Perceptual maps to aggregate assessments from different rating profiles: A hesitant fuzzy linguistic approach

Abstract

In decision making environments under uncertainty, assessments are frequently expressed in linguistic terms. When people express their opinions using linguistic terms, the meanings ascribed to these terms may not always align. This phenomenon is captured by the concept of a linguistic perceptual map, which draws from the established lattice of hesitant fuzzy linguistic term sets. Each individual or group of people (referred to as a ’profile’) possesses their own distinct perceptual map. By projecting and aggregating the opinions of these individuals or groups onto a common perceptual map, an average opinion and a level of consensus are derived. This paper extensively studies the mathematical properties of the projection function. We prove that it is a monomorphism between lattices, preserving crucial order relations. Additionally, we progress beyond existing research by introducing an interpretation function. This function facilitates the translation of the aggregated result (referred to as the ’centroid’) from the common perceptual map to each individual’s perceptual map. The properties of the interpretation function are also subject to analysis, demonstrating its role in preserving previous order relations, despite not being a morphism. To illustrate the practicality of these concepts, we propose a methodology that we apply to a real-world case study involving data in the form of ratings from the Amazon books platform. The results obtained highlight that utilizing distinct perceptual maps for each user profile statistically enhances the degree of consensus compared to scenarios where perceptual maps are not differentiated.