An approach for consensual analysis on typical hesitant fuzzy sets via extended aggregations and fuzzy implications based on admissible orders

Abstract

The Typical Hesitant Fuzzy Logic (THFL) is founded on the theory of the Typical Hesitant Fuzzy Sets (THFS), which are defined by considering as membership degrees the finite and non-empty subsets of the unit interval, which are called as Typical Hesitant Fuzzy Elements (THFE). In such logical approach, not only a number but also subintervals, in the unitary interval are also THFE-representations, which can be applied in the decision-making process based on multiple criteria involving many specialists (ME-MCDM). In this context, THFL provides the modelling for situations where there exists not only data uncertainty, but also indecision or hesitation among experts about the possible values for preferences regarding collections of objects. In order to reduce the information collapse for comparison and/or ranking of alternatives in the preference relationships, this thesis firstly develops new ideas about THFL’s logical connectives, which are investigated within the scope of three admissible orders. In the set H of all hesitant fuzzy values, consider: (i) the lexicographic orders hH; Lex1i and hH; Lex2i, related to the occurrence of the smallest/largest element in an ascending/descending ordered THFS, respectively; (ii) the relevant class of order hH; Ai, satisfying the injective cardinality property. In particular, properties of negations and aggregations are studied, as t-norms and OWA operators, with special interest in the axiomatic structures defining the implications and preserving their algebraic properties and representability. Thus, these theoretical results are submitted to the ME-MCDM problem, in order to select the better support for multiple software alternatives. As a main contribution, in this thesis, we discuss consensus measures on THFE and present a model that formally builds consensus measures through extended aggregation functions and fuzzy negation, using admissible orders for comparison and further, differentiating an analysis of consistency over preference matrices. The action of automorphisms provides the generation of new conjugate operators, preserving the main consensual properties as proposed in the Beliakov’s research, including unanimity, minimum consensus, maximum dissension, symmetry and invariance for replication. The new CCAI-method is presented, by applying admissible orders and promoting the use of fuzzy consensus measures based on multi-valued fuzzy logics, and, then, this work enables comparisons even between THFE with different cardinalities. These new theoretical results are then applied to another ME-MCDM problem, obtaining CCAI-consensus in a group of experts which consider typical hesitant fuzzy sets to provide classifications for multiple styles of craft beers.