# Selection of an Effective Hand Sanitizer to Reduce COVID-19 Effects and Extension of TOPSIS Technique Based on Correlation Coefficient under Neutrosophic Hypersoft Set

Authors: Abdul Samad; Rana Muhammad Zulqarnain; Emre Sermutlu; Rifaqat Ali; Imran Siddique; Fahd Jarad; Thabet Abdeljawad
Journal: Complexity
 (2021)

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## Abstract

Correlation coefficients are used to tackle many issues that include indistinct as well as blurred information excluding is not able to deal with the general fuzziness along with obscurity of the problems that have various information. The correlation coefficient (CC) between two variables plays an important role in statistics. Likewise, the accuracy of relevance assessment depends on the information in a set of discourses. The data collected for numerous statistical studies is full of exceptions. The concept of the neutrosophic hypersoft set (NHSS) is a parameterized family that deals with the subattributes of the parameters and is a proper extension of the neutrosophic soft set to accurately assess the deficiencies, anxiety, and uncertainty in decision-making. Compared with existing research, NHSS can accommodate more uncertainty, which is the most significant technique for describing fuzzy information in the decision-making process. The core objective of follow-up research is to develop the concept and characteristics of CC and the weighted correlation coefficient (WCC) of NHSS. We also introduced some aggregation operators in the considered environment, which can help us establish a prioritization technique for order preference by similarity to the ideal solution (TOPSIS) based on CC and WCC under NHSS. A decision-making strategy is established to solve multicriteria group decision-making (MCGDM) problems utilizing developed methodology. Moreover, the proposed method is utilized for the selection of an effective hand sanitizer during the COVID-19 pandemic to ensure the validity of the proposed approach. The practicality, effectivity, and flexibility of the current approach are proved through comparative analysis with the assistance of some existing studies.


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## Body

## 1. Introduction

Correlation plays an essential role in statistics and engineering. Employing correlation analysis, the joint relationship of two variables can be employed to assess the interdependency. In addition to utilizing probabilistic methods to such a lot of practical engineering complications, you also can locate several barriers to probabilistic strategies. Incidentally, the possibility of this process relies on a large amount of information, which may be random. However, structures have various uncertainties that are difficult to elevate, and it is difficult to obtain exact outcomes. So, due to unexplainable extensive information, the consequences of probability theory cannot provide experts with suitable information. Also, in real-world problems, there are no adequate grounds to proceed with the renowned statistical data. The consequences of probability theory are not in favor of experts because of initial hurdles. So, the probability theory has insufficient competency to resolve the uncertainty stated in the data. Various investigators in the world have predetermined and recommended different methods to solve such issues which involve anxiety. Zadeh suggested the idea of fuzzy sets (FS) [1] to solve complex issues which included anxiety as well as uncertainty. The fuzzy set theory allows modern ratings of the Mem of elements in the set. This is represented by the Mem function, and the effective unit interval of the Mem function is [0, 1]. The fuzzy set is the generalization of the classical set because the indicator function of the classic set is a special case of the Mem function of the fuzzy set if the latter only takes the value 0 or 1. In the fuzzy set theory, the classical bivalent set is usually called the crisp set. Fuzzy set theory can be used in a wide range of fields with incomplete or imprecise information.In some circumstances, decision makers consider the Mem and nonmembership (Nmem) values of objects. In such cases, Zadeh’s FS is unable to handle the imprecise and vague information. Atanassov [2] developed the notion of intuitionistic fuzzy sets (IFS) to deal the abovementioned difficulties. The IFS accommodates the imprecise and inaccurate information using Mem and Nmem values. Atanassov IFS is unable to solve those problems in which decision makers considered the membership degree (MD) and nonmembership degree (NMD) such as MD = 0.5 and NDM = 0.8; then, 0.5+0.8 ≰ 1. Yager [3, 4] extended the notion of IFS to Pythagorean fuzzy sets (PFSs) to overcome the above-discussed complications by modifying MD + NMD≤1 to MD2 + NMD2≤1. After the development of PFSs, Zhang and Xu [5] proposed operational laws for PFSs and established a DM approach to resolving the MCDM problem. Wei and Lu [6] proposed Pythagorean fuzzy power aggregation operators (AOs) and established the DM methodologies to solve multiattribute decision-making problems (MADM) using their developed AOs. Wang and Li [7] presented interaction operational laws and power Bonferroni mean operators for PFSs with their basic properties. Gao et al. [8] presented several aggregation operators by considering the interaction and proposed a DM approach to solving MADM difficulties by utilizing the developed operators. Wei [9] developed the interaction operational laws for Pythagorean fuzzy numbers (PFNs) by considering interaction and established interaction aggregation operators by using the developed interaction operations. Zhang [10] developed the accuracy function and presented a DM approach to solve multiple criteria group decision-making (MCGDM) problems using PFNs. Wang et al. [11] extended the PFSs and introduced interactive Hamacher operation with some novel AOs. They also established a DM method to solve MADM problems by using their proposed operators. Wang and Li [12] developed some operators for interval-valued PFSs and utilized their operators to resolve multiattribute group decision-making (MAGDM) problems. Peng and Yuan [13] established some novel operators such as Pythagorean fuzzy point operators and developed DM techniques using their proposed operators. Peng and Yang [14] introduced some operations with their desirable properties under PFSs and planned DM methodology to solve the MAGDM problem. Garg [15] developed the logarithmic operational laws for PFSs and proposed some AOs. Arora and Garg [16] presented the operational laws for linguistic IFS and developed prioritized AOs. Ma and Xu [17] presented some innovative AOs for PFSs, and they also developed the score and accuracy functions for PFNs.Abovementioned theories and their DM methodologies have been used in several fields of life. However, these theories are unable to deal with the parametrization of the alternatives. Molodtsov [18] developed the soft sets (SS) to overcome the abovementioned complications. Molodtsov’s SS competently deals with imprecise, vague, and unclear information of objects considering their parametrization. Maji et al. [19] prolonged the concept of SS and introduced basic operations with their properties. Maji et al. [20] established a DM technique using their developed operations for SS. They also merged two well-known theories such as FS and SS and established the concept of fuzzy soft sets (FSS) [21]. They also proposed the notion of an intuitionistic fuzzy soft set (IFSS) [22] and discussed their basic operations. Garg and Arora [23] extended the notion of IFSS and presented a generalized form of IFSS with AOs. They also planned a DM technique to resolve undefined and inaccurate information under IFSS information. Garg and Arora [24] presented the correlation and weighted correlation coefficients for IFSS and extended the TOPSIS technique using developed correlation measures. Zulqarnain et al. [25] introduced some AOs and correlation coefficients for interval-valued IFSS. They also extended the TOPSIS technique using their developed correlation measures and utilized them to solve the MADM problem. Peng et al. [26] proposed the Pythagorean fuzzy soft sets (PFSSs) and presented fundamental operations of PFSSs with their desirable properties by merging PFS and SS. Athira et al. [27] extended the notion of PFSSs and proposed entropy measures for PFSSs. They also presented some distance measures for PFSSs and utilized their developed distance measures to solve DM [28] issues. Zulqarnain et al. [29] introduced operational laws for Pythagorean fuzzy soft numbers (PFSNs) and developed AOs such as Pythagorean fuzzy soft weighted average and geometric by using defined operational laws for PFSNs. They also planned a DM approach to solve MADM problems with the help of presented operators. Riaz et al. [30] prolonged the idea of PFSSs and developed the m polar PFSSs. They also established the TOPSIS method under-considered hybrid structure and proposed a DM methodology to solve the MCGDM problem. Riaz et al. [31] developed the similarity measures for PFSS with their fundamental properties. Han et al. [32] protracted the TOPSIS method under PFSSs’ environment and utilized their developed approach to solving the MAGDM problem. Zulqarnain et al. [33] planned the TOPSIS methodology in the PFSS environment based on the correlation coefficient. They also established a DM methodology to resolve the MCGDM concerns and utilized the developed approach in green supply chain management.All the above studies only deal the insufficient data considering membership and nonmembership values; however, these theories cannot handle the overall incompatible as well as imprecise information. To address such incompatible as well as imprecise records, the idea of the neutrosophic set (NS) was developed by Smarandache [34]. Maji [35] offered the idea of a neutrosophic soft set (NSS) with necessary operations and properties. Karaaslan [36] developed the idea of the possibility NSS and introduced a possibility of neutrosophic soft DM method to solve those problems which contain uncertainty based on And-product. Broumi [37] developed the generalized NSS with some operations and properties and used the projected concept for DM. To solve MCDM problems with single-valued neutrosophic numbers (SVNNs) presented by Deli and Subas in [38], they constructed the concept of cut sets of SVNNs. Based on the correlation of IFS, the term correlation coefficient (CC) of SVNSs [39] was introduced. Ye [40] introduced the simplified NSs with some operational laws and AOs such as weighted arithmetic and weighted geometric average operators and constructed an MCDM method based on his proposed AOs. Masooma et al. [41] progressed a new concept through combining the multipolar fuzzy set and neutrosophic set which is known as the multipolar neutrosophic set, and they also established various characterization and operations with examples. Zulqarnain et al. [42] presented the generalized neutrosophic TOPSIS and used their presented technique for supplier selection in the production industry.Smarandache [43] protracted the idea of SS to hypersoft sets (HSS) by substituting the one-parameter function f to a multiparameter (subattribute) function. Samarandache claimed that the established HSS is competently dealing with uncertain objects comparative to SS. Nowadays, HSS theory and its extensions have been arising unexpectedly. Several investigators go through progressed distinctive operators along with characteristics under HSS and its extensions [44, 45]. Zulqarnain et al. [46] presented the IFHSS which is the generalized version of IFSS. They established the TOPSIS method to resolve the MADM problem utilizing the developed correlation coefficient. The authors of [47] introduced the Pythagorean fuzzy hypersoft sets with some basic operations and their properties. They also established a decision-making technique to deal with decision-making complications. Zulqarnain et al. [48] proposed the Pythagorean fuzzy hypersoft sets with AOs and correlation coefficients. They also established the TOPSIS technique using their developed correlation coefficient and utilized the presented approach for the selection of appropriate antivirus face masks. Zulqarnain et al. [49] presented some novel operations for interval-valued Pythagorean fuzzy hypersoft sets and discussed their desirable properties. They also developed the correlation coefficient and weighted correlation coefficient and developed a decision-making technique to solve decision-making issues utilizing their developed correlation measures [50] because the above work is considered to examine the environment of linear inequality between the MD and NMD of subattributes of the considered attributes. However, all existing studies only deal with the scenario by using MD and NMD of subattributes of the considered attributes. If any decision maker considers the truthiness, falsity, and indeterminacy of any subattribute of the alternatives, then, clearly, we can see that it cannot be handled by the abovementioned theories. To overcome the above limitations, we proposed some AOs for NHSS such as neutrosophic hypersoft weighted average and neutrosophic hypersoft weighted geometric operators. The core objective of the following scientific research is to develop novel AOs for the NHSS environment and processing mechanism, which can also follow the assumptions of NHSNs. Furthermore, the TOPSIS technique to solve the MCGDM problem was developed and a numerical illustration to justify the effectiveness of the proposed approach under the NHSS environment was presented.The rest of the article can be summarized as follows. In Section2, we introduced the fundamental notions such as SS, NSS, and NHSS, which can help us to build the subsequent research structure. In Section 3, we planned the correlation and relationship with NHSS informational energies and used the correlation and informational energies to develop CC, WCC, and their characteristics. Also, we use the planned CC to establish the TOPSIS method under-considered environment and propose some aggregation operators. To solve the MCGDM problem, an algorithm is established by using the presented TOPSIS approach, and a numerical explanation is provided in Section 4. Moreover, the planned DM method is used for the selection of multipurpose hand sanitizer in the COVID-19 pandemic. Also, we apply some available techniques to propose a comparative analysis of our planned approach. Similarly, the benefits of the planned algorithmic rule, superiority, tractability, and effectivity are presented. We will briefly discuss and equate the proposed strategy along with available methodologies in Section 5.


## 2. Preliminaries

In the following section, we recalled fundamental concepts that help us to develop the structure of the current article such as SS, NS, NSS, HSS, FHSS, and NHSS.Definition 1.
(see [18]). Let U and E be the universe of discourse and set of attributes, respectively. Let PU be the power set of U and A⊆E. A pair (ℱ,A) is called a SS over U, and its mapping is expressed as follows:(1)ℱ:A⟶PU.
Also, it can be defined as follows:(2)ℱ,A=ℱe∈PU:e∈E,ℱe=∅ife∉A.
Maji et al. [21] explored the theory of FS and SS and planned a more generalized version to handle the uncertainty compared with the existing FS and SS along with its unique features. This is generally known as a fuzzy soft set, which is a combination of FS and SS.Definition 2.
(see [21]). Let U and E be a universe of discourse and set of attributes, respectively, and ℱU be a power set of U. Let A ⊆ E; then, (ℱ,A) is an FSS over U, and its mapping can be expressed as follows:(3)ℱ:A⟶ℱU.Definition 3.
(see [34]). Let U be a universe and A be an NS on U which is defined as A = v,TAv,ℐAv,ℭAv:v∈U, where T, ℐ, ℭ: U ⟶ ]0−, 1+[ and 0− ≤ TAv+ℐAv+ℭAv ≤ 3+.
Maji et al. [35] established the notion of the neutrosophic soft set by merging the two existing theories such as NS and SS with some basic operations and their properties.Definition 4.
(see [35]). Let U be the universal set and E be the set of attributes concerning U. Let PU be the set of neutrosophic values of U andA⊆E. A pair ℱ,A is called a neutrosophic soft set over U and its mapping is given as(4)ℱ:A⟶PU.Definition 5.
(see [43]). Let U be a universe of discourse and P(U) be a power set of U and k = {k1, k2, k3,...,kn}, (n ≥ 1) and Ki represented the set of attributes and their corresponding subattributes such as Ki ∩ Kj = φ, where i ≠ j for each n ≥ 1 and i, j ∈ {1, 2, 3, …, n}. Assume K1×K2×K3×…×Kn=A⃛=a1h×a2k×⋯×anl is a collection of subattributes, where 1 ≤ h ≤ α, 1 ≤ k ≤ β, and 1 ≤ l ≤ γ, and α, β, γ ∈ ℕ. Then, the pair ℱ,K1×K2×K3×…×Kn=A⃛ is known as HSS defined as follows:(5)ℱ:K1×K2×K3×⋯×Kn=A⃛⟶PU.
It is also defined as(6)ℱ,A⃛=aˇ,ℱA⃛aˇ:aˇ∈A⃛,ℱA⃛aˇ∈PU.Definition 6.
(see [43]). Let U be a universe of discourse and P(U) be a power set of U and k = {k1, k2, k3,...,kn}, (n ≥ 1) be a set of attributes, and set Ki as a set of corresponding subattributes of ki, respectively, with Ki ∩ Kj = φ for n ≥ 1 for each i, j ∈ {1, 2, 3, …, n} and i ≠ j. Assume K1 × K2 × K3 × … × Kn = A⃛ = a1h×a2k×⋯×anl is a collection of subattributes, where 1 ≤ h .. α, 1 ≤ k ≤ β, and 1 ≤ l ≤ γ, and α, β, and γ ∈ ℕ, and IFSU is a collection of all intuitionistic fuzzy subsets over U. Then, the pair (ℱ, K1 × K2 × K3 × … × Kn = A⃛) is said to be IFHSS over U, and its mapping is defined as(7)ℱ:K1×K2×K3×⋯×Kn=A⃛⟶IFSU.
It is also defined as(8)ℱ,A⃛=aˇ,ℱA⃛aˇ:aˇ∈A⃛,ℱA⃛aˇ∈IFSU∈0,1,where ℱA⃛aˇ = δ,σℱaˇδ,τℱaˇδ:δ∈U, where σℱaˇδ and τℱaˇδ represent the membership and nonmembership values of the attributes such as σℱaˇδ, τℱaˇδ ∈ 0,1 and 0 ≤ σℱaˇδ + τℱaˇδ ≤ 1.
Simply an intuitionistic fuzzy hypersoft number (IFHSN) can be expressed asℱ = σℱaˇδ,τℱaˇδ, where 0 ≤σℱaˇδ+τℱaˇδ≤ 1.
The abovementioned IFHSS only deals with the MD and NMD of subattributes, and it is unable to accommodate the indeterminacy of the multi-subattributes of the considered attributes. To overcome such complications, the concept of NHSS has been developed by Smarandache.Definition 7.
(see [43]). Let U be a universe of discourse and P(U) be a power set of U and k = {k1, k2, k3,...,kn}, (n ≥ 1), and Ki represented the set of attributes and their corresponding subattributes such as Ki ∩ Kj = φ, where i ≠ j for each n ≥ 1 and i, j ∈ {1, 2, 3, …, n}. Assume K1×K2×K3×…×Kn=A⃛=a1h×a2k×⋯×anl is a collection of subattributes, where 1 ≤ h ≤ α, 1 ≤ k ≤ β, and 1 ≤ l ≤ γ, and α, β, γ ∈ ℕ, and NSU represents neutrosophic subsets over U. Then, ℱ,K1×K2×K3×⋯×Kn=A⃛ is called NHSS, and its mapping can be expressed as follows:(9)ℱ:K1×K2×K3×⋯×Kn=A⃛⟶NSU.
It is also defined asℱ,A⃛=aˇ,ℱA⃛aˇ:aˇ∈A⃛,ℱA⃛aˇ∈NSU, where ℱA⃛aˇ=v,Tℱaˇv,ℐℱaˇv,ℭℱaˇv:v∈U, where Tℱaˇv, ℐℱaˇv, and ℭℱaˇv represent the truth, indeterminacy, and falsity grades of the attributes such as Tℱaˇv, ℐℱaˇv, ℭℱaˇv ∈0,1, and 0 ≤ Tℱaˇv + ℐℱaˇv + ℭℱaˇv ≤ 3.
Simply a neutrosophic hypersoft number (NHSN) can be expressed asℱ = Tℱaˇv,ℐℱaˇv,ℭℱaˇv, where 0 ≤Tℱaˇv+ℐℱaˇv+ℭℱaˇv≤ 3.Example 1.
ConsiderU = v1,v2 is a universe of discourse and L = L1=Teaching methdology,L2=Subjects,L3=Classes are the considered attributes, and their corresponding n-tuple subattributes are given as follows: L1 = a11=project base,a12=class discussion, L2 = a21=Mathematics,a22=Computer Science,a23=Statistics, and Classes = L3 = a31=Masters,a32=Doctorol. Let Α⃛=L1×L2×L3 be a set of attributes(10)Α⃛=L1×L2×L3=a11,a12×a21,a22,a23×a31,a32=a11,a21,a31,a11,a21,a32,a11,a22,a31,a11,a22,a32,a11,a23,a31,a11,a23,a32,a12,a21,a31,a12,a21,a32,a12,a22,a31,a12,a22,a32,a12,a23,a31,a12,a23,a32,,Α⃛=aˇ1,aˇ2,aˇ3,aˇ4,aˇ5,aˇ6,aˇ7,aˇ8,aˇ9,aˇ10,aˇ11,aˇ12.
Then, the NHSS overU is given as follows:(11)F,Α⃛=a⌣1,δ1,.6,.3,.8,δ2,.9,.3,.5,a⌣2,δ1,.5,.2,.7,δ2,.7,.1,.5,a⌣3,δ1,.5,.2,.8,δ2,.4,.3,.4,a⌣4,δ1,.2,.5,.6,δ2,.5,.1,.6,a⌣5,δ1,.8,.4,.3,δ2,.2,.3,.5,a⌣6,δ1,.9,.6,.4,δ2,.7,.6,.8,a⌣7,δ1,.6,.5,.3,δ2,.4,.2,.8,a⌣8,δ1,.8,.2,.5,δ2,.6,.8,.4,a⌣9,δ1,.7,.4,.9,δ2,.7,.3,.5,a⌣10,δ1,.8,.4,.6,δ2,.7,.2,.9,a⌣11,δ1,.8,.4,.5,δ2,.4,.2,.5,a⌣5,δ1,.7,.5,.8,δ2,.7,.5,.9.
For simplicity, we will expressℱviaˇj = Tℱaˇvi,ℐℱaˇvi,ℭℱaˇvi|vi∈U as Jaˇij = Tℱaˇij,Jℱaˇij,ℭℱaˇij which is called NHSN, where 0≤Tℱaˇij+Jℱaˇij+ℭℱaˇij≤3 and Tℱaˇij,Jℱaˇij,ℭℱaˇij∈0,1. The score function for NHSNs Jaˇij is defined as follows:(12)SJaˇij=Tℱaˇij−ℭℱaˇij,where SJaˇij ∈ −1,1, and sometimes the score function is unable to compare any two NHSNs. Like Jaˇ11 = .6,.2,.2 and Jaˇ12 = .5,.1,.1, in such cases, it is difficult to choose which alternative is more appropriate. To handle such types of information, we need to introduce the accuracy function for NHSNs such as follows:(13)HJdˇij=Tℱaˇij+Jℱaˇij+ℭℱaˇij,HJaˇij∈0,1.
In the following, we present the comparison laws to compare NHSNsJaˇij and Taˇij such as(i)
IfSJaˇij > STaˇij, then Jaˇij > Taˇij.(ii)
IfSJaˇij = STaˇij, then(1)
IfHJaˇij > HTaˇij, then Jaˇij > Taˇij(2)
IfHJaˇij = HTaˇij, then Jaˇij = Taˇij.


## 3. Correlation Coefficient for Neutrosophic Hypersoft Set

In the subsequent section, we are going to present the notions of CC and WCC with their necessary properties over NHSS. Also, some aggregation operators for NHSNs will be introduced.Definition 8.
Let℘,A⃛=vi,T℘d⌣kvi,ℐ℘dˇkvi,C℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, their informational neutrosophic energies can be defined as(14)ςNHSS℘,A⃛=∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2,ςNHSSQ,B⃛=∑k=1m∑i=1nTQdˇkvi2+ℐQdˇkvi2+CQdˇkvi2.Definition 9.
Let℘,A⃛=vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛=vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, the correlation among them can be defined as(15)CNHSS℘,A⃛Q,B⃛=∑k=1m∑i=1nT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi.Proposition 1.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs and CNHSS℘,A⃛, Q,B⃛ be a correlation between them and satisfied the following properties:(1)
CNHSS℘,A⃛,℘,A⃛ = ςNHSS℘,A⃛(2)
CNHSSQ,B⃛,Q,B⃛ = ςNHSSQ,B⃛Proof.
By utilizing equation (15), it easily can be proved.Definition 10.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs; then, CC between them is given as δNHSS℘,A⃛,Q,B⃛ and expressed as follows:(16)δNHSS℘,A⃛,Q,B⃛=CNHSS℘,A⃛,Q,B⃛℘NHSS℘,A⃛∗℘NHSSQ,B⃛,(17)δNHSS℘,A⃛,Q,B⃛=∑k=1m∑i=1nT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1m∑i=1nTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.Theorem 1.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs; then, CC between them satisfies the following properties:(1)
0≤ δNHSS℘,A⃛,Q,B⃛ ≤ 1(2)
δNHSS℘,A⃛,Q,B⃛ = δNHSSQ,B⃛,℘,A⃛(3)
If℘,A⃛ = Q,B⃛, that is, ∀i, k, ℑ℘dˇkvi=ℑQdˇkvi, ℐ℘dˇkvi=ℐQdˇkvi, and ℭ℘dˇkvi=ℭQdˇkvi, then δNHSS℘,A⃛,Q,B⃛ = 1Proof.
δNHSS℘,A⃛,Q,B⃛ ≥ 0 is trivial; here, we only need to prove that δNHSS℘,A⃛,Q,B⃛ ≤ 1.
From equation (15), we have(18)δNHSS℘,A‥,Q,B‥=∑k=1m∑i=1nT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi=∑k=1mT℘dˇkv1∗TQdˇkv1+ℐ℘dˇkv1∗ℐQdˇkv1+ℭ℘dˇkv1∗ℭQdˇkv1+∑k=1mT℘dˇkv2∗TQdˇkv2+ℐ℘dˇkv2∗ℐQdˇkv2+ℭ℘dˇkv2∗ℭQdˇkv2+⋮+∑k=1mT℘dˇkvn∗TQdˇkvn+ℐ℘dˇkvn∗ℐQdˇkvn+ℭ℘dˇkvn∗ℭQdˇkvn,δNHSS℘,A‥,Q,B‥=T℘dˇ1v1∗TQdˇ1v1+ℐ℘dˇ1v1∗ℐQdˇ1v1+ℭ℘dˇ1v1∗ℭQdˇ1v1+T℘dˇ2v1∗TQdˇ2v1+ℐ℘dˇ2v1∗ℐQdˇ2v1+ℭ℘dˇ2v1∗ℭQdˇ2v1+⋮+T℘dˇmv1∗TQdˇmv1+ℐ℘dˇmv1∗ℐQdˇmv1+ℭ℘dˇmv1∗ℭQdˇmv1+T℘dˇ1vn∗TQdˇ1vn+ℐ℘dˇ1vn∗ℐQdˇ1vn+ℭ℘dˇ1vn∗ℭQdˇ1vn+T℘dˇ2vn∗TQdˇ2vn+ℐ℘dˇ2vn∗ℐQdˇ2vn+ℭ℘dˇ2vn∗ℭQdˇ2vn+⋮+T℘dˇmvn∗TQdˇmvn+ℐ℘dˇmvn∗ℐQdˇmvn+ℭ℘dˇmvn∗ℭQdˇmvnT℘dˇ1v2∗TQdˇ1v2+ℐ℘dˇ1v2∗ℐQdˇ1v2+ℭ℘dˇ1v2∗ℭQdˇ1v2+T℘dˇ2v2∗TQdˇ2v2+ℐ℘dˇ2v2∗ℐQdˇ2v2+ℭ℘dˇ2v2∗ℭQdˇ2v2+⋮+T℘dˇmv2∗TQdˇmv2+ℐ℘dˇmv2∗ℐQdˇmv2+ℭ℘dˇmv2∗ℭQdˇmv2∑k=1mT℘dˇkv1∗TQdˇkv1+T℘dˇkv2∗TQdˇkv2+⋯+T℘dˇkvn∗TQdˇkvn+∑k=1mℐ℘dˇkv1∗ℐQdˇkv1+ℐ℘dˇkv2∗ℐQdˇkv2+⋯+ℐ℘dˇkvn∗ℐQdˇkvn+∑k=1mℭ℘dˇkv1∗ℭQdˇkv1+ℭ℘dˇkv2∗ℭQdˇkv2+⋯+ℭ℘dˇkvn∗ℭQdˇkvn.
By using Cauchy–Schwarz inequality,(19)δNHSS℘,A‥,Q,B‥2≤∑k=1mT℘dˇkv12+T℘dˇkv22+…+T℘dˇkvn2+ℐ℘dˇkv12+ℐ℘dˇkv22+⋯+ℐ℘dˇkvn2+ℭ℘dˇkv12+ℭ℘dˇkv22+⋯+ℭ℘dˇkvn2×∑k=1mTQdˇkv12+TQdˇkv22+…+TQdˇkvn2+ℐQdˇkv12+ℐQdˇkv22+⋯+ℐQdˇkvn2+ℭQdˇkv12+ℭQdˇkv22+⋯+ℭQdˇkvn2,δNHSS℘,A⃛,Q,B⃛2≤∑k=1m∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2×∑k=1m∑i=1nTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2δNHSS℘,A⃛,Q,B⃛2≤ςNHSS℘,A⃛×ςNHSSQ,B⃛.
Therefore,δNHSS℘,A⃛,Q,B⃛2≤ςNHSS℘,A⃛×ςNHSSQ,B⃛. Hence, by using Definition 9, we have δNHSS℘,A⃛,Q,B⃛ ≤ 1. So, 0 ≤ δNHSS℘,A⃛,Q,B⃛ ≤ 1.Proof.
The proof is straightforward.Proof.
From equation (17), we have(20)δNHSS℘,A⃛,Q,B⃛=∑k=1m∑i=1nT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1m∑i=1nTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.
As we know thatT℘dˇkvi=TQdˇkvi, ℐ℘dˇkvi=ℐQdˇkvi, and ℭ℘dˇkvi=ℭQdˇkvi∀i, k, we obtain(21)δNHSS℘,A⃛,Q,B⃛=∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2,δNHSS℘,A⃛,Q,B⃛=1.
Thus, prove the required result.Definition 11.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, their correlation coefficient is given as δNHSS1℘,A⃛,Q,B⃛ and defined as follows:(22)δNHSS1℘,A⃛,Q,B⃛=CNHSS℘,A⃛,Q,B⃛maxςNHSS℘,A⃛,ςNHSSQ,B⃛,δNHSS1℘,A⃛,Q,B⃛=∑k=1m∑i=1nT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvimax∑k=1m∑i=1nT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2,∑k=1m∑i=1nTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2,δWNHSS℘,A⃛,Q,B⃛=∑k=1mΩk∑i=1nγiT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1mΩk∑i=1nγiTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.Theorem 2.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, CC between them satisfies the following properties:(1)
0≤ δNHSS1℘,A⃛,Q,B⃛ ≤ 1(2)
δNHSS1℘,A⃛,Q,B⃛ = δNHSS1Q,B⃛,℘,A⃛(3)
If℘,A⃛=Q,B⃛, that is, ∀i, k, T℘dˇkvi=TQdˇkvi, ℐ℘dˇkvi=ℐQdˇkvi, and ℭ℘dˇkvi=ℭQdˇkvi, then δNHSS1℘,A⃛,Q,B⃛=1Proof.
It is similar to Theorem1.
It is very important to deliberate the weight of NHSS for practical fortitudes nowadays. Whenever experts regulate distinctive weights for every alternative, the choice might be dissimilar. So, it is, precisely, to plot the weights for experts preceding assembling a decision. Assume the weights of experts can be expressed asΩ=Ω1,Ω2,Ω3,…,ΩmT, where Ωk >0, ∑k=1mΩk = 1. Similarly, assume the weights for subattributes as follows γ=γ1,γ2,γ3,…,γnT, where γi > 0, ∑i=1nγi = 1.Definition 12.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, WCC among them is expressed as δWNHSS℘,A⃛,Q,B⃛ and defined as follows:(23)δWNHSS℘,A⃛,Q,B⃛=CWNHSS℘,A⃛,Q,B⃛ςWNHSS℘,A⃛∗ςWNHSSQ,B⃛,δWNHSS℘,A⃛,Q,B⃛=∑k=1mΩk∑i=1nγiT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1mΩk∑i=1nγiTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.Definition 13.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs. Then, WCC among them is also given as δWNHSS1℘,A⃛,Q,B⃛ and defined as follows:(24)δWNHSS1℘,A⃛,Q,B⃛=CWNHSS℘,A⃛,Q,B⃛maxGWNHSS℘,A⃛,GWNHSSQ,B⃛,δWNHSS1℘,A⃛,Q,B⃛=∑k=1mΩk∑i=1nγiT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvimax∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2,∑k=1mω‵k∑k=1mΩk∑i=1nγiTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.
If we considerΩ = {1/m, 1/m,…, 1/m} and γ = {1/n, 1/n,…, 1/n}, then δWNHSS℘,A⃛,Q,B⃛ and δWNHSS1℘,A⃛,Q,B⃛ are reduced to δWNHSS℘,A⃛,Q,B⃛ and δWNHSS1℘,A⃛,Q,B⃛, respectively.Theorem 3.
Let℘,A⃛ = vi,T℘dˇkvi,ℐ℘dˇkvi,ℭ℘dˇkvi|vi∈U and Q,B⃛ = vi,TQdˇkvi,ℐQdˇkvi,ℭQdˇkvi|vi∈U be two NHSSs; then, WCC between them satisfies the following properties:(1)
0≤ δWNHSS℘,A⃛,Q,B⃛ ≤ 1(2)
δWNHSS℘,A⃛,Q,B⃛ = δWNHSSQ,B⃛,℘,A⃛(3)
If℘,A⃛ = Q,B⃛, that is, ∀i, k, T℘dˇkvi=TQdˇkvi, ℐ℘dˇkvi=ℐQdˇkvi, and ℭ℘dˇkvi=ℭQdˇkvi, then δWNHSS℘,A⃛,Q,B⃛ = 1Proof.
δWNHSS℘,A⃛,Q,B⃛ ≥ 0 is trivial, and here, we only need to prove that δWNHSS℘,A⃛,Q,B⃛ ≤ 1.
We have(25)CWNHSS℘,A‥,Q,B‥=∑k=1mΩk∑i=1nγiT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi=∑k=1mΩkγ1T℘dˇkv1∗TQdˇkv1+ℐ℘dˇkv1∗ℐQdˇkv1+ℭ℘dˇkv1∗ℭQdˇkv1+∑k=1mΩkγ2T℘dˇkv2∗TQdˇkv2+ℐ℘dˇkv2∗ℐQdˇkv2+ℭ℘dˇkv2∗ℭQdˇkv2+⋮+∑k=1mΩkγnT℘dˇkvn∗TQdˇkvn+ℐ℘dˇkvn∗ℐQdˇkvn+ℭ℘dˇkvn∗ℭQdˇkvn=Ω1γ1T℘dˇ1v1∗TQdˇ1v1+ℐ℘dˇ1v1∗ℐQdˇ1v1+ℭ℘dˇ1v1∗ℭQdˇ1v1+Ω2γ1T℘dˇ2v1∗TQdˇ2v1+ℐ℘dˇ2v1∗ℐQdˇ2v1+ℭ℘dˇ2v1∗ℭQdˇ2v1+⋮+Ωmγ1T℘dˇmv1∗TQdˇmv1+ℐ℘dˇmv1∗ℐQdˇmv1+ℭ℘dˇmv1∗ℭQdˇmv1+Ω1γ2T℘dˇ1v2∗TQdˇ1v2+ℐ℘dˇ1v2∗ℐQdˇ1v2+ℭ℘dˇ1v2∗ℭQdˇ1v2+Ω2γ2T℘dˇ2v2∗TQdˇ2v2+ℐ℘dˇ2v2∗ℐQdˇ2v2+ℭ℘dˇ2v2∗ℭQdˇ2v2+⋮+Ωmγ2T℘dˇmv2∗TQdˇmv2+ℐ℘dˇmv2∗ℐQdˇmv2+ℭ℘dˇmv2∗ℭQdˇmv2=Ω1γ1T℘dˇ1v1∗Ω1γ1TQdˇ1v1+Ω1γ1ℐ℘dˇ1v1∗Ω1γ1ℐQdˇ1v1+Ω1γ1ℭ℘dˇ1v1∗Ω1γ1ℭQdˇ1v1+Ω2γ1T℘dˇ2v1∗Ω2γ1TQdˇ2v1+Ω2γ1ℐ℘dˇ2v1∗Ω2γ1ℐQdˇ2v1+Ω2γ1ℭ℘dˇ2v1∗Ω2γ1ℭQdˇ2v1+⋮+Ωmγ1T℘dˇmv1∗Ωmγ1TQdˇmv1+Ωmγ1ℐ℘dˇmv1∗Ωmγ1ℐQdˇmv1+Ωmγ1ℭ℘dˇmv1∗Ωmγ1ℭQdˇmv1+Ω1γ2T℘dˇ1v2∗Ω1γ2TQdˇ1v2+Ω1γ2ℐ℘dˇ1v2∗Ω1γ2ℐQdˇ1v2+Ω1γ2ℭ℘dˇ1v2∗Ω1γ2ℭQdˇ1v2+Ω2γ2T℘dˇ2v2∗Ω2γ2TQdˇ2v2+Ω2γ2ℐ℘dˇ2v2∗Ω2γ2ℐQdˇ2v2+Ω2γ2ℭ℘dˇ2v2∗Ω2γ2ℭQdˇ2v2+⋮+Ωmγ2T℘dˇmv2∗Ωmγ2TQdˇmv2+Ωmγ2ℐ℘dˇmv2∗Ωmγ2ℐQdˇmv2+Ωmγ2ℭ℘dˇmv2∗Ωmγ2ℭQdˇmv2=Ω1γnT℘dˇ1vn∗TQdˇ1vn+ℐ℘dˇ1vn∗ℐQdˇ1vn+ℭ℘dˇ1vn∗ℭQdˇ1vn+Ω2γnT℘dˇ2vn∗TQdˇ2vn+ℐ℘dˇ2vn∗ℐQdˇ2vn+ℭ℘dˇ2vn∗ℭQdˇ2vn+⋮+ΩmγnT℘dˇmvn∗TQdˇmvn+ℐ℘dˇmvn∗ℐQdˇmvn+ℭ℘dˇmvn∗ℭQdˇmvn+Ω1γ1T℘dˇ1v1∗γ1TQdˇ1v1+γ1ℐ℘dˇ1v1∗γ1ℐQdˇ1v1+γ1ℭ℘dˇ1v1∗γ1ℭQdˇ1v1+Ω2γ1T℘dˇ2v1∗γ1TQdˇ2v1+γ1ℐ℘dˇ2v1∗γ1ℐQdˇ2v1+γ1ℭ℘dˇ2v1∗γ1ℭQdˇ2v1+⋮+Ωmγ1T℘dˇmv1∗γ1TQdˇmv1+γ1ℐ℘dˇmv1∗γ1ℐQdˇmv1+γ1ℭ℘dˇmv1∗γ1ℭQdˇmv1Ω1γ2T℘dˇ1v2∗γ2TQdˇ1v2+γ2ℐ℘dˇ1v2∗γ2ℐQdˇ1v2+γ2ℭ℘dˇ1v2∗γ2ℭQdˇ1v2+Ω2γ2T℘dˇ2v2∗γ2TQdˇ2v2+γ2ℐ℘dˇ2v2∗γ2ℐQdˇ2v2+γ2ℭ℘dˇ2v2∗γ2ℭQdˇ2v2+⋮+Ωmγ2T℘dˇmv2∗γ2TQdˇmv2+γ2ℐ℘dˇmv2∗γ2ℐQdˇmv2+γ2ℭ℘dˇmv2∗γ2ℭQdˇmv2+⋮+Ω1γnT℘dˇ1vn∗γnTQdˇ1vn+γnℐ℘dˇ1vn∗γnℐQdˇ1vn+γnℭ℘dˇ1vn∗γnℭQdˇ1vn+Ω2γnT℘dˇ2vn∗γnTQdˇ2vn+γnℐ℘dˇ2vn∗γnℐQdˇ2vn+γnℭ℘dˇ2vn∗γnℭQdˇ2vn+⋮+ΩmγnT℘dˇmvn∗γnTQdˇmvn+γnℐ℘dˇmvn∗γnℐQdˇmvn+γnℭ℘dˇmvn∗γnℭQdˇmvn.
By using Cauchy Schwarz inequality, we obtain(26)CWNHSS℘,A⃛,Q,B⃛2≤Ω1γ1T℘dˇ1v12+J℘dˇ1v12+ℭ℘dˇ1v12+Ω2γ1T℘dˇ2v12+J℘dˇ2v12+ℭ℘dˇ2v12+⋯++Ωmγ1T℘dˇmv12+J℘dˇmv12+ℭ℘dˇmv12Ω1γ2T℘dˇ1v22+J℘dˇ1v22+ℭ℘dˇ1v22+Ω2γ2T℘dˇ2v22+J℘dˇ2v22+ℭ℘dˇ2v22+⋯++Ωmγ2T℘dˇmv22+J℘dˇmv22+ℭ℘dˇmv22⋮+Ω1γnT℘dˇ1vn2+J℘dˇ1vn2+ℭ℘dˇ1vn2+Ω2γ2T℘dˇ2vn2+J℘dˇ2vn2+ℭ℘dˇ2vn2+⋯++ΩmγnT℘dˇmvn2+J℘dˇmvn2+ℭ℘dˇmvn2×Ω1γ1TQdˇ1v12+JQdˇ1v12+ℭQdˇ1v12+Ω2γ1TQdˇ2v12+JQdˇ2v12+ℭQdˇ2v12+⋯++Ωmγ1TQdˇmv12+JQdˇmv12+ℭQdˇmv12Ω1γ2TQdˇ1v22+JQdˇ1v22+ℭQdˇ1v22+Ω2γ2TQdˇ2v22+JQdˇ2v22+ℭQdˇ2v22+⋯++Ωmγ2TQdˇmv22+JQdˇmv22+ℭQdˇmv22⋮+Ω1γnTQdˇ1vn2+JQdˇ1vn2+ℭQdˇ1vn2+Ω2γ2TQdˇ2vn2+JQdˇ2vn2+ℭQdˇ2vn2+⋯++ΩmγnTQdˇmvn2+JQdˇmvn2+ℭQdˇmvn2,(27)CWNHSS℘,A⃛,Q,B⃛2≤∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2×∑k=1mΩk∑i=1nγiTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2CWNHSS℘,A⃛,Q,B⃛2≤ςWNSS℘,A⃛∗ςWNHSSQ,B⃛.
Therefore,CWNHSS℘,A⃛,Q,B⃛≤ςWNHSS℘,A⃛∗ςWNHSSQ,B⃛; hence, 0≤δWNHSS℘,A⃛,Q,B⃛≤1.Proof.
The proof is straightforward.Proof.
From equation (23), we have(28)δWNHSS℘,A⃛,Q,B⃛=∑k=1mΩk∑i=1nγiT℘dˇkvi∗TQdˇkvi+ℐ℘dˇkvi∗ℐQdˇkvi+ℭ℘dˇkvi∗ℭQdˇkvi∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1mΩk∑i=1nγiTQdˇkvi2+ℐQdˇkvi2+ℭQdˇkvi2.
As we know thatT℘dˇkvi=TQdˇkvi, ℐ℘dˇkvi=ℐQdˇkvi, and ℭ℘dˇkvi=ℭQdˇkvi. ∀i, k. We obtain(29)δWNHSS℘,A⃛,Q,B⃛=∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2∑k=1mΩk∑i=1nγiT℘dˇkvi2+ℐ℘dˇkvi2+ℭ℘dˇkvi2,δWNHSS℘,A⃛,Q,B⃛=1.Definition 14.
LetJdˇk = Tℱdˇij,Jℱdˇij,ℭℱdˇij, Jdˇ11 = Tℱdˇ11,Jℱdˇ11,ℭℱdˇ11, and Jdˇ12 = Tℱdˇ12,Jℱdˇ12,ℭℱdˇ12 be three NHSNs and α be a positive real number; by algebraic norms, we have(1)
Jdˇ11⊕Jdˇ12 = <Tℱdˇ11+Tℱdˇ12−Tℱdˇ11Tℱdˇ12,Jℱdˇ11Jℱdˇ12,ℭℱdˇ11ℭℱdˇ12>(2)
Jdˇ11⊗Jdˇ12 = <Tℱdˇ11Tℱdˇ12,Jℱdˇ11+Jℱdˇ12−Jℱdˇ11Jℱdˇ12,ℭℱdˇ11+ℭℱdˇ12−ℭℱdˇ11ℭℱdˇ12>(3)
αJdˇk = <1−1−Tdˇkα,Jdˇkα,ℭdˇkα>(4)
Jdˇkα = <Tdˇkα,1−1−Jdˇkα,1−1−ℭdˇkα>
We develop some AOs based on the above-defined operational laws for the collection of NHSNs ∆.Definition 15.
LetJdˇk = Tℱdˇij,Jℱdˇij,ℭℱdˇij be an NHSN, and Ωi and γj are weight vector for expert’s and subattributes of the considered attributes correspondingly along with specified circumstances Ωi > 0, ∑i=1nΩi = 1, γj > 0, ∑j=1mγj = 1. Then, NHSWA operator is defined as follows.
NHSWA:Δn⟶Δ is defined as follows:(30)NHSWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕j=1mγj⊕i=1nΩiJdˇij=1−∏j=1m∏i=1n1−TdˇijΩiγj,∏j=1m∏i=1nJdˇijΩiγj∏j=1m∏i=1nℭdˇijΩiγj.Definition 16.
LetJdˇk = Tℱdˇij,Jℱdˇij,ℭℱdˇij be an NHSN, and Ωi and γj are weight vector for expert’s and subattributes of the considered attributes correspondingly along with specified circumstances Ωi > 0, ∑i=1nΩi = 1, and γj > 0, ∑j=1mγj = 1. Then, NHSWG operator is defined as follows.
NHSWG:Δn⟶Δ defined as follows:(31)NHSWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1m⊗i=1nJdˇnmΩiγjNHSWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m∏i=1nTdˇijΩiγj,1−∏j=1m∏i=1n1−JdˇijΩiγj,1−∏j=1m∏i=1n1−ℭdˇijΩiγj.


## 4. TOPSIS Approach on NHSS for MCGDM Problem Based on the Correlation Coefficient

In this section, we are going to develop a methodology for solving DM issues through continuing the TOPSIS technique for NHSS information according to CC. Hwang and Yoon [51] progressed the TOPSIS methodology and utilized it to promote the order of evaluation components of the PIA and NIA for DM complications. Through the use of the presented approach, we will find the best possible choices which have the smallest and largest distances to PIS and NIS. The TOPSIS method ensures that the correlation measure can be utilized to differentiate PIA and NIA by choosing positions. Generally, investigators are using the TOPSIS approach to find CC along with distinctive distance forms. The TOPSIS method along with CC is superior for locating closeness coefficients instead of distance as well as similarity measures. Since the correlation measure holds the direct association among the aspects restrained, by utilizing the established CC, a TOPSIS method is presented to choose the most appropriate choice.


### 4.1. Proposed Decision-Making Approach

LetX = X1,X2,X3,…,Xs be a collection of “s” alternatives for assessment under the group of experts Q = Q1,Q2,Q3,…,Qn with weights Ω = Ω1,Ω1,…,ΩnT and Ωi>0, ∑i=1nΩi = 1. Consider L=d1,d2,…,dm be a set of attributes with their corresponding subattributes such as L′=d1ρ×d2ρ×⋯×dmρfor allρ∈1,2,…,t with weights γ=γ1ρ,γ2ρ,γ3ρ,…,γmρT such as γρ>0, ∑ρ=1tγρ=1, and can be stated as L′=dˇ∂:∂∈1,2,…,k. The group of experts {Qi: i = 1, 2, …, n} evaluate the alternatives {Xz: z = 1, 2, …, s} under the desired subattributes of the considered parameters dˇ∂:∂=1,2,…,k given in the form of NHSNs such as ℵdˇikzn×∂ = Tdˇikz,ℐdˇikz,ℭdˇikzn×∂, where 0 ≤ Tdˇikz,ℐdˇikz,ℭdˇikz ≤ 1 and 0 ≤ Tdˇikz+ℐdˇikz+ℭdˇikz ≤ 3 for all i, k.Step 1: development of decision matrices for each alternative {Xz: z = 1, 2, …, s} by using subattributes of the given attributes in the form of NHSNs such as follows:(32)Step 2: normalize the aggregated decision matrices by transforming cost type subattributes to benefit type subattribute using the normalization rule:(33)Xij=Jdˇijc,cost type parameter,Jdˇij,benefit type parameter,whereJdˇijc = 1−ℭdˇij,Jdˇij,Tdˇij, which represents the complement of <Tdˇij,Jdˇij,ℭdˇij>.Step 3: development of weighted decision matrix for each alternativeX¯z=J¯dˇijzn×∂, where(34)J¯dˇijz=γjΩiJdˇijz1−∏j=1m∏i=1n1−TdˇijΩiγj,∏j=1m∏i=1nJdˇijΩiγj,∏j=1m∏i=1nℭdˇijΩiγj=T¯dˇijz,J¯dˇijz,ℭ¯dˇijz.Here,Ωi and γj expressed the weights of experts and subattributes.Step 4: to determine the PIA and NIA compute indices by utilizing equation (12), such as argmaxzθijz and gij = arg minzθijz,(35)ℒ+=Tdˇij+,Jdˇij+,ℭdˇij+n×∂=T¯dˇijℏij,J¯dˇijℏij,ℭ¯dˇijℏij,(36)ℒ−=Tdˇij−,Jdˇij−,ℭdˇij−n×∂=T¯dˇijgij,J¯dˇijgij,ℭ¯dˇijgij.Step 5: considerX¯z and PIA ℒ+ analyze the CC such as follows:(37)pz=δNHSSX¯z,ℒ+=CNHSSX¯z,ℒ+ςNHSSXz∗ςNHSSℒ+=∑j=1m∑i=1mT¯dˇijz∗Tdˇij++J¯dˇijz∗Jdˇij++ℭ¯dˇijz∗ℭdˇij+∑j=1m∑i=1nT¯dˇijz2+J¯dˇijz2+ℭ¯dˇijz2∑j=1m∑i=1nTdˇij+2+Jdˇij+2+ℭdˇij+2.Step 6: considerX¯z and NIA ℒ− analyze the CC such as follows:(38)qz=δNHSSX¯z,ℒ−=CNHSSX¯z,ℒ−ςNHSSXz∗ςNHSSℒ−=∑j=1m∑i=1nT¯dˇijz∗Tdˇij−+J¯dˇijz∗Jdˇij−+ℭ¯dˇijz∗ℭdˇij−∑j=1m∑i=1nT¯dˇijz2+J¯dˇijz2+ℭ¯dˇijz2∑j=1m∑i=1nTdˇij−2+Jdˇij−2+ℭdˇij−2.Step 7: for the alternatives, ranking computes the closeness coefficient:(39)ℛz=KX¯z,ℒ−KX¯z,ℒ++KX¯z,ℒ−.where(40)KX¯z,ℒ−=1−qzandKX¯z,ℒ+=1−pz.Step 8: pick the alternate with the largest closeness coefficient.Step 9: rank the alternatives.Graphical representation of the above presented TOPSIS algorithm is given in Figure1.Figure 1
Graphical representation of developed TOPSIS technique for NHSS.


### 4.2. Application of the Proposed MCGDM Technique for the Selection of Hand Sanitizer

#### 4.2.1. Case Study

Sanitization is the most important process because direct contact along with airborne microorganism droplets caused by sneezing and sneezing can without problems be infected. Particularly, when a pandemic breaks out, it is necessary to interrupt the overall chain of transmitting of the virus by proper hand sterilization trials. This will be accomplished by employing the right to use isolation as well as accommodating emergency care possessions, such as handling sterilizing in psychiatric divisions as well as public places. The assumption of sanitizers simply relies on the use of multipurpose sanitizers developed in various types and varieties, nominative antibacterial soaps and water-based or alcohol-based sanitizers, which have been common in hospitals. Until now, the most helpful hand sanitizer product is the alcohol-based formulation, which could include 62% to 95% alcohol, since it can convert microbic proteins and have the ability to inactivate infections. The reassessment of this system of rules is expounded to the information in PubMed. It is going to examine the variety of available sanitizers along with their efficiency plus formulation features, contrary effects, and approvals to improve preparation proficiency along with safety. Furthermore, this article concentrates on the efficiency of alcohol-based sanitizers against disease. Up to now, the most effective hand sanitizer overall has been formulated based on alcoholic brews, which conquer 62% to 95% of alcoholic beverages because it can formalize microbial proteins and [52, 53] can be disabled. Due to its intoxicated alcohol content, the current expression gives some rival as well as concerns about safety disadvantages along with skin cytotoxicity [54]. Current research objectives scrutinize a variety of accessible sanitizers, as well as their effectiveness against the redeeming coronavirus, besides strategies and findings as well as ideas for reinforcing the formation of saviors from accessible hands.Hand sanitizers may be a liquid or gel, mainly used to cut down transmittable diseases on the hands. In most organizations, alcohol-based sanitizers are superior for cleaning. The overall concentrates for risk reduction and prevention propose that person washing their hands with detergent to restrict the spread of virus infection and cut down the risk of unhealthiness. Due to the lack of soap and water, the CDC recommends that people employ alcohol-based (at least 60%) hand sanitizer. According to scientific research by the world health organization (WHO), in the context of this coronavirus pandemic, hygienic as well as physical distancing are the best methods to protect ourselves and every person around us from COVID-19 infection. The virus is spread by people with this disease, and it can also be spread out via contact with sick people. We cannot treat the coronavirus with complete caution. Hence, good hand sanitization may be the ultimate obstacle between us and the virus. The WHO endorses using alcohol-based hand sanitizer to eliminate the new COVID-19 virus. Alcohol-based hand sanitizer can prevent microbial proteins (together with bacteria and certain viruses) from operating usually. Hand sanitizer needs to comprehend ethanol, isopropanol, n-propanol, or a mixture of the alcohols. All of those can appropriately refuse to accept infections such as the new coronavirus. In that serious COVID-19 situation, the inquiry for hand sanitizer seems to have exaggerated. Due to exaggerated demand, it is difficult to obtain fine quality as well as multipurpose sanitizers in the overall market. The rise in demand also led to inferior sanitizers coming into the market. Akram et al. [55] utilized the GFFYWA operator for the selection of an effective hand sanitizer. The most important determinant of the current application is to reduce the spread of coronavirus by utilizing the TOPSIS technique under the NHSS scenario.


### 4.3. Numerical Example

LetX = X1,X2,X3,X4 be a collection of four hand sanitizers (alternatives) such as X1 = Megababe Squeaky Clean Sanitizer, X2 = Purell Advanced Sanitizer Gel, X3 = Touchland power mist citrus, and X4 = Lemon sanitizer spray. A team of four medical experts such as Q = Q1,Q2,Q3,Q4 having weight vector .1,.25,.35,.3T for the selection of an appropriate alternative (sanitizer), which is most helpful to protect the COVID-19 effects, is hired. The group of medical experts chooses the set of attributes for the selection of an effective sanitizer such as L = ℓ1=alcohol ratio,ℓ2=ingredients,ℓ3=Quality with their corresponding subattribute: alcohol ratio = ℓ1 = d11=less than40%,d12=more than60%, ingredients = ℓ2 = d21=hydrogen  peroxide,d22=glycerol, and quality = ℓ3 = d31=effectiveness. Let L' = ℓ1 × ℓ2 × ℓ3 show the multi-subattributes:(41)L′=ℓ1×ℓ2×ℓ3=d11,d12×d21,d22×d31=d11,d21,d31,d11,d22,d31,d12,d21,d31,d12,d22,d31.L' = dˇ1,dˇ2,dˇ3,dˇ4 with weights 0.2,0.1,0.4,0.3T. Each expert will evaluate the ratings of hand sanitizer in the form of NHSNs for each subattribute .Table 1
Decision matrix for alternativeX1.
X1dˇ1dˇ2dˇ3dˇ4Q10.9,0.2,0.10.3,0.3,0.70.6,0.4,0.20.7,0.1,0.3Q20.8,0.3,0.20.6,0.2,0.60.8,0.3,0.10.2,0.6,0.8Q30.6,0.1,0.30.6,0.1,0.30.8,0.2,0.10.6,0.3,0.4Q40.9,0.1,0.10.9,0.1,0.10.8,0.1,0.10.9,0.1,0.2Table 2
Decision matrix for alternativeX2.
X2dˇ1dˇ2dˇ3dˇ4Q10.3,0.3,0.70.9,0.2,0.10.6,0.1,0.30.3,0.6,0.2Q20.8,0.2,0.10.8,0.3,0.20.9,0.1,0.10.8,0.3,0.1Q30.6,0.3,0.40.8,0.1,0.20.9,0.1,0.10.2,0.3,0.8Q40.9,0.1,0.20.8,0.1,0.10.7,0.1,0.30.6,0.3,0.4Table 3
Decision matrix for alternativeX3.
X3dˇ1dˇ2dˇ3dˇ4Q10.6,0.3,0.40.2,0.3,0.80.3,0.6,0.20.3,0.6,0.2Q20.9,0.1,0.10.9,0.1,0.10.9,0.1,0.10.8,0.3,0.2Q30.8,0.3,0.20.9,0.2,0.10.9,0.1,0.10.2,0.3,0.8Q40.3,0.3,0.70.9,0.1,0.20.7,0.1,0.30.6,0.3,0.4Table 4
Decision matrix for alternativeX4.
X4dˇ1dˇ2dˇ3dˇ4Q10.9,0.1,0.10.9,0.1,0.20.8,0.2,0.10.3,0.6,0.2Q20.8,0.2,0.10.8,0.2,0.10.6,0.3,0.40.8,0.3,0.2Q30.8,0.1,0.10.8,0.1,0.20.9,0.1,0.10.3,0.6,0.2Q40.9,0.1,0.20.3,0.3,0.70.8,0.3,0.20.9,0.1,0.1Table 5
Weighted decision matrixX¯1.
X¯1dˇ1dˇ2dˇ3dˇ4Q10.7354,0.9835,0.50210.5691,0.9778,0.95760.6791,0.9762,0.84910.6694,0.9848,0.8491Q20.7200,0.9872,0.90090.7038,0.9728,0.94130.7955,0.9729,0.54760.9009,0.9301,0.9835Q30.7443,0.9438,0.49590.6422,0.9513,0.92250.7270,0.9798,0.82130.7386,0.9465,0.9622Q40.6328,0.9891,0.47840.5034,0.9786,0.98880.6829,0.9909,0.78660.6377,0.9863,0.9945Table 6
Weighted decision matrixX¯2.
X¯2dˇ1dˇ2dˇ3dˇ4Q10.6867,0.9878,0.80730.6178,0.9676,0.98890.7920,0.9862,0.79200.6694,0.9794,0.9862Q20.7599,0.9754,0.80330.6109,0.9517,0.99150.7068,0.9529,0.70680.9009,0.9301,0.9933Q30.7849,0.9513,0.77490.7379,0.9513,0.94980.7819,0.9911,0.90830.7386,0.9465,0.9377Q40.7229,0.9891,0.78660.4402,0.9908,0.99590.6829,0.9880,0.76310.5644,0.9821,0.9931Table 7
Weighted decision matrixX¯3.
X¯3dˇ1dˇ2dˇ3dˇ4Q10.7559,0.9835,0.85510.6178,0.9754,0.99090.7920,0.9862,0.59870.6694,0.9729,0.9699Q20.7599,0.9754,0.90700.6109,0.9517,0.98930.7068,0.9529,0.46610.9009,0.9301,0.9236Q30.8678,0.9257,0.85270.6955,0.9362,0.96830.9083,0.9529,0.68970.7386,0.9465,0.9408Q40.7229,0.9891,0.84570.5034,0.9786,0.99740.7408,0.9949,0.61950.6049,0.9863,0.9919Table 8
Weighted decision matrixX¯4.
X¯4dˇ1dˇ2dˇ3dˇ4Q10.6867,0.9915,0.82410.6572,0.9754,0.98400.7311,0.9818,0.50210.7048,0.9645,0.9605Q20.7599,0.9754,0.90700.6109,0.9517,0.98210.9085,0.9333,0.65090.7781,0.9301,0.9553Q30.8678,0.9438,0.85270.7379,0.9513,0.96830.8886,0.9377,0.64150.8376,0.9465,0.9225Q40.7229,0.9891,0.84570.4402,0.9786,0.99650.7152,0.9931,0.65750.6667,0.9821,0.9506


#### 4.3.1. Selection of an Effective Hand Sanitizer by Using the TOPSIS Method

Step 1: decision makers will investigate the situation in the case of NHSNs, and there are only four alternatives. The subattributes of the considered attributes, along with a summary of their scores are given in Tables1–4.Step 2: there is no need for normalization since all attributes are identical.Step 3: weighted decision matrix for each alternativeX¯z = ℒ¯ijzn×∂ is developed by using equation (30). The values of the weighted decision matrices for each alternative are given in Tables 5–8.Step 4: by utilizing equations (35) and (36), compute the PIA and NIA such as follows:(42)ℒ+=0.6867,0.9878,0.80730.6178,0.9676,0.98890.7311,0.9818,0.50210.6694,0.9729,0.96990.7599,0.9754,0.90700.6109,0.9517,0.99150.7068,0.9529,0.70680.9009,0.9301,0.98350.7849,0.9513,0.77490.7379,0.9513,0.94980.7270,0.9798,0.82130.8376,0.9465,0.92250.7229,0.9891,0.84570.5034,0.9786,0.99740.6829,0.9909,0.78660.6049,0.9863,0.9919,ℒ−=0.7354,0.9835,0.50210.6572,0.9754,0.98400.6791,0.9762,0.84910.7048,0.9645,0.96050.7200,0.9872,0.90090.7038,0.9728,0.94130.7955,0.9729,0.54760.9009,0.9301,0.92360.8678,0.9438,0.85270.7379,0.9513,0.96830.9083,0.9529,0.68970.7386,0.9465,0.96220.6328,0.9891,0.47840.4402,0.9908,0.99590.7152,0.9931,0.65750.6667,0.9821,0.9506.Step 5: by using equation (37), compute the CC such as p1 = 0.99915, p2 = 0.99811, p3 = 0.99746, and p4 = 0.99787.Step 6: by using equation (38), compute the CC such as q1 = 0.99742, q2 = 0.99806, q3 = 0.99870, and q4 = 0.99870.Step 7: for the alternatives ranking, compute the closeness coefficient by using equation (39):(43)ℛ1=0.75219,ℛ2=0.50653,ℛ3=0.33854,ℛ4=0.37901.Step 8: pick the alternate with the largest closeness coefficientℛ1 = 0.75219, so X1 will be the best alternative.Step 9: analyze the ranking:ℛ1 > ℛ2>ℛ4 > ℛ3, so the ranking of the alternatives is X1 > X2>X4 > X3.


## 4.1. Proposed Decision-Making Approach

LetX = X1,X2,X3,…,Xs be a collection of “s” alternatives for assessment under the group of experts Q = Q1,Q2,Q3,…,Qn with weights Ω = Ω1,Ω1,…,ΩnT and Ωi>0, ∑i=1nΩi = 1. Consider L=d1,d2,…,dm be a set of attributes with their corresponding subattributes such as L′=d1ρ×d2ρ×⋯×dmρfor allρ∈1,2,…,t with weights γ=γ1ρ,γ2ρ,γ3ρ,…,γmρT such as γρ>0, ∑ρ=1tγρ=1, and can be stated as L′=dˇ∂:∂∈1,2,…,k. The group of experts {Qi: i = 1, 2, …, n} evaluate the alternatives {Xz: z = 1, 2, …, s} under the desired subattributes of the considered parameters dˇ∂:∂=1,2,…,k given in the form of NHSNs such as ℵdˇikzn×∂ = Tdˇikz,ℐdˇikz,ℭdˇikzn×∂, where 0 ≤ Tdˇikz,ℐdˇikz,ℭdˇikz ≤ 1 and 0 ≤ Tdˇikz+ℐdˇikz+ℭdˇikz ≤ 3 for all i, k.Step 1: development of decision matrices for each alternative {Xz: z = 1, 2, …, s} by using subattributes of the given attributes in the form of NHSNs such as follows:(32)Step 2: normalize the aggregated decision matrices by transforming cost type subattributes to benefit type subattribute using the normalization rule:(33)Xij=Jdˇijc,cost type parameter,Jdˇij,benefit type parameter,whereJdˇijc = 1−ℭdˇij,Jdˇij,Tdˇij, which represents the complement of <Tdˇij,Jdˇij,ℭdˇij>.Step 3: development of weighted decision matrix for each alternativeX¯z=J¯dˇijzn×∂, where(34)J¯dˇijz=γjΩiJdˇijz1−∏j=1m∏i=1n1−TdˇijΩiγj,∏j=1m∏i=1nJdˇijΩiγj,∏j=1m∏i=1nℭdˇijΩiγj=T¯dˇijz,J¯dˇijz,ℭ¯dˇijz.Here,Ωi and γj expressed the weights of experts and subattributes.Step 4: to determine the PIA and NIA compute indices by utilizing equation (12), such as argmaxzθijz and gij = arg minzθijz,(35)ℒ+=Tdˇij+,Jdˇij+,ℭdˇij+n×∂=T¯dˇijℏij,J¯dˇijℏij,ℭ¯dˇijℏij,(36)ℒ−=Tdˇij−,Jdˇij−,ℭdˇij−n×∂=T¯dˇijgij,J¯dˇijgij,ℭ¯dˇijgij.Step 5: considerX¯z and PIA ℒ+ analyze the CC such as follows:(37)pz=δNHSSX¯z,ℒ+=CNHSSX¯z,ℒ+ςNHSSXz∗ςNHSSℒ+=∑j=1m∑i=1mT¯dˇijz∗Tdˇij++J¯dˇijz∗Jdˇij++ℭ¯dˇijz∗ℭdˇij+∑j=1m∑i=1nT¯dˇijz2+J¯dˇijz2+ℭ¯dˇijz2∑j=1m∑i=1nTdˇij+2+Jdˇij+2+ℭdˇij+2.Step 6: considerX¯z and NIA ℒ− analyze the CC such as follows:(38)qz=δNHSSX¯z,ℒ−=CNHSSX¯z,ℒ−ςNHSSXz∗ςNHSSℒ−=∑j=1m∑i=1nT¯dˇijz∗Tdˇij−+J¯dˇijz∗Jdˇij−+ℭ¯dˇijz∗ℭdˇij−∑j=1m∑i=1nT¯dˇijz2+J¯dˇijz2+ℭ¯dˇijz2∑j=1m∑i=1nTdˇij−2+Jdˇij−2+ℭdˇij−2.Step 7: for the alternatives, ranking computes the closeness coefficient:(39)ℛz=KX¯z,ℒ−KX¯z,ℒ++KX¯z,ℒ−.where(40)KX¯z,ℒ−=1−qzandKX¯z,ℒ+=1−pz.Step 8: pick the alternate with the largest closeness coefficient.Step 9: rank the alternatives.Graphical representation of the above presented TOPSIS algorithm is given in Figure1.Figure 1
Graphical representation of developed TOPSIS technique for NHSS.


## 4.2. Application of the Proposed MCGDM Technique for the Selection of Hand Sanitizer

### 4.2.1. Case Study

Sanitization is the most important process because direct contact along with airborne microorganism droplets caused by sneezing and sneezing can without problems be infected. Particularly, when a pandemic breaks out, it is necessary to interrupt the overall chain of transmitting of the virus by proper hand sterilization trials. This will be accomplished by employing the right to use isolation as well as accommodating emergency care possessions, such as handling sterilizing in psychiatric divisions as well as public places. The assumption of sanitizers simply relies on the use of multipurpose sanitizers developed in various types and varieties, nominative antibacterial soaps and water-based or alcohol-based sanitizers, which have been common in hospitals. Until now, the most helpful hand sanitizer product is the alcohol-based formulation, which could include 62% to 95% alcohol, since it can convert microbic proteins and have the ability to inactivate infections. The reassessment of this system of rules is expounded to the information in PubMed. It is going to examine the variety of available sanitizers along with their efficiency plus formulation features, contrary effects, and approvals to improve preparation proficiency along with safety. Furthermore, this article concentrates on the efficiency of alcohol-based sanitizers against disease. Up to now, the most effective hand sanitizer overall has been formulated based on alcoholic brews, which conquer 62% to 95% of alcoholic beverages because it can formalize microbial proteins and [52, 53] can be disabled. Due to its intoxicated alcohol content, the current expression gives some rival as well as concerns about safety disadvantages along with skin cytotoxicity [54]. Current research objectives scrutinize a variety of accessible sanitizers, as well as their effectiveness against the redeeming coronavirus, besides strategies and findings as well as ideas for reinforcing the formation of saviors from accessible hands.Hand sanitizers may be a liquid or gel, mainly used to cut down transmittable diseases on the hands. In most organizations, alcohol-based sanitizers are superior for cleaning. The overall concentrates for risk reduction and prevention propose that person washing their hands with detergent to restrict the spread of virus infection and cut down the risk of unhealthiness. Due to the lack of soap and water, the CDC recommends that people employ alcohol-based (at least 60%) hand sanitizer. According to scientific research by the world health organization (WHO), in the context of this coronavirus pandemic, hygienic as well as physical distancing are the best methods to protect ourselves and every person around us from COVID-19 infection. The virus is spread by people with this disease, and it can also be spread out via contact with sick people. We cannot treat the coronavirus with complete caution. Hence, good hand sanitization may be the ultimate obstacle between us and the virus. The WHO endorses using alcohol-based hand sanitizer to eliminate the new COVID-19 virus. Alcohol-based hand sanitizer can prevent microbial proteins (together with bacteria and certain viruses) from operating usually. Hand sanitizer needs to comprehend ethanol, isopropanol, n-propanol, or a mixture of the alcohols. All of those can appropriately refuse to accept infections such as the new coronavirus. In that serious COVID-19 situation, the inquiry for hand sanitizer seems to have exaggerated. Due to exaggerated demand, it is difficult to obtain fine quality as well as multipurpose sanitizers in the overall market. The rise in demand also led to inferior sanitizers coming into the market. Akram et al. [55] utilized the GFFYWA operator for the selection of an effective hand sanitizer. The most important determinant of the current application is to reduce the spread of coronavirus by utilizing the TOPSIS technique under the NHSS scenario.


## 4.2.1. Case Study

Sanitization is the most important process because direct contact along with airborne microorganism droplets caused by sneezing and sneezing can without problems be infected. Particularly, when a pandemic breaks out, it is necessary to interrupt the overall chain of transmitting of the virus by proper hand sterilization trials. This will be accomplished by employing the right to use isolation as well as accommodating emergency care possessions, such as handling sterilizing in psychiatric divisions as well as public places. The assumption of sanitizers simply relies on the use of multipurpose sanitizers developed in various types and varieties, nominative antibacterial soaps and water-based or alcohol-based sanitizers, which have been common in hospitals. Until now, the most helpful hand sanitizer product is the alcohol-based formulation, which could include 62% to 95% alcohol, since it can convert microbic proteins and have the ability to inactivate infections. The reassessment of this system of rules is expounded to the information in PubMed. It is going to examine the variety of available sanitizers along with their efficiency plus formulation features, contrary effects, and approvals to improve preparation proficiency along with safety. Furthermore, this article concentrates on the efficiency of alcohol-based sanitizers against disease. Up to now, the most effective hand sanitizer overall has been formulated based on alcoholic brews, which conquer 62% to 95% of alcoholic beverages because it can formalize microbial proteins and [52, 53] can be disabled. Due to its intoxicated alcohol content, the current expression gives some rival as well as concerns about safety disadvantages along with skin cytotoxicity [54]. Current research objectives scrutinize a variety of accessible sanitizers, as well as their effectiveness against the redeeming coronavirus, besides strategies and findings as well as ideas for reinforcing the formation of saviors from accessible hands.Hand sanitizers may be a liquid or gel, mainly used to cut down transmittable diseases on the hands. In most organizations, alcohol-based sanitizers are superior for cleaning. The overall concentrates for risk reduction and prevention propose that person washing their hands with detergent to restrict the spread of virus infection and cut down the risk of unhealthiness. Due to the lack of soap and water, the CDC recommends that people employ alcohol-based (at least 60%) hand sanitizer. According to scientific research by the world health organization (WHO), in the context of this coronavirus pandemic, hygienic as well as physical distancing are the best methods to protect ourselves and every person around us from COVID-19 infection. The virus is spread by people with this disease, and it can also be spread out via contact with sick people. We cannot treat the coronavirus with complete caution. Hence, good hand sanitization may be the ultimate obstacle between us and the virus. The WHO endorses using alcohol-based hand sanitizer to eliminate the new COVID-19 virus. Alcohol-based hand sanitizer can prevent microbial proteins (together with bacteria and certain viruses) from operating usually. Hand sanitizer needs to comprehend ethanol, isopropanol, n-propanol, or a mixture of the alcohols. All of those can appropriately refuse to accept infections such as the new coronavirus. In that serious COVID-19 situation, the inquiry for hand sanitizer seems to have exaggerated. Due to exaggerated demand, it is difficult to obtain fine quality as well as multipurpose sanitizers in the overall market. The rise in demand also led to inferior sanitizers coming into the market. Akram et al. [55] utilized the GFFYWA operator for the selection of an effective hand sanitizer. The most important determinant of the current application is to reduce the spread of coronavirus by utilizing the TOPSIS technique under the NHSS scenario.


## 4.3. Numerical Example

LetX = X1,X2,X3,X4 be a collection of four hand sanitizers (alternatives) such as X1 = Megababe Squeaky Clean Sanitizer, X2 = Purell Advanced Sanitizer Gel, X3 = Touchland power mist citrus, and X4 = Lemon sanitizer spray. A team of four medical experts such as Q = Q1,Q2,Q3,Q4 having weight vector .1,.25,.35,.3T for the selection of an appropriate alternative (sanitizer), which is most helpful to protect the COVID-19 effects, is hired. The group of medical experts chooses the set of attributes for the selection of an effective sanitizer such as L = ℓ1=alcohol ratio,ℓ2=ingredients,ℓ3=Quality with their corresponding subattribute: alcohol ratio = ℓ1 = d11=less than40%,d12=more than60%, ingredients = ℓ2 = d21=hydrogen  peroxide,d22=glycerol, and quality = ℓ3 = d31=effectiveness. Let L' = ℓ1 × ℓ2 × ℓ3 show the multi-subattributes:(41)L′=ℓ1×ℓ2×ℓ3=d11,d12×d21,d22×d31=d11,d21,d31,d11,d22,d31,d12,d21,d31,d12,d22,d31.L' = dˇ1,dˇ2,dˇ3,dˇ4 with weights 0.2,0.1,0.4,0.3T. Each expert will evaluate the ratings of hand sanitizer in the form of NHSNs for each subattribute .Table 1
Decision matrix for alternativeX1.
X1dˇ1dˇ2dˇ3dˇ4Q10.9,0.2,0.10.3,0.3,0.70.6,0.4,0.20.7,0.1,0.3Q20.8,0.3,0.20.6,0.2,0.60.8,0.3,0.10.2,0.6,0.8Q30.6,0.1,0.30.6,0.1,0.30.8,0.2,0.10.6,0.3,0.4Q40.9,0.1,0.10.9,0.1,0.10.8,0.1,0.10.9,0.1,0.2Table 2
Decision matrix for alternativeX2.
X2dˇ1dˇ2dˇ3dˇ4Q10.3,0.3,0.70.9,0.2,0.10.6,0.1,0.30.3,0.6,0.2Q20.8,0.2,0.10.8,0.3,0.20.9,0.1,0.10.8,0.3,0.1Q30.6,0.3,0.40.8,0.1,0.20.9,0.1,0.10.2,0.3,0.8Q40.9,0.1,0.20.8,0.1,0.10.7,0.1,0.30.6,0.3,0.4Table 3
Decision matrix for alternativeX3.
X3dˇ1dˇ2dˇ3dˇ4Q10.6,0.3,0.40.2,0.3,0.80.3,0.6,0.20.3,0.6,0.2Q20.9,0.1,0.10.9,0.1,0.10.9,0.1,0.10.8,0.3,0.2Q30.8,0.3,0.20.9,0.2,0.10.9,0.1,0.10.2,0.3,0.8Q40.3,0.3,0.70.9,0.1,0.20.7,0.1,0.30.6,0.3,0.4Table 4
Decision matrix for alternativeX4.
X4dˇ1dˇ2dˇ3dˇ4Q10.9,0.1,0.10.9,0.1,0.20.8,0.2,0.10.3,0.6,0.2Q20.8,0.2,0.10.8,0.2,0.10.6,0.3,0.40.8,0.3,0.2Q30.8,0.1,0.10.8,0.1,0.20.9,0.1,0.10.3,0.6,0.2Q40.9,0.1,0.20.3,0.3,0.70.8,0.3,0.20.9,0.1,0.1Table 5
Weighted decision matrixX¯1.
X¯1dˇ1dˇ2dˇ3dˇ4Q10.7354,0.9835,0.50210.5691,0.9778,0.95760.6791,0.9762,0.84910.6694,0.9848,0.8491Q20.7200,0.9872,0.90090.7038,0.9728,0.94130.7955,0.9729,0.54760.9009,0.9301,0.9835Q30.7443,0.9438,0.49590.6422,0.9513,0.92250.7270,0.9798,0.82130.7386,0.9465,0.9622Q40.6328,0.9891,0.47840.5034,0.9786,0.98880.6829,0.9909,0.78660.6377,0.9863,0.9945Table 6
Weighted decision matrixX¯2.
X¯2dˇ1dˇ2dˇ3dˇ4Q10.6867,0.9878,0.80730.6178,0.9676,0.98890.7920,0.9862,0.79200.6694,0.9794,0.9862Q20.7599,0.9754,0.80330.6109,0.9517,0.99150.7068,0.9529,0.70680.9009,0.9301,0.9933Q30.7849,0.9513,0.77490.7379,0.9513,0.94980.7819,0.9911,0.90830.7386,0.9465,0.9377Q40.7229,0.9891,0.78660.4402,0.9908,0.99590.6829,0.9880,0.76310.5644,0.9821,0.9931Table 7
Weighted decision matrixX¯3.
X¯3dˇ1dˇ2dˇ3dˇ4Q10.7559,0.9835,0.85510.6178,0.9754,0.99090.7920,0.9862,0.59870.6694,0.9729,0.9699Q20.7599,0.9754,0.90700.6109,0.9517,0.98930.7068,0.9529,0.46610.9009,0.9301,0.9236Q30.8678,0.9257,0.85270.6955,0.9362,0.96830.9083,0.9529,0.68970.7386,0.9465,0.9408Q40.7229,0.9891,0.84570.5034,0.9786,0.99740.7408,0.9949,0.61950.6049,0.9863,0.9919Table 8
Weighted decision matrixX¯4.
X¯4dˇ1dˇ2dˇ3dˇ4Q10.6867,0.9915,0.82410.6572,0.9754,0.98400.7311,0.9818,0.50210.7048,0.9645,0.9605Q20.7599,0.9754,0.90700.6109,0.9517,0.98210.9085,0.9333,0.65090.7781,0.9301,0.9553Q30.8678,0.9438,0.85270.7379,0.9513,0.96830.8886,0.9377,0.64150.8376,0.9465,0.9225Q40.7229,0.9891,0.84570.4402,0.9786,0.99650.7152,0.9931,0.65750.6667,0.9821,0.9506


### 4.3.1. Selection of an Effective Hand Sanitizer by Using the TOPSIS Method

Step 1: decision makers will investigate the situation in the case of NHSNs, and there are only four alternatives. The subattributes of the considered attributes, along with a summary of their scores are given in Tables1–4.Step 2: there is no need for normalization since all attributes are identical.Step 3: weighted decision matrix for each alternativeX¯z = ℒ¯ijzn×∂ is developed by using equation (30). The values of the weighted decision matrices for each alternative are given in Tables 5–8.Step 4: by utilizing equations (35) and (36), compute the PIA and NIA such as follows:(42)ℒ+=0.6867,0.9878,0.80730.6178,0.9676,0.98890.7311,0.9818,0.50210.6694,0.9729,0.96990.7599,0.9754,0.90700.6109,0.9517,0.99150.7068,0.9529,0.70680.9009,0.9301,0.98350.7849,0.9513,0.77490.7379,0.9513,0.94980.7270,0.9798,0.82130.8376,0.9465,0.92250.7229,0.9891,0.84570.5034,0.9786,0.99740.6829,0.9909,0.78660.6049,0.9863,0.9919,ℒ−=0.7354,0.9835,0.50210.6572,0.9754,0.98400.6791,0.9762,0.84910.7048,0.9645,0.96050.7200,0.9872,0.90090.7038,0.9728,0.94130.7955,0.9729,0.54760.9009,0.9301,0.92360.8678,0.9438,0.85270.7379,0.9513,0.96830.9083,0.9529,0.68970.7386,0.9465,0.96220.6328,0.9891,0.47840.4402,0.9908,0.99590.7152,0.9931,0.65750.6667,0.9821,0.9506.Step 5: by using equation (37), compute the CC such as p1 = 0.99915, p2 = 0.99811, p3 = 0.99746, and p4 = 0.99787.Step 6: by using equation (38), compute the CC such as q1 = 0.99742, q2 = 0.99806, q3 = 0.99870, and q4 = 0.99870.Step 7: for the alternatives ranking, compute the closeness coefficient by using equation (39):(43)ℛ1=0.75219,ℛ2=0.50653,ℛ3=0.33854,ℛ4=0.37901.Step 8: pick the alternate with the largest closeness coefficientℛ1 = 0.75219, so X1 will be the best alternative.Step 9: analyze the ranking:ℛ1 > ℛ2>ℛ4 > ℛ3, so the ranking of the alternatives is X1 > X2>X4 > X3.


## 4.3.1. Selection of an Effective Hand Sanitizer by Using the TOPSIS Method

Step 1: decision makers will investigate the situation in the case of NHSNs, and there are only four alternatives. The subattributes of the considered attributes, along with a summary of their scores are given in Tables1–4.Step 2: there is no need for normalization since all attributes are identical.Step 3: weighted decision matrix for each alternativeX¯z = ℒ¯ijzn×∂ is developed by using equation (30). The values of the weighted decision matrices for each alternative are given in Tables 5–8.Step 4: by utilizing equations (35) and (36), compute the PIA and NIA such as follows:(42)ℒ+=0.6867,0.9878,0.80730.6178,0.9676,0.98890.7311,0.9818,0.50210.6694,0.9729,0.96990.7599,0.9754,0.90700.6109,0.9517,0.99150.7068,0.9529,0.70680.9009,0.9301,0.98350.7849,0.9513,0.77490.7379,0.9513,0.94980.7270,0.9798,0.82130.8376,0.9465,0.92250.7229,0.9891,0.84570.5034,0.9786,0.99740.6829,0.9909,0.78660.6049,0.9863,0.9919,ℒ−=0.7354,0.9835,0.50210.6572,0.9754,0.98400.6791,0.9762,0.84910.7048,0.9645,0.96050.7200,0.9872,0.90090.7038,0.9728,0.94130.7955,0.9729,0.54760.9009,0.9301,0.92360.8678,0.9438,0.85270.7379,0.9513,0.96830.9083,0.9529,0.68970.7386,0.9465,0.96220.6328,0.9891,0.47840.4402,0.9908,0.99590.7152,0.9931,0.65750.6667,0.9821,0.9506.Step 5: by using equation (37), compute the CC such as p1 = 0.99915, p2 = 0.99811, p3 = 0.99746, and p4 = 0.99787.Step 6: by using equation (38), compute the CC such as q1 = 0.99742, q2 = 0.99806, q3 = 0.99870, and q4 = 0.99870.Step 7: for the alternatives ranking, compute the closeness coefficient by using equation (39):(43)ℛ1=0.75219,ℛ2=0.50653,ℛ3=0.33854,ℛ4=0.37901.Step 8: pick the alternate with the largest closeness coefficientℛ1 = 0.75219, so X1 will be the best alternative.Step 9: analyze the ranking:ℛ1 > ℛ2>ℛ4 > ℛ3, so the ranking of the alternatives is X1 > X2>X4 > X3.


## 5. Discussion and Comparative Analysis

In the subsequent section, we are going to talk over the effectiveness, simplicity, and tractability along with the benefits of the planned approach. We also performed a brief comparison of the undermentioned: the planned approach along with some existing techniques.


### 5.1. Comparative Analysis

Through the current exploration along with comparative analysis in Table9, it can be concluded that the results acquired by the proposed approach overlap with the accessible methodologies. However, in connection with accessible decision-making techniques, the main benefit of the proposed method is that it comprises additional information utilizing the membership, nonmembership, and indeterminacy of subattributes of considered attributes to address uncertainty in data. Among them, the information related to the object can be interpreted more accurately and objectively. It is also a useful tool for solving inaccurate and imprecise data in the DM process. Also, the calculation process of the developed method is different from the available techniques. This is due to the determination of PIA and NIA. In preceding techniques, PIA and NIA were conceptualized according to a given distance and similarity measures. Therefore, the inspiration for the score value corresponding to each parameter will not affect other parameters, so predictable information loss will occur in the process. On the contrary, in the proposed method, PIA and NIA are calculated based on the inspiration of the maximum CC on a given substitution level, so there is no serious loss of information in the process. After all, the intensity of each ideal correlation measure is obtained from the assessment explanations, so to find the superlative result compute the correlation between them. The benefit of the planned TOPSIS method along with related measures over existing methods is that it notices not only the degree of discrimination but also the degree of similarity between observations so that avoiding decisions based on negative reasons. Therefore, it is a suitable tool to combine inaccurate as well as uncertain info in the DM process. Hence, our suggested approach is effective, flexible, simple, and in advance of distinctive hybrid structures of FS.Table 9
Comparison between NHSS and some existing techniques.
SetTruthinessindeterminacyFalsityParametrizationAttributesSubattributesZadeh [1]FS✓×××✓×Atanassov [2]IFS✓×✓×✓×Maji et al. [21]FSS✓××✓✓×Maji et al. [22]IFSS✓×✓✓✓×Peng et al. [26]PFSS✓×✓✓✓×Zulqarnain et al. [46]IFHSS✓×✓✓✓✓Zulqarnain et al. [48]PFHSS✓×✓✓✓✓Proposed approachNHSS✓✓✓✓✓✓


## 5.1. Comparative Analysis

Through the current exploration along with comparative analysis in Table9, it can be concluded that the results acquired by the proposed approach overlap with the accessible methodologies. However, in connection with accessible decision-making techniques, the main benefit of the proposed method is that it comprises additional information utilizing the membership, nonmembership, and indeterminacy of subattributes of considered attributes to address uncertainty in data. Among them, the information related to the object can be interpreted more accurately and objectively. It is also a useful tool for solving inaccurate and imprecise data in the DM process. Also, the calculation process of the developed method is different from the available techniques. This is due to the determination of PIA and NIA. In preceding techniques, PIA and NIA were conceptualized according to a given distance and similarity measures. Therefore, the inspiration for the score value corresponding to each parameter will not affect other parameters, so predictable information loss will occur in the process. On the contrary, in the proposed method, PIA and NIA are calculated based on the inspiration of the maximum CC on a given substitution level, so there is no serious loss of information in the process. After all, the intensity of each ideal correlation measure is obtained from the assessment explanations, so to find the superlative result compute the correlation between them. The benefit of the planned TOPSIS method along with related measures over existing methods is that it notices not only the degree of discrimination but also the degree of similarity between observations so that avoiding decisions based on negative reasons. Therefore, it is a suitable tool to combine inaccurate as well as uncertain info in the DM process. Hence, our suggested approach is effective, flexible, simple, and in advance of distinctive hybrid structures of FS.Table 9
Comparison between NHSS and some existing techniques.
SetTruthinessindeterminacyFalsityParametrizationAttributesSubattributesZadeh [1]FS✓×××✓×Atanassov [2]IFS✓×✓×✓×Maji et al. [21]FSS✓××✓✓×Maji et al. [22]IFSS✓×✓✓✓×Peng et al. [26]PFSS✓×✓✓✓×Zulqarnain et al. [46]IFHSS✓×✓✓✓✓Zulqarnain et al. [48]PFHSS✓×✓✓✓✓Proposed approachNHSS✓✓✓✓✓✓


## 6. Discussion

By utilizing the methodology of Zadeh [1], we can process the MD of the alternative, but the FS has no information about the NMD of the considered parameters. Atanassov [2] deals with the uncertainty by using MD and NMD. However, these theories cannot deal with the situation when the sum of MD and NMD exceeds one; on the other hand, our presented notion competently deals with such difficulties. Meanwhile, these theories have no information about the parametrization of the attributes. To overcome such difficulties, Maji et al. [21] presented the notion of FSS to deal with the parametrization of the objects which contains uncertainty by considering the MD of the attributes. However, the presented FSS provides no information about the NMD of the object. To overcome the presented drawback, Maji et al. [22] offered the concept of IFSS, the presented notion handles the uncertain object more accurately by utilizing the MD and NMD of the attributes with their parametrization, and the sum of MD and NMD does not exceed 1. To handle this scenario, Peng et al. [26] proposed the notion of PFSS by modifying the condition MD+ NMD≤1 to MD2+ NMD2≤1 with their parametrization. However, all the abovementioned studies have no information about the subattributes of the considered attributes; when attributes have their corresponding subattributes, then all abovementioned theories fail to handle the situation. Zulqarnain et al. [46] presented the TOPSIS technique under the IFHSS environment to deal with uncertain problems by using MD and NMD in which the sum of subattributes of the considered parameters cannot exceed one. However, IFHSS cannot provide any information on the indeterminacy of the subattribute of the considered attribute. Zulqarnain et al. [48] proposed the more generalized notion of PFHSS comparative to IFHSS. All existing hybrid structures of FS cannot handle the indeterminacy of subattributes of considered n-tuple attributes. To overcome the above limitations, we prolonged the IFHSS to NHSS. Instead, the method we developed is a progressive approach that can handle alternatives with multiple subattribute info by using truthiness, indeterminacy, and falsity object with the following condition 0 ≤Tℱaˇv+ℐℱaˇv+ℭℱaˇv≤3. Consequently, relying on the obtained results, it can be confidently concluded that the proposed methodology indicates higher stability and usability for decision makers in the DM procedure.


## 7. Conclusion

This study focuses on NHSS to solve the problem of insufficient data, ambiguity, and inconsistency by considering the degree of truthiness, falsity, and indeterminacy on then-tuple subattributes of the considered attributes. This research puts forward the new concepts and properties of CC and WCC in the NHSS environs. Based on the proposed correlation measure, by considering the n-tuple subattributes as well as decision makers, the TOPSIS method is extended in the neutrosophic hypersoft environment. To get the overall top-ranking of possible choices, we characterized the closeness coefficient under the constructed TOPSIS approach. We have also defined NHSWA and NHSWG operators and proposed the DM strategy according to the developed TOPSIS technique. To resolve the MCGDM problem, a numerical illustration has been described by utilizing the recommended TOPSIS approach for the selection of more suitable hand sanitizer (alternative) during the COVID-19 pandemic. Moreover, a comparison has been conducted for the justification along with effectivity of the approach. Consequently, relying upon the obtained outcomes, it can be confidently concluded that the proposed methodology indicates greater constancy as well as usability for decision makers in the DM process. Future research will surely concentrate on presenting several other operators to solve decision-making complications utilizing NHSS. Many other structures such as topological, algebraic, and ordered structures can be developed and discussed under-considered environment. The suggested idea can be applied to quite a lot of issues in real life, including the medical profession, robotics, artificial intelligence, pattern recognition, and economics. We are sure this article will open new vistas for investigators in this field.



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*Source: 5531830-2021-06-04.xml*