# Interactive Multiobjective Fuzzy Random Linear Programming through Fractile Criteria

Authors: Hitoshi Yano; Masatoshi Sakawa
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

We propose an interactive fuzzy decision making method for multiobjective fuzzy random linear programming problems through fractile criteria optimization. In the proposed method, it is assumed that the decision maker has fuzzy goals for not only objective functions but also permissible probability levels in a fractile optimization model, 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 an extended Pareto optimal solution set through the interaction with the decision maker. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.


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## Body

## 1.   Introduction

In the real world decision making situations, we often have to make a decision under uncertainty. In order to deal with decision problems involving uncertainty, stochastic programming approaches [1–4] and fuzzy programming approaches [5–7] have been developed. Recently, mathematical programming problems with fuzzy random variables [8] have been proposed [9–11] whose concept includes both probabilistic uncertainty and fuzzy one simultaneously. Extensions to multiobjective fuzzy random linear programming problems (MOFRLP) have been done and interactive methods to obtain the satisfactory solution for the decision maker have been proposed [11–13]. In their methods, it is required in advance for the decision maker to specify permissible possibility levels in a probability maximization 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, a fuzzy approach to MOFRLP, in which the decision maker specifies the membership functions for the fuzzy goals of both objective functions and permissible probability levels has been proposed [14]. In the proposed method, it is assumed that the decision maker adopts the fuzzy decision [6] 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 can not be applied in the real-world decision situation.As one kind of the multiobjective programming problems in the real-world decision making problems, the agricultural resource management problems have been formulated and investigated to obtain satisfactory solution [15, 16]. Unfortunately, in the agricultural resource management problems, there are many kinds of uncertainty, such as future profits for crops and the various influence of the future weather. In order to deal with such uncertainty, Itoh et al. [17] formulated the agricultural resource management problem as a stochastic programming problem, and proposed a fuzzy approach to obtain the optimal cultivation areas for several kinds of vegetables. Toyonaga et al. [18] formulated the agricultural resource management problem as a fuzzy random programming problem, where the profit coefficients of the crops are defined as fuzzy random variables.Under these circumstances, in this paper, we propose an interactive fuzzy decision making method for MOFRLP to obtain the satisfactory solution from among an extended Pareto optimal solution set. Using a concept of a possibility measure [19], MOFRLP is transformed into multiobjective stochastic programming ones. Through a fractile optimization model, they are further transformed into usual multiobjective programming problems. An extended Pareto optimal concept is introduced and the minmax problem is formulated to obtain an extended Pareto optimal solution. An interactive algorithm is introduced to obtain the satisfactory solution from among an extended Pareto optimal solution set by solving the minmax problem on the basis of the linear programming technique. In order to illustrate the proposed method, a two-objective fuzzy random linear programming problem is formulated as a numerical example, and the interactive processes under the hypothetical decision maker are demonstrated.


## 2.   Multiobjective Fuzzy Random Linear Programming Problems

Throughout this paper, assuming that coefficients of the objective functions are fuzzy random variables, we focus on the following multiobjective fuzzy random linear programming problem (MOFRLP):[MOFRLP](1)minimizeC-~x=(c-~1x,…,c-~kx),subjecttox∈X=def{x∈Rn∣Ax≤b,x≥0},
where x=(x1,x2,…,xn)T is an n dimensional decision variable column vector, A is an (m×n) coefficient matrix, and b=(b1,…,bm)T is an m dimensional column vector. c-~i=(c-~i1,…,c-~in),i=1,…,k are coefficient vectors of objective function c-~ix, whose elements are fuzzy random variables [8, 11, 20], and the symbols “−” and “~” mean randomness and fuzziness, respectively.In order to deal with the objective functionsc-~ix,i=1,…,k as a special version of a fuzzy random variable, an LR-type fuzzy random variable has been proposed [11–13]. Under the occurrence of each elementary event ω, c-~ij(ω) is a realization of an LR-type fuzzy random variable c-~ij, which is an LR fuzzy number [19] whose membership function is defined as follows:
(2)μc-~ij(ω)(s)={L(d-ij(ω)-sα-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)=defmax{0,l(t)} is a real-valued continuous function from [0,∞) to [0,1], and l(t) is a strictly decreasing continuous function satisfying l(0)=1 [19]. Also, R(t)=defmax{0,r(t)} satisfies the same conditions. d-ij,α-ij,β-ij are random variables expressed by d-ij=dij1+t-idij2, α-ij=αij1+t-iαij2, and β-ij=βij1+t-iβij2. t-i is a random variable whose distribution function is denoted by Ti(·) which is strictly increasing and continuous, and dij1,dij2, αij1,αij2,βij1,βij2 are constants. These parameters are set by the specialist who formulates MOFRLP (1).Using a concept of a possibility measure [19], MOFRLP has been transformed into a multiobjective stochastic programming problem (MOSP) [11–13]. As shown in [11–13], the realizations c-~i(ω)x becomes an LR fuzzy number characterized by the following membership functions on the basis of the extension principle [19].
(3)μc-~i(ω)x(y)={L(d-i(ω)x-yα-i(ω)x)y≤d-i(ω)x,R(y-d-i(ω)xβ-i(ω)x)y>d-i(ω)x.
For the realizations c-~i(ω)x,i=1,…,k it is assumed that the decision maker has fuzzy goals G~i,i=1,…,k [6], whose membership functions μG~i(y), i=1,…,k are continuous and strictly decreasing for minimization problems. By using a concept of a possibility measure [19], a degree of possibility [11] that the objective function value c-~ix satisfies the fuzzy goal G~i is expressed as follows. 						(4)Πc-~ix(G~i)=defsupymin{μc-~ix(y),μG~i(y)},i=1,…,k.
Using a possibility measure, MOFRLP can be transformed into the following multiobjective stochastic programming problem (MOSP) [11–13]:[MOSP](5)maximizex∈X(Πc-~1x(G~1),…,Πc-~kx(G~k)).
Through a probability maximization model and a fractile maximization model, MOSP (5) has been transformed into usual multiobjective programming problems and interactive algorithms to obtain a satisfactory solution has been proposed [11–13]. In their methods, the decision maker must specify permissible probability levels or permissible possibility levels for the objective functions in advance. However, it seems to be very difficult to specify appropriate permissible levels because they have a great influence on the objective function values or distribution function values. In the following sections, by assuming that the decision maker has fuzzy goals for permissible probability levels, we propose an interactive fuzzy decision making method for MOFRLP to obtain a satisfactory solution.


## 3. Possibility-Based Fractile Optimization Model

For the objective functions of MOSP (5), if we adopt a fractile optimization model, we can transform MOSP into the following multiobjective programming problem (MOP), where the decision maker specifies permissible probability levels p^i,i=1,…,k in a subjective manner [11]:[MOP(p^)]
(6)maximizex∈X,hi∈[0,1],i=1,…,k(h1,…,hk),subjecttopi(x,hi)≥p^i,i=1,…,k,
where p^=(p^1,…,p^k) is a vector of permissible probability levels.Since a distribution functionTi(·) is continuous and strictly increasing, the constraints of MOP(p^) (6) can be transformed to the following form:
(7)p^i≤pi(x,hi)=Ti(μG~i-1(hi)-(di1x-L-1(hi)αi1x)di2x-L-1(hi)αi2x),⟺μG~i-1(hi)≥(di1x-L-1(hi)αi1x)+Ti-1(p^i)·(di2x-L-1(hi)αi2x).
Let us define the right-hand side of the inequality (7) as follows:
(8)fi(x,hi,p^i)=def(di1x-L-1(hi)αi1x)+Ti-1(p^i)·(di2x-L-1(hi)αi2x).
It is clear that fi(x,hi,p^i) is strictly increasing with respect to p^i. Then, MOP(p^) (6) can be equivalently transformed into the following form:[MOP(p^)]
(9)maximizex∈X,hi∈[0,1],i=1,…,k(h1,…,hk),subjecttoμG~i(fi(x,hi,p^i))≥hi,i=1,…,k.
In the inequality constraints of (9), fi(x,hi,p^i) is continuous and strictly increasing with respect to hi for any x∈X. This means that the left-hand-side of the inequality constraints of (9) is continuous and strictly decreasing with respect to hi for any x∈X. Since the right-hand-side of the inequality constraints of (9) is continuous and strictly increasing with respect to hi, the inequality constraints of (9) must always satisfy the active condition, that is, μG~i(fi(x,hi,p^i))=hi, i=1,…,k at the optimal solution. From such a point of view, MOP(p^) (9) is equivalently expressed as the following form:[MOP(p^)]
(10)maximizex∈X,hi∈[0,1],i=1,…,k(μG~1(f1(x,h1,p^1)),…,μG~k(fk(x,hk,p^k))),subjecttoμG~i(fi(x,hi,p^i))=hi,i=1,…,k.
In MOP (10), we can define Pareto optimal solutions, which depend on the permissible probability levels p^i,i=1,…,k.Definition 1.
For the given permissible probability levelsp^i,i=1,…,k, x*∈X,hi*∈[0,1],i=1,…,k is said to be a F-Pareto optimal solution to MOP (10), if and only if there does not exist another x∈X,hi∈[0,1],i=1,…,k such that μDGi(x,hi,p^i)≥μDGi(x*,hi*,p^i),i=1,…,k with strict inequality holding for at least one i, where μG~i(fi(x*,hi*,p^i))=hi*, μG~i(fi(x,hi,p^i))=hi,i=1,…,k.Katagiri et al. [21] have proposed an interactive decision making method for MOP (10) to obtain a satisfactory solution from among an F-Pareto optimal solution set. However, in their proposed method, the decision maker must specify permissible probability levels p^ in advance. From the inequalities in MOP (9), it is obvious that the less values of permissible probability levels p^i,i=1,…,k give the larger values of membership functions μG~i(y), i=1,…,k. Figure 1 shows the image of F-Pareto optimal solutions for two kinds of permissible probability levels p^1=(p^11,p^21) and p^2=(p^12,p^22), where p^i1≤p^i2,i=1,2. From Figure 1, there exists an infinite number of Pareto optimal solution sets corresponding to permissible probability levels. In the interactive decision making method proposed by Katagiri et al. [21], it is assumed that the decision maker can specify appropriate permissible probability levels. However, because of a conflict between permissible probability levels p^i,i=1,…,k and the corresponding values of the membership functions μG~i(y), i=1,…,k, it seems to be difficult for the decision maker to specify permissible probability levels appropriately. In general, the decision maker seems to prefer not only the larger value of a permissible probability level but also the larger value of the corresponding membership functions μG~i(·). From such a point of view, we consider the following multiobjective programming problem which can be regarded as a natural extension of MOP(p^):Figure 1
F-Pareto optimal solution sets forp^1≤p^2.[MOP](11)maximizex∈X,hi∈[0,1],p^i∈(0,1),i=1,…,k(μG~1(f1(x,h1,p^1)),…,μG~k(fk(x,hk,p^k)),p^1,…,p^k),subjecttoμG~i(fi(x,hi,p^i))=hi,i=1,…,k.
It should be noted in MOP that permissible probability 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 has a fuzzy goal for each permissible probability level. Such a fuzzy goal can be quantified by eliciting the corresponding membership function. Let us denote a membership function of a permissible probability levelp^i as μp^i(p^i). Then, MOP can be transformed as the following multiobjective programming problem:[MOP](12)maximizex∈X,hi∈[0,1],p^i∈(0,1),i=1,…,k(μG~1(f1(x,h1,p^1)),…,μG~k(fk(x,hk,p^k)),μp^1(p^1),…,μp^k(p^k)),subjecttoμG~i(fi(x,hi,p^i))=hi,i=1,…,k.
In order to elicit the membership functions appropriately, we suggest the following procedures. First of all, the decision maker sets the intervals Pi=[pimin,pimax],i=1,…,k, where pimin is an unacceptable maximum value of p^i and pimax is a sufficiently satisfactory minimum value of p^i. Throughout this section, we make the following assumption.Assumption 1.
μ
p
^
i
(
p
^
i
)
,
i
=
1
,
…
,
k are strictly increasing and continuous with respect to p^i∈Pi, and μp^i(pimin)=0, μp^i(pimax)=1.
Corresponding to the intervalPi, the interval of hi, which is defined as Hi(Pi)=[himin,himax], can be obtained as follows. The maximum value himax can be obtained by solving the following problem:
(13)minimizex∈X,hi∈[0,1]fi(x,hi,pimin),subjecttohi=μG~i(fi(x,hi,pimin)).
This is equivalent to the following problem:
(14)himax=defmaximizex∈X,hi∈[0,1]hi,subjecttoμG~i-1(hi)=(di1x-L-1(hi)αi1x)+Ti-1(pimin)·(di2x-L-1(hi)αi2x).
The optimal solution x*,hi*,i=1,…,k of the above problem can be obtained by combined use of the bisection method with respect to hi∈[0,1] and the first-phase of the two-phase simplex method of linear programming. In order to obtain himin, we first solve the following k linear programming problems:
(15)minimizex∈X,hi∈[0,1]fi(x,hi,pimax),subjecttohi=μG~i(fi(x,hi,pimax)).
Let (xi*,hi*),i=1,…,k be the above optimal solution. Using the optimal solutions (xi*,hi*),i=1,…,k, himin can be obtained as follows:
(16)himin=defminℓ=1,…,k,ℓ≠iμG~i(fi(xℓ*,hℓ*,pimax)).
It should be noted here that, μG~i(fi(x,hi,p^i)) is strictly decreasing with respect to p^i. If the decision maker adopts the fuzzy decision [6] to integrate μG~i(fi(x,hi,p^i)) and μp^i(p^i), MOP can be transformed into the following form:
[MOP](17)maximizex∈X,p^i∈Pi,hi∈Hi(Pi),i=1,…,k(μDG1(x,h1,p^1),…,μDGk(x,hk,p^k)),subjecttoμG~i(fi(x,hi,p^i))=hi,i=1,…,k,
where
(18)μDGi(x,hi,p^i)=defmin{μp^i(p^i),μG~i(fi(x,hi,p^i))}.
In order to deal with MOP (17), we introduce the following concept of an MPF-Pareto optimal solution, which can be viewed as an M-Pareto optimal solution through possibility-based fractile models.Definition 2.
x
*
∈
X
,
p
^
i
*
∈
P
i
,
h
i
*
∈
H
i
(
P
i
)
,
i
=
1
,
…
,
k is said to be an MPF-Pareto optimal solution to MOP (17), if and only if there does not exist another x∈X,p^i∈Pi,hi∈Hi(Pi),i=1,…,k such that μDGi(x,hi,p^i)≥μDGi(x*,hi*,p^i*),i=1,…,k with strict inequality holding for at least one i, where μG~i(fi(x*,hi*,p^i*))=hi*, μG~i(fi(x,hi,p^i))=hi,i=1,…,k.
For generating a candidate of a satisfactory solution which is alsoMPF-Pareto optimal, the decision maker is asked to specify the reference membership values [6]. Once the reference membership values μ^=(μ^1,…,μ^k) are specified, the corresponding MPF-Pareto optimal solution is obtained by solving the following minmax problem:
[MINMAX(μ^)]
(19)minimizex∈X,p^i∈Pi,hi∈Hi(Pi),i=1,…,k,λ∈Λλ,subjecttoμ^i-μp^i(p^i)≤λ,i=1,…,k,μ^i-hi≤λ,i=1,…,k,μG~i(fi(x,hi,p^i))=hi,i=1,…,k,
where
(20)Λ=[maxi=1,…,kμ^i-1,mini=1,…,kμ^i].
In the constraints of (19), it holds that
(21)hi=μG~i(fi(x,hi,p^i))≥μ^i-λ,⟺μG~i-1(hi)=fi(x,hi,p^i)≤μG~i-1(μ^i-λ)⟺μG~i-1(hi)=(di1x-L-1(hi)αi1x)+Ti-1(p^i)·(di2x-L-1(hi)αi2x)≤μG~i-1(μ^i-λ).
In the right hand side of (21), because of L-1(hi)≤L-1(μ^i-λ) and αi1x+Ti-1(p^i)αi2x>0, it holds that
(22)(di1x-L-1(hi)αi1x)+Ti-1(p^i)·(di2x-L-1(hi)αi2x)=(di1x+Ti-1(p^i)di2x)-L-1(hi)(αi1x+Ti-1(p^i)αi2x)≥(di1x+Ti-1(p^i)di2x)-L-1(μ^i-λ)×(αi1x+Ti-1(p^i)αi2x).
Using (21) and (22), it holds that
(23)μG~i-1(μ^i-λ)≥(di1x+Ti-1(p^i)di2x)-L-1(μ^i-λ)(αi1x+Ti-1(p^i)αi2x)=(di1x-L-1(μ^i-λ)αi1x)+Ti-1(p^i)·(di2x-L-1(μ^i-λ)αi2x).
From the inequality μ^i-μp^i(p^i)≤λ in (19) and Assumption 1, it holds that p^i≥μp^i-1(μ^i-λ). Therefore, (23) can be transformed into the following form:
(24)Ti(μG~i-1(μ^i-λ)-(di1x-L-1(μ^i-λ)αi1x)di2x-L-1(μ^i-λ)αi2x)≥p^i≥μp^i-1(μ^i-λ),⟺μG~i-1(μ^i-λ)≥(di1x-L-1(μ^i-λ)αi1x)+Ti-1(μp^i-1(μ^i-λ))·(di2x-L-1(μ^i-λ)αi2x).
Therefore, MINMAX (19) can be reduced to the following minmax problem:
[MINMAX(μ^)]
(25)minimizex∈X,λ∈ΛλsubjecttoμG~i-1(μ^i-λ)≥(di1x-L-1(μ^i-λ)αi1x)+Ti-1(μp^i-1(μ^i-λ))·(di2x-L-1(μ^i-λ)αi2x),i=1,…,k.
The relationships between the optimal solution (x*,λ*) of MINMAX(μ^) (25) and MPF-Pareto optimal solutions can be characterized by the following theorem.Theorem 3.
Ifx*∈X,λ*∈Λ is a unique optimal solution of MINMAX(μ^) (25), then x*,p^i*=μp^i-1(μ^i-λ*),hi*=μ^i-λ*,i=1,…,k is an MPF-Pareto optimal solution.
Ifx*∈X,p^i*∈Pi,hi*∈Hi(Pi),i=1,…,k is an MPF-Pareto optimal solution, then x*,λ*=μ^i-μp^i(p^i*)=μ^i-μG~i(fi(x*,hi*,p^i*)),i=1,…,k is an optimal solution of MINMAX(μ^) (25) for some reference membership values μ^=(μ^1,…,μ^k).Proof.
(1)  From the inequality constraints of (25), it holds that
(26)μ^i-λ*≤μG~i(fi(x*,μ^i-λ*,μp^i-1(μ^i-λ*))),
and it is obvious that μ^i-λ*=μp^i(μp^i-1(μ^i-λ*)). Assume that x*∈X,μ^i-λ*∈Hi(Pi),μp^i-1(μ^i-λ*)∈Pi,i=1,…,k is not an MPF-Pareto optimal solution. Then, there exist x∈X,p^i∈Pi,hi∈Hi(Pi),i=1,…,k such that
(27)μDGi(x,hi,p^i)=min{μp^i(p^i),μG~i(fi(x,hi,p^i))}≥μDGi(x*,μ^i-λ*,μp^i-1(μ^i-λ*))=μ^i-λ*,i=1,…,k,
with strict inequality holding for at least one i, and μG~i(fi(x,hi,p^i))=hi,i=1,…,k. Then it holds that
(28)μp^i(p^i)≥μ^i-λ*,i=1,…,k,μG~i(fi(x,hi,p^i))≥μ^i-λ*,i=1,…,k.
From Assumption 1 and (8),  can be transformed as follows:
(29)p^i≥μp^i-1(μ^i-λ*),i=1,…,k,p^i≤Ti(μG~i-1(μ^i-λ*)-(di1x-L-1(hi)αi1x)di2x-L-1(hi)αi2x),i=1,…,k.
Because of L-1(hi)≤L-1(μ^i-λ*),i=1,…,k, there exists x∈X such that
(30)μG~i-1(μ^i-λ*)-(di1x-L-1(hi)αi1x)≥Ti-1(μp^i-1(μ^i-λ*))·(di2x-L-1(hi)αi2x),⟺μG~i-1(μ^i-λ*)≥(di1x+Ti-1(μp^i-1(μ^i-λ*))·di2x)-L-1(hi)(αi1x+Ti-1(μp^i-1(μ^i-λ*))·αi2x),⟺μG~i-1(μ^i-λ*)≥(di1x+Ti-1(μp^i-1(μ^i-λ*))·di2x)-L-1(μ^i-λ*)(αi1x+Ti-1(μp^i-1(μ^i-λ*))·αi2x),i=1,…,k.
This contradicts the fact that x*∈X,λ*∈Λ is a unique optimal solution to MINMAX(μ^) (25).
(2) Assume thatx*∈X,λ*∈Λ is not an optimal solution to MINMAX(μ^) (25) for any reference membership values μ^=(μ^1,…,μ^k) which satisfy the equalities
(31)μ^i-λ*=μp^i(p^i*)=μG~i(fi(x*,hi*,p^i*)),i=1,…,k.
Then, there exists some x∈X,λ<λ* such that
(32)μG~i-1(μ^i-λ)-(di1x-L-1(μ^i-λ)αi1x)≥Ti-1(μp^i-1(μ^i-λ))·(di2x-L-1(μ^i-λ)αi2x),⟺μG~i(fi(x,μ^i-λ,μp^i-1(μ^i-λ)))≥μ^i-λ,i=1,…,k,
Because of (31), (32) and μ^i-λ>μ^i-λ*,i=1,…,k, the following inequalities hold:
(33)μp^i(p^i)>μp^i(p^i*),i=1,…,k,μG~i(fi(x,h^i,p^i))>μG~i(fi(x*,hi*,p^i*)),i=1,…,k,
where p^i=μp^i-1(μi-λ)∈Pi, h^i=μ^i-λ∈Hi(Pi),i=1,…,k. This means that there exists some x∈X,p^i∈Pi,h^i∈Hi(Pi),i=1,…,k such that μDGi(x,h^i,p^i)>μDGi(x*,hi*,p^i*),i=1,…,k. This contradicts the fact that x*∈X,p^i*∈Pi,hi*∈Hi(Pi),i=1,…,k is an MPF-Pareto optimal solution.


## 4.   Pareto Optimality Test

From Theorem3, it is not guaranteed that the optimal solution (x*,λ*) of MINMAX(μ^) (25) is MPF-Pareto optimal, if it is not unique. In order to guarantee the MPF-Pareto optimality, we first assume that k constraints of MINMAX(μ^) (25) are active at the optimal solution (x*,λ*), that is,
(34)μG~i-1(μ^i-λ*)-(di1x*-L-1(μ^i-λ*)αi1x*)=Ti-1(μp^i-1(μ^i-λ*))·(di2x*-L-1(μ^i-λ*)αi2x*),i=1,…,k.
If the jth constraint of MINMAX(μ^) (25) is inactive, that is,
(35)μG~j-1(μ^j-λ*)-(dj1x*-L-1(μ^j-λ*)αj1x*)>Tj-1(μp^j-1(μ^j-λ*))·(dj2x*-L-1(μ^j-λ*)αj2x*),⟺μG~j-1(μ^j-λ*)>fj(x*,μ^j-λ*,μp^j-1(μ^j-λ*)),
we can convert the inactive constraint (35) into the active one by applying the bisection method for the reference membership value μ^j∈[λ*,λ*+1].For the optimal solution(x*,λ*) of MINMAX(μ^) (25), where the active conditions (34) are satisfied, we solve the MPF-Pareto optimality test problem defined as follows:[
M
P
F-Pareto optimality test problem](36)maximizex∈X,ϵi≥0,i=1,…,kw=∑i=1k‍ϵi,subjecttoTi-1(μp^i-1(μ^i-λ*))·(di2x-L-1(μ^i-λ*)αi2x)+(di1x-L-1(μ^i-λ*)αi1x),+ϵiTi-1(μp^i-1(μ^i-λ*))·(di2x*-L-1(μ^i-λ*)αi2x*)+(di1x*-L-1(μ^i-λ*)αi1x*),i=1,…,k.
For the optimal solution of the above test problem, the following theorem holds.Theorem 4.
For the optimal solutionxˇ,ϵˇi,i=1,…,k of the test problem (36), if w=0 (equivalently, ϵˇi=0,i=1,…,k), x*∈X,μp^i-1(μ^i-λ*)∈Pi,μ^i-λ*∈Hi(Pi),i=1,…,k is an MPF-Pareto optimal solution.Proof.
From the active conditions (34), it holds that
(37)μ^i-λ*=μG~i(fi(x*,μ^i-λ*,μp^i-1(μ^i-λ*))),
and it is obvious that μ^i-λ*=μp^i(μp^i-1(μ^i-λ*)),i=1,…,k. Assume that x*∈X,μp^i-1(μ^i-λ*)∈Pi,μ^i-λ*∈Hi(Pi),i=1,…,k is not an MPF-Pareto optimal solution. Then, there exists some x∈X,p^i∈Pi,hi∈Hi(Pi),i=1,…,k such that
(38)μDGi(x,hi,p^i)=min{μp^i(p^i),μG~i(fi(x,hi,p^i))}≥μDGi(x*,μ^i-λ*,μp^i-1(μ^i-λ*))=μ^i-λ*,i=1,…,k,
with strict inequality holding for at least one i, and μG~i(fi(x,hi,p^i))=hi,i=1,…,k. This means that the following inequalities hold:
(39)μp^i(p^i)≥μ^i-λ*,i=1,…,k,μG~i(fi(x,hi,p^i))≥μ^i-λ*,i=1,…,k.
Because of Assumption 1, (8), and L-1(hi)≤L-1(μ^i-λ*),  can be transformed into the following forms
(40)p^i≥μp^i-1(μ^i-λ*),i=1,…,k,p^i≤Ti(μG~i-1(μ^i-λ*)-(di1x-L-1(hi)αi1x)di2x-L-1(hi)αi2x)≤Ti(μG~i-1(μ^i-λ*)-(di1x-L-1(μ^i-λ*)αi1x)di2x-L-1(μ^i-λ*)αi2x)i=1,…,k.
From (37), there exists some x∈X such that
(41)μG~i-1(μ^i-λ*)-(di1x-L-1(μ^i-λ*)αi1x)≥Ti-1(μp^i-1(μ^i-λ*))·(di2x-L-1(μ^i-λ*)αi2x),⟺Ti-1(μp^i-1(μ^i-λ*))·(di2x*-L-1(μ^i-λ*)αi2x*)+(di1x*-L-1(μ^i-λ*)αi1x*)≥Ti-1(μp^i-1(μ^i-λ*))·(di2x-L-1(μ^i-λ)αi2x)+(di1x-L-1(μ^i-λ)αi1x),i=1,…,k,
with strict inequality holding for at least one i. This contradicts the fact that w=0.


## 5. Interactive Algorithm

Following the above discussions, we are now ready to present an interactive algorithm in order to derive a satisfactory solution from among anMPF-Pareto optimal solution set.[Interactive algorithm].Step 1.
Calculateminx∈XE[d-i]x,i=1,…,k, where d-i(ω)x are central values of LR fuzzy numbers c-~i(ω)x. Let us denote xi as the optimal solution of the optimization problem minx∈XE[d-i]x. Then, define fimin=defE[d-i]xi, and fimax=defmaxj=1,…,k,j≠iE[d-i]xj. The decision maker determines the membership functions μG~i(y),i=1,…,k in his/her subjective manner, where μG~i(fimin)=1,μG~i(fimax)=0.Step 2.
The decision maker specifies the intervalPi=[pimin,pimax],i=1,…,k, where pimin is an unacceptable maximum value of p^i and pimax is a sufficiently satisfactory minimum value of p^i. For the interval Pi, the decision maker determines the membership function μp^i(p^i) according to Assumption 1.Step 3.
Set the initial reference membership values asμ^i=1,i=1,…,k.Step 4.
Solve MINMAX(μ^) 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 optimal solution (x*,λ*), the corresponding MPF-Pareto optimality test problem (36) is formulated and solved.Step 5.
If the decision maker is satisfied with the current values of theMPF-Pareto optimal solution μDGi(x*,hi*,p^i*),i=1,…,k, where p^i*=μp^i-1(μ^i-λ*),hi*=μ^i-λ*,i=1,…,k, then stop. Otherwise, the decision maker updates the reference membership values μ^i,i=1,…,k, and returns to Step 4.


## 6.  Numerical Example

In order to demonstrate the feasibility of the proposed method, consider the following two-objective fuzzy random linear programming problem [11, 12]:
(42)minimizec-~1x=c-~11x1+c-~12x2+c-~13x3,minimizec-~2x=c-~21x1+c-~22x2+c-~23x3,subjectto2x1+6x2+3x3≤150,6x1+3x2+5x3≤175,5x1+4x2+2x3≤160,2x1+2x2+3x3≥90,xi≥0,i=1,2,3.
It is assumed here that a realization c-~ij(ω) of an LR-type fuzzy random variable c-~ij is an LR fuzzy number whose membership function is defined as follows:
(43)μc-~ij(s)={L(dij1+t-i(ω)dij2-sαij1+t-i(ω)αij2)(s≤d-ij(ω)),R(s-dij1+t-i(ω)dij2βij1+t-i(ω)βij2)(s>d-ij(ω)),
where L(t)=R(t)=max{0,1-t}, and the parameters dij1,dij2,αij1,αij2,βij1,βij2 are given in Table 1 [11, 12].Table 1
Parameter values for LR-type fuzzy random variablesc¯~ij.
j
1
2
3
d
1
j
1
2
1
3
d
2
j
1
-
7
-
7
-
9
α
1
j
1
0.5
0.4
0.5
α
2
j
1
0.3
0.5
0.4
β
1
j
1
0.6
0.5
0.6
β
2
j
1
0.4
0.5
0.5
d
1
j
2
1.3
1.1
1.2
d
2
j
2
1.1
1.2
1.1
α
1
j
2
0.05
0.04
0.05
α
2
j
2
0.05
0.04
0.05
β
1
j
2
0.06
0.05
0.06
β
2
j
2
0.06
0.06
0.05Moreover,t-i,i=1,2 are Gaussian random variables defined as t-i~N(0,1).At Step1, calculate minx∈XE[d-i]x,i=1,2, and we obtain f1min=75,f1max=96.42857 and f2min=-332.143,f2max=-285. We assume that the decision maker adopts the linear membership functions μG~i(y),i=1,…,k defined as follows:
(44)μG~i(fi(x,hi,p^i))=fimax-fi(x,hi,p^i)fimax-fimin,i=1,2.
At Step 2, the decision maker specifies the intervals P1=[p1min,p1max]=[0.401066,0.714968] and P2=[p2min,p2max]=[0.213304,0.812859]. Then, we assume that the decision maker adopts the linear membership functions μp^i(p^i) defined as follows:
(45)μp^i(p^i)=p^i-pimin(pimax-pimin),i=1,2.
At Step 3, set the initial reference membership values as (μ^1,μ^2)=(1,1). Solve MINMAX(μ^) 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 MPF-Pareto optimal solution (x*,λ*) (Step 4). Consider
(46)μG~1(f1(x*,h1*,p^1*))=μp^1(p^1*)=0.564271,μG~2(f2(x*,h2*,p^2*))=μp^2(p^2*)=0.564271.
The hypothetical decision maker is not satisfied with the current value of the MPF-Pareto optimal solution (x*,λ*), and, in order to improve μDG2(·)=min{μG~2(·),μp^2(·)} at the expense of μDG1(·)=min{μG~1(·),μp^1(·)}, he/she updates his/her reference membership values as (μ^1,μ^2)=(0.5,0.6) (Step 5). Then, the corresponding MPF-Pareto optimal solution is obtained by solving MINMAX(μ^) (Step 4). Consider
(47)μG~1(f1(x*,h1*,p^1*))=μp^1(p^1*)=0.514421,μG~2(f2(x*,h2*,p^2*))=μp^2(p^2*)=0.614421.
For the current value of the MPF-Pareto optimal solution, the hypothetical decision maker updates his/her reference membership values (μ^1,μ^2)=(0.52,0.59) in order to improve μDG1(·) at the expense of μDG2(·) slightly (Step 5). The corresponding MPF-Pareto optimal solution is obtained by solving MINMAX(μ^) (Step 4). Consider
(48)μG~1(f1(x*,h1*,p^1*))=μp^1(p^1*)=0.529412,(49)μG~2(f2(x*,h2*,p^2*))=μp^2(p^2*)=0.599412.
Then, since the hypothetical decision maker is satisfied with the current value of the MPF-Pareto optimal solution, stop the interactive processes (Step 5). The interactive processes under the hypothetical decision maker are summarized in Table 2.Table 2
Interactive processes.
1
2
3
μ
^
1
1
0.5
0.52
μ
^
2
1
0.6
0.59
μ
D
G
1(x*,h1*,p^1*)
0.564271
0.514421
0.529412
μ
D
G
2(x*,h2*,p^2*)
0.564271
0.614421
0.599412
p
^
1
*
0.578193
0.562545
0.567250
p
^
2
*
0.551616
0.581684
0.572685
f
1(x*,h1*,p^1*)
84.3370
85.4053
85.0840
f
2(x*,h2*,p^2*)
-
311.601
-
313.966
-
313.258In order to compare our proposed approach with the previous ones, let us consider multiobjective programming problem MOP(p^) (10), which is formulated through a fractile optimization model. We obtain one of the Pareto optimal solutions of MOP(p^) (10), which are defined in membership space, that is, μG~i(fi(x,hi,p^i)),i=1,…,k. Similar to MINMAX(μ^) (19), we can formulate the following minmax problem to obtain the Pareto optimal solution of MOP(p^) (10), where p^i,i=1,2 are not decision variables but constant values:[MINMAX(p^,μ^)]
(50)minx∈X,hi∈[0,1],i=1,…,k,λ∈Λλ,subjecttoμ^i-μG~i(fi(x,hi,p^i))≤λ,i=1,…,k,μG~i(fi(x,hi,p^i))=hi,i=1,…,k.
In MINMAX(p^,μ^), we assume that the decision maker sets his/her permissible probability levels as p^1=p^2=0.75, and the reference membership values as μ^1=μ^2=1. Then, the corresponding Pareto optimal solution is obtained as follows:
(51)f1(x*,h1*,0.75)=94.0338,f2(x*,h2*,0.75)=-290.269,μG~i(fi(x*,hi*,0.75))=0.11176,p^i=0.75,i=1,2.
In our proposed algorithm, by solving MINMAX(μ^) (25) for the reference membership values μ^1=μ^2=1, the MPF-Pareto optimal solution is obtained as follows (see the first iteration of Table 2):
(52)f1(x*,h1*,p^1*)=84.3370,f2(x*,h2*,p^2*)=-311.601,μDGi(x*,hi*,p^i*)=0.564271,i=1,2,p^1*=0.578193,p^2*=0.551616.
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.  Conclusion

In this paper, we have proposed an interactive fuzzy decision making method for multiobjective fuzzy random linear programming problems to obtain a satisfactory solution from among an extended Pareto optimal solution 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. An extended Pareto optimal concept calledMPF-Pareto optimal was introduced. The satisfactory solution can be obtained by updating the reference membership values and solving the corresponding minmax problem based on the linear programming technique. An illustrative numerical example demonstrated the feasibility of the proposed method. In the near future, applications of the proposed method to the real-world decision making situations as well as extensions to more general cases will be required.



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*Source: 521080-2012-10-22.xml*