On admissible pairs of aggregation functions based on quasi-linear means and related families
Michał Boczek, Marek Kaluszka, Jakub Łompieś
TL;DR
The paper addresses the construction of admissible orders on interval endpoints via pairs of aggregation functions, extending beyond the classical weighted arithmetic mean pair. It develops a general method to build admissible AF pairs and applies it to quasi-linear means, Archimedean t-norms/t-conorms, and strictly Schur-convex/concave functions, analyzing their relation to the alpha-beta order and identifying cases where admissible orders diverge from the alpha-beta framework. Key contributions include explicit admissibility criteria for these classes, a unifying Schur-convexity perspective, and demonstrations of both coinciding and non-coinciding instances with the alpha-beta order. These results broaden the toolkit for interval-valued operators in decision making, machine learning, and information fusion, and open questions for complete classification of admissible AFs and their induced orders.
Abstract
Admissible orders play a key role in ranking subintervals of the unit interval. In 2013, Bustince et al. proposed constructing such relations by means of admissible pairs of aggregation functions. The only significant example in the literature is a pair of weighted arithmetic means with different weights. In this paper, we present a method for constructing admissible pairs of aggregation functions, which allows us to verify the admissibility of various function classes, including quasi-linear means, Archimedean t-norms (and t-conorms), and certain strictly Schur-convex (or Schur-concave) functions. Furthermore, we examine the relationship between admissible orders generated by admissible pairs of aggregation functions and the (α, \b{eta})-order, identifying cases where these two notions do not coincide.