Hesitant fuzzy sets

By Vicenç Torra. International Journal of Intelligent Systems, 2010

1,835 citations recorded by the source corpus.

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

Subfields: Artificial Intelligence, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/47b76c7577a3e8ec1b1ac877720f744cdf98373b

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Citations in the corpus

Hesitant Fuzzy Sets: State of the Art and Future Directions (2014, background): “To use HFS properly, it is recommended to consider ﬁnite and nonempty HFS, that is, THFS.”
A supply chain transparency and sustainability technology appraisal model for blockchain technology (2019, background): “Let X be a reference set, a hesitant fuzzy set (HFS) A on X is deﬁned in terms of a function h A ( x ) , when applied to X returns a ﬁnite subset of [0,1] (Torra 2010): where function h A ( x ) is a set of values within [0,1], denoting the possible membership degrees of the element x ∈ X to A .”
Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making (2013, background): “Torra and Narukawa [36] proposed the following aggregation principle for HFEs.”
Hesitant fuzzy entropy and cross‐entropy and their use in multiattribute decision‐making (2012, background): “Recently, Torra, 40 and Torra and Narukawa 41 gave another generalization of fuzzy set, hesitant fuzzy set, describing the hesitant situation when people make a decision.”
A VIKOR-based method for hesitant fuzzy multi-criteria decision making (2013, background): “Torra and Narukawa (2009), Torra (2010) firstly introduced this new extension of fuzzy set and gave some basic operations on it.”
Pythagorean Fuzzy Choquet Integral Based MABAC Method for Multiple Attribute Group Decision Making (2016, methodology): “Future studies will extend the proposed method to the MAGDM problems with hesitant fuzzy sets 50 or interval-valued Pythagorean fuzzy set.”
A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets (2013, background): “Much research has been developed on GDM problems [18,30,31], being one of the main topics, the managing and modelling of uncertainty by different forms of information, used by experts to provide their preferences, such as utility vectors [5], fuzzy preference relations [21], linguistic variables [14”
Application of decision-making techniques in supplier selection: A systematic review of literature (2013, background): “A new generalization of fuzzy sets called hesitant fuzzy sets has recently been proposed by Torra (2010), it’s the theoretical basis of which has received considerable attention (e.g. Rodriguez, Martinez, & Herrera, 2012; Wei, 2012; Xu & Xia, 2011; Xu & Xia, 2012).”
Spherical fuzzy sets and spherical fuzzy TOPSIS method (2018, neutral): “1 [16, 17, 22, 24, 26, 29, 30, 34, 35, 39].”
TOPSIS for Hesitant Fuzzy Linguistic Term Sets (2013, background): “For any x ∈ X, the value B ( x ) is called the degree of membership of x in B. Torra 40 introduced an extension for fuzzy sets to manage those situations in which several values are possible for the deﬁnition of a membership function of a fuzzy set.”
Some Hesitant Fuzzy Aggregation Operators with Their Application in Group Decision Making (2011, methodology): “Torra and Narukawa (2009) and Torra (2010) also discussed the relationships between hesitant fuzzy set and other three kinds of fuzzy sets (intuitionistic fuzzy set, type-2 fuzzy set and fuzzy multiset).”
Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems (2016, background): “Furthermore, in order to handle the situations where people are hesitant when expressing their preferences regarding objects in the decision-making process, hesitant fuzzy sets were introduced by Torra and Narukawa (2009) and Torra (2010).”
Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information (2013, background): “Hesitant fuzzy set (HFS) [13] [14], which has been introduced by Torra and Narukawa as an extension of fuzzy set [15], describes the situations that permit the membership of an element to a given set having a few different values, which is a useful means to describe and deal with uncertain informati”
On distance and correlation measures of hesitant fuzzy information (2011, neutral): “D EFINITION 1 (Torra and Narukawa, 26 Torra 31 ) .”
A Historical Account of Types of Fuzzy Sets and Their Relationships (2016, background): “2010 Torra introduces the notion of HFS [137].”
Distance and similarity measures for hesitant fuzzy sets (2011, methodology): “Distance and similarity measures for HFSs Hesitant fuzzy sets (HFSs) were first introduced by Torra [31] and Torra and Narukawa [32], it permits the membership degree of an element to a set to be represented as several possible values between 0 and 1.”
Fuzzy Multicriteria Decision-Making: A Literature Review (2015, positive): “…[9], extend fuzzy sets by an additional degree, which is called the degree of uncertainty; fuzzy multisets based on multisets that allow repeated elements in the set [10]; hesitant fuzzy sets (HFS) that have been recently introduced in [11] and provide a very interesting extension of fuzzy sets.”