A fast method for solving the linear optimization p roblem subjected to simplified Dombi fuzzy relational equations

TL;DR

This paper introduces a fast, modified branch-and-bound method for solving linear optimization problems constrained by simplified Dombi fuzzy relational equations, addressing the non-convexity and NP-hardness of the feasible solution set.

Contribution

It presents a novel, efficient solution technique based on theoretical properties and domain characterization for fuzzy relation equations involving Dombi t-norms.

Findings

The method effectively reduces the number of feasible paths to find solutions.

The approach is demonstrated with a concrete example.

It improves computational efficiency for complex fuzzy relational systems.

Abstract

In this paper, an optimization model with a linear objective function subjected to a system of fuzzy relation equations (FRE) is studied where the feasible region is defined by the Dombi t-norm. Dombi family of t-norms includes a parametric family of continuous strict t-norms, whose members are increasing functions of the parameter. This family of t-norms covers the whole spectrum of t-norms when the parameter is changed from zero to infinity. Since the feasible solutions set of FREs is non-convex and 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, based on some theoretical properties of the problem, a modified branch-and-bound solution technique is presented, which solves the problem by considering a few number of feasible paths. After presenting our solution procedure, a concrete example is included for illustration purpose.