Upper Bounded Minimal Solution of the Max-Min Fuzzy Relation Inequality System

Abstract

Resolution of the minimal solutions plays an important role in the research on fuzzy relation equations or inequalities system. Most of the existing works focused on the general minimal solutions or some specific minimal solutions that optimize particular objective functions. In a recently published work, the restricted minimal solution of fuzzy relation inequalities with addition-min composition was studied. Motivated by such an idea, we investigate the so-called upper bounded minimal solution of fuzzy relation inequalities with max-min composition in this work. The upper bounded minimal solution is defined as the minimal solution that is less than or equal to a given vector. Here, the given vector can be viewed as the upper bound. The major content in this work consists of two components: the existence and the resolution of the upper bounded minimal solution. First, we provide some necessary and sufficient conditions to determine whether the upper bounded minimal solution exists with respect to a given vector. Second, when it exists, we further develop two algorithms to search for the upper bounded minimal solution in a step-by-step approach. The validity of our proposed Algorithms I and II is formally proved in theory. The computational complexities of Algorithms I and II are O $(mn)$ and O $(mn^{2})$ , respectively. Moreover, our proposed algorithms are illustrated by some numerical examples.

Keywords

• Fuzzy relation inequality
• fuzzy relation equation
• max-min composition
• minimal solution
• upper bounded