On the resolution and Linear programming problems subjected by Aczel-Alsina Fuzzy relational equations

TL;DR

This paper investigates linear programming problems constrained by Aczel-Alsina fuzzy relational equations, analyzing their solution space, and proposes an algorithm to find all optimal solutions despite the non-convexity and NP-hardness of the problem.

Contribution

It introduces a novel algorithm for solving linear optimization problems with Aczel-Alsina fuzzy relational constraints, addressing the challenge of non-convex feasible regions.

Findings

The feasible region is non-convex and complex.

An algorithm is developed to find all optimal solutions.

The approach is illustrated with a practical example.

Abstract

Aczel-Alsina t-norm belongs to the family of strict t-norms that are the most applied fuzzy operators in various fuzzy modelling problems. In this paper, we study a linear optimization problem where the feasible region is formed as a system of fuzzy relational equations (FRE) defined by the Aczel-Alsina t-norm. Since the feasible solutions set of FREs is non-convex and the finding of all minimal solutions is an NP-hard problem, conventional methods may not be directly employed. The resolution of the feasible region is completely investigated. Based on some theoretical properties of the problem, an algorithm is presented to find all the optimal solutions, and finally an example is described to illustrate this algorithm.