Matrix Aggregation Operators
TL;DR
This paper analyzes the decomposability and symmetry properties of MAOs, showing that certain operators cannot be expressed in decomposable form and examining several notions of symmetry, and introduces a family of MAOs termed maximum entropy global coverage indices (MEGCIs).
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.
