A Two-Fold Linear Programming Model with Fuzzy Data

A Two-Fold Linear Programming Model with Fuzzy Data

Saber Saati (Islamic Azad University, North Tehran Branch, Iran), Adel Hatami-Marbini (Université catholique de Louvain, Belgium), Madjid Tavana (La Salle University, USA) and Elham Hajiahkondi (Payame Noor University, Iran)
Copyright: © 2012 |Pages: 12
DOI: 10.4018/ijfsa.2012070101
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Abstract

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.
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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.

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