ENVIRONMENTAL STOCHASTIC INPUT-OUTPUT MODEL
AND ITS DETERMINISTIC EQUIVALENT
Ruslan Biloskurskyy, Vasyl Grygorkiv
Chernivtsi National University
1. Introduction
Actual problems of ecological and economic cooperation and sustainable
development have been the object of research of many domestic and foreign
scientists. Important place in creating and solving the theoretical foundations of
sustainable development are, in particular, labors of Vernadsky, Duchin,
Liashenko, Leontief, Ford, Ryumina, Nomura, Kostanzy, Gul, Kay, Dzhampetro,
Daily, Nash, Forrester, Hubacek, Suh, Giljum and many others.
However, at present, there is no evidence that the problem of sustainable
development explored enough. Their complexity and diversity at the regional and
global levels require further study to develop economic and social mechanisms for
optimal balancing economic, environmental and social components. The first thing
to do in this respect science to create an effective instrumental and methodological
apparatus of systems analysis and modeling of sustainable development and
decision-making that will ensure in practice its real functioning.
To achieve the desired environmental and economic balance, the public should
actively carry out the regulation of production and the environment. The main
methods of this regulation are legal rules, economic mechanisms and technical or
technological support. Relationship and interconditionality these methods are
obvious, but to achieve practical effect require them timely and optimal
implementation. Theoretical foundations and algorithms for efficient interaction of
nature with industrial and economic processes to create a science. In this structure
the methodological and methodical approaches to the regulation includes macro-,
meso-and micro-level, which in practice under the same global, regional and local
levels.
At the macro level, the main task is to develop a theoretical framework for
environmental protection and the functioning of human society. Lawyers are here
to create rules of coexistence that would ensure protection of material, spiritual and
all other human values that are the heritage of humanity. Research economists
should be focused on maximum satisfaction of human needs with limited natural
resources. Representatives of scientific and technical units to engage further the
knowledge of the laws of nature in order to use the obtained knowledge in the
management of nature and society.
Meso level provides adaptation tasks to specific regions and economies. It is
formed principles of individual territories, states, units of State and others. Most
conflicts over natural resources and development of production occurs at this level.
To solve these contradictions can only using the fundamental theoretical arsenal of
methods of regulation.
At the micro level of theory and methodology implemented regulations
designed for macro-and a level. It is considered a separate person or an individual
household, which is the main economic agent or participants. In other words,
established at the macro and a level control device, at the micro level is specific
and tested in a sense check the adequacy of its appointment.
Thus, the role of science in the resolution of problems of sustainable development
is extremely high, but a special place among the scientific methodology here is
economic-mathematical modeling of ecological and economic interactions.
Difficulties arising separately in the mathematical modeling of economic and
ecological systems, greatly increasing in the case of construction of ecological and
economic models. This was a prerequisite for the formation of two main areas of
ecological-economic modeling:
1) taking into account environmental considerations in economic and
mathematical models;
2) consideration of human actions in ecological models.
A striking example of the first direction is the interbranch balance model
Leontief - Ford, and the second - Model Mono-Jerusalemskyy (model of optimal
exploitation of natural resources or the best "harvest"). In balance models in
economics and mathematical modeling also distinguish two large classes of
models: optimization and simulation. For example, the famous project of Forrester
is one of the simulations (it is studied the interaction of three systems: the
demographic, industrial, agricultural). The aim of this project is to maintain
sustainable development (ie a balance between society and nature) globally. As for
the optimization of ecological and economic models, then to them belongs a
considerable range of static and dynamic models, based on certain balances.
However, many problems of ecological-economic modeling still waiting to be
solved.
2. Stochastic Leontief - Ford model
Given the purpose of this work, back to the static Leontief - Ford model,
which is devoted to the study of many works. This model generalizes the static
Leontief inter-model on two groups of industries (production): primary production
(branches of material production) and auxiliary production (industry, dealing with
the destruction of pollutants). Mathematical models of such record:
 x (1)  Ax (1)  Bx ( 2 )  y (1)
 ( 2)
 x  Cx (1)  Dx ( 2 )  y ( 2 ) ,
(1)
where x (1)  ( x1(1) ,..., x n(1) ) T - column vector of gross output; x ( 2)  ( x1( 2) ,..., xm( 2) )T - column
vector deleted pollutants; y (1)  ( y1(1) ,..., yn(1) )T - column vector of the final product;
y ( 2)  ( y1( 2 ) ,..., ym( 2) )T - a vector-column volumes of pollutants neznyschenyh;
А  (аij )in, j 1 - square matrix production i costs in production j; B  (bil ) in,,lm1 - a
rectangular matrix of costs of production units i to destroy contaminants l;
С  (сlj ) lm, ,jn1 - rectangular matrix release of contaminants l during the production
unit j, D  (dls )lm, s 1 - square matrix of release of pollutants l during the destruction of
unit pollutant s (T - transpose operation).
The current scientific status of construction, analysis and application of
optimization and balance ecological and economic models based on the LeontiefFord model, to verify the real richness of content and usefulness of these models in
solving problems of economic analysis and forecasting, taking into account
ecological and economic balance. However, in most cases, these models are
deterministic, ie with a high degree of idealization reflect the real situation. In this
regard, there is a need to explore and stochastic models intersectoral environmental
and economic balance and their generalization to more adequately adapted to the
random (probability) character of the initial information about the importance of
technological characteristics, volumes and destroyed neznyschenoho pollution,
resources and income (or gains) and others. Specification of random values of
model parameters through their dependence on the elementary events a probability
space , S, P, where  - the space of elementary events, S - algebra of events, P the likelihood or numerical function. Models, parameters which are random
variables is called stochastic models.
Here we consider stochastic analogues of one of the optimization models of
cross-industry environmental and economic balance. Before formulating
optimization model, we note that one of the important criteria of ecological and
economic system is to maximize the income from that part of the main products
remaining after certain costs for destruction of complex pollutants. In a sense, this
criterion is tantamount to maximize total revenue at all, but in terms of ecological
and economic equilibrium described by equations of the ecological and economic
balance or the appropriate balance generalizing inequalities. Assume that the total
costs of primary products on the functioning of primary and secondary industries,
which according to equation (1) is the difference between gross and final releases
do not exceed the specified resource limits R ( R  n  dimensional vector), and all
the pollution produced, ie the total value of destroyed and neznyschenoho
contamination does not exceed some set limit Z ( Z  m  dimensional vector).
Vectors R and Z can be obtained as a result of certain predictive calculations.
Eventually, they may be some theoretical expertise or even strictly regulated
estimated in decision making. In addition, denote respectively the vectors c (1) and
c ( 2 ) specific ratings of main and auxiliary products. Then we get the deterministic
version of the optimization model:
с (1) , x (1)  c 2 , x ( 2 )  max,
 Ax (1)  Bx ( 2 )  R,
 (1)
Cx  Dx ( 2 )  Z ,
x (1)  0, x ( 2)  0,
(2)
(3)
(4)
where zero indicated vectors corresponding dimensions, and  .,.  - the operation
of scalar multiplication. Let


A B
,
H 


C D


R
q    ,
Z 
 c (1) 
p   ( 2 )  ,
 c 
from (2) - (4) move to a model
p, x  max,
 x (1) 
x   ( 2 )  ,
x 
(5)
Hx  q,
(6)
x  0,
(7)
where 0  (n  m) - dimensional zero vector. In mathematical terms the model (2) (4) or (5) - (7) is a problem of linear programming. Note also that in practice the
vectors of gross output and destroyed by pollutants, of course, is limited from
~
~
above by some given vectors ~x (1) and ~x ( 2) and the bottom - some given vectors ~x (1)
and ~x ( 2 ) , therefore, in addition to constraints (4) and (7) is content to consider the
restrictions
~ (1)
~
x (1)  x (1)  ~
x
~ ( 2)
~ ( 2)
 x  x ( 2)  ~
x
~
~
x x~
x
~ ~ ~
where ~x  (~x (1) , ~x ( 2) ) T , ~x  ( ~x (1) , ~x ( 2) )T .
(8)
(9)
In other words, except for model (2) - (4) or (5) - (7), should also use the
model (2), (3), (8) or (5), (6), (9).
If the components of vectors p, q and matrices H are random variables, then
we will deal with stochastic analogs of the above models. Stochastic analogue (5) (7) can be written using the relations (7) and
p, x  max,
(10)
PHx  q   ,
(11)
where p  M ( p) - the vector of mathematical expectations of random vectors p ,
PHx  q - the likelihood of vector inequality Hx  q,  - given level of
probability (scalar quantity) adopted for each of the inequalities of limitations.
Similarly to (7), (10), (11) stochastic model P - setting formalized relations (7),
(11) and the ratio
P p, x  r max,
(12)
where r - some preset value. Note that in many real problems of restriction (11)
should be replaced by constraints
nm

P   hij x j  qi    i , i  1, n  m ,
 j 1

where  i - given the level of probability for the i-th restriction, and hij - elements
of the matrix N. However, further assume  i   , i  1, n  m , that no narrowing
of generality the following assumptions.
3. Deterministic equivalent of stochastic Leontief - Ford model
The mathematical structure of models (7), (10), (11) and (7), (11), (12) such
that they can not be directly solved. One possible approach to solving them is the
image of these models as appropriate deterministic equivalent. Thus we note that it
can be done only if the random parameters p , q, H have a specific
distribution M (hij )  hij , M (qi )  qi , D(hij )   ij2 , D(q i )   i2 . Then
nm

n m

=
=
P   hij x j  qi 
P   hij x j  qi  0 
 j 1

 j 1

n m
n m
 n m

=
P (  hij x j   hij x j )  ( qi  qi )   hij x j  qi  0
j 1
j 1
 j 1

nm
nm
  nm


(
h
x

h
x
)

(
q

q
)
q

h
x
   ij j  ij j

i
i 

i
ij j
  j 1

j 1
j 1

P

  P i  ti  = Fi (ti )   .
1
1
nm
nm

(  x 2j  ij2  i2 ) 2
(  x 2j  ij2  i2 ) 2 


j 1
j 1


Since the distribution function is monotonically nondecreasing, then the last
inequality will be carried out if ti  t , where t  Fi 1 ( ) . Thus, the deterministic
equivalent model (7), (10), (11) is a model which can be rewritten in expanded
form as follows:
nm
p x
j 1
j
j
max,
(13)
n m
h x
j 1
ij
nm
j
1
2 2
i
 t  (  x    )  qi , i  1, n  m,
j 1
2
j
2
ij
x j  0, j  1, n  m.
(14)
(15)
Applied value is also a model (9), (13), (14). Similarly to (13) - (15) and (9),
(13), (14) can write the equivalent deterministic model (7), (11), (12). In particular,
the
nm

=
P  p j x j  r 
P p, x  r
 j 1

n

m
n

m
n

m


P (  p j x j   p j x j )   p j x j  r  0  =
j 1
j 1
 j 1

nm
nm
  n m


(
p
x

p
x
)
  j j  j j  r   pjxj 
 j 1

j 1
j 1

P
1
1 =
nm
nm

(  x 2j  2j ) 2
(  x 2j  ij2 ) 2 


j 1
j 1


=
n m

P  p j x j  r  0
 j 1

=
Pr  w = 1 – Pr  w =
1 – F (u) ,
where  j2  D( p j ) , F (u) - the value of the distribution function of a random
variable  that depends on the parameters x j , j  1, n  m .
The task of minimizing the function F (u) on the set (14), (15) is the
deterministic equivalent model (7), (11), (12).
Consider in more detail (13) - (15) and (9), (13), (14). If, for example p , q ,
H have a normal distribution, then
1
Fi (t ) 
2
t
e
2

2
d

as a random variable i has a normalized normal distribution. Note that in this case
equivalent to the equation Fi (t )   :
a) equation (t )    0.5 , if   0.5 ;
b) the equation t  t1 where  (t1 )  1   , when   0.5 .
1
Function (t ) 
2
t
e
2

2
d is tabbed.

In mathematical terms the problem (13) - (15) and (9), (13), (14) is a
nonlinear programming tasks, as in the general case can be solved by known
methods of nonlinear programming. However, these problems can still be written
in more convenient for the practical solution of separable form. For example,
n m
1
2 2
i
defining yi  (  x    ) , the problem (9), (13), (14) can be rewritten as:
j 1
2
j
2
ij
nm
max,
 p j x j
 j 1
nm
  hij x j  t  yi  qi , i  1, n  m,
 j 1
 2 n m 2 2
2
 yi  (  x j  ij  i )  0, i  1, n  m,
j 1

 x  x  x , j  1, n  m,
j
j
 j
 y  y  y , i  1, n  m,
i
i
 i


nm
1
2 2
i
(16)
1
2 2
i
where yi  (  x    ) , yi  ( x    ) .
j 1
2
j
2
ij
2
j
2
ij
j
Separable problem (16) can be solved using a standard nonlinear
programming software, and construction of its approximation to the problem of
linear programming. However, in the latter case, the solution will be close.
Analysis of solutions of problem (16) and the deterministic problem (5), (6), (9)
represents the effects of stochastic information on simulation results.
4. Numerical example
Here are examples of problems (5), (6), (9) and (16) (deterministic


 0 .3 0 .4 
,
equivalent of stochastic model (9), (10), (11)). Let n  2 , m  1 , A  


 0 .2 0 .3 


 0.2 
(1)
T
(2)
(2)
T
B    , C  (0.3 0.2) , D  (0.2) , c  (10,15) , c  c2  5 , R  (100,150) ,
 0.3 
Z  Z1  30 ,
x(1)  (11,13)T ,
x(2)  x1(2)  4 ,


 0.25 0.3 
,
A 


 0.3 0.32 


 0.18 
 ,
B  
 0.22 
C  (0.26 0.21) , x(1)  (100,200)T , x(2)  x1(2)  20 , D  (0.24) , c (1)  (12,17)T ,
c (2)  c1(2)  4.5 , R  (97,145)T , Z  Z1  27 ,   0.7 , 1  8 ,  2  7 , 3  2 ,
 11  0.05 ,  12  0.02 ,  13  0.03 ,  21  0.04 ,  22  0.06 ,  23  0.03 ,  31  0.05 ,
 32  0.07 ,  33  0.05 .
The problem (5), (6), (9) in this case is follows:
10 x1(1)  15 x2(1)  5 x (2)
max,

(1)
(1)
(2)
0.3 x1  0.4 x2  0.2 x  100,
0.2 x (1)  0.3 x (1)  0.3x (2)  150,
1
2


(1)
(1)
(2)
0.3 x1  0.2 x2  0.2 x  30,

(1)
11

x
 100,
1

13  x2(1)  200,

4  x (2)  20.
(17)
Because t  0.52 , y1  8.024 , y2  7.058 , y3  2.274 , y1  10.265 ,
2
y2  14.469 , y3  15.033 ,  112  0.0025 , 122  0.0004 , 132  0.0009 ,  21
 0.0016 ,
 222  0.0036 ,  232  0.0009 ,  312  0.0025 ,  322  0.0049 ,  332  0.0025 , 12  64 ,
 22  49 , 32  4 , the problem (16) will look like:
12 x1(1)  17 x2(1)  4.5 x (2)
max,

(1)
(1)
(2)
0.25 x1  0.3 x2  0.18 x  0.52 y1  97,
0.3 x (1)  0.32 x (1)  0.22 x (2)  0.52 y  145,
1
2
2

(1)
(1)
(2)
0.26 x1  0.21x2  0.24 x  0.52 y3  27,
 2
 y1  0.0025( x1(1) ) 2  0.0004( x2(1) ) 2  0.0009( x (2) ) 2  64   0,

 y22  0.0016( x1(1) ) 2  0.0036( x2(1) ) 2  0.0009( x (2) ) 2  49   0,

2
(1) 2
(1) 2
(2) 2
 y3  0.0025( x1 )  0.0049( x2 )  0.0025( x )  4   0,

(1)
11  x1  100,
13  x (1)  200,
2

4  x (2)  20,

8.024  y1  10.265,
7.058  y  14.469,
2

2.274  y3  15.033.

(18)
Having solved (17), we get x1(1)  11, x2(1)  129.5 , x (2)  4 .
After solving the problem (18) we obtain x1(1)  11, x2(1)  93.39071 ,
x (2)  4 , y1  8,23441 , y2  8,97812 , y3  6,86145 .
5. Conclusions
Concluding the main statement of this work, we note that in reality is a
useful analysis of compatible tasks (5), (6), (9) and (16), which reduced the above
optimizations or cross- industry models of ecological and economic balance. This
analysis allows to determine the additional amounts of inputs required to guarantee
execution of production tasks, the desired range of products the main production
and destruction of possible contaminants in manufacturing subsidiary, and the
adequacy of the values of model parameters that are realizations of random
variables. Summarizing the above presented results we make some conclusions.
First, the problem of ecological economics and sustainable development now is
relevant to human society, which requires active scientific research solutions.
Second, economic and mathematical tools research environmental and economic
problems requires consideration of their specificity and their further development
and improvement. Thirdly, the construction of new economic and mathematical
models, including models of cross-industry environmental and economic balance
has scientific and practical value and also often value in the educational process in
economics, ecology, applied mathematics, and others. Fourth, proposed in this
paper ecological and economic models directly aimed at the clarification and
formalization of nature is not sufficiently studied the conditions of inter- industry
environmental and economic balance, which is one of the problems of sustainable
development. Fifth, research conducted above reveals a number of other issues of
economic and mathematical modeling, which in our view is also positive.
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