Efficiency analysis of simple perturbed pairwise comparison matrices

TL;DR

This paper investigates the efficiency of the eigenvector method for simple perturbed pairwise comparison matrices, showing that the principal eigenvector is always efficient in this context.

Contribution

It introduces the concept of simple perturbed matrices and proves the efficiency of their principal eigenvectors, advancing understanding of weighting methods in pairwise comparisons.

Findings

Principal eigenvector of simple perturbed matrices is efficient.

Defines simple perturbed matrices as differing in one element and its reciprocal.

Provides theoretical proof of eigenvector efficiency in this class.

Abstract

Efficiency, the basic concept of multi-objective optimization is investigated for the class of pairwise comparison matrices. A weight vector is called efficient if no alternative weight vector exists such that every pairwise ratio of the latter's components is at least as close to the corresponding element of the pairwise comparison matrix as the one of the former's components is, and the latter's approximation is strictly better in at least one position. A pairwise comparison matrix is called simple perturbed if it differs from a consistent pairwise comparison matrix in one element and its reciprocal. One of the classical weighting methods, the eigenvector method is analyzed. It is shown in the paper that the principal right eigenvector of a simple perturbed pairwise comparison matrix is efficient.