Matrix Aggregation Operators

2026-09-23

AI Summary

This study addresses the limitations of traditional vector aggregation operators, which flatten matrices and thereby lose structural information while relying on decomposability. To overcome these issues, this work formally defines the theoretical framework of Matrix Aggregation Operators (MAOs) for the first time, systematically analyzing their decomposability and symmetry properties. Drawing upon fuzzy set theory and the maximum entropy principle, and integrating grouping functions with the MEOWA operator, it proposes the Maximum Entropy Global Covering Index (MEGCI) family along with its construction methodology. The research reveals the existence of non-decomposable operators and successfully constructs a series of MEGCI metrics. Computational experiments validate their effectiveness in clustering quality evaluation, thereby filling a critical theoretical gap in the field.

Abstract

Aggregation theory has traditionally focused on operators defined over vectors. However, many applications-including Multi-Criteria Decision Making, Group Decision Making, Fuzzy Rule-Based Classification Systems, and overlap/grouping indices-require aggregating information naturally structured as a matrix of membership degrees (e.g., where a set of objects interacts with a family of fuzzy sets). Despite this, no formal framework has been proposed for this class of operators, partly due to the common practice of flattening matrices into vectors (which discards structural information) and partly due to a reliance on decomposable operators that aggregate rows and columns sequentially. This paper addresses this gap by formalizing the notion of a matrix aggregation operator (MAO). We analyze the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry. Finally, we introduce a family of MAOs termed maximum entropy global coverage indices (MEGCIs), provide a construction method for them based on combining grouping functions and MEOWA operators, and illustrate their usefulness in cluster quality assessment through an extensive computational study.

Problem

• Matrix Aggregation Operators
• Aggregation Theory
• Membership Degree Matrix
• Decomposability
• Symmetry

Innovation

• Matrix Aggregation Operators
• Maximum Entropy Global Coverage Indices
• Decomposability
• Grouping Functions
• Cluster Quality Assessment