Analysis of T-Spherical Fuzzy Matrix and Their Application in Multiattribute Decision-Making Problems

e aim of this study is to introduce an innovative concept of T-spherical fuzzy matrix, which is a hybrid structure of fuzzy matrix and T-spherical fuzzy set. is article introduces the square T-spherical fuzzy matrix and constant T-spherical fuzzy matrix and discusses related properties. Determinant and the adjoint of a square T-spherical fuzzy matrix are also established, and some related properties are investigated. An application of the T-spherical fuzzy matrix in decision-making problem with an illustrative example is discussed here. en, in the end, to check capability and viability, a practical demonstration of the planned approach has also been explained.


Introduction
In real life, sometimes it is necessary to compare two different things from di erent perspectives while dealing with di erent problems related to machine learning, namely, decision-making (DM), and image processing. An abundance of data is available in fuzzy and nonfuzzy situations concerned with the application. Di erent comparative measures may be applicable for various problems. e under-consideration article, for the most part, is related to the multi-attribute decision-making (MADM) problems. MADM is the important problem of deciding science, whose objective is to get the best choice from the group of similar choices. Originally in DM, one needed to evaluate the alternate options by many other categories. e non-cooperative behavior management for personalized individual semantic-based social network group decision-making is developed in [1], the group consensus-based travel destination evaluation method with online reviews is discussed in [2], and the comprehensive star rating approach for cruise ships based on interactive group DM with personalized individual semantics is performed in [3]. In order to regulate it, the concept of a fuzzy set (FS) was initiated by Zadeh [4]. It was a helpful tool to deal with uncertainties in real-life problems. Some prominent developments in these directions are mentioned. e fundamental theory of fuzzy sets with illustrative examples has been discussed in [5], some aggregation procedures, choice problems, and treatment of attributes are examined in [6], DM approaches to vowel and speaker recognition are studied in [7], multiple objective DM is discussed in [8], and fuzzy sets and fuzzy decisionmaking are discussed by Li and Yen [9]. Following this new direction in fuzzy theory, the idea of a fuzzy matrix (FM) was initiated in [10]. Later on, some operations and generalizations on FM such that FM with row and column have been developed in [11], and interval-valued FM with rows and columns is discussed by Pal [12]. e study of bipolar FM has been developed in [13]. e generalized FMs are discussed in [14]. Pradhan and Pal [15] developed the concept of the triangular FM norm and its properties. Ragab discussed the adjoint and determinant of square FM in [16], and he further developed the concept of min-max composition of FMs [17].
e FS theory has not been able to deliver in some conditions. In particular, in clear information, the complement of the participation degree (PD) is equal to the nonparticipation degree (NPD). In such cases, the NPD is not the complement of the PD. In this situation, the PD and NPD are needed. To handle the situation, Atanassov introduced the concept of intuitionistic fuzzy set (IFS) [18], which describes the PD and the NPD of an element or object. Following this new direction in fuzzy theory, the idea of an intuitionistic fuzzy determinant was initiated in [19]. Later on, some operations and generalizations on intuitionistic FM (IFM) such that interval-valued IFM have been examined by Khan and Pal [20], the concept of generalized inverse of block IFM is discussed in [21], and intuitionistic fuzzy incline matrix and determinant have been developed in [22]. Furthermore, Sriram and Murugadas [23] developed the concept of semiring of IFM, and he also studies the α-cut of IFM [24].
An IFS is a better tool than Zadeh's FS as it describes the NPD as well. But IFS has not been able to deliver in some conditions. For example, if a person is given 0.7 PD and 0.5 NPD, in that condition IFS will be unable to manage it, i.e., 0.7 + 0.5 � 1.2 > 1. In that condition, IFS has not been kept in mind. In the same way, some problems were faced in reallife matters, where the IFS was also deviated. To handle the situation, Yager [25,26] initiated the system of Pythagorean FSs (PyFSs), having the condition (PD) 2 + (NPD) 2 ∈ [0, 1]. Following this new direction in fuzzy theory, the idea of Pythagorean FM (PyFM) was initiated in [27]. Later on, some operations and generalizations on PyFM were developed in [28,29].
In various fields of real life, it turns out that to represent a physical phenomenon two components are not enough. For example, a disease may have three aspects: positive, neutral, and negative. To handle such type of data, the IFS model is not sufficient. To overcome these limitations, Cuong initiated the concept of picture fuzzy set (PFS) in [30,31], which described the PD, abstained degree (AD), and NPD of an element or object. Some picture fuzzy operators are discussed in [32,33]. In the generalization of PFSs, the new concept of picture fuzzy matrix (PFM) was introduced in [34].
e PFSs extend the model of FSs and IFSs, but there is still a limitation in the structure. For example, if a person is given 0.6 PD, 0.4 AD, and 0.3 NPD, in that condition PFS will be unable to manage it, i.e., 0.6 + 0.4 + 0.3 � 1.3 > 1. In that condition, PFS has not been kept in mind. In the same way, some problems were faced in real-life matters, where the PFS was also deviated. To handle the situation, the concept of T-spherical fuzzy set (TSFS,) which rectifies these limitations, was proposed in [35] having the condition (PD) n + (AD) n + (NPD) n ∈ [0, 1]. Some new similarity measures for TSFSs have been developed by Ullah et al. [36] and Saad and Rafiq [37]. e divergence measure of TSFSs with their applications in pattern recognition has been discussed in [38]. A study of correlation coefficients based on TSFSs has been examined in [39]. Algorithms based on improved interactive aggregation operators are discussed in [40], and immediate probabilistic interactive averaging aggregation operators are discussed in [41]. With application in MADM problems, the Einstein hybrid aggregation operators based on TSFS are discussed in [42]. Wu et al. [38] discuss the divergence measure of TSFSs and their applications. Quek et al. [43] discussed the generalized T-spherical fuzzy weighted aggregation operators on neutrosophic sets. Based on TSFSs, the shortest path problem and the DM approach are discussed by Zedam et al. [44].
Several studies then explore the concepts of DM in FM and the PFM model. But there are some limitations, as we are not independent to assign the values to all participation grades while investigating the data are in picture fuzzy form [34]. In this situation, we needed a structure in the FM theory that is independent to assign the values of different grades that are involved in it. Also, we are forbidden to treat the data in T-spherical fuzzy (TSF) context. Keeping in view the importance of the matrix theory, the fuzzy matrix, and the broad domain of TSFSs, our aim is to develop the hybrid structure of FM and TSFSs named as T-spherical fuzzy matrix (TSFM). e following point shows the importance of the proposed work. A lot of objectives are under consideration to emphasize the need to build this model. Some of these objectives are mentioned as follows: (1) e foremost aim to build this model is to overcome the research loopholes that are found in the existing methodologies. e FM and TSFS may also be involved together in decision analysis. algorithm to solve the decision-making problems, the approach has been illustrated with a numerical example.
is article is further separated into various sections. Section 2 reviews some of the essentials of the developed work. Section 3 introduces a new concept as TSFM and its features. In section 4, we initiated the decision-making algorithm for solving the problems and provided the numerical examples for justification. Section 5 provides a comparative study of the work with the existing studies. Finally, Section 6 concludes the paper.

Preliminaries
Here, the notions discussed provided a foundation for our work. From now onward, we use t, i, and f that act as PD, AD, and NPD, respectively. Furthermore, m xyt , m xyi , and m xyf mentioned the PD of the xy th element of M, AD of the xy th element of M, and NPD of the xy th element of M, respectively. Furthermore, P j denotes the set of permutations on 1, 2, . . . , j , P j y j x (P j x j y ) is a set of all permutations of a set j x over j y (j x over j y ), and X acts as a universal set.
Definition 1 (see [18]). An IFS is of the form where t and f are functions from X to an element in the unit interval [0, 1] with a restriction 0 ≤ t + f ≤ 1, and r(x) � 1 − (t + f) is the refusal degree (RD) of x in A. Here, (t, f) is an intuitionistic fuzzy number (IFN).
Definition 3 (see [34]). A PFS is of the form where t, i, and f are functions from X to an element in the unit interval [0, 1] with a restriction 0 ≤ t + i + f ≤ 1, and r(x) Definition 4 (see [34]) Remarks 1 (see [34]) Here, u runs from 1 to j.
Definition 6 (see [34]). For a SPFM M � (m xyt , m xyi , m xyf ) of order j, the |M| is defined as follows: Here, j x � 1, 2, . . . , j − x { }. e PFSs extend the model of FSs and IFSs, but there is still a limitation in the structure. For example, if a person is given 0.6 PD, 0.4 AD, and 0.3 NPD, in that condition PFS will be unable to manage it, i.e., 0.6 + 0.4 + 0.3 � 1.3 > 1. In that condition, PFS has not been kept in mind. In the same way, some problems were faced in real-life matters, where the PFS was also deviated. To handle the situation, the concept of spherical fuzzy set (SFS) and TSFS, which rectifies these limitations, was proposed in [30] having the conditions is shows the importance and advantages of TSFSs over existing fuzzy structures.

T-Spherical Fuzzy Matrix
Here, we will define a novel concept TSFM, in the generalization of PFM.

Some Properties on Square T-Spherical Fuzzy Matrix.
Here, we will discuss some ground properties of STSFM.

Proof
(1) For two STSFM M 1 � (m 1xyt , m 1xyi , m 1xyf ) and M 2 � (m 2xyt , m 2xyi , m 2xyf ) of order j, Taking M 1 � (0.50, 0.46, 0.64), It is clear that m n 1xyt ∨m n 2xyt ≥ m n 1xyt ∧m n 2xyt , m n 1xyi ∧ m n 2xyi ≥ m n 1xyi ∧m n 2xyi , and m n 1xyf ∧m n 2xyf ≤ m n 1xyf ∨ m n 2xyf . So, While investigating the TSFSs, the convex combination (CC) defined in [34] has found some limitations, as we are not independent to assign the values to the PD, RD, and NPD. To overcome the Mathematical Problems in Engineering problem, we will define the CC of two matrixes in a T-spherical fuzzy context.
So, it is observed that the CC of two STSFM is the CC of their entries.

Determinant of STSFM.
Here, we will define determinants and some related results along with a numerical example.
Definition 18. For a TSFM M of order j, the |M| is defined as follows: To find determinant, it is necessary to find all permutations on 1, 2, 3 { }.
e participation degree of |M| is as follows:  Mathematical Problems in Engineering e refusal degree of |M| is as follows:  e nonparticipation degree of |M| is as follows:  Proof. It is trivial, so we omit here. TSFN z � (z 1 , z 2 , z 3 ), then

Proposition 5. Let M be a STSFM. If a row is multiplied by a
Proof. It is trivial, so we omit here.

Adjoint of STSFM.
Here, we will define adjoint and some related results on it.

Proposed Decision-Making Algorithm and Illustration
e TSFM is the most generalized idea in fuzziness; it is applicable for various DM problems. Let promotion test is passed by the n administrative officers (AOs). Over goal is to pick m out of n based on the AO's approach to government (Govt) performance, because all members of Govt are from different groups. e solution of this problem is to find out how close the AO's ideology is to the Govt performance. e performance is a linguistic term and has no special meaning. We use fuzzy logic to handle such conditions, more specifically T-spherical fuzzy logic. is is the most generalized fuzzy structure in the existing fuzzy theory. e choice of AOs must meet certain conditions, and many counts are required. e algorithm defined below finds the appropriate AOs among many candidates. e proposed algorithm is depicted in Figure 1, as a flowchart. e step-by-step explanation of the proposed algorithm is given as Algorithm 1.
Example 3. Let AA 1 , AA 2 , and AA 3 are three political parties coming from different Govts. G 1 , G 2 , and G 3 . e A 1 , A 2 , A 3 , A 4 , and A 5 are AOs qualified for promotion. Now, a TSFM M 1 � (m 1xyt , m 1xyi , m 1xyf ) of order 5 × 3, which shows the view of AOs to the party-backed Govt.
e works performed by the Govt and their commitments are in M 2 during the election period, followed by the party. Aim: Obtain a picked list of AOs based on the AO's approach to Govt performance. Input: From two given TSFM, first indicates the AO's view of the Govt by the political party, and the second one "the work done by the Govt. during election period." Output: For different Govts, the selected list of AOs.

Mathematical Problems in Engineering
Step 1: Extract all TSFSs from the given TSFM over the set of parties.
Step 2: Using the distance formula between two TSFSs, compute a distance matrix as follows: , and s u ′ � 1 − x u ′ − y u ′ − z u ′ are PD, RD, NPD, and HD of a u : u � 1, 2, . . . , p in A 1 and A 2 , respectively, where is a universal set understudy?
Step 3: Arrange the degree of closeness (DOCs) in descending order based on their distance to find selected AOs.
In M 1 and M 2 , AA 1 , AA 2 , and AA 3 represent the three columns, respectively.
e TSFM is extracted as follows: Step 2. e distance matrix D is computed, by applying the distance formula for n � 3.
where columns show the G 1 , G 2 , and G 3 , respectively.
Step 3. e results made from the matrix D are as follows: As a result, the selected list of AOs is presented in Table 1.
For first government, A 2 and A 4 , for second A 5 and A 1 , and for third A 1 and A 4 are selected. Also, note that A 1 and A 4 are selected for more than one Govt.

Comparative Study
Here, we will analyze the proposed work with existing work and compare it in the light of suitable examples.
Another advantage of our proposed work is that it can be used where all existing structures failed to find the results. Considering Example 3, the sum of all grades of the data given in the matrix M 1 exceeds from the unit interval [0, 1] for n � 1, in Table 2, so the information is not in picture fuzzy form and the method proposed in [34] is unable to handle the information. By observing Table 3, it is seen that the sum of all grades of the data given in M 1 is also rose above from the unit interval [0, 1] for n � 2, so the information is not spherical fuzzy form and the so far proposed methods are unable to handle the information. From Table 4, it is observed that the data are in T-spherical fuzzy form for n � 3. e proposed method is only to handle such type of data, which shows the importance of the proposed article.
Sum of all grades of the data given in M 1 for n � 2 is given in Table 3.
Sum of all grades of the data given in M 1 for n � 3 is given in Table 4.
From all the above discussion, it is clear that TSFM is the most generalized in all the existing fuzzy structures.

Conclusions
In this paper, a concept of T-spherical fuzzy matrix is presented by taking the importance of the matrix theory, fuzzy matrix, and the T-spherical fuzzy sets. e key findings of the present study are listed as below as follows: (1) e concept of T-spherical fuzzy matrix is introduced, which is an extension of matrix and fuzzy matrix. problems is presented to solve the decision-making problems. (5) A numerical example is solved using the developed algorithm, where the appropriate AOs among many candidates are selected. A comparative study has been made to show the importance and novelty of the proposed work.
In our next study, our aim is to explore the concept. In further, our aim is to extend the proposed work to develop some applicable results in the matrix theory in the context of T-spherical fuzzy matrix and to utilize them in decisionmaking problems.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.

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