Multicriteria Decision-Making Problem via Weighted Cosine Similarity Measure and Several Characterizations of Hypergroup and (Weak) Polygroups under the Triplet Single-Valued Neutrosophic Structure

Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Punjab, Pakistan Department of Mathematics, Islamia University Bahawalpur, Rahim Yar Khan Campus, Punjab 64200, Pakistan Department of Mathematics, University of Ghana, P. O. Box LG 62 Legon, Accra, Ghana Plant Production Department (Horticulture-pomology), Faculty of Agriculture, Saba Basha Alexandria University, Alexandria, Egypt


Introduction
e classical methods of mathematical analysis are unable to make sense of the ambiguities that exist in the universe. As a consequence of this, these structures need to be rethought in order to take into account the possibility of uncertainty. In 1965, Zadeh [1] proposed a fuzzy set. A fuzzy set is a mathematical model of ambiguity in which things belong to a speci c set to some degree.
is degree is generally a number that falls within the unit range of [0, 1].
In later years, as an extension of the fuzzy set, Sambuc [2] presented the notion of an interval-valued fuzzy set in 1975, Atanassov [3] provided the idea of an intuitionistic fuzzy set in 1984, Yager [4] initiated the concept of fuzzy multiset in 1986, Smarandache [5] presented the premise of a neutrosophic set (NS) in 1998, Molodstov [6] introduced the idea of soft sets in 1999, and Torra [7] developed a hesitant fuzzy set in 2010. Feng et al. [8] broadened soft sets by integrating them with fuzzy and rough sets, Aktas and Cagman [9] investigated soft groups, and Acar et al. [10] developed soft rings.
Marty [11] was the rst to propose algebraic hyperstructures, which are an overarching concept of classical algebraic structures. He broadened the de nition of a group to include the concept of a hypergroup. e resultant of two elements in a classical algebraic structure is an element.
However, the resultant of two elements in an algebraic hyperstructure is a set. Algebraic hyperstructures have been used in a wide range of subjects over the years, including hypergraphs, binary relations, cryptography, codes, median algebras, relation algebras, artificial intelligence, geometry, convexity, automata, combinatorial coloring problems, lattice theory, Boolean algebras, and logic probabilities. Hypergroups have mostly been used in the context of special subclasses.
Polygroups, which are spectacular subclasses of hypergroups, are developed by Ioulidis in [12] and employed to examine color algebras by Comer in [13,14]. Comer showed the effectiveness of polygroups by exploring their connections to graphs, relations, Boolean, and cylindric algebras. e theory of algebraic hyperstructures has since been investigated and expanded by a number of scholars. Many scholars working in these domains have been drawn to the combination of fuzzy sets and algebraic hyperstructures, as well as neutrosophic sets and algebraic hyperstructures, resulting in the creation of new branches of research, namely fuzzy algebraic hyperstructures and neutrosophic algebraic hyperstructures.
Comer developed quasi-canonical hypergroups in [15] as an extension of canonical hypergroups, which were presented in [16]. In [17], Comer introduced a number of algebraic and combinatorial properties. In [18], Davvaz and Poursalavati introduced matrix representations of polygroups over hyperrings and the idea of a polygroup hyperring, which expanded the concept of a group ring. Davvaz devised permutation polygroups and topics connected to them, employing the notion of generalized permutation [19]. We refer to some important and recent innovative work relative to the fuzzy structures and polygroups in  for further information.
Neutrosophy is a new subfield of philosophy that investigates the origin, nature, and multitude of neutralities, as well as their interactions with other ideological spectrums, which was first proposed by Smarandache in 1995. In the neutrosophic set, indeterminacy is quantified explicitly and truth-membership, indeterminacy membership, and falsitymembership are independent. In a neutrosophic set, truth (T), indeterminacy (I), and falsity (F) are the three types of membership functions. In this work, we develop set theoretic operators on a special kind of the neutrosophic set known as the single-valued neutrosophic set. A single-valued neutrosophic set (SVNS) is a type of NS that may be employed to address intellectual and technical problems in the real world. As a result, the study of SVNSs and their attributes is essential in terms of applications as well as comprehending the principles of uncertainty.
In this article, first we define the generalized concept (η, ξ, φ)-SVNS and then apply this concept to hypergroups and polygroups. For decision-making problems, a weighted cosine similarity measure (WCSM) is applied to each alternative, and the ideal alternative is used to rank the alternatives and choose the best option. In addition, we compared our strategy to current approaches and demonstrated its superiority. In conclusion, an example scenario illustrates how the suggested D-M technique may be implemented. In comparison, existing fuzzy multicriteria decision-making (M-CDM) strategies are incapable of tackling the decision-making difficulty stated in this paper. e suggested single-valued neutrosophic (SVN) decision-making technique has the benefit of being able to cope with ambiguous and inconsistent information, both of which are typical in real-world circumstances. e motivation of the proposed concept is explained as follows: to present a more generalized concept, i.e., (1) (η, ξ, φ)-single-valued neutrosophic hypergroups. (2) (η, ξ, φ)-single-valued neutrosophic polygroups. (3) (η, ξ, φ)-antisingle-valued neutrosophic polygroups. (4) Single-valued neutrosophic multicriteria decision-making method. Note that, clearly Υ Ω � Ω, Υ ∅ � ∅ , which shows that our proposed definition can be converted into a single-valued neutrosophic set. e purpose of this paper is to present the study of single-valued neutrosophic hypergroups and single-valued neutrosophic polygroups, and anti-single-valued neutrosophic polygroups under the triplet structure as a generalization of hypergroups, polygroups, and anti-polygroups as a powerful extension of single-valued neutrosophic sets.

Preliminaries
is section covers basic definitions related to SVNSs. In this section, we also present fundamental properties and relationships between SVNSs.
Definition 12 (see [45]). A subset K of H is called as subhypergroup if 〈K,°〉 is a hypergroup.
Mathematical Problems in Engineering 5 e proof is simple for readers.
And we get Mathematical Problems in Engineering 7 Similarly we can show that max ι ξ And we get Similarly we can show that max ι ξ And we get Similarly we can show that max ι ξ
Let H be a nonempty set, and P * (H) be the collection of all nonempty subsets of H. " * " should be formulated as follows: * : H × H ⟶ P * (H)(u, v)u * v en (H, * ) becomes a hypergroupoid and " * " is a hyperoperation.
Weak polygroups are generalization of polygroups and they are defined in the same way as polygroups but instead of (44) in Definition 14, we have In a (weak) polygroup P, (u − 1 ) − 1 � u, ∀u ∈ P.

Remark 2. Every group is a (weak) polygroup.
We present examples on polygroups that are not groups.
Definition 17 (see [48]). Let (P, * ) be a polygroup and Υ be a fuzzy set with a degree of membership m over P. en, Υ is considered a fuzzy polygroup over P if the followings conditions are satisfied ∀u, v ∈ P.
where e is the identity in P.
□ Definition 22 (see [48]). Let Υ be a fuzzy set over a polygroup (P, * ) with membership function m. en Υ is called the anti-fuzzy polygroup over P if ∀u, v ∈ P, the following conditions are fulfilled.
is a subpolygroup of P.
Proof 18. Let Q be a subpolygroup of P, consider t � (τ 1 , τ 2 , τ 3 ), where 0 < τ 1 , τ 2 < 1, and 0 < τ 3 < 1. Define the (η, ξ, φ)-SVNS over P as follows: (0, 0, 1), otherwise. (94) We represent a SVNS by δ ij � 〈ρ ij , ϱ ij , σ ij 〉. An SVNS is often synthesized from the evaluation of an alternative R i with regard to a criteria Y j in implementation using a score law and data processing. As a result, we may derive a SVN decision matrix D � (δ ij ) r×y . e notion of ideal point has been intended to assist discover the optimal option in a M-CDM scenario. Although the perfect alternative does not exist in the real world, it does give a valuable theoretical framework against which alternatives may be evaluated. e notion of optimum point has been achieved by involving to discover the optimal option in a M-CDM context. Although the perfect alternative somehow does not persist in the everyday life, it does give a valuable theoretical framework against which alternatives may be evaluated.
As a reason, the ideal alternative R @ is defined as the SVNS δ j � 〈ρ @ j , ϱ @ j , σ @ j 〉 � 〈1, 0, 0〉 for j � 1, 2, . . . y. e WCSM between an alternative R i and the ideal alternative R * represented by the SVNSs is defined by en, the higher the WCSM value, the better the option. e measure values can produce the ranking order of all alternatives and the best option by using (98).

Application
is section demonstrates an overview of a M-CDM issue with choices to exemplify the relevance and efficacy of the offered D-M strategy. Consider the paradox of D-M. ere is an investment firm that wants to put money into the finest choice. ere is a panel with four potential financing options: (1) R 1 is a manufacturer of automobiles; (2) R 2 is a manufacturer of electronics; (3) R 3 is a vacation rentals; and (4) R 4 is an industrial 3D printing builder company. e investment firm must make a judgement based on the three criteria listed below: (1) Y 1 is the financial, risk, and sensitivities; (2) Y 2 is the progress assessment; and (3) Y 3 is the environmental and location assessment. e criteria's weight vector is hence specified by w � (0.30, 0.25, 0.45). e questionnaire of a professional expert is used to appraise an alternative R i (i � 1, 2, 3, 4) in relation to a criteria Y j (j � 1, 2, 3).
When asked to experts of their opinion on a potential alternative R 1 corresponding to Y 1 , for instance, an expert might respond that there is a 0.6 chance that the statement is superb, a 0.2 chance that it is low, and a 0.1 chance that they are unsure. It may be written as δ 11 � 〈0.6, 0.2, 0.1〉 using the neutrosophic notation. e following SVN decision matrix D may be obtained when the expert evaluates the four potential options in light of the aforementioned three criteria: By employing (98), we can also give the following values of WCSM Q i (R i , R @ )(i � 1, 2, 3, 4) as e four options are thus ranked as follows: R 4 , R 1 , R 2 , and R 3 .
According to the order described by the rank matrix, industrial 3D printing builder company is turn out to be the best investment firm to put money into the finest choice whereas vacation rentals is the worst as per the criteria described.

Superiority of the Proposed Approach.
rough this analysis and comparison, it was possible to conclude that the Table 7: Comparison between SVNS and some existing approaches.

Set
Truth Indeterminacy Falsity Attributes Zadeh [1] Fuzzy set √ × × Atanassov [3] Intuitionistic fuzzy set √ × √ √ Yager [49] Pythagorean fuzzy set √ × √ √ Chen et al. [50] m-polar fuzzy set √ × × √ Naeem et al. [51] Pythagorean m-polar fuzzy set √ × √ √ Maji et al. [52] Fuzzy soft set √ × × √ Maji et al. [53] Intuitionistic fuzzy soft set √ × √ √ Peng et al. [54] Pythagorean fuzzy soft set √ × √ √ Zulqarnain et al. [55] Intuitionistic fuzzy hyper soft set √ × √ √ Zulqarnain et al. [56] Pythagorean fuzzy hyper soft set Mathematical Problems in Engineering proposed procedure has produced more frequent results than either of the alternatives. In general, the D-M approach associated with prevalent D-M methods permits additional data to alleviate hesitancy. In the D-M process, it is thus acceptable to propagate false and unclear information. erefore, the proposed method is reasonable, modest, and ahead of the fuzzy set's characteristic structures. e general information associated with the object could be stated precisely and analytically, as shown in Table 7.

Conclusion
is paper presented an algebraic hyperstructure of (η, ξ, φ)-SVNSs in the form of (η, ξ, φ)-SVN hypergroup, (η, ξ, φ)-SVNPs, and (η, ξ, φ)-ASVNPs. Several intriguing properties of the newly defined notions were discussed. e findings of this article can be thought of as a generalization of prior research on fuzzy hypergroups and fuzzy polygroups. We also discussed in this section a M-CDM system developed in an SVN environment using WCSM. WCSM between each option and the ideal alternative may be used to establish the ranking order of all alternatives and to readily identify the greatest alternative. Finally, an instructive example demonstrated how the new technique may be used. As a result, the proposed SVN M-CDM technique is more suited for real-world scientific and engineering applications since it can manage not only inadequate information but also indeterminate and inconsistent information, both of which are typical in real-world scenarios.
e strategy suggested in this study enhances previous D-M methods and offers decision-makers with an useable method.
is work provided an algebraic hyperstructure of (η, ξ, φ)-SVNSs as (η, ξ, φ)-SVN hypergroup, (η, ξ, φ)-SVNPs, and (η, ξ, φ)-ASVNPs. Several remarkable characteristics of the newly formed concepts were addressed. e results of this article can be seen as a generalization of previous research on fuzzy hypergroups and fuzzy polygroups. In this part, we also described an M-CDM system constructed in an SVN environment utilizing WCSM. WCSM between each option and the best option may be used to define the ranking order of all options and quickly discover the best choice. Finally, an illustrative illustration explained how the new method may be implemented. Consequently, the suggested SVN M-CDM approach is more suitable for real-world scientific and engineering applications, since it can handle not only insufficient information but also indeterminate and inconsistent information, both of which are characteristic of real-world settings. is research proposes an approach that advances earlier D-M methods and provides decision-makers with a practical method.
(i) Researchers will continue to work on complex D-M issues with uncertain weights of criteria, as well as other disciplines such as expert systems, information fusion systems, biochemistry, epidemiology, geology, entomology, and biomedical engineering.
In the realm of algebraic structure theory, it possesses a fantastic novel idea that has the potential to be utilized in the future for the solution of a variety of algebraic issues. (ii) Using the algebraic structure of multi-polygroup in terms of intuitionistic fuzzy set theory, this method may be readily extended to the intuitionistic fuzzy multi-polygroups. Connecting intuitionistic fuzzy multiset theory, set theory, and polygroup theory may provide a novel notion of polygroup that may be used to illustrate the effect of intuitionistic fuzzy multisets on a polygroup's structure. Using this concept, researchers may study intuitionistic fuzzy normal multi-subpolygroups along with their characterizations and algebraic characteristics. Additionally, the homomorphisms of intuitionistic fuzzy multi-polygroups and some of their structural properties may be addressed. Additionally, this idea may be used to investigate intuitionistic fuzzy quotient multi-polygroups. (iii) Researchers may expand this concept to include various neutrosophic multi-topological group structures. For this, they can introduce the definition of semi-open neutrosophic multiset, semiclosed neutrosophic multiset, neutrosophic multiregularly open set, neutrosophic multi-regularly closed set, neutrosophic multi-continuous mapping. In addition, since the idea of the almost topological group is so novel, they may utilize the definition of neutrosophic multi almost topological group to define neutrosophic multi almost topological group. (iv) is idea can be used to the development of the neutrosophic multi almost topological group of the neutrosophic multi-vector spaces, etc. is notion can be expanded to soft neutrosophic polygroups, weak soft neutrosophic polygroups, strong soft neutrosophic polygroups, soft neutrosophic polygroup homomorphism, and soft neutrosophic polygroup isomorphism. Furthermore, scholars might explore the homological properties of these polygroups.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.