Download Decision Making Under Fuzzy Risk Using Finite level Fuzzy Numbers

Decision Making Under Fuzzy Risk Using Finite level
Fuzzy Numbers
)‫اتخاذ القرار في حالة المخاطر الضبابيه( الغامضه‬
Dr.Samah W. Najib
Department of Computer Information System, College of Information Technology,
Arab Academy for Banking and Financial Science.
E-Mail: [email protected]
Abstract:
In this paper, a generalization of the classical probability space to the fuzzy
probability space will be introduced, and their possible applications in methods of
decision – making under risk will be described. Uncertain probabilities of states of nature
and the expected cost (profit) in the decision matrix will be estimated using a special kind
of fuzzy numbers called finite level fuzzy numbers.
1. Introduction
Models of decision –making under risk are more realistic when the mathematical
theory of the fuzzy sets is applied. The states of the world ( nature) in decision matrices
are supposed to be set exactly and their probabilities to be known. However in practice,
the states of the world are often specified only Vaguely and their probabilities are based
on experts’ estimations.
In this work, fuzzy numbers will be used in the decision matrix to model the decision
maker’s risk in order to define a more complete decision making solution.
2. Preliminaries
First we describe the general concepts of fuzzy sets theory and fuzzy numbers without
considering risk attitudes.
2.1 Fundamental of Fuzzy Sets
In ordinary (crisp) sets we say that an element x is either belong to a set A on a
universal set U or it does not. This relationship can be represented using a membership
function  A as:
1
0
 A ( x)  
x A
x A
iff
iff
 A : U  {0,1}
……………(1)
…………….. (2)
A fuzzy set is a class that admits the possibility pf partial membership in it [1]. So
equation(2) can be extended in fuzzy sets theory to be :
 A : U  [0,1]
………………(3)
So, fuzzy set is a vague boundary set ‘comparing with ordinary sets’.
A fuzzy set A is completely characterized by the set of pairs
A  x,  A ( x), x U 
……………..(4)
Fuzzy numbers is a special case of fuzzy sets. In another word, a fuzzy set must satisfies
some conditions in order to be a fuzzy number.
10
Fig.1 Fuzzy number 10.
In particular, an intuitively easy and effective approach to capturing the expert’s
uncertainty about the value of an unknown number is by using fuzzy numbers.
2
Definition 2.1 A finite level fuzzy number A is defined by the set of pairs
A  xh ,  i N
………………..(5)
Where N is an integer number, and 1 , 2 ,  , N  [0,1] such that
1   2       n1   n  1
 N   N 1       n1   n  1
and
x1  x2      x N .
Operations on this kind of fuzzy numbers can be defined using extension principle as
follows:
If we have two fuzzy numbers A  xi , i N , B   yi , i N , then Z = A(*)B is
defined as
 A B ( z )  i if
z  xi  yi
…………….(6)
where
 A B ( z )  1 if
z  x n  yn
Where (*) is any binary operation. Maximum and minimum of A and B denoted by
 or  are defined in same way:
 A B ( z )   i if z i  max( xi , yi ), i  1,2,..., n
…..(7)
3. Fuzzy Probability
3.1. Fuzzy Event
When dealing with the ordinary probability theory, an event has its precise
boundary. For instance, if an event is A = {1,2,3}, its boundary is sharp and thus it can
be represented as a crisp set. When we deal with an event whose boundary is not sharp,
it can be considered as a fuzzy set, that is, a fuzzy event. For example,
B = “ small integer ” = {(1,0.9),(2,0.5), (3,0.3)}.
3
3.2 Fuzzy probability of Fuzzy Event
Talasova in [3] and [4] extended the classical probability to deal with fuzzy quantity,
and proved that the fuzzy probability has properties representing a generalization of
axioms of the classical one.
Given the following fuzzy event in the sample space S,
A  x, ,  A ( x) x  S
The α-cut set of the event (set) A is given as the following crisp set;
A  x  A (x)   
……………..(8)
The probability of the α-cut event is defined by :
P( A ) 
 p ( x)
..……………(9)
xA
For the probability of the α-cut event, we can say that:
‘ The possibility of the probability of set A to be P( A ) is α’.
Now, the probability of the fuzzy event A, is defined as follows”
~
P ( A)  P( A ),    0,1
……………(10)
If we use finite level fuzzy number to represent the fuzzy event A, equation(10) can be
rewritten as:
~
P ( A)   p( xi ,  i N
……………(11)
Example3.1
Assume the probability of each element in the sample space S ={a,b,c,d}as follows:
P(a) = 0.2, P(b) = 0.3, P(c) = 0.4, P(d) = 0.1.
A fuzzy event A is given by :
A  xi ,  i 4
4
Where
x1  a, x2  b, x3  c, x4  d
 1  0.5,  2  0.8,  3  1.0,  4  0.3
~
P ( A)  P( xi ),  i 4
 P( x1 ),  1 P( x 2 ),  2 P( x3 ),  3 P( x 4 ),  4 
 0.2,0.5, 0.3,0.8, 0.4,1.0, 0.1,0.3.
0.4
0.3
0.2
0.1
a
b
c
d
Fig.2. Sample space S.
1
0.8
0.5
0.3
a
b
c
d
Fig.3.Fuzzy Event.
5
1
0.8
0.5
0.3
0.1
0.2
0.3
0.4
Fig.4. Fuzzy Probability.
4- Main Result
Let us consider a problem of decision – making under risk that is described by the
following fuzzy decision matrix:

s1
s2
~
p1
~
p2

~
pr
EFV
a1
v~11
v~12

v~1r
EFV1
a2
v~21
v~22

v~2 r
EFV2


~
v n1

~
vn 2


~
v nr

an

sr
EFVn
Where a1 , a2 , ...., an are alternatives, s1 , s2 , ...., sr states of the nature, ~
p1 , ~
p2 , ...., ~
pr their
~
fuzzy probabilities, and vij , i  1, 2, ..., n, j  1, 2, ..., r are fuzzy costs (fuzzy profits) in
which the alternative a i satisfies the given decision objective if the state s j occurs.
6
We will assume that fuzzy probabilities of states of nature and the fuzzy costs(profit) are
finite level fuzzy numbers. Fuzzy numbers FEVi express expected fuzzy monetary
values of alternatives a i for i = 1,2, …, n;. The expected fuzzy monetary values are
computed using equation(6). This means that they are calculated according to the
following formula:
 i  1, 2, ..., n
r
EFVi   ~
p j  v~ij
……………………(12)
j 1
The best of the alternative will be chosen by the rule of the maximum (minimum)
expected fuzzy monetary value.
Example4.1
Let us consider the problem of investing JD15,000 in the stock market by buying
shares in one of two companies: A or B. Shares in company A could yield a 50% return
on investment during the next year. If the stock market conditions are not favorable, the
stock may lose 20% of its value. Company B provides safe investment with 15% return in
a ‘bull’ market and only 5% in a ‘ bear ’ market. An expert estimates probabilities of
occurrence of states of stock by fuzzy numbers ~
p1 , ~
p2 and as illustrated in the following
fuzzy decision matrix:
Fuzzy Prob.
Alternative
“Bull” Market(JD)
“Bear” market(JD)
Company A stock
~
p1
v~
11
~
p2
v~
12
FEV1
Company B stock
v~21
v~22
FEV2
The estimated values of fuzzy numbers v~ij will be as follows:
v~11  (7100,0.2), (7300,0.3), (7500,1), (7650,0.4)
v~12  (3450,0.2), (3250,0.3), (3000,1), (2800,0.4)
v~  (1850,0.2), (2000,0.3), (2250,1), (2400,0.4)
21
v~22  (400,0.2), (600.0.3), (750,1), (800,0.4)
7
And the fuzzy probabilities will have the following values:
~
p1  (0.5,0.2), (0.55,0.3), (0.6,1), (0.7,0.4)
~
p 2  (0.5,0.2), (0.45,0.3), (0.4,1), (0.3,0.4)
Now, we will compute the values of EFVi , i  1,2 ; according to equation(12), we have:
~ ~
~
~
EFV1  P1  V11  P2  V12
 (3550,0.2), (4015,0.3), (4500,1.0), (5355,0.4)
(1725,0.2), (1462.5,0.3), (1200,1.0), (840,0.4)
 (1825,0.2), (2552.5,0.3), (3300,1.0), (4515,0.4)
~ ~
~
~
EFV2  P1  V21  P2  V22
 (925,0.2), (1100,0.3), (1350,1.0), (1680,0.4)
(200,0.2), (270,0.3), (300,1.0), (240,0.4)
 (1125,0.2), (1370,0.3), (1650,1.0), (1920,0.4)
Using equation(6), we can find the maximum expected fuzzy monetary value MEFV as:
MEFV  (1825,0.2), (2552.5,0.3), (3300,1.0), (4515,0.4)
Based on these computations, our decision is to invest in stock A.
8
Reference:
1. D.Dubois, H. Prade, “Fuzzy Sets and Systems: Theory and Applications",
Academic Press, INC., 1980.
2. Adil M. Mahmood, Samah W. Najib, “Fuzzy Operator”, To be appeared Soon.
3. Talasova, J. “ Fuzzy Methods of Multi-criteria Evaluation and DecisionMaking”, Publishing Hose of Placky University, Olomouc 2003.
4. Talasova, J. “Fuzzy Sets in Decision – Making Under Risk”, Proceedings of the
4th Conference Aplimat Bratislava, February 1-4, 2005, 532-544.
5. Wayen L. Winston, “ Operations Research: Applications and Algorithms”, PWSKENT 1991.
9