Application of the Generalized Reachable Sets Method to Water Resources Problems in the Southern Peel Region of the Netherlands

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

APPLICATION OF THE GENERALIZED REACHABLE SETS METHOD TO WATER RESOURCES PROBLEMS IN THE SOUTHERN PEEL REGION OF THE NETHERLANDS

G .K. Kamenev A.V. Lotov

P.E.V. van Walsum

June 1986 CP-86-19

CoLLaborative P a p e r s r e p o r t work which has not been performed solely at t h e International Institute f o r Applied Systems Analysis and which has received only limited review. Views o r opinions e x p r e s s e d h e r e i n do not necessarily r e p r e s e n t those of t h e Insti- t u t e , i t s National Member Organizations, o r o t h e r organizations supporting t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREFACE

This work i s an interesting example of t h e application of t h e GRS method f o r multiobjective analysis t o one of t h e c a s e studies of t h e Regional Water Policies Project at IIASA. The simplification of models and computations f o r this example were conducted at t h e Computing Centre of t h e USSR Academy of Sciences in Moscow by t h e a u t h o r of t h e GRS method, A.V. Lotov, and his colleague G.K. Kamenev. P.E.V. van Walsum, a member of t h e IIASA p r o j e c t w a s at this end of t h e collaboration, and s a w to t h e qualitative significance of t h e r e s u l t s by comparing them with t h e r e s u l t s g e n e r a t e d at IIASA using o t h e r methods.

S.A. Orlouski P r o j e c t Leader

Regional Water Policies P r o j e c t

CONTENTS

1. Introduction 2. The GRS Method 3. The Original Model

3.1 Agricultural Technologies 3.2 Animal S l u r r y By-products 3.3 Labour Requirements 3.4 lncome

3.5 Water Quantity P r o c e s s e s

3.6 Fertilization, Mineralization of Organic-N 3.7 Public Water Supply

3.8 Natural Ecosystems 4. The Aggregation

4.1 Aggregation of "Flow" Type Variables 4.2 Aggregation of Groundwater Levels 4.3 Aggregation of Coefficients

5. Construction of t h e GRS

6. The Results f o r t h e "Model Subregion"

7. The Five Cluster Model

7.1 Animal S l u r r y By-products 7.2 lncome

7.3 Water Quantity P r o c e s s e s 7.4 Public Water Supply 7.5 Water Quality

8. The Results for Cluster Model

8.1 The Cross-Sections Containing lncome R e f e r e n c e s

Appendix 1 Appendix 2

APPLICATION OF THE GENERALIZED REACHABLE SETS METHOD TO WATER RESOURCES PROBLEMS IN THE SOUTHERN PEEL REGION OF THE NETHEELANDS

G .K. Kamenev, A.V. Lotov and P.E.V. van Walsum

1. INTRODUCTION

This r e s e a r c h w a s c a r r i e d out as a p a r t of investigations undertaken by t h e International Institute of Applied Systems Analysis (IIASA) on t h e methods and p r o c e d u r e s t h a t c a n assist t h e design of policies aimed at pro- viding f o r t h e rational use of water and r e l a t e d r e s o u r c e s taking into account economic, environmental and institutional a s p e c t s . One of t h e c a s e studies which w a s used f o r testing t h e s e methods i s t h e problem of applica- tion of water in a g r i c u l t u r a l production in t h e Southern P e e l region of t h e Netherlands. The problem w a s discussed in t h e p a p e r by Orlovski and Van Walsum (1984). H e r e we apply t o t h e investigation of water problems of t h e t-egion under study a new a p p r o a c h

-

t h e generalized r e a c h a b l e sets (GRS) method. This a p p r o a c h makes it possible t o p r e s e n t t h e information con- tained in a model implicitly in a n aggregated explicit form. Within t h e framework of decision s u p p o r t systems t h e GRS a p p r o a c h c a n b e applied f o r analysis of simplified models on screening s t a g e of investigation. The method allows us t o d e s c r i b e t h e set of all indicator values which are r e a c h - a b l e (accessible) under feasible alternatives.

The decision makers ( o r e x p e r t s ) study two-dimensional cross-sections of t h e accessible set p r e s e n t e d in dialogue regime on t h e s c r e e n of t h e com- p u t e r , choose some "interesting" combinations of indicators and corresponding decisions. These decisions are checked a f t e r w a r d s in simu- lation experiments with more adequate models.

This a p p r o a c h gives t o decision makers, e x p e r t s a n d t h e i n t e r e s t e d public t h e understanding of potential possibilities of t h e system u n d e r study. I t helps t o formulate s c e n a r i o s and decisions f o r simulation e x p e r i - ment and t h e r e f o r e helps t o overcome t h e main disadvantage of simulation, consisting in t h e difficulty of choosing s c e n a r i o s and decisions t o b e checked in simulation runs.

The generalized r e a c h a b l e sets method was applied at IIASA f o r inves- tigation of water allocation in t h e Southwestern Skane region of Sweden (Bushenkov et al., 1982). This time t h e method i s applied t o w a t e r problems of t h e S o u t h e r n P e e l r e g i o n in t h e Netherlands.

Two s t a g e s of t h e investigation are described herein. In t h e f i r s t s t a g e a "modal subregion1' of t h e region i s studied. In t h e second s t a g e t h e region i s d e s c r i b e d as a combination of five economic c l u s t e r s .

F i r s t of a l l w e s h a l l discuss t h e mathematical formulation of t h e GRS method.

2. THE GRS PETHOD

The generalized r e a c h a b l e sets (GRS) method of investigation of con- trolled systems was developed f o r t h e analysis of t h e models with exogenous variables. The basic idea of t h e method consists of t h e following. The pro- p e r t i e s of t h e model u n d e r study are investigated using a g g r e g a t e d vari- ables. The investigation i s based on numerical construction of a set of a l l combinations of values of a g g r e g a t e d v a r i a b l e s which are r e a c h a b l e ( o r accessible) using feasible combinations of original v a r i a b l e s of t h e model.

This set i s called t h e GRS and for t h i s r e a s o n t h e method i s called t h e GRS a p p r o a c h . In multiple c r i t e r i a decision making t h e GRS a p p r o a c h employs a n explicit r e p r e s e n t a t i o n of t h e set of all accessible values of objectives o r performance indicators.

The mathematical formulation of t h e method is as follows. Let t h e mathematical model of t h e system under study b e

where R n i s n-dimensional l i n e a r s p a c e of v a r i a b l e s x (controls), G, is t h e set of feasible values of v a r i a b l e s x , A i s a given matrix, b i s a given vec- tor. Let f E R m b e t h e v e c t o r of a g g r e g a t e d v a r i a b l e s ( c r i t e r i a ) . The vec- tor -t' i s connected with v a r i a b l e s x by l i n e a r mapping d e s c r i b e d by t h e given matrix

F ,

i.e.

The GRS f o r t h e model (1) with t h e mapping (2) i s defined as

The GRS a p p r o a c h consists of t h e construction ( o r approximation) of t h e set Gf in t h e form

G~ = [ f E R ~ : D ~ s ~ ~

and in t h e f u r t h e r analysis of t h e set Gf.

Note t h a t t h e description of t h e set Gf h a s a form of a n intersection of a finite number of hemispaces. This makes i t possible to provide a decision maker with any two-dimensional cross-section of t h e set s h o r t l y a f t e r his r e q u e s t for it.

The p r e s e n t e d mathematical formalization of t h e a p p r o a c h relates to finite dimensional models. Nevertheless i t c a n easily b e reformulated f o r l i n e a r functional s p a c e s of g e n e r a l t y p e (see Lotov 1981a, 1981b). In t h e latter case t h e feasible set and t h e mapping must b e approximated by finite dimensional analogues.

The construction of t h e GRS i s d e s c r i b e d in Section 5. The GRS a p p r o a c h w a s used f o r various purposes: f o r evaluation of potential possi- bilities of economic systems (Lotov 1981b, Bushenkov et al. 1982) f o r a g g r e - gation of economic models (Lotov 1982), f o r coordination of economic models (Lotov 1983). The a p p r o a c h c a n a l s o b e effectively applied to multi- ple c r i t e r i a decision making (MCDM).

Application of t h e GRS a p p r o a c h applied t o MCDM problems usually s e r v e s t o provide t h e decision maker (DM) with t h e information on t h e effective set in objective s p a c e which i s a p a r t of t h e boundary of t h e GRS.

In t h i s c a s e t h e GRS a p p r o a c h is r e l a t e d t o so-called generating interactive methods (Cohon 1978) which inform t h e DM on t h e possibilities of t h e system under study while t h e p r o c e s s of choosing a compromise between competing c r i t e r i a i s l e f t t o t h e DM.

3.

THE

ORIGINAL MODEL

The model of t h e water r e s o u r c e s of t h e Southern Peel Region w a s introduced and discussed in Orlovski and Van Walsum (1984). H e r e w e use t h e aggregated versions of t h e same model t o study possible development of t h e a g r i c u l t u r a l and water systems of t h e region during one y e a r .

The model by Orlovsky and Van Walsum i s a simplified model which links submodels of a g r i c u l t u r a l production, water quantity and quality p r o c e s s e s and soil nitrogen processes. In t h e model a y e a r i s split into two p a r t s :

"summer" which starts on April 1 and "winter" which starts on October 1.

The y e a r i s t a k e n from t h e beginning of winter.

The Southern P e e l region i s divided into 31 subregions. This division i s based on classes of groundwater conditions and soil physical units.

The a g r i c u l t u r a l production in t h e model i s described by means of

"technologies".

3.1. Agricultural Technologies

The t e r m "technology" i s used f o r a combination of agricultural activi- t i e s involved in growing and processing of a c e r t a i n c r o p and/or livestock.

I t i s assumed t h a t technologies differ from e a c h o t h e r by t h e i r outputs and a l s o by t h e inputs r e q u i r e d t o produce t h e s e outputs. For convenience, a distinction i s made between agricultural technologies t h a t use land and t h o s e t h a t do not. The s e t of t h e f o r m e r is denoted by

JX,

t h e set of t h e latter by

JZ.

I t i s a l s o convenient t o f u r t h e r subdivide t h e set

JX

into t h e subset

JXL

of land-use technologies involving livestock and t h e subset

JXD

of land-use technologies not involving livestock. A similar subdivision of t h e set JZ into t h e subsets J Z and JZD i s made.

All technologies considered a r e explicitly c h a r a c t e r i z e d by t h e follow- ing t y p e s of inputs (resources): labour, capital, water. Land-use technolo- gies of t h e s e t X are additionally c h a r a c t e r i z e d by t h e input of nitrogen supplied by fertilization.

Each technology i s a l s o c h a r a c t e r i z e d by t h e output o r production of t h e r e s p e c t i v e goods ( c r o p yields, livestock products). Technologies t h a t involve livestock are additionally c h a r a c t e r i z e d by outputs of animal slur- r i e s produced as byproducts.

The use of a g r i c u l t u r a l technologies is described in terms of t h e i r intensities. For land-use technologies intensities have t h e meaning of areas of land allocated t o t h e s e technologies. F o r technologies t h a t do not use land and t h a t involve livestock (from s e t JZL) intensities have t h e meaning of a number of livestock-heads; f o r a technology from t h e s e t J D , t h e intensity may have t h e meaning of f o r instance t h e amount of pig s l u r r y t r a n s p o r t e d t o outside t h e region.

I t i s assumed t h a t such inputs as l a b o r and c a p i t a l f o r e v e r y technology c a n b e r e p r e s e n t e d by corresponding quantities p e r unit of i t s intensity.

(For example, amount of l a b o r p e r unit area of land f o r a technology from set X . ) I t i s a l s o assumed t h a t t h e water inputs f o r technologies not using land can b e quantified in t h e same normative way (amount p e r unit inten- sity).

But t h e situation i s different with describing water inputs and t h e corresponding outputs f o r land-use technologies. One r e a s o n f o r t h i s difference i s t h a t both t h e water availability and t h e output of land-use technologies depend on weather conditions. Another r e a s o n i s t h a t t h e availability of water is a l s o influenced by activities in t h e region, especially pumping of groundwater. In o r d e r t o t a k e into account t h e r e s p e c t i v e pos- sible variations in t h e performance of land-use technologies a finite number of options f o r e a c h such technology are considered, which c o v e r a suitable v a r i e t y of typical water availability situations in e a c h subregion. F o r t h e s a k e of brevity t h e term subtechnology i s used t o r e f e r t o such a n option.

Each of subtechnologies k is c h a r a c t e r i z e d by t h e c r o p productivity qk , by t h e corresponding seasonal a v e r a g e s of t h e soil moisture v r k and of a c t u a l evapotranspiration e a t , as well as by t h e t o t a l nitrogen requirement nrk (all amounts p e r unit area of land). The value v r k i s t r e a t e d in t h e model as t h e "demand" for soil moisture, t h e satisfaction of which ( t o g e t h e r with t h e satisfaction of t h e requirement for nitrogen) g u a r a n t e e s obtaining t h e c r o p productivity not lower t h a n qk

.

The following notation i s used for intensities of technologies and sub- technologies (r-subregions, j-technology, k-subtechnology):

x ( r , j )

-

_{area of} land allocated to technology j E JX,

mu ( r , j ,k )

-

area of land allocated to subtechnology k of technology j E J X ,

z ( r , j )

-

intensity of technology j E

JZ.

In t h e model used in t h i s r e p o r t t h e r e were 1 0 land-use technologies divided into 3 subtechnologies and 5 technologies which d o not use land. The follow- ing technologies are used in t h e model. F o r j E

JXL

w e h a v e

glasshouse h o r t i c u l t u r e intensive field h o r t i c u l t u r e extensive field h o r t i c u l t u r e p o t a t o e s

c e r e a l s

maize with low nitrogen application maize with medium nitrogen application maize with high nitrogen application grassland with high cow density grassland with low cow density

F o r j E

JZ

_{w e} have:

j

=

1 : beef c a l v e s j

=

2: pigs for feeding j

=

3: pigs for breeding j

=

4: egg-laying chickens

j

=

5 : b r o i l e r s W e obviously have

f o r all r

=

1 ,

. . . ,

31 and j

=

1

, . . .

, 10. The t o t a l area of a g r i c u l t u r a l land in subregion r i s denoted by za ( r ). Wehave

W e also have c o n s t r a i n t s o n land allocated t o c e r t a i n g r o u p s of c r o p s :

where C1, L

=

1

, . . . ,

8, are s u b s e t s of

JX

and z m a z ( r ,L ) are exogene- ously fixed. We have Cj

= t

j

1

f o r j

=

1 ,

. . . ,

5 ; C6

=

[6,7,8

1,

^C7

=

[ g j, C,

=

[ l o ] .

3.2. Animal Slurry By-products

The technologies t h a t involve livestock p r o d u c e animal s l u r r i e s as byproducts. These s l u r r i e s are used as f e r t i l i z e r s f o r land-use technolo- gies in t h e region. From t h e environmental viewpoint t h e s l u r r i e s produced during t h e summer and t h e winter c a n b e s t b e temporarily s t o r e d in t a n k s till t h e n e x t s p r i n g a n d only t h e n applied t o t h e land. The s t o r a g e must not e x c e e d s t o r a g e capacities:

where m x w ( j , m ) i s t h e winter production of s l u r r y m p e r unit technology j ^E LXZ, m z ( j ,m ) i s t h e y e a r production of s l u r r y m p e r unit technology j ^E

JZ,

m a ( r ,L ,m ) is t h e autumn application of s l u r r y m t o t h e L-th direc- tion of application, m c ( r ) i s t h e s t o r a g e capacity. In t h e model five kinds

of s l u r r i e s are described:

m = 1 : c a t t l e s l u r r y , m

=

2: beef calf s l u r r y , m

=

3: pigs s l u r r y m

=

4: chicken s l u r r y , m

=

5: b r o i l e r manure.

Two d i r e c t i o n s of s l u r r y application are described:

L

=

1 : on t h e a r a b l e land.

L

=

2: on t h e grassland

W e suppose t h a t a f t e r s p r i n g application of s l u r r i e s t h e t a n k s are empty:

where ms ( r ,L ,m ) i s t h e s p r i n g application of s l u r r y m in t h e L-th direction of application.

3.3. Labour Requirements

Labour requirements are d e s c r i b e d by t h e equality

where Lp i s t h e amount of l a b o u r in t h e region, Lu i s t h e unemployment, Lh i s t h e amount of l a b o u r h i r e d from outside of t h e region. T h e r e are t h e res- t r i c t i o n s

LA S Lhmax , _('7)

Lu S Lumax , ( 8 )

3.4. Income

The income y is calculated by t h e following equation

where f s ( r , l )

-

i s t h e amount of chemical f e r t i l i z e r nitrogen applied t o land type l , L

=

1,2

i s ( r )

-

i s t h e amount of sprinkling from surf ace water

i g ( r )

-

is t h e amount of sprinkling from groundwater

vz

( r , j ,k

1,

Y z ( r , j > , r m c ( r ) , p f , p e i s ( r ) , p e i g ( r ) , r s c ( r ) , r g c ( r ) , plh are t h e

corresponding incomes (costs) p e r unit of t h e respective variables.

3.5. Water Quantity Processes

The water quantity p r o c e s s e s are described in t h e following manner.

Let hs ( r ) b e t h e groundwater level in t h e beginning of t h e summer, hw ( r )

b e t h e groundwater level at t h e end of t h e summer, where index r stands f o r subregion. Let

hs =

(As ( r ) , r

=

1 ,

. . .

, 31) and h y

=

(hw ( r ) , r

=

1 ,

. . .

^{, 31)} b e corresponding vectors. Let gw

- =

(gw ( r ) ,

r

=

1 , .

. .

^{, 31)} b e t h e v e c t o r of public water supply extractions during

winter. Then in v e c t o r notation w e have

- -

h s

=

h s o

-

~qw ,

where h x

=

(Aso ( r ) , r

=

1 ,

. . .

, 31) is t h e v e c t o r of groundwater levels t h a t would o c c u r if t h e r e were no extractions, A is 31 x 31 matrix with non- negative elements describing influence of extractions.

The groundwater level at t h e end of t h e summer is d e s c r i b e d in a simi- lar manner:

h w = h w o

-

- B g 7 E - & T + h i i , ( 1 1 )

where v e c t o r notation i s used f o r t h e influence matrices B , C and D and t h e v e c t o r of amounts of subirrigation

= ⁼

( u s ( r ) , r

=

1 ,

. . .

, 3 1 ) . T h e r e are t h e r e s t r i c t i o n s

describing hydrogeologic circumstances. The amount of sprinkling i s res- t r i c t e d by sprinkling c a p a c i t i e s

Sprinkling from s u r f a c e water is connected with subirrigation in t h e follow- ing manner

where p i s a proportionality constant. The s u r f a c e water supply capacity c a n b e limited

U S ( ? + i s ( r ) S s , , ( r ) , r = I , .

. .

, 31

.

( 2 0 ) The supply of water f o r t h e whole region is r e s t r i c t e d too:

The moisture content of t h e r o o t zone in t h e middle of summer i s d e s c r i b e d by t h e following equations

where t h e total amount of moisture required f o r subtechnology zu, ( r

,

j ,k ) i s r e s t r i c t e d by t h e moisture content in t h e beginning of summer (the f i r s t addendum, i.e. Lvs ( r )

.

hs ( r )) plus t h e change of soil moisture content in t h e middle of summer (@ = 1 / 2 ) . H e r e icpvs ( t ) and Lvs ( r ) are t h e coeffi- cient and proportionality constant describing t h e influence of t h e level of groundwater at t h e end of winter on t h e moisture content of t h e rootzone;

Lvz ( r ) i s t h e coefficient of a piecewise l i n e a r function describing t h e influ- e n c e of a r i s e of t h e level of t h e groundwater at t h e end of summer on t h e capillary r i s e of moisture t o t h e rootzone, and vz,, i s t h e maximal amount of capillary r i s e of moisture t o t h e rootzone in t h e c a s e when t h e groundwa- ter level r i s e s above a c e r t a i n c r i t i c a l level.

In o u r model meteorological p a r a m e t e r s of t h e y e a r 1976 w e r e used.

This i s t h e y e a r with low precipitation.

3.6. Fertilization. Mineralization of Organic-N

Each technology j t h a t uses land h a s a specified level nr ( r , j ) of t h e amount of nitrogen t h a t i s r e q u i r e d f o r c r o p growth. This nitrogen can come from different s o u r c e s

-

chemical f e r t i l i z e r and various types of animal s l u r r i e s . The simplified r e p r e s e n t a t i o n of t h e constraints p r e s c r i b - ing t h e satisfaction of nitrogen requirements of technologies h a s t h e follow- ing form ( L = 1 , 2 )

z

5 [ e m u (L ,m )ma ( r ,L ,m )

+

ems ( L ,n )ms ( r ,L ,m

11 + I s

( r ,I )

=

_{( 2 4 )}

where emu ( L , m ) i s nitrogen effectivity of s l u r r y m applied in autumn in L -th direction;

e m s ( 1 , m ) i s t h e same, b u t applied in summer.

The following r e s t r i c t i o n d e s c r i b e s a minimum amount of chemical f e r t i l i z e r nitrogen applied in s p r i n g ( 1

=

1 3 )

where r f s ( L , j ) i s t h e requirement of technology of I-th t y p e ( p e r unit a r e a ) .

The leaching of n i t r a t e to groundwater w a s not d e s c r i b e d in t h e f i r s t d r a f t of t h e model, b u t i t i s included into t h e modified version of t h e model which i s d e s c r i b e d later in Section 7.

3.7. Public Water Supply

If t h e demands of public water supply in winter q p w and in summer q p s t h e n t h e total of t h e e x t r a c t i o n s in t h e subregions must satisfy r e s p e c t i v e l y f o r t h e winter and summer period

where w x w ( j ) i s t h e w a t e r u s e p e r unit of x ( r , j ) during winter, w x s ( j ) i s t h e same during summer

wmu ( j ) i s t h e w a t e r use p e r unit of z ( r , j ) during winter, wzs ( j ) i s t h e same during summer.

3.8. N a t u r a l E c o s y s t e m s

The r e s t r i c t i o n s on groundwater levels in some subregions are given

4. THE AGGREGATION

In o r d e r to make possible t h e application of t h e Generalized Reachable S e t s Approach t h e model d e s c r i b e d in t h e previous section was a g g r e g a t e d . In t h e f i r s t s t a g e of t h e investigation a "modal subregion" of t h e region was obtained. This "modal subregion" was used as a " c h a r a c t e r i s t i c r e p r e s e n t a - tive" of t h e whole region in a preliminary calculation of potential possibili- t i e s of t h e region.

In t h e second s t a g e of t h e investigation t h e region was divided into economic c l u s t e r s , e a c h of them consisting of a number of subregions. Let t h e t o t a l number of economic c l u s t e r s b e S. Let I, b e t h e set of subregions belonging to t h e s - t h c l u s t e r . W e suppose t h a t any subregion belongs to o n e and only one c l u s t e r . The "modal subregion" c a n b e t r e a t e d as a n economic c l u s t e r containing a l l subregions. T h e r e f o r e aggregation in both s t a g e s of investigation c a n b e d e s c r i b e d simultaneously (in t h e f i r s t s t a g e S

=

1 and on t h e second s t a g e S

+

^{1, in o u r case S}

=

5).

4.1.

A g g r e g a t i o n o f "Flow" T y p e V a r i a b l e s

The v a r i a b l e s x , no, y , q w , s , u s , i s , i g , m a , m s , mc have been a g g r e g a t e d in t h e following way:

F o r t h e c a p a c i t i e s (i.e. u p p e r limits) of t h e s e v a r i a b l e s t h e same scheme of aggregation was used.

4.2. A g g r e g a t i o n o f G r o u n d w a t e r L e v e l s

The levels of groundwater have been aggregated according t o t h e fol- lowing scheme

The initial groundwater levels h s o , hwo in equations ( l o ) , (11) have been aggregated in t h e same manner.

4.3. A g g r e g a t i o n o f C o e f f i c i e n t s

The coefficients of t h e equations have been aggregated according t o t h e t y p e of aggregation of variables presented in t h e s e equations.

(a) If t h e equation is of t h e t y p e

f o r example, equations (9), (22), (23) where

then t h e following method i s suggested: To obtain coefficients a g ( s ), ah (S ), B ( s ) in

u*(s)*(s)

+

a h ( s ) h s ( s ) = B ( s ) we use t h e formulae

where 19,,,(r) i s t h e maximal value (capacity) of 19(r), and h s o ( s )

=

(b) If t h e equation is of t h e t y p e

f o r example, equations ( l o ) , ( l l ) , then t h e following method is suggested: To obtain coefficient a ( s ,q ) in equation

w e u s e t h e formula

This scheme h a s been used f o r aggregation of matrices A , B

,

C ,

D .

5. CONSTRUCTION OF THE GRS

In t h i s Section, w e briefly discuss numerical methods f o r t h e construc- tion of t h e GRS. W e suppose t h a t t h e model of t h e system under study h a s t h e form

where A i s a given matrix, b is a given v e c t o r . The v e c t o r of c r i t e r i a

p

E Rm is connected with v a r i a b l e s by l i n e a r mapping

P =PX

^I (32)

where F i s a given matrix. The GRS which is defined implicitly as

should b e c o n s t r u c t e d in t h e form

G,

= [ p

€ R m :

D f s d ] .

To c o n s t r u c t t h e GRS in t h e form (34), at t h e Computing Center of t h e USSR Academy of Sciences a group of numerical methods w a s developed.

These methods were combined into t h e software system POTENTIAL f o r t h e computer BESM-6. The f i r s t version of t h e system POTENTIAL w a s published in (Bushenkov and Lotov, 1980), t h e second one w a s described in (Bushenkov and Lotov, 1982 and 1984).

The methods included into t h e system POTENTIAL are based on t h e con- s t r u c t i o n of projections of finite dimensional polyhedral sets into sub- spaces. Suppose we have some polyhedral set M belonging t o

O,

+ q ) - dimensional l i n e a r s p a c e R P + Q . Suppose t h i s set is described in t h e form of solution of finite system of l i n e a r inequalities

where v E R P , w E R Q , t h e matrices A and B as w e l l as t h e v e c t o r c are given. The projection of t h e set

M

into t h e s p a c e RQ of v a r i a b l e s w i s defined as t h e set Mw of all points w E Rq f o r which t h e r e e x i s t s such a point v E RP t h a t [ v ,w

1

^E RP ^{+ Q} belongs t o M . An example of t h e two dimen- sional set and i t s projection into one dimensional s p a c e is p r e s e n t e d in Fig- u r e 1.

To c o n s t r u c t t h e GRS f o r t h e system (31)-(32) l e t s consider t h e set

The GRS i s t h e projection of t h e set Z into t h e s p a c e Rm of t h e c r i t e r i a f

.

The methods of t h e system POTENTIAL give t h e possibility t o c o n s t r u c t pro- jections of polyhedral sets in t h e form of solution of a system of l i n e a r ine- qualities. Therefore, using t h e system POTENTIAL i t i s possible t o c o n s t r u c t t h e GRS in t h e form (34).

The f i r s t method of t h e construction of projections of polyhedral sets described as solutions of systems of l i n e a r inequalities w a s introduced by F o u r i e r (1826). This method w a s based on exclusion of variables by combi- nation of inequalities. The application of t h e method given by F o u r i e r t o t h e construction of t h e projection of t h e set given in Figure 1 is described in (Lotov 1981a).

Figure 1

The idea suggested by F o u r i e r w a s used in more effective methods of construction of projections of polyhedral sets ( s e e Motzkin et al. 1953, as w e l l as Chernikov 1965) based on exclusion of v a r i a b l e s by combination of inequalitiess. The experimental study of t h e s e methods proved t h a t t h e methods of t h i s kind are effective for small systems only (n

-

10-30). For mathematical models (31) with hundreds of v a r i a b l e s new methods were sug- gested. The most effective of them a t t h e moment i s t h e method of

improvement of t h e approximations of t h e projection (IAP). The IAP method gives t h e possibility t o approximate t h e projection of t h e set 2 f o r all t h e models (31) f o r which l i n e a r optimization problems c a n b e solved. The IAP method i s described in s h o r t below. For more detailed information see Bushenkov and Lotov (1982) as well as Bushenkov (1985).

The g e n e r a l idea of t h e IAP method consists of combining methods based on exclusion of variables with t h e optimization methods of construc- 'tion of t h e GRS which are usually ineffective f o r m

>

2. The IAP method

consists of iterations having t h e following form.

Before t h e k-th iteration two polyhedral sets Pk and P' should b e given while i t holds

The set Pk which is t h e internal approximation of t h e set Gf should b e given in t h e two following forms simultaneously:

1 ) as t h e solution of a system of linear inequalities, i.e.

where cj a r e t h e v e c t o r s and dj a r e t h e numbers calculated on previous iterations,

2) as t h e convex combination of points (vertices) f ,

. . .

, f r k :

The description of t h e polyhedral set using both forms i s called double description (Motzkin et al. 1953). I t i s necessary t o note t h a t t h e conver- sion from one form to a n o t h e r is a v e r y difficult t a s k . Only if t h e number of inequalities sk in t h e f i r s t form o r t h e number of points r k in t h e second one a r e r a t h e r small c a n t h i s task b e solved numerically. This i s why on t h e z e r o iteration w e c o n s t r u c t a simple approximation P I f o r which t h e

conversion between forms c a n b e fulfilled easily. Subsequently, s t e p by s t e p t h e internal approximation i s improved: from t h e set P I w e obtain P 2 and s o on. On k -th i t e r a t i o n we c o n s t r u c t t h e set Pk f o r which

Each form of t h e presentation of t h e set Pk+l i s calculated on t h e basis of t h e same form f o r t h e set Pk

.

To obtain t h e set Pk+l on t h e basis of t h e set Pk w e add t o t h e s e t Pk a new v e r t e x FTk+l. This v e r t e x is chosen in t h e following manner. I t i s sup- posed t h a t f o r any v e c t o r c j in t h e f i r s t form of presentation t h e following optimiation problem w a s solved:

{ ( c j , f ) ^-B

7

max

=

Fz ,

Az

S b

Let {z; , f;

1

b e t h e optimal solution of t h i s problem. Let

where

lbjll

is t h e norm of cj. The values of Aj d e s c r i b e descrepancy between t h e i n t e r n a l approximation Pk and t h e e x t e r n a l approximation Pk which i s described as

Since

t h e value of max {Aj: j

=

¹ ,

. . .

, Sk

1

can b e used as estimation of descrepancy between t h e sets Pk and Gf

.

In Figure 2 f o r m

=

2 we have t h e internal approximation Pk (its v e r - t i c e s are A , B , C and D), t h e e x t e r n a l approximation Pk (its v e r t i c e s are

K ,

_L

,

M and N) and t h e set Gf which i s unknown f o r t h e r e s e a r c h e r . The points E , F , G and H a r e t h e solutions of optimization problems (36) f o r ine- qualities describing t h e set Pk

.

Figure 2

The set Pk i s obtained by inclusion into t h e second form of descrip- tion of t h e i n t e r n a l approximation t h e new v e r t e x J'rk f o r which i s chosen one of t h e points

J',:

obtained in optimization problems (36). I t i s r a t h e r effective to u s e t h e point with maximal value of A; b u t d i f f e r e n t s t r a t e g i e s c a n b e applied as well. The description of t h e approximation of t h e set Pk in t h e second form i s found. Note t h a t to obtain J'rk t h e f i r s t form of t h e description of t h e set Pk was used. N o w w e will c o n s t r u c t t h e f i r s t form of description of t h e set Pk

In Figure 2 t h e point with maximal value of A, i s t h e point G . I t is included into t h e description of t h e s e t which t h e r e f o r e h a s v e r t i c e s A , B , C , D , and G. To obtain t h e description of t h e Pk in t h e f i r s t form in t w o dimensional case p r e s e n t e d in Figure 2 i t i s sufficient to exclude from t h e description t h e inequality corresponding to t h e line passing through t h e points D and C and to include in t h e description t w o new inequalities corresponding to t h e lines passing through points D and G as w e l l as C and G . In g e n e r a l (m >2) t h e problem of constructing t h e description of t h e set

Pk+l in t h e f i r s t form is not s o simple and c a n b e solved by using methods of excluding of variables in t h e system of l i n e a r inequalities.

F i r s t of all, we find t h e inequalities which a r e violated by t h e point

f r k + , . These inequalities a r e excluded from t h e system. Let

P I ,

f ^{2 ,}

. . .

^{, f N k} be t h e v e r t i c e s belonging t o t h e excluded inequalities. Let us

consider t h e cone with t h e v e r t e x

Irk+,

and t h e edges f 1 -

P 2 -

^grk+l

. . . INk -

i.e. t h e cone

W e shall p r e s e n t t h i s cone in t h e form of solution of finite number of ine- qualities. For t h i s r e a s o n w e consider t h e set

and c o n s t r u c t i t s projection into t h e s p a c e

R m

of v a r i a b l e s f

.

This projec- tion coincides with t h e cone

K

and c a n b e constructed by means of methods of excluding of v a r i a b l e s in systems of l i n e a r inequalities. Note t h a t t h i s problem h a s small dimensionality and c a n b e solved easily. The obtained system of l i n e a r inequalities we include into t h e description of t h e approxi- mation. The set Pk in t h e f i r s t form i s constructed. Then we solve optimi- zation problems (36) f o r new inequalities. The e x t e r n a l approximation p k

i s constructed as well. The k-th i t e r a t i o n i s finished.

After a finite number of iterations t h e polyhedral set Gf could b e con- s t r u c t e d . Usually if t h e system (31) is l a r g e enough i t i s necessarily t o ful- fill millions of i t e r a t i o n s t o c o n s t r u c t Gf precisely. T h e r e f o r e in p r a c t i c a l problems i t i s reasonable t o find a good approximation of t h e set GI. The good approximation i s usually found a f t e r a small number of iterations. For example, f o r m

=

4 t h e set Gf i s approximated with 1% precision a f t e r 15-20 iterations. Note t h a t w e c o n s t r u c t both internal and e x t e r n a l approximations of t h e set G f ; so i t i s possible t o decide a f t e r e a c h iteration t o s t o p o r not t o stop t h e p r o c e s s on t h e basis of graphical presentation of both approximations of t h e set.

I t is n e c e s s a r y t o note t h a t t h e IAP method coincides in some details with non-inferior s e t estimation (NISE) method introduced by J. Cohon (1978). The main f e a t u r e of t h e IAP method consists of using t h e double description of t h e polyhedral set and application of t h e methods of exclu- sion of variables from t h e systems of l i n e a r inequalities f o r t h e construc- tion of t h e double description.

The IAP method w a s applied f o r t h e construction of t h e GRS f o r t h e model of P e e l region.

6. THE RESULTS FOR THE "MODEL SUBREGION"

The GRS method w a s applied f o r investigation of t h e "model subregion"

described e a r l i e r . Four indicators were chosen:

(1) income y ; i t s maximal value equals t o 2340 x

lo5

^fl;

(2) level of groundwater above t h e minimal level h w

-

h u m i n ; i t s maximal value equals t o 50.2 cm; hwmin

=

-2OOcm ;

(3) public water supply e x t r a c t i o n during winter qw; its maximal value equals t o 51 X 106m3;

(4) public water supply e x t r a c t i o n during summer q s ; i t s maximal value equals t o 51 x lo6m3.

The GRS in t h e s p a c e [y , h w

-

^{hwmin, qw , q s j w a s} constructed. This s e t w a s studied in dialogue using presentation of two dimensional cross-sections (slices) of t h e s e t on t h e s c r e e n of t h e computer. Some of t h e c r o s s - sections are presented in t h i s p a p e r . A number of interesting points belonging t o t h e GRS were chosen. The values of t h e variables in this points a r e p r e s e n t e d as well. The values of indicators corresponding t o maximal income a r e pointed out by t h e s t a r : [y ' , h w '

-

^hwmin,

q w * , q s ' j .

Figure 3 shows t h e slices of t h e GRS in t h e s p a c e [ h w

-

^{hw,,,, q s}

1.

The value qw i s fixed on t h e level corresponding t o maximal income, i.e.

qw

=

qw '

=

11.75 x lo6m3. The value of income y i s changing from s l i c e t o slice. The slice with low value of y

=

^{1500 X} lo5fl contains points 1 , 3 , 3' and 1'. While t h e value of y i s increasing t h e s l i c e i s getting smaller and smaller. Changes of s l i c e caused by increment of t h e value of y a r e shown

by t h e arrows. The r e l a t e d value of y is p r e s e n t e d n e a r t h e boundary of t h e slice. For maximal value of y'

=

2340 x lo5fl t h e slice consists of one

6 3

point hw'

-

^hwmin

⁼

^{48.8 cm, qs}

⁼

^{11.8 x 1 0 m} (point 2).

I I

I I

I I

b

-.

10 qsW 1 20 30 40 50 qs

Y qw fixe

Figure 3

The analysis of slices presented in Figure 3 r e s u l t s in t h e following p r o p e r t i e s of t h e system under study. F i r s t of all point 1 with t h e maximal groundwater level can b e achieved only if t h e income is r a t h e r s m a l l (in Fig- u r e 3 we have y

=

1500 X lo5fl). If t h e value of y i s increased t h e point 1 does not belong t o t h e slice. But t h e fall of t h e maximum groundwater level belonging t o t h e slice i s r a t h e r small

-

only about 1 centimeter. On t h e smallest slice r e l a t e d t o value y'

=

2340 X lo5fl w e have hw'

-

^hwmin

=

48.8 cm. I t means t h a t t h e r e exists a conflict between income and groundwa- ter level but t h i s conflict i s limited t o small changes in level of groundwa- t e r .

The second f e a t u r e of t h e system under study which r e s u l t s from analysis of Figure 3 i s t h e l i n e a r trade-off between hw and qs described by t h e line between points 1 and 3. Note t h a t this boundary i s t h e s a m e f o r all slices and does not depend on t h e value of y . S o we have evident conflict between groundwater level and summer e x t r a c t i o n s of water.

Figure 4 p r e s e n t s five slices of t h e GRS in t h e s p a c e iy , qs j. These slices show t h e same p r o p e r t i e s of t h e system but from a different viewpoint. The value of qw i s chosen t o b e technologically connected with t h e value of qs and t o b e optimal in this sense. The value of hw is changed from slice t o slice. On t h e slice A t h e value of hw i s minimal, i.e., hw

=

hwmin

=

-200 c m . The possible values of y on this slice are not high, not g r e a t e r than 2200 X 1 0 f l . If t h e level of groundwater 5 hw i s increasing, t h e u p p e r boundary of y i n c r e a s e s as well (see slice B). This p r o p e r t y is connected with t h e f a c t t h a t t h e groundwater level hw and t h e income y have no conflict if t h e values of hw and y a r e r a t h e r small. On t h e slice C where t h e value of hw i s h i g h e r (hw

-

^{hw mi,}

=

40cm) t h e conflict between hw and qs a r i s e s : t o p r e s e r v e t h i s value of hw one h a s t o e x t r a c t less than 2 3 X l o 6 m 3 of water during t h e summer.

On t h e slice

D

_{w e} have hw

=

hw'

=

-151.2 cm. In t h i s c a s e water e x t r a c t i o n during t h e summer is limited by qs'

=

11.8 X lo6m3. The rnax- imum value of income y *

=

2340 X l o 5 f l c a n b e achieved on t h i s slice. On t h e slice E r e l a t e d t o maximal value of hw , i.e., hw

=

h w d n

+

50.2 cm

=

-179.8 c m , we have only limited possibilities of obtaining income which i s not h i g h e r than 1840 X l o 5 f l . Note t h a t t h e conflict between income and groundwater level a r i s e s if t h e value of hw i s h i g h e r than hw *

=

-151.2 cm.

Figures 5 and 6 p r e s e n t t h e same p i c t u r e but t h e s c a l e i s changed. The new slice F is r e l a t e d t o t h e value of hw which i s less than hw,,, but higher than hw

.

^{In Figure} 7 t h r e e slices of t h e GRS in t h e s p a c e f y , qw j a r e presented. All slices a r e r e l a t e d t o income-optimal value of groundwa- ter level hw '

.

The value of qs changes from slice t o slice:

Figure 4 hw = -151.2 cm

6 3

s l i c e A i s r e l a t e d to qs

=

1 0 X 1 0 m ,

---__-

hw = hwmi, = -200 cm

---

I I '

11 I I

1840

--

_{11 I I} ^I

I1 ' I

--

^{ti-+ hw =} -149.8 cm I

I ' I

E

/ j

^I

1720

--

^I

I I

f !

^I

--

^I

1 I ^I

I ' I

1600

-- i

_I

'

_'

~ ~ - - - 1 - - -

^A

s l i c e B i s r e l a t e d to qs

=

10.3 x lo6m3,

3 B A

s l i c e C i s r e l a t e d to qs

=

qs ' 11.82 X lo6m3.

I . . I I

I a ' I ^{I I I I}

10

d*

^{20 I} 30 40 50 qs ^b

Figure 7 shows t h a t if t h e value of q w i s minimally (qru

=

1 0 . 1 0 ~ m ~ ) * , t h e n t h e value of income c a n b e not g r e a t e r t h a n 1700 X l o 5 f l but a small additional value of q w makes t h e u p p e r boundary of t h e income jump up.

The boundary is kinked: a f t e r a s h a r p i n c r e a s e of possible income t h e boundary becomes horizontal. I t means t h a t t h e additional value of qru i s used ineffectively. Note t h a t t h e jump of t h e income n e a r t h e minimal value 8 M , j n ~ l values of qw and qr are connected with extractions for public water supply of

10 m

.

I I,-'' I I I 1

l o q s * 14

I I I I I

I I I

18

b

22 _qs

Figure 5

of qw depends on t h e value of q s . This means t h a t qs and qw are "techno- logically" connected in t h e model.

Figure 8 p r e s e n t s t h e same effect. W e have t h e s a m e s p a c e ly ,qw j but five slices p r e s e n t e d in Figure 8 c o r r e s p o n d t o d i f f e r e n t values of h w . The slice C i s t h e same as f o r Figure 7. The correspondence between slices and t h e values of qs and hw i s t h e following:

7 3

s l i c e A : qs = q s m i , = 1 0 m , h w = h w m , , = 1 4 9 . 8 c m ,

6 3

slice B: qs

=

10.27 x 10 m , hw

=

-150 cm,

Figure 6

6 3

slice C: qs

=

q s s

=

11.82 X 10 m , h w

=

h w

=

-151.2 cm,

6 3

slice D: qs

=

35 X 10 m , h w

=

-170 cm,

6 3

s l i c e E : qs = 5 l X l O m , h w = - 1 9 2 c m ,

2 hw = -151.2

cm_,-r0C7~

4- I

,//-

^{I I}

/' I

// I

/ I

/ I

/ / I

/ I

/ I

I I

(F

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 j

^{I I}

I

i ---.--- l-

^--,-,-J ^---------

2250

2200

2150

2100

--

--

2050 --

--

2000

--

I t is interesting t o note t h a t high extractions of groundwater during winter

6 3

and summer (qs

=

35 x 10 m , qw

=

15 x l o 6 m 3 ) can r e s u l t in sufficient high income ( y

=

2 3 0 0 ) . However, in this c a s e t h e groundwater level in n a t u r e areas i s very low ( h w

=

-170 cm).

T 1

^{I I I I I I I I , 1 I}

_

^. 10.5 qs* 10.9 11.3 11.7 q s

-- --

-

-- -- --

Figure 9 presents t h e r e s u l t s on cross-sections in t h e s p a c e f y , h w

-

h w m i n j . W e have 9 slices h e r e a r r a n g e d in t h r e e groups. The group A corresponds t o a minimal value of t h e summer extraction

7 3

qs

=

qsmin

=

10 m

.

group B corresponds t o qs

=

q s s

=

11.82 x

l o 6 ,

group

Figure 7

6 3

C c o r r e s p o n d s t o qs

=

qs,,,

=

5 1 x 1 0 m

.

The slices in o n e g r o u p are dis- tinguished by t h e value of q w . I t is clear t h a t t h e maximal value of hw which c o r r e s p o n d s t o g r o u p A is in s h a r p conflict with t h e income. The maximal value of income c a n b e achieved by a small i n c r e a s e of t h e summer water e x t r a c t i o n (group B). The f u r t h e r increasing of summer water e x t r a c t i o n (group C) r e s u l t s in low groundwater levels (not h i g h e r t h a n -180 cm) and r e a s o n a b l e d r o p s in incomes.

Figure 1 0 shows slices in t h e same s p a c e f y

,

hw

-

h w d ,

1,

but t h e s c a l e is d i f f e r e n t .

Figure 8

On the figures described above some points are presented (points 1-7).

The controls resulting in these points a r e described in Appendix 1. The indicators related t o points 1-7 are presented in Table 1.

Table 1

7 . THE

FIYE

CLUSTER MODEL

On t h e second s t a g e of o u r investigation a more complicated model w a s studied. The model consists of five economic c l u s t e r s (see Figure 11). I t w a s obtained on t h e b a s e of t h e model consisting of 3 1 subregions. The con- s t r u c t i o n of five c l u s t e r s model w a s based on aggregation described in Sec- tion 4, while S

=

5. The original 3 1 subregional model described in Section 3 was slightly modified. The modifications are described f o r t h e c l u s t e r model.

N

B.1. INVESTMENT AND REORGANISATION OF THE ECONOMIC STRUCTURE The modified model t a k e s into account t h e possibilities of reorganisa- tion of t h e economic s t r u c t u r e of t h e region. For t h i s purpose investment i s used. The volume of investment needed i s described in t h e following manner:

indicators (10

%

fl) hw-hw sub min

(cm)

Cbw

³

(10 m

(10 m 3

10

inv

= x

C p z i ( j ) . z i ( j )

-

^0.3 p z i ( j ) . z d ( j ) l +

j = I 1

170.3

50.2

10

10

2

234.6

46.6

11.75

11.82

5 5 5 2

+ x

p z i ( j )

.

z i ( j )

-

^{0 . 2 -} p z i ( j ) z d ( j ) ]

+ x

^Cpsto

x x

m s ( s , k , L ) ]

+

j =I j =I k = l 1 =1

+

scinv * sci

+

gcinv gci

-

0.9 scinv

.

^scd

-

^{0.5gc inv}

.

^gcd , 3

228.2

19.8

5 1

5 1

4

228.0

19.9

11.77

20.61 5

234.5

48.6

11.75

11.82 6

170.2

48.8

10

11.82 7

233.5

48.8

5 1

11.82

where

A : qs = qsmin

B: qs-qs*

C: qs = qs,,

--

2230

--

2140

-- --

2050

-- --

1960

--

Figure 9 1870

1780

--

inv i s investment,

zi ( j ) , zd ( j )

-

increments and decrements of intensi- t i e s of technologies j E

JZ,

z i ( j ) , z d ( j ) - the same for j E

JZ,

.

2

-- --

A

-.--s=-

/ - s ~ = l - c

qw = 51 X

lo6

m 3 -:3. ^B

--s+--T

e-

s.s/

e

_----IC

.-

# - C -

qw = 11.8X

lo6

^rn3

7 3 q w = 10 m

--

1690

-- --

1

______---

1600-- ] ]

I [

l , < # + + , t # # l

I 1 I I

I I 1 I I

0

t ^{l o}

^{20 30 I 40 I 50 hw- b}

hw*

-

^hwmin

I ^{I . ,} _{l " l r}^.,_l ^#_{r l}^I_r^,_r^,_l^,I _I ^I I _{I b} I

25 30 35 4 0 45 50 hw

-

hwmi,

Figure

10

sc* , scd

-

increment and decrement of sprinkling capacities from surface water,

get , gcd

-

the same from groundwater p z i (j ), pzi ( j ), sc inv, gc inv$ psto

-

prices.

Increments and decrements a r e connected with initial capacities

-

³³

-

THE SOUTHERN PEEL AREA

I Cluster: 1-6 Subregions Natural zones:

I I Cluster: 7-14 Subregions (N) 10 Subregion 1 1 1 Cluster: 15-19 Subregions (N) 16 Subregion I V Cluster: 20-26 Subregions 27 Subregion V Cluster: 27-31 Subregions (N)

Figure 11

The maximal possible value of investment i s given by inv,,,.

I t i s supposed that the total area of agricultural land i s used

7.1. Animal Slurry By-products

I t is supposed t h a t in addition to s p r i n g application of s l u r r y (see Sec- tion 3) autumn application is possible. In t h i s case t h e autumn application of manure of k - t h t y p e in t h e s - t h c l u s t e r should not e x c e e d t h e half-year production. The possibility of manure t r a n s p o r t between c l u s t e r s i s t a k e n i n t o account. W e obtain

where mt, ( s ,k )

-

e x p o r t of manure mtf ( s , k )

-

import of manure

The manure s t o r a g e b e f o r e s p r i n g application should not exceed s t o r a g e c a p a c i t i e s while in s p r i n g t h e t o t a l amount of manure should b e applied to land:

T h e r e is n o e x p o r t of manure outside t h e region

7.2. Income

The equation of income (9) should b e modified in a c c o r d a n c e with modif- ication of t h e model. F o r t h i s r e a s o n from income y , calculated by (9), t h e

5

cost of manure t r a n s p o r t

z

p m t mt, ( s .k ) is s u b t r a c t e d .

j =I

F u r t h e r , t h e t o t a l income should e x c e e d consumption

where l p i s t h e amount of l a b o u r in t h e region.

7.3. Water Quantity Processes

In t h e modified model i t was supposed t h a t

q w ( s ) = 0 . 8 . q s ( s ) , s = 1

, . . . ^,

⁵ ,

Introducing t h e fall of groundwater level in c l u s t e r s from t h e level hs O(s )

A h s ( s )

=

hs ( s )

-

^{h s o ( s )}

we obtain

For t h e fall of groundwater level in winter we obtain

The description of moisture content of t h e rootzone i s slightly changed while t h e p r o c e s s of deep percolation during t h e summer i s taken into account. The amount of d e e p percolation during t h e summer in c l u s t e r s denoted by d s ( s ) i s described as follows:

The soil moisture requirements a r e modified as well: