Interactive Multiobjective Fuzzy Random Linear Programming through Fractile Criteria

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.

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] minimize𝐂𝐱=𝐜̃₁𝐱,…,𝐜̃_𝑘𝐱,subjectto𝐱∈𝑋=^def𝐱∈𝑅^𝑛∣𝐴𝐱≤𝐛,𝐱≥𝟎, (1) where 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 is an 𝑛 dimensional decision variable column vector, 𝐴 is an (𝑚×𝑛) coefficient matrix, and 𝐛=(𝑏₁,…,𝑏_𝑚)^𝑇 is an 𝑚 dimensional column vector. 𝐜̃‌_𝑖=(𝑐̃_𝑖1,…,𝑐̃_𝑖𝑛),𝑖=1,…,𝑘 are coefficient vectors of objective function 𝐜̃‌_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 functions 𝐜̃‌_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, 𝑐̃‌_𝑖𝑗(𝜔) is a realization of an LR-type fuzzy random variable 𝑐̃‌_𝑖𝑗, which is an LR fuzzy number [19] whose membership function is defined as follows: 𝜇̃𝑐_{𝑖𝑗(𝜔)}(𝑠)=𝐿𝑑_𝑖𝑗(𝜔)−𝑠𝛼_𝑖𝑗(𝜔)𝑠≤𝑑_𝑖𝑗(𝜔),𝑅∀𝜔𝑠−𝑑_𝑖𝑗(𝜔)𝛽_𝑖𝑗(𝜔)𝑠>𝑑_𝑖𝑗(𝜔)∀𝜔, (2) 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=𝑑¹_𝑖𝑗+𝑡_𝑖𝑑²_𝑖𝑗, 𝛼_𝑖𝑗=𝛼¹_𝑖𝑗+𝑡_𝑖𝛼²_𝑖𝑗, and 𝛽_𝑖𝑗=𝛽¹_𝑖𝑗+𝑡_𝑖𝛽²_𝑖𝑗. 𝑡_𝑖 is a random variable whose distribution function is denoted by 𝑇_𝑖(⋅) which is strictly increasing and continuous, and 𝑑¹_𝑖𝑗,𝑑²_𝑖𝑗, 𝛼¹_𝑖𝑗,𝛼²_𝑖𝑗,𝛽¹_𝑖𝑗,𝛽²_𝑖𝑗 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 𝐜̃‌_𝑖(𝜔)𝐱 becomes an LR fuzzy number characterized by the following membership functions on the basis of the extension principle [19]. 𝜇̃𝐜_{𝑖(𝜔)𝐱}𝑦=𝐿𝐝_𝑖(𝜔)𝐱−𝑦𝜶_𝑖(𝜔)𝐱𝑦≤𝐝_𝑖(𝜔)𝑅𝐱,𝑦−𝐝_𝑖(𝜔)𝐱𝜷_𝑖(𝜔)𝐱𝑦>𝐝_𝑖(𝜔)𝐱. (3) For the realizations 𝐜̃‌_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 𝐜̃‌_𝑖𝐱 satisfies the fuzzy goal 𝐺_𝑖 is expressed as follows. Π̃𝐜_𝑖𝐱𝐺=_𝑖^defsup_𝑦min𝜇̃𝐜_𝑖𝐱𝑦𝐺,𝜇_𝑖𝑦,𝑖=1,…,𝑘. (4) Using a possibility measure, MOFRLP can be transformed into the following multiobjective stochastic programming problem (MOSP) [11–13]:

[MOSP] maximize_𝐱∈𝑋Π̃𝐜_1𝐱𝐺₁,…,Π̃𝐜_𝑘𝐱𝐺_𝑘. (5) 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)] maximize_{𝐱∈𝑋,ℎ𝑖∈[]0,1,𝑖=1,…,𝑘}ℎ₁,…,ℎ_𝑘,subjectto𝑝_𝑖𝐱,ℎ_𝑖≥𝑝_𝑖,𝑖=1,…,𝑘, (6) where 𝐩=(𝑝₁,…,𝑝_𝑘) is a vector of permissible probability levels.

Since a distribution function 𝑇_𝑖(⋅) is continuous and strictly increasing, the constraints of MOP(𝐩) (6) can be transformed to the following form: 𝑝_𝑖≤𝑝_𝑖𝐱,ℎ_𝑖=𝑇_𝑖𝜇⁻¹𝐺_𝑖ℎ_𝑖−𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱,⟺𝜇⁻¹𝐺_𝑖ℎ_𝑖≥𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱. (7) Let us define the right-hand side of the inequality (7) as follows: 𝑓_𝑖𝐱,ℎ_𝑖,𝑝=_𝑖^def𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱. (8) It is clear that 𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖) is strictly increasing with respect to 𝑝_𝑖. Then, MOP(𝐩) (6) can be equivalently transformed into the following form:

[MOP(𝐩)] maximize_{𝐱∈𝑋,ℎ𝑖∈[]0,1,𝑖=1,…,𝑘}ℎ₁,…,ℎ_𝑘,𝐺subjectto𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥ℎ_𝑖,𝑖=1,…,𝑘. (9) In the inequality constraints of (9), 𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖) is continuous and strictly increasing with respect to ℎ_𝑖 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 ℎ_𝑖 for any 𝐱∈𝑋. Since the right-hand-side of the inequality constraints of (9) is continuous and strictly increasing with respect to ℎ_𝑖, the inequality constraints of (9) must always satisfy the active condition, that is, 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖))=ℎ_𝑖, 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)] maximize_{𝐱∈𝑋,ℎ𝑖∈[]0,1,𝑖=1,…,𝑘}𝜇𝐺₁𝑓₁𝐱,ℎ₁,𝑝₁𝜇𝐺,…,_𝑘𝑓_𝑘𝐱,ℎ_𝑘,𝑝_𝑘,𝐺subjectto𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘. (10) 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 levels 𝑝_𝑖,𝑖=1,…,𝑘, 𝐱^∗∈𝑋,ℎ^∗_𝑖∈[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,ℎ_𝑖∈[0,1],𝑖=1,…,𝑘 such that 𝜇_𝐷𝐺𝑖(𝐱,ℎ_𝑖,𝑝_𝑖)≥𝜇_𝐷𝐺𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝_𝑖),𝑖=1,…,𝑘 with strict inequality holding for at least one 𝑖, where 𝜇𝐺_𝑖(𝑓_𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝_𝑖))=ℎ^∗_𝑖, 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖))=ℎ_𝑖,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 𝐩¹=(𝑝¹₁,𝑝¹₂) and 𝐩²=(𝑝²₁,𝑝²₂), where 𝑝¹_𝑖≤𝑝²_𝑖,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 for𝐩1≤𝐩2.

[MOP] maximize_{𝑥∈𝑋,ℎ𝑖∈[],0,1}𝑝_{𝑖∈()0,1,𝑖=1,…,𝑘}𝜇𝐺₁𝑓₁𝐱,ℎ₁,𝑝₁𝜇𝐺,…,_𝑘𝑓_𝑘𝐱,ℎ_𝑘,𝑝_𝑘,𝑝₁,…,𝑝_𝑘,𝐺subjectto𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘. (11) 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 level 𝑝_𝑖 as 𝜇𝑝_𝑖(𝑝_𝑖). Then, MOP can be transformed as the following multiobjective programming problem:

[MOP] maximize_{𝐱∈𝑋,ℎ𝑖∈[],0,1}𝑝_{𝑖∈()0,1,𝑖=1,…,𝑘}𝜇𝐺₁𝑓₁𝐱,ℎ₁,𝑝₁𝜇𝐺,…,_𝑘𝑓_𝑘𝐱,ℎ_𝑘,𝑝_𝑘,𝜇𝑝₁𝑝₁,…,𝜇𝑝_𝑘𝑝_𝑘,𝐺subjectto𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘. (12) In order to elicit the membership functions appropriately, 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 unacceptable maximum value of 𝑝_𝑖 and 𝑝_𝑖max is a sufficiently satisfactory minimum value of 𝑝_𝑖. Throughout this section, we make the following assumption.

Assumption 1. 𝜇𝑝_𝑖(𝑝_𝑖),𝑖=1,…,𝑘 are strictly increasing and continuous with respect to 𝑝_𝑖∈𝑃_𝑖, and 𝜇𝑝_𝑖(𝑝_𝑖min)=0, 𝜇𝑝_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 obtained as follows. The maximum value ℎ_𝑖max can be obtained by solving the following problem: minimize_{𝐱∈𝑋,ℎ𝑖∈[]0,1}𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖min,subjecttoℎ_𝑖𝐺=𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖min. (13) This is equivalent to the following problem: ℎ=_𝑖max^defmaximize_{𝐱∈𝑋,ℎ𝑖∈[]0,1}ℎ_𝑖,subjectto𝜇⁻¹𝐺_𝑖ℎ_𝑖=𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖min⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱. (14) The optimal solution 𝐱^∗,ℎ^∗_𝑖,𝑖=1,…,𝑘 of the above problem can be obtained by combined use of the bisection method with respect to ℎ_𝑖∈[0,1] and the first-phase of the two-phase simplex method of linear programming. In order to obtain ℎ_𝑖min, we first solve the following 𝑘 linear programming problems: minimize_{𝐱∈𝑋,ℎ𝑖∈[]0,1}𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖max,subjecttoℎ_𝑖𝐺=𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖max. (15) Let (𝐱^∗_𝑖,ℎ^∗_𝑖),𝑖=1,…,𝑘 be the above optimal solution. Using the optimal solutions (𝐱^∗_𝑖,ℎ^∗_𝑖),𝑖=1,…,𝑘, ℎ_𝑖min can be obtained as follows: ℎ=_𝑖min^defmin_{ℓ=1,…,𝑘,ℓ≠𝑖}𝜇𝐺_𝑖𝑓_𝑖𝐱^∗_ℓ,ℎ^∗_ℓ,𝑝_𝑖max. (16) It should be noted here that, 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖)) is strictly decreasing with respect to 𝑝_𝑖. If the decision maker adopts the fuzzy decision [6] to integrate 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖)) and 𝜇𝑝_𝑖(𝑝_𝑖), MOP can be transformed into the following form:
[MOP] maximize_𝐱∈𝑋,𝑝_{𝑖∈𝑃𝑖,ℎ𝑖∈𝐻𝑖𝑃𝑖,𝑖=1,…,𝑘}𝜇_𝐷𝐺1𝐱,ℎ₁,𝑝₁𝜇,…,_𝐷𝐺𝑘𝐱,ℎ_𝑘,𝑝_𝑘,𝐺subjectto𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘, (17) where 𝜇_𝐷𝐺𝑖𝐱,ℎ_𝑖,𝑝=_𝑖^defmin𝜇𝑝_𝑖𝑝_𝑖𝐺,𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖. (18) In order to deal with MOP (17), we introduce the following concept of an 𝑀_𝑃𝐹-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 𝑀_𝑃𝐹-Pareto optimal solution to MOP (17), if and only if there does not exist another 𝐱∈𝑋,𝑝_𝑖∈𝑃_𝑖,ℎ_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 such that 𝜇_𝐷𝐺𝑖(𝐱,ℎ_𝑖,𝑝_𝑖)≥𝜇_𝐷𝐺𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖),𝑖=1,…,𝑘 with strict inequality holding for at least one 𝑖, where 𝜇𝐺_𝑖(𝑓_𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖))=ℎ^∗_𝑖, 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖))=ℎ_𝑖,𝑖=1,…,𝑘.
For generating a candidate of a satisfactory solution which is also 𝑀_𝑃𝐹-Pareto optimal, the decision maker is asked to specify the reference membership values [6]. Once the reference membership values 𝝁=(𝜇₁,…,𝜇_𝑘) are specified, the corresponding 𝑀_𝑃𝐹-Pareto optimal solution is obtained by solving the following minmax problem:
[MINMAX(𝝁)] minimize_𝐱∈𝑋,𝑝_{𝑖∈𝑃𝑖,ℎ𝑖∈𝐻𝑖}^𝑃_𝑖^{,𝑖=1,…,𝑘,𝜆∈Λ}𝜇𝜆,subjectto_𝑖−𝜇𝑝_𝑖𝑝_𝑖𝜇≤𝜆,𝑖=1,…,𝑘,_𝑖−ℎ_𝑖𝜇𝐺≤𝜆,𝑖=1,…,𝑘,_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘, (19) where Λ=max_{𝑖=1,…,𝑘}𝜇_𝑖−1,min_{𝑖=1,…,𝑘}𝜇_𝑖. (20) In the constraints of (19), it holds that ℎ_𝑖𝐺=𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥𝜇_𝑖−𝜆,⟺𝜇⁻¹𝐺_𝑖ℎ_𝑖=𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≤𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆⟺𝜇⁻¹𝐺_𝑖ℎ_𝑖=𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱≤𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆. (21) In the right hand side of (21), because of 𝐿⁻¹(ℎ_𝑖)≤𝐿⁻¹(𝜇_𝑖−𝜆) and 𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹(𝑝_𝑖)𝜶²_𝑖𝐱>0, it holds that 𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱=𝐝¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝜶²_𝑖𝐱≥𝐝¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖×𝜶−𝜆_1𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝜶²_𝑖𝐱. (22) Using (21) and (22), it holds that 𝜇⁻¹𝐺_𝑖𝜇_𝑖≥𝐝−𝜆_1𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖𝜶²_𝑖𝐱=𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝑝_𝑖⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱. (23) From the inequality 𝜇_𝑖−𝜇𝑝_𝑖(𝑝_𝑖)≤𝜆 in (19) and Assumption 1, it holds that 𝑝_𝑖≥𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆). Therefore, (23) can be transformed into the following form: 𝑇_𝑖𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆−𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱≥𝑝_𝑖≥𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆,⟺𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆≥𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱. (24) Therefore, MINMAX (19) can be reduced to the following minmax problem:
[MINMAX(𝝁)] minimize_{𝐱∈𝑋,𝜆∈Λ}𝜆subjectto𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆≥𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱,𝑖=1,…,𝑘. (25) The relationships between the optimal solution (𝐱^∗,𝜆^∗) of MINMAX(𝝁) (25) and 𝑀_𝑃𝐹-Pareto optimal solutions can be characterized by the following theorem.

Theorem 3. If 𝐱^∗∈𝑋,𝜆^∗∈Λ is a unique optimal solution of MINMAX(𝝁) (25), then 𝐱^∗,𝑝^∗_𝑖=𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗),ℎ^∗_𝑖=𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘 is an 𝑀_𝑃𝐹-Pareto optimal solution.
If 𝐱^∗∈𝑋,𝑝^∗_𝑖∈𝑃_𝑖,ℎ^∗_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 is an 𝑀_𝑃𝐹-Pareto optimal solution, then 𝐱^∗,𝜆^∗=𝜇_𝑖−𝜇𝑝_𝑖(𝑝^∗_𝑖)=𝜇_𝑖𝐺−𝜇_𝑖(𝑓_𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖)),𝑖=1,…,𝑘 is an optimal solution of MINMAX(𝝁) (25) for some reference membership values 𝝁=(𝜇₁,…,𝜇_𝑘).

Proof. (1) From the inequality constraints of (25), it holds that 𝜇_𝑖−𝜆^∗𝐺≤𝜇_𝑖𝑓_𝑖𝐱^∗,𝜇_𝑖−𝜆^∗,𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗, (26) and it is obvious that 𝜇_𝑖−𝜆^∗=𝜇𝑝_𝑖(𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗)). Assume that 𝐱^∗∈𝑋,𝜇_𝑖−𝜆^∗∈𝐻_𝑖(𝑃_𝑖),𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗)∈𝑃_𝑖,𝑖=1,…,𝑘 is not an 𝑀_𝑃𝐹-Pareto optimal solution. Then, there exist 𝐱∈𝑋,𝑝_𝑖∈𝑃_𝑖,ℎ_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 such that 𝜇_𝐷𝐺𝑖𝐱,ℎ_𝑖,𝑝_𝑖=min𝜇𝑝_𝑖𝑝_𝑖𝐺,𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥𝜇_𝐷𝐺𝑖𝐱^∗,𝜇_𝑖−𝜆^∗,𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗=𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘, (27) with strict inequality holding for at least one 𝑖, and 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖))=ℎ_𝑖,𝑖=1,…,𝑘. Then it holds that 𝜇𝑝_𝑖𝑝_𝑖≥𝜇_𝑖−𝜆^∗𝜇𝐺,𝑖=1,…,𝑘,_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘. (28) From Assumption 1 and (8), can be transformed as follows: 𝑝_𝑖≥𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗𝑝,𝑖=1,…,𝑘,_𝑖≤𝑇_𝑖𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱,𝑖=1,…,𝑘. (29) Because of 𝐿⁻¹(ℎ_𝑖)≤𝐿⁻¹(𝜇_𝑖−𝜆^∗),𝑖=1,…,𝑘, there exists 𝐱∈𝑋 such that 𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱≥𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱,⟺𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗≥𝐝¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝜶²_𝑖𝐱,⟺𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗≥𝐝¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱+𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝜶²_𝑖𝐱,𝑖=1,…,𝑘. (30) This contradicts the fact that 𝐱^∗∈𝑋,𝜆^∗∈Λ is a unique optimal solution to MINMAX(𝝁) (25).
(2) Assume that 𝐱^∗∈𝑋,𝜆^∗∈Λ is not an optimal solution to MINMAX(𝝁) (25) for any reference membership values 𝝁=(𝜇₁,…,𝜇_𝑘) which satisfy the equalities 𝜇_𝑖−𝜆^∗=𝜇𝑝_𝑖𝑝^∗_𝑖𝐺=𝜇_𝑖𝑓_𝑖𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖,𝑖=1,…,𝑘. (31) Then, there exists some 𝐱∈𝑋,𝜆<𝜆^∗ such that 𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆−𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱≥𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱,𝐺⟺𝜇_𝑖𝑓_𝑖𝐱,𝜇_𝑖−𝜆,𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆≥𝜇_𝑖−𝜆,𝑖=1,…,𝑘, (32) Because of (31), (32) and 𝜇_𝑖−𝜆>𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘, the following inequalities hold: 𝜇𝑝_𝑖𝑝_𝑖>𝜇𝑝_𝑖𝑝^∗_𝑖𝜇𝐺,𝑖=1,…,𝑘,_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖𝐺>𝜇_𝑖𝑓_𝑖𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖,𝑖=1,…,𝑘, (33) where 𝑝_𝑖=𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆)∈𝑃_𝑖, ℎ_𝑖=𝜇_𝑖−𝜆∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘. This means that there exists some 𝐱∈𝑋,𝑝_𝑖∈𝑃_𝑖,ℎ_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 such that 𝜇_𝐷𝐺𝑖(𝐱,ℎ_𝑖,𝑝_𝑖)>𝜇_𝐷𝐺𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖),𝑖=1,…,𝑘. This contradicts the fact that 𝐱^∗∈𝑋,𝑝^∗_𝑖∈𝑃_𝑖,ℎ^∗_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 is an 𝑀_𝑃𝐹-Pareto optimal solution.

4. Pareto Optimality Test

From Theorem 3, it is not guaranteed that the optimal solution (𝐱^∗,𝜆^∗) of MINMAX(𝝁) (25) is 𝑀_𝑃𝐹-Pareto optimal, if it is not unique. In order to guarantee the 𝑀_𝑃𝐹-Pareto optimality, we first assume that 𝑘 constraints of MINMAX(𝝁) (25) are active at the optimal solution (𝐱^∗,𝜆^∗), that is, 𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱^∗=𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱^∗,𝑖=1,…,𝑘. (34) If the 𝑗th constraint of MINMAX(𝝁) (25) is inactive, that is, 𝜇⁻¹𝐺_𝑗𝜇_𝑗−𝜆^∗−𝐝¹_𝑗𝐱^∗−𝐿⁻¹𝜇_𝑗−𝜆^∗𝜶¹_𝑗𝐱^∗>𝑇_𝑗⁻¹𝜇𝑝⁻¹_𝑗𝜇_𝑗−𝜆^∗⋅𝐝²_𝑗𝐱^∗−𝐿⁻¹𝜇_𝑗−𝜆^∗𝜶²_𝑗𝐱^∗,⟺𝜇⁻¹𝐺_𝑗𝜇_𝑗−𝜆^∗>𝑓_𝑗𝐱^∗,𝜇_𝑗−𝜆^∗,𝜇𝑝⁻¹_𝑗𝜇_𝑗−𝜆^∗, (35) 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 𝑀_𝑃𝐹-Pareto optimality test problem defined as follows:

[𝑀_𝑃𝐹-Pareto optimality test problem]maximize_{𝐱∈𝑋,𝜖𝑖≥0,𝑖=1,…,𝑘}𝑤=^𝑘_𝑖=1𝜖_𝑖,subjectto𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱+𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱,+𝜖_𝑖𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱^∗+𝐝¹_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱^∗,𝑖=1,…,𝑘. (36) For the optimal solution of the above test problem, the following theorem holds.

Theorem 4. For the optimal solution 𝐱̌,𝜖̌_𝑖,𝑖=1,…,𝑘 of the test problem (36), if 𝑤=0 (equivalently, 𝜖̌_𝑖=0,𝑖=1,…,𝑘), 𝐱^∗∈𝑋,𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗)∈𝑃_𝑖,𝜇_𝑖−𝜆^∗∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 is an 𝑀_𝑃𝐹-Pareto optimal solution.

Proof. From the active conditions (34), it holds that 𝜇_𝑖−𝜆^∗𝐺=𝜇_𝑖𝑓_𝑖𝐱^∗,𝜇_𝑖−𝜆^∗,𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗, (37) and it is obvious that 𝜇_𝑖−𝜆^∗=𝜇𝑝_𝑖(𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗)),𝑖=1,…,𝑘. Assume that 𝐱^∗∈𝑋,𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗)∈𝑃_𝑖,𝜇_𝑖−𝜆^∗∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 is not an 𝑀_𝑃𝐹-Pareto optimal solution. Then, there exists some 𝐱∈𝑋,𝑝_𝑖∈𝑃_𝑖,ℎ_𝑖∈𝐻_𝑖(𝑃_𝑖),𝑖=1,…,𝑘 such that 𝜇_𝐷𝐺𝑖𝐱,ℎ_𝑖,𝑝_𝑖=min𝜇𝑝_𝑖𝑝_𝑖𝐺,𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥𝜇_𝐷𝐺𝑖𝐱^∗,𝜇_𝑖−𝜆^∗,𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗=𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘, (38) with strict inequality holding for at least one 𝑖, and 𝜇𝐺_𝑖(𝑓_𝑖(𝐱,ℎ_𝑖,𝑝_𝑖))=ℎ_𝑖,𝑖=1,…,𝑘. This means that the following inequalities hold: 𝜇𝑝_𝑖𝑝_𝑖≥𝜇_𝑖−𝜆^∗𝜇𝐺,𝑖=1,…,𝑘,_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖≥𝜇_𝑖−𝜆^∗,𝑖=1,…,𝑘. (39) Because of Assumption 1, (8), and 𝐿⁻¹(ℎ_𝑖)≤𝐿⁻¹(𝜇_𝑖−𝜆^∗), can be transformed into the following forms 𝑝_𝑖≥𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗𝑝,𝑖=1,…,𝑘,_𝑖≤𝑇_𝑖𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶¹_𝑖𝐱𝐝²_𝑖𝐱−𝐿⁻¹ℎ_𝑖𝜶²_𝑖𝐱≤𝑇_𝑖𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱𝑖=1,…,𝑘. (40) From (37), there exists some 𝐱∈𝑋 such that 𝜇⁻¹𝐺_𝑖𝜇_𝑖−𝜆^∗−𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱≥𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱,⟺𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶²_𝑖𝐱^∗+𝐝¹_𝑖𝐱^∗−𝐿⁻¹𝜇_𝑖−𝜆^∗𝜶¹_𝑖𝐱^∗≥𝑇_𝑖⁻¹𝜇𝑝⁻¹_𝑖𝜇_𝑖−𝜆^∗⋅𝐝²_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶²_𝑖𝐱+𝐝¹_𝑖𝐱−𝐿⁻¹𝜇_𝑖−𝜆𝜶¹_𝑖𝐱,𝑖=1,…,𝑘, (41) 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 an 𝑀_𝑃𝐹-Pareto optimal solution set.

[Interactive algorithm].

Step 1. Calculate min_𝐱∈𝑋𝐸[𝐝_𝑖]𝐱,𝑖=1,…,𝑘, where 𝐝_𝑖(𝜔)𝐱 are central values of LR fuzzy numbers 𝐜̃‌_𝑖(𝜔)𝐱. Let us denote 𝐱_𝑖 as the optimal solution of the optimization problem min_𝐱∈𝑋𝐸[𝐝_𝑖]𝐱. Then, define 𝑓=_𝑖min^def𝐸[𝐝_𝑖]𝐱_𝑖, and 𝑓=_𝑖max^defmax_{𝑗=1,…,𝑘,𝑗≠𝑖}𝐸[𝐝_𝑖]𝐱_𝑗. The decision maker determines the membership functions 𝜇𝐺_𝑖(𝑦),𝑖=1,…,𝑘 in his/her subjective manner, where 𝜇𝐺_𝑖(𝑓_𝑖min𝐺)=1,𝜇_𝑖(𝑓_𝑖max)=0.

Step 2. The decision maker specifies the interval 𝑃_𝑖=[𝑝_𝑖min,𝑝_𝑖max],𝑖=1,…,𝑘, where 𝑝_𝑖min is an unacceptable maximum value of 𝑝_𝑖 and 𝑝_𝑖max is a sufficiently satisfactory minimum value of 𝑝_𝑖. For the interval 𝑃_𝑖, the decision maker determines the membership function 𝜇𝑝_𝑖(𝑝_𝑖) according to Assumption 1.

Step 3. Set the initial reference membership values as 𝜇_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 𝑀_𝑃𝐹-Pareto optimality test problem (36) is formulated and solved.

Step 5. If the decision maker is satisfied with the current values of the 𝑀_𝑃𝐹-Pareto optimal solution 𝜇_𝐷𝐺𝑖(𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖),𝑖=1,…,𝑘, where 𝑝^∗_𝑖=𝜇𝑝⁻¹_𝑖(𝜇_𝑖−𝜆^∗),ℎ^∗_𝑖=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]: minimize𝐜̃₁𝐱=𝑐̃₁₁𝑥₁+𝑐̃₁₂𝑥₂+𝑐̃₁₃𝑥₃,minimize𝐜̃₂𝐱=𝑐̃₂₁𝑥₁+𝑐̃₂₂𝑥₂+𝑐̃₂₃𝑥₃,subjectto2𝑥₁+6𝑥₂+3𝑥₃≤150,6𝑥₁+3𝑥₂+5𝑥₃≤175,5𝑥₁+4𝑥₂+2𝑥₃≤160,2𝑥₁+2𝑥₂+3𝑥₃𝑥≥90,_𝑖≥0,𝑖=1,2,3. (42) It is assumed here that a realization 𝑐̃‌_𝑖𝑗(𝜔) of an LR-type fuzzy random variable 𝑐̃‌_𝑖𝑗 is an LR fuzzy number whose membership function is defined as follows: 𝜇_{̃‌𝑐𝑖𝑗}(𝑠)=𝐿𝑑¹_𝑖𝑗+𝑡_𝑖(𝜔)𝑑²_𝑖𝑗−𝑠𝛼¹_𝑖𝑗+𝑡_𝑖(𝜔)𝛼²_𝑖𝑗𝑠≤𝑑_𝑖𝑗(𝜔),𝑅𝑠−𝑑¹_𝑖𝑗+𝑡_𝑖(𝜔)𝑑²_𝑖𝑗𝛽¹_𝑖𝑗+𝑡_𝑖(𝜔)𝛽²_𝑖𝑗𝑠>𝑑_𝑖𝑗(𝜔), (43) where 𝐿(𝑡)=𝑅(𝑡)=max{0,1−𝑡}, and the parameters 𝑑¹_𝑖𝑗,𝑑²_𝑖𝑗,𝛼¹_𝑖𝑗,𝛼²_𝑖𝑗,𝛽¹_𝑖𝑗,𝛽²_𝑖𝑗 are given in Table 1 [11, 12].

Table 1 

Parameter values for LR-type fuzzy random variables 𝑐̃‌_𝑖𝑗.

Moreover, 𝑡_𝑖,𝑖=1,2 are Gaussian random variables defined as 𝑡_𝑖∼𝑁(0,1).

At Step 1, calculate min_𝐱∈𝑋𝐸[𝐝_𝑖]𝐱,𝑖=1,2, and we obtain 𝑓_1min=75,𝑓_1max=96.42857 and 𝑓_2min=−332.143,𝑓_2max=−285. We assume that the decision maker adopts the linear membership functions 𝜇𝐺_𝑖(𝑦),𝑖=1,…,𝑘 defined as follows: 𝜇𝐺_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=𝑓_𝑖max−𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖𝑓_𝑖max−𝑓_𝑖min,𝑖=1,2. (44) At Step 2, the decision maker specifies the intervals 𝑃₁=[𝑝_1min,𝑝_1max]=[0.401066,0.714968] and 𝑃₂=[𝑝_2min,𝑝_2max]=[0.213304,0.812859]. Then, we assume that the decision maker adopts the linear membership functions 𝜇𝑝_𝑖(𝑝_𝑖) defined as follows: 𝜇𝑝_𝑖𝑝_𝑖=𝑝_𝑖−𝑝_𝑖min𝑝_𝑖max−𝑝_𝑖min,𝑖=1,2. (45) At Step 3, set the initial reference membership values as (𝜇₁,𝜇₂)=(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 𝑀_𝑃𝐹-Pareto optimal solution (𝐱^∗,𝜆^∗) (Step 4). Consider 𝜇𝐺₁𝑓₁𝐱^∗,ℎ^∗₁,𝑝^∗₁=𝜇𝑝₁𝑝^∗₁𝜇𝐺=0.564271,₂𝑓₂𝐱^∗,ℎ^∗₂,𝑝^∗₂=𝜇𝑝₂𝑝^∗₂=0.564271. (46) The hypothetical decision maker is not satisfied with the current value of the 𝑀_𝑃𝐹-Pareto optimal solution (𝐱^∗,𝜆^∗), and, in order to improve 𝜇_𝐷𝐺2𝐺(⋅)=min{𝜇₂(⋅),𝜇𝑝₂(⋅)} at the expense of 𝜇_𝐷𝐺1𝐺(⋅)=min{𝜇₁(⋅),𝜇𝑝₁(⋅)}, he/she updates his/her reference membership values as (𝜇₁,𝜇₂)=(0.5,0.6) (Step 5). Then, the corresponding 𝑀_𝑃𝐹-Pareto optimal solution is obtained by solving MINMAX(𝝁) (Step 4). Consider 𝜇𝐺₁𝑓₁𝐱^∗,ℎ^∗₁,𝑝^∗₁=𝜇𝑝₁𝑝^∗₁𝜇𝐺=0.514421,₂𝑓₂𝐱^∗,ℎ^∗₂,𝑝^∗₂=𝜇𝑝₂𝑝^∗₂=0.614421. (47) For the current value of the 𝑀_𝑃𝐹-Pareto optimal solution, the hypothetical decision maker updates his/her reference membership values (𝜇₁,𝜇₂)=(0.52,0.59) in order to improve 𝜇_𝐷𝐺1(⋅) at the expense of 𝜇_𝐷𝐺2(⋅) slightly (Step 5). The corresponding 𝑀_𝑃𝐹-Pareto optimal solution is obtained by solving MINMAX(𝝁) (Step 4). Consider 𝜇𝐺₁𝑓₁𝐱^∗,ℎ^∗₁,𝑝^∗₁=𝜇𝑝₁𝑝^∗₁𝜇𝐺=0.529412,₂𝑓₂𝐱^∗,ℎ^∗₂,𝑝^∗₂=𝜇𝑝₂𝑝^∗₂=0.599412. (48)(49) Then, since the hypothetical decision maker is satisfied with the current value of the 𝑀_𝑃𝐹-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.

In 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(𝑓_𝑖(𝐱,ℎ_𝑖,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)] min_{𝐱∈𝑋,ℎ𝑖∈[]0,1,𝑖=1,…,𝑘,𝜆∈Λ}𝜇𝜆,subjectto_𝑖𝐺−𝜇_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖𝜇𝐺≤𝜆,𝑖=1,…,𝑘,_𝑖𝑓_𝑖𝐱,ℎ_𝑖,𝑝_𝑖=ℎ_𝑖,𝑖=1,…,𝑘. (50) In MINMAX(𝐩,𝝁), we assume that the decision maker sets his/her permissible probability levels as 𝑝₁=𝑝₂=0.75, and the reference membership values as 𝜇₁=𝜇₂=1. Then, the corresponding Pareto optimal solution is obtained as follows: 𝑓₁𝐱^∗,ℎ^∗₁𝑓,0.75=94.0338,₂𝐱^∗,ℎ^∗₂𝜇𝐺,0.75=−290.269,_𝑖𝑓_𝑖𝐱^∗,ℎ^∗_𝑖,0.75=0.11176,𝑝_𝑖=0.75,𝑖=1,2. (51) In our proposed algorithm, by solving MINMAX(𝝁) (25) for the reference membership values 𝜇₁=𝜇₂=1, the 𝑀_𝑃𝐹-Pareto optimal solution is obtained as follows (see the first iteration of Table 2): 𝑓₁𝐱^∗,ℎ^∗₁,𝑝^∗₁=84.3370,𝑓₂𝐱^∗,ℎ^∗₂,𝑝^∗₂𝜇=−311.601,_𝐷𝐺𝑖𝐱^∗,ℎ^∗_𝑖,𝑝^∗_𝑖𝑝=0.564271,𝑖=1,2,_∗1=0.578193,𝑝^∗₂=0.551616. (52) 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 called 𝑀_𝑃𝐹-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.