Using Ranking Functions to Extend of Hungarian algorithm in solving

World Applied Sciences Journal 17 (4): 497-501, 2012 ISSN 1818-4952 © IDOSI Publications, 2012 Corresponding Author: R. Maddahi, Department of Mathematics, Najaf Abad Branch, Islamic Azad University, Najaf Abad, Iran. 497 Using Ranking Functions to Extend of Hungarian Algorithm in Solving Assignment Problem with Fuzzy Costs R. Maddahi and A. Ebrahimnejad1 2 Department of Mathematics, Najaf Abad Branch, Islamic Azad University, Najaf Abad, Iran1 Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran2 Abstract: Linear programming with fuzzy cost

coefficients is one of the most frequently applied fuzzy linear programming techniques. In this paper, we consider a special kind of such problems for a kind of the assignment problem (AP), which is a basic model in network structured linear programming problems. The aim of this model is to find the minimal fuzzy cost of assignment individuals to jobs. Here, we generalize the Hungarian algorithm which is a direct implementation of the primal-dual algorithm for solving the assignment problem in fuzzy environment. Key words: Assignment problem % Fuzzy numbers % Duality INTRODUCTION numbers. Mahdavi-Amiri and Nasseri [3] showed that the Many authors considered various types of fuzzy Then they obtained some new results leading to a new linear programming (FLP) problems and proposed several dual algorithm for solving the FVLP problem without the approaches for solving these problems. Since the need of any auxiliary problem. However, the primal simplex fuzziness may appear in a linear programming problem in algorithm and the dual simplex algorithm, proposed by many ways, the definition of fuzzy linear programming Maleki et al. [1] and Nasseri and Ebrahimnejad [2], problem is not unique. In general, fuzzy linear respectively, is not efficient for solving linear programming problems are first converted into equivalent programming with fuzzy cost coefficients when some or all crisp linear or nonlinear programs, which are then solved variables are restricted into lie within lower and upper by standard methods. In effect, most convenient methods bound. So, Ebrahimnejad et al. [4] and Ebrahimnejad and are based on the concept of comparison of fuzzy numbers Nasseri [5] developed the above mentioned algorithms for by use of ranking functions which is desired in this paper. this situation. In continue, Ebrahimnejad et al. [6] Instead of discussing the general type, we focus on that proposed another efficient method for solving linear kind of FLP problems in which the coefficient matrix and programming problems with fuzzy costs based on primal- right-hand-side vector are crisp, while the cost dual algorithm. In this paper, we consider a special kind of coefficients of objective function are fuzzy numbers. linear programming problem with fuzzy cost coefficients Maleki et al. [1] developed the primal simplex algorithm for knows as assignment problem (AP) with fuzzy costs. We solving such problems without converting them to the also extend the Hungarian algorithm which is a direct crisp linear programming problems. In a similar manner, implementation of the...

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