Efficient vectors for simple perturbed consistent matrices

Henrique F. da Cruz, Rosário Fernandes, Susana Furtado

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)54-68
Number of pages15
JournalInternational Journal of Approximate Reasoning
Publication statusPublished - Dec 2021


  • Consistent matrix
  • Decision-making
  • Pairwise comparison matrix
  • Vector efficiency


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