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-
hand-side numbers
We consider the linear programming problem 5.34-5.36 with fuzzy technological coefficients and fuzzy right-hand-side numbers
max X
ϕ
=
∑
= n
j j
j
X c
1
5.58
subject to
i mT
n j
j ij
b X
a ~
~
1
≤
∑
+ =
,
2 ,
1 +
+ =
m mT
i
, 5.59
≥
j
X
,
mT n
j +
= ,1
. 5.60
Assumption 5.3.
ij
a ~
and
i
b ~
are fuzzy numbers for any i and j. In this case we consider the following linear membership functions:
i 1. For
1 ,
, 1
, 1
− =
+ +
= T
K m
K Km
i
and
n j
, 1
=
,
⎪ ⎩
⎪ ⎨
⎧ +
− ≥
+ −
≤ −
− +
− −
=
+ ,
1
, ,
1
ij ij
ij ij
ij ij
ij ij
j k
a
d r
t if
d r
t r
if d
t d
r r
t if
t
ij
μ
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
m K
m mT
mT i
, 1
, 2
, 3
= +
+ +
=
and
TK n
K T
n j
+ −
+ =
, 1
,
⎪ ⎩
⎪ ⎨
⎧ +
≥ +
≤ −
+ =
,
, ,
1
ij ij
ij ij
ij ij
ij ij
ij a
d p
t if
d p
t p
if d
t d
p p
t if
t
ij
μ
and
89
⎪⎩ ⎪
⎨ ⎧
+ ≥
+ ≤
− +
=
i i
i i
i i
i i
i b
p b
t if
p b
t b
if p
t p
b b
t if
t
i
, ,
1
μ
where
R t
∈
and
ij
d
for all
2 ,
1 ,
, 1
+ +
= =
m mT
i n
j
. For defuzzification of this problem 5.58-5.60, we first calculate the lower and upper bounds of the optimal
values. The optimal values
l
z
and
u
z
can be defined by solving the following standard linear programming problems, for which we assume that all they have the finite optimal
values. Now defuzzification of this problem 5.58-5.60. first we 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
=
5.61 subject
to
i j
ij n
j ij
b X
d r
≤ +
−
∑
= 1
,
2 ,
1 +
+ =
m mT
i
5.62
≥
j
X
,
n j
, 1
=
, 5.63
and
max
2
X z
ϕ
=
5.64 subject
to
i i
j n
j ij
p b
X r
+ ≤
−
∑
=1
, 5.65
≥
j
X
,
n j
, 1
=
, 5.66
and
max
3
X z
ϕ =
5.67 subject
to
i i
j n
j ij
ij
p b
X d
r +
≤ +
−
∑
=1
, 5.68
≥
j
X
,
n j
, 1
=
, 5.69
and
90
max
4
X z
ϕ
=
5.70 subject
to:
i j
n j
ij
b X
r ≤
−
∑
=1
, 5.71
≥
j
X
,
n j
, 1
=
. 5.72
Let
, ,
, min
4 3
2 1
z z
z z
z =
l
and
, ,
, max
4 3
2 1
z z
z z
z
u
=
. The objective function takes values between
l
z
and
u
z
while technological coefficients take values between
ij
r −
and
ij ij
d r
+ −
and the right-hand side numbers take values between
i
b
and
i i
p b
+
. Then, the fuzzy set of optimal values, G, which is a subset of
mT n
R
+
, is defined by
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
≥ ≤
≤ −
− =
∑ ∑
∑ ∑
= =
= =
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
5.73
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
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
+ +
− ≥
+ +
− ≤
− +
+ −
=
∑ ∑
∑ ∑
∑ ∑
= =
= =
= =
n j
i j
ij ij
i n
j n
j i
j ij
ij i
j ij
n j
n j
i j
ij j
ij i
n j
j ij
i C
p X
d r
b p
X d
r b
X r
p X
d X
r b
X r
b X
i
1 1
1 1
1 1
. ,
1 ,
, ,
, μ
5.74 2. For
m K
Km i
1 ,
1 +
+ =
and
1 ,
− = T
K
91 ⎪
⎪ ⎪
⎩ ⎪
⎪ ⎪
⎨ ⎧
+ +
− ≥
+ +
− ≤
− +
+ −
=
∑ ∑
∑ ∑
∑ ∑
+ =
+ =
+ =
+ =
= +
=
, 1
, 1
, 1
, 1
1 ,
1
. 1
, 1
, 1
, ,
,
K i
n n
j i
j ij
i K
i n
n j
K i
n n
j i
j ij
i j
K i
n n
j n
j i
j ij
j i
K i
n n
j j
i C
p X
d b
p X
d b
X p
X d
X b
X b
X
i
μ
5.75 where
1 1
, +
+ −
− +
= K
Km i
T n
K i
n
. ii For
2 ,
3 +
+ +
= m
mT mT
i
and
m K
, 1
= X
i
C
μ
=
⎪ ⎪
⎪ ⎩
⎪ ⎪
⎪ ⎨
⎧
+ +
≥ +
+ ≤
+ −
∑ ∑
∑ ∑
∑ ∑
+ −
+ =
+ −
+ =
+ −
+ =
+ −
+ =
+ −
+ =
+ −
+ =
TK n
K T
n j
i j
ij ij
i TK
n K
T n
j TK
n K
T n
j i
j ij
ij i
j ij
TK n
K T
n j
TK n
K T
n j
i j
ij j
ij i
TK n
K T
n j
j ij
i
p X
d p
b p
X d
p b
X p
p X
d X
p b
X p
b
1 1
1 ,
1 1
1
. ,
1 ,
, ,
5.76 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
=
. In this case the an optimal fuzzy decision is a solution of the problem
min ,
min max
max X
X X
j
C j
G X
D X
μ μ
μ
≥ ≥
=
. Consequently, the problem 5.58-5.60 can be written as to the following optimization
problem
λ
max
5.77
λ μ
≥ X
G
5.78
2 ,
1 ,
+ +
= ≥
m mT
i X
i
C
λ μ
5.79
≥ X
,
1 ≤
≤
λ
. 5.80
92
By using the method of defuzzification as for the problem 5.50-5.53, we get the following theorem.
Theorem 5.2 The problem 5.58-4.60 is reduced to one of the following crisp
problems :
λ
max
5.81
1 1
1 2
≤ +
− −
∑
=
z X
c z
z
n j
j j
λ
, 5.82
∑
=
≤ −
+ +
−
n j
i i
j ij
ij
b p
X d
r
1
λ λ
,
m K
Km i
1 ,
1 +
+ =
and
1 ,
− = T
K
5.83
≥
j
X
,
n j
, 1
=
,
1 ≤
≤
λ
; 5.84
λ
max
5.85
1 1
1 2
≤ +
− −
∑
=
z X
c z
z
n j
j j
λ
, 5.86
∑
+ =
≤ −
+ +
−
, 1
1
K i
n n
j i
i j
ij
b p
X d
λ λ
,
m K
Km i
1 ,
1 +
+ =
and
1 ,
− = T
K
5.87
≥
j
X
,
, ,
1 K
i n
n j
+ =
,
1 ≤
≤
λ
, 5.88
where ni,K=n+Ti-Km-1+K+1;
λ
max
5.89
1 1
1 2
≤ +
− −
∑
=
z X
c z
z
n j
j j
λ
, 5.90
∑
+ −
+ =
≤ −
+ +
TK n
K T
n j
i i
j ij
ij
b p
X d
p
1
λ
,
2 ,
3 +
+ +
= m
mT mT
i
and
m K
, 1
=
5.91
≥
j
X
,
TK n
K T
n j
+ +
+ =
, 1
,
1 ≤
≤
λ
. 5.92
Notice that, the problem given in this theorem are also nonconvex programming problems, similar for the problem 5.77-5.80.
93
5.6 The modified subgradient method
