On fuzzy linearization approaches for solving multi-objective linear fractional programming problems

Abstract

This study first surveys fuzzy linearization approaches for solving multi-objective linear fractional programming (MOLFP) problems. In particular, we review different existing methods dealing with fuzzy objectives on a crisp constraint set. Those methods transform the given MOLFP problem into a linear or a multi-objective linear programming (LP or MOLP) problem and obtain one efficient or weakly efficient solution of the main MOLFP problem. We show that one of these popular existing methods has shortcomings, and we modify it to be able to find efficient solutions. The main idea of LP-based methods is optimizing a weighted sum of numerators and the negative form of denominators of the given fractional objective function over the feasible set. We prove there is no weight region to guarantee the efficiency of the optimal solutions of such LP-based methods whenever the interior of the feasible set is nonempty. Moreover, MOLP-based methods obtain an equivalent MOLP problem to the main MOLFP problem using fuzzy set techniques. We prove MOLFP problems with a non-closed efficient set are not equivalent to MOLP ones whenever the equivalency mapping is continuous.