Efficient vectors for double perturbed consistent matrices

Abstract

Efficiency, also called Pareto optimality, is a fundamental concept in multi-criteria decision making. In this paper we describe all the efficient vectors for an $n\times n$ double perturbed consistent matrix A, that is, a pairwise comparison matrix obtained from a consistent one by perturbing two entries above the main diagonal and the corresponding reciprocal entries. We also give conditions under which, when deleting a certain entry of an efficient vector for A, we obtain an efficient vector for the corresponding $(n-1)\times (n-1)$ principal submatrix of A. As a simple consequence of our work, we obtain the result by Ábele-Nagy et al. (2018) which states that the principal eigenvector of a double perturbed consistent matrix is efficient. This paper extends the recent paper by Cruz et al. (2021) in which the description of the efficient vectors for simple perturbed consistent matrices is given.

Keywords:

• Consistent matrix
• decision making
• pairwise comparison matrix
• perturbation
• vector efficiency

2020 Mathematics Subject Classifications:

• 90B50
• 91B06
• 15A18
• 05C20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of this author was supported by FCT- Fundação para a Ciência e Tecnologia, under project UIDB/04721/2020.