W O R K I N G P A P E R
LINKAGE OF REGImAL MODELS
Murat Albegov Alexander Umnov
July 1981 WP-81-85
I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis
NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
LINKAGE OF REGIONAL MODELS
Murat Albegov A l e x a n d e r Umnov J u l y 1981
TC-81-85
W o r k i n g P a p e r s a r e i n t e r i m r e p o r t s on work o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s and have r e c e i v e d o n l y l i m i t e d r e v i e w . Views o r o p i n i o n s e x p r e s s e d h e r e i n do 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 o s e o f t h e I n s t i t u t e o r o f i t s N a t i o n a l Member O r g a n i z a t i o n s .
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , A u s t r i a
For s e v e r a l y e a r s t h e a c t i v i t i e s o f t h e Regional Development Task a t the I n t e r n a t i o n a l I n s t i t u t e f o r Applied Systems A n a l y s i s
(IIASA) have been d i r e c t e d towards t h e development o f a system of r e g i o n a l models, t h e e l e m e n t s o f which were e l a b o r a t e d o v e r . t h e p e r i o d 1977-1979. The f i n a l s t a g e of t h e work, which i n v o l v e s t h e c o o r d i n a t i o n of t h e s e i n d i v i d u a l l y developed models, i s now n e a r i n g completion. However, b e f o r e t h i s system can become f u l l y o p e r a t i o n a l , t h r e e major problems have t o be overcome. They con- c e r n t h e modeling approach, l e v e l o f a g g r e g a t i o n , and method o f c o o r d i n a t i o n t o be used. The l i n k a g e problem i s examined i n t h i s p a p e r .
LINKAGE OF REGIONAL MODELS Murat Alhegov
*
Alexander Umnov
INTRODUCTION
Regions-have complex economic s t r u c t u r e s and, i n most c a s e s , a s p e c i f i c s e t o f f u t u r e development problems. The number o f
a g g r e g a t e d s e c t o r s o f a r e g i o n a l economy can i n c l u d e 10 upwards.
I f s e c t o r a l development i s c o n s i d e r e d i n m u l t i d i m e n s i o n a l t e r m s , it r e q u i r e s t h e s o l u t i o n o f numerous problems. I t i s , t h e r e f o r e , i m p o s s i b l e t o d e s c r i b e a s y s t e m o f models t h a t e m b r a c e s a l l p r o - blems and i s a p p r o p r i a t e f o r a l l r e g i o n s . However, i n g e n e r a l o n l y a small number o f key s e c t o r s of t h e r e g i o n a l economy i n f l u - e n c e i t s f u t u r e development.
An approach t h a t seems t o b e s u i t a b l e f o r d e a l i n g w i t h a l l t y p e s o f r e g i o n s i s one t h a t i n c l u d e s module-type d e s c r i p t i o n s o f a l l t h e more i m p o r t a n t s e c t o r s o f a r e g i o n a l economy. A l i m i t e d number o f t h e s e modules c a n t h e n be s e l e c t e d , a d a p t e d , and l i n k e d t o form a s y s t e m of models t h a t r e f l e c t s t h e u r g e n t development problems o f t h e r e g i o n under a n a l y s i s . T h i s a p p r o a c h i m p l i e s t h a t e a c h module s h o u l d be s u f f i c i e n t l y g e n e r a l t o be w i d e l y a p p l i c a b l e and y e t a t t h e same t i m e f l e x i b l e enough f o r a d a p t a t i o n t o t h e s p e c i f i c problems o f d i f f e r e n t r e g i o n s .
* P r o f e s s o r Murat Albegov l e d t h e R e g i o n a l Development Task a t IIASA from 1977 t o 1980. H e i s c u r r e n t l y a t t h e C e n t r a l Economics and Mathematics I n s t i t u t e , Moscow.
**In t h i s p a p e r t h e r e g i o n i s t r e a t e d a s a u n i f i e d t e r r i t o r y , which i s homogeneous w i t h r e s p e c t t o economic, s o c i a l , e n v i r o n - m e n t a l , and i n s t i t u t i o n a l problems.
The advantage of a d o p t i n g t h e approach d e s c r i b e d above i s c l e a r : t h e g e n e r a l s e c t o r a l modules s e r v e a s a b a s i s f o r d e v e l o p i n g w i d e l y a p p l i c a b l e models. The model s y s t e m s h o u l d t h e n be formed o n l y from t h o s e modules t h a t a r e e s s e n t i a l f o r s o l v i n g t h e problems o f t h e given r e g i o n .
GENERAL APPROACH
There a r e many approaches t o r e g i o n a l development modeling.
One p o s s i b l e method o f c l a s s i f y i n g t h e s e approaches i s t o . examine t h e ' sequence of a n a l y s i s adopted. E x t e r n a l o r i n t e r n a l problems a r e g e n e r a l l y t h e s t a r t i n g p o i n t s f o r a n i t e r a t i v e p r o c e d u r e . T h e r e a f t e r , t h e 'bottom-up' approach i s used ( f o r d e t a i l s , s e e Andersson and P h i l i p o v 1979, pp. 33-69)
.
T h i s approach i s b a s e d on t h e assumption t h a t t h e m a r g i n a l c o s t s f o r commodities produced and r e s o u r c e s u s e d ' a s w e l l a s t h e d a t a f o r d e t e r m i n i n g r e g i o n a l i n - and o u t - m i g r a t i o n f l o w s ( a v e r a g e n a t i o n a l s a l a r y , d w e l l i n g s p a c e p e r c a p i t a , e t c . ) i s known ( F i g u r e 1 ) . The s t a r t i n g p o i n t i s t h e a n a l y s i s o f t h e r e g i o n a l s p e c i a l i z a - t i o n problem. A t l e v e l 2 , i n t r a r e g i o n a l l o c a t i o n problems a r e
s o l v e d , f o l l o w e d by an a n a l y s i s o f l a b o r and f i n a n c i a l b a l a n c e problems a t l e v e l 3. F i n a l l y , a t l e v e l 4 , problems c o n n e c t e d w i t h e n v i r o n m e n t a l q u a l i t y c o n t r o l a s w e l l a s s e t t l e m e n t s and s e r v i c e p r o v i s i o n a r e c o n s i d e r e d . I n t h i s scheme, c o o r d i n a t i o n between l e v e l s I and I1 and l e v e l s I11 and I V i s e s s e n t i a l l y t h a t o f e s t i m a t i n g f u t u r e r e g i o n a l economic growth and t h e s i z e o f t h e l a b o r f o r c e .
A s c a n be s e e n from F i g u r e 1 , t h e scheme i n c l u d e s many b l o c k s (models) and i s r a t h e r c o m p l i c a t e d t o compute. However, t h e
number o f u r g e n t problems t o be s o l v e d i n a g i v e n r e g i o n i s u s u a l l y f a i r l y small. For example, a d i s c u s s i o n between IIASA
members and l o c a l d e c i s i o n makers f o r t h e S i l i s t r a r e g i o n ( B u l g a r i a ) r e v e a l e d t h a t t h e r e a r e o n l y s i x o b j e c t i v e s f o r t h i s r e g i o n :
1. To maximize r e g i o n a l a g r i c u l t u r a l p r o d u c t i o n . T h i s s h o u l d i n v o l v e n o t o n l y t h e maximization o f meat and g r a i n p r o d u c t i o n , f o r which t h e a r e a i s p a r t i c u l a r l y w e l l s u i t e d , b u t a l s o t h e i n c r e a s e of l o c a l c r o p pro- d u c t i o n ( a p r i c o t s , g r a p e s , and v e g e t a b l e s )
.
1
1
I1
.
SPECIALIZATION ILOCATION OF I
I SERVICES I
LOCATION OF INDUSTRY AGRICULTURE AND
PUBLIC S E R V ~ E S
GROWTH AND GROWTH AND LOCATION OF LOCATION OF
INDUSTRY SYSTEM SUPPLY
POPULATION MIGRATION
3. LABOR, CAPITAL
RESOURCES INVESTMENT
.
IURBAN
SETTLEMENTS POLLUTION
EXTERNAL DATA
--* ITERATIVE PRECISION FEEDBACK DATA
-
INFORMATION FLOWFIXED DIRECTION OF SPECIALIZATION
F i g u r e 1 . T h e bottom-up approach.
2 . To develop an i r r i g a t i o n system t h a t w i l l e n a b l e l o c a l a g r i c u l t u r e t o a c h i e v e o p t i m a l p r o d u c t i o n e f f i c i e n c y
.
3 . To develop l o c a l i n d u s t r y t o complement l o c a l a g r i - c u l t u r e . This s h o u l d i n c l u d e t h e developnent of some branches o f i n d u s t r y t h a t have t h e p o t e n t i a l f o r growth i n t h e r e g i o n and t h a t would h e l p t o b a l a n c e l a b o r demand and s u p p l y .
4 . To maximize t h e p r o d u c t i v e use o f l a b o r r e s o u r c e s i n l o c a l a g r i c u l t u r e , t h e r e b y r e s t r i c t i n g r u r a l - u r b a n m i g r a t i o n .
5 . To develop a system of s e t t l e m e n t s and p u b l i c s e r v i c e s . Above a l l , t h e e x i s t i n g s t o c k o f r u r a l d w e l l i n g s should be f u l l y u t i l i z e d and t h e road network, t h e h e a l t h
c a r e system, e t c . s h o u l d be improved.
6 . To develop l o c a l a g r i c u l t u r e and i n d u s t r y such t h a t no s e r i o u s environmental problems r e s u l t and t o c r e a t e a r e c r e a t i o n a l a r e a i n t h e r e g i o n .
Thus, i t i s c l e a r t h a t f o r t h e S i l i s t r a r e g i o n it i s essen- t i a l t o c o o r d i n a t e a n a l y s e s o f r e g i o n a l a g r i c u l t u r e , i n d u s t r y , water-supply, s e r v i c e s , and m i g r a t i o n . I t s h o u l d be remembered t h a t t h e d e c i s i o n maker may wish t o change t h e d i s t r i b u t i o n o f some r e s o u r c e s ( f o r example, c a p i t a l i n v e s t m e n t s ) i n t h e model o r t o a s s e s s ( u s i n g t h e computer) t h e consequences o f d i f f e r e n t p o l i c i e s , e t c .
The scheme p r e s e n t e d i n F i g u r e 2 shows t h e i n d i v i d u a l r e g i o n a l models t h a t were l i n k e d t o f o r n a system. This scheme a l l o w s
t h e g a i n from i n d u s t r i a l and a g r i c u l t u r a l a c t i v i t i e s t o be maxi?
mized a f t e r d i f f e r e n t t y p e s o f r e s o u r c e s ( i n c l u d i n g e x t e r n a l
i n v e s t m e n t s ) and p r o d u c t i v e a c t i v i t i e s have been b a l a n c e d . Three t y p e s o f r e s o u r c e s a r e i n c l u d e d :
--
c a p i t a l i n v e s t m e n t s (which a r e s h a r e d between p r o d u c t i o n and s e r v i c e s ;--
l a b o r r e s o u r c e s ( f o r which e q u i l i b r i u m can be achieved by r e g u l a t i n g t h e s h a r e o f s e r v i c e s ) ;--
w a t e r r e s o u r c e s , which s h o u l d s a t i s f y a g r i c u l t u r a l and i n d u s t r i a l demands ( w a t e r consumption i n t h e s e t t l e m e n t system i s f i x e d ) .E x t e r n a l
i n v e s t m e n t s ( c )
i?
Local= v e c t o r s o f i n t e r r e g i o n a l d i s t r i b u t i o n o f c a p i t a l investments t o t h e a g r i c u l t u r a l ,
i n d u s t r i a l , water-supply, and s e r v i c e s e c t o r s ;
zA,
EI,&, ES = v e c t o r s o f i n t e r r e g i o n a l d i s t r i b u t i o n o f l a b o r t o the a g r i c u l t u r a l , i n d u s t r i a l , water-supply, and s e r v i c e s e c t o r s ;-
C s
A g r i c u l t u r a l
Ew, 6,
I n d u s t r i a lmodel mode 1
- I
L
Water
-
supply model
r - - - - - 1
, = s u b r e g i o n a l w a t e r flows t o a g r i c u l t u r e and i n d u s t r y ;
I
I
-
L = v e c t o r o f s u b r e g i o n a l l a b o r ;= i n f o r m a t i o n flows.
I
F i g u r e 2 . A s i m p l i f i e d system of r e g i o n a l models.
I
I I
- - - - - - - - - - - - - a
P o p u l a t i o n 4 Migration
I
4 I
I n t h i s scheme, a l t h o u g h t h e i n d i v i d u a l b l o c k s may be s u f f i - c i e n t l y d e t a i l e d t o d e s c r i b e r e a l s e c t o r a l problems, it i s g e n e r a l l y n e c e s s a r y t o choose an a p p r o p r i a t e l e v e l o f a g g r e g a t i o n f o r e a c h one i n o r d e r t o make t h e whole system o p e r a t i o n a l . Both d e t a i l e d and a g g r e g a t e d s e c t o r a l models may be used t o complement t h i s scheme.
A s can be seen from F i g u r e 2 , it i s n e c e s s a r y f o r t h e l o c a l d e c i s i o n maker t o s u p p l y t h e models w i t h i n t r a r e g i o n a l d a t a on t h e d i s t r i b u t i o n o f c a p i t a l investments, l a b o r , w a t e r r e s o u r c e s , p r o d u c t i o n p a t t e r n s , e t c . The d i s t r i b u t i o n f o r a given c a s e
obviously depends on t h e n a t u r a l and economic c o n d i t i o n s i n t h e r e g i o n , b u t i n g e n e r a l t h e r e g i o n s h o u l d be d i v i d e d i n t o 10-20 s u b r e g i o n s .
T h i s scheme i s r e l a t i v e l y f l e x i b l e because i t p e r m i t s changes t o be made t o t h e r e s o u r c e a l l o c a t i o n , t h e a d d i t i o n of c o n s t r a i n t s , o r v a r i a t i o n o f t h e o b j e c t i v e f u n c t i o n :
1 . The s h a r e s o f t h e p r o d u c t i v e and s e r v i c e s e c t o r s may be changed by t h e d e c i s i o n maker.
2. The o b j e c t i v e f u n c t i o n c o e f f i c i e n t s may b e weighted i n accordance w i t h t h e d e c i s i o n maker's p r e f e r e n c e s .
3 . C o n s t r a i n t s on r e s o u r c e consumption by c e r t a i n s e c t o r s c o u l d be i n c l u d e d .
4 . The s p e c i f i c a t i o n of c e r t a i n goods produced i n
some s e c t o r s can e a s i l y be i n t r o d u c e d ( f o r example, t o a t t a i n t h e predetermined p r o d u c t i o n t a r g e t s )
.
5. The scheme and/or t h e c o o r d i n a t i o n p r o c e d u r e s c o u l d be changed t o c o r r e s p o n d t o t h e s p e c i f i c s e t o f problems o f a g i v e n r e g i o n .
Requirement 5 i m p l i e s t h a t each module i n t h e s e t r e p r e s e n t s a g e n e r a l d e s c r i p t i o n o f a p a r t i c u l a r s e c t o r o f t h e r e g i o n a l economy.
COMPLETED MODULES
Work on g e n e r a l l y a p p l i c a b l e d e s c r i p t i o n s o f t h e most impor- t a n t s e c t o r s o f t h e r e g i o n a l economy began i n 1 9 7 7 . S i n c e t h i s work c o u l d n o t be f u l f i l l e d by IIASA's Regional Development Task
(RD) a l o n e , Task members, w h i l e c o n t i n u i n g t h e i r own a c t i v i t i e s ,
made an a t t e m p t t o f i n d s u i t a b l e models completed by o t h e r g r o u p s a t I I S A o r o u t s i d e t h e I n s t i t u t e . Themain c r i t e r i a f o r t h e s e l e c t i o n was t h a t t h e models s h o u l d b e g e n e r a l l y a p p l i c a b l e and s u p p l i e d w i t h t h e n e c e s s a r y computer s o f t w a r e .
A s a r e s u l t t h e f o l l o w i n g c o m b i n a t i o n o f models was used:
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G e n e r a l i z e d Regional A g r i c u l t u r a l Model (GRIIM),
e l a b o - r a t e d i n RD;--
Regional Water-Supply Model, e l a b o r a t e d j o i n t l y i n RD and t h e Resources and Environment Area (IIASA);--
M i g r a t i o n Model, e l a b o r a t e d j o i n t l y i n RD and t h e Human S e t t l e m e n t s and S e r v i c e s Area (IIASA) ;--
Model o f P o p u l a t i o n Growth, e l a b o r a t e d i n HSS;--
G e n e r a l i z e d I n d u s t r i a l Model, e l a b o r a t e d i n Moscow a t t h e C e n t r a l Economics and Mathematics I n s t i t u t e(CEMI)
.
Although o n l y t h e s e f i v e models were i n c l u d e d i n t h e s y s t e m o f r e g i o n a l models, it i s p o s s i b l e t o add o t h e r s a s r e q u i r e d . G e n e r a l i z e d R e g i o n a l A g r i c u l t u r a l Model
The G e n e r a l i z e d R e g i o n a l A g r i c u l t u r a l Model (GRAM) h a s a l r e a d y been p r e s e n t e d i n d e t a i l i n Albegov ( 1 9 7 9 )
,
t h e r e f o r e o n l y i t s main f e a t u r e s a r e d i s c u s s e d below.GRAM was o r i g i n a l l y d e v e l o p e d f o r i n t r a r e g i o n a l a g r i c u l t u r a l problem a n a l y s i s i n R D ' s S i l i s t r a and Notec c a s e s t u d i e s . I t i s a g e n e r a l model and i s n o t i n t e n d e d t o r e p l a c e s p e c i a l i z e d a g r i c u l t u r a l models d e s i g n e d t o s o l v e s p e c i f i c problems. R a t h e r i t s h o u l d be t r e a t e d a s a t o o l f o r examining g e n e r a l a g r i c u l t u r a l problems i n t h e framework o f comprehensive r e g i o n a l a n a l y s i s . The c h a r a c t e r o f t h i s model i s r e v e a l e d i n t h e v a r i a b l e s it c o n t a i n s , a s g i v e n below:
'iprl = volume o f c r o p i p u r c h a s e d f o r a n i m a l f e e d on market 1 by p r o p e r t y p i n s u b r e g i o n r;.
Q i p r l = volume o f c r o p i p u r c h a s e d f o r human consumption on market 1 by p r o p e r t y p i n s u b r e g i o n r;
Q m p r l = volume o f l i v e s t o c k p r o d u c t m p u r c h a s e d f o r human consumption on market 1 by p r o p e r t y p i n s u b r e g i o n r ;
R i p r l = volume o f c r o p i s o l d on market 1 by p r o p e r t y
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p i n s u b r e g i o n r ;R m p r l = volume o f l i v e s t o c k p r o d u c t m s o l d on market 1 by p r o p e r t y p i n s u b r e g i o n r ;
'ipr = human consumption o f c r o p i on p r o p e r t y p i n s u b r e g i o n r;
W = human consumption o f l i v e s t o c k p r o d u c t m on mpr p r o p e r t y p i n s u b r e g i o n r ;
'iprsa = volume o f f i r s t h a r v e s t o f c r o p i on p r o p e r t y p on l a n d a i n s u b r e g i o n r , when t e c h n o l o g y s i s used;
' j k p r s l = number of l i v e s t o c k j of s p e c i a l i z a t i o n k on p r o p e r t y p i n s u b r e g i o n r , when t e c h n o l o g y s ' i s u s e d ;
'iprscr = volume o f second h a r v e s t o f c r o p i on p r o p e r t y p on l a n d a i n s u b r e g i o n r , when t e c h n o l o g y s i s used;
'ipr = consumption by l i v e s t o c k o f c r o p i on p r o p e r t y p i n s u b r e g i o n r;
Z = consumption by l i v e s t o c k o f l i v e s t o c k p r o d u c t rn mpr on p r o p e r t y p i n s u b r e g i o n r .
The s e t o f c o n s t r a i n t s c o n t a i n e d i n GRAM r e l a t e s t o l a n d - u s e c o n d i t i o n s , t h e f o r a g e b a l a n c e , human consumption, p r o d u c t i o n l i m i t s , e t c . ( T a b l e 1 )
.
Each group o f c o n s t r a i n t s c o n s i s t s i n s e v e r a li n e q u a l i t i e s ; t a k e , f o r example, l a n d u s e ( a , b , c , d l e ) :
--
c o n s t r a i n t on a r a b l e l a n d f o r t h e r e g i o n a s a whole;--
c o n s t r a i n t on a r a b l e l a n d a c c o r d i n g t o t y p e s o f p r o - p e r t y ;--
c o n s t r a i n t on area o f l a n d o c c u p i e d by p l a n t s ;--
c o n s t r a i n t on a r e a o f l a n d t h a t c a n be improved by i r r i g a t i o n , t e r r a c i n g , e t c . ;--
c o n s t r a i n t on a r e a o f p a s t u r e s and meadows.The model can b e u s e d t o a n a l y z e t h e f o l l o w i n g problems:
--
r e g i o n a l a g r i c u l t u r a l s p e c i a l i z a t i o n ;--
d i f f e r e n t t y p e s o f p r o d u c t i o n ( c r o p , l i v e s t o c k , market g a r d e n i n g , e t c.
) i n d i s a g g r e g a t e d f o m ;Table 1 . L i s t o f c o n s t r a i n t s i n c l u d e d i n GRAM.
Notation t o Table 1 .
maximum amount of l a b o r a v a i l a b l e i n t h e whole region;
maximum amount of l a b o r a v a i l a b l e on p r o p e r t y p i n subregion r;
t o t a l ( e x t e r n a l and i n t e r n a l ) c a p i t a l i n v e s t - ment a v a i l a b l e f o r r e g i o n a l a g r i c u l t u r e ; t o t a l ( e x t e r n a l and i n t e r n a l ) c a p i t a l i n v e s t - ment a v a i l a b l e f o r a g r i c u l t u r e f o r p r o p e r t y p i n subregion r;
maximum annual water supply a v a i l a b l e i n t h e whole r e g i o n ;
maximum water supply a v a i l a b l e a t peak p e r i o d s in t h e whole r e g i o n ;
maximum annual water supply a v a i l a b l e f o r p r o p e r t y p i n subregion r;
maximum water supply a v a i l a b l e a t peak p e r i o d s f o r p r o p e r t y p in subregion r;
maximum amount of a g r i c u l t u r a l machinery a v a i l a b l e f o r t h e whole r e g i o n ;
consumption of c r o p i and l i v e s t o c k product m, r e s p e c t i v e l y , i n t h e whole region;
production of crop i on p r o p e r t y g in subregion r :
maximum volume of f e r t i l i z e r f a v a i l a b l e i n t h e whole r e g i o n ;
maximum volume of f e r t i l i z e r f a v a i l a b l e f o r p r o p e r t y p in subregion r;
maximum volume of. e x t e r n a l purchases of crop i on market 1 f o r l i v e s t o c k i n t h e whole r e g i o n ; maximum volume of e x t e r n a l purchases o f crop i and l i v e s t o c k p r o d u c t m, r e s p e c t i v e l y , on market 1 f o r human consumption i n t h e whole r e g i o n ;
s a l e l i m i t a t i o n of c r o p i and l i v e s t o c k p r o d u c t m, r e s p e c t i v e l y , on market 1;
L = a r e a of l a n d ( s t a t e , c o l l e c t i v e , o r p r i v a t e ) L i p r l mpr
t h a t , in accordance with c r o p r o t a t i o n , c o u l d be used f o r c r o p 'i and l i v e s t o c k product m, r e s p e c t i v e l y , on p r o p e r t y p i n subregion r ;
= maximum a r e a of a r a b l e l a n d on p r o p e r t y p i n subregion r;
= a r e a of l a n d a a v a i l a b l e on p r o p e r t y p i n subregion r;
= minimum and maximum a r e a of l a n d a on p r o p e r t y p i n subregion r t h a t can be improved using technology s;
= production of l i v e s t o c k j on p r o p e r t y p i n subregion r;
= minimum wage l e v e l p e r c a p i t a on p r o p e r t y p .
--
l a n d - u s e problems, w i t h r e f e r e n c e t o i r r i g a t i o n , d r a i n - a g e , e t c . ;--
c h o i c e o f a n i m a l - f e e d c o m p o s i t i o n s ( p r o t e i n , rough and g r e e n f o r a g e , e t c . ) ;--
c h o i c e o f c r o p - r o t a t i o n c o n d i t i o n s ;--
a v a i l a b i l i t y o f r e g i o n a l s u p p l i e s o f l a b o r , c a p i t a l i n v e s t m e n t , f e r t i l i z e r s , w a t e r , e t c .S p e c i a l l y e l a b o r a t e d growth o p e r a t o r s h e l p t o g e n e r a t e GRAM'S m a t r i x , which i n c l u d e s h u n d r e d s o f i n e q u a l i t i e s .
R e g i o n a l I n d u s t r i a l Model
The model d e v e l o p e d by teams a t t h e C e n t r a l I n s t i t u t e o f Economics and Mathematics i n Moscow was u s e d a s a p r o t o t y p e f o r
t h e G e n e r a l i z e d I n d u s t r i a l Model (Mednitsky 1 9 7 8 ) . D e s c r i p t i o n s o f many r e s o u r c e s and f i n a l p r o d u c t s , n o n l i n e a r d e p e n d e n c i e s o f c o s t s on p r o d u c t i o n s c a l e , t r a n s p o r t a t i o n o f d i f f e r e n t p r o d u c t s , e t c . may be i n c l u d e d i n t h i s model.
To d e s c r i b e t h e main i d e a s o f t h e i r model, which i s m o d i f i e d w i t h r e s p e c t t o i n t r a r e g i o n a l problems, t h e f o l l o w i n g n o t a t i o n s w e r e i n t r o d u c e d :
i = i n d e x o f p r o d u c t ;
1 = p o s s i b l e l o c a t i o n o f p r o d u c t i o n u n i t s w i t h i n t h e r e g i o n u n d e r a n a l y s i s ;
s = p o i n t s where demand i s c o n c e n t r a t e d ( w i t h i n t h e r e g i o n and on t h e b o u n d a r i e s ) ;
r = v a r i a n t s i n p r o d u c t i o n u n i t c a p a c i t y ; E = r a t e o f r e t u r n on c a p i t a l i n v e s t m e n t ; I1 = set o f t r a n s p o r t a b l e commodities;
I 2 = s e t o f n o n t r a n s p o r t a b l e commodities;
0
'i = f i n a l demand ( w i t h i n and o u t s i d e t h e r e g i o n ) f o r p r o d u c t i ;
1
ai = f i x e d demand f o r t r a n s p o r t a b l e commodities i a t p o i n t 1;
C i
l r = u n i t c o s t a t p o i n t 1 f o r p r o d u c t i o n o f commodity i under v a r i a n t r;
i
K l r = c a p i t a l i n v e s t m e n t p e r u n i t o f p a r t i c u l a r commo- d i t y i a t p o i n t 1 under v a r i a n t r;
i
l r = l o c a l r e s o u r c e s a v a i l a b l e f o r p r o d u c i n g commodity i under v a r i a n t r a t p o i n t 1;
i
Tls = c o s t o f t r a n s p o r t a t i o n ( o f p a r t i c u l a r commodity i ) from p o i n t 1 t o p o i n t s;
Z: = consumption of l o c a l r e s o u r c e s i a t p o i n t 1;
A i r = matrix o f i n p u t s of t r a n s p o r t a b l e commodities;
= matrix o f i n p u t s o f n o n t r a n s p o r t a b l e commodities;
i
" ~ r = m a t r i x o f o u t p u t s o f c o ~ m o d i t i e s ; i
'lr = v e c t o r of i n t e n s i t y of p r o d u c t i o n of commodity i under v a r i a n t r a t p o i n t 1;
uis
= v e c t o r o f volume o f t r a n s p o r t o f commodity i from p o i n t 1 t o p o i n t s;0 i
l r = i n t e g e r v a r i a b l e s t h a t i n d i c a t e whether v a r i a n t r s h o u l d be used a t p o i n t 1 f o r p r o d u c i n g commodity i.
The f o l l o w i n g c o n s t r a i n t s a r e i n c l u d e d i n t h e model. Demand f o r t r a n s p o r t a b l e r e s o u r c e s w i t h i n o r o u t s i d e t h e r e g i o n under a n a l y s i s s h o u l d be s a t i s f i e d :
Local demand f o r n o n t r a n s p o r t a b l e r e s o u r c e s s h o u l d a l s o be s a t i s f i e d :
The t r a n s p o r t volume must c o r r e s p o n d t o t h e amount o f t r a n s - p o r t a b l e commodities a t e a c h p o i n t o f p r o d u c t i o n :
Fixed demand f o r t r a n s p o r t a b l e commodities and a d d i t i o n a l demand from new e n t e r p r i s e s a t e a c h p o i n t s h o u l d be s a t i s f i e d :
Local consumption of n o n t r a n s p o r t a b l e r e s o u r c e s s h o u l d be c o n f i n e d t o t h e a v a i l a b l e s u p p l y :
V a r i a b l e a r e n o n n e g a t i v e and some a r e i n t e g e r s :
I t i s p o s s i b l e t o modify t h e o b j e c t i v e f u n c t i o n ; t h e m o d i f i - c a t i o n most f r e q u e n t l y used w i l l be m i n i m i z a t i o n o f p r o d u c t i o n and t r a n s p o r t a t i o n c o s t s :
The model ( 1 )
-
( 7 ) may be u s e f u l i n s e v e r a l c a s e s . But if. . . . . - - - - - . .
i t i s i n c o n v e n i e n t f o r a p a r t i c u l a r c a s e , a s p e c i a l model may be developed f o r i n c l u s i o n i n t h e model system.
Water-Supply Model
The Water-Supply Model was d e s c r i b e d i n d e t a i l i n Albegov and C h e r n y a t i n ( 1 9 7 8 ) , t h e r e f o r e o n l y i t s main c h a r a c t e r i s t i c s a r e p r e s e n t e d below. The p r i n c i p a l assumptions a r e :
1. The w a t e r r e q u i r e m e n t s , which a r e d i s t r i b u t e d o v e r t i m e (by s e a s o n s ) and s p a c e , a r e p r e d e t e r m i n e d by t h e l o c a - t i o n o f i n d u s t r i a l and a g r i c u l t u r a l a c t i v i t i e s .
2 . Water r e s o u r c e s f o r t h e w a t e r - d i s t r i b u t i o n systems a r e u n l i m i t e d (mainstream w a t e r r e g u l a t i o n i s n o t a n a l y z e d h e r e )
.
3 . ~ l l w a t e r u s e r s consume w a t e r r e s o u r c e s i r r e v e r s i b l y . 4 . Only w i t h i n - y e a r w a t e r - r e s o u r c e r e g u l a t i o n i s c o n s i d e r e d . 5 . Time d e l a y s f o r w a t e r t r a n s i t a r e n o t t a k e n i n t o a c c o u n t . The main g o a l of t h e model i s t o meet w a t e r r e q u i r e m e n t s
f o r a g i v e n p e r i o d w i t h minimum c o s t s . W a t e r - q u a l i t y problems a r e n o t c o n s i d e r e d . The e q u a t i o n s o f t h i s model a r e d e r i v e d by a p p l y i n g a mass b a l a n c e f o r e v e r y node and e v e r y r e s e r v o i r , upper and lower bounds f o r nodes, r e s e r v o i r s , pumping s t a t i o n s , and c a n a l s a r e s p e c i f i e d . The o b j e c t i v e f u n c t i o n i s t o minimize t h e sum o f reduced c o s t s f o r c o n s t r u c t i o n and o p e r a t i o n .
This model h a s t h e f o l l o w i n g a d v a n t a g e s :
1 . Any c o n f i g u r a t i o n s o f t h e s y s t e m c a n be c o n s i d e r e d . 2. Regional s p a c e may be r e p r e s e n t e d by a number of sub-
r e g i o n s .
3 . The model t a k e s i n t o a c c o u n t s e a s o n a l i r r e g u l a r i t i e s i n w a t e r consumption.
4 . The m a t r i x growth o p e r a t o r f a c i l i t a t e s i m p l e m e n t a t i o n of t h e model.
Pcpulation-Growth and M i g r a t i o n Models
Because o f t h e i n t r a r e g i o n a l c h a r a c t e r o f t h e a n a l y s i s , sequen- t i a l l a b o r - f o r c e a n a l y s i s i s r e q u i r e d a t a r e g i o n a l a s w e l l a s a t a s u b r e g i o n a l l e v e l . For this r e a s o n , t h e f o l l o w i n g s e t o f c a l c u l a -
t i o n s s h o u l d be performed, u s i n g t h e p o p u l a t i o n and m i g r a t i o n models:
--
c a l c u l a t i o n of i n - and o u t - m i g r a t i o n f o r t h e r e g i o n a s a whole ;--
c a l c u l a t i o n o f t h e f u t u r e p o p u l a t i o n f o r t h e r e g i o n a s a whole;--
c a l c u l a t i o n of t h e f u t u r e p o p u l a t i o n f o r t h e m u l t i s u b r e - g i o n a l system;--
c a l c u l a t i o n of f u t u r e r e g i o n a l and s u b r e g i o n a l l a b o r . The p o p u l a t i o n - g r o w t h and m i g r a t i o n models, e l a b o r a t e d i n HSS ( W i l l e k e n s and Rogers 1 9 7 8 ) , a r e r a t h e r g e n e r a l . Never- t h e l e s s , t h e m i g r a t i o n model may r e q u i r e c e r t a i n m o d i f i c a - t i o n s depending on t h e c o n d i t i o n s of t h e r e g i o n under a n a l y s i s .For example, a f t e r some i n v e s t i g a t i o n s (Andersson and P h i l i p o v
1979) i t was d e c i d e d t h a t t h e m i g r a t i o n model used f o r t h e S i l i s t r a r e g i o n i n B u l g a r i a should t a k e t h e form
-
J- -
IP i j
-
exp ( v i ) +exp ( v j )e x p ( v -v ) + 1 I
i j
where
j = p r o b a b i l i t y of moves from r e g i o n i t o r e g i o n j ;
v i t V j = u t i l i t i e s f o r r e g i o n i and j, r e s p e c t i v e l y . The form of t h e f u n c t i o n v s u g g e s t e d i s
where
*ik = c h a r a c t e r i s t i c s o f r e g i o n i;
aik I Bi = c o e f f i c i e n t s t o be e s t i m a t e d by an e c o n o m e t r i c approach.
The r e s u l t s o f t h e r e g i o n a l m i g r a t i o n model can be plugged i n t o t h e r e g i o n a l population-growth model t o o b t a i n a f o r e c a s t o f t h e t o t a l r e g i o n a l p o p u l a t i o n . The r e g i o n a l m i g r a t i o n r a t e can be changed a n n u a l l y , depending on t h e r e s u l t s o f t h e m i g r a t i o n model r u n s . The age and s e x s t r u c t u r e of m i g r a n t s can be assumed t o be t h e same a s f o r t h e p r e v i o u s l y observed p e r i o d .
Taking t h e d a t a on r e g i o n a l p o p u l a t i o n growth a s g i v e n , i n t r a r e g i o n a l p o p u l a t i o n growth can be a n a l y z e d . The Willekens/
Rogers model (1978) can be used f o r t h i s purpose:
where
{ K t }
-
= age and s u b r e g i o n a l d i s t r i b u t i o n of t h e popula- t i o n a t time t ;-
G = m u l t i r e g i o n a l ( i n t h i s c a s e , m u l t i s u b r e g i o n a l m a t r i x growth o p e r a t o r o r g e n e r a l i z e d L e s l i e m a t r i x ) ;t + l = t i m e p e r i o d f o l l o w i n g t ( u s u a l l y 5 - y e a r p e r i o d s a r e a n a l y z e d )
.
From t h e r e s u l t s f o r e a c h t i m e p e r i o d and e a c h s u b r e g i o n , t h e p o p u l a t i o n number and i t s a g e s t r u c t u r e (and i f n e c e s s a r y , i t s s e x s t r u c t u r e ) c a n be o b t a i n e d .
R e g i o n a l and s u b r e g i o n a l p o p u l a t i o n and i t s a g e / s e x s t r u c - t u r e forms t h e b a s i s f o r a s s e s s i n g t h e l a b o r f o r c e . S u b r e g i o n a l l a b o r c a n e a s i l y b e c a z c u l a t e d by a c c o u n t i n g f o r the p o s s i b l e c h a n g e s i n t h e p r o p o r t i o n o f t h e t o t a l p o p u l a t i o n c o n s t i t u t e d by t h e l a b o r f o r c e and s h o u l d b e c o n s i d e r e d as a c o n s t r a i n t - ' o n r e g i o n a l growth.
MODEL LINKAGE
The i d e a o f a m o d e l - l i n k a g e p r o c e d u r e w a s d e s c r i b e d i n Umnov (1979) a n d , t h e r e f o r e , i t w i l l o n l y b r i e f l y b e d i s c u s s e d h e r e t o
a i d t h e r e a d e r ' s g e n e r a l u n d e r s t a n d i n g o f t h e c a l c u l a t i o n p r o c e - d u r e and t h e p o s s i b i l i t i e s o f f e r e d by the u s e o f this method.
The l i n k a g e models a r e f o r m a l l y d e s c r i b e d by two s e t s o f numbers. The f i r s t s e t , ' v a r i a b l e s ,
'
p r e s e n t s t h e s t a t e o f t h e s u b j e c t t o be modeled. The s e c o n d set, ' p a r a m e t e r s , ' g i v e s t h e e x t e r n a l c o n d i t i o n s o f t h e s u b j e c t , i . e . t h e s t a t e o f i t s ' e n v i r o n - m e n t . ' Only f i n i t e - d i m e n s i . o n a 1 o p t i m i z a t i o n models are c o n s i -d e r e d . It i s assumed t h a t t h e r e i s a common c r i t e r i o n , which i s e x p r e s s e d by t h e v a r i a b l e s and p a r a m e t e r s , f o r a l l models. The a i m i s t o f i n d v a l u e s o f the v a r i a b l e s a n d p a r a m e t e r s t h a t are o p t i m a l f o r t h e common o b j e c t i v e , s u b j e c t t o t h e models t h a t a r e t o be u s e d as i n d e p e n d e n t s o f t w a r e u n i t s .
The main i d e a o f the l i n k a g e p r o c e d u r e may b e f o r m u l a t e d as f o l l o w s . S i n c e t h e o p t i m a l s t a t e o f the model ( i n t h e s e n s e o f i t s o b j e c t i v e ) g e n e r a l l y depends on t h e v a l u e s o f i t s p a r a - m e t e r s , it i s p o s s i b l e t o assume ( w i t h some a d d i t i o n a l c o n d i t i o n s ) t h a t t h e r e e x i s t p a r a m e t e r v a l u e s t h a t w i l l p r o v i d e a l l
t h e models w i t h t h e o p t i m a l s t a t e f o r t h e common o b j e c t i v e ; f o r example, when t h e common c r i t e r i o n i s a convex f u n c t i o n o f t h e o b j e c t i v e s o f t h e models.
L e t us assume t h a t t h e l i n k e d models can b e w r i t t e n i n t h e form:
k Minimize w i t h r e s p e c t t o xk E E"
s u b j e c t t o
where
xk = v a r i a b l e v e c t o r of t h e model k:
v = l i n k a g e p a r a m e t e r v e c t o r (common f o r a l l m o d e l s ) ; rnk = number o f c o n s t r a i n t s o f model k ;
* *
L = number of l i n k a g e p a r a m e t e r s .
k k
W e s h a l l a l s o assume t h a t a l l f u n c t i o n s f k ( x k , v ) and Ys (x , v ) a r e d e f i n e d and a r e s u f f i c i e n t l y t i m e d i f f e r e n t i a b l e w i t h r e s p e c t t o a l l t h e i r arguments.
I t i s p o s s i b l e t h a t a s e t of c o n s t r a i n t s f o r components of t h e l i n k a g e v e c t o r v e x i s t s . L e t i t be
where M i s t h e number o f c o n s t r a i n t s t h a t w e r e f e r t o a s common c o n s t r a i n t s .
*The s y s t e m o f c o n s t r a i n t s may a l s o c o n t a i n e q u a l i t i e s , b u t t h i s d o e s n o t p r e s e n t any problems.
**Vector v c o n t a i n s o n l y p a r a m e t e r s t h a t a r e u s e d f o r l i n k a g e .
A s mentioned above, t h e common c r i t e r i o n must be a convex f u n c t i o n o f t h e models' o b j e c t i v e s . However, w i t h o u t l o s i n g t h e g e n e r a l c h a r a c t e r o f t h e scheme, w e can c o n s i d e r t h e common o b j e c - t i v e a s a l i n e a r combinati.on o f a l l t h e s e o b j e c t i v e s , which h a s p o s i t i v e w e i g h t c o e f f i c i e n t s i n t h e form
where N i s t h e number o f l i n k e d models.
W e can now f o r m u l a t e t h e m a t h e m a t i c a l programming problem.
Minimize w i t h r e s p e c t t o a l l xk and v
s u b j e c t t o
*k
*
t h e s o l u t i o n o f which { x , v g i v e s u s t h e d e s i r a b l e v a l u e s o f t h e v a r i a b l e s and p a r a m e t e r s .
N k N k
Problem (15) h a s L
+
C n v a r i a b l e s and M+
C m con-k= 1 k= 1
s t r a i n t s . Thus, i t s d i m e n s i o n s a r e s u f f i c i e n t l y l a r g e even f o r t h e s i m p l e s t p r a c t i c a l c a s e . O u r aim i s t o t r y t o s i m p l i f y problem
( 15) a s f a r a s p o s s i b l e , u s i n g t h e s o l u t i o n s t o problems ( 1 1 ) f o r t h e f i x e d v a l u e s o f t h e l i n k a g e p a r a m e t e r s .
L e t t h e d e p e n d e n c i e s o f t h e o p t i m a l xk o f v b e e x p r e s s e d by x * ~ ( v )
.
S u b s t i t u t i n g them i n t o (15),
w e have a new problem.Minimize w i t h r e s p e c t v E E~ ( o n l y )
s u b j e c t t o
C o n s t r a i n t s k Ys ( x * ~ ( v )
,
v ) 2 0 a r e o m i t t e d h e r e b e c a u s e xtk ( v ) a r e t h e s o l u t i o n s o f ( 1 1 ) and a l l c o n s t r a i n t s o f t h e problems a r e t o be s a t i s f i e d by t h e i r s o l u t i o n s .A s G e o f f r i o n (1970) h a s shown, v
*
i s t h e s o l u t i o n o f problem (18).
I n o t h e r words, t h e f o l l o w i n g r e l a t i o n s a r e v a l i d :Thus, w e can i n d e p e n d e n t l y o b t a i n a l l o p t i m a l ( i n t h e u s u a l s e n s e ) p o i n t s f o r models ( 1 1 )
,
a s soon a s we f i n d t h e s o l u t i o n t o problem ( 18).
Problem ( 1 8 ) i s t h e c e n t r a l c o n s i d e r a t i o n . The p r o c e d u r e f o r s o l v i n g t h i s problem was p r e v i o u s l y r e f e r r e d t o a s t h e
l i n k a g e p r o c e s s . T h e r e f o r e , t h e method o f s o l u t i o n w i l l d e f i n e t h e c o n t e n t and volume of i n f o r m a t i o n a l exchange between t h e l i n k e d models.
Although problem ( 18) h a s f o r m a l l y fewer dimensions t h a n ( 1 5 ) , there a r e d i f f i c u l t i e s ( i n a d d i t i o n t o t h e u s u a l problems e n c o u n t e r e d ) t h a t p r e v e n t us from u s i n g s t a n d a r d schemes f o r s o l u t i o n :
1. I t i s i m p o s s i b l e t o f i n d e x p l i c i t e x p r e s s i o n s f o r -
x t k ( v )
,
e x c e p t p e r h a p s i n some c a s e s o f no p r a c t i c a l i n t e r e s t .2 . F u n c t i o n s xtk ( v ) a r e n o t d e f i n e d f o r any v s a t i s f y i n g (12)
,
s i n c e problems ( 1 1 ) c a n have no f e a s i b l e s o l u t i o n f o r some v.3 . F u n c t i o n s x * ~ ( v ) a r e n o t d i f f e r e n t i a b l e a t some p o i n t s o f E L
.
I t i s n e c e s s a r y t o emphasize t h a t a l l t h e d i f f i c u l t i e s
r e s u l t from t h e d i s t r i b u t e d scheme. The s i m p l e s t way o f a v o i d i n g them i s t o merge a l l models ( 1 1 )
.
However, w e c o n s i d e r a s i t u a - t i o n i n which t h i s i s u n r e a s o n a b l e o r , s i m p l y , i m p o s s i b l e . There-f o r e , w e need t o f i n d a n o t h e r approach t o s o l v i n g problem ( 1 8 ) .
There have been a t t e m p t s t o s o l v e such t y p e s o f problems, where e a c h of t h e d i f f i c u l t i e s mentioned above was overcome by
d i f f e r e n t meL\ods ( G e o f f r i o n 1970, Ermoliev 19 80)
.
Here a n o t h e r a p p r o a c h , which p e r m i t s u s t o r e s o l v e a l l t h e o b s t a c l e s by -.means o f a s i n g l e method, w i l l be c o n s i d e r e d .
The approach c o n s i s t s i n s u b s t i t u t i n g i n t o ( I S ) , i n s t e a d of x * ~ ( v )
,
new f u n c t i o n s x - ~ ( v ),
which:--
a r e d e f i n e d a t any v E E L ;--
a r e d i f f e r e n t i a b l e f o r a l l v E EL;--
have v a l u e s , which a r e c l o s e t o v a l u e s o f x * ~ ( v ) f o r a l l v , where x i k ( v ) e x i s t s .I n s t e a d of u s i n g f u n c t i o n s -k x ( v )
,
w e c a n u s e t h e s o l u t i o n s o f pro- blems ( 1 1 ),
which a r e found by employing a 'smooth' v e r s i o n o f t h e P e n a l t y F u n c t i o n Method, o r t h e SUMT (see, f o r example, F i a c c o and McCormick 1 9 6 8 ) . The method r e p l a c e s problems (11) by an uncon- s t r a i n e d m i n i m i z a t i o n of t h e f o l l o w i n g a u x i l i a r y f u n c t i o n :where P ( T t a ) i s t h e p e n a l t y f u n c t i o n , which s a t i s f i e s t h e r e l a t i o n l i m 0 , f o r a > 0
T-+O P ( T , a ) =
+ a , f o r a < 0
I n o t h e r words,
-
~ ( v ) i s a p o i n t a t which f u n c t i o n ( 2 1 ) h a s a minimum.From t h e p r o p e r t i e s o f t h e SUMT, ~ ~ ( v ) a r e d e f i n e d f o r a l l v E E L t b e c a u s e t h e a u x i l i a r y f u n c t i o n s (21) have t h e minimum b o t h f o r f e a s i b l e and i n f e a s i b l e problems ( 1 1 )
.
For a l l p o i n t s , where xik ( v ) e x i s t , t h e f o l l o w i n g e q u a l i t y i s v a l i d :
l i m -k *k
T++O x ( T , v ) = x ( v )
.
T h e r e f o r e , v a l u e s o f x ( v ) and x i k ( v ) a r e c l o s e . -k
The s t a t i o n a r y c o n d i t i o n f o r f u n c t i o n ( 2 1 ) i s k -k
g r a d
. .
E ( x , v ) = O.
k
k k k k
I f a l l f u n c t i o n s f (x
,
v,
YS (X,
V ),
and P ( T , a ) a r e s u f f i c i e n t l y smooth, i t would be p o s s i b l e t o a p p l y t h e well-known i m p l i c i tf u n c t i o n ' s theorem t o e q u a t i o n ( 2 4 ) and t o d i s c o v e r t h a t ~ ~ ( v ) i s d i f f e r e n t i a b l e f o r any v E EL'
I t i s a s d i f f i c u l t t o f i n d t h e e x p l i c i t form o f z k ( v ) a s i t i s t o f i n d t h e e x p l i c i t form o f x l k ( v )
.
Hence, w e u s e a n u m e r i c a l a l g o r i t h m t h a t d o e s n o t r e q u i r e x * ~ ( v ) t o be s t a t e d e x p l i c i t l y , b u t n e e d s o n l y some n u m e r i c a l e v a l u a t i o n s ( s u c h a s v a l u e s o f func- t i o n s and t h e i r d e r i v a t i v e s ) , t o s o l v e problem ( 1 8 ) . F i n a l l y , w e o b t a i n a new problem.Minimize w i t h r e s p e c t t o v E E L
s u b j e c t t o
where $ ( v ) a r e t h e s o l u t i o n s o f ( 2 4 ) .
The d i r e c t s o l u t i o n o f t h e problem may r e q u i r e 'know-how' t o c a l c u l a t e v a l u e s o f t h e f i r s t ( a n d p e r h a p s t h e s e c o n d ) d e r i - v a t i v e s o f x ( v ) . -k I t i s a d i f f i c u l t c o m p u t a t i o n a l problem, b u t t h e r e i s a way o f s i m p l i f y i n g t h e p r o c e d u r e s l i g h t l y .
L e t u s r e t u r n t o problem ( 15)
,
which w e a l s o s o l v e by t h e same 'smooth' v e r s i o n o f t h e SUMT. The a u x i l i a r y f u n c t i o n i n t h i s c a s e w i l l be w r i t t e n a sN k
N m
E = *
c
; i k 8 ( x k t v )+ c
;ik L P ( T , % ( X ~ , V ) )k= 1 k=l s = l
~ u l t i p l y i n g t h e p e n a l t y terms o f E~ by p o s i t i v e number ;iK does n o t change t h e s i t u a t i o n .
L e t us s u b s t i t u t e x ( v ) i n t o -k E t o reduce i t s dimensions:
k -k
~ ( T , v ) = w ( T t v )
+
Z AkE ( T ~ x ( v ) , v) t k= 1and l e t
S ( T )
be a minimum p o i n t o f ( 2 8 ).
Thenl i m
-
T +O v ( T ) = V*
,
i. e .
,
we can f i n d t h e optimum ( w i t h some small e r r o r s ) by mini- mizing f u n c t i o n ( 2 8 ) . The problem o f a c c u r a c y i s t h e o r e t i c a l l y i n t e r e s t i n g and d i f f i c u l t ( s e e Umnov 1974, 1975, and 1979),
b u t h a s l i t t l e p r a c t i c a l v a l u e i n t h e scheme.Now we g i v e t h e formula f o r c a l c u l a t i n g t h e f i r s t and second p a r t i a l d e r i v a t i v e s o f Q i t h r e s p e c t t o components o f v.
F o r t h e f i r s t d e r i v a t i v e s :
But t a k i n g ( 2 4 ) i n t o c o n s i d e r a t i o n , we o b t a i n simply
aE - -
- -
aEF r
f o r a l l r = 1,L.
avr
I n a n analogous way, f o r t h e second p a r t i a l d e r i v a t i v e s :
Once a g a i n u s i n g ( 2 4 ) and t a k i n g i n t o a c c o u n t t h a t , a f t e r d i f f e r e n t i a t i o n o f ( 2 4 ) w i t h r e s p e c t t o v w e w i l l have t h e r e l a t i o n P'
n k a2Ek
5
3 2 ~ k-
=
- ,
f o r a l l p = l , Lk k a v k
-
j = l ax. ax
1 j P av axi a n d K = I , N
,
P w e t h e n f i n d t h a t
- -
f o r a l l r = 1 , L and p = l , L b e c a u s e
The f o r m u l a e ( 3 1 ) and ( 3 4 ) a l l o w u s t o minimize ( 2 8 ) by any o f t h e s t a n d a r d p r o c e d u r e s , u s i n g p a r t i a l d e r i v a t i v e s o f t h e f i r s t and s e c o n d o r d e r o f
E.
However, t h e s e p r o c e d u r e r e q u i r e u s e o f t h e f i r s t o r d e r o n l y t o o b t a i n v a l u e s o f components f o r z k ( v ) a n d t h e m a t r i x,
which i s c a l l e d t h e s e n s i t i v e m a t r i x .av
The l i n k a g e scheme c o n s i s t s i n t h e f o l l o w i n g p r o c e d u r e . For
L -k
a s t a r t i n g p o i n t v E E
,
a l l models i n d e e n d e n t l y g e n e r a t e x ( v ),
k -k ax
-R
E ( x ( v ) , v )
,
and t h e s e n s i t i v e m a t r i x-
av On t h e b a s i s o f t h i s i n f o r m a t i o n , t h e c e n t r a l p r o c e s s o r f i n d s a new p o i n t i n E La c c o r d i n g t o t h e o p t i m i z a t i o n a l p r o c e d u r e s e l e c t e d . U s u a l l y , t h e i t e r a t i o n c a n b e w r i t t e n a s
where
vo = the s t a r t i n g p o i n t ; v = t h e new ' b e t t e r ' p o i n t ;
w = t h e d i r e c t i o n o f m i n i m i z a t i o n ; s = t h e l e n g t h o f t h e s t e p a l o n g w.
s and w a r e c a l c u l a t e d on t h e b a s i s o f v a l u e s o f t h e f u n c t i o n minimized, i t s g r a d i e n t , and p e r h a p s e v e n , i t s g e s s i a n . I f t h e t e r m i n a t i n g c o n d i t i o n i s n o t s a t i s f i e d a t p o i n t v , t h e i t e r a t i o n i s r e p e a t e d .
P r o c e d u r e s s u c h a s ( 3 6 ) a r e u s u a l f o r o p t i m i z a t i o n a l schemes.
However, t h e d i s t r i b u t e d n a t u r e o f t h e problem makes u s r e - e v a l u a t e some o f t h e s t a n d a r d views on t h e s e p r o c e d u r e s . W e a l s o have t o t a k e i n t o a c c o u n t some o f t h e s p e c i f i c f e a t u r e s o f t h e l i n k e d models.
L e t u s c o n s i d e r t h e scheme g i v e n f o r t h e c a s e o f l i n e a r o p t i m i z a t i o n a l models. W e assume t h a t :
1 . The common c o n s t r a i n t s a r e l i n e a r w i t h r e s p e c t t o v . 2 . The l i n k a g e p a r a m e t e r s a r e i n c l u d e d o n l y i n t h e f r e e
t e r m s o f t h e i n t e r n a l c o n s t r a i n t s o f t h e models.
3. There i s n o s o f t w a r e t h a t e n a b l e s a v e r s i o n o f t h e . SUMT t o be u s e d ; o n l y s t a n d a r d s i m p l e x p r o c e d u r e s a r e a v a i l a b l e .
Then, e a c h o f t h e l i n k e d models may be f o r m u l a t e d a s : k
Minimize w i t h r e s p e c t t o xk E E"
s u b j e c t t o
where
a k = c o n s t a n t s f o r a l l v a l u e s o f t h e i r P i t bst C s ' s j
i n d i c e s ;
P ( k , s ) = a n i n d e x f u n c t i o n , which e q u a l s t h e i n d e x o f t h e component o f v c o n t a i n e d i n t h e s - t h c o n s t r a i n t o f model k ;
The s e t o f common c o n s t r a i n t s i s
-
-
-F q
+
r = I I D q r r v-
' 0,
9 = 1tM twhere F and D a r e g i v e n c o n s t a n t s .
9 q r
I t i s t h e n n e c e s s a r y t o choose a n a l g o r i t h m f o r scheme ( 3 6 ) . L e t i t be a m o d i f i c a t i o n o f t h e well-known Newton method, which r e q u i r e s p a r t i a l d e r i v a t i v e s o f t o be c a l c u l a t e d f o r the f i r s t and second o r d e r .
F o r m a l l y , u s i n g f o r m u l a (31 )
,
w e havewhere 6 i j i s e q u a l t o 1 , i f i i s e q u a l t o j , and i s z e r o o t h e r w i s e . I n t h i s e x p r e s s i o n it i s n e c e s s a r y t o d e t e r m i n e t h e t e r m s
-
apa;
;
since o n l y they a r e r e l a t e d t o t h e 'smooth' v e r s i o n o f t h e SUMT f o r problems ( 3 7 ) - ( 3 8 )
.
T h i s may be done w i t h t h e h e l p o f t h e F i a c c o and McCormick theorem ( 1 968).
I f g r a d i e n t s o f a c t i v e c o n s t r a i n i n g f u n c t i o n s ( i . e . y:(xk,v)) a r e l i n e a r l y i n d e p e n d e n t , t h e n t h e r e l a t i o n
t a k e s p l a c e , where u i i s t h e Lagrangian m u l t i p l i e r f o r c o n s t r a i n t s s o f model k .
I n t h e o p p o s i t e c a s e , w e s h o u l d use s e v e r a l i t e r a t i o n s o f t h e Newton method t o s o l v e problems (24)
,
w i t h t h e s t a r t i n g p o i n t sg i v e n by t h e Simplex method, and t o c a l c u l a t e 7i ap d i r e c t l y . W e w i l l now d e s c r i b e how e l e m e n t s o f t h e s e n s i t i v e m a t r i x c a n b e ays
found
.
I n t h e l i n e a r c a s e
a
k Ek k --
m C k a . a k k a 2 paxi ax S= 1 s i s j k 2
'
j
a
(yS)Taking i n t o c o n s i d e r a t i o n proved r e l a t i o n s (Umnov 1975) :
l i m a L p = + m
T++O k 2 f o r a c t i v e s
,
a(yS
and
lim =
o ,
f o r n o n a c t i v e s,
~ + + O k 2 3:(ys)
we o b t a i n t h e systems o f l i n e a r e q u a t i o n s f o r d e s i r a b l e compo-
a x k .
n e n t s o f
.
-
kf o r a l l i = l , n and r =
,
where Qk i s t h e s e t o f i n d i c e s f o r t h e a c t i v e c o n s t r a i n t s o f model k .
For t h e p a r t i a l d e r i v a t i v e s o f t h e second o r d e r we c a n use t h e same i d e a s w i t h o u t any t h e o r e t i c a l i n n o v a t i o n s .
To complete o u r c o n s i d e r a t i o n o f t h e d i s t r i b u t e d system of l i n e a r models, w e have t o s o l v e t h e problem o f how t o choose t h e l e n g t h o f t h e s t e p a l o n g t h e minimizing d i r e c t i o n w . AS men- t i o n e d above, it i s n o t r e a s o n a b l e t o use methods t h a t a r e b a s e d on t e s t i n g a l a r g e number o f sample p o i n t s . I t i s b e t t e r t o u s e no sample p o i n t s .
I n t h e proposed approach we o n l y have t h e p o s s i b i l i t y of e v a l u a t i n g t h e l e n g t h on t h e b a s i s o f i n f o r m a t i o n a l r e a d y o b t a i n e d . The d e s i r a b l e e v a l u a t i o n i s e q u a l t o t h e minimum of t h e f o l l o w i n g t h r e e numbers:
--
t h e norm o f w , i . e .)I
w11
;--
t h e v a l u e of t h e s t e px,
by which a t l e a s t one of t h e n o n a c t i v e common c o n s t r a i n t s becomes a c t i v e ;--
t h e v a l u e of t h e s t e pG ,
by which a t l e a s t one of t h e n o n a c t i v e c o n s t r a i n t s b e l o n g i n g t o t h e l i n k e d models becomes a c t i v e .T h i s e v a l u a t i o n e n s u r e s a d e c r e a s e i n t h e v a l u e o f
a
a n d ,hence, t h e convergence o f t h e whole p r o c e d u r e ( P s h e n i t s h n i j and D a n i l i n 1975)
.
The v a l u e s of
g
and may be found i n t h e f o l l o w i n g way.L e t v = v
+
&A, t h e n0
Hence
We c o n s i d e r o n l y t h o s e q f o r which Rq(v0) > 0 and E D w c 0
.
r= 1 q r r
min R (v0)
8 =
q LE D w
\ r = l q r r
I n an analogous way, w e can o b t a i n
where, f o r example, i n t h e c a s e of independence, t h e g r a d i e n t s of a c t i v e c o n s t r a i n t s
I n o t h e r c a s e s @: can be e v a l u a t e d by u s i n g a t e s t p o i n t along t h e d i r e c t i o n w.
This i m p l i e s t h a t
f o r which
Yr(vo) k > 0 and
mk
T < 0.
TEST CASE
A s p e c i a l example s u p p l i e d w i t h good s y n t h e t i c d a t a was pre- pared i n o r d e r t o t e s t t h e model system. The r e g i o n under a n a l y s i s was d i v i d e d i n t o t h r e e s u b r e g i o n s , which c o n t a i n e d t h e following s e c t o r s : a g r i c u l t u r e , i n d u s t r y , w a t e r s u p p l y , and l a b o r .
C o o r d i n a t i o n of t h e s e c t o r s i s shown i n Figure 3 , which d i f f e r s from F i g u r e 2 i n t h e f o l l o w i n g way. A l a b o r model r e p l a c e s t h e p o p u l a t i o n and m i g r a t i o n models (which i n Figure 2 depend on. c a p i t a l and l a b o r a l l o c a t i o n ) . I n t h e l a b o r model, t h e number of employees i s dependent only on c a p i t a l investments
d i r e c t e d t o t h e s e r v i c e s e c t o r . The c h a r a c t e r i s t i c s of each s e c t o r a r e d i s c u s s e d below.
A g r i c u l t u r e
For each of t h e t h r e e s u b r e g i o n s f o u r t y p e s of c r o p and two t y p e s of technology ( w i t h water-consumption v a r i a n t s p e r c r o p u n i t ) a r e c o n s i d e r e d . The f o l l o w i n g c o n s t r a i n t s a r e a l s o assumed:
--
c o n s t r a i n t on t h e l a n d a v a i l a b l e f o r a g r i c u l t u r e ;--
c o n s t r a i n t on t h e c h o i c e of technology;--
c o n s t r a i n t on w a t e r a v a i l a b l e f o r b a s i c consumption;--
c o n s t r a i n t on w a t e r a v a i l a b l e f o r peak-period consumption;.External
investments (c)
I
Capital andI
cA, EI,
G,
ES = vectors o f i n t e r r e g i o n a l d i s t r i b u t i o n o f c a p i t a l investments t o the a g r i c u l t u r a l , i n d u s t r i a l , water-supply, and s e r v i c e s e c t o r s ; EA, EI, Q, LS = v e c t o r s o f i n t e r r e g i o n a l d i s t r i b u t i o n o f labort o the a g r i c u l t u r a l , i n d u s t r i a l , water-supply, and s e r v i c e s e c t o r s ;
%,GI = subregional water flows t o a g r i c u l t u r e and industry :
E = vector o f subregional labor;
+
= information f l o w s .Figure 3. The tested system of regional models.
