# Some T-Spherical Fuzzy Einstein Interactive Aggregation Operators and Their Application to Selection of Photovoltaic Cells

Authors: Shouzhen Zeng; Muhammad Munir; Tahir Mahmood; Muhammad Naeem
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this article, it is pointed out that the existing intuitionistic fuzzy and T-spherical fuzzy Einstein averaging and geometric operators have some limitations. To overcome these limitations, we proposed some new averaging and geometric operators in the T-spherical fuzzy environment instead of the intuitionistic fuzzy environment because the T-spherical fuzzy set is the most generalized form and the proposed operators can be particularized to the intuitionistic fuzzy environment. First, new operational laws for T-spherical fuzzy information are defined, on the basis of which Einstein geometric interaction operators and Einstein averaging interactive aggregation operators are then proposed. Some basic properties and advantages of proposed aggregation operators are also discussed. Moreover, the proposed operators are applied to the MADM problem to check their reliability. The superiority of the proposed operators over existing work is checked with the help of an example.


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## Body

## 1. Introduction

To deal with uncertainty, a tool called fuzzy set (FS) was introduced by Zadeh [1]. FS tells the degree of membership (MD) of an object. Intuitionistic fuzzy set (IFS), a generalization of FS, was introduced by Atanassov [2, 3]. IFS has two functions, a membership and a nonmembership function, and both functions map on a closed unit interval from a nonempty set. As the name indicates that they tell the MD and nonmembership degree (NMD) of an object, respectively, the IFS has a condition that the sum of both degrees must belong to [0, 1]. Whenever the sum of both degrees exceeds from 1, IFS fails to handle that information. To overcome this issue, Yager [4, 5] relaxed the condition that the square sum of both degrees must belong to [0, 1] and named that structure as Pythagorean fuzzy set (PyFS). In PyFS, the decision maker has more options for giving values to an object but the decision maker is not free to give any value because there are some limitations. To overcome this problem, Yager [6] proposed an improved fuzzy structure called q-rung orthopair fuzzy set (q-ROPFS) in which there is no limitation overall.The notion picture fuzzy set (PFS) was introduced by Coung [7]. In PFS, there are four functions which express the MD, NMD, abstinence (AD), and refusal with a condition that the sum of MD, AD, and NMD must belong to [0, 1]. PFS fails when their sum exceeds one. Mahmood et al. [8] overcame this problem by introducing the concept of spherical fuzzy set in which they relaxed the condition that the square sum of MD, AD, and NMD must belong to [0, 1]. In [8], they also proposed the concept of a T-spherical fuzzy set in which the decision makers have to find the power for which the sum lies in the unit’s closed interval.All the fuzzy structures discussed above are used to solve many problems in the field of decision making, similarity measures, etc. Xu and Yager [9] proposed weighted geometric operators for IFS. Zeng et al. [10] proposed some probabilistic averaging operators for IFS and studied their application to group decision making. Garg [11] investigated the MADM problem using interval-valued IF robust geometric aggregation operators. Zhang et al. [12] discussed the IF MULTIMOORA approach for the MADM problem. He et al. [13] investigated the MADM problem using interactive geometric aggregation operators for IFS. Zhao and Wei [14] used Einstein t-norm and t-conorm to introduce IF Einstein geometric aggregation (IFEWA) operators and IF Einstein averaging aggregation (IFEWG) operators to solve the MADM problem. Mu et al. [15] investigated the MADM problem using Zhenyuan aggregation operators for interval-valued IFSs. Some MADM problems are solved for IFSs in [16–18]. Garg [19] investigated the MADM problem using Einstein norms for PyFSs. Wei and Lu [20] used power operators for PyFSs to solve the MADM problem. Yang and Chang [21] solved the MADM problem using interval-valued Pythagorean normal fuzzy aggregation operators. Zeng et al. [22] investigated the MADM problem using the novel aggregation method for PyFS. Zeng et al. [23] investigated the MADM problem using PyF confidence aggregation operators. By using PyFSs, some problems are solved in [24, 25]. Peng et al. [26] investigated the MADM problem by using exponential aggregation operators for q-ROPFS. Yang et al. [27] solved the MADM problem using q-ROPF interaction Heronian mean operators. Yang et al. [28] proposed the q-ROPF multicriteria decision algorithm and used it to investigate multiple heterogeneous relationships. Hussain et al. [29] investigated the MADM problem using a covering-based q-ROPF rough set model. Hussain et al. [30] used q-ROPF soft averaging operators to solve the MADM problem. Some MADM problems are studied using q-ROPFSs in [31–33].Wei [34] proposed picture fuzzy Hamacher aggregation operators and studied their application in the MADM problem. Khan et al. [35] investigated the MADM problem using some logarithmic aggregation operators for PFS. Some MADM problems for PFS are studied in [36–38]. Quek et al. [39] studied the MADM problem using T-spherical fuzzy weighted aggregation operators. Garg [40] proposed interactive aggregation operators for T-spherical fuzzy sets and solved the MADM problem using the proposed operators. Zeng et al. [41] proposed immediate probabilistic averaging aggregation operators for T-spherical fuzzy sets and investigated their application in the selection of solar cells. Munir et al. [42] solved the MADM problem using Einstein aggregation operators for T-spherical fuzzy set. By using spherical fuzzy sets and T-spherical fuzzy sets, some MADM problems are studied in [43–46].The existing Einstein operations [14, 42] fail under some circumstances, e.g., if I1=mA,0 and I2=0,nA are intuitionistic fuzzy numbers (IFNs), then IFEWA operators aggregate these IFNs as some value,0 and IFEWG operators aggregates these IFNs as 0,some value. From the above example, it is easy to notice that the IFEWA operator cannot aggregate the nonmembership value when nonmembership value of anyone IFN becomes zero and similarly the IFEWG operator cannot aggregate the membership value when nonmembership value of anyone IFN becomes zero. This issue motivates us to propose some improved Einstein aggregation operators which will overcome these problems.The purposes of this manuscript are as follows:(1)
To develop some new interactive Einstein operational laws(2)
To develop new T-spherical fuzzy Einstein weighted geometric interaction operators using the new proposed interactive operational laws(3)
To develop new T-spherical fuzzy Einstein weighted averaging interaction operators based on the new proposed interactive operations(4)
To present an algorithm for solving the MADM problem(5)
To check the applicability of the proposed operators through a numerical exampleTo achieve these aims, the rest of the article is summarized as follows. Section2 has some notions that will be helpful for the readers in further sections. In Section 3, new operational laws are proposed for T-spherical fuzzy sets, and in Section 4, T-spherical fuzzy Einstein interaction geometric operators are proposed and some basic properties are also proved. In Section 5, T-spherical fuzzy Einstein interaction averaging operators are defined and some of their basic properties are also discussed. The advantages of the proposed work and comparative analysis between the existing and proposed operators are developed in Section 6. In Section 7, an algorithm for solving a MADM problem is developed and also applied in an example. A brief conclusion of the manuscript is included in Section 8.


## 2. Preliminaries

Some basic definitions over the setX are defined in this section. These basic notions will help the readers to understand the proposed work.Definition 1.
(see [8]). A T-spherical fuzzy set on X is defined as(1)P=x,mx,ix,nxx∈X,where m,i,n:X⟶0,1 are a membership, abstinence, and nonmembership function, respectively, with the condition 0≤mtx+itx+ntx≤1,∀x∈X,t∈ℕ.
rx=1−mtx+itx+ntxt is called the refusal degree of x in P, and m,i,n is known as T-spherical fuzzy number (T-SFN).Definition 2.
(see [8]). Let P=m,i,n∈ T-spherical fuzzy set. Then, the score value of P is defined as SCP=mtx−ntx, and the accuracy value of P is defined as ACP=mtx+itx+ntx. A T-SFN with a greater score is superior to others. If the score of any two T-SFNs becomes equal, then their superiority will be checked with the help of their accuracy values. The number which has greater accuracy value will be superior as compared to the others. If accuracy values also become equal, then both numbers are considered similar.Remark 1.
(1)
Definitions1 and 2 can be reduced for spherical fuzzy set if we consider t=2(2)
Definitions1 and 2 can be reduced for PFS if we consider t=1(3)
Definitions1 and 2 can be reduced for q-ROFS if we consider i=0(4)
Definitions1 and 2 can be reduced for PyFS if we consider t=2 and i=0(5)
Definitions1 and 2 can be reduced for IFS if we consider t=1 and i=0(6)
Definition1 can be reduced for FS if we consider t=1, i=0, and n=0Definition 3.
(see [14]). Some operations for any two IFNs I1=m1,n1 and I2=m2,n2 are defined as(2)I1⊗I2=m1m21+1−m11−m2,n1+n21+n1n2,I1⊕I2=m1+m21+m1m2,n1n21+1+n11−n2,τI1=1+m1τ+1−m1τ1+m1τ−1−m1τ,2n1τ2−n1τ+n1τ,τ>0,I1τ=2m1τ2−m1τ+m1τ,1+n1τ+1−n1τ1+n1τ−1−n1τ,τ>0.


## 3. Einstein Interaction Operations for T-Spherical Fuzzy Set

Existing Einstein operations have some limitations that they fail under some conditions. So, we proposed some new Einstein operations on which we define some new aggregation operators. LetP1=m1,i1,n1 and P2=m2,i2,n2 be two T-spherical fuzzy sets; then, their Einstein operations are as follows:(3)P1⊗P2=21−n1t−i1t1−n2t−i2t−1−m1t−i1t−n1t1−m2t−i2t−n2t1+n1t1+n2t+1−n1t1−n2tt,1+i1t1+i2t−1−i1t1−i2t1+i1t1+i2t+1−i1t1−i2tt,1+n1t1+n2t−1−n1t1−n2t1+n1t1+n2t+1−n1t1−n2tt,P1⊕P2=1+m1t1+m2t−1−m1t1−m2t1+m1t1+m2t+1−m1t1−m2tt,1+i1t1+i2t−1−i1t1−i2t1+i1t1+i2t+1−i1t1−i2tt,21−m1t−i1t1−m2t−i2t−1−m1t−i1t−n1t1−m2t−i2t−n2t1+m1t1+m2t+1−m1t1−m2tt,τP1=1+m1tτ−1−m1tτ1+m1tτ+1−m1tτt,1+i1tτ−1−i1tτ1+m1tτ+1−m1tτt,21−m1t−i1tτ−1−m1t−i1t−n1tτ1+m1tτ+1−m1tτt,τ>0,P1τ=21−n1t−i1tτ−1−m1t−i1t−n1tτ1+n1tτ+1−n1tτt,1+i1tτ−1−i1tτ1+i1tτ+1−i1tτt,1+n1tτ−1−n1tτ1+n1tτ+1−n1tτt,τ>0.Remark 2.
(1)
The defined operations will be reduced to spherical fuzzy sets fort=2(2)
The defined operations will be reduced to PFSs fort=1(3)
The defined operations will be reduced to q-ROFSs fori=0(4)
The defined operations will be reduced to PyFSs fort=2 and i=0(5)
The defined operations will be reduced to IFSs fort=1 and i=0(6)
The defined operations will be reduced to FSs fort=1, i=0, and n=0


## 4. T-Spherical Fuzzy Einstein Geometric Interaction Operators

In this section, on the basis of new proposed Einstein operations, we defined geometric interaction operators in the environment of T-spherical fuzzy set and some of its basic properties such as monotonicity, boundedness, and idempotency are also discussed. The validity of the proposed work is checked with the help of an example.Definition 4.
For any collection of T-SFNs,Pj=mj,ij,njj=1,2,3,…,k. The mapping(4)T−SFEWIGwP1,P2,…,Pk=⊗j=1kPjwj=2∏j=1k1−njt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjt,∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt,∏j=1k1+njtwj−∏j=1k1−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjt,is called the T-spherical fuzzy Einstein weighted interactive geometric (T-SFEWIG) operator, where w=w1,…,wkT is the weight vector of Pj with wj∈0,1 and ∑j=1kwj=1.Theorem 1.
If allPj=P0, then T−SFEWIGwP1,P2,…,Pk=P0.Proof.
LetPj=P0=m0,i0,n0 for all j; then,(5)T−SFEWIGwP1,P2,…,Pk=2∏j=1k1−njt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjt,∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt,∏j=1k1+njtwj−∏j=1k1−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjtT−SFEWIGwP1,P2,…,Pk=21−njt−ijt∑j=1kwj−1−mjt−ijt−njt∑j=1kwj1+njt∑j=1kwj+1−njt∑j=1kwjt,1+ijt∑j=1kwj−1−ijt∑j=1kwj1+ijt∑j=1kwj+1−ijt∑j=1kwjt,1+njt∑j=1kwj−1−njt∑j=1kwj1+njt∑j=1kwj+1−njt∑j=1kwjt=m0,i0,n0=P0.Theorem 2.
Consider a collection of T-SFNsPjj=1,2,…,k with PL=minjPj and PU=maxjPj. Then,(6)PL≤T−SFEWIGwP1,P2,…,Pk≤PU.Proof.
Straightforward.Theorem 3.
Consider any two T-SFNsPj=mj,ij,nj and Pj′=mj′,ij′,nj′j=1,2,…,k such that Pj≤Pj′ for all j. Then,(7)T−SFEWIGwP1,P2,…,Pk≤T−SFEWIGwP1′,P2′,…,Pk′.Proof.
LetPj≤Pj′; then, mj≤mj′, ij≤ij′, and nj≥nj′. Then, by using basic information,(8)2∏j=1k1−njt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjt≤2∏j=1k1−nj′t−ij′twj−∏j=1k1−mj′t−ij′t−nj′twj∏j=1k1+nj′twj+∏j=1k1−nj′twjt∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt≤∏j=1k1+ij′twj−∏j=1k1−ij′twj∏j=1k1+ij′twj+∏j=1k1−ij′twjt∏j=1k1+njtwj−∏j=1k1−njtwj∏j=1k1+njtwj+∏j=1k1−njtwjt≥∏j=1k1+nj′twj−∏j=1k1−nj′twj∏j=1k1+nj′twj+∏j=1k1−nj′twjt.
This shows that(9)T−SFEWIGwP1,P2,…,Pk≤T−SFEWIGwP1,P2,…,Pk.Definition 5.
For any collection of T-SFNs,Pj=mj,ij,njj=1,2,3,…,k. The mapping(10)T−SFEOWIGωP1,P2,…,Pk=⊗j=1kPσjωj=2∏j=1k1−nσjt−iσjtwj−∏j=1k1−mσjt−iσjt−nσjtwj∏j=1k1+nσjtwj+∏j=1k1−nσjtwjt,∏j=1k1+iσjtwj−∏j=1k1−iσjtwj∏j=1k1+iσjtwj+∏j=1k1−iσjtwjt,∏j=1k1+nσjtwj−∏j=1k1−nσjtwj∏j=1k1+nσjtwj+∏j=1k1−nσjtwjt,T−SFEOWIGω is called the T-spherical fuzzy Einstein ordered weighted interactive geometric (T-SFEOWIG) operator, where ω=ω1,…,ωkT is the associated weight vector of Pj with ωj∈0,1 and ∑j=1kωj=1 and σj is any permutation of 1,2,…,k such that P˜σj−1≥P˜σj.Theorem 4.
If allPj=P0, then T−SFEOWIGωP1,P2,…,Pk=P0.Proof.
Same as in Theorem1.Theorem 5.
Consider a collection ofT−SFNsPjj=1,2,…,k with PL=minjPj and PU=maxjPj. Then,(11)PL≤T−SFEOWIGωP1,P2,…,Pk≤PU.Proof.
Straightforward.Theorem 6.
For any two T-SFNs,Pj=mj,ij,nj and Pj′=mj′,ij′,nj′ for all j=1,2,…,k such that Pj≤Pj′ for all j. Then,(12)T−SFEOWIGωP1,P2,…,Pk≤T−SFEOWIGωP1′,P2′,…,Pk′.Proof.
Directly follows from the proof of Theorem3.Definition 6.
For any collection of T-SFNs,Pj=mj,ij,njj=1,2,3,…,k. The mapping(13)T−SFEHIGw,ωP˜1,P˜2,…,P˜k=⊗j=1kP˜σjωj=2∏j=1k1−n˜σjt−i˜σjtwj−∏j=1k1−m˜σjt−i˜σjt−n˜σjtwj∏j=1k1+nσjtwj+∏j=1k1−nσjtwjt,∏j=1k1+i˜σjtwj−∏j=1k1−i˜σjtwj∏j=1k1+i˜σjtwj+∏j=1k1−i˜σjtwjt,∏j=1k1+n˜σjtwj−∏j=1k1−n˜σjtwj∏j=1k1+n˜σjtwj+∏j=1k1−n˜σjtwjt,T−SFEHIGw,ωis called the T-spherical fuzzy Einstein hybrid interactive geometric (T-SFEHIG) operator, where P˜j=Pjkwj. Let w=w1,…,wkT is the weight vector and ω=ω1,…,ωkT is the associated weight vector of Pj with a condition that both weight and associated weight vectors belong to closed unit interval and their sum is equal to 1.
Hybrid aggregation operators first aggregate the given data considering their attributes and then rearrange them in a specific order. After that, they aggregate the data considering their order. This means that hybrid operators are a generalization of weighted and ordered weighted operators. So, the T-SFEHIG operator will satisfy idempotency, monotonicity, and bounded property.Example 1.
Consider T-SFNsP1=0.7,0.3,0.2, P2=0.9,0.1,0.6, P3=0.4,0.6,0.8, P4=0.1,0.5,0.7, and P5=0.0,0.0,0.8 with a weight vector w=0.25,0.20,0.15,0.18,0.22T.Solution 1.
First of all, we find the aggregated value of these T-SFNs by using the T-spherical fuzzy Einstein hybrid geometric aggregation (T-SFEHG) operator [42] to find out the drawbacks of the given operators. For this purpose, first, we have to calculate the value of t for which the given data lie in T-SF information.
As0.9+0.1+0.6=1.6,Fort=2, 0.92+0.12+0.62=1.18t=3, 0.93+0.13+0.63=0.946
Similarly, fort=3, all the given data lie in the T-spherical fuzzy information.
By using T-spherical fuzzy Einstein weighted geometric operator, we shall be able to find these values:(14)P1=0.6388,0.3232,0.5381,P2=0.9,0.1,0.6,P3=0.4163,0.5464,0.7370,P4=0.1050,0.4829,0.6776,P5=0.0,0.0,0.8206.
Their scores values will be(15)SCP1=0.1048,SCP2=0.5130,SCP3=−0.3282,SCP4=−0.3099,SCP5=−0.5525.
Now using the score value, the aggregated values obtained by using T-SFEWG operators are rearranged in descending order. Then, these ordered values are again aggregated by using the T-SFEHG operator with associated weight vector beingω=0.112,0.236,0.304,0.236,0.112:(16)P˜σ1=0.9,0.1,0.6,P˜σ2=0.6388,0.3232,0.5381,P˜σ3=0.1050,0.4829,0.6776,P˜σ4=0.4163,0.5464,0.7370,P˜σ5=0.0,0.0,0.8206.
Now, employ the T-SFEHG operator as follows:(17)T−SFEHGw,ωP1,…,P5=0.0,0.8525,0.9882.
From the above result, it is noticed that when abstinence or nonmembership value of one T-SFN is zero, then the T-SFEHG operator cannot aggregate the whole membership value. This shows a big flaw in the T-SFEHG operator. This means that the results obtained from T-SFEHG operators are not reliable. Now, by using the T-SFEHIG operator, we shall show that the proposed operator will overcome this drawback.
By using the T-SFEWIG operator, we shall be able to find(18)P1=0.7393,0.3232,0.2154,P2=0.9,0.1,0.6,P3=0.4131,0.5464,0.7370,P4=0.0988,0.4829,0.6776,P5=0.0,0.0,0.8206.
Their scores values will be(19)SCP1=0.3941,SCP2=0.5130,SCP3=−0.3299,SCP4=−0.3101,SCP5=−0.5525.
Now using the score value, the aggregated values obtained by using T-SEWIG operators are rearranged in descending order. Then, these ordered values are again aggregated by using the T-SFEHIG operator with the associated weight vector beingω=0.112,0.236,0.304,0.236,0.112:(20)P˜σ1=0.9,0.1,0.6,P˜σ2=0.7393,0.3232,0.2154,P˜σ3=0.0988,0.4829,0.6776,P˜σ4=0.4131,0.5464,0.7370,P˜σ5=0.0,0.0,0.8206.
Now, employ the T-SFEHG operator as follows:(21)T−SFEHIGω,wP˜1,…,P˜5=0.6878,0.4329,0.6591.
This shows that the T-SFEIG operator aggregates the membership value.


## 5. T-Spherical Fuzzy Einstein Hybrid Interaction Averaging Operators

In this section, on the basis of new proposed Einstein operations, we define averaging interaction operators in T-spherical fuzzy environment and some basic properties are also discussed.Definition 7.
Consider a collection of T-SFNPj=mj,ij,njj=1,2,3,…,k. Then,(22)T−SFEWIAwP1,P2,…,Pk=⊕j=1kwjPj=∏j=1k1+mjtwj−∏j=1k1−mjtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjt,∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt,2∏j=1k1−mjt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjt,T−SFEWIAw is called T-spherical fuzzy Einstein weighted interactive averaging (T-SFEWIA) operator with weight vector w=w1,w2,…,wkT of Pj with wj∈0,1 and ∑j=1kwj=1.Theorem 7.
If allPj=P0, then T−SFEWIAwP1,P2,…,Pk=P0.Proof.
LetPj=P0=m0,i0,n0 for all j; then,(23)T−SFEWIAwP1,P2,…,Pk=∏j=1k1+mjtwj−∏j=1k1−mjtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjt,∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt,2∏j=1k1−mjt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjtT−SFEWIAwP1,P2,…,Pk=1+mjt∑j=1kwj−1−mjt∑j=1kwj1+mjt∑j=1kwj+1−mjt∑j=1kwjt,1+ijt∑j=1kwj−1−ijt∑j=1kwj1+ijt∑j=1kwj+1−ijt∑j=1kwjt,21−mjt−ijt∑j=1kwj−1−mjt−ijt−njt∑j=1kwj1+mjt∑j=1kwj+1−mjt∑j=1kwjt=m0,i0,n0=P0.Theorem 8.
Consider a collection of T-SFNsPjj=1,2,…,k with PL=minjPj and PU=minjPj. Then,(24)PL≤T−SFEWIAwP1,P2,…,Pk≤PU.Proof.
Straightforward.Theorem 9.
For any two T-SFNs,Pj=mj,ij,nj and Pj′=mj′,ij′,nj′ such that Pj≤Pj′ for all j. Then,(25)T−SFEWIAwP1,P2,…,Pk≤T−SFEWIAwP1′,P2′,…,Pk′.Proof.
LetPj≤Pj′; then, mj≤mj′, ij≤ij′, and nj≥nj′. Then, by using this basic information,(26)2∏j=1k1−mjt−ijtwj−∏j=1k1−mjt−ijt−njtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjt≤2∏j=1k1−mj′t−ij′twj−∏j=1k1−mj′t−ij′t−nj′twj∏j=1k1+mj′twj+∏j=1k1−mj′twjt,∏j=1k1+ijtwj−∏j=1k1−ijtwj∏j=1k1+ijtwj+∏j=1k1−ijtwjt≤∏j=1k1+ij′twj−∏j=1k1−ij′twj∏j=1k1+ij′twj+∏j=1k1−ij′twjt,∏j=1k1+mjtwj−∏j=1k1−mjtwj∏j=1k1+mjtwj+∏j=1k1−mjtwjt≥∏j=1k1+mj′twj−∏j=1k1−mj′twj∏j=1k1+mj′twj+∏j=1k1−mj′twjt.
This shows that(27)T−SFEWIAwP1,P2,…,Pk≤T−SFEWIAwP1′,P2′,…,Pk′.Definition 8.
Consider a collection of T-SFNsPj=mj,ij,njj=1,2,3,…,k. Then,(28)T−SFEOWIAωP1,P2,…,Pk=⊕j=1kωjPσj=∏j=1k1+mσjtωj−∏j=1k1−mσjtωj∏j=1k1+mσjtωj+∏j=1k1−mσjtωjt,∏j=1k1+iσjtωj−∏j=1k1−iσjtωj∏j=1k1+iσjtωj+∏j=1k1−iσjtωjt,2∏j=1k1−mσjt−iσjtωj−∏j=1k1−mσjt−iσjt−nσjtωj∏j=1k1+mσjtωj+∏j=1k1−mσjtωjtT−SFEOWIAω is called the T-spherical fuzzy Einstein ordered weighted interactive averaging (T-SFEOWIA) operator with associated weight vector ω=ω1,ω2,…,ωkT of Pj with ωj∈0,1 and ∑j=1kωj=1. Here, σj is any permutation of 1,2,…,k such that P˜σj−1≥P˜σj.Theorem 10.
If for allPj=P0, then T−SFEOWAωP1,P2,…,Pk=P0.Proof.
Same as in Theorem7.Theorem 11.
Consider a collection of T-SFNsPjj=1,2,…,k with PL=minjPj and PU=minjPj. Then,(29)PL≤T−SFEOWIAωP1,P2,…,Pk≤PU.Proof.
Straightforward.Theorem 12.
Consider any twoT−SFNsPj=mj,ij,nj and Pj′=mj′,ij′,nj′ such that Pj≤Pj′ for all j. Then,(30)T−SFEOWIAωP1,P2,…,Pk≤T−SFEOWIAωP1′,P2′,…,Pk′.Proof.
Same as in Theorem9.Definition 9.
Consider a collection of T-SFNsPj=mj,ij,njj=1,2,3,…,k. The mapping(31)T−SFEHIAw,ωP˜1,P˜2,…,P˜k=⊕j=1kωjP˜σj=∏j=1k1+m˜σjtωj−∏j=1k1−m˜σjtωj∏j=1k1+m˜σjtωj+∏j=1k1−m˜σjtωjt,∏j=1k1+i˜σjtωj−∏j=1k1−i˜σjtωj∏j=1k1+i˜σjtωj+∏j=1k1−i˜σjtωj,t2∏j=1k1−m˜σjt−i˜σjtωj−∏j=1k1−m˜σjt−i˜σjt−nσjtωj∏j=1k1+m˜σjtωj+∏j=1k1−m˜σjtωjt,is called the T-spherical fuzzy Einstein hybrid interactive averaging (T-SFEHIA) operator, where P˜j=kwjPj and wj,ωj∈0,1, and ∑j=1kwj=1 and ∑j=1kωj=1.
Hybrid aggregation operators first aggregate the given data considering their attributes and then rearrange them in a specific order. After that, they aggregate the data considering their order. This means that hybrid operators are a generalization of weighted and ordered weighted operators. So, the T-SFEHIA operator will satisfy idempotency, monotonicity, and bounded property.Example 2.
Consider five T-SFNsP1=0.9,0.3,0.4, P2=0.6,0.3,0.2, P3=0.3,0.8,0.6, P4=0.4,0.5,0.8, and P5=0.6,0.0,0.0 with a weight vector w=0.25,0.20,0.15,0.18,0.22T.solution 1.
First of all, we find the aggregated value of these T-SFNs by using the T-spherical fuzzy Einstein hybrid averaging aggregation (T-SFEHA) operator [42] to find out the drawbacks of given operators. For this purpose, first, we have to find the value of t for which the given data lie in T-spherical fuzzy environment.
As0.3+0.8+0.6=1.7,Fort=2, 0.32+0.82+0.62=1.09Fort=3, 0.33+0.83+0.63=0.755
Similarly, fort=3, all the given data lie in the T-spherical fuzzy environment.
By using T-spherical fuzzy Einstein weighted averaging operator, we shall be able to find these values:(32)P1=0.9362,0.2104,0.3029,P2=0.6,0.3,0.2,P3=0.2726,0.8527,0.6984,P4=0.3862,0.5437,0.8211,P5=0.6187,0.0,0.0.
Their scores values will be(33)SCP1=0.7927,SCP2=0.2080,SCP3=−0.3203,SCP4=−0.4961,SCP5=0.2368.
Now using the score value, the aggregated values obtained by using T-SFEWA operators are rearranged in descending order. Then, these ordered values are again aggregated by using the T-SFEHA operator with associated weight vector beingω=0.112,0.236,0.304,0.236,0.112:(34)P˜σ1=0.9362,0.2104,0.3029,P˜σ2=0.6187,0.0,0.0,P˜σ3=0.6,0.3,0.2,P˜σ4=0.2726,0.8527,0.6984,P˜σ5=0.3862,0.5437,0.8211.
Now, employ the T-SFEHA operator as follows:(35)T−SFEHAw,ωP˜1,…,P˜5=0.6187,0.0,0.0.
From the above result, it is noticed that when abstinence or nonmembership value of one T-SFN is zero, then the T-SFEHA operator cannot aggregate the whole abstinence and nonmembership value. This shows a big flaw in the T-SFEHA operator. This means that the results obtained from T-SFEHA operators are not reliable. Now, by using the T-SFEIA operator, we shall show that the proposed operator will overcome this drawback.
By using T-SFEIA operator, we shall be able to find(36)P1=0.9362,0.4308,0.2739,P2=0.6,0.3,0.2,P3=0.2726,0.7370,0.5956,P4=0.3862,0.4829,0.7888,P5=0.6187,0.0,0.0.
Their scores values will be(37)SCP1=0.7999,SCP2=0.2080,SCP3=−0.1910,SCP4=−0.4333,SCP5=0.2368.
Now using the score value, the aggregated values obtained by using T-SFEWIA operators are rearranged in descending order. Then, these ordered values are again aggregated by using the T-SFEHIA operator, with associated weight vector beingω=0.112,0.236,0.304,0.236,0.112:(38)P˜σ1=0.9362,0.4308,0.2739,P˜σ2=0.6187,0.0,0.0,P˜σ3=0.6,0.3,0.2,P˜σ4=0.2726,0.7370,0.5956,P˜σ5=0.3862,0.4829,0.7888.
Now, employ the T-SFEHIA operator as follows:(39)T−SFEHIAw,ωP˜1,…,P˜5=0.6372,0.5055,0.3978.
This shows that the T-SFEIA operator aggregates the membership value.


## 6. Advantages

In this section, we prove that our work is more generalized than the existing work. In our proposed work, experts are free in giving the values to alternatives according to the given attributes. Not only this, the proposed work is also valid under those conditions where the existing work fails. Here, we reduced the proposed work to intuitionistic, Pythagorean, q-rung orthopair, picture, and spherical fuzzy environments. This proves that the proposed work is valid for all those environments.Consider the T-SFEHIA defined as follows:(1)
Fort=2, equation (31) reduces to SF Einstein hybrid interaction averaging operators (SFEHIA operators), i.e.,(40)T−SFEHIAw,ωP˜1,P˜2,…,P˜k=∏j=1k1+m˜σj2ωj−∏j=1k1−m˜σj2ωj∏j=1k1+m˜σj2ωj+∏j=1k1−m˜σj2ωj,∏j=1k1+i˜σj2ωj−∏j=1k1−i˜σj2ωj∏j=1k1+i˜σj2ωj+∏j=1k1−i˜σj2ωj2∏j=1k1−m˜σj2−i˜σj2ωj−∏j=1k1−m˜σj2−i˜σj2−nσjt2ωj∏j=1k1+m˜σj2ωj+∏j=1k1−m˜σj2ωj.(2)
Fort=1, equation (31) reduces to PF Einstein hybrid interaction averaging operators (PFEHIA operators), i.e.,(41)PFEHAw,ωP˜1,P˜2,…,P˜k=∏j=1k1+m˜σjωj−∏j=1k1−m˜σjωj∏j=1k1+m˜σjωj+∏j=1k1−m˜σjωj,∏j=1k1+i˜σjωj−∏j=1k1−i˜σjωj∏j=1k1+i˜σjωj+∏j=1k1−i˜σjωj2∏j=1k1−m˜σjωj−∏j=1k1−m˜σj−i˜σjnσjωj∏j=1k1+m˜σjωj+∏j=1k1−m˜σjωj.(3)
Fori=0, equation (31) reduces to q-ROPF Einstein hybrid interaction averaging operators (q-ROFEHIA operators), i.e.,(42)q−ROFEHIAw,ωP˜1,P˜2,…,P˜k=∏j=1k1+m˜σjtωj−∏j=1k1−m˜σjtωj∏j=1k1+m˜σjtωj+∏j=1k1−m˜σjtωjt,2∏j=1k1−m˜σjtωj−∏j=1k1−m˜σjt−nσjtωj∏j=1k1+m˜σjtωj+∏j=1k1−m˜σjtωjt.(4)
Fort=2 and i=0, equation (31) reduces to PyF Einstein hybrid interaction averaging operators (PyFEHIA operators), i.e.,(43)PyFEHIAw,ωP˜1,P˜2,…,P˜k∏j=1k1+m˜σj2ωj−∏j=1k1−m˜σj2ωj∏j=1k1+m˜σj2ωj+∏j=1k1−m˜σj2ωj,2∏j=1k1−m˜σj2ωj−∏j=1k1−m˜σj2−n˜σj2ωj∏j=1k1+m˜σj2ωj+∏j=1k1−m˜σj2ωj.(5)
Fort=1 and i=0, equation (31) reduces to IF Einstein hybrid interaction averaging operators (IFEHA operators), i.e.,(44)IFEHIAw,ωP˜1,P˜2,…,P˜k=∏j=1k1+m˜σjωj−∏j=1k1−m˜σjωj∏j=1k1+m˜σjωj+∏j=1k1−m˜σjωj,2∏j=1k1−m˜σjωj−∏j=1k1−m˜σj−n˜σjωj∏j=1k1+m˜σjωj+∏j=1k1−m˜σjωj.Similarly, we can reduce T-SFEWIA operator, T-SFEOWIA operator, T-SFEWIG operator, T-SFEOWIG operator, and T-SFEHIG operator.The proposed aggregation operators can aggregate the data given in FS, IFS, PyFS, q-ROPFS, PFS, and spherical fuzzy set environments, but the converse is not possible. Here, with the help of an example, it is proved that the proposed aggregation operator can aggregate the data given in IFSs.Example 4.
Let IFNsP1=0.2,0.5, P2=0.7,0.1, P3=0.3,0.4, P4=0.6,0.2, and P5=0.5,0.5 with a weight vector w=0.25,0.20,0.15,0.18,0.22T.Solution 1.
We can write these IFNs in the form of T-SFNs asP1=0.2,0,0.5, P2=0.7,0,0.1, P3=0.3,0,0.4, P4=0.6,0,0.2, and P5=0.5,0,0.5. Then, by using the T-SFEWIA operator, we shall be able to find these values:(45)P1=0.2481,0,0.5312,P2=0.7,0,0.1,P3=0.2281,0,0.3630,P4=0.5538,0,0.2071,P5=0.5401,0,0.4599.
Their scores values will be(46)SCP1=−0.2831,SCP2=0.6,SCP3=−0.1350,SCP4=0.3467,SCP5=0.0801.
Now, using the score value, the aggregated values obtained by using T-SFEWIA operators are rearranged in descending order. Then, these ordered values are again aggregated by using the T-SFEHIA operator with the associated weight vector beingω=0.112,0.236,0.304,0.236,0.112:(47)P˜σ1=0.7,0,0.1,P˜σ2=0.5538,0,0.2071,P˜σ3=0.5401,0,0.4599,P˜σ4=0.2281,0,0.3630,P˜σ5=0.2481,0,0.5312.
Now, utilize the T-SFEHIA operator as follows:(48)T−SFEHIAw,ωP˜1,P˜2,…,P˜5=0.4709,0,0.2645.
Here, it is proved that the information given in IFNs can be solved by using the T-SFEHIA operator. Similarly, we can solve the information given in IFNs by using the T-SEHIG operator and the information given in any other fuzzy structure can also be aggregated using the proposed operators.


## 7. An Algorithm for MADM with T-Spherical Fuzzy Information

Consider a set of alternativesD=d1,d2,d3,…,dl and a set of attributes M=m1,m2,m3,…,mk having a weight vector w=w1,w2,w3,…,wl, where wj∈0,1 and ∑m=1lwm=1. For making a decision, we have to follow these steps:Step 1. Calculate t for which the values lie in T-spherical informationStep 2. Aggregate the given alternatives according to attributes by T-SFEWIA (or T-SFEWIG) operators using some weight vectorsStep 3. Find scores values and with the help of score value, we reorder them in descending orderStep 4. Aggregate these ordered values using T-SFEHIA (or T-SFEHIG) operatorStep 5. Using score values find out the best optionExample 3.
The board of governors of a company decided to reduce their expenses for maximizing the profit. They observe that the cost of electricity is one of the major expense and they can reduce it if they started to generate electricity using solar energy. They have three options of photovoltaic cells that they may use in their solar plant:(i)
d1: monocrystalline photovoltaic cell(ii)
d2: polycrystalline photovoltaic cell(iii)
d3: thin Film photovoltaic cell
They assess the given photovoltaic cell on the basis of the following attributes.(i)
M1: heat tolerance(ii)
M2: cost(iii)
M3: reliability(iv)
M4: efficiency(v)
M5: ability of charge separationStep 1. As 0.8+0.3+0.7=1.8∉0,1, 0.82+0.32+0.72=1.22∉0,1 and 0.83+0.33+0.73=0.882∈0,1. Similarly, we found that all values in Table 1 belong to 0,1 for t=3.Step 2. By taking w=0.25,0.20,0.15,0.18,0.22T, we find T-SFEWIA values of given data, as listed in Table 2.Step 3. Scores of each alternative with respect to all attributes are shown in Table 3.By comparing the score values, we have(49)SCP15>SCP12>SCP13>SCP14>SCP11,SCP21>SCP23>SCP25>SCP22>SCP24,SCP34>SCP32>SCP33>SCP35>SCP31.Based on above score analysis, the data are arranged in descending order and the aggregated values of ordered data are as listed in Table4.Step 4. The associated weight vector will be ω=0.112,0.236,0.304,0.236,0.112T, and by using T-SFEHA operators, we have(50)P˜1=0.6596,0.5227,0.4668,P˜2=0.6176,0.5291,0.5276,P˜3=0.5826,0.7075,0.2290.Step 5. Now, we have to find the score values:(51)SCP˜1=0.1853,SCP˜2=0.0887,SCP˜3=0.1858,SCP˜3>SCP˜1>SCP˜2.Since the score value ofd3 is highest, the thin film photovoltaic cell is the best option.Now, we check their validity by using Einstein hybrid geometric interaction operators.By takingw=0.25,0.20,0.15,0.18,0.22T, we find T-SFEWG values of given data, as listed in Table 5.Scores of each alternative with respect to all attributes are shown in Table6.By comparing the score values, we have(52)SCP15>SCP13>SCP12>SCP14>SCP11,SCP23>SCP21>SCP25>SCP22>SCP24,SCP34>SCP33>SCP32>SCP35>SCP31.Based on the above score analysis, the data are arranged in descending order and the aggregated values of ordered data are as listed in Table7.Associated weight vector will beω=0.112,0.236,0.304,0.236,0.112T, and by using T-SFEHG operators, we have(53)P˜1=0.5445,0.5217,0.5419,P˜2=0.6830,0.5376,0.5913,P˜3=0.7556,0.6879,0.2780.Step 6. Now, we have to find the score values:(54)SCP˜1=0.0023,SCP˜2=0.1080,SCP˜3=0.4099,SCP˜3>SCP˜2>SCP˜1.
Here again, the score value of alternatived3 is high. So, the thin film photovoltaic cell is the best option.Table 1
Decision matrix.
M1M2M3M4M5d10.4,0.1,0.70.5,0.2,0.40.8,0.3,0.70.4,0.8,0.50.9,0.5,0.2d20.7,0.4,0.30.2,0.4,0.70.9,0.3,0.60.3,0.2,0.80.4,0.7,0.5d30.4,0.7,0.50.6,0.6,0.10.6,0.9,0.20.8,0.1,0.10.5,0.6,0.2Table 2
Aggregated values.
M1M2M3M4M5d10.4308,0.1077,0.73670.5,0.2,0.40.7370,0.2726,0.71720.3862,0.7770,0.50220.9164,0.5160,0.1906d20.4308,0.4308,0.30610.2,0.4,0.70.8440,0.2726,0.66770.2896,0.1931,0.78200.4129,0.7206,0.5037d30.4308,0.7485,0.50640.6,0.6,0.10.5464,0.8440,0.24300.7770,0.0965,0.08290.5160,0.6187,0.2031Table 3
Score values.
M1M2M3M4M5d1−0.31990.06100.0314−0.06900.7626d20.3906−0.33500.3035−0.4538−0.0574d3−0.04990.21500.14880.46850.1290Table 4
Ordered aggregated values.
M1M2M3M4M5d10.9164,0.5160,0.19060.5,0.2,0.40.7370,0.2726,0.71720.3862,0.7770,0.50220.4308,0.1077,0.7367d20.7485,0.4308,0.30610.8440,02726,0.66770.4129,0.7206,0.50370.2,0.4,0.70.2896,0.1931,0.7820d30.7770,0.0965,0.08290.6,0.6,0.10.5464,0.8440,0.24300.5160,0.6187,0.20310.4308,0.7485,0.5064Table 5
Aggregated values.
M1M2M3M4M5d10.4117,0.1077,0.74850.5,0.2,0.40.7998,0.2726,0.63980.4008,0.7770,0.48290.9051,0.5160,0.2064d20.7347,0.4308,0.32320.2,0.4,0.70.9071,0.2726,0.54640.2984,0.1931,0.77700.4034,0.7206,0.5160d30.4064,0.7485,0.53810.6,0.6,0.10.6434,0.8440,0.18170.7807,0.0965,0.09650.5103,0.6187,0.2064Table 6
Score values.
M1M2M3M4M5d1−0.34950.06100.2497−0.04820.7327d20.3628−0.33500.5833−0.4425−0.0717d3−0.08870.21500.26030.47490.1241Table 7
Ordered aggregated values.
M1M2M3M4M5d10.9051,0.5160,0.20640.7998,0.2726,0.63980.5,0.2,0.40.4008,0.7770,0.48290.4117,0.1077,0.7485d20.9071,0.2726,0.54640.7347,0.4308,0.32320.4034,0.7206,0.51600.2,0.4,0.70.2984,0.1931,0.7770d30.7807,0.0965,0.09650.6434,0.8440,0.18170.6,0.6,0.10.5103,0.6187,0.20640.4064,0.7485,0.5381


## 8. Conclusion

In this paper, it is pointed out that the existing work [14, 42] fails under some conditions such as in Einstein averaging operators, if one nonmembership value of an IFN is zero, then the NMD of aggregated value will also become zero and neglect the other nonmembership values. Similarly, if the membership value of one IFN becomes zero, then the MD of the aggregated value of the Einstein geometric operator will also become zero and neglect the other membership values. So, new interactive operational laws are proposed in this article. On the basis of these operational laws, T-spherical Einstein interactive geometric operators and T-spherical Einstein interactive averaging operators are proposed. After that, some conditions are discussed under which the proposed operators can reduce to other fuzzy frameworks. A comparison of proposed and existing work is also established and explained using an example. We validate the proposed operators with the help of an application in MADM. In the future, we have a plan to propose some power aggregation operations for T-spherical fuzzy sets and try to use them in the MADM process.



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