The solution set of fuzzy relation equations with addition-min composition

TL;DR

This paper presents algorithms to find all minimal and maximal solutions of fuzzy relation equations with addition-min composition, and characterizes the entire solution set between these bounds.

Contribution

It introduces a method to completely describe the solution set of fuzzy relation equations with addition-min composition by identifying all extremal solutions.

Findings

Algorithms for minimal and maximal solutions are provided.

The entire solution set is characterized between minimal and maximal solutions.

Numerical examples illustrate the algorithms and theoretical results.

Abstract

This paper deals with the resolutions of fuzzy relation equations with addition-min composition. When the fuzzy relation equations have a solution, we first propose an algorithm to find all minimal solutions of the fuzzy relation equations and also supply an algorithm to find all maximal solutions of the fuzzy relation equations, which will be illustrated, respectively, by numeral examples. Then we prove that every solution of the fuzzy relation equations is between a minimal solution and a maximal one, so that we describe the solution set of the fuzzy relation equations completely.