Efficient vectors for simple perturbed consistent matrices

In the Analytic Hierarchy process, a method used in Decision Making, it may be important to approximate a pairwise comparison matrix (PC matrix) by a consistent one. In this context, the notion of efficient vector for a PC matrix arises. In this paper we describe all efficient vectors for an n×n com...

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Bibliographic Details
Published in:International journal of approximate reasoning 2021-12, Vol.139, p.54-68
Main Authors: da Cruz, Henrique F., Fernandes, Rosário, Furtado, Susana
Format: Article
Language:English
Subjects:
Consistent matrix
Decision-making
Pairwise comparison matrix
Vector efficiency
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Summary:In the Analytic Hierarchy process, a method used in Decision Making, it may be important to approximate a pairwise comparison matrix (PC matrix) by a consistent one. In this context, the notion of efficient vector for a PC matrix arises. In this paper we describe all efficient vectors for an n×n comparison pairwise matrix obtained from a consistent one by perturbing one entry above the main diagonal, and the corresponding reciprocal entry. As a consequence, we give a simple proof of the result obtained by K. Ábele-Nagy and S. Bozóki (2016) that states that the principal eigenvector of a simple perturbed consistent matrix is efficient. In addition, we consider a set of non-efficient vectors associated with the simple perturbed consistent matrix and describe all the efficient vectors that dominate each vector in that set.
ISSN:0888-613X
1873-4731
DOI:10.1016/j.ijar.2021.09.007