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The effect of uncertainty in irrigation development Nyumbu, Inyambo Liyambila 1976

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EFFECT OF UNCERTAINTY. IN IRRIGATION DEVELOPMENT by INYAMBO LIYAMBILA NYUMBU B.Eng., U n i v e r s i t y of Zambia, 1972 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES (The Department of C i v i l Engineering) Ue accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March", 1976 (c) Inyambo Liy a m b i l a Nyumbu, 1976 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a n d s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f CAOA C^ V^ -Cy,^  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 A B S T R A C T U n c e r t a i n t y i n i r r i g a t i o n development i s of s i g n i f i c a n t concern, more so nou than b e f o r e , because marginal p r o j e c t s are being developed. Piecemeal e f f o r t s to account f o r u n c e r t a i n t y do not i n d i c a t e i t s r e l a t i v e importance i n design d e c i s i o n s ; i t s only uhen u n c e r t a i n t y i n a l l the input f u n c t i o n s to the system i s p r o p e r l y accounted f o r that i t s s i g n i f i c a n c e can be r e a l i s e d . T h i s t h e s i s presents a method of a n a l y s i n g an i r r i g a -t i o n p r o j e c t i n which u n c e r t a i n t y was d i r e c t l y taken i n t o account. The design d e c i s i o n problem i n v o l v e d the choice of the c a p a c i t y of an i r r i g a t i o n system using r e g u l a t e d s t r e a m f l o u . The a n a l y s i s showed that the optimal s i z e of the i r r i g a t i o n p r o j e c t and the expected u t i l i t y value decreases 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 s , and i t a l s o depends on the a b i l i t y of the owners to s u r v i v e poor h a r v e s t s . i i TABLE OF CONTENTS Chapter . Page 1. INTRODUCTION 1 2. DESCRIPTION OF THE PROBLEM 4 2.1 Types of U n c e r t a i n t y i n Uater Resources P r o j e c t s . 4 2.2 Schematic P r e s e n t a t i o n of the Problem . . . . . . 6 2.3 A v a i l a b l e Data 8 :T.3. METHOD OF SOLUTION 21 3.1 B a s i c Assumptions 21 3.2 D e r i v a t i o n of Basic R e l a t i o n s h i p s . . . . . . . . 22 3.3 The Method of A n a l y s i s 28 4. DECISION CRITERION BASED ON MONETARY VALUE . . . 35 4.1 D e f i n i t i o n of B e n e f i t s 35 4.2 D e r i v a t i o n of B e n e f i t s . . . . . . 36 4.3 Computed Values of Monetary B e n e f i t s . . . . . . 38 4.4 E f f e c t of U n c e r t a i n t y i n I r r i g a t i o n Uater Requirement. 38 5. DECISION CRITERION BASED ON UTILITY VALUE. . . . 46 5.1 I n t r o d u c t i o n of U t i l i t y Value 46 5.2 Expected U t i l i t y D e c i s i o n s . . . . . . . . . . . 47 5.3 D e r i v a t i o n of U t i l i t y Function f o r I r r i g a t i o n Investment . 49 5.4 U t i l i t y F u n c t i o n . 50 5.5 Computation of Expected U t i l i t y 52 5.6 Values of Expected U t i l i t y 53 6. THE EFFECT OF UNCERTAINTY ON OPTIMAL DESIGN. . . 62 6.1 I n t r o d u c t i o n 62 6.2 R e v i s i o n of P r o b a b i l i t i e s 63 6.3 A p p l i c a t i o n to Problem of I r r i g a t i o n Development . . . . . . . . . . . . . . 6 4 6.4 Re s u l t s 69 7. DISCUSSION AND CONCLUSIONS 78 7.1 D i s c u s s i o n of Method of A n a l y s i s 78 7.2 Comparison of Results 79 7.3 Contrast between Expected Monetary Value and Expected U t i l i t y 82 7.4 Co n c l u s i o n s . 85 i i i C h a p t e r P a g e R E F E R E N C E S . 8 9 A P P E N D I X 91 P r o g r a m A . . . 9 3 P r o g r a m B 9 4 LIST OF TABLES Table Page 2.1 Powers Creek, Annual Streamflow 11 2.2 Frequency A n a l y s i s of Powers Creek Flow. . . . . 12 3.1 Okanagan R i v e r , Streamflow Record. . 27 3.2 Combining Two Sets of Costs 30 4.1 Optimum Design C o n d i t i o n s 40 •4.2 Intermediate Values of Optimum Con d i t i o n s On Basis of D o l l a r b e n e f i t s , I r r i g a t i o n water Requirement = 3 f e e t / y e a r 41 4.3 U n c e r t a i n t y i n Uater Requirement v 42 4.4 Intermediate Values of Optimum C o n d i t i o n s , Integrated I r r i g a t i o n Uater Requirement 43 5.1 Maximum Expected U t i l i t i e s . . . . . . . . . . . 55 5.2 Intermediate Values of Maximum Expected U t i l i t y , Using D i f f e r e n t U t i l i t y F u nctions; I r r i g a t i o n Uater Requirement = 3 f e e t / y e a r 56 5.3 Intermediate Values of Expected U t i l i t y , I n t egrated I r r i g a t i o n Uater Requirement 57 6.1 Data Used i n I n v e s t i g a t i n g the E f f e c t of Un c e r t a i n t y 66 6.2 Maximum Expected B e n e f i t with B e t t e r Information I r r i g a t i o n Uater Requirement = 3 f e e t / y e a r . . . 71 6.3 Summary of Results 72 v LIST OF FIGURES AND ILLUSTRATIONS F i g u r e Page 2.1 Schematic R e p r e s e n t a t i o n of t h e system 9 2.2 Streamflow Frequency D i s t r i b u t i o n . . . . . . . . 13 2.3 Crop Y i e l d as a F u n c t i o n of Percentage A v a i l a b l e / D e s i g n U ater Requirement 15 2.4 R e s e r v o i r Cost. 18 2.5 Land Development Cost 19 2.6 Annual Farm Maintenance Cost 20 3.1 H y p o t h e t i c a l F u n c t i o n Y = f(x) 24 3.2 "Skew Normal" D i s t r i b u t i o n 24 3.3 D e c i s i o n Tree 33 4.3 . D e c i s i o n T ree: U n c e r t a i n t y i n U a t e r Requirement . 44 4.2 Expected Economic B e n e f i t from I r r i g a t e d Area . . 45 5.1 H y p o t h e t i c a l U t i l i t y F u n c t i o n of a Farmer . . . . 51 5.2 D e c i s i o n Tree w i t h U t i l i t y Values 58 5.3 Expected U t i l i t y as a F u n c t i o n of I r r i g a t e d Area Slope S 2 = 2 59 5.4 Expected U t i l i t y as a F u n c t i o n of I r r i g a t e d A r e a , S 2 = 3 60 5.5 Expected U t i l i t y as a F u n c t i o n of I r r i g a t e d A r e a , S 2 = 5 61 6.1 Frequency D i s t r i b u t i o n of Powers Creek Annual Flow; U n c e r t a i n t y l i m i t s f i t t e d S t a t i s t i c a l l y and S u b j e c t i v e l y 68 6.2 P r o b a b i l i t y M a t r i x of Stream Flow 70 6.3 C u m u l a t i v e P r o b a b i l i t y of Annual Flow of Powers Creek. 73 v i F i g u r e Page 6.4 D e c i s i o n Tree with B e t t e r i n f o r m a t i o n 74 6.5 Expected B e n e f i t ; D o l l a r s and U t i l i t i e s ; B e t t e r i n f o r m a t i o n 75 6.6 Expected Economic Benefit., Varying Degrees of U n c e r t a i n t y 76 6.7 Expected U t i l i t y , Varying Degrees of U n c e r t a i n t y . 77 7.1 Summary of B e n e f i t Functions 88a v i i ACKNOWLEDGEMENT I n t h e p r e p a r a t i o n o f t h i s t h e s i s I have had t h e o p p o r t u n i t y t o g e t a s s i s t a n c e f r o m s e v e r a l p e o p l e i n C i v i l E n g i n e e r i n g D e p a r t m e n t . However, I w o u l d l i k e t o a c k n o w l e d g e most s i n c e r e l y t h e c o o p e r a t i o n , a s s i s t a n c e and g u i d a n c e I r e c e i v e d f r o m P r o f . S . O . R u s s e l l , my s u p e r v i s o r i n t h i s w o r k ; a l s o P r o f . U . F . C a s e l t o n has been v e r y h e l p f u l . T h a n k s a l s o t o Mr. Ron U n g e l s , f o r a s s i s t a n c e i n c o m p u t e r p r o g r a m m i n g , and t o Mr. R i c h a r d B r u n f o r d o i n g t h e d r a w i n g s . F i n a l l y I w o u l d l i k e t o t h a n k t h e D e p a r t m e n t o f C i v i l E n g i n e e r i n g f o r f i n a n c i a l a s s i s t a n c e f o r t h e p r e p a r a t i o n o f t h e t h e s i s , and f o r use o f t h e c o m p u t i n g f a c i l i t i e s . v i i i Chapter 1 INTRODUCTION Many of the most f a v o u r a b l e i r r i g a t i o n p r o j e c t s have a l r e a d y been developed. The remaining ones are o b v i o u s l y not fa v o u r a b l e - perhaps because they are marginal at best, perhaps because of l a c k of i n f o r m a t i o n that'., would permit t h e i r e f f e c t i v e e v a l u a t i o n . U i t h i r r i g a t i o n p r o j e c t s u n c e r t a i n t y i s a major f a c t o r : they are c a p i t a l i n t e n s i v e and they have a long payoff p e r i o d , t h e r e f o r e unsound d e c i s i o n s can be very costly,; p r i c e s of products vary and cannot: be p r e d i c t e d with c e r t a i n t y over the long l i f e of the p r o j e c t ; p r i c e s of in p u t s such as f e r t i l -i s e r s , labour and maintenance c o s t s vary; many past p r o j e c t s have not: l i v e d up to e x p e c t a t i o n s , hence i t i s d i f f i c u l t to p r e d i c t how new ones w i l l be accepted. In a d d i t i o n there i s the problem of the s t o c h a s t i c nature: of water: supply to. the system; crop y i e l d s are s u b j e c t to the water supply and to the other v a g a r i e s of weather.; a l s o estimates of f u t u r e demands of water, future; o p e r a t i n g p o l i c i e s , t e c h n o l o g i c a l developments and f u t u r e p o l i t i c a l d e c i s i o n s cannot be p r e d i c t e d with c e r t a i n t y . The degree of u n c e r t a i n t y that can be accepted depends on the a b i l i t y of the owners of the i r r i g a t i o n system to s u r v i v e bad years; t h i s a b i l i t y a l s o changes with time over the l i f e of the p r o j e c t . If u n c e r t a i n t y i s not considered the a n a l y s i s could e i t h e r lead to o v e r d e s i g n , which can r u i n the owner economically, or to 1 2 underdesign which i s wa s t e f u l of sca r c e r e s o u r c e s . Hence methods of a n a l y s i n g i r r i g a t i o n p r o j e c t s should d i r e c t l y take i n t o account the u n c e r t a i n t y i n the system. Although i t i s g e n e r a l l y recognised that there i s some degree of u n c e r t a i n t y that must be taken i n t o c o n s i d e r a t i o n i n water resources p r o j e c t p l a n n i n g , design and o p e r a t i o n , most eng-i n e e r i n g and economic analyses do not c o n s i d e r u n c e r t a i n t y d i r e c t l y . Sometimes u n c e r t a i n t y i s taken i n t o account i n d i r e c t l y - by design c r i t e r i a such as d e s i g n i n g f o r the flow that w i l l be ex-ceeded a c e r t a i n percentage of the time, and by s e n s i t i v i t y s. a n a l y s e s . T h i s t h e s i s presents a method of a n a l y s i n g a p r o j e c t i n which u n c e r t a i n t y was taken i n t o account d i r e c t l y . The a n a l y s i s was a p p l i e d to a h y p o t h e t i c a l p r o j e c t using s i m p l i f i e d but r e a l i s t i c d ata. The problem i n the p r o j e c t i n v o l v e d the ch o i c e of the c a p a c i t y of an i r r i g a t i o n system using r e g u l a t e d stream flow. Powers Creek Basin, a small water resources system i n the Okanagan Ba s i n , i n the i n t e r i o r of B r i t i s h Columbia was chosen to provide the necessary data r e q u i r e d i n a n a l y s i n g the hypo-t h e t i c a l i r r i g a t i o n p r o j e c t . Powers Creek Basin has a high a g r i c u l t u r a l p o t e n t i a l , but t h i s can only be r e a l i s e d with i r r i g a t i o n . The essence of the method of a n a l y s i s adopted i s that u n c e r t a i n t i e s , represented by p r o b a b i l i t i e s , can be combined together i n a methodical manner which permits a proper d e c i s i o n of the s i z e of the p r o j e c t to be made. The optimum s i z e of the p r o j e c t was t h a t which y i e l d s j the maximum value of expected b e n e f i t s . 3 The b e n e f i t s from the system were measured i n terms of net economic v a l u e , and i n terms of u t i l i t y value to the owner of the i r r i g a t i o n system. The p r o b a b i l i t i e s were d e r i v e d from a v a i l a b l e i n f o r m a t i o n , where p o s s i b l e , or they were s u b j e c t i v e assessments d e r i v e d on the b a s i s of e n g i n e e r i n g judgement. Bayesian d e c i s i o n theory provides the mechanics of working with such p r o b a b i l i t i e s and making a r a t i o n a l d e c i s i o n . The s p e c i f i c c h a r a c t e r i s t i c s of the data - cost c a p a c i t y r e l a t i o n s h i p s , streamflow and crop y i e l d - were not of concern i n themselves so long as they are c o n s i s t e n t with r e a l i t y . The main emphasis was on the method of a n a l y s i s , i t s a p p l i c a b i l i t y , and general c o n c l u s i o n s d e r i v e d from the t r i a l a n a l y s i s . The analyses were f i r s t made on the b a s i s of expected monetary b e n e f i t s and then on'the b a s i s of expected u t i l i t y . The l a t t e r showed t h a t the optimal s i z e of the p r o j e c t decreased 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 s and as the ownerfs a b i l i t y to s u r v i v e a drought decreased. The p r o j e c t i s d e s c r i b e d i n Chapter 2, and Chapter 3 presents the method of a n a l y s i s . Chapters 4 and 5 present the r e s u l t s of the analyses using a d e c i s i o n . c r i t e r i a of expected monetary value and expected u t i l i t y r e s p e c t i v e l y . Chapter. 6 demonstrates the e f f e c t of u n c e r t a i n t y on the optimal d e s i g n . In Chapter 7 the r e s u l t s are d i s c u s s e d and c o n c l u s i o n s drawn. The computer programs used i n the a n a l y s i s are given i n the Appendix. C h a p t e r 2 D E S C R I P T I O N OF THE PROBLEM 2 . 1 . T y p e s o f U n c e r t a i n t y i n U a t e r R e s o u r c e s P r o j e c t s M o s t d a t a r e q u i r e d i n w a t e r r e s o u r c e s s y s t e m d e s i g n a n d o p e r a t i o n c a n b e m e a s u r e d w i t h a r e a s o n a b l e d e g r e e o f a c c u r a c y . H o w e v e r , i n s p i t e o f s u c h a c c u r a c y , t h e m a g n i t u d e s a n d t i m i n g o f f u t u r e f l o w s c a n n o t be p r e d i c t e d w i t h a h i g h d e g r e e o f c e r t a i n t y . I n a d d i t i o n o u r k n o w l e d g e o f t h e u n d e r l y i n g h y d r o l o g i c a l p r o c e s s e s i s n o t a d e q u a t e ; a l s o t h e m a t h e m a t i c a l m o d e l s o f t h e p r o c e s s h a v e l i m i t e d d e g r e e o f s u i t a b i l i t y . I t i s n e c e s s a r y t o r e c o g n i s e t h e d i f f e r e n t t y p e s o f u n c e r t a i n t y w h i c h s h o u l d b e d e a l t w i t h i n w a t e r r e s o u r c e s s y s t e m d e s i g n a n d o p e r a t i o n . T h e r e a r e t h r e e b a s i c t y p e s - p r o c e s s u n c e r t a i n t y , s t a t i s t i c a l u n c e r t a i n t y a n d f u n d a m e n t a l u n c e r t a i n t y , ( B e n j a m i n a n d C o r n e l l , 1 9 7 0 ) , b u t t h e r e i s a f o u r t h t y p e w h i c h r e s u l t s f r o m c i r c u m s t a n c e s e x t e r n a l t o t h e s y s t e m . ( l ) P r o c e s s U n c e r t a i n t y - t h i s r e s u l t s f r o m l i m i t e d k n o w l e d g e o f t h e a c t u a l p r o c e s s , f o r e x a m p l e , t h e h y d r o l o g i c a l p r o c e s s . I n a n e f f o r t t o p r e d i c t s u c h n a t u r a l p r o c e s s e s , m a t h e m a t i c a l m o d e l s a r e o f t e n u s e d , b u t , a s a r e s u l t o f i g n o r a n c e o f t h e t r u e : p r o c e s s e s i n v o l v e d , t h e s e p r e d i c t i o n s may b e o f q u e s t i o n a b l e a c c u r a c y , ( i i ) S t a t i s t i c a l U n c e r t a i n t y - e v e n i f t h e p r o c e s s was w e l l u n d e r s t o o d a n d t h e a p p r o p r i a t e m o d e l s w e r e u s e d , t h e u n c e r t a i n t y o f t h e l i k e l i h o o d s s t i l l r e m a i n s . 4 5 because of limited s t a t i s t i c a l records. The data may be insufficient and relatively inaccurate; consequently when s t a t i s t i c a l '-parameters can be predicted, they s t i l l contain- some uncertainty, (iii . ) Fundamental Uncertainty - regardless of what data is available, future states of the process cannot be predicted uith absolute certainty. For example, neither the magnitude nor the sequence of hydro-logical events can be predicted easily, (iv) Uncertainty brought, about: by external circumstances -future operating policies of the uater resources system are not knoun uith: certainty. Such policies are even: more d i f f i c u l t to estimate i f the project is a multipurpose one. Future p o l i t i c a l decisions, future technological developments and social changes, and overall economic conditions a l l affect the system. These conditions cannbt be predicted uith certainty over the long time period of most, i r r igat ion projects, Most often these uncertainties are: interrelated and there is no indication as to uhich type of uncertainty is pre-dominant. Usually the type, of data u i l l give a clue to uhich type of uncertainty governs the si tuation. If better information is required then i t is important to recognise the type of un-certainty and hou i t . can be reduced. More data u i l l tend to reduce the magnitude of process and s t a t i s t i c a l uncertainty; fundamental uncertainty remains unchanged by the number of past experiments; the fourth type of uncertainty is more d i f f i c u l t to 6 p r e d i c t b e c a u s e i t h a s e l e m e n t s o f t h e f o r m e r t h r e e . T h e a d v a n t a g e o f d e c i s i o n t h e o r y i n - a c c o u n t i n g f o r . u n c e r t a i n t y i s t h e p o s s i b i l i t y o f b e i n g a b l e t o a s s i g n p r o b a b i -l i t i e s t o t h e u n c e r t a i n e v e n t s . P r o b a b i l i t i e s c a n b e d e r i v e d e i t h e r , f r o m o b s e r v a t i o n s o r f r o m e n g i n e e r i n g j u d g e m e n t . T h e p r o b a b i l i t i e s c a n t h e n b e i n c o r p o r a t e d i n a d e c i s i o n p r o c e s s . I n t h e f o l l o w i n g p r o b l e m t h e p r o b a b i l i t i e s a s s o c i a t e d u i t h t h e p o s s i b l e o u t c o m e s a r e d e r i v e d p a r t l y f r o m l i m i t e d d a t a a n d p a r t l y f r o m t h e . e n g i n e e r i n g j u d g e m e n t , o f e x p e r t s i n t h e i r r e s p e c t i v e f i e l d s . 2 . 2 . S c h e m a t i c P r e s e n t a t i o n o f t h e P r o b l e m T h e p h y s i c a l c o n f i g u r a t i o n t o u h i c h d e c i s i o n t h e o r y i s a p p l i e d i s d e s c r i b e d b e l o w a n d s h o w n s c h e m a t i c a l l y , o n F i g . 2.1 A s i n g l e r e s e r v o i r s u p p l i e s i r r i g a t i o n w a t e r : t o a n a g r i c u l t u r a l a r e a A ^ . T h e s i z e o f t h e a r e a u n d e r : i r r i g a t i o n i s d e p e n d e n t o n a v a i l a b l e w a t e r f r o m t h e r e s e r v o i r u h i c h i n t u r n d e p e n d s o n t h e i n f l o w 1 ^ . T h e o u t f l o w CK i s t h e a m o u n t o f w a t e r r e q u i r e d t o m a i n t a i n n o r m a l f l o w c o n d i t i o n s d o w n s t r e a m o f t h e r e s e r v o i r . A l l q u a n t i t i e s o f f l o u a r e b a s e d o n a n n u a l v o l u m e s . T h e a n n u a l i n f l o w 1 ^ i s k n o w n f r o m s t r e a m f l o w r e c o r d s , h o w e v e r , t h e l e n g t h o f : r e c o r d s i s v e r y l i m i t e d , s o t h a t i t h a s c o n s i d e r a b l e u n c e r t a i n t y . T h e a n n u a l i r r i g a t i o n d i v e r s i o n r e q u i r e m e n t i s k n o w n o n c e t h e a r e a u n d e r i r r i g a t i o n i s d e t e r -m i n e d , b u t i t a l s o v a r i e s w i t h w e a t h e r c o n d i t i o n s . T h e c o s t s o f b u i l d i n g t h e r e s e r v o i r a n d a l l t h e a t t e n d a n t s t r u c t u r e s , t h e c o s t s o f l a n d d e v e l o p m e n t f o r i r r i g a t i o n , t h e c o s t s o f c o n v e y i n g a n d d i s t r i b u t i n g t h e w a t e r , a n d t h e m a i n t e n a n c e 7 c o s t s , a l l can be estimated from previous p r o j e c t s i n the ar e a . The crop y i e l d from i r r i g a t e d land depends on the a v a i l -a ble u a t e r . The crop y i e l d can be estimated with some accuracy; these estimates are knoun from past y i e l d f i g u r e s and a l s o from a g r i c u l t u r a l knowledge from other a r e a s . The s e l l i n g p r i c e of a g r i c u l t u r a l products can be estimated w i t h i n c e r t a i n l i m i t s . I t i s known t h a t p r i c e s w i l l vary with the l e v e l of p r o d u c t i o n i n the area and w i l l a l s o vary from time to time as a r e s u l t of e x t e r n a l i n f l u e n c e ? . The main d e c i s i o n problem i s to determine the l e v e l of investment ( p h y s i c a l s i z e s of the components of the system) which w i l l y i e l d the maximum d e s i r e d b e n e f i t s . The co s t s of i n v e s t -ment i n c l u d e the r e s e r v o i r , land development and a l l a u x i l i a r y f a c i l i t i e s . The dilemma i s t h a t , i f the r e s e r v o i r i s small and the area under c u l t i v a t i o n i s a l s o s m a l l , s u f f i c i e n t use of the p o s s i b l e water supply may not be f u l l y r e a l i s e d . On the other hand, having a l a r g e r e s e r v o i r and a l a r g e area under i r r i g a t i o n , the i n f l o w volumes may not be s u f f i c i e n t to support such c u l t i -v a t i o n hence net r e t u r n s w i l l be reduced and the investment wasted. Within these two extremes-.i there i s a whole range of p o s s i b l e combinations of r e s e r v o i r s i z e and area under i r r i g a t i o n . The optimum design i s that combination of r e s e r v o i r s i z e and area under i r r i g a t i o n which produces a maximum value of expected b e n e f i t : Expected B e n e f i t = Expected Revenue - Expected Cost. E(B)= E(R(D.))- E(C(D.)) where R(D^) repr e s e n t s the revenue obtained from using 8 i r r i g a t i o n water C(D^) represents the c o s t s of the system ( R e s e r v o i r , Area under i r r i g a t i o n ) a s s o c i a t e d with s u p p l y i n g volume ; of water f o r i r r i g a t i o n . The c a l c u l a t i o n s of these q u a n t i t i e s should take i n t o account the 'imperfect' or u n c e r t a i n knowledge of c o s t s , p r i c e s , crop y i e l d and stream flow. 2.3. A v a i l a b l e Data The p e r t i n e n t i n f o r m a t i o n used i n i t h e a n a l y s i s was obtained from Powers Creek Basin, a sub basin of the Okanagan Basin i n the i n t e r i o r of B r i t i s h Columbia. The Okanagan Basin has 22 inches average annual p r e c i p i t a t i o n ; l o c a l v a r i a t i o n i n p r e c i p i t a t i o n range from 10 to 15 inches i n the v a l l e y bottoms to 35 to 40 inches i n the mountain areas; summers are f a i r l y warm with low humidity, and w i n t e r s have o c c a s s i o n a l l y very low temperatures. The basin i s an important a g r i c u l t u r a l area, but i t . i s h e a v i l y dependent on i r r i g a t i o n . However, there i s sparse h y d r o m e t e o r o l o g i c a l data i n most areas of the b a s i n , and i n s p i t e of the important need f o r developing the area, d e c i s i o n s w i l l s t i l l be based on t h i s kind of data ( Canada - Okanagan Basin Agreement 1974). Powers Creek Basin i s a s m a l l watershed (56 square m i l e s ) on the western s i d e of the Okanagan B a s i n . S e v e r a l upland lakes on the Creek have been harnessed to r e g u l a t e water flow, p r i m a r i l y f o r a g r i c u l t u r a l purposes. T h i s Basin has been chosen to provide data r e q u i r e d i n the a n a l y s i s because i t i s t y p i c a l of those basins earmarked f o r a g r i c u l t u r a l development i n s p i t e of the 9 Stream Flow FIG. 2. I SCHEMATIC REPRESENTATION OF THE SYSTEM 10 inadequacy of h y d r o m e t e o r o l o g i c a l and a g r i c u l t u r a l data on i t . One of the: important forage crops grown with i r r i g a t i o n i n the area i s a l f a l f a . T h i s crop i s used to represent the a g r i c u l t u r a l y i e l d from i r r i g a t i o n i n the f o l l o w i n g a n a l y s i s . The a v a i l a b l e data are given below. 2.3.1. Stream Flow Records Continuous stream flow records of Powers Creek are a v a i l a b l e f o r the l a s t e i g h t year© (up to 1974); p r i o r to th a t the stream flow was measured only during f r e s h e t season (water survey of Canada, 1973)'. Powers Creek and most creeks i n the Okanagan Basin lta.y:e l e s s than f i f t e e n years continuous stream flow r e c o r d . The main r i v e r i t s e l f , the Okanagan R i v e r , has f i f t y years continuous stream flow record at two l o c a t i o n s i n the. b a s i n . The stream flow r e c o r d f o r Powers Creek i s shown i n table; 2.1 and 2.2. F i g . 2.2 shows the frequency d i s t r i b u t i o n of the stream flow. 2.3.2. Crop Y i e l d The p r o d u c t i o n r e l a t i o n s h i p between water and crop y i e l d i s complex. Many v a r i a b l e s ( s o i l c h a r a c t e r i s t i c s , moisture supply, a i r temperature, wind, crop d i s e a s e s e t c ) , some of which are d i f f i c u l t to c o n t r o l , a f f e c t the p r o d u c t i o n . How-ever an estimate of the pro d u c t i o n r e l a t i o n s h i p was made a f t e r examining h i s t o r i c y i e l d f i g u r e s and water consumption. An a g r i c u l t u r a l i s t who i s very f a m i l i a r with the area gave an estimate ( F i g . 2.3.) of the r e l a t i o n s h i p between crop y i e l d and a v a i l a b l e i r r i g a t i o n water. The: upper and lower l i m i t s TABLE 2.1 POWERS CREEK ANNUAL STREAM FLOW S t a t i o n # 08NM 059 Belou Uest Bank D i v e r s i o n Drainage Area = 53.6 sq. miles YEAR TOTAL ANNUAL DISCHARGE Acre-Feet 1966 7200 1967 13200 1968 17300 1969 16400 1970 7450 1971 20600 1972 26500 1973 7900 Mean Annual Flou = 14,570 A c r e - f e e t Standard d e v i a t i o n = 6,980 A c r e - f e e t 11 T A B L E 2 . 2 F R E Q U E N C Y A N A L Y S I S OF P O W E R S C R E E K F L O W F L u U Y E A R R A N K R E T U R N % F L O U A c r e - f e e t m P E R I O D E Q U A L L E D OR t r = n + 1 E X C E E D E D m 2 6 , 5 0 0 1 9 7 2 1 9 11 2 0 , 6 0 0 1 9 7 1 2 4 . 5 2 2 1 7 , 3 0 0 1 9 6 8 3 3 . 0 3 3 1 6 , 4 0 0 1 9 6 9 4 2 . 2 5 4 4 1 3 , 2 0 0 1 9 6 7 5 1 . 8 0 5 6 7 , 9 0 0 1 9 7 3 6 1 . 5 0 6 7 7 , 4 5 0 1 9 7 0 7 1 . 2 9 7 8 7 , 2 0 0 1 9 6 6 8 1 . 1 2 5 8 9 1 2 30000 - 2 5 0 0 0 0) U < 20000 3 o ° 15000 c c • • • < ° 10000 5000 100 90 80 70 60 50 40 30 20 Percentage Time Indicated Flow Equalled or Exceeded 10 FIG.2.2 STREAM FLOW FREQUENCY DISTRIBUTION 14 bound the e r r o r s i n the e s t i m a t e . Uhat F i g . 2.3 shows i s that there i s a minimum y i e l d uhich can be obtained without i r r i g a t i o n - t h i s corresponds to the n a t u r a l y i e l d . I f more i r r i g a t i o n .,: water i s a v a i l a b l e the y i e l d i n c r e a s e s and there i s a l i m i t i n g value above which the y i e l d w i l l s t a r t to drop - u s u a l l y when accumulation of water leads to ponding. The crop y i e l d f u n c t i o n used i n the a n a l y s i s i s that which l i e s between the maximum and minimum values of y i e l d . 2.3.3 I r r i g a t i o n Uater Requirement The i r r i g a t i o n d i v e r s i o n requirement i n the Okanagan Basin v a r i e s from 6.0 f e e t / y e a r i n the southern end of the b a s i n to 1.75 f e e t / y e a r i n the north (Canada- B r i t i s h Columbia Oka-nagan Basin Agreement 1974). L o c a l requirements i n any area depend on l o c a l weather c o n d i t i o n s , s o i l c o n d i t i o n s , type of crop, topography and e l e v a t i o n of the l a n d . The B r i t i s h Columbia I r r i g a t i o n Guide suggests water requirements v a r y i n g from 2.0 f e e t / y e a r to 4.4 f e e t / y e a r depending on the number of f r o s t f r e e days i n an a r e a . The same source a l s o i n d i c a t e s t h a t the t o t a l e v a p o - t r a n s p i r a t i o n f o r pasture v a r i e s each year, and over a p e r i o d of 11 years the mean i s 1.6 f e e t / y e a r , with minimum and maximum values of 75% and 115% of the mean r e s p e c t i v e l y . The i r r i g a t i o n d i v e r s i o n requirement i n Powers Creek has been estimated to be 3.03 f e e t / y e a r (Canada- B r i t i s h Columbia Okanagan Basin Agreement). In the a n a l y s i s the i r r i g a t -ion requirement of Powers Creek Basin was assumed to be 3.0 f e e t / year but s e v e r a l p o s s i b l e values i n the range from a minimum value of 2.0 f e e t / y e a r to a maximum value of 4.0 f e e t / y e a r were 0 10 20 30 40 50 60 70 80 90 100 % Available Water/Design Water Requirement FIG.2.3. CROP YIELD AS A FUNCTION OF WATER. 16 a l s o used f o r purpose of comparison. 2.3.4 Costs of Development P r e l i m i n a r y estimates of the r e s e r v o i r and land deve-lopment c o s t s f o r i r r i g a t i o n uere roughly estimated by a u a t e r resources engineer u i t h experience i n the ar e a . These c o s t - s i z e r e l a t i o n s h i p s are shoun i n Fig;' 2.4 ( R e s e r v o i r ) , F i g . 2.5 (Land Development) and F i g . 2.6 (Maintenance on i r r i g a t e d a r e a ) . The upper and lower l i m i t s bound the u n c e r t a i n t y i n the estimates. The r e s e r v o i r c o s t s i n c l u d e a l l a u x i l i a r y s t r u c t u r e s - s p i l l w a y , i n t a k e , e t c . The land development c o s t s i n c l u d e the cost of con-veying and d i s t r i b u t i o n s t r u c t u r e s ( i r r i g a t i o n channels, l a t e r a l s , p i p i n g , pumps, e t c . ) and a l s o the cost of c l e a r i n g and l e v e l l i n g the land f o r a g r i c u l t u r a l use. The annual c a p i t a l c o s t s (repayment of lo a n , i n t e r e s t on c a p i t a l e t c ) and the annual o p e r a t i n g and maintenance costs of the f i x e d f a c i l i t i e s ( r e s e r v o i r , i r r i g a t i o n networks) uas assumed to be ten percent of the t o t a l c a p i t a l c o s t . " T h i s .'.'is.^equi-v a l e n t to assuming a 20 year l i f e of the p r o j e c t u i t h 7% i n t e r e s t on c a p i t a l and 3% of t o t a l c a p i t a l cost f o r the average annual o p e r a t i n g and maintenance c o s t s of the system. The t o t a l annual cost i s the sum of the annual c a p i t a l cost and the annual o p e r a t i n g and maintenance c o s t s given i n F i g . 2.6. 2.3.5 P r i c e s of Crops P r i c e s may vary each year depending on the laus of supply and demand c o n s t r a i n e d by e x t e r n a l i n f l u e n c e s such as c o n t r a c t u a l agreements, and u o r l d economic c o n d i t i o n s . A f i x e d 1 p r i c e of $100 per ton was used i n the a n a l y s i s . The a n a l y s i s uas a l s o done u s i n g two p r i c e s , | 9 0 per ton and ©110 per ton fo purposes of comparison. 0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 3900 4200 R e s e r v o i r Vo lume in A c r e - f e e t m FIG.2.4 RESERVOIR COST 1400, 1200 • I0Q0 < CD °- 800 o . . o Q 600 u 400 2 00 Cost includes land clearance and levelling, water distribution and drainage works. 100 200 300 400 500 600 Irrigated Land in Acres 700 800 900 IOOO FIG.2.5 LAND DEVELOPMENT COSTS. 400 360 240 200 " H i g h -obable Low 0 100 200 300 400 500 600 700 800 900, 1000 Irrigated Land in Acres FIG.2.6 ANNUAL FARM MAINTENANCE COST Chapter 3 METHOD OF SOLUTION 3.1 B a s i c Assumptions. The f o l l o w i n g assumptions about- the system were made. 3.1.1. R e s e r v o i r ( i ) The r e s e r v o i r i s to be used p r i m a r i l y f o r s t o r i n g i r r i g a t i o n water, ( i i ) There i s no c a r r y over storage from season to season. A l l the water comes at the beginning of the season. Excess flow above the i r r i g a t i o n requirement i s s p i l l e d as outflow, ( i i i ) The i n f l o w i n t o the r e s e r v o i r i s n a t u r a l flow from the upstream b a s i n . The e f f e c t of other storage r e s e r v o i r s upstream i s i g n o r e d , ( i v ) Losses of water from the r e s e r v o i r through e v a p o r a t i o n , seepage i n t o the embankment, and other secondary e f f e c t s are assumed to have 1 bseh: deducted from the i n f l o w . Therefore the r e s e r v o i r s i z e i s the net s i z e r e q u i r e d to supply the nescessary i r r i g a t i o n requirements. 3.1.2. Area Under I r r i g a t i o n and Uater Use ( i ) The t o t a l i r r i g a t i o n requirement v a r i e s between 2.0 f e e t / y e a r and 4.0 f e e t / y e a r . T h i s amount a l s o i n c l u d e s the water l o s s e s i n c u r r e d on the farm and the i r r i g a t i o n networks, ( i i ) Only one crop under i r r i g a t i o n . 21 22 ( i i i ) The maximum area under i r r i g a t i o n i s assumed to be 1000 a c r e s . ( i v ) The maximum p e r m i s s i b l e withdrawal from the creek i s set: at one tenth of annual flow. 3.1.3. Crop Y i e l d . The crop y i e l d v a r i e s with the a v a i l a b l e i r r i g a t i o n water. U i t h o u t any i r r i g a t i o n the p o s s i b l e y i e l d i s around 2.0 Tons/acre. I t i s assumed t h a t the farm wat8r a p p l i c a t i o n r a t e s w i l l be c o n t r o l l e d so as to avoid ponding and to obt a i n maximum yielid p o s s i b l e with the a v a i l a b l e water. 3.1.4. Stream Flow ( i ) Annual volumes of stream flow were used i n the a n a l y s i s , ( i i ) Only 10% of the annual flow could be d i v e r t e d f o r i r r i g a t i o n . The r e s t of the flow was assumed to be used to meet other requirements - f o r example to maintain normal flow c o n d i t i o n s downstream f o r other water u s e r s . 3.2. D e r i v a t i o n of B a s i c R e l a t i o n s h i p s 3.2.1. P r o b a b i l i t y of Costs and Y i e l d . The p r o b a b i l i t y ^ e q u i v a l e n t to u n c e r t a i n t y ) of each of the co s t s and the y i e l d were d e r i v e d as f a l l o w s . ( i ) A computer l i b r a r y program was used which approximates each f u n c t i o n ( c o s t , y i e l d ) by a c u b i c s p l i n e f u n c t i o n and i n t e r p o l a t e s between given data p o i n t s . F i g . 2.3. 2.4, 2.5, and 2.6 were drawn using the data d e r i v e d by such a program. 23 ( i i ) U i t h the f u n c t i o n s d e r i v e d as above the p r o b a b i l i t y of the c o s t s and the y i e l d were de r i v e d by another computer program. F i g . 3.1. shows a h y p o t h e t i c a l f u n c t i o n Y = f ( X ) . For any value of X there i s a set of p o s s i b l e values of Y bounded by the upper and lower l i m i t s . The t r u e value of Y l i e s somewhere between the upper and lower l i m i t s . The u n c e r t a i n t y about the t r u e value of Y can be d e s c r i b e d by a 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 , and thus each value of Y has a p r o b a b i l i t y attached to i t . The values of Y were assumed to be d i s t r i b u t e d a c c o r d i n g to a "skew normal" d i s t r i b u t i o n , see Fig.3.2, (Hershman 1974). A computer program has been d e r i v e d (Higgins 1975) which c a l c u l a t e s the p r o b a b i l i t y a s s o c i a t e d with any value of Y by i n t e g r a t i n g o v e r the 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 bounded by the upper and lower l i m i t s of the f u n c t i o n Y = f ( X ) . T h i s program i s d e s c r i b e d i n Appendix, Program A. T h i s procedure was used i n d e r i v i n g the p r o b a b i l i -t i e s of c o s t s , crop y i e l d and stream flow. The "skew normal" d i s t r i b u t i o n was o r i g i n a l l y d e r i v e d f o r i hydro-l o g i c data but i n t h i s a n a l y s i s i t was used f o r economic data ( c o s t s ) as w e l l as crop y i e l d . T h i s i s one manner of accounting f o r u n c e r t a i n t y i n knowledge. Bogardi and Szidarovsky (1974) have used a normal 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 to account f o r i n f o r m a t i o n u n c e r t a i n t y stemming from l i m i t e d data i n h y d r o l o g i c d e s i g n . 24 Xi X FIG. 3.1 HYPOTHET ICAL FUNCTION Y= f ( x ) . 25 However , the "skew normal" d i s t r i b u t i o n seems to be an a p p r o p r i a t e r e p r e s e n t a t i o n of most p r a c t i c a l s i t u a t i o n s : v a r i a b l e s are measured from a base ( i . e . lower l i m i t ) and a l s o possess an upper bound. 3.2.2. P r o b a b i l i t y of Stream flow. The d e r i v a t i o n of a stream flow d i s t r i b u t i o n when there i s only eight, years data i s , at best, a guess. The r e l a t i v e l y s h o r t h i s t o r i c flow record does not c l e a r l y d e f i n e the p r o p e r t i e s and parameters of the d i s t r i b u t i o n ; t h e r e f o r e e r r o r s could be made i n s e l e c t i n g the u n d e r l y i n g d i s t r i b u t i o n . For the sake of keeping the a n a l y s i s simple at t h i s stage, i t . was assumed that the stream flow was d i s t r i b u t e d a c c o r d i n g to a "skew normal" d i s t r i b u t i o n and the same program mentioned i n the previous s e c t i o n was used to d e r i v e the p r o b a b i l i t y of v a r i o u s magnitudes of stream flow. For a t h e o r e t i c a l "skew normal" d i s t r i b u t i o n the." lower l i m i t i s -*<5 and the upper l i m i t i s + 0 0 . However, i n r e a l i t y t h ere e x i s t s the upper and lower l i m i t s , and Q^, of stream flow p o s s i b l e i n a given b a s i n . The l i m i t i n g values of flow have to be determined from the h i s t o r i c flow r e c o r d s . U i t h respect to Powers Creek, the only way to determine Q u and i s to examine the stream flow records of some of the r i v e r s i n the same major b a s i n ( i . e . Okanagan Basin) and get a comparison of the l i m i t i n g f l o w s . However the other sub-basins i n the Okanagan Basin have r e l a t i v e l y short, stream flow r e c o r d s , hence the only comparison p o s s i b l e i s with the Okanagan R i v e r i t s e l f . Table 3.1 shows the stream flow records of the 26 Okanagan River at a hydro-metric s t a t i o n where more than f i f t y years continuous record i s a v a i l a b l e . The three d r i e s t years were 1929, 1930 and 1931; the three wettest years were 1928, 1948 and 1972: Average minimum flow = "i (83,200 + 58,600 + 36,900)/3 = 59^567 a c r e - f e e t Average maximum flow = -(966,000 + 719,000 + 844,000)/3 = 843,000 a c r e - f e e t Mean annual flow = 380,000 a c r e - f e e t . T h e r e f o r e : minimum flow = 59.567 = 0.16 mean flow 380,000 maximum flow mean flow 843.000 380,000 2.22 These same r a t i o s of flow were used i n e s t i m a t i n g the l i m i t i n g values of flow f o r Powers Creek. T h e r e f o r e : -minimum flow = 0.16 x 14,570 = 2,330 a c r e - f e e t maximum flow = 2.22 x 14,570 = 32,345 acre f e e t With Q u and Q 1 d e f i n e d , the p r o b a b i l i t i e s of other flow magnitudes were determined by the computer program r e f e r e d to p r e v i o u s l y . 3.2.3. Matrix Format A f t e r determining the p r o b a b i l i t i e s of v a r i o u s values of Y f o r each given value of X (See F i g . 3.1), d i s c r e t e p i e c e s Table 3.1 OKANAGAN RIVEfc, STREAMFLOW RECORD OKANAGAN RIVER AI OKANAGAN FALLS - 6TATION NO. 08NM002 ANNUAL EXTREMES OF DISCHARGE IN CFS AND ANNUAL TOTAL DISCHARGE IN AC-FT FOR THE PERIOD OF RECORD YEAR MAXIMUM INSTANTANEOUS DISCHARGE 19 1 S — I M S 19 17 1916 19 13 —— -1920 19 21 — -1922 . . . 1923 ---1920 — ~ — 1925 1926 1927 ---192B — -1929 — - — 193 0 19 J 1 19 12 1933 1931 1935 10 1936 1000 c r s AT 1000 P5T ON JUN 1937 1080 CFS AT 1030 PST ON JUN ] 193B 1270 c r s AT 0900 PST ON MAY 28 1939 50 J CFS AT 1500 PST ON HAY 8 19 0 0 300 c r s AT 2000 PST ON JUN 11 19" 1 8 2 0 c r s AT 1300 PST ON DEC 23 19»2 1 J 10 c r s AT 1900 PST ON JUL 0 1903 865 CFS AT 0900 PST ON MAY 9 19M 802 CFS AT 1300 PST ON JUN « 19« 5 1250 c r s AT 1100 PST ON JUN 7 19"6 1360 CFS AT 1800 PST ON MAY 1 J 1907 696 c r s AT 1130 PST ON JAN 1 1906 1550 c r s AT 1200 PST ON JUN 18 1909 1350 CFS AT 1000 PST ON MAY 16 1950 1310 CFS AT 1300 PST ON JUN 15 1951 1000 CFS AT 1230 PST ON MAY 1J 1952 10 10 CFS AT 1700 PST ON MAY 20 1953 1010 CFS AT 1730 PST ON HAY 22 1950 ---1955 CFS 1956 1500 AT 1015 PST ON JUL IS 1957 1670 CFS AT 2130 PST ON AUG 27 1958 2790 CFS AT 0800 PST ON APR 25 1959 2 150 CFS AT 090 J PST ON MAY 25 1960 . 1160 c r s AT 0010 PST ON JAN 1 196 1 1970 c r s AT 1230 PST ON JUN J 1952 1310 CFS AT 1005 PST ON APR 30 1953 58 1 c r s AT 1000 PST ON AUG 16 1961 1380 CFS AT 0915 PST ON JUN 16 1965 1600 c r s AT 0930 PST ON JUN 6 1966 062 CFS AT 1500 PST ON APR 12 1967 . . . 1968 1760 CFS AT 1030 PST ON JUN 12 1969 1600 CFS AT 1556 PST ON MAY 31 1970 1030 CFS AT 0930 PST ON HAY 13 197 1 1970 crs AT 0900 PST OH JUN 5 1972 2700 c r s AT 2 130 PST ON MAY 10 1973 1200 CFS AT 07 30 PST ON JUN 2 MAXIMUM DAILY DISCHARGE MINIMUM DAILY DISCHARGE 1 160 CFS ON MAY 29 too CFS ON MAR 23 1300 CFS ON JUL 2 — 20 1100 CFS ON JUN 29 190 CFS ON NOV 1100 CFS ON JUN 10 160 CFS ON MAR 6 1200 CFS ON JUN 5 150 CFS ON JAN 7 1 100 CFS ON JUL 10 100 CFS ON APR 19 2500 CFS ON JUN 10 105 CFS ON MAR 0 950 CFS ON MAY 19 130 CFS ON SEP 19 1150 CFS ON 0 UN 22 390 CFS ON DEC 2 558 CFS ON FED 2 53.0 CFS ON NOV 28 1160 CFS ON MAY 20 05. 0 CFS OH JAN 27. 705 CFS ON MAR 16 30. 0 CFS ON OCT 1 3 1030 CFS ON DEC 28 30. 0 CFS ON JAN 2680 CFS ON JUN 10 • 03. 0 c r s ON DEC 28 713 CFS ON MAY 31 . 8. 1 CFS ON DEC 0 002 CFS ON HAY 10 5. 3 c r s ON JAN 31 106 CFS ON HAY 7 0 . 6 CFS ON MAR 10 970 CFS ON HAY 7 7. 7 CFS ON JAN 9 1300 CFS ON MAY 30 118 CFS ON FEB 25 1160 CFS ON APR. 20 208 CFS ON SEP 7 1 110 CFS ON HAY 20 057 CFS ON JAN 2 1000 CFS ON J1IN IB 178 CFS ON APR 13 1000 CFS ON JUH 3 96 0 CFS ON JAN 5 1200 CFS ON MAY 28 150 CFS ON DEC 30 532 CFS ON HAY 0 117 CFS ON FEB IB 263 CFS ON JUN 12 52 0 CFS ON FEB 20 798 CFS ON DEC 20 63 0 CFS OH MAY 27 1280 CFS ON JUL 1 123 c r s ON HAR 29 798 CFS ON HAY 9 130 CFS OH OCT 19 795 CFS OH JUN 3 70 0 CFS ON HAY 4 1200 CFS ON JUH 7 257 CFS ON SEP 10 1360 CFS ON HAY .10 092 CFS ON FEB 16 689 CFS ON JAN 1 110 CFS ON MAR 1 1530 CFS ON JUN 18 100 CFS ON MAR 21 1330 CFS ON HAY 16 355 CFS ON DEC 22 1100 CFS ?M JUN 6 . 321 CFS ON JAN 30 1000 CFS ON HAY 13 590 CFS ON JAN 3 1 10 10 CFS ON HAT 20 160 CFS ON NOV 29 900 CFS ON MAY 23 152 c r s ON JAN 16 1190 CFS ON JUL 1 1 05 0 CFS ON OCT 19 1030 CFS ON JUN 10 276 CFS ON DEC 19 1500 CFS ON JUL 15 282 CFS ON JAN 1 1080 CFS ON AUG 28 223 CFS ON APR 13 2560 c r s ON APR 26 315 CFS OH OCT 3 2120 CFS ON MAY 26 126 c r s ON JAN 13 1150 CFS ON JAN 1 • 315 CFS ON JUN • 1790 c r s ON JUN 6 202 CFS ON FEB 15 1250 CFS ON APR 28 229 c r s ON JAN 28 308 c r s ON AUG 16 10 1 CFS ON MAY 15 1360 CFS ON JUN 17 163 CFS ON JAN 1 1620 cr« ON JUN 6 216 c r s ON NOV 9 055 c r s ON APR 10 153 CFS ON OCT 18 1360 c r s ON JUN 5 * — 1730 CFS ON JUN 12 -— 12 1(60 CFS ON MAY 31 269 c r s ON DEC 1020 CFS ' ON MAY 13 97 . 1 c r s ON DEC 13 1930 CFS ON JUN 6 60.7 CFS ON FEB 19 2600 CFS ON HAY 11 203 c r s ON OCT 17 680 CFS ON AUG 7 AO .6 CFS ON NOV 21 TOTAL DISCHARGE 030000 AC-TT 330000 AC-FT 278000 AC-FT 303000 AC-FT 201000 AC-FT 012000 AC-FT 335000 AC-FT 037000 AC-FT 170000 AC-FT 205000 AC-FT 105000 AC-FT 323000 AC-FT 9C6000 AC-FT 83200 AC-FT 58600 36900 389000 5 1 0 000 070000 505000 367000 330000 337000 216000 122000 288000 560000 288000 190000 O57O0O 620000 212000 7 1 9000 516000 AC-FT AC-FT AC-FT AC-FT AC-FT AC-FT AC-FT AC-TT AC-FT AC-FT AC-FT AC-FT AC-FT AC-FT AC-FT i AC-FT AC-FT AC-FT AC-FT AC-FT" pv.np.v g p r n a n r n ran TUP D r n t n n fir nr.rnnn 091000 AC-FT 607000 AC-FT 507000 AC-FT 331000 AC-FT 520000 AC-FT 073000 AC-FT 537000 AC-FT 033000 AC-TT 3B5000 AC-TT 608000 AC-FT 380000 AC-TT 300000 AC-FT 309000 AC^FT 165000 AC-TT 010000 AC-F1 178000 AC-FT 226000 AC-FT 096000 AC-FT 211000 AC-FT 357000 AC-FT 800000 AC-FT 186000 AC-FT .180000 AC-FT 0 05 YEAR 1915 19 16 19 17 1918 1919 1920 1921 1922 1923 1920 1925 1926 13 27 1920 1929 1930 193 1.. 1932 1933 1930 1935 1936 1937 1938 1939 1900 190 1 1902 1903 1900 1905 1906 1907 1908 1909 1950 1951 1952 1953 1950 1955 1950 1957 1958 1959 1960 1961 1962 1963 1960 1965 • 1966 1967 1968 1969 1970 197 1 1972 1973 HEAN N 3 ^ 3 28 of data are obtained - d i s c r e t e p r o b a b i l i t i e s f o r d i s c r e t e values of Y and X. Therefore the e n t i r e f u n c t i o n can be broken i n t o d i s c r e t e q u a n t i t i e s . A very u s e f u l method of handling t h i s data i s to s t o r e i t i n a matrix format. For example, a matrix of r e s e -r v o i r c o s t s , a matrix of land development c o s t s and a matrix of crop y i e l d . U i t h r e s p e c t to the f u n c t i o n i n F i g . 3.1, the f i r s t row of matrix would represent d i s c r e t e l e v e l s of X, the f i r s t column would represent d i s c r e t e l e v e l s of Y, and the other elements of the matrix represent the p r o b a b i l i t i e s of values of Y at s p e c i f i c l e v e l s of X. T h i s method of r e p r e s e n t i n g data has two advantages: f i r s t l y , the e n t i r e f u n c t i o n ( c o s t , y i e l d , e t c . ) can be broken down i n t o d i s c r e t e l e v e l s and the f u n c t i o n can be used i n i t s normal mode without recourse to mathematical appro-ximation; secondly, a d d i t i o n , m u l t i p l i c a t i o n (or any other o p e r a t i o n ) of matrices can be e a s i l y c a r r i e d out using computer programs that are a v a i l a b l e i n most computer l i b r a r i e s . The number of d i s c r e t e l e v e l s of each f u n c t i o n depends on the d e s i r e d accuracy. This method of data r e p r e s e n t a t i o n was used i n the a n a l y s i s given below. 3'i. 3 The Method of A n a l y s i s 3.3.1 Combining Tup F u n c t i o n s . The premise f o r combining any two cost f u n c t i o n s i s that they are independent events i . e . the co s t s of the r e s e r v o i r are not r e l a t e d at a l l with the co s t s of land development and the co s t s on the farm. The r e f o r e the b a s i c axioms of p r o b a b i l i t y can 29 b e a p p l i e d i n c o m b i n i n g t h e p r o b a b i l i t i e s ( u n c e r t a i n t y ) a s s o c i a t e d w i t h e a c h o f t h e c o s t s . A c c o r d i n g l y , t h e p r o b a b i l i t y o f t h e j o i n t o c c u r e n c e o f t u o i n d e p e n d e n t e v e n t s A a n d B i s t h e p r o d u c t o f t h e i r p r o b a b i l i t i e s , a n d t h e p r o b a b i l i t y o f t h e u n i o n o f t u o m u t u a l l y e x c l u s i v e e v e n t s C a n d D i s t h e sum o f t h e i r p r o b a b i l i t i e s , A s s u m e t h a t t u o c o s t e s t i m a t e s h a v e t o b e c o m b i n e d , a n d t h a t t h e c o s t s h a v e a l r e a d y b e e n p r e s e n t e d i n a m a t r i x f o r m a t . T h e r e a r e t h e n t u o i n d e p e n d e n t s e t s o f d a t a a s s h o u n i n T a b l e 3 . 2 ; e a c h c o s t e s t i m a t e h a s a p r o b a b i l i t y a s s o c i a t e d u i t h i t . S e t A may r e p r e s e n t t h e c o s t o f a r e s e r v o i r o f a p a r t i c u l a r s i z e , a n d S e t B r e p r e s e n t s t h e c o s t o f d e v e l o p i n g l a n d o f a c e r t a i n a c r e a g e . A n y c o s t i n S e t A c a n c o m b i n e u i t h a n y c o s t i n S e t B a n d t h e p r o b a b i l i t y o f t h e c o m b i n e d v a l u e i s t h e p r o d u c t o f t h e i n d i v i d u a l p r o b a b i l i t i e s . T h i s may l o o k l i k e a v e r y l a r g e n u m b e r o f c o m b i n a t i o n s , b u t i t i s n o t n e c e s s a r i l y s o ; s o m e o f t h e c o m b i n e d v a l u e s a r e n o t u h j i q u e . C o n s i d e r t h e m i n i m u m a n d maximum v a l u e s o f t o t a l c o s t : M i n i m u m t o t a l c o s t = 1 5 0 + 2 7 0 = 4 2 0 Maximum t o t a l c o s t = 1 9 0 + 3 0 0 = 4 9 0 T h e i n t e r v a l b e t u e e n t h e m i n i m u m a n d maximum v a l u e s i s d i v i d e d i n t o i n t e r v a l s o f 10 j u s t l i k e t h e c o s t i t e m s o f S e t s A a n d B. T h e n u m b e r o f c o s t c o m b i n a t i o n s r e q u i r e d t o p r o d u c e a t o t a l c o s t o f 4 2 0 , f o r e x a m p l e , i s l i m i t e d - o n l y t h e sum o f 1 5 0 a n d 2 7 0 c a n g i v e . a v a l u e o f 4 2 0 . T u o s e t s o f c o s t s c a n p r o d u c e a v a l u e o f 4 3 0 , a sum o f 1 5 0 a n d 2 8 0 , a n d a sum o f 1 6 0 a n d 2 7 0 . T h e p r o b a b i l i t y - o f 4 3 0 i s t h e sum o f t h e p r o d u c t s o f t h e i n d i v i d u a l 30 TABLE 3.2 COMBINING TUO SETS OF COSTS (a) SET A SET B UALUE . PROBABILITY 150 a 160 b 170 c 180 d 190 e VALUE PROBABILITY 270 r 280 s 290 t 300 u (b) Combined Value No. TOTAL VALUE POSSIBLE PAIRS OF VALUES TO GIVE TOTAL VALUE PROBABILITY OF TOTAL VALUE 1 420 (150,270) ar 2 430 (150,280),(160,270) as + br 3 440 (150,290),(160,280), (170,270) at+bs+cr • • • • * • • • n 490 (190,300) eu 3 i p r o b a b i l i t i e s . T h e p r o c e d u r e i s i l l u s t r a t e d i n T a b l e 3.2 ( b ) . T h i s m e t h o d c a n b e a p p l i e d t o a n y n u m b e r o f s e t s o f d a t a , e v e n i f t h e s i z e s o f s e t s a r e d i f f e r e n t , p r o v i d e d o n l y t w o s e t s o f d a t a a r e c o m b i n e d a t a t i m e . T h e c o s t i n t e r v a l s b e t w e e n t h e e l e m e n t s o f t h e s e t s ( e . g . 10 i n S e t s A a n d B) n e e d n o t b e t h e s a m e . H o w e v e r t h e c o m b i n a t i o n i s e a s i e r a n d m o r e a c c u r a t e i f t h e i n t e r v a l s a r e e q u a l , W h e r e t h e t w o i n t e r v a l s a r e n o t e q u a l a n a v e r a g e v a l u e c o u l d b e u s e d . T h e c o m b i n a t i o n n e e d n o t b e a d d i t i o n o n l y , o t h e r o p e r a t i o n s c o u l d b e u s e d a s w e l l . By r e p r e s e n t i n g t h e f u n c t i o n s i n a m a t r i x f o r m a t t h e a b o v e o p e r a t i o n s a r e e a s i e r t o do b y c o m p u t e r , a n d t h e c o m b i n e d f u n c t i o n i s a l s o r e p r e s e n t e d a s a m a t r i x . 3 . 3 . 2 . E x p e c t e d B e n e f i t s . T h e p r e c e d i n g a c t i v i t i e s p r e p a r e d t h e d a t a f o r u s e i n t h e d e c i s i o n p r o c e s s . T h e d e c i s i o n p r o b l e m i s t o d e t e r m i n e t h e c o m b i n a t i o n o f r e s e r v o i r s i z e a n d l a n d u n d e r i r r i g a t i o n w h i c h y i e l d s t h e maximum e x p e c t e d b e n e f i t : E x p e c t e d B e n e f i t = E x p e c t e d R e v e n u e - E x p e c t e d C o s t U h e r e E x p e c t e d v a l u e = SX^P (X^ ) f o r d i s c r e t e v a r i a b l e s X^ i s d i s c r e t e v a r i a b l e P(X^) i s t h e p r o b a b i l i t y o f v a r i a b l e X^ T h e d e c i s i o n p r o b l e m c a n b e r e p r e s e n t e d i n t h e f o r m o f a d e c i s i o n t r e e , a s i n d i c a t e d i n F i g . 3 . 3 , a n d t h e p r o b a b i l i t i e s s h o w n i n d i c a t e t h e u n c e r t a i n t y a s s o c i a t e d w i t h t h e d a t a . T h e r e a r e t w o t y p e s o f e v e n t s s h o w n o n t h e d e c i s i o n t r e e . T h o s e 32 events over which the d e c i s i o n maker has choice of a c t i o n ( f o r example, choice of r e s e r v o i r s i z e and choice of acreage to i r r i g a t e ) are c a l l e d d e c i s i o n events; those events which depend on chance or n a t u r a l circumstances ( f o r example, stream flow) are c a l l e d chance events. The main d e c i s i o n problem i s how to choose between a l t e r n a t i v e l e v e l s of investment when there i s u n c e r t a i n t y about each outcome. The use of expected values -c o s t s b e n e f i t s , y i e l d - reduces the d i s s i m i l a r magnitudes to a common denominator from which they can a l l be compared. The procedure i s then to c a l c u l a t e the expected values at each branch. The optimum d e c i s i o n i s the one which has the maxi-mum expected b e n e f i t . The b e n e f i t can be measured e i t h e r by monetary value or by u t i l i t y v a l u e . The d e c i s i o n procedure i s summarised below; a l s o r e f e r to F i g . 3.3. For a value of T o t a l I r r i g a t i o n Uater requirement 1. Choose area to i r r i g a t e . 2. Choose R e s e r v o i r s i z e to supply water. 3. Determine c o s t s a s s o c i a t e d with the choice - Annual C a p i t a l Cost - Maintenance Cost -: T o t a l Annual Cost and the expected c o s t . 4. Determine a v a i l a b l e q u a n t i t y of water from stream flow r e c o r d s . 5. Determine p o s s i b l e crop y i e l d s and Revenues. 6. Determine net b e n e f i t and expected b e n e f i t . 7. Repeat 4 to 6 f o r a l l p o s s i b l e f l o w s . c ice) Unit Cost Probability that true Revenue Net Benefit B, 1 B; B, D Decision Event O Natural Occurence FIG. 3 . 3 DECISION TREE 8. R e p e a t 2 t o 7 f o r a l l p o s s i b l e r e s e v o i r s . 9. R e p e a t 2 t o 8 f o r a l l p o s s i b l e a c r e a g e s . 1 0 . C a l c u l a t e t h e maximum o p t i o n . R e p e a t f o r o t h e r v a l u e s o f i r r i g a t i o n u a t e r r e q u i r e m e n t a n d c h o o s e t h e o p t i m u m d e c i s i o n . Chapter 4 DECISION CRITERION BASED ON MONETARY VALUE 4.1 D e f i n i t i o n of B e n e f i t s The determination of b e n e f i t s from an i r r i g a t i o n system must, of n e c e s s i t y , f i r s t of a l l d e f i n e the b e n e f i c i a r y , the co s t s to such b e n e f i c i a r y and the b e n e f i t s a c c r u i n g . If the b e n e f i c i a r y i n v o l v e s the whole s o c i e t y or a l a r g e community both the c o s t s of the system and the b e n e f i t s from i t are wide spread i n t o the community; a l s o these c o s t s and b e n e f i t s may be d i r e c t (e.g. c a p i t a l c o s t s of b u i l d i n g the system) or they may be i n d i r e c t , and to a l a r g e degree, d i f f i c u l t to q u a n t i f y . Besides, a monetary measure of b e n e f i t (or c o s t s ) may not n e c e s s a r i l y r e f l e c t the net worth to s o c i e t y . I f the b e n e f i c i a r y i s reduced to a s i n g l e farmer, though some of the c o s t s and b e n e f i t s from an i r r i g a t i o n system may a f f e c t t h i r d p a r t i e s , n e v e r t h e l e s s the co s t s and b e n e f i t s to the farmer are l o c a l i s e d and r e l a t i v e l y e a s i e r to measure. The l a t t e r approach was adopted i n the a n a l y s i s , the b e n e f i c i a r y was assumed to be a s i n g l e farmer, who was a l s o assumed to bear a l l c o s t s and to r e c e i v e a l l b e n e f i t s . Another way to look at the problem of a b e n e f i c i a r y i s to con s i d e r s o c i e t y or a community as an e n t i t y ( i n s t e a d of a conglomerate) and hence c o s t s and b e n e f i t s are considered with respect to one body i n s t e a d of spreading them among i n d i v i d u a l s . T h i s avoids the controversy of who a c t u a l l y bears the cost and who r e c e i v e s the b e n e f i t s . The b e n e f i t s were measured i n terms of economic values -35 36 t h a t i s , the i n c r e a s e i n p r o f i t s obtained by d i r e c t use of the u a t e r . I n d i r e c t b e n e f i t s to the farmer such as enhanced land values and e f f i c i e n t u t i l i z a t i o n of resources uere not considered at t h i s j u n c t u r e . Nonetheless i n d i r e c t b e n e f i t s can be i n c o r p -orated i n the u t i l i t y value a farmer attaches to the investment a l t e r n a t i v e s . T h i s aspect, of d e c i s i o n making i s considered i n Chapter 5. 4.2. D e r i v a t i o n of B e n e f i t s The b a s i c equation f o r determining expected monetary b e n e f i t i s : -Expected B e n e f i t = Expected Revenue - Expected Cost Hence knouing the c o s t - c a p a c i t y r e l a t i o n s h i p s , the stream f l o u and the crop y i e l d , the optimum c a p a c i t y of the p r o j e c t i s one uhich produces the maximum net value*.' Costs are p r o p o r t i o n a l to the land under i r r i g a t i o n and the r e s e r v o i r volume; each co s t item has a p r o b a b i l i t y ( u n c e r t a i n t y ) a s s o c i a t e d u i t h i t . Revenues are p r o p o r t i o n a l to the l e v e l of development ( i . e . R e s e r v o i r and l a n d ) , the stream f l o u , the consumptive use, the crop y i e l d and the p r i c e ; a l s o each q u a n t i t y has p r o b a b i l i t y ( u n c e r t a i n t y ) a s s o c i a t e d u i t h i t . Since there i s a uhole range of p o s s i b l e combinations of r e s e r v o i r s i z e s u i t h land under i r r i g a t i o n , each combination y i e l d s a unique set of b e n e f i t s . I t i s e a s i e r to f o l l o u the sequence of a c t i o n s to be taken by examining F i g . 3.3. The c a c u l a t i o n s are i l l u s t r a t e d belou and are shoun i n d e t a i l i n computer Program B-1 i n the Appendix. 1. Choose land area to i r r i g a t e , A^ ,, and r e s e r v o i r s i z e R m. 37 T o t a l C o s t = A. ( c , f j ) + R ( c , 6 ) = T C ( c , e ) u h e r e c = c o s t Q = p r o b a b i l i t y t h a t t r u e c o s t i s c E x p e c t e d C o s t = c x 6; c a n d 8 a r e d e t e r m i n e d a s i n T a b l e 3.2 ( b ) . T o t a l E x p e c t e d C o s t = 2 c S 2. F o r f l o u Q. u i t h p r o b a b i l i t y p ( Q j ) t h e r e i s a s e t o f c r o p y i e l d s p o s s i b l e , Y^, u i t h p r o b a b i l i t y P ( Y ^ ) , u h e r e i = 1 , 2 , . . . . n 3. R e v e n u e = P x A x Y^ P = p r i c e , A = A r e a 4. N e t B e n e f i t s c o r r e s p o n d i n g t o t h e y i e l d s B 1 = P A Y 1 - C B 2 . P A Y 2 - C B. = PAY. - C i i B = PAY - C n n 5. E x p e c t e d B e n e f i t a t N o d e 1 E ( B . ) = P ( Y . ) ( P A Y : . - C ) 6. T o t a l E x p e c t e d B e n e f i t a t N o d e 1 n = £ P ( - Y i ) ( P A Y . - C) i 7 . E x p e c t e d E x p e c t e d B e n e f i t g i v e n f l o u Q . n J » P ( Q j ) 2 P ( Y i ) ( P A Y . - C ) 1 8. T o t a l E x p e c t e d E x p e c t e d B e n e f i t f o r t h e c h o s e n a l t e r n a t i v e 1 n E E ( B ) = 2 P ( Q j ) 2 P C Y ^ C P A Y . - C) j i T h i s i s t h e e x p e c t e d e x p e c t e d b e n e f i t f o r o n e c h o i c e o f L a n d a n d R e s e r v o i r s i z e . S e v e r a l c o m b i n a t i o n s a r e t r i e d a n d t h e 38 maximum o f t h e s e v a l u e s g i v e s t h e o p t i m u m d e s i g n c o n d i t i o n s s u b j e c t t o u n c e r t a i n t y i n c o s t s , y i e l d , a n d s t r e a m f l o u , 4 . 3 , C o m p u t e d v a l u e s O f M o n e t a r y B e n e f i t s . T h e r e s u l t s o f t h e c o m p u t a t i o n s a r e s u m m a r i s e d i n T a b l e 4 , 1 . T h e maximum e x p e c t e d b e n e f i t s v a r y u i t h t h e v a l u e o f t o t a l u a t e r r e q u i r e m e n t ; t h e l o u e s t u a t e r d e m a n d y i e l d s t h e maximum b e n e f i t s f r o m t h e s y s t e m . T h e v a l u e s o f maximum b e n e f i t o b t a i n a b l e f r o m e a c h i r r i g a t e d a r e a i s s h o w n i n F i g , 4 , 2 . T h e maximum b e n e f i t p r o g r e s s i v e l y d e c r e a s e s u i t h a n i n c r e a s e i n u a t e r r e q u i r e m e n t . T h e o p t i m u m r e s e r v o i r v o l u m e c o r r e s p o n d i n g t o e a c h a r e a i s d i r e c t l y p r o p o r t i o n a l t o t h e t o t a l u a t e r r e q u i r e -m e n t f o r s m a l l e r a c r e a g e s , s e e T a b l e 4 . 2 , h o u e v e r f o r l a r g e r a c r e a g e s i t s e e m s e c o n o m i c a l l y b e t t e r t o p r o v i d e s m a l l e r r e s e r v o i r s r a t h e r t h a n t o h a v e t h o s e l a r g e e n o u g h t o p r o v i d e a l l t h e u a t e r . A s a n e x a m p l e , f o r a n a r e a o f 7 0 0 a c r e s , u i t h t o t a l u a t e r r e q u i r e m e n t o f 3.0 f e e t / a c r e , i t i s m o r e p r o f i t a b l e t o p r o v i d e a s m a l l e r r e s e r v o i r ( 1 8 0 0 a c r e - f e e t ) t h a n a l a r g e r o n e ( 2 1 0 0 a c r e - f e e t ) t h a t may s e e m a p p r o p r i a t e . 4 . 4 . E f f e c t o f U n c e r t a i n t y i n I r r i g a t i o n U a t e r R e q u i r e m e n t I n t h e p r e v i o u s a n a l y s i s d i f f e r e n t v a l u e s o f u a t e r r e q u i r e m e n t u e r e u s e d t o d e m o n s t r a t e h o u t h e e x p e c t e d b e n e f i t s a r e a f f e c t e d b y t h e u a t e r r e q u i r e m e n t . H o w e v e r i t i s b e t t e r t o t a k e i n t o a c c o u n t t h e u n c e r t a i n t y a s s o c i a t e d w i t h t h e w a t e r r e q u i r e m e n t . T h e u n c e r t a i n t y a b o u t t h e t r u e v a l u e o f w a t e r r e -q u i r e m e n t i s r e p r e s e n t e d b y a s s o c i a t i n g ( s u b j e c t i v e l y ) a p r o b a -b i l i t y t o e a c h v a l u e a s s h o w n i n T a b l e 4 . 3 . 39 The d e c i s i o n sequence i s shoun i n F i g . 4.1, and the t o t a l expected expected b e n e f i t i s K i n 2 P ( l k ) £ P(Qj) 2 P(Y.)(PAY i - C) The r e s u l t s are summarised i n Table 4.4 and the optimum design i s I r r i g a t e d Land = 400 acres Optimum R e s e r v o i r = 1200 a c r e - f e e t Maximum B e n e f i t = $65,600 The r e s u l t s summarised i n Table 4.4 uere p l o t t e d on the same f i g u r e i n F i g . 4.2 - shoun as " I n t e g r a t e d Curve". Comparison of the b e n e f i t curve f o r 3.0 f e e t / y e a r uater requirement and that of the i n t e g r a t e d curve shous that u h i l e the optimum s i z e of the p r o j e c t i s u n a f f e c t e d ( i . e . i r r i g a t e d land = 400 a c r e s , r e s e r v o i r s i z e = 1200 a c r e s ) by a ccounting f o r u n c e r t a i n t y i n uater requirement, the expected b e n e f i t i s reduced by 10%, TABLE 4.1 OPTIMUM DESIGN CONDITIONS ANNUAL IRRIGATION REQUIREMENT f e e t / y e a r OPTIMUM ACREAGE UNDER IRRIGATION acres OPTIMUM RESERVOIR SIZE a c r e - f e e t MAXIMUM EXPECTED BENEFITS $ 2.0 600 1200 107,400 2.5 500 1200 78,300 3.0 400 1200 72,900 3.5 300 1200 54,900 4.0 300 1200 51,900 Optimum R e s e r v o i r S i z e = 1200 a c r e - f e e t Optimum Acreage = 600 acres Optimum Uater R e q u i r e m e n t s 2.0 f e e t / y e a r Maximum Expected b e n e f i t s = $107,400 40 TABLE 4.2 INTERMEDIATE VALUES OF OPTIMUM CONDITIONS ON BASIS OF DOLLAR BENEFITS, TOTAL IRRIGATION REQUIREMENT = 3 FT/YR I irrigated Land Acres Optimum Re s e r v o i r S i z e Acre-Feet Maximum Expected D o l l a r B e n e f i t s * 100 300 20,400 200 600 42,300 300 900 62,500 400 1200 72,900 500 1500 64,200 600 1800 39,600 700 1800 - 5,000 800 1800 -48,300 900 21 00 -101 ,800 1000 1800 -149,300 * Taken to nearest |100 41 TABLE 4.3 UNCERTAINTY IN UATER REQUIREMENT T o t a l Uater P r o b a b i l i t y Requirement Assessment Feet/acre per year 2.0 0.05 2.5 0.10 3.0 0.60 3.5 0.20 4.0 0.05 42 TABLE 4.4 INTERMEDIATE VALUES OF OPTIMUM CONDITIONS, INTEGRATED IRRIGATION UATER REQUIREMENT. I r r i g a t e d L a nd Optimum R e s e r v o i r S i z e Maximum E x p e c t e d B e n e f i t s * A c r e s A c r e - F e e t $ 100 300 1 8300 200 900 39200 300 1200 57100 400 1200 65600 500 1500 57000 600 1800 32800 700 1800 - 6400 800 1800 - 48300 900 1800 -100000 1000 1 800 -147200 * N e a r e s t S100 43 Land Reservoir Water Requirement _ 3 . 0 _ 0 . 6 0 "O Flow Yield PCYjH V Benefit PCYnD B n FIG. 4. I DECISION T R E E = UNCERTAINTY IN WATER REQUIREMENT J> 45 FIG.4.2 EXPECTED ECONOMIC BENEFIT FROM IRRIGATED AREA .. C h a p t e r 5 D E C I S I O N C R I T E R I O N B A S E D ON; U T I L I T Y V A L U E 5.1 I n t r o d u c t i o n o f U t i l i t y V a l u e I n t h e p r e v i o u s C h a p t e r t h e b e n e f i t a c c r u i n g f r o m a n i r r i g a t i o n s y s t e m u e r e m e a s u r e d i n t e r m s o f m o n e t a r y v a l u e s . I t u a s a s s u m e d , f o r e x a m p l e , t h a t a s l o n g , a s t h e r e u a s e x c e s s r e v e n u e o v e r c o s t , t h e i n v e s t m e n t u a s j u s t i f i e d a n d i t u a s b e n e -f i c i a l . T h i s e c o n o m i c c o n s i d e r a t i o n h i d e s t h e u n d e r l y i n g p r i n c i p l e t h a t b e n e f i t s a n d c o s t s a r e d e r i v a t i v e s o f s o c i a l a n d c u l t u r a l v a l u e s o f a p a r t i c u l a r s o c i e t y - i n o t h e r u o r d s , t h e t r u e u o r t h o f b e n e f i t s a n d c o s t s v a r i e s f r o m o n e g r o u p o f p e o p l e t o a n o t h e r . I n a s m u c h a s u a t e r r e s o u r c e s d e v e l o p m e n t i s a m e a n s o f e n h a n c i n g s o c i a l u e l f a r e , t h e n t h e s o c i a l v a l u e s a n d a t t i t u d e s a s o c i e t y h o l d s r e g a r d i n g d e s i r a b l e d e v e l o p m e n t m u s t b e c o n s i d e r e d i n d e c i s i o n m a k i n g . T h e q u e s t i o n u h e t h e r a h i g h e r b e n e f i t - c o s t r a t i o i n d i c a t e s a g o o d p r o j e c t c a n o n l y b e e x -a m i n e d i n t h e c o n t e x t o f a u n i q u e s e t o f s o c i a l a n d c u l t u r a l c o n d i t i o n s . F o r a f a r m e r , i n t h e w e s t e r n s o c i e t i e s , c o n c e r n e d u i t h p a y m e n t o f l o a n s , i n t e r e s t , t a x e s a n d d e p r e c i a t i o n o f m a c h i n e r y , a h i g h e r b e n e f i t - c o s t r a t i o i s n o t o n l y d e s i r a b l e , i t i s a n e c e s s i t y . F o r a r u r a l v i l l a g e r , i n a d e v e l o p i n g c o u n t r y u h e r e f a r m i n g i s l a b o u r i n t e n s i v e , t h e a c t u a l c o s t i s a m i n o r c o n s i -d e r a t i o n c o m p a r e d t o h i s d e s i r e t o p r o d u c e m o r e f o o d f o r t h e i m m e d i a t e f a m i l y . 46 47 These t u o j p o i n t s of v i e u serve to emphasize the need to t i e the s c a l e of c o s t s and b e n e f i t s to some form of i n t r i n s i c values to the people f o r uhich the investment i s made. The de-gree to uhich these values are i n c o r p o r a t e d i n d e c i s i o n making i s a determining f a c t o r of the acceptance of the p r o j e c t . The " t r u e " value of a p r o j e c t v a r i e s from one i n d i v i d u a l to another even i n one c u l t u r a l group. It u i l l a l s o vary u i t h time and circumstances uhen the p r o j e c t i s undertaken. For most s i t u a t i o n s an ' investment i s considered u o r t h u h i l e i f i t s y i e l d i s above a t h r e s h o l d v a l u e . T h i s value i s i n f l u e n c e d by s e v e r a l f a c t o r s such as r e l a t i v e u e a l t h of the i n v e s t o r , the t i m i n g of the investment, the degree of r i s k ( u n d e s i r a b l e consequence) and the a t t i t u d e of the i n v e s t o r touards r i s k . It u i l l a l s o depend on hou the i n v e s t o r vieus h i s r o l e to s o c i e t y - i . e . uhether - he i s more concerned u i t h d i r e c t personal g a i n , or u i t h the value of investment to h i s s o c i e t y . These f a c t o r s are important and i t i s proper that the measure of b e n e f i t s and c o s t s should be an assessment that takes such f a c t o r s i n t o c o n s i d e r a t i o n . The complex i n t e r e l a t i o n s h i p s and values i n v o l v e d can only be assessed i n a s u b j e c t i v e manner. Thus i t i s necessary to r e l a t e the d e s i r e d b e n e f i t s to another s c a l e that measures the s u b j e c t i v e value that a d e c i s i o n maker att a c h e s to the d i f f e r e n t l e v e l s of investment. A " u t i l i t y v a l u e " or i n t r i n s i c uorth provides such f a c i l i t y ( H a l t e r and Dean 1971). The numerical assessment of u t i l i t y value of each outcome i s done v i a a u t i l i t y f u n c t i o n . 5.2. Expected U t i l i t y D e c i s i o n s Uhen a d e c i s i o n maker p r e f e r s one outcome to another 48 he has, i m p l i c i t l y or e x p l i c i t l y , attached a u t i l i t y value to each outcome. U s u a l l y the u t i l i t y value i s r e l a t i v e to other d e s i r a b l e and u n d e s i r a b l e outcomes. The concept of the u t i l i t y f u n c t i o n and of r a t i o n a l choice i s based on axioms of coherence (Luce and Raiffa,1957) uhich can be summarised as f o l l o u s : -1. Faced u i t h tuo a l t e r n a t i v e s A and B a d e c i s i o n maker e i t h e r p r e f e r s A to B, B to A,or i s i n d i f f e r e n t . 2. I f A i s p r e f e r r e d to B and B to C then A i s p r e f e r r e d to C. 3. I f a d e c i s i o n maker i s i n d i f f e r e n t betueen a l t e r n a t i v e s A and B, they can be s u b s t i t u t e d f o r each other. 4. An a l t e r n a t i v e D uhich has a set of p o s s i b l e outcomes , D 2 and u i t h p r o b a b i l i t i e s p^, p 2 and p^ r e s p e c t -i v e l y i s e q u i v a l e n t to the sum of t h e i r expected v a l u e s . U(D) = P 1U(D 1) + p 2U(D 2) + P 3U(D 3) U(D),U(D / |) etc are u t i l i t y v a l u e s . 5. I f tuo a l t e r n a t i v e s lead to the same consequence the a l t e r n a t i v e i n uhich the p r e f e r r e d outcome has the g r e a t e r p r o b a b i l i t y u i l l be chosen. 6. I f A i s p r e f e r r e d to B and B to C then there must be some set of odds on A and C such that a d e c i s i o n maker i s i n d i f f e r e n t betueen choosing B u i t h c e r t a i n t y and a " l o t t e r y " u i t h odds on A and C. i . e . U(B) = pU(A) + ( l - p ) u ( C ) uhere U(B), U(A), U(C) are u t i l i t y / v a l u e s of A, B and C. p = p r o b a b i l i t y of A 49 H o w e v e r t h e r e a r e t u o b a s i c p r o b l e m s i n t h e p r o c e s s : h o u t o a s s e -s s u t i l i t y v a l u e ( i . e . t o d e t e r m i n e u t i l i t y f u n c t i o n ) , a n d u h o s e d e c i s i o n v a l u e t o u s e i n a d e c i s i o n s i t u a t i o n - t h e u t i l i t y v a l u e o f o n e i n d i v i d u a l d o e s n o t n e c e s s a r i l y r e p r e s e n t t h e u t i l i t i e s o f a l l p e o p l e u h o u i l l b e a f f e c t e d b y a p a r t i c u l a r o u t c o m e . A s s e s s i n g u t i l i t y v a l u e i s a s u b j e c t i v e p r o c e s s i n u h i c h t h e d e c i s i o n m a k e r i s f o r c e d t o c o n s i d e r e x p l i c i t l y t h e f a c t o r s i n v o l v e d i n a p r o b l e m a n d t h e c o n s e q u e n c e s o f t h e d e c i s i o n . n T j i i s i n d e p t h a n a l y s i s o f a s i t u a t i o n i s a n i m p o r t a n t i n g r e d i e n t o f t h e o v e r a l l d e c i s i o n p r o c e s s . O n c e t h e u t i l i t i e s h a v e b e e n a s s i g n e d t o o u t c o m e s i n t h e a b o v e m a n n e r t h e n m a x i m i z i n g e x p e c t e d u t i l i t y u i l l b e c o n s i s t e n t u i t h t h e d e c i s i o n m a k e r ' s p r e f e r e n c e ; t h e r e f o r e i t i s a v a l i d c r i t e r i o n f o r c h o o s i n g b e t u e e n c o m p e t i n g a l t e r n a t i v e s ( B e n j a m i n a n d C o r n e l l 1 9 7 0 ) . 5 . 3 . D e r i v a t i o n o f U t i l i t y F u n c t i o n f o r I r r i g a t i o n I n v e s t m e n t T h e u t i l i t y f u n c t i o n o f a f a r m e r u a s u s e d t o m e a s u r e t h e b e n e f i t s f r o m a n i r r i g a t i o n s y s t e m . T h e d e r i v a t i o n o f t h e u t i l i t y f u n c t i o n u a s b a s e d o n t h e f o l l o u i n g c o n d i t i o n s . 1. O b j e c t i v e : T o m a x i m i z e t h e c r o p y i e l d a n d t h e n e t e c o n o m i c b e n e f i t f r o m t h e i r r i g a t i o n s y s t e m . 2. A s s u m p t i o n s : ( i ) T h e i n v e s t m e n t i s u n d e r t a k e n s o l e l y b y t h e f a r m e r , ( i i ) T h e f a r m e r g r o u s o n l y o n e c r o p , ( i i i ) T h e f a r m e r h a s t o m a k e a m i n i m u m r e t u r n n e c e s s a r y t o s u p p o r t h i m s e l f - m e e t " h i s f i n a n c i a l o b l i g a t i o n s , c o s t 50 of p r o d u c t i o n , and f o r the welfare of h i s f a m i l y . Below t h i s minimum r e t u r n on investment th;e farmer w i l l begin to face h a r d s h i p s . The most f a v o u r a b l e outcome to the farmer i s an annual bumper y i e l d , and the worst consequence i s when the y i e l d i s zero. Hence assuming t h a t the maximum a v a i l a b l e acreage i s 1000 acres and the maximum y i e l d i s 6.0 Tons/acre Most f a v o u r a b l e outcome = 1000 x 6 = 6000 Tons Maximum revenue = 6000 x 100 = $600,000 Least f a v o u r a b l e outcome = 1000 x 0 = 0 Tons Minimum Revenue = 0 x 100 = $0 While the l i k e l i h o o d of the above outcomes i s very very s m a l l , they serve to h i g h l i g h t the l i m i t i n g c o n d i t i o n s . Obviously the outcome which y i e l d s $600,000 has the highest u t i l i t y v alue. The farmer's expected u t i l i t y value f o r any d e c i s i o n w i l l vary between these two b a s i c a l t e r n a t i v e s depending on the p r o b a b i l i t i e s and u t i l i t y value of the outcome. Thus i t i s p o s s i b l e to a t t a c h an expected u t i l i t y value to each branch of the d e c i s i o n t r e e i n F i g . 5.2. The optimum d e c i s i o n choice i s the one uhich y i e l d s the maximum expected u t i l i t y . 5.4 U t i l i t y F u n ction A s i m p l i f i e d u t i l i t y f u n c t i o n of the farmer i s shown i n F i g 5.1. The f u n c t i o n i s assumed to be a f u n c t i o n of net economic b e n e f i t . T h i s u t i l i t y f u n c t i o n w i l l vary from one d e c i s i o n maker to another, i t w i l l a l s o vary with d i f f e r e n t circumstances surrounding the d e c i s i o n . The a c t u a l shape i s not 52 c r i t i c a l to the methodology presented here. The general p r i n c i p l e d e p i c t e d i n F i g . 5.1 i s t h a t there i s a minimum u t i l i t y value belou uhich the u t i l i t y i s d i m i n i s h i n g more r a p i d l y than the expected v a l u e , and above uhich the farmer's u t i l i t y i s synonmous u i t h monetary value. The value of B corresponds to the minimum r e t u r n the farmer needs to c a r r y on h i s o p e r a t i o n . The slope S 2 of the u t i l i t y f u n c t i o n r e p r e s e n t s the farmer's circumstances and a t t i t u d e touards r i s k . If the r e t u r n i s Bg the farmer has to use h i s reserve of u e a l t h or borrou money i n o r d e r to meet hi s o p e r a t i n g expenses. This u t i l i t y f u n c t i o n can be l i k e n e d to that of a d e c i s i o n maker r e s p o n s i b l e f o r food production i n a developing country. An a g r i c u t u r a l p r o j e c t should not only produce enough food but i t must a l s o be economically v i a b l e i n o r d e r to pay back loans on c a p i t a l . Hence the u t i l i t y value i s a f u n c t i o n of net economic b e n e f i t . The value of U c represents the u t i l i t y attached to the minimum comfortable l e v e l of food p r o d u c t i o n . Food prod-u c t i o n l e v e l s belou B cause i n c r e a s i n g hardships i n the s o c i e t y , c and f o r values belou B = 0 the s i t u a t i o n i s hopeless - there i s l e s s food and not enough money to s u s t a i n loan payments. Hence the u t i l i t y corresponding to values l e s s than B = 0 have very high negative v a l u e s . It i s reasonable to assume that the 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 uould be improved by some form of c h a r i t a b l e guarantee or support i n case of bad harvest, f o r example food r e l i e f a i d from the United Nations O r g a n i s a t i o n . 5.5 Computation of Expected U t i l i t y " Reference to F i g . 5.2 i n d i c a t e s that there i s a u t i l i t y 53 value a s s o c i a t e d u i t h each d e c i s i o n branch. These u t i l i t i e s uere c a l c u l a t e d by a s s o c i a t i n g each net b e n e f i t value u i t h a u t i l i t y v i a the u t i l i t y f u n c t i o n i n F i g 5.1. Net b e n e f i t s uere c a l c u l a t e d i n the same manner as i n Chapter 4. Thus each value of y i e l d and consequent b e n e f i t has a unique u t i l i t y v a l u e . The d e c i s i o n procedure then uas to c a l c u l a t e the expected u t i l i t y at each d e c i s i o n a l t e r n a t i v e . For f l o u Q. u i t h p r o b a b i l i t y P(Q,)» expected u t i l i t y at Node 2. n EU Jj = P ( Q j ) 2 P(Y.)U(Y.) 1 L/here U(Y^) « u t i l i t y a s s o c i a t e d u i t h each y i e l d o b t a i n a b l e . u i t h f l o u Qj P ( Y i ) = P r o b a b i l i t y of y i e l d Y i For a given choice of land and r e s e r v o i r , the expected expected u t i l i t y m n EEU = 2 P ( Q j ) 1 P(Y.)U(Y.) j i Taking i n t o account the u n c e r t a i n t y i n i r r i g a t i o n uater r e q u i r e -ment, then K m n EEU = £ P d k ) 2 P(QJ-) 2 P t Y . M Y . ) k j i The land and r e s e r v o i r s i z e s a s s o c i a t e d u i t h the maximum expected expected u t i l i t y are the optimum design v a l u e s . The uorking procedure i s given i n d e t a i l i n Program B-2 i n the Appendix. 5,6. Values of Expected U t i l i t y The maximum values of expected expected u t i l i t y f o r d i f f e r e n t demand l e v e l s of u a t e r requirement are shoun i n Table 5.1. The u t i l i t i e s vary u i t h the u t i l i t y f u n c t i o n , hence the 54 computed values are shoun f o r 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 . F i g 5.3, 5.4 and 5.5 shou t h i s v a r i a t i o n g r a p h i c a l l y . The i n -t e g r a t e d uater demand curve i s a l s o superimposed on each graph. Tables 5.2 and 5.3 shou a summary of i n t e r m e d i a t e values of maxi-mum u t i l i t y both f o r a t o t a l uater requirement of 3.0 f e e t / y e a r and f o r the c o n d i t i o n uhen u n c e r t a i n t y i n uater demand i s taken i n t o c o n s i d e r a t i o n . The d e c i s i o n maker u i l l u s u a l l y a s s i g n very high d i s -u t i l i t y values to those options uhich y i e l d negative b e n e f i t s . In t h i s a n a l y s i s i t uas assumed that f o r a l l negative net b e n e f i t s the u t i l i t y i s - 600,000 uhatever the shape of the f u n c t i o n . Note that because of the preponderance of high d i s -u t i l i t y values u i t h s t e e p l y s l o p i n g u t i l i t y f u n c t i o n s the expected u t i l i t y values are p r o g r e s s i v e l y l o u e r going doun Table 5.1. In p a r t i c u l a r the high d i s u t i l i t y value of • -600,000 ueighed doun h e a v i l y on the expected v a l u e s . Uhen the u t i l i t y f u n c t i o n i s synonmous u i t h monetary b e n e f i t , the expected b e n e f i t i s the same as i n Table 4.1, Chapter 4. TABLE 5.1 Maximum Expected U t i l i t i e s Slope S 2 I r r i g a t - Optimum of i o n Uater I r r i g a t e d U t i l i t y Require- Area Function ment FT/YR Acres Optimum Re s e r v o i r S i z e Acre-Feet Maximum Expected Expected U t i l i t y 1 2.0 600 2.5 500 3.0 400 3.5 300 4.0 300 1200 107,400 1200 78,300 1200 72,900 1200 54,900 1200 51,900 2 2.0 400 2.5 400 3.0 300 3.5 300 4.0 200 900 79,200 1200 60,900 900 50,000 1200 28,500 900 12,300 2.0 400 900 78,400 2.5 400 1200 59,300 3.0 300 900 46,200 3.5 300 1200 22,000 4.0 300 1200 400 5 2.0 400 900 76,700 2.5 400 1200 56,100 3.0 300 900 38,700 3.5 300 1200 8,800 4.0 • 300 1 200 -13,700 55 Table 5.J INTERMEDIATE VALUES OF MAXIMUM EXPECTED UTILITY m , T«P n WATER REQUIREMENT = 3 . 0 f e e t / y e a r 0 " M " ™*"»»> "RIOATION Ir r i g a t e d Land Acres 100 200 300 400 500 600 700 800 900 1000 Optimum" Reservoir Acre-feet 300 600 900 1200 1500 1500 1800 2100 2100 2400 Nearest 100 Expectec U t i l i t y -19 ,500 23,100 49,900 42,200 -22,100 -120,700 -260,200 -^01,500 -490,700 -557,200 Optimum Reservoijl Acre-300" 600 900 1200 1500 1500 1800 2100 2100 2400 Expected U t i l i t y * fee* -597300" 5,700 46,200 38,900 -26,900 -127.300 - 2 6 8 , 0 0 0 -fK>5,*>0 -495,400 - 5 6 0 , 0 0 0 Optimum Reservoir Acre-feet 300 600 900 1200 1500 1500 1800 2100 2100 2400 Expected U t i l i t y * - 29,200 38,700 32,200 - 36,500 -140,400 -283,700 -413,600 -504,700 -565,900 Table 5.3 INTERMEDIATE VALUES OF EXPECTED UTILITY, INTEGRATED IRRIGATION WATER REQUIREMENT. Irr i g a t e d Land U t i l i t y Function with slope S 2 = 2 Optimum Reservoir Acres Acre 100 300 200 900 300 900 400 1200 500 1500 600 1800 700 1800 800 2100 900 2100 1000 2400 Expected U t i l i t y * - 24,000 15,500 37,600 27,700 - 35,600 -152,200 -268,300 -408,200 -487,300 -544,900 Slope S 2 a 3 Optimum Reservoir 300 900 1200 1200 1500 1800 1800 2100 2100 2400 Expected U t i l i t y * - 65,700 - 4,900 31,500 22,100 - 42,100 -157,700 -276,400 -411,900 -490,700 -547,300 Slope S 2 ss 5 Optimum Reservoir 300 900 1200 1200 1500 1800 1800 2100 2100 2400 Expected U t i l i t y * -148,900 - 45,600 20,300 10,700 - 55,100 -168,800 -292,800 -419,400 -497,600 •552,200 * Nearest 100 © © Choose Land Choose Reservoir vjo e Natural Stream Flow 0\ Crop Yield Yield Y, •PCY,D PCYj 3 V PCYnD Utility U(Y,) U(Yj) U (Yn ) CJl CD FIG. 5.2 DECISION T R E E WITH UTILITY V A L U E S 59 0 100 200 300 400 500 600 700 800 '900 1000 I r r i g a t e d L a n d i n A c r e s FIG.5. 3 E X P E C T E D UTILITY AS FUNCTION OF IRRIGATED A R E A . 60 FIG.5.4 E X P E C T E D UTILITY AS FUNCTION OF IRRIGATED A R E A . 6T UTILITY FUNCTION SLOPE S 2 =5 100 200 300 400 500 600 700 800 9 0 0 1000 Irrigated Land in Acres F I G . 5 . 5 E X P E C T E D UTIL ITY A S FUNCT ION OF IRRIGATED A R E A . / Chapter 6 THE EFFECT, OF UNCERTAINTY ON OPTIMAL DESIGN 6.1 I n t r o d u c t i o n U h i l e the value of observed f a c t u a l i n f o r m a t i o n i s un-d e n i a b l e only time can provide the lengths of records r e q u i r e d f o r adequate p r o b a b i l i t y a n a l y s i s of i n f o r m a t i o n necessary f o r planning and design of water resources systems. S t a r t i n g from a s t a t e of no I n f o r m a t i o n , as more data becomes a v a i l a b l e the g r e a t e r the c e r t a i n t y ue can a t t a c h to the gathered i n f o r m a t i o n . The more data needed the higher the costs of o b t a i n i n g the data. At a c e r t a i n p o i n t i n time ue have to ask o u r s e l v e s the q u e s t i o n : "What i s the value of b e t t e r i n f o r m a t i o n ? " or "Uhat i s the pote-n t i a l b e n e f i t f o r reducing u n c e r t a i n t y ? " T h i s means that no data g a t h e r i n g process should be considered i f i t c o s t s more than the p o t e n t i a l savings gained from use of b e t t e r i n f o r m a t i o n . I t i s necessary at t h i s p o i n t to d e f i n e uhat i s meant by "Value of P e r f e c t Information." I t i s d e f i n e d to be "the d i f f e r e n c e betueen (1) the p r i o r e x p e c t a t i o n 2U (e^) P(8^), i n uhiph U(fJ\) i s the u t i l i t y a s s o c i a t e d u i t h the best choice of a c t i o n given that i s knoun u i t h c e r t a i n t y that tK i s the t r u e s t a t e , and (2) the expected u t i l i t y a s s o c i a t e d u i t h no e x p e r i -ment" (Benjamin and C o r n e l l , 1970). There are s e v e r a l methods a v a i l a b l e to a d e c i s i o n maker f o r o b t a i n i n g b e t t e r i n f o r m a t i o n : o b s e r v a t i o n over long periods of time; f i e l d experiments, e.g. c o n t r o l l e d experimentation i n 62 63 an a g r i c u l t u r a l s t a t i o n may y i e l d b e t t e r estimates of y i e l d of a crop; computer m o d e l l i n g , e.g. s i m u l a t i o n of n a t u r a l stream f l o u . Whichever method i s adopted, the r e s u l t i n g data u i l l have some degree of u n c e r t a i n t y a s s o c i a t e d u i t h i t . Hence before a d e c i s i o n can be made, i t i s necessary to update the p r i o r p r o b a b i l i t y " o f each piece of i n f o r m a t i o n i n the l i g h t of the neuly a v a i l a b l e i n f o r m a t i o n , 6.2. R e v i s i o n of P r o b a b i l i t i e s The process of combining neu i n f o r m a t i o n u i t h p r i o r p r o b a b i l i t i e s to d e r i v e p o s t e r i o r p r o b a b i l i t i e s i s done through Bayes' Theorem: P(Bj|A) = P(A|B .)P(B .) f P ( A | B i ) P ( B i ) i Uhere B^, i = 1,2, n i s a set of mutually e x c l u s i v e and c o l l e c t i v e l y exhaustive events, A i s a set of events t h a t i s a l s o d e f i n e d over the same sample space as B^. T h i s theorem can be a p p l i e d to y i e l d 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 of events given the e x i s t e n c e of experiments to estimate those events. The equation can then be u r i t t e n as P ( e i l Z k ) - P ( Z k l e j ) P < e i ) •2 P ( z k | « i ) P ( e i ) i Z i s the set of experimental outcomes Q i s the set of s t a t e s of nature ^ P (e^) i s p r i o r p r o b a b i l i t y of fIL 64 P ( Z ^ | ) i s sample l i k e l i h o o d of given the s t a t e 8^ 2 P(Z^ |tK )P(£K ) i s a n o r m a l i z i n g f a c t o r . A d e c i s i o n a n a l y s i s i n uhich o n l y i t h e p r i o r p r o b a b i l i t i e s are used i s c a l l e d a p r i o r a n a l y s i s ; uhereas ; a t e r m i n a l a n a l y s i s uses the neu p o s t e r i o r p r o b a b i l i t y obtained by i n c o r p o r a t i n g neu i n f o r m a t i o n about the s t a t e s . The d e c i s i o n analyses done i n Chapter 4 and 5 are examples of p r i o r a n a l y s i s , they used p r i o r estimates and p r o b a b i l i t i e s . A p o s t e r i o r a n a l y s i s i n v o l v e s making a t e r m i n a l a n a l y s i s of each experimental outcome to o b t a i n the u t i l i t y a s s o c i a t e d u i t h each outcome. Consequently an experiment i s chosen i f i t s expected value of b e t t e r i n f orm-a t i o n i s more than the cost of o b t a i n i n g such i n f o r m a t i o n . In p r a c t i c e i t may not be necessary to apply Bayes' theorem r i g o r o u s l y i n o r d e r to d e r i v e p o s t e r i o r p r o b a b i l i t i e s . Besides, the i n t e r e l a t i o n s h i p s betueen events are u s u a l l y so complex t h a t i t i s d i f f i c u l t to d e r i v e c o n d i t i o n a l p r o b a b i l i t i e s . Thus the p o s t e r i o r p r o b a b i l i t i e s are d e r i v e d d i r e c t l y from neu (and a l s o cumulative) data t h a t i s a v a i l a b l e . Once the p o s t e r i o r p r o b a b i l i t y has been d e r i v e d , the d e c i s i o n a n a l y s i s proceeds i n a s i m i l a r manner as i n p r i o r a n a l y s i s . The same u t i l i t y f u n c t i o n as i n p r i o r a n a l y s i s can be used because the d e c i s i o n maker's preference i s not a l t e r e d by the p r o b a b i l i t y a s s i g n -ments to each pi e c e of i n f o r m a t i o n . 6.3 A p p l i c a t i o n to Problem of I r r i g a t i o n Development As more i n f o r m a t i o n accumulates and as more knouledge about the system grous the o v e r a l l u n c e r t a i n t y i n the input f u n c t i o n s u i l l be reduced. I f the c o s t - c a p a c i t y r e l a t i o n s h i p s 65 uere known u i t h c e r t a i n t y , the f u n c t i o n s could be represented by the "probable" curves only i n F i g . 2.4, 2.5 and 2.6. S i m i l a r l y i f the p h y s i c a l c o n d i t i o n s a f f e c t i n g the crop y i e l d could.be a s c e r t a i n e d the crop y i e l d obtained from use of any amount of uater could be knoun e x a c t l y . Longer and more r e l i a b l e stream f l o u records uould enable the d e r i v a t i o n of a more ac c u r a t e f l o u d i s t r i b u t i o n . The e f f e c t of u n c e r t a i n t y on the design d e c i s i o n uas i n v e s t i g a t e d by a n a l y s i n g the i r r i g a t i o n system u i t h data of v a r y i n g degrees of u n c e r t a i n t y . The main d e c i s i o n model uas run using the data represented i n Table 6.1. The u n c e r t a i n t y ihi the data uas .".removed" by using only the middle or "probable" curve of each f u n c t i o n i n s t e a d of uorking u i t h i n a set of l e v e l s bounded by the upper and l o u e r curves. Uhere stream f l o u uas assumed to be knoun u i t h c e r t a i n t y , the f l o u d i s t r i b u t i o n d e r i v e d i n s e c t i o n 3.3.2 uas used. The u n c e r t a i n t y i n stream f l o u uas accounted f o r as given i n S e c t i o n 6.3.1 belou. 6.3.1 Accounting f o r U n c e r t a i n t y i n Stream F l o u S t a t i s t i c a l model parameters d e r i v e d from h i s t o r i c r e -cords have a c o n s i d e r a b l e degree of u n c e r t a i n t y . According to Benson (i960) the frequency d i s t r i b u t i o n of 40 samples (draun from a f i n i t e p o p u l a t i o n of 1,000 f l o o d d i s c h a r g e s ) each of a 25 year s i z e of the maximum f l o o d discharges shoued a very.uide s c a t t e r e s p e c i a l l y f o r long (more than 200 years) r e t u r n p e r i o d s . T h i s s c a t t e r i n d i c a t e s the e r r o r i n e s t i m a t i n g p o p u l a t i o n cha-r a c t e r i s t i c s from a sample. The process u n c e r t a i n t y , s t a t i s t i c a l u n c e r t a i n t y and fundamental u n c e r t a i n t y are a l l q u i t e s i g n i f i c a n t TABLE 6.1 DATA USED IN INVESTIGATING THE EFFECT OF UNCERTAINTY Set Stream Number Flow I r r i g a t -ion Uater Require-ment Costs of Develop-ment Crop Y i e l d 1 C r j - J H f r C C 2 C C . C U 3 C c u C 4 C c u U 5 C u u U 6 U c u u C - Data assumed to be known with c e r t a i n t y U s U n c e r t a i n t y i n Data * A n a l y s i s done i n Chapter 4 and 5 ** I r r i g a t i o n Uater Requirement assumed to be 3.0 f e e t / y ear. 66 67 where short h i s t o r i c records are involved. The uncertainty about the true values can be incorporated by defining con-fidence l i m i t s on the model parameters and on the data, or by a subjective estimate of the l i m i t i n g values of the data. A frequency d i s t r i b u t i o n of Powers Creek annual flow uas obtained by f i t t i n g a Log Pearson Type III d i s t r i b u t i o n to ' the data. Confidence l i m i t s to the d i s t r i b u t i o n were then estimated s t a t i s t i c a l l y . A graphical estimation of the s t a t i s t i c a l procedure (Yevjevich, 1972) was used to put 90% confidence l i m i t s on the d i s t r i b u t i o n . The subjective estimate of l i m i t s of flou uas based on the assumption that the true value of maximum flous were 20 to 30% higher, and the minimum flous uere 30% lower than the h i s t o r i c record shous. These figures uere derived from analysis of the streamflou records of r i v e r basins of comparable size to Pouers Creek in the Okanagan Basin. In any p r a c t i c a l s i t u a t i o n the subjective estimates u i l l be based on the decision maker's knouledge of the area he i s uorking u i t h . Fig. 6.1 shows the frequency d i s t r i b u t i o n of Powers Creek annual flow uith s t a t i s t i c a l and subjective estimate of the uncertainty l i m i t s . Once uncertainty l i m i t s have been derived, the uncert-ainty about the true value of the flou at a given probability can be defined by a probability d i s t r i b u t i o n bounded by the l i m i t s of f l o u in Fig. 6.1. The probability d i s t r i b u t i o n uas assumed to be a "3keu normal" defined previously. A pr o b a b i l i t y matrix of stream f l o u , Fig.6.2, uas determined in the same manner out-lined in previous chapters. A probability density function can be derived from the 68 Qi Q> I o < 5 o o 3 C C < o o 2 0 0 , 0 0 0 0 0 , 0 0 0 5 0 , 0 0 0 10 ,000 5 , 0 0 0 I , 0 0 0 : I • I: I I I S t a t i s t i c a l Es t imate of 9 0 % Confidence limits Range of Flows at given probabi l i ty Sub jec t i ve E s t imate of Outer limits J I I 99.99 99.9 99 9 0 50 20 10 4 2 I 0.5 0.1 Percent Chance of O c c u r e n c e NOTE = Flow P r o b a b i l i t y A n a l y s i s log Pearson Type III Dist. FIG. 6.1 F R E Q U E N C Y DISTRIBUTION OF POWERS CREEK A N N U A L FLOW ; U N C E R T A I N T Y LIMITS FITTED S T A T I S T I C A L L Y AND S U B J E C T I V E L Y . 69 matrix of Fig 6.2. The probability of flou Q\^ . i s P(Qij) - S A P j a i j uhere APj = probability i n t e r v a l on horizontal axis a^j = probability that the true value of flou Q. i s Q. ., defined uith respect to p. The p r o b a b i l i t y density function of flou derived above uas used in the main decision model. Fig 6.3 shous a graphical represent-ation of the cumulative probability function of flou "obtained from s t a t i s t i c a l and subjective estimates of uncertainty l i m i t s . 6.4. Results Fig. 6,4. shous the decision tree for the situation uhen a l l input functions are knoun uith certainty. The t o t a l cost of the system i s nou a simple summation of the costs of the ind i v i d u a l components of the system; there i s only one value of crop y i e l d obtainable uith a given f l o u . Table 6.2 gives a summary of results obtained from analysing the system uith better information; and F i g . 6.5 i s a graphical representation of the results for a uater requirement of 3.0 feet/year. Fig 6.6 and 6.7 shou the values of expected economic benefit and expected u t i l i t y respectively for varying degrees of uncertainty in the input functions. A l l the results are summarised in Table 6.3. The results shou a d e f i n i t e increase in expected benefits uith better information. The potential benefit of reducing uncertainty can then be compared to the cost incurred in obtaining better information and a decision can be made of the r e l a t i v e importance of each input function. 70 Pj . 1.0 Probabil ity Flow Equalled or Exceeded Qj, Qj6 are possible flows at given proba bi lity Pj a ( a 6 probabilities that the true value of Qj is Qj, , Q i 2 ........Q i 6 FIG.6.2 PROBABILITY MATRIX OF STREAM FLOW . Table 6.2 MAXIMUM EXPECTED BENEFIT WITH BETTER INFORMATION. IRRIGATION WATER REQUIREMENT » 3.0 feet/year I r r i g a t e d Land Acres Expected Economic Benefit $ Expected U t i l i t y Slope S 2 = 2 Expected U t i l i t y Slope S 2 = 3 > Expected U t i l i t y Slope S 2 = 5 100 21,300 17,500 - 56,200 -133,700 200 44,400 28,300 12,800 - 18,200 300 65,000 58,300 56,600 53,400 400 76,800 5^,700 51,800 45,800 500 70,400 13,400 6,700 - $,900 600 45,200 - 72,000 - 80,200 - 96,400 700 4,000 -214,400 -241,700 -274,900 800 - 36 ,300 -367 ,500 -371,600 -379,800 900 -85,500 -457,500 -462,800 -473,400 1000 -131,300 -524,100 -531,000 -544,900 Table 6.3 SUMMARY OF RESULTS Uncertainty i n Data shown Decision C r i t e r i o n Optimum Area Acres f> Change Over Perfect Data Range of fo Change I r r i g a t e d Over Area Perfect Data Acres Expected Ben e f i t , D o l l a r s , and U t i l i t y Costs Crop Y i e l d Stream Flow U t i l i t y S 2 % Change Over Perfect Data I . 0 0 1 2 0 T o t a l Annual Flow in 1 0 0 0 Ac re - feet FIG.6.3 CUMULATIVE PROBABILITY OF ANNUAL FLOWS OF POWERS CREEK. Choose to Gather more Data Choose Land Choose Reservoir Volume R .Cost Natural Stream Flow Net Benefit and Uti I ity Value U(Y, "I U (Y2'). U (Yi) U(Y n ) -This branch of. the Tree is simi lar to that in FIG. 5.2 FIG'. 6,4" DECISION T R E E WITH BETTER INFORMATION 75 FIG.6.5 E X P E C T E D B E N E F I T S , D O L L A R S AND U T I L I T I E S - B E T T E R INFORMATION . 76 9 0 , 8 0 I i — i — r — j — r — Total Irrigation Water Requirement : 3 . 0 f t . / Y r . Uncertainty in Costs of Developement Uncertainty in Crop Yield Uncertainty in Costs , Yield,] Water Requirement Uncertainty in Costs,-Yield and Stream Flow 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 0 I O O O I r r i g a t e d L a n d i n A c r e s FIG.6.6 EXPECTED ECONOMIC BENEF IT , VARYING DEGREES OF UNCERTAINTY. 77 ~ r — n — i • i : i . — i — n — Total Irr igation Water Requirement =' 3.0ft ./Yr . Slope S 2 = 2 Slope S 2 - 5 Uncertainty in Costs of Development Uncertainty Crop Yield Uncertainty in Costs .Yield and Stream Flow J L 100 2 0 0 . 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1000 I r r i g t e d L a n d in A c r e s FIG. 6.7 E X P E C T E D UTILITY . V A R Y I N G D E G R E E S OF U N C E R T A I N T Y . Chapter 7 D I S C U S S I O N A N D C O N C L U S I O N S 7.1 D i s c u s s i o n of Method of A n a l y s i s The p r o b a b i l i t y model used to present u n c e r t a i n t y i n the data i s not n e c e s s a r i l y unique to the system; other models could have been used and the a n a l y s i s would have y i e l d e d s l i g h t l y d i f f e r e n t d e c i s i o n s . Uood and Rodriguez-Iturbe (1975) have shown t h a t d i f f e r e n t f l o o d frequency models lead to d i f f e r e n t b e n e f i t f u n c t i o n s . T h e r e f o r e the c o n c l u s i o n s given below should be viewed i n the l i g h t of the u n d e r l y i n g assumptions and the model used. The e f f e c t of model u n c e r t a i n t y on the d e c i s i o n was not i n v e s t i g a t e d i n t h i s a n a l y s i s . The manner of a t t a c h i n g u n c e r t a i n t y bounds to' the data was a s u b j e c t i v e process. U h i l e i t could be c r i t i c i s e d on s t a -t i s t i c a l grounds, i t i s , nonetheless, a p r a c t i c a l process; v a r i -ances are not always symme t r i c a l l y d i s t r i b u t e d about the mean, as i s the usual assumption i n s t a t i s t i c a l d e r i v a t i o n of confidence l i m i t s . Besides most data i n p r a c t i c e i s measured from a lower bound to an upper bound, and determining these l i m i t i n g values s t a t i s t i c a l l y from a short h i s t o r i c record i s not always easy. The s u b j e c t i v e u n c e r t a i n t y bounds allow the d e c i s i o n maker to pool together a l l the a v a i l a b l e i n f o r m a t i o n , be i t h i s t o r i c a l or s u b j e c t i v e . Expected values are long term averages of the f u n c t i o n s , t h e r e f o r e i t can be i n f e r r e d that the d e c i s i o n chosen i s the 78 79 optimum over a long time provided that the u n d e r l y i n g assump-t i o n s are v a l i d over the same time h o r i z o n . The u n c e r t a i n t y about f u t u r e c o n d i t i o n s i s s t i l l p resent. The method of a n a l y s i s does not i n d i c a t e the d e c i s i o n s i t u a t i o n i n the face of short term, t r e n d s , f o r example, a three year drought. I t i s p o s s i b l e to analyse f l o u sequences uhich can i n c l u d e such events, but t h i s uould have complicated the a n a l y s i s and uas not considered i n the present study. Uhen tuo f u n c t i o n s are combined together i n the manner o u t l i n e d i n Chapter 3, some accuracy can be l o s t i f there i s a l a r g e d i f f e r e n c e i n magnitudes of the d i s c r e t e i n t e r v a l s of the f u n c t i o n s . Therefore j u d i c i o u s choice of the d i s c r e t e i n t e r v a l s as u e l l as the s i z e of the m a t r i c e s - i s e s s e n t i a l i n order to preserve the d e s i r e d accuracy and a l s o to save computation time and c o s t s . 7.2 Comparison of Re s u l t s A l l the comparison of r e s u l t s , uhether of d e c i s i o n c r i -t e r i a , magnitude of b e n e f i t s , or optimum design c o n d i t i o n s uere made using 3.0^feet/year as the i r r i g a t i o n uater requirement. U i t h r e s p e c t to the r e s u l t s , the "optimum" s i z e or c a p a c i t y i s the s i z e of the system uhich y i e l d s the maximum r e t u r n s - given any p a r t i c u l a r d e c i s i o n c r i t e r i o n . The " t o t a l i r r i g a b l e area" r e f e r s to the area under i r r i g a t i o n uhich y i e l d s p o s i t i v e r e t u r n s . T h i s i s a measure of the t o t a l p r e f e r r e d area that can be developed at the p r e v a i l i n g c o n d i t i o n s . For both d e c i s i o n c r i t e r i a - maximizing expected economic 80 b e n e f i t , m a x i m i z i n g the expected u t i l i t y - the optimum d e c i s i o n uas s e n s i t i v e to the i r r i g a t i o n uater requirements. The r e s u l t s of Chapter 4 and 5 shoued t h a t l o u uater demands l e d to the high e s t expected r e t u r n s , and high uater demands produced the l o u e s t expected r e t u r n s . For example reducing the uater demand by 30% from 3.0 f e e t / y e a r to 2.0 f e e t / y e a r , the optimum i r r i g a t e d area i n c r e a s e d by 40% and the expected r e t u r n s uent up by 50% f o r both d e c i s i o n c r i t e r i a ; i n c r e a s i n g the u a t e r requirement by 15% to 3.5 f e e t / y e a r reduced the optimum i r r i g a t e d area and expected r e t u r n s by more than 20%. These f i g u r e s serve to i n d i c a t e the importance of accounting f o r u n c e r t a i n t y i n water requirements as u e l l as other i n p u t s to the system. I f an e r r o r i s made i n e s t i m a t i n g the uater requirement, the expected b e n e f i t s u i l l be a l t e r e d by q u i t e a high order of magnitude. 7.2.1 C r i t e r i o n of Maximizing Expected Economic B e n e f i t The d e c i s i o n choice uas not s i g n i f i c a n t l y a f f e c t e d by the u n c e r t a i n t y i n the input data. The maximum expected b e n e f i t and the optimum c a p a c i t y were obtained from using data that uas knoun u i t h c e r t a i n t y . Uhen u n c e r t a i n t y uas int r o d u c e d i n t o the data, the optimum d e c i s i o n d i d not change s i g n i f i c a n t l y (see Table 6.3) f o r u n c e r t a i n t y i n costs of development, crop y i e l d and water requirement; however the expected b e n e f i t uas reduced s l i g h t l y u i t h more u n c e r t a i n t y i n the'system. Uhen u n c e r t a i n t y i n stream flow was considered as w e l l , the optimum c a p a c i t y was reduced by 18% and the maximum b e n e f i t was reduced by 24%. 81 7.2.2 C r i t e r i o n of Maximizing Expected U t i l i t y Value The d e c i s i o n choice uas more s e n s i t i v e to u n c e r t a i n t y uhen u t i l i t y value uas used as a d e c i s i o n c r i t e r i o n . The a n a l y s i s y i e l d e d theVmaximum expected u t i l i t y and the l a r g e s t optimum s i z e uhen a l l the input data to the system uere knoun u i t h c e r t a i n t y . Uhen there uas u n c e r t a i n t y i n the costs of development but a l l other data uere knoun u i t h c e r t a i n t y , the optimum c a p a c i t y and the de r i v e d b e n e f i t s uere m a r g i n a l l y reduced. This r e s u l t i s probably due to the f a c t t h a t expected values of c o s t s uere used i n the a n a l y s i s , and these uere not s i g n i f i c a n t l y d i f f e r e n t from the cost f u n c t i o n d e f i n e d by the "probable" curve on the graphs of u n i t c o s t s . U n c e r t a i n t y i n the crop y i e l d alone ( a l l other data knoun u i t h c e r t a i n t y ) had a more s i g n i f i c a n t e f f e c t on the de-c i s i o n . The optimum i r r i g a t e d area uas reduced by 5%, the t o t a l i r r i g a b l e area by "\8% and the expected u t i l i t y by 24/2, on an average of the u t i l i t y f u n c t i o n s used. U s u a l l y the steeper the u t i l i t y f u n c t i o n , the l a r g e r the r e d u c t i o n i n expected v a l u e s , i . e . the more c o n s e r v a t i v e the d e c i s i o n maker the more s e n s i t i v e he i s to u n c e r t a i n t y . U n c e r t a i n t y i n both the c o s t s of development and the crop y i e l d l e d to almost the same d e c i s i o n as uhen only u n c e r t a i n t y i n crop y i e l d uas co n s i d e r e d ; houever the t o t a l i r r i g a b l e area uas reduced by 19 to 23% and the expected u t i l i t y by 16 to 29%, The more c o n s e r v a t i v e the d e c i s i o n maker ( i . e . steeper u t i l i t y f u n c t i o n ) the higher the r e d u c t i o n . 82 By c o n s i d e r i n g u n c e r t a i n t y i n water requirement as w e l l as i n co s t s of development and crop y i e l d , the r e d u c t i o n i n optimum c a p a c i t y uas 7%. Houever the g r e a t e s t change uas i n the maximum expected u t i l i t y and the t o t a l i r r i g a b l e area - i t uas reduced by 23 to 38%, u h i l e the expected u t i l i t y value uas reduced by 36 to 60% depending on the u t i l i t y f u n c t i o n . U n c e r t a i n t y i n stream f l o u had the g r e a t e s t e f f e c t on the .decision. The optimal c a p a c i t y and the maximum expected u t i l i t y v alue uere reduced s i g n i f i c a n t l y uhen stream f l o u u n c e r t a i n t y uas accounted f o r . Using a u t i l i t y f u n c t i o n u i t h slope = 2, the optimal c a p a c i t y uas reduced by 28% and the maximum expected u t i l i t y value uas reduced by 59%. The e f f e c t of v a r i a t i o n i n p r i c e of crops uas i n v e s t i -gated by using tuo p r i c e s uhich uere 10% higher and lou e r than the standard p r i c e . It uas found that the optimum s i z e of the system and the expected r e t u r n s i n c r e a s e d u i t h i n c r e a s e i n p r i c e and decreased u i t h decrease i n p r i c e . 7.3 Contrast betueen Expected Monetary Walue and Expected U t i l i t y The e f f e c t of u n c e r t a i n t y i n input data on the d e c i s i o n uas more n o t i c e a b l e uhen u t i l i t y value uas used as a d e c i s i o n c r i t e r i o n . Table 6.3 shous the d i f f e r e n t d e c i s i o n choices obtained by using the tuo d e c i s i o n c r i t e r i a . Uhatever l e v e l of u n c e r t a i n t y uas con s i d e r e d , the c r i t e r i o n of maximizing expected u t i l i t y value uas more s e n s i t i v e to the l e v e l of u n c e r t a i n t y . U i t h i n c r e a s e i n u n c e r t a i n t y i n the system the optimum area chosen on the b a s i s of maximizing economic b e n e f i t uas reduced by 3 , 83 to 18% u h i l e the expected b e n e f i t uas reduced by 2 to 24%. For the c r i t e r i o n of maximizing expected u t i l i t y value the r e d u c t i o n i n area uas 3 to 28%, u h i l e the expected u t i l i t y uas reduced by 2 to 60%. At each l e v e l of u n c e r t a i n t y uhich uas considered the optimum area chosen on the b a s i s of maximizing expected u t i l i t y value uas at l e a s t 20% l e s s than the area chosen on the b a s i s of maximizing economic b e n e f i t . The c r i t e r i o n of maximizing expected economic b e n e f i t i s not s e n s i t i v e to the r i s k s i n v o l v e d i n each outcome. The process of maximizing expected monetary values does not d i s c r i -minate betueen high l o s s e s uhich the d e c i s i o n maker cannot absorb, and high b e n e f i t s uhich 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. This c r i t e r i o n i s o f t e n equated to the gambler's r u i n s i t u a t i o n ( H a l l and Dracup, 1970). This i s c l e a r l y n o t i c e a b l e by examining the l i m i t i n g values of i r r i g a b l e l a n d . On the b a s i s of economic value the l o u e s t l i m i t of development i s zero acreage and the maximum l i m i t i s near the a v a i l a b l e acreage. A more c a r e f u l " o b s e r v a t i o n i n d i c a t e s that marginal b e n e f i t s from very l a r g e areas uere very small i n comparison to the cost i n c u r r e d ; a l s o s m a l l e r developments are j u s t not u o r t h u h i l e . A g r i c u l t u r a l development i s u s u a l l y considered to be a high r i s k undertaking because of the u n c e r t a i n t i e s i n v o l v e d . Consequently d e c i s i o n making i n t h i s f i e l d must take i n t o account these f a c t o r s , and a l s o the i n i t i a l circumstances (or u e a l t h ) of the d e c i s i o n maker as u e l l as h i s a t t i t u d e towards r i s k . The u t i l i t y value has the advantage of being able to take i n t o 84 c o n s i d e r a t i o n such r e l e v a n t f a c t o r s . A steeper u t i l i t y f u n c t i o n i n d i c a t e s a more c o n s e r v a t i v e a t t i t u d e touards r i s k . Poor r e t u r n s are assigned very high negative u t i l i t y v a l u e s , and the minimum i r r i g a b l e area i s f a r g r e a t e r than zero; the magnitude i n c r e a s e s , t h e more c o n s e r v a t i v e the d e c i s i o n maker. The t o t a l i r r i g a b l e area decreases u i t h i n c r e a s i n g u n c e r t a i n t y i n the data, and u i t h more c o n s e r v a t i v e u t i l i t y f u n c t i o n s . It i s n a t u r a l f o r people to act i n a c o n s e r v a t i v e manner uhen faced u i t h u n c e r t a i n t y i n outcomes. In engineering a n a l y s i s t h i s conservatism i s d i s p l a y e d by use of l i b e r a l f a c t o r s of s a f e t y . Houever the o v e r a l l e f f e c t of u n c e r t a i n t y cannot be adequately estimated by f a c t o r s of s a f e t y i n s p i t e of t h e i r u s e f u l n e s s and s i m p l i c i t y i n use. I t i s only by a d i r e c t comp-rehensive c o n s i d e r a t i o n of u n c e r t a i n t y i n a d e c i s i o n t h a t the e f f e c t of u n c e r t a i n t y can be e f f e c t i v e l y e valuated. Knouledge of these ef f ectsj ; can then lead to c o r r e c t i v e measures. The c r i t e r i o n used i n d e c i s i o n making should be such that as to b r i n g out the fundamental a t t i t u d e s of the d e c i s i o n maker touards the d e s i r e d development. The importance of t h i s human a t t r i b u t e of conservatism should not be overlooked i n development s t u d i e s . People u s u a l l y d e s i r e to uork u i t h i n the framework of a system that they are comfortable u i t h . As such they u i l l r e s i s t development plans uhich they i n t u i t i v e l y f e e l are loaded u i t h u n c e r t a i n t y . T h i s i s p a r t i c u l a r l y important i n developing c o u n t r i e s . Long time proven t r a d i t i o n a l methods of farming are hard to change, people are s l o u to accept expansive development programs i f they c o n s i d e r the costs and the r i s k s to be too high, or i f they place lou 85 u t i l i t y v a l u e on t h e p r o d u c t s . F a i l u r e t o r e c o g n i s e t h i s a s p e c t o f human n a t u r e u s u a l l y l e a d s t o f r u s t r a t i o n i n a t t e m p t i n g t o p e r s u a d e d e v e l o p i n g c o u n t r i e s t o a d o p t c e r t a i n modes o f a g r i c u l t u r a l d e v e l o p m e n t . As F r a n c e s c h i ( 1 9 7 2 ) p o i n t e d o u t , one o f t h e b a s i c p r o b l e m s o f d e v e l o p m e n t i n d e v e l o p i n g c o u n t r i e s i s one o f c h a n g i n g human a t t i t u d e s f r o m a p a s s i v e a c c e p t a n c e o f t r a d i t i o n a l e v e n t s t o an a c t i v e s t r u g g l e a g a i n s t c h a l l e n g i n g c i r c u m s t a n c e s . U h i l e u t i l i t y v a l u e c r i t e r i o n i s a b e t t e r p r i n c i p l e , i t i s n o t a l w a y s e a s y t o d e t e r m i n e t h e u t i l i t y f u n c t i o n . F o r an i r r i g a t i o n p r o j e c t t o be r e a d i l y a c c e p t a b l e i t must m a x i m i z e t h e u t i l i t y v a l u e s o f t h e f a r m e r s ; however o n l y t h e u t i l i t y v a l u e o f t h e m a i n d e c i s i o n maker i s u s e d i n t h e a n a l y s i s . The a d v a n t a g e o f t h i s t y p e o f a n a l y s i s i s t h a t t h e d e c i s i o n maker has t h e o p p o r t u n i t y t o s y n t h e s i z e h i s u t i l i t y v a l u e f r o m know-l e d g e o f t h e u t i l i t y v a l u e o f t h e p e o p l e f o r whom t h e p r o j e c t i s p l a n n e d , 7.4. C O N C L U S I O N S . T h i s t h e s i s has shown t h e i m p o r t a n c e o f d i r e c t l y a c c o u n t i n g f o r u n c e r t a i n t y i n w a t e r r e s o u r c e s p r o j e c t s . T h i s a p p r o a c h has t h e a d v a n t a g e o f f o c u s s i n g t h e a t t e n t i o n o f t h e d e -c i s i o n maker on t h e i m p o r t a n t i n p u t s t o t h e s y s t e m . The a n a l y s i s has shown t h a t i f u n c e r t a i n t y i s d i r e c t l y t a k e n i n t o c o n s i d e r a t i o n , t h e a n a l y s i s w o u l d y i e l d l o w e r optimum d e s i g n c o n d i t i o n s t h a n when t h a n when t h e a n a l y s i s i s c a r r i e d o u t w i t h o u t c o n s i d e r i n g u n c e r t a i n t y . The g r e a t e r t h e u n c e r t a i n t y t h e l e s s t h e optimum c o n d i t i o n s o b t a i n e d . The a n a l y s i s h a s shown t h a t h y d r o l o g i c a l 86 u n c e r t a i n t y i s t h e m o s t i m p o r t a n t c o n s i d e r a t i o n t o t h e s y s t e m , i t t e n d s t o r e d u c e t h e o p t i m u m c a p a c i t y a n d t h e e x p e c t e d b e n e f i t s f r o m t h e s y s t e m . T h e u n c e r t a i n t y i n e c o n o m i c i n p u t s a n d i n c r o p y i e l d u i l l s i g n i f i c a n t l y r e d u c e t h e e x p e c t e d r e t u r n s f r o m t h e s y s t e m . T h e r e s u l t s o f t h e a n a l y s i s a r e s u m m a r i s e d i n F i g . 7 . 1 . T h e r e a l i s a t i o n o f t h e r e l a t i v e i m p o r t a n c e o f e a c h o f t h e i n p u t s h e l p s t o d e f i n e t h e a r e a s o f c o n c e n t r a t e d a c t i v i t y i n a n y e f f o r t t o i m p r o v e t h e s y s t e m . I n t h i s r e s p e c t , t h e c o m p r e h e n s i v e a n a l y s i s o f u n c e r t a i n t y i s f a r s u p e r i o r t o t h e u s e o f m e r e f a c t o r s o f s a f e t y o r f a c t o r s o f i g n o r a n c e . T h e a n a l y s i s h a s s h o u n q u a n t i t a t i v e l y t h e e f f e c t s o f a c o n s e r v a t i v e a t t i t u d e o n d e c i s i o n m a k i n g u n d e r u n c e r t a i n t y . I n t h e f a c e o f u n c e r t a i n t y p e o p l e u i l l t e n d t o m ake d e c i s i o n s u h i c h m i n i m i z e t h e u n d e s i r a b l e c o n s e q u e n c e s . T h e n a t u r e o f s u c h d e c i s i o n s d e p e n d s o n t h e p r e v a i l i n g c i r c u m s t a n c e s a t t h e t i m e t h e d e c i s i o n i s made - i . e . t h e d e g r e e o f r i s k i n v o l v e d , t h e l i k e l i h o o d o f s u c h r i s k , a n d t h e d e c i s i o n m a k e r ' s r e l a t i v e u e a l t h o r a b i l i t y t o s u r v i v e u n d e s i r a b l e c o n s e q u e n c e s . U i t h r e s p e c t t o i r r i g a t i o n d e v e l o p m e n t , t h e m o r e u n c e r t a i n t y i n t h e s y s t e m t h e s m a l l e r t h e d e s i r e d d e v e l o p m e n t . T h e i m p o r t a n c e o f t h e s e a t t i t u d e s e s p e c i a l l y i n a g r i c u l t u r a l d e v e l o p m e n t e x p l a i n s some o f t h e p r o b l e m s e n c o u n t e r e d i n i m p l e m e n t i n g d e v e l o p m e n t s c h e m e s i n t h e d e v e l o p i n g c o u n t r i e s . T h e d e g r e e t o u h i c h t h e i n t r i n s i c v a l u e s a n d a t t i t u d e s u h i c h t h e d e c i s i o n m a k e r a t t a c h e s t o t h e p r o j e c t c a n b e i n c o r p o r a t e d i n a d e c i s i o n p r o c e s s d e p e n d s o n t h e d e c i s i o n c r i -t e r i o n u s e d . A u t i l i t y v a l u e , c a n , t o a l a r g e e x t e n t , r e f l e c t 8 7 the t r u e uorth a d e c i s i o n maker at t a c h e s to the p r o j e c t . I t s measurement i s not always easy but through the use of s u b j e c t i v e p r o b a b i l i t y theory i t i s p o s s i b l e to pool together a l l r e l e v a n t i n f o r m a t i o n i n t o one u s e f u l form. Once the u t i l i t y value has been determined, the d e c i s i o n c r i t e r i o n of maximizing expected u t i l i t y value u i l l be c o n s i s t e n t u 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 . T h i s i s the essence of development - to maximize the u t i l i t y v alue. The method used to account f o r u n c e r t a i n t y i n stream f l o u i s a u s e f u l a n a l y s i s u h i c h i s o f t e n ignored. Unless i t can be proved that the model parameters estimated from a short h i s t o r i c sample are knoun u i t h a d e s i r a b l e degree of c e r t a i n t y , the parameters should have u n c e r t a i n t y l i m i t s on them to r e f l e c t l a c k of knouledge of the true values.- Most h i s t o r i c records are so s h o r t t h a t the u n c e r t a i n t y about the t r u e parameters i s q u i t e h i g h , t h e r e f o r e they should be analysed u i t h u n c e r t a i n t y ^ l i m i t s on them. The method of a n a l y s i s given i n t h i s t h e s i s i s u s e f u l i n comparing, s e l e c t i n g and ranking \? d i f f e r e n t p r o j e c t s necessary to meet a given demand. Each p r o j e c t has i t s oun problems and i t s oun u n c e r t a i n t y . By a n a l y s i n g the p r o j e c t s as given i n t h i s t h e s i s , the d e s i r a b i l i t y of each p r o j e c t u i l l be i n f l u e n c e d by the magnitude of i t s output as u e l l as the l i k e l i -hood of a t t a i n i n g such an output. Some improvements on the method of a n a l y s i s are p o s s i b l e . The p r o b a b i l i t y model used to represent u n c e r t a i n t y need not be the "skeu normal" d i s t r i b u t i o n alone, other models such as normal d i s t r i b u t i o n could be i n v e s t i g a t e d . Uhat i s needed i s to determine 88 uhich i s the most s u i t a b l e model to use i n a p a r t i c u l a r s i t u a t i o n . The a n a l y s i s should a l s o be c a r r i e d out using stream f l o u sequences that i n c l u d e the most adverse short term f l o u c o n d i t i o n s , f o r example using 5-year or 10-year sequences of f l o u s obtained by stream f l o u g e n e r a t i o n , provided model parameters have been analysed f o r u n c e r t a i n t y . In the end, t h i s method of a n a l y s i s has one s u b t l e advantage, the designer can knou u i t h some degree of c e r t a i n t y that the e f f e c t : of u n c e r t a i n t y i n the system has been accounted f o r . N O T E = U t i l i t y F u n c t i o n u s e d h a s S l o p e S 2 = 2 Expected Bene f i t s in S; A l l Input Data known with Certa inty Expected Benef i t s in B, Uncerntainty in Costs and Crop Y i e l d Expected Benef i t s in 8, Uncertainty in C o s t s , Y i e l d and Streamf low Expected U t i l i t y ; AM Input Data known with C e r t a i n t y Expec ted Ut i l i ty j Uncerta inty in Costs and Y i e l d Expected Ut i l i ty ; Uncerta inty in Costs , Crop Y ie ld , I r r igat ion Water Requ i rement Expected Ut i l i t y ; Uncertainty in Costs, Crop Y ie ld and S t r e a m f l o w 100 200 300 400 500 600 700 800 900 I r r i g a t e d L a n d i n A c r e s FIG.7.1 S U M M A R Y OF B E N E F I T F U N C T I O N S GO CO OJ 89 REFERENCES 1. Benjamin, 3.R., and C . A . C o r n e l l : " P r o b a b i l i t y , S t a t i s t i c s , and D e c i s i o n s f o r C i v i l Engineers," McGrau-Hill Book company, Neu York, 1970. 2. Benson, M.A.: / / C h a r a c t e r i s t i c s of Frequency Curves Based on a T h e o r e t i c a l T,000-year Record; Flood Frequency A n a l y s i s , " U.S. G e o l o g i c a l Survey Uater Supply Paper 1543-A, 1960. 3. Borgadi, I; and F. Szidarovsky: "Induced Safety Algorithm f o r H y d r o l o g i c a l Design". Uater Resources Research, Volume 10, No. 4, A p r i l 1974. 4. " B r i t i s h Columbia I r r i g a t i o n Guide", Department of A g r i c u l -t u r e , V i c t o r i a , B r i t i s h Columbia. 5. "Canada - B r i t i s h Columbia Okanagan Basin Agreement, 1974; T e c h n i c a l Supplement I", B r i t i s h Columbia Uater Resources S e r v i c e , Parliament B u i l d i n g s , V i c t o r i a B.C. 6. F r a n c e s c h i , L.E.: "Uater Resources U t i l i z a t i o n i n Developing C o u n t r i e s " , Proceedings of the F i r s t I n t e r n a t i o n a l Conference on 'Transfer of Uater Resources Knouledge' September 1972, Fort C o l l i n s , Colorado, U.S.A. 7. H a l l , U.A., and J.A. Dracup: "Uater Resources Systems Engin-e e r i n g " , McGrau-Hill Book Company, Neu York, 1970. 8. H a l t e r , A., and G.Dean: " D e c i s i o n s Under U n c e r t a i n t y u i t h Research ApplicationsV,South-uestern P u b l i s h i n g Co., C i n c i n n a t i , Ohio, 1971. 9. Hershman, 5.: " A p p l i c a t i o n of D e c i s i o n Theory to Uater Q u a l i t y Management," Master's T h e s i s , 1974, U n i v e r s i t y of B r i t i s h Columbia, Vancouver, B.C. 10. H i g g i n s , R., - personal communication, 1975. 11. Luce, R., and H. R a i f f a : "Games and D e c i s i o n s " , 3ohn U i l e y & Sons, Inc., Neu York, 1957. 12. " H i s t o r i c a l Stream f l o u Summary; B r i t i s h Columbia to 1973", Inland Uaters D i r e c t o r a t e , Uater Resources Branch, Uater Survey of Canada, Department of the Environment, Ottaua, Canada. 90 13. Y e v j e v i c h , V. : " P r o b a b i l i t y and S t a t i s t i c s i n Hydrology", Uater Resources P u b l i c a t i o n s , Fort C o l l i n s , Colorado, U.S.A. 1972. 14. Uood, F.E., and I. Rodriguez-Iturbe: "Bayesian Inference and D e c i s i o n Making f o r Extreme Hydrologic Events", Uater Resources Research, Volume 11 No. 4, August 1975. 91 APPENDIX There.are;, tuo b a s i c computer programs necessary i n the a n a l y s i s . The f i r s t program, designated as Program A prepares a l l the data f o r the main program, c a l l e d Program B, uhich has v a r i a n t s B-1, B-2 and B-3. Program A It has tuo complementary f u n c t i o n s uhich together prepare the data and s t o r e s i t i n a Matrix Format ready f o r use i n the main program. 1. For a given f u n c t i o n such as Crop y i e l d versus Percentage A v a i l a b l e Uater, the program approximates the f u n c t i o n by a p i e c e u i s e c u b i c polynomial. I t a l s o s u p p l i e s i n t e r -p o l a t e d data betueen the given data p o i n t s . T h i s curve f i t t i n g and i n t e r p o l a t i o n i s done f o r upper, mid and lower curves of the r e l a t i o n s h i p , r e p r e s e n t i n g the upper, probable and l o u e r estimates. 2. For any value of the a b s c i s s a i t i s assumed that the o r d i n a t e s betueen the upper and l o u e r l i m i t s f o l l o u a "skeu normal" d i s t r i b u t i o n . The program then c a l c u l a t e s the p r o b a b i l i t i e s of values of o r d i n a t e s by i n t e g r a t i n g the assumed 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 . T h i s leads to values of o r d i n a t e s each u i t h a p r o b a b i l i t y a s s o c i a t e d u i t h i t . The r e s u l t s are s t o r e d as a matrix u i t h the f i r s t rou and f i r s t column r e p r e s e n t i n g the a b s c i s s a and o r d i n a t e axes of the f u n c t i o n , and the other elements of the matrix the p r o b a b i l i t i e s of values of o r d i n a t e s at given values of the a b s c i s s a . The b a s i c procedure i n the program i s summarised i n subsequent s e c t i o n s of t h i s Appendix. Program B T h i s i s the main program used i n the d e c i s i o n model, i t 92 computes the expected economic b e n e f i t and expected u t i l i t i e s . Most of the input f u n c t i o n s ( c o s t , y i e l d e t c ) are s u p p l i e d i n a matrix format. The v a r i a n t s of the program are Program B-1: C a l c u l a t e s the Expected B e n e f i t s using monetary values as a measure of b e n e f i t s . Program B-2: C a l c u l a t e s the B e n e f i t s i n terms of u t i l i t y v a l u e . Program B-3: T h i s i s s l i g h t l y s i m i l a r to B-2, here the u n c e r t a i n attached to the i n p u t f u n c t i o n has been e l i m i n a t e d . These t h r e e programs are q u i t e s i m i l a r , but Program B-2 i s more general because i t co n t a i n s subprograms to determine the u t i l i t i e s a s s o c i a t e d u i t h each outcome. Only Program B-2 u i l l be d e s c r i b e d i n d e t a i l . 93 PROGRAM A (By Richard Higgins) TO DETERMINE THE PROBABILITY MATRICES OF INPUT FUNCTIONS 1. S t a r t the Program 2. Read i n the Data R e s e r v o i r Cost versus S i z e Land Development Cost Annual Farm Maintenance Cost Crop y i e l d versus Percentage A v a i l a b l e / D e s i g n uater Requirement. The Data f o r each input f u n c t i o n c o n s i s t s of the o r d i n a t e s and a b s c i s s a e of the upper, probable and l o u e r estimates of each r e l a t i o n s h i p , 3. Enter Data of R e s e r v o i r Cost versus S i z e . 4. C a l l a Cubic S p l i n e Program to Approximate the f u n c t i o n and to I n t e r p o l a t e e x t r a data p o i n t s on graphs. 5. Choose One L e v e l of S i z e e.g. R e s e r v o i r Volume 6. C a l l a Program to F i t a "Skeu Normal" 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 between the Upper and Louer l i m i t s of graph 7. C a l l a Program to I n t e g r a t e the "Skeu Normal" D i s t r i b u t i o n and to Compute the P r o b a b i l i t y of chosen l e v e l s of u n i t c o s t s betueen Upper and Louer l i m i t s 8. Repeat Steps 4 , 5 , 6 and 7 f o r other R e s e r v o i r S i z e s . 9. Enter the R e s u l t s as Elements of a Matrix; F i r s t Rou gives the S i z e of R e s e r v o i r , F i r s t Column gives the Unit Costs, and A l l other Elements are P r o b a b i l i t i e s of Costs at v a r i o u s R e s e r v o i r S i z e s . 10. Store Matrix of R e s e r v o i r Costs 11. Repeat 3 to 10 f o r other Data 12. Stop. $COMPILE C 94 C PROGRAM B—2 C C THIS PROGRAM CALCULATES THE EXPECTED BENEFITS FROM A C SINGLE PURPOSE RESERVOIR BUILT TO SUPPLY IRRIGATION C WATER TO AN AGRICULTURAL AREA C BENEFITS ARE CALCULATED IN * UTILITY VALUE* C UTILITY IS A FUNCTION OF NET ECONOMIC VALUE C SEVERAL VALUES OF IRRIGATION WATER REQUIREMENTS ARE C TRIED 2«0, 2.5, 3„0, 3.5, 4.0 FEET/YEAR C REZCST=THE MATRIX OF RESERVOIR COSTS C LNDCST=THE MATRIX OF COSTS OF LAND DEVELOPMENT C MAIN=MATR IX OF MAINTENANCE AND OPERATING COST C YIELD=MATRIX OF CROP YIELD C STFL=MATRIX OF STREAM FLOW C FLPRB=MATRIX OF STREAM FLOW PROBABILITY C REVN=MATRIX OF TOTAL REVENUE FROM SALE OF CROPS C EEY=EXPECTEO YIELD OF CROPS C BNFT=EXPECTED BENEFITS C YCRP=CROP YIELD (TONS/ACRE) C YAXIS= PERCENT AVAILABLE WATER OVER DESIGN WATER C REQUIREMENT C C NRR=NUMBER OF ROWS OF MATRIX *REZCST» C NCR=NUMBER OF COLUMNS OF MATRIX *REZCST» C LRL=NO. OF ROWS OF •LNDCST' C LCL=NOo OF COLUMNS OF •LNDCST* C MRM=N0o OF ROWS OF * MA IN' C MCM=NO. OF COLUMNS OF *MAIN» C KRY=NOo OF ROWS OF • YIELD* C KCY=NO. OF COLUMNS OF »YIELD* C C PRICE OF CROPS IS ASSUMED TO BE $100*00 PER TON C 1 REAL REZCSTI60,30),LNDC ST( 60,30),MAIN(60,30), 1YIELD<60,30) 2 DIMENSION WREQD(5),SPB(60),REVNC20),BNFT(20),EXP(20) 3 DIMENSION CST(60),FST(60),SCCST(60),SCND(60),SUM(60), 1SMPB160) 4 DIMENSION AREA(20),XLAKE(20),VALUE(20),XLND(20), 1RZV(10),RETNU0) 5 DIMENSION STFL(60),FLPRB(60),YAXISC 25),YCRPC25) 6 DIMENSION XX(60,30),AB(30),XPC(30),EXPNSC60),UNCT(60) 7 DIMENSION PRUT(20),TRUE(20) 8 INTEGER LRZ(20),HRZ(20),LLD(20),HLD(20) 9 INTEGER LYD(30),HYD(30),LMN(30),HMN(30) 10 INTEGER SML(30>,BIG(30) 11 INTEGER COLM,ROW C 12 COMMON XX,SML,8IG 13 COMMON EXPNS,UNCT 14 COMMON STFL,FLPRB,YAXIS,YCRP 15 C0MM0N/BLK1/SUM,SMPB 16 COMMON/BLK2/CST,FST,SCCST,SCND/BLK3/NI,N2,NC0MB,DELTA 17 COMMON/BLK4/LRZ,HRZ,LLD,HLD,LMN,HMN 18 COMMON/BLK5/EEY,REVN,BNFT,EXP C C:::READ IN DIMENSIONS OF MATRICES 19 READ 10,NRR,NCR,LRL,LCL,MRM,MCM,KRY,KCY FORMAT (818) READ 15,BMAX,BC,YC,SLP 9; FORMAT*4F10.2) READ 20,PRC FORMAKF 10.2) NCOMB=0 UNITIALSE THE MATRICES TO ZERO CALL GSET (REZCST,60,30,60,OoO) CALL GSET (LNDCST,60,30,60,0*0) CALL GSET (MAIN,60,30,60,0.0) CALL GSET 4YIELD,60, 30,60,0. 0) :READ FROM FILES AND STORE IN MATRICES SPECIFIED 00 40 I=1,NRR READ<4,35) (REZCST<I»J)»J=1»NCR) FGRMATCF10i3,llF9.3/F10.3,3F9.3) CONTINUE DO 50 I=1,LRL RE AD(3,45) {LNDC ST(I,J),J=l,LCL) F0RMATIF10.3, 11F9.3) CONTINUE DO 60 1=1,MRM READ{8,55) C MAIN(I , J ), J~1,MCM) FORMAT<F10*3,11F9.3) CONTINUE DO 65 1=1,KRY REA0i7,64) <YIELDU,J),J=1,KCY) FORMAT (F10. 3, 1 IF 9. 3/F10. 3, 9F9. 3) CONTINUE DO 2 K=l,40 READ(2,67) STFL*K),FLPRBCK) F0RMAT{F10o2,F15o5) FL0W=STFH K) /10.0 STFLlK)=FLOW CONTINUE READ IRRIGATION WATER REQUIREMENT READ 70, {WREGDUR),IR=1,5I FORMAT( 5F 10. 2) PRINT 71 PRINT 72 FORMAT{* 1 9 X,'CONSUMPTIVE USE* ,9X,•IRRIGATED LAND 1• ,10X,'OPTIMUM RESERVOIR SIZE*,8X,'EXPECTED 1BENEFITS') FORMAT(1 «,9X,«FEET PER YEAR * *13X,*ACRE •,20X, 1»ACRE-FEET«,21X,'UTILITY') FOR EACH OF THE MATRICES DETERMINE THE ROWS CONTAINING NON-ZERO VALUES CALL GCOPY(REZCST,XX,NRR,NCR,60,60) CALL PREP(XX »NRR,NCR,SML,BIG I CALL GCOPYISML,LRZ,NCR,1,30,20) CALL GC0PY(BIG,HRZ,NCR,1,30,20) CALL GSET {XX,60,30,60,0.0) CALL GC0PY(LNDCST,XX,LRL,LCL,60,60) CALL PREP(XX,LRL,LCL,SML,BIG) CALL GC0PYCSML,LLD,LCL,l,30,20) CALL GCGPY{BIG,HLD,LCL,1,30, 20) CALL GSET iXX ,60,30,60,0.0) 68 CALL GC0PY(MAIN,XX,MRM,MCM,60,60) 69 CALL PREP(XX,MRM,MCM,$ML,BIG) 70 CALL GCOPY(SML,LMN,MCM,1,30,20) 71 CALL GC0PY(BIG,HMN,MCM,1,3Q,20) 72 CALL GSET <XX,60,30,60,0„0) C 73 KKL=KCY-l 74 CALL GCOPY(YIELD,XX,KRY,KCY,60,60) 75 CALL EXPECT(XX,KRY,KCY,AB,XPC) 76 CALL PREP(XX,KRY,KCY,SML,BIG) 77 CALL GCOPY(AB,YAXIS,KKL,1,30,25) 78 CALL GC0PY(SML,LYD,KCY,1,30,30) 79 CALL GC0PY(BIG,HYD,KCY,1,30,30) 80 CALL GSET (XX,60,30,60,0©0) C C:::CH00SE ONE VALUE OF IRRIGATION WATER REQUIREMENT 81 DO 900 IR=1,5 82 PRINT 701 83 701 FORMAT(*—*,10X»'NEW CONSUMPTIVE USE') 84 OEW=WREQD(IR) C:::CHOOSE ACREAGE TO BE IRRIGATED 85 DO 800 L=3,LCL 86 CALL GSET (EXPNS,60,1,60,0.0) 87 CALL GSET (UNCT,60,i,60,0.0) 88 S0IL=LNDCST(1,L) 89 V0LM=LNDCST(1,L)*WREQD(IR) 90 MIN=LLD{L) 91 LARG£=HLD(L) 92 DELl=(LNDCSTUMIN+l)tl)-LNDCST(MIN,l))*LNDCST(l,L ) 93 TEMP=DEL 1 C:::DEL1 IS THE COST INTERVAL FOR LAND COSTS 94 N1=0 95 DO 100 I=MIN,LARGE 96 N1=N1+1 97 EXPNS(N1)=LNDCST(I,1)*LNDCST(1,L) 98 UNCT(N1)=LNDCST(I,L) 99 N0RM=N1 100 100 CONTINUE C C:::CH00SE THE RESERVOIR SIZE 101 DO 700 N=2,NCR 102 CALL GCOPYCEXPNS,CST,NORM,1,60,60) 103 CALL GCOPY(UNCT,FST,NORM,1,60,60) 104 N1=N0RM 105 NB=N-1 106 DEL1=TEMP 107 QA=R£ZCSTU,N) 108 MIN=LRZ(N) 109 LARGE=HRZ1N) 110 D0D=REZCST((MIN+1),1)-REZCST(MIN,1) 111 DEL2=DDD*REZCST(l,N)/3.0 112 N2=0 113 DO 120 I=MIN,LARGE 114 N2=N2-H 115 SCCST(N2)=(REZCST(I,1)*REZCST(1,N))/3 o0 116 SCND(N2)=REZCST(I,N) 117 120 • CONTINUE C:::CHOOSE COST COMBINATION INTERVAL 118 DELTA=(DE L1+DEL2)/2«0 c C:::START COST COMBINATIONS 97 C:::CALL A PROGRAMME TO COMBINE TWO MATRICES • 119 CALL CMBINE(CST,FST,SCCST,SCND,N1,N2,DELTA, 4NC0MB,SUM,SMPB) 120 CALL GSET (CST,60,1,60, 0.0) 121 CALL GSET (FST,60*1,60,0.0) 122 CALL GSET (SCCST,60,1,60,0.0) 123 CALL GSET (SCND,60,1,60,0.0) C:::ANNUAL OPERATING C0ST=10? OF TOTAL CAPITAL COST 124 DO 150 J=1,NC0MB 125 CST(J)=SUM(J)/10„0 126 150 FST(J)=SMPBIJ) 127 CALL GSET CSUM,60,1,60,0.0) 128 CALL GSET (SMPB,60,1,60,0.0) C:::COMBINE ANNUAL OPERATING COST WITH MAINTENANCE AND C OPERATING COST OF IRRIGATION SYSTEM 129 DEL1=CST(2)-CST(1) 130 N1=NC0MB 131 MIN=LMN(L) 132 LARGE=HMN(L) 133 DEL2=(MAIN((MIN+1),1)-MAIMMIN,1))*MAIN(1,L) 134 N2=0 135 DO 220 I=MIN,LARGE 136 N2=N2+1 137 SCCSTtN2)=MAIN(I,l)*MAIN(l,L) 138 SCND(N2)=MAIN(I ,L) 139 220 CONTINUE 140 DELTA=(DELl+D£L2)/2.0 C 141 CALL CMBINECCST,FST,SCCST,SCND,N1,N2,DELTA, 4NC0MB,SUM,SMPB) C::DETERMINE THE EXPECTED ANNUAL COST 142 COST=EXPCST(NCOMB,SUM,SMPB) C C::: CALL A SUBROUTINE TO MATCH STREAM INFLOW WITH C IRRIGATION WATER REQUIREMENT AND TO CALCULATE THE C EXPECTEO CROP YIELD AND EXPECTED UTILITY 143 CALL WATER(YlELD,STFL,FLPRB,YAXIS,LYD,HYD,SOIL, 6PRC,V0LM,QA,C0ST,SLP,YC, BCEEU), 144 TRUE(NB)=EEU C 145 CALL GSET (CST,60,1,60,0.0) 146 CALL GSET IFST,60,1,60,0.0) 147 CALL GSET (SCCST,60,1,60,0.0) 148 CALL GSET (SCND,60,1,60,0.0) 149 CALL GSET (SUM,60,1,60,0.0) 150 CALL GSET CSMPB,60,1,60,0.0) 151 700 CONTINUE C C::CALCULATE THE MAXIMUM BENEFIT 152 CALL MXBNFT(TRUE»NB,NL»XMAX) 153 LM=L^2 154 AREACLM)=LNDCST(1,LI 155 XLAKE(LM)=REZCST(1, (NL-H) ) 156 VALUE(LM)=XMAX 157 PRINT 750,DEW,AREA(LMI,XLAKE(LM),VALUE(LM) 158 750 FORMAT(• •,10X,F12.2,4X,F12.2,6X,F12.2,8X,F20 o2) 159 800 CONTINUE C C:::CALCULATE THE MAXIMUM OF MAXIMUM BENEFITS FROM EACH 98 C QUANTITY OF IRRIGATION WATER USE C 160 CALL MXBNFT(VALUE,LM,NL»XMAX) 161 XLNDU R)=AREA(NL) 162 RZV(IR)=XLAKE(NL) 163 RETNH R) = XMAX C 164 900 CONTINUE C C:::CALCULATE THE MAXIMUM OF ALL THE MAXIMUMS 165 CALL MXBNFKRETN, 5,NL, XMAX) 166 CULT=XLND(NL) 167 DAM=RZVfNL) 168 GRT=XMAX C:::PRINT OUT MAXIMUM VALUES 169 DO 905 IR=1,5 170 PRINT 903,WREQD(IR),XLNDUR),RZV(IR),RETN{IR) 171 903 FORMAT!'-' ,9X,F10.2,13X,F10.2,18X,F10.2, 16X.F12.2) 172 905 CONTINUE 173 PRINT 910,WREODINL) 174 910 FORMAT^-' ,10X,'OPTIMUM CONSUMPTIVE USE =',F5.2,4X, 1•FEET PER YEAR * ) 175 PRINT 920,CULT 176 920 FORMAT!'-' ,10X,'OPTIMUM ACREAGE =*,F12.2,4X,'ACRES«) 177 PRINT 930,DAM 178 930 F0RMATC-',10X,'OPTIMUM RESERVOIR SIZE -',F12. 2, 4X 6,'ACRE-FEET'j 179 PRINT 940,GRT 180 940 FORMATC-' ,1GX,'EXPECTED BENEFIT =«,F15.2 t4X, 6'UTILITY') 181 STOP 182 END C 183 SUBROUTINE PREPiXX,ROW,COLM,SML,BIG) 184 REAL XXI60,30) 185 INTEGER SML<30),BIGC30) 186 INTEGER COLM.RGW 187 SML<1)=0 188 BIG(1)=0 189 DO 200 J=2,C0LM 190 DO 100 I=2,ROW 191 IF(XX(I,J)«GTo0»0) GO TO 120 192 100 CONTINUE 193 120 SML(J)=I 194 LF=I+1 195 DO 130 I=LF,ROW 196 IFCXXi I , J-UEQ.O..O) GO TO 140 197 130 CONTINUE 198 140 BIG(J)=I-1 199 200 CONTINUE 200 RETURN 201 END C 202 SUBROUTINE EXPECT(XX,NR0W,K0L,AB,XPC) 203 REAL XX<60,30),ABC30),XPC(30) 204 KKL=KOL—1 205 DO 100 JJ=1,KKL 206 I=JJ+1 207 100 AB(JJ) = XX(1,1) 99 208 KS=1 209 DO 400 K=2,K0L 210 DO 200 I=2,NR0W 211 LL=I 212 IF(XX(I,K)oGTo0 o0) GO TO 210 213 200 CONTINUE 214 210 SUM=0.0 215 DO 300 L=LL»NROW 216 TEMP=XX(L,K)*XX(L,1) 217 SUM=SUM-»-TEMP 218 IF(TEMP oEQ.0 o0) GO TO 350 219 300 CONTINUE 220 350 XPC(KS)=SUM 221 KS=KS+1 222 400 CONTINUE 223 RETURN 224 END C C SUBROUTINE TO COMBINE TWO MATRICES 225 SUBROUTINE CMBINE(CST,FST,SCCST,SCND,N1,N2,DELTA, 6NC0MB,SUM,SMPB) 226 DIMENSION CST(60),FST(60),SCCST(60),SCNDC60) 227 DIMENSION SUM(60),SMPB(60) 228 STAT=(CSTl1)+SCCST(1))-DELTA 229 T0P=CST(N1)+SCCST(N2) 230 PRL=CST(1) + SCCST(1) 231 IF(PRLoEQ.CST(1)) GO TO 100 232 GO TO 140 233 100 DO 120 K=1,N1 234 SUM(K) = CST(K ) 235 120 SMPB(K)=FST(K) 236 NC0MB=N1 237 GO TO 500 238 140 DO 400 1=1,60 239 PROB=0«.O 240 SMCST=0„0 241 PP=0„0 242 SUMCI)=STAT+(DELTA*I) 243 IF((SUM(I)-TOP).GE„DELTA) GO TO 410 244 DO 3 00 J=l,N1 245 00 200 M=1,N2 246 SMCST=CST(J)+SCCST(M) 247 PP=FST(J)*SCND(M) 248 IF(SMCST«LEe(SUM<I)-DELTA)) GO TO 200 249 IF(SMCST.LE.SUMd)) GO TO 160 250 IF(SMCST.GT.SUM(I)) GO TO 300 251 PRINT 150 252 150 FORMAT(* HELP I AM L0ST») 253 160 PR0B=PR0B+PP 254 200 CONTINUE 255 300 CONTINUE 256 SMPB(I)=PROB 257 400 CONTINUE 258 410 NC0MB=I-1 259 500 RETURN 260 END C 100 C***SUBROUTINE TO MATCH STREAM INFLOW WITH IRRIGATION C WATER REQUIREMENTS 261 SUBROUTINE WATERiYIELD,STFL,FLPRB,YAXIS,LYD,HYD, 6S0IL,PRC,V0LM,QA,C0ST,SLP,YC,BC, EEU ) 262 REAL YIELD{60,30),STFLt60),FLPRB(60),YAXIS{25) 263 INTEGER LYD(30),HYD(30) 264 MAX=21 265 SS=0. 0 266 IF(V0LM.EQo0.0) GO TO 710 267 DO 700 J=l ,40 268 M=2l 269 IF(STFL{JJoGEoQA) GO TO 500 270 PECNT=100.0*STFL{J J/VOLM 271 GO TO 540 272 500 PECNT=100o0*QA/V0LM 273 540 IF(PECNT.GTwlOO.O) GO TO 645 274 DO 600 K= 1,21 275 IF((YAXIS(M)-PECNT).GE.5.0) GO TO 550 276 IF((YAXIS(M)-PECNT).LE.2.5) GO TO 650 277 550 M=21-K 278 600 CONTINUE 279 645 M=MAX 280 650 K0L=M+1 C$$CALL A SUBROUTINE TO LOOK UP YIELD AND TO COMPUTE C NET ECONOMIC BENEFIT AND EXPECTED UTILITY 281 CALL PRODCE(YIELD,LYD,HYD,SOIL,PRC,SLP,YC,BC,KOL, 6C0ST,EU) 282 SS=SS+EU*FLPRB(J) 283 700 CONTINUE 284 710 EEU=SS 285 RETURN 286 END C C C SUBROUTINE TO DETERMINE EXPECTED UTILITY 287 SUBROUTINE PRODCEIYIELD,LYD,HYD,SOIL,PRC,SLP.YC,BC 6K0L,C0ST,EU) 288 REAL YIELD(60,30),BNFT(20),REVN(20),PRUT(20) 289 INTEGER LYD(30),HYD(30) 290 MIN=LYD(KOL) 291 LAGE=HYDIKOL) 292 NZ=0 293 DO 100 I=MIN,LAGE 294 NZ=NZ+1 295 REVN(NZ)=YIEL0(I,1)*S0IL*PRC 296 BNFT(NZ)=REVN(NZ)-COST 297 100 CONTINUE C***CALL A SUBROUTINE TO DETERMINE UTILITIES ASSOCIATED C WITH ABOVE ECONOMIC BENEFITS 298 CALL UT1LIT(BNFT,NZ,SLP,YC,BC,PRUT) 299 L=0 300 EU=0.0 301 DO 200 IN=MIN,LAGE 302 L=L+1 303 EU=EU+PRUT(L)*YIELD(IN,KOL) 304 200 CONTINUE 305 RETURN 306 END c C***SUBROUTINE TO CALCULATE EXPECTED UTIL IT IES 101 307 SUBROUTINE UT I L IT (BNFT ,N , SLP»YC ,BC ,PRUT J 308 REAL BNFT150 ), PRUT(50) 309 DO 600 1 = 1, N 310 I F (BNFTtDoLEoOoO) GO TO 400 311 I F (BNFT (D .GToBC ) GO TO 500 312 Y=YC-SLP*(BC-BNFT(1)3 313 GO TO 550 314 400 Y=-600000«0 315 GO TO 550 316 500 Y=BNFT(I) 317 550 PRUT(I)=Y 318 600 CONTINUE 319 RETURN 320 END C C SUBROUTINE TO CALCULATE THE EXPECTED COST 321 FUNCTION EXPCST(NCOMB,SUM,SMPB) 322 REAL SUM(60) ,SMPB<60) 323 £XPCST=0. 0 324 00 100 1=1, NCOMB 325 100 EXPCST=EXPCST+(SUM(I ) *SMPBU)) 326 RETURN 327 END C C SUBROUTINE TO CALCULATE MAXIMUM VALUES 328 SUBROUTINE MXBNFT(BNFT,NUM,NL » XMAX) 329 REAL BNFT(20) 330 NL=1 331 XMAX=-loOE 8 332 DO 100 1=1,NUM 333 IF iBNFTt IJoLEoXMAX) GO TO 100 334 XMAX=BNFT(I) 335 NL=I 336 100 CONTINUE 337 RETURN 338 END C C C C C C c C C C C c C C C C C SDATA 

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