MAN-MACHINE PROCEDURE FOR MULTIOBJECTIVE

CONTROL I N WATER RESOURCE SYSTEMS

I l y a V. Gouevsky

Ivan P . Popchev

June 1975

Research R e p o r t s are p u b l i c a t i o n s r e p o r t i n g on t h e work o f t h e a u t h o r s . Any views o r c o n c l u s i o n s a r e t h o s e o f t h e a u t h o r s , and do n o t n e c e s s a r i l y r e f l e c t t h o s e o f IIASA.

Man-Machine Procedure f o r M u l t i o b j e c t i v e

C o n t r o l i n Water Resource Svstems

I l y a V. ~ o u e v s k ~ ' and Ivan P. Popchev 2

A b s t r a c t

The f o r m u l a t i o n o f a p rocedure f o r m u l t i o b j e c t i v e c o n t r o l is p r e s e n t e d . The p rocedure h a s been developed under t h e assumpt ion t h a t t h e r e a r e two s u b s e t s of o b j e c t i v e s : t h e pr imary o b j e c t i v e and t h e secondary o b j e c t i v e s . The p r imary o b j e c t i v e i s used f o r d e f i n i n g a s c a l a r - v a l u e d o p t i m i z a t i o n problem. The secondary o b j e c t i v e s a r e b e i n g improved by a DM th rough changing a set o f p a r a m e t e r s i n t h e s c a l a r - v a l u e d o p t i m i z a t i o n problem. An i l l u s t r a t i v e example p r e s e n t e d , i n c l u d e s t h e p a r t o f t h e I s c a r R i v e r , B u l g a r i a . R e s u l t s a r e v e r y s a t i s f a c t o r y when compared w i t h t h e e x i s t i n g c o n t r o l p o l i c y .

1 . I n t r o d u c t i o n

Water r e s o u r c e sys tems c r e a t e s p e c i a l problems t h a t make

t h e a p p l i c a t i o n o f c l a s s i c a l d e c i s i o n making t e c h n i q u e s q u i t e

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

i n s i g h t , two q u i t e i m p o r t a n t c h a r a c t e r i s t i c s o f t h e s e sys tems

s t i l l s t r o n g l y i n f l u e n c e t h e i r management.

The f i r s t c h a r a c t e r i s t i c c r e a t e s t h e problem o f c o n t r o l

under u n c e r t a i n t y and r i s k i n v i r t u a l l y a l l w a t e r r e s o u r c e

d e c i s i o n s . Over t h e p a s t t h i r t y y e a r s , many p a p e r s have been

devo ted t o i n v e s t i g a t i n g t h e s t o c h a s t i c n a t u r e o f t h e p r o c e s s e s

i n w a t e r c o n t r o l sys tems and many models have been s u g g e s t e d

l ~ n t e r n a t i o n a l I n s t i t u t e f o r Appl ied Systems A n a l y s i s , S c h l o s s Laxenburg, A-2361 Laxenburg, A u s t r i a .

' ~ n s t i t u t e o f E n g i n e e r i n g C y b e r n e t i c s , B u l g a r i a n Academy of S c i e n c e s , S o f i a , B u l g a r i a .

The second characteristic creates the problem of control

under multiobjectives. This problem has become quite interesting

during the past ten years. The existence of a great number of

noncommensurable objectives in water resource systems makes

decision making more complex and requires applying the special

techniques of the multiobjective optimization.

In multiobjective optimization, two basic approaches exist

[131. The first approach comes out of the assumption that one

criterion dominates over the remaining criteria. Being primary,

this criterion can be used in classical scalar optimization

problem, where secondary objectives can be taken into account

through constraints.

The second approach deals with criteria which could not be

ordered (or at least to be divided into primary and secondary).

The techniques used when this approach is being applied could

be divided approximately into the following groups:

1) noninferior vector's technique [ 4 ,5,24,28], etc.;

2 ) ideal vector's technique [3,7,29], etc.;

3) the utility theory approaches [7,11,171, etc.;

4) the game theory approaches [1,14,19,271, etc.;

5) techniques using man-machine procedures [2,9110118, 25,261, etc.

For the time being, these five techniques have been

insufficiently implemented in water control systems. There

are several examples but they are still more methodologically

oriented studies concerning the development of the water

resources systems.

In this paper, an attempt is made to use an approach for

multiobjective optimization when there is one primary criterion.

In order to take into consideration the stochastic nature of

the processes in an implicit way, and to reduce computational

difficulties arising from treating the secondary criteria as

constraints, a man-machine procedure is used which is described

in detail below.

2. Basic Notations

Suppose, a decision maker (DM) in a water control system

has to solve the following optimization problem:

min {ql (XI, $2 (XI 1 . f$v(x) 1 X

subject to

where x is an n-dimensional vector of decision variables;

$Ji(x) , i = 1,2, ..., v are v objective functions; and G.(x), 3

j = 1, ..., m are constraint functions. The interrelation (3) denotes that the criterion $J1(x)

dominates over all the remaining criteria, but there is no

order between the criteria J12 (x) . . . $, (x) . The described

optimization problem could be solved by the techniques already

mentioned or by reducing it to the following scalar optimization

problem:

min Ql (XI X

subject to

where ti are the maximum admissible values of the criteria

Qi(x) i = 2 , , v The values of ti should be determined by the

DM on the basis of the DM'S preferences about the satisfaction

level of Qi(x), past experience, etc.

In many cases, especially with high dimensional

optimization problems and nonlinear functions G.(x) and I

Qi(x), the DM would encounter a great number of conputational

difficulties. Furthermore, better decision making is needed

to evaluate the sensitivity of the solution.

This means that the problem has to be solved many times,

and if the criteria Qi(x), i = 2.v are far from their bounds

ti, and at the same time Q1(x) has comparatively large value,

a compromize between these values should be found.

So as to reduce the computational difficulties, a

man-machine procedure is suggested in this paper. This

procedure is based on the following general assumptions:

A) The 3M could determine the values of Si, i = 2,3, ..., v in advance, or if values of Si have been suggested to the DM, he

can choose those which satisfy him.

B) The DM could state quantitative considerations which would

improve t h e v a l u e s o f a g i v e n c r i t e r i o n q i ( x ) , i = G.

T h i s means t h a t i n t h e problem, t h e r e i s a s e t o f p a r a m e t e r s

{pel which c o u l d be changed and t h e DM a t l e a s t knows t h e

d i r e c t i o n t h e d e c i s i o n h a s moved when {P,} i s changed.

C ) The DM c o u l d b e t t e r e v a l u a t e t h e d e c i s i o n i n t h e

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

D ) The DM h a s t o have t h e p o s s i b i l i t y t o f i n d a t l e a s t two

d i f f e r e n t d e c i s i o n s o b t a i n e d by d i f f e r e n t v a l u e s of some of

t h e p a r a m e t e r s P i n o r d e r t o make s u r e t h e d e c i s i o n i s

P a r e t o o p t i m a l .

3 . D e s c r i p t i o n o f t h e Procedure EVAL

The d e c i s i o n made by f o l l o w i n g t h i s p r o c e d u r e i s c a l l e d

" r a t i o n a l d e c i s i o n . " The main i d e a f o r f i n d i n g it i s based

on t h e p o s s i b i l i t y f o r i n t e r a c t i v e d i a l o g u e between a DM and

a computer .

The p r o c e d u r e c o n s i s t s o f t h e f o l l o w i n g s e v e r a l s t e p s .

1 ) A t t h e f i r s t s t a g e , o p t i m i z a t i o n problem ( 7 ) and ( 8 ) i s

s o l v e d :

s u b j e c t t o

(remember t h a t (x) i s t h e domina t ing c r i t e r i o n ) a s a r e s u l t

of t h i s s t a g e , t h e v e c t o r x1 i s o b t a i n e d .

2) The v a l u e o f a l l c r i t e r i a $ i ( ~ ) , i = 2 , 3 , ..., v a r e

-1 computed u s i n g t h e v e c t o r x , o b t a i n e d i n s t e p 1) above.

3 ) Choose t h e v a l u e of S i t i = 1 , v . I f t h e DM has no

i d e a a b o u t t h e magnitude of E i he c a n c o n t i n u e t h e p rocedure

1 assuming t h a t S i = i i ( x 1 .

1 4 ) The v a l u e s i i ( x ) and E i f o r i = 2 , . . . , v a r e compared:

I I f i i ( x ) 5 E i f o r a l l i = 2 , . . . , v t h e DM e i t h e r can d e c i d e

t o a c c e p t it a s a r a t i o n a l d e c i s i o n , o r go t o s t e p 5) f o r

1 improving it; i f i i ( x ) > E i f o r any i = 2, ..., v go t o s t e p 5 ) .

1 5) Analyzing t h e v a l u e s o f t h e f u n c t i o n i i ( x ) , i = 2 , . . . , v

t h e DH changes some o f t h e pa ramete r s o f t h e s e t {pe l i n such

a way t h a t he b e l i e ~ ~ e s t h e v a l u e s cou ld b e improved.

6) The DM d e t e r m i n e s t h e maximum v a l u e of c o n c e s s i o n toward

t h e v a l u e o f t h e f u n c t i o n i 1 ( x ) . L e t t h i s v a l u e b e denoted

w i t h E l . GO t o s t e p 1 ) .

T h i s p rocedure w i l l be implemented below f o r y e a r s

c o n t r o l o f a m u l t i p u r p o s e r e s e r v o i r .

4 . C o n t r o l o f Mul t ipurpose R e s e r v o i r

L e t us c o n s i d e r a r e s e r v o i r c r e a t e d f o r s u p p l y i n g w a t e r

f o r i n d u s t r i a l , munic ipa l and i r r i g a t i o n needs , a s w e l l a s f o r u s e

i n g e n e r a t i n g h y d r o e l e c t r i c power and f o r p r o v i d i n g f a c i l i t i e s f o r

f i s h i n g and r e c r e a t i o n . The whole p r o c e s s i n v o l v i n g c o n t r o l of a

r e s e r v o i r i s c o n s i d e r e d i n i n t e g e r t i m e s = 1 , ..., N . The s t r u c t u r e

o f t h e r e s e r v o i r f o r any s t a g e (month) s i s shown i n F i g u r e 1.

C o n t r o l v a r i a b l e s , i n t h e i n v e s t i g a t e d r e s e r v o i r ' s n ~ d e ,

S a r e t h e amount o f w a t e r yk a l l o c a t e d t o t h e kth u s e r a t t h e

sth s t a g e , k = I , n + l : s = and t h e amount of w a t e r i n t h e

r e s e r v o i r a t t h e sth s t a g e .

Upon t h e c o n t r o l v a r i a b l e s , denoted by t h e v e c t o r

S x = {ykf z s = m; k = lfn+l}, the following constraints s '

are imposed:

n+ 1 - Zs = min (MI Rs + yS Zs-1 - I YE) I s = lfN (10)

k= 1

where R is an input into the reservoir at the sth month. S

For determining the R = I R ~ } additional model, taking into

account that the stochastic nature of the input is needed,

ys is the evaporation coefficient at the sth stage, o < ys < 1

Gby y other kinds of losses could also be taken into consideration), S

M is the capacity of the reservoir;

Mo is the minimum admissible amount of water in the

I reservoir (under the value the released water i could not be used for municipal supply);

S v is the mandatory release for the kth user at the s th k

stage. This release allows for technological, I

S

'n+l i s t h e demand f o r d r i n k i n g w a t e r .

C o n s t r a i n t ( 9 ) means t h a t t h e amount o f r e l e a s e d w a t e r

a t t h e sth s t a g e c a n n o t b e i n e x c e s s of w a t e r a v a i l a b l e a t t h i s

s t a g e . C o n s t r a i n t (10) r e f l e c t s t h e r e s t r i c t i o n c a p a c i t y o f

t h e r e s e r v o i r a t e v e r y s t a g e .

C o n s t r a i n t s ( 9 ) , ( 1 0 ) , ( 1 1 ) , ( 1 2 ) and ( 1 3 ) form t h e

se t G ( x ) and t h e s e t c o m p r i s e s a l l a d m i s s i b l e s o l u t i o n s . The main

1 p roblem o f t h e DM i s t o c h o o s e s u b s e t G ( x ) o p t i m a l i n t h e s e n s e

of c r i t e r i a Q i ( x ) , i = 1 , V d e s c r i b e d below and a f t e r t h a t t o

1 * t r y t o r e d u c e t h e s u b s e t G ( x ) i n t o a s i n g l e d e c i s i o n x .

I t i s assumed t h a t t h e DM c a n d i v i d e a l l o f t h e c r i t e r i a

i n t o two g r o u p s . The f i r s t g r o u p c o n t a i n s o n l y t h e c r i t e r i o n

Q 1 ( x ) , w h i l e t h e s econd g r o u p c o n s i s t s o f s i x c r i t e r i a

F o r c o n v e n i e n c e i n compar ing t h e r e s u l t s , a l l c r i t e r i a

a r e n o r m a l i z e d i n t h e i n t e r v a l ( 0 1 ) Because o f t h i s ,

t h e mos t d e s i r a b l e l e v e l o f a l l c r i t e r i a i s 1 .

The a n a l y t i c a l e x p r e s s i o n of t h e c r i t e r i o n Q 1 ( x ) , . . . , Q 7 ( ~ )

i s a s f o l lows : N n 7 7 f S c t

where f E ( y E ) d e n o t e s t h e r e l a t i o n s h i p be tween t h e amount o f

w a t e r d i s t r i b u t e d t o t h e kth u s e r a t t h e sth s t a g e and t h e l o s s

( i n mone ta ry u n i t s ) o b t a i n e d b y t h e u s e r . The l o s s d e f i n e d a s

S above i s f ( v S ) , when yk = v s k k k I t i s assumed i n t h e p a p e r t h a t

the functions fS(yS) , for all k and s, are convex and piece-wise k k S nonlinear in the interval (vkI pz).

The second group of secondary objectives are more

interesting for the DM when a single reservoir is operated and thus

is described below.

$J~(x) - Users' Priority This criterion is based on the following assumptions. Let

us denote by tk the average degree of satisfying the

demand p , s = 1 of the kth user

and to each user an integer Ck& (l,n+l), is ascribed:

L R = 1, if max 5, = 5, k = 1 ,n+l

C = 2, if max q

- k = 1 ,n+l 5k - 5,

k # ,

C = n, if max (5 5,) = 5 P P' P

If the order of the users concerning the average degree 0

of satisfying the demand, preferably for the DM, is Ck, k = m,

t h e n t h e c r i t e r i o n Q2(x) cou ld be expressed i n t h e form

$ ? ( X I - E q u a b i l i t y o f S a t i s f y i n g t h e Users' Demand

Because o f a number o f u s e r s ' t e c h n o l o g i c a l p e c u l i a r i t i e s ,

n o t o n l y i s t h e amount of wa te r a l l o c a t e d t o a g iven u s e r o f

g r e a t importance , b u t a l s o t h a t t h i s amount shou ld be

even ly d i s t r i b u t e d . Q u a n t i t y e v a l u a t i o n of t h e s e requ i rements

cou ld b e made by d e f i n i n g t h e f u n c t i o n $ 3 ( x ) i n t h e fo l l owing

manner :

n+ 1 S s

(max gk - min gk ) k=l s S

QU(x ) - F l e x i b i l i t y o f t h e Decis ion

With t h i s c r i t e r i o n , t h e r e i s a p o s s i b i l i t y f o r r e a l l o c a t i o n

( i . e . changing t h e s o l u t i o n of t h e problem (7 ) and (8) b e f o r e

making t h e f i n a l d e c i s i o n ) i n t h e amount of wa t e r among t h e u s e r s

when some of t h e paramete rs { P ~ ] a r e changed. T h i s i n d i c a t e s

t h a t t h e DM does n o t have t o r e s t r a i n t h e se t G(x) v e r y much, i . e .

S t o p u t a h igh boundary on vk and Mo ( t h e h i g h e r v z and Mo t h e

more r e s t r a i n e d G ( x ) ) .

I f qk i s t h e number o f s t a g e s a t which t h e amount of

S wate r yk d i s t r i b u t e d t o t h e kth u s e r i s equa l t o t h e lower

S boundary v k , t hen q 4 ( x ) can be ob ta ined by t h e express ion :

$ c - S t a b i l i t y of t h e Decis ion

I t i s assumed t h a t f o r s o l v i n g t h e problem ( 7 ) and (8),

t h e mean va lue of t h e i n p u t R = {Rs) ob ta ined on t h e b a s i s of

h i s t o r i c a l d a t a i s used. By means of q 5 ( x ) t h e p o s s i b i l i t y f o r

implementation of t h i s d e c i s i o n i s eva lua t ed when v e c t o r R i s a

W W s t o c h a s t i c v a r i a b l e w i t h a number of va lues RW = ( R 1 , . . . ,RS,. . . ,$I ,

w = 1 , ..., 0 . 0 may simply e i t h e r denote t h e number of h i s t o r i c a l

d a t a a v a i l a b l e , o r it can be determined us ing a p p r o p r i a t e d a t a

p roces s ing .

To each v e c t o r RW an admis s ib l e se t GW(x) determined by

(9) , (1 0 ) , ( 1 1) , (1 2 ) , and ( 1 3 ) corresponds. I t i s assumed t h a t

t h e d e c i s i o n x i s admis s ib l e f o r t h e i n p u t RW i f x E GW(x) . I f

t h e v a r i a b l e hW i s in t roduced ,

then t h e c r i t e r i o n q 5 ( x ) could be determined a s fo l lows:

$ (x) - End S t a t e of t h e Reservoir

I n order t o meet f u t u r e demands it i s of s u b s t a n t i a l

i n t e r e s t t o allow f o r t h e end r e s e r v o i r ' s s t a t e . So t o

eva lua te t h i s c r i t e r i o n t h e following funct ion i s introduced:

where

Z N i s t h e r e s e r v o i r ' s s t a t e a t t h e Nth s t age .

$,(XI - Ecological E f f e c t of the Decision

For a region where t h e r e s e r v o i r i s t h e main water source,

t h e following f a c t o r s inf luence b a s i c a l l y t h e eco log ica l

equi l ibr ium (from t h e p o i n t of view of water resources o n l y ) :

a ) r e s e r v o i r l e v e l , o r , which i s t h e equ iva len t , t h e -

amount of water Z s , s = 1 , N .

b) t h e amount of waste water 9; d ischarged by

t h e kth use r i n t h e r i v e r below the r e s e r v o i r a t

t h e sth s t a g e , and t h e l e v e l A; of i t s p o l l u t i o n

( k € I W I where Iw is a s e t of indexes of use r s

d ischarging waste water ; 9' i s t h e c o e f f i c i e n t k

between o and 1) . c ) t h e i r r i g a t i o n regime of crops area- - insuff ic iency

of water could d i s t u r b the eco log ica l equi l ibr ium of

t h e r e g i o n . Th i s regime depends on t h e v a l u e o f

g; I a deg ree o f s a t i s f y i n g u s e r s ' demand f o r i r r i g a t i o n

( k c Iir, where Iir i s a se t of i ndexes of t h e c r o p s ) .

Taking i n t o accoun t t h e above f a c t o r s , t h e DM cou ld

e s t i m a t e t h e e c o l o g i c a l e f f e c t o f any a d m i s s i b l e d e c i s i o n

x th rough t h e f u n c t i o n $ ( x ) f o rmu la t ed , f o r i n s t a n c e , i n t h e Y

f o l l owing manner:

where 0 O s O s Z s , Xn and yn a r e r e s p e c t i v e l y t h e nece s sa ry v a l u e s ,

from t h e e c o l o g i c a l e q u i l i b r i u m p o i n t

S S - o f view, o f Z s , X n , and yk , s = 1,N, k r I i r ;

A i s t h e l e v e l o f p o l l u t i o n , o < hS < 1 , - k -

de te rmined fo l l owing [8] ( t h e most

p o l l u t e d wa t e r h a s a l e v e l h: = 0 ) ;

b l , b2 , b3 a r e c o e f f i c i e n t s i n d i c a t i n g t h e i n t e n s i t y

of t h e above t h r e e t e r m s on t h e e c o l o g i c a l 3

e q u i l i b r i u m ; C br = 1; br > O f o r a l l r . r = l

The s t a t e d seven c r i t e r i a do n o t cover t h e g r e a t v a r i e t y

of p o s s i b l e c r i t e r i a . I f neces sa ry , t h e DM could e i t h e r

complement t h i s set of c r i t e r i a o r d e f i n e ano the r one, t a k i n g

i n t o account t h e s p e c i f i c c o n d i t i o n s of t h e i n v e s t i g a t e d system.

The o p t i m i z a t i o n problem t h e DM encounte rs when he i s

seek ing t h e b e s t d e c i s i o n can be formal ized as a m u l t i o b j e c t i v e

non l inea r problem, i . e .

max ( X I , $2 ( X I , Y~ ( X I Y~ ( X I 1 Y~ ( X I 1 Y6 ( X I 1 $7 (XI ) (22)

X E G ( X )

where G(x) i s determined by ( 9 ) , (10) , (1 1 ) , (12) and (13) . Although it i s p o s s i b l e t o use convent iona l t echniques

f o r s o l v i n g t h e problem mentioned i n the beginning, t h e

complexity of c r i t e r i a , t h e i r s u b s t a n t i a l n o n l i n e a r c h a r a c t e r

and t h e need o f many r u n s of t h e problem s o as t o o b t a i n a

P a r e t o s u r f a c e , make t h e usage of t h e s e techniques very

d i f f i c u l t . For t h e s e reasons an a t t empt i s made below t o s o l v e

t h e problem (22) by t h e desc r ibed man-machine procedure .

5. I l l u s t r a t i v e Example

The d e s c r i b e d procedure f o r making t h e r a t i o n a l d e c i s i o n by

i n t e r a c t i v e man-machine d i a logue has been a p p l i e d f o r c o n t r o l of

a mul t ipurpose r e s e r v o i r f o r a t i m e pe r iod of one y e a r d iv ided

i n t o twelve months. The amount of w a t e r r e l e a s e d from t h e

r e s e r v o i r i s d i s t r i b u t e d t o t h e fo l lowing u s e r s : i n d u s t r i a l

wate r supply ( u s e r s No. 1 and No. 2 ) ; i r r i g a t i o n of d i f f e r e n t

c rops ( u s e r No. 3--wheat, No. 4--barley, No. 5--corn, No. 6--

corn f o r fodder , No. 7--vegetables , No. 8--lucerne, No. 9--

meadows and p a s t u r e s ) ; an a d d i t i o n a l amount of wate r g iven

f o r power gene ra t ion ( u s e r No. 10) ( excep t t h a t one d i s t r i -

buted t o t h e u s e r s No. 1 - No. 9 and used a l r e a d y f o r power

gene ra t ion ) ; and d r i n k i n g water supply ( u s e r No. 1 1 ) .

The d a t a needed f o r c a r r y i n g o u t t h e i t e m s 1 and 2 of

t h e procedure EVAL a r e a s fol lows:

1) i n p u t i n t h e r e s e r v o i r R = {Rs 1, s = 1,12

(Table 1 ) ; t h e choice of t h e v e c t o r R has been

done assuming t h e comparat ively b i g sho r t age

w i l l t a k e p l a c e du r ing t h e o p t i m i z a t i o n pe r iod ;

2 ) v e c t o r s RW = { R ~ I , s = 1 , 1 2 ; w = 1,10 , r e p r e s e n t

t h e h i s t o r i c a l d a t a a v a i l a b l e f o r t h e i n p u t R

of t h e r e s e r v o i r (Table 1) ;

S 3) lower vk and upper us bounds f o r a l l s = 1 , I 2 k

and k = (Table 3) ;

evapora t ion and o t h e r l o s s e s Y = { Y 1 a r e i n t h e S

r e s e r v o i r (Table 1) ; 0 0

4 ) t h e c o n s t a n t s M , M o t Z s f A k f e k f b l , b 2 , b3, ck ,

k = 1 , I 1 a r e shown i n Table 2 ;

S -s 5) t h e l o s s f u n c t i o n s f ( y ) , a l l k and s a r e p iece- k k

w i s e n o n l i n e a r and concave.

where

The c o e f f i c i e n t s o f t h e s e f u n c t i o n s a r e shown i n Tab le 3.

The se t o f p a r a m e t e r s {PE} used by t h e DM i n t h e p rocedure

S EVAL a r e : lower vS and upper p k b o u n d a r i e s and sometimes t h e k

S -S c o e f f i c i e n t s o f t h e l o s s f u n c t i o n s f k ( y k ) and t h e i n p u t R .

A l so a l l of t h e rest v a r i a b l e s and c o e f f i c i e n t s c a n be i n c l u d e d

i n {PE} i f t h e y a r e c o n s i d e r e d a s i n a c c u r a t e l y d e f i n e d .

Using t h e method d e s c r i b e d i n [ 1 8 ] , t h e f o l l o w i n g r e s u l t s

have been o b t a i n e d :

a ) t h e optimum amount o f w a t e r , a c c o r d i n g t o t h e f i r s t

i t e m o f EVAL, which s h o u l d be d i s t r i b u t e d among t h e

u s e r s d u r i n g t h e o p t i m i z a t i o n p e r i o d . I n F i g u r e s

2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 and 12 t h e g r a p h i c a l

r e s u l t s a r e shown ( u s i n g r i g h t h a t c h ) . The a n a l y t i c a l

s o l u t i o n is shown i n T a b l e 4 .

b ) t h e s t a t e o f t h e r e s e r v o i r d u r i n g t h e o p t i m i z a t i o n

p e r i o d ( F i g u r e 1 3 ) . The s t a t e o f t h e r e s e r v o i r when

t h e v e c t o r RW, w = 1,10 i s changed i s a l s o shown i n

t h i s f i g u r e . T h i s s t a t e i s computed under t h e

assumpt ions t h a t t h e a l l o c a t i o n which have been

a l r e a d y a c c e p t e d c o u l d n o t b e changed, i . e . t h e

q u e s t i o n i s answered, what w i l l happen i f RW changes

b u t v e c t o r x, and r e s p e c t i v e l y i t s components yS k t

does n o t .

1 1 C ) t h e v a l u e s o f t h e c r i t e r i a + ( X , . . . ,+ ( X ) ( t h e

1 b i g g e r v a l u e o f + ( x ) t h e more a c c e p t a b l e v e c t o r

x l ) ( T a b l e 6 ) . Having a l l t h i s i n f o r m a t i o n , a s w e l l a s some of t h e i n t e r m e d i a t e

c o m p u t a t i o n a l r e s u l t s , t h e DM goes t o i t e m t h r e e o f t h e

p r o c e d u r e EVAL, namely e v a l u a t i o n of t h e r e s u l t s which have

a l r e a d y been o b t a i n e d .

What do t h e r e s u l t s o b t a i n i n g a t t h e f i r s t s t a g e of t h e

p r o c e d u r e EVAL f o r t h i s p a r t i c u l a r sys tem show?

1) Users N1 and N2 ( i n d u s t r i a l w a t e r s u p p l y ) o b t a i n i n

t h e months I , 11, I11 and X t h e whole demanded w a t e r . During

t h e remain ing months t h e maximum demand d e v i a t i o n i s .98%.

2) User N11 ( d r i n k i n g w a t e r s u p p l y ) i s f u l l y s a t i s f i e d

a l l t h e t i m e .

3) U s e r N3 g e t s 18.86% a s i t s demand o c c u r s o n l y i n

Oc tober . User N9 does n o t g e t w a t e r a t a l l . There a r e a t

l e a s t two r e a s o n s f o r t h i s : a ) t h e s e c r o p s a r e n o t i m p o r t a n t

f o r t h e sys tem c o n s i d e r e d ; b ) i f a ) i s n o t t r u e , t h e n t h e

o b j e c t i v e f u n c t i o n s o f t h e s e c r o p s w e r e n o t de te rmined p r e c i s e l y .

4 ) Some o f t h e u s e r s , i .e . N 4 and N5, do n o t have a

uni form s a t i s f a c t i o n o f t h e i r demands.

5 ) The end s t a t e of t h e r e s e r v o i r e q u a l s t h e minimum

a d m i s s i b l e amount of w a t e r i n it.

6 ) F i g u r e 13 i n d i c a t e s t h e i n f l u e n c e of t h e s t o c h a s t i c

i n p u t r e p r e s e n t e d by t h e v e c t o r R ~ , w = 1 , l o . I f w = 3 , 4 , 6 , 7 , 8

and 9 , t h e n a f t e r August t h e r e s e r v o i r w i l l be empty. I f

= 1 , 2 , 5 and 10, t h e n t h e r e w i l l be enough wa t e r even f o r

complete s a t i s f a c t i o n of t h e u s e r s ' demands i n t h e r e s e r v o i r .

7) I t can be s een t h a t t h e r e i s a d i f f e r e n c e between

a p rede te rmined o r d e r o f t h e u s e r s and t h i s one a l r e a d y

o b t a i n e d concern ing t h e u s e r s Nos. 3 , 4 , 5 , 6 , 7 and 9 .

Such a d i f f e r e n c e may be caused by t h e fo l l owing :

a ) t h e p rede te rmined o r d e r r e f l e c t i n g t h e s u b j u n c t i v e

o p i n i o n o f t h e DM i s wrong;

b ) t h e o b j e c t i v e f u n c t i o n o f t h e u s e r s a r e n o t

de te rmined e x a c t l y .

8) User No. 10 (hydro power g e n e r a t i o n ) o b t a i n s

6 3 20.5475 . 10 m a d d i t i o n a l wa t e r . T h i s a d d i t i o n a l amount of

wa t e r c anno t be used by t h e o t h e r u s e r s d u r i n g t h e " n o n i r r i -

ga t ed" months and i n t h e c o n d i t i o n o f s h o r t a g e cou ld be " a

p e c u l i a r l uxu ry" t o produce energy by w a t e r .

Taking i n t o c o n s i d e r a t i o n t h e r e s u l t s o b t a i n e d a t t h e

f i r s t s t a g e o f p rocedure EVAL, t h e DM h a s dec ided f i r s t o f

S a l l t o reduce t h e upper bound p10 of u s e r No. 10. The new

s o l u t i o n i s shown i n Tab le 5 and i n F i g u r e s 2 , 3 , 4 , 5 , 6 , 7 ,

8 , 9 , 10 , 11 , 12 and 14 u s ing l e f t h a t c h . A t t h a t , t h e

second s o l u t i o n i s b e t t e r w i t h r e s p e c t t o t h e c r i t e r i o n b ( x ) , i . e .

e q u a b i l i t y i n s a t i s f y i n g u s e r s ' demands h a s been improved.

Using o t h e r pa ramete r s of t h e set {P ), t h e DM cou ld R

improve t h e s o l u t i o n of t h e problem. Th is p r o c e s s cou ld go

on u n t i l t h e DM f i n d s t h e s a t i s f y i n g compromise among t h e

v a l u e s o f t h e c r i t e r i a (x) , . . . , $7 (x) .

6. Computational Time

For carrying out the procedure EVAL a computer program was

developed. The program consists of three subroutines: pre-

parative subroutine, subroutine for optimization and subroutine

S S for introducing the functions fk(yk). The latter permits the

introduction of piece-wise, concave, nonlinear functions of

any type.

A single reservoir's optimization model comprising 120

variables was executed in about 0.25 minutes on CDC 6600. This

included all computations needed for carrying out the first and

the second items in the procedure EVAL.

7. Conclusions

The procedure EVAL presented here can be used when multi-

objectives are divided into two subsets: a primary objective

and the secondary objectives. In many practical applications

in water resource systems, economic efficiency usually can be

asserted as being primary and the secondary objectives imply

additional goals have to be achieved.

Using this procedure, instead of conventional techniques

for multiobjective programming, a closer contact between a

system investigator (SI) and the DM can be established.

Furthermore, such an interactive approach enables both SI and

DM to define parameters more precisely in the optimization

model. Another advantage of this approach is that the complexity

of an optimization problem does not depend on the mathematical

structure of the secondary objectives at all.

The p rocedure c a n a l s o be implemented i n a m u l t i r e s e r v o i r

w a t e r r e s o u r c e s y s t e m and i n m o r e g e n e r a l gaming s i t u a t i o n s

between d i f f e r e n t D M ' S who a g r e e t h a t one of t h e c r i t e r i a i s

p r imary . A f t e r chang ing p a r a m e t e r s o f t h e o p t i m i z a t i o n model

i n a d i r e c t i o n t h e y t h i n k i s a p p r o p r i a t e and a f t e r e v a l u a t i o n

o f r e s u l t s , a compromise between them c o u l d be a c h i e v e d .

Ta

ble

1

.

Rs

- in

pu

t i

n t

he

re

se

rvo

ir

1 R

I

..

.I

R1

- in

pu

t i

n t

he

re

se

rvo

ir

(av

ail

ab

le h

isto

ric

al

da

ta)

s -

ev

ap

ora

tio

n

co

ef

fic

ien

t 0 Z

s -

de

sir

ab

le s

tate

of

the

re

se

rvo

ir

0 7

m

03

I.

In

3

m

CV

7

.q

0 0 03

7 0

0 0 w 7 0

7

0 0

0 w

0

k ar

0 c, w

0 S ar c, II)

0 w

S a

0 ar m k rb A

0 U In cn

-4 0

w a

0 7

a G a

0 0 CV w 7 m k

In ar 0 0 II c,

0 N

S X X .. a

0 0 ar In 7 rl rl I. w -4

a rl rl a 0 rl o o rb rd 7 G a

o o a ~1 k k II 0 -4 !Y

c, k

0 0 m 0 cn a, w w n cv !Y U)

0 0 d .. d G 3

CV 03 CV CV II ar CV 7 X 7 X 7 0 $ ar

o o 4 a m a A c0 II II c, c,

0 w ar II . . 0 .Q w N IO

0 rl 0

0 In II II A 7

0 m m cnX cnX ,. ar -4 X

7 4 a c, G 0 0 II II CV m a, Id rb

CV I. d k k 3

0 I I I II II II II

cn X m'X X Y X 7 X 7 cnx r n ~ X 4 ou 4 a n G 4 a ou

o m m

3 r r- 0 w m o

o m m I

o m m I

3 C

r- 0 w m 0

o m m

o m m I

f 7

r- 0 w m 0

o m m I

3 7

r- 0 w m o

o m m I

0 m m I

f r I- 0 w m 0

o m m I

3 -

m m P. w OD ln

m N I

m ln m r- co w l n

m N I

o m m I

w w w w w w 0 1 0 0 0 0 0 - - - - - - X X X X X X

W N m N m N V1 N I n N m N t n c W m a W I > & l

I I I I I I I I I

- m o m - m

m , 0 ; ; 2 r m N

? O O O O m r

0 " o d d d d o N t

2 I I

L n o N m N - t m w w w o l - t o r O O r O O O W r

. O . 0 0 0 0 0 0 o m I I

I I I I I I I I I

I I I I I I I I I

X X X X X X X mm mm m o m o m o corn m o m o m o P I m w m c 3 u > ! 4 7

I l l 1 I I I I I

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I

7 7 r - - - r X X X X X X X

rna m * m t rna m a r n t c o t a t m a P I m w m ? 3 u > & x

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I

X X X X X X X mm mm mm m m mm mm mm mm mm & a L o c a a u ; ) & l

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I

I I I l l I I I I

I I I I I I I I I

I I I I I I I I I 3 m o o o 0 0 o m 3 0 0 0 0 o m - ~ N N V) 0 l - n r 3 w m o o r - o

c o o o o o o 7 7

0 I I U

. - 7 - - x - 7

X X X X X X m p m l - m l - m l - m l - m l - m l - cnl- m l - p . r ~ s r n t ) w > k z

I I I I I I I I I

I I I I I I I I I

I I I I I I I I I I

I I I I I I I I I

w w w w w w w 1 0 0 1 0 0 0 0 0 7 7 7 - 7 - 7

X X X X X X X m a 3 m a 3 m a 3 m a 3 m w m w m a 3 m a 3 m w P l r J w r n a w 2 k 3

I I I I

I I I I

m m w w a = t m u l * O W N 0 0 0 0

0 0 0 0 I I

0 - N 0 f I n . - 0 w - w - - o m

w O O O I I I O I I

I I I I

I I I I

I I I I

Ta

ble

4

.

N 7

7

7

0 -

cn

03

b

0 m 7 03 m

* I I I I I I I w N 0 7

r

0 r 7 b N

* 1 1 1 1 1 1 1 W N 0 7

7

W N m 0 3 0 w 0 3 0 w 0 N O 0 3 1 I l l l W l n

w N 0 0 7 7

m 03 7 0 0 3 0 3 7 3 0 - 0 m 0 3 0 c n m m o 0 0 0 m I b 0 3 m I V ) N

w C\1 0 3 0 m 0 7

7

m I. m o o c n cn cn 0 0 0 m 7

3 b N o m m o o o N O 1 l W 3 N K - l 0 Ln

W N N W - N 0 7

7

QI CT\ 3 0 0 0 CT\ Q\ 3 0 0 0 7

3 I- w 0 m 0 0 0 0 N O 1 I c n m N m 0 m

w N o m - m o 7

7

03 m m o cn o o m 0 m 7

m 03 m o m 0 0 0 0 0 3 1 I I O W O W . a F 7 7 C*l 0 0

7

w m m rn cn m - N 7 m m 03 0 b m m 0 0 QI 3 1 I b W I V ) O

W 7 0 7 N 0 7 7

0 3 N w C\1 N L n C X ) O N 0 CT\ 03 b I I I I l a w

m - 0 0 0

0 - cn b 0 * I 1 1 1 1 1 1

w 7 0 7

7

0 m b 03 cn

I I I 1 1 1 1 m 7 0 cn

0 - cn cn 0 * 1 1 1 1 1 1 1

w 7 0 7

7

O F - N m a U l W b Q c n r -

water supply of d i f f e r e n t

Zs-l 4 I z - - - -Ccs

A

F i g u r e 1.

y:

c r o p s 10 - hydro power s t a t i o n 11 - d r i n k i n g water supply

y ;

E l

I - r e s e r v o i r 1 and 2 - i n d u s t r i a l

3 . 4 , ..., 9 - i r r i g a t i o n

k demand

o p t i m a l d i s t r i b u t i o n ( I r u n )

o p t i m a l d i s t r i b u t i c n (IT r u n )

+ s ( m o n t h s )

s t F i g u r e 2 . Demand and d i s t r i b u t e d amount o f w a t e r t o t h e 1 u s e r

F i g u r e 3 . Ijcmand and d i s t r u b u t e d amount o f w a t e r t o t h e 2nd u s e r

r LI F i g u r e 4 . Demand a n d d i s t r u b u t e d amount of w a t e r t o t h e 3 u s e r

t t1 F i r 5 . I)i.rnand a n d d i s t r i b u t e d amount o f w a t e r t o tilt, 4 c lser

demand

o p t io ic l l d i s t ~ - i b l ~ ~ i o n

( I r u n )

O ~ L i n u 1 (1 i sLr i I I U L i 0 1 1

( I I t-1112)

figure. h . Demnncl al~tl d i s t r i b u t e d amount o f w a t e r t o t h e 5 t h r1st.r

s ( m o n t h s )

-. . t I1 1 1 , . Drmnnd and d i s t r i b u t e d amount of w a t e r t o t h c h u s e r

s (mont 11s) 2 3 4 5 6 7 8 9 I 0 11 I 2

t h F i g u r e 8 . Demand a n d d i s t r i b u t e d a m o u n t o f w a t e r t o t h e 7 u s e r

s s 6 3 P, yaxLO [m I

d enia nd

o p t i m a l d i s t r i b n r i o r ( I r u n )

o p t i m a l d i s t r i b u t i o r ( T I r u n )

F i g u r e 9 . Demand a n d d i s t r i b u t e d amoun t o f w a t e r t o

t 11 F i g u r e 10. Demand a n d d i s t r i b u t e d a m o u n t of w a t e r t o t h e 9 u s e r

demand

o p t i m a l d i s t r i b u t i o ~ i ( I r u n )

o p t i m a l d i s t r i b u t i o n (I1 r u n )

F i g u r e 11. Demand and d i s t r i b u t e d amount of w a t e r t o t h e 10th u s e r

s (months) I I 2 3 4 5 6 7 8 9 T n T T I 2

F i g u r e 1 2 . Demand and d i s t r i b u t e d amount of w a t e r t o t h e lrth u s e r

- s (months)

Figure 1 4 .

s ( m o n t h s )

References

[ I ] Belenson, S.M. and Kapur, C.K. "An Algor i thm f o r S o l v i n g M u l t i c r i t e r i o n Programming Problem w i t h Examples." O p e r a t i o n a l Research Q u a r t e r l y , 24, 1 (1973) .

[ 2 ] Benayoun, R. and Terguy, J. " C r i t e r e s M u l t i p l e s e t Programat ion Mathematique: une S o l u t i o n dans l e c a s l i n e a i r e . " Rev. F ranc . In fo rm. e t Rech. Oper . , 3 , - V-2 (1969) .

[ 3 ] B o r i s o v , V . I . "On Vector-Valued O p t i m i z a t i o n Problems." ( I n R u s s i a n ) . P roceed ings of t h e 1 s t O p e r a t i o n Research Conference . Minsk, 1972.

[ 4 ] Boychuck, L.M. and Ovchinnikov, V.O. " B a s i c Approaches f o r S o l v i n g M u l t i o b j e c t i v e O p t i m i z a t i o n Problems (Survey) ." ( I n Russ ian) . Automat ica , - 1 (1973) .

[5 ] Ching, C . K . "On t h e N o n i n f e r i o r S e t o f t h e Systems w i t h Vector-Valued O b j e c t i v e Func t ion . " IEEE T r a n s a c t i o n s on Automatic C o n t r o l , - 15, 5 ( 1 9 7 0 ) .

[ 6 ] C l a r k e , R.T. "Mathemat ica l Models i n Hydrology, Food and A g r i c u l t u r e O r g a n i z a t i o n o f t h e Uni ted Na t ions . " Rome, 1973.

[ 7 ] Dinkelbach, W . " ~ b e r e i n e n Losungsansatz zum Vector- maximumproblem." L e c t . Notes Oper. R e s . and Math. S y s t . , 50 ( 1 9 7 1 ) .

[ 8 ] Dinus, S.H. " S o c i a l Accounting System f o r E v a l u a t i n g Water Resources . " Water Resources Resea rch , - 8 , 5 (1972) .

[ 9 ] Ducks te in , L . , Monarchi, D . , and K i s i e l , C .C . " I n t e r a c t i v e ~ u l t i - O b j e c t i v e D e c i s i o n Making Under U n c e r t a i n t y . " NATO Conference on "The Role and E f f e c t i v e n e s s of D e c i s i o n T h e o r i e s i n P r a c t i c e . " Luxembourg, August 1973.

[ l o ] E n g l i s h , R . , K i s i e l , C . C . , and Ducks te in , L . "An I n t e r - a c t i v e Algor i thm f o r t h e Mul t ipurpose C o n t r o l o f a N a t u r a l Lake." Symposium on t h e Hydrology o f Lakes. H e l s i n k i , 1973.

[ I l l F i s c h b u r n , P.C. U t i l i t y Theory f o r ~ e c i s i o n Making. N e w York, Academic P r e s s , 1970.

[I21 Gouevsky, I. "Decomposition Algorithm for a Class of Nonlinear Programming Problems." Proceedings of the Institute of Engineering Cybernetics, Bulgarian Academy of Sciences, Vol. 18, Sofia, 1974.

[I31 Haimes, Y.Y. and Hall, W.A. "Multiobjectives in Water Resource Systems Analysis: The Surragate Worth Trade Off Method." Water Resources Research, - 10, 4 (August 1974).

[I41 Haurie, A. "On Pareto Optimal Decisions for a Coalition of a Subset of Players." IEEE Transactions on Automatic Control, AC-18, 2 (1973).

[15] Hufschmidt, M.M. and Fiering, M.B. Simulation Techniques for Design of Water Resource Systems. Cambridge, Mass., Harvard University Press, 1966.

[16] Kaczmarek, Z. "Some Problems of Stochastic Storage with Correlated Inflow. Proceedings of the International Symposium on Mathematical Models in Hydrology. 26-31 July 1971. Vol. 1, Part 1. Warsaw.

[I71 Kapur, K.C. "Mathematical Methods of Optimization for - Multiob jective Transportation systems. " Socio. Econ. Plan. Sci.. 4. 4 (1970).

[18] Larichev, 0.1. "Man-Machine Procedure for Decision Making (Survey) ." (In Russian). Automation and Telemechanics, 12 (1 971) .

[I91 Luce, R.D. and Raiffa, H. Games and Decisions. New York, John Wiley & Sons, 1958.

[20] Moran, P.A.P. The Theory of Storage. London, Methuen, 1959.

[21] Polyak, R.A. "An Approach to ~ultiobjective Programming Problems." (In Russian). Proceedings of the 4th Winter School on Mathematical Programming. Vol. 2, 1971.

[ 221 Prabhu, N .U. "Time-Dependent Results in Storage Theory. 'I Methuen's Review Series in Applied Probability, Vol. 1 , 1964.

[23] Proceedings of the International Symposium on Uncertainties in Hydrology and Water Resource Systems. December 11-14, 1972. Vol. 3, Tucson, Arizonq, University of Arizona.

[24] Reid , R.W. and Vemuri, V. "On t h e N o n i n f e r i o r Index Approach t o Large-Scale M u l t i - C r i t e r i a Systems." J. F r a n k l i n I n s t . , - 291, 2 ( 1 9 7 1 ) .

[25] Roy, B . "Classement e t c h o i x e n p r e s e n c e d e p o i n t s d e vue m u l t i p l e s ( l a methode ELECTRE) ." Rev. F ranc . Inform. e t Rech. Operat., - 2 (1968) .

[26] S a v i r , D . " M u l t i o b j e c t i v e L i n e a r Programming." O p e r a t i o n s Research C e n t e r , U n i v e r s i t y oG C a l i f o r n i a , November 1 966.

[271 Schrni tendorf , W.E. "Co-opera t ive G a m e s and Vector- Valued Cr i t e r i a ~ r o b i e m s . " IEEE T r a n s a c t i o n s on Automat ic C o n t r o l , AC-18, 2 ( 1 9 7 3 ) .

[28] Zadeh, L.A. " O p t i m a l i t y and Non-Scalar-Valued Performance criteria. "- IEEE ~ r a n s a c t i o n s on Automatic C o n t r o l , AC-8, 1 ( 1 9 6 3 ) .

[29] Zak, F.A. "Models and Methods f o r F i n d i n g N o n i n f e r i o r S o l u t i o n s i n Vector-Valued O p t i m i z a t i o n Problems." ( I n Russ ian) . C y b e r n e t i c s , 4 (1 972) .