Einstein Weighted Geometric Operator for Pythagorean Fuzzy Hypersoft with Its Application in Material Selection

1  Introduction

The solution to the problems in our daily lives is based on the classification of information, data, the collection of facts, etc. The critical question in decision analysis is the absence of accurate facts. This statistical difference is bridged by taking a scientific model and applying the appropriate DM. DM’s ideas can support the manufacturing enterprise, assemble, and categorize multiple priorities from best to worst viable alternative. As a result, it is a tool to help us select, categorize, and establish our prospects and comprehensively evaluate alternatives. MS is intense in enterprise and product development. The material chosen affects the manufacturer’s success and affordability [1]. The manufacturing enterprise suffers from legislation, cost, and penetrating global goals, often inadequate content. The persistence of the manufactured equipment strategy is to select components with state-of-the-art light design standards while providing the best offer at the lowest reasonable price [2]. However, these goals and obstacles are common in conflict situations, so it is essential to address which feature is more important. Suppose the appropriate method is not ready for the design approach. The design method’s funding and resource aspects cannot be used in the restructuring or industrial agenda section [3]. Identifying the best materials is essential because design concerns are not correct. Eliminate inappropriate alternatives and manage high quality. Variables that interfere with selecting specific material engineering applications should use logical and straightforward applications [4].

MCDM has deliberated the best applicable procedure for the verdict and the best adequate alternative from all possible choices, ensuing criteria, or attributes. In real-life circumstances, most decisions are taken when the objectives and limitations are usually indefinite or ambiguous. To overcome such ambiguities and anxieties, Zadeh offered the idea of the fuzzy set (FS) [5], a prevailing tool to handle the obscurities and uncertainties in DM. Such a set allocates to all objects a membership value ranging from 0 to 1. Experts mainly consider membership and non-membership value in the DM process that FS cannot handle. Atanassov [6] introduced the generalization of the FS, the idea of the intuitionistic fuzzy set (IFS), to overcome the constraint mentioned above. In 2011, Wang et al. [7] presented numerous operations on IFS, such as Einstein product, Einstein sum, etc., and constructed two aggregation operators (AOs). They also discussed some essential properties of these operators and utilized their proposed operator to resolve multi-attribute decision making (MADM) for the IFS information.

The models mentioned above have been well-recognized by specialists. Still, the existing IFS cannot handle the inappropriate and vague data because it is deliberate to envision the linear inequality concerning the membership and non-membership grades. For example, if decision-makers choose membership and non-membership values 0.9 and 0.6, respectively, then 0.9+0.6≥1. The IFS mentioned above theory cannot be applied to this data. To resolve the limitation described above, Yager [8] presented the idea of the Pythagorean fuzzy set (PFS) by improving the basic circumstance a+b≤1 to a2+b2≤1 and developed some results associated with score function and accuracy function. Ejegwa [9] extended the notion of PFS and presented a decision-making technique. Rahman et al. [10] formed the Pythagorean fuzzy Einstein weighted geometric operator and presented a multi-attribute group decision making (MAGDM) methodology utilizing the proposed operator. Zhang et al. [11] developed some basic operational laws and prolonged the TOPSIS method to resolve MCDM complications for PFS information. Pythagorean fuzzy power AOs along with essential characteristics were introduced by Wei et al. [12]. They also recommended a DM technique to resolve MADM difficulties based on presented operators. Wang et al. [13] offered the interaction operational laws for PFNs, and developed power Bonferroni mean operators under the PFS environment. They also discussed some definite cases of developed operators and their basic characteristics. IIbahar et al. [14] offered the Pythagorean fuzzy proportional risk assessment technique to assess professional health risk. Zhang [15] proposed a novel decision-making (DM) approach based on similarity measures to resolve multi-criteria group decision making (MCGDM) difficulties for the PFS information.

Peng et al. [16] introduced the division and subtraction operations for Pythagorean fuzzy numbers (PFNs), proved their basic properties, and presented a superiority and inferiority ranking approach under the considered environment. Garg [17] introduced operational laws based on Einstein norms for PFNs, proposed weighted average and ordered weighted average operators, and then utilized these operators for DM. Garg [18] presented the series of generalized geometric AOs for PFS. Garg [19] introduced logarithmic operational laws for the PFS and constructed various weighted operators based on the proposed logarithm operational laws. Gao et al. [20] developed numerous interaction AOs under the PFS environment. Wang et al. [21] offered the interactive Hamacher operations for the PFS and settled on a DM method to solve MCDM difficulties. Wang et al. [22] utilized the interval-valued PFS, presented some novel PFS operators, and offered a DM approach to resolve the MCGDM complications. Moreover, to deal with the MCDM complexities. Peng et al. [23] explored some new inequalities for AOs under PFS. They introduced some point operators under the PFS environment. They combined the Pythagorean fuzzy point operators with the generalized AOs and offered a MADM approach based on settled operators. Moreover, Arora et al. [24] presented basic operational laws and suggested several selected AOs for linguistic IFSs. Ma et al. [25] modified the existing score function and accuracy function for PFNs and defined novel AOs for PFS.

All the methods mentioned above have too many applications in many fields. But due to their inefficiency, these methods have many limitations in terms of parameterization tools. In presenting the solution to obscurity and ambiguity, Molodtsov [26] introduced the basic notions of soft sets (SS) and debated some elementary operations with their possessions. Maji et al. [27] protracted the idea of SS and defined several basic operations for SS. Maji et al. [28] combined two prevailing notions, such as FS and SS. They developed the idea of FSS, which is a more robust and reliable tool. They also presented basic operations and established and applied this concept in DM. Maji et al. [29] demonstrated the intuitionistic fuzzy soft set (IFSS) theory and offered some basic operations with their essential properties. Arora et al. [30] developed the AOs for IFSS and discussed their basic properties. Nowadays, the conception and application consequences of soft sets and the earlier-mentioned several research developments are evolving speedily. Peng et al. [31] established the concept of PFSS by merging two prevailing models, PFS and SS. Athira et al. [32] established entropy measures for the PFSS. They also offered Euclidean distance and hamming distance for the PFSS and utilized their methods for DM [33]. Naeem et al. [34] developed the TOPSIS and VIKOR methods for PFSNs and presented an approach to the stock exchange investment problem. Zulqarnain et al. [35] introduced the AOs under the PFSS environment and presented an application for green supplier chain management. Zulqarnain et al. [36,37] formed the Einstein-ordered weighted average and geometric AOs for PFSS. They also proposed the MAGDM techniques using their developed operators for sustainable supplier selection and a business to finance money.

Smarandache [38] proposed the idea of the hypersoft set (HSS), which penetrates multiple sub-attributes in the parameter function f, which is a characteristic of the cartesian product with the n attribute. Compared with SS and other existing concepts, Samarandche HSS is the most suitable theory which handles the multiple sub-attributes of the considered parameters. Rahman et al. [39] settled the DM techniques based on similarity measures for IFHSS. Zulqarnain et al. [40] prolonged the notion of IFHSS to PFHSS with fundamental operations and their properties. Zulqarnain et al. [41] expanded the AOs under the IFHSS environment and developed a DM approach based on their presented AOs. Zulqarnain et al. [42] extended the PFSS to interval-valued PFSS and developed the AOs for interval-valued PFSS. They developed the MAGDM approach to resolve DM complications. The method designated in [43] is inadequate to examine the data with a reflective intellect for higher commencement and perfect decisions. For example, O={O1,O2} be a set of two professionals and d1,d2 are two parameters with their corresponding sub-attributes d1={d11,d12} and d2={d21}. Then d1×d2={d11,d12}×{d21}={(d11,d21),(d12,d21)}={dˇ1,dˇ2}, where H an alternative, then preferences of experts be can be summarized as H=[(0,0.7)(0.6,0.7)(0.8,0.7)(0.7,0.2)]. Let θi=(0.7,0.3)T and λj=(0.4,0.6)T indicate the weights of experts and sub-parameters, respectively. Then, we attained the aggregated assessment expending the PFHSWG [43] operator is ⟨0,0.6638⟩. This clearly shows that there is no influence on the collective result μe. Because aF(dˇk)=aF(dˇ11)=0,aF(dˇ12)=0.8,aF(dˇ21)=0.6,andaF(dˇ22)=0.7, which is unreasoning. PFHSS is a hybrid intellectual structure of PFSS. An enhanced sorting process fascinates investigators to crack baffling and inadequate information. Rendering to the investigation outcomes, PFHSS plays a vital role in decision-making by collecting numerous sources into a single value. According to the most generally known knowledge, the emergence of PFSS and hypersoft set (HSS) hybridization has not been combined with the PFSS background. Therefore, to inspire the current research of PFHSS, we will state AO based on rough data, the fundamental objectives of the following study are given as follows:

•   The PFHSS competently deals the complex issues considering the multi sub-attributes of the considered parameters in the DM process. To keep this advantage in mind, we establish the AO for PFHSS.

•   The Einstein operator is a well-known charming guesstimate AO. It is noticed that the prevailing Einstein AOs look unenthusiastic in marking the exact judgment through the DM procedure in some circumstances. To overwhelm these particular difficulties, these AOs need to be modified. We demonstrate advanced operational laws based on Einstein norms for Pythagorean fuzzy hypersoft numbers (PFHSNs).

•   Establish the PFHSEWG operator using the above-mentioned Einstein operational laws with fundamental properties.

•   A novel MCDM technique was established based on the proposed PFHSEWG to cope with DM issues under the PFHSS environment.

•   MS is a significant aspect of engineering as it sees the practical standards of all constituents. MS is a time-consuming but significant step in the enterprise procedure. The industrialist’s productivity, effectiveness, and character will suffer as an outcome deprived of material selection.

•   Comparative analysis of the developed MCDM technique is proposed with current approaches to deliberate the practicality and supremacy of the planned model.

This study is systematized as follows: Basic knowledge of some important notions like SS, HSS, IFHSS, PFHSS, and Einstein norms have been deliberate in Section 2. Section 3 demarcated some basic operational laws for PFHSNs based on Einstein norms and established the PFHSEWG operator. Also, the planned operator’s dynamic properties will be present in the same section. An MCDM approach is introduced using the PFHSEWG operator in Section 4. In the same section, a case study has been presented for material selection in the manufacturing industry. In Section 5, a comparison with some standing approaches is provided.

2  Preliminaries

This section remembers some essential concepts such as SS, HSS, IFHSS, and PFHSS.

Definition 2.1 [26] Let X and N be the universe of discourse and set of attributes, respectively. Let P(X) be the power set of X and A⊆N. A pair (Ω,A) is called a SS over X, and its mapping is expressed as follows:

Ω:A→P(X)

Also, it can be defined as follows:

(Ω,A)={Ω(e)∈P(X):e∈N,Ω(e)=∅ife∉A}

Definition 2.2 [38] Let X be a universe of discourse and P(X) be a power set of X and k={k1,k2,k3,…,kn},(n≥1) andKi represented the set of attributes and their corresponding sub-attributes, such as Ki∩Kj=φ, where i ≠ j for each n≥1andi,jϵ{1,2,3,…,n}. Assume K1×K2×K3⋯×Kn=A⃛={d1h×d2k×⋯×dnl} is a collection of sub-attributes, where 1≤h≤α, 1≤k≤β, and 1≤l≤γ, and α,β,γ∈N. Then the pair (F,K1×K2×K3×⋯×Kn=(Ω,A⃛) is known as HSS and defined as follows:

Ω:K1×K2×K3×…×Kn=A⃛→P(X).

It is also defined as

(Ω,A⃛)={dˇ,ΩA⃛(dˇ):dˇ∈A⃛,ΩA⃛(dˇ)∈P(X)}.

Definition 2.3 [38] Let X be a universe of discourse and P(X) be a power set of X and k={k1,k2,k3,…,kn},(n≥1) andKi represented the set of attributes and their corresponding sub-attributes, such as Ki∩Kj=φ, where i ≠ j for each n≥1andi,jϵ{1,2,3,…,n}. Assume K1×K2×K3⋯×Kn=A⃛={d1h×d2k×⋯×dnl} is a collection of sub-attributes, where 1≤h≤α, 1≤k≤β, and 1≤l≤γ, and α,β,γ∈N, and IFSX be a collection of all fuzzy subsets over X. Then the pair (Ω,K1×K2×K3×…×Kn=(Ω,A⃛) is known as IFHSS and defined as follows:

Ω:K1×K2×K3×⋯×Kn=A⃛→IFSX.

It is also defined as

(Ω,A⃛)={(dˇ,ΩA⃛(dˇ)):dˇ∈A⃛,ΩA⃛(dˇ)∈IFSX∈[0,1]}, where ΩA⃛(dˇ)={⟨δ,aΩ(dˇ)(δ),bΩ(dˇ)(δ)⟩:δ∈X}, where aΩ(dˇ)(δ) and bΩ(dˇ)(δ) signifies the Mem and NMem values of the attributes:

aΩ(dˇ)(δ),bΩ(dˇ)(δ)∈[0,1],and0≤aΩ(dˇ)(δ)+bΩ(dˇ)(δ)≤1.

Definition 2.4 [40] Let U be a universe of discourse and P(U) be a power set of U and k={k1,k2,k3,…,kn},(n≥1) andKi represented the set of attributes and their corresponding sub-attributes, such as Ki∩Kj=φ, where i ≠ j for each n ≥ 1 and i, j ϵ{1,2,3,…,n}. Assume K1×K2×K3×…×Kn=A⃛={d1h×d2k×⋯×dnl} is a collection of sub-attributes, where 1≤h≤α, 1≤k≤β, and 1≤l≤γ, and α,β,γ∈N. and PFSU be a collection of all fuzzy subsets over U. Then the pair (F,K1×K2×K3×⋯×Kn=(F,A⃛)) is known as PFHSS and defined as follows:

F:K1×K2×K3×⋯×Kn=A⃛→PFSU.

It is also defined as (F,A⃛)={(dˇ,FA⃛(dˇ)):dˇ∈A⃛,FA⃛(dˇ)∈PFSU∈[0,1]}, where FA⃛(dˇ)={⟨δ,aF(dˇ)(δ),bF(dˇ)(δ)⟩:δ∈U}, where aF(dˇ)(δ)andbF(dˇ)(δ) signifies the Mem and NMem values of the attributes:

aF(dˇ)(δ),bF(dˇ)(δ)∈[0,1],and0≤(aF(dˇ)(δ))2+(bF(dˇ)(δ))2≤1.

A Pythagorean fuzzy hypersoft number (PFHSN) can be stated as F={(aF(dˇ)(δ),bF(dˇ)(δ))}, where 0≤(aF(dˇ)(δ))2+(bF(dˇ)(δ))2≤1.

Remark 2.1 If (aF(dˇ)(δ))2+(bF(dˇ)(δ))2andaF(dˇ)(δ)+bF(dˇ)(δ)≤1 both are holds. Then, PFHSS is condensed to IFHSS [41].

For readers’ aptness, the PFHSN Fδi(dˇj)={(aF(dˇj)(δi),bF(dˇj)(δi))|δi∈U} can be written as Jdˇij=⟨aF(dˇij),bF(dˇij)⟩. The score function [43] for Jdˇij is expressed as follows:

S(Jdˇij)=aF(dˇij)2−bF(dˇij)2,S(Jdˇij)∈[−1,1](1)

But, in some cases, the above-defined score function cannot handle the scenario. For example, if we consider the two PFHSNs, such as Jdˇ11=⟨.4,.7⟩ and Jdˇ12=⟨.5,.8⟩. The score function cannot deliver relevant results to subtract the PFHSNs. So, in such situations, it is tough to achieve the most suitable alternative S(Jdˇ11)=.3= S(Jdˇ12). The accuracy function [43] had been developed.

H(Jdˇij)=aF(dˇij)2+bF(dˇij)2,H(Jdˇij)∈[0,1](2)

The consequent comparative laws will be used Jdˇij and Tdˇij.

1.    If S(Jdˇij)>S(Tdˇij),thenJdˇij>Tdˇij.

2.    If S(Jdˇij)=S(Tdˇij), then

•   If H(Jdˇij)>H(Tdˇij),thenJdˇij>Tdˇij

•   If H(Jdˇij)=H(Tdˇij),thenJdˇij=Tdˇij.

Definition 2.5 Einstein’s sum ⊕ε and Einstein product ⊗ε are good alternatives of algebraic t-norm and t-conorm, respectively, given as follows:

a⊕εb=a+b1+(a⋅b)anda⊗εb=a⋅b1+(1−a)⋅(1−b),∀(a,b)∈[0,1]2

Under the Pythagorean fuzzy environment, Einstein sum ⊕ε and Einstein product ⊗ε are defined as:

a⊕εb=a2+b21+(a2⋅b2),a⊗εb=a⋅b1+(1−a2)⋅(1−b2),∀(a,b)∈[0,1]2

where a⊕εb and a⊗εb is known as t-norm and t-conorm, respectively, satisfying the bounded, monotonicity, commutativity, and associativity properties.

3  Einstein Weighted Geometric Aggregation Operator for Pythagorean Fuzzy Hypersoft Set

This section will introduce a novel Einstein-weighted AO such as the PFHSEWG operator for PFHSNs with essential properties.

3.1 Operational Laws for PFHSNs

Definition 3.1 [44] Let Jdˇk=(adˇk,bdˇk),Jdˇ11=(adˇ11,bdˇ11),andJdˇ12=(adˇ12,bdˇ12) represents the PFHSNs and ∂ is a positive real number. Then, operational laws for PFHSNs based on Einstein norms can be expressed as follows:

1.    Jdˇ11⊗εJdˇ12=⟨(1+adˇ1j2)−(1−adˇ1j2)(1+adˇ1j2)+(1−adˇ1j2),2bdˇ1j2(2−bdˇ1j2)+bdˇ1j2⟩

2.    Jdˇ11⊗εJdˇ12=⟨2αdˇ1j2(2−αdˇ1j2)+αdˇ1j2,(1+bdˇ1j2)−(1−bdˇ1j2)(1+bdˇ1j2)+(1−bdˇ1j2)⟩

3.    ∂Jdˇk=⟨(1+adˇk2)∂−(1−adˇk2)∂(1+adˇk2)∂+(1−adˇk2)∂,2(bdˇk2)∂(2−bdˇk2)∂+(bdˇk2)∂⟩

4.    Jdˇk∂=⟨2(adˇk2)∂(2−adˇk2)∂+(adˇk2)∂,(1+bdˇk2)∂−(1−bdˇk2)∂(1+bdˇk2)∂+(1−bdˇk2)∂⟩

Definition 3.2 Let Jdˇij=(adˇij,bdˇij) be a collection of PFHSNs. Then the PFHSEWG operator is defined as

PFHSEWG(Jdˇ11,Jdˇ12,…,Jdˇnm)=⊗j=1mλj(⊗i=1nθiJdˇij)(3)

where (i=1,2,…,n),(j=1,2,…,m) andθi,λj signify the weighted vectors such as θi>0, ∑i=1nθi=1 and λj>0, ∑j=1nλj=1.

Theorem 3.1 Let Jdˇij=⟨(adˇij,bdˇij)⟩ be a collection of PFHSNs, then the aggregated value attained by Eq. (3) is given as

PFHSEWG(Jdˇ11,Jdˇ12,…,Jdˇnm)=⊗j=1mλj(⊗i=1nθiJdˇij)=⟨2∏j=1m(∏i=1n(αdˇij2)θi)λj∏j=1m(∏i=1n(2−αdˇij2)θi)λj+∏j=1m(∏i=1n(αdˇij2)θi)λj,∏j=1m(∏i=1n(1+bdˇij2)θi)λj−∏j=1m(∏i=1n(1−bdˇij2)θi)λj∏j=1m(∏i=1n(1+bdˇij2)θi)λj+∏j=1m(∏i=1n(1−bdˇij2)θi)λj⟩(4)

where (i=1,2,…,n),(j=1,2,…,m) andθi,λj signify the weight vectors such that θi>0, ∑i=1nθi=1 and λj>0, ∑j=1nλj=1.

Proof: We will use mathematical induction to demonstrate the above result.

For n = 1, we get θi=1.

PFHSEWG(Jdˇ11,Jdˇ12,…,Jdˇnm)=⊗j=1mλjJdˇ1j=⟨2∏j=1m(αdˇ1j2)λj∏j=1m(2−αdˇ1j2)λj+∏j=1m(αdˇ1j2)λj,∏j=1m(1+bdˇ1j2)λj−∏j=1m(1−bdˇ1j2)λj∏j=1m(1+bdˇ1j2)λj+∏j=1m(1−bdˇ1j2)λj⟩=⟨2∏j=1m(∏i=11(αdˇij2)θi)λj∏j=1m(∏i=11(2−αdˇij2)θi)λj+∏j=1m(∏i=11(αdˇij2)θi)λj,∏j=1m(∏i=11(1+bdˇij2)θi)λj−∏j=1m(∏i=11(1−bdˇij2)θi)λj∏j=1m(∏i=11(1+bdˇij2)θi)λj+∏j=1m(∏i=11(1−bdˇij2)θi)λj⟩

For m = 1, we get λj=1.

PFHSEWG(Jdˇ11,Jdˇ12,…,Jdˇnm)=⊗i=1nθiJdˇi1=⟨2∏i=1n(αdˇi12)θi∏i=1n(2−αdˇi12)θi+∏i=1n(αdˇi12)θi,∏i=1n(1+bdˇi12)θi−∏i=1n(1−bdˇi12)θi∏i=1n(1+bdˇi12)θi+∏i=1n(1−bdˇi12)θi⟩=⟨2∏j=11(∏i=1n(αdˇij2)θi)λj∏j=11(∏i=1n(2−αdˇij2)θi)λj+∏j=11(∏i=1n(αdˇij2)θi)λj,∏j=11(∏i=1n(1+bdˇij2)θi)λj−∏j=11(∏i=1n(1−bdˇij2)θi)λj∏j=11(∏i=1n(1+bdˇij2)θi)λj+∏j=11(∏i=1n(1−bdˇij2)θi)λj⟩

So, Eq. (4) true for n=1, m =1.

Assume that the equation grasps for n=δ2, m =δ1+1 and for n=δ2+1,m =δ1

⊗j=1δ1+1λj(⊗i=1δ2θiJdˇij)=⟨2∏j=1δ1+1(∏i=1δ2(αdˇij2)θi)λj∏j=1δ1+1(∏i=1δ2(2−αdˇij2)θi)λj+∏j=1δ1+1(∏i=1δ2(αdˇij2)θi)λj,∏j=1δ1+1(∏i=1δ2(1+bdˇij2)θi)λj−∏j=1δ1+1(∏i=1δ2(1−bdˇij2)θi)λj∏j=1δ1+1(∏i=1δ2(1+bdˇij2)θi)λj+∏j=1δ1+1(∏i=1δ2(1−bdˇij2)θi)λj⟩

⊗j=1δ1+1λj(⊗i=1δ2+1θiJdˇij)=⟨2∏j=1δ1(∏i=1δ2+1(αdˇij2)θi)λj∏j=1δ1(∏i=1δ2+1(2−αdˇij2)θi)λj+∏j=1δ1(∏i=1δ2+1(αdˇij2)θi)λj,∏j=1δ1(∏i=1δ2+1(1+bdˇij2)θi)λj−∏j=1δ1(∏i=1δ2+1(1−bdˇij2)θi)λj∏j=1δ1(∏i=1δ2+1(1+bdˇij2)θi)λj+∏j=1δ1(∏i=1δ2+1(1−bdˇij2)θi)λj⟩

Now we show the Eq. (4) for m =δ1+1 and n=δ2+1

⊗j=1δ1+1λj(⊗i=1δ2+1θiJdˇij)=⊗j=1δ1+1λj(⊗i=1δ2θiJdˇij⊗θi+1Jdˇ(δ2+1)j)=(⊗j=1δ1+1⊗i=1δ2θiλjJdˇij)(⊗j=1δ1+1λjθi+1Jdˇ(δ2+1)j)=⟨2∏j=1δ1+1(∏i=1δ2(αdˇij2)θi)λj∏j=1δ1+1(∏i=1δ2(2−αdˇij2)θi)λj+∏j=1δ1+1(∏i=1δ2(αdˇij2)θi)λj⊗2∏j=1δ1+1((αdˇ(δ2+1)j2)θδ2+1)λj∏j=1δ1+1((2−αdˇ(δ2+1)j2)θδ2+1)λj+∏j=1δ1+1((αdˇ(δ2+1)j2)θδ2+1)λj,∏j=1δ1+1(∏i=1δ2(1+bdˇij2)θi)λj−∏j=1δ1+1(∏i=1δ2(1−bdˇij2)θi)λj∏j=1δ1+1(∏i=1δ2(1+bdˇij2)θi)λj+∏j=1δ1+1(∏i=1δ2(1−bdˇij2)θi)λj⊗∏j=1δ1+1((1+bdˇ(δ2+1)j2)θδ2+1)λj−∏j=1δ1+1((1−bdˇ(δ2+1)j2)θδ2+1)λj∏j=1δ1+1((1+bdˇ(δ2+1)j2)θδ2+1)λj+∏j=1δ1+1((1−bdˇ(δ2+1)j2)θδ2+1)λj⟩=⟨2∏j=1δ1+1⁡(∏i=1δ2+1⁡(αdij2)θi)λj∏j=1δ1+1⁡(∏i=1δ2+1⁡(2−αdij2)θi)λj+∏j=1δ1+1⁡(∏i=1δ2+1⁡(αdij2)θi)λj,∏j=1δ1+1⁡(∏i=1δ2+1⁡(1+bdij2)θi)λj−∏j=1δ1+1⁡(∏i=1δ2+1⁡(1−bdij2)θi)λj∏j=1δ1+1⁡(∏i=1δ2+1⁡(1+bdij2)θi)λj+∏j=1δ1+1⁡(∏i=1δ2+1⁡(1−bdij2s)θi)λj⟩=⊗j=1δ1+1λj(⊗i=1δ2+1θiJdˇij)