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
Based on definition of the set of feasible parameters; the solvability set and the stability set of the first kind SSK1 of problem FMOFP via problem
α
-MOFP. Let
⎭ ⎬
⎫ ⎩
⎨ ⎧
= ∈
=
∑ ∑
= ∈
= p
i i
i b
A X
x p
i i
i n
x F
x F
R x
E
1 ,
1
max λ
λ λ
be the set of
α
-optimal solutions of problem
λ
P
.
Definition 5.6 The set of feasible parameter of problem
α
-MOFP, which is denoted b, which is denoted by U, is defined by:
{ }
φ
α
≠ =
∈ ∈
= ,
and ,
1 ,
~ b
A X
m r
b L
b b
U
r r
m
R
.
Definition 5.7 The solvability set of problem
α
-MOFP, which is denoted by V, is defined by:
{ }
where ,
solution efficient
an has
Problem
λ α
α
E x
x MOFP
U b
V ∈
− −
∈ =
.
Definition 5.8 The stability set of the first kind. Supose that
V b
∈
with the corresponding
α
-efficient solution
x
of problem
α
-MOFP such that
λ E
x ∈
, then the stability set of the first kindSSK1 of problem
α
-MOFP, which is denoted by
x S
, is defined by:
126
{ }
problem of
solution efficientn
- an
is MOFP
x V
b x
S −
∈ =
α α
.
5.9.4. Utilization of Kuhn-Tucker conditions corresponding to problem
λ
P
Problem
λ
P
can be written in the followings form:
λ
P
:
, max
1
∑
= p
i i
i
x F
λ
subject to
⎪ ⎪
⎪ ⎩
⎪⎪ ⎪
⎨ ⎧
= ≥
= ≤
≤ =
≤ =
∑
=
. ,
1 ,
, ,
1 ,
, ,
1 ,
,
1
n j
x m
r H
b h
m r
b x
a b
x
j r
r r
n j
r j
rj r
r
ψ
It is clear that the constraint
~ b
L b
α
∈
in the problem P has been replaced by the equivalent constraint
m r
H b
h
r r
r
, 1
, =
≤ ≤
in problem
λ
P
, where
r
h
and
r
H
, are lower and upper bounds on
r
b
, respectively. Therefor, the Kuhn-Tucker necessary optimality conditions corresponding to the
maximization problem
λ
P
we have the following form:
∑ ∑ ∑
∑ ∑
∑ ∑
= =
= =
= =
=
= +
− +
∂ ∂
= +
∂ ∂
− ∂
∂
m r
m r
m r
r r
m r
r j
r r
r p
i m
r n
j j
j r
r r
j i
i
x b
x x
b x
x x
F
1 1
1 1
1 1
1
, ,
, ,
η γ
ξ ψ
ξ β
ψ ξ
λ
r r
r
b b
x ≤
,
ψ
,
,
r r
b h
≤ ,
r r
H b
≤
, ≥
j
x
, ]
, [
= −
r r
r r
b b
x
ψ ξ
, =
−
r r
r
b h
γ
127
, =
−
r r
r
H b
η =
j j
x β
≥
r
ξ
,
, ≥
j
β
, ≥
r
γ
, ≥
r
η
where
} ,...,
1 {
m I
r =
∈
and
} ,...,
1 {
n J
j =
∈
. In addition, all the expressions of Kuhn- Tucker conditions are evaluated at the
α
-optimal solution
x
of problem
λ
P
.
Furthermore,
r r
j r
η γ
β ξ
, ,
,
are the Lagrange multipliers. The first two together with the last four relations of the above system of the Kuhn-Tucker
conditions represent a Polytope in
ξβγη
-space for which its vertices can be determined using any algorithm based upon the simplex method, for example, Balinski 1961.
According to whether any of the variables
} ,...,
1 {
, ,
, ,
m I
r
r r
j r
= ∈
η γ
β ξ
and
} ,...,
1 {
, n
J j
j
= ∈
β
are zero positive, then the set of parameters
m r
b
r
, 1
, =
for which the
α
-efficient solution
x
for one vector of parameters
V b
∈
rests efficient for all parameters
V b
∈
.
128
C
HAPTER 6
A POSSIBILISTIC APPROACH FOR A PORTFOLIO SELECTION PROBLEM
6.1. Introduction
