Efficient weight vectors from pairwise comparison matrices

TL;DR

This paper introduces algorithms to test and compute efficient weight vectors from pairwise comparison matrices, improving decision-making accuracy in multi-criteria analysis.

Contribution

It presents linear programming methods to verify and find efficient or weakly efficient weight vectors, including implementation in an online calculator.

Findings

Principal eigenvector is always weakly efficient.

Numerical examples show eigenvector can be inefficient.

Algorithms effectively identify and improve weight vectors.

Abstract

Pairwise comparison matrices are frequently applied in multi-criteria decision making. A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and strictly better in at least one position. A weight vector is weakly efficient if the pairwise ratios cannot be improved in all non-diagonal positions. We show that the principal eigenvector is always weakly efficient, but numerical examples show that it can be inefficient. The linear programs proposed test whether a given weight vector is (weakly) efficient, and in case of (strong) inefficiency, an efficient (strongly) dominating weight vector is calculated. The proposed algorithms are implemented in Pairwise Comparison Matrix Calculator, available at pcmc.online.