Solving a Fully Fuzzy Linear Programming Problem through Compromise Programming

. In the current literatures, there are several models of fully fuzzy linear programming (FFLP) problems where all the parameters and variables were fuzzy numbers but the constraints were crisp equality or inequality. In this paper, an FFLP problem with fuzzy equality constraints is discussed, and a method for solving this FFLP problem is also proposed. We first transform the fuzzy equality constraints into the crisp inequality ones using the measure of the similarity, which is interpreted as the feasibility degree of constrains, and then transform the fuzzy objective into two crisp objectives by considering expected value and uncertainty of fuzzy objective. Since the feasibility degree of constrains is in conflict with the optimal value of objective function, we finally construct an auxiliary three-objective linear programming problem, which is solved through a compromise programming approach, to solve the initial FFLP problem. To illustrate the proposed method, two numerical examples are solved.


Introduction
Linear programming (LP) has important applications in many areas of engineering and management.In these applications, since the real-world problems are very complex, the parameters of LP are usually represented by fuzzy numbers.Therefore, many researchers have shown interest in the area of fuzzy linear programming (FLP).
Recently fuzzy set theory has been applied in many research regions, since fuzzy set theory is effective to solve the decision-making problems with imprecise data [1][2][3].Several kinds of the FLP problems have appeared in the literature [4][5][6][7][8][9][10][11][12][13][14][15][16].Delgado et al. [4] have proposed a general model for the FLP problems in which constraints are fuzzy inequality and the parameters of constraints are fuzzy numbers but the parameters of the objective function are crisp.Rommelfanger [5] has also proposed a general model for the FLP problems and the main difference compared with [4] is that here parameters of the objective function are also fuzzy numbers.Considering the different hypotheses, researchers [6][7][8][9][10][11][12][13] have proposed some particular FLP problems, which can be deduced from the general model.In order to solve these FLP problems, different approaches have been proposed too.Some methods are based on the concepts of the superiority and inferiority of fuzzy numbers [7], the degrees of feasibility [8], the satisfaction degree of the constraints [10], and the statistical confidence interval [11].Other kinds of methods are multiobjective optimization method [6], penalty method [12], and semi-infinite programming method [13].Similar other interesting works also can be found in the literature [14][15][16].Mahdavi-Amiri and Nasseri [14] develop a new dual algorithm for solving the FLP problem directly.Ganesan and Veeramani [15] propose a method for solving fuzzy linear programming problems without converting them to crisp linear programming problems.Maleki et al. [16] propose a good method for solving an FLP problem, and an auxiliary problem is introduced in their model.
In recent years, several kinds of the fully fuzzy linear programming (FFLP) problems in which all the parameters and variables are represented by fuzzy numbers have appeared in the literature [17][18][19][20][21].Some authors [17,18] have discussed FFLP problems with crisp inequality constraints and obviously different methods for solving them have been proposed.In these methods the fuzzy optimal solutions of the FFLP problems are obtained by converting FFLP problem into crisp linear programming (CLP) problem.Other authors [19,20] have discussed FFLP problems with crisp equality constraints and different methods for solving them have been proposed too.In their methods, the method proposed by Lotfi et al. [19] can only obtain the approximate solution of the FFLP problems, but the method proposed by Kumar et al. [20] can find the fuzzy optimal solution which satisfies the constraints exactly.Guo and Shang [21] propose the computing model to the positive fully fuzzy linear matrix equation, and the fuzzy approximate solution is obtained by using pseudoinverse.However, in most of previous literatures, all constraints of FFLP problems have the crisp form.
In this paper, an FFLP problem with fuzzy equality constraints has been considered.In order to solve it, we first transform the FFLP problem into a crisp three-objective LP model by considering expected value and uncertainty of fuzzy objective and the feasibility degree of fuzzy constrains.Then we solve it using a compromise programming (CP) approach.
This paper is organized as follows.In Section 2 some basic definitions, arithmetic operations, and comparison operations between two triangular fuzzy numbers are reviewed.In Section 3 the processes for transforming the FFLP problem with fuzzy equality constraints into crisp problem are described.In Section 4 a crisp three-objective LP model to find the fuzzy optimal solution of the FFLP problem is built, and the model is solved through CP in Section 5.In Section 6 a numerical example is given.Conclusions are discussed in Section 7.

Preliminaries
2.1.Basic Definitions.In this paper, the triangular fuzzy numbers are considered because this form of fuzzy numbers is very simple and popular.Moreover, we can express and estimate many other types of fuzzy numbers with triangular fuzzy number [7].The triangular fuzzy numbers are defined as follows.

Presentation of the Problem
The FFLP problem with fuzzy equality constraints "≅" is written as follows: Max Z = C ⊗ X, X is a nonnegative fuzzy vector, (P1) and c , x , ã , b ∈ ().The symbols "≅" denote the fuzzified versions of "=" and can be read as "approximately equal to." P1) may be written as follows: Also, (P2) can be expressed as follows: Here the symbols "≲ and ≳" denote the fuzzified versions of "⩽ and ⩾" and can be read as "approximately less/greater than or equal to." As the decision maker (DM) knows that all the parameters and variables in each constraint of (P3) are fuzzy numbers, he may allow some violation of the right hand fuzzy number in each constraint.This violation can also be considered as a fuzzy number.Let p and q ,  = 1, 2, . . ., , be fuzzy numbers determined by the DM giving his allowed maximum violation in the accomplishment of the th constraint and the  + th constraint of (P3), respectively.It means that the DM tolerates violations in each constraint of (P2) up the value b + p and down the value b − q ,  = 1, 2, . . ., , respectively.Based on these ideas, according to the resolution methods proposed in [4], (P3) will become where the symbols " ⃝ ≤ and ⃝ ≥" are relations between fuzzy numbers which preserve the ranking when fuzzy numbers are multiplied by positive scalars, and they can be anyone the DM chooses.Different kind of relation ⃝ ≤ and ⃝ ≥ will lead to different models of CLP problems.In this paper, we assume that the relation ⃝ ≤ and ⃝ ≥ will be determined by using the comparison operations defined in Section 2.3.
Here ( b , b + p ) is the similarity between two triangular fuzzy numbers b and   + p ,  = 1, 2, . . ., , ( b , b − q ) is the similarity between two triangular fuzzy numbers b and b − q ,  = 1, 2, . . ., , and  is the allowed similarity level given by the DM.
According to transforming processes of (P1) to (P7), we may set the following proposition.Proposition 6.A decision matrix  0 () is -fuzzy-feasible solution of (P1) if and only if  0 () is -feasible solution of (P7).
In the following, the set of the -feasible solution of (P7) will be denoted by ℵ(), and it is evident that Then, (P7) can be rewritten as Max ( (1) ,  (2) ,  (3) ( (1)    (1)   ,  (2)    (2)   ,  (3)    (3)  ) In order to transform the fuzzy objective into crisp one, we should consider expected value and uncertainty of fuzzy objective.We use a ranking function to define the expected value of fuzzy objective.Many ranking functions can be found in the literatures, and we choose the same ranking function, which is defined in Definition 4, used by Kumar et al. [20].The uncertainty of fuzzy objective is measured using the difference between upper bound and lower bound of fuzzy objective value.Therefore, (P7) may be transformed into the following crisp problem: ( (1)    (1)   + 2 (2)    (2)   +  (3)    (3)  ) (1)    (1)    s.t.
(P8) (P8) is an -parametric crisp biobjective LP model.Giving the value of , and we can solve the -efficient solution, which is defined as follows.
Definition 7. Giving the value of ,  * () ∈ ℵ() is said to be -efficient solution to the problem (P8) if there does not exist another  0 () ∈ ℵ() such that where at least one of these inequalities is strict.
From Definition 7, we have the following proposition.
From Proposition 8, we can obtain the -fuzzy-optimal solutions to the problem (P1) by solving (P8).

The Auxiliary Three-Objective LP Model
From (12), a bigger value of  implies that the DM allows a smaller violation of the right hand fuzzy number in each constraint.So  can be interpreted as the feasibility degree of constrains, the bigger the value of  is; the higher the feasibility degree of constrains will be.However, from ( 13) and (P7), the bigger the value of  is, the worst the objective value will be.Therefore, we want to find a balance solution between two goals: to improve the objectives function values and to increase the feasibility degree of constrains.According to the previous analysis, (P8) can be transformed into the following auxiliary three-objective LP model: ( (1)    (1)   + 2 (2)    (2)   +  (3)    (3)  ) , (1)    (1)   , where   is the allowed minimum similarity level, and it is specified by the DM according to his interests.The three objectives of (P9) represent the DM's preference for the alternative with the higher expected value, less uncertainty of objective, and the higher feasibility degree of constrains, respectively.
In the following, the set of the feasible solutions of (P9) will be denoted by ℵ.Definition 9. ( * ,  * ) ∈ ℵ is said to be an efficient solution to the problem (P9) if there does not exist another ( 0 ,  0 ) ∈ ℵ such that where at least one of these inequalities is strict.
From Definitions 7 and 9 and Proposition 8, we have the following proposition.Proposition 10.All efficient solutions ( * ,  * ) to the problem (P9) are  * -fuzzy-optimal solutions to the problem (P1) and reciprocally.

Compromise Solutions
CP is a Multiple Criteria Decision Making approach which ranks alternatives according to their closeness to the ideal point.The best alternative is the one whose point is at the least distance from an ideal point in the set of efficient solutions [26].
In order to apply the CP approach to solve the problem (P9), we need to obtain the pay-off matrix.For this, we optimize each objective separately, calculating the values reached by the objectives on the optimal solution, respectively.Let  *  ,  = R, Δ, , be the optimal solutions for each objective, and R  , Δ  ,   the values reached by three objectives on the optimal solution  *  ,  = R, Δ, , respectively.Then the obtained pay-off matrix can be expressed as in Table 1.
The distance between each objective value and the corresponding ideal point is As the objectives are measured with different units, it is necessary to homogenize the distances as The distance measure used in CP is the family of  metrics given as where   is the weight or relative importance attached to the th objective and  is the topological metric that is a real number belonging to the closed interval [1, ∞].
A compromise solution is the one which minimizes   .Therefore, we have Obviously the solution of (P10) depends on the chosen metric.The most commonly obtained compromise solutions are for metrics  = 1 and  = ∞ because for other metrics the nonlinear programming algorithms are needed [27].Also, in the biobjective case they are the bounds of the whole compromise set [28,29].
For  = 1, the compromise solution closest to the ideal solution can be obtained by solving the following LP problem: (( (ℎ)  )
A composite form of CP for  = 1 and  = ∞ can be obtained by minimizing a linear combination between  1 and  ∞ ; that is, where  ∈ [0,1]. can be interpreted as a trade-off or marginal rate of substitution between  1 and  ∞ .When  = 1, problem (P10-4) gives the compromise solution for  = 1, and for  = 0, (P10-4) gives the compromise solution for  = ∞.For any set of positive weights and any  > 0, the solutions of problem (P10-4) are efficient [36].
In order to obtain the pay-off matrix, we optimize each objective separately, calculating the values reached by the objectives on the optimal solution, respectively.The obtained pay-off matrix is shown in Table 2.
(i) For  = 1, the compromise solution closest to the ideal solution can be obtained by solving the following LP problem: Min  1 = 0.35

Table 1 :
The pay-off matrix.
In this section two numerical examples are given to illustrate the proposed model and method.