یک روش جدید برای حل مسائل برنامه ریزی خطی با اعداد فازی ذوزنقه ای متقارن

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.

Linear programming (LP) is a mathematical technique for optimal allocation of scarce resources to several competing activities on the basis of given criteria of optimality. Precise data are fundamentally indispensable in conventional LP problems. However, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been used to handle imprecise data in LP by generalizing the notion of membership in a set. Essentially, each element in a fuzzy set is associated with a point-value selected from the unit interval [0, 1]. The fundamental challenge in fuzzy LP (FLP) is to construct an optimization model that can produce the optimal solution with imprecise data. The theory of fuzzy mathematical programming was first proposed by Tanaka et al. Tanaka et al. [1] based on the fuzzy decision framework of Bellman and Zadeh [2]. Zimmerman [3] introduced the first formulation of FLP to address the impreciseness of the parameters in LP problems with fuzzy constraints and objective functions. Zimmerman [3] constructed a crisp model of the problem and obtained its crisp results using an existing algorithm. He then used Bellman and Zadeh [2] interpretation that a fuzzy decision is a union of goals and constraints and fuzzified the problem by considering subjective constants of admissible deviations for the goal and the constraints. Finally, he defined an equivalent crisp problem using an auxiliary variable that represented the maximization of the minimization of the deviations on the constraints. FLP is by far the most widely used method by practitioners for constrained optimization problems with fuzzy data [4], [5], [6], [7], [8], [9] and [10]. We propose a simplified new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We show that the optimal solution of the FLP problem can be found simply by solving an equivalent crisp LP problem. The remainder of this paper is organized as follows. We review the relevant FLP literature in Section 2. In Section 3, we review some necessary concepts and backgrounds on fuzzy arithmetic. We then formulate the FLP problem proposed by Ganesan and Veeramani [11] in Section 4. In Section 5, we present our proposed FLP method. We present our conclusions and future research directions in Section 6.

A large number of LP models with different levels of sophistication have been proposed in the literature. However, some of these models have limited real-life applications because the conventional LP models generally assume crisp data for the coefficients of the objective function, the values of the right-hand-side, and the elements of the coefficient matrix. Contrary to the conventional LP methods, we consider imprecise data in the real-life LP problems and develop an alternative FLP method that is simple and yet addresses these shortfalls in the existing models in the literature. In the FLP method proposed in this study, the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. The optimal solution of the FLP problem is simply found by solving an equivalent crisp LP problem. The FLP problem is converted into a crisp equivalent LP problem and the crisp LP problem is solved with the standard primal simplex method. We showed that the method proposed in this study requires less arithmetic operations as opposed to the FLP method proposed by Ganesan and Veeramani [11]. In addition, the proposed method produces a fuzzy solution by solving an equivalent crisp problem without increasing the number of constraints and variables of the original problem as opposed to Mehar’s method proposed by Kumar and Kaur [43]. Future research could focus on the comparison of the results obtained from the method proposed in this study with those that could be obtained with other competing methods. In addition, based on Definition 3.3, we defined a ranking for each symmetric trapezoidal fuzzy number for comparison purposes. If View the MathML sourceÃ=(aL,aU,α,α) is a symmetric trapezoidal fuzzy number, its ranking is defined as View the MathML sourceR(Ã)=aL+aU2 (from the decision maker’s point of view). It is obvious that if we have View the MathML sourceR(Ã)=R(B̃), we can’t guarantee the equality of View the MathML sourceÃ=B̃. Therefore, further research on introducing a new ranking method for solving FLP problems satisfying this property (space) is an interesting stream of future research. Also, the proposed method is not applicable to FLP problems with non-symmetric trapezoidal fuzzy numbers. The generalization of the proposed method to overcome this shortcoming is left to future research in FLP problems with non-symmetric trapezoidal fuzzy numbers. Finally, we point out that the FLP method proposed in this study does not consider fuzzy cost coefficients and a fuzzy constraint matrix. Developing a full fuzzy version of the proposed method and overcoming this limitation is an interesting stream of future research.