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Then the membership functions about
a
are formulated by (2) and the cor-
responding constraints about
α
-level set are denotes as (3). The four individual
optimum values of each objective function for
α
=0and
α
= 1 are computed
according to Table 1. Then the aspiration values and the tolerant limits of the
two objectives are respectively (-20, 40) and (-45, 83). The corresponding mem-
bership functions under certain
α
are presented as
μ
f
1
(
x,a
)=(40
−
a
1
x
1
−
3
x
2
−
a
2
x
3
)
/
60
1)
2
3)
2
2)
2
)
/
128
μ
f
2
(
x,a
)=(83
−
(
x
1
−
−
2(
x
2
−
−
a
3
(
x
3
−
Therefore, the reformulated first model is written in the following expression
max
λ
s.t. μ
f
i
(
x,a
)
⎫
⎬
≥
λ,μ
f
i
(
x,a
)
≤
1
,i
=1
,
2
μ
f
1
(
x,a
)=(40
−
a
1
x
1
−
3
x
2
−
a
2
x
3
)
/
60
1)
2
3)
2
2)
2
)
/
128
μ
f
2
(
x,a
)=(83
−
(
x
1
−
−
2(
x
2
−
−
a
3
(
x
3
−
⎭
x
1
+
x
2
+
x
3
≤
10
0
≤
x
1
,x
2
,x
2
≤
10
a
∈
(
a
)
α
The second model is
min
γ
s.t. μ
f
i
(
x,a
)
⎫
⎬
λ
∗
−
≥
Δδ, i
=1
,
2
μ
f
1
(
x,a
)
−
μ
f
2
(
x,a
)
≤
γ
μ
f
1
(
x,a
)=(40
−
a
1
x
1
−
3
x
2
−
a
2
x
3
)
/
60
μ
f
2
(
x,a
)=(83
−
(
x
1
−
1)
2
−
2(
x
2
−
3)
2
−a
3
(
x
3
−
2)
2
)
/
128
⎭
x
1
+
x
2
+
x
3
≤
10
0
≤ x
1
,x
2
,x
2
≤
10
−
1
≤
γ
≤
1
a
∈
(
a
)
α
According to the algorithms, the above modes are solved iteratively. The corre-
sponding optimization results are given in Table 2.
Tabl e 2.
Optimization Results
α
δ
γ
f
1
(
x, a
)
f
2
(
x,a
)
μ
f
1
(
x,a
)
μ
f
2
(
x,a
)
0.15
-0.1788
11.74
-0.1773
0.4710
0.6498
0.25
-0.2819
17.74
-0.5663
0.3710
0.6529
0.8
0.35
-0.4023
23.74
-3.1832
0.2710
0.6733
0.15
-0.1757
11.344
-06185
0.4776
0.6533
0.25
-0.2863
17.344
-1.9778
0.3776
0.6639
0.6
0.35
-0.4378
23.344
-8.5726
0.2776
0.7154
From the various results listed in Table 2, it is known that the alternation
of the priority variable
γ
conforms to that of the releasing parameter
Δδ
,and
the values of
γ
are always less than 0. This means that all the results satisfy
the preemptive priority requirement. Then DM can choose one from them as the
preferred solution according to his requirement.
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