69
fuzzy, and this set changes during the problem solving process; ii the DM does not exist as an active entity, and the preferences consist of badly formulated beliefs, which are
riddled with conflicts and contradictions; iii data on preferences are imprecise, and iv a decision should be good or bad not only in relation to some model, but in relation to the
actual context. These problems have initiated active and fast-growing research on the use of fuzzy set theory in solving multiple criteria decision problems Carlsson [28, 29],
Takeda [191] and Zimmermann [222]. The problem of optimizing a portfolio of finitely many assets is a classical problem in
theoretical and computational finance. Since the seminal work of Markowitz [112] it is generally agreed that portfolio performance should be measured in two distinct
dimensions: the mean describing the expected return, and the risk which measures the uncertainty of the return. As one of theoretical approach to the portfolio selection
problem is that of stochastic dominance see Rockafellar and Uryasev [152]
. In the context of choosing optimal portfolios that target returns above the risk-free rate
for certain market scenarios while at the same time guaranteeing a minimum rate of return, fuzzy decisions theory provides an excellent framework for analysis. This is
because the nature of the problem requires one to examine various market scenarios, and each such scenario will in turn give rise to an objective function. In the last section, we
will describe a multi-objective linear programming problem formulation where the objective functions are considered to be fuzzy.
5.2 Fuzzy linear programming models
Empirical surveys reveal that linear programming LP is one of most frequently applied operations research techniques is real-word problems. However, given the power of LP
one could have expected oven more applications. This might be due to the fact that LP requiresmunch well-defined and precise data which involves high-information cost. In
real-word applications certainty, reliability and precision of data is often illusory. Furthermore the optimal solution of LP only depends on a limited number of constraints
and, thus, much of the information collected has little inpact on the solution. Fuzz linear programming, propused by Bellman and Zadeh, is an extention of LP with both objective
functions and constraints represented by fuzzy sets.
70
Now we can defined a LP problem with crisp of fuzzy resource constraints, and a crisp or fuzzy objective as:
subject to
, ,
1 ,
~ ,
~ max
1
≥ =
≤ =
∑
=
X p
i b
X a
X c
Z
i j
m j
ij T
5.1
where fuzzy resources
i b
i
∀ ,
~
have the same form of membership function. We may also consider the following fuzzy inequality constraints:
subject to
, ,
1 ,
~ ~
, ~
max
1
≥ =
≤ =
∑
=
X p
i b
X a
X c
Z
i j
m j
ij T
5.2
even though 5.1 and 5.2 are defferent in some points of view, we can use the same approach to handle them under the pre-assumtion of the membership functions of the
fuzzy available resources and fuzzy inequality constraints. The difference between crisp and fuzzy constraints is that in case of crisp constraints the
decision maker can strictly differentiate between feasibility and infeasibility; in case of fuzzy constraints he wants to consider a certain degree of feasibility in the interval
Werners, 1987. Now, we consider some approaches for fuzzy linear programming models.
Correspondingly we could build portfolio models as sections 5.5, 5.6 and 57.
First approach: The resources can be determined precisely, a traditional LP problem is
consider as :
subject to
, ,
~ ,
max
1
≥ ∀
≤ =
∑
=
X i
b X
a X
c Z
i j
m j
ij T
5.3
where
i b
a c
i ij
∀ ,
and ,
are precisely given. The optimal solution of 5.3 ia a unique optimal solution.
71
Second approach Chanas and Verdegay: A decision maker wishes to make a
postoptimization analysis. Thus, a parametric programming problem is formulated as:
subject to
, ],
1 ,
[ ,
~ ,
max
1
≥ ∈
+ ≤
=
∑
=
X p
b X
a X
c Z
i j
m j
ij T
θ θ
5.4
where
i p
b a
c
i i
ij
∀ ,
and ,
,
are precisely given and
θ
is a parameter,
i p
i
∀ ,
are maximum tolerances which are always positive. The solution
θ Z
of 5.4 are function of
θ
. That is, for each
θ
we can obtain an optimal solution. On the other hand, the available resources may be fuzzy. Then the LP problem with fuzzy
resources becomes:
subject to
, ,
~ ,
max
1
≥ ∀
≤ =
∑
=
X i
b X
a X
c Z
i j
m j
ij T
5.5
It is possible to determine the maximum tolerance
i
p
of the fuzzy resources
i b
i
∀ ,
. Then we can construct the membership functions
i
μ
assumed linear for each fuzzy constraints, as follows:
⎪ ⎪
⎪ ⎪
⎪
⎩ ⎪
⎪ ⎪
⎪ ⎪
⎨ ⎧
+ +
≤ ≤
− −
≤
=
∑ ∑
∑ ∑
= =
= =
. if
, if
1 ,
if 1
1 1
1 1
i i
m j
j ij
i i
m j
j ij
i i
i m
j j
ij i
m j
j ij
i
p b
X a
p b
X a
b p
b X
a b
X a
μ
5.6
Verdegay and Chanas, propuse that 5.5 and 5.6, however, are equivalent to 5.4, a parametric LP where
i p
b a
c
i i
ij
∀ ,
and ,
,
are given, by use of the
λ
-level cut concept.
72
For each
λ
-level cut of the fuzzy constraint set 5.5 becomes a traditional LP problem. That is,
subject to
]. 1
, [
, ,
, {
, ,
max ∈
≥ ∀
≥ =
∈ =
λ λ
μ
λ λ
X and
i X
X X
X X
c Z
i T
5.7
and equivalent to:
subject to
, and
] 1
, [
, ,
1 ,
~ max
1
≥ ∈
∀ −
+ ≤
=
∑
=
X i
p b
X a
X c
Z
i i
m j
j ij
T
λ λ
5.8
where
i p
b a
c
i i
ij
∀ ,
and ,
,
are precisely given. Now, if we set
θ λ
− =1
, then equation given by 5.8 will be the same as 5.4. Then a solution table is presented to the decision
maker to determine the satisfying solution.
θ Z
,
] 1
, [
∈ θ
is the fuzzy solution corresponding to Verdegay’s approach.
Third approach Weners’s approach: A decision maker may want to solve a FLP
problem with a fuzzy objective and fuzzy constraints, while the goal
b
, is not given. That is:
subject to
, ,
, ~
, ~
max
1
≥ ∀
≤ =
∑
=
X i
b X
a X
c Z
i m
j j
ij T
5.9
which is equivalent to:
subject to
, and
] 1
, [
, ,
~ ,
~ max
1
≥ ∈
∀ +
≤ =
∑
=
X i
p b
X a
X c
Z
i i
n j
j ij
T
λ θ
5.10
where
i p
b a
c
i i
ij
∀ ,
and ,
,
are given, but the goal of the fuzzy objective is not given. To solve 5.10 by use of Werners’s approach, let us first define
Z
and
1
Z
as follows:
73
, max
inf =
= =
∈
θ
Z X
c Z
T X
X
5.11
1 max
sup
1
= =
=
∈
θ
Z X
c Z
T X
X
5.12 where
}. and
], 1
, [
, ,
{
1
≥ ∈
∀ +
≤ =
∑
=
X i
p b
X a
X
i i
n j
j ij
θ θ
X
Then, we can obtain Werners’s membership function
μ
of the fuzzy objective. That is:
⎪ ⎪
⎩ ⎪
⎪ ⎨
⎧ ≤
≤ −
− −
= .
if ,
if 1
, if
1
1 1
1 1
Z X
c Z
X c
Z Z
Z X
c Z
Z X
c
T T
T T
μ
5.13
The membership functions
i
i
∀ ,
μ
, of the fuzzy constraints are defined as 5.6. By use of the min-operator proposed by Bellman and Zadeh, we can obtain the decision space D
which is defined by its membership function
D
μ
where,
,..., min
p D
μ μ
μ =
. 5.14 It is reasonable to choose the decision where
D
μ
is maximal as the optimal solution of 5.9. Therfore, 5.9 is equivalent to:
λ max
, ],
1 ,
[ and
, ,
,
≥ ∀
∈ ≥
≥
X i
i i
μ μ
λ λ
μ λ
μ
5.15
where
i p
b a
c
i i
ij
∀ ,
and ,
,
are given, and
,..., min
m D
μ μ
μ λ
= =
. Let
θ λ
− =1
. Then the problem given by 5.15 will be equivalent to:
θ max
subject to
, and
] 1
, [
, ,
,
1 1
≥ ∈
∀ +
≤ −
− ≥
X i
p b
X a
Z Z
Z X
c
i i
i ij
T
θ θ
θ
5.16
74
where
i p
b a
c
i i
ij
∀ ,
and ,
,
are given and
θ
is fraction of
1
Z Z
−
for the first constraint and a fraction of maximum tolerance for others. The solution is a unique
optimal solution.
Forth approach Zimmermann’s approach: A decision maker may want to solve a FLP
problem with a fuzzy objective and fuzzy constraints, when the goal
b
of the fuzzy objective and its minimum tolerance are given. That is,
subject to
, ,
, ~
, ~
max
1
≥ ∀
+ ≤
=
∑
=
X i
p b
X a
X c
Z
i i
n j
j ij
T
θ
5.17
where
i ij
b p
b a
c ,
, ,
,
and
i p
i
∀ ,
are given. The problem given by 5.17 is actually equivalent to:
Find X,
subject to
, ,
, ,
~ max
1
≥ ∀
+ ≤
=
∑
=
X i
p b
X a
X c
Z
i i
m j
j ij
T
θ
5.18
with the membership function of the fuzzy constraints as previously described in 5.6 and the membership function of the fuzzy objective
μ
as follows:
⎪ ⎪
⎩ ⎪
⎪ ⎨
⎧
− ≤
≤ −
− −
− =
. if
, if
1 ,
if 1
1
p b
X c
b X
c p
b Z
Z X
c b
b X
c
T T
T T
μ
5.19
Thus by use of the maximum concept, 5.18 is actually equivalent to:
λ max
subject to
, ],
1 ,
[ and
, ,
, and
i
≥ ∀
∈ ∀
≥ X
i i
i
μ μ
λ λ
μ μ
5.20
75
where
i ij
b p
b a
c ,
, ,
,
and
i p
i
∀ ,
are given. Let
θ λ
− =1
. Then 5.15 will be equivalent to:
subject to
, and
] 1
, [
, ,
, ,
max
1
≥ ∈
∀ +
≤ +
≥
∑
=
X i
p b
X a
p b
X c
i i
m j
j ij
i T
θ θ
θ θ
5.21
where
i ij
b p
b a
c ,
, ,
,
and
i p
i
∀ ,
are given and
θ
is a fraction of the maximum tolerances. The optimal solution 5.21 is unique.
When a fuzzy objective is assumed, what Zimmermann and Werner’s approaches are asumming is essentially a performance function on the objective,
] 1
, [
∈ =
X c
F X
f
T
. Thn, in all cases, if
] 1
, [
, ∈
θ θ
Z
, is the fuzzy solution to the problem, the corresponding point solution for each fuzzy objective performance function associated to
Zimmermann’s or Werner’s approach to be considered, can be obtained by solving the point-fix equation. It is showed that in the application.
Fifth approach: A decision maker may want to solve a FLP problem with a fuzzy
objective and fuzzy constraints, while only the goal
b
of the fuzzy objective is given, but its tolerance
p
is not given. That is,
subject to
, ,
, ~
, x
~ ma
~ max
1
≥ ∀
≤ =
∑
=
X i
b X
a X
c Z
i n
j j
ij T
5.22
where
i ij
b b
a c
, ,
, ,
and
i p
i
∀ ,
are given, but
p
is not given. While
p
is not given, we do know that
p
should be in between 0 and
Z b
−
. For each
] ,
[ Z
b p
− ∈
, we can obtain the membership function of the fuzzy objective as 5.19. Sice in a high-
productivity system the objective value should be larger then
Z
at
=
θ
, there is no meaning to given a positive grade of membership for those which are less than
Z
.
76
The difference between this problem and Zimmermann is that
p
is not initially given in this problem. Therefore, we may assume a set of
s p
, where
] ,
[ Z
b p
− ∈
. Then, the problem of each
p
given is a Zimmermann problem. The decision maker may choose a refined
p
among the solution for this given set of
s p
. Then a Zimmermann problem with the decision maker’s refined
p
is solved. This solution will be the final optimal solution for 5.22.
5.3 Interactive fuzzy linear programming
