Since the problems of everyday life are relative, so far various tools such as fuzzy sets, intuitive fuzzy sets, etc. have been expressed to express these ambiguities in mathematical modeling. In 2009, Torra introduced a new horizon for the discussion of Hesitant fuzzy sets to discuss issues that are uncertain about decision making. In the course of his work, the quantitative and qualitative expansion of uncertain fuzzy sets is discussed. In this article, for the purpose of introducing more researchers to Hesitant fuzzy sets, we review the types of Hesitant fuzzy sets such as dual uncertain fuzzy sets, generalized Hesitant fuzzy sets, and so on.

The main aim of this paper is to present a novel method for ranking Hesitant fuzzy sets (HFSs) based on transforming HFSs into fuzzy sets (FSs). The idea behind the method is an interesting HFS decomposition which is referred here to as the horizontal representation in the current study. To show the validity of the proposed ranking method, we apply it to solve a multi-attribute decision-making problem under Hesitant fuzzy environment. Interestingly, the results show that the proposed method gives the most accepted precedence of alternatives in comparison with the other existing methods.

In this article, after the definitions of the reduced Hesitant L-fuzzy automaton (RHLFA) and the minimal Hesitant L-fuzzy automaton; we convert a Hesitant L-fuzzy automaton (HLFA) to an RHLFA by reducing the number of its states such that its language is equal to the original HLFA language. Then, by defining an equivalence relation on the monoid X*,we construct an HLFA whose language is equal to the language of the transformed RHLFA, and we show that this HLFA is minimal. In conclusion, we delineate the criteria under which, the number of states in the minimal HLFA is equal to the number of states in the RHLFA

In this paper, an effective method for regression is presented in which a variety of fuzzy clustering methods and concepts of Hesitant fuzzy sets are used. First, the fuzzy clustering algorithm is applied to the data, and after projecting the cluster membership function on different features, the number of clusters of fuzzy sets is obtained on each dimension (or feature). We then consider these fuzzy sets as a Hesitant fuzzy set on each feature, and we obtain the Hesitant Fuzzy Correlation Coefficient Matrix (HFCCM) for the attributes. Subsequently, a nonlinear mapping based on the principal components analysis of the HFCCM is used to convert the dataset's features into new features. Finally, the new extracted features are assigned to the fuzzy clustering algorithm and a Sugeno fuzzy regression system is fitted. The proposed method was compared with some other methods to several regression datasets. The results of the experiments indicate that the proposed method is successful in extracting and reducing the characteristics, as well as increasing the regression accuracy. Also, the number of rules of the fuzzy regression model in the proposed method is fairly low.

This paper presents the introduction of two novel equation types: the partial Hesitant fuzzy equation and the half Hesitant fuzzy equation‎. Additionally, ‎ an efficient method is proposed to solve these equations by defining four solution categories: Controllable‎, ‎Tolerable Solution set (TSS)‎, Controllable ‎Solution set (CSS)‎, ‎and Algebraic Solution set (ASS)‎. ‎ Furthermore, ‎ the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of Hesitant fuzzy solutions‎. ‎ The practicality of the proposed method is demonstrated through numerical examples.

Probabilistic Hesitant fuzzy set (PHFS) is a fruitful concept that adds to Hesitant fuzzy set (HFS) the term of probability which is able to retain more information than the usual HFS. Here, we demonstrate that the existing definitions of PHFS are not still reasonable, and therefore, we first improve the PHFS definition. By endowing the set and algebraic operations with a new re-definition of PHFS, we propose a class of T-norm-based and S-norm-based operations for PHFSs together with a number of aggregation operators. Eventually, on the basis of the new operators, the effectiveness and practicality of re-defined PHFS will be tested using three multiple criteria decision making (MCDM) problems concerning the automotive industry safety evaluation, the evaluation of Chines hospitals and the evaluation of alternatives in an investment company.

In the real world, in most cases, such as industry, management, and even in daily life, we encounter optimization and decision-making problems that require the opinions of experts and masters on the problem to be able to make the best decision. In these cases, it is necessary to use an optimization problem with Hesitant fuzzy parameters.There are few studies on Hesitant fuzzy linear programming (HFLP) problems. Therefore,in this paper, we  consider such problems.Especially, we study HFLP problems with Hesitant cost coefficients. For this purpose,we propose the simplex  method to solve the introduced optimization problems and draw a flowchart of the proposed  method.Finally, by solving two illustrative examples with Hesitant fuzzy information, we examine the applicability of the proposed method.

The Hesitant fuzzy soft set (HFSS), as a combination of Hesitant fuzzy and soft sets, is regarded as a useful tool for dealing with the uncertainty and ambiguity of real-world problems. In HFSSs, each element is defined in terms of several parameters with arbitrary membership degrees. In addition, distance and similarity measures are considered as the important tools in different areas such as pattern recognition, clustering, medical diagnosis, and the like. For this purpose, the present study aimed to evaluate the distance and similarity measures for HFSSs by using well-known Hamming, Euclidean, and Minkowski distance measures. Further, some examples were used to demonstrate that these measures fail to perform well in some applications. Accordingly, new distance and similarity measures were proposed by considering a hesitance index for HFSSs and the effect of considering hesitance index was shown by using an example of pattern recognition. Finally, the application of the proposed measures and hesitance index was investigated in the clustering and decision-making problem, respectively. In conclusion, the use of the proposed measures in clustering and hesitance index in decision-making can provide better and more reasonable results.

High dimensional microarray datasets are difficult to classify since they have many features with small number of instances and imbalanced distribution of classes. This paper proposes a filter-based feature selection method to improve the classification performance of microarray datasets by selecting the significant features. Combining the concepts of rough sets, weighted rough set, fuzzy rough set and Hesitant fuzzy sets for developing an effective algorithm is the main contribution of this paper. The mentioned method has two steps, in the first step, four discretization approaches are applied to discretize continuous datasets and selects a primary subset of features by combining of weighted rough set dependency degree and information gain via Hesitant fuzzy aggregation approach. In the second step, a significance measure of features (defined by fuzzy rough concepts) is employed to remove redundant features from primary set. The Wilcoxon Signed Ranked tes (A Non-parametric statistical test) is conducted for comparing the presented method with ten feature selection methods across seven datasets. The results of experiments show that the proposed method is able to select a significant subset of features and it is an effective method in the literature in terms of classification performance and simplicity.

Through this study, we are going to introduce a new aspect of cut sets for Hesitant fuzzy sets. Compared to the rare researches in regard of cut sets for Hesitant fuzzy sets, not only this study does not deal with extra parameters, but also the proposed concept of cut set conforms with the accepted principles, and further, it is extension of existing cut set concept of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets and three-dimensional fuzzy sets. In the sequel, using proposed concept of cut sets for Hesitant fuzzy sets, we extend the corresponding theorems of decomposition and representation for Hesitant fuzzy sets. The current research can be a basis for the future work on the application of cut sets in multiple criteria decision making problems under Hesitant fuzzy environment.