Interactive Fuzzy Decision Making for Multiobjective Fuzzy Random
Linear Programming Problems
Hitoshi Yano
1
and Masatoshi Sakawa
2
1
School of Humanities and Social Sciences, Nagoya City University, 467-8501 Nagoya, Japan
2
Department of System Cybernetics, Graduate School of Engineering, Hiroshima University,
739-8511 Higashi-Hiroshima, Japan
Keywords:
Fuzzy Random Variable, A Probability Maximization Model, A Fractile Criterion Optimization Model,
Satisfactory Solution, Interactive Decision Making.
Abstract:
In this paper, we propose an interactive fuzzy decision making method for multiobjective fuzzy random linear
programming problems (MOFRLP), in which the criteria of probability maximization and fractile optimiza-
tion are considered simultaneously. In the proposed method, it is assumed that the decision maker has fuzzy
goals for not only objective functions of MOFRLP but also permissible probability levels in a fractile opti-
mization model for MOFRLP, and such fuzzy goals are quantified by eliciting the corresponding membership
functions. Using the fuzzy decision, such two kinds of membership functions are integrated. In the integrated
membership space, the satisfactory solution is obtained from among a Pareto optimal solution set through the
interaction with the decision maker.
1 INTRODUCTION
In the real world decision making situations, we of-
ten have to make a decision under uncertainty. In
order to deal with decision problems involving uncer-
tainty, stochastic programming approaches (Birgeand
Louveaux, 1997; Charnes and Cooper,1959; Dantzig,
1955; Kall and Mayer, 2005) and fuzzy programming
approaches (Lai and Hwang, 1992; Sakawa, 1993;
Zimmermann, 2011) have been developed. Recently,
mathematical programming problems with fuzzy ran-
dom variables (Kwakernaak, 1978) have been pro-
posed (Katagiri et al., 1997; Luhandjula and Gupta,
1996; Wang and Qiao, 1993) whose concept includes
both probabilistic uncertainty and fuzzy ones simul-
taneously. For multiobjective fuzzy random linear
programming problems (MOFRLP), (Sakawa et al.,
2011) formulated and proposed interactive methods
to obtain the satisfactory solution. In their methods, it
is required in advance for the decision maker to spec-
ify permissible possibility levels in a probability max-
imization model or permissible probability levels in a
fractile optimization model. However, it seems to be
very difficult for the decision maker to specify such
permissible levels appropriately. From such a point of
view, (Yano and Matsui, 2011) have proposed a fuzzy
approach for MOFRLP, in which the decision maker
specifies the membership functions for the fuzzy
goals of both objective functions of MOFRLP and
permissible probability levels. In the proposed
method, it is assumed that the decision maker adopts
the fuzzy decision (Sakawa, 1993) to integrate the
membership functions. However, the fuzzy decision
can be viewed as one special operator to integrate the
membership functions. If the decision maker would
not adopt the fuzzy decision, the proposed method
cannot be applied in the real-world decision situation.
In this paper,we propose an interactive fuzzy decision
making method for MOFRLP to obtain the satisfac-
tory solution from among a Pareto optimal solution
set. In section 2, MOFRLP is formulated by using a
concept of a possibility measure (Dubois and Prade,
1980). In section 3, through a probability maximiza-
tion model, the D
p
-Pareto optimal concept is intro-
duced in order to deal with MOFRLP, and the minmax
problem is formulated to obtain a D
p
-Pareto optimal
solution, which can be solved on the basis of the linear
programming technique. In section 4, through a frac-
tile optimization model, the D
G
-Pareto optimal con-
cept is introduced and the minmax problem is formu-
lated to obtain a D
G
-Pareto optimal solution. In sec-
tion 5, we propose an interactive algorithm to obtain
the satisfactory solution from among a Pareto optimal
solution set by solving the minmax problem on the ba-
319
Yano H. and Sakawa M..
Interactive Fuzzy Decision Making for Multiobjective Fuzzy Random Linear Programming Problems.
DOI: 10.5220/0004087003190328
In Proceedings of the 4th International Joint Conference on Computational Intelligence (FCTA-2012), pages 319-328
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

sis of the linear programming technique. In section 5,
in order to demonstrate the interactive processes un-
der the hypothetical decision maker, a two-objective
fuzzy random linear programming problem, as a nu-
merical example, is formulated and solved by using
the proposed interactive algorithm. Finally, in section
7, we conclude this paper.
2 MULTIOBJECTIVE FUZZY
RANDOM LINEAR
PROGRAMMING PROBLEMS
In this section, we focus on multiobjective program-
ming problems involving fuzzy random variable coef-
ficients in objective functions, which is called multi-
objective fuzzy random linear programming problem
(MOFRLP).
[MOFRLP]
min
e
Cx = (
e
c
1
x,··· ,
e
c
k
x)
subject to
x ∈ X
def
= {x ∈ R
n
| Ax ≤ b, x ≥ 0}
where x = (x
1
,x
2
,··· ,x
n
)
T
is an n dimensional deci-
sion variable column vector, A is an (m × n) coeffi-
cient matrix, b = (b
1
,··· ,b
m
)
T
is an m dimensional
column vector.
e
c
i
= (
e
c
i1
,··· ,
e
c
in
),i = 1,··· , k, are
coefficient vectors of objective function
e
c
i
x, whose
elements are fuzzy random variables (Kwakernaak,
1978; Puri and Ralescu, 1986; Sakawa et al., 2011),
and the symbols
"-" and "˜" mean randomness and
fuzziness respectively.
In order to deal with the objective functions
e
c
i
x,i = 1, ··· , k, (Sakawa et al., 2011) proposed an
LR-type fuzzy random variable which can be re-
garded as a special version of a fuzzy random vari-
able. Under the occurrence of each elementary event
ω,
e
c
ij
(ω) is a realization of an LR-type fuzzy random
variable
e
c
ij
, which is an LR fuzzy number (Dubois
and Prade, 1980) whose membership function is de-
fined as follows.
µ
e
c
ij
(ω)
(s) =



L

¯
d
ij
(ω)−s
¯
α
ij
(ω)

(s ≤
¯
d
ij
(ω) ∀ω),
R

s−
¯
d
ij
(ω)
¯
β
ij
(ω)

(s >
¯
d
ij
(ω) ∀ω),
where the function L(t)
def
= max{0, l(t)} is a real-
valued continuous function from [0, ∞) to [0, 1], and
l(t) is a strictly decreasing continuous function sat-
isfying l(0) = 1. Also, R(t)
def
= max{0, r(t)} satisfies
the same conditions.
¯
d
ij
,
¯
α
ij
,
¯
β
ij
are random variables
expressed by
¯
d
ij
= d
1
ij
+
¯
t
i
d
2
ij
,
¯
α
ij
= α
1
ij
+
¯
t
i
α
2
ij
and
¯
β
ij
= β
1
ij
+
¯
t
i
β
2
ij
.
¯
t
i
is a random variable whose dis-
tribution function is denoted by T
i
(·) which is strictly
increasing and continuous, and d
1
ij
,d
2
ij
, α
1
ij
,α
2
ij
,β
1
ij
,β
2
ij
are constants.
(Sakawa et al., 2011) transformed MOFRLP into
a multiobjective stochastic programming problem
(MOSP) by using a concept of a possibility measure
(Dubois and Prade, 1980). As shown in (Sakawa
et al., 2011), the realizations
e
c
i
(ω)x becomes an LR
fuzzy number characterized by the following mem-
bership functions on the basis of the extension princi-
ple (Dubois and Prade, 1980).
µ
e
c
i
(ω)x
(y) =







L

¯
d
i
(ω)x−y
¯
α
i
(ω)x

y ≤
¯
d
i
(ω)x
R

y−
¯
d
i
(ω)x
¯
β
i
(ω)x

y >
¯
d
i
(ω)x
For the realizations
e
c
i
(ω)x,i = 1,··· ,k, it is assumed
that the decision maker has fuzzy goals
e
G
i
,i = 1, ··· ,k
(Sakawa, 1993), whose membership functions µ
e
G
i
(y),
i = 1, ··· , k are continuous and strictly decreasing for
minimization problems. By using a concept of a pos-
sibility measure (Dubois and Prade, 1980), a degree
of possibility that the objective function value
e
c
i
x sat-
isfies the fuzzy goal
e
G
i
is expressed as follows (Kata-
giri et al., 1997).
Π
e
c
i
x
(
˜
G
i
)
def
= sup
y
min{µ
e
c
i
x
(y),µ
˜
G
i
(y)} (1)
Using a possibility measure, MOFRLP can be trans-
formed into the following multiobjective stochastic
programming problem (MOSP).
[MOSP]
max
x∈X
(Π
e
c
1
x
(
˜
G
1
),··· ,Π
e
c
k
x
(
˜
G
k
)) (2)
(Sakawa et al., 2011) transformed MOSP into the
usual multiobjective programming problems through
a probability maximization model and a fractile max-
imization model, and proposed interactive algorithms
to obtain a satisfactory solution. In their methods, the
decision maker must specify permissible probability
levels or permissible possibility levels for the objec-
tive functions in advance. However, it seems to be
very difficult to specify appropriate permissible lev-
els because they have a great influence on the objec-
tive function values or distribution function values.
In the following sections, by assuming that the deci-
sion maker has fuzzy goals for permissible probabil-
ity levels and permissible possibility levels, we pro-
pose an interactive fuzzy decision making method for
MOFRLP to obtain a satisfactory solution.
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
320

3 A FORMULATION THROUGH
A PROBABILITY
MAXIMIZATION MODEL
For the objective function of MOSP, if the decision
maker specifies the permissible possibility level h
i
∈
[0, 1], then MOSP can be formulated as the following
multiobjective programming problem through a prob-
ability maximization model.
[MOP1(h)]
max
x∈X
(Pr(ω | Π
e
c
1
(ω)x
(
˜
G
1
) ≥ h
1
),··· ,
Pr(ω | Π
e
c
k
(ω)x
(
˜
G
k
) ≥ h
k
))
where Pr(·) is a probability measure, h = (h
1
,··· , h
k
)
is a vector of permissible possibility levels. In
MOP1(h), the inequality Π
e
c
i
(ω)x
(
e
G
i
) ≥ h
i
can be
equivalently transformed into the following form.
sup
y
min{µ
e
c
i
x
(y),µ
˜
G
i
(y)} ≥ h
i
,
⇔ (
¯
d
i
(ω) − L
−1
(h
i
)
¯
α
i
(ω))x ≤ µ
−1
˜
G
i
(h
i
)
where L
−1
(·) and R
−1
(·) are pseudo-inverse func-
tions. Therefore, using the distribution function T
i
(·)
of the random variable
¯
t
i
, the objective functions in
MOP1(h) can be expressed as the following form.
Pr(ω | Π
e
c
i
(ω)x
(
˜
G
i
) ≥ h
i
)
= T
i
µ
−1
˜
G
i
(h
i
) − (d
1
i
x− L
−1
(h
i
)α
1
i
x)
d
2
i
x− L
−1
(h
i
)α
2
i
x
!
def
= p
i
(x, h
i
) (3)
where it is assumed that (d
2
i
− L
−1
(0)α
2
i
)x > 0, i =
1, ··· , k for any x ∈ X. As a result, using p
i
(x,h
i
),i =
1, ··· , k, MOP1(h) can be transformed to the follow-
ing simple form (Sakawa et al., 2011).
[MOP2(h)]
max
x∈X
(p
1
(x,h
1
),··· , p
k
(x, h
k
))
In MOP2(h), the decision maker seems to prefer not
only the larger value of a permissible possibility level
h
i
but also the larger value of the corresponding dis-
tribution function p
i
(x,h
i
). Since these values conflict
with each other, the larger value of a permissible pos-
sibility level h
i
results in the less value of the corre-
sponding distribution function p
i
(x,h
i
). From such a
point of view, we consider the following multiobjec-
tive programming problem which can be regarded as
a natural extension of MOP2(h).
[MOP3]
max
x∈X,h
i
∈[0,1],i=1,···,k
(p
1
(x,h
1
),· · · , p
k
(x, h
k
),
h
1
,· · · ,h
k
)
It should be noted in MOP3 that permissible possi-
bility levels h
i
,i = 1,··· , k are not the fixed values
but the decision variables. Considering the imprecise
nature of the decision maker’s judgment, it is natu-
ral to assume that the decision maker has fuzzy goals
for p
i
(x,h
i
),i = 1,· · · , k. In this section, we assume
that such fuzzy goals can be quantified by eliciting
the corresponding membership functions. Let us de-
note a membership function of a distribution function
as µ
p
i
(p
i
(x, h
i
)). Then, MOP3 can be transformed to
the following multiobjective programming problem.
[MOP4]
max
x∈X,h
i
∈[0,1],i=1,···,k
(µ
p
1
(p
1
(x,h
1
)),· · · ,
µ
p
k
(p
k
(x, h
k
)),h
1
,· · · ,h
k
)
In order to elicit the membership functions
µ
p
i
(p
i
(x,h
i
)),i = 1,··· ,k appropriately, we suggest
the following procedures. First of all, the decision
maker sets the intervals H
i
= [h
imin
,h
imax
] for per-
missible possibility levels, where h
imin
is a maximum
value of an unacceptable levels and h
imax
is a mini-
mum value of a sufficiently satisfactory levels. For
the interval H
i
, the corresponding interval of p
i
(x,
ˆ
h
i
)
can be defined as P
i
(H
i
) = [p
imin
, p
imax
] = {p
i
(x, h
i
) |
x ∈ X, h
i
∈ H
i
}. p
imax
can be obtained by solving the
following optimization problem.
p
imax
def
= max
x∈X
p
i
(x, h
imin
) (4)
In order to obtain p
imin
, we first solve the optimization
problems max
x∈X
p
i
(x,h
imax
),i = 1,··· , k, and denote
the corresponding optimal solutions as x
i
,i = 1, ··· , k.
Using the optimal solution x
i
,i = 1, ··· , k, p
imin
can
be obtained as the following minimum value.
p
imin
def
= min
ℓ=1,···,k,ℓ6=i
p
i
(x
ℓ
,h
imax
) (5)
For the membership functions µ
p
i
(p
i
(x,h
i
)),i =
1, ··· , k defined on P
i
(H
i
), we make the following as-
sumption.
Assumption 1.
µ
p
i
(p
i
(x,h
i
)),i = 1,· · · , k are strictly increasing and
continuous with respect to p
i
(x,h
i
) ∈ P
i
(H
i
), and
µ
p
i
(p
imin
) = 0, µ
p
i
(p
imax
) = 1.
It should be noted here that µ
p
i
(p
i
(x,h
i
)) is strictly
decreasing with respect to h
i
∈ H
i
. If the decision
maker adopts the fuzzy decision (Sakawa, 1993) to
integrate µ
p
i
(p
i
(x,h
i
)) and h
i
, MOP4 can be trans-
formed into the following form.
[MOP5]
max
x∈X,h
i
=H
i
,i=1,···,k

µ
D
p
1
(x,h
1
),· · · , µ
D
p
k
(x,h
k
)

where
µ
D
p
i
(x, h
i
)
def
= min{h
i
,µ
p
i
(p
i
(x,h
i
))} (6)
InteractiveFuzzyDecisionMakingforMultiobjectiveFuzzyRandomLinearProgrammingProblems
321

In order to deal with MOP5, we introduce a D
p
-Pareto
optimal solution concept.
Definition 1.
x
∗
∈ X, h
∗
i
∈ H
i
,i = 1, · · · , k is said to be a D
p
-Pareto
optimal solution to MOP5, if and only if there does
not exist another x ∈ X, h
i
∈ H
i
,i = 1,·· · ,k such that
µ
D
p
i
(x,h
i
) ≥ µ
D
p
i
(x
∗
,h
∗
i
) i = 1, ··· , k with strict in-
equality holding for at least one i.
For generating a candidate of a satisfactory so-
lution which is also D
p
-Pareto optimal, the decision
maker is asked to specify the reference membership
values (Sakawa, 1993) in membership space. Once
the reference membership values ˆµ = (ˆµ
1
,· · · , ˆµ
k
) are
specified, the corresponding D
p
-Pareto optimal so-
lution is obtained by solving the following minmax
problem.
[MINMAX1(ˆµ)]
min
x∈X,h
i
∈H
i
,i=1,···,k,λ∈Λ
λ (7)
subject to
ˆµ
i
− µ
p
i
(p
i
(x, h
i
)) ≤ λ, i = 1, ··· , k (8)
ˆµ
i
− h
i
≤ λ, i = 1, ··· , k (9)
where
Λ = [ max
i=1,···,k
ˆµ
i
− 1, min
i=1,···,k
ˆµ
i
]. (10)
From Assumption 1, the inequality constraints (8) can
be transformed into the following form.
ˆµ
i
− µ
p
i
(p
i
(x,h
i
)) ≤ λ
⇔ µ
−1
˜
G
i
(h
i
) ≥ (d
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))d
2
i
x)
−L
−1
(h
i
)(α
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))α
2
i
x)
(11)
In (11), because of ˆµ
i
− λ ≤ h
i
and Assumption 1,
it holds that µ
−1
˜
G
i
(h
i
) ≤ µ
−1
˜
G
i
(ˆµ
i
− λ) and L
−1
(h
i
) ≤
L
−1
(ˆµ
i
− λ). Since it is guaranteed that (α
1
i
x +
T
−1
i
(µ
−1
p
i
(ˆµ
i
−λ)) α
2
i
x) > 0, the following inequalities
can be derived.
(d
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))d
2
i
x)
−L
−1
(h
i
)(α
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))α
2
i
x)
≥ (d
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))d
2
i
x)
−L
−1
(ˆµ
i
− λ)(α
1
i
x+ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ))α
2
i
x)
= (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x)
(12)
From (11) and (12), it holds that
µ
−1
˜
G
i
(ˆµ
i
− λ) ≥ µ
−1
˜
G
i
(h
i
)
≥ (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x).
Therefore, MINMAX1(ˆµ) can be reduced to the
following minmax problem.
[MINMAX2(ˆµ)]
min
x∈X,λ∈Λ
λ (13)
subject to
µ
−1
˜
G
i
(ˆµ
i
− λ) ≥ (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x),
i = 1,·· · ,k (14)
It should be noted here that the constraints (14) can be
reduced to a set of linear inequalities for some fixed
value λ ∈ Λ. This means that an optimal solution
(x
∗
,λ
∗
) of MINMAX2(ˆµ) is obtained by combined
use of the bisection method with respect to λ ∈ Λ and
the first-phase of the two-phase simplex method of
linear programming. The relationships between the
optimal solution (x
∗
,λ
∗
) of MINMAX2(ˆµ) and D
p
-
Pareto optimal solutions can be characterized by the
following theorem.
Theorem 1.
(1) If x
∗
∈ X, λ
∗
∈ Λ is a unique optimal solution of
MINMAX2(ˆµ), then x
∗
∈ X, ˆµ
i
− λ
∗
∈ H
i
,i = 1,·· · , k
is a D
p
-Pareto optimal solution.
(2) If x
∗
∈ X, h
∗
i
∈ H
i
,i = 1, ··· , k is a D
p
-Pareto
optimal solution, then x
∗
∈ X, λ
∗
= ˆµ
i
− h
∗
i
= ˆµ
i
−
µ
p
i
(p
i
(x
∗
,h
∗
i
)),i = 1,· · · ,k is an optimal solution of
MINMAX2(ˆµ) for some reference membership val-
ues ˆµ = (ˆµ
1
,· · · , ˆµ
k
).
(Proof)
(1) From (14), it holds that ˆµ
i
− λ
∗
≤ µ
p
i
(p
i
(x
∗
, ˆµ
i
−
λ
∗
)),i = 1, ··· , k. Assume that x
∗
∈ X, ˆµ
i
−λ
∗
∈ H
i
,i =
1, ··· , k is not a D
p
-Pareto optimal solution. Then,
there exist x ∈ X,h
i
∈ H
i
,i = 1, ··· , k such that
µ
D
p
i
(x,h
i
) = min{h
i
,µ
p
i
(p
i
(x,h
i
))}
≥ µ
D
p
i
(x
∗
, ˆµ
i
− λ
∗
)
= ˆµ
i
− λ
∗
,i = 1, · · · , k,
with strict inequality holding for at least one i. Then
it holds that
h
i
≥ ˆµ
i
− λ
∗
,i = 1, · · · , k (15)
µ
p
i
(p
i
(x,h
i
)) ≥ ˆµ
i
− λ
∗
,i = 1, · · · , k (16)
From Assumption 1, (3) and L
−1
(h
i
) ≤ L
−1
(ˆµ
i
− λ
∗
),
(15) and (16) can be transformed as follows.
µ
−1
˜
G
i
(h
i
) ≤ µ
−1
˜
G
i
(ˆµ
i
− λ
∗
),i = 1, ··· , k
µ
−1
˜
G
i
(h
i
) ≥ (d
1
i
x− L
−1
(ˆµ
i
− λ
∗
)α
1
i
x)
+T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ
∗
))
·(d
2
i
x− L
−1
(ˆµ
i
− λ
∗
)α
2
i
x),
i = 1,·· · ,k
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322

As a result, there exists x ∈ X such that
µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) − (d
1
i
x− L
−1
(ˆµ
i
− λ
∗
)α
1
i
x)
≥ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ
∗
)) · (d
2
i
x− L
−1
(ˆµ
i
− λ
∗
)α
2
i
x),
i = 1,·· · ,k,
which contradicts the fact that x
∗
∈ X, λ
∗
∈ Λ is a
unique optimal solution to MINMAX2(ˆµ).
(2) Assume that x
∗
∈ X,λ
∗
∈ Λ is not an optimal solu-
tion to MINMAX2(ˆµ) for any reference membership
values ˆµ = (ˆµ
1
,· · · , ˆµ
k
), which satisfy the equalities
ˆµ
i
− λ
∗
= h
∗
i
= µ
p
i
(p
i
(x
∗
,h
∗
i
)),i = 1, ··· , k. (17)
Then, there exists some x ∈ X, λ < λ
∗
such that
µ
−1
˜
G
i
(ˆµ
i
− λ) − (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
≥ T
−1
i
(µ
−1
p
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x),
⇔ µ
p
i
(p
i
(x, ˆµ
i
− λ)) ≥ ˆµ
i
− λ, i = 1,··· ,k (18)
Because of (17),(18) and ˆµ
i
−λ > ˆµ
i
−λ
∗
,i = 1, ··· , k,
the following inequalities hold.
µ
p
i
(p
i
(x, h
i
)) > µ
p
i
(p
i
(x
∗
,h
∗
i
)),i = 1, ··· , k
where h
i
= ˆµ
i
− λ ∈ H
i
. Then, because of h
i
> h
∗
i
,
there exists x ∈ X, h
i
∈ H
i
,i = 1, · · · , k such that
µ
D
p
i
(x,h
i
) > µ
D
p
i
(x
∗
,h
∗
i
),i = 1, ··· , k.
This contradicts the fact that x
∗
∈ X, h
∗
i
∈ H
i
,i =
1, ··· , k is a D
p
-Pareto optimal solution.
4 A FORMULATION THROUGH
A FRACTILE OPTIMIZATION
MODEL
If we adopt a fractile optimization model for the ob-
jective functions of MOSP, we can convert MOSP to
the following multiobjective programming problem,
where the decision maker specifies permissible prob-
ability levels ˆp
i
,i = 1, ··· , k in his/her subjectiveman-
ner (Sakawa et al., 2011).
[MOP6( ˆp)]
max
x∈X,h
i
∈[0,1],i=1,···,k
(h
1
,· · · ,h
k
) (19)
subject to
p
i
(x,h
i
) ≥ ˆp
i
,i = 1, · · · , k (20)
where ˆp = ( ˆp
1
,· · · , ˆp
k
) is a vector of permissible
probability levels. Since a distribution function T
i
(·)
is continuous and strictly increasing, the constraints
(20) can be transformed to the following form.
ˆp
i
≤ p
i
(x,h
i
)
⇔ µ
−1
˜
G
i
(h
i
) ≥ (d
1
i
x− L
−1
(h
i
)α
1
i
x)
+T
−1
i
( ˆp
i
) · (d
2
i
x− L
−1
(h
i
)α
2
i
x) (21)
Let us define the right-hand side of the inequality (21)
as follows.
f
i
(x, h
i
, ˆp
i
)
def
= (d
1
i
x− L
−1
(h
i
)α
1
i
x)
+T
−1
i
( ˆp
i
) · (d
2
i
x− L
−1
(h
i
)α
2
i
x)
(22)
Then, MOP6( ˆp) can be equivalently transformed into
the following form.
[MOP7( ˆp)]
max
x∈X,h
i
∈[0,1],i=1,···,k
(h
1
,· · · ,h
k
) (23)
subject to
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) ≥ h
i
,i = 1, ··· , k (24)
In MOP7( ˆp), let us pay attention to the inequal-
ities (24). f
i
(x, h
i
, ˆp
i
) is continuous and strictly in-
creasing with respect to h
i
for any x ∈ X. This
means that the left-hand-side of (24) is continuous
and strictly decreasing with respect to h
i
for any x ∈
X. Since the right-hand-side of (24) is continuous and
strictly increasing with respect to h
i
, the inequalities
(24) must always satisfy the active condition, that is,
µ
e
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
, i = 1, · · · , k at the optimal solu-
tion. From such a point of view, MOP7( ˆp) is equiva-
lently expressed as the following form.
[MOP8( ˆp)]
max
x∈X,h
i
∈[0,1],i=1,···,k
(µ
˜
G
1
( f
1
(x,h
1
, ˆp
1
)),· · · ,
µ
˜
G
k
( f
k
(x, h
k
, ˆp
k
))) (25)
subject to
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k (26)
In order to deal with MOP8( ˆp), the decision
maker must specify permissible probability levels ˆp
in advance. However, in general, the decision maker
seems to prefer not only the larger value of a permis-
sible probability level but also the larger value of the
corresponding membership functions µ
˜
G
i
(·). From
such a point of view, we consider the following mul-
tiobjective programming problem which can be re-
garded as a natural extension of MOP8( ˆp).
[MOP9]
max
x∈X,h
i
∈[0,1], ˆp
i
∈(0,1),i=1,···,k
(µ
˜
G
1
( f
1
(x, h
1
, ˆp
1
)),
·· · ,µ
˜
G
k
( f
k
(x, h
k
, ˆp
k
)), ˆp
1
,· · · , ˆp
k
)
subject to
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k (27)
It should be noted in MOP9 that permissible proba-
bility levels are not the fixed values but the decision
variables.
Considering the imprecise nature of the decision
maker’s judgment, we assume that the decision maker
InteractiveFuzzyDecisionMakingforMultiobjectiveFuzzyRandomLinearProgrammingProblems
323

has a fuzzy goal for each permissible probability
level. Such a fuzzy goal can be quantified by elic-
iting the corresponding membership function. Let us
denote a membership function of a permissible prob-
ability level ˆp
i
as µ
ˆp
i
( ˆp
i
). Then, MOP9 can be trans-
formed as the following multiobjective programming
problem.
[MOP10]
max
x∈X,h
i
∈[0,1], ˆp
i
∈(0,1),i=1,···,k
(µ
˜
G
1
( f
1
(x,h
1
, ˆp
1
)),
·· · ,µ
˜
G
k
( f
k
(x,h
k
, ˆp
k
)),µ
ˆp
1
( ˆp
1
),· · · , µ
ˆp
k
( ˆp
k
))
subject to
µ
˜
G
i
( f
i
(x, h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k (28)
In order to elicit the membership functions appro-
priately, we suggest the following procedures. First
of all, the decision maker sets the intervals P
i
=
[p
imin
, p
imax
],i = 1, · · · , k, where p
imin
is an unaccept-
able maximum value of ˆp
i
and p
imax
is a sufficiently
satisfactory minimum value of ˆp
i
. Throughout this
section, we make the following assumption.
Assumption 2.
µ
ˆp
i
( ˆp
i
),i = 1, ··· , k are strictly increasing and con-
tinuous with respect to ˆp
i
∈ P
i
, and µ
ˆp
i
(p
imin
) = 0,
µ
ˆp
i
(p
imax
) = 1.
Corresponding to the interval P
i
, the interval of h
i
,
which is defined as H
i
(P
i
) = [h
imin
,h
imax
], can be ob-
tained as follows. The maximum value h
imax
can be
obtained by solving the following problem.
min
x∈X,h
i
∈[0,1]
f
i
(x,h
i
, p
imin
) (29)
subject to h
i
= µ
˜
G
i
( f
i
(x,h
i
, p
imin
)) (30)
This is equivalent to the following problem.
h
imax
def
= max
x∈X,h
i
∈[0,1]
h
i
(31)
subject to
µ
−1
˜
G
i
(h
i
) = (d
1
i
x− L
−1
(h
i
)α
1
i
x)
+T
−1
i
(p
imin
) · (d
2
i
x− L
−1
(h
i
)α
2
i
x)
(32)
The optimal solution x
∗
,h
∗
i
,i = 1, ··· , k of the above
problem can be obtained by combined use of the bi-
section method with respect to h
i
∈ [0,1] and the first-
phase of the two-phase simplex method of linear pro-
gramming. In order to obtain h
imin
, we first solve the
following k linear programming problems.
min
x∈X,h
i
∈[0,1]
f
i
(x,h
i
, p
imax
) (33)
subject to h
i
= µ
˜
G
i
( f
i
(x,h
i
, p
imax
)) (34)
Let (x
∗
i
,h
∗
i
),i = 1, · ·· , k be the aboveoptimal solution.
Using the optimal solutions (x
∗
i
,h
∗
i
),i = 1, ··· ,k, h
imin
can be obtained as follows.
h
imin
def
= min
ℓ=1,···,k,ℓ6=i
µ
˜
G
i
( f
i
(x
∗
ℓ
,h
∗
ℓ
, p
imax
)) (35)
It should be noted here that, µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) is
strictly decreasing with respect to ˆp
i
. If the decision
maker adopts the fuzzy decision (Sakawa, 1993) to
integrate µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) and µ
ˆp
i
( ˆp
i
), MOP10 can be
transformed into the following form.
[MOP11]
max
x∈X, ˆp
i
∈P
i
,h
i
∈H
i
(P
i
),i=1,···,k

µ
D
G
1
(x,h
1
, ˆp
1
),· · · , µ
D
G
k
(x,h
k
, ˆp
k
)

(36)
subject to
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k (37)
where
µ
D
G
i
(x,h
i
, ˆp
i
)
def
= min{µ
ˆp
i
( ˆp
i
),µ
˜
G
i
( f
i
(x, h
i
, ˆp
i
))}
(38)
In order to deal with MOP11, we introduce a D
G
-
Pareto optimal solution concept.
Definition 2.
x
∗
∈ X, ˆp
∗
i
∈ P
i
,h
∗
i
∈ H
i
(P
i
),i = 1, · · · , k is said to
be a D
G
-Pareto optimal solution to MOP11, if and
only if there does not exist another x ∈ X, ˆp
i
∈
P
i
,h
i
∈ H
i
(P
i
),i = 1,· · · , k such that µ
D
G
i
(x, h
i
, ˆp
i
) ≥
µ
D
G
i
(x
∗
,h
∗
i
, ˆp
∗
i
),i = 1, ··· , k with strict inequality
holding for at least one i, where µ
˜
G
i
( f
i
(x
∗
,h
∗
i
, ˆp
∗
i
)) =
h
∗
i
, µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k.
For generating a candidate of a satisfactory solu-
tion which is also D
G
-Pareto optimal, the decision
maker is asked to specify the reference membership
values (Sakawa, 1993). Once the reference member-
ship values ˆµ = (ˆµ
1
,· · · , ˆµ
k
) are specified, the corre-
sponding D
G
-Pareto optimal solution is obtained by
solving the following minmax problem.
[MINMAX3(ˆµ)]
min
x∈X, ˆp
i
∈P
i
,h
i
∈H
i
(P
i
),i=1,···,k,λ∈Λ
λ (39)
subject to
ˆµ
i
− µ
ˆp
i
( ˆp
i
) ≤ λ,i = 1, ··· , k, (40)
ˆµ
i
− h
i
≤ λ,i = 1, ··· , k, (41)
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k. (42)
where
Λ = [ max
i=1,···,k
ˆµ
i
− 1, min
i=1,···,k
ˆµ
i
]. (43)
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
324

In the constraints (41) and (42), it holds that
h
i
= µ
˜
G
i
( f
i
(x, h
i
, ˆp
i
)) ≥ ˆµ
i
− λ,
⇔ µ
−1
˜
G
i
(h
i
) = f
i
(x,h
i
, ˆp
i
) ≤ µ
−1
˜
G
i
(ˆµ
i
− λ)
⇔ µ
−1
˜
G
i
(h
i
) = (d
1
i
x− L
−1
(h
i
)α
1
i
x)
+T
−1
i
( ˆp
i
) · (d
2
i
x− L
−1
(h
i
)α
2
i
x)
≤ µ
−1
˜
G
i
(ˆµ
i
− λ). (44)
In the right hand side of (44), because of L
−1
(h
i
) ≤
L
−1
(ˆµ
i
− λ) and α
1
i
x+ T
−1
i
( ˆp
i
)α
2
i
x > 0, it holds that
(d
1
i
x− L
−1
(h
i
)α
1
i
x)
+T
−1
i
( ˆp
i
) · (d
2
i
x− L
−1
(h
i
)α
2
i
x)
≥ (d
1
i
x+ T
−1
i
( ˆp
i
)d
2
i
x)
−L
−1
(ˆµ
i
− λ)

α
1
i
x+ T
−1
i
( ˆp
i
)α
2
i
x

. (45)
Using (44) and (45), it holds that
µ
−1
˜
G
i
(ˆµ
i
− λ)
≥ (d
1
i
x+ T
−1
i
( ˆp
i
)d
2
i
x)
−L
−1
(ˆµ
i
− λ)

α
1
i
x+ T
−1
i
( ˆp
i
)α
2
i
x

= (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
( ˆp
i
) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x). (46)
Moreover, because of ˆp
i
≥ µ
−1
ˆp
i
(ˆµ
i
− λ), (46) can be
transformed into the following form.
T
i
µ
−1
˜
G
i
(ˆµ
i
− λ) − (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x
!
≥ ˆp
i
≥ µ
−1
ˆp
i
(ˆµ
i
− λ),
⇔ µ
−1
˜
G
i
(ˆµ
i
− λ) ≥ (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x)
(47)
Therefore, MINMAX3(ˆµ) can be reduced to the fol-
lowing minmax problem.
[MINMAX4(ˆµ)]
min
x∈X,λ∈Λ
λ (48)
subject to
µ
−1
˜
G
i
(ˆµ
i
− λ) ≥ (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
+T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x),
i = 1,·· · ,k (49)
It should be noted here that MINMAX4(ˆµ) is equiva-
lent to MINMAX2(ˆµ). The relationships between the
optimal solution (x
∗
,λ
∗
) of MINMAX4(ˆµ) and D
G
-
Pareto optimal solutions can be characterized by the
following theorem.
Theorem 2.
(1) If x
∗
∈ X, λ
∗
∈ Λ is a unique optimal solution
of MINMAX4(ˆµ), then x
∗
∈ X, ˆp
∗
i
= µ
−1
ˆp
i
(ˆµ
i
− λ
∗
) ∈
P
i
,h
∗
i
= ˆµ
i
− λ
∗
∈ H
i
(P
i
),i = 1, ··· , k is a D
G
-Pareto
optimal solution.
(2) If x
∗
∈ X, ˆp
∗
i
∈ P
i
,h
∗
i
∈ H
i
(P
i
),i = 1, ··· , k is a
D
G
-Pareto optimal solution, then x
∗
∈ X, λ
∗
= ˆµ
i
−
µ
ˆp
i
( ˆp
∗
i
) = ˆµ
i
−µ
˜
G
i
( f
i
(x
∗
,h
∗
i
, ˆp
∗
i
)),i = 1, ··· ,k is an op-
timal solution of MINMAX4(ˆµ) for some reference
membership values ˆµ = (ˆµ
1
,· · · , ˆµ
k
).
(Proof)
(1) From (49), it holds that
ˆµ
i
− λ
∗
≤ µ
˜
G
i
( f
i
(x
∗
, ˆµ
i
− λ
∗
,µ
−1
ˆp
i
(ˆµ
i
− λ
∗
))),
and it is obvious that ˆµ
i
− λ
∗
= µ
ˆp
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)).
Assume that x
∗
∈ X, ˆµ
i
− λ
∗
∈ H
i
(P
i
),µ
−1
ˆp
i
(ˆµ
i
− λ
∗
) ∈
P
i
,i = 1, · ·· , k is not a D
G
-Pareto optimal solution.
Then, there exist x ∈ X, ˆp
i
∈ P
i
,h
i
∈ H
i
(P
i
),i = 1, ··· , k
such that
µ
D
G
i
(x,h
i
, ˆp
i
) = min{µ
ˆp
i
( ˆp
i
),µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
))}
≥ µ
D
G
i
(x
∗
, ˆµ
i
− λ
∗
,µ
−1
ˆp
i
(ˆµ
i
− λ
∗
))
= ˆµ
i
− λ
∗
,i = 1, ··· , k,
with strict inequality holding for at least one i, and
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k. Then it holds that
µ
ˆp
i
( ˆp
i
) ≥ ˆµ
i
− λ
∗
,i = 1, ··· , k, (50)
µ
˜
G
i
( f
i
(x, h
i
, ˆp
i
)) ≥ ˆµ
i
− λ
∗
,i = 1, ··· , k. (51)
From Assumption 2 and (22), (50) and (51) can be
transformed as follows.
ˆp
i
≥ µ
−1
ˆp
i
(ˆµ
i
− λ
∗
),i = 1, ··· , k
ˆp
i
≤ T
i
µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) − (d
1
i
x− L
−1
(h
i
)α
1
i
x)
d
2
i
x− L
−1
(h
i
)α
2
i
x
!
,
i = 1,·· · ,k
Because of L
−1
(h
i
) ≤ L
−1
(ˆµ
i
− λ
∗
),i = 1, · · · ,k, there
exists x ∈ X such that
µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) − (d
1
i
x− L
−1
(h
i
)α
1
i
x)
≥ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · (d
2
i
x− L
−1
(h
i
)α
2
i
x),
⇔ µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) ≥
(d
1
i
x+ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · d
2
i
x)
−L
−1
(h
i
)(α
1
i
x+ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · α
2
i
x),
⇔ µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) ≥ (d
1
i
x+ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · d
2
i
x)
−L
−1
(ˆµ
i
− λ
∗
)(α
1
i
x+ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · α
2
i
x)
i = 1, · ·· , k.
InteractiveFuzzyDecisionMakingforMultiobjectiveFuzzyRandomLinearProgrammingProblems
325

This contradicts the fact that x
∗
∈ X, λ
∗
∈ Λ is a
unique optimal solution to MINMAX4(ˆµ).
(2) Assume that x
∗
∈ X,λ
∗
∈ Λ is not an optimal solu-
tion to MINMAX4(ˆµ) for any reference membership
values ˆµ = (ˆµ
1
,· · · , ˆµ
k
) which satisfy the equalities
ˆµ
i
− λ
∗
= µ
ˆp
i
( ˆp
∗
i
) = µ
˜
G
i
( f
i
(x
∗
,h
∗
i
, ˆp
∗
i
)),
i = 1, ··· ,k. (52)
Then, there exists some x ∈ X, λ < λ
∗
such that
µ
−1
˜
G
i
(ˆµ
i
− λ) − (d
1
i
x− L
−1
(ˆµ
i
− λ)α
1
i
x)
≥ T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ)) · (d
2
i
x− L
−1
(ˆµ
i
− λ)α
2
i
x),
⇔ µ
˜
G
i
( f
i
(x, ˆµ
i
− λ, µ
−1
ˆp
i
(ˆµ
i
− λ)) ≥ ˆµ
i
− λ
i = 1, ··· ,k. (53)
Because of (52), (53) and ˆµ
i
−λ > ˆµ
i
−λ
∗
,i = 1, · · · , k,
the following inequalities hold.
µ
ˆp
i
( ˆp
i
) > µ
ˆp
i
( ˆp
∗
i
),i = 1, ··· , k
µ
˜
G
i
( f
i
(x,
ˆ
h
i
, ˆp
i
)) > µ
˜
G
i
( f
i
(x
∗
,h
∗
i
, ˆp
∗
i
)),
i = 1, ··· , k
where ˆp
i
= µ
−1
ˆp
i
(µ
i
− λ) ∈ P
i
,
ˆ
h
i
= ˆµ
i
− λ ∈ H
i
(P
i
),i =
1, ··· , k. This means that there exists some x ∈ X, ˆp
i
∈
P
i
,
ˆ
h
i
∈ H
i
(P
i
),i = 1,· · · , k such that µ
D
G
i
(x,
ˆ
h
i
, ˆp
i
) >
µ
D
G
i
(x
∗
,h
∗
i
, ˆp
∗
i
),i = 1, ··· , k. This contradicts the fact
that x
∗
∈ X, ˆp
∗
i
∈ P
i
,h
∗
i
∈ H
i
(P
i
),i = 1, ··· , k is a D
G
-
Pareto optimal solution.
5 AN INTERACTIVE
ALGORITHM
In this section, we propose an interactive algorithm
to obtain a satisfactory solution from among a D
G
-
Pareto optimal solution set. From Theorem 2, it
is not guaranteed that the optimal solution (x
∗
,λ
∗
)
of MINMAX4(ˆµ) is D
G
-Pareto optimal, if it is not
unique. In order to guarantee the D
G
-Pareto op-
timality, we first assume that k constraints (49) of
MINMAX4(ˆµ) are active at the optimal solution
(x
∗
,λ
∗
), i.e.,
µ
−1
˜
G
i
(ˆµ
i
− λ
∗
) − (d
1
i
x
∗
− L
−1
(ˆµ
i
− λ
∗
)α
1
i
x
∗
)
= T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
))
·(d
2
i
x
∗
− L
−1
(ˆµ
i
− λ
∗
)α
2
i
x
∗
),
i = 1,·· · ,k. (54)
If the j-th constraint of (49) is inactive, i.e.,
µ
−1
˜
G
j
(ˆµ
j
− λ
∗
) − (d
1
j
x
∗
− L
−1
(ˆµ
j
− λ
∗
)α
1
j
x
∗
)
> T
−1
j
(µ
−1
ˆp
j
(ˆµ
j
− λ
∗
))
·(d
2
j
x
∗
− L
−1
(ˆµ
j
− λ
∗
)α
2
j
x
∗
),
⇔ µ
−1
˜
G
j
(ˆµ
j
− λ
∗
) > f
j
(x
∗
, ˆµ
j
− λ
∗
,µ
−1
ˆp
j
(ˆµ
j
− λ
∗
)),
(55)
we can convert the inactive constraint (55) into the
active one by applying the bisection method for the
reference membership value ˆµ
j
∈ [λ
∗
,λ
∗
+ 1].
For the optimal solution (x
∗
,λ
∗
) of
MINMAX4(ˆµ), where the active conditions (54)
are satisfied, we solve the D
G
-Pareto optimality test
problem defined as follows.
[D
G
-Pareto Optimality Test Problem.]
max
x∈X,ε
i
≥0,i=1,···,k
w =
k
∑
i=1
ε
i
(56)
subject to
T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · (d
2
i
x− L
−1
(ˆµ
i
− λ
∗
)α
2
i
x)
+(d
1
i
x− L
−1
(ˆµ
i
− λ
∗
)α
1
i
x) + ε
i
= T
−1
i
(µ
−1
ˆp
i
(ˆµ
i
− λ
∗
)) · (d
2
i
x
∗
− L
−1
(ˆµ
i
− λ
∗
)α
2
i
x
∗
)
+(d
1
i
x
∗
− L
−1
(ˆµ
i
− λ
∗
)α
1
i
x
∗
),i = 1, ··· , k (57)
For the optimal solution of the above test problem, the
following theorem holds.
Theorem 3.
For the optimal solution ˇx,
ˇ
ε
i
,i = 1, ··· , k of the
test problem (56)-(57), if w = 0 (equivalently,
ˇ
ε
i
=
0, i = 1,· · · ,k), x
∗
∈ X, µ
−1
ˆp
i
(ˆµ
i
− λ
∗
) ∈ P
i
, ˆµ
i
− λ
∗
∈
H
i
(P
i
),i = 1, ··· , k is a D
G
-Pareto optimal solution.
Now, following the above discussions, we can
present the interactive algorithm in order to derive a
satisfactory solution from among a D
G
-Pareto optimal
solution set.
[An Interactive Algorithm.]
Step 1: The decision maker sets the membership
functions µ
˜
G
i
(y),i = 1, ··· , k for the fuzzy goals of the
objective functions in MOFRLP.
Step 2: The decision maker sets his/her membership
function µ
ˆp
i
( ˆp
i
).
Step 3: Set the initial reference membership values
as ˆµ
i
= 1, i = 1, ··· , k.
Step 4: Solve MINMAX4(ˆµ) by combined use of
the bisection method λ ∈ Λ and the first-phase of the
two-phase simplex method of linear programming,
and obtain the optimal solution (x
∗
,λ
∗
). For the op-
timal solution (x
∗
,λ
∗
), The corresponding D
G
-Pareto
optimality test problem (56)-(57) is formulated and
solved.
Step 5: If the decision maker is satisfied with
the current values of the D
G
-Pareto optimal solu-
tion µ
D
G
i
(x
∗
,h
∗
i
, ˆp
∗
i
),i = 1,··· ,k where ˆp
∗
i
= µ
−1
ˆp
i
(ˆµ
i
−
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
326

Table 1: The parameters for LR-type fuzzy random vari-
ables
e
c
ij
.
j 1 2 3 j 1 2 3
d
1
1j
2 1 3 d
2
1j
1.3 1.1 1.2
d
1
2j
-7 -7 -9 d
2
2j
1.1 1.2 1.1
α
1
1j
0.5 0.4 0.5 α
2
1j
0.05 0.04 0.05
α
1
2j
0.3 0.5 0.4 α
2
2j
0.05 0.04 0.05
β
1
1j
0.6 0.5 0.6 β
2
1j
0.06 0.05 0.06
β
1
2j
0.4 0.5 0.5 β
2
2j
0.06 0.06 0.05
λ
∗
),h
∗
i
= ˆµ
i
− λ
∗
,i = 1,· · · , k, then stop. Otherwise,
the decision maker updates his/her reference member-
ship values ˆµ
i
,i = 1, ··· , k, and return to Step 4.
6 A NUMERICAL EXAMPLE
We consider the following two-objective fuzzy ran-
dom linear programming problem to demonstrate the
feasibility of the proposed method under the hypo-
thetical decision maker.
[MOFRLP]
min
x∈X
e
c
1
x =
e
c
11
x
1
+
e
c
12
x
2
+
e
c
13
x
3
min
x∈X
e
c
2
x =
e
c
21
x
1
+
e
c
22
x
2
+
e
c
23
x
3
where X = {(x
1
,x
2
,x
3
) ≥ 0 | 2x
1
+ 6x
2
+ 3x
3
≤
150, 6x
1
+ 3x
2
+ 5x
3
≤ 175, 5x
1
+ 4x
2
+ 2x
3
≤
160, 2x
1
+ 2x
2
+ 3x
3
≥ 90}, and it is assumed that a
realization
e
c
ij
(ω) of an LR-type fuzzy random vari-
able
e
c
ij
is an LR fuzzy number whose membership
function is defined as follows.
µ
e
c
ij
(ω)
(s) =







L

d
1
ij
+
¯
t
i
(ω)d
2
ij
−s
α
1
ij
+
¯
t
i
(ω)α
2
ij

(s ≤
¯
d
ij
(ω)),
R

s−d
1
ij
+
¯
t
i
(ω)d
2
ij
β
1
ij
+
¯
t
i
(ω)β
2
ij

(s >
¯
d
ij
(ω)),
where L(t) = R(t) = max{0, 1 − t}, and the parame-
ters d
1
ij
,d
2
ij
, α
1
ij
,α
2
ij
,β
1
ij
,β
2
ij
are given in Table 1.
Moreover,
¯
t
i
,i = 1, 2 are Gaussian random variables
defined as
¯
t
i
∼ N(0,1).
In MOFRLP, let us assume that the hypothetical
decision maker sets the membership functions as fol-
lows (Step 1, 2).
µ
˜
G
1
( f
1
(x,h
1
, ˆp
1
)) =
96.42857− f
1
(x,h
1
, ˆp
1
)
96.42857− 75
µ
˜
G
2
( f
2
(x,h
2
, ˆp
2
)) =
(−285) − f
2
(x,h
2
, ˆp
2
)
(−285) − (−332.143)
µ
ˆp
1
( ˆp
1
) =
ˆp
1
− 0.401066
(0.714968− 0.401066)
µ
ˆp
2
( ˆp
2
) =
ˆp
2
− 0.213304
(0.812859− 0.213304)
Set the initial reference membership values as
(ˆµ
1
, ˆµ
2
) = (1, 1) (Step 3), and solve MINMAX4(ˆµ) by
combined use of the bisection method with respect to
λ and the first-phase of the two-phase simplex method
of linear programming to obtain the corresponding
D
G
-Pareto optimal solution (x
∗
,λ
∗
) (Step 4).
µ
˜
G
1
( f
1
(x
∗
,h
∗
1
, ˆp
∗
1
)) = µ
ˆp
1
( ˆp
∗
1
) = 0.564271
µ
˜
G
2
( f
2
(x
∗
,h
∗
2
, ˆp
∗
2
)) = µ
ˆp
2
( ˆp
∗
2
) = 0.564271
The hypothetical decision maker is not satisfied with
the current value of the D
G
-Pareto optimal solu-
tion (x
∗
,λ
∗
), and, in order to improve µ
D
G
2
(·) =
min{µ
˜
G
2
(·),µ
ˆp
2
(·)} at the expense of µ
D
G
1
(·) =
min{µ
˜
G
1
(·),µ
ˆp
1
(·)}, he/she updates his/her reference
membership values as (ˆµ
1
, ˆµ
2
) = (0.5,0.6) (Step 5).
Then, the corresponding D
G
-Pareto optimal solution
is obtained by solving MINMAX4(ˆµ) (Step 4).
µ
˜
G
1
( f
1
(x
∗
,h
∗
1
, ˆp
∗
1
)) = µ
ˆp
1
( ˆp
∗
1
) = 0.514421
µ
˜
G
2
( f
2
(x
∗
,h
∗
2
, ˆp
∗
2
)) = µ
ˆp
2
( ˆp
∗
2
) = 0.614421
For the current value of the D
G
-Pareto optimal solu-
tion, the hypothetical decision maker updates his/her
reference membership values (ˆµ
1
, ˆµ
2
) = (0.52, 0.59)
in order to improve µ
D
G
1
(·) at the expense of µ
D
G
2
(·)
slightly (Step 5). The corresponding D
G
-Pareto op-
timal solution is obtained by solving MINMAX4(ˆµ)
(Step 4).
µ
˜
G
1
( f
1
(x
∗
,h
∗
1
, ˆp
∗
1
)) = µ
ˆp
1
( ˆp
∗
1
) = 0.529412
µ
˜
G
2
( f
2
(x
∗
,h
∗
2
, ˆp
∗
2
)) = µ
ˆp
2
( ˆp
∗
2
) = 0.599412
Then, since the hypothetical decision maker is satis-
fied with the current value of the D
G
-Pareto optimal
solution, stop the interactive processes (Step 5). The
interactive processes under the hypothetical decision
maker are summarized in Table 2.
In order to compare our proposed approach with the
previous ones, let us obtain one of the Pareto optimal
solutions of MOP8( ˆp), which is defined in member-
ship space, i.e., µ
˜
G
i
( f
i
(x, h
i
, ˆp
i
)),i = 1, ··· , k. Simi-
lar to MINMAX3(ˆµ), we can formulate the following
minmax problem to obtain the Pareto optimal solution
of MOP8( ˆp).
[MINMAX5( ˆp, ˆµ)]
min
x∈X,h
i
∈[0,1],i=1,···,k,λ∈Λ
λ
subject to
ˆµ
i
− µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) ≤ λ,i = 1, ··· , k,
µ
˜
G
i
( f
i
(x,h
i
, ˆp
i
)) = h
i
,i = 1, ··· , k.
InteractiveFuzzyDecisionMakingforMultiobjectiveFuzzyRandomLinearProgrammingProblems
327

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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