A lexicographically optimal completion for pairwise comparison matrices with missing entries

TL;DR

This paper introduces a new algorithm for estimating missing entries in pairwise comparison matrices, ensuring lexicographical optimality by reducing the most inconsistent triads first, and guarantees uniqueness under a simple connectivity condition.

Contribution

The study proposes a novel, measure-independent algorithm for completing pairwise comparison matrices with missing data, based on inconsistency reduction inspired by the nucleolus concept.

Findings

The algorithm guarantees a unique lexicographically optimal completion when the comparison graph is connected.

It does not depend on arbitrary inconsistency measures, using a single reasonable triad inconsistency index.

The method is applicable to multi-criteria decision making, especially in the Analytic Hierarchy Process.

Abstract

Estimating missing judgements is a key component in many multi-criteria decision making techniques, especially in the Analytic Hierarchy Process. Inspired by the Koczkodaj inconsistency index and a widely used solution concept of cooperative game theory called the nucleolus, the current study proposes a new algorithm for this purpose. In particular, the missing values are substituted by variables, and the inconsistency of the most inconsistent triad is reduced first, followed by the inconsistency of the second most inconsistent triad, and so on. The necessary and sufficient condition for the uniqueness of the suggested lexicographically optimal completion is proved to be a simple graph-theoretic notion: the undirected graph associated with the pairwise comparisons, where the edges represent the known elements, should be connected. Crucially, our method does not depend on an arbitrarily chosen measure of inconsistency as there exists essentially one reasonable triad inconsistency index.