Interval Valued T-Spherical Fuzzy Soft Average Aggregation Operators and Their Applications in Multiple-Criteria Decision Making

: This paper deals with uncertainty, asymmetric information, and risk modelling in a complex power system. The uncertainty is managed by using probability and decision theory methods . Multiple-criteria decision making (MCDM) is a very effective and well-known tool to investigate fuzzy information more effectively. However, the selection of houses cannot be done by utilizing symmetry information, because enterprises do not have complete information, so asymmetric information should be used when selecting enterprises. In this paper, the notion of soft set (𝑆 (cid:3033)(cid:3047) 𝑆) and interval-valued T-spherical fuzzy set (IVT-SFS) are combined to produce a new and more effective notion called interval-valued T-spherical fuzzy soft set (𝐼𝑉𝑇 − 𝑆𝐹𝑆 (cid:3033)(cid:3047) 𝑆) . It is a more general concept and provides more space and options to decision makers (DMs) for making their decision in the field of fuzzy set theory. Moreover, some average aggregation operators like interval-valued T-spherical fuzzy soft weighted average (𝐼𝑉𝑇 − 𝑆𝐹𝑆 (cid:3033)(cid:3047) 𝑊𝐴) operator, interval-valued T-spherical fuzzy soft ordered weighted average (𝐼𝑉𝑇 − 𝑆𝐹𝑆 (cid:3033)(cid:3047) 𝑂𝑊𝐴) operator, and interval-valued T-spherical fuzzy soft hybrid average (𝐼𝑉𝑇 − 𝑆𝐹𝑆 (cid:3033)(cid:3047) 𝐻𝐴) operators are explored. Furthermore, the properties of these operators are discussed in detail. An algorithm is developed and an application example is proposed to show the validity of the present work. This manuscript shows how to make a decision when there is asymmetric information about an enterprise. Further, in comparative analysis, the established work is compared with another existing method to show the advantages of the present work.


Introduction
Multi-criteria decision making (MCDM) is a process that can give the ranking results for the finite alternatives according to the attribute values of different alternatives, and it is an important aspect of decision sciences. In recent years, the development of enterprises and social decision making in all aspects is related to the issue of MCDM, so it is widely applied in all kinds of fields. In the real decision-making process, an important problem is how to express the attribute value more efficiently and accurately. In the real world, because of the complexity of decision-making problems and the fuzziness of decisionmaking environments, it is not enough to express attribute values of alternatives by exact values. For this, the concept of fuzzy set (FS) was proposed by Zadeh [1], and many extensions have been established by researchers and many new notions were developed over time. Since FS only deals with membership grade (MG) " " with the condition that 0 ≤ ≤ 1, which is the limited idea, so the idea of FS was further generalized into an interval-valued fuzzy set [2] (IVFS). In many practical examples, we have to deal not only with MG but also consider the non-membership grade (NMG) " ". Since in FS the NMG is not under consideration, which is a drawback of FS, the concept of intuitionistic fuzzy set (IFS) was established by Atanassov [3] having the characteristics that 0 ≤ + ≤ 1. In addition, some prioritized IF aggregation operators are discussed in [4]. Moreover, IF interaction aggregation operators and IF hybrid arithmetic and geometric aggregation operators are established in [5,6]. To provide more space to DMs, Atanassov [7] generalized IFS into IVIFS, and some IVIF aggregation operators are given in [8]. Aggregation operators are a valuable tool to deal with the fuzzy information because it converts the whole data into a single value which is helpful in the decision-making process. When DMs provide "0.6" as MG and "0.5" as NMG, then IFS fails to deal with such types of information. To overcome this issue, the idea of IFS was further extended into Pythagorean fuzzy set [9] having the condition that 0 ≤ + ≤ 1. It is a stronger apparatus and it can tackle fuzzy information more effectively. Based on Einstein's t-norm and t-co norm, some generalized fuzzy geometric aggregation operators are given by Garg et al. [10]. This idea is further extended into and some aggregation operators are provided in [11].
s also limited notion because when DMs provide 0.7 as MG and "0.9" as NMG, then cannot tackle this type of data. To overcome this complexity, this notion is further generalized into q-rung orthopair fuzzy set (q-ROFS) established by Yager [12] having the necessary condition that 0 ≤ + ≤ 1. Some q-ROF point weighted aggregation operators are explored in [13]. Some IVq-ROF Archimedean Muirhead Mean operators are discussed in [14]. Molodtsov [15] established the idea of a soft set which is a parameterization structure to deal with uncertainty in data. Maji et al. [16] explored some new operations and proposed application of . Ali et al. [17] explored the application of in decision-making problems. Since the idea of has been established, some new notions are established like a fuzzy soft set established by Maji et al. [18], which is the combination of FS and . Some considerable extensions have been developed keeping in view the idea of and then IVFS and are combined by Yang et al. [19] to introduce the new idea called . Since is a limited structure, so notions of IF soft set [20] have been developed. Moreover, generalized and group-based generalized intuitionistic fuzzy soft sets with their applications in decision making have been explored in [21,22]. In addition, due to the drawback of , the further idea of has been extended into a Pythagorean fuzzy soft set [23]. Further q-rung orthopair fuzzy soft set ( − ) proposed by Hussain et al. [24] developed the notion of and also explored some − , − and − operators. From the mentioned literature, it is clear that all the fuzzy information deals with only MG and NMG. Sometimes, DMs consider the obstinacy grade AG " " along with MG " " and NMG " " in their information, and there are many practical examples which can be provided in this regard, so due to this reason, the idea of picture fuzzy set (PFS) [25] has been developed, which also considers the AG, which is more general information and provides more space to deal with vagueness in data with condition that 0 ≤ + + ≤ 1. Similarly, as the idea of IFS is generalized into , the notion of PFS set is extended into the spherical fuzzy set (SFS) by Mahmood et al. [26] with condition that 0 ≤ + + ≤ 1. Moreover, Ashraf et al. [27] established the spherical fuzzy Dombi aggregation and proposed their application in group decision-making problems. SFS is a limited idea because if DMs provide "0.9" as an MG, 0.8 as an NMG, and 0.7 as an AG, then both PFS and SFS fail to deal with such types of information, so to overcome this complexity, the notion of T-spherical fuzzy set (T-SFS) has been established by Ullah et al. [28] with condition that 0 ≤ + + ≤ 1 and exploring some similarities measures based on T-SFNs. Some T-SF power Muirhead mean operators based on novel operational law have been developed in [29]. Further, Quek et al. [30] established the generalized T-spherical fuzzy weighted aggregation operators on neutrosophic sets. Correlation coefficients for T-SFS and their application in clustering and multi-attribute decision making have been established by Ullah et al. [31] and a note on geometric aggregation operators in the T-SF environment is given in [32]. Furthermore, Ullah et al. [33] proposed T-SF Hamacher aggregation operators. Some T-SF Einstein hybrid aggregation operators and their application in multi-attribute decision-making problems have been proposed by Munir et al. [34]. Based on improved interactive aggregation operators, an algorithm for T-SF multi-attribute decision making has been established by Garg et al. [35]. The idea of T-SFS has been extended to interval-valued T-spherical fuzzy set (IVT-SFS) established by Ullah et al. [36] and they have explored the evaluation of investment policy based on multi-attribute decision making using IVT-SF aggregation operators. Keeping in view the idea of , , and − , the notion of PF soft set has been proposed by Yang et al. [37], which generalizes all the above literature due to parameterization structure. The idea of a multi-valued picture fuzzy soft set was proposed by Jan et al. [38]. The study of aggregation operators and their application in decision making can be seen in [39,40]. Perveen et al. [41] extended the idea of into the spherical fuzzy soft set , which is the combination of and SFS. Since T-SFS is more general than SFS, so the concept of is further extended into a T-spherical fuzzy soft set − proposed by Guleria et al. [42]. Moreover, some new operations on interval-valued picture fuzzy soft set ( ) are discussed in [43] and interval-valued spherical fuzzy weighted arithmetic means (IVSFWAM) and interval-valued spherical fuzzy weighted geometric mean (IVSFWGM) operators are established in [44]. The notion of interval-valued T-spherical fuzzy sets and soft sets is very closely related to the notion of symmetry. Based on symmetry, we can talk about the mixture of both theories. We can extend the notion of interval-valued T-spherical fuzzy to intervalvalued T-spherical fuzzy soft sets, especially when determining the aggregate intervalvalued T-spherical fuzzy soft number estimated by several experts and in a situation where there is imperfect knowledge (when one party has different information to another).
MCDM is a very effective and well-known tool to investigate fuzzy information more effectively. Thus, from the mentioned literature, it is clear that the interval-valued structures are more general and gain more attention in decision-making problems. To the best of our knowledge, there is no work on combining the notion of IVT-SFS and . Hence, in this paper, the notion of and IVT-SFS are combined to produce a new notion called the − . It is a more general concept and provides more space to DMs for making their decision in the field of fuzzy set theory. Moreover, some new average aggregation operators like − operator and − operators are explored.
− can only find the − values and − weight the ordered position. Hence, due to this drawback, the − operators are explored, as they can account for both aspects. Furthermore, the properties of these operators are discussed in detail. An algorithm is developed, and an application example is proposed to show the validity of the proposed work. In a comparative analysis, the present work is compared with another existing method to show the advantages the present work offers. The manuscript is structured as follows: Section 2 deals with basic notions of PFS, SFS, T-SFS, , , and − . Moreover, their operations are discussed. Section 3 deals with the basic notion of − and some fundamental operations on this notion are discussed in detail. In Section 4, we have established some new operators called − , − and − operator. In Section 5, we have established an algorithm and an illustrative example is given to show the validity of the present work. In addition, we have provided a comparative analysis of the present work to demonstrate its advantages compared to the approaches from the literature. Finally, Section 6 provides concluding remarks.

Preliminaries
This section deals with the basic notion of SFS, T-SFS, , and − . Moreover, their basic properties are discussed which will help us in further sections.    is NMG with the condition that 0 ≤ ( ) + ( ) + ( ( )) ≤ 1.       Table 1.

Interval-Valued T-Spherical Fuzzy Soft Average − Aggregation Operator
In this section, the detailed study of − , − and − operators is discussed and further, we will discuss the properties of these operators.

Interval-Valued T-Spherical Fuzzy Soft Weighted Average − Aggregation Operators
Here, we discuss the detailed structure of − operators and their properties are discussed in detail.  Proof. We will use the mathematical induction method to prove this result. We know by the operational laws that First of all, we will show that Equation (1) is true for = 2 and = 2, so we have Hence the result is true for = 2 and = 2.
Further, suppose that Equation (1) is true for = + 1 and It is clear from the above expression that − is again an − .
Hence it is true for all , ≥ 1. □

operator reduces to an interval-valued T-spherical fuzzy weighted average (IVT-SFWA) operator.
Hence it is clear that , , , and IVT-SFWA operators are the special cases of − operator. The present work is more general.  Table 2. By using Equation (1), we have − , , … , Hence IVT − SFS WA F , F , … , F = F .

(Boundedness
Now for each = 1, 2, … , and = 1, 2, … , , we have Moreover, for each = 1, 2, … , and = 1, 2, … , , we have Therefore from Equations (2)-(4), it is clear that according to the definition of score function given in Definition 11, we obtain According to this condition, we have the following cases Case i. If ( ) < and ( ) > , then by Definition 12, we have Then by using the above inequalities, we get Case iii. If ( ) = , then and , ≥ , , , ⇒ , Moreover, Hence the result is proved. □

Operator
From the above discussion, it is clear that − operator only weighted the value of − . However, on the other hand, the − operator weights the ordered position by scoring the − values. Here, we will discuss the − operator and also its properties.   Table 3.  .

(Homogeneity). For any real number
Proof. The proof is simple and follows from Theorem 3. □

Interval-Valued T-Spherical Fuzzy Soft Hybrid Aggregation − Operator
In this section, we will discuss interval-valued T-spherical fuzzy soft hybrid aggregation operator which can deal with both aspects like measuring the values of − and also considering the ordered position by "SF" of − values.
Moreover, we will discuss the properties related to these operators.   Table 5.  − ,

(Homogeneity). For any real number
Proof. The proof is simple and follows from Theorem 3. □

An Algorithm for MCDM based on − Information
MCDM approach is a well-known and very effective technique for the selection of the best alternative among the given and this approach has been used in different fields of fuzzy sets theory for the selection of the best alternative. About an alternative, the decision makers keep many aspects in their mind, such as the flexibility of the alternative, benefits, different features, and drawbacks. After the evaluation of all these aspects, they could decide which alternative is best and reach the best result. In this section, we will propose a stepwise algorithm for MCDM under the environment of  = ( , , ). Lastly, we will use the formula of score function for over aggregated − for alternatives and rank them according to their order and choose the best result.
The stepwise algorithm for overall above discussion is given as follows: Step 1. Accumulate the evaluation information of all experts for each alternative according to their parameters and arrange it to construct an overall decision matrix = × given by Step 2. Normalize the given information by interchanging of cost type parameter into the benefit type parameter if it is needed. The formula is given below: Step 3 . Aggregate the  −  ,  =  ,  ,  ,  ,  ,  by   using the proposed aggregation operators for each alternative ( = 1, 2, … , ) to get the  aggregated  −  =  ,  ,  ,  , , .
Step 4. Calculate the score values for each " " by using Definition 11.

Application Steps for the Proposed Method
In this section, we will provide an example of the present work in detail to show its validity and advantages.
Let By using − operators: Step 1. The experts present their information of each alternative in the shape of − according to their resultant parameters. This information is given in Tables 6-9 correspondingly Step 2. There is no requirement f or normalization of − matrix since all the parameters are of a similar kind.
− matrix for alternative .  Step 5. Rank the score values and select the best alternative. Hence, we obtain the ranking result as Hence, from the above discussion, it is clear that " " is the best alternative. By using − operators: Step 1. Same as above.
Step 2. Same as above.
Step 3. The information of each expert for each alternative ( = 1, 2, 3, 4) is aggregated by using Equation (8) Step 5. Rank the score values and select the best alternative. Hence, we obtain the ranking result as

( ) > ( ) > ( ) > ( )
Hence, it is noted that the aggregated result for − operator is the same as the result obtained for − operator. Hence " " is the best alternative.
By using − operators: Step 1. Same as above.
Step 2. Same as above.
Step 3. The information of each expert for each alternative ( = 1, 2, 3, 4) is to be aggregated by using Equation (8)  Step 5. Rank the score values and select the best alternative. Hence, we obtain the ranking result as Hence, it is noted that the aggregated result for − operator is the same as the result obtained for − and − operator. Hence " " is the best alternative.

Comparative Analysis
Here in this section, we will propose the comparative analysis of established work with other existing methods to prove the superiority of the present work. We will compare the present work with IVPFWA, IVPFOWA, IVPFHA, , IVSFWA, IVSFOWA, IVSFHA, , IVT-SFWA [36], IVT-SFOWA [36], IVT-SFHA [36], IVSFWAM [44], IVSFWGM [44], and [43].  Table 10. We use IVPFWA, IVPFOWA, IVPFHA, , IVSFWA, IVSFOWA, IVSFHA, , IVT-SFWA [36], IVT-SFOWA [36], IVT-SFHA [36], IVSFWAM [44], IVSFWGM [44], and [43] operators to compare with the present work and the evaluation results are shown in Table 11.  Table 11, we can see that we can use different methods to get different results under the same evaluation data. Notice that" " is the best alternative in all cases that shows the validity of proposed work. Moreover, proposed operators can consider the parameterization structure while the operators given as IVPFWA, IVPFOWA, IVPFHA, IVSFWA, IVSFOWA, IVSFHA, IVT-SFWA [36], IVT-SFOWA [36], IVT-SFHA [36], IVSF-WAM [44], IVSFWGM [44] cannot consider the parameterization structure. From the above analysis, it is clear that the present work is more general than existing methods.  Table 12. IVSFHA, , IVT-SFWA [36], IVT-SFOWA [36], IVT-SFHA [36], IVSFWAM [44], IVSFWGM [44], and [43] to compare with the present work and the evaluation results are shown in Table 13.  [36], IVT-SFOWA [36], IVT-SFHA [36], IVSFWAM [44], IVSFWGM [44], and [43] to compare with proposed work.  [30]. Similarly, if data given in Tables 6-9 are considered, then all the above-given methods fail to handle all this information, while the present work along with the method given in [30] can easily handle this type of information. Hence, it is clear that the present work provides more space to DMs in making their decisions for MCDM problems. Hence, the present work is more general. For this, − are aggregated and the overall decision matrix for different mobile phone brands ; = 1, 2, 3, 4 by using WVs = 0.28, 0.25, 0.23, 0.24 is given in Table 14. From Table 14, it is clear that all the information consists of − and this information cannot be tackled by all the above-given methods, so we cannot calculate the score values for all the above given operators, while the presented operators can tackle this information along with the method given in [30] and also we can calculate the score values for all data given in Table 14. Now using this information, a comparative evaluation of all the above given aggregation operators with the present work is given together with their results in Table 15.   Table 15, note that " 3 " is the best alternative, which shows the validity of the proposed work. Further, the characteristic evaluation of the present approach with all the above operators is given in Table 16. Hence, it is clear that IVPFWA, IVPFOWA, IVPFHA, IVSFWA, IVSFOWA, IVSFHA, IVSFWAM [44], IVSFWGM [44], IVT-SFWA [36], IVT-SFOWA [36], IVT-SFHA [36] cannot consider the parameterization structure. The main advantage of the present work is that it provides more space to DMs, generalizes many existing structures, and also considers parameterization structures to deal with real-life problems. Hence, the present work can be used in MCDM problems rather than using it for other operators in the − environment.  [44] Yes No IVSFWGM [44] Yes No IVT-SFWA [36] Yes No IVT-SFOWA [36] Yes No IVT-SFHA [36] Yes

Scientifitic Decision of the Proposed Works
The idea of − is an important technique to cope with complicated and uncertain information in real-life issues. The idea of − is the mixture of two different ideas such as − and , which contains the grade of truth, abstinence, and falsity with a rule that the sum of the upper parts of the q-powers of all grades is restricted to unit interval. The advantages of the proposed − are discussed below: 1. If we choose the value of = 2, then the proposed − is converted for interval-valued spherical fuzzy soft sets. 2. If we choose the value of = 1, then the proposed − is converted for interval-valued picture fuzzy soft sets. 3. If we choose the value of abstinence is zero, then the proposed − is converted for interval-valued q-rung orthopair fuzzy soft sets. 4. If we choose the value of abstinence is zero with = 2, then the proposed − is converted for interval-valued Pythagorean fuzzy soft sets. 5. If we choose the value of abstinence is zero with = 1, then the proposed − is converted for interval-valued intuitionistic fuzzy soft sets.
Similarly, in future, we will extend the proposed work − for the following ideas: In future, this work will be used in the environment of image segmentation, pattern recognition, medical diagnosis, and determination of the dangers of brain cancers.

Conclusions
MCDM approach is a well-known and very effective technique for the selection of the best alternative among the given and this approach has been used in different fields of fuzzy sets theory. Aggregation operators are an effective tool to deal with fuzzy information and desirable results for real-life problems can be obtained by these means. Here in this paper, we have combined two notions, IVT-SFS and SS, to generate the new notion called − . It is a strong apparatus to deal with fuzzy information and also generalized many previous ideas such as , , − , and . Moreover, inspired by the parameterization property of soft set, we have established the operators such as − , − and − operators and also their properties are discussed in detail. An algorithm is developed and an application example is proposed to show the validity and superiority of the proposed work. Further, in comparative analysis, the established work is compared with another existing method to show the superiority of the present work.
In the future, one can combine − and − to introduce a new notion called cubic T-spherical fuzzy soft set − . In addition, this notion can be used in many MCDM approaches and desirable results can be obtained. Moreover, numerous scholars have introduced the hybrid notion of rough set and other fuzzy sets theories and applied these notions to multi-attribute decision-making problems as given in [45][46][47][48]. Therefore, one can also use the established structure and rough set to introduce new hybrid notions like interval-valued T-spherical fuzzy soft rough set and soft rough interval-valued T-spherical fuzzy set, and then this notion can be used in many decisionmaking problems.
In future, we will extend the proposed idea to bipolar soft sets [49], complex T-spherical fuzzy sets [50,51], and complex neutrosophic sets [52]. This work will also be utilized in the environment of image segmentation [53], pattern recognition [54], medical diagnosis, and determination of the dangers of brain cancers.