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 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.