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In this paper, the linear programming problem subject to the Bipolar Fuzzy Relation Equation (BFRE) constraints with the max-parametric hamacher composition operators is studied. The structure of its feasible domain is investigated and its feasible solution set determined. Some necessary and sufficient conditions are presented for its solution existence. Then the problem is converted to an equivalent programming problem. Some rules are proposed to reduce the dimensions of problem. Under these rules, some of the optimal variables are found without solving the problem. An algorithm is then designed to find an upper bound for its optimal objective value. With regard to this algorithm, a modified branch and bound method is extended to solve the problem. We combine the rules, the algorithm, and the modified branch and bound method in terms of an algorithm to solve the original problem.

Linear optimization with bipolar max-parametric hamacher fuzzy relation equation constraints