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as far as possible until
γ<
0 and DM is satisfactory. If
γ>
0, the solution does
not satisfy priority. Then the maximum overall satisfactory degree
λ
∗
needs to be
relaxed continuously through changing the parameter
Δδ
and solve the second
model again, or
α
is regulated.
4 Algorithms
The corresponding algorithms of the proposed method is summarized as follows
Step 1.
Initialization: Calculate the individual minimum
f
min
i
and maximum
f
max
i
oftheobjectivefunction
f
i
(
x, a
), (
i
=1
,
···
,k
), under the given constraints
for
α
=0and
α
=1.
Step 2.
Determine the desirable target and the tolerance, construct the mem-
bership functions of the objectives in
α
-FMOO problem, and ask DM to select
the initial value of
α
.
Step 3.
Solve (9), and get the maximum overall satisfactory degree
λ
∗
.
Step 4.
Let the initial releasing parameter
Δδ
= 0, and formulate the second
model (10) according to the priority order.
Step 5.
Solve (10). If there is no feasible solution, then go to step 7. On the
contrary, continue.
Step 6.
Judge: if
γ>
0, go to next step. If
γ
0 but not satisfactory, go to
step 7. Otherwise stop optimization, and the satisfactory solution is acquired.
Step 7.
Relax the maximum overall satisfactory degree
λ
∗
by increasing the
releasing parameter
Δδ
and back to step 5; or decrease
α
and back to step 3.
≤
5 Numerical Example
We demonstrate for the effectiveness of the proposed optimization method by
the following numerical example
⎫
⎬
min f
1
(
x, a
)=
a
1
x
1
+3
x
2
+
a
2
x
3
min f
2
(
x, a
)=(
x
1
−
1)
2
+2(
x
2
−
3)
2
+
a
3
(
x
3
−
2)
2
s.t. x
1
+
x
2
+
x
3
≤
10
⎭
0
≤
x
1
,x
2
,x
2
≤
10
a
=(
a
1
,a
2
,a
3
) are fuzzy parameters, whose distributions are given in Table 1.
Tabl e 1.
Fuzzy Parameters
(
c,c
∗
, c
)
c
a
1
(-2, -1, 0)
a
2
(2,3,4)
a
3
(-1, 0, 1)
The preemptive priority requirement is that
f
2
(
x, a
) is higher than
f
1
(
x, a
).
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