Solving Linear Optimization Problems Subject to Bipolar Fuzzy Relational Equalities Defined with Max-Hamacher Family of t-Norms

Ghodousian, Amin; Chopannavaz, Mohammad Sedigh

doi:10.311581/JAMDA.2512.1026.1.4.5

Solving Linear Optimization Problems Subject to Bipolar Fuzzy Relational Equalities Defined with Max-Hamacher Family of t-Norms

Document Type : Research Article

Authors

Amin Ghodousian ¹

Mohammad Sedigh Chopannavaz ²

¹ Faculty of Engineering Science, College of Engineering, University of Tehran, P.O.Box 11365-4563, Tehran, Iran.

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

10.311581/JAMDA.2512.1026.1.4.5

Abstract

This paper considers linear objective function optimization under the bipolar system of fuzzy relation equations constraints defined by the max-Hamacher family of t-norms, which is a parametric family of continuous strict t-norms whose members are decreasing functions of the parameters. It is demonstrated that the feasible solution set is represented as the union of a finite number of closed convex cells that are not necessarily connected. In order to determine the feasibility of the proposed system, some necessary and sufficient conditions are derived based on the bipolar FRE constraint defined by the max-Hamacher t-norm. Therefore, the feasible solution set for the problem is completely identified. Also, some simplification techniques have been introduced to accelerate the solution of the current problem, and an algorithm has been developed accordingly in order to identify feasible regions. To further clarify the approach presented in the paper, a step-by-step example is presented in several sections.

Keywords

Bipolar fuzzy relational equations

Hamacher max-t-norms

Strict t-norm

Local optimal solution

Global optimization

Linear optimization