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On the resolution and linear optimization problems subject to a system of bipolar fuzzy relational equalities defined with continuous Archimedean t-norms

Document Type : Research Paper

Authors

1 Faculty of Engineering Science, College of Engineering, University of Tehran

2 Department of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran

3 Department of Electrical and Computer Engineering, University of Alberta

Abstract

Bipolar fuzzy relational equations are an interesting generalization of fuzzy relational equations that occur prominently in information processing, possibility theory, and preference modeling. This paper considers the linear objective function optimization with respect to a more general class of bipolar fuzzy relational equations, where the fuzzy compositions are defined by an arbitrary continuous Archimedean t-norm. In addition, a faster method for finding a global optimum is proposed that, unlike the previous work, does not require obtaining all local optimal solutions and classifying the constraints (and therefore, it does not require applying different approaches to check the feasibility of constraints in different groups and the optimality of solutions in their feasible regions). Analytical concepts and properties of the Archimedean bipolar fuzzy equations are investigated and two necessary conditions are presented to conceptualize the feasibility of the problem. It is shown that the feasible solution set can be resulted by a union of the finite number of compact sets, where each compact set is obtained by a function (called admissible function in this paper). Moreover, to accelerate identification of the mentioned compact sets (and therefore, to speed up solution finding), four simplification techniques are presented, which are based on either omitting redundant constraints and/or eliminating unknowns by assigning them a fixed value. Also, three additional simplification techniques are given to reduce the search domain by removing some parts of the feasible region that do not contain optimal solutions. Subsequently, a method is proposed to find an optimal solution for the current linear optimization problems. The proposed method consists of two accelerative strategies that are used during the problem solving process. By the first strategy, the method neglects some candidate solutions that are not optimal, by considering only a subset of admissible functions (called modified functions in this paper). As for the second strategy, a branch-and-bound method is used to delete non-optimal branches. Then, the method is summarized in an algorithm that represents all essential steps of the solution and finally, the whole method is applied in an example that has been chosen in such a way that the various situations are illustrated.

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Iranian Journal of Fuzzy Systems
Volume 22, Issue 4
July and August 2025
Pages 137-160
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