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

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.

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,   . Yager [3, 4] extended the notion of IFS to Pythagorean fuzzy sets (PFSs) to overcome the above-discussed complications by modifying  +  to  + . 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 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 Section 2, 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 and be the universe of discourse and set of attributes, respectively. Let be the power set of and . A pair () is called a SS over , and its mapping is expressed as follows:Also, it can be defined as follows: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 and be a universe of discourse and set of attributes, respectively, and be a power set of . Let   ; then, () is an FSS over , and its mapping can be expressed as follows:

Definition 3. (see [34]). Let be a universe and be an NS on which is defined as  = , where , , :   ], [ and  ≤  ≤ .
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 be the universal set and be the set of attributes concerning . Let be the set of neutrosophic values of and. A pair is called a neutrosophic soft set over and its mapping is given as

Definition 5. (see [43]). Let be a universe of discourse and () be a power set of and  = {, , ,...,}, (n ≥ 1) and represented the set of attributes and their corresponding subattributes such as  ∩  = φ, where i ≠ j for each n ≥ 1 and i, j  {1, 2, 3, …, n}. Assume is a collection of subattributes, where 1    , 1    , and 1    , and , ,   ℕ. Then, the pair is known as HSS defined as follows:It is also defined as

Definition 6. (see [43]). Let be a universe of discourse and () be a power set of and  = {, , ,...,}, (n ≥ 1) be a set of attributes, and set as a set of corresponding subattributes of , respectively, with  ∩  = φ for n ≥ 1 for each i, j  {1, 2, 3, …, n} and i ≠ j. Assume  ×  ×  × … ×  =  =  is a collection of subattributes, where 1   .. , 1    , and 1    , and , , and   ℕ, and is a collection of all intuitionistic fuzzy subsets over . Then, the pair (,  ×  ×  × … ×  = ) is said to be IFHSS over , and its mapping is defined asIt is also defined aswhere  = , where and represent the membership and nonmembership values of the attributes such as ,    and 0   +   1.
Simply an intuitionistic fuzzy hypersoft number (IFHSN) can be expressed as  = , where 0  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 be a universe of discourse and () be a power set of and  = {, , ,...,}, (n ≥ 1), and represented the set of attributes and their corresponding subattributes such as  ∩  = φ, where i ≠ j for each n ≥ 1 and i, j  {1, 2, 3, …, n}. Assume is a collection of subattributes, where 1    , 1    , and 1    , and , ,   ℕ, and represents neutrosophic subsets over . Then, is called NHSS, and its mapping can be expressed as follows:It is also defined as , where , where , , and represent the truth, indeterminacy, and falsity grades of the attributes such as , ,  , and 0   +  +   3.
Simply a neutrosophic hypersoft number (NHSN) can be expressed as  = , where 0  3.

Example 1. Consider  =  is a universe of discourse and  =  are the considered attributes, and their corresponding n-tuple subattributes are given as follows:  = ,  = , and Classes =  = . Let be a set of attributesThen, the NHSS over is given as follows:For simplicity, we will express  = as  = which is called NHSN, where and . The score function for NHSNs is defined as follows:where   , and sometimes the score function is unable to compare any two NHSNs. Like  =  and  = , 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:In the following, we present the comparison laws to compare NHSNs and such as(i)If   , then   .(ii)If   , then(1)If   , then   (2)If   , then   .

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 and  =  be two NHSSs. Then, their informational neutrosophic energies can be defined as

Definition 9. Let and be two NHSSs. Then, the correlation among them can be defined as

Proposition 1. Let  =  and  =  be two NHSSs and , be a correlation between them and satisfied the following properties:(1) = (2) = 

Proof. By utilizing equation (15), it easily can be proved.

Definition 10. Let  =  and  =  be two NHSSs; then, CC between them is given as and expressed as follows:

Theorem 1. Let  =  and  =  be two NHSSs; then, CC between them satisfies the following properties:(1)0    1(2) = (3)If  = , that is, , , , , and , then  = 1

Proof.   0 is trivial; here, we only need to prove that   1.
From equation (15), we haveBy using Cauchy–Schwarz inequality,Therefore, . Hence, by using Definition 9, we have   1. So, 0    1.

Proof. The proof is straightforward.

Proof. From equation (17), we haveAs we know that , , and , , we obtainThus, prove the required result.

Definition 11. Let  =  and  =  be two NHSSs. Then, their correlation coefficient is given as and defined as follows:

Theorem 2. Let  =  and  =  be two NHSSs. Then, CC between them satisfies the following properties:(1)0    1(2) = (3)If , that is, , , , , and , then

Proof. It is similar to Theorem 1.
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 as , where  0,  = 1. Similarly, assume the weights for subattributes as follows , where   0,  = 1.

Definition 12. Let  =  and  =  be two NHSSs. Then, WCC among them is expressed as and defined as follows:

Definition 13. Let  =  and  =  be two NHSSs. Then, WCC among them is also given as and defined as follows: