Some T-Spherical Fuzzy Einstein Interactive Aggregation Operators andTheir Application to Selection of Photovoltaic Cells

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. (e superiority of the proposed operators over existing work is checked with the help of an example.


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. e 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 MULTI-MOORA 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 intervalvalued IFSs. Some MADM problems are solved for IFSs in [16][17][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][32][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][37][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][44][45][46]. e existing Einstein operations [14,42] fail under some circumstances, e.g., if I 1 � (m A , 0) and I 2 � (0, n A ) 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.
is issue motivates us to propose some improved Einstein aggregation operators which will overcome these problems. e 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 example To achieve these aims, the rest of the article is summarized as follows. Section 2 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. e 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.

Preliminaries
Some basic definitions over the set X are defined in this section.
ese 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 where m, i, n: X ⟶ [0, 1] are a membership, abstinence, and nonmembership function, respectively, with the con- is called the refusal degree of x in P, and (m, i, n) is known as T-spherical fuzzy number (T-SFN).
en, the score value of P is defined as SC(P) � m t (x) − n t (x), and the accuracy value of P is defined as AC(P) � m t (x) + i t (x) + n t (x). 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. e 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. Definition 3 (see [14]). Some operations for any two IFNs I 1 � (m 1 , n 1 ) and I 2 � (m 2 , n 2 ) are defined as , n 1 + n 2 1 + n 1 n 2 ,

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. Let P 1 � (m 1 , i 1 , n 1 ) and P 2 � (m 2 , i 2 , n 2 ) be two T-spherical fuzzy sets; then, their Einstein operations are as follows:

Remark 2
(1) e defined operations will be reduced to spherical fuzzy sets for t � 2 (2) e defined operations will be reduced to PFSs for t � 1 (3) e defined operations will be reduced to q-ROFSs for i � 0 (4) e defined operations will be reduced to PyFSs for t � 2 and i � 0 (5) e defined operations will be reduced to IFSs for t � 1 and i � 0 Mathematical Problems in Engineering 3 (6) e defined operations will be reduced to FSs for t � 1, i � 0, and n � 0

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. e validity of the proposed work is checked with the help of an example.
Proof. Same as in eorem 1. □ Theorem 5. Consider a collection of T − SFNs P j (j � 1, 2, . . . , k) with P L � min j P j and P U � max j P j . en, Proof. Straightforward. □ Theorem 6. For any two T-SFNs, P j � (m j , i j , n j ) and P j ′ � (m j ′ , i j ′ , n j ′ ) for all (j � 1, 2, . . . , k) such that P j ≤ P j ′ for all j. en,

Mathematical Problems in Engineering
Proof. Directly follows from the proof of eorem 3. □ Definition 6. For any collection of T-SFNs, P j � (m j , i j , n j )(j � 1, 2, 3, . . . , k). e mapping T is the associated weight vector of P j 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. is 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.

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.
By using T-spherical fuzzy Einstein weighted geometric operator, we shall be able to find these values:     Mathematical Problems in Engineering 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. is shows a big flaw in the T-SFEHG operator. is 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 eir scores values will be SC P 1 � 0.3941, Now using the score value, the aggregated values obtained by using T-SEWIG operators are rearranged in descending order.
is shows that the T-SFEIG operator aggregates the membership value.

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.
Proof. Let P j � P 0 � (m 0 , i 0 , n 0 ) for all j; then, Mathematical Problems in Engineering 7 Theorem 8. Consider a collection of T-SFNs P j (j � 1, 2, . . . , k) with P L � min j P j and P U � min j P j . en, Proof. Straightforward. □ Theorem 9. For any two T-SFNs, P j � (m j , i j , n j ) and P j ′ � (m j ′ , i j ′ , n j ′ ) such that P j ≤ P j ′ for all j. en, Proof. Let P j ≤ P j ′ ; then, m j ≤ m j ′ , i j ≤ i j ′ , and n j ≥ n j ′ . en, by using this basic information, 8 Mathematical Problems in Engineering is shows that T − SFEWIA w P 1 , P 2 , . . . , P k ≤ T − SFEWIA w P 1 ′ , P 2 ′ , . . . , P k ′ .
Proof. Same as in eorem 7. □ Theorem 11. Consider a collection of T-SFNs P j (j � 1, 2, . . . , k) with P L � min j P j and P U � min j P j . en, Proof. Straightforward. □ Theorem 12. Consider any two T − SFNs P j � (m j , i j , n j ) and P j ′ � (m j ′ , i j ′ , n j ′ ) such that P j ≤ P j ′ for all j. en, Proof. Same as in eorem 9.
□ Definition 9. Consider a collection of T-SFNs P j � (m j , i j , n j )(j � 1, 2, 3, . . . , k). e mapping is called the T-spherical fuzzy Einstein hybrid interactive averaging (T-SFEHIA) operator, where P j � kw j P j and w j , ω j ∈ [0, 1], and k j�1 w j � 1 and k j�1 ω j � 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. is 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. 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.
As 0.3 + 0.8 + 0.6 � 1.7, Similarly, for t � 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: 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.
is shows a big flaw in the T-SFEHA operator.
is 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

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is shows that the T-SFEIA operator aggregates the membership value.

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. is proves that the proposed work is valid for all those environments.

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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.

An Algorithm for MADM with T-Spherical Fuzzy Information
Consider a set of alternatives D � d 1 , d 2 , d 3 , . . . , d l and a set of attributes M � m 1 , m 2 , m 3 , . . . , m k having a weight vector w � w 1 , w 2 , w 3 , . . . , w l , where w j ∈ [0, 1] and l m�1 w m � 1. For making a decision, we have to follow these steps: Step 1. Calculate t for which the values lie in T-spherical information Step 2. Aggregate the given alternatives according to attributes by T-SFEWIA (or T-SFEWIG) operators using some weight vectors Step 3. Find scores values and with the help of score value, we reorder them in descending order Step 4. Aggregate these ordered values using T-SFEHIA (or T-SFEHIG) operator Step 5. Using score values find out the best option Example 3. e board of governors of a company decided to reduce their expenses for maximizing the profit. ey 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. ey have three options of photovoltaic cells that they may use in their solar plant:  Table 1 belong to [0, 1] for t � 3.
Step 3. Scores of each alternative with respect to all attributes are shown in Table 3 Based on above score analysis, the data are arranged in descending order and the aggregated values of ordered data are as listed in Table 4.
Step 4. e associated weight vector will be ω � (0.112, 0.236, 0.304, 0.236, 0.112) T , and by using T-SFEHA operators, we have (50) Step 5. Now, we have to find the score values: Since the score value of d 3 is highest, the thin film photovoltaic cell is the best option. Now, we check their validity by using Einstein hybrid geometric interaction operators. By taking w � (0.25, 0.20, 0.15, 0.18, 0.22) T , 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 Table 6. By comparing the score values, we have SC P 15 > SC P 13 > SC P 12 > SC P 14 > SC P 11 , SC P 23 > SC P 21 > SC P 25 > SC P 22 > SC P 24 , SC P 34 > SC P 33 > SC P 32 > SC P 35 > SC P 31 .

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Here again, the score value of alternative d 3 is high. So, the thin film photovoltaic cell is the best option.

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.
Data Availability e data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.