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.

In recent years, industrial clusters have received considerable attention from economists and industry analysts because they are seen as the main reason for certain economic regions' economic growth and success. For many Industrial States Organization, the selection of industrial clusters has become a critical strategic consideration due to the Budget allocation priority. In this paper, an extended qualitative flexible multiple (QUALIFLEX) methods is used to solve problems regarding the priority among this cluster using probability Hesitant fuzzy information, which can lead to allocating the budget for industrial clusters more effectively. For more accuracy, we have applied a Hesitant fuzzy Topsis for prioritizing. Both rankings have been aggregated by the Copeland method. From our research results, the Larestan Muscat is of great importance, and Abade Inlaid Wood, Citrus packaging, Shiraz Marquetry, and Niriz stone have ranked respectively.

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.

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.

A Hesitant fuzzy number (HFN) is important as a generalization of the fuzzy number for Hesitant fuzzy analysis and takes some applications that were discussed in recent literature. In this paper, we develop the Hesitant fuzzy arithmetic, which is based on the extension principle for Hesitant fuzzy sets. Employing this principle, standard arithmetic operations on fuzzy numbers are extended to HFNs and we show that the outcome of these operations on two HFNs are an HFN. Also we use the extension principle in HFSs for the ranking of HFNs, which may be an interesting topic. In this paper, we show that the HFNs can be ordered in a natural way. To introduce a meaningful ordering of HFNs, we use a new lattice operation on HFNs based upon extension principle and de ning the Hamming distance on them. Finally, the applications of them are explained on optimization and decision-making problems.

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

Here are many situations in real applications of decision making where we deal with uncertain conditions. Due to the di erent sources of uncertainty, since its original de nition of fuzzy sets in 1965 [45], di erent generaliza-tions and extensions of fuzzy sets have been introduced: Type-2 fuzzy sets [11, 39], Intuitionistic fuzzy sets [1], fuzzy multi-sets [44] and etc. However, in such cases, it is suitable for experts to provide their preferences or assessments by using linguistic information rather than quantitative values.

The objective of this paper is to develop matrix games with pay-off of triangular Hesitant fuzzy elements (THFEs). To solve such games, a new methodology has been derived based on the notion of weighted average operator and score function of THFEs. Firstly, we formulate two non-linear programming problems with THFEs. Then applying the score function of THFEs, we transform these two problems into two non-linear multi-objective programming problems with triangular fuzzy numbers (TFNs). Finally, the Lexicographic method is used to solve these two multi-objective programming problems. A market share problem is considered to show the validity and applicability of the proposed methodology.

fuzzy Time Series Forecasting (TSF) is an approach for dealing with uncertainty in time series data that uses fuzzy logic. The Hesitant fuzzy Set (HFS) theory better emphasizes the chances of capturing fuzziness and uncertainty due to randomness than the classic fuzzy set theory. This study aims to improve the previously identified Hesitant fuzzy TSF models by including various degrees of hesitation to improve forecasting performance. The goal is to deal with the issue of identifying a common membership grade when several fuzzification methods are available to fuzzify time series data. The proposed method utilizes trapezoidal and bell-shaped fuzzy membership functions for constructing HFSs. AHesitant fuzzy weighted averaging operator is then applied to the Hesitant fuzzy Elements (HEFs) to create fuzzy logical relations. The suggested technique is employed to forecast enrollment in the University of Alabama and Cancer Incidence Rates (CIRs) in India. The efficiency of the proposed forecasting approach is determined by rigorously comparing it to various computational fuzzy TSF methods in terms of error measurements like Root Mean Square Error (RMSE), Average Forecasting Error (AFE), and Mean Absolute Deviation (Mad). The validity of the proposed forecasting model is verified by using correlation coefficients, coefficients of determination, Tracking Signals (TSs), and Performance Parameters (PPs). The significance of improved accuracy in forecasted results is also confirmed using the two-tailed t-test. The study results revealed that the enhanced Hesitant fuzzy Time Series (FTS) model is more effective and accurate in forecasting the university enrolment of Alabama and the CIRs of India.

In a clustering task, choosing a proper clustering algorithm and obtaining qualified clusters are crucial issues. Sometimes, a clustering algorithm is chosen based on the data distribution, but data distributions are not known beforehand in real world problems. In this case, we hesitate which clustering algorithm to choose. In this paper, this hesitation is modeled by a Hesitant fuzzy multi criteria decision making problem {\small (HFMCDM)} in which some clustering algorithms play the role of experts. Here, we consider fuzzy {\footnotesize C}-means {\small (FCM)} and agglomerative clustering algorithms as representative of two popular categories of clustering algorithms partitioning and hierarchical clustering methods, respectively.Then, we propose a new clustering procedure based on Hesitant fuzzy decision making approaches {\small (HFC)} to decide which of the {\small FCM} family or hierarchical clustering algorithms is suitable for our data. This procedure ascertains a good clustering algorithm using neutrosophic {\small FCM} ({\small NFCM}) through a two phases process. The {\small HFC} procedure not only makes a true decision about applying partitioning clustering algorithms, but also improves the performance of {\small FCM} and evolutionary kernel intuitionistic fuzzy c-means clustering algorithm ({\small EKIFCM}) with construction Hesitant fuzzy partition {\small (HFP)} conveniently. Experimental results show that the clustering procedure is applicable and practical. According to {\small HFC} procedure, it should be mentioned that it is possible to replace the other clustering algorithms that belong to any partitioning and hierarchical clustering methods. Also, we can consider other categories of clustering algorithms.