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
A solution algorithm to solve fuzzy multiobjective fractional programming problem FMOFP is described in a series of steps. The suggested algorithm can be summarized in
the following manner : Saad [159]
Step 1. Start with an initial level set
= =
α α
.
Step 2. Determine point
, ,
,
4 3
2 1
b b
b b
for the vector of fuzzy parameters
b ~
in problem FMOFP to elicit a membership function
~
b
b
μ
satisfying assumptions 5.148-5.153 in Definition 5.1.
Step 3. Convert problem FMOFP into its nonfuzzy version
MOFP −
α
.
Step 4. Rewrite problem
MOFP −
α
in the form of problem
λ
P
of single-objective function.
123
Step 5. Choose
=
i i
λ λ
and
1
1
=
∑
= p
i i
λ
with fixed values of
, 1
, p
i
i i
= =
θ θ
and use GINO software package Lieberman, et al [102] to find the
α
-optimal solution
x
of prolem
λ
P
.
Step 6. Set
] 1
, [
∈ +
= step
α α
and go to step 1.
Step 7. Repeat again the above procedure until the interval [0,1] is fully exhausted. Then,
stop. Example. Saad [160] In what follows we provide a numerical example to clarify the
solution algorithm suggested above. Let
2 2
2 2
1 1
2 1
1
2 ,
2 ,
2 1
, 1
x x
g x
f x
x g
x x
f −
= =
+ =
− =
So
. 2
2 ,
2 1
1
2 2
2 2
2 1
2 1
1 1
1
x x
g x
f x
F x
x x
g x
f x
F −
= =
+ −
= =
Consider the followings fuzzy bicriterion practionl programming problem FBFP
, ,
max
2 1
x F
x F
x F
=
subject to
. ,
~
2 1
2 2
2 1
≥ ≤
+ x
x b
x x
where
b ~
is a fuzzy parameter and is characterized by the following fuzzy numbers:
5 ,
3 ,
1 ,
~ = b
Assume that the membership function of these fuzzy numbers in the following form:
124 ⎪
⎪ ⎪
⎪ ⎪
⎩ ⎪⎪
⎪ ⎪
⎪
⎨ ⎧
≥ ≤
≤ ⎟⎟⎠
⎞ ⎜⎜⎝
⎛ −
− −
≤ ≤
≤ ≤
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− −
− ≤
=
4 4
3 2
3 4
3 3
2 2
1 2
2 1
2 1
~
, ,
1 ,
1 ,
1 ,
b b
b b
b b
b b
b b
b b
b b
b b
b b
b b
b
b
b
μ
Let
19 .
=
α , for example, then we get:
8 .
4 1
. ≤
≤ b
Choosing
1 =
b
, then non-fuzzy α -bicriterion fractional programming problem
α -BFPbecomes:
, ,
max
2 1
x F
x F
x F
=
subject to
. ,
1
2 1
2 2
2 1
≥ ≤
+ x
x x
x
observe that point ,
= x
is an α
-efficient solution of problem
α
-BFP since, for each feasible x and then we have
2 1
2 1
2 1
1
2 2
2 2
2 1
2 2
2 1
1 1
≤ +
+ −
= −
+ −
= −
x x
x x
x x
F x
F
, and
2 1
2 2
2 2
2 2
2
≥ −
= −
− =
− x
x x
x F
x F
, and there is no other feasible point for which
1 ,
1 ,
2 1
≥ =
x F
x F
x F
. We now consider the case when
1 ,
2 =
= j i
in the definition of a properly efficient solution and therefore it can be seen that
, =
x
is also properly
α
-efficient solution. When
2 2
x F
x F
−
we have
2
2 2
− x x
; that is,
2
x
.
125
Then
2 1
2
2 2
2 2
2 1
1 1
+ +
= −
x x
x x
F x
F
and
2 2
− x
F x
F
.
2 2
2 1
2 2
2 1
2 2
2 2
+ −
+ =
x x
x x
x M
. We have
1 1
2 2
x F
x F
M x
F x
F −
≤ −
. So that point
, =
x
is properly
α
-efficient solution for problem
α
-BFP with the corresponding
α
-level set equals 0.19.
5.9.3. Basic stability notions for problem FMOFP
