Download Solving Some Economic Model with Fuzzy and Random Data Theory:

Solving Some Economic Model with Fuzzy and
Random Data Theory
Sara Ebrahimi* and Hamid Reza Sahebi**
The method described in this paper offers an opportunity for combining
both fuzzy and random data in an Economic Model by considering them
as degenerate fuzzy random variables. The approach described in this
paper focuses on discrete fuzzy random variables and on combining
fuzzy number with discrete random variables. defaults in developing
countries.
JEL Codes: 23A43,F45
1. Introduction:
The notion of fuzzy subset of a set was introduced by Zadeh [9]. the problem of
combining both uncertainty and imprecision in an optimization setting has not
received the attention it merits. See for instance Czogala [1], [2] where the
concept of probabilistic set is exploited, Yazenin [3], Roubens and Teghem [4]
where comparison between fuzzy and stochastic approaches is discussed
without any attempts for integration and Luhandjula [5] where a laconic
discussion on flexible programming with random data is presented. The purpose
of this paper is to describe an approach for solving a linear program with fuzzy
random variabte coefficients. The main idea behind this approach is to convert
the original problem into a stochastic program so that techniques of stochastic
optimization can apply. The conversion is made via semi-infinite optimization as
described in Luhandjula [6].
2. Fuzzy random variable
In the following, we restrict on notations used in [8].
Definition.1 Consider a probability space (, , P) . A fuzzy random variable on
this
space is a fuzzy set-valued mapping:
~
X :   F0 ( R),

X
______________________
* Sara Ebrahimi, Department of Mathematics, Islamic Azad University, Ashtian, Iran.
Email : [email protected]
**Dr. Hamid Reza Sahebi, Department of Mathematics, Islamic Azad University, Ashtian, Iran
Email: [email protected]
such that for each Borel set B and for every   (0,1) :
~
~
X 1 ( B)  {   / X   B}  
~
where F0 ( R) and X  stand for the set of fuzzy numbers with compact supports
~
and the  -level set of the fuzzy set X  respectively. The following result is
crucial for the remaining of this paper.
~
~
Theorem.2 X is a fuzzy random variable if and only if given    , X  is a
random interval for all   (0,1) .
3. Metamorphosis
Consider the following mathematical program:
Max cx
 (a
S .to :
) x j  (bi )
ij 
i  1, m (1)
x  R n ,   .
~
Where (a~ij )  and (bi )  are fuzzy random variables defined on a probability space
~
(, , P) the possibility distribution of which are: (a~ ) ,
(b ) respectively.

ij 

i 
(1) can be merely written:
Max cx
~
~
S .to : A x  b
x  0,   .
~
~
Where A and b are m n and m 1 matrices, the joint possibility distribution of
which are

~
A
and

~
b
respectively.
~
~
T ,T
In the sequel 1 2 denote support of A , support of b respectively. Put :
T  T1  T2 and defined : u : T  [0,1] By formula :
u (t ( ))  u (t1 ( ), t 2 ( ))
 min( A~ (t1 ( )), b~ (t 2 ( )).


It is clear that u(t ( )) is the degree of compatibility of t1 ( )  T1 and t 2 ( )  T2 ,
with restrictions defined by

~
A
and

~
b
respectively. Consider now a
subdivision T i , i  1,2,..., p  1 of T defined as follows:
Ti  {t ( ) T : i 1  u(t )   i }
Where  i (i  1,2,..., p  1) are real number such that
Observe that we have:
0   0  1  ....   p   p1  1.
T k  S  k  k / S  k
~
~
Where S i  Ai  bi ./,  denote ensemblist difference and Cartesian product
~  is the  -level
respectively and m
~
~ i
~ . Furthermore, by Theorem 1, ( A
cut of the fuzzy set m
) and (b  i ) are families

i

i
of Cartesian product of random intervals and random variables respectively.
( )
Consider real numbers i i 1, 2,...,p 1 such that:
0   p 1   p  ....   1
chosen to allow some leeways on constraints containing t ( )  T which are less
compatible
with restrictions defined by  A~ and  b~ .


Consider now the following mathematical program:
Max cx

S .to : t 1 ( )x  t 2 ( )   pm1; (t 1 ( ), t 2 ( )) T  p 1

t 1 ( )x  t 2 ( )   pm ; (t 1 ( ), t 2 ( )) T  p
(2)
t 1 ( )x  t 2 ( )  1m ; (t 1 ( ), t 2 ( )) T 1
x  0,   .
where
 1 
 
 . 
m
j  
.
 
 
 j
A feasible action for (2) meets feasibility equirements of the original program (1)

for most favourable circumstances ( t ( )  T p 1 ). In addition, strong violations of
feasibility requirements are tolerated for less favourable circumstances
( t ( )  T1 ). Intermediary
situations are also taken into account by accepting more or less violations of the
feasibility according to the value of u(t ( )) .So (2) can be regarded as a
stochastic counterpart of the original fuzzy stochastic program.
Theorem 3. The mathematical program (2) is equivalent to the following
program:
Max cx
S .to :
t

t
 p 1
1ij
1ij
p

( )x j   p 1  t 2i p 1 ( ); i  1, m

( )x j   p  t 2i p ( );
i  1, m
.....
(3)
.......
 t  ( )x
1
1ij
j
 1  t 2i 1 ( );
i  1, m
x j  0,   .

where t1ijk ( ) and t 2i k ( ) are right and left endpoints of random intervals:
(a~ ) k  {s /
( s)  

ij 
k
( a~ij )
And
~
(bij )k  {s /  ( s)   k
~
( bij )
respectively. The proof of this theorem is similar to that of Proposition 6 in [6].
Observe that (3) is a stochastic linear program and according as previsional or
decisional option is chosen, the wait and see or the here and now attitude can be
adopted.
Assume now that we have a decisional option in sight and that
t1ijk ( ), k  1,2,..., p1 , i  1, m; j  1, n
are deterministic, then we can apply the
chance constrained program main approach. In this case (3) can be written:
Max cx
Pr ob [ t 1ij k ( )x j   k  t 2i k ( )]   ;
S .to :
i  1, m ; k  1,...,  p 1
(4)
x j  0,   .
(4) is equivalent to the following program ([7]):
Max cx
S .to :
Pr ob[ t1ijk ( ) x j   k1  Bi(1k  ) ;
i  1, m; k  1,...,  p 1
x j  0,
Where Bi(1k )  ( Fi k ) 1 (1   )
Fi k

denoting the repartition function of the random variable t 2i k ( ) and  some
prescribed level. The resulting problem is then a linear program and existing
packages may be used to find a solution of the problem.
For the general case, the two stage programming approach seems more
appropriate. The idea behind this approach ([7]) is to pay a penalty for any
violation of a constraints.
Let g i k be the penality for a unit of discrepancy between:

t 2i k ( ) and t1ijk ( ) x j   k .
Introduce now the following variables:

  t1ijk ( ) x j   k  t 2i k ( ) quantity positive
k
yi ( )  
0
otherwise

The resulting problem is then:
Max cx  E ( g i k y i k ( ))
S .to : y i k ( )   t 1ij k ( )x j   k  t 2i k ( ) (5)
x  0, y i k ( )  0
i  1, m ; k  1,.., p  1
where E denotes the mathematical epectation.
4. Numerical example in Economic
Consider the mathematical program:
Max 3x1  2 x2
~
~
S .to : A x  b
x  0.
~
~
Where A and b are 2  2 and 21 matrices respectively. Components of these
matrices are fuzzy random variables defined on a discrete sample   (1 , 2 )
as follows:
~ ~
 ~ ~
 ~
1 1   3 

( A1 , b1 )   ~ ~ ,  ~ 

 
~ ~ 
 2 1   4 
( A x, b )  
~ ~
 ~
1 3   5 
 ~ ~

~ ,  ~ 
( A1 , b1 )   ~
1
2   4 


~ denote a triangular fuzzy number with
Where p(1 )  0.25, p(2 )  0.75 and m
the following membership function:
0
x  m 1

 x  (m  1) x  (m  1, m]

u m ( )  
 x  (m  1) x  (m, m  1]

0
x  m 1
Let 0   0  0.4  1  0.6   2  0.8   3  1   4 and
0   4  0.01   3  0.02   2  0.03  1
And suppose :
g i3  1; g i 2  2; g i1  3. The program (5) corresponding to this example is then:
Min
3 x1  2 x 2  y113  y 213  0.75 y112 
0.75 y 212  0.5 y111  0.5 y 211  3 y123
 3 y 223  2.25 y122  2.25 y 222 
1.5 y121  1.5 y 221
y113  1.2 x1  1.2 x 2  2.81;
S .to :
y 213  2.2 x1  1.2 x 2  3.81;
y123  1.2 x1  3.2 x 2  4.81;
y 223  1.2 x1  2.2 x 2  2.62;
y112  1.4 x1  1.4 x 2  4.62;
y 222  1.4 x1  2.4 x 2  3.62;
y111  1.6 x1  1.6 x 2  2.43;
y 211  2.6 x1  1.6 x 2  3.43;
y121  1.6 x1  3.6 x 2  4.43;
x j  0, y ilk  y  k i ( l )  0; l  1,2; i  1,2; k  1,3.
The solution of this linear program is
y113  0
y 213  0.34
y123  0
y 223  0
y112  0.658
y 222  0.658
y111  1.31
y 211  1.65
y121  1.316
y 212  1
y122  .658
y 221  1.316
x1  1.34
x2  1
Then (1.34,0) may be considered as a solution of the original problem according
to desiderata of the Decider expressed by his choice of  i .
5. Conclusions
The method described in this paper offers an opportunity for combining both
fuzzy and random data in an optimization context by considering them as

degenerate fuzzy random variables. The choice of k can be thought of as a
way to discretize the interval [0, 1]; the more p is large the more we have a fine
discretization and the more we have a faithful description of the reality.) have
been ravaged by persistence loan default and loan loss since mid-1980s.
References:
[1] Czogala E., “On distribution (unction of probability sets and its applications in
decision making”, Fuzzy Sets and Systems 10 (1983)21-29.
[2] Czogala E., “Probability Sets in Decision Making and Control”, Verlag TUV
(1984).
[3] Luhandjula M.K., “Linear programming under randomness and fuzziness”,
Fuzzy Sets and Systems 10 (1983) pp.45-55.
[4] Luhandjula M.K., Ichihashi H.,” Inuiguchi M., Fuzzy and Semi-infinite
Optimization”,Inf. Sciences 61 (1992) pp.233-250.
[5] Yazenin A.V., “Fuzzy and stochastic programming”, Fuzzy Sets and Systems
22 (1987)pp.171-188.
[6] Roubens M. and Teghem J.,“Comparison between multiobjectif fuzzy linear
programming and multiobjective stochastic programming”, in Lecture Notes
Econ. Math. Syst. 310 (1988) pp.240-265.
[7] Vajda S., “Probabilistic Programming “(Academic Press, New York and
London,(1972).
[8] Wang G. and Zhang Y., “The theory of fuzzy stochastic processes”, Fuzzy
Sets and Systems 51 (1992) 161-178.
[9] Zadeh, L.A.,” Fuzzy Sets”, Inform. And Control 8 (1965), 338-353.