# Multicriteria Decision-Making Approach for Pythagorean Fuzzy Hypersoft Sets’ Interaction Aggregation Operators

Authors: Rana Muhammad Zulqarnain; Imran Siddique; Rifaqat Ali; Fahd Jarad; Aiyared Iampan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we examine the multicriteria decision-making (MCDM) difficulties for Pythagorean fuzzy hypersoft sets (PFHSSs). The PFHSSs are a suitable extension of the Pythagorean fuzzy soft sets (PFSSs) which deliberates the parametrization of multi-subattributes of considered parameters. It is a most substantial notion for describing fuzzy information in the decision-making (DM) procedure to accommodate more vagueness comparative to existing PFSSs and intuitionistic fuzzy hypersoft sets (IFHSSs). The core objective of this study is to plan some innovative operational laws considering the interaction for Pythagorean fuzzy hypersoft numbers (PFHSNs). Also, based on settled interaction operational laws, two aggregation operators (AOs) i.e., Pythagorean fuzzy hypersoft interaction weighted average (PFHSIWA) and Pythagorean fuzzy hypersoft interaction weighted geometric (PFHSIWG) operators for PFHSSs operators have been presented with their fundamental properties. Furthermore, an MCDM technique has been established using planned interaction AOs. To ensure the strength and practicality of the developed MCDM method, a mathematical illustration has been presented. The usefulness, influence, and versatility of the developed method have been demonstrated via comparative analysis with the help of some conventional studies.


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## Body

## 1. Introduction

Multicriteria decision-making (MCDM) is a prerequisite for decision science. The goal is to distinguish between the most essential of the possible choices. The decision maker must assess the selection specified by different types of diagnostic circumstances such as intervals and numbers. However, in numerous circumstances, it is difficult for one person to do it because of various uncertainties within the data. One is because of the shortcoming of professional knowledge or contraventions. Hence, to measure given hazards and think about the method, a series of theories have been proposed. Zadeh presented the theory of fuzzy sets (FSs) [1] to resolve the complex problem of anxiety along with ambiguity. Usually, we need to observe membership as a nonmembership degree to indicate objects for which FSs cannot handle. To conquer the current concern, Atanassov anticipated the concept of intuitionistic fuzzy sets (IFSs) [2]. Atanassov’s IFSs competently deal with insufficient data because of membership and nonmembership values, but IFSs are not able to influence incompatible and imprecise information. The theories declared over had been fairly advised by specialists, along with the sum up of two membership and nonmembership values cannot overreach one because the above work is regarded as to visualize the environment of linear inequality between the degree of membership (MD) and the degree of nonmembership (NMD). If the experts considered the MD and NMD such as MD = 0.4 and NDM = 0.7, then 0.4+0.7≰1 and IFSs cannot handle the situation. Yager [3, 4] prolonged the idea of IFSs to Pythagorean fuzzy sets (PFSs) to overcome the above-discussed difficulties by amending MD+NMD≤1 to MD2+NMD2≤1. Succeeding the construction of PFSs, Zhang and Xu [5] planned operational rules for PFSs and set up the DM strategy to address the MCDM problem. Sanam et al. [6] presented the induced intuitionistic fuzzy Einstein hybrid AOs and discussed their desired properties. Wang and Li [7] offered some novel operational laws and AOs for PFSs considering the interaction with their desirable properties. Gao et al. [8] prolonged the notion of PFSs and developed numerous AOs considering the interaction. They also established a multiattribute decision-making (MADM) approach based on their established operators.Wei [9] developed some novel operational laws for Pythagorean fuzzy numbers (PFNs) considering the interaction and proposed AOs for PFSs based on their developed operational laws. Talukdar et al. [10] utilized the linguistic PFSs for medical diagnoses and introduced some distance measures and accuracy function. They also proposed a DM technique to solve multiple criteria group decision-making (MCGDM) complications utilizing PFNs. Wang et al. [11] extended the concept of PFSs, proposed the interactive Hamacher AOs, and established a MADM method to resolve DM complications. Ejegwa et al. [12] established a correlation measure for IFSs and presented an MCDM approach. Peng and Yang [13] offered various essential operations for PFSs along with their basic characteristics. Garg [14] proposed some AOs for PFSs based on his developed logarithmic operational laws. Arora and Garg [15] introduced prioritized AOs for linguistic IFSs based on their developed operational laws. Ma and Xu [16] established novel AOs for PFSs and offered the comparison laws for PFNs. Current theories and their progressed DM strategies have been utilized in various aspects of life. However, these theories fail to cope with the parameters of alternatives.The above-presented theories with their DM techniques are used in many fields of life such as medical diagnoses, artificial intelligence, and economics. But these theories have some limitations because of their inability with the parameterization tool. Molodtsov [17] introduced the notion of soft sets (SSs) to accommodate the abovementioned drawbacks considering the parameterization of the alternatives. Maji et al. [18] prolonged the idea of SSs with several necessary operations along with their appropriate possessions and established a DM method to resolve DM issues utilizing their developed operations [19]. Maji et al. [20] merged the two existing theories such as FSs and SSs and offered the concept of fuzzy soft sets (FSSs) with some elementary operations and their desired properties. Maji et al. [21] extended the notion of FSSs and proposed the idea of intuitionistic fuzzy soft sets with some operations and properties. Xu [22] introduced a method for IFSs to compare intuitionistic fuzzy numbers utilizing score and accuracy functions. Xu and Yager [23] proposed the weighted average and ordered weighted average operators for IFSs with their examples and properties. They also presented a DM approach to solve MADM complications utilizing their developed operators. Garg and Arora [24] proposed the generalized form of IFSSs with AOs and established a DM methodology based on their developed AOs to resolve DM issues. Garg and Arora [25] developed the correlation coefficient (CC) and weighted correlation coefficient (WCC) for IFSSs. They also presented the TOPSIS methodology to resolve MADM issues utilizing their developed correlation measures. Zulqarnain et al. [26] extended the notion of interval-valued IFSSs and proposed AOs for interval-valued IFSSs. They also presented the CC and WCC for interval-valued IFSSs and constructed the TOPSIS approach to resolve the MADM complications based on their presented correlation measures.Peng et al. [27] introduced the theory of PFSSs by merging two existing theories such as PFSs and SSs. They also presented some fundamental operations of PFSSs and discussed their desirable properties. Athira et al. [28] extended the notion of PFSSs, introduced some novel distance measures for PFSSs, and established a DM method based on presented distance measures to solve complicated problems. Zulqarnain et al. [29] developed the operational laws for Pythagorean fuzzy soft numbers (PFSNs) and proposed the AOs for PFSNs. They also presented a MADM method to resolve DM concerns using their developed AOs. Riaz et al. [30] defined the concept of m polar PFSSs and developed the TOPSIS method to solve MCGDM problems. Riaz et al. [31] presented the similarity measures for PFSSs and discussed their essential properties. They also proposed the weighted AOs for m-polar PFSs [32] and established a decision-making approach to solve DM concerns. Zulqarnain et al. [33] extended the idea of PFSSs and developed the TOPSIS method based on the CC. They also presented an MCGDM approach and utilized their developed approach for the selection of suppliers in green supply chain management. Mehmood et al. [34] proposed the AOs for T-spherical fuzzy sets and developed a DM approach to solving MADM issues. Wang and Garg [35] introduced some novel operational laws considering the interaction and established the AOs based on their developed rules. Batool et al. [36] introduced the TOPSIS method for Pythagorean probabilistic hesitant fuzzy sets and entropy measures under considered environment. Ullah et al. [37] developed the complex PFSs with some novel distance measures and their desirable properties. Hussain et al. [38] introduced the soft rough PFSs and Pythagorean fuzzy soft rough set with some necessary operators and properties.The existing studies are unable to accommodate the situation when any parameters of a set of attributes have corresponding subattributes. Smarandache [39] developed the concept of hypersoft sets (HSSs) which replace the function f of a parameter with a multi-subattribute, that is, characterized on the Cartesian product of n attributes. The developed HSS competently deals with the uncertainty and vagueness comparative to SS. He also presented many other extensions of HSS such as crisp HSS, fuzzy HSS, intuitionistic fuzzy HSS, neutrosophic HSS, and plithogenic HSS. Zulqarnain et al. [40] developed the theory of neutrosophic hypersoft matrices with some logical operators. They also proposed the MADM approach to solve DM concerns. The authors presented the generalized AOs for NHSSs [41]. Zulqarnain et al. [42] developed the CC and WCC for IFHSSs and proposed the TOPSIS method using developed CC. Zulqarnain et al. [43] proposed some AOs and CC for PFHSSs and discussed their properties. They also developed the TOPSIS approach for PFHSSs based on their presented CC. However, the above-discussed theories only deal with the uncertainty utilizing MD and NMD of subattributes. If experts consider MD = 0.6 and NDM = 0.7, then 0.6+0.7≥1 of any subattribute of the alternatives. We will check that it cannot be addressed by the above strategies. To overwhelm the above restrictions, we introduced some AOs for PFHSSs by modifying the condition Tℱdˇδ+Jℱdˇδ≤1 to Tℱdˇδ2+Jℱdˇδ2≤1. The main purpose of the succeeding study is to originate new AOs for the PFHSSs considering interactions, which may also observe the assertions of PFHSs. Moreover, an MCDM method with a numerical example has been presented which shows the effectiveness of the planned methodology.Supplier selection and valuation are a crucial prospect of business routine. Due to variations in management strategies, the selection of suppliers is considered from multiple perspectives, which included environmental and social necessities. Therefore, in the literature, this query is stated as a reference question for MCGDM as a sustainable supplier selection. Continuing, there are several papers [44–47] that carried the MCDM approach for the selection of sustainable suppliers according to relevant data and considerations that appropriately reflect the preferences of decision makers. However, all the above methods are not appropriate for summarizing the abovementioned methodologies and cannot deliberate the interaction among Mem and NMem functions. Particularly, we can say that the influence of other levels of Mem or NMem on the conforming geometric or average AOs does not have any influence on the aggregation process. In addition, it has been stated from the above-discussed models that the overall Mem (NMem) function level is independent of its corresponding NMem (Mem) function level. So, the consequences corresponding to those models are not favorable, so no reasonable order of preference is given for alternatives. Therefore, how to add these PFHSNs through interaction relations is an interesting topic. To solve this problem, in this article, we are going to develop some interaction AOs such as PFHSIWA and PFHSIWG operators for PFHSSs. An algorithm is planned to resolve the DM problem based on our established operators. A numerical example has been presented to ensure the practicality of the developed DM approach.The rest of the research can be summarized as follows: In Section2, we presented the necessary concepts such as SSs, FSSs, HSSs, IFHSSs, and PFHSSs which can support us to construct the subsequent research organization. In Section 3, we defined some novel operational laws for PFHSSs considering interaction and developed some AOs based on interaction operational laws such as PFHSIWA and PFHSIWG operators using presented operational laws with their desirable properties. In Section 4, an MCDM method is developed utilizing the proposed operators. A numerical example is provided to ensure the implementation of the setup MCDM method. Moreover, we used some of the existing methods to present comparative analysis with our planned approach. Also, we present the benefits, simplicity, flexibility, as well as effectiveness of the planned method in Section 5, and we organized a comprehensive debate and comparison among some available techniques and our established methodology.


## 2. Preliminaries

In this section, we recollect some fundamental notions such as SSs, FSSs, HSSs, IFHSSs, and PFHSSs.Definition 1 (see [17]).
LetU 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 SSs 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.Definition 2 (see [20]).
LetU 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 FSSs over U, and its mapping can be stated as follows:(3)Ϝ:A⟶FU.Definition 3 (see [39]).
LetU be a universe of discourse, PU be a power set of U, k=k1,k2,k3,…,kn, n ≥ 1, and Ki represents the set of attributes and their corresponding subattributes such as Ki∩Kj=φ, where i ≠ j for each n ≥ 1 and iandj∈1,2,3,…,n. Assume K1×K2×K3×⋯×Kn=A⋯=d1h×d2k×⋯×dnl is a collection of subattributes, where 1≤h≤α, 1≤k≤β, 1≤l≤γ, and α,β,andγ∈ℕ. Then, the pair (ℱ,K1×K2×K3×⋯×Kn=ℱ,A⋯ is known as HSSs, defined as follows:(4)ℱ:K1×K2×K3×⋯×Kn=A⋯⟶PU.
It is also defined as(5)ℱ,A⋯=dˇ,ℱA⋯dˇ:dˇ∈A⋯,ℱA⋯dˇ∈PU.Definition 4 (see [39]).
LetU be a universe of discourse, PU be a power set of U, k=k1,k2,k3,…,kn, n ≥ 1, and Ki represents the set of attributes and their corresponding subattributes such as Ki∩Kj=φ where i ≠ j for each n ≥ 1 and iandj∈1,2,3,…,n. Assume K1×K2×K3×⋯×Kn=A⋯=d1h×d2k×⋯×dnl is a collection of subattributes, where 1≤h≤α, 1≤k≤β, 1≤l≤γ, and α,β,andγ∈ℕ, and let IFSU be a collection of all fuzzy subsets over U. Then, the pair (ℱ,K1×K2×K3×⋯×Kn=ℱ,A⋯ is known as IFHSSs, defined as follows:(6)ℱ:K1×K2×K3×⋯×Kn=A⋯⟶IFSU.
It is also defined asℱ,A⋯=dˇ,ℱA⋯dˇ:dˇ∈A⋯,ℱA⋯dˇ∈IFSU∈0,1, where ℱA⋯dˇ=δ,Tℱdˇδ,Jℱdˇδ:δ∈U, where Tℱdˇδ and Jℱdˇδ signify the Mem and NMem values of the attributes:(7)Tℱdˇδ,Jℱdˇδ∈0,1,0≤Tℱdˇδ+Jℱdˇδ≤1.Remark 1.
IfTFdˇδ2+JFdˇδ2≤1 and Tℱdˇδ+Jℱdˇδ≤1 are satisfied, then PFHSSs are reduced to IFHSSs [42].
The PFHSNsℱδidˇj=Tℱdˇjδi,Jℱdˇjδi|δi∈U can be express as Jdˇij=Tℱdˇij,Jℱdˇij. To compute the alternatives, ranking score function of Jdˇij can be defined as follows:(8)SJdˇij=Tℱdijˇ2−Jℱdijˇ2,SJdˇij∈−1,1.
But, sometimes the scoring function such asJdˇ11=0.4,0.7 and Jdˇ12=0.5,0.8 cannot compare two PFHSNs. It is impossible to claim that which alternative is most suitable SJdˇ11=0.3=SJdˇ12. To overcome such difficulties, we need to introduce the accuracy function as follows:(9)HJdˇij=Tℱdijˇ2+Jℱdijˇ2,HJdˇij∈0,1.
Hence, some rules have been introduced in the following for the comparison among two PFHSNsJdˇij and Tdˇij.(1)
IfSJdˇij>STdˇij, then Jdˇij>Tdˇij(2)
IfSJdˇij=STdˇij, thenIfHJdˇij>HTdˇij, then Jdˇij>TdˇijIfHJdˇij=HTdˇij, then Jdˇij=Tdˇij
Observe that the overall difference between PFHSNs and IFHSNs lies in their distinguishing limits. The Pythagorean membership degree area is larger than either the intuitionistic membership degree area. PFHSNs cannot only model IFHSNs’ ability to capture DM scenarios anywhere the sum of Mem as well as NMem of subattributes of the considered parameters is equal to or less than 1 but it is also unable to handle the circumstances where IFHSNs are not able to characterize the sum of Mem as well as NMem of multi-subattributes of the considered attributes exceeding 1. On the contrary, PFHSNs accommodate more uncertainty considering Mem as well as NMem of multi-subattributes of the considered attributes, and the sum of their squares is equal to or less than 1.Definition 5 (see [43]).
LetJdˇk=Tdˇk,Jdˇk, Jdˇ11=Tdˇ11,Jdˇ11, and Jdˇ12=Tdˇ12,Jdˇ12 be three PFHSNs and α be a positive real number; by algebraic norms, we have(1)
Jdˇ11⊕Jdˇ12=Tdˇ112+Tdˇ122−Tdˇ112Tdˇ122,Jdˇ11Jdˇ12(2)
Jdˇ11⊗Jdˇ12=Tdˇ11Tdˇ12,Jdˇ112+Jdˇ122−Jdˇ112Jdˇ122(3)
αJdˇk=1−1−Tdˇk2α,Jdˇkα(4)
Jdˇkα=Tdˇkα,1−1−Jdˇk2α
For the collection of PFHSNsJdˇij, where i=1,2,…,nandj=1,2,…,m, Ω=Ω1,Ω1,…,ΩnT and γ=γ1,γ2,γ3,…,γmT be weight vectors for experts and attributes Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1, respectively. Zulqarnain et al. [43] presented the averaging and geometric AOs as follows:(10)PFHSWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏I=1n1−Tdˇij2Ωiγj,∏j=1m∏I=1nJdˇijΩiγj,(11)PFHSWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m∏I=1nTdˇijΩiγj,1−∏j=1m∏I=1n1−Jdˇij2Ωiγj.


## 3. Interaction Aggregation Operators for Pythagorean Fuzzy Hypersoft Numbers

In this section, we introduce interaction AOs for PFHSNs. In it, some fundamental properties have been discussed based on defined interaction PFHSIWA and PFHSIWG operators for PFHSNs.


### 3.1. Interaction Operational Laws for PFHSNs

Definition 6.
LetJdˇk=Tdˇk,Jdˇk, Jdˇ11=Tdˇ11,Jdˇ11, and Jdˇ12=Tdˇ12,Jdˇ12 be three PFHSNs and α be a positive real number; by algebraic norms, considering the interaction, we have(1)
Jdˇ11⊕Jdˇ12=Tdˇ112+Tdˇ122−Tdˇ112Tdˇ122,Jdˇ112+Jdˇ122−Jdˇ112Jdˇ122−Jdˇ112Tdˇ122−Tdˇ112Jdˇ122(2)
Jdˇ11⊗Jdˇ12=Tdˇ112+Tdˇ122−Tdˇ112Tdˇ122−Tdˇ112Jdˇ122−Jdˇ112Tdˇ122,Jdˇ112+Jdˇ122−Jdˇ112Jdˇ122(3)
αJdˇk=1−1−Tdˇk2α,1−Tdˇk2α−1−Tdˇk2+Jdˇk2α(4)
Jdˇkα=1−Jdˇk2α−1−Tdˇk2+Jdˇk2α,1−1−Jdˇk2α
Based on the above-defined operational laws, now we introduce some interaction AOs for PFHSNs’Δ.Definition 7.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs and Ωi and γj represent the weights of expert’s and multi-subattributes along with stated conditions Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1. Then, PFHSIWA: Δn⟶Δ is defined as follows:(12)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕j=1mγj⊕i=1nΩiJdˇij.Theorem 1.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs, where i=1,2,…,nandj=1,2,…,m. Then, the attained aggregated values using equation (12) is also a PFHSN and(13)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2Ωiγj,∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,where Ωi and γj represent the expert’s and subattributes’ weights with certain circumstances Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1.Proof.
The PFHSIWA operator can be proved using the principle of mathematical induction as follows:Forn=1, we get Ω1=1. Then, we have(14)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕j=1mγjJdˇij,PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m1−Tdˇij2γj,∏j=1m1−Tdˇij2γj−∏j=1m1−Tdˇij2+Jdˇij2γj,=1−∏j=1m∏i=111−Tdˇij2Ωiγj,∏j=1m∏i=111−Tdˇij2Ωiγj−∏j=1m∏i=111−Tdˇij2+Jdˇij2Ωiγj.Form=1, we get γ1=1. Then, we have(15)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕i=1nΩiJdˇ11=1−∏i=1n1−Tdˇi12Ωi,∏1=1n1−Tdˇi12Ωi−∏i=1n1−Tdˇi12+Jdˇi12Ωi=1−∏j=11∏i=1n1−Tdˇij2Ωiγj,∏j=11∏i=1n1−Tdˇij2Ωiγj−∏j=11∏i=1n1−Tdˇij2+Jdˇij2Ωiγj.
The above justification shows that equation (13) holds for n=1 and m=1. Now, assume that equation (13) also holds for m=β1+1, n=β2, m=β1, and n=β2+1:(16)⊕j=1β1+1γj⊕i=1β2ΩiJdˇij=1−∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj,∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj,⊕j=1β1γj⊕i=1β2+1ΩiJdˇij=1−∏j=1β1∏i=1β2+11−Tdˇij2Ωiγj,∏j=1β1∏i=1β2+11−Tdˇij2Ωiγj−∏j=1β1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj.
Form=β1+1 and n=β2+1, we have(17)⊕j=1β1+1γj⊕i=1β2+1ΩiJdˇij=⊕j=1β1+1γj⊕i=1β2ΩiJdˇij⊕Ωβ2+1Jdˇβ2+1jJdˇβ2+1j,=⊕j=1β1+1⊕i=1β2γjΩiJdˇij⊕j=1β1+1γjΩβ2+1Jdˇβ2+1j,=1−∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj⊕1−∏j=1β1+11−Tdˇβ2+1j2Ωβ2+1γj∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj⊕∏j=1β1+11−Tdˇβ2+1j2Ωβ2+1γj−∏j=1β1+11−Tdˇβ2+1j2+Jdˇβ2+1j2Ωβ2+1γj,=1−∏j=1β1+1∏i=1β2+11−Tdˇij2Ωiγj,∏j=1β1+1∏i=1β2+11−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj.
Hence, it is true form=β1+1 and n=β2+1.Example 1.
LetUu1,u2,u3 be a set of experts whose weights are given as Ωi=0.243,0.514,0.343T. Experts evaluate the beauty of a house under a considered set of attributes J′=d1=lawn,d2=security system with their corresponding subattributes lawn=d1=d11=with grass,d12=without grass and security systemd2=d21=guards,d22=cameras. Let J′=d1×d2 be a set of subattributes J′=d1×d2=d11,d12×d21,d22=d11,d21,d11,d22,d12,d21,d12,d22 and J′=dˇ1,dˇ2,dˇ3,dˇ4 represent the set subattributes with weights γj=0.25,0.15,0.2,0.4T. Experts’ opinion for each multi-subattribute in the form of PFHSNs J,J′=Tdˇij,Jdˇij3×4 is given as follows:(18)J,J′=0.3,0.80.4,0.60.3,0.60.5,0.60.8,0.30.7,0.40.7,0.30.4,0.80.3,0.60.5,0.70.6,0.50.5,0.4.
By using equation (13),(19)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=14∏i=131−Tdˇij2Ωiγj,∏j=14∏i=131−Tdˇij2Ωiγj−∏j=14∏i=131−Tdˇij2+Jdˇij2Ωiγj,=1−0.910.2430.36.5140.91.3430.250.840.2430.510.5140.750.3430.150.910.2430.510.5140.640.343.20.750.2430.840.5140.750.343.40.910.2430.360.5140.910.3430.250.840.2430.510.5140.750.3430.150.910.2430.510.5140.640.3430.20.750.2430.840.5140.750.3430.4−0.270.2430.270.5140.550.3430.250.480.2430.350.5140.26.3430.150.550.2430.420.5140.390.3430.20.390.2430.200.5140.590.3430.4,=0.58759,0.58241.
Hence, some fundamental properties utilizing the planned PFHSIWA operator for the collection of PFHSNs are established based on Theorem1.


### 3.2. Properties of PFHSIWA Operator

#### 3.2.1. Idempotency

IfJdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(20)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇ.Proof.
As we know that allJdˇij=Jdˇ=Tdˇij,Jdˇij, using equation (13), we have(21)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2Ωiγj,∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj=1−1−Tdˇij2∑i=1nΩi∑j=1mγj,1−Tdˇij2∑i=1nΩi∑j=1mγj−1−Tdˇij2+Jdˇij2∑i=1nΩi∑j=1mγj=1−1−Tdˇij2,1−Tdˇij2−1−Jdˇij2+Tdˇij2=Tdˇij,Jdˇij=Jdˇ.


#### 3.2.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then, Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.Proof.
As we know thatJdˇij=Tdˇij,Jdˇij is a collection of PFHSNs, then(22)minjminiTdˇij2≤Tdˇij2≤maxjmaxiTdˇij2⇒1−maxjmaxiTdˇij2≤1−Tdˇij2≤1−minjminiTdˇij2⇔1−maxjmaxiTdˇij2Ωi≤1−Tdˇij2≤1−minjminiTdˇij2Ωi⇔1−maxjmaxiTdˇij2∑i=1nΩi≤∏i=1n1−Tdˇij2Ωi≤1−minjminiTdˇij2∑i=1nΩi⇔1−maxjmaxiTdˇij2∑j=1mγj≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2∑j=1mγj⇔1−maxjmaxiTdˇij2≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2⇔minjminiTdˇij2≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij2⇔minjminiTdˇij≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij.
Similarly,(23)minjminiJdˇij≤∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−JdˇijΩiγj≤maxjmaxiJdˇij.
LetPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=TδandJδ=Jδ; then, inequalities (22) and (23) can be changed into the subsequent arrangement minjminiTdˇij≤Tδ≤maxjmaxiTdˇij and minjminiJdˇij≤Jδ≤maxjmaxiJdˇij, respectively.
Operating equation (8), we get(24)SJδ=Tδ2−Jδ2≤maxjmaxiTdˇij−minjminiJdˇij=SJdˇij+,SJδ=Tδ2−Jδ2≥minjminiTdˇij−maxjmaxiJdˇij=SJdˇij−.
Then, by order relation among two PFSNs, we have(25)Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


#### 3.2.3. Homogeneity

Prove thatPFHSIWAαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.Proof.
LetJdˇij be a collection of PFHSNs and >0; then, by using Definition 6 (10), we have(26)αJdˇij=1−1−Tdˇij2α,1−Tdˇij2α−1−Tdˇij2+Jdˇij2α.
So,(27)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2αΩiγj,∏j=1m∏i=1n1−Tdˇij2αΩiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2αΩiγj=1−∏j=1m∏i=1n1−Tdˇij2Ωiγjα,∏j=1m∏i=1n1−Tdˇij2Ωiγjα−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγjα=αPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm.
The proof is completed.Definition 8.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs and Ωi and γj represent the weights of expert’s and multi-subattributes along with stated conditions Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1. Then, PFHSIWG: Δn⟶Δ is defined as follows:(28)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1m⊗i=1nJdˇnmΩiγj.Theorem 2.
LetJdˇij=Tdˇij,Jdˇij be a collection of PFHSNs, where i=1,2,…,nandj=1,2,…,m. Then, utilizing equation (28), we get PFHSN and(29)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m∏i=1n1−Jdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=1n1−Jdˇij2Ωiγj,where Ωi and γj represent the expert’s and subattributes’ weights with certain circumstances Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1.Proof.
The PFHSIWG operator can be proved using the principle of mathematical induction as follows:Forn=1, we get Ω1=1. Then, we have(30)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1mJdˇ1jγj,PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m1−Jdˇ1j2γj−∏j=1m1−Tdˇ1j2+Jdˇ1j2γj,1−∏j=1m1−Jdˇ1j2γj,=∏j=1m∏i=111−Jdˇij2Ωiγj−∏j=1m∏i=111−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=111−Jdˇij2Ωiγj.Form=1, we get γ1=1. Then, we have(31)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗i=1nΩiJdˇ11=∏1=1n1−Jdˇi12Ωi−∏i=1n1−Tdˇi12+Jdˇi12Ωi,1−∏i=1n1−Jdˇi12Ωi=∏j=11∏i=1n1−Jdˇij2Ωiγj−∏j=11∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=11∏i=1n1−Jdˇij2Ωiγj.
The above justification shows that equation (10) holds for n=1 and m=1. Now, assume that equation (10) also holds for m=β1+1, n=β2, m=β1, and n=β2+1:(32)⊗j=1β1+1γj⊗i=1β2ΩiJdˇij=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj,⊗j=1β1γj⊗i=1β2+1ΩiJdˇij=∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj.
Form=β1+1 and n=β2+1, we have(33)⊗j=1β1+1γj⊗i=1β2+1ΩiJdˇij=⊗j=1β1+1γj⊗i=1β2ΩiJdˇij⊗Ωβ2+1Jdˇβ2+1j=⊗j=1β1+1⊗i=1β2γjΩiJdˇij⊗j=1β1+1γjΩβ2+1Jdˇβ2+1j=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj⊕∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj−∏j=1β1+11−Tdˇβ2+1j2+Jdˇβ2+1j2Ωβ2+1γj1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj⊕1−∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj=∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj.
Hence, it is true form=β1+1 and n=β2+1.Example 2.
LetUu1,u2,u3 be a set of experts whose weights are given as Ωi=0.243,0.514,0.343T. Experts evaluate the beauty of a house under a considered set of attributes J′=d1=lawn,d2=security system with their corresponding subattributes lawn =  d1=d11=with grass,d12=without grass and security system =  d2=d21=guards,d22=cameras. Let J′=d1×d2 be a set of subattributes J′=d1×d2=d11,d12×d21,d22=d11,d21,d11,d22,d12,d21,d12,d22 and J′=dˇ1,dˇ2,dˇ3,dˇ4 represents the set subattributes with weights γj=0.25,0.15,0.2,.4T. Experts’ opinion for each multi-subattribute in the form of PFHSNs J,J′=Tdˇij,Jdˇij3×4 is given as follows:(34)J,J′=0.3,0.80.4,0.60.3,0.60.5,0.60.8,0.30.7,0.40.7,0.30.4,0.80.3,0.60.5,0.70.6,0.50.5,0.4.
By using equation (13),(35)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=14∏i=131−Jdˇij2Ωiγj−∏j=14∏i=131−Tdˇij2+Jdˇij2Ωiγj,1−∏j=14∏i=131−Jdˇij2Ωiγj,=0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4−0.270.2430.270.5140.550.3430.250.480.2430.350.5140.260.3430.150.550.2430.420.5140.390.3430.20.390.2430.200.5140.590.3430.41−0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4,0.53653,0.62976.
Hence, some basic properties for PFHSNs using the PFHSWG operator are established using Theorem2.


### 3.3. Properties of PFHSIWG Operator

#### 3.3.1. Idempotency

Jdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(36)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇδ.


#### 3.3.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then,(37)Jdˇij−≤PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


#### 3.3.3. Homogeneity

Prove thatPFHSIWGαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.


## 3.1. Interaction Operational Laws for PFHSNs

Definition 6.
LetJdˇk=Tdˇk,Jdˇk, Jdˇ11=Tdˇ11,Jdˇ11, and Jdˇ12=Tdˇ12,Jdˇ12 be three PFHSNs and α be a positive real number; by algebraic norms, considering the interaction, we have(1)
Jdˇ11⊕Jdˇ12=Tdˇ112+Tdˇ122−Tdˇ112Tdˇ122,Jdˇ112+Jdˇ122−Jdˇ112Jdˇ122−Jdˇ112Tdˇ122−Tdˇ112Jdˇ122(2)
Jdˇ11⊗Jdˇ12=Tdˇ112+Tdˇ122−Tdˇ112Tdˇ122−Tdˇ112Jdˇ122−Jdˇ112Tdˇ122,Jdˇ112+Jdˇ122−Jdˇ112Jdˇ122(3)
αJdˇk=1−1−Tdˇk2α,1−Tdˇk2α−1−Tdˇk2+Jdˇk2α(4)
Jdˇkα=1−Jdˇk2α−1−Tdˇk2+Jdˇk2α,1−1−Jdˇk2α
Based on the above-defined operational laws, now we introduce some interaction AOs for PFHSNs’Δ.Definition 7.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs and Ωi and γj represent the weights of expert’s and multi-subattributes along with stated conditions Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1. Then, PFHSIWA: Δn⟶Δ is defined as follows:(12)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕j=1mγj⊕i=1nΩiJdˇij.Theorem 1.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs, where i=1,2,…,nandj=1,2,…,m. Then, the attained aggregated values using equation (12) is also a PFHSN and(13)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2Ωiγj,∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,where Ωi and γj represent the expert’s and subattributes’ weights with certain circumstances Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1.Proof.
The PFHSIWA operator can be proved using the principle of mathematical induction as follows:Forn=1, we get Ω1=1. Then, we have(14)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕j=1mγjJdˇij,PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m1−Tdˇij2γj,∏j=1m1−Tdˇij2γj−∏j=1m1−Tdˇij2+Jdˇij2γj,=1−∏j=1m∏i=111−Tdˇij2Ωiγj,∏j=1m∏i=111−Tdˇij2Ωiγj−∏j=1m∏i=111−Tdˇij2+Jdˇij2Ωiγj.Form=1, we get γ1=1. Then, we have(15)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=⊕i=1nΩiJdˇ11=1−∏i=1n1−Tdˇi12Ωi,∏1=1n1−Tdˇi12Ωi−∏i=1n1−Tdˇi12+Jdˇi12Ωi=1−∏j=11∏i=1n1−Tdˇij2Ωiγj,∏j=11∏i=1n1−Tdˇij2Ωiγj−∏j=11∏i=1n1−Tdˇij2+Jdˇij2Ωiγj.
The above justification shows that equation (13) holds for n=1 and m=1. Now, assume that equation (13) also holds for m=β1+1, n=β2, m=β1, and n=β2+1:(16)⊕j=1β1+1γj⊕i=1β2ΩiJdˇij=1−∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj,∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj,⊕j=1β1γj⊕i=1β2+1ΩiJdˇij=1−∏j=1β1∏i=1β2+11−Tdˇij2Ωiγj,∏j=1β1∏i=1β2+11−Tdˇij2Ωiγj−∏j=1β1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj.
Form=β1+1 and n=β2+1, we have(17)⊕j=1β1+1γj⊕i=1β2+1ΩiJdˇij=⊕j=1β1+1γj⊕i=1β2ΩiJdˇij⊕Ωβ2+1Jdˇβ2+1jJdˇβ2+1j,=⊕j=1β1+1⊕i=1β2γjΩiJdˇij⊕j=1β1+1γjΩβ2+1Jdˇβ2+1j,=1−∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj⊕1−∏j=1β1+11−Tdˇβ2+1j2Ωβ2+1γj∏j=1β1+1∏i=1β21−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj⊕∏j=1β1+11−Tdˇβ2+1j2Ωβ2+1γj−∏j=1β1+11−Tdˇβ2+1j2+Jdˇβ2+1j2Ωβ2+1γj,=1−∏j=1β1+1∏i=1β2+11−Tdˇij2Ωiγj,∏j=1β1+1∏i=1β2+11−Tdˇij2Ωiγj−∏j=1β1+1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj.
Hence, it is true form=β1+1 and n=β2+1.Example 1.
LetUu1,u2,u3 be a set of experts whose weights are given as Ωi=0.243,0.514,0.343T. Experts evaluate the beauty of a house under a considered set of attributes J′=d1=lawn,d2=security system with their corresponding subattributes lawn=d1=d11=with grass,d12=without grass and security systemd2=d21=guards,d22=cameras. Let J′=d1×d2 be a set of subattributes J′=d1×d2=d11,d12×d21,d22=d11,d21,d11,d22,d12,d21,d12,d22 and J′=dˇ1,dˇ2,dˇ3,dˇ4 represent the set subattributes with weights γj=0.25,0.15,0.2,0.4T. Experts’ opinion for each multi-subattribute in the form of PFHSNs J,J′=Tdˇij,Jdˇij3×4 is given as follows:(18)J,J′=0.3,0.80.4,0.60.3,0.60.5,0.60.8,0.30.7,0.40.7,0.30.4,0.80.3,0.60.5,0.70.6,0.50.5,0.4.
By using equation (13),(19)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=14∏i=131−Tdˇij2Ωiγj,∏j=14∏i=131−Tdˇij2Ωiγj−∏j=14∏i=131−Tdˇij2+Jdˇij2Ωiγj,=1−0.910.2430.36.5140.91.3430.250.840.2430.510.5140.750.3430.150.910.2430.510.5140.640.343.20.750.2430.840.5140.750.343.40.910.2430.360.5140.910.3430.250.840.2430.510.5140.750.3430.150.910.2430.510.5140.640.3430.20.750.2430.840.5140.750.3430.4−0.270.2430.270.5140.550.3430.250.480.2430.350.5140.26.3430.150.550.2430.420.5140.390.3430.20.390.2430.200.5140.590.3430.4,=0.58759,0.58241.
Hence, some fundamental properties utilizing the planned PFHSIWA operator for the collection of PFHSNs are established based on Theorem1.


## 3.2. Properties of PFHSIWA Operator

### 3.2.1. Idempotency

IfJdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(20)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇ.Proof.
As we know that allJdˇij=Jdˇ=Tdˇij,Jdˇij, using equation (13), we have(21)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2Ωiγj,∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj=1−1−Tdˇij2∑i=1nΩi∑j=1mγj,1−Tdˇij2∑i=1nΩi∑j=1mγj−1−Tdˇij2+Jdˇij2∑i=1nΩi∑j=1mγj=1−1−Tdˇij2,1−Tdˇij2−1−Jdˇij2+Tdˇij2=Tdˇij,Jdˇij=Jdˇ.


### 3.2.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then, Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.Proof.
As we know thatJdˇij=Tdˇij,Jdˇij is a collection of PFHSNs, then(22)minjminiTdˇij2≤Tdˇij2≤maxjmaxiTdˇij2⇒1−maxjmaxiTdˇij2≤1−Tdˇij2≤1−minjminiTdˇij2⇔1−maxjmaxiTdˇij2Ωi≤1−Tdˇij2≤1−minjminiTdˇij2Ωi⇔1−maxjmaxiTdˇij2∑i=1nΩi≤∏i=1n1−Tdˇij2Ωi≤1−minjminiTdˇij2∑i=1nΩi⇔1−maxjmaxiTdˇij2∑j=1mγj≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2∑j=1mγj⇔1−maxjmaxiTdˇij2≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2⇔minjminiTdˇij2≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij2⇔minjminiTdˇij≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij.
Similarly,(23)minjminiJdˇij≤∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−JdˇijΩiγj≤maxjmaxiJdˇij.
LetPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=TδandJδ=Jδ; then, inequalities (22) and (23) can be changed into the subsequent arrangement minjminiTdˇij≤Tδ≤maxjmaxiTdˇij and minjminiJdˇij≤Jδ≤maxjmaxiJdˇij, respectively.
Operating equation (8), we get(24)SJδ=Tδ2−Jδ2≤maxjmaxiTdˇij−minjminiJdˇij=SJdˇij+,SJδ=Tδ2−Jδ2≥minjminiTdˇij−maxjmaxiJdˇij=SJdˇij−.
Then, by order relation among two PFSNs, we have(25)Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


### 3.2.3. Homogeneity

Prove thatPFHSIWAαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.Proof.
LetJdˇij be a collection of PFHSNs and >0; then, by using Definition 6 (10), we have(26)αJdˇij=1−1−Tdˇij2α,1−Tdˇij2α−1−Tdˇij2+Jdˇij2α.
So,(27)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2αΩiγj,∏j=1m∏i=1n1−Tdˇij2αΩiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2αΩiγj=1−∏j=1m∏i=1n1−Tdˇij2Ωiγjα,∏j=1m∏i=1n1−Tdˇij2Ωiγjα−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγjα=αPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm.
The proof is completed.Definition 8.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs and Ωi and γj represent the weights of expert’s and multi-subattributes along with stated conditions Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1. Then, PFHSIWG: Δn⟶Δ is defined as follows:(28)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1m⊗i=1nJdˇnmΩiγj.Theorem 2.
LetJdˇij=Tdˇij,Jdˇij be a collection of PFHSNs, where i=1,2,…,nandj=1,2,…,m. Then, utilizing equation (28), we get PFHSN and(29)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m∏i=1n1−Jdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=1n1−Jdˇij2Ωiγj,where Ωi and γj represent the expert’s and subattributes’ weights with certain circumstances Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1.Proof.
The PFHSIWG operator can be proved using the principle of mathematical induction as follows:Forn=1, we get Ω1=1. Then, we have(30)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1mJdˇ1jγj,PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m1−Jdˇ1j2γj−∏j=1m1−Tdˇ1j2+Jdˇ1j2γj,1−∏j=1m1−Jdˇ1j2γj,=∏j=1m∏i=111−Jdˇij2Ωiγj−∏j=1m∏i=111−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=111−Jdˇij2Ωiγj.Form=1, we get γ1=1. Then, we have(31)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗i=1nΩiJdˇ11=∏1=1n1−Jdˇi12Ωi−∏i=1n1−Tdˇi12+Jdˇi12Ωi,1−∏i=1n1−Jdˇi12Ωi=∏j=11∏i=1n1−Jdˇij2Ωiγj−∏j=11∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=11∏i=1n1−Jdˇij2Ωiγj.
The above justification shows that equation (10) holds for n=1 and m=1. Now, assume that equation (10) also holds for m=β1+1, n=β2, m=β1, and n=β2+1:(32)⊗j=1β1+1γj⊗i=1β2ΩiJdˇij=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj,⊗j=1β1γj⊗i=1β2+1ΩiJdˇij=∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj.
Form=β1+1 and n=β2+1, we have(33)⊗j=1β1+1γj⊗i=1β2+1ΩiJdˇij=⊗j=1β1+1γj⊗i=1β2ΩiJdˇij⊗Ωβ2+1Jdˇβ2+1j=⊗j=1β1+1⊗i=1β2γjΩiJdˇij⊗j=1β1+1γjΩβ2+1Jdˇβ2+1j=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj⊕∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj−∏j=1β1+11−Tdˇβ2+1j2+Jdˇβ2+1j2Ωβ2+1γj1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj⊕1−∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj=∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj.
Hence, it is true form=β1+1 and n=β2+1.Example 2.
LetUu1,u2,u3 be a set of experts whose weights are given as Ωi=0.243,0.514,0.343T. Experts evaluate the beauty of a house under a considered set of attributes J′=d1=lawn,d2=security system with their corresponding subattributes lawn =  d1=d11=with grass,d12=without grass and security system =  d2=d21=guards,d22=cameras. Let J′=d1×d2 be a set of subattributes J′=d1×d2=d11,d12×d21,d22=d11,d21,d11,d22,d12,d21,d12,d22 and J′=dˇ1,dˇ2,dˇ3,dˇ4 represents the set subattributes with weights γj=0.25,0.15,0.2,.4T. Experts’ opinion for each multi-subattribute in the form of PFHSNs J,J′=Tdˇij,Jdˇij3×4 is given as follows:(34)J,J′=0.3,0.80.4,0.60.3,0.60.5,0.60.8,0.30.7,0.40.7,0.30.4,0.80.3,0.60.5,0.70.6,0.50.5,0.4.
By using equation (13),(35)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=14∏i=131−Jdˇij2Ωiγj−∏j=14∏i=131−Tdˇij2+Jdˇij2Ωiγj,1−∏j=14∏i=131−Jdˇij2Ωiγj,=0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4−0.270.2430.270.5140.550.3430.250.480.2430.350.5140.260.3430.150.550.2430.420.5140.390.3430.20.390.2430.200.5140.590.3430.41−0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4,0.53653,0.62976.
Hence, some basic properties for PFHSNs using the PFHSWG operator are established using Theorem2.


## 3.2.1. Idempotency

IfJdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(20)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇ.Proof.
As we know that allJdˇij=Jdˇ=Tdˇij,Jdˇij, using equation (13), we have(21)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2Ωiγj,∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj=1−1−Tdˇij2∑i=1nΩi∑j=1mγj,1−Tdˇij2∑i=1nΩi∑j=1mγj−1−Tdˇij2+Jdˇij2∑i=1nΩi∑j=1mγj=1−1−Tdˇij2,1−Tdˇij2−1−Jdˇij2+Tdˇij2=Tdˇij,Jdˇij=Jdˇ.


## 3.2.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then, Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.Proof.
As we know thatJdˇij=Tdˇij,Jdˇij is a collection of PFHSNs, then(22)minjminiTdˇij2≤Tdˇij2≤maxjmaxiTdˇij2⇒1−maxjmaxiTdˇij2≤1−Tdˇij2≤1−minjminiTdˇij2⇔1−maxjmaxiTdˇij2Ωi≤1−Tdˇij2≤1−minjminiTdˇij2Ωi⇔1−maxjmaxiTdˇij2∑i=1nΩi≤∏i=1n1−Tdˇij2Ωi≤1−minjminiTdˇij2∑i=1nΩi⇔1−maxjmaxiTdˇij2∑j=1mγj≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2∑j=1mγj⇔1−maxjmaxiTdˇij2≤∏j=1m∏i=1n1−Tdˇij2Ωiγj≤1−minjminiTdˇij2⇔minjminiTdˇij2≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij2⇔minjminiTdˇij≤1−∏j=1m∏i=1n1−Tdˇij2Ωiγj≤maxjmaxiTdˇij.
Similarly,(23)minjminiJdˇij≤∏j=1m∏i=1n1−Tdˇij2Ωiγj−∏j=1m∏i=1n1−JdˇijΩiγj≤maxjmaxiJdˇij.
LetPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=TδandJδ=Jδ; then, inequalities (22) and (23) can be changed into the subsequent arrangement minjminiTdˇij≤Tδ≤maxjmaxiTdˇij and minjminiJdˇij≤Jδ≤maxjmaxiJdˇij, respectively.
Operating equation (8), we get(24)SJδ=Tδ2−Jδ2≤maxjmaxiTdˇij−minjminiJdˇij=SJdˇij+,SJδ=Tδ2−Jδ2≥minjminiTdˇij−maxjmaxiJdˇij=SJdˇij−.
Then, by order relation among two PFSNs, we have(25)Jdˇij−≤PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


## 3.2.3. Homogeneity

Prove thatPFHSIWAαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.Proof.
LetJdˇij be a collection of PFHSNs and >0; then, by using Definition 6 (10), we have(26)αJdˇij=1−1−Tdˇij2α,1−Tdˇij2α−1−Tdˇij2+Jdˇij2α.
So,(27)PFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm=1−∏j=1m∏i=1n1−Tdˇij2αΩiγj,∏j=1m∏i=1n1−Tdˇij2αΩiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2αΩiγj=1−∏j=1m∏i=1n1−Tdˇij2Ωiγjα,∏j=1m∏i=1n1−Tdˇij2Ωiγjα−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγjα=αPFHSIWAJdˇ11,Jdˇ12,…,Jdˇnm.
The proof is completed.Definition 8.
LetJdˇij=Tdˇij,Jdˇij be PFHSNs and Ωi and γj represent the weights of expert’s and multi-subattributes along with stated conditions Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1. Then, PFHSIWG: Δn⟶Δ is defined as follows:(28)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1m⊗i=1nJdˇnmΩiγj.Theorem 2.
LetJdˇij=Tdˇij,Jdˇij be a collection of PFHSNs, where i=1,2,…,nandj=1,2,…,m. Then, utilizing equation (28), we get PFHSN and(29)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m∏i=1n1−Jdˇij2Ωiγj−∏j=1m∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=1n1−Jdˇij2Ωiγj,where Ωi and γj represent the expert’s and subattributes’ weights with certain circumstances Ωi>0, ∑i=1nΩi=1, γj>0, and ∑j=1mγj=1.Proof.
The PFHSIWG operator can be proved using the principle of mathematical induction as follows:Forn=1, we get Ω1=1. Then, we have(30)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗j=1mJdˇ1jγj,PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=1m1−Jdˇ1j2γj−∏j=1m1−Tdˇ1j2+Jdˇ1j2γj,1−∏j=1m1−Jdˇ1j2γj,=∏j=1m∏i=111−Jdˇij2Ωiγj−∏j=1m∏i=111−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1m∏i=111−Jdˇij2Ωiγj.Form=1, we get γ1=1. Then, we have(31)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=⊗i=1nΩiJdˇ11=∏1=1n1−Jdˇi12Ωi−∏i=1n1−Tdˇi12+Jdˇi12Ωi,1−∏i=1n1−Jdˇi12Ωi=∏j=11∏i=1n1−Jdˇij2Ωiγj−∏j=11∏i=1n1−Tdˇij2+Jdˇij2Ωiγj,1−∏j=11∏i=1n1−Jdˇij2Ωiγj.
The above justification shows that equation (10) holds for n=1 and m=1. Now, assume that equation (10) also holds for m=β1+1, n=β2, m=β1, and n=β2+1:(32)⊗j=1β1+1γj⊗i=1β2ΩiJdˇij=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj,⊗j=1β1γj⊗i=1β2+1ΩiJdˇij=∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1∏i=1β2+11−Jdˇij2Ωiγj.
Form=β1+1 and n=β2+1, we have(33)⊗j=1β1+1γj⊗i=1β2+1ΩiJdˇij=⊗j=1β1+1γj⊗i=1β2ΩiJdˇij⊗Ωβ2+1Jdˇβ2+1j=⊗j=1β1+1⊗i=1β2γjΩiJdˇij⊗j=1β1+1γjΩβ2+1Jdˇβ2+1j=∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β21−Tdˇij2+Jdˇij2Ωiγj⊕∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj−∏j=1β1+11−Tdˇβ2+1j2+Jdˇβ2+1j2Ωβ2+1γj1−∏j=1β1+1∏i=1β21−Jdˇij2Ωiγj⊕1−∏j=1β1+11−Jdˇβ2+1j2Ωβ2+1γj=∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj−∏j=1β1+1∏i=1β2+11−Tdˇij2+Jdˇij2Ωiγj,1−∏j=1β1+1∏i=1β2+11−Jdˇij2Ωiγj.
Hence, it is true form=β1+1 and n=β2+1.Example 2.
LetUu1,u2,u3 be a set of experts whose weights are given as Ωi=0.243,0.514,0.343T. Experts evaluate the beauty of a house under a considered set of attributes J′=d1=lawn,d2=security system with their corresponding subattributes lawn =  d1=d11=with grass,d12=without grass and security system =  d2=d21=guards,d22=cameras. Let J′=d1×d2 be a set of subattributes J′=d1×d2=d11,d12×d21,d22=d11,d21,d11,d22,d12,d21,d12,d22 and J′=dˇ1,dˇ2,dˇ3,dˇ4 represents the set subattributes with weights γj=0.25,0.15,0.2,.4T. Experts’ opinion for each multi-subattribute in the form of PFHSNs J,J′=Tdˇij,Jdˇij3×4 is given as follows:(34)J,J′=0.3,0.80.4,0.60.3,0.60.5,0.60.8,0.30.7,0.40.7,0.30.4,0.80.3,0.60.5,0.70.6,0.50.5,0.4.
By using equation (13),(35)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=∏j=14∏i=131−Jdˇij2Ωiγj−∏j=14∏i=131−Tdˇij2+Jdˇij2Ωiγj,1−∏j=14∏i=131−Jdˇij2Ωiγj,=0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4−0.270.2430.270.5140.550.3430.250.480.2430.350.5140.260.3430.150.550.2430.420.5140.390.3430.20.390.2430.200.5140.590.3430.41−0.360.2430.910.5140.640.3430.250.640.2430.840.5140.510.3430.150.640.2430.910.5140.750.3430.20.640.2430.360.5140.840.3430.4,0.53653,0.62976.
Hence, some basic properties for PFHSNs using the PFHSWG operator are established using Theorem2.


## 3.3. Properties of PFHSIWG Operator

### 3.3.1. Idempotency

Jdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(36)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇδ.


### 3.3.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then,(37)Jdˇij−≤PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


### 3.3.3. Homogeneity

Prove thatPFHSIWGαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.


## 3.3.1. Idempotency

Jdˇij=Jdˇ=Tdˇij,Jdˇij,∀i,j, then(36)PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm=Jdˇδ.


## 3.3.2. Boundedness

LetJdˇij be a collection of PFHSNs, Jdˇij−=minjminiTdˇij,maxjmaxiJdˇij, and Jdˇij+=maxjmaxiTdˇij,minjminiJdˇij; then,(37)Jdˇij−≤PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm≤Jdˇij+.


## 3.3.3. Homogeneity

Prove thatPFHSIWGαJdˇ11,αJdˇ12,…,αJdˇnm=α⋅PFHSIWGJdˇ11,Jdˇ12,…,Jdˇnm for any positive real number α.


## 4. An MCDM Approach Based on Interaction Aggregation Operators for PFHSSs

An MCDM approach is established here under the developed operators and presented a comprehensive comparative analysis to prove the usefulness and practicality of our established method.


### 4.1. Proposed MCDM Approach

Considerχ=χ1,χ2,χ3,…,χs to be a set of s alternatives and U=δ1,δ2,δ3,…,δn to be a set n experts. The weights of experts are given as Ω=Ω1,Ω1,…,ΩnT and Ωi>0, ∑i=1nΩi=1. Let L=d1,d2,…,dm represent the set attributes with their corresponding multi-subattributes such as J′=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 J′=dˇ∂:∂∈1,2,…,m. The group of experts κi:i=1,2,…,n assess the alternatives ℵz:z=1,2,…,s under the chosen subattributes dˇ∂:∂=1,2,…,k in the form of PFHSNs such as χdˇikzn×m=Tdˇij,Jdˇijn×m where 0≤Tdˇij,Jdˇij≤1, and 0≤Tdˇij2+Jdˇij2≤1 for all i and k. Utilizing the proposed PFHSIWA, PFHSIWG operators develop aggregated PFHSNs ℒϕ for each alternative according to the expert’s preferences. Finally, utilizing equation (8), compute the score function. The above-presented approach can be concise as follows:Step 1. Develop decision matrices for each alternative Dz:z=1,2,…,s as follows:(38)χz,J′n×∂=δ1δ2⋮δnTdˇ11z,ℐdˇ11zTdˇ12z,ℐdˇ12z⋯Tdˇ1∂z,ℐdˇ1∂zTdˇ21z,ℐdˇ21zTdˇ22z,ℐdˇ22z⋯Tdˇ2∂z,ℐdˇ2∂z⋮⋮⋮⋮Tdˇn1z,ℐdˇn1zTdˇn2z,ℐdˇn2z⋯Tdˇn∂z,ℐdˇn∂z.Step 2. Obtain normalized decision matrices for alternatives utilizing the normalization rule:(39)Hij=Jdˇijc;cost type parameter,Jdˇij;benefit type parameter.Step 3. Establish a collective decision matrix ℒk for each alternative using developed AOsStep 4. Using equation (8), compute the score values for each alternativeStep 5. Select the most suitable alternative with the maximum score valueStep 6. Rank the alternativesThe graphical representation of the presented approach can be expressed in following Figure1.Figure 1
Flowchart of presented PFHSIWA or PFHSIWG operators.


### 4.2. Case Study

The problem of supplier selection is an essential part at both a logical and practical level. This is an ongoing problem for the organization because the most suitable choice of suppliers is the basis for effective supply chain management and also the basis of reasonable benefit, which includes environmental management standards and includes more features of sustainable improvement in environmental management standards and supplier selection procedures. Depending on the visible horizon of substantial or social activities, supplier selection is typically known as “sustainable supplier selection” in the literature. This is a multidimensional consequence along with conflicting specifications. The self-assessment process needs to deliberate several features. From these perspectives, the issue of supplier selection is often considered a “reference” issue in the literature, with a wide range of methods used to support incorporative decisions. The problem of choosing and assessing a sustainable supplier is solved in lots of the best ways. This example of sustainable supplier selection results in a set of five parameters, using the analysis of [44–53]. These are d1, supremacy of service, d2, delivery, d3, environmental efficiently, d4, troposphere, and d5, corporate social concern.Considerχ1,χ2,χ3,χ4,χ5 to be a set of alternatives, and L  = {d1=supremacy, d2=delivery, d3=environmental efficiently, d4=troposphere, d5=corporate societal concern} represents the collection of considered parameters prearranged as supremacy  =  d1  =  d11=national level,d12=international level, delivery =  d2  =  d21=by carriar,d22=by hand, environmental efficiently =  d3  =  d31=environmental efficiently, troposphere =  d4  =  d41=friendly,d42=non serious, and corporate social concern =  d5  =  d51=corporate social concern. Let J′=d1×d2×d3×d4×d5 be a set of subattributes:(40)J′=d1×d2×d3×d4×d5=d11,d12×d21,d22×dd31,d32×d41×d51,=d11,d21,d31,d41,d51,d11,d21,d32,d41,d51,d11,d22,d31,d41,d51,d11,d22,d32,d41,d51,d12,d21,d31,d41,d51,d12,d21,d32,d41,d51,d12,d22,d31,d41,d51,d12,d22,d32,d41,d51,where J′=dˇ1,dˇ2,dˇ3,dˇ4,dˇ5,dˇ6,dˇ7,dˇ8 represents the set of all subattributes with weights 0.12,0.18,0.1,0.15,0.05,0.22,0.08,0.1T. Let u1,u2,u3 be a set of experts with weights 0.243,0.514,0.343T to evaluate the optimum alternative. Specialists contribute their predilections in the form of PFHSNs under multi-subattributes of considered attributes.


#### 4.2.1. By Using PFHSIWA Operator

Step 1. Experts access the matters to illustrate the PFHSN. A summary of the many subattributes of the perceived attributes as well as their score values is given in Tables 1–3 .Step 2. All attributes are of the same type, so no need to normalize them.Step 3. Experts’ opinion can be summarized utilizing equation (13) as follows:ℒ1=0.6009,0.4342, ℒ2=0.6499,0.4078, ℒ3=0.6179,0.3506, ℒ4=0.6076,0.3527, and ℒ5=0.5493,0.4345.Step 4. Compute the score values using equation (8):Sℒ1=0.1667, Sℒ2=0.2421, Sℒ3=0.2673, Sℒ4=0.2549, and Sℒ5=0.1148.Step 5. χ3 has greatest score value, so χ3 is the finest option.Step 6. Alternatives’ ranking using the PFHSIWA operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.Table 1
PFHS decision matrix foru1.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.3,0.80.7,0.30.6,0.70.5,0.40.2,0.40.4,0.60.5,0.80.9,0.3χ20.6,0.70.4,0.60.3,0.40.9,0.20.3,0.80.2,0.40.7,0.50.4,0.5χ30.7,0.30.2,0.50.1,0.60.3,0.40.4,0.60.8,0.40.6,0.70.2,0.5χ40.8,0.40.2,0.90.2,0.40.4,0.60.6,0.50.5,0.60.4,0.50.8,0.3χ50.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5Table 2
PFHS decision matrix foru2.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.7,0.60.3,0.40.6,0.50.3,0.90.5,0.40.4,0.60.7,0.50.4,0.8χ20.8,0.50.7,0.40.9,0.20.7,0.40.4,0.50.9,0.30.2,0.70.3,0.8χ30.3,0.70.4,0.50.4,0.80.3,0.40.6,0.70.3,0.40.9,0.20.7,0.2χ40.5,0.40.7,0.60.9,0.30.8,0.50.9,0.20.2,0.40.4,0.60.6,0.5χ50.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4Table 3
PFHS decision matrix foru3.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5χ20.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4χ30.6,0.80.4,0.50.6,0.50.6,0.40.7,0.50.8,0.40.5,0.80.4,0.5χ40.5,0.70.9,0.30.3,0.50.5,0.70.3,0.50.8,0.50.7,0.50.2,0.5χ50.5,0.40.4,0.80.5,0.60.3,0.40.7,0.60.7,0.50.4,0.90.5,0.2


#### 4.2.2. By Using PFHSIWG Operator

Step 1 and Step 2. They are the same as Section 4.2.1.Step 3. Experts’ opinion can be summarized utilizing equation (29) as follows:ℒ1=0.4679,0.5590, ℒ2=0.5157,0.5289, ℒ3=0.4892,0.4387, ℒ4=0.4910,0.4751, and ℒ5=0.4440,0.6407Step 4. Compute the score values using equation (8):Sℒ1=−0.0911, Sℒ2=−0.0132, Sℒ3=0.0505, Sℒ4=0.0159, and Sℒ5=−0.1967Step 5. χ3 has the greatest score value, so χ3 is the finest optionStep 6. Alternatives’ ranking using the PFHSIWG operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.


## 4.1. Proposed MCDM Approach

Considerχ=χ1,χ2,χ3,…,χs to be a set of s alternatives and U=δ1,δ2,δ3,…,δn to be a set n experts. The weights of experts are given as Ω=Ω1,Ω1,…,ΩnT and Ωi>0, ∑i=1nΩi=1. Let L=d1,d2,…,dm represent the set attributes with their corresponding multi-subattributes such as J′=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 J′=dˇ∂:∂∈1,2,…,m. The group of experts κi:i=1,2,…,n assess the alternatives ℵz:z=1,2,…,s under the chosen subattributes dˇ∂:∂=1,2,…,k in the form of PFHSNs such as χdˇikzn×m=Tdˇij,Jdˇijn×m where 0≤Tdˇij,Jdˇij≤1, and 0≤Tdˇij2+Jdˇij2≤1 for all i and k. Utilizing the proposed PFHSIWA, PFHSIWG operators develop aggregated PFHSNs ℒϕ for each alternative according to the expert’s preferences. Finally, utilizing equation (8), compute the score function. The above-presented approach can be concise as follows:Step 1. Develop decision matrices for each alternative Dz:z=1,2,…,s as follows:(38)χz,J′n×∂=δ1δ2⋮δnTdˇ11z,ℐdˇ11zTdˇ12z,ℐdˇ12z⋯Tdˇ1∂z,ℐdˇ1∂zTdˇ21z,ℐdˇ21zTdˇ22z,ℐdˇ22z⋯Tdˇ2∂z,ℐdˇ2∂z⋮⋮⋮⋮Tdˇn1z,ℐdˇn1zTdˇn2z,ℐdˇn2z⋯Tdˇn∂z,ℐdˇn∂z.Step 2. Obtain normalized decision matrices for alternatives utilizing the normalization rule:(39)Hij=Jdˇijc;cost type parameter,Jdˇij;benefit type parameter.Step 3. Establish a collective decision matrix ℒk for each alternative using developed AOsStep 4. Using equation (8), compute the score values for each alternativeStep 5. Select the most suitable alternative with the maximum score valueStep 6. Rank the alternativesThe graphical representation of the presented approach can be expressed in following Figure1.Figure 1
Flowchart of presented PFHSIWA or PFHSIWG operators.


## 4.2. Case Study

The problem of supplier selection is an essential part at both a logical and practical level. This is an ongoing problem for the organization because the most suitable choice of suppliers is the basis for effective supply chain management and also the basis of reasonable benefit, which includes environmental management standards and includes more features of sustainable improvement in environmental management standards and supplier selection procedures. Depending on the visible horizon of substantial or social activities, supplier selection is typically known as “sustainable supplier selection” in the literature. This is a multidimensional consequence along with conflicting specifications. The self-assessment process needs to deliberate several features. From these perspectives, the issue of supplier selection is often considered a “reference” issue in the literature, with a wide range of methods used to support incorporative decisions. The problem of choosing and assessing a sustainable supplier is solved in lots of the best ways. This example of sustainable supplier selection results in a set of five parameters, using the analysis of [44–53]. These are d1, supremacy of service, d2, delivery, d3, environmental efficiently, d4, troposphere, and d5, corporate social concern.Considerχ1,χ2,χ3,χ4,χ5 to be a set of alternatives, and L  = {d1=supremacy, d2=delivery, d3=environmental efficiently, d4=troposphere, d5=corporate societal concern} represents the collection of considered parameters prearranged as supremacy  =  d1  =  d11=national level,d12=international level, delivery =  d2  =  d21=by carriar,d22=by hand, environmental efficiently =  d3  =  d31=environmental efficiently, troposphere =  d4  =  d41=friendly,d42=non serious, and corporate social concern =  d5  =  d51=corporate social concern. Let J′=d1×d2×d3×d4×d5 be a set of subattributes:(40)J′=d1×d2×d3×d4×d5=d11,d12×d21,d22×dd31,d32×d41×d51,=d11,d21,d31,d41,d51,d11,d21,d32,d41,d51,d11,d22,d31,d41,d51,d11,d22,d32,d41,d51,d12,d21,d31,d41,d51,d12,d21,d32,d41,d51,d12,d22,d31,d41,d51,d12,d22,d32,d41,d51,where J′=dˇ1,dˇ2,dˇ3,dˇ4,dˇ5,dˇ6,dˇ7,dˇ8 represents the set of all subattributes with weights 0.12,0.18,0.1,0.15,0.05,0.22,0.08,0.1T. Let u1,u2,u3 be a set of experts with weights 0.243,0.514,0.343T to evaluate the optimum alternative. Specialists contribute their predilections in the form of PFHSNs under multi-subattributes of considered attributes.


### 4.2.1. By Using PFHSIWA Operator

Step 1. Experts access the matters to illustrate the PFHSN. A summary of the many subattributes of the perceived attributes as well as their score values is given in Tables 1–3 .Step 2. All attributes are of the same type, so no need to normalize them.Step 3. Experts’ opinion can be summarized utilizing equation (13) as follows:ℒ1=0.6009,0.4342, ℒ2=0.6499,0.4078, ℒ3=0.6179,0.3506, ℒ4=0.6076,0.3527, and ℒ5=0.5493,0.4345.Step 4. Compute the score values using equation (8):Sℒ1=0.1667, Sℒ2=0.2421, Sℒ3=0.2673, Sℒ4=0.2549, and Sℒ5=0.1148.Step 5. χ3 has greatest score value, so χ3 is the finest option.Step 6. Alternatives’ ranking using the PFHSIWA operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.Table 1
PFHS decision matrix foru1.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.3,0.80.7,0.30.6,0.70.5,0.40.2,0.40.4,0.60.5,0.80.9,0.3χ20.6,0.70.4,0.60.3,0.40.9,0.20.3,0.80.2,0.40.7,0.50.4,0.5χ30.7,0.30.2,0.50.1,0.60.3,0.40.4,0.60.8,0.40.6,0.70.2,0.5χ40.8,0.40.2,0.90.2,0.40.4,0.60.6,0.50.5,0.60.4,0.50.8,0.3χ50.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5Table 2
PFHS decision matrix foru2.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.7,0.60.3,0.40.6,0.50.3,0.90.5,0.40.4,0.60.7,0.50.4,0.8χ20.8,0.50.7,0.40.9,0.20.7,0.40.4,0.50.9,0.30.2,0.70.3,0.8χ30.3,0.70.4,0.50.4,0.80.3,0.40.6,0.70.3,0.40.9,0.20.7,0.2χ40.5,0.40.7,0.60.9,0.30.8,0.50.9,0.20.2,0.40.4,0.60.6,0.5χ50.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4Table 3
PFHS decision matrix foru3.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5χ20.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4χ30.6,0.80.4,0.50.6,0.50.6,0.40.7,0.50.8,0.40.5,0.80.4,0.5χ40.5,0.70.9,0.30.3,0.50.5,0.70.3,0.50.8,0.50.7,0.50.2,0.5χ50.5,0.40.4,0.80.5,0.60.3,0.40.7,0.60.7,0.50.4,0.90.5,0.2


### 4.2.2. By Using PFHSIWG Operator

Step 1 and Step 2. They are the same as Section 4.2.1.Step 3. Experts’ opinion can be summarized utilizing equation (29) as follows:ℒ1=0.4679,0.5590, ℒ2=0.5157,0.5289, ℒ3=0.4892,0.4387, ℒ4=0.4910,0.4751, and ℒ5=0.4440,0.6407Step 4. Compute the score values using equation (8):Sℒ1=−0.0911, Sℒ2=−0.0132, Sℒ3=0.0505, Sℒ4=0.0159, and Sℒ5=−0.1967Step 5. χ3 has the greatest score value, so χ3 is the finest optionStep 6. Alternatives’ ranking using the PFHSIWG operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.


## 4.2.1. By Using PFHSIWA Operator

Step 1. Experts access the matters to illustrate the PFHSN. A summary of the many subattributes of the perceived attributes as well as their score values is given in Tables 1–3 .Step 2. All attributes are of the same type, so no need to normalize them.Step 3. Experts’ opinion can be summarized utilizing equation (13) as follows:ℒ1=0.6009,0.4342, ℒ2=0.6499,0.4078, ℒ3=0.6179,0.3506, ℒ4=0.6076,0.3527, and ℒ5=0.5493,0.4345.Step 4. Compute the score values using equation (8):Sℒ1=0.1667, Sℒ2=0.2421, Sℒ3=0.2673, Sℒ4=0.2549, and Sℒ5=0.1148.Step 5. χ3 has greatest score value, so χ3 is the finest option.Step 6. Alternatives’ ranking using the PFHSIWA operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.Table 1
PFHS decision matrix foru1.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.3,0.80.7,0.30.6,0.70.5,0.40.2,0.40.4,0.60.5,0.80.9,0.3χ20.6,0.70.4,0.60.3,0.40.9,0.20.3,0.80.2,0.40.7,0.50.4,0.5χ30.7,0.30.2,0.50.1,0.60.3,0.40.4,0.60.8,0.40.6,0.70.2,0.5χ40.8,0.40.2,0.90.2,0.40.4,0.60.6,0.50.5,0.60.4,0.50.8,0.3χ50.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5Table 2
PFHS decision matrix foru2.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.7,0.60.3,0.40.6,0.50.3,0.90.5,0.40.4,0.60.7,0.50.4,0.8χ20.8,0.50.7,0.40.9,0.20.7,0.40.4,0.50.9,0.30.2,0.70.3,0.8χ30.3,0.70.4,0.50.4,0.80.3,0.40.6,0.70.3,0.40.9,0.20.7,0.2χ40.5,0.40.7,0.60.9,0.30.8,0.50.9,0.20.2,0.40.4,0.60.6,0.5χ50.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4Table 3
PFHS decision matrix foru3.
dˇ1dˇ2dˇ3dˇ4dˇ5dˇ6dˇ7dˇ8χ10.5,0.70.8,0.50.7,0.40.4,0.30.4,0.90.2,0.40.8,0.40.7,0.5χ20.8,0.50.7,0.40.8,0.50.5,0.20.5,0.70.7,0.50.7,0.60.6,0.4χ30.6,0.80.4,0.50.6,0.50.6,0.40.7,0.50.8,0.40.5,0.80.4,0.5χ40.5,0.70.9,0.30.3,0.50.5,0.70.3,0.50.8,0.50.7,0.50.2,0.5χ50.5,0.40.4,0.80.5,0.60.3,0.40.7,0.60.7,0.50.4,0.90.5,0.2


## 4.2.2. By Using PFHSIWG Operator

Step 1 and Step 2. They are the same as Section 4.2.1.Step 3. Experts’ opinion can be summarized utilizing equation (29) as follows:ℒ1=0.4679,0.5590, ℒ2=0.5157,0.5289, ℒ3=0.4892,0.4387, ℒ4=0.4910,0.4751, and ℒ5=0.4440,0.6407Step 4. Compute the score values using equation (8):Sℒ1=−0.0911, Sℒ2=−0.0132, Sℒ3=0.0505, Sℒ4=0.0159, and Sℒ5=−0.1967Step 5. χ3 has the greatest score value, so χ3 is the finest optionStep 6. Alternatives’ ranking using the PFHSIWG operator is given as follows:Sℒ3>Sℒ4>Sℒ2>Sℒ1>Sℒ5. So, χ3>χ4>χ2>χ1>χ5.


## 5. Comparative Analysis and Discussion

In the following section, we will discuss quality, naivety, and tractability by means of the planned method. We also gave a brief overview of the following: the proposed approach with some existing methods.


### 5.1. Superiority of the Proposed Method

Through this study, along with association, it is resolute that the concerns attained with the proposed method are rather extrageneral than either technique. However, the developed MCDM approach has been provided more information to cope with the hesitancy in the DM procedure related to the existing MCDM strategies. Besides, the numerous mixed structures of FSs have grown into a unique feature of PFHSSs; after adding some proper conditions, the general facts about the component can be stated precisely and logically, as shown in Table4. It is observed that the obtained results deliver extrainformation comparative to existing studies. The developed PFHSSs accurately accommodate more information considering the multi-subattributes of the parameters. It is quite an easy tool to mix inexact and unsure information within the DM process. Hence, the proposed methodology is pragmatic, diffident, and distinctive from available hybrid structures of fuzzy sets.Table 4
Comparison of PFHSSs with some prevailing models.
SetTruthinessFalsityParametrizationAttributesSubattributesZadeh [1]FS✓××✓×Atanassov [2]IFS✓✓×✓×Maji et al. [21]IFSS✓✓✓✓×Peng et al. [27]PFSS✓✓✓✓×Zulqarnain et al. [42]IFHSS✓✓✓✓✓Proposed approachPFHSS✓✓✓✓✓


### 5.2. Discussion

Zadeh’s [1] FSs only addressed the rough and vague facts using MD considering the subattributes for each alternative. But, the FSs are unable to deal with the NMD of parameters. Atanassov’s [2] IFSs accommodate the unclear and undefined objects using MD and NMD. However, IFSs are unable to handle the circumstances when MD+NMD>1; on the contrary, our presented idea expertly compacts with such complications. Maji et al. [21] proposed the theory of IFSSs; the presented idea conducts the anxiety of the object in which the characteristics of MD and NMD can be used appropriately along with their parameterization with the following condition MD +NMD≤1. To handle these consequences, Peng et al. [27] suggested the idea of PFSSs by amending the condition MD+NMD≤1 to MD2+NMD2≤1 with their parametrization. But there is no information about the subattributes of the attributes under consideration in all the above studies. Therefore, the above theories are unable to address the situation when their subattributes are associated with the attributes. All prevailing hybrid structures of FSs cannot handle the NMem values of subattributes of considered n-tuple attributes. Zulqarnain et al. [42] extended the IFSSs to IFHSSs and proposed the CC and WCC for IFHSSs in which MD+NMD≤1 for each subattribute. But IFHSSs cannot provide any information on the Mem and NMem values of the subattribute of the considered attribute when MD+NMD≥1. It can be seen the finest choice of the projected approach simulates itself and ensures the success of the developed method as well as the responsibility.


### 5.3. Comparative Analysis

We endorse a new algorithmic rule for PFHSSs using developed PFHSIWA and PFHSIWG operators within the succeeding section. Consequently, we used the proposed algorithmic rule for any veridical problem, that is to say, supplier selection in SSCM. Results demonstrate that algorithmic governance is effective as well as sensible. From the above calculation, it can be observed thatχ3 supplier is the premium alternative for SSCM. From the exploration, it is terminated that the results attained from the proposed viewpoint are more than the consequences of the planned theories. Thus, compared to available techniques, established AOs addressed unsure and unclear information efficiently. However, under available MCDM methods, the most important benefit of the proposed approach is that it can serve more information in the data than the available methodology. The comparison between existing AOs and our developed operators is given in following Table 5. The presented approach contemplates the interaction among the Mem and NMem function of PFHSNs, which can attain the more realistic decision effects considering the parametric values of the multi-subattributes of the parameters.Table 5
Comparative analysis with existing operators.
MethodScore values for alternativesRanking orderχ1χ2χ3χ4χ5PFIWA [8]0.553740.339010.600190.520070.36813χ3>χ1>χ4>χ5>χ2PFIWG [9]0.493250.418370.730000.489060.46524χ3>χ1>χ4>χ5>χ2PFSWA [10]0.211730.220170.332150.270080.21893χ3>χ4>χ2>χ5>χ1PFSWG [10]0.205870.230660.329020.254620.21727χ3>χ4>χ2>χ5>χ1PFEWA [54]0.516860.548330.604670.590210.51235χ3>χ4>χ2>χ1>χ5PFEWG [54]0.542190.565970.621900.593810.52209χ3>χ4>χ2>χ1>χ5SPFWA [16]0.081580.076740.147620.099590.07985χ3>χ4>χ1>χ5>χ2IFHSWA [55]0.498300.417350.409350.461750.43247χ3>χ2>χ4>χ1>χ5IFHSWG [55]0.426150.361750.356350.407900.40635χ3>χ2>χ4>χ1>χ5Proposed PFHSIWA operator0.16670.24210.26730.25490.1148χ3>χ4>χ2>χ1>χ5Proposed PFHSIWG operator−0.0911−0.01320.05050.0159−0.1967χ3>χ4>χ2>χ1>χ5The existing PFIWA [8], PFIWG [9], PFEWA, PFEWG [54], and SPFWA [16] operators are not capable of dealing with the parametrization of the alternatives. The PFSWA and PFSWG [10] operators handle the parametric values of the alternatives but these operators cannot accommodate the multi-subattributes of the considered parameters. The prevailing IFHSWA and IFHSWG [55] operators competently deals the multi-sub attributes of the parameters comparative to above discuss operators. But IFHSSs cannot handle the situation when the sum of Mem and NMem values of the subattribute of the considered attribute exceeds 1. On the contrary, our proposed PFHSIWA and PFHSIWG operators competently accommodate the abovementioned shortcomings. So, we claim that our established operators are extraordinary than existing operators to solve imprecise as well as vague facts in DM procedure. The assistance of the deliberated approach along with related measures over present approaches is evading conclusions grounded on adverse reasons. Therefore, it is a useful tool for combining inaccurate and uncertain information in the DM process.


## 5.1. Superiority of the Proposed Method

Through this study, along with association, it is resolute that the concerns attained with the proposed method are rather extrageneral than either technique. However, the developed MCDM approach has been provided more information to cope with the hesitancy in the DM procedure related to the existing MCDM strategies. Besides, the numerous mixed structures of FSs have grown into a unique feature of PFHSSs; after adding some proper conditions, the general facts about the component can be stated precisely and logically, as shown in Table4. It is observed that the obtained results deliver extrainformation comparative to existing studies. The developed PFHSSs accurately accommodate more information considering the multi-subattributes of the parameters. It is quite an easy tool to mix inexact and unsure information within the DM process. Hence, the proposed methodology is pragmatic, diffident, and distinctive from available hybrid structures of fuzzy sets.Table 4
Comparison of PFHSSs with some prevailing models.
SetTruthinessFalsityParametrizationAttributesSubattributesZadeh [1]FS✓××✓×Atanassov [2]IFS✓✓×✓×Maji et al. [21]IFSS✓✓✓✓×Peng et al. [27]PFSS✓✓✓✓×Zulqarnain et al. [42]IFHSS✓✓✓✓✓Proposed approachPFHSS✓✓✓✓✓


## 5.2. Discussion

Zadeh’s [1] FSs only addressed the rough and vague facts using MD considering the subattributes for each alternative. But, the FSs are unable to deal with the NMD of parameters. Atanassov’s [2] IFSs accommodate the unclear and undefined objects using MD and NMD. However, IFSs are unable to handle the circumstances when MD+NMD>1; on the contrary, our presented idea expertly compacts with such complications. Maji et al. [21] proposed the theory of IFSSs; the presented idea conducts the anxiety of the object in which the characteristics of MD and NMD can be used appropriately along with their parameterization with the following condition MD +NMD≤1. To handle these consequences, Peng et al. [27] suggested the idea of PFSSs by amending the condition MD+NMD≤1 to MD2+NMD2≤1 with their parametrization. But there is no information about the subattributes of the attributes under consideration in all the above studies. Therefore, the above theories are unable to address the situation when their subattributes are associated with the attributes. All prevailing hybrid structures of FSs cannot handle the NMem values of subattributes of considered n-tuple attributes. Zulqarnain et al. [42] extended the IFSSs to IFHSSs and proposed the CC and WCC for IFHSSs in which MD+NMD≤1 for each subattribute. But IFHSSs cannot provide any information on the Mem and NMem values of the subattribute of the considered attribute when MD+NMD≥1. It can be seen the finest choice of the projected approach simulates itself and ensures the success of the developed method as well as the responsibility.


## 5.3. Comparative Analysis

We endorse a new algorithmic rule for PFHSSs using developed PFHSIWA and PFHSIWG operators within the succeeding section. Consequently, we used the proposed algorithmic rule for any veridical problem, that is to say, supplier selection in SSCM. Results demonstrate that algorithmic governance is effective as well as sensible. From the above calculation, it can be observed thatχ3 supplier is the premium alternative for SSCM. From the exploration, it is terminated that the results attained from the proposed viewpoint are more than the consequences of the planned theories. Thus, compared to available techniques, established AOs addressed unsure and unclear information efficiently. However, under available MCDM methods, the most important benefit of the proposed approach is that it can serve more information in the data than the available methodology. The comparison between existing AOs and our developed operators is given in following Table 5. The presented approach contemplates the interaction among the Mem and NMem function of PFHSNs, which can attain the more realistic decision effects considering the parametric values of the multi-subattributes of the parameters.Table 5
Comparative analysis with existing operators.
MethodScore values for alternativesRanking orderχ1χ2χ3χ4χ5PFIWA [8]0.553740.339010.600190.520070.36813χ3>χ1>χ4>χ5>χ2PFIWG [9]0.493250.418370.730000.489060.46524χ3>χ1>χ4>χ5>χ2PFSWA [10]0.211730.220170.332150.270080.21893χ3>χ4>χ2>χ5>χ1PFSWG [10]0.205870.230660.329020.254620.21727χ3>χ4>χ2>χ5>χ1PFEWA [54]0.516860.548330.604670.590210.51235χ3>χ4>χ2>χ1>χ5PFEWG [54]0.542190.565970.621900.593810.52209χ3>χ4>χ2>χ1>χ5SPFWA [16]0.081580.076740.147620.099590.07985χ3>χ4>χ1>χ5>χ2IFHSWA [55]0.498300.417350.409350.461750.43247χ3>χ2>χ4>χ1>χ5IFHSWG [55]0.426150.361750.356350.407900.40635χ3>χ2>χ4>χ1>χ5Proposed PFHSIWA operator0.16670.24210.26730.25490.1148χ3>χ4>χ2>χ1>χ5Proposed PFHSIWG operator−0.0911−0.01320.05050.0159−0.1967χ3>χ4>χ2>χ1>χ5The existing PFIWA [8], PFIWG [9], PFEWA, PFEWG [54], and SPFWA [16] operators are not capable of dealing with the parametrization of the alternatives. The PFSWA and PFSWG [10] operators handle the parametric values of the alternatives but these operators cannot accommodate the multi-subattributes of the considered parameters. The prevailing IFHSWA and IFHSWG [55] operators competently deals the multi-sub attributes of the parameters comparative to above discuss operators. But IFHSSs cannot handle the situation when the sum of Mem and NMem values of the subattribute of the considered attribute exceeds 1. On the contrary, our proposed PFHSIWA and PFHSIWG operators competently accommodate the abovementioned shortcomings. So, we claim that our established operators are extraordinary than existing operators to solve imprecise as well as vague facts in DM procedure. The assistance of the deliberated approach along with related measures over present approaches is evading conclusions grounded on adverse reasons. Therefore, it is a useful tool for combining inaccurate and uncertain information in the DM process.


## 6. Conclusion

In this article, PFHSSs consider solving the complexities of information related to unsatisfactory, instability, and deviation by considering MD and NMD on then-tuple subattributes of the suggested attributes. The core objective of this research is to propose novel operational laws considering the interaction. We also presented interaction aggregation operators, i.e., PFHSIWA and PFHSIWG, utilizing developed operational laws and discussed their desirable properties. Furthermore, based on developed interaction AOs, an MCDM approach has been established to solve real-life complications. To certify the applicability and practicality of our anticipated method, we planned an ephemeral comparative analysis of our developed methodology with some existing studies. From the obtained results, it can be decided absolutely that the predetermined methodology indicates that the experts have high stability and accessibility in the process of DM. The subsequent study will clarify the presentation of DM techniques using a number of other initiatives under PFHSSs, such as entropy and similarity measures. Furthermore, many other structures can be established and proposed, such as topological structure, algebraic structure, and configurable structure. In the future, PFHSSs can be extended to q-rung orthopair fuzzy hypersoft sets and spherical and T-spherical fuzzy hypersoft sets with their several AOs and decision-making approaches.



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*Source: 9964492-2021-08-13.xml*