Concept Of Hesitant Fuzzy Set Research Articles

An (α,β)-Hesitant Fuzzy Set Approach to Ideal Theory in Semigroups

The aim of this manuscript is to introduce the \((\alpha,\beta)\)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of \((\alpha,\beta)\)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study \((\alpha,\beta)\)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of \((\alpha,\beta)\)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of \((\alpha,\beta)\)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of \((\alpha,\beta)\)-hesitant fuzzy left ideals and \((\alpha,\beta)\)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of \((\alpha,\beta)\)-hesitant ideals are given.

Correlation coefficient of T-spherical type-2 hesitant fuzzy sets and their applications in clustering analysis

The fuzzy set (FS) theory has a significant role in the modelling of uncertainties. However, using the fuzzy set remain incapable in the modelling of the problems, when the decision-makers do not have the same opinion about the membership degree of an element or a decision-maker has different opinion related to membership degree of an element. To cope with this problem, the concept of a hesitant fuzzy set (HFS) was introduced by Torra and Narukawa. Recently, many generalizations of the FSs have been worked by researchers. One of them is type-2 fuzzy set (T2FS) and this set is an important role for dealing with problems involving uncertainty and linguistic variables. To take advantages of T2FSs and HFSs in the modelling of some problems into consideration, the concept of type-2 hesitant fuzzy sets (T2HFS) was defined. Another generalization of the FSs is spherical fuzzy set (SFS) and T-spherical fuzzy set (T-SFS) which is a generalization of SFS. In this research paper, to take advantages of T2HFSs and T-SFSs in the modelling of some problems into consideration, the concept of T-spherical type-2 hesitant fuzzy sets (T-ST2HFS) is defined and operations of union and intersection between two T-ST2HFSs. The basic idea behind of the set introduced in this paper is the assignment of T2HF elements for the components of a T-SFS. Then, a method is given to equalize lengths of the T2HF elements of which lengths are different from each other based on the optimistic and pessimistic approaches. The correlation coefficient is a very useful tool for dealing with some situations requiring the calculation of the relation between the two objects. With this motivation, correlation coefficients and weighted correlation coefficients are defined between proposed sets and some of their properties are obtained. Furthermore, a clustering analysis method, based on the put forward correlation coefficient formula is presented. Finally, to tackle a problem involving clustering the alternatives for a firm that wants to invest with a large amount of money, an application of the proposed method is given.

A remark on hesitant fuzzy ideals in ordered semigroups

We investigate the concept of hesitant fuzzy sets. This concept is a generalized notion of fuzzy sets. The concept of fuzzy sets can be applied to the study of ordered semigroups. It is well-known that in regular ordered semigroups, any fuzzy bi-ideal and fuzzy quasi-ideal coincide. In this paper, we illustrate that this fact does not hold for hesitant fuzzy sets. Furthermore, we provide an example illustrating that in a regular ordered semigroup, there is a hesitant fuzzy bi-ideal which is not a hesitant fuzzy quasi-ideal. Moreover, we present a class of hesitant fuzzy sets in which hesitant fuzzy bi-ideals and hesitant fuzzy quasi-ideals coincide.

AN ADAPTIVE HESITANT FUZZY SETS BASED GROUP RECOMMENDATION SYSTEM

Accurate group movie recommendation systems are a need in society today. We find that people tend to watch movies in groups rather than by themselves. However, the groups of people that tend to watch movies together are very diverse. In the existing methods, the characteristics of individual users are simply aggregated to obtain the group’s attributes and most of the time useful data is not utilized. This can be improved upon by ensuring the utilization of all the data that we are presented with from the scenario. The method proposed in this paper is termed integrated as we weighed in the individual traits of each user in the group when predicting the group’s rating for a movie. We used the concept of Hesitant Fuzzy Sets (HFS) in order to keep track of the characteristics of each of the users individually. The method we proposed in this paper employs Matrix Factorisation (MF) based Collaborative Filtering (CF) along with hesitant fuzzy sets. The way we performed MF based CF for a group is that we found the factors first and then formed the groups. The ratings were then predicted for these groups. The groups we have considered are of three sizes - 3 users, 5 users, and 10 users.

Hesitant fuzzy parameterized soft sets and their applications in decision making

The fuzzy set theory plays an important role in the modeling of the problems involving uncertain data. Some extensions of the fuzzy sets are needed due to the variety of problems encountered in real life. The concept of a hesitant fuzzy set is one of these extensions. Also, soft set theory, which is free from the difficulties of determining the membership function in fuzzy sets, plays an important role in dealing with uncertainty. In this study, we introduce the concept of hesitant fuzzy parameterized soft set as a generalization of the fuzzy parameterized soft sets. Then we define set-theoretical operations of the hesitant fuzzy parameterized soft sets and obtain some of their properties. We also improve a decision-making algorithm under the hesitant fuzzy parameterized soft environment and give an example to show the process of the algorithm. Finally, we compare the proposed decision-making algorithm with methods existing in the literature.

Some distance measures for type 2 hesitant fuzzy sets and their applications to multi-criteria group decision-making problems

The fuzzy set has an important role in the modeling of uncertainties. However, the fuzzy set is not sufficient in modeling of the problems, when the decision makers do not have the same opinion about membership degree of an element. To overcome this problem, the concept of hesitant fuzzy set was defined by Torra and Narukawa. Recently, the concept of the type 2 hesitant fuzzy set was defined by Feng and a ranking method among elements of a type 2 hesitant fuzzy element was given. In this paper, firstly, we point out some shortcomings in the ranking method given by Feng and then we give a new ranking method among elements of a type 2 hesitant fuzzy element. The distance and similarity measures are the effective mathematical tools to solve the problems such as medical diagnosis, decision making, pattern recognition and marketing strategy selection. Therefore, we introduce some distance measure methods between two type 2 hesitant fuzzy sets based on Hamming, Euclidean and Hausdorff distance measures. We obtain some properties of the proposed distance measure methods. We also develop a multi-criteria group decision-making method by integrating the TOPSIS method and the proposed distance measure methods under the type 2 hesitant fuzzy environment. Furthermore, we present a numerical example of multi-criteria group decision-making problem to choose the best alternative among firms to invest in order to illustrate the process and validate of the proposed method.

Hesitant relations: Novel properties and applications in three-way decisions

Since Torra introduced the concept of hesitant fuzzy sets, it has seen wide applications in decision-making problems, cluster analysis, medical diagnosis, personnel appraisal, information retrieval and so on. And, nowadays, there arise many discussions related to hesitant fuzzy sets. In particular, recently, Hu extends the truth values set of hesitant fuzzy sets and hesitant fuzzy relations from the family of subsets of unit interval to the family of nonempty subsets of any linearly ordered set with an involutive negator and calls them hesitant sets and hesitant relations, respectively. This paper continues to consider this topic and mainly intends to give a systematical discussion to hesitant relations on their properties and potential applications in three-way decisions. First, this paper shows some novel properties of the operations on linearly ordered set with an involutive negator. Second, this paper investigates some vital properties of hesitant relations, especially for the compositions, projections, reflexivity, symmetry, *-transitivity and so on. Third, this paper proposes various types of ordered hesitant relations. Finally, this paper discusses some applications of hesitant relations in the constructions of decision evaluation functions in three-way decision spaces.

A New Model for Deriving the Priority Weights from Hesitant Triangular Fuzzy Preference Relations

Fuzzy preference relation is a common tool to express the uncertain preference information of decision maker in the process of decision making. However, the traditional fuzzy preference relation will fail under hesitant fuzzy environment as the membership has a single value. In addition, it is very difficult to obtain the precise membership values. Therefore, a new model of fuzzy preference relation is proposed in this paper. Firstly, the concept of hesitant triangular fuzzy preference relation is defined and its properties are investigated based on the concepts of hesitant fuzzy set, hesitant triangular fuzzy set, fuzzy preference relation, and hesitant fuzzy preference relation. Then, the steps of applying this novel model are offered for the case of determining the weights of failure modes. Finally, an example is used to illustrate the proposed model.

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.

Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making

In this paper, we investigate multiple attribute decision making (MADM) problems based on Frank triangular norms, in which the attribute values assume the form of hesitant fuzzy information. Firstly, some basic concepts of hesitant fuzzy set (HFS) and the Frank triangle norms are introduced. We develop some hesitant fuzzy aggregation operators based on Frank operations, such as hesitant fuzzy Frank weighted average (HFFWA) operator, hesitant fuzzy Frank ordered weighted averaging (HFFOWA) operator, hesitant fuzzy Frank hybrid averaging (HFFHA) operator, hesitant fuzzy Frank weighted geometric (HFFWG) operator, hesitant fuzzy Frank ordered weighted geometric (HFFOWG) operator, and hesitant fuzzy Frank hybrid geometric (HFFHG) operator. Some essential properties together with their special cases are discussed in detail. Next, a procedure of multiple attribute decision making based on the HFFHWA (or HFFHWG) operator is presented under hesitant fuzzy environment. Finally, a practical example that concerns the human resource selection is provided to illustrate the decision steps of the proposed method. The result demonstrates the practicality and effectiveness of the new method. A comparative analysis is also presented.

A position and perspective analysis of hesitant fuzzy sets on information fusion in decision making. Towards high quality progress

The necessity of dealing with uncertainty in real world problems has been a long-term research challenge which has originated different methodologies and theories. Recently, the concept of Hesitant Fuzzy Sets (HFSs) has been introduced to model the uncertainty that often appears when it is necessary to establish the membership degree of an element and there are some possible values that make to hesitate about which one would be the right one. Many researchers have paid attention on this concept who have proposed diverse extensions, relationships with other types of fuzzy sets, different types of operators to compute with this type of information, applications on information fusion and decision-making, etc.Nevertheless, some of these proposals are questionable, because they are straightforward extensions of previous works or they do not use the concept of HFSs in a suitable way. Therefore, this position paper studies the necessity of HFSs and provides a discussion about current proposals including a guideline that the proposals should follow and some challenges of HFSs.

ELECTRE method for multiple attributes decision making problem with hesitant fuzzy information

The ELECTRE method utilizes the concordance and discordance set. And then use concordance and discordance indices to construct concordance and discordance matrices. Both concordance and discordance indices have to be calculated for every pair of alternatives. Based on the aggregate matrix, a decision graph can be depicted to determine which alternative is preferable, incomparable or indifferent. It first uses concordance and discordance indices to analyze outranking relations among alternatives from real-world applications. In this paper, an extension of ELECTRE method, a multiple attributes hesitant fuzzy decision making technique, to a decision environment is investigated, where the individual decision value are provided by the hesitant fuzzy information. We first introduce some concepts of hesitant fuzzy set. Then, the hesitant fuzzy ELECTRE method is proposed in detail. Finally, an illustrative example for evaluating the performance of the industry zones is used to illustrate the

Correlation coefficient of dual hesitant fuzzy sets and its applications

Correlation between two variables or parameters plays a very significant role in statistics. Furthermore, the accuracy in the measurement of the correlation depends upon the data collected for the set of discourse. It is quite evident that in many cases the data collected for various statistical measures are full of uncertainties. A dual hesitant fuzzy set (DHFS) is a generalized form of a hesitant fuzzy set (HFS) and negates the effects of uncertainty inherent in the collected data. In the present paper, the concept of HFS has been replaced with DHFS and the correlation between two DHFSs is obtained. A formula for the correlation coefficient between two DHFSs has been derived. The proposed method was used to determine the coefficient of correlation between different parameters of water in four different lakes in Rajasthan, India.

Hesitant fuzzy linguistic ordered weighted distance operators for group decision making

Since the concept of hesitant fuzzy sets was put forward, different types of extensions have been proposed to deal with actual problems. A hesitant fuzzy linguistic term set provides a linguistic and computational basis to increase the flexibility and richness of linguistic elicitation based on the fuzzy linguistic approach. In this paper, we consider the concept of distance operator and develop a hesitant fuzzy linguistic ordered weighted distance (HFLOWD) operator. The HFLOWD operator is very suitable to deal with the uncertain situations with linguistic information. Moreover, it is also a new aggregation operator that provides parameterized families of distance aggregation operators between the minimum and the maximum distance. Some of its main properties and different families of HFLOWD operators are investigated. Finally, an application of the new approach is offered and comparative analyses are also provided to show the advantages over existing methods.

A novel SIR method for multiple attributes group decision making problem under hesitant fuzzy environment

In this paper, an extension of SIR method, a multiple attributes hesitant fuzzy group decision making technique, to a group decision environment is investigated, where the individual decision value and the weights of attributes and experts are provided as hesitant fuzzy information. We first introduce some concepts of hesitant fuzzy set and interval-valued hesitant fuzzy set. Then, the hesitant fuzzy superiority and inferiority ranking HF-SIR method and the interval-valued hesitant fuzzy superiority and inferiority ranking IVHF-SIR method are proposed in detail. Furthermore, the approach to MAGDM problem based on the HF-SIR and IVHF-SIR method is given under hesitant fuzzy environment. Finally, a comparative analysis is illustrated with a numerical example.

A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method

Hesitancy is the most common problem in decision making, for which hesitant fuzzy set can be considered as a suitable means allowing several possible degrees for an element to a set. And, the VIKOR method is an effective tool to determine a compromise solution by providing a maximum “group utility” for the “majority” and a minimum “individual regret” for the “opponent” in decision making, particularly in a situation where the decision maker is not able or does not know to express his/her preference. In many practical situations, the inter-dependent or interactive characteristics among criteria and preference of decision makers should be taken into account, we generally utilize the Choquet integral to depict the correlations and interactions in the decision process. When the Choquet integral is used to solve the correlative decision making problems, we ignore the importance of the ordered position of the element. In fact, each element has the same probability to be drawn and all permutations have the same probability. In this paper, we introduce the Shapley value to solve correlative problem, which is used to be interpreted as a kind of average value of the contribution of an element alone in all coalitions with the same position probability. Firstly, we present some concepts of hesitant fuzzy set and define the Shapley value-based Lp --metric SLp,μ --metric. With the SLp,μ --metric, an extended VIKOR method is developed to deal with the correlative multiple criteria decision making MCDM problem under hesitant fuzzy environment. To comparative analysis, we also apply the TOPSIS method to solve the problem based on the Shapley value. Finally, a comparative analysis of the two methods is illustrated with a numerical example.

Interval-valued hesitant preference relations and their applications to group decision making

We introduce a new type of fuzzy preference structure, called interval-valued hesitant preference relations, to describe uncertain evaluation information in group decision making (GDM) processes. Moreover, it allows decision makers to offer all possible interval values that are not accounted for in current preference structure types when one compares two alternatives. We generalize the concept of hesitant fuzzy set (HFS) to that of interval-valued hesitant fuzzy set (IVHFS) in which the membership degrees of an element to a given set are not exactly defined, but denoted by several possible interval values. We give systematic aggregation operators to aggregate interval-valued hesitant fuzzy information. In addition, we develop an approach to GDM based on interval-valued hesitant preference relations in order to consider the differences of opinions between individual decision makers. Numerical examples are provided to illustrate the proposed approach.