Linear optimization with mixed fuzzy relation inequality constraints using the pseudo-t-norms and its application

Abstract

This paper studies the minimization problem of a linear objective function subject to mixed fuzzy relation inequalities (MFRIs) over finite support with regard to max-T 1 and max-T 2 composition operators, where T 1 and T 2 are two pseudo-t-norms. We first determine the structure of its feasible domain and then show that the solution set of a MFRI system is determined by a maximum solution and a finite number of minimal solutions. Moreover, sufficient and necessary conditions are proposed to check whether the feasible domain of the problem is empty or not. The MFRI path is defined to determine the minimal solutions of its feasible domain. The resolution process of the optimization problem is also designed based on the structure of its feasible domain. Procedures are proposed to reduce the size of the problem. With regard to the above points and the procedures, an algorithm is designed to solve the problem. Its application is expressed in the area of investing and covering. Finally, the algorithm is compared with other approaches.

Linear optimization with bipolar max-parametric hamacher fuzzy relation equation constraints

“…The constraint part of problem (1) is to find a set of solution vectors x ∈ [0, 1] n for the following system of bipolar fuzzy relation equations…”

“…After it, we explain the modified branch and bound (B&B) method with the jump-tracking technique to solve the optimization problem (11). Then we present an algorithm for resolution of problem (1). At first, we need to express the following theorem.…”

“…Then a polynomial-time algorithm was proposed for its resolution. Abbasi Molai [1] studied the linear optimization problem with the Mixed Fuzzy Relation Inequality (MFRI) constraints with two max-pseudo t-norms. He investigated to the structure of its feasible domain and designed an algorithm to solve the problem.…”

“…With regard to the importance of MFREs [1,10,17] and BFREs [11,20,21], we consider the linear optimization problem with BFRE constraints with max-parametric hamacher composition operators and bipolar variables, simultaneously. In this problem, the composition operator of each its constraint can be associated to a different members of the parametric hamacher family.…”