{"title":"Lexicographic minimum solutions of fuzzy relation inequalities with product-max-min composition","authors":"Zhining Wang , Xue-ping Wang","doi":"10.1016/j.fss.2025.109363","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"510 ","pages":"Article 109363"},"PeriodicalIF":3.2000,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011425001022","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.