Pseudo Overlap Functions, Fuzzy Implications and Pseudo Grouping Functions with Applications

Abstract

Overlap and grouping functions are important aggregation operators, especially in information fusion, classification and decision-making problems. However, when we do more in-depth application research (for example, non-commutative fuzzy reasoning, complex multi-attribute decision making and image processing), we find overlap functions as well as grouping functions are required to be commutative (or symmetric), which limit their wide applications. For the above reasons, this paper expands the original notions of overlap functions and grouping functions, and the new concepts of pseudo overlap functions and pseudo grouping functions are proposed on the basis of removing the commutativity of the original functions. Some examples and construction methods of pseudo overlap functions and pseudo grouping functions are presented, and the residuated implication (co-implication) operators derived from them are investigated. Not only that, some applications of pseudo overlap (grouping) functions in multi-attribute (group) decision-making, fuzzy mathematical morphology and image processing are discussed. Experimental results show that, in many application fields, pseudo overlap functions and pseudo grouping functions have greater flexibility and practicability.

Some Construction Methods for Pseudo-Overlaps and Pseudo-Groupings and Their Application in Group Decision Making

“…In recent independent works, Batista [14] and Zhang and Liang [10] have removed the commutativity requirement from the properties of overlap and grouping functions, leading to pseudo-overlap and pseudo-grouping functions. These functions have been studied in terms of their related properties and applications.…”

“…These functions have been studied in terms of their related properties and applications. Furthermore, Liang and Zhang [10,22] have generalized the notion of interval-valued and n-dimensional pseudooverlap functions.…”

“…Since in many practical situations, commutativity is an undesired property, researchers have investigated noncommutative fuzzy operators and discussed their applications [23][24][25][26]. In this regard, pseudo-overlap and pseudo-grouping functions have proven useful in multi-attribute decision-making, fuzzy mathematical morphology, and image processing [10].…”

“…However, a key assumption in the existing definitions of overlap and grouping functions is commutativity [10]. In real-world applications, such as decision-making, criteria or experts often possess varying importance, rendering the assumption of commutativity unreasonable [11][12][13].…”

“…In real-world applications, such as decision-making, criteria or experts often possess varying importance, rendering the assumption of commutativity unreasonable [11][12][13]. To address this limitation, researchers have introduced pseudooverlaps [14] and pseudo-groupings [10] as non-commutative generalizations of overlaps and groupings, respectively. This paper focuses on the development of new construction methods for pseudooverlaps and pseudo-groupings.…”