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Fuzzy relational inequalities with fuzzy constraints (FRI-FC) are thegeneralized form of fuzzy relational inequalities (FRI) in which fuzzyinequality replaces ordinary inequality in the constraints. Fuzzy constraintsenable us to attain optimal points (called super-optima) that are bettersolutions than those resulted from the resolution of the similar problems withordinary inequality constraints. This paper considers the linear objectivefunction optimization with respect to max-product FRI-FC problems. It is provedthat there is a set of optimization problems equivalent to the primal problem.Based on the algebraic structure of the primal problem and its equivalentforms, some simplification operations are presented to convert the main probleminto a more simplified one. Finally, by some appropriate mathematicalmanipulations, the main problem is transformed into an optimization model whoseconstraints are linear. The proposed linearization method not only provides asuper-optimum (that is better solution than ordinary feasible optimalsolutions) but also finds the best super-optimum for the main problem. Thecurrent approach is compared with our previous work and some well-knownheuristic algorithms by applying them to random test problems in differentsizes.

An exact algorithm for linear optimization problem subject to max-product fuzzy relational inequalities with fuzzy constraints