Lexicographic minimum solutions of fuzzy relation inequalities with product-max-min composition

Abstract

With the consideration of the potential line faults in a Peer-to-Peer (P2P) network, the success rates of the lines have already been incorporated into the P2P network. A novel system of fuzzy relation inequalities (FRIs) with product-max-min composition has been proposed for characterizing such kind of P2P network. In this article, some essential properties of characteristic matrices (resp. discriminant matrices and submatrices) of the system are first examined and, as a by-product, the structure theorem for the entire solution set of the system is established by relying on the submatrices. In particular, it is proved that for every solution of the system there is a minimal solution that is less than or equal to the solution. Moreover, in order to alleviate the network congestion based on a given priority level of the nodes, the concept of a lexicographic minimum solution of the system is introduced and an algorithm named a CAD-based resolution algorithm is proposed for calculating such solutions. Several examples are given to demonstrate the algorithm.