Neutrosophic Semiopen Hypersoft Sets with an Application to MAGDM under the COVID-19 Scenario

Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, Vellore 635 601, Tamilnadu, India Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Suvarnabhumi, Nonthaburi 11000, ,ailand Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University, Phuket 83000, ,ailand Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, ,ailand


Introduction
Fuzzy set theory [1] is an important tool for dealing with vagueness and incomplete data and is much more evolving and applied in different fields. Fuzzy set, which is an extension of general sets, has elements with membership function within the interval [0, 1]. In view of other options of human thinking, fuzzy set along with some conditions is extended to the intuitionistic fuzzy set [2]. e intuitionistic fuzzy set assigns membership and nonmembership functions to each object which satisfies the constraint that the sum of both membership functions is between 0 and 1.
Fuzziness was improved and extended from intuitionistic sets to neutrosophic sets. Smarandache [3] proposed neutrosophic sets, an essential mathematical tool which deals with incomplete, indeterminant, and inconsistent information. Neutrosophic set is characterized by the elements with truth, indeterminacy, and false membership functions which assume values within the range of 0 and 1. Wang et al. [4] proposed the concept of single-valued neutrosophic sets, a generalization of intuitionistic sets and a subclass of neutrosophic sets, which comprise elements with three membership functions which they belong to interval [0,1]. Under this neutrosophic environment, many researchers have worked on their extensions and developed many applications and results. A ranking approach based on the outranking relations of simplified neutrosophic numbers is developed in order to solve MCDM problems. Practical examples are provided to illustrate the proposed approach with a comparison analysis [5]. A comparison analysis is performed for this method with two examples [6], and the developed singlevalued neutrosophic TOPSIS extension is demonstrated on a numerical illustration of the evaluation and selection of e-commerce development strategies [7].
Molodtsov [8] introduced the idea of soft theory as a new approach to dealing with uncertainty, and now, there is a rapid growth of soft theory along with applications in various fields. Maji et al. [9] defined various basic concepts of soft theory, and the study of soft semirings by using the soft set theory has been initiated, and the notions of soft semirings, soft sub-semirings, soft ideals, idealistic soft semirings, and soft semiring homomorphisms with several related properties are investigated [10,11]. Maji et al. [12] developed the fuzzy soft set theory, which is a combination of soft and fuzzy sets. e idea of soft sets was generalized into hypersoft sets by Smarandache [13] by transforming the argument function F into a multiargument function. He also introduced many results on hypersoft sets. Saqlain et al. [14] utilized this notion and proposed a generalized TOPSIS method for decision-making. Neutrosophic sets [15], from their very introduction, have seen many such extensions and have been very successful in applications. A new hybrid methodology for the selection of offshore wind power station location combining the Analytical Hierarchy Process and Preference Ranking Organization Method for Enrichment Evaluations methods in the neutrosophic environment has been proposed [16], a neutrosophic preference ranking organization method for enrichment evaluation technique for multicriteria decision-making problems to describe fuzzy information efficiently was proposed and applied to a real case study to select proper security service for FMEC in the presence of fuzzy information [17], and a model is proposed based on a plithogenic set and is applied to differentiate between COVID-19 and other four viral chest diseases under the uncertainty environment [18].
In 2019, Rana et al. [19] introduced the plithogenic fuzzy hypersoft set (PFHS) in the matrix form and defined some operations on the PFHS. Single-and multivalued neutrosophic hypersoft sets were proposed by Saqlain et al. [20], who also defined tangent similarity measure for singlevalued sets and an application of the same in a decisionmaking scenario. In another effort, Saqlain et al. [21] also introduced aggregation operators for neutrosophic hypersoft sets. A recent development in this area of research is the introduction of basic operations on hypersoft sets in which hypersoft points in different fuzzy environments are also introduced [22].
Fuzzy topology, a collection of fuzzy sets fulfilling the axioms, was defined by Chang [23]. A new definition of fuzzy space compactness and observed to have α−compactness along with a Tychonoff theorem for an arbitrary product of α−compact fuzzy spaces and a 1-point compactification [24], filters in the lattice I X , where I is the unit interval and X an arbitrary set, have all been studied and using this study the convergence is defined in fuzzy topological space which leads to characterise fuzzy continuity and compactness [25]. en, the basic concepts of intuitionistic fuzzy topological spaces were constructed, and the definitions of fuzzy continuity, fuzzy compactness, fuzzy connectedness, and fuzzy Hausdorff space and some characterizations concerning fuzzy compactness and fuzzy connectedness were defined [26]. Neutrosophic topological spaces were introduced by Salama and Alblowi [27], and further concepts such as connectedness, semiclosed sets, and generalized closed sets [28] were developed. e concept of fuzzy soft topology and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, and fuzzy soft subspace topology were studied [29]. e soft topological spaces, soft continuity of soft mappings, soft product topology, and soft compactness, as well as properties of soft projection mappings, have all been defined [30], and a relationship between a fuzzy soft set's closure and its fuzzy soft limit points has been constructed on fuzzy soft topological spaces [31]. Subspace, separation axioms, compactness, and connectedness on intuitionistic fuzzy soft topological spaces were defined along with some base theorems [32], some important properties of intuitionistic fuzzy soft topological spaces and intuitionistic fuzzy soft closure and interior of an intuitionistic fuzzy soft set were introduced, and an intuitionistic fuzzy soft continuous mapping with structural characteristics was studied [33]. A topology on a neutrosophic soft set was constructed, neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, and neutrosophic soft boundary were introduced, some of their basic properties were studied, and the concept of separation axioms on the neutrosophic soft topological space was introduced [34]. e concept of fuzzy hypersoft sets was applied to fuzzy topological spaces, and fuzzy hypersoft topological spaces were presented by Ajay and Charisma in [35]. In the same work, fuzzy hypersoft topology has been extended to intuitionistic and neutrosophic hypersoft topological spaces along with their properties. In this paper, we define the idea of semiopen sets in fuzzy hypersoft topological spaces with their characterization and extend to semiopen sets in intuitionistic and neutrosophic hypersoft topological spaces. e paper is structured as follows: Section 2 recalls few basic terminologies and definitions of fuzzy hypersoft topological spaces. In Section 3, we define semiopen sets in fuzzy hypersoft topological spaces along with some of their properties. Sections 4 and 5 elaborate the logical extension of fuzzy hypersoft semiopen sets to intuitionistic and neutrosophic hypersoft semiopen sets. In Section 6, we present an application of the neutrosophic hypersoft open set and topology in an MAGDM and conclude in Section 7.
If the above axioms are satisfied, then τ is a fuzzy hypersoft topology (FHT) on (ϖ, X) · (X ϖ , τ) which is called a fuzzy hypersoft topological space (FHTS). Every member of τ is called an open fuzzy hypersoft set (OFHS). A fuzzy hypersoft set is said to be a closed fuzzy hypersoft set (CFHS) if its complement is OFHS. 4 and the attributes be E 1 � a 1 , a 2 , E 2 � a 3 , a 4 , and E 3 � a 5 , a 6 . en, the fuzzy hypersoft set is Let us consider this fuzzy hypersoft as (ϖ, X). en, the subfamily τ � ∅ X , (ϖ, X), · a 1 , a 3 , a 5 , of P(ϖ, X) is a FHT on (ϖ, X).
If the above axioms are satisfied, then τ is a neutrosophic hypersoft topology (NHT) on (ϖ, X) · (X ϖ , τ) which is called a neutrosophic hypersoft topological space (NHTS). Every member of τ is called an open neutrosophic hypersoft set (ONHS). A neutrosophic hypersoft set is called a closed neutrosophic hypersoft set (CNHS) if its complement is an ONHS.
en, (χ, B) is said to be an interior neutrosophic hypersoft set (INHS) of (Θ, J) if and only if (Θ, J) is a neighbourhood of (χ, B). e union of the whole INHS of (Θ, J) is named the interior of (Θ, J) and denoted as (Θ, J) ∘ or FHint(Θ, J).

Fuzzy Semiopen and Closed Hypersoft Sets
e fuzzy hypersoft topological space is τ: e fuzzy hypersoft set is a FSOHS.
Definition. 13An intuitionistic hypersoft set (Θ, J) in the IHST space is an intuitionistic semiclosed hypersoft set (ISCHS) if and only if its complement (Θ, J) C is ISOHS. e class of ISCHS is denoted by ISCHS(X).
Example 3. Let X � y 1 , y 2 , y 3 and the attributes be e intuitionistic hypersoft set is ISOHS.

Application
In this section, we present a multiattribute group decisionmaking (MAGDM) application of the NHS and NHS topology using two different algorithms, and the results of both algorithms are compared at the end. e algorithms proposed in [36] are considered, and some of their techniques are followed. Hypersoft sets are more feasible than soft sets and are more advantageous to use for applications since they can be dealt with more uncertainties. ere are many methods proposed for multiattribute group decisionmaking applications, but the proposed method is feasible than the methods which were proposed beforehand and done by using the more advanced recent work.

Numerical Example.
We propose to analyse the risk of COVID-19 by two MAGDM methods described by Algorithms 1 and 2 based on neutrosophic hypersoft sets and topology. We have all been affected by the current COVID-19 pandemic. However, the impact and consequences of the pandemic vary depending on our status as individuals and members of the society. We all find it difficult to be treated in hospitals because COVID affects everyone regardless of age. As a result, determining who should be treated first and assisting the most affected in becoming cured are difficult. e following method proposes methods for reducing the risk and treating patients based on their high risk of virus infection. Suppose that a committee of doctors have to give a report on patients having risk of COVID-19 in a particular area or hospital.
Let X � p 1 , p 2 , p 3 , p 4 , p 5 be the patients reported to the hospital. Suppose that the doctors consider the following set of attributes: E � e 1 , e 2 , e 3 , e 4 , e 5  is the person having most common symptoms (fever, dry cough, and tiredness), e 42 is the person having less common symptoms (aches and pain, sore throat, diarrhoea, headache, and loss of taste or smell), and e 43 is the person having serious symptoms (shortness of breath, chest pain, and loss of speech or movement).
Doctors divide the criteria into two subsets, A (category 1, for higher risk) and B (category 2, for medium risk). First, we solve the problem by using the NHS-MAGDM method as described in Algorithm 1.
Step 1: two NHSs, namely, (f, A) and (g, B) over X, are constructed after receiving all the required data from the committee.  Tables 1 and 2. Step 2: we are now constructing the NHS topology given by τ � ∅, X, (f, A), (g, B) , where ∅, X are NHS empty and full sets. e neutrosophic hypersoft open set (f, A) and (g, B) are formed in Tables 3 and  4, respectively, by taking the average for each element  from Tables 1 and 2. Step 3: the score matrix of NHS sets (f, A) and (g, B) is calculated in Tables 5 and 6, respectively.
Step 4: we are now calculating the decision table of (f, A) and (g, B) by averaging the score values correspondingly. Table 7 gives the decision values of (f, A) and (g, B).
Step 5: now, by adding the decision values of (f, A) and (g, B), we find the final decision value. Table 8 is the  required final decision table. Step 6: using Table 8, the final ranking of the patients is given by We see that patient 2 has the maximum value. So, patient 2 is selected for the immediate treatment.

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Journal of Mathematics Using Algorithm 2, we are now solving the same problem.
Steps 1, 2, and 3 are identical to those in Algorithm 1.
Step 4: the cardinal is computed by the formula given above in the algorithm. e cardinal for (f, A) is Similarly, the cardinal for (g, B) is and the cardinal for empty and full sets is completely 0 and 1, respectively.
Step 7: by using the optimal decision function max (f, A) * + (g, B) * (p) , we have the ranking of the patients who are of high risk of COVID-19. e final ranking according to Algorithm 2 is given by 6.2. Comparison Analysis. Using NHS, cardinal sets, score matrices, and aggregate fuzzy sets, we produced two MAGDM techniques. Table 9 provides a comparison of both algorithms, showing the optimal alternative and results. Both algorithms provide the same optimum decision, as can be seen in the comparison table.

Conclusion
e idea of hypersoft sets is a newly emerging technique in dealing with problems in the real world. Herein, we have defined the new concept of semi-hypersoft sets of the fuzzy hypersoft topological space. en, it has been extended to intuitionistic and neutrosophic semisets of intuitionistic and neutrosophic hypersoft topological spaces along with basic characterizations. Also, a real-life example in the current scenario of COVID-19 to make decision on the critical stage of medical diagnosis has been projected in MAGDM. is hypersoft topological space will also be extended to Pythagorean hypersoft topological spaces, as well as various forms of open sets, and more fuzzy topological space properties will be investigated.
e concept of open sets introduced in this work may be extended to pre-, alpha-open neutrosophic hypersoft sets and strong open neutrosophic hypersoft sets based on which more such applications to real-world problems can be explored.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
N.B. contributed to funding acquisition. G.R. and N.B. conceptualized the study, contributed to software, performed formal analysis, developed the methodology, wrote the original draft, and validated the study. G.R., P.H., J.J.C., and D.A. supervised the study. G.R., D.A., and N.B. reviewed and edited the article. All authors read and agreed to the published version of the manuscript.