Linear optimization on Hamacher-fuzzy relational inequalities

In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated where the feasible region is formed as the intersection of two inequality fuzzy systems and Hamacher family of t-norms is considered as fuzzy composition. Hamacher family of t-norms is a parametric family of continuous strict t-norms, whose members are decreasing functions of the parameter. The resolution of the feasible region of the problem is firstly investigated when it is defined with max-Hamacher composition. Based on some theoretical results, a necessary and sufficient condition and three other necessary conditions are derived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented. It is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. A method is proposed to generate random feasible max-Hamacher fuzzy relational inequalities and an algorithm is presented to solve the problem. Finally, an example is described to illustrate these algorithms.