Convexity of hesitant fuzzy sets

Janiš, Vladimír; Montes, Susana; Renčová, Magdaléna

doi:10.3233/jifs-162204

Cited by 1 publication

(6 citation statements)

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“…As we commented previously, the arithmetic mean is an aggregation function between minimum and maximum such that the intersection of two convex hesitant fuzzy sets need not be convex. This was already commented in [14] for their definition, but it is also true now for the new one, as we can see from the following example.…”

Section: The Case Between Minimum and Maximumsupporting

confidence: 58%

“…This is a very important requirement, since it makes the collection of convex sets very important for applications as, for instance, optimization (see [2] or [30]). As it was shown in [14], the concept of convexity introduced by Rashid and Beg does not preserve this property. In that paper, a definition of convexity for hesitant fuzzy sets was given such that it fulfills the natural conditions: it extends the concept of convexity for fuzzy sets, it preserves convexity under intersections and equivalence of the convexity for cuts.…”

Section: Any α-Cut Of a Is Convex Crisp Setmentioning

confidence: 99%

“…Taking into account these properties, they arrived to a natural definition of convexity for hesitant fuzzy sets based on the maximum of the membership values at any point. More precisely, Definition 6 [14] Let X be a finite linear space and let A be a hesitant fuzzy set on X.…”

Section: Any α-Cut Of a Is Convex Crisp Setmentioning

confidence: 99%

“…Therefore, A ∩ B is also A-convex. This proof is inspired by a similar result in [14]. However, in that case the definition of convexity was slightly different, and it is here adapted to the new definition.…”

Section: The Case Of the Maximum/minimummentioning

confidence: 99%

“…As far as we know, the first attempt to define convexity for hesitant fuzzy sets has been done by Rashid and Beg in 2016 [26]. That definition had some problems, which were solved by Janiš et al in 2018 [14]. These two definitions seem to be quite different.…”

Section: Introductionmentioning

confidence: 99%

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Convexity of hesitant fuzzy sets based on aggregation functions

et al. 2021

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Convexity is one of the most important geometric properties of sets and a useful concept in many fields of mathematics, like optimization. As there are also important applications making use of fuzzy optimization, it is obvious that the studies of convexity are also frequent. In this paper we have extended the notion of convexity for hesitant fuzzy sets in order to fulfill some necessary properties. Namely, we have found an appropriate definition of convexity for hesitant fuzzy sets on any ordered universe based on aggregation functions such that it is compatible with the intersection, that is, the intersection of two convex hesitant fuzzy sets is a convex hesitant fuzzy set and it fulfills the cut worthy property.

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Section: The Case Between Minimum and Maximumsupporting

confidence: 58%

Section: Any α-Cut Of a Is Convex Crisp Setmentioning

confidence: 99%

Section: Any α-Cut Of a Is Convex Crisp Setmentioning

confidence: 99%

Section: The Case Of the Maximum/minimummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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