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Fig. 1.
Fuzzy number
c
⎧
⎨
⎩
0
c
≥
c
c
∗
c
−
c
∗
≤ c ≤ c
1
−
c
−
c
∗
c
=
c
∗
μ
c
(
c
)=
1
(2)
c
∗
−
c
c
∗
1
−
c
≤
c
≤
c
∗
−
c
0
otherwise
Then
α
-level set of
a,b
is defined as
⎧
⎨
μ
a
id
(
a
id
)
≥
α,μ
b
je
(
b
je
)
≥
α
a,b
α
=
(
a,b
)
(3)
i
=1
,
2
,
···
,k
;
d
=1
,
2
,
···
,r
i
⎩
j
=1
,
2
,
···
,m
;
e
=1
,
2
,
···
,s
j
Then
α
-MOO problem with fuzzy parameters is presented as follows
⎫
⎬
min (
f
1
(
x,a
)
,
···
,f
k
(
x,a
))
s.t.
x
∈
G
(
b
)=
{
x
|
g
j
(
x,b
)
≤
0
,j
=1
,
···
,m
}
a,b
α
(4)
⎭
(
a,b
)
∈
where the parameters
a
and
b
are treated as the decision variables about
α
.
In a fuzzy environment, DM usually gives all objectives the implicit tar-
gets. For minimization problem, DM permits the objective value
f
i
(
x, a
), (
i
=
1
,
,k
) are more than aspiration level up
f
i
to stated tolerant limit
f
ma
i
.
The triangle-like membership function under
α
-level set is defined for the fuzzy
objective.
···
⎧
⎨
f
i
(
x,a
)
≤ f
i
1
f
i
(
x,a
)
−
f
i
f
i
≤
f
max
i
μ
f
i
(
x,a
)=
1
−
f
i
(
x,a
)
≤
(5)
f
max
i
f
i
⎩
−
f
max
i
0
f
i
(
x,a
)
≥
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