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However, it is seen that the comparison is too strict. If (7) is taken as the
constraint, the feasible or satisfactory solution maybe can not be obtained. In
addition, the bigger difference between the objectives is not reflected from (7).
Thus the priority variable
γ
is used to release the order of satisfactory degrees.
Then the released preemptive priority requirement is reformulated as
γ,s,s
∈{
=
s
μ
f
s
(
x, a
)
−
μ
f
s
(
x, a
)
≤
1
,
···
,k
}
,s
(8)
3.2 The First Step
For preemptive priority requirement,
α
-FMOO problem (6) is divided into two
models. They include the preliminary optimization model and the priority model.
The first step is to solve the former to get the maximum overall satisfactory
degree of all objectives by max-min decisions regardless of priority. Thus the
preliminary optimization model is equivalent to
⎫
⎬
max
λ
s.t. μ
f
i
(
x,a
)
≥
λ,i
=1
,
···
,k
μ
f
i
(
x,a
)
≤
1
(9)
⎭
x
∈
G
(
b
)=
{
x
|
g
j
(
x,b
j
)
≤
0
,j
=1
,
···
,m
}
a,b
(
a,b
)
∈
α
By means of (9), all objectives can be optimized simultaneously as much as
possible. The optimization result, i.e. maximum overall satisfactory degree
λ
∗
under certain
α
-level set will be treated as the given condition of the next step
optimization.
3.3 The Second Step
After optimization of the first step, preemptive priority requirement needs to be
considered. On basis of the first step, the second model is utilized to balance
optimization and the priority order. The maximum overall satisfactory degree
λ
∗
about
α
-level set is relaxed by means of the releasing parameter
Δδ
(
Δδ
0),
which is determined by the interaction between DM and the analyzer. And
the comparing inequality is incorporated as the constraint. Then the second
optimization model is constructed in the following expression
≥
⎫
⎬
min
γ
s.t. μ
f
i
(
x,a
)
λ
∗
−
≥
Δδ, i
=1
,
···
,k
γ, s,s
=1
,
=
s
μ
f
s
(
x,a
)
−
μ
f
s
(
x,a
)
≤
···
,k,s
(10)
−
1
≤
γ
≤
1
⎭
x
∈
G
(
b
)=
{
x
|
g
j
(
x,b
j
)
≤
0
,j
=1
,
···
,m
}
a,b
(
a,b
)
∈
α
By the releasing parameter
Δδ
, the feasibility of (10) can be ensured. And the
preemptive priority requirement is realized by minimizing the priority variable
γ
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