Incomplete pairwise comparison matrices and their applications

TL;DR

This paper explores incomplete pairwise comparison matrices, their graph representations, algorithms for completion, and applications, aiming to optimize resource use and decision-making accuracy.

Contribution

It introduces new completion algorithms, discusses inconsistency measures, and demonstrates practical benefits of incomplete comparisons in decision processes.

Findings

Proposes incomplete eigenvector and logarithmic least squares methods.

Discusses inconsistency thresholds based on dominant eigenvalues.

Highlights applications showing advantages of incomplete comparisons.

Abstract

Incomplete pairwise comparison matrices are increasingly employed to save resources and reduce cognitive load by collecting only a subset of all possible pairwise comparisons. We present their graph representation and some completion algorithms, including the incomplete eigenvector and incomplete logarithmic least squares methods, as well as a lexicographical minimisation of triad inconsistencies. The issue of ordinal violations is discussed for matrices generated by directed acyclic graphs and the best--worst method. We also show a reasonable approach to generalise the inconsistency threshold based on the dominant eigenvalue to the incomplete case, and state recent results on the optimal order of obtaining pairwise comparisons. The benefits of using incomplete pairwise comparisons are highlighted by several applications.