Efficiency analysis for the Perron vector of a reciprocal matrix

• Type: Article in International Scientific Journal
• Date: 2024

Authors

Susana Borges Furtado (Author) FEP; Johnson, CR (Author) Other

Journal

Title: Applied Mathematics and Computation

Vol. 480

ISSN: 0096-3003

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 5 Citations

Scopus - 5 Citations

Other information

Authenticus ID: P-010-P39

DOI: 10.1016/j.amc.2024.128913

Abstract (EN): In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is necessary to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. One of the most used ranking methods employs the (right) Perron eigenvector of the reciprocal matrix as vector of weights. It is known that the Perron vector may not be efficient. Here, we focus extending arbitrary reciprocal matrices and show, constructively, that two different extensions any fixed size always exist for which the Perron vector is inefficient and for which it is efficient, with the following exception. If B is consistent, any reciprocal matrix obtained from B by adding one row and one column has efficient Perron vector. As a consequence of our results, we obtain families of reciprocal matrices for which the Perron vector is inefficient. These include known classes of such matrices and many more. We also characterize the 4-by-4 reciprocal matrices with inefficient Perron vector. Some prior results are generalized or completed.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 14