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A clustering approach for pairwise comparison matrices

Kolos Csaba Ágoston, Sándor Bozóki, László Csató

TL;DR

This paper addresses clustering in large-scale group decision making where opinions are expressed as pairwise comparison matrices (PCMs). It introduces a $k$-medoids clustering framework with a linear programming formulation, ensuring cluster centres are actual PCMs and enabling constraints on cluster inconsistency, while accommodating incomplete data. Using dissimilarities such as $D_1$ and $D_3$, the approach detects data errors (e.g., typos) and reveals common sources of inconsistency without relying on a single weighting scheme. Applied to Bozoki et al. data, the method yields balanced clusters largely robust to the chosen dissimilarity measure, and demonstrates practical benefits such as outlier detection and nuanced pattern discovery in both objective and subjective PCM samples.

Abstract

We consider clustering in group decision making where the opinions are given by pairwise comparison matrices. In particular, the k-medoids model is suggested to classify the matrices since it has a linear programming problem formulation that may contain any condition on the properties of the cluster centres. Its objective function depends on the measure of dissimilarity between the matrices but not on the weights derived from them. Our methodology provides a convenient tool for decision support, for instance, it can be used to quantify the reliability of the aggregation. The proposed theoretical framework is applied to a large-scale experimental dataset, on which it is able to automatically detect some mistakes made by the decision-makers, as well as to identify a common source of inconsistency.

A clustering approach for pairwise comparison matrices

TL;DR

This paper addresses clustering in large-scale group decision making where opinions are expressed as pairwise comparison matrices (PCMs). It introduces a -medoids clustering framework with a linear programming formulation, ensuring cluster centres are actual PCMs and enabling constraints on cluster inconsistency, while accommodating incomplete data. Using dissimilarities such as and , the approach detects data errors (e.g., typos) and reveals common sources of inconsistency without relying on a single weighting scheme. Applied to Bozoki et al. data, the method yields balanced clusters largely robust to the chosen dissimilarity measure, and demonstrates practical benefits such as outlier detection and nuanced pattern discovery in both objective and subjective PCM samples.

Abstract

We consider clustering in group decision making where the opinions are given by pairwise comparison matrices. In particular, the k-medoids model is suggested to classify the matrices since it has a linear programming problem formulation that may contain any condition on the properties of the cluster centres. Its objective function depends on the measure of dissimilarity between the matrices but not on the weights derived from them. Our methodology provides a convenient tool for decision support, for instance, it can be used to quantify the reliability of the aggregation. The proposed theoretical framework is applied to a large-scale experimental dataset, on which it is able to automatically detect some mistakes made by the decision-makers, as well as to identify a common source of inconsistency.
Paper Structure (14 sections, 3 theorems, 33 equations, 6 figures, 4 tables)

This paper contains 14 sections, 3 theorems, 33 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Pairwise comparison matrices and their similarity
  3. Pairwise comparison matrices
  4. Quantifying the similarity of pairwise comparison matrices
  5. Cluster analysis and the $k$-medoids problem
  6. Numerical results
  7. Analysing a sample of objective type
  8. Analysing a sample of subjective type
  9. The appropriate number of clusters
  10. Typical patterns in pairwise comparison matrices
  11. Discussion
  12. The potential role of cluster analysis in the study of pairwise comparison matrices
  13. Incorporating demographic factors into the cluster analysis
  14. Conclusions

Key Result

Lemma 1

Dissimilarity index $D_3$ does not satisfy the triangle inequality.

Figures (6)

  • Figure 1: Distribution of inconsistency ratios $\mathit{CR}$, dataset M4, $k=2$ clusters
  • Figure 2: Approximated coordinates of PCMs, dataset M4, measure $D_3$ Notes: The two clusters are represented by red dots and blue squares. A darker mark is associated with a higher inconsistency. The cluster centers are green and have double sizes.
  • Figure 3: Distribution of inconsistency ratios $\mathit{CR}$, dataset S4, $k=4$ clusters
  • Figure 4: The optimum as a function of the number of clusters, dataset S4
  • Figure 5: Dendrograms, sample S4
  • ...and 1 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof