Triple perturbed consistent matrix and the efficiency of its principal right eigenvector

IF 3.2 3区计算机科学 Q2 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

International Journal of Approximate Reasoning Pub Date : 2024-04-18 DOI:10.1016/j.ijar.2024.109204

Rosário Fernandes , Susana Palheira

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引用次数: 0

Abstract

Let A be a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, $x, y, z$ , and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing the three perturbed entries and not containing a diagonal entry. In this paper we describe the relations among $x, y, z$ with which A always has its principal right eigenvector efficient. Previously, and only for a few cases of this problem, R. Fernandes and S. Furtado (2022) proved the efficiency of the principal right eigenvector of A. In this paper, we continue to use the strong connectivity of a certain digraph associated with A and its principal right eigenvector to characterize the vector efficiency. For completeness, we show that the existence of a sink in this digraph is equivalent to the inefficiency of the principal right eigenvector of A.

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三重扰动一致矩阵及其右主特征向量的效率

假设 A 是一个成对比较矩阵，它是通过扰动主对角线上方的三个条目 x、y、z 以及相应的倒数条目而从一致矩阵中得到的，其方式是存在一个大小为 2 的子矩阵，其中包含三个扰动条目，且不包含对角线条目。在本文中，我们将描述 x、y、z 之间的关系，A 的主右特征向量总是有效的。此前，R. Fernandes 和 S. Furtado（2022 年）仅针对该问题的少数情况证明了 A 的主右特征向量的效率。在本文中，我们继续使用与 A 及其主右特征向量相关的某个数图的强连接性来描述向量效率。为完整起见，我们证明了该图中汇的存在等价于 A 的主右特征向量的无效率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Approximate Reasoning 工程技术-计算机：人工智能

CiteScore

6.90

自引率

12.80%

发文量

170

审稿时长

67 days

期刊介绍： The International Journal of Approximate Reasoning is intended to serve as a forum for the treatment of imprecision and uncertainty in Artificial and Computational Intelligence, covering both the foundations of uncertainty theories, and the design of intelligent systems for scientific and engineering applications. It publishes high-quality research papers describing theoretical developments or innovative applications, as well as review articles on topics of general interest. Relevant topics include, but are not limited to, probabilistic reasoning and Bayesian networks, imprecise probabilities, random sets, belief functions (Dempster-Shafer theory), possibility theory, fuzzy sets, rough sets, decision theory, non-additive measures and integrals, qualitative reasoning about uncertainty, comparative probability orderings, game-theoretic probability, default reasoning, nonstandard logics, argumentation systems, inconsistency tolerant reasoning, elicitation techniques, philosophical foundations and psychological models of uncertain reasoning. Domains of application for uncertain reasoning systems include risk analysis and assessment, information retrieval and database design, information fusion, machine learning, data and web mining, computer vision, image and signal processing, intelligent data analysis, statistics, multi-agent systems, etc.