , . 5 1 . . 9 8. 1 3 1 23 . e 2 . 3 2. . 0 2 . C --10.5 10 . 9_ 2 8. i 1. C 3. .3 2 . 3 O . 4 JKAINAGE AREA J SC MILES NATURAL FLOW ANNUAL EXTREMES OF DISCHAR MAXIMUM INSTANTANEOUS DISCHARGE MOORE CREEK NEAR QUILCHENA - STATION NO. IN CFS AND ANNUAL TOTAL DISCHARGE 08LG0 11 IN AC- FT FOR THE PERIOD OF S£".-.5 MAXIMUM DAILY DISCHARGE MINIMUM DAILY DISCHARGE :A. D:SC:-:ASJ£ 3.0 CFS ON SE? 5 3.3 CFS CN OCT 8 260 CFS ON MAY 28 3.0 CFS ON JUL 23 — 52.0 CFS ON J UN 3 0.-0 CFS ON JUL 23 • 130 CFS ON MAY 1 5 2.0 CFS ON JUL 9 — 41.0 CFS ON MAY 17 1.0 CFS ON AUG 4 268 CFS ON HAY 20 • 1.0 CFS ON JJL 14 • - EXTREME RECORDED FOR THE PERIOD OF RECORD 277 TABLE B.l (c) QUILCHENA CREEK AT_0'JILCHENA - STATION NO. 08LG0 17 MONTHLY AND ANNUAL KEAN DISCHARGES IN CUBIC FEET pr.? SErnNn FCR THE PERIOD OF RECORD FEB MAR APR MA^  JUN JUL — .... - OCT NOV DEC MEAN - - - 155 32.7 3 . 0 1. 2 12.9 36.7 — : : : 5 1 . 6 96 54 5" 2 l t 1 10 26 . 0 231 292 12.1 19.6 12 . 3 27.2 1 7 . 6 . 6 . 4 . 030 1 1 1 . 6. 2. 7 . 10. 20 9 1 0 : : : : : : : : : : : : : : : ,13. 20. .61. 23 . 1 3 3 1 ' 52 e 19 . 6 162 3 1 6 3 C . 2 1 . 2 28 . 5 58.7 3 . 1 0 . 0 . 15. 0 . 1 1 76 l 2 1 0. 1 . 9 . 0 . 4 1 33 . 3C : : : : : : - - - . — - - - — 18 . 1 1 1 - - . . 15 3 2 1.7 5 . 9 5. 3 12.9 36.7 — — LAT LCN3 50 120 09 1 5 N 30 05 w -f AREA ;oi so MILES YEAR 1931 1965 1966 1967 1968 1969 '1970 19T1 1972 1973 CUILCHSNA CHEEK AT QUILCHENA - STATION NO. 08LG017 ANNVAL EXTREMES OF DISCHARGE IN CFS AND ANNUAL TOTAL DISCHARGE IN AC-FT FOR THE PERIOD OF RECORD INSTANTANEOUS DISCHARGE HAXIM'JM DAILY DISCHARGE MINIMUM DAILY DISCHARGE TOTAL DISCHARGE 261 CFS ON JUN 8 1.5 CFS ON AUG 19 1933 — - 196S 120 CFS ON MAY 1 5 1.7 CFS ON SEP 9 1966 50 1 CFS ON JUN 6 1.8 CFS ON SEP 2 1 1967 639 CFS CN JUN 1 1 • 1 . 9 CFS ON AUG 20 — 1968 27 1 CFS ON MAY 17 1 .2 CFS ON AUG 29 1969 48 . 2 CFS ON MAY 3 1 0 CFS ON JUL 17 • 1970 330 CFS ON JUN F, 0. 13 CFS ON AUG 18 — 1971 597 CFS ON JUN 1 3.0 CFS ON SEP 1 ; - — 1972 1 13 CFS ON MAY 21 0.05 CFS ON AUG 15 1973 EXTREME RECORDED FOR THE PERIOD OF RECORD — Ac-rr MEAN 278 TABLE B.2 SUMMARY OF IRRIGATION SEASON MONTHLY RAINFALL NICOLA VALLEY REGION M o n t h, .1. y R a i. n f a 1 .1 (mm) Total I r r i g a t i o n Season R a i n f a l l (mm) Year May June July August September .1070 37.00 .1.4 .70 36.10 7.60 9. 90 106.17 18 79 28.20 61. 00 81.00 27.70 5.80 203.71 18 80 38.60 8.10 42.90 56.10 39.60 185.42 1881 27.90 27.20 13.50 40.40 •42.70 151.64 1882 15.20 50.00 21.80 28.40 17.80 133.35 1883 30.00 16.50 3.00 8.90 9.90 68. 33 1884 7.10 33.00 47.00 23.90 44.50 155.45 1885. 56.10 37.30 39.90 21.10 33.30 187.71 1886 7.40 13.20 16.30 17.00 6.10 59.94 1887 14.00 50. 00 17. 50 11.90 17.80 111.2-5 1888 58.40 60.50 14.00 11.20 3.80 147.83 1889 46.50 5.60 16.00 7.90 20.80 96.77 1890 38.90 57.40 . 24.90 42.20 9.40 172.72 1891 13.00 72.60 12.70 31. 20 18. 30 147.83 1892 15.70 27.70 16. 30 9.40 11.20 80.26 1893 37.10 17.50 10.90 8.60 24.60 98.81 1894 26.70 46. 70 7. 60 2.80 19. 30 103.12 1895 72.10 6.90 22.10 10.20 29.50 140.72 1896 39. 60 5.10 0.00 2.50 25.40 72.64 1897 12.45 78.23 44.20 6.10 12.45 153.42 1898 33.86 31. 75 9.91 3. 05 17.78 101.35 1899 25.91 37.59 45.72 87.88 27.94 225.04 1900 17.53 57.66 30. 99 82.80 32. 26 221.23 1901 16.51 62.23 25.91 0.76 28.45 133.86 1902 46.48 31.50 28.19 6.60 23.88 136.65 1903 4.06 35.81 80.77 69.95 65.28 255.78 1904 7.11 22.86 8.64 3.56 14.48 56.64 1905 47.50 25.15 35. 31 6.10 54. 36 168.40 1906 57. 66 34.54 7 . 62 3. 05 19.56 122.43 1907 13.46 25.40 27.18 64.01 58.42 188.47 1908 28.19 6.10 9. 91 39.12 6.60 89. 92 1909 27.43 20. 57 36. 83 4.83 21. 34 111.00 1910 30.73 43.43 7.11 22. 37 37.08 141.73 1911 23. 62 11. 68 7.11 38. 86 29. 72 111.00 1912 21.08 22. 35 40.13 51.56 24.64 158.00 1913 19. 56 57.31 21.59 37.85 18.54 148.84 1914 100.33 11.43 1.02 1.27 6.35 120.40 1915 55.88 41.40 39.88 14.48 21. 08 172.72 1916 33.53 60.71 24.89 17.78 13.46 150.37 1917 30. 99 28.70 7.11 4.57 14.73 86.11 1918 19. 30 1.02 24.13 24. 89 14.73 84.07 Number of Years of Record = 88 years 279 Table B.2 (continued) Total M o n t h 1 y R a i n f a l l (mm) I r r i g a t i o n Season R a i n f a l l Year May June July August September (mm) 1919 16.00 24. 64 15. 24 1. 02 15.75 72.64 1920 4.57 12.19 16.76 20. 32 18. 29 72.14 1921 39. 88 10.16 6.10 10.16 16.26 57.15 1922 1. 27 0. 00 5.59 22. 61 23.88 53. 34 1923 11.17 45. 21 11.43 59. 69 9. 91 137.41 1924 2.03 12. 95 13.46 19.30 15.49 63. 25 1925 2.03 12.70 10.41 26. 92 8. 64 60.71 1926 7. 87 9.40 0. 00 12.19 10. 67 40.13 1927 34. 54 14.48 17.02 25.40 17.02 108.46 1928 3.56 21. 84 5.33 12.19 9.14 52. 07 1929 26.16 17.78 0.00 2. 54 12.95 59.44 193 0 11.43 15.24 5.08 8.8 9 13. 21 53.85 1931 36.83 20. 83 5.08 7.11 25.40 95.25 1932 16. 51 5.84 6. 35 24.89 22.86 76.45 19 33 16.76 8. 64 24. 38 20.32 20.57 90. 68 1934 16.51 13. 97 4. 57 9. 91 16. 26 61.21 1935 7 . 62 3.56 14.48 11.43 15.75 52. 83 1936 21. 08 26. 92 1. 02 21. 34 14. 73 85. 09 1937 12.19 20. 07 6.60 15.24 9. 91 64. 01 1938 0.00 7 .62 2 0.32 15.24 59.94 103.12 1939 7.8 7 92.46 4 . 57 0.51 14. 99 120.40 1940 23.62 2. 03 24. 89 2.29 6.86 59. 69 1941 4 8. 01 10.41 17.02 26.92 32. 26 134.62 1942 24.64 22. 61 17. 53 7.87 4.06 76.71 1943 39.88 28.19 2.79 21. 34 3.30 95.50 1944 33.02 28 .70 32.51 14.73 28.45 137 .41 1945 26.42 21. 84 13.46 7.62 25.40 94. 74 1946 7. 37 41. 66 2.54 5.59 3.05 60.20 1947 13.46 34.54 15.24 22.86 11. 94 98 . 04 1948 17. 53 23.62 36.58 61.47 7 . 37 146.56 1949 10.67 21.08 13.72 5.59 29.46 80.52 1950 16. 00 9.14 9.14 4. 32 4 . 32 42.93 1951 10.41 24.13 10.16 42.67 • 18. 54 105.92 1962 27. 94 18. 54 36.58 40. 32 11. 94 135.89 1963 8.13 34.54 21. 84 24. 64 20.57 v 109.73 1964 6. 35 4 8.51 18.80 13. 97 62.23 149.86 1965 8. 64 5. 08 13.46 55.12 10. 67 92.96 1966 .7 .37 11. 94 59. 69 10.41 11. 68 .101. 09 .1.9 67 2. . '3 4 1 7 . *> 3 4 . r/ .1 2 . 1 •> '). no 4 5 . ')7 1968 58.93 35.81 10.92 2 5. G 5 14.22 145.54 19 69 9. 91 2 6.42 25.40 3. 05 .1.8 . 0 3 82.80 197 0 6. 60 8.64 3.30 26.67 '). 4 0 5 4 . fi 1. 1971 17 .78 22. 10 1.27 2.29 8.13 51.56 1972 9.14 56. 90 7.62 7. 62 30.73 112.01 1973 5.33 17. 02 0. 51 9. 91 12.45 45. 21 1974 11.18 5.33 25.40 19.81 6.60 68.33 1975 36. 07 17. 02 9.14 20. 57 5.59 88. 39 280 TABLE B .3 QUILCHENA CREEK AT QUILCHENA - STATION NO. 08I.O0J_7 DAILY DISCHARGE IN CUBIC FEET PER S J C O N D J O B • 1S7« - •- . . - —. — —- — I JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC DAY 20.5 100 272 £ 58 . 4 4 . 6 12. 5 15.8 . - — 1 23.0 1 4 0 257 E 4 1 . 8 4 . 4 12. 5 16.3 • 2 20 . 0 131 254 E 4 1 . 8 4 . 3 1 1 . 9 15.a .- 3 ' 20.0 1 1 3 25 1 E 34 . 5 4 . 5 1 1 . 4 15 . S — - a > 1 . 5 1 0 1 25 3 34 . 5 3 . 6, ? i . 7 15.5 5 25.0 96.0 255 r 34 . 5 16 . 3 12. 8 15.3 S 25 . 0 106 256 £ 29. 5 15 . 8 13. 3 15.0 —— 7 21 . 0 1 1 7 243 £ 34 . 5 15. 3 13. 5 15.0 - - - 8 20.0 130 25 1 E 3 l . 2 E 7 . 8 15 . 8 15.0 --• - 9 --- --- 23.0 149 233 E 27 . 8 E 7 . 7 16. 3 15.3 10 --- 23.5 1*6 226 t 24 . 5 8 . 2 15 . I 15.3 . 11 25.0 • 1 4 0 2 C 3 E 24 . 5 8 . 6 14. 8 15.0 - - - 12 25.C 120 226 E 24 . 5 3 . 6 1 4 . 6 15.0 • - - 13 25.0 105 263 r 1 9 . 5 8 . 8 14 . a 15.0 14 25 . 0 96 .0 3 19 17 . 3 E 8 . 2 14 . 3 i2.a — . •15 --. 25 . 0 5u . 0 335 £ 15 . ,0 8. . 1 13 . a 12.3 IS 25.5 75 . IS 357 E 15. .0 8 . . 2 1 3 . T" T2 . 5 - — — - 17 2 3.5 72.3 E 38 1 E 1 1 , . 5 8 . . 1 1 3 . 5 12.5 — - IE 23.5 £9.2 37 3 E 1 1 . . 5 8 . . 0 . 1 3 . 5 12.5 . 19 2 3.0 57 . 4 349 - 1 1 . 5 3 . .2 1 3 . ; 12.5 20 23.5 55.5 327 E 1 1 . 5 8 . r 1 2 . . 0 12.5 21 - __ 20 . C 5 3.9 290 E 8 . 8 8 .2 . 13 . 0 16.3 • - - 22 • 30.0 59.3 262 E 8 . 8 8 . 2 12. . 0 14.3 23 32.5 75.5 233 E 8 . 3 5 . 2 12. . I E ' 15.8 — - — - 24 • .. 1 C 63.0 220 E 7 . 0 8 . 2 12. .2 E 15.8 - 25 J T . £ • - 193 6 . 8 8 . 2 12 . 2 *^ ~* 15.5 26 i . i 139 1 ! .2' S . 2 1 3 . . 0 ' E 15 . £ - — 27 3 :. 0 l 5 2 96 . C 6 . 5 8 . 2 13. . 7 E 15.S •—_ 23 203 76.1 3 . 7 8 . 3 1 4 . » E 15.5 —- — 29 ! o 2 1 1 E 53.3 3 . 5- 3 . r~- 15 . 1 E '5.3 • • — • - - 30 ; . — 221 E 4 . 5 1 1 . 3 16.0 —:— 31 .9 8.9 3 5 9 3.0 7 4 7 6." 623 . 9 '.6 1 . 2 404 . 4 4 53.5 TOTAL 3 3.3 ! 1 S 249 20 . 1 8 . 4 1 3 .5 14.8 : HE AN •• 3 0 7-10 14800 1240 5 18 802 .03 — - ' AC-FT *^ * 37 . 0 22 1 38 1 sa . 4 16 . 3 1 i . 3 16.3 --- MAX < — 20 . 0 53.9 59 . 3 3 . 5 3 . 6 1 1 . 4 12.5 MIN 'JtARY FOR THE YEAR : 3 71 TY? t CF GAUGE - MANUAL MAXIM VM DA:L Y DISCHARCE. 3i 11 CFS ON JVS 13 LCC AT ION - LAT 50 09 15 N E-ESTIMAT ED MINIM Y DISCHARGE, 3 . .5 CFS CS 7"JL 3 0 LONG 120 30 05 w AGE AREA 334 SQ M ES FEGULATTl 12 -•IC-IS -uj- -12- " iC -8-k -4- -AUGUST TOTAL RAIN = 0.78 ins, (3 days) SEPTEMBER TOTAL RAIN = 0.26 ins, (3 days) a, b, c, d, e, f indicate" days with measurable summer r a i n f a l l Runoff from 5th Aug. to 4th Sept. i s caused by 0.78 i n s . of r a i n JL 2S ~3 & i> ' JULY 19 74 AUGUST 19 74 So to IS . -io SEPTEMBER 197 4 Figure B .l Discharge Hydrograph Employed i n Establishing the Rainfall-Runoff Relationship for Quilchena Creek CO 282 TABLE B.4 TOTAL RAINFALL AND CORRESPONDING TOTAL ESTIMATED INFLOW FOR QUILCHENA CREEK DRAINAGE AREA DURING THE IRRIGATION SEASON Year Total I r r i g a t i o n Total I r r i g a t i o n Season R a i n f a l l Season R a i n f a l l (mm) (103m3) 1878 106.17 4180 1879 203.71 8020 1880 185.42 7300 1881 151.64 5970 1882 133.35 5250 1883 68.33 2690 18 8.4 155.45 6120 1885 187.71 7390 18 8 6 59. 94 2 3 6 0 18 87 111.25 4380 1888 147.83 5820 1889 96.77 3810 1890 172.72 6800 1891 147.83 5 82 0 1892 80.26 3160 1893 98.81 3890 1894 103.12 4060 1895 140.72 5540 1896 72. 64 2860 189 7 153.42 6040 1898 101.35 3990 189 9 225.04 8860 1900 221.23 8710 1901 133.86 5270 1902 136.65 5380 1903 255.78 10070 1904 56.64 2230 1905 1.6 8.4 0 6630 1906 122.43 4820 1907 188.47 7420 1908 89.92 3540 1909 111.00 4370 1910 141.73 55 8 0 1911 111.00 4370 1912 158.00 6290 1913 148.84 5860 1914 120.40 4740 1915 172.72 6800 283 Table B.4 (continued) Year Total I r r i g a t i o n Total I r r i g a t i o n Season R a i n f a l l Season R a i n f a l l (mm) 3 3 (10 m) 1916 150.37 59 2 0 1917 . 86.11 3390 19.18 84.07 3310 1919 72.64 2860 1920 72.14 2840 1921 57.15 2250 1922 53. 34 2100 1923 137.41 5410 1924 63.25 2490 1925 60.71 2390 1926 40.13 1580 1927 108.46 4270 1928 52.07 2050 1929 59.44 2340 1930 53.85 2120 1931 95.25 3750 1932 76.45 3010 1933 90.68 3570 1934 61.21 2410 1935 52.83 2080 1936 85.09 3350 1937 64.01 2520 1938 103.12 4060 1939 120.40 4740 1940 59. 69 2350 1941 134.62 5300 1942 76.71 3020 1943 95.50 3760 1944 137.41 5410 1945 94.74 3730 1946 60.20 2370 1947 98.04 3860 1948 146.56 5770 1949 80.52 3170 1950 42.93 1690 1951 105.92 417 0 1952 135.89 5350 1963 • 109.73 4320 1964 149.86 5900 1965 92.96 3660 1966 101.09 3980 1967 45.97 1810 284 Table B.4 (continued) Year Total I r r i g a t i o n Season R a i n f a l l (mm) Total I r r i g a t i o n Season R a i n f a l l 3 3 (10 m ) 1968 145.54 5730 1969 82.80 3260 1970 54.61 2150 1971 51.56 2030 1972 112.01 4410 1973 45.21 1780 1974 68. 33 2690 1975 88.39 3480 285 TABLE B.5 ESTIMATED TOTAL IRRIGATION SEASON INFLOW FROM RAINFALL AND SNOW RUNOFF FOR QUILCHENA CREEK DRAINAGE AREA Yoar Estimated Total I r r i g a t i o n Season Inflow (.10 3 m3) Year Estimated Total I r r i g a t i o n Season Inflow 1 3 (10 m ) 1878 16,720 1922 8,400 1879 32,000 1923 21,640 1880 29,200 1924 9,960 1881 23,880 1925 9 , 560 1882 21,000 1926 6,320 1883 10,760 1927 17,080 1884 24,4 80 1928 8,2 00 1885 29,560 1929 9,360 1886 9,440 1930 8,480 1887 17,520 1931 15,000 1888 23,280 1932 12,040 1889 15,240 1933 14,280 1890 27,200 1934 9,640 1891 23,280 1935 8,320 1892 12,640 1936 13,400 1893 15,560 1937 10,080 1894 16,240 1938 16,240 1895 22,160 1939 18,960 1896 11,440 1940 9,400 1897 24,160 1941 21,200 1898 15,960 1942 12,080 1899 35,440 1943 15,040 1900 34,840 1944 21,640 1901 21,080 1945 14,920 1902 21,520 1946 9,480 1903 40,280 1947 15,440 1904 8,920 1948 23,080 1905 26,520 1949 12,680 1906 19,280 1950 6,760 1907 29,680 1951 16,680 1908 14,160 1962 21,400 1909 17,4 80 1963 17,280 1910 22,320 1964 23,600 1911 17,480 1965 14,640 1912 25,160 1966 15,920 1913 23,440 1967 7,240 1914 18,960 1968 22,920 1915 27,200 1969 13,040 1916 23,680 1970 8,600 1917 13,560 1971 8,120 1918 13,240 197 2 17,640 1919 11,440 1973 7,120 1920 11,360 1974 10,760 1921 9,000 1975 13,920 286 TABLE B.6 RANKING OF DATA FROM TABLE B..5 AND FREQUENCY ANALYSIS FOR QUILCHENA CREEK'S FLOW Year Flow 3 3 (10 m) Rank (m) Return Period T - N + 1 (Years) R m % Flow Equalled or Exceeded 100/T R 1903 40,280 1 89.0 1.124 1899 35,440 2 44.5 2.247 1900 34,840 3 29.7 3. 367 1879 32,000 4 22.3 4.484 1907 29,680 5 17. 8 5. 618 1885 29,560 6 14.8 6.757 1880 29,200 7 12.7 7.874 1890 27,200 8 11.1 9. 009 1915 27,190 9 9.9 10.101 1905 26,520 10 8.9 11.236 1912 25,160 11 8.1 12.346 1884 24,480 12 7.4 13.514 1897 24,160 13 6.8 14.706 1881 23,880 14 6.4 15.625 1916 23,680 15 5.9 16.949 1964 23,600 16 5.6 17.8 57 1913 23,440 17 5.2 19.231 188 8 23,280 18 4.9 20.408 1891 23,270 19 4.7 21.277 1948 23,080 20 4.5 22.222 1968 22,920 21 4.2 2 3.810 1910 22,320 22 4.0 25.000 1895 22,160 23 3.9 25.641 1923 21,640 24 3.7 27.027 1944 21,630 25 3.6 27.778 1902 21,520 26 3.4 29.412 1962 21,400 27 3.3 30.303 1941 21,200 28 3.2 31.250 1901 21,080 29 3.1 32.258 1882 21,000 30 3.0 33.333 1906 19,280 31 2.9 34.483 1914 18,960 32 2.8 35.714 1939 18,950 33 2.7 37.037 1972 17,640 34 . 2.6 38.462 1887 17,520 35 2. 54 39.370 1909 17,480 36 2.47 40.48 6 1911 17,470 37 2.41 41.494 1963 17,280 38 2. 34 42.735 1927 17,080 39 2.28 4 3.8 60 287 Table B.6 (continued) % Flow Return Period Equalled Flow Rank or Exceeded M 1 1 Year (10Jm3) (m) T„ - N 1 (Years) R in 100/T R 10 7 8 16,720 40 2.23 44.843 1951 16,680 41 2.17 46.083 1894 16,240 42 2.12 47.170 1938 16,230 43 2. 07 48.309 1898 15,960 44 2.02 49.505 1966 15,920 45 1.98 50.505 1893 15,560 46 1.93 51.813 1947 15,440 47 1.89 52.910 1889 15,240 48 1. 85 54.054 194 3 15,040 49 1.82 54.945 1931 15,000 50 1.78 56.180 1945 14 ,920 51 1.75 57.143 1965 14,640 52 1.71 58.480 1933 14,280 53 1.68 59.524 1908 14 ,160 54 1. 65 60.606 1975 13,920 55 1. 62 61.728 1917 13,560 56 1. 60 62.500 1936 13,400 57 1.56 64.103 1918 13,240 58 1.53 65.359 1969 13,040 59 1.51 66.225 1949 12,680 60 1.48 67.568 18 9 2 12 ,640 61 1.46 68.493 1942 12,080 62 1.44 69.444 1932 12,040 6 3 1.41 70.922 1896 11,440 64 1.39 71.942 1919 11,430 65 1. 37 72.993 1920 11,360 66 1. 35 74.074 1883 10,760 67 1. 33 75.188 1974 10,750 68 1. 31 76.336 1937 10,080 69 1.29 77.519 1924 9,960 70 1.27 78 ,740 1934 9, 640 71 1.25 80.000 1925 9,560 72 1.24 80.645 1946 9,48 0 73 .1.22 81.967 18 86 9,440 7 4 1.20 83.333 1940 9,400 75 1.19 84.034 19 29 9, 360 7 6 1.17 85.470 19 21 9, 000 77 1. 16 86.207 19 04 8,92 0 78 1.14 87.719 .197 0 0,600 7 9 1.13 88.4 96 19 3 0 8,480 00 1.11 90.090 1922 8,400 81 1.10 90.909 .288 Table B.6 (continued) % Flow Flow Equalled Rank or Exceeded T ' •— N + 1 (Years) R m Year (10 3m 3) (m) 1 0 0 / T R 1 9 3 5 8 , 3 2 0 82 1 . 0 9 9 1 . 7 4 3 1 9 2 8 8 , 2 0 0 83 1 . 07 9 3 . 4 5 8 1 9 7 1 8 , 1 2 0 84 1 . 0 6 9 4 . 3 4 0 1 9 6 7 7 , 2 4 0 85 1 . 05 9 5 . 2 3 8 1 9 7 3 7 , 1 2 0 86 1 . 0 3 9 7 . 0 8 7 1 9 5 0 6 , 7 6 0 87 1 . 02 9 8 . 0 3 9 1 9 2 6 6 , 3 2 0 88 1 . 01 9 9 . 0 1 0 100 5 10 15 20 25 30 35 . 4 0 45 6 3 T o t a l A v a i l a b l e S u p p l y ( i n f l o w ) D u r i n g the I r r i g a t i o n Season (10 m ) £J F i g u r e B.2 C u m u l a t i v e P r o b a b i l i t y D i s t r i b u t i o n o f T o t a l I r r i g a t i o n Season Flow f o r Q u i l c h e n a Creek ^ 290 TABLE B.7 WEEKLY EVAPOTRANSPIRATION RECORDED AT MERRITT AND DOUGLAS LAKE IN 1967, USING OGOPOGO EVAPORIMETER Week Ending Evapotranspiration (mm) Merritt Douglas Lake A p r i l 10 16.26. 15.75 17 18.03 16.26 24 17.78 14.73 May 1 17.53 14.48 8 26.67 24.38 15 21.84 16.76 22 32.51 26.42 29 26.16 18.54 June 5 30.23 27.69 12 26.16 22.61 19 39.12 36. 07 26 29. 97 26.42 July 3 42. 67 37.85 10 34.29 30.23 17 42.16 39.12 24 33.53 26.67 31 38.86 31.75 August 7 30.48 23. 37 14 38.35 32.26 21 4 7.24 25.40 28 35.56 2 5.65 September 4 35. 81 18.80 11 23.11 11.18 18 23.88 11.43 25 24.89 11.43 30 13.46 12.45 Total .766.55 571.03 291 TABLE B.8 MONTHLY NET POTENTIAL EVAPOTRANSPIRATION, (mm), AT MERRITT AND DOUGLAS LAKE USING OGOPOGO EVAPORIMETER, 1968 AND 1969 Monthly Not Pet Values (mm) Location and Year - -~-• -• •-• -~ -~ ._ . .— May June July August September Merritt, 1968 Pet 103.63 112.52 140.21 96. 01 49.28 Rain 46.23 37. 59 0. 00 18.03 10. 92 Net Pet 57.40 74.93 140.21 77. 98 38. 35 Douglas Lake, 1968 Pet 94.49 97. 03 107.70 62. 99 48.51 Rain 41.40 60.96 3. 30 39.12 19.30 Net Pet 53. 09 36. 07 104.40 23.87 29. 21 Merritt, 1969 Pet - 119.38 169.67 14 5.2.9 79.76 Rain - 18. 03 22.86 1.52 0.00 Net Pet 101.35 146.81 14 3.77 79. 76 Douglas Lake, 1969 Pet 108.20 144.53 131.57 121.92 67 . 06 Rain 0.00 28. 96 24 .13 6. 60 32.00 Net Pet 108.20 115.57 107.44 115.32 35.06 TABLE B.9 SEASONAL EVAPOTRANSPIRATION OF ALFALFA AT MERRITT AS ESTIMATED BY DIFFERENT METHODS Source Year ET at Merritt May-September (mm) Hedke 1924 383.54 Lowry-Johnson 1942 589.28 Thornwaite 1944 637.54 Blaney-Criddle 1950 718.82 Hargreaves 1956 574.04 Manson 1962 500.38 Ogopogo Evaporimeter 1967 7 6 6.55 Ogopogo Evaporimeter 1968 501.65 Ogopogo Evaporimeter 1969 609.22 TABLE B.10 TOTAL SEASONAL EVAPOTRANSPIRATION FOR PASTURE BASED ON A DAILY WATER BUDGET - CANADA DEPARTMENT OF A GUI CULTURE, AGASSI/. Year Total ET (mm) 1955 414.02 1956 513.08 1957 533.40 1958 571.50 1959 500.00 196 0 508.00 1961 533.40 1962 444.50 1963 510.54 1964 373.38 .1965 563.88 

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