Similarity measures, penalty functions, and fuzzy entropy from new fuzzy subsethood measures

Abstract

In this study, we discuss a new class of fuzzy subsethood measures between fuzzy sets. We propose a new definition of fuzzy subsethood measure as an intersection of other axiomatizations and provide two construction methods to obtain them. The advantage of this new approach is that we can construct fuzzy subsethood measures by aggregating fuzzy implication operators which may satisfy some properties widely studied in literature. We also obtain some of the classical measures such as the one defined by Goguen. The relationships with fuzzy distances, penalty functions, and similarity measures are also investigated. Finally, we provide an illustrative example which makes use of a fuzzy entropy defined by means of our fuzzy subsethood measures for choosing the best fuzzy technique for a specific problem.

The Null Space of Fuzzy Inclusion Measures

“…More specifically, these authors demand: σ(A ∪ B, C) = min{σ(A, C), σ(B, C)} (1) σ(A, B ∩ C) = min{σ(A, B), σ(A, C)}, ∀ A, B, C. 2The above constraints influence the functional expression of σ, and, as a consequence, they implicitly restrict the conditions under which a pair (A, B) satisfies the equality σ(A, B) = 0, as we will prove in this paper. In particular, we will show that Equations 1 and 2 are incompatible with some of the proposals about σ(A, B) = 0 alternatively required by some authors such as Santos et al [11], for example.…”

“…In the near future, we intend to study the relationships between the properties of inclusion measures and other functional expressions not based on the minimum function. There has already been a precedent for this study in Santos et al [11], where some relationships between the properties of alternative aggregation operators and the axioms of the inclusion measures are studied.…”

“…Some other authors as Bustince et al [8], Vlachos et al [9] or Zhang and Zhang [10] have provided additional discussions around those axiomatic definitions, or applied them in practical situations. A further axiomatic definition has been recently proposed by Santos et al [11]. Most of those definitions coincide in requiring that the inclusion measure σ assigns the value σ(A, B) = 1 to any pair of fuzzy sets A and B satisfying the relation A ⊆ B.…”

Strategies on admissible total orders over typical hesitant fuzzy implications applied to decision making problems

“…The study of typical hesitant fuzzy connectives plays an indispensable role in logical systems, by enabling to model solutions for the hesitant fuzzy multicriteria decision‐making (MCDM) problems, such as hesitant fuzzy linguistic information 8 and hesitant fuzzy linguistic terms sets 9,10 . As basic logical operations, implication operations have been widely used in the narrow sense to formalize concepts, such as similarity degrees, 11,12 coefficient correlation, 13,14 consensus measures, 15 and integrated to fuzzy subsethood measures 16–18 . In the broad sense, when a large amount of data are considered under hesitant fuzzy environments, the detailed study of fuzzy implication properties and their main classes provides a lot of different meanings and distinct applications.…”

Entropy Measures for Plithogenic Sets and Applications in Multi-Attribute Decision Making

“…On the other hand, Kang and Deng [34] outlined the general patterns for which the formula for entropy measures could be formed, thus also applicable for entropy measures to various generalizations of fuzzy sets. Santos et al [35] and Cao and Lin [36] derived their entropy formulas for data processing based on those for fuzzy entropy with applications in image thresholding and electroencephalogram.…”