A branch and bound technique for finding the minimal solutions of the linear optimization problems subjected to Lukasiewicz

TL;DR

This paper introduces a branch-and-bound method to find minimal solutions of linear optimization problems constrained by fuzzy relation equations using Lukasiewicz t-norm, addressing the NP-hardness of the problem.

Contribution

It presents a novel branch-and-bound algorithm tailored for minimal solutions in fuzzy relation constrained linear optimization, expanding solution techniques for NP-hard problems.

Findings

Characterized the feasible domain under Lukasiewicz t-norm.

Developed a modified branch-and-bound solution procedure.

Illustrated the method with a concrete example.

Abstract

In this paper, an optimization model with a linear objective function subject to a system of fuzzy relation equations (FRE) is studied where the feasible region is defined by the Lukasiewicz t-norm. Since the finding of all minimal solutions is an NP-hard problem, designing an efficient solution procedure for solving such problems is not a trivial job. Firstly, the feasible domain is characterized and then the problem is solved with a modified branch-and-bound solution technique based on a new solution set that includes the minimal solutions. After presenting our solution procedure, a concrete example is included for illustration purposes.