2024/04/21 by Susana Furtado, Furtado, Susana, Charles R. Johnson +1 · 1 citation
Chemistry · Computer Science · Decision Sciences · #05C20 #15A18 #15B48 #90B50 #91B06 #Combinatorics (math.CO) #FOS: Mathematics #History and advancements in chemistry #Intuitionistic Fuzzy Systems Applications #Multi-Criteria Decision Making
paper · pdf · doi:10.48550/arxiv.2404.13713
openalex publication_date 2024/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the vector of weights. It is known that the Perron vector may not be efficient. Here, we focus on extending arbitrary reciprocal matrices and show, constructively, that two different extensions of 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.