Two-sided fuzzy relation inequalities with addition-min composition

Abstract

Considering the bilateral requirements of the terminals in a P2P network system, we aim to study the two-sided fuzzy relation inequalities with addition-min composition in this work. Each solution of such a two-sided fuzzy relation system is indeed a feasible flow control scheme for the corresponding P2P network system. The major content includes three aspects: (i) finding a minimal solution less than or equal to a given solution; (ii) finding a maximal solution more than or equal to a given solution; (iii) constructing the structure of the solution set to the fuzzy relation system. The purpose of (i) or (ii) is to find some specific minimal or maximal solutions of the two-sided system. We develop two resolution algorithms, i.e., Algorithm I and II, to find the specific minimal and maximal solutions. The computation complexities of both Algorithm I and II are polynomial. Their effectiveness is illustrated by some numerical examples. It is found that the complete solution set of the two-sided system could be fully determined by all the minimal solutions and maximal solutions. Moreover, the solution set might be non-convex.