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

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 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.

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.

Hesitant fuzzy sets are very useful for dealing with group decision-making problems when experts have hesitation among several possible memberships for an element in a set. Recently, many researchers have used these concepts in clustering analysis and decision-making. In this article, the notions of Hesitant hybrid subsemigroups and Hesitant hybrid left(resp., right) ideals in a semigroup are introduced, and several properties are investigated. The concept of Hesitant hybrid product is also introduced, and properties of Hesitant hybrid subsemigroups and hybrid left (resp., right) ideals are considered using the notion of Hesitant hybrid product. Relations between Hesitant hybrid intersection and Hesitant hybrid product are exhibited. Additionally, we establish Hesitant hybrid interior ideals in semigroups and study theirproperties. Furthermore, we prove that in regular and intra-regular semigroups, the Hesitant hybrid ideal and the Hesitant hybrid interior ideals are coincide.

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.

This research attained to identify the subjectivity of Hesitant customer in circumstance that they want decide to purchase. For this aim they Q-method that is one of qualiquantological approach was used. Thirty three diction that gathered from interview and literature constitute the discourse circumscription. These diction ordering by 66 person and then analyzed by SPSS and Q-factor analyzing. The reliability of instrument tested by Cronbach test and attained by 0.952. The validity of discourse asked from participant at the end of questionnaire with an open question and approved. At last 9 subjectivity about purchase decision identified. Paying attention to recommendation of other persons, quality, brand, mode, pleasure, price, knowledge, country of origin, habit were these subjectivities between Hesitant customers when they encountered an acatalepsy in purchase decision making condition.

In this paper, the concepts of somewhat fuzzy automata continuous functions and somewhat fuzzy automata open functions in fuzzy automata topological spaces are introduced and some interesting properties of these functions are studied. In this connection, the concepts of fuzzy automata resolvable spaces and fuzzy automata irresolvable spaces are also introduced and their properties are studied.