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.

The standard model capacitated facility location (CPLP / CWLP) to locate the facility on a network of customer demand until total cost of the allocation minimize. The purposes of this paper cover all the demands of specified number of facilities to reduce the cost. In this paper closer to the real world, parameters of model is fuzzy and objective function is multi objective at the same time the total establishment cost and travel time is minimized in accordance with the constraints. The innovation is combining hybrid fuzzy simplex with genetic algorithm approach to accelerate near optimization process to change multi-objective to a single objective with LP metric. Simulated annealing algorithm also compared to reflect significant differences in performance and response time. Its application in locating facilities include warehouses, factories and services sector.

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.

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

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.