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

Authors: Harish Garg; Muhammad Saad; Ayesha Rafiq
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

The 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. This 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. Then, in the end, to check capability and viability, a practical demonstration of the planned approach has also been explained.


---

## Body

## 1. Introduction

In real life, sometimes it is necessary to compare two different things from different perspectives while dealing with different 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. Different comparative measures may be applicable for various problems. The 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. The 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. The 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 decision-making 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]. The study of bipolar FM has been developed in [13]. The 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].The 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 given0.7 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 real-life matters, where the IFS was also deviated. To handle the situation, Yager [25, 26] initiated the system of Pythagorean FSs (PyFSs), having the condition PD2+NPD2∈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].The PFSs extend the model of FSs and IFSs, but there is still a limitation in the structure. For example, if a person is given0.6 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 PDn+ADn+NPDn∈0,1. Some new similarity measures for TSFSs have been developed by Ullah et al. [36] and Saad and Rafiq [37]. The 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). The 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)
The foremost aim to build this model is to overcome the research loopholes that are found in the existing methodologies. The FM and TSFS may also be involved together in decision analysis.(2)
To discuss the concepts of a square T-spherical fuzzy matrix, constant T-spherical fuzzy matrix and constant square T-spherical fuzzy matrix and study their related properties.(3)
To present multiattribute decision-making (MADM) algorithm to solve the decision-making problems, the approach has been illustrated with a numerical example.This article is further separated into various sections. Section2 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.


## 2. Preliminaries

Here, the notions discussed provided a foundation for our work. From now onward, we uset, i, and f that act as PD, AD, and NPD, respectively. Furthermore, mxyt,mxyi, and mxyf mentioned the PD of the xyth element of M, AD of the xyth element of M, and NPD of the xyth element of M, respectively. Furthermore, Pj denotes the set of permutations on 1,2,…,j, Pjyjx (Pjxjy) is a set of all permutations of a set jx over jy (jx over jy), and X acts as a universal set.Definition 1 (see [18]).
An IFS is of the formA=x,tAx,fAx|x∈X, 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 rx=1−t+f is the refusal degree (RD) of x in A. Here, t,f is an intuitionistic fuzzy number (IFN).Definition 2 (see [19]).
An IFMM of order j×k is of the form M=mxyt,mxyf, where mxyt and mxyf∈0,1 with the condition 0≤mxyt+mxyf≤1 for x=1,2,…,j and y=1,2,…,k. An IFM is said to be square IFM (SIFM) if j=k.Definition 3 (see [34]).
A PFS is of the formA=x,tAx,iAx,fAx|x∈X, 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 rx=1−t+i+f is the RD of x in A, where t,i,f is a picture fuzzy number (PFN).Definition 4 (see [34]).
A PFMM of order j×k is of the form M=mxyt,mxyi,mxyf, where mxyt,mxyi, and mxyf∈0,1 with the condition 0≤mxyt+mxyi+mxyf≤1 for x=1,2,…,j and y=1,2,…,k.Remarks 1 (see [34]).
(1)
A PFM is said to be square PFM (SIFM) ifj=k(2)
An identity PFMI of order j is SPFM with all diagonal entries 1,0,0 and others 0,1,1(3)
A null PFM of orderj is the SPFM with all entries 0,1,1Definition 5 (see [34]).
For two SPFMM1=m1xyt,m1xyi,m1xyf and M2=m2xyt,m2xyi,m2xyf, the product is defined as M=M1×M2=mxyt,mxyi,mxyf, where mxyt=∨um1xut∧m2uyt, mxyi=∨um1xui∧m2uyi, and mxyi=∧um1xui∨m2uyi for x,y=1,2,…,j. Here, u runs from 1 to j.Definition 6 (see [34]).
For a SPFMM=mxyt,mxyi,mxyf of order j, the M is defined as follows:(1)M=∨δ∈Pjm1δ1t∧m2δ2t∧…∧mjδjt∧δ∈Pjm1δ1i∧m2δ2i∧…∧m1δji∧δ∈Pjm1δ1f∨m2δ2f∨…∨mjδjf.Definition 7 (see [34]).
For a SPFMM=mxyt,mxyi,mxyf of order j, the adjM is defined as adjM=R=rxyt,rxyi,rxyf, where(2)rxyt=∨δ∈Pjyjx∧u∈jxmuδut,rxyi=∧δ∈Pjyjx∧u∈jxmuδui,rxyf=∧δ∈Pjyjx∨u∈jxmuδuf.
Here,jx=1,2,…,j−x.
The PFSs extend the model of FSs and IFSs, but there is still a limitation in the structure. For example, if a person is given0.6 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 PD2+HD2+NPD2∈0,1 and PDn+HDn+NPDn∈0,1, respectively. This shows the importance and advantages of TSFSs over existing fuzzy structures.Definition 8 (see [35]).
A SFS is of the formA=x,tAx,iAx,fAx|x∈X, where t, i, and f are functions from X to an element in the unit interval 0,1 with a restriction of 0≤t2+i2+f2≤1, and rx=1−t2+i2+f2 is the RD of x in A, where t,i,f is a spherical fuzzy number (SFN).Definition 9 (see [35]).
A TSFS is of the formA=x,tAx,iAx,fAx|x∈X, where t, i, and f are function from X to an element in the unit interval 0,1 with a restriction of 0≤tn+in+fn≤1 for n∈Z, and rx=1−tn+in+fnn is the RD of x in A, where t,i,f is a T-spherical fuzzy number (TSFN).


## 3. T-Spherical Fuzzy Matrix

Here, we will define a novel concept TSFM, in the generalization of PFM.Definition 10.
A TFMM of order j×k is of the form M=mxyt,mxyi,mxyf, where mxyt,mxyi, and mxyf∈0,1 with the condition 0≤mxytn+mxyin+mxyfn≤1,n∈Z for x=1,2,…,j and y=1,2,…,k.Remarks 2.
(1)
Forn=2, a TSFM becomes spherical fuzzy matrix(2)
A TSFM is said to be square TSFM (SIFM) ifj=k(3)
An identity TSFMI of order j is STSFM with all diagonal entries 1,0,0 and others 0,1,1(4)
A null TSFM of orderj is the STSFM with all entries 0,1,1Definition 11.
For two STSFMM1=m1xyt,m1xyi,m1xyfand M2=m2xyt,m2xyi,m2xyf, the product is defined as follows: M=M1×M2=mxyt,mxyi,mxyf, where mxytn=∨km1xktn∧m2kytn, mxyin=∧km1xkin∧m2kyin, and mxyin=∧km1xktn∨m2kytn for x,y=1,2,…,j. Here, k runs from 1 to j.Definition 12.
For two STSFMM1=m1xyt,m1xyi,m1xyf and M2=m2xyt,m2xyi,m2xyf, M1≤M2 if m1xytn≤m2xytn, m1xyin≤m2xyin and m1xyfn≥m2xyfn for x,y=1,2,…,j.Definition 13.
For a STSFMM=mxyt,mxyi,mxyf, the multiplication by a TSFN z=z1,z2,z3 is defined as z.M=z1n.mxytn,z2n.mxyin,z3n.mxyfn.Definition 14.
Let a STSFMM=mxyt,mxyi,mxyf and TSFN z=z1,z2,z3, for mxyt,mxyi,mxyf≥z1,z2,z3 means that, mxytn≥z1n, mxyin≥z2n and mxyfn≤z3n.Definition 15.
For two STSFMM1=m1xyt,m1xyi,m1xyf and M2=m2xyt,m2xyi,m2xyf, the operations of product, supremum, infimum, transpose, and complement are defined as follows:(1)
M1⊙M2=m1xytn.m2xytn,m1xyin.m2xyin,m1xyfn.m2xyfn(2)
M1∨M2=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn(3)
M1∧M2=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn(4)
M1t=m1yxtn,m1yxin,m1yxfn(5)
M1c=m1xyfn,m1xyin,m1xytn


### 3.1. Some Properties on Square T-Spherical Fuzzy Matrix

Here, we will discuss some ground properties of STSFM.Proposition 1.
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,m3xyi,m3xyf, 1 to 7 holds.(1)
M1∧M2=M2∧M1(2)
M1∨M2=M2∨M1(3)
M1tt=M1(4)
M1tt=M1tc(5)
M1∧M2∨M3=M1∧M2∨M1∧M3(6)
M1∨M2∧M3=M1∨M2∧M1∨M3(7)
z.M1t=z.M1tProof.
(1)
We have(3)M1∧M2=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn=m2xytn∧m1xytn,m2xyin∧m1xyin,m2xyfn∨m1xyfn=M2∧M1.(2)
We have(4)M1∨M2=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn=m2xytn∨m1xytn,m2xyin∧m1xyin,m2xyfn∧m1xyfn=M2∨M1.(3)
We have(5)M1t=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn,M1tt=m1yxtn,m1yxin,m1yxfnt=m1xytn,m1xyin,m1xyfn=M1,M1c=m1xyfn,m1xyin,m1xytn,M1ct=m1xyfn,m1xyin,m1xytnt=m1yxfn,m1yxin,m1yxtn.Now,(6)M1t=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn,M1tc=m1yxtn,m1yxin,m1yxfnc=m1yxfn,m1yxin,m1yxtn.Thus,M1tt=M1tc.(4)
We have(7)M1∧M2∨M3=m1xytn,m1xyin,m1xyfn∧m2xytn∨m3xytn,m2xyin∧m3xyin,m2xyfn∧m3xyfn=m1xytn∧m2xytn∨m3xytn,m1xyin∧m2xyin∧m3xyin,m1xyfn∨m2xyfn∧m3xyfn=m1xytn∧m2xytn∨m2xytn∧m3xytn,m1xyin∧m2xyin∧m2xyin∧m3xyin,m1xyfn∨m2xyfn∧m2xyfn∨m3xyfn=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn∨m2xytn∧m3xytn,m2xyin∧m3xyin,m2xyfn∨m3xyfn=M1∧M2∨M1∧M3.(5)
We have(8)M1∨M2∧M3=m1xytn,m1xyin,m1xyfn∨m2xytn∧m3xytn,m2xyin∧m3xyin,m2xyfn∨m3xyfn=m1xytn∨m2xytn∧m3xytn,m1xyin∧m2xyin∧m3xyin,m1xyfn∧m2xyfn∨m3xyfn=m1xytn∨m2xytn∧m2xytn∨m3xytn,m1xyin∧m2xyin∧m2xyin∧m3xyin,m1xyfn∧m2xyfn∨m2xyfn∧m3xyfn=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn∧m2xytn∨m3xytn,m2xyin∧m3xyin,m2xyfn∧m3xyfn=M1∨M2∧M1∨M3,z.M1t=z1n,z2n,z3n.m1xytn,m1xyin,m1xyfnt=z1n∧m1xytn,z2n∧m1xyin,z3n∨m1xyfnt=z1n∧m1yxtn,z2n∧m1yxin,z3n∨m1yxfn=z1n,z2n,z3n.m1yxtn,m1yxin,m1yxfn=z.M1t.Proposition 2.
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,\m3xyi,m3xyf,(1)
M1∨M2≥M1∧M2(2)
M1∨M3≥M2∨M3, when M1≥M2(3)
M1∧M3≤M2∧M3, when M1≤M2Proof.
(1)
For two STSFMM1=m1xyt,m1xyi,m1xyf and M2=m2xyt,m2xyi,m2xyf of order j,(9)M1∨M2=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn,M1∧M2=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn.Taking(10)M1=0.50,0.46,0.64,M2=0.52,0.49,0.36forn=3,(11)M1∨M2=0.13∨0.14,0.10∧0.12,0.26∧0.05=0.14,0.10,0.05,M1∧M2=0.13∧0.14,0.10∧0.12,0.26∨0.05=0.13,0.10,0.26.It is clear thatm1xytn∨m2xytn≥m1xytn∧m2xytn, m1xyin∧m2xyin≥m1xyin∧m2xyin, and m1xyfn∧m2xyfn≤m1xyfn∨m2xyfn. So,(12)M1∨M2≥M1∧M2.(2)
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,m3xyi,m3xyf of order j,M1≥M2⇒m1xytn≥m2xytn, m1xyin≥m2xyin and m1xyfn≤m2xyfnWhenm1xytn≥m2xytn≥m3xytn, m1xytn∨m3xytn=m1xytn and m2xytn∨m3xytn=m2xytn.Whenm1xytn≥m3xytn≥m2xytn, m1xytn∨m3xytn=m1xytn and m2xytn∨m3xytn=m2xytn.Whenm3xytn≥m1xytn≥m2xytn, m1xytn∨m3xytn=m3xytn and m2xytn∨m3xytn=m3xytn.Now, whenm1xyin≥m2xyin≥m3xyin, m1xyin∧m3xyin=m3xyin and m2xyin∧m3xyin=m3xyin.Whenm1xyin≥m3xyin≥m2xyin, m1xyin∧m3xyin=m3xyin and m2xyin∧m3xyin=m2xyin.Whenm3xyin≥m1xyin≥m2xyin, m1xyin∧m3xyin=m1xyin and m2xyin∧m3xyin=m2xyin.Also, whenm1xyfn≤m2xyfn≤m3xyfn, m1xyfn∧m3xyfn=m1xyfn and m2xyfn∧m3xyfn=m2xyfn.Whenm1xyfn≤m3xyfn≤m2xyfn, m1xyfn∧m3xyfn=m1xyfn and m2xyfn∧m3xyfn=m3xyfn.Whenm3xyfn≤m1xyfn≤m2xyfn, m1xyfn∧m3xyfn=m3xyfn and m2xyfn∧m3xyfn=m3xyfn.Therefore,⇒m1xytn∨m3xytn≥m2xytn∨m3xytn, m1xytn∧m3xytn≥m2xytn∧m3xytn and m1xytn∧m3xytn≤m2xytn∧m3xytn for x,y=1,2,…,j. Consequently, M1∨M3≥M2∨M3.(3)
The proof is similar to2.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 problem, we will define the CC of two matrixes in a T-spherical fuzzy context.Definition 16.
For two STSFMM1 and M2 of order j, the CC is denoted and defined as follows:(13)M1∗M2=mxyt,mxyi,mxyf,where mxytn=δm1xytn+1−δm2xytn,mxyin=δm1xyin+1−δm2xyin, and mxyfn=δm1xyfn+1−δm2xyfn for x,y=1,2,…,j, 0≤δ≤1.
So, it is observed that the CC of two STSFM is the CC of their entries.Definition 17.
A STSFMM=mxyt,mxyi,mxyf is called idempotent w.r.t. some operation “. ⃘” if M∘M=M.Proposition 3.
Every STSFM is idempotent concerning CC “∗.”Proof.
For an STSFMM=mxyt,mxyi,mxyf of order j, let M∗M=m1xyt,m1xyi,m1xyf, where m1xytn=δmxytn+1−δmxytn=mxytn(14)mx1yin=δmxyin+1−δmxyin=mxyin.
m1xyfn=δmxyfn+1−δmxyfn=mxyfn for x,y=1,2,…,j, 0≤δ≤1.
Therefore,M∗M=M.


### 3.2. Determinant of STSFM

Here, we will define determinants and some related results along with a numerical example.Definition 18.
For a TSFMM of order j, the M is defined as follows:(15)M=∨δ∈Pjm1δ1tn∧m2δ2tn∧…∧mjδjtn∧δ∈Pjm1δ1in∧m2δ2in∧…∧mjδjin∧δ∈Pjm1δ1fn∨m2δ2fn∨…∨mjδjin.Example 1.
LetM be a STSFM of order 3, for n=3(16)M=0.63,0.53,0.310.63,0.43,0.530.53,0.33,0.510.41,0.26,0.470.42,0.18,0.440.25,0.46,0.430.51,0.38,0.450.71,0.34,0.340.50,0.50,0.50.
To find determinant, it is necessary to find all permutations on1,2,3.(17)δ1=123123,δ2=123132,δ3=123213,δ4=123231,δ5=123312,δ6=123321.
The participation degree ofM is as follows:(18)m1δ11t3∧m2δ12t3∧m3δ13t3∨m1δ21t3∧m2δ22t3∧m3δ23t3∨m1δ31t3∧m2δ32t3∧m3δ33t3∨m1δ41t3∧m2δ42t3∧m3δ43t3∨m1δ51t3∧m2δ52t3∧m3δ53t3∨m1δ61t3∧m2δ62t3∧m3δ63t3=m11t3∧m22t3∧m33t3∨m11t3∧m23t3∧m32t3∨m12t3∧m21t3∧m33t3∨m12t3∧m23t3∧m31t3∨m13t3∧m21t3∧m32t3∨m13t3∧m22t3∧m31t3=0.2500∧0.0741∧0.1250∨0.2500∧0.0156∧0.3589∨0.2500∧0.0689∧0.1250∨0.2500∧0.0156∧0.1327∨0.1489∧0.0689∧0.3579∨0.1489∧0.0741∧0.1327=0.0741∨0.0156∨0.0689∨0.0156∨0.0689∨0.0741=0.0741.
The refusal degree ofM is as follows:(19)m1δ11i3∧m2δ12i3∧m3δ13i3∧m1δ21i3∧m2δ22i3∧m3δ23i3∧,m1δ31i3∧m2δ32i3∧m3δ33i3∧m1δ41i3∧m2δ42i3∧m3δ43i3∧,m1δ51i3∧m2δ52i3∧m3δ53i3∧m1δ61i3∧m2δ62i3∧m3δ63i3,=m11i3∧m22i3∧m33i3∧m11i3∧m23i3∧m32i3∧m12i3∧m21i3∧m33i3∧,m12i3∧m23i3∧m31i3∧m13i3∧m21i3∧m32i3∧m13i3∧m22i3∧m31i3,=0.1489∧0.0058∧0.1250∧0.1489∧0.0973∧0.0393,∧0.0795∧0.0176∧0.1250∧0.0795∧0.0973∧0.0549,∧0.0359∧0.0176∧0.0393∧0.0359∧0.0058∧0.0549,=0.0058∧0.0393∧0.0176∧0.0549∧0.0176∧0.0058,=0.0058.
The nonparticipation degree ofM is as follows:(20)m1δ11f3∨m2δ12f3∨m3δ13f3∧m1δ21f3∨m2δ22f3∨m3δ23f3∧,m1δ31f3∨m2δ32f3∨m3δ33f3∧m1δ41f3∨m2δ42f3∨m3δ43f3∧,m1δ51f3∨m2δ52f3∨m3δ53f3∧m1δ61f3∨m2δ62f3∨m3δ63f3,=m11f3∨m22f3∨m33f3∧m11f3∨m23f3∨m32f3∧,m12f3∨m21f3∨m33f3∧m12f3∨m23f3∨m31f3∧,m13f3∨m21f3∨m32f3∧m13f3∨m22f3∨m31f3,=0.0298∨0.0852∨0.1250∧0.0298∨0.0795∨0.0393,∧0.1489∨0.1038∨0.1250∧0.1489∨0.0795∨0.0911,∧0.1327∨0.1038∨0.0393∧0.1327∨0.0852∨0.0911,∧0.1327∨0.1038∨0.0393∧0.1327∨0.0852∨0.0911,=0.1250∧0.0795∧0.1489∧0.1489∧0.1327∧0.1327,=0.0795.
From equations (18)–(20),(21)M=0.0741,0.0058,0.0795.Proposition 4.
LetM be a STSFM, and Mt is a transpose of M. Then, Mt=M.Proof.
It is trivial, so we omit here.Proposition 5.
LetM be a STSFM. If a row is multiplied by a TSFN z=z1,z2,z3, then(22)z.M=z.M.Proof.
It is trivial, so we omit here.


### 3.3. Adjoint of STSFM

Here, we will define adjoint and some related results on it.Definition 19.
For a TSFMM of order j, the adjM is defined as follows: adjM=R=rxyt,rxyi,rxyf, where(23)rxytn=∨δ∈Pjyjx∧u∈jxmuδutn,rxyin=∧δ∈Pjyjx∧u∈jxmuδuin,rxyfn=∧δ∈Pjyjx∨u∈jxmuδufn.
Here,jx=1,2,…,j−x.Example 2.
LetM be a STSFM of order 3, for n=3,(24)M=0.40,0.60,0.300.30,0.30,0.300.30,0.40,0.500.60,0.40,0.300.40,0.40,0.400.40,0.50,0.400.30,0.30,0.300.40,0.50,0.400.40,0.60,0.30.
Forx=1 and y=1, jy=1,2,3−1=2,3 and jx=1,2,3−1=2,3. Then, Pjyjx is as follows: 2323 and 2332.
So,(25)m22tn∧m33tn∨m23tn∧m32tn=0.0640∧0.0640∨0.0640∧0.0640=0.0640∨0.0640=0.0640,m22in∧m33in∨m23in∧m32in=0.0640∧0.2160∨0.1250∧0.1250=0.0640∨0.1250=0.1250,m22fn∨m33fn∧m23fn∨m32fn=0.0640∨0.0270∧0.0640∨0.0640=0.0640∧0.0640=0.0640.
Forx=1 and y=2, jy=1,2,3−1=2,3 and jx=1,2,3−2=1,3. Then, Pjyjx is as follows: 1323 and 1332
So,(26)m12tn∧m33tn∨m13tn∧m32tn=0.0270∧0.2160∨0.0270∧0.0640=0.0270∨0.0270=0.0270,m12in∧m33in∨m13in∧m32in=0.0270∧0.2160∨0.0640∧0.1250=0.0270∨0.0640=0.0640,m12fn∨m33fn∧m13fn∨m32fn=0.0270∨0.0270∧0.1250∨0.0640=0.0270∧0.1250=0.0270.
Calculating similarly,(27)adjM=0.0640,0.1250,0.06400.0270,0.0640,0.02700.0270,0.0270,0.06400.0640,0.0640,0.02700.0640,0.2160,0.02700.0640,0.1250,0.06400.0640,0.0640,0.06400.0640,0.1250,0.02700.0640,0.0640,0.0270.Proposition 6.
LetM be a STSFM. Then, adjMt=adjMt.Proof.
LetM be a STSFM of order j and adjM=rxyt,rxyi,rxyf, then(28)Mt=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn.
Then, by definition of adjoint, we have(29)adjMt=ryxt,ryxi,ryxf,where(30)ryxtn=∨δ∈Pjxjy∧u∈jymuδutn,ryxin=∧δ∈Pjxjy∧u∈jymuδuin,ryxfn=∧δ∈Pjxjy∨u∈jymuδufn.
Here,jy=1,2,…,j−y.
Hence,(31)adjM=rxyt,rxyi,rxyf,where(32)rxytn=∨δ∈Pjyjx∧u∈jxmuδutn,rxyin=∧δ∈Pjyjx∧u∈jxmuδun,rxyfn=∧δ∈Pjyjx∨u∈jxmuδufn.
Here,jx=1,2,…,j−x.(33)adjMt=ryxt,ryxi,ryxf,where(34)ryxtn=∨δ∈Pjxjy∧u∈jymuδutn,ryxin=∧δ∈Pjxjy∧u∈jymuδuin,ryxfn=∧δ∈Pjxjy∨u∈jymuδufn.
Here,jy=1,2,…,j−y.
From equations (29) and (33),(35)adjMt=adjMt.Definition 20.
A STSFMM of order j is called a CSTSFM if mxktn,mxkin,mxkfn=myktn,mykin,mykfn for x,y,k=1,2,…j.Proposition 7.
For a CSTSFMM, the adjMt is constant.Proof.
It is trivial, so we omit here.Proposition 8.
For a CSTSFMM, the M.adjM is constant.Proof.
It is trivial, so we omit here.


## 3.1. Some Properties on Square T-Spherical Fuzzy Matrix

Here, we will discuss some ground properties of STSFM.Proposition 1.
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,m3xyi,m3xyf, 1 to 7 holds.(1)
M1∧M2=M2∧M1(2)
M1∨M2=M2∨M1(3)
M1tt=M1(4)
M1tt=M1tc(5)
M1∧M2∨M3=M1∧M2∨M1∧M3(6)
M1∨M2∧M3=M1∨M2∧M1∨M3(7)
z.M1t=z.M1tProof.
(1)
We have(3)M1∧M2=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn=m2xytn∧m1xytn,m2xyin∧m1xyin,m2xyfn∨m1xyfn=M2∧M1.(2)
We have(4)M1∨M2=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn=m2xytn∨m1xytn,m2xyin∧m1xyin,m2xyfn∧m1xyfn=M2∨M1.(3)
We have(5)M1t=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn,M1tt=m1yxtn,m1yxin,m1yxfnt=m1xytn,m1xyin,m1xyfn=M1,M1c=m1xyfn,m1xyin,m1xytn,M1ct=m1xyfn,m1xyin,m1xytnt=m1yxfn,m1yxin,m1yxtn.Now,(6)M1t=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn,M1tc=m1yxtn,m1yxin,m1yxfnc=m1yxfn,m1yxin,m1yxtn.Thus,M1tt=M1tc.(4)
We have(7)M1∧M2∨M3=m1xytn,m1xyin,m1xyfn∧m2xytn∨m3xytn,m2xyin∧m3xyin,m2xyfn∧m3xyfn=m1xytn∧m2xytn∨m3xytn,m1xyin∧m2xyin∧m3xyin,m1xyfn∨m2xyfn∧m3xyfn=m1xytn∧m2xytn∨m2xytn∧m3xytn,m1xyin∧m2xyin∧m2xyin∧m3xyin,m1xyfn∨m2xyfn∧m2xyfn∨m3xyfn=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn∨m2xytn∧m3xytn,m2xyin∧m3xyin,m2xyfn∨m3xyfn=M1∧M2∨M1∧M3.(5)
We have(8)M1∨M2∧M3=m1xytn,m1xyin,m1xyfn∨m2xytn∧m3xytn,m2xyin∧m3xyin,m2xyfn∨m3xyfn=m1xytn∨m2xytn∧m3xytn,m1xyin∧m2xyin∧m3xyin,m1xyfn∧m2xyfn∨m3xyfn=m1xytn∨m2xytn∧m2xytn∨m3xytn,m1xyin∧m2xyin∧m2xyin∧m3xyin,m1xyfn∧m2xyfn∨m2xyfn∧m3xyfn=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn∧m2xytn∨m3xytn,m2xyin∧m3xyin,m2xyfn∧m3xyfn=M1∨M2∧M1∨M3,z.M1t=z1n,z2n,z3n.m1xytn,m1xyin,m1xyfnt=z1n∧m1xytn,z2n∧m1xyin,z3n∨m1xyfnt=z1n∧m1yxtn,z2n∧m1yxin,z3n∨m1yxfn=z1n,z2n,z3n.m1yxtn,m1yxin,m1yxfn=z.M1t.Proposition 2.
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,\m3xyi,m3xyf,(1)
M1∨M2≥M1∧M2(2)
M1∨M3≥M2∨M3, when M1≥M2(3)
M1∧M3≤M2∧M3, when M1≤M2Proof.
(1)
For two STSFMM1=m1xyt,m1xyi,m1xyf and M2=m2xyt,m2xyi,m2xyf of order j,(9)M1∨M2=m1xytn∨m2xytn,m1xyin∧m2xyin,m1xyfn∧m2xyfn,M1∧M2=m1xytn∧m2xytn,m1xyin∧m2xyin,m1xyfn∨m2xyfn.Taking(10)M1=0.50,0.46,0.64,M2=0.52,0.49,0.36forn=3,(11)M1∨M2=0.13∨0.14,0.10∧0.12,0.26∧0.05=0.14,0.10,0.05,M1∧M2=0.13∧0.14,0.10∧0.12,0.26∨0.05=0.13,0.10,0.26.It is clear thatm1xytn∨m2xytn≥m1xytn∧m2xytn, m1xyin∧m2xyin≥m1xyin∧m2xyin, and m1xyfn∧m2xyfn≤m1xyfn∨m2xyfn. So,(12)M1∨M2≥M1∧M2.(2)
For three STSFMM1=m1xyt,m1xyi,m1xyf, M2=m2xyt,m2xyi,m2xyf, and M3=m3xyt,m3xyi,m3xyf of order j,M1≥M2⇒m1xytn≥m2xytn, m1xyin≥m2xyin and m1xyfn≤m2xyfnWhenm1xytn≥m2xytn≥m3xytn, m1xytn∨m3xytn=m1xytn and m2xytn∨m3xytn=m2xytn.Whenm1xytn≥m3xytn≥m2xytn, m1xytn∨m3xytn=m1xytn and m2xytn∨m3xytn=m2xytn.Whenm3xytn≥m1xytn≥m2xytn, m1xytn∨m3xytn=m3xytn and m2xytn∨m3xytn=m3xytn.Now, whenm1xyin≥m2xyin≥m3xyin, m1xyin∧m3xyin=m3xyin and m2xyin∧m3xyin=m3xyin.Whenm1xyin≥m3xyin≥m2xyin, m1xyin∧m3xyin=m3xyin and m2xyin∧m3xyin=m2xyin.Whenm3xyin≥m1xyin≥m2xyin, m1xyin∧m3xyin=m1xyin and m2xyin∧m3xyin=m2xyin.Also, whenm1xyfn≤m2xyfn≤m3xyfn, m1xyfn∧m3xyfn=m1xyfn and m2xyfn∧m3xyfn=m2xyfn.Whenm1xyfn≤m3xyfn≤m2xyfn, m1xyfn∧m3xyfn=m1xyfn and m2xyfn∧m3xyfn=m3xyfn.Whenm3xyfn≤m1xyfn≤m2xyfn, m1xyfn∧m3xyfn=m3xyfn and m2xyfn∧m3xyfn=m3xyfn.Therefore,⇒m1xytn∨m3xytn≥m2xytn∨m3xytn, m1xytn∧m3xytn≥m2xytn∧m3xytn and m1xytn∧m3xytn≤m2xytn∧m3xytn for x,y=1,2,…,j. Consequently, M1∨M3≥M2∨M3.(3)
The proof is similar to2.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 problem, we will define the CC of two matrixes in a T-spherical fuzzy context.Definition 16.
For two STSFMM1 and M2 of order j, the CC is denoted and defined as follows:(13)M1∗M2=mxyt,mxyi,mxyf,where mxytn=δm1xytn+1−δm2xytn,mxyin=δm1xyin+1−δm2xyin, and mxyfn=δm1xyfn+1−δm2xyfn for x,y=1,2,…,j, 0≤δ≤1.
So, it is observed that the CC of two STSFM is the CC of their entries.Definition 17.
A STSFMM=mxyt,mxyi,mxyf is called idempotent w.r.t. some operation “. ⃘” if M∘M=M.Proposition 3.
Every STSFM is idempotent concerning CC “∗.”Proof.
For an STSFMM=mxyt,mxyi,mxyf of order j, let M∗M=m1xyt,m1xyi,m1xyf, where m1xytn=δmxytn+1−δmxytn=mxytn(14)mx1yin=δmxyin+1−δmxyin=mxyin.
m1xyfn=δmxyfn+1−δmxyfn=mxyfn for x,y=1,2,…,j, 0≤δ≤1.
Therefore,M∗M=M.


## 3.2. Determinant of STSFM

Here, we will define determinants and some related results along with a numerical example.Definition 18.
For a TSFMM of order j, the M is defined as follows:(15)M=∨δ∈Pjm1δ1tn∧m2δ2tn∧…∧mjδjtn∧δ∈Pjm1δ1in∧m2δ2in∧…∧mjδjin∧δ∈Pjm1δ1fn∨m2δ2fn∨…∨mjδjin.Example 1.
LetM be a STSFM of order 3, for n=3(16)M=0.63,0.53,0.310.63,0.43,0.530.53,0.33,0.510.41,0.26,0.470.42,0.18,0.440.25,0.46,0.430.51,0.38,0.450.71,0.34,0.340.50,0.50,0.50.
To find determinant, it is necessary to find all permutations on1,2,3.(17)δ1=123123,δ2=123132,δ3=123213,δ4=123231,δ5=123312,δ6=123321.
The participation degree ofM is as follows:(18)m1δ11t3∧m2δ12t3∧m3δ13t3∨m1δ21t3∧m2δ22t3∧m3δ23t3∨m1δ31t3∧m2δ32t3∧m3δ33t3∨m1δ41t3∧m2δ42t3∧m3δ43t3∨m1δ51t3∧m2δ52t3∧m3δ53t3∨m1δ61t3∧m2δ62t3∧m3δ63t3=m11t3∧m22t3∧m33t3∨m11t3∧m23t3∧m32t3∨m12t3∧m21t3∧m33t3∨m12t3∧m23t3∧m31t3∨m13t3∧m21t3∧m32t3∨m13t3∧m22t3∧m31t3=0.2500∧0.0741∧0.1250∨0.2500∧0.0156∧0.3589∨0.2500∧0.0689∧0.1250∨0.2500∧0.0156∧0.1327∨0.1489∧0.0689∧0.3579∨0.1489∧0.0741∧0.1327=0.0741∨0.0156∨0.0689∨0.0156∨0.0689∨0.0741=0.0741.
The refusal degree ofM is as follows:(19)m1δ11i3∧m2δ12i3∧m3δ13i3∧m1δ21i3∧m2δ22i3∧m3δ23i3∧,m1δ31i3∧m2δ32i3∧m3δ33i3∧m1δ41i3∧m2δ42i3∧m3δ43i3∧,m1δ51i3∧m2δ52i3∧m3δ53i3∧m1δ61i3∧m2δ62i3∧m3δ63i3,=m11i3∧m22i3∧m33i3∧m11i3∧m23i3∧m32i3∧m12i3∧m21i3∧m33i3∧,m12i3∧m23i3∧m31i3∧m13i3∧m21i3∧m32i3∧m13i3∧m22i3∧m31i3,=0.1489∧0.0058∧0.1250∧0.1489∧0.0973∧0.0393,∧0.0795∧0.0176∧0.1250∧0.0795∧0.0973∧0.0549,∧0.0359∧0.0176∧0.0393∧0.0359∧0.0058∧0.0549,=0.0058∧0.0393∧0.0176∧0.0549∧0.0176∧0.0058,=0.0058.
The nonparticipation degree ofM is as follows:(20)m1δ11f3∨m2δ12f3∨m3δ13f3∧m1δ21f3∨m2δ22f3∨m3δ23f3∧,m1δ31f3∨m2δ32f3∨m3δ33f3∧m1δ41f3∨m2δ42f3∨m3δ43f3∧,m1δ51f3∨m2δ52f3∨m3δ53f3∧m1δ61f3∨m2δ62f3∨m3δ63f3,=m11f3∨m22f3∨m33f3∧m11f3∨m23f3∨m32f3∧,m12f3∨m21f3∨m33f3∧m12f3∨m23f3∨m31f3∧,m13f3∨m21f3∨m32f3∧m13f3∨m22f3∨m31f3,=0.0298∨0.0852∨0.1250∧0.0298∨0.0795∨0.0393,∧0.1489∨0.1038∨0.1250∧0.1489∨0.0795∨0.0911,∧0.1327∨0.1038∨0.0393∧0.1327∨0.0852∨0.0911,∧0.1327∨0.1038∨0.0393∧0.1327∨0.0852∨0.0911,=0.1250∧0.0795∧0.1489∧0.1489∧0.1327∧0.1327,=0.0795.
From equations (18)–(20),(21)M=0.0741,0.0058,0.0795.Proposition 4.
LetM be a STSFM, and Mt is a transpose of M. Then, Mt=M.Proof.
It is trivial, so we omit here.Proposition 5.
LetM be a STSFM. If a row is multiplied by a TSFN z=z1,z2,z3, then(22)z.M=z.M.Proof.
It is trivial, so we omit here.


## 3.3. Adjoint of STSFM

Here, we will define adjoint and some related results on it.Definition 19.
For a TSFMM of order j, the adjM is defined as follows: adjM=R=rxyt,rxyi,rxyf, where(23)rxytn=∨δ∈Pjyjx∧u∈jxmuδutn,rxyin=∧δ∈Pjyjx∧u∈jxmuδuin,rxyfn=∧δ∈Pjyjx∨u∈jxmuδufn.
Here,jx=1,2,…,j−x.Example 2.
LetM be a STSFM of order 3, for n=3,(24)M=0.40,0.60,0.300.30,0.30,0.300.30,0.40,0.500.60,0.40,0.300.40,0.40,0.400.40,0.50,0.400.30,0.30,0.300.40,0.50,0.400.40,0.60,0.30.
Forx=1 and y=1, jy=1,2,3−1=2,3 and jx=1,2,3−1=2,3. Then, Pjyjx is as follows: 2323 and 2332.
So,(25)m22tn∧m33tn∨m23tn∧m32tn=0.0640∧0.0640∨0.0640∧0.0640=0.0640∨0.0640=0.0640,m22in∧m33in∨m23in∧m32in=0.0640∧0.2160∨0.1250∧0.1250=0.0640∨0.1250=0.1250,m22fn∨m33fn∧m23fn∨m32fn=0.0640∨0.0270∧0.0640∨0.0640=0.0640∧0.0640=0.0640.
Forx=1 and y=2, jy=1,2,3−1=2,3 and jx=1,2,3−2=1,3. Then, Pjyjx is as follows: 1323 and 1332
So,(26)m12tn∧m33tn∨m13tn∧m32tn=0.0270∧0.2160∨0.0270∧0.0640=0.0270∨0.0270=0.0270,m12in∧m33in∨m13in∧m32in=0.0270∧0.2160∨0.0640∧0.1250=0.0270∨0.0640=0.0640,m12fn∨m33fn∧m13fn∨m32fn=0.0270∨0.0270∧0.1250∨0.0640=0.0270∧0.1250=0.0270.
Calculating similarly,(27)adjM=0.0640,0.1250,0.06400.0270,0.0640,0.02700.0270,0.0270,0.06400.0640,0.0640,0.02700.0640,0.2160,0.02700.0640,0.1250,0.06400.0640,0.0640,0.06400.0640,0.1250,0.02700.0640,0.0640,0.0270.Proposition 6.
LetM be a STSFM. Then, adjMt=adjMt.Proof.
LetM be a STSFM of order j and adjM=rxyt,rxyi,rxyf, then(28)Mt=m1xytn,m1xyin,m1xyfnt=m1yxtn,m1yxin,m1yxfn.
Then, by definition of adjoint, we have(29)adjMt=ryxt,ryxi,ryxf,where(30)ryxtn=∨δ∈Pjxjy∧u∈jymuδutn,ryxin=∧δ∈Pjxjy∧u∈jymuδuin,ryxfn=∧δ∈Pjxjy∨u∈jymuδufn.
Here,jy=1,2,…,j−y.
Hence,(31)adjM=rxyt,rxyi,rxyf,where(32)rxytn=∨δ∈Pjyjx∧u∈jxmuδutn,rxyin=∧δ∈Pjyjx∧u∈jxmuδun,rxyfn=∧δ∈Pjyjx∨u∈jxmuδufn.
Here,jx=1,2,…,j−x.(33)adjMt=ryxt,ryxi,ryxf,where(34)ryxtn=∨δ∈Pjxjy∧u∈jymuδutn,ryxin=∧δ∈Pjxjy∧u∈jymuδuin,ryxfn=∧δ∈Pjxjy∨u∈jymuδufn.
Here,jy=1,2,…,j−y.
From equations (29) and (33),(35)adjMt=adjMt.Definition 20.
A STSFMM of order j is called a CSTSFM if mxktn,mxkin,mxkfn=myktn,mykin,mykfn for x,y,k=1,2,…j.Proposition 7.
For a CSTSFMM, the adjMt is constant.Proof.
It is trivial, so we omit here.Proposition 8.
For a CSTSFMM, the M.adjM is constant.Proof.
It is trivial, so we omit here.


## 4. Proposed Decision-Making Algorithm and Illustration

The TSFM is the most generalized idea in fuzziness; it is applicable for various DM problems. Let promotion test is passed by then 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. The solution of this problem is to find out how close the AO’s ideology is to the Govt performance. The performance is a linguistic term and has no special meaning. We use fuzzy logic to handle such conditions, more specifically T-spherical fuzzy logic. This is the most generalized fuzzy structure in the existing fuzzy theory. The choice of AOs must meet certain conditions, and many counts are required. The algorithm defined below finds the appropriate AOs among many candidates.The proposed algorithm is depicted in Figure1, as a flowchart. The step-by-step explanation of the proposed algorithm is given as Algorithm 1.Figure 1
Flowchart of the proposed algorithm.Algorithm 1:
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.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:γA1,A2=1/3p∑u=1pxu−xu′2+yu−yu′2+zu−zu′2+su−su′2,wherexu=tA1nau, yu=iA1nau, zu=fA1nau, and su=1−xu−yu−zu,xu′=tA2nau, yu′=iA2nau, zu′=fA2nau, and su′=1−xu′−yu′−zu′ are PD, RD, NPD, and HD of au:u=1,2,…,p in A1 and A2, 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.Example 3.
LetAA1, AA2, and AA3 are three political parties coming from different Govts. G1, G2, and G3. The A1, A2, A3, A4, and A5 are AOs qualified for promotion. Now, a TSFM M1=m1xyt,m1xyi,m1xyf of order 5×3, which shows the view of AOs to the party-backed Govt.(36)M1=A1A2A3A4A50.70,0.60,0.650.70,0.80,0.300.60,0.60,0.600.64,0.64,0.400.46,0.30,0.800.70,0.60,0.700.50,0.70,0.500.70,0.50,0.600.64,0.50,0.400.46,0.30,0.600.60,0.50,0.400.46,0.40,0.800.46,0.30,0.800.40,0.70,0.800.50,0.70,0.50.
The works performed by the Govt and their commitments are inM2 during the election period, followed by the party.(37)M2=G1G2G30.30,0.60,0.800.80,0.70,0.400.70,0.40,0.700.70,0.70,0.400.40,0.34,0.500.40,0.80,0.400.50,0.60,0.400.50,0.40,0.800.40,0.34,0.50.
InM1 and M2, AA1, AA2, and AA3 represent the three columns, respectively.Step 1.
The TSFM is extracted as follows:(38)A1=AA1,0.70,0.60,0.65,AA2,0.70,0.80,0.30,AA3,0.60,0.60,0.60,A2=AA1,0.64,0.64,0.40,AA2,0.46,0.30,0.80,AA3,0.70,0.60,0.70,A3=AA1,0.50,0.70,0.50,AA2,0.70,0.50,0.60,AA3,0.64,0.50,0.40,A4=AA1,0.46,0.30,0.60,AA2,0.60,0.50,0.40,AA3,0.46,0.40,0.80,A5=AA1,0.46,0.30,0.80,AA2,0.40,0.70,0.80,AA3,0.50,0.70,0.50,G1=AA1,0.30,0.60,0.80,AA2,0.80,0.70,0.40,AA3,0.70,0.40,0.70,G2=AA1,0.70,0.70,0.40,AA2,0.40,0.34,0.50,AA3,0.40,0.80,0.40,G3=AA1,0.50,0.60,0.40,AA2,0.50,0.40,0.80,AA3,0.40,0.34,0.50.Step 2.
The distance matrixD is computed, by applying the distance formula for n=3.(39)D=A1A2A3A4A50.17890.33390.33840.31350.27260.27310.24350.25860.18080.29710.30190.28160.26950.35450.2685,where columns show the G1, G2, and G3, respectively.Step 3.
The results made from the matrixD are as follows:(40)DOCA2,G1>DOCA4,G1>DOCA5,G1>DOCA3,G1>DOCA1,G1,DOCA5,G2>DOCA1,G2>DOCA4,G2>DOCA2,G2>DOCA3,G2,DOCA1,G3>DOCA4,G3>DOCA2,G3>DOCA5,G3>DOCA3,G3.
As a result, the selected list of AOs is presented in Table1.
For first government,A2 and A4, for second A5 and A1, and for third A1 and A4 are selected. Also, note that A1 and A4 are selected for more than one Govt.Table 1
Selected AOs.
Govts.Selected asG1A2, A4G2A5, A1G3A1, A4


## 5. Comparative Study

Here, we will analyze the proposed work with existing work and compare it in the light of suitable examples.Remarks 3.
We consider Definition10:(1)
Forn=2, TSFM becomes SFM(2)
Forn=1, TSFM becomes PFM, developed in [34](3)
Forn=2 and mxyi=0, TSFM becomes PyFM, developed in [27](4)
Forn=1 and mxyi=0, TSFM becomes IFM, developed in [19](5)
Forn=1, mxyi=0, and mxyf=0, TSFM becomes FM, developed in [10]
From the above remarks, it is clear that TSFM is most generalized among all existing fuzzy matrix structures.
Another advantage of our proposed work is that it can be used where all existing structures failed to find the results. Considering Example3, the sum of all grades of the data given in the matrix M1 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 M1 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. The 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 inM1 for n=2 is given in Table 3.
Sum of all grades of the data given inM1 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.Table 2
Sum of all grades ofM1 for n=1.
AA1AA2AA3A11.951.801.80A21.681.562.00A31.701.801.54A41.361.501.66A51.561.901.70Table 3
Sum of all grades ofM1 for n=2.
AA1AA2AA3A11.271.221.08A20.980.941.34A30.991.100.82A40.660.771.01A50.941.290.99Table 4
Sum of all grades ofM1 for n=3.
AA1AA2AA3A10.830.880.65A20.590.640.90A30.590.680.45A40.340.410.67A50.640.920.59


## 6. 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. The key findings of the present study are listed as below as follows:(1)
The concept of T-spherical fuzzy matrix is introduced, which is an extension of matrix and fuzzy matrix.(2)
The concepts of a square T-spherical fuzzy matrix, constant T-spherical fuzzy matrix, and constant square T-spherical fuzzy matrix with their related properties are defined, and their related properties are investigated with examples.(3)
Determinant and adjoint of square T-spherical fuzzy matrix with their related results are discussed.(4)
An algorithm for multiattribute decision-making 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 decision-making problems.



---

*Source: 2553811-2022-09-27.xml*