Resolution and simplification of Dombi-fuzzy relational equations and latticized optimization programming on Dombi FREs

TL;DR

This paper introduces a method for solving complex fuzzy relational equations based on Dombi t-norms, providing conditions for feasibility and an algorithm to find exact solutions in non-convex, NP-hard optimization problems.

Contribution

It develops a novel approach for resolving Dombi-fuzzy relational equations and latticized optimization programming, including feasibility conditions and an exact solution algorithm.

Findings

Feasibility conditions for Dombi-fuzzy relational equations derived.

Feasible solutions characterized as finite convex cells.

Algorithm successfully finds exact solutions in examples.

Abstract

In this paper, we introduce a type of latticized optimization problem whose objective function is the maximum component function and the feasible region is defined as a system of fuzzy relational equalities (FRE) 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. Some necessary and sufficient conditions are derived to determine the feasibility of the problem. The feasible solution set is characterized in terms of a finite number of closed convex cells. An algorithm is presented for solving this nonlinear problem. It is proved that the algorithm can find the exact optimal solution and an example is presented to illustrate the proposed algorithm.