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

TL;DR

This paper introduces an exact algorithm for solving max-product fuzzy relational inequalities with fuzzy constraints, transforming the problem into a linear optimization model to find super-optimal solutions.

Contribution

It presents a novel linearization method that converts complex fuzzy relational inequalities into a linear form, enabling the computation of super-optima.

Findings

The method finds the best super-optimum solutions.

It outperforms previous approaches and heuristics on test problems.

The approach simplifies complex fuzzy problems into linear optimization models.

Abstract

Fuzzy relational inequalities with fuzzy constraints (FRI-FC) are the generalized form of fuzzy relational inequalities (FRI) in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy constraints enable us to attain optimal points (called super-optima) that are better solutions than those resulted from the resolution of the similar problems with ordinary inequality constraints. This paper considers the linear objective function optimization with respect to max-product FRI-FC problems. It is proved that there is a set of optimization problems equivalent to the primal problem. Based on the algebraic structure of the primal problem and its equivalent forms, some simplification operations are presented to convert the main problem into a more simplified one. Finally, by some appropriate mathematical manipulations, the main problem is transformed into an optimization model whose constraints are linear. The proposed linearization method not only provides a super-optimum (that is better solution than ordinary feasible optimal solutions) but also finds the best super-optimum for the main problem. The current approach is compared with our previous work and some well-known heuristic algorithms by applying them to random test problems in different sizes.