A Two-Fold Linear Programming Model with Fuzzy Data

Linear programming (LP) is the most widely used optimization technique for solving real-life problems because of its simplicity and efficiency. Although LP models require well-suited information and precise data, managers and decision makers dealing with optimization problems often have a lack of information on the exact values of some parameters used in their models. Fuzzy sets provide a powerful tool for dealing with this kind of imprecise, vague, uncertain or incomplete data. In this paper, the authors propose a two-fold model which consists of two new methods for solving fuzzy LP (FLP) problems in which the variables and the coefficients of the constraints are characterized by fuzzy numbers. In the first method, the authors transform their FLP model into a conventional LP model by using a new fuzzy ranking method and introducing a new supplementary variable to obtain the fuzzy and crisp optimal solutions simultaneously with a single LP model. In the second method, the authors propose a LP model with crisp variables for identifying the crisp optimal solutions. The authors demonstrate the details of the proposed method with two numerical examples. DOI: 10.4018/ijfsa.2012070101 2 International Journal of Fuzzy System Applications, 2(3), 1-12, July-September 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. For example, sometimes coefficient variables are not known precisely, other times constraints satisfaction limits may be vague. The challenge in FLP is to construct an optimization model that can produce the optimal solution with subjective professional judgments. In this study, we propose a two-fold model for solving FLP problems in which the variables and the coefficients of the constraints are characterized by fuzzy numbers. We transform a FLP model into a conventional LP model by applying a new fuzzy ranking method and obtaining the fuzzy and crisp optimal solutions. This paper is organized as follows: The next section presents a brief review of the existing literature followed by some primary definitions of fuzzy sets. We then introduce a mathematical model for FLP. Following this introduction, we illustrate the details of the proposed framework followed by a numerical example to demonstrate the applicability of the proposed method. Finally, we finish the paper with our conclusions and future research directions. LITERATURE REVIEW The theory of fuzzy mathematical programming was first proposed by Tanaka et al. (1974) based on the fuzzy decision framework of Bellman and Zadeh (1970) to address the impreciseness and vagueness of the parameters in problems with fuzzy constraints and objective functions. Zimmermann (1978) introduced the first formulation of FLP. He constructed a crisp model of the problem and obtained its crisp results using an existing algorithm. He then used the crisp results 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. Zimmermann (1978, 1987) used Bellman and Zadeh’s (1970) interpretation that a fuzzy decision is a union of goals and constraints. In the past decade, researchers have discussed various properties of FLP problems and proposed an assortment of models (Luhandjula, 1989). Zhang et al. (2003) proposed a FLP with fuzzy numbers for the coefficients of objective functions. They introduced a number of optimal solutions to the FLP problems and developed a number of theorems for converting the FLP problems to multi-objective optimization problems with four-objective functions. Stanciulescu (2003) proposed a FLP model with fuzzy coefficients for the objectives and the constraints. He used fuzzy decision variables with a joint membership function instead of crisp decision variables and linked the decision variables together to sum them up to a constant. He considered lower-bounded fuzzy decision variables that set up the lower bounds of the decision variables. He then generalized the method to lower–upper-bounded fuzzy decision variables that set up also the upper bounds of the decision variables. Ganesan and Veeramani (2006) proposed a FLP model with symmetric trapezoidal fuzzy numbers. They proved fuzzy analogues of some important LP theorems and obtained some interesting results which in turn led to the solution for FLP problems without converting them into crisp LP problems. Ebrahimnejad (2011) showed that the method proposed by Ganesan and Veermani (2006) stops in a finite number of iterations and proposed a revised version of their method that was more efficient and robust in practice. He also proved the absence of degeneracy and showed that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it also has an optimal fuzzy basic solution. Hosseinzadeh Lotfi et al. (2009) considered full FLP problems where all parameters and variables were triangular fuzzy numbers. They pointed out that there is no method in the literature for finding the fuzzy optimal solution of full FLP problems and proposed a new method to find the fuzzy optimal solution of full FLP problems with equality constraints. They used the concept of the symmetric triangular fuzzy numbers and introduced an approach to International Journal of Fuzzy System Applications, 2(3), 1-12, July-September 2012 3 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. defuzzify a general fuzzy quantity. They first approximated the fuzzy triangular numbers to its nearest symmetric triangular numbers, with the assumption that all decision variables were symmetric triangular. They then converted every FLP model into two crisp complex LP models and used a special ranking for fuzzy numbers to transform their full FLP model into a multiobjective linear programming where all variables and parameters were crisp. Kumar et al. (2011) further studied the full FLP problems with equality introduced by Hosseinzadeh Lotfi et al. (2009) and proposed a new method for finding the fuzzy optimal solution in these problems. Mahdavi-Amiri and Nasseri (2006) proposed a FLP model where a linear ranking function was used to order trapezoidal fuzzy numbers. They established the dual problem of the LP problem with trapezoidal fuzzy variables and deduced some duality results to solve the FLP problem directly with the primal simplex tableau. Ebrahimnejad (2010) introduced a new primal-dual algorithm for solving FLP problems by using the duality results proposed by Mahdavi-Amiri and Nasseri (2007). Ebrahimnejad (2011) has also generalized the concept of sensitivity analysis in FLP problems by applying fuzzy simplex algorithms and using the general linear ranking functions on fuzzy numbers. AN OVERVIEW OF FUZZY SETS Let  be the set of all real numbers. The fuzzy subset  A is defined by μ  A x ( ) [ , ] → 0 1 , which is called a membership function. Definition 1 (Fuzzy number). The fuzzy number  A is a normal and convex fuzzy subset of X and is defined as Normality: ∃ ∈ ∨ = x x x A , ( ) μ 1 Convexity: μ λ λ μ μ λ A A A x y x y x y ( ( ) ) min( ( ), ( )), , , [ , ] + − ≥ ∀ ∈ ∀ ∈ 1