{"title":"Geometric interpretation of efficient weight vectors","authors":"Zsombor Szádoczki, Sándor Bozóki","doi":"10.1016/j.knosys.2024.112403","DOIUrl":null,"url":null,"abstract":"<div><p>Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.</p></div>","PeriodicalId":49939,"journal":{"name":"Knowledge-Based Systems","volume":"303 ","pages":"Article 112403"},"PeriodicalIF":7.2000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0950705124010372/pdfft?md5=d2357383b0b1fd8f0f64eeccf147c178&pid=1-s2.0-S0950705124010372-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Knowledge-Based Systems","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0950705124010372","RegionNum":1,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Pairwise comparison matrices (PCMs) are frequently used in different multicriteria decision making problems. A weight vector is said to be efficient if no other weight vector is at least as good in estimating the elements of the PCM, and strictly better in at least one position. Understanding the efficient weight vectors is crucial to determine the appropriate weight calculation technique for a given problem. In this paper we study the set of efficient weight vectors for three and four dimensions (alternatives) from a geometric viewpoint, which is a complementary to the algebraic approach used in the literature. Besides providing well-interpretable demonstrations, we also draw attention to the particular role of weight vectors calculated from spanning trees. Weight vectors corresponding to line graphs are vertices of the (polyhedral, but usually nonconvex) set of efficient weight vectors, while weight vectors corresponding to other spanning trees are also on the boundary.
期刊介绍:
Knowledge-Based Systems, an international and interdisciplinary journal in artificial intelligence, publishes original, innovative, and creative research results in the field. It focuses on knowledge-based and other artificial intelligence techniques-based systems. The journal aims to support human prediction and decision-making through data science and computation techniques, provide a balanced coverage of theory and practical study, and encourage the development and implementation of knowledge-based intelligence models, methods, systems, and software tools. Applications in business, government, education, engineering, and healthcare are emphasized.