An Algorithm for Solving Linear Optimization Problems Subjected to the Intersection of Two Fuzzy Relational Inequalities Defined by Frank Family of T-Norms

Abstract

Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated. The feasible region is formed as the intersection of two inequality fuzzy systems defined by frank family of t-norms is considered as fuzzy composition. First, the resolution of the feasible solutions set is studied where the two fuzzy inequality systems are defined with max-Frank composition. Second, some related basic and theoretical properties are derived. Then, a necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem. Subsequently, it is shown that a lower bound is always attainable for the optimal objective value. Also, it is proved that the optimal solution of the problem is always resulted from the unique maximum solution and a minimal solution of the feasible region. Finally, an algorithm is presented to solve the problem and an example is described to illustrate the algorithm. Additionally, a method is proposed to generate random feasible max-Frank fuzzy relational inequalities. By this method, we can easily generate a feasible test problem and employ our algorithm to it. Fuzzy relation, fuzzy relational inequality, linear optimization, fuzzy compositions and t-norms. KEYWORDS

A modified PSO algorithm for linear optimization problem subject to the generalized fuzzy relational inequalities with fuzzy constraints (FRI-FC)

“…In fact, the same result holds true for a more general t-norms instead of the minimum and product operators [3,29,30]. Over the past decades, the solvability of FRE which is defined using different max-t compositions have been investigated by many researchers [15,16,19,39,43,46,47,50,52]. Moreover, some other researchers have worked on introducing novel concept and at times improving some of the existing theoretical aspects and applications of fuzzy relational inequalities (FRI) [13,17,18,20,21,27,53].…”

“…Over the past decades, the solvability of FRE which is defined using different max-t compositions have been investigated by many researchers [15,16,19,39,43,46,47,50,52]. Moreover, some other researchers have worked on introducing novel concept and at times improving some of the existing theoretical aspects and applications of fuzzy relational inequalities (FRI) [13,17,18,20,21,27,53]. Li and Yang [27] studied FRI with addition-min composition and presented an algorithm to search for minimal solutions.…”

“…They also introduced some rules for reducing the problems and presented an algorithm for solving optimization problems using FRI constraints. The optimization problem subject to FRE and FRI is one of the most interesting and on-going research topics amongst similar problems [1,8,12,13,[15][16][17][18][19][20][21][22][23]26,31,40,44,48,53]. Many methods were designed based on the translation of the main problem into an integer linear programming problem which is then solved using well-developed techniques.…”

“…Also, in [23] some necessary conditions to test the feasibility and simplification techniques were presented in order to solve FRE with max-product composition. Moreover, generalizations of the linear optimization problem with respect to FRE have been studied; this was done through replacement of max-min and maxproduct compositions with different fuzzy compositions such as max-average composition [48] or max-t-norm composition [15,16,19,22,26,44]. For example, Li and Fang solved the linear optimization problem subjected to a system of sup-t equations by reducing it to a 0-1 integer optimization problem [26].…”

“…In a separate, the foregoing bipolar linear optimization problem was solved by an analytical method based on the resolution and some structural properties of the feasible region (using a necessary condition for characterizing an optimal solution and a simplification process for reducing the problem) [31]. The optimization problem subjected to various versions of FRI is widely available in the literature as well [12,13,17,18,20,21,53,54]. In [13], the authors studied the linear optimization with constraints formed by…”