minimizing a polynomial function subject to bipolar fuzzy relation equation constraints

A polynomial programming problem subject to the max-product bipolar fuzzy relation equation constraints is studied in this paper. its structure of feasible domain is firstly investigated using the lower and upper bound of its feasible solutions. some sufficient conditions are proposed to determine the optimal solution of the problem based on the covering concept. moreover, a value matrix is created to display the candidate components for optimality. a matrix-based algorithm is designed to solve the problem.