Efficiency of the convex hull of the columns of certain triple perturbed consistent matrices

TL;DR

This paper investigates the efficiency of convex hull vectors of certain perturbed reciprocal matrices in decision making, providing conditions under which all convex combinations are efficient and comparing them to geometric mean vectors.

Contribution

It offers necessary and sufficient conditions for the efficiency of convex hull vectors in specific perturbed matrices, extending previous results on the Perron vector.

Findings

All vectors in the convex hull are efficient under certain perturbation conditions.

The paper generalizes known efficiency conditions for the Perron vector.

Numerical examples compare convex hull vectors with geometric mean vectors.

Abstract

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A, a classical proposal for the weight vector, and the Perron eigenvector of AA^{T} (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.