Vreke, J.

NARCIS 

Outline of a procedure to generate the target state

Wageningen Yield 

NN31545.1B48 
BIBLIOTHEEK 
TARINGGEBOUW S 
ICW note 1648 
Zuidelijk Peelgebied 39 
October 1985 
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OUTLINE OF A PROCEDURE TO GENERATE THE TARGET STATE 
BIBLIOTHEEK BE RAAFF 
Droevendaalsesteeg 3a 
Postbus 241 
6700 AE Wageningen 
drs. J. Vreke 
1 6 FEB. 1998 
A nota ( working paper ) of the Institute is a way of internal 
communication and not a publication. As such their contents vary 
strongly, from a sinple presentation of data to a discussion of 
preliminary research results with tentative conclusions. Some working 
papers are confidential and when so indicated are not available to third 
parties. 
CENTRALE LANDBOUWCATALOGUS 
0000 0941 2707 
ncjo3S,tf| 
contents 
page 
1. Introduction 1 
2. The Linear model 2 
3. The selection of the target state A 
A. The reachability of the target state 6 
page 1 
1. Introduction. 
In the Scenario Generating System a target scenario for the 
development of a region is selected. In the selection a 
linear model is uspd. This model describes the hydrological 
system, the agricultural system and the interactions between 
these systems. The target state is considered to be a 
stationary state. It is selected by maximizing income in 
agriculture taking into account constraints with respect to 
environment and public water supply. When a target state has 
been selected it has to be shown that it can be reached from 
the current situation in the region. 
In this paper an outline is given of a procedure to 
select the targetstate and to verify its reachability. The 
procedure is not finished yet 
and can probably be improved. The linear model is described 
very briefly in section 2. In section 3 an approach to 
select the target state is outlined. The reachability of 
this target state is the subject of section A. 
page 2 
2. The linear model. 
The linear model describes the hydrological system, the 
agricultural system and the interactions between these 
systems. Both the hydrological system and the agricultural 
system depend on the weather conditions which are 
stochastic. So the linear model is stochastic, but it has a 
deterministic component. The restrictions in the model can 
be split up into stochastic (i.e. weather dependent) 
restrictions and deterministic restrictions. The objective 
function is stochastic. For year t the model can be stated 
as: 
(1) max d.x(t) + c2.y(t) income in 
x(t),y(t) agriculture 
m.(x(t)-x(t-1)> - R2.y(t-1) <= b1 mutations in x 
B1.x(t) + B2.y(t) <= b2 deterministic 
constraints 
D1.x(t) • D2.y(t) <= b3 weather dependent 
constraints 
x(t),y(t) >= 0 
with d,c2,D1,D2 and b3 vectors or matrices with 
coefficients that depend on 
the weather. This means that 
when the weather is known then 
the coefficients are known too. 
B1,fl2,B1,B2,b1 and b2 vectors or matrices with 
coefficients that do not depend 
on the weather 
x the state vector with the allocation or the 
intensity of the technologies, the capacity of 
sprinkling etc.. The vector x is fixed during the 
year, 
y the vector with variables that depend on the 
weather (i.e. that can be changed during the 
year). Examples of variables are subtechnologies, 
irrigation and the quantity applied for the 
different types of manure. 
In (1) the general form of the model for one year is 
presented in the case that the weather is known. In reality 
the weather is not known. It is assumed that the climate can 
be described by a finite number of weather years each having 
its own probability of occurence. In this case the weather 
dependent constraints can be replaced by chance constraints. 
These chance constraints have to be met with a specified 
probability p. Or in other words the chance constraints have 
to be met in 100.p percent of the years. The objective 
function also depends on the weather. This can be handled by 
introducing the minimum earned income YDE5. 
page 3 
If an income Less then p is allowed with a probability (1-p), 
then the objective function of (1) can be replaced 
by (2) and (3) 
(2) max YDE5 
x(t),y(t) 
and the chance constraint ( Prl } indicates the probability 
that { ) ) 
(3) Pr<c1.x(t) + c2.y(t) - YDE5 >= 0} >= p 
Inclusion of YDE5 in y and of c1#c2 and -1 in D1 and D2 and 
introduction of chance constraints leads to the following 
reformulation of (1) 
(4) max YDE5 
x(t),y(t) 
R1.<x(t)-x(t-
Bl.xCt) + B2 
Pr<D1.x(t) + 
-1)> - R2.y(t-1) < = 
y(t) <= 
D2.y(t) <= b3} >= 
x(t),y(t) >= 
b1 
b2 
P 
0 with YDE5 
included in y(t) 
It is also possible to introduce subsets of chance 
constraints with different probabilities. Introduction of 
these subsets into the model and into the proposed procedure 
is very simple. However the model stated in (4) is used for 
notational reasons. 
page 4 
3. The selection of the target state. 
The target state selected in the Scenario Generating System 
has to be such that: 
- restrictions stemming from the environment and the public 
water supply are met with a specified probability p. 
- income in agriculture is as high as possible and equals or 
exceeds the income YDE5 with probability p 
- the state can be reached starting from the current state 
within T years 
The reachability of the target state is the subject of 
section 4. In the present section the selection of the 
target state is treated. The target state is considered to 
be a stationary state. In this concept the intensities of 
the technologies the capacities of sprinkling etc remain 
constant for an infinite number of years (i.e. the vector x) 
and there are no investments other then replacement 
investments. The vector y depends on the weather and changes 
from year to year. 
Because the target state is a stationary state the 
constraints in (4) concerning the year to year mutations can 
be left out and the problem can be stated as 
(5) max YDES 
x,y 
B1.x 4 B2.y <= b2 
PHD1.X + D2.y <= b3) >= p 
x,y >= 0 with YDES included in y 
This problem can be solved using Deterministic Equivalents 
for the chance constraints. One of the disadvantages of the 
linear model is that it results in specialisation. It is the 
allocation of land to a minimum number of technologies and a 
specialisation with respect to the technologies that do not 
use land. This is caused by the fact that minor income 
differences between technologies have the same influence 
with respect to allocation as very large differences. This 
leads to a large difference between current state and target 
state. 
In order to avoid this problem as much as possible the 
following procedure is proposed: 
STEP 1 select a desired minimum income YDES* by solving (S) 
and reduce this income by a small percentage rho. 
STEP 2 select the allocations and intensities of 
technologies which guarantee with probability p at 
least the the reduced optimum income 
( (1-rho).YDES* ), which meets the other conditions 
stated in (5) and which minimizes the changes 
(especially investments) with respect to the current 
state. In doing this the reachability of the target 
state is enlarged and the specialisation is less 
strong then in the optimum solution generated in 
STEP 1. 
page 5 
The second step can be considered as a kind of a sensitivity 
analysis of the optimum solution in 5TEP 1. In STEP 2 the 
income variable YDE5 is replaced by by the constant 
(l-rho).YDES* , with YDE5* the income generated in STEP 1. 
Let fCO,T] represent the changeswith respect to the current 
state. Find the limitations for the changes in a period of T 
years by extrapolating the mutation constraints 
R1.<x(t)-x(t-1)> - G2.y(t-1) <= b1 
The cumulative savings are not incorporated in the 
extrapolation because at this moment only technical 
constraints are considered. The cumulative savings are 
treated in section 4. The extrapolation results in the 
following set of constraints 
(6) PT.<x-x(0)) <= bT 
The problem that has to be solved in STEP 2 can be stated as 
( 7 ) min f [ 0 , T 3 
x , y 
flT.<x-xCO)} <= bT 
B1.x + B2.y <= b? 
Pr<D1.x • D2.y <= b 3 ) >= p 
x,y >= 0 with the variable 
YDE5 replaced by the 
constant (1-rho).YDES* 
When deterministic equivalents are used the problem 
becomes deterministic and can be solved. 
page 6 
A. The reachability of the target state. 
When the target state is selected it has to be shown that it 
can be reached from the current state within T years. In 
this section an outline of an approach based on Markov 
chains is described. This approach can be divided into 
1. the selection of changes in the state vector x. It is 
assumed that changes take place at the end of each period 
of for instance 5 years. The state vector is independent 
of the weather. The changes are limited by the income 
(via cumulative savings), by the constraints on the 
mutations and by the hydrological model. 
2. the analysis of the behaviour of the system within the 
period of 5 years. This means the selection of the vector 
y that depends on the weather conditions. The vector y is 
choosen in such a way that, for a known state vector x, 
the income in agriculture is maximized conditional to the 
weather conditions. This is stated in (6) 
(8) max c2.y(t) 
yCt) 
B2.y(t) <= b2 - B1.x(t) 
D2.y(t) <= b3 - D1.x(t) 
y(t) >= 0 
with c2, D1, D2 and b3 weather dependent 
coefficients. It can be derived from (8) that the value 
of y does not depend on the years y(t-1) etc. it only 
depends on x and on the weather. 5o for a known state 
vector x each weather year corresponds with a unique 
solution y, income (d.x + c2.y) and savings (income 
minus normative consumption). The sum of the savings 
during a period limits the possibility to invest at the 
end of the period. 
For both problems a procedure has to be formulated. It has 
been shown that the analysis of the behaviour within the 5 
years period can be reduced to seperate analysis for each of 
the five years. These yearshave the same mutually 
independent probability distribution for the the weather 
conditions. Because the savings in one year are directly 
linked to the weather year it is possible to generate the 
probability function for the cumulative savings in a period 
as the convolution over five years of the probability 
distribution of the weather years. The distribution of the 
cumulatove savings depends on the state vector x. 
For the changes in the state vector x a decision rule has 
to be formulated, fl possibility for this decision rule is 
minimisation of the differences between the target state and 
the actual state, taking into account the limitations posed 
by the hydrological system, the limitations with respect to 
the mutations and the cumulative savings. 
page 7 
Let ftt,T] represent the difference between the actual state 
and the target state (x(T)). Then the decision rule can be 
formulated as (the cumulative savings of period t-1 are 
included in y(t-1) ) 
(9) min flt,T3 
x(t) 
B1.<x(t)-x(t-1)> - 02.y(t-1) <= b1 
B1.x(t) • B2.y(t) <= b2 
Pr<D1.x(t) + D2.y(t) <= b3) >= p 
x(t),y(t) >= 0 
For the desired income a rather low value can be choosen, 
for instance the normative consumption. The problem stated 
in (9) can be solved when deterministic equivalents are 
used. Each value of the cumulative savings leads to a unique 
solution of the problem. This means that x(t+1) can be 
linked directly to the cumulative savings in period t (for 
x(t) known). Because the probability distribution of the 
cumulative savings (conditional to x(t) ) is given it is 
easy to generate the probability distribution of x(t+1) 
(conditional to x(t)). 
The procedure described corresponds to an analysis based 
on Markov chains, that is x(t+1) depends on x(t) and not on 
x(t-1),x(t-2), . Moreover the probability that x(t+1)=x1 
given that x(t)=xO is equal to the probability that 
x(t+n)=x1 when x(t+n-1)=x0. This because the probability 
distribution of the mutations in a period depends only on 
the statevector at the begin of the period. 
Whether this procedure can be applied depends on the 
number of potential future states. If this number is rather 
small then the procedure can be applied. The Markov chain 
contains two absorbing states i.e. the target state and the 
'infeasible state'. 0 reduction in the number of states can 
be obtained by classifying the cumulative savings into a 
small number of classes. This is a subject for further 
investigations. 


Outline of a procedure to generate the target state

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