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Fig. 2.
Membership function
μ
f
i
(
x,a
)
The Fig. 2 shows the shape of this membership function.
The value of the membership function about one solution is also called the
satisfactory degree of the objective.
2.2 Preemptive Priority Requirement
Preemptive priority requires all the objectives be optimized in an order. Usually,
there are one or several objectives in one level, which means all the objectives are
grouped according to priority. Then
α
-FMOO problem with preemptive priority
is formulated as
max
P
1
μ
f
1
(
x,a
)
,
(
x,a
)
,
,P
L
μ
f
1
(
x,a
)
,
,μ
f
l
L
(
x,a
)
⎬
···
,μ
f
l
1
···
···
s.t. x
∈
G
(
b
)=
{
x
|
g
j
(
x,b
)
≤
0
,j
=1
,
···
,m
}
(6)
a,b
α
⎭
(
a,b
)
∈
where
P
j
is the priority factor.
f
1
(
x, a
)
,
,f
lj
(
x, a
) represent some objectives
···
in
j
th the priority level.
For example, suppose the priority of
f
s
(
x,a
) is higher than that of
f
s
(
x, a
),
(
s,s
∈{
1
,
···
,k
}
,s
=
s
). Then
f
s
(
x, a
) should be optimized before
f
s
(
x,a
).
3 Enhanced Two-Step Satisfactory Method
3.1 Order of Satisfactory Degrees
According to the assumption in Section 2,
f
s
(
x, a
) has the higher priority than
f
s
(
x, a
). That means the former objective has the higher satisfactory degree
than the latter. Then the preemptive priority requirement can transformed into
the order of the satisfactory degrees, i.e.
μ
f
s
(
x, a
)
,s,s
∈{
=
s
μ
f
s
(
x, a
)
≤
1
,
···
,k
}
,s
(7)
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