# An Intelligent Multiattribute Decision-Support Framework Based on Parameterization of Neutrosophic Hypersoft Set

Authors: Atiqe Ur Rahman; Muhammad Saeed; Alhanouf Alburaikan; Hamiden Abd El-Wahed Khalifa
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

Hypersoft set is a novel area of interest which is able to tackle the real-world scenarios where classification of parameters into their respective sub-parametric values in the form of overlapping sets is mandatory. It employs a new approximate mapping which considers such sets in the form of sub-parametric tuples as its domain. The existing soft set-like structures are insufficient to tackle such kind of situations. This research intends to establish a novel concept of parameterization of fuzzy set under hypersoft set environment with uncertain components of intuitionistic fuzzy set and neutrosophic set. Two novel structures, i.e., fuzzy parameterized intuitionistic fuzzy hypersoft set (fpifhs-set) and fuzzy parameterized neutrosophic hypersoft set (fpnhs-set), are developed by employing algebraic techniques like theoretic, analytical, pictorial, and algorithmic techniques. After characterizing the elementary properties and set-theoretic operations of fpifhs-set and fpnhs-set, two novel algorithms are proposed to solve real-life decision-making COVID-19 problem. The results of both algorithms are compared with related already established models through certain evaluating features to judge the advantageous aspects of the proposed study. The generalization of the proposed models is discussed by describing some of their particular cases.


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## Body

## 1. Introduction

The conventional logic is not always significant in real-life state of affairs, where the accessible information is unclear or indefinite. To cope with such sort of circumstances, a specific class of sets recognized as fuzzy set (f-set) (initiated by Zadeh [1]) is regarded as apt. In such set, a belonging grade within [0, 1] is assigned to each element of the initial universe. Nevertheless, to manage more complex and uncertain real-life situations, f-set was found incapable, and therefore Atanassov [2] established a novel model, i.e., intuitionistic fuzzy set (if-set) which is more effective in tackling vagueness of data. It assigns belonging and nonbelonging grades to each element of the initial universe subject to the condition that their sum lies within [0, 1]. Moreover, if-sets are proficient at emulating the available information more precisely and rationally. Many researchers made contributions regarding the extensions of f-sets and if-sets, but the contributions of Deli and Keleş [3], Mahmood et al. [4], Ünver et al. [5], and Wang and Garg [6] are more significant. They introduced novel extensions of if-sets and applied them in real-world problems through algorithm-based decision-making techniques. Both f-sets and if-sets were observed to be incapable of considering the grade of indeterminacy, so neutrosophic set (n-set) was instigated by Smarandache [7] to deal with such kind of limitations. The n-set is more proficient at preserving the impreciseness in the contents of perceived data and may assist rough reasoning behavior meticulously. Although the expressive capacity of n-set is higher than that of the conventional f-set and if-set due to the consideration of a neutral grade, they have reasonably higher computational complication over f-set and if-set.The above described structures showed some drawbacks with respect to the consideration of parameterization tool. To manage such insufficiency, Molodtsov [8] developed soft set (s-set) as a new mathematical tool. In s-set, an approximate mapping is used to map a set of parameters to power set of universe. In order to enhance the applicability of s-set in real-world uncertain scenarios, some of its hybridized structures like fuzzy soft set (fs-set) [9, 10], intuitionistic fuzzy soft set (ifs-set) [11, 12], and neutrosophic soft set (ns-set) [13] were developed. Ali et al. [14], Li [15], Maji et al. [16], Pei and Miao [17], and Sezgin and Atagün [18] discussed the rudiments of s-set with numerical examples. Babitha and Sunil [19, 20] introduced the concept of relations, functions, and orders under soft set environment. Broumi et al. [21] and Deli [22] developed hybridized structures of n-set with s-set with interval settings. They described some of their rudimentary properties and applied different methods for their implementation in certain cases.Table 1
Literature review on the parameterization of fuzzy set-like models under soft set.
AuthorsStructureDomain parameterizationRange settingAdam and Hassan [40]MultiQ-fuzzy parameterized soft setMultiQ-fuzzy setSoft setAlkhazaleh et al. [41]Fuzzy parameterized interval-valued fuzzy soft setFuzzy setSoft setAydın and Enginoğlu [42]Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft setInterval-valued intuitionistic fuzzy setInterval-valued intuitionistic fuzzy soft setBroumi et al. [21]Neutrosophic parameterized soft set (nps-set)Neutrosophic setSoft setÇağman et al. [43]Fuzzy parameterized fuzzy soft set (fpfs-set)Fuzzy setFuzzy soft setDeli and Çağman [44]Intuitionistic fuzzy parameterized soft set (ifps-set)Intuitionistic fuzzy setSoft setHassan and Al-Qudah [45]Fuzzy parameterized complex multi-fuzzy soft set (fpcmfs-set)Fuzzy setComplex multi-fuzzy soft setHazaymeh et al. [46]Fuzzy parameterized fuzzy soft expert set (fpfse-set)Fuzzy setFuzzy soft expert setJoshi et al. [47]Intuitionistic fuzzy parameterized fuzzy soft set (ifpfs-set)Intuitionistic fuzzy setFuzzy soft setKaraaslan [48]Intuitionistic fuzzy parameterized intuitionistic fuzzy soft set (ifpifs-set)Intuitionistic fuzzy setIntuitionistic fuzzy soft setRiaz and Hashmi [49]Fuzzy parameterized fuzzy soft set (fpfs-set)Fuzzy setFuzzy soft setZhu and Zhan [50]Fuzzy parameterized fuzzy soft set (fpfs-set)Fuzzy setFuzzy soft setIn various real-life decision-making situations, it is observed that nonoverlapping sets having sub-parametric values corresponding to parameters are required to be considered, but thes-set and its hybrids are not projected for such situations. Therefore, Smarandache [23] presented the notion of hypersoft set (hs-set) which is capable of handling such situations by using a new approximate mapping having multiargument domain. In order to utilize hs-set in other fields of study, its various basic properties, relations, and matrices have been investigated in [24] with the help of supporting examples. Debnath [25] discussed the decision-making application based on weightage operators of fuzzy hypersoft set. Deli [26] developed a novel hybrid of hs-set called “neutrosophic valued n-attribute neutrosophic hypersoft set” which generalizes most of the existing fuzzy set-like structures for dealing with uncertainty, vagueness, and indeterminacy. Kamacı and Saqlain [27] discussed the validity of hs-set for the entitlement of multidecisive opinions under expert set environment. Martin and Smarandache [28, 29] discussed the hybridization of hypersoft set with plithogenic set and discussed its applications. Rahman et al. [30–33] studied various notional properties of complex set, convex cum concave sets, and parameterization by using the environment of hs-set. They also solved real-world decision-making problems with the help of algorithms based on such hybrids. Saeed et al. [34–36] extended the concept by introducing the mappings and graphs of hs-set with indeterminate settings and discussed its implementation in product selection and medical science. Saqlain et al. [37, 38] discussed aggregation operators, distance, and similarity measures of neutrosophic hypersoft sets with application in decision-making. Zulqarnain et al. [39] investigated generalized aggregate operators for neutrosophic hypersoft sets with some results.Figure 1
Comparison of soft set model and hypersoft set model.


### 1.1. Research Gap and Motivation

(1)
The phrase “parameterization of fuzzy set” is actually meant for dispensing the fuzzy value to each attribute/subattribute in the domain of single-argument/multiargument approximate function (maa-function).(2)
Many researchers discussed the parameterization of fuzzy set-like structures under soft set environment with fuzzy set-like settings. The literature review of most relevant models is presented in Table1. It can be observed easily that the above-mentioned models focused on a single set of attributes and used ordinary soft approximate mapping to study the parameterized nature of parametric domain. They are incapable of dealing with situations having compulsion partitioning of attributes into their respective attribute-valued nonoverlapping sets.(3)
The incapability of soft set-like models leads to the demand for new structure; therefore,hs-set is initiated to manage such situations. As hs-set emphasizes deep observation of attributes, hs-set can be considered as flexible and reliable model to have unbiased decision-support system. In Figure 1, the difference between s-set and hs-set is presented with the help of an example for product selection through decision-making.(4)
Motivated by the above-mentioned literature in general and [43, 44, 47–50] in specific, we construct novel structures of fuzzy parameterized intuitionistic fuzzy hypersoft set (fpifhs-set) and fuzzy parameterized neutrosophic hypersoft set (fpnhs-set) and characterize them with the help of algorithm-based decision-support systems.


### 1.2. Main Contributions

Main contributions of this research are given below:(1)
The structures like those in [43, 44, 47–50] are made capable with the entitlement of maa-function by characterizing fpifhs-sets and fpnhs-sets.(2)
The situations with sub-parametric-valued sets are managed for the environments ofif-set and n-set by using fpifhs-sets and fpnhs-sets,(3)
Some essential rudiments, i.e., properties, elementary laws, and aggregation operations offpifhs-set and fpnhs-set are investigated.(4)
Two algorithms based onfpifhs-set and fpnhs-set are proposed to deal with daily-life decision-making problems having uncertain data/information in COVID-19 scenario.(5)
In order to judge the distinction of this research, the computed algorithm-based results are compared with most relevant models by considering suitable evaluating indicators.(6)
The particular cases of the proposed models are discussed along with graphical representation to show their generalized aspects.(7)
The paper is summarized with the description of its merits and future scope.


### 1.3. Paper Organization

The rest of the paper is organized as follows: In Section2, some essential definitions of elementary nature are reviewed from literature for better understanding of the main study. Theories of fpifhs-set and fpnhs-set are developed along with their decision-support systems in Sections 3 and 4, respectively. In order to observe the advantageous aspect of the proposed study, a comparison is presented with some existing relevant models in Section 5, and Section 6 presents the discussion on generalization and merits of proposed work. The paper is summarized with future directions in the last section.


## 1.1. Research Gap and Motivation

(1)
The phrase “parameterization of fuzzy set” is actually meant for dispensing the fuzzy value to each attribute/subattribute in the domain of single-argument/multiargument approximate function (maa-function).(2)
Many researchers discussed the parameterization of fuzzy set-like structures under soft set environment with fuzzy set-like settings. The literature review of most relevant models is presented in Table1. It can be observed easily that the above-mentioned models focused on a single set of attributes and used ordinary soft approximate mapping to study the parameterized nature of parametric domain. They are incapable of dealing with situations having compulsion partitioning of attributes into their respective attribute-valued nonoverlapping sets.(3)
The incapability of soft set-like models leads to the demand for new structure; therefore,hs-set is initiated to manage such situations. As hs-set emphasizes deep observation of attributes, hs-set can be considered as flexible and reliable model to have unbiased decision-support system. In Figure 1, the difference between s-set and hs-set is presented with the help of an example for product selection through decision-making.(4)
Motivated by the above-mentioned literature in general and [43, 44, 47–50] in specific, we construct novel structures of fuzzy parameterized intuitionistic fuzzy hypersoft set (fpifhs-set) and fuzzy parameterized neutrosophic hypersoft set (fpnhs-set) and characterize them with the help of algorithm-based decision-support systems.


## 1.2. Main Contributions

Main contributions of this research are given below:(1)
The structures like those in [43, 44, 47–50] are made capable with the entitlement of maa-function by characterizing fpifhs-sets and fpnhs-sets.(2)
The situations with sub-parametric-valued sets are managed for the environments ofif-set and n-set by using fpifhs-sets and fpnhs-sets,(3)
Some essential rudiments, i.e., properties, elementary laws, and aggregation operations offpifhs-set and fpnhs-set are investigated.(4)
Two algorithms based onfpifhs-set and fpnhs-set are proposed to deal with daily-life decision-making problems having uncertain data/information in COVID-19 scenario.(5)
In order to judge the distinction of this research, the computed algorithm-based results are compared with most relevant models by considering suitable evaluating indicators.(6)
The particular cases of the proposed models are discussed along with graphical representation to show their generalized aspects.(7)
The paper is summarized with the description of its merits and future scope.


## 1.3. Paper Organization

The rest of the paper is organized as follows: In Section2, some essential definitions of elementary nature are reviewed from literature for better understanding of the main study. Theories of fpifhs-set and fpnhs-set are developed along with their decision-support systems in Sections 3 and 4, respectively. In order to observe the advantageous aspect of the proposed study, a comparison is presented with some existing relevant models in Section 5, and Section 6 presents the discussion on generalization and merits of proposed work. The paper is summarized with future directions in the last section.


## 2. Preliminaries

This portion of the paper presents some elementary terms and definitions by reviewing the existing literature for clear understanding of the proposed study. Throughout the rest of the paper, the symbolsU,ℙU, and I denote universe of discourse, power set of U, and closed unit interval. Furthermore, FU, IFU, and ℕU denote the collection of f-sets, if-sets, and n-sets over U, respectively. In 1965, Zadeh [1] initiated the concept of fuzzy set as a generalization of classical set (crisp set) to deal with uncertain nature of data. This set employs a membership function which maps set of objects (alternatives) to unit closed interval.Definition 1.
[1].
A fuzzy setP is defined as(1)P=u^,TPu^|u^∈U,
such thatTP:U⟶I where TPu^ represents the membership grade of u^∈P.
Fuzzy set emphasizes degree of membership only for dealing with uncertain scenarios, but there are many situations where it is necessary to consider nonmembership degree; therefore, into adequate fuzzy set with such situation, Atanassov [2] introduced intuitionistic fuzzy set as a generalization of fuzzy set. It provides due status to both membership and nonmembership degrees of an alternative.Definition 2.
[2].
An intuitionistic fuzzy setQ is defined as(2)Q=u^,<TQu^,FQu^>|u^∈U,
such thatTQ:U⟶I and FQ:U⟶I, where TQu^ and FQu^ denote the membership and nonmembership grades of u^∈U subject to the condition that(3)0≤TQu^+FQu^≤1.
Both fuzzy set and intuitionistic fuzzy set are insufficient to tackle the various uncertain scenarios where the consideration of indeterminate grades is necessary. In order to manage such situations, Smarandache [7] characterized neutrosophic set which not only considers membership and nonmembership grades but also provides due status to degree of indeterminacy of an alternative.Definition 3.
[7].
A neutrosophic setℛ is defined as(4)ℛ=u^,<Tℛu^,Iℛu^,Fℛu^>|u^∈U,
such thatTℛu^,Iℛu^,Fℛu^:U⟶0−,1+,where Tℛu^,Iℛu^, and Fℛu^ represent the grades of membership, indeterminacy, and nonmembership of u^∈U subject to the condition that(5)0−≤Tℛu^+Iℛu^+Fℛu^≤3+.
Fuzzy set, intuitionistic fuzzy set, and neutrosophic set depict some kind of insufficiency regarding the consideration of parameterization tool. In order to manage this limitation, Molodtsov [8] developed soft set as a new mathematical tool to tackle uncertainties and vagueness in the data.Definition 4.
[8].
A soft set overU is a pair ℱS,G, where ℱS:G⟶ℙU and G⊆E (a set of parameters).
More details on soft set and its operations can be seen in [10, 16]. In many real-world scenarios, the classification of attributes into sub-attributive values in the form of sets is necessary. The existing concept of soft set is not sufficient and is incompatible with such scenarios, so Smarandache [23] introduced hypersoft sets to address the insufficiency of soft set and to handle the situations with multiargument approximate function (MAAF).Definition 5.
[23].
A hypersoft set overU is a pair W,ℋ, where ℋ is the Cartesian product of ℋi,i=1,2,3,…,n,ℋi∩ℋj=∅ for all i≠j having attribute values of h^i,i=1,2,3,…,n,h^i≠h^j,i≠j, respectively, and W:ℋ⟶ℙU.
For the sake of proper use of hypersoft set under uncertain environments, the hybrids ofs-sets like fs-sets, ifs-sets, and ns-sets are further modified for hypersoft set environment so the following novel structures are conceptualized.Definition 6.
[23].
A hypersoft setW,ℋ is called fuzzy hypersoft set, intuitionistic fuzzy hypersoft set, and neutrosophic hypersoft set if ℙU in W:ℋ⟶ℙU is replaced with FU, IFU, and ℕU, respectively.
For more details on hypersoft set and its operations, please see [23, 24].
Before presenting main results, first necessity of the proposed study is discussed. In various daily-world scenarios like product selection, supplier selection, and medical diagnosis, the decision-makers face the following problems:(1)
The chosen parameters (attributes) need to be further partitioned into their respective attribute-valued disjoint sets.(2)
A new type of function (mapping), i.e., multiargument approximated function, is needed to tackle the attribute-valued disjoint sets collectively as its domain.(3)
The computed tuples have uncertain nature and need a fuzzy membership value to depict their uncertainty.(4)
The approximate elements of multi-approximate function are assessed in the form of intuitionistic fuzzy values or neutrosophic values.
In order to manage all above problems collectively, parameterization of hypersoft set with fuzzy setting is required under intuitionistic fuzzy set and neutrosophic set environments which lead to the demand for the proposed study.


## 3. Fuzzy Parameterized Intuitionistic Fuzzy Hypersoft Set (fpifhs-Set) with Application

In various real-world decision-making scenarios, the decision-makers are uncertain about the nature of selected parameters underhs-set environment with approximate elements having intuitionistic fuzzy values. Such type of parameters is known as fuzzy parameters, and such scenarios are tackled with the help of fpifhs-set. The methodology of the proposed study is explained in Figure 2. In this section, firstly theory of fpifhs-set is constructed, and then decision-making problem is discussed with the help of proposed algorithm.Figure 2
Methodology of the proposed study.Definition 7.
LetX1, X2, X3,…, Xn be parameter-valued sets with Xi∩Xj=∅,i≠j respective to parameters x1,x2,x3,…,xn,xi≠xj,i≠j, respectively. A fpifhs-set Ψℬ over U is defined as(6)Ψℬ=Lℬg^g^,ψℬg^:g^∈G,ψℬg^∈IFU,where(i)
G=X1×X2×X3×….×Xn.(ii)
ℬ∈FU with Lℬ:G⟶I.(iii)
ψℬ:G⟶IFU is known as approximate function of fpifhs-set.
Note that collection of allfpifhs-sets is represented by ⊎fpifhsU. The vivid comparison of the existing model fpifs-set and the proposed fpifhs-set model with the help of Example 2 can be viewed in Figure 3.Figure 3
Comparison offpifs-set model and fpifhs-set model with the help of Example 2.Definition 8.
LetΨℬ∈⊎fpifhsU. If ψℬg^=∅,Lℬg^=0 for all g^∈G, then Ψℬ is called ℬ-empty fpifhs-set, denoted by ΨΦℬ. If ℬ=∅, then ℬ-empty fpifhs-set is called an empty fpifhs-set, denoted by ΨΦ.Definition 9.
LetΨℬ∈⊎fpifhsU. If ψℬg^=U,Lℬg^=1 for all g^∈G, then Ψℬ is called ℬ-universal fpifhs-set, denoted by Ψℬ˜. If ℬ=G, then the ℬ-universal fpifhs-set is called universal fpifhs-set, denoted by ΨG˜.Example 1.
Consider
U=u^1,u^2,u^3,u^4,u^5 and Y=Y1,Y2,Y3 with Y1=y^11,y^12, Y2=y^21,y^22,Y3=y^31; then, G=Y1×Y2×Y3, G=y^11,y^21,y^31,y^11,y^22,y^31,y^12,y^21,y^31,y^12,y^22,y^31G=g^1,g^2,g^3,g^4.Case 1. Whenℬ1=0.2/g^2,0.0/g^3,1.0/g^4, ψℬ1g^2=0.2,0.4/u^2,0.3,0.5/u^4, ψℬ1g^3=∅, and ψℬ1g^4=U, then Ψℬ1=0.2/g^2,0.2,0.4/u^2,0.3,0.5/u^4,0.0/g^3,∅,1.0/g^4,U.Case 2. Whenℬ2=0.0/g^2,0.0/g^3,ψℬ2g^2=∅ and ψℬ2g^3=∅, then Ψℬ2=ΨΦℬ2.Case 3. Whenℬ3=∅ for all elements of G, then Ψℬ3=ΨΦ.Case 4. Whenℬ4=1.0/g^1,1.0/g^2,ψℬ4g^1=U, and ψℬ4g^2=U, then Ψℬ4=Ψℬ4˜.Case 5. Whenℬ5=U for all elements of G, then Ψℬ5=ΨG˜.Definition 10.
LetΨℬ1, Ψℬ2∈⊎fpifhsU; then, Ψℬ1 is an fpifhs-subset of Ψℬ2, denoted by Ψℬ1⊆˜fΨℬ2 ifLℬ1g^≤Lℬ2g^ and ψℬ1g^⊆ifψℬ2g^ for all g^∈G. Here, ⊆˜f and ⊆if represent fuzzy subset and intuitionistic fuzzy subset notations, respectively.Definition 11.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, Ψℬ1 and Ψℬ2 are fpifhs-equal, represented as Ψℬ1=Ψℬ2, if and only if Lℬ1g^=Lℬ2g^ and ψℬ1g^=ifψℬ2g^ for all g^∈G.Definition 12.
LetΨℬ∈⊎fpifhsU; then, complement of Ψℬ (i.e., Ψℬc˜) is an fpifhs-set given as Pℬc˜g^=1−Lℬg^ and ψℬc˜g^=U∖ifψℬg^.Proposition 1.
LetΨℬ∈⊎fpifhsU; then,(1)
Ψℬc˜c˜=Ψℬ.(2)
Ψϕc˜=ΨG˜.Definition 13.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, union of Ψℬ1 and Ψℬ2, denoted by Ψℬ1∪˜fΨℬ2, is a fpifhs-set defined by(i)
Lℬ1∪˜ℬ2g^=maxLℬ1x,Lℬ2g^.(ii)
ψℬ1∪˜ℬ2g^=ψℬ1g^∪ifψℬ2g^, for all g^∈G.Definition 14.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, intersection of Ψℬ1 and Ψℬ2, denoted by Ψℬ1∩˜fΨℬ2, is a fpifhs-set defined by(i)
Lℬ1∩˜ℬ2g^=minLℬ1x,Lℬ2g^.(ii)
ψℬ1∩˜ℬ2g^=ψℬ1g^∩ifψℬ2g^, for all g^∈G.Remark 1.
ForΨℬ∈⊎fpifhsU. If Ψℬ≠fΨG˜, then Ψℬ∪˜fΨℬc˜≠fΨG˜ and Ψℬ∩˜fΨℬc˜≠fΨΦ.Proposition 2.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, the following De Morgan laws are valid:(1)
Ψℬ1∪˜fΨℬ2c˜=Ψℬ1c˜∩˜fΨℬ2c˜.(2)
Ψℬ1∩˜fΨℬ2c˜=Ψℬ1c˜∪˜fΨℬ2c˜.Proof.
For allg^∈G:(1)
Since(7)Lℬ1∪˜ℬ2c˜g^=1−Lℬ1∪˜ℬ2g^=1−maxLℬ1g^,Lℬ2g^=min1−Lℬ1g^,1−Lℬ2g^=minPℬ1c˜g^,Pℬ2c˜g^=Pℬ1∩˜ℬ2c˜g^,and(8)ψℬ1∪˜ℬ2c˜g^=U∖ifψℬ1∪˜ℬ2g^=U∖ifψℬ1g^∪ifψℬ2g^=U∖ifψℬ1g^∩ifU∖ifψℬ2g^=ψℬ1c˜g^∩if˜ψℬ2c˜g^=ψℬ1∩˜ℬ2c˜g^,
similarly, (2) can be proved easily.Proposition 3.
LetΨℬ1,Ψℬ2,Ψℬ3∈⊎fpifhsU; then,(1)
Ψℬ1∪˜fΨℬ2∩˜fΨℬ3=Ψℬ1∪˜fΨℬ2∩˜fΨℬ1∪˜fΨℬ3.(2)
Ψℬ1∩˜fΨℬ2∪˜fΨℬ3=Ψℬ1∩˜fΨℬ2∪˜fΨℬ1∩˜fΨℬ3.Proof.
For allg^∈G:(1)
Since(9)Lℬ1∪˜ℬ2∩˜ℬ3g^=maxLℬ1g^,Lℬ2∩˜ℬ3g^=maxLℬ1g^,minLℬ2g^,Lℬ3g^=minmaxLℬ1g^,Lℬ2g^,maxLℬ1g^,Lℬ3g^=minLℬ1∪˜ℬ2g^,Lℬ1∪˜ℬ3g^=Lℬ1∪˜ℬ2∩˜ℬ1∪˜ℬ3g^,and(10)ψℬ1∪˜ℬ2∩˜ℬ3g^=ψℬ1g^∪ifψℬ2∩˜ℬ3g^=ψℬ1g^∪ifψℬ2g^∩ifψℬ3g^=ψℬ1g^∪ifψℬ2g^∩ifψℬ1g^∪ifψℬ3g^=ψℬ1∪˜ℬ2g^∩ifψℬ1U˜ℬ3g^=ψℬ1∪˜ℬ2∩˜ℬ1∪˜ℬ3g^,
in the same way, (2) can be proved.Definition 15.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, OR-operation of Ψℬ1 and Ψℬ2, denoted by Ψℬ1⊻˜Ψℬ2, is an fpifhs-set defined by(i)
Lℬ1⊻˜ℬ2g^1,g^2=maxLℬ1g^1,Lℬ2g^2,(ii)
ψℬ1⊻˜ℬ2g^1,g^2=ψℬ1g^1∪ifψℬ2g^2, for all g^1,g^2∈ℬ1×ℬ2.Definition 16.
LetΨℬ1,Ψℬ2∈⊎fpifhsU; then, AND-operation of Ψℬ1 and Ψℬ2, denoted by Ψℬ1⊼˜Ψℬ2, is an fpifhs-set defined by(i)
Lℬ1⊼˜ℬ2g^1,g^2=minLℬ1g^1,Lℬ2g^2.(ii)
ψℬ1⊼˜ℬ2g^1,g^2=ψℬ1g^1∩ifψℬ2g^2, for all g^1,g^2∈ℬ1×ℬ2.Proposition 4.
LetΨℬ1,Ψℬ2,Ψℬ3∈⊎fpifhsU; then,(1)
Ψℬ1⊼˜ΨΦ=ΨΦ.(2)
Ψℬ1⊼˜Ψℬ2⊼˜Ψℬ3=Ψℬ1⊼˜Ψℬ2⊼˜Ψℬ3.(3)
Ψℬ1⊻˜Ψℬ2⊻˜Ψℬ3=Ψℬ1⊻˜Ψℬ2⊻˜Ψℬ3.


### 3.1. Fuzzy Decision Set offpifhs-Set

In this part, fuzzy decision-support system based onfpifhs-set will be established.Definition 17.
LetΨℬ∈⊎fpifhsU; then, a fuzzy decision set of Ψℬ (i.e., ΨℬD) is
represented as(11)ΨℬD=TℬDu^u^:u^∈U,where TℬD:U⟶I and(12)TℬDu^=1U∑v∈SℬTℬvΓψℬvu^,where • denotes set cardinality with(13)Γψℬvu^=Tψℬu^−Fψℬu^,u^∈Γψℬv,0,u^∉Γψℬv.
With the establishment ofΨℬD, optimal single selection from the set of alternatives can easily be evalauated. Therefore, the following algorithm is proposed to make appropriate decision.
Figure4 presents the flowchart of Algorithm 1.Algorithm 1: Optimal selection of hand sanitizer by using fpifhs-set.
StartInput:(1)
ConsiderU as universe of discourse(2)
ConsiderB as set of n parameters(3)
Classify n parameters into disjoint parametric-valued setsB1,B2,B3,…,BnConstruction:(4)
Determineℚ=B1×B2×B3×⋯×B5(5)
Takeℝ⊆ℚComputation:(6)
Determineℬ=<Tℬqi>/qi:Tℬqi∈I,qi∈ℝ,(7)
Findψℬqi,(8)
ComputeΨℬ over U,(9)
ComputeΨℬD,Output:(10)
Choose the maximum ofTℬDu^,EndFigure 4
Decision-making algorithm forfpifhs-set.


#### 3.1.1. Problem Scenario

Hand sanitizer is a fluid or gel that is utilized to kill germs on the hands. As per the World Health Organization (WHO), excellent sterilization and actual detachment are simply the most ideal approaches to rescue the individuals from COVID-19 in the ebb and flow illness circumstance. This infection is communicated by reaching somebody who is debilitated. We cannot be excessively cautious with this infection on the off chance that we do not isolate ourselves totally. Accordingly, great disinfection can be the best way to guard us against the infection. WHO suggests alcohol-based hand sanitizers (ABH-sanitizers) for eliminating the novel corona virus. ABH-sanitizers prevent germ proteins, like microscopic organisms and infections, from working ordinarily. There are two kinds of formulations (see Figure5) recommended by WHO for the production of large quantities of hand sanitizer from chemicals available in developing countries, where commercial hand sanitizer may not be available. The interest in hand sanitizers has expanded drastically in light of COVID-19’s critical situation. Thus, discovering great and powerful hand sanitizers in local markets is troublesome. Because of the expanded interest, inferior quality hand sanitizers have additionally been dispatched. The fundamental objective of this application is to utilize fpifhs-set and fpnhs-set theories to track down a successful sanitizer to stop the spread of COVID-19.Figure 5
Formulations of hand sanitizers, figure source: Wikipedia (https://en.wikipedia.org/wiki/Hand_sanitizer).


#### 3.1.2. Operational Role of Constituents of ABH-Sanitizers

It is clear from Figure5 that ABH-sanitizers have four major constituents that are given below with their brief operation role in sanitation process:(1)
Ethanol: It is an organic chemical compound and majorly used in medication as antiseptic, antidote, and medicinal solvent. It is the major constituent of ABH-sanitizers, and it ranges from 70% to 80%V/V as an active ingredient.(2)
Glycerol: It is a nontoxic viscous liquid having no color, no odor, and sweet taste. It is mildly antimicrobial and antiviral. It prevents skin dryness with its moisturizing properties. It draws moisture up through skin layers and slows or prevents excessive drying and evaporation. It is added to ABH-sanitizers to prevent drying of the skin, and it ranges from 1.35% to 1.45%V/V as an active ingredient.(3)
Hydrogen peroxide: It is a very pale blue colored chemical compound having viscosity slightly more than water. Usually as a dilute solution (3–6% by weight) in water, it is used as an oxidizer, bleaching agent, and antiseptic for consumer use. It can be used for industrial purposes with high concentrations. It is used in ABH-sanitizers as sterilizer, and it ranges from 0.120% to 0.125%V/V as an active ingredient.(4)
Distilled water: It is an essential ingredient of ABH-sanitizers. It ranges from 15.425% to 23.425%V/V as an active ingredient. It is used as cleansing agent to prevent the interaction of chlorine and other contaminants with the alcohol and other ingredients in the sanitizer.


#### 3.1.3. Statement of the Problem

Professor John is the head of an educational institution. He is very concerned about the health of the students as well as the faculty members of his institution in view of the current coronary epidemic. He wants to buy good and useful hand sanitizers for his institution, but he is also worried about the nonstandard hand sanitizers available in the market. Therefore, he decides to call for bids from different potential suppliers for this purchase to fulfill the departmental official compliances and to avoid any expected loss. Some suppliers are scrutinized by adopting proper procedure already framed by relevant department. For the sake of satisfaction, he constitutes a committee consisting of some staff members with good procurement experience to evaluate the items (hand sanitizers) offered by scrutinized suppliers. The following example elaborates the whole procedure of such evaluation:Table 2
Degrees of membershipTℬqi.
TℬqiDegreeTℬqiDegreeTℬqiDegreeTℬqiDegreeTℬq10.1Tℬq50.5Tℬq90.9Tℬq130.35Tℬq20.2Tℬq60.6Tℬq100.16Tℬq140.75Tℬq30.3Tℬq70.7Tℬq110.25Tℬq150.65Tℬq40.4Tℬq80.8Tℬq120.45Tℬq160.85Table 3
Approximate functionsψℬqi.
qiψℬqiqiψℬqiq10.2,0.1/ℍ1,0.3,0.2/ℍ2q90.4,0.3/ℍ2,0.6,0.4/ℍ7,0.5,0.4/ℍ8q20.1,0.2/ℍ1,0.5,0.4/ℍ2,0.1,0.4/ℍ3q100.2,0.1/ℍ6,0.6,0.4/ℍ7,0.4,0.3/ℍ8q30.4,0.3/ℍ2,0.5,0.4/ℍ3,0.6,0.3/ℍ4q110.5,0.4/ℍ2,0.6,0.3/ℍ4,0.7,0.2/ℍ6q40.6,0.2/ℍ4,0.7,0.3/ℍ5,0.8,0.1/ℍ6q120.7,0.2/ℍ2,0.8,0.1/ℍ3,0.9,0.1/ℍ6q50.2,0.1/ℍ6,0.1,0.2/ℍ7,0.4,0.3/ℍ8q130.2,0.1/ℍ3,0.4,0.3/ℍ5,0.6,0.1/ℍ7q60.4,0.2/ℍ2,0.3,0.4/ℍ3,0.4,0.5/ℍ4q140.2,0.5/ℍ1,0.5,0.4/ℍ3,0.6,0.2/ℍ5q70.2,0.3/ℍ1,0.3,0.4/ℍ3,0.4,0.3/ℍ5q150.6,0.3/ℍ5,0.4,0.3/ℍ7,0.2,0.4/ℍ8q80.1,0.4/ℍ2,0.3,0.5/ℍ3,0.5,0.4/ℍ7q160.3,0.6/ℍ4,0.5,0.4/ℍ5,0.7,0.1/ℍ6Table 4
Membership valuesTℬDℍi.
ℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍ10.0406ℍ30.1006ℍ50.1006ℍ70.0728ℍ20.0950ℍ40.0800ℍ60.1676ℍ80.0358Figure 6
Graphical representation of Table2.Figure 7
Fuzzy decision system onfpifhs-set.Example 2.
Input and Construction Stages (1–5):Suppose there are eight kinds of hand sanitizer (options) which form the set of discourse(14)U=U1,U2,U3,whereU1=ℍ1,ℍ2,ℍ3, U2=ℍ4,ℍ5,ℍ6, and U3=ℍ7,ℍ8 are the collection of hand sanitizers made by manufacturers X1, X2, and X3, respectively. With their mutual consensus, the committee members (experts) agreed on a set of parameters after observing various attributes for this evaluation. The finalized evaluating attributes are b1  manufacturer; b2  quantity of ethanol (percentage); b3  quantity of distilled water (percentage); b4  quantity of glycerol (percentage); and b5  quantity of hydrogen peroxide (percentage). After observing the opinions of various professionals and other relevant sources on the composition of hand sanitizers, these attributes are further classified into attribute-valued sets which are given as(15)B1=b11=X1,b12=X2,b13=X3,B2=b21=75.15,b22=80,B3=b31=23.425,b32=18.425,B4=b41=1.30,b42=1.45,B5=b51=0.125.Then,ℚ=B1×B2×B3×B4×B5,ℚ=q1,q2,q3,q4,…..,q24, where each qi,i=1,2,…,24, is a 5-tuple element. For convenience, take ℝ=q1,q2,q3,q4,…..,q16⊆ℚ.Computation Stage (6–9):(6)
The graphical depiction of Table2 can be seen in Figure 6.From Table2, we can construct ℬ as(16)ℬ=0.1q1,0.2q2,0.3q3,0.4q4,0.5q5,0.6q6,0.7q7,0.8q8,0.9q9,0.16q100.25q110.45q12,0.35q13,0.75q14,0.65q15,0.85q16.(7)
Table3 presents ψℬqi corresponding to each element of G.(8)
Ψℬ can be constructed with the support of step 6 and step 7 as performed previously.(9)
From Table4, we can construct ΨℬD as(17)ΨℬD=0.0406ℍ1,0.0950ℍ2,0.1006ℍ3,0.0800ℍ4,0.1006ℍ5,0.1676ℍ6,0.0728ℍ7,0.0358ℍ8.For its pictorial representation, please see Figure7.Output Stage:(10)
Since maximum ofTℬDℍi is 0.1676, hand sanitizer ℍ6 is selected.


## 3.1. Fuzzy Decision Set offpifhs-Set

In this part, fuzzy decision-support system based onfpifhs-set will be established.Definition 17.
LetΨℬ∈⊎fpifhsU; then, a fuzzy decision set of Ψℬ (i.e., ΨℬD) is
represented as(11)ΨℬD=TℬDu^u^:u^∈U,where TℬD:U⟶I and(12)TℬDu^=1U∑v∈SℬTℬvΓψℬvu^,where • denotes set cardinality with(13)Γψℬvu^=Tψℬu^−Fψℬu^,u^∈Γψℬv,0,u^∉Γψℬv.
With the establishment ofΨℬD, optimal single selection from the set of alternatives can easily be evalauated. Therefore, the following algorithm is proposed to make appropriate decision.
Figure4 presents the flowchart of Algorithm 1.Algorithm 1: Optimal selection of hand sanitizer by using fpifhs-set.
StartInput:(1)
ConsiderU as universe of discourse(2)
ConsiderB as set of n parameters(3)
Classify n parameters into disjoint parametric-valued setsB1,B2,B3,…,BnConstruction:(4)
Determineℚ=B1×B2×B3×⋯×B5(5)
Takeℝ⊆ℚComputation:(6)
Determineℬ=<Tℬqi>/qi:Tℬqi∈I,qi∈ℝ,(7)
Findψℬqi,(8)
ComputeΨℬ over U,(9)
ComputeΨℬD,Output:(10)
Choose the maximum ofTℬDu^,EndFigure 4
Decision-making algorithm forfpifhs-set.


### 3.1.1. Problem Scenario

Hand sanitizer is a fluid or gel that is utilized to kill germs on the hands. As per the World Health Organization (WHO), excellent sterilization and actual detachment are simply the most ideal approaches to rescue the individuals from COVID-19 in the ebb and flow illness circumstance. This infection is communicated by reaching somebody who is debilitated. We cannot be excessively cautious with this infection on the off chance that we do not isolate ourselves totally. Accordingly, great disinfection can be the best way to guard us against the infection. WHO suggests alcohol-based hand sanitizers (ABH-sanitizers) for eliminating the novel corona virus. ABH-sanitizers prevent germ proteins, like microscopic organisms and infections, from working ordinarily. There are two kinds of formulations (see Figure5) recommended by WHO for the production of large quantities of hand sanitizer from chemicals available in developing countries, where commercial hand sanitizer may not be available. The interest in hand sanitizers has expanded drastically in light of COVID-19’s critical situation. Thus, discovering great and powerful hand sanitizers in local markets is troublesome. Because of the expanded interest, inferior quality hand sanitizers have additionally been dispatched. The fundamental objective of this application is to utilize fpifhs-set and fpnhs-set theories to track down a successful sanitizer to stop the spread of COVID-19.Figure 5
Formulations of hand sanitizers, figure source: Wikipedia (https://en.wikipedia.org/wiki/Hand_sanitizer).


### 3.1.2. Operational Role of Constituents of ABH-Sanitizers

It is clear from Figure5 that ABH-sanitizers have four major constituents that are given below with their brief operation role in sanitation process:(1)
Ethanol: It is an organic chemical compound and majorly used in medication as antiseptic, antidote, and medicinal solvent. It is the major constituent of ABH-sanitizers, and it ranges from 70% to 80%V/V as an active ingredient.(2)
Glycerol: It is a nontoxic viscous liquid having no color, no odor, and sweet taste. It is mildly antimicrobial and antiviral. It prevents skin dryness with its moisturizing properties. It draws moisture up through skin layers and slows or prevents excessive drying and evaporation. It is added to ABH-sanitizers to prevent drying of the skin, and it ranges from 1.35% to 1.45%V/V as an active ingredient.(3)
Hydrogen peroxide: It is a very pale blue colored chemical compound having viscosity slightly more than water. Usually as a dilute solution (3–6% by weight) in water, it is used as an oxidizer, bleaching agent, and antiseptic for consumer use. It can be used for industrial purposes with high concentrations. It is used in ABH-sanitizers as sterilizer, and it ranges from 0.120% to 0.125%V/V as an active ingredient.(4)
Distilled water: It is an essential ingredient of ABH-sanitizers. It ranges from 15.425% to 23.425%V/V as an active ingredient. It is used as cleansing agent to prevent the interaction of chlorine and other contaminants with the alcohol and other ingredients in the sanitizer.


### 3.1.3. Statement of the Problem

Professor John is the head of an educational institution. He is very concerned about the health of the students as well as the faculty members of his institution in view of the current coronary epidemic. He wants to buy good and useful hand sanitizers for his institution, but he is also worried about the nonstandard hand sanitizers available in the market. Therefore, he decides to call for bids from different potential suppliers for this purchase to fulfill the departmental official compliances and to avoid any expected loss. Some suppliers are scrutinized by adopting proper procedure already framed by relevant department. For the sake of satisfaction, he constitutes a committee consisting of some staff members with good procurement experience to evaluate the items (hand sanitizers) offered by scrutinized suppliers. The following example elaborates the whole procedure of such evaluation:Table 2
Degrees of membershipTℬqi.
TℬqiDegreeTℬqiDegreeTℬqiDegreeTℬqiDegreeTℬq10.1Tℬq50.5Tℬq90.9Tℬq130.35Tℬq20.2Tℬq60.6Tℬq100.16Tℬq140.75Tℬq30.3Tℬq70.7Tℬq110.25Tℬq150.65Tℬq40.4Tℬq80.8Tℬq120.45Tℬq160.85Table 3
Approximate functionsψℬqi.
qiψℬqiqiψℬqiq10.2,0.1/ℍ1,0.3,0.2/ℍ2q90.4,0.3/ℍ2,0.6,0.4/ℍ7,0.5,0.4/ℍ8q20.1,0.2/ℍ1,0.5,0.4/ℍ2,0.1,0.4/ℍ3q100.2,0.1/ℍ6,0.6,0.4/ℍ7,0.4,0.3/ℍ8q30.4,0.3/ℍ2,0.5,0.4/ℍ3,0.6,0.3/ℍ4q110.5,0.4/ℍ2,0.6,0.3/ℍ4,0.7,0.2/ℍ6q40.6,0.2/ℍ4,0.7,0.3/ℍ5,0.8,0.1/ℍ6q120.7,0.2/ℍ2,0.8,0.1/ℍ3,0.9,0.1/ℍ6q50.2,0.1/ℍ6,0.1,0.2/ℍ7,0.4,0.3/ℍ8q130.2,0.1/ℍ3,0.4,0.3/ℍ5,0.6,0.1/ℍ7q60.4,0.2/ℍ2,0.3,0.4/ℍ3,0.4,0.5/ℍ4q140.2,0.5/ℍ1,0.5,0.4/ℍ3,0.6,0.2/ℍ5q70.2,0.3/ℍ1,0.3,0.4/ℍ3,0.4,0.3/ℍ5q150.6,0.3/ℍ5,0.4,0.3/ℍ7,0.2,0.4/ℍ8q80.1,0.4/ℍ2,0.3,0.5/ℍ3,0.5,0.4/ℍ7q160.3,0.6/ℍ4,0.5,0.4/ℍ5,0.7,0.1/ℍ6Table 4
Membership valuesTℬDℍi.
ℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍ10.0406ℍ30.1006ℍ50.1006ℍ70.0728ℍ20.0950ℍ40.0800ℍ60.1676ℍ80.0358Figure 6
Graphical representation of Table2.Figure 7
Fuzzy decision system onfpifhs-set.Example 2.
Input and Construction Stages (1–5):Suppose there are eight kinds of hand sanitizer (options) which form the set of discourse(14)U=U1,U2,U3,whereU1=ℍ1,ℍ2,ℍ3, U2=ℍ4,ℍ5,ℍ6, and U3=ℍ7,ℍ8 are the collection of hand sanitizers made by manufacturers X1, X2, and X3, respectively. With their mutual consensus, the committee members (experts) agreed on a set of parameters after observing various attributes for this evaluation. The finalized evaluating attributes are b1  manufacturer; b2  quantity of ethanol (percentage); b3  quantity of distilled water (percentage); b4  quantity of glycerol (percentage); and b5  quantity of hydrogen peroxide (percentage). After observing the opinions of various professionals and other relevant sources on the composition of hand sanitizers, these attributes are further classified into attribute-valued sets which are given as(15)B1=b11=X1,b12=X2,b13=X3,B2=b21=75.15,b22=80,B3=b31=23.425,b32=18.425,B4=b41=1.30,b42=1.45,B5=b51=0.125.Then,ℚ=B1×B2×B3×B4×B5,ℚ=q1,q2,q3,q4,…..,q24, where each qi,i=1,2,…,24, is a 5-tuple element. For convenience, take ℝ=q1,q2,q3,q4,…..,q16⊆ℚ.Computation Stage (6–9):(6)
The graphical depiction of Table2 can be seen in Figure 6.From Table2, we can construct ℬ as(16)ℬ=0.1q1,0.2q2,0.3q3,0.4q4,0.5q5,0.6q6,0.7q7,0.8q8,0.9q9,0.16q100.25q110.45q12,0.35q13,0.75q14,0.65q15,0.85q16.(7)
Table3 presents ψℬqi corresponding to each element of G.(8)
Ψℬ can be constructed with the support of step 6 and step 7 as performed previously.(9)
From Table4, we can construct ΨℬD as(17)ΨℬD=0.0406ℍ1,0.0950ℍ2,0.1006ℍ3,0.0800ℍ4,0.1006ℍ5,0.1676ℍ6,0.0728ℍ7,0.0358ℍ8.For its pictorial representation, please see Figure7.Output Stage:(10)
Since maximum ofTℬDℍi is 0.1676, hand sanitizer ℍ6 is selected.


## 3.1.1. Problem Scenario

Hand sanitizer is a fluid or gel that is utilized to kill germs on the hands. As per the World Health Organization (WHO), excellent sterilization and actual detachment are simply the most ideal approaches to rescue the individuals from COVID-19 in the ebb and flow illness circumstance. This infection is communicated by reaching somebody who is debilitated. We cannot be excessively cautious with this infection on the off chance that we do not isolate ourselves totally. Accordingly, great disinfection can be the best way to guard us against the infection. WHO suggests alcohol-based hand sanitizers (ABH-sanitizers) for eliminating the novel corona virus. ABH-sanitizers prevent germ proteins, like microscopic organisms and infections, from working ordinarily. There are two kinds of formulations (see Figure5) recommended by WHO for the production of large quantities of hand sanitizer from chemicals available in developing countries, where commercial hand sanitizer may not be available. The interest in hand sanitizers has expanded drastically in light of COVID-19’s critical situation. Thus, discovering great and powerful hand sanitizers in local markets is troublesome. Because of the expanded interest, inferior quality hand sanitizers have additionally been dispatched. The fundamental objective of this application is to utilize fpifhs-set and fpnhs-set theories to track down a successful sanitizer to stop the spread of COVID-19.Figure 5
Formulations of hand sanitizers, figure source: Wikipedia (https://en.wikipedia.org/wiki/Hand_sanitizer).


## 3.1.2. Operational Role of Constituents of ABH-Sanitizers

It is clear from Figure5 that ABH-sanitizers have four major constituents that are given below with their brief operation role in sanitation process:(1)
Ethanol: It is an organic chemical compound and majorly used in medication as antiseptic, antidote, and medicinal solvent. It is the major constituent of ABH-sanitizers, and it ranges from 70% to 80%V/V as an active ingredient.(2)
Glycerol: It is a nontoxic viscous liquid having no color, no odor, and sweet taste. It is mildly antimicrobial and antiviral. It prevents skin dryness with its moisturizing properties. It draws moisture up through skin layers and slows or prevents excessive drying and evaporation. It is added to ABH-sanitizers to prevent drying of the skin, and it ranges from 1.35% to 1.45%V/V as an active ingredient.(3)
Hydrogen peroxide: It is a very pale blue colored chemical compound having viscosity slightly more than water. Usually as a dilute solution (3–6% by weight) in water, it is used as an oxidizer, bleaching agent, and antiseptic for consumer use. It can be used for industrial purposes with high concentrations. It is used in ABH-sanitizers as sterilizer, and it ranges from 0.120% to 0.125%V/V as an active ingredient.(4)
Distilled water: It is an essential ingredient of ABH-sanitizers. It ranges from 15.425% to 23.425%V/V as an active ingredient. It is used as cleansing agent to prevent the interaction of chlorine and other contaminants with the alcohol and other ingredients in the sanitizer.


## 3.1.3. Statement of the Problem

Professor John is the head of an educational institution. He is very concerned about the health of the students as well as the faculty members of his institution in view of the current coronary epidemic. He wants to buy good and useful hand sanitizers for his institution, but he is also worried about the nonstandard hand sanitizers available in the market. Therefore, he decides to call for bids from different potential suppliers for this purchase to fulfill the departmental official compliances and to avoid any expected loss. Some suppliers are scrutinized by adopting proper procedure already framed by relevant department. For the sake of satisfaction, he constitutes a committee consisting of some staff members with good procurement experience to evaluate the items (hand sanitizers) offered by scrutinized suppliers. The following example elaborates the whole procedure of such evaluation:Table 2
Degrees of membershipTℬqi.
TℬqiDegreeTℬqiDegreeTℬqiDegreeTℬqiDegreeTℬq10.1Tℬq50.5Tℬq90.9Tℬq130.35Tℬq20.2Tℬq60.6Tℬq100.16Tℬq140.75Tℬq30.3Tℬq70.7Tℬq110.25Tℬq150.65Tℬq40.4Tℬq80.8Tℬq120.45Tℬq160.85Table 3
Approximate functionsψℬqi.
qiψℬqiqiψℬqiq10.2,0.1/ℍ1,0.3,0.2/ℍ2q90.4,0.3/ℍ2,0.6,0.4/ℍ7,0.5,0.4/ℍ8q20.1,0.2/ℍ1,0.5,0.4/ℍ2,0.1,0.4/ℍ3q100.2,0.1/ℍ6,0.6,0.4/ℍ7,0.4,0.3/ℍ8q30.4,0.3/ℍ2,0.5,0.4/ℍ3,0.6,0.3/ℍ4q110.5,0.4/ℍ2,0.6,0.3/ℍ4,0.7,0.2/ℍ6q40.6,0.2/ℍ4,0.7,0.3/ℍ5,0.8,0.1/ℍ6q120.7,0.2/ℍ2,0.8,0.1/ℍ3,0.9,0.1/ℍ6q50.2,0.1/ℍ6,0.1,0.2/ℍ7,0.4,0.3/ℍ8q130.2,0.1/ℍ3,0.4,0.3/ℍ5,0.6,0.1/ℍ7q60.4,0.2/ℍ2,0.3,0.4/ℍ3,0.4,0.5/ℍ4q140.2,0.5/ℍ1,0.5,0.4/ℍ3,0.6,0.2/ℍ5q70.2,0.3/ℍ1,0.3,0.4/ℍ3,0.4,0.3/ℍ5q150.6,0.3/ℍ5,0.4,0.3/ℍ7,0.2,0.4/ℍ8q80.1,0.4/ℍ2,0.3,0.5/ℍ3,0.5,0.4/ℍ7q160.3,0.6/ℍ4,0.5,0.4/ℍ5,0.7,0.1/ℍ6Table 4
Membership valuesTℬDℍi.
ℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍiTℬDℍiℍ10.0406ℍ30.1006ℍ50.1006ℍ70.0728ℍ20.0950ℍ40.0800ℍ60.1676ℍ80.0358Figure 6
Graphical representation of Table2.Figure 7
Fuzzy decision system onfpifhs-set.Example 2.
Input and Construction Stages (1–5):Suppose there are eight kinds of hand sanitizer (options) which form the set of discourse(14)U=U1,U2,U3,whereU1=ℍ1,ℍ2,ℍ3, U2=ℍ4,ℍ5,ℍ6, and U3=ℍ7,ℍ8 are the collection of hand sanitizers made by manufacturers X1, X2, and X3, respectively. With their mutual consensus, the committee members (experts) agreed on a set of parameters after observing various attributes for this evaluation. The finalized evaluating attributes are b1  manufacturer; b2  quantity of ethanol (percentage); b3  quantity of distilled water (percentage); b4  quantity of glycerol (percentage); and b5  quantity of hydrogen peroxide (percentage). After observing the opinions of various professionals and other relevant sources on the composition of hand sanitizers, these attributes are further classified into attribute-valued sets which are given as(15)B1=b11=X1,b12=X2,b13=X3,B2=b21=75.15,b22=80,B3=b31=23.425,b32=18.425,B4=b41=1.30,b42=1.45,B5=b51=0.125.Then,ℚ=B1×B2×B3×B4×B5,ℚ=q1,q2,q3,q4,…..,q24, where each qi,i=1,2,…,24, is a 5-tuple element. For convenience, take ℝ=q1,q2,q3,q4,…..,q16⊆ℚ.Computation Stage (6–9):(6)
The graphical depiction of Table2 can be seen in Figure 6.From Table2, we can construct ℬ as(16)ℬ=0.1q1,0.2q2,0.3q3,0.4q4,0.5q5,0.6q6,0.7q7,0.8q8,0.9q9,0.16q100.25q110.45q12,0.35q13,0.75q14,0.65q15,0.85q16.(7)
Table3 presents ψℬqi corresponding to each element of G.(8)
Ψℬ can be constructed with the support of step 6 and step 7 as performed previously.(9)
From Table4, we can construct ΨℬD as(17)ΨℬD=0.0406ℍ1,0.0950ℍ2,0.1006ℍ3,0.0800ℍ4,0.1006ℍ5,0.1676ℍ6,0.0728ℍ7,0.0358ℍ8.For its pictorial representation, please see Figure7.Output Stage:(10)
Since maximum ofTℬDℍi is 0.1676, hand sanitizer ℍ6 is selected.


## 4. Fuzzy Parameterized Neutrosophic Hypersoft Set (fpnhs-Set) with Application

fpifhs-set is quite better to tackle real-world decision-making scenarios where the decision-makers are uncertain about the nature of selected parameters under hs-set environment with approximate elements having intuitionistic fuzzy values. However, it is still insufficient to manage the situation when approximate elements are needed to be assessed with indeterminacy degree; therefore, this shortcoming demands another structure to deal with these situations. Therefore, in this section, the theory of fpnhs-set is developed with the characterization of some of its essential rudiments and decision-support system.Definition 18.
LetS1,S2,S3,….,Sn be parameter-valued sets with Si∩Sj=∅,i≠j respective to parameterss1,s2,s3,…,sn,si≠sj,i≠j, respectively. A fpnhs-set ΨD over U is defined as(18)ΨD=ADg^g^,ψDg^:g^∈G,ψDg^∈ℕU,where(i)
G=S1×S2×S3×….×Sn.(ii)
D∈FU with LD:G⟶I.(iii)
ψD:G⟶ℕU is known as approximate function of fpnhs-set.
Note that collection of allfpnhs-sets is represented by ⊎fpnhsU. The pictorial self-explanatory comparison of the existing model fpns-set and the proposed fpnhs-set model with the help of Example 4 can be viewed in Figure 8.Figure 8
Comparison offpns-set model and fpnhs-set model with the help of Example 4.Definition 19.
LetΨD∈⊎fpnhsU. If ψDg^=∅,ADg^=0 for all g^∈G, then ΨD is called D-empty fpnhs-set, denoted by ΨΦD. If D=∅, then D-empty fpnhs-set is called an empty fpnhs-set, denoted by ΨΦ.Definition 20.
LetΨD∈⊎fpnhsU. If ψDg^=U,ADg^=1 for all g^∈G, then ΨD is called D-universal fpnhs-set, denoted by ΨD˜. If D=G, then the D-universal fpnhs-set is called universal fpnhs-set, denoted by ΨG˜.Example 3.
Consider
U=u^1,u^2,u^3,u^4,u^5 and Z=Z1,Z2,Z3 with Z1=z^11,z^12, Z2=z^21,z^22,Z3=z^31; then, G=Z1×Z2×Z3, G=z^11,z^21,z^31,z^11,z^22,z^31,z^12,z^21,z^31,z^12,z^22,z^31, G=g^1,g^2,g^3,g^4.Case 1. WhenD1=0.2/g^2,0.0/g^3,1.0/g^4, ψD1g^2=0.2,0.4,0.6/u^2,0.3,0.5,0.7/u^4, ψD1g^3=ϕ, and ψD1g^4=U, then(19)ΨD1=0.2g^2,0.2,0.4,0.6u^2,0.3,0.5,0.7u^4,0.0g^3,∅,1.0g^4,U.Case 2. WhenD2=0.0/g^2,0.0/g^3,ψD2g^2=∅, and ψD2g^3=∅, then ΨD2=ΨΦD2.Case 3. WhenD3=∅ for all members of G, then ΨD3=ΨΦ.Case 4. WhenD4=1.0/g^1,1.0/g^2,ψD4g^1=U, and ψD4g^2=U, then ΨD4=ΨD4˜.Case 5. WhenD5=U for all members of G, then ΨD5=ΨG˜.Definition 21.
LetΨD1, ΨD2∈⊎fpnhsU; then, ΨD1 is an fpnhs-subset of ΨD2, denoted by ΨD1⊆˜fΨD2 ifAD1g^≤AD2g^ and ψD1g^⊆nψD2g^ for all g^∈G.Proposition 5.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then,(1)
ΨD1⊆˜fΨG˜.(2)
ΨΦ⊆˜fΨD1.(3)
ΨD1⊆˜fΨD1.(4)
IfΨD1⊆˜fΨD2 and ΨD2⊆˜fΨD3, thenΨD1⊆˜fΨD3.Definition 22.
LetΨD1,ΨD2∈⊎fpnhsU; then, ΨD1 and ΨD2 are fpnhs-equal, represented as ΨD1=ΨD2, if and only if AD1g^=AD2g^ and ψD1g^=nψD2g^ for all g^∈G.Proposition 6.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then:(1)
IfΨD1=ΨD2 and ΨD2=ΨD3, thenΨD1=ΨD3.(2)
IfΨD1⊆˜fΨD2 and ΨD2⊆˜fΨD1thenΨD1=ΨD2.Definition 23.
LetΨD∈⊎fpnhsU; then, complement of ΨD (i.e., ΨDc˜) is an fpnhs-set given as PDc˜g^=1−ADg^ and ψDc˜g^=U∖nψDg^.Proposition 7.
LetΨD∈⊎fpnhsU; then,(1)
ΨDc˜c˜=ΨD.(2)
Ψϕc˜=ΨG˜.Proof.
The verification of the above parts can be easily obtained from Definition23.Definition 24.
LetΨD1,ΨD2∈⊎fpnhsU; then, union of ΨD1 and ΨD2, denoted by ΨD1∪˜ΨD2, is a fpnhs-set defined by(i)
AD1∪˜D2g^=maxAD1g^,AD2g^.(ii)
ψD1∪˜D2g^=ψD1g^∪˜nψD2g^, for all g^∈G.Proposition 8.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then,(1)
ΨD1∪˜ΨD1=ΨD1,(2)
ΨD1∪˜ΨΦ=ΨD1,(3)
ΨD1∪˜ΨG˜=ΨG˜,(4)
ΨD1∪˜ΨD2=ΨD2∪˜ΨD1,(5)
ΨD1∪˜ΨD2∪˜ΨD3=ΨD1∪˜ΨD2∪˜ΨD3.Proof.
(1)
TakingD1=D2 in Definition 24, we have AD1∪˜D1g^=maxAD1g^,AD1g^=AD1g^ and ψD1∪˜D1g^=ψD1g^∪˜nψD1g^, for all g^∈G, which implies ψD1∪˜D1g^=ψD1g^.Parts (2)–(4) can be proved in a similar manner.(5)
Since(20)AD1∪˜D2∪˜D3g^=maxAD1g^,AD2∪˜D3g^=maxAD1g^,maxAD2g^,AD3g^=maxAD1g^,AD2g^,AD3g^=maxAD1g^,AD2g^,AD3g^=maxmaxAD1g^,AD2g^,AD3g^=maxAD1∪˜D2g^,AD3g^=AD1∪˜D2∪˜D3g^,and(21)ψD1∪˜nD2∪˜nD3g^=ψD1g^∪˜nψD2∪˜nD3g^=ψD1g^∪˜nψD2g^∪˜nψD3g^=ψD1g^∪˜nψD2g^∪˜nψD3g^=ψD1∪˜nD2g^∪˜nψD3g^=ψD1∪˜nD2∪˜nD3g^.Definition 25.
LetΨD1,ΨD2∈⊎fpnhsU; then, intersection of ΨD1 and ΨD2, denoted by ΨD1∩˜ΨD2, is a fpnhs-set defined by(i)
AD1∩˜D2g^=minAD1g^,AD2g^.(ii)
ψD1∩˜D2g^=ψD1g^∩˜nψD2g^, for all g^∈G.Proposition 9.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then(1)
ΨD1∩˜ΨD1=ΨD1.(2)
ΨD1∩˜ΨΦ=ΨΦ.(3)
ΨD1∩˜ΨG˜=ΨD1˜.(4)
ΨD1∩˜ΨD2=ΨD2∩˜ΨD1.(5)
ΨD1∩˜ΨD2∩˜ΨΨD3=ΨD1∩˜ΨD2∩˜ΨΨD3.Proof.
(1)
TakingD1=D2 in Definition 25, we have AD1∩˜D1g^=minAD1g^,AD1g^=AD1g^ and ψD1∩˜D1g^=ψD1g^∩˜nψD1g^, for all g^∈G, which implies ψD1∩˜D1g^=ψD1g^.Parts (2)–(4) can also be shown in a similar manner.(5)
Since(22)AD1∩˜D2∩˜D3g^=minAD1g^,AD2∩˜D3g^=minAD1g^,minAD2g^,AD3g^=minAD1g^,AD2g^,AD3g^=minAD1g^,AD2g^,AD3g^=minminAD1g^,AD2g^,AD3g^=minAD1∩˜D2g^,AD3g^=AD1∩˜D2∩˜D3g^,and(23)ψD1∩˜nD2∩˜nD3g^=ψD1g^∩˜nψD2∩˜nD3g^=ψD1g^∩˜nψD2g^∩˜nψD3g^=ψD1g^∩˜nψD2g^∩˜nψD3g^=ψD1∩˜nD2g^∩˜nψD3g^=ψD1∩˜nD2∩˜nD3g^.
Note. It is pertinent to mention here that Propositions5, 6, 8, and 9 are also valid for elements of ⊎fpifhsU.Remark 2.
LetΨD∈⊎fpnhsU. If ΨD≠ΨG˜, then ΨD∪˜ΨDc˜≠ΨG˜ and ΨD∩˜ΨDc˜≠ΨΦ.Proposition 10.
LetΨD1,ΨD2∈⊎fpnhsU; then, the following De Morgan laws are valid:(1)
ΨD1∪˜ΨD2c˜=ΨD1c˜∩˜ΨD2c˜.(2)
ΨD1∩˜ΨD2c˜=ΨD1c˜∪˜ΨD2c˜.Proof.
For allg^∈G:(1)
Since(24)AD1∪˜D2c˜g^=1−AD1∪˜D2g^=1−maxAD1g^,AD2g^=min1−AD1g^,1−AD2g^=minPD1c˜g^,PD2c˜g^=PD1∩˜D2c˜g^,and(25)ψD1∪˜nD2c˜g^=U∖nψD1∪˜nD2g^=U∖nψD1g^∪˜nψD2g^=U∖nψD1g^∩˜nU∖nψD2g^=ψD1c˜g^∩n˜ψD2c˜g^=ψD1∩˜nD2c˜g^.
By adopting the same method, the second part may easily be proved.Proposition 11.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then:(1)
ΨD1∪˜ΨD2∩˜ΨD3=ΨD1∪˜ΨD2∩˜ΨD1∪˜ΨD3.(2)
ΨD1∩˜ΨD2∪˜ΨD3=ΨD1∩˜ΨD2∪˜ΨD1∩˜ΨD3.Proof.
For allg^∈G:(1)
Since(26)AD1∪˜D2∩˜D3g^=maxAD1g^,AD2∩˜D3g^=maxAD1g^,minAD2g^,AD3g^=minmaxAD1g^,AD2g^,maxAD1g^,AD3g^=minAD1∪˜D2g^,AD1∪˜D3g^=AD1∪˜D2∩˜D1∪˜D3g^,and(27)ψD1∪˜nD2∩˜nD3g^=ψD1g^∪˜nψD2∩˜nD3g^=ψD1g^∪˜nψD2g^∩˜nψD3g^=ψD1g^∪˜nψD2g^∩˜nψD1g^∪˜nψD3g^=ψD1∪˜nD2g^∩˜nψD1∪˜nD3g^=ψD1∪˜nD2∩˜nD1∪˜nD3g^.
The same method may be used to prove the second part.Definition 26.
LetΨD1,ΨD2∈⊎fpnhsU; then, OR-operation of ΨD1 and ΨD2, denoted by ΨD1⊕˜ΨD2, is a fpnhs-set defined by(i)
AD1⊕˜D2g^1,g^2=maxAD1g^1,AD2g^2.(ii)
ψD1⊕˜D2g^1,g^2=ψD1g^1∪˜nψD2g^2, for all g^1,g^2∈D1×D2.Definition 27.
LetΨD1,ΨD2∈⊎fpnhsU; then, AND-operation of ΨD1 and ΨD2, denoted by ΨD1⊗˜ΨD2, is a fpnhs-set defined by(i)
AD1⊗˜D2g^1,g^2=minAD1g^1,AD2g^2.(ii)
ψD1⊗˜D2g^1,g^2=ψD1g^1∩˜nψD2g^2, for all g^1,g^2∈D1×D2.Proposition 12.
LetΨD1,ΨD2,ΨD3∈⊎fpnhsU; then,(1)
ΨD1⊗˜ΨΦ=ΨΦ.(2)
ΨD1⊗˜ΨD2⊗˜ΨD3=ΨD1⊗˜ΨD2⊗˜ΨD3.(3)
ΨD1⊕˜ΨD2⊕˜ΨD3=ΨD1⊕˜ΨD2⊕˜ΨD3.Proof.
The above parts can easily be verified with the help of Definitions26 and 27.


### 4.1. Fuzzy Decision Set offpnhs-Set

This section presents the conceptualization of fuzzy decision set forfpnhs-set to solve decision-making problems via proposed algorithm and example.Definition 28.
LetΨD∈⊎fpnhsU; then, a fuzzy decision set of ΨD (i.e., ΨDD) is represented as(28)ΨDD=TDDu^u^:u^∈U,where TDD:U⟶I and(29)TDDu^=1U∑v∈SDTDvΓψDvu^,where • denotes set cardinality with(30)ΓψDvu^=TψDu^+IψDu^−FψDu^,u∈ΓψDv,0,u^∉ΓψDv.
WheneverΨDD has been established, it might be relevant to select the suitable single substitute from the available options. Therefore, the following algorithm may lead to the final decisive selection.
Figure9 presents the flowchart of Algorithm 2.Algorithm 2: Optimal selection of hand sanitizer by using fpnhs-set.
StartInput:(1)
ConsiderU as universe of discourse(2)
ConsiderB as set of n parameters(3)
Classify n parameters into disjoint parametric-valued setsB1,B2,B3,…,BnConstruction:(4)
Determineℚ=B1×B2×B3×⋯×B5(5)
Takeℝ⊆ℚComputation:(6)
DetermineD=TDpi/pi:TDpi∈I,pi∈ℝ,(7)
FindψDpi,(8)
ComputeΨD over U,(9)
ComputeΨDD,Output:(10)
Choose the maximum ofTDDu^,EndTable 5
Degrees of membershipTDpi.
TDpiDegreeTDpiDegreeTDpiDegreeTDpiDegreeTDp10.1TDp50.5TDp90.9TDp130.35TDp20.2TDp60.6TDp100.16TDp140.75TDp30.3TDp70.7TDp110.25TDp150.65TDp40.4TDp80.8TDp120.45TDp160.85Table 6
Approximate functionsψDpi.
piψDpipiψDpip10.2,0.1,0.2/ℍ1,0.3,0.2,0.1/ℍ2p90.4,0.3,0.2/ℍ2,0.6,0.4,0.3/ℍ7,0.5,0.4,0.3/ℍ8p20.1,0.2,0.1/ℍ1,0.5,0.4,0.3/ℍ2,0.1,0.4,0.3/ℍ3p100.2,0.1,0.2/ℍ6,0.6,0.4,0.5/ℍ7,0.4,0.3,0.2/ℍ8p30.4,0.3,0.1/ℍ2,0.5,0.4,0.3/ℍ3,0.6,0.3,0.2/ℍ4p110.5,0.4,0.3/ℍ2,0.6,0.3,0.2/ℍ4,0.7,0.2,0.3/ℍ6p40.6,0.2,0.3/ℍ4,0.7,0.3,0.4/ℍ5,0.8,0.1,0.4/ℍ6p120.7,0.2,0.5/ℍ2,0.8,0.1,0.5/ℍ3,0.9,0.1,0.7/ℍ6p50.2,0.1,0.1/ℍ6,0.1,0.2,0.1/ℍ7,0.4,0.3,0.1/ℍ8p130.2,0.1,0.2/ℍ3,0.4,0.3,0.2/ℍ5,0.6,0.1,0.4/ℍ7p60.4,0.2,0.3/ℍ2,0.3,0.4,0.3/ℍ3,0.4,0.5,0.3/ℍ4p140.2,0.5,0.4/ℍ1,0.5,0.4,0.6/ℍ3,0.6,0.2,0.5/ℍ5p70.2,0.3,0.4/ℍ1,0.3,0.4,0.4/ℍ3,0.4,0.3,0.4/ℍ5p150.6,0.3,0.3/ℍ5,0.4,0.3,0.4/ℍ7,0.2,0.4,0.5/ℍ8p80.1,0.4,0.3/ℍ2,0.3,0.5,0.6/ℍ3,0.5,0.4,0.7/ℍ7p160.3,0.6,0.5/ℍ4,0.5,0.4,0.8/ℍ5,0.7,0.1,0.6/ℍ6Table 7
Membership valuesTDDℍi.
ℍiTDDℍiℍiTDDℍiℍiTDDℍiℍiTDDℍiℍ10.0431ℍ30.1588ℍ50.1656ℍ70.1588ℍ20.1825ℍ40.1606ℍ60.0964ℍ80.1231Figure 9
Decision-making algorithm forfpnhs-set.Figure 10
Graphical representation of Table5.Figure 11
Fuzzy decision system on FPNHS-set.Example 4.
Input and Construction Stages (1–5):Considering the data given in Example2, we haveComputation Stage (6–9):(6)
The graphical depiction of Table5 can be seen in Figure 10.From Table5, we can construct D as(31)D=0.1p1,0.2p2,0.3p3,0.4p4,0.5p5,0.6p6,0.7p7,0.8p8,0.9p9,0.16p100.25p11,0.45p12,0.35p13,0.75p14,0.65p15,0.85p16.(7)
Table6 presents ψDpi corresponding to each element of G.(8)
ΨD can be constructed with the help of step 6 and step 7 in the same way as step 8 of Section 3.(9)
From Table7, we can construct ΨDD as(32)ΨDD=0.0431ℍ1,0.1825ℍ2,0.1588ℍ3,0.1606ℍ4,0.1656ℍ5,0.0964ℍ6,0.1588ℍ7,0.1231ℍ8.Please see Figure11 for its pictorial depiction.Output Stage:(10)
Since maximum ofTDDℍi is 0.1825, the hand sanitizer ℍ2 is selected.


## 4.1. Fuzzy Decision Set offpnhs-Set

This section presents the conceptualization of fuzzy decision set forfpnhs-set to solve decision-making problems via proposed algorithm and example.Definition 28.
LetΨD∈⊎fpnhsU; then, a fuzzy decision set of ΨD (i.e., ΨDD) is represented as(28)ΨDD=TDDu^u^:u^∈U,where TDD:U⟶I and(29)TDDu^=1U∑v∈SDTDvΓψDvu^,where • denotes set cardinality with(30)ΓψDvu^=TψDu^+IψDu^−FψDu^,u∈ΓψDv,0,u^∉ΓψDv.
WheneverΨDD has been established, it might be relevant to select the suitable single substitute from the available options. Therefore, the following algorithm may lead to the final decisive selection.
Figure9 presents the flowchart of Algorithm 2.Algorithm 2: Optimal selection of hand sanitizer by using fpnhs-set.
StartInput:(1)
ConsiderU as universe of discourse(2)
ConsiderB as set of n parameters(3)
Classify n parameters into disjoint parametric-valued setsB1,B2,B3,…,BnConstruction:(4)
Determineℚ=B1×B2×B3×⋯×B5(5)
Takeℝ⊆ℚComputation:(6)
DetermineD=TDpi/pi:TDpi∈I,pi∈ℝ,(7)
FindψDpi,(8)
ComputeΨD over U,(9)
ComputeΨDD,Output:(10)
Choose the maximum ofTDDu^,EndTable 5
Degrees of membershipTDpi.
TDpiDegreeTDpiDegreeTDpiDegreeTDpiDegreeTDp10.1TDp50.5TDp90.9TDp130.35TDp20.2TDp60.6TDp100.16TDp140.75TDp30.3TDp70.7TDp110.25TDp150.65TDp40.4TDp80.8TDp120.45TDp160.85Table 6
Approximate functionsψDpi.
piψDpipiψDpip10.2,0.1,0.2/ℍ1,0.3,0.2,0.1/ℍ2p90.4,0.3,0.2/ℍ2,0.6,0.4,0.3/ℍ7,0.5,0.4,0.3/ℍ8p20.1,0.2,0.1/ℍ1,0.5,0.4,0.3/ℍ2,0.1,0.4,0.3/ℍ3p100.2,0.1,0.2/ℍ6,0.6,0.4,0.5/ℍ7,0.4,0.3,0.2/ℍ8p30.4,0.3,0.1/ℍ2,0.5,0.4,0.3/ℍ3,0.6,0.3,0.2/ℍ4p110.5,0.4,0.3/ℍ2,0.6,0.3,0.2/ℍ4,0.7,0.2,0.3/ℍ6p40.6,0.2,0.3/ℍ4,0.7,0.3,0.4/ℍ5,0.8,0.1,0.4/ℍ6p120.7,0.2,0.5/ℍ2,0.8,0.1,0.5/ℍ3,0.9,0.1,0.7/ℍ6p50.2,0.1,0.1/ℍ6,0.1,0.2,0.1/ℍ7,0.4,0.3,0.1/ℍ8p130.2,0.1,0.2/ℍ3,0.4,0.3,0.2/ℍ5,0.6,0.1,0.4/ℍ7p60.4,0.2,0.3/ℍ2,0.3,0.4,0.3/ℍ3,0.4,0.5,0.3/ℍ4p140.2,0.5,0.4/ℍ1,0.5,0.4,0.6/ℍ3,0.6,0.2,0.5/ℍ5p70.2,0.3,0.4/ℍ1,0.3,0.4,0.4/ℍ3,0.4,0.3,0.4/ℍ5p150.6,0.3,0.3/ℍ5,0.4,0.3,0.4/ℍ7,0.2,0.4,0.5/ℍ8p80.1,0.4,0.3/ℍ2,0.3,0.5,0.6/ℍ3,0.5,0.4,0.7/ℍ7p160.3,0.6,0.5/ℍ4,0.5,0.4,0.8/ℍ5,0.7,0.1,0.6/ℍ6Table 7
Membership valuesTDDℍi.
ℍiTDDℍiℍiTDDℍiℍiTDDℍiℍiTDDℍiℍ10.0431ℍ30.1588ℍ50.1656ℍ70.1588ℍ20.1825ℍ40.1606ℍ60.0964ℍ80.1231Figure 9
Decision-making algorithm forfpnhs-set.Figure 10
Graphical representation of Table5.Figure 11
Fuzzy decision system on FPNHS-set.Example 4.
Input and Construction Stages (1–5):Considering the data given in Example2, we haveComputation Stage (6–9):(6)
The graphical depiction of Table5 can be seen in Figure 10.From Table5, we can construct D as(31)D=0.1p1,0.2p2,0.3p3,0.4p4,0.5p5,0.6p6,0.7p7,0.8p8,0.9p9,0.16p100.25p11,0.45p12,0.35p13,0.75p14,0.65p15,0.85p16.(7)
Table6 presents ψDpi corresponding to each element of G.(8)
ΨD can be constructed with the help of step 6 and step 7 in the same way as step 8 of Section 3.(9)
From Table7, we can construct ΨDD as(32)ΨDD=0.0431ℍ1,0.1825ℍ2,0.1588ℍ3,0.1606ℍ4,0.1656ℍ5,0.0964ℍ6,0.1588ℍ7,0.1231ℍ8.Please see Figure11 for its pictorial depiction.Output Stage:(10)
Since maximum ofTDDℍi is 0.1825, the hand sanitizer ℍ2 is selected.


## 5. Comparison Analysis

There are many cases where consideration of attributes only is not sufficient; all available distinct attributes are further partitioned into their respective disjoint attributive sets. Decision-making techniques based on existing soft set-like models are inadequate for such cases. Therefore, our proposed models not only emphasize the due status of such partitioning of attributes, but also enable the decision-makers to deal with daily-life problems with great ease. Tables8 and 9 present a clear comparison of our proposed models with some existing models under the evaluating indicators MD (membership degree), NMD (nonmembership degree), ID (indeterminacy degree), SAAF (single-argument approximate function), and MAAF (multiargument approximate function). Figure 12 depicts the comparison of reduced fuzzy values obtained from our proposed structures.Table 8
Comparison with existing models under consideration of data given in Examples 2, 4, Definitions 7, and 18.
AuthorsStructureDomain parameterizationRange approximationType of approximate functionRanking underhs-set environmentÇağman et al. [43]fpfs-setFuzzy set parameterizationFuzzy soft setSingle-argumentInadequateDeli and Çağman [44]ifps-setIntuitionistic fuzzy set parameterizationFuzzy soft setSingle-argumentInadequateJoshi et al. [47]ifpfs-setIntuitionistic fuzzy set parameterizationFuzzy soft setSingle-argumentInadequateKaraaslan [48]ifpifs-setIntuitionistic fuzzy set parameterizationIntuitionistic fuzzy soft setSingle-argumentInadequateRiaz and Hashmi [49]fpfs-setFuzzy set parameterizationFuzzy soft setSingle-argumentInadequateZhu and Zhan [50]fpfs-setFuzzy set parameterizationFuzzy soft setSingle-argumentInadequateProposed modelfpifhs-setFuzzy set parameterizationIntuitionistic fuzzy hypersoft setMultiargumentH6>H3=H5>H2>H4>H7>H1>H8Proposed modelfpnhs-setFuzzy set parameterizationNeutrosophic hypersoft setMultiargumentH2>H5>H4>H3=H7>H8>H6>H1Table 9
Comparison with existing models under appropriate features.
AuthorsStructureMDNMDIDSAAFMAAFÇağman et al. [43]fpfs-set✔××✔×Deli and Çağman [44]ifps-set✔✔×✔×Joshi et al. [47]ifpfs-set✔✔×✔×Karaaslan [48]ifpifs-set✔✔×✔×Riaz and Hashmi [49]fpfs-set✔××✔×Zhu and Zhan [50]fpfs-set✔××✔×Proposed modelfpifhs-set✔✔×✔✔Proposed modelfpnhs-set✔✔✔✔✔Figure 12
Comparison of membership values of fuzzy decision sets onfpifhs-set and fpnhs-set.


## 6. Discussion

Decision-makers consistently face a type of uncertainty and the decision taken by overlooking uncertainty might have some degree of tendency. Indeterminacy and uncertainty are both interconnected and interrelated. The proposed work has elaborated (for example, see Figure12) how the results are affected by the omission or consideration of some uncertain components while dealing with the real-world problems under any decisive system. Two models fpifhs-set and fpnhs-set have been proposed and characterized. The latter one is very useful, flexible, and reliable in dealing with problems having uncertain information and data. The following discussion shows the generalization of fpnhs-set as it fulfills all the characteristics, features, and properties of many existing soft set-like models. In Definition 18:(i)
IfℕU is replaced with IFU, it becomes fpifhs-set.(ii)
IfℕU is replaced with FU, it behaves as fpfhs-set.(iii)
IfℕU is replaced with ℙU, it converts to fphs-set.(iv)
IfG=S1×S2×S3×⋯⋅×Sn is omitted and only set of parameters is considered, then it is called fpns-set.(v)
If only set of parameters is considered after omittingG=S1×S2×S3×⋯⋅×Sn and replacing ℕU with IFU, then it is called fpifs-set.(vi)
If only set of parameters is considered after omittingG=S1×S2×S3×⋯⋅×Sn and replacing ℕU with FU, then it is known as fpfs-set.(vii)
IfADg^/g^ is replaced with g^, then it is called nhs-set.(viii)
IfADg^/g^ is replaced with g^, and ℕU is replaced with IFU, then it is called ifhs-set.(ix)
IfADg^/g^ is replaced with g^, and ℕU is replaced with FU, then it is called fhs-set.Similarly,fpnhs-set transforms to s-set by ignoring all uncertain components and considering only set of attributes. From Figure 13, it is obvious that the proposed structure is the most generalized structure satisfying all the main characteristics of existing relevant structures.Figure 13
Generalization of the proposed structure.


### 6.1. Advantages of Proposed Research

Now, some advantages of the proposed structure (i.e.,fpnhs-set) are presented below:(i)
The proposed technique took the importance of the concept of parameterization along with thenhs-set for dealing with decision-making problems. The considered parameterized fuzzy degree mirrors the possibility of the existence of the level of acknowledgment and excusal; along these lines, this association has marvelous prospective in the legitimate representation inside the space of computational incursions.(ii)
Due to focus on deep observation of parameters and respective attribute-valued sets, the proposed model may assist the decision-makers to have flexible and reliable decisions through decision-making.(iii)
It validates all the qualities and properties of the predefined models so it is not irrational to consider it as the generalized version of these models.The advantage of the proposed structure can easily be judged from Tables8 and 9. The comparison is evaluated on the basis of two different aspects:(1)
Main features discussed in the study (see Table8).(2)
Features like MD (membership degree), NMD (nonmembership degree), ID (indeterminacy degree), SAAF (single-argument approximate function), and MAAF (multiargument approximate function) (see Table9).


## 6.1. Advantages of Proposed Research

Now, some advantages of the proposed structure (i.e.,fpnhs-set) are presented below:(i)
The proposed technique took the importance of the concept of parameterization along with thenhs-set for dealing with decision-making problems. The considered parameterized fuzzy degree mirrors the possibility of the existence of the level of acknowledgment and excusal; along these lines, this association has marvelous prospective in the legitimate representation inside the space of computational incursions.(ii)
Due to focus on deep observation of parameters and respective attribute-valued sets, the proposed model may assist the decision-makers to have flexible and reliable decisions through decision-making.(iii)
It validates all the qualities and properties of the predefined models so it is not irrational to consider it as the generalized version of these models.The advantage of the proposed structure can easily be judged from Tables8 and 9. The comparison is evaluated on the basis of two different aspects:(1)
Main features discussed in the study (see Table8).(2)
Features like MD (membership degree), NMD (nonmembership degree), ID (indeterminacy degree), SAAF (single-argument approximate function), and MAAF (multiargument approximate function) (see Table9).


## 7. Conclusion

The key features of this work can be summarized as follows:(1)
The fundamental theory offpifhs-set is developed, and its decision-support system is constructed. A real-world problem (an optimum selection of hand sanitizer) is discussed with the support of the proposed algorithm and decision system of fpifhs-set.(2)
The rudiments offpnhs-set are characterized, and its decision-making based system is developed. A real-life problem (an optimum selection of hand sanitizer) is studied with the help of proposed algorithm and decision system of fpnhs-set.(3)
Both proposed models,fpifhs-set and fpnhs-set, are compared professionally via inter-cum-intra-strategy with some existing relevant models in view of important evaluating features.(4)
The particular cases of the proposed modelsfpifhs-set and fpnhs-set are discussed with the generalization of these structures.(5)
Although the proposed structure is generalized and reliable for decision-making scenarios, it is flexible due to some limitations; for example, it is inadequate for the situation when the inclusion of nonmembership grade and indeterminate grade in the domain of multiargument approximate function is mandatory. Therefore, future work may include the following:(i)
The extension of this study to develop intuitionistic fuzzy parameterized neutrosophic hypersoft set and neutrosophic parameterized neutrosophic hypersoft set to tackle the situations having mandatory inclusion of nonmembership grade and indeterminate grade in the domain of multiargument approximate function.(ii)
The application of this study to discuss real-world problems using multicriteria decision-making approaches like TOPSIS, PROMETHEE, and MULTIMORA.(iii)
The application of this study to discuss real-world problems (e.g., pattern recognition) using similarity measures, like cosine similarity, cotangent similarity, and Dice similarity, and entropy measures. [51].



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*Source: 6229947-2022-03-24.xml*