TY - JOUR

T1 - Efficiency of the principal eigenvector of some triple perturbed consistent matrices

AU - Fernandes, Rosário

AU - Furtado, Susana

N1 - Funding Information:
The authors would like to thank the referees for the valuable comments that helped improving the first version of this paper.

PY - 2022/5/1

Y1 - 2022/5/1

N2 - In this paper we study the efficiency of the principal eigenvector of a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing three of the perturbed entries and not containing a diagonal entry. Our approach uses the characterization of vector efficiency based on the strongly connection of a certain digraph associated with the matrix. By imposing some conditions on the magnitudes of the perturbations, we identify a class of matrices for which the principal eigenvector is efficient. This class is the union of two subclasses such that the digraph of all the matrices in the same subclass contains a common cycle with all vertices. The matrices we analyse include the simple perturbed consistent matrices, as well as a subfamily of the double perturbed consistent matrices, whose principal eigenvector efficiency has been studied in the literature. For completeness, we also include some examples of triple perturbed matrices for which the principal eigenvector is not efficient. Explicit expressions for the characteristic polynomial and the principal eigenvector of the triple perturbed consistent matrices we consider are also given.

AB - In this paper we study the efficiency of the principal eigenvector of a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing three of the perturbed entries and not containing a diagonal entry. Our approach uses the characterization of vector efficiency based on the strongly connection of a certain digraph associated with the matrix. By imposing some conditions on the magnitudes of the perturbations, we identify a class of matrices for which the principal eigenvector is efficient. This class is the union of two subclasses such that the digraph of all the matrices in the same subclass contains a common cycle with all vertices. The matrices we analyse include the simple perturbed consistent matrices, as well as a subfamily of the double perturbed consistent matrices, whose principal eigenvector efficiency has been studied in the literature. For completeness, we also include some examples of triple perturbed matrices for which the principal eigenvector is not efficient. Explicit expressions for the characteristic polynomial and the principal eigenvector of the triple perturbed consistent matrices we consider are also given.

KW - Consistent matrix

KW - Decision processes

KW - Pairwise comparison matrix

KW - Principal eigenvector efficiency

KW - Strongly connected graph

UR - http://www.scopus.com/inward/record.url?scp=85114362168&partnerID=8YFLogxK

U2 - 10.1016/j.ejor.2021.08.012

DO - 10.1016/j.ejor.2021.08.012

M3 - Article

AN - SCOPUS:85114362168

SN - 0377-2217

VL - 298

SP - 1007

EP - 1015

JO - European Journal of Operational Research

JF - European Journal of Operational Research

IS - 3

ER -