08-04-2016 | Original Article | Issue 5/2017

# Lexicography minimum solution of fuzzy relation inequalities: applied to optimal control in P2P file sharing system

Important notes

Supported by the Innovation and Building Strong School Project of Colleges of Guangdong Province (2015KQNCX094); Special funds of public welfare research and capacity building of Guangdong Province (2015A010103015); Natural Science Foundation of Guangdong Province (2014A030307014); High Level Talents in Colleges of Guangdong Province (Guangdong Finance Education [2013] No. 246).

## Abstract

A peer-to-peer (P2P) file sharing system can be reduced into a system of addition-min fuzzy relation inequalities. Concept of lexicography minimum solution is introduced and applied to such system. It is found that the unique lexicography minimum solution can be selected from the minimal solution set of the corresponding fuzzy relation inequalities. However it is difficult to find the minimal solution set which is probably infinite. In order to avoid such difficulty, we propose a so-called Circulation Algorithm to find the unique lexicography minimum solution. The algorithm is developed step by step and illustrated by a numerical application example.