Decomposition theorems and extension principles for hesitant fuzzy sets

We prove a decomposition theorem for hesitant fuzzy sets, which states that every typical hesitant fuzzy set on a set can be represented by a well-structured family of fuzzy sets on that set. This decomposition is expressed by the novel concept of hesitant fuzzy set associated with a family of hesitant fuzzy sets, in terms of newly defined families of their cuts. Our result supposes the first representation theorem of hesitant fuzzy sets in the literature. Other related representation results are proven. We also define two novel extension principles that extend crisp functions to functions that map hesitant fuzzy sets into hesitant fuzzy sets

Ideal Theory in BCK/BCI-algebras in the Frame Of Hesitant Fuzzy Set Theory

Several generalizations and extensions of fuzzy sets have been introduced in the literature, for example, Atanassov’s intuitionistic fuzzy sets, type 2 fuzzy sets and fuzzy multisets, etc. Using the Torra’s hesitant fuzzy sets, the notions of Sup-hesitant fuzzy ideals in BCK/BCI-algebras are introduced, and its properties are investigated. Relations between Sup-hesitant fuzzy subalgebras and Sup-hesitant fuzzy ideals are displayed, and characterizations of Sup-hesitant fuzzy ideals are discussed

Ranked hesitant fuzzy sets for multi-criteria multi-agent decisions

This paper introduces and investigates ranked hesitant fuzzy sets, a novel extension of hesitant fuzzy sets that is less demanding than both probabilistic and proportional hesitant fuzzy sets. This new extension incorporates hierarchical knowledge about the various evaluations submitted for each alternative. These evaluations are ranked (for example by their plausibility, acceptability, or credibility), but their position does not necessarily derive from supplementary numerical information (as in probabilistic and proportional hesitant fuzzy sets). In particular, strictly ranked hesitant fuzzy sets arise when no ties exist, i.e., when for any fixed alternative, each submitted evaluation is either strictly more plausible or strictly less plausible than any other submitted evaluation. A detailed comparison with similar models from the literature is performed. Then in order to produce a natural strategy for multi-criteria multi-agent decisions with ranked hesitant fuzzy sets, canonical representations, scores and aggregation operators are designed in the framework of ranked hesitant fuzzy sets. In order to help implementation of this model, Mathematica code is provided for the computation of both scores and aggregators. The decision-making technique that is prescribed is tested with a comparative analysis with four methodologies based on probabilistic hesitant fuzzy information. A conclusion of this numerical exercise is that this methodology is reliable, applicable and robust. All these evidences show that ranked hesitant fuzzy sets are an intuitive extension of the hesitant fuzzy set model designed by V. Torra, that can be implemented in practice with the aid of computationally assisted algorithms.Junta de Castilla y León y European Regional Development Fun

Scores of hesitant fuzzy elements revisited: “Was sind und was sollen”

[EN] This paper revolves around the notion of score for hesitant fuzzy elements, the constituent parts of hesitant fuzzy sets. Scores allow us to reduce the level of uncertainty of hesitant fuzzy sets to classical fuzzy sets, or to rank alternatives characterized by hesitant fuzzy information. We propose a rigorous and normative definition capable of encapsulating the characteristics of the most important scores introduced in the literature. We systematically analyse different types of scores, with a focus on coherence properties based on cardinality and monotonicity. The hesitant fuzzy elements considered in this analysis are unrestricted. The inspection of the infinite case is especially novel. In particular, special attention will be paid to the analysis of hesitant fuzzy elements that are intervals

Himpunan Kabur Hesitant Ganda yang Diperluas (Expanded Dual Hesitant Fuzzy Sets)

Konsep Expanded Dual Hesitant Fuzzy Sets merupakan suatu pengembangan dari konsep Hesitant Fuzzy Sets. Berdasarkan konsep Hesitant Fuzzy Sets telah diperoleh beberapa konsep seperti Dual Hesitant Fuzzy Sets, Expanded Hesitant Fuzzy Sets, dan Extended Hesitant Fuzzy Sets yang menjadi konsep dasar pengembangan Expanded Dual Hesitant Fuzzy Sets. Kemudian akan dikaji beberapa operasi-operasi terkait Expanded Dual Hesitant Fuzzy Sets, skor pada Expanded Dual Hesitant Fuzzy Sets, hukum perbandingan, dan penerapan konsep Expanded Dual Hesitant Fuzzy Sets dalam masalah pengambilan keputusan

Partial orderings for hesitant fuzzy sets

New partial orderings =o=o, =p=p and =H=H are defined, studied and compared on the set HH of finite subsets of the unit interval with special emphasis on the last one. Since comparing two sets of the same cardinality is a simple issue, the idea for comparing two sets A and B of different cardinalities n and m respectively using =H=H is repeating their elements in order to obtain two series with the same length. If lcm(n,m)lcm(n,m) is the least common multiple of n and m we can repeat every element of A lcm(n,m)/mlcm(n,m)/m times and every element of B lcm(n,m)/nlcm(n,m)/n times to obtain such series and compare them (Definition 2.2). (H,=H)(H,=H) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H,=H)(H,=H) have a lattice structure. Moreover in the set BB of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder =B=B can be defined in a similar way as =H=H in HH and considering the quotient set View the MathML sourceB¿=B/~ of BB by the equivalence relation ~ defined by A~BA~B when A=BBA=BB and B=BAB=BA, View the MathML source(B¿,=B) is a lattice and (H,=H)(H,=H) can be naturally embedded into it.Peer ReviewedPostprint (author's final draft

Expanded hesitant fuzzy sets and group decision making: slides for FUZZ-IEEE 2017

[EN]We define expanded hesitant fuzzy sets, which incorporate all available information of the decision makers that provide the membership degrees that define a hesitant fuzzy set. We show how this notion relates to hesitant fuzzy set and extended hesitant fuzzy set. We define various scores for this setting, which generalize popular scores for hesitant fuzzy elements. Finally, a group decision making procedure is presented and illustrated with an example

Modeling group assessments by means of hesitant fuzzy linguistic term sets

Hesitant linguistic term sets have been introduced to capture the human way of reasoning using linguistic expressions involving different levels of precision. In this paper, a lattice structure is provided to the set of hesitant fuzzy linguistic term sets by means of the operations intersection and connected union. In addition, in a group decision making framework, hesitant fuzzy linguistic descriptions are defined to manage situations in which decision makers are assessing different alternatives by means of hesitant fuzzy linguistic term sets. Based on the introduced lattice structure, two distances between hesitant fuzzy linguistic descriptions are defined. These metric structures allow distances between decision makers to be computed. A centroid of the decision making group is proposed for each distance to model group representatives in the considered group decision making framework.Peer ReviewedPostprint (author's final draft

Dual Hesitant Fuzzy Set

Konsep dual hesitant fuzzy set merupakan suatu pengembangan dari konsep fuzzy set. Berdasarkan konsep fuzzy sets telah diperoleh beberapa konsep seperti Intuitionistic fuzzy sets dan Hesitant Fuzzy Sets yang menjadi konsep dasar pengembangan dual hesitant fuzzy set. Kemudian akan dikaji beberapa operasi-operasi terkait dual hesitant fuzzy set, pembuktian teorema-teorema yang ada dan dikaji juga tentang penerapan konsep dual hesitant fuzzy set dalam masalah pengambilan keputusan

Intuitionistic Hesitant Fuzzy Filters in BE-Algebras

The notions of hesitant fuzzy filters and hesitant implicative filter was introduced. In this paper, we introduce the notion of intuitionistic hesitant fuzzy filters (IHFF) and intuitionistic hesitant implicative filters (IHIFB) and several properties are investigated. Also, we defined ?-level sets, and we show the relation between IHFF, IHIFF and ?-Level