# Analysis of Social Networks by Using Pythagorean Cubic Fuzzy Einstein Weighted Geometric Aggregation Operators

Authors: Tehreem; Amjad Hussain; Jung Rye Lee; Muhammad Sajjad Ali Khan; Dong Yun Shin
Journal: Journal of Mathematics
 (2021)

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## Abstract

Pythagorean cubic set (PCFS) is the combination of the Pythagorean fuzzy set (PFS) and interval-valued Pythagorean fuzzy set (IVPFS). PCFS handle more uncertainties than PFS and IVPFS and thus are more extensive in their applications. The objective of this paper is under the PCFS to establish some novel operational laws and their corresponding Einstein weighted geometric aggregation operators. We describe some novel Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operators to handle multiple attribute group decision-making problems. The desirable relationship and the characteristics of the proposed operator are discussed in detail. Finally, a descriptive case is given to describe the practicality and the feasibility of the methodology established.


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## Body

## 1. Introduction

Multicriteria decision-making (MCDM) is a process that can give the ranking result of finite alternatives according to the attribute value of different alternatives, and it is an important aspect of decision sciences. A significant part of the decision-making model that has been commonly used in human impacts is MCDM (or MCGDM) [1]. The assessment information is generally fuzzy because the real decision-making issues have always been created from a complicated context. In general, fuzzy data take two models: one quantitatively and one qualitatively. Fuzzy set (FS) [2], intuitionistic fuzzy set (IFS) [3], Pythagorean fuzzy set (PFS) [4], and so on, can express quantitative fuzzy knowledge. The theory of FS suggested by Zadeh [2] was used to explain fuzzy quantitative knowledge containing only a degree of membership. On this basis, Atanassov [5] proposed the idea of IFS as a generalization of FS; the important aspect is that it has two fuzzy values: the first is called membership grade and the second is called nonmembership grade. Sometimes, meanwhile, the two degrees do not satisfy the limit, so the square sum is less than or equal to one. The PFS was introduced by Yager [4] in which the sum of squares of membership and nonmembership is equal to or less than one. In certain conditions, PFS is capable of expressing the fuzzy data compared to the IFS. For instance, PFS improved the concept of IFS by enlarging its domain. To define this decision information, IFS is invalid, but it can be efficiently defined by PFS. In the Pythagorean fuzzy set, Peng et al. [6] introduced some characteristics, which are division, subtraction, and other significant properties.To understand multicriteria problems in group decision-making in the Pythagorean fuzzy setting, authors are concerned with the methods of dominance and a ranking of dependencies. For multicriteria decision-making based on Pythagorean fuzzy sets, Khan et al. established prioritized aggregation operators in [7]. Peng et al. [8] advanced linguistic Pythagorean fuzzy sets (LPFSs) and the Pythagorean fuzzy linguistic numbers’ operating laws and score function. An optimizing variance technique was developed by Wei et al. [9] to clarify problems involving decision-making depending on Pythagorean fuzzy environments valued at intervals. The Pythagorean fuzzy numbers (PFNs) subtraction and division acts were intended by Gou et al. [10]. The notion of the obvious concept of the Pythagorean fuzzy distance degree was provided by Pend et al. [11], which is categorized by a Pythagorean fuzzy number that will minimize a drawback of data additionally proceeding to provide imaginative proof. The well-known definition of the novel score function is also well defined. Liang et al. [12] introduced the Bonferroni weighted Pythagorean fuzzy geometric (BWPFG) operator.In [13], Garg introduced an interval-valued Pythagorean fuzzy geometric (IVPFG) operator and discussed a new precision function. Khan et al. improved the definition of the multiattribute decision-making TOPSIS system as well as established the integral Choquet method of TOPSIS on the basis of IVPFNs [14]. In [15], Khan suggested the GRA method for making multicriteria decisions under the Pythagorean fuzzy condition valued at intervals. The authors first developed the Choquet integral average interval-valued Pythagorean operator and then developed a system for making multiattribute decisions dependent on the GRA technique. An Einstein geometric intuitionistic fuzzy (EGIF) operator was introduced by Wang [16] and an ordered weighted Einstein geometric intuitionistic fuzzy (OWEGIF) operator.The definition of the intuitionistic fuzzy Einstein weighted averaging operator was introduced by Wang and Liu [17] and an ordered weighted Einstein average intuitionistic fuzzy (OWEAIF) operator. Einstein operations can be divided into two categories: Einstein sum and product. In [18], Garg implemented the Einstein sum definition of the Pythagorean fuzzy mean aggregation operators such as the average operator of Pythagorean fuzzy Einstein, the weighted average operator of Pythagorean fuzzy Einstein, the geometric operator of Pythagorean fuzzy Einstein, and the ordered geometric weighted operator of Pythagorean fuzzy Einstein. For more related work, one may refer to [19–39].We will use the Einstein product in this article and present the Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator definition. Under Pythagorean fuzzy data, these two are new decision-making methods, but the Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator is more reliable than mean aggregation operators.This paper is composed of nine sections. We begin with a brief overview relevant to the literature review in Section1. We provide essential concepts and consequences in Section 2 that we can include in the following aspects. In Section 3, we define the Pythagorean cubic fuzzy number and their properties. We propose Pythagorean cubic fuzzy Einstein operations in Section 4 and examine some excellent features of the suggested operations. We present a Pythagorean cubic fuzzy Einstein weighted geometric aggregation operator (PCFEWG) in Section 5. With Pythagorean cubic fuzzy data, we apply the (PCFEWG) operator to MADM in Section 6 and we also give a case of numerical development (PFEWG) operator in Section 7. In Section 8, the comparative analysis is given and the conclusion is in Section 9.


## 2. Preliminaries

We introduce a basic definition and essential characteristics in this section.Definition 1.
(see [8]). Let X^ be a universal set, then the fuzzy set (FS) F^ is defined as follows:(1)F^=x^,μ^F^x^|x^∈X^,where μ^F^x^ is a mapping from X^ to [0,1] and μ^F^x^ is known as the membership function of x^∈X^.Definition 2.
(see [3]). Let X^ be a universal set, then the intuitionistic fuzzy set (IFS) I^ is defined as follows:(2)I^=x^,μ^I^x^,ν^I^x^|x^∈X^,where μ^I^x^ and ν^I^x^ are a mapping from X^ to [0,1] also satisfy the condition 0≤μ^I^≤1,0≤ν^I^≤1 for all x^∈X^ and represent the membership and nonmembership function of x^ in X^.Definition 3.
(see [19]). Let X^ be a universal set, then the Pythagorean fuzzy set (PFS) P^ is defined as follows:(3)P^=x^,μ^P^x^,ν^P^x^|x^∈X^,where μ^P^ and ν^P^ are a mapping from X^ to [0,1] also satisfying the conditions 0≤μ^P^≤1,0≤ν^P^≤1, and 0≤μ^P^2≤1,0≤ν^P^2≤1, for all x^∈X^ and characterize the membership and nonmembership degree to set P^. Let π^P^x^=1−μ^P^2−ν^P^2, then it is known as the Pythagorean fuzzy index of x^∈X^ to set P^, representing the degree of indeterminacy of P^. Also, for every x^∈X^, we represent the Pythagorean fuzzy number (PFN) by B˜=ΛB˜,ΓB˜.Definition 4.
(see [19]). Let B˜1=ΛB˜1,ΓB˜1,B˜2=ΛB˜2,ΓB˜2, and B˜=ΛB˜,ΓB˜be three (PFNs) and λ^>0, then we have(1)
B˜1⊕B˜2=μ^B˜12+μ^B˜22−μ^B˜12μ^B˜22,ν^B˜1ν^B˜2;(2)
B˜1⊗B˜2=μ^B˜1μ^B˜2,ν^B˜12+ν^B˜22−ν^B˜12ν^B˜22;(3)
λ^B˜=1−1−μ^B˜2λ^,ν^B˜λ^;(4)
B˜λ^=μ^B˜λ^,1−1−ν^B˜2λ^;(5)
B˜c=ν^B˜,μ^B˜.Definition 5.
(see [20]). Let X^ be a universal set, then the object with the following formulation is an IVPFS set R^:(4)R^=x^,μ^R^x^,ν^R^x^|x^∈X^,Where μ^R^x^=μ^R^Lx^,μ^R^Lx^⊆0,1 and ν^R^x^=ν^R^Lx^,ν^R^Lx^⊆0,1 are the intervals, and μ^R^Lx^=infμ^R^x^ and μ^R^Ux^=Supμ^R^x^; similarly, ν^R^Lx^=infν^R^x^ and ν^R^Ux^=Supν^R^x^, for all x^∈X^.Also, 0≤μ^R^Ux^2+ν^R^Ux^2≤1. Let π^R^x^=π^R^Lx^,π^R^Lx^, for all x^∈X^, then it is known as the interval-valued Pythagorean fuzzy index of x^ to R^, where π^R^Lx^=1−μ^R^Lx^2+ν^R^Lx^2 and π^R^Ux^=1−μ^R^Ux^2+ν^R^Ux^2 which meet the requirements of the following relationship:(1)
Ifμ^R^Lx^=μ^R^Ux^ and ν^R^Lx^=ν^R^Ux^, then an IVPFS set becomes a PFS set.(2)
Ifμ^R^Ux^+ν^R^Ux^≤1, then an IVPFS becomes an IVIFS.Definition 6.
(see [21]). Let A^=μ^A^L,μ^A^U,ν^A^L,ν^A^U, A^1=μ^A^1L,μ^A^1U,ν^A^1L,ν^A^1U, and A^2=μ^A^2L,μ^A^2U,ν^A^2L,ν^A^2U arethree IVPFNs and λ^>0, then we have the following:(1)
λ^A^=1−1−μ^A^Lλ^,1−1−μ^A^Uλ^,ν^A^L,ν^A^U,(2)
A^λ^=μ^A^L,μ^A^U,1−1−ν^A^Lλ^,1−1−ν^A^Uλ^,(3)
A^1⊕A^2=μ^A^1L2+μ^A^2L2−μ^A^1L2μ^A^2L2,μ^A^1U2+μ^A^2U2−μ^A^1U2μ^A^2U2,ν^A^1Lν^A^2L,ν^A^1Uν^A^2U,(4)
A^1⊗A^2=μ^A^1Lμ^A^2L,μ^A^1Uμ^A^2U,ν^A^1L2+ν^A^2L2−ν^A^1L2ν^A^2L2,ν^A^1U2+ν^A^2U2−ν^A^1U2ν^A^2U2.Definition 7.
(see [21]). Let A^=μ^A^L,μ^A^U,ν^A^L,ν^A^U; the score function of A^ can be defined as follows using the IVPFN A^:(5)SA^=12μ^A^L2+μ^A^U2−ν^A^L2−ν^A^U2,whereSA^∈0,1.Definition 8.
(see [23]). Let A^=μ^A^L,μ^A^U,ν^A^L,ν^A^U; the accuracy function of A^ can be defined as follows using the IVPFN A^:(6)HA^=12μ^A^L2+μ^A^U2+ν^A^L2+ν^A^U2,whereHA^∈0,1.Definition 9.
(see [21]). Let A^=μ^A^L,μ^A^U,ν^A^L,ν^A^U and A^1=μ^A^1L,μ^A^1U,ν^A^1L,ν^A^1U be two IVPFNs, then(7)SA^=12μ^A^L2+μ^A^U2−ν^A^L2−ν^A^U2,SA^1=12μ^A^1L2+μ^A^1U2−ν^A^1L2−ν^A^1U2are the score of A^ and A^1, separately, while(8)HA^=12μ^A^L2+μ^A^U2+ν^A^L2+ν^A^U2,HA^1=12μ^A^1L2+μ^A^1U2+ν^A^1L2+ν^A^1U2are the accuracy of A and A1, separately, which meet the following criteria:(1)
IfSA^<SA^1, then A^<A^1;(2)
IfSA^>SA^1, then A^>A^1;(3)
IfSA^=SA^1, we have the following:(a)
IfHA^=HA^1, then A^=A^1,(b)
IfHA^<HA^1, then A^<A^1,(c)
IfHA^>HA^1, then A^>A^1.Definition 10.
(see [22]). Let X^ be a universal set. Then, a cubic set can be stated:(9)C=x^,μ¯Cx^,ν^Cx^|x^∈X^,where μ¯C is an interval-valued fuzzy set in X^ and ν^C is a fuzzy set in X^.Definition 11.
(see [19]). Let p1 and p2 be two PFNs, then the distance between p1 and p2 can be described as(10)dp1,p2=12μ^p12−μ^p22+ν^p12−ν^p22+π^p12−π^p22.Definition 12.
(see [23]). Let pi=ai,bi,ci,di,i=1,2, be two IVPFNs, then the distance between p1and p2is defined as follows:(11)dp1,p2=14a12−a22+b12−b22+c12−c22+d12−d22+π12−π22+ψ12−ψ22,where π1,ψ1=1−a12−c12,1−b12−d12 and π2,ψ2=1−a22−c22,1−b22−d22.Definition 13.
(see [24]). Let pi=ai,bi,ci,di,i=1,2,3,…,n, be the collection of IVPFNs, then IVPFWG operator is defined as(12)IVPFWGwp1,p2,…,pn=∏i=1naiwi,∏i=1nbiwi,1−∏i=1n1−ci2wi,1−∏i=1n1−di2wi,where w=w1,w2,…,wnT is the weight vector of pii=1,2,3,…,n and wi∈0,1 and ∑i=1nwi=1.Definition 14.
(see [24]). Let pi=ai,bi,ci,di,i=1,2,3,…,n, be the collection of IVPFNs, then IVPFOWG operator is defined as(13)IVPFOWGwp1,p2,…,pn=∏i=1naσiwi,∏i=1nbσiwi,1−∏i=1n1−cσi2wi,1−∏i=1n1−dσi2wi,where pσi is the i-th largest value and w=w1,w2,…,wnT is the weight vector of pii=1,2,3,…,n.Definition 15.
(see [24]). Let pi=ai,bi,ci,di,i=1,2,3,…,n, be the collection of IVPFNs, then IVPFHWG operator is defined as(14)IVPFHGwp1,p2,…,pn=∏i=1naτiwi,∏i=1nbτiwi,1−∏i=1n1−cτi2wi,1−∏i=1n1−dτi2wi,where w=w1,w2,…,wnT is the weight vector of pii=1,2,3,…,n.Definition 16.
(see [25, 26]). Let pi=ai,bi,ci,di,i=1,2,3,…,n, be the collection of IVPFNs, and δ>0, then the following operational laws are satisfied:(1)
δp1=1−1−a12δ,1−1−b12δ,c1δ,d1δ,(2)
p1δ=a1δ,b1δ,1−1−c12δ,1−1−d12δ,(3)
p1⊗p2=a1a2,b1b2,c12+c22−c12c22,d12+d22−d12d22,(4)
p1⊕p2=a12+a22−a12a22,b12+b22−b12b22,c1c2,d1d2.


## 3. Pythagorean Cubic Fuzzy Numbers and Their Characteristics

In this unit, we define some new concepts of the Pythagorean cubic fuzzy set and discuss the characteristics of the Pythagorean cubic fuzzy set that is not an intuitionistic cubic fuzzy set with the help of illustrations. In this article,pc stands for a Pythagorean cubic fuzzy set.Definition 17.
(see [27]). Let X^ be a fixed set, then a Pythagorean cubic fuzzy set can be defined as(15)pc=x,μc1x,νc1x|x∈X^,where(16)μc1x=Ax,λx,νc1x=A˜x,μx,0≤μc1x2+νc1x2≤1^,1.
The preceding condition may also be written as follows:(17)0≤supAx2+supA˜x2≤1,0≤λ2x+μ2x≤1.
For a Pythagorean cubic set, the degree of indeterminacy is classified as(18)πpc=1−supAx2−supA˜x2,1−λ2x−μ2x.
For simplicity, we callμc1,νc2 a Pythagorean cubic fuzzy number (PCFN) denoted by Pc=μc1,νc2.Example 1.
LetX=x1,x2,x3 be a fixed set and consider a set in X by(19)pc=x1,0.5,0.6,0.7,0.6,0.7,0.5,x2,0.4,0.7,0.6,0.5,0.4,0.6,x3,0.4,0.6,0.7,0.5,0.2,0.6.
Then, also0.72+0.52=0.49+0.25=0.74<1; similarly, we can calculate the other cases. Thus, pc1,pc2, and pc3 are (PCFNs). Therefore, pc are PCFS.Definition 18.
Letpc1=A1,λ1,A˜1,μ1,pc2=A2,λ2,A˜2,μ2 and pc=A,λ,A˜,μ be three PCFNs and δ>0, where A1=a1,b1,A˜1=a˜1,b˜1,A2=a2,b2,A˜2=a˜2,b˜2,A=a,b, and A˜=a˜,b˜; the operational laws are as follows:(1).
pc1⊕pc2=a12+a22−a12a22,b12+b22−b12b22,λ12+λ22−λ12λ22,a˜1,a˜2,μ1μ2,(2).
pc1⊗pc2=a1,a2,λ1λ2,a˜12+a˜22−a˜12a˜22,b˜12+b˜22−b˜12b˜22,μ12+μ22−μ12μ22,(3).
δpc1=1−1−a12δ,1−1−b12δ,1−1−λ12δ,a˜1a˜2δ,b˜1b˜2δ,μ1μ2δ,(4).
pc1δ=a1a2δ,b1b2δ,λ1λ2δ,1−1−a˜12δ,1−1−b˜12δ,1−1−μ12δ,(5).
pc1c=A˜1,μ1,A1,λ1.Theorem 1.
Letpc1=A1,λ1,A˜1,μ1,pc2=A2,λ2,A˜2,μ2, and pc=A,λ,A˜,μ be three PCFNs and δ>0,δ1>0, and δ2>0, where A1=a1,b1,A˜1=a˜1,b˜1,A2=a2,b2,A˜2=a˜2,b˜2,A=a,b,andA˜=a˜,b˜, then the following will hold:(1).
pc1⊕pc2=pc2⊕pc1,(2).
pc1⊗pc2=pc2⊗pc1,(3).
δpc1⊕pc2=δpc1⊕δpc2,(4).
δ1+δ2pc=δ1pc⊕δ2pc,(5).
pq1⊗pc2δ=pq1δ⊗pc2δ,(6).
pcδ1+δ2=pcδ1⊗pcδ2.Proof.
The proof is obvious.
We describe a score function and its basic properties to equate two PCFNs.Definition 19.
Letpc=A1,λ1,A˜1,μ1 be a PCFN, where A1=a1,b1,A˜1=a˜1,b˜1. We can introduce the score function of pc as(20)Spc=a1+b1−λ132−a˜1+b˜1−μ132,where Spc∈−1,1.Definition 20.
Letpc1=A1,λ1,A1,μ1 and pc2=A2,λ2,A˜2,μ2 be two PCFNs, Spc1be the score function of pc1, and Spc2 be the score function of pc2. Then,(1)
IfSpc1<Spc2, then pc1<pc2.(2)
IfSpc1>Spc2, then pc1>pc2.(3)
IfSpc1=Spc2, then pc1∼pc2.Example 2.
Letpc1=0.5,0.7,0.9,0.1,0.5,0.6,pc2=0.4,0.7,0.6,0.2,0.4,0.6, andpc3=0.03,0.8,0.9,0.0,0.3,0.7 be three PCFNs. Then, by Definition 18, we have Spq=0.01,Spc2=0.027 and Spc3=−0.0173. Thus, Spc2>Spc1>Spc3. Let pc1=0.5,0.7,0.9,0.1,0.5,0.6 and pc2=0.4,0.7,0.6,0.2,0.4,0.7 be two PCFNs. Then by Definition 19, we have Spq=0.01 and Spc2=0.01 Thus, Spq=Spc2.
Therefore, by Definition20, we cannot get information from Pc1 and Pc2. Usually, such a case grows in preparation. It is clear from Definition 20 that we are unable to consider the requirement that two PCFNs have the same ranking. On the other side, deviancy may be changed. The consistency property of all the components to the average number in a PCFNs returns that they may accept. For the comparison of two PCFNs, we present a definition of accuracy degree.Definition 21.
Letpc=A1,λ1,A1,μ1 be a PCFN. Then, we define the accuracy degree of pc which is denoted by αpc, where A1=a1,b1,A˜1=a˜1,b˜1 can be defined as(21)αpc=a1+b1−λ132+a˜1+b˜1−μ132,where αpc∈0,1.Definition 22.
Letpc1=A1,λ1,A1,μ1 and pc2=A2,λ2,A2,μ2 be two PCFNs, αpc1be the accuracy degree of pc1, and αpc2 be the accuracy degree of pc2. Then,(1)
Ifαpc1<αpc2, then pc1<pc2.(2)
Ifαpc1>αpc2, then pc1>pc2.(3)
Ifαpc1=αpc2, then pc1−pc2.Example 3.
From example2, since Spc1=0.01 and Spc2=0.01, thus, Spc1=Spc2..So, we have αpc1=0.01 and αpc2=0.044. Thus, αpc1>αpc2. Hence, pc1>pc2. As a result, the condition when two PCFNs have the same score has been resolved.Definition 23.
LetPc1 and pc2 be any two PCFNs on a set X=x1,x2,…,xn. The following is a definition of the distance measure between Pc1 and Pc2:(22)Dpc1,pc2=16a12−a12+b12−b12+a˜12−a˜12+b˜12−b˜12+λ12−λ12+μ12−μ12.Example 4.
Letpc1=0.6,0.7,0.3,0.5,0.7,0.8 and pc20.5,0.6,0.4,0.4,0.7,0.5be two PCFNs. Then,(23)Dpc1,pc2=160.36−0.25+0.49−0.36+0.25−0.16+0.49−0.49+0.9−0.16+0.64−0.25,Dpc1,pc2=160.11+0.13+0.09+0+0.74+0.39=0.2433.


## 4. Einstein Operations of Pythagorean Cubic Fuzzy Sets

In this section, we defined the Einstein productpc1⊗εpc2 and the Einstein sum pc1⊕εpc2on two PCFSs pc1 and pc2 which can be defined in the following forms.Definition 24.
Letpc1=A1,λ1,A˜1,μ1 and pc2=A2,λ2,A˜2,μ2 be two PCFNs, where A1=a1,b1,A˜1=a˜1,b˜1,A2=a2,b2,andA˜2=a˜2,b˜2, then(24)pc1⊗εpc2=a12,b12,λI2⋅a22,b22,λ221+1−a12,b12,λI21−a12,b12,λ12,a˜12,b˜12,μ12+a˜22,b˜22,μ221+a˜12,b˜12,μ12a˜22,b˜22,μ2,pc1⊕εpc2=a12,b12,λ12+a22,b22,λ221+a12,b12,λ12⋅a12,b12,λ12,a˜12,b˜12,μ2⋅a˜22,b˜22,μ221+1−a˜12,b˜12,μ12⋅1−a˜22,b˜22,μ2.Theorem 2.
Letn be any positive integer and pc1 is a PCFS, then the exponentiation operation pc1∧εn is a mapping from Z+×⋄⟶⋄:(25)pc1∧εn=2a12,b12,λ42n2−a12,b12,λ12n+a12,b12,λ12n,1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+1−a˜12,b˜12,μ12n,where pc1∧εn=pc1⊗εpc1⊗εpc1⊗ε,…,pc1⊗εpc1︷n. Moreover, pc1∧εn is a Pythagorean cubic fuzzy set PCFS, even if n∈ℝ+.Proof.
We may prove that equation (25) holds for all positive integers n using mathematical induction. First, it holds for n=1.(26)pc1∧ε1=2a12,b12,λ1212−a12,b12,λ121+q12,b12,λ121,1+a˜12,b˜12,μ121−1−a˜12,b˜12,μ1211+a˜12,b˜12,μ121+1−a˜12,b˜12,μ121.
Taking the left-hand side of the equation above,(27)pc1∧ε1=pc1=x,μpq,vpc|x∈X.
Taking the right-hand side of the equation above,(28)=2a12,b12,λ1212−a12,b12,λ121+a12,b12,λ1211+a˜12,b˜12,μ121−1−a˜12,b˜12,μ1211+a˜12b˜12,μ121+1−a˜12,b˜12,μ121=pc1=x,μpq,vpq|x∈X.
From equations (25) and (27), we have equation (25) which holds for n=1. Next, we show that equation (25) holds for n=k. If equation (25) holds for n=k, then equation (25) also holds for n=k+1.(29)2a12,b12,λ12k2−a12,b12,λ12k+a12,b12,λ12k,1+a˜12,b˜12,μ12k−1−a˜12,b˜12,μ12k1+a˜12b˜12,μ12k+1−a˜12b˜12,μ12k⊗εpc1=x,μp1,vpq|x∈X=2a12,b12,λ12k+12−a12,b12,λ12k+1+a12,b12,λ12k+11+a˜12,b˜12,μ12k+1−1−a˜12,b˜12,μ12k+11+a˜12b˜12,μ12k+1+1−a˜12,b˜12,μ12k+1.
Now, we’ll show that equation (25) is valid for every positive integer n,(30)0≤2a12,b12,λ12n2−a12,b12,λ12n+a12,b12,λ12n2,1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12b˜12,μ12n+1−a˜12,b˜12,μ12n2≤1,even if n∈ℝ+. Since 0≤μpq2x≤1,0≤vpq2x≤1,0≤μpq12x+vpq2x≤1, then 1−μpc12x≥vpc12x≥0,.so(31)0≤2a12,b12,λ12n2−a12,b12,λ12n+a12,b12,λ12n2,1+a˜12,b˜12,μ12w1n−1−a˜12,b˜12,μ12w1n1+a˜12,b˜12,μ12w1n+1−a˜12,b˜12,μ12m1n2≤10≤2a12,b12,λ12n2−a12,b12,λ12n+a12,b12,λ12n,1+a˜12,b˜12,μ12v1n−1−a˜12,b˜12,μ12v1n1+a˜12,b˜12,μ12v1n+1−a˜12,b˜12,μ12v1n≤1.
Since(32)2a12,b12,λ12n2−a12,b12,λ12n+a12,b12,λ12n=2a12,b12,λ12n1+1−a12,b12,λ12n+a12,b12,λ12n≤2a12,b12,λ12n1+1−a˜12,b˜12,u12n+a12,b12,λ12n≤2a12,b12,λ12n2a12,b12,λ12n2−a12,b12,λ12n+a12,b12,λ12n≤11+1−a˜12,b˜12,u12n+a12,b12,λ12n,again(33)1−vppc12x≥μpc12x≥01+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+1−a˜12,b˜12,μ12n=1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+1−a˜12,b˜12,μ12n≤1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+a12,b12,λ12n.
Thus,(34)1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+1−a˜12,b˜12,μ12n≤1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+a12,b12,λ12n.
From equations (14) and (34), we have(35)=2a12,b12,λ12n1+1−a˜12,b˜12,u12n+a12,b12,λ12n+1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+a12,b12,λ12n=2a12,b12,λ12n1+1−a˜12,b˜12,u12n+a12,b12,λ12n+1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+a12,b12,λ12n≤2a12,b12,λ12n+1+a˜12,b˜12,μ12n−a12,b12,λ12n1+1−a˜12,b˜12,u12n+a12,b12,λ12n≤1.
Thus,(36)0≤2a12,b12,λ12n1+1−a˜12,b˜12,u12n+a12,b12,λ12n+1+a˜12,b˜12,μ12n−1−a˜12,b˜12,μ12n1+a˜12,b˜12,μ12n+a12,b12,λ12n≤1.
Thus, a PCFSpciε defined above is a PCFS for any n∈ℝ+.


## 5. Pythagorean Cubic Fuzzy Einstein Weighted Geometric Aggregation Operator

Definition 25.
Letpcj=μpcj,vpj,j=1,…,m, be the collection of PCFVS with ≤L, then a PCFWGε operator of dimension n is a mapping PCFWGw˜ε:ϕm⟶ϕ, and(37)PCFWGw˜εpc1,pc2,…,pcm=pc1εw˜1⊗εpc2ε2⊗ε,…,⊗εpcmεm,where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of Pcjj=1,…,n such that w˜j∈0,1and ∑j=1mw˜j=1.Theorem 3.
Letpcj=μpcj,vptj,j=1,…,n, be the collection of PCFVs with ≤L, then their aggregated value by using the PCFWGε operator is also a PCFV, and let pc1=A1,λ1,A˜1,μ1 and pc2=A2,λ2,A˜2,μ2, where A1=a1,b1,A˜1=a˜1,b˜1,A2=a2,b2,andA˜2=a˜2,b˜2, then(38)PCFWGw˜εpc1,pc2,…,pcm=2∏j=1maj2w˜j∏j=1m2−aj2w˜j+∏j=1maj2w˜j,2∏j=1mbj2w˜j∏j=1m2−bj2w˜j+∏j=1mbj2w˜j,2∏j=1mλj2w˜j∏j=1m2−λj2w˜j+∏j=1mλj2w˜j;∏j=1m1+a˜j2w˜j+∏j=1m1−a˜j2w˜j∏j=1m1+a˜j2w˜j+∏j=1m1−a˜j2w˜k,∏j=1m1+b˜j2w˜j−∏j=1m1−b˜j2w˜j∏j=1m1+b˜j2w˜j+∏j=1m1−b˜j2w˜j,∏j=1m1+μj2w˜j−∏j=1m1−μj2w˜k∏j=1m1+μj2w˜j+∏j=1m1−μj2w˜k,where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of Pcj (j = 1,...,n) such that w˜j∈0,1 and ∑j=1mw˜j=1.Proof.
Mathematical induction may be used to prove this theorem. To begin, we prove that equation (38) holds for m = 1. Taking the left side,(39)PCFWGw˜εpc1,pc2,…,pcm=PCFWGw˜εpc=pcεw˜.
Now, taking right-hand side,(40)=2∏j=1maj2w˜j∏j=1m2−aj2w˜j+∏j=1maj2w˜j,2∏j=1mbj2w˜j∏j=1m2−bj2w˜j+∏j=1mbj2w˜j,2∏j=1mλj2w˜j∏j=1m2−λj2w˜j+∏j=1mλj2w˜j;∏j=1m1+a˜j2w˜j−∏j=1m1−a˜j2w˜j∏j=1m1+a˜j2w˜j+∏j=1m1−a˜j2w˜k,∏j=1m1+b˜j2w˜j−∏j=1m1−b˜j2w˜j∏j=1m1+b˜j2w˜j+∏j=1m1−b˜j2w˜j,∏j=1m1+μj2w˜j−∏j=1m1−μj2w˜k∏j=1m1+μj2w˜j+∏j=1m1−μj2w˜k.
From equations (39) and (40), we have equation (38) which holds for m = 1. Now, we show that equation (38) holds for m = k.(41)PCFWGw˜εpc1,pc2,…,pck=2∏j=1kaj2w˜j∏j=1k2−aj2w˜j+∏j=1kaj2w˜j,2∏j=1kbj2w˜j∏j=1k2−bj2w˜j+∏j=1kbj2w˜j,2∏j=1kλj2w˜j∏j=1k2−λj2w˜j+∏j=1kλj2w˜j;∏j=1k1+a˜j2w˜j−∏j=1k1−a˜j2w˜j∏j=1k1+a˜j2w˜j+∏j=1k1−a˜j2w˜k,∏j=1k1+b˜j2w˜j−∏j=1k1−b˜j2w˜j∏j=1k1+b˜j2w˜j+∏j=1k1−b˜j2w˜j,∏j=1k1+μj2w˜j−∏j=1k1−μj2w˜k∏j=1k1+μj2w˜j+∏j=1k1−μj2w˜k.
Next, we are going to show that equation (38) holds for m = k + 1.(42)PCFWGw˜εpc1,pc2,…,pck+1=2∏j=1kaj2w˜j∏j=1k2−aj2w˜j+∏j=1kaj2w˜j,2∏j=1kbj2w˜j∏j=1k2−bj2w˜j+∏j=1kbj2w˜j,2∏j=1kλj2w˜j∏j=1k2−λj2w˜j+∏j=1kλj2w˜j;∏j=1k1+a˜j2w˜j−∏j=1k1−a˜j2w˜j∏j=1k1+a˜j2w˜j+∏j=1k1−a˜j2w˜j,∏j=1k1+b˜j2w˜j−∏j=1k1−b˜j2w˜j∏j=1k1+b˜j2w˜j+∏j=1k1−b˜j2w˜j,∏j=1k1+μj2w˜j−∏j=1k1−μj2w˜k∏j=1k1+μj2w˜j+∏j=1k1−μj2w˜k⊗ε2∏j=1k+1aj2w˜j∏j=1k+12−aj2w˜j+∏j=1k+1aj2w˜j,2∏j=1k+1bj2w˜j∏j=1k+12−bj2w˜j+∏j=1k+1bj2w˜j,2∏j=1k+1λj2w˜j∏j=1k+12−λj2w˜j+∏j=1k+1λj2w˜j;∏j=1k+11+a˜j2w˜j−∏j=1k+11−a˜j2w˜j∏j=1k+11+a˜j2w˜j+∏j=1k+11−a˜j2w˜j,∏j=1k+11+b˜j2w˜j−∏j=1k+11−b˜j2w˜j∏j=1k+11+b˜j2w˜j+∏j=1k+11−b˜j2w˜j,∏j=1k+11+μj2w˜j+∏j=1k+11−μj2w˜j∏j=1k+11+μj2w˜j+∏j=1k+11−μj2w˜j.
Let(43)s1=2∏j=1kaj2w˜j∏j=1k2−aj2w˜j+∏j=1kaj2w˜j,2∏j=1kbj2w˜j∏j=1k2−bj2w˜j+∏j=1kbj2w˜j,s2=2∏j=1kλj2w˜j∏j=1k2−λj2w˜j+∏j=1kλj2w˜j,s3=∏j=1k1+a˜j2w˜j−∏j=1k1−a˜j2w˜j∏j=1k1+a˜j2w˜j+∏j=1k1−a˜j2w˜j,∏j=1k1+b˜j2w˜j−∏j=1k1−b˜j2w˜j∏j=1k1+b˜j2w˜j+∏j=1k1−b˜j2w˜j,s4=∏j=1k1+μj2w˜j−∏j=1k1−μj2w˜j∏j=1k1+μj2w˜j+∏j=1k1−μj2w˜j,u1=2∏j=1k+1aj2w˜j∏j=1k+12−aj2w˜j+∏j=1k+1aj2w˜j,2∏j=1k+1bj2w˜j∏j=1k+12−bj2w˜j+∏j=1k+1bj2w˜j,u2=2∏j=1k+1λj2w˜j∏j=1k+12−λj2w˜j+∏j=1k+1λj2w˜j,u3=∏j=1k+11+a˜j2w˜j−∏j=1k+11−a˜j2w˜j∏j=1k+11+a˜j2w˜j+∏j=1k+11−a˜j2w˜j,∏j=1k+11+b˜j2w˜j−∏j=1k+11−b˜j2w˜j∏j=1k+11+b˜j2w˜j+∏j=1k+11−b˜j2w˜j,u4=∏j=1k+11+μj2w˜j−∏j=1k+11−μj2w˜j∏j=1k+11+μj2w˜j+∏j=1k+11−μj2w˜j.
Now, putting these values in equation (40), we have(44)PCFWGwεpc1,pc2,…,pck+1=s1s2,s3s4⊗zu1u2,u3u4s1u12s22u22+s12u12−s22u12−s12u22.
Now, putting the values in equation (42), we have(45)PCFWGw˜εpc1,pc2,…,pck+1=2∏j=1kaj2w˜j∏j=1k2−aj2w˜j+∏j=1kaj2w˜j,2∏j=1kbj2w˜j∏j=1k2−bj2w˜j+∏j=1kbj2w˜j,2∏j=1kλj2w˜j∏j=1k2−λj2w˜j+∏j=1kλj2w˜j;∏j=1k1+a˜j2w˜j−∏j=1k1−a˜j2w˜j∏j=1k1+a˜j2w˜j+∏j=1k1−a˜j2w˜j,∏j=1k1+b˜j2w˜j−∏j=1k1−b˜j2w˜j∏j=1k1+b˜j2w˜j+∏j=1k1−b˜j2w˜j,∏j=1k1+μj2w˜j−∏j=1k1−μj2w˜k∏j=1k1+μj2w˜j+∏j=1k1−μj2w˜k⊗ε2∏j=1k+1aj2w˜j∏j=1k+12−aj2w˜j+∏j=1k+1aj2w˜j,2∏j=1k+1bj2w˜j∏j=1k+12−bj2w˜j+∏j=1k+1bj2w˜j,2∏j=1k+1λj2w˜j∏j=1k+12−λj2w˜j+∏j=1k+1λj2w˜j∏j=1k+11+a˜j2w˜j−∏j=1k+11−a˜j2w˜j∏j=1k+11+a˜j2w˜j+∏j=1k+11−a˜j2w˜j,∏j=1k+11+b˜j2w˜j−∏j=1k+11−b˜j2w˜j∏j=1k+11+b˜j2w˜j+∏j=1k+11−b˜j2w˜j,∏j=1k+11+μj2w˜j+∏j=1k+11−μj2w˜j∏j=1k+11+μj2w˜j+∏j=1k+11−μj2w˜j.
Equation (38) holds for m = k + 1. Thus, equation (38) holds for all m.Lemma 1.
Letpcj>0,w˜j>0,j=1,…,n and ∑j=1mw˜j=1. Then,(46)∏j=1mpcjw˜j≤∑j=1mw˜jpcj,where the equality holds if and only if pc1=pc2=⋯=pcm.Theorem 4.
Letpcj=μpcj,vptj,j=1,…,n, be the collection of PCFVs with ≤L, then(47)PCFWGw˜pc1,pc2,…,pcm≤PCFWGw˜εpc1,pc2,…,pcm,where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of pcj (j = 1,..., n) such that w˜j∈0,1 and ∑j=1mw˜j=1.ProofStraight.
forward.Theorem 5.
Letpcj(j = 1,...,n) be the collection of PCFVs with ≤L, where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of (j = 1,...,n) such that w˜j∈0,1 and i∑j=1mw˜j=1. Then,(1).
Idempotency: if allPcj (j = 1,...,n) are equal, i.e., Pcj (j = 1,...,n) =  Pcj, then(48)PCFWGw˜εpc1,pc2,…,pcm=pcj.(2).
Boundary:(49)pmin≤PCFWGw˜εpc1,pc2,…,pcm≤pmaxfor everyw˜.(3).
Monotonicity: letpcj∗=μpcj∗,vpcj∗,j=1,…,n be the collection of PCFVs with ≤L, and μpcj≤μpcj∗, vpcj≤vpcj∗, for all, then(50)PCFWGw˜εpc1,pc2,…,pcm≤PCFWGw˜εpc1∗,p2∗,…,pcm∗for everyw˜.Proof.
(1)
Idempotency: since(51)PCFWGw˜εpc1,pc2,…,pcm=2∏j=1maj2w˜j∏j=1m2−aj2w˜j+∏j=1maj2w˜j,2∏j=1mbj2w˜j∏j=1m2−bj2w˜j+∏j=1mbj2w˜j,2∏j=1mλj2w˜j∏j=1m2−λj2w˜j+∏j=1mλj2w˜j;∏j=1m1+a˜j2w˜j−∏j=1m1−a˜j2w˜j∏j=1m1+a˜j2w˜j+∏j=1m1−a˜j2w˜k,∏j=1m1+b˜j2w˜j−∏j=1m1−b˜j2w˜j∏j=1m1+b˜j2w˜j+∏j=1m1−b˜j2w˜j,∏j=1m1+μj2w˜j−∏j=1m1−μj2w˜k∏j=1m1+μj2w˜j+∏j=1m1−μj2w˜k,For simplicity, we use the notation of PCFSs. Letpcj=μpcj,vptj where μpq=A1,λ1,A˜1,μ1 and μpc2=A2,λ2,A˜2,μ2, where A1=a1,b1,A˜1=a˜1,b˜1,A2=a2,b2,andA˜2=a˜2,b˜2, then the above equation can be written in the following form:(52)PCFWGw˜εpc1,pc2,…,pcm=2∏j=1mμpcj2w˜j∏j=1m2−μpcj2w˜j+∏j=1mμpcj2w˜j,∏j=1m1+ν˜pcj2w˜j−∏j=1m1−ν˜pcj2w˜j∏j=1m1+ν˜pcj2w˜j+∏j=1m1−ν˜cjw˜j.Now,pcj(j = 1,...,n) = pc. Then, equation (26) can be written as(53)PCFWGw˜εpc1,pc2,…,pcm=2μpcj22−μpcj2+μpcj2,1+ν˜pcj2−1−ν˜pcj21+ν˜pcj2+1−ν˜cj.(2)
Boundary:(54)pmin≤PCFWGw˜εpc1,pc2,…,pcm≤pmaxfor everyw˜,wherepmin=minpcj and pmax=maxpcj. Let fx=2−x2/x2,x∈0,1, then fx=−2/x3x2/2−x2<0. So, fx is the decreasing function on 0,1. Since μpcmin≤μpcj≤μpcmax for all j, then fμpcmax≤fμpc≤fμpcminj=1,…n2−μpcmax2/μpcmax2≤2−μpc2/μpc2≤2−μpcmin2/μpcmin2where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of pcjj=1,…,n such that w˜j∈0,1 and ∑j=1mw˜j=1. Then, we have(55)⟺∏j=1m2−μpcmax2μpcmax2w˜j≤∏j=1m2−μpc2μpc2w˜j≤∏j=1m2−μpcmin2μpcmin2w˜j⇔∏j=1m2−μpcmax2μpcmax2∑j=1mw˜j≤∏j=1m2−μpc2μpc2∑j=1mw˜j≤∏j=1m2−μpcmin2μpcmin2∑j=1mw˜j⇔νpcmin≤∏j=1mμpc2w˜j∏j=1m2−μpc2w˜j+∏j=1mμpc2w˜j≤μpcmax.Again, lethy=1−y2/1+y2,x∈0,1, thenhy=−2y/1−y321+y2/1−y2<0. So, hy is the decreasing function on 0,1. Since vpcmin≤vpcj≤vpcmaxfor all j.Then, hvpcmax≤hvpc≤hvpcmin for all j1−vpcmax2/1+vpcmax2≤1−vpc2/vpc2≤1−vpcmin2/vpcmin2, where w˜=w˜1,w˜2,…,w˜mT is the weighted vector of pcjj=1,…,n such that w˜j∈0,1and ∑j=1mw˜j=1. Then, we have(56)⇔∏j=1m1−νpcmax21+νpcmax2w˜j≤∏j=1m1−νpc21+νpc2w˜j≤∏j=1m1−νpcmin21+νpcmin2w˜j⇔∏j=1m1−νpcmax21+νpcmax2∑j=1mw˜j≤∏j=1m1−νpc21+νpc2∑j=1mw˜j≤∏j=1m1−νpcmin21+νpcmin2∑j=1mw˜j⇔νpcmax≤∏j=1m1+νpc2w˜j−∏j=1m1−νpc2w˜j∏j=1m1−νpc2w˜j+∏j=1m1+νpc2w˜j≤μpcmin.LetPCFWGw˜εpc1,pc2,…,pcm=pc. Then, equations (52) and (55) can be written as μpcmin≤μpcj≤μpcmax and vpcmin≤vpcj≤vpcmax, respectively. Thus, Spc=μpc2−vpc2≤μpcmax2−vpcmax2=Spcmax and Spc=μpc2−vpc2≥μpcmin2−vpcmin2=Spcmin.IfSpc<Spcmax and Spc>Spcmin, then(57)pcmin<PCFWGw˜εpc1,pc2…,pcm<pcmaxfor allw˜.IfSpc=Spcmax, then μpc2=μ2pccmax and vpc2=vpcmax2.Thus,Hpc=μpc2+vpc2=μpcmax2+vpcmax2=Hpcmax. Then, we have(58)PCFWGw˜εpc1,pc2,…,pcm=pcmax,for everyw˜.IfSpc=Spcmin, then μpc2−vpc2=μpcmin2−vpcmin2, then μpc2=μpcmin2 and vpc2=vpcmin2. Thus, Hpc=μpc2+vpc2=μpcmin2+vpcmin2=Hpcmin. Then, we have(59)PCFWGw˜εpc1,pc2,…,pcm<pcminfor everyw˜.Thus, from equations (55) to (57), we have pcmin<PCFWGw˜εpc1,pc2,…,pcm<pcmax for every w˜.(3)
Monotonicity:
The proof follows from (2).


## 6. An Application of the Pythagorean Cubic Fuzzy Einstein Weighted Geometric (PCFEWG) Aggregation Operator to Group Decision-Making Problems

In this unit, we develop an application of Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator to multicriteria decision-making problem.Algorithm. Let F=F1,F2,…,Fn be the set of n alternatives, H=H1,H2,…,Hm be the set of m attributes, and D⟶=D⟶1,D⟶2,…,D⟶k be the set of k decision makers. Let w=w1,w2,…,wmT be the weighted vector of the attributes Hii=1,2,…,m, such that wi∈0,1 and ∑i=1mwi=1. Let η⟶=η⟶1,η⟶2,η⟶3,…,η⟶kT be the weighted vector of the decision makers D⟶ss=1,2,…,k, such that η⟶s∈0,1 and ∑s=1kη⟶s=1. This method has the following steps:Step 1. In this step, we construct the Pythagorean cubic fuzzy decision-making matrices, D⟶s=α⟶jisn×ms=1,2,…,k. If the criteria have two types, such as benefit criteria and cost criteria, then the Pythagorean cubic fuzzy decision matrices, D⟶s=α⟶jisn×m can be converted into the normalized Pythagorean cubic fuzzy decision matrices, R⟶s=εsm×n, where εijs. If all the criteria have the same type, then there is no need of normalization.Step 2. We use the Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator to aggregate all the individual normalized Pythagorean cubic fuzzy decision matrices, R⟶s=εsm×ns=1,2,…,k, into the single Pythagorean cubic fuzzy decision matrix, R⟶=εijm×n.Step 3. We aggregate all the preference values εijj=1,2,3,…,n,i=1,2,…,m by using the PCFEWG operator and get the overall preference values εjj=1,2,…,n corresponding to the alternatives Fjj=1,…,n.Step 4. We calculate the scores of εijj=1,2,3,…,n. If there is no difference between two or more than two scores, then we must have to find out the accuracy degrees of the collective overall preference values.Step 5. We arrange the scores of all the alternatives in the form of descending order and select that alternative which has the highest score function.


## 7. Numerical Example

In Pakistan’s stock exchange, listed Internet companies play an important role. The performance of listed companies affects capital market resource allocation and has become a common concern of shareholders, creditors, government bodies, and other stakeholders. An investment firm would like to invest a sum of money in stocks on the Internet. So, the investment bank employs three kinds of experts to determine the possible investment value: market maker, dealer, and finder. Three Internet stocks are chosen in which the earnings ratio is higher than other stocks: (1) is PTCL; (2) is NayaTel; (3) is Wi-Tribe out of three characteristics: (1) is the trend in the stock market; (2) is in the course of policy; (3) is the annual results. About the attributes Aj (j = 1, 2, 3), the three experts test Internet stocks xi (I = 1, 2, 3) and create the following three Pythagorean cubic fuzzy decision matrices in Table 1. Tablesss 2 and 3 display the expert weights and attribute weights, which all take the form of PCFEs, respectively. Then, to get the most desirable alternative(s), which includes the following steps, we use the approach developed in Section 6:Step 1. The decision maker gives his decision in Tables 1–3.Step 2. We apply the Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator to aggregate all the individual normalized Pythagorean cubic fuzzy decision matrices R⟶s=εsm×n, s=1,2,…,k, into the single Pythagorean cubic fuzzy decision matrix, R⟶=εijm×n.Aggregated Pythagorean cubic fuzzy decision matrixD1,(60)X1=.4660,.6120,.7323;.5760,.7056,.5726,X2=.5780,.7306,.5360;.4810,.6235,.7283,X3=.6662,.7908,.5691;.4661,.5657,.7381.Aggregated Pythagorean cubic fuzzy decision matrixD2,(61)X1=.5700,.6742,.6364;.5532,.7443,.5563,X2=.5095,.6564,.5333;.5972,.6994,.5637,X3=.6733,.7002,.6155;.4523,.5812,.5810.Aggregated Pythagorean cubic fuzzy decision matrixD3,(62)X1=.6204,.7207,.5655;.4523,.5810,.5703,X2=.6784,.7788,.7024;.4233,.5232,.5434,X3=.6291,.7306,.4948;.4765,.6040,.6994.Step 3. We aggregate all the preference values, which are(63)X1=.6808,.7692,.7472;.4468,.5859,.4777,X2=.7041,.8039,.7070;.4312,.5310,.5221,X3=.7591,.8173,.6866;.3902,.4940,.5743.Step 4. We calculate the scores of Xjj=1,2,3.(64)SX1=.0207,SX2=.0532,SX3=.0773.Step 5. We organize the scores of the alternatives in descending order and choose the highest score function. Hence, X3>X2>X1. Thus, the most wanted alternative is X3.Table 1
Decision matrixCby decision maker D1.
A1A2A3A4X1.5,.7;.5.4,.5;.6.3,.4;.8.8,.9;.6.5,.7;.8.4,.6;.6.6,.7;.8.5,.6;.5X2.5,.6;.8.4,.7;.6.6,.7;.6.5,.6;.7.6,.8;.2.5,.6;.6.6,.8;.7.5,.6;.6X3.8,.9;.3.3,.4;.9.7,.8;.9.5,.6;.3.6,.8;.5.5,.6;.7.6,.7;.6.5,.6;.8Table 2
Decision matrixCby decision maker D2.
A1A2A3A4X1.5,.6;.8.7,.8;.5.4,.5;.8.7,.9;.5.7,.8;.4.4,.5;.7.7,.8;.6.3,.6;.5X2.7,.8;.4.4,.5;.7.4,.5;.7.7,.8;.5.5,.8;.7.6,.7;.4.5,.6;.4.5,.7;.6X3.7,.8;.6.4,.5;.5.6,.7;.7.5,.6;.6.7,.5;.7.5,.7;.6.7,.8;.5.4,.5;.6Table 3
Decision matrixCby decision maker D3.
A1A2A3A4X1.7,.8;.4.4,.5;.7.6,.7;.4.5,.6;.7.6,.7;.8.5,.6;.4.6,.7;.7.4,.6;.4X2.6,.7;.7.5,.6;.6.7,.8;.5.4,.5;.7.7,.8;.7.4,.5;.5.7,.8;.9.4,.5;.3X3.5,.6;.6.6,.7;.5.6,.7;.5.5,.6;.8.7,.8;.4.4,.5;.7.7,.8;.5.4,.6;.7


## 8. Comparison Analysis

The same numerical example is solved by using other aggregation operators, including IFEWG (intuitionistic fuzzy Einstein weighted geometric) operator, IFEOWG (intuitionistic fuzzy Einstein ordered weighted geometric) operator, PFEWG (picture fuzzy Einstein weighted geometric) operator, PFEOWG (picture fuzzy Einstein ordered weighted geometric) operator, PyFEWG (Pythagorean fuzzy Einstein weighted geometric) operator, PyFEOWG (Pythagorean fuzzy Einstein ordered weighted geometric) operator, ICFEWG (intuitionistic cubic fuzzy Einstein weighted geometric) operator, ICFEOWG (intuitionistic cubic fuzzy Einstein ordered weighted geometric) operator, CPFEWG (cubic picture fuzzy Einstein weighted geometric) operator, and CPFEOWG (cubic picture fuzzy Einstein ordered weighted geometric) operator to demonstrate the efficiency and eminent benefits of the proven aggregation operators, by ignoring the additional preference matrix in some existing operators. Different aggregation operators have distinct strategic classifications so that, in compliance with their consultation, they may retain a small disparity. By contrast, the appropriate choice developed by any aggregation operator is important and recognizes the proposed solution’s feasibility and effectiveness of aggregation operators. Table4 gives a comparative study of the final rankings of all aggregation operators.Table 4
Final ranking comparative study with existing aggregation operators.
ModelsAggegation operatorsRankingIFEWGIntuitionistic fuzzy Einstein weighted geometric [18]X3>X1>X2IFEOWGIntuitionistic fuzzy Einstein ordered weighted geometric [18]X3>X1>X2PFEWGPicture fuzzy Einstein weighted geometric [28]X3>X2>X1PFEOWGPicture fuzzy Einstein ordered weighted geometric [28]X3>X2>X1PyFEWGPythagorean fuzzy Einstein weighted geometric [29]X3>X2>X1PyFEOWGPythagorean fuzzy Einstein ordered weighted geometric [29]X3>X2>X1ICFEWGIntuitionistic cubic fuzzy Einstein weighted geometric [22]X3>X2>X1ICFEOWGIntuitionistic cubic fuzzy Einstein ordered weighted geometric [22]X3>X2>X1PCFEWGPythagorean cubic fuzzy Einstein weighted geometric (proposed)X3>X2>X1PCFEOWGPythagorean cubic fuzzy Einstein ordered weighted geometric (proposed)X3>X2>X1


## 9. Conclusion

We introduced the Pythagorean cubic fuzzy set, which is a generalization of the interval-valued Pythagorean fuzzy set, in this paper. Einstein’s Pythagorean cubic fuzzy weighted geometric operator has been described (PCFEWG). We also discussed some of the fundamental properties of this operator, such as idempotency, boundary, and monotonicity. The Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG) operator was then used to deal with different parameters for decision-making problems under Pythagorean cubic fuzzy details. We developed a multicriteria decision-making algorithm for Pythagorean cubic fuzzy Einstein weighted geometric problems (PCFEWG). Finally, we put together a numerical example of a decision-making problem.In future, we can extend this concept for spherical cubic fuzzy sets and their application in multicriteria group decision-making, pattern recognition, and cluster analysis. We can also extend Pythagorean cubic fuzzy sets for various aggregation operators such as Hamacher, Dombi, Haronian mean, Bonferroni mean, TOPSIS, and their applications in group decision-making.



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