Characteristics of the Feasible Solution Set of Optimization Problems with Bipolar Fuzzy Relational Equality Constraints Defined by the Max--Einstein-Product Composition

Ghodousian, Amin; Chopannavaz, Mohammad Sedigh

Characteristics of the Feasible Solution Set of Optimization Problems with Bipolar Fuzzy Relational Equality Constraints Defined by the Max--Einstein-Product Composition

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.

Abstract

This paper studies the structure of feasible solution sets for optimization problems subject to bipolar fuzzy relational equality constraints defined by the max--Einstein-product composition, where the Einstein product is a continuous strict t-norm. We derive explicit one-dimensional feasible sets for the individual bipolar constraints and use them to obtain necessary and sufficient feasibility conditions for the complete system. The feasible solution set is represented as the union of finitely many compact sets; equivalently, it can be decomposed into a finite union of closed convex cells and need not be connected. An admissible-function representation is developed to describe the complete feasible region, and an algorithm is given to implement the resulting resolution procedure. A step-by-step numerical example illustrates the construction.

Keywords

Bipolar fuzzy relational equations

Einstein Product max-t-norms

Strict t-norm

Feasible solution sets

Optimization problem