83 ⎪
⎩ ⎪
⎨ ⎧
+ −
+ +
= −
= +
+ =
− −
= +
+ =
= −
= otherwise
i T
n n
j and
T K
m K
Km i
T K
m K
Km i
n j
r a
ij ij
, 1
1 ,
1 ,
1 ,
, 1
, 1
, 1
1 ,
, 1
, 1
, ,
1 ,
⎩ ⎨
⎧ +
= =
= otherwise
mT i
n j
a
ij
, 1
, ,
1 ,
1
⎩ ⎨
⎧ +
= =
− =
otherwise mT
i n
j a
ij
, 2
, ,
1 ,
1
⎩ ⎨
⎧ +
+ +
= =
+ +
− +
= =
− −
−
otherwise m
mT mT
i m
K TK
n K
T n
j p
a
K T
n j
ij
, 2
, 3
, ,
1 ,
, 1
1 ,
1
In the next section we extended this result to fuzzy decisions theory.
5.5 Case of fuzzy technological coefficients and fuzzy right-hand side numbers
5.5.1 Case of fuzzy technological coefficients
In this section presents an approach to portfolio selection using fuzzy decisions theory. We consider the problem 5.34 – 5.36 with fuzzy technological coefficients Gasimov
[57].
max X
ϕ
=
∑
= n
j j
j
X c
1
5.37
subject to
i mT
n j
j ij
b X
a ≤
∑
+ =1
~
,
2 ,
1 +
+ =
m mT
i
, 5.38
≥
j
X
,
mT n
j +
= ,1
. 5.39
Assumption 5.1.
ij
a ~
is a fuzzy number for any i and j. In this case we consider the following membership functions:
i 1. For
1 ,
, 1
, 1
− =
+ +
= T
K m
K Km
i
and
n j
, 1
=
⎪ ⎩
⎪ ⎨
⎧ +
− ≥
+ −
≤ −
− +
− −
= .
, ,
1
ij ij
ij ij
ij ij
ij ij
ji a
d r
t if
d r
t r
if d
t d
r r
t if
t
ij
μ
84
2. For
1 ,
, 1
, 1
− =
+ +
= T
K m
K Km
i
and j=n+Ti-Km-1+K+1
⎪ ⎩
⎪ ⎨
⎧ +
− ≥
+ −
≤ −
− +
− −
=
,
1 ,
1 1
1 1
1
ij ij
ij ij
a
d t
if d
t if
d t
d t
if t
ij
μ
ii For
2 ,
3 +
+ +
= m
mT mT
i
,
m K
, 1
=
and
TK n
K T
n j
+ +
− +
= ,
1 1
⎪ ⎩
⎪ ⎨
⎧ +
≥ +
≤ −
+ =
− −
− −
− −
− −
− −
− −
− −
− ,
1
1 1
1 1
, ,
1
ij ij
ij ij
K T
n j
a
d p
t if
d p
t p
if d
t d
p p
t if
t
K T
n j
K T
n j
K T
n j
K T
n j
ij
μ
where
R t
∈
and
ij
d
for all
2 ,
1 +
+ =
m mT
i
,
1 ,
− =
T K
and
mT n
j +
= ,1
. For defuzzification of this problem, we first fuzzify the objective function. This is done
by calculating the lower and upper bound of the optimal values first. The bounds of the optimal values
l
z
and
u
z
are obtained by solving the standard linear programming problems
max
1
X z
ϕ
=
5.40 subject
to
i j
mT n
j ij
b X
a ≤
∑
+ =
1
,
2 ,
1 +
+ =
m mT
i
, 5.41
≥
j
X
,
mT n
j +
= ,1
, 5.42
and
max
2
X z
ϕ
=
5.43 subject
to
i j
mT n
j ij
b X
a ≤
∑
+ =
1
ˆ
,
2 ,
1 +
+ =
m mT
i
, 5.44
≥
j
X
,
mT n
j +
= ,1
, 5.45
where
85 ⎪
⎩ ⎪
⎨ ⎧
− =
+ −
+ +
= +
+ =
+ −
− =
+ +
= =
+ −
= otherwise
d T
K and
i T
n n
j m
Km Km
i d
T K
and m
K Km
i n
j d
r a
ij ij
ij ij
ij
, 1
, ,
1 1
, 1
, 1
, 1
, 1
1 ,
1 ,
1 ,
, 1
, ˆ
⎪⎩ ⎪
⎨ ⎧
+ =
= +
= otherwise
d mT
i n
j d
a
ij ij
ij
, 1
, ,
, 1
, 1
ˆ
⎪⎩ ⎪
⎨ ⎧
+ =
= +
− =
otherwise d
mT i
n j
d a
ij ij
ij
, 2
, ,
1 ,
1 ˆ
⎪⎩ ⎪
⎨ ⎧
+ +
+ =
= +
+ −
+ =
+ =
− −
−
otherwise d
m mT
mT i
and m
K TK
n K
T n
j d
p a
ij ij
K T
n j
ij
, 2
, 3
, ,
1 ,
, 1
1 ,
ˆ
1
The objective function takes values between
1
z
and
2
z
while technological coefficients vary between
ij
a
and
ij ij
d a
+
. Let
, min
2 1
z z
z =
l
and
, max
2 1
z z
z =
u
. Then
l
z
and
u
z
are called the lower and upper bounds of the optimal values, respectively.
Assumption 5.2. The linear crisp problems 5.40- 5..42 and 5.43-5.45 have finite
optimal values. In this case the fuzzy set of optimal values, G, which is subset of
mT n
R
+
, is defined as Klir and Yuan [84 ]
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
≥ ≤
≤ −
− =
∑ ∑
∑ ∑
= =
= =
n j
j j
n j
j j
n j
j j
n j
j j
G
z X
c if
z X
c z
if z
z z
X c
z X
c if
X
1 1
1 1
1
u u
l l
u l
l
μ
5.46
The fuzzy set of the ith constraint,
i
C
, which is a subset of
mT n
R
+
, is defined by i 1. For
m K
Km i
1 ,
1 +
+ =
and
1 ,
− = T
K
86 X
i
C
μ
=
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
+ −
≥ +
− ≤
− +
−
∑ ∑
∑ ∑
∑ ∑
= =
= =
= =
n j
j ij
ji i
n j
n j
j ij
ij i
j ij
n j
n j
j ij
j ij
i n
j j
ji i
X d
r b
X d
r b
X r
X d
X r
b X
r b
1 1
1 1
1 1
, 1
, ,
5.47
2. For
m K
Km i
1 ,
1 +
+ =
and
1 ,
− = T
K
= X
i
C
μ ⎪
⎪ ⎪
⎩ ⎪
⎪ ⎪
⎨ ⎧
+ −
≥ +
− ≤
− +
−
∑ ∑
∑ ∑ ∑
∑
+ =
+ =
+ =
+ =
= +
=
, 1
, 1
, 1
, 1
1 ,
1
, 1
, 1
, 1
, ,
,
K i
n n
j j
ij i
K i
n n
j K
i n
n j
j ij
i j
K i
n n
j n
j j
ij j
i K
i n
n j
j i
X d
b X
d b
X X
d X
b X
b
5.48
where ni,K=n+Ti-Km-1+K+1
ii For
, 2
, 3
+ +
+ =
m mT
mT i
and
m K
, 1
=
X
i
C
μ
=
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
+ ≥
+ ≤
−
∑ ∑
∑ ∑
∑ ∑
+ −
+ =
− −
− +
− +
= +
− +
= −
− −
− −
− +
− +
= +
− +
= −
− −
+ −
+ =
− −
−
TK n
K T
n j
j ij
K T
n j
i TK
n K
T n
j TK
n K
T n
j j
ij K
T n
j i
j K
T n
j TK
n K
T n
j TK
n K
T n
j j
ij j
K T
n j
i TK
n K
T n
j j
K T
n j
i
X d
p b
X d
p b
X p
X d
X p
b X
p b
1 1
1 1
1 1
1 1
1 1
1
. ,
1 ,
, ,
,
5.49 By using the definition of the fuzzy decisions proposed by Bellman and Zadeh
[9], we have
min ,
min X
X X
j
C j
G D
μ μ
μ
=
. i.e.
min ,
min max
max X
X X
j
C j
G X
D X
μ μ
μ
≥ ≥
=
87
Consequently, the problem 5.37-5.39 can be written as
λ
max
5.50
, λ
μ ≥
X
G
5.51
2 ,
1 ,
+ +
= ≥
m mT
i X
i
C
λ μ
, 5.52
≥
j
X
,
1 ≤
≤
λ
,
mT j
, 1
=
. 5.53
By using 5.46 and 5.47-5.53, we obtain the following theorem.
Theorem 5.1 The portfolio problem with fuzzy technological coefficient can be reduced
to the following problem
λ
max
5.54
2 1
2 1
≤ +
− −
∑
=
z X
c z
z
n j
j j
λ
, 5.55
∑
+ =
≤ −
mT n
j i
j ij
b X
a
1
ˆ
λ
,
2 ,
1 +
+ =
m mT
i
, 5.56
≥
j
X
,
1 ≤
≤
λ
,
mT n
j +
= ,1
. 4.57 where
⎪ ⎩
⎪ ⎨
⎧ +
− +
= −
= +
+ =
+ −
− =
+ +
= =
+ −
= otherwise,
, ,
1 1
, 1
1 ,
, 1
, 1
, 1
, 1
, ,
1 ,
1 ,
, 1
, ˆ
ij ij
ij ij
ij
d i
T n
j and
T K
m K
Km i
d T
K and
m K
Km i
n j
d r
a
λ λ
λ λ
⎪⎩ ⎪
⎨ ⎧
+ =
= +
= ,
otherwise ,
, 1
, ,
1 ,
1 ˆ
ij ij
ij
d mT
i n
j d
a
λ λ
λ
⎪⎩ ⎪
⎨ ⎧
+ =
= +
− =
otherwise, ,
, 2
, ,
1 ,
1 ˆ
ij ij
ij
d mT
i n
j d
a
λ λ
λ
⎪⎩ ⎪
⎨ ⎧
+ +
+ =
+ +
− +
= +
=
− −
−
. otherwise
, 2
, 3
, ,
1 1
, ˆ
1 ij
ij K
T n
j ij
d m
mT mT
i TK
n K
T n
j d
p a
λ λ
λ
Notice that, the constraints in problem 5.54-5.57 containing the cross product term
j
X λ
are not convex. Therefore the solution of this problem requires the special approach
88
adopted for solving general nonconvex optimization problem Rockafellar and Wets [153] and White [209].
5.5.2 Portfolio problems with fuzzy technological coefficients and fuzzy right-