Table 2: Interactive processes.
@ 1 2 3
ˆµ
1
1 0.5 0.52
ˆµ
2
1 0.6 0.59
µ
D
G
1
(x
∗
,h
∗
1
, ˆp
∗
1
) 0.564271 0.514421 0.529412
µ
D
G
2
(x
∗
,h
∗
2
, ˆp
∗
2
) 0.564271 0.614421 0.599412
ˆp
∗
1
0.578193 0.562545 0.567250
ˆp
∗
2
0.551616 0.581684 0.572685
f
1
(x
∗
,h
∗
1
, ˆp
∗
1
) 84.3370 85.4053 85.0840
f
2
(x
∗
,h
∗
2
, ˆp
∗
2
) -311.601 -313.966 -313.258
In MINMAX5( ˆp, ˆµ), it is assumed that the
decision maker sets his/her permissible probabil-
ity levels as ˆp
1
= ˆp
2
= 0.75, and the reference
membership values as ˆµ
1
= ˆµ
2
= 1. Then, the
corresponding Pareto optimal solution is obtained
as f
1
(x
∗
,h
∗
1
,0.75) = 94.0338, f
2
(x
∗
,h
∗
2
,0.75) =
−290.269, µ
˜
G
i
( f
i
(x
∗
,h
∗
i
,0.75)) = 0.11176, i= 1, 2. In
our proposed algorithm, by solving MINMAX4(ˆµ)
for the reference membership values ˆµ
1
= ˆµ
2
=
1, the D
G
-Pareto optimal solution is obtained as
f
1
(x
∗
,h
∗
1
, ˆp
∗
1
) = 84.3370, f
2
(x
∗
,h
∗
2
, ˆp
∗
2
) = −311.601,
ˆp
∗
1
= 0.578193, ˆp
∗
2
= 0.551616 (see the ﬁrst iteration
of Table 2). This means that a proper balance between
permissible probability levels and the corresponding
objective functions in a fractile optimization model is
attained in membership space.
7 CONCLUSIONS
In this paper, we have proposed an interactive fuzzy
decision making method for multiobjectivefuzzy ran-
dom linear programming problems to obtain a satis-
factory solution from among a Pareto optimal solu-
tion set. In the proposed method, the decision maker
is required to specify the membership functions for
the fuzzy goals of not only objective functions but
also the permissible probability levels. Pareto optimal
concepts called D
p
-Pareto optimal and D
G
-Pareto op-
timal are introduced. The satisfactory solution can be
obtained by updating the reference membership val-
ues and solving the corresponding minmax problem
based on the linear programming technique. At the
optimal solution of MINMAX2(ˆµ) or MINMAX4(ˆµ),
it is expected that a proper balance between permis-
sible possibility levels for a probability maximization
model and permissible probability levels for a fractile
optimization model is attained. In general, in order to
deal with MOFRLP, the decision maker must specify
many parameters in advance. Fuzzy operators such as
the fuzzy decision will lighten his/her burden to spec-
ify such parameters as ﬁxed values.
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