Arbitrary-term-absent max-product fuzzy relation inequalities and its lexicographic minimal solution

Abstract

Recently the max-product fuzzy relation inequalities were applied to describe the wireless communication station system. However, when an arbitrary line fault appears between the emission basic station and the signal testing point, the classical max-product system is no longer effective. Considering an arbitrary line fault in the wireless communication station system, we introduce and investigate the so-called arbitrary-term-absent (ATA) max-product fuzzy relation inequalities in this paper. In order to reduce the damage caused by the electromagnetic radiation from the basic stations, a minimal solution is usually desired. However, the structure of solution set to the ATA max-product fuzzy relation inequalities shows that the minimal solutions are not unique in most cases. Hence, we further introduce the lexicographic order and define the relevant concept of lexicographic minimal solution. A lexicographic minimal solution is indeed an optimal scheduling, minimizing the electromagnetic radiations in a lexicographic order. An efficient algorithm with polynomial complexity is developed for obtaining the unique lexicographic minimal solution of the ATA max-product system. At last, our proposed resolution algorithm is compared to the existing one and illustrated by a numerical example.