Linear fractional programming problem with bipolar max-product fuzzy relation equation constraints

There are two interesting methods, in the literature, for solving fuzzy linear programming problems in which the elements of coefficient matrix of the constraints are represented by real numbers and rest of the parameters are represented by symmetric trapezoidal fuzzy numbers.In this paper, the linear fractional programming problem subject to a system of bipolar fuzzy relation equations with max-product composition operator is studied. The structure of its feasible domain is investigated. Some sufficient conditions are then given that under them, one of the optimal solutions of the problem can be determined directly. Also, under other sufficient conditions, some components of an optimal solution can be characterized without solving the problem. Then the problem is transformed into a linear programming problem by reformulating the system of bipolar fuzzy relation equations as a system of 0-1 mixed integer inequalities and using the Charnes and Cooper’s method. Finally, an algorithm is suggested to solve the problem based on the above reductions. The first method, named as fuzzy primal simplex method, assumes an initial primal basic feasible solution is at hand. The second method, named as fuzzy dual simplex method, assumes an initial dual basic feasible solution is at hand. In this paper, the shortcomings of these methods are pointed out and to overcome these shortcomings, a new method is proposed to determine the fuzzy optimal solution of such fuzzy problems.