116
Moreover, a solution algorithm has been described to solve the BINOLFP. Furthermore, a solution algorithm has been proposed by Saad and Abd-Rabo [162] was based upon the
chance-constrained programming technique Seppala [168] along with the branch-and-
bound method Ammar 1988 .
Recently, Saad and Sharif developed a solution method to solve integer linear fractional program with chance-constraints and having statistically
independent random parameters Dutta 1992. Pareto-optimality for multiobjective linear fractional programming problems with fuzzy parameter has been discussed by Sakawa
and Yano [165]. Programming with linear fractional functions was introduced into the literature by Charnes and Cooper [30]. Since we can use a fuzzy multiobjective fractional
portfolio models, in this section we give some recently results on fuzzy multiobjective fractional programming problem.
5.9.1 Problem formulation and the solution concept
The problem to be considered in this paper is the following fuzzy multiobjective fractional programming problem:
FMOFP
, ,...,
max
1 1
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
= ⎥⎦
⎤ ⎢⎣
⎡ x
g x
f x
g x
f x
g x
f
p p
subject to
} ,
~ {
~ ,
≥ ≤
∈ =
∈ x
b Ax
R x
b A
X x
n
, where A is an
n m
×
-matrix
b ~
is an m-vector of fuzzy parameters and we suppose that they are given by fuzzy numbers, estimated from the information provided by the
decision maker. Moreover
, 1
, p
i x
g
i
=
for all x in the feasible region of problem FMOFP.
Definition 5.1 Dubois and Prade [44] It is apropiate to recall that a real fuzzy number
a ~
is a continuous fuzzy subset from the real R whose membership function
~
a
a
μ
is defined by:
1 A continuous mapping function R to the closed interval [0,1],
2
~
= a
a
μ
for all
] ,
1
a a
−∞ ∈
, 3
~
a
a
μ
is strictly increasing on
] ,
[
2 1
a a
,
117
4
1
~
= a
a
μ
for all
] ,
3 2
a a
a ∈
, 5
~
a
a
μ
is strictly decreasing on
] ,
4 3
a a
a ∈
, 6
~
= a
a
μ
for all
, [
4
∞ ∈ a
a
, Figure 5.2 Illustrates the graph of possible shape of a membership function of a fuzzy
number
a ~
. Here, the vector of fuzzy parameters
b ~
involved in problem FMOFP is a vector of fuzzy numbers whose membership function is
~
b
b
μ
.
~
a
a
μ
1
1
a
2
a
3
a
4
a a
Fig. 5.2 Membership function of a fuzzy number
a ~
. In what follows, we give the definition of the
α
-level set or
α
-cut of the fuzzy vector
] ~
,..., ~
[ ~
1
m
b b
b =
.
Definition 5.2 [44] The
α
-level set of the vector of fuzzy parameters
b ~
in problem FMOFP is defined as the ordinary set
} {
~
~
α μ
α
≥ ∈
= b
R b
b L
b m
. For a certain degree
α α =
in [0,1], estimated by the decision maker, the FMOFP can be understood as the following nonfuzzy
α
-multiobjective fractional programming problem
α
-MOFP:
α
-MOFP:
}. ~
, ,
{ ~
, ,
,..., max
1 1
b L
b x
b AX
R x
b A
X x
to subject
x g
x f
x g
x f
x g
x f
n p
p α
∈ ≥
≤ ∈
= ∈
⎥ ⎥
⎦ ⎤
⎢ ⎢
⎣ ⎡
= ⎥⎦
⎤ ⎢⎣
⎡
It should be emphasized here in the
α
-MOFP above that the vector of parameters
b
is treated as a vector of decision variables rather than constants. Problem
α
-MOFP can be reformulated in the following form:
118
P
,..., max
1
x F
x F
x F
p
=
, subject to
, ,
b A
X x
∈
where
x g
x x
f x
F
i i
i i
θ −
=
and
, 1
, ,
p i
x g
x f
r r
r i
= =
≥ θ
θ
are fixed parameters and for their specification, Singh and Hanson [174].
Based on Definition 2 of the
α
-level set of the vector of fuzzy numbers
b ~
, we introduce the concept of
α
-efficient solution of problem P above as follows:
Definition 5.3 Sakawa and Yano [165] A point
, b
A X
x ∈
is said to be an
α
- efficient solution of problem P if only if there exists no other
~ ,
, b
L b
b A
X x
α
∈ ∈
such that
, 1
; p
i x
F x
F
i i
= ≤
with strictly inequality holding for at least one i, where the corresponding values of parameters
, 1
m r
b
r
=
are called the
α
-level optimal parameters.
Now, consider
λ
is a p-dimensional strictly positive fixed vector, then problem P can be written again in a problem of scalar single-objective function
λ
P
in the following form:
λ
P
:
. ,
, max
1
b A
X x
to subject
x F
p i
i i
∈
∑
=
λ
Let
, b
A X
denote the set of feasible solutions of problem
α
-MOFP or P or
λ
P
. We assume that
≥ x
f
,
x g
, for all
, b
A X
x ∈
. We further assume that f,-g are concave functions and
, b
A X
is a convex set. It follows that F is concave Singh and Hanson [174].
Problem
λ
P
can be solved at
1 =
=
i i
λ λ
with the corresponding fixed parameters
, 1
, p
i
i i
= =
θ θ
using any available nonlinear programming package, for example, GINO Lieberman, et al [102], to find the
α
-optimal solution
x
together with the optimal parameters
, 1
m r
b
r
=
.
119
It should be noted from Singh and Hanson [174] that
x
is an
α
-efficient solution to problem
α
-MOFP or problem P with the corresponding
α
-level optimal parameters
, 1
m r
b
r
=
if there exists
≥ λ
such that
x
solves problem
λ
P
and either one of the following conditions holds:
i.
=
i i
λ λ
for all
, 1
p i
=
. ii.
x
is the unique maximizer of problem
λ
P
.
Definition 5.4 Geofrion [60] Consider the multiobjective programming problem
,..., max
1
x x
x
k
φ φ
φ =
, subject
to
n
R S
x ⊆
∈
. We say that
S x
∈
is efficient if ond only if there exists no
S x
∈
such that
x x
φ φ
≤
.
Definition 5.5 Geofrion [60] For the multiobjective programming problem in
Definition 4, we say that an efficient solution
x
is properly efficient if only if for each i and
S x
∈
, there exists a positive real number M and a j such that
− x
x
j j
φ φ
and
x x
M x
x
j j
i i
φ φ
φ φ
− ≤
−
, whenever
− x x
i i
φ φ
. Before we go any further, the reader is reminded that for multiobjective linear
fractional programming, when the emphasis is on finding efficient solution, there is no general method for finding all the efficient solutions but Choo and Atkins [34] have
developed an algorithm, using row parameters, for solving the bicriterion linear fractional programming problem BLFP. Choo [33] has also shown that if
x
is an efficient solution to BLFP then
x
is properly efficient Geofrion [60]. The nonnegativity of
i
θ
is needed to establish part b of Theorem 5.7 bellow.
Theorem 5.7 a If
x
is an
α
-optimal solution of
λ
P
, then
x
is properly an
α
-efficient for
P
. b If f and -g are concave and
x
is properly an
α
-efficient for P, then it is an
α
- optimal for
λ
P
.
120
To prove Theorem 5.7 above, the reader is referred to Geofrion [60].
Theorem 5.8 The point
, b
A X
x ∈
is an
α
-efficient solution of
α
-MOFP if it is an
α
-efficient of P with
= x
F
.
Proof Suppose
, b
A X
x ∈
is an
α
-efficient solution of
α
-MOFP. Then by Definition4, there is no
, b
A X
x ∈
such that
p i
x g
x f
x g
x f
i i
i i
, 1
, =
∀ ≤
.
Letting
x g
x f
i i
i
= θ
for
p i
, 1
=
, we see from the above inequality that there does not exists an
, b
A X
x ∈
such that
p i
x F
x g
x f
i i
i i
, 1
, =
∀ =
− ≤
θ
Since
p i
x F
x g
x f
i i
i i
, 1
, =
= −
=
θ
, we see that there exists no x in
, b
A X
such that
p i
x F
x F
i i
, 1
, =
≤
. Therefore,
x
is an
α
-efficient of P with
= x
F
. Conversely, suppose that
x
is an
α
-efficient solution of P with
x g
x f
x F
θ −
= =
. That means, by Definition 5.4, there exists no
, b
A X
x ∈
such that
p i
x g
x f
x F
x F
i i
i
, 1
, =
∀ −
= ≤
=
θ
. That is, there exists no
, b
A X
x ∈
such that
p i
x g
x f
x g
x f
i i
i i
i
, 1
, =
∀ ≤
= θ
Hence,
x
is an
α
-efficient solution of
α
-MOFP. ■
For the development that follows, we assume that there exists real numbers
, K
k
such that
K x
g k
i
for all i. Applying Definition 5.5 of the proper efficiency to
121
problem
α
-MOFP, we note that an
α
-efficient solution
x
of problem
α
-MOFP is properly
α
-efficient if there exists a real number
M
such that for each i, we have
] [
x g
x f
x g
x f
M x
g x
f x
g x
f
j j
j j
i i
i i
− ≤
−
for some j such that
x g
x f
x g
x f
j j
j j
whenever
, b
A X
x ∈
and
x g
x f
x g
x f
i i
i i
. Or, rewriting these inequalities slightly differently, we say an
α
-efficient solution
x
of problem
α
-MOFP is properly
α
-efficient if there exists a real number
M
such that for each i, we have
] [
] [
x g
x g
x f
x g
x f
M x
g x
g x
f x
g x
f
j j
j j
j i
i i
i i
− ≤
−
, 5.148
where
k MK
M =
for some j such that
x f
x g
x g
x f
j j
j
5.149 Whenever
, b
A X
x ∈
and
] [
− x
g x
f x
g x
f
i i
i i
. 5.150
To link proper
α
-efficiently of problem
α
-MOFP and P, we prove the following theorem.
Theorem 5.9 The point
, b
A X
x ∈
is a properly
α
- efficiently solution of problem
α
-MOFP if and only if it is a properly
α
- efficiently solution on P with
= x
F
. Proof. Supose
x
is a properly
α
- efficiently solution of problem
α
-MOFP. Then by Theorem 5.8, we know its an
α
- efficiently solution on P with
= x
F
. Now
x
is a properly
α
- efficiently solution of problem P if there exists a positive real number M such that for each i,
x F
x F
M x
F x
F
j j
i i
− ≤
−
. 5.151 for some j such that
.
− x
F x
F
j j
5.152 whenever
, b
A X
x ∈
and
− x
F x
F
i i
. 5.153
122
Or [in view of the fact that
= x
F
for all i and
x g
x f
x F
i i
i i
θ −
=
with
x g
x f
i i
i
= θ
for
p i
, 1
=
], the result holds if and only if there exists an
M
such for each i,
] [
] ]
[ x
g x
g x
f x
g x
f M
x g
x g
x f
x g
x f
j j
j j
j i
i i
i i
− ≤
−
5.154 for some j such that
− x
g x
f x
g x
f
j j
j j
5.155 whenever
, b
A X
x ∈
and
− x
g x
f x
g x
f
i i
i i
5.156 Relation 5.154-5.156 hold by 5.148-5.150 with
M M
=
. Conversely, suppose
x
is a properly
α
-efficient solution of P with
= x
F
. Then by Definition 5.5, relation 5.151-5.153 hold for some M and eacj i and
, b
A X
x ∈
. From this it follows that 5.154-5.156 hold which are 5.148-5.150 with
M M
=
. ■
5.9.2. Solution algorithm
