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A lexicographically optimal completion for pairwise comparison matrices with missing entries

Kolos Csaba Ágoston, László Csató

TL;DR

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.

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.

A lexicographically optimal completion for pairwise comparison matrices with missing entries

TL;DR

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.

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.
Paper Structure (7 sections, 1 theorem, 31 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 7 sections, 1 theorem, 31 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A lexicographically optimal completion of the missing elements
  4. A necessary and sufficient condition for the uniqueness of the lexicographically optimal completion
  5. The case of independent missing entries
  6. Numerical experiment: the similarity of completion methods
  7. Conclusions

Key Result

Theorem 1

The lexicographically optimal completion is unique if and only if the graph $G$ associated with the incomplete pairwise comparison matrix is connected.

Figures (3)

  • Figure 1: The graph representation of the pairwise comparison matrix $\mathbf{A}$ in Example \ref{['Examp1']}
  • Figure 2: Triad inconsistencies as a function of unknown entries in Example \ref{['Examp5']}
  • Figure 3: Comparison of the lexicographically optimal filling and the eigenvector method for incomplete pairwise comparison matrices

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 1
  • Example 2
  • Definition 3.1
  • ...and 7 more