# Some Generalized T-Spherical and Group-Generalized Fuzzy Geometric Aggregation Operators with Application in MADM Problems

Authors: Yujuan Chen; Muhammad Munir; Tahir Mahmood; Azmat Hussain; Shouzhen Zeng
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, the generalized parameter is involved in T-spherical fuzzy set (TSFS), and with the help of this generalized parameter, some generalized geometric aggregation operators for TSFSs are proposed. Then these operators are extended for group-generalized parameter. By using proposed operators, an algorithm is developed for the MADM problem. To check the validity of proposed operators, a numerical example is also investigated. In a comparative analysis, it is discussed that, under some conditions, the proposed work can be reduced to other fuzzy structures. An example is also solved in which it is shown that our proposed technique is superior to the existing technique.


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## Body

## 1. Introduction

The concept of fuzzy set (FS) was introduced by Zadeh [1] which tells the membership grade (MG) of an object. FS plays an important role in solving problems in imprecise and uncertain environment. A generalization of the FS called an intuitionistic fuzzy set (IFS) was introduced by Atanassov [2], which tells the MG and nonmembership grade (NMG) of an object with a restriction that the sum of MG and NMG mu0,1st belong to . IFS fails when the sum of MG and NMG exceeds 1. To overcome this issue, an extension of IFS was introduced by Yager [3] called Pythagorean fuzzy set (PyFS). In PyFS, the condition was relaxed to that the sum of squares of MG and NMG must belong to 0,1.IFS and PyFS fail when the third degree of abstinence is involved. To deal with this type of data, the idea of picture fuzzy set (PFS) was given by Cuong [4]. In PFS, there are four grades known as MG, abstinence, NMG, and refusal. PFS has a restriction that the sum of MG, abstinence, and NMG belongs to [0, 1]. PFS fails when their sum exceeds 1. To overcome this issue, Mahmood et al. [5] proposed the notions of spherical fuzzy set (SFS) and TSFS. SFS has a restriction that the square sum of MG, abstinence, and NMG must belong to 0,1, and in TSFS, the experts have the flexibility that the sum of any integral power of MG, abstinence, and NMG must belong to 0,1.Many authors defined different aggregation operators for these tools of uncertainty. Xu [6] defined intuitionistic fuzzy (IF) averaging operators. Xu and Yager [7] defined IF geometric operators and applied them to solve the MADM problem. Liu and Chen [8] proposed Heronian operators for IFSs. Liu [9] proposed several intuitionistic fuzzy power Heronian operators. Hayat et al. [10] proposed some aggregation operators on group-based generalized intuitionistic fuzzy soft sets. Based on the conception of entropy, some IF power operators are proposed by Jiang et al. [11]. Some MADM problems were investigated using IFSs in [12–16]. Jana et al. [17] proposed Pythagorean fuzzy Dombi aggregation operators and investigated their usefulness in the MADM. Teng et al. [18] introduced some power Maclaurin symmetric mean aggregation operators for PyFS. Liu et al. [19] extended Bonferroni mean operators to study the MADM problem for PyFSs. Jana et al. [20] proposed some Dombi aggregation operators for q-rung orthopair fuzzy set and investigated the MADM problem. Joshi [21] proposed group-generalized averaging aggregation operators for PyFSs and solved the MADM problem. Some MADM problems were solved using PyFSs in [22–25].Wei [26] proposed averaging and geometric aggregation operators for PFSs and studied their usefulness in MADM. Garg [27] investigated decision-making problem using averaging operators for PFSs. Jana et al. [28] investigated the MADM problem by utilizing picture fuzzy Dombi aggregation operators. Some MADM problems were investigated using PFSs in [29–31]. Zeng et al. [32] investigated the decision-making problem by utilizing the idea of spherical fuzzy covering-based rough set model. Jin et al. [33] introduced logarithmic aggregation operators for SFSs. Donyatalab et al. [34] proposed harmonic mean aggregation operators and investigated their applications in MADM. Munir et al. [35] studied the MADM problem using TSF Einstein operators. Guleria and Bajaj [36] studied the MADM problem using aggregation operators for T-spherical fuzzy soft sets. Gündoğdu and Kahraman [37] investigated the MADM problem for SF VIKOR method. More studies on MADM problems with complex fuzzy tools can be found in [38–40].If a pharmacist suggests a medicine only on symptoms provided by the patient, then he may not be cured because a patient may have more than one disease due to which he may not be able to express the symptoms more clearly. For example, pain is the main symptom of a heart attack if a patient suffering from a congenital disease has a heart attack, then he is unable to express pain. If junior doctors give the treatment only on symptoms provided by a patient without consulting specialist/senior doctor, then the patient may lead to death. So, it is necessary to consult with some specialist/senior doctor for good treatment. Another example in which expert opinion is involved is the construction of a house/building. If labor constructs a house/building only following the instructions of the owner, then it may be beautiful but not durable. So, for making a house more durable, an opinion of the engineer is necessary. By keeping this type of problem in mind, some generalized and group-generalized geometric aggregation operators are proposed in which the opinion of an expert is also involved in decision making.In this article, by utilizing the most generalized fuzzy structure called TSFS, some aggregation operators based on generalized and group-generalized parameters are proposed. In these aggregation operators, the decision makers have a huge space for assigning the values to membership, abstinence, and nonmembership grades. In these aggregation operators, the opinion of an expert is also involved due to which these aggregation operators are more reliable.The purposes of writing this manuscript are as follows:(i)
To define generalized parameter (GP) for TSFSs(ii)
To propose generalized geometric aggregation operators for TSFSs(iii)
To propose group-generalized geometric aggregation operators for TSFSs(iv)
To develop an algorithm for solving MADM problem using proposed operators(v)
To discuss the advantages of proposed operatorsThe manuscript can be concluded as follows. Section2 reviews some basic definitions. In Section 3, a GP is defined for TSFSs. In Section 4, some generalized geometric operators are proposed for TSFSs. In Section 5, some group-generalized geometric operators are proposed for TSFSs. In Section 6, an approach to solve the MADM problem is proposed. In Section 7, a comparative analysis is developed in which it is described that the newly defined operators can be reduced to other fuzzy structures by using some conditions. The whole article is concluded in Section 8.


## 2. Preliminaries

In this section, some basic notions will be discussed which help in further study.Definition 1.
(see [4]). For a nonempty set X, TSFS is(1)T=x,hx,ox,sx:x∈X,where h,o,s:X⟶0,1 having a condition that 0≤htx+otx+stx≤1 for any positive integer t and the refusal degree will be rx=1−htx+otx+stxt.Remark 1.
Definition1 can be reduced to SFSs, PFSs, PyFSs, IFSs, and FSs by using the following conditions:(i)
t=2 reduced it to SFSs(ii)
t=1 reduced it to PFSs(iii)
t=2, o=0 reduced it to PyFSs(iv)
t=1, o=0 reduced it to IFSs(v)
t=1, o=0, s=0 reduced it to FSsDefinition 2.
(see [4]). Consider any two TSFNs T1=h1,o1,s1 and T2=h2,o2,s2, then some operations on these will be defined as follows:(i)
T1⊕T2=1−1−h1t1−h2tt,o1o2,s1s2(ii)
T1⊗T2=h1h2,1−1−o1t1−o2tt,1−1−s1t1−s2tt(iii)
τT1=1−1−h1tτt,o1τ,s1τ, τ>0(iv)
T1τ=h1τ,1−1−o1tτt,1−1−s1tτt, τ>0Definition 3.
(see [4]). For any collection of TSFNs Tj=hj,oj,sjj=1,2,…,m, TSFWG operator is defined as follows:(2)TSFWGϖT1,T2,…,Tm=∏j=1mhjϖj,1−∏j=1m1−ojtϖjt,1−∏j=1m1−sjtϖjt,where the weight vector ϖ=ϖ1,ϖ2,…,ϖm satisfies ϖ∈0,1 and ∑j=1mϖj=1.Definition 4.
The score and accuracy functions for any TSFNT=h,o,s are defined as follows:(3)ScT=ht−ot−st,AcT=ht+ot+st.(1)
IfScT1<ScT2, then T2 is greater than T1(2)
IfScT1=ScT2, then we have to check accuracy, if, then T2 is greater than T1, and if again AcT1=AcT2, then both numbers will be equal


## 3. Generalized Parameter for T-Spherical Fuzzy Sets

In a medical diagnosis problem, a patient goes to a doctor and provides the symptoms based on his perception. If a disease is only diagnosed on symptoms provided by the patient, then he may not be cured, e.g., if a person who is also a patient of congenital disease (not feeling pain) has a stress. The preferences of the patient will be(4)T=0.7,0.2,0.4Low energy,0.8,0.4,0.3Upset stomach,0.0,0.1,0.5Pains,0.7,0.3,0.1Insomnia.Here, the patient gives 0 membership value to pains because he does not feel pain. If a doctor provides a treatment, then he may not be cured. So for a better treatment, it is required to get an opinion from an expert. To achieve this concept, generalized parameter is proposed.Definition 5.
For a nonempty setX, generalized T-spherical fuzzy set (GTSFS) is defined as follows:(5)T=x,hx,ox,sxhg,og,sg:x∈X,where h,o,s:X⟶0,1 having a condition that 0≤htx+otx+stx≤1 for any positive integer t and hg,og,sg∈0,1 denote the expert opinion with a condition 0≤hgt+ogt+sgt≤1. hg,og,sg is called GP for TSFS.


## 4. Generalized T-Spherical Fuzzy Geometric Aggregation Operators

In this section, the generalized TSF weighted geometric (GTSFWG) operator, generalized TSF ordered weighted geometric (GTSFOWG) operator, and generalized TSF hybrid geometric (GTSFHG) operator are defined. Some basic results of these operators are also proved.


### 4.1. Generalized T-Spherical Fuzzy Weighted Geometric Operator

Definition 6.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFWG operator is defined as(6)GTSFWGT1,T2,…,Tm,Tg=Tg⊕TSFWGT1,T2,…,Tm.Theorem 1.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the GTSFWG operator is given by(7)GTSFWGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mTjϖj=hgt+1−hgt∏j=1mhjtϖjt,og.1−∏j=1m1−ojtϖjt,sg.1−∏j=1m1−sjtϖjt.Proof.
By using mathematical induction, this proof can be done.
Form=2,(8)GTSFWGT1,T2,Tg=Tg⊕T1ϖ1⊗T2ϖ2=hg,og,sg⊕h1ϖ1,1−1−o1tϖ1t,1−1−s1tϖ1t⊗h2ϖ2,1−1−o2tϖ2t,1−1−s2tϖ2t=hg,og,sg⊕h1ϖ1h2ϖ2,1−1−o1tϖ11−o2tϖ2t,1−1−s1tϖ11−s2tϖ2t=hgt+h1tϖ1h2tϖ2−hgth1tϖ1h2tϖ2t,og.1−1−o1tϖ11−o2tϖ2t,sg.1−1−s1tϖ11−s2tϖ2t=hgt+1−hgth1tϖ1h2tϖ2t,og.1−1−o1tϖ11−o2tϖ2t,sg.1−1−s1tϖ11−s2tϖ2t.
This shows that results hold form=2. Let us consider that the result is true for m=l,(9)GTSFWGT1,T2,…,Tl,Tg=hgt+1−hgt∏j=1lhjtϖjt,og.1−∏j=1l1−ojtϖjt,sg.1−∏j=1l1−sjtϖjt.
Now(10)GTSFWGT1,T2,…,Tl,Tl+1,Tg=hgt+1−hgthl+1tϖk+1∏j=1lhjtϖjt,og.1−1−ol+1tϖk+1∏j=1l1−ojtϖjt,sg.1−1−sl+1tϖk+1∏j=1l1−sjtϖjt,GTSFWGT1,T2,…,Tl+1,Tg,=hgt+1−hgt∏j=1l+1hjtϖjt,og.1−∏j=1l+11−ojtϖjt,sg.1−∏j=1l+11−sjtϖjt.Theorem 2.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFWGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFWGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFWGT1,T2,…,Tm,Tg≤GTSFWGT1′,T2′,…,Tm′,TgProof.
(i)
IfTj=T0=h0,o0,s0 for all j=1,2,…,m, then from the definition of GTSFWG operator(11)GTSFWGT1,T2,…,Tm,Tg=hgt+1−hgt∏j=1mhjtϖjt,og.1−∏j=1m1−ogtϖjt,sg.1−∏j=1m1−sgtϖjt=hgt+1−hgthjt∑j=1mϖjt,og.1−1−ojt∑j=1mϖjt,sg.1−1−sjt∑j=1mϖjt=hgt+1−hgth0tt,og.1−1−o0tt,sg.1−1−s0tt=hgt+h0t−h0thgtt,og.o0,sg.s0.(ii)
ConsiderTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕TjwhereminhTg⊕Tj=hgt+1−hgtminhjtt, maxhTg⊕Tj=hgt+1−hgtmaxhjtt, minoTg⊕Tj=og.minoj, maxoTg⊕Tj=og.maxoj, minsTg⊕Tj=sg.minsj, and maxsTg⊕Tj=sg.maxsj. Then, for every j=1,2,…,m(12)minhj≤hj≤maxhj,minhjt≤∏j=1mhjtϖj≤maxhjt.For every0≤hg≤1(13)1−hgtminhjt≤1−hgt∏j=1mhjtϖj≤1−hgtmaxhjt,hgt+1−hgtminhjt≤hgt+1−hgt∏j=1mhjtϖj≤hgt+1−hgtmaxhjt,hgt+1−hgtminhjtt≤hgt+1−hgt∏j=1mhjtϖjt≤hgt+1−hgtmaxhjtt,minhTg⊕Tj≤hgt+1−hgt∏j=1mhjtϖjt≤maxhTg⊕Tj.Furthermore,minoj≤oj≤maxoj(14)minojt≤ojt≤maxojt,1−maxojt≤1−ojt≤1−minojt,∏j=1m1−maxojtϖj≤∏j=1m1−ojtϖj≤∏j=1m1−minojtϖj,1−maxojt∑j=1mϖj≤∏j=1m1−ojtϖj≤1−minojt∑j=1mϖj,1−1−minojt≤1−∏j=1m1−ojtϖj≤1−1−maxojt,1−1−minojtt≤1−∏j=1m1−ojtϖjt≤1−1−maxojtt,minoj≤1−∏j=1m1−ojtϖjt≤maxoj.For every0≤og≤1,(15)ogminoj≤og.1−∏j=1m1−ojtϖjt≤ogmaxoj,minoTg⊕Tj≤og.1−∏j=1m1−ojtϖjt≤maxoTg⊕Tj.Now,minsj≤sj≤maxsj(16)minsjt≤sjt≤maxsjt,1−maxsjt≤1−sjt≤1−minsjt,∏j=1m1−maxsjtϖj≤∏j=1m1−sjtϖj≤∏j=1m1−minsjtϖj,1−maxsjt∑j=1mϖj≤∏j=1m1−sjtϖj≤1−minsjt∑j=1mϖj,1−1−minsjt≤1−∏j=1m1−sjtϖj≤1−1−maxsjt,1−1−minsjtt≤1−∏j=1m1−sjtϖjt≤1−1−maxsjtt,minsj≤1−∏j=1m1−sjtϖjt≤maxsj.For every0≤sg≤1,(17)sgminsj≤sg.1−∏j=1m1−sjtϖjt≤sgmaxsj,minsTg⊕Tj≤sg.1−∏j=1m1−sjtϖjt≤maxsTg⊕Tj.(iii)
This can be proved by following part (ii).


### 4.2. Generalized T-Spherical Fuzzy Ordered Weighted Geometric Operator

Definition 7.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFOWG operator is defined as(18)GTSFOWGT1,T2,…,Tm,Tg=Tg⊕TSFOWGT1,T2,…,Tm.Theorem 3.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a associated weight vector ω=ω1,ω2,…,ωmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the GTSFOWG operator is given by(19)GTSFOWGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mTςjωj=hgt+1−hgt∏j=1mhςjtωjt,og.1−∏j=1m1−oςjtωjt,sg.1−∏j=1m1−sςjtωjt.Theorem 4.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a associated weight vector ω=ω1,ω2,…,ωmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFOWGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFOWGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFOWGT1,T2,…,Tm,Tg≤GTSFOWGT1′,T2′,…,Tm′,TgProof.
The proof is as in Theorem2.


### 4.3. Generalized T-Spherical Fuzzy Hybrid Weighted Geometric Operator

In this section, the GTSFHG operator which weights both TSFNs and their ordered positions is proposed. Some of its basic properties are also proved.Definition 8.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFHG operator is defined as(20)GTSFHGT1,T2,…,Tm,Tg=Tg⊕TSFHGT1,T2,…,Tm.Theorem 5.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT such that ϖ,ω∈0,1, ∑j=1mϖj=1 and ∑j=1mωj=1, then the GTSFHG operator is given by(21)GTSFHGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mT˜ςjωj=hgt+1−hgt∏j=1mh˜ςjtωjt,og.1−∏j=1m1−o˜ςjtωjt,sg.1−∏j=1m1−s˜ςjtωjt,where T˜ςj=Tjmϖjj=1,…,m is the permutation and m is the balancing coefficient.Theorem 6.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmTand associated weight vector ω=ω1,ω2,…,ωmT such that ϖ,ω∈0,1, and ∑j=1mωj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFHGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFHGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFHGT1,T2,…,Tm,Tg≤GTSFHGT1′,T2′,…,Tm′,TgProof.
The proof is as in Theorem2.


## 4.1. Generalized T-Spherical Fuzzy Weighted Geometric Operator

Definition 6.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFWG operator is defined as(6)GTSFWGT1,T2,…,Tm,Tg=Tg⊕TSFWGT1,T2,…,Tm.Theorem 1.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the GTSFWG operator is given by(7)GTSFWGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mTjϖj=hgt+1−hgt∏j=1mhjtϖjt,og.1−∏j=1m1−ojtϖjt,sg.1−∏j=1m1−sjtϖjt.Proof.
By using mathematical induction, this proof can be done.
Form=2,(8)GTSFWGT1,T2,Tg=Tg⊕T1ϖ1⊗T2ϖ2=hg,og,sg⊕h1ϖ1,1−1−o1tϖ1t,1−1−s1tϖ1t⊗h2ϖ2,1−1−o2tϖ2t,1−1−s2tϖ2t=hg,og,sg⊕h1ϖ1h2ϖ2,1−1−o1tϖ11−o2tϖ2t,1−1−s1tϖ11−s2tϖ2t=hgt+h1tϖ1h2tϖ2−hgth1tϖ1h2tϖ2t,og.1−1−o1tϖ11−o2tϖ2t,sg.1−1−s1tϖ11−s2tϖ2t=hgt+1−hgth1tϖ1h2tϖ2t,og.1−1−o1tϖ11−o2tϖ2t,sg.1−1−s1tϖ11−s2tϖ2t.
This shows that results hold form=2. Let us consider that the result is true for m=l,(9)GTSFWGT1,T2,…,Tl,Tg=hgt+1−hgt∏j=1lhjtϖjt,og.1−∏j=1l1−ojtϖjt,sg.1−∏j=1l1−sjtϖjt.
Now(10)GTSFWGT1,T2,…,Tl,Tl+1,Tg=hgt+1−hgthl+1tϖk+1∏j=1lhjtϖjt,og.1−1−ol+1tϖk+1∏j=1l1−ojtϖjt,sg.1−1−sl+1tϖk+1∏j=1l1−sjtϖjt,GTSFWGT1,T2,…,Tl+1,Tg,=hgt+1−hgt∏j=1l+1hjtϖjt,og.1−∏j=1l+11−ojtϖjt,sg.1−∏j=1l+11−sjtϖjt.Theorem 2.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFWGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFWGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFWGT1,T2,…,Tm,Tg≤GTSFWGT1′,T2′,…,Tm′,TgProof.
(i)
IfTj=T0=h0,o0,s0 for all j=1,2,…,m, then from the definition of GTSFWG operator(11)GTSFWGT1,T2,…,Tm,Tg=hgt+1−hgt∏j=1mhjtϖjt,og.1−∏j=1m1−ogtϖjt,sg.1−∏j=1m1−sgtϖjt=hgt+1−hgthjt∑j=1mϖjt,og.1−1−ojt∑j=1mϖjt,sg.1−1−sjt∑j=1mϖjt=hgt+1−hgth0tt,og.1−1−o0tt,sg.1−1−s0tt=hgt+h0t−h0thgtt,og.o0,sg.s0.(ii)
ConsiderTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕TjwhereminhTg⊕Tj=hgt+1−hgtminhjtt, maxhTg⊕Tj=hgt+1−hgtmaxhjtt, minoTg⊕Tj=og.minoj, maxoTg⊕Tj=og.maxoj, minsTg⊕Tj=sg.minsj, and maxsTg⊕Tj=sg.maxsj. Then, for every j=1,2,…,m(12)minhj≤hj≤maxhj,minhjt≤∏j=1mhjtϖj≤maxhjt.For every0≤hg≤1(13)1−hgtminhjt≤1−hgt∏j=1mhjtϖj≤1−hgtmaxhjt,hgt+1−hgtminhjt≤hgt+1−hgt∏j=1mhjtϖj≤hgt+1−hgtmaxhjt,hgt+1−hgtminhjtt≤hgt+1−hgt∏j=1mhjtϖjt≤hgt+1−hgtmaxhjtt,minhTg⊕Tj≤hgt+1−hgt∏j=1mhjtϖjt≤maxhTg⊕Tj.Furthermore,minoj≤oj≤maxoj(14)minojt≤ojt≤maxojt,1−maxojt≤1−ojt≤1−minojt,∏j=1m1−maxojtϖj≤∏j=1m1−ojtϖj≤∏j=1m1−minojtϖj,1−maxojt∑j=1mϖj≤∏j=1m1−ojtϖj≤1−minojt∑j=1mϖj,1−1−minojt≤1−∏j=1m1−ojtϖj≤1−1−maxojt,1−1−minojtt≤1−∏j=1m1−ojtϖjt≤1−1−maxojtt,minoj≤1−∏j=1m1−ojtϖjt≤maxoj.For every0≤og≤1,(15)ogminoj≤og.1−∏j=1m1−ojtϖjt≤ogmaxoj,minoTg⊕Tj≤og.1−∏j=1m1−ojtϖjt≤maxoTg⊕Tj.Now,minsj≤sj≤maxsj(16)minsjt≤sjt≤maxsjt,1−maxsjt≤1−sjt≤1−minsjt,∏j=1m1−maxsjtϖj≤∏j=1m1−sjtϖj≤∏j=1m1−minsjtϖj,1−maxsjt∑j=1mϖj≤∏j=1m1−sjtϖj≤1−minsjt∑j=1mϖj,1−1−minsjt≤1−∏j=1m1−sjtϖj≤1−1−maxsjt,1−1−minsjtt≤1−∏j=1m1−sjtϖjt≤1−1−maxsjtt,minsj≤1−∏j=1m1−sjtϖjt≤maxsj.For every0≤sg≤1,(17)sgminsj≤sg.1−∏j=1m1−sjtϖjt≤sgmaxsj,minsTg⊕Tj≤sg.1−∏j=1m1−sjtϖjt≤maxsTg⊕Tj.(iii)
This can be proved by following part (ii).


## 4.2. Generalized T-Spherical Fuzzy Ordered Weighted Geometric Operator

Definition 7.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFOWG operator is defined as(18)GTSFOWGT1,T2,…,Tm,Tg=Tg⊕TSFOWGT1,T2,…,Tm.Theorem 3.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a associated weight vector ω=ω1,ω2,…,ωmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the GTSFOWG operator is given by(19)GTSFOWGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mTςjωj=hgt+1−hgt∏j=1mhςjtωjt,og.1−∏j=1m1−oςjtωjt,sg.1−∏j=1m1−sςjtωjt.Theorem 4.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a associated weight vector ω=ω1,ω2,…,ωmT such that ϖ∈0,1 and ∑j=1mϖj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFOWGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFOWGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFOWGT1,T2,…,Tm,Tg≤GTSFOWGT1′,T2′,…,Tm′,TgProof.
The proof is as in Theorem2.


## 4.3. Generalized T-Spherical Fuzzy Hybrid Weighted Geometric Operator

In this section, the GTSFHG operator which weights both TSFNs and their ordered positions is proposed. Some of its basic properties are also proved.Definition 8.
Considering the GPTg=hg,og,sg for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GTSFHG operator is defined as(20)GTSFHGT1,T2,…,Tm,Tg=Tg⊕TSFHGT1,T2,…,Tm.Theorem 5.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT such that ϖ,ω∈0,1, ∑j=1mϖj=1 and ∑j=1mωj=1, then the GTSFHG operator is given by(21)GTSFHGT1,T2,…,Tm,Tg=Tg⊕⊗j=1mT˜ςjωj=hgt+1−hgt∏j=1mh˜ςjtωjt,og.1−∏j=1m1−o˜ςjtωjt,sg.1−∏j=1m1−s˜ςjtωjt,where T˜ςj=Tjmϖjj=1,…,m is the permutation and m is the balancing coefficient.Theorem 6.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m with GP Tg=hg,og,sg having a weight vector ϖ=ϖ1,ϖ2,…,ϖmTand associated weight vector ω=ω1,ω2,…,ωmT such that ϖ,ω∈0,1, and ∑j=1mωj=1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m, then GTSFHGT1,T2,…,Tm,Tg=Tg⊕T0(ii)
IfTjL=minhTg⊕Tj,minoTg⊕Tj,maxsTg⊕Tj and TjU=maxhTg⊕Tj,maxoTg⊕Tj,minsTg⊕Tj, then TjL≤GTSFHGT1,T2,…,Tm,Tg≤TjU(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then GTSFHGT1,T2,…,Tm,Tg≤GTSFHGT1′,T2′,…,Tm′,TgProof.
The proof is as in Theorem2.


## 5. Group-Generalized T-Spherical Fuzzy Geometric Aggregation Operators

In this section, the group-generalized TSF weighted geometric (GGTSFWG) operator, group-generalized TSF ordered weighted geometric (GGTSFOWG) operator, and group-generalized TSF hybrid geometric (GGTSFHG) operator are defined. Some basic results of these operators are also discussed.


### 5.1. Group-Generalized T-Spherical Fuzzy Weighted Geometric Operator

Definition 9.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFWG operator is defined as(22)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFWGT1,T2,…,Tm.Theorem 7.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with a weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT, then the GGTSFWG operator is given by(23)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mTjϖj=∏k=1nhgktϖk′+∏j=1mhjtϖj−∏k=1nhgktϖk′.∏j=1mhjtϖjt,1−∏k=1n1−ogktϖk′t.1−∏j=1m1−ojtϖjt,1−∏j=1m1−sjtϖk′t.1−∏j=1m1−sjtϖjt.Proof.
By using mathematical induction, this proof can be done.
Form=2,(24)GGTSFWGT1,T2,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕T1ϖ1⊗T2ϖ2=∏k=1nhgkϖk′,1−∏k=1n1−ogktϖk′t,1−∏k=1n1−sgktϖk′t⊕h1ϖ1,1−1−o1tϖ1t,1−1−s1tϖ1t⊗h2ϖ2,1−1−o2tϖ2t,1−1−s2tϖ2t=∏k=1nhgkϖk′,1−∏k=1n1−ogktϖk′t,1−∏k=1n1−sgktϖk′t⊕h1ϖ1h2ϖ2,1−1−o1tϖ11−o2tϖ2t,1−1−s1tϖ11−s2tϖ2t=∏k=1nhgktϖk′+h1tϖ1h2tϖ2−∏k=1nhgktϖk′h1tϖ1h2tϖ2t,1−∏k=1n1−ogktϖk′t·1−1−o1tϖ11−o2tϖ2t,1−∏k=1n1−sgktϖk′t.1−1−s1tϖ11−s2tϖ2t.
This shows that results hold form=2. Let us consider that result is true for m=l,(25)GGTSFWGT1,T2,…,Tl,Tg1,Tg2,…,Tgn=∏k=1nhgktϖk′+∏j=1lhjtϖj−∏k=1nhgktϖk′.∏j=1lhjtϖjt,1−∏k=1n1−ogktϖk′t.1−∏j=1l1−ojtϖjt,1−∏k=1n1−sgktϖk′t.1−∏j=1l1−sjtϖjt.
Now(26)GGTSFWGT1,T2,…,Tl,Tl+1,Tg1,Tg2,…,Tgn=1−∏k=1nhgktϖk′+hl+1tϖl+1∏j=1lhl+1tϖj−∏k=1nhgktϖk′⋅hl+1tϖl+i∏j=1lhl+1tϖj∏k=1nhgktϖk′+hl+1tϖl+i∏j=1lhl+1tϖj−∏k=1nhgktϖk′⋅hl+1tϖl+i∏j=1lhl+1tϖjtt,1−∏k=1n1−ogktϖk′t⋅1−1−ol+1tϖl+1∏j=1l1−ojtt,1−∏k=1n1−sgktϖk′t⋅1−1−sl+1tϖl+1∏j=1l1−sjtϖjt,GGTSFWGT1,T2,…,Tl,Tl+1,Tg1,Tg2,…,Tgn=∏k=1nhgktϖk′+∏j=1l+1hjtϖj−∏k=1nhgktϖk′.∏j=1l+1hjtϖjt,1−∏k=1n1−ogktϖk′t⋅1−∏j=1l+11−ojtϖjt,1−∏k=1n1−sgktϖk′t.1−∏j=1l+11−sjtϖjt.
Thus, results hold for allm.Theorem 8.
Consider a collection of TSFNsTj=hj,oj,sjj=1,2,…,m and the expert preferences Tgk=hgk,ogk,sgkk=1,2,…,n having a weight vectors ϖ=ϖ1,ϖ2,…,ϖmT and ϖ′=ϖ1′,ϖ2′,…,ϖn′T, respectively. Then, the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then (27)TjL≤GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(28)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFWGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


### 5.2. Group-Generalized T-Spherical Fuzzy Ordered Weighted Geometric Operator

Definition 10.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFOWG operator is defined as(29)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFOWGT1,T2,…,Tm.Theorem 9.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having associated weight vector ω=ω1,ω2,…,ωmT, then the GGTSFOWG operator is given by(30)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mTςjωj=∏k=1nhgktϖk′+∏j=1mhςjtωj−∏k=1nhgktϖk′.∏j=1mhςjtωjt⋅1−∏k=1n1−ogktϖk′t.1−∏j=1m1−oςjtωjt,1−∏k=1n1−sgktϖk′t.1−∏j=1m1−sςjtωjt.Theorem 10.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m and the expert preferences Tgk=hgk,ogk,sgkk=1,2,…,n having associated weight vectors ω=ω1,ω2,…,ωmT and weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T, respectively, with a condition that each weight vector must belong to 0,1 and the sum of all weights must be equal to 1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then(31)TjL≤GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(32)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFOWGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


### 5.3. Group-Generalized T-Spherical Fuzzy Hybrid Weighted Geometric Operator

In this section, the GGTSFHG operator which weights both TSFNs and their ordered positions is proposed. Some of its basic properties are also discussed.Definition 11.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFHG operator is defined as(33)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFHGT1,T2,…,Tm.Theorem 11.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT, GGTSFHG operator is given by(34)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mT˜ςjωj=∏k=1nhgktϖk′+∏j=1mh˜ςjtωj−∏k=1nhgktϖk′.∏j=1mh˜ςjtωjt,1−∏k=1n1−ogktϖk′t.1−∏j=1m1−o˜ςjtωjt1−∏k=1n1−sgktϖk′t.1−∏j=1m1−s˜ςjtωjt,where T˜ςj=Tjmϖjj=1,…,m is the permutation and m is the balancing coefficient.Theorem 12.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T˜0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then(35)TjL≤GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(36)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFHGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


## 5.1. Group-Generalized T-Spherical Fuzzy Weighted Geometric Operator

Definition 9.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFWG operator is defined as(22)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFWGT1,T2,…,Tm.Theorem 7.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with a weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT, then the GGTSFWG operator is given by(23)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mTjϖj=∏k=1nhgktϖk′+∏j=1mhjtϖj−∏k=1nhgktϖk′.∏j=1mhjtϖjt,1−∏k=1n1−ogktϖk′t.1−∏j=1m1−ojtϖjt,1−∏j=1m1−sjtϖk′t.1−∏j=1m1−sjtϖjt.Proof.
By using mathematical induction, this proof can be done.
Form=2,(24)GGTSFWGT1,T2,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕T1ϖ1⊗T2ϖ2=∏k=1nhgkϖk′,1−∏k=1n1−ogktϖk′t,1−∏k=1n1−sgktϖk′t⊕h1ϖ1,1−1−o1tϖ1t,1−1−s1tϖ1t⊗h2ϖ2,1−1−o2tϖ2t,1−1−s2tϖ2t=∏k=1nhgkϖk′,1−∏k=1n1−ogktϖk′t,1−∏k=1n1−sgktϖk′t⊕h1ϖ1h2ϖ2,1−1−o1tϖ11−o2tϖ2t,1−1−s1tϖ11−s2tϖ2t=∏k=1nhgktϖk′+h1tϖ1h2tϖ2−∏k=1nhgktϖk′h1tϖ1h2tϖ2t,1−∏k=1n1−ogktϖk′t·1−1−o1tϖ11−o2tϖ2t,1−∏k=1n1−sgktϖk′t.1−1−s1tϖ11−s2tϖ2t.
This shows that results hold form=2. Let us consider that result is true for m=l,(25)GGTSFWGT1,T2,…,Tl,Tg1,Tg2,…,Tgn=∏k=1nhgktϖk′+∏j=1lhjtϖj−∏k=1nhgktϖk′.∏j=1lhjtϖjt,1−∏k=1n1−ogktϖk′t.1−∏j=1l1−ojtϖjt,1−∏k=1n1−sgktϖk′t.1−∏j=1l1−sjtϖjt.
Now(26)GGTSFWGT1,T2,…,Tl,Tl+1,Tg1,Tg2,…,Tgn=1−∏k=1nhgktϖk′+hl+1tϖl+1∏j=1lhl+1tϖj−∏k=1nhgktϖk′⋅hl+1tϖl+i∏j=1lhl+1tϖj∏k=1nhgktϖk′+hl+1tϖl+i∏j=1lhl+1tϖj−∏k=1nhgktϖk′⋅hl+1tϖl+i∏j=1lhl+1tϖjtt,1−∏k=1n1−ogktϖk′t⋅1−1−ol+1tϖl+1∏j=1l1−ojtt,1−∏k=1n1−sgktϖk′t⋅1−1−sl+1tϖl+1∏j=1l1−sjtϖjt,GGTSFWGT1,T2,…,Tl,Tl+1,Tg1,Tg2,…,Tgn=∏k=1nhgktϖk′+∏j=1l+1hjtϖj−∏k=1nhgktϖk′.∏j=1l+1hjtϖjt,1−∏k=1n1−ogktϖk′t⋅1−∏j=1l+11−ojtϖjt,1−∏k=1n1−sgktϖk′t.1−∏j=1l+11−sjtϖjt.
Thus, results hold for allm.Theorem 8.
Consider a collection of TSFNsTj=hj,oj,sjj=1,2,…,m and the expert preferences Tgk=hgk,ogk,sgkk=1,2,…,n having a weight vectors ϖ=ϖ1,ϖ2,…,ϖmT and ϖ′=ϖ1′,ϖ2′,…,ϖn′T, respectively. Then, the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then (27)TjL≤GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(28)GGTSFWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFWGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


## 5.2. Group-Generalized T-Spherical Fuzzy Ordered Weighted Geometric Operator

Definition 10.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFOWG operator is defined as(29)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFOWGT1,T2,…,Tm.Theorem 9.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having associated weight vector ω=ω1,ω2,…,ωmT, then the GGTSFOWG operator is given by(30)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mTςjωj=∏k=1nhgktϖk′+∏j=1mhςjtωj−∏k=1nhgktϖk′.∏j=1mhςjtωjt⋅1−∏k=1n1−ogktϖk′t.1−∏j=1m1−oςjtωjt,1−∏k=1n1−sgktϖk′t.1−∏j=1m1−sςjtωjt.Theorem 10.
Considering a collection of TSFNsTj=hj,oj,sjj=1,2,…,m and the expert preferences Tgk=hgk,ogk,sgkk=1,2,…,n having associated weight vectors ω=ω1,ω2,…,ωmT and weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T, respectively, with a condition that each weight vector must belong to 0,1 and the sum of all weights must be equal to 1, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then(31)TjL≤GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(32)GGTSFOWGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFOWGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


## 5.3. Group-Generalized T-Spherical Fuzzy Hybrid Weighted Geometric Operator

In this section, the GGTSFHG operator which weights both TSFNs and their ordered positions is proposed. Some of its basic properties are also discussed.Definition 11.
Considering theTgk=hgk,ogk,sgkk=1,…,n be the expert preferences for the TSFNs Tj=hj,oj,sjj=1,2,…,m, then the GGTSFHG operator is defined as(33)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=TSFWGTg1,Tg2,…,Tgn⊕TSFHGT1,T2,…,Tm.Theorem 11.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT, GGTSFHG operator is given by(34)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=⊗k=1nTgkϖk′⊕⊗j=1mT˜ςjωj=∏k=1nhgktϖk′+∏j=1mh˜ςjtωj−∏k=1nhgktϖk′.∏j=1mh˜ςjtωjt,1−∏k=1n1−ogktϖk′t.1−∏j=1m1−o˜ςjtωjt1−∏k=1n1−sgktϖk′t.1−∏j=1m1−s˜ςjtωjt,where T˜ςj=Tjmϖjj=1,…,m is the permutation and m is the balancing coefficient.Theorem 12.
Considering aTgk=hgk,ogk,sgkk=1,2,…,n with weight vector ϖ′=ϖ1′,ϖ2′,…,ϖn′T be the expert preferences for TSFNs Tj=hj,oj,sjj=1,2,…,m having a weight vector ϖ=ϖ1,ϖ2,…,ϖmT and associated weight vector ω=ω1,ω2,…,ωmT, then the following properties hold:(i)
IfTj=T0 for all j=1,2,…,m and Tgk=Tg0 for all k=1,2,…,n, then GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=Tg0⊕T˜0(ii)
IfTjL=minhTgk⊕Tj,minoTgk⊕Tj,maxsTgk⊕Tj and TjU=maxhTgk⊕Tj,maxoTgk⊕Tj,minsTgk⊕Tj, then(35)TjL≤GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤TjU.(iii)
Considering a collection of TSFNsTj′=hj′,oj′,sj′j=1,2,…,m such that hj≤hj′, oj≤oj′ and sj≥sj′ for all j, then(36)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn≤GGTSFHGT1′,T2′,…,Tm′,Tg1,Tg2,…,Tgn.Proof.
The proof is as in Theorem2.


## 6. Approach to MADM Problem Using Proposed Operators

LetT1,T2,…,Tl be the set of alternatives and G1,G2,…,Gm be the set of attributes with a weight vector ϖ=ϖ1,ϖ2,…,ϖmT satisfying ϖj∈0,1 and ∑j=1mϖj=1. A group of experts Tg1,Tg2,…,Tgn with a weight vector ϖ′=ϖ1′,ϖ2′,…,ϖm′T satisfying ϖk′∈0,1 and ∑k=1nϖk′=1 evaluates each alternative against each attribute. Each expert rates alternatives in the form of TSFN. Then, an algorithm for solving the MADM problem is proposed as follows:Step 1: the expert evaluates the alternatives by considering the attributes in terms of TSFNs and summarizes them in the decision matrix as(37)T=h11,o11,s11⋯h1m,o1m,s1m⋮⋱⋮hl1,ol1,sl1⋯hlm,olm,slm.Step 2: convert cost type data into benefit type data and normalize the decision matrix by using(38)τm=hlm,olm,slm,if attribute is of benefit type,slm,olm,hlm,if attribute is of cost type.Step 3: calculatet for which the given information lies in TSF environment.Step 4: aggregate the given information using GGTSFWG (GTSFWG) operator with weight vectorϖ and ϖ′.Step 5: order the aggregated values in descending order with respect to score function.Step 6: aggregate the ordered information using GGTSFHG (GTSFHG) operator with associated weight vectorω=ω1,ω2,…,ωm.Step 7: find out the best option using score function.Example 1.
A construction company wants to construct new apartments. The company wants to select a place from the setT1,T2,T3,T4 on the basis of following attributes G1,G2,G3,G4, where G1: cost of land, G2: surroundings, G3: technological, G4: rental value, with a weight vector0.2,0.1,0.3,0.4T. An expert evaluates all given alternatives on the basis of given attributes as given in Table 1.
The normalized decision matrix is shown in Table2.
A group of senior expertsTg1,Tg2,Tg3 with weight vector 0.3,0.3,0.4T assesses the alternatives listed in Table 3.
Table4 is obtained by combining Tables 2 and 3.
As0.83+0.44+0.35=1.62∉0,1, 0.832+0.442+0.352=1.005∉0,1, 0.833+0.443+0.353=0.6999∈0,1. Similarly, all values in Table 3 belong to 0,1 fort=3.
After aggregating the values of Table4 by utilizing GGTSFWG operators, the results will be as shown in Table 5.
The corresponding scores of aggregated values of Table5 are as shown in Table 6.
In Table7, the aggregated values are ordered on the basis of descending order of score function.
Aggregated values of Table7 by utilizing the GGTSFHG operators will be as follows:(39)T˜ς1=0.7821,0.1739,0.2230,T˜ς2=0.6903,0.0892,0.1369,T˜ς3=0.5944,0.0584,0.2448,T˜ς4=0.8594,0.1008,0.1131.
The score values of these aggregated values are(40)SCT˜ς1=0.4620,SCT˜ς2=0.3257,SCT˜ς3=0.1951,SCT˜ς4=0.6322.
The score value ofT4 is highest. So T4 is the best option for a company to construct new apartments.Table 1
Information given by an expert in TSFSs.
G1G2G3G4T10.43,0.20,0.610.54,0.35,0.630.81,0.62,0.110.18,0.33,0.66T20.14,0.32,0.740.26,0.17,0.260.77,0.23,0.550.61,0.34,0.57T30.75,0.12,0.410.59,0.29,0.130.56,0.22,0.360.11,0.14,0.45T40.35,0.44,0.830.91,0.12,0.490.63,0.11,0.270.31,0.36,0.84Table 2
Normalized decision matrix.
G1G2G3G4T10.61,0.20,0.430.54,0.35,0.630.81,0.62,0.110.66,0.33,0.18T20.74,0.32,0.140.26,0.17,0.260.77,0.23,0.550.57,0.34,0.61T30.41,0.12,0.750.59,0.29,0.130.56,0.22,0.360.45,0.14,0.11T40.83,0.44,0.350.91,0.12,0.490.63,0.11,0.270.84,0.36,0.31Table 3
Information given by senior experts in TSFSs.
Tg1Tg2Tg3T10.71,0.30,0.400.58,0.21,0.790.49,0.52,0.43T20.74,0.41,0.250.34,0.24,0.230.44,0.19,0.28T30.32,0.29,0.690.67,0.35,0.210.56,0.22,0.36T40.78,0.46,0.390.87,0.13,0.170.53,0.21,0.37Table 4
Combination of Tables1 and 2.
G1G2G3G4Tg1Tg2Tg3T10.61,0.2,0.430.54,0.35,0.630.81,0.62,0.110.66,0.33,0.180.71,0.30,0.400.58,0.21,0.790.49,0.52,0.43T20.74,0.32,0.140.26,0.17,0.260.77,0.23,0.550.57,0.34,0.610.74,0.41,0.250.34,0.24,0.230.44,0.19,0.28T30.41,0.12,0.750.59,0.29,0.130.56,0.22,0.360.45,0.14,0.110.32,0.29,0.690.67,0.35,0.210.56,0.22,0.36T40.83,0.44,0.350.91,0.12,0.490.63,0.11,0.270.84,0.36,0.310.78,0.46,0.390.87,0.13,0.170.53,0.21,0.37Table 5
Aggregated values by utilizing the GGTSFWG operator.
G1G2G3G4Tg1Tg2Tg3T10.6734,0.1857,0.40030.7816,0.2509,0.47730.7766,0.6532,0.11690.5144,0.3846,0.21040.7347,0.2898,0.38660.6125,0.2028,0.77050.4248,0.5499,0.4557T20.7859,0.2974,0.13000.5834,0.1253,0.19190.7308,0.2443,0.58110.4068,0.3961,0.69630.7626,0.3963,0.24140.3787,0.2318,0.22210.3734,0.2019,0.2973T30.4900,0.1114,0.70800.8097,0.2142,0.09580.4987,0.2337,0.38200.2787,0.1637,0.12860.3586,0.2801,0.67030.6974,0.3382,0.20280.4987,0.2337,0.3820T40.8615,0.4097,0.32540.9630,0.0884,0.36550.5744,0.1169,0.28670.7566,0.4191,0.36150.7996,0.4449,0.37690.8822,0.1255,0.16420.4668,0.2231,0.3925Table 6
Score values.
G1G2G3G4Tg1Tg2Tg3T10.23480.35130.18800.06990.3145−0.2360−0.1842T20.45700.18960.1795−0.33240.36720.03090.0175T3−0.23860.52020.05550.0151−0.27710.29220.0555T40.53620.84350.16430.31220.36970.68020.0301Table 7
Ordered aggregated values.
G1G2G3G4Tg1Tg2Tg3Tς10.7816,0.2509,0.47730.6734,0.1857,0.40030.7766,0.6532,0.11690.5144,0.3846,0.21040.7347,0.2898,0.38660.4248,0.5499,0.45570.6125,0.2028,0.7705Tς20.7859,0.2974,0.13000.5834,0.1253,0.19190.7308,0.2443,0.58110.4068,0.3961,0.69630.7626,0.3963,0.24140.3787,0.2318,0.22210.3734,0.2019,0.2973Tς30.8097,0.2142,0.09580.4987,0.2337,0.38200.2787,0.1637,0.12860.4900,0.1114,0.70800.6974,0.3382,0.20280.4987,0.2337,0.38200.3586,0.2801,0.6703Tς40.9630,0.0884,0.36550.8615,0.4097,0.32540.7566,0.4191,0.36150.5744,0.1169,0.28670.8822,0.1255,0.16420.7996,0.4449,0.37690.4668,0.2231,0.3925


## 7. Comparative Analysis

In this section, some conditions are studied under which the defined work can be reduced to other fuzzy structures, and the significance of proposed operators is proved by solving an example of the existing literature by using proposed operators.Consider(41)GGTSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=∏k=1nhgktϖk′+∏j=1mh˜ςjtωj−∏k=1nhgktϖk′.∏j=1mh˜ςjtωjt,1−∏k=1n1−ogktϖk′t.1−∏j=1m1−o˜ςjtωjt,1−∏k=1n1−sgktϖk′t.1−∏j=1m1−s˜ςjtωjt.(i)
Fort=2, (1) can be reduced to group-generalized spherical fuzzy hybrid geometric (GGSFHG) operators(42)GGSFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=∏k=1nhgk2ϖk′+∏j=1mh˜ςj2ωj−∏k=1nhgk2ϖk′.∏j=1mh˜ςj2ωj,1−∏k=1n1−ogk2ϖk′.1−∏j=1m1−o˜ςj2ωj,1−∏k=1n1−sgk2ϖk′.1−∏j=1m1−s˜ςj2ωj.(ii)
Fort=1, (1) can be reduced to group-generalized picture fuzzy hybrid geometric (GGPFHG) operators(43)GGPFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=∏k=1nhgkϖk′+∏j=1mh˜ςjωj−∏k=1nhgkϖk′.∏j=1mh˜ςjωj,1−∏k=1n1−ogkϖk′.1−∏j=1m1−o˜ςjωj,1−∏k=1n1−sgkϖk′.1−∏j=1m1−s˜ςjωj.(iii)
Fort=2 and o=0, (1) can be reduced to group-generalized Pythagorean fuzzy hybrid geometric (GGPyFHG) operators(44)GGPyFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=∏k=1nhgk2ϖk′+∏j=1mh˜ςj2ωj−∏k=1nhgk2ϖk′.∏j=1mh˜ςj2ωj,1−∏k=1n1−sgk2ϖk′.1−∏j=1m1−s˜ςj2ωj.(iv)
Fort=1 and o=0, (1) can be reduced to group-generalized intuitionistic fuzzy hybrid geometric (GGIFHG) operators(45)GGIFHGT1,T2,…,Tm,Tg1,Tg2,…,Tgn=∏k=1nhgkϖk′+∏j=1mh˜ςjωj−∏k=1nhgkϖk′.∏j=1mh˜ςjωj,1−∏k=1n1−sgkϖk′.1−∏j=1m1−s˜ςjωj.Similarly, all other defined operators can be reduced to other fuzzy structures by using these conditions.Example 2.
ConsiderTg=0.4,0.7 be the GP of T1=0.4,0.5, T2=0.6,0.6, T3=0.8,0.3 and T4=0.7,0.6 have a weight vector ϖ=0.4,0.3,0.1,0.2T. Then find the aggregated value by using GTSFWG operator.
Solution. The given information can be written in TSF environment as Tg=0.4,0.0,0.7, T1=0.4,0.0,0.5, T2=0.6,0.0,0.6, T3=0.8,0.0,0.3, and T4=0.7,0.0,0.6.
As0.4+0.0+0.7=1.1∉0,1, 0.62+0.02+0.52=0.65∈0,1. Similarly, all values lie in TSF environment fort=2.(46)hg2+1−hg2∏j=14hj2ϖj=0.42+1−0.420.420.40.620.30.820.10.720.2=0.6374,og.1−∏j=141−og2ϖjt=0.0,sg.1−∏j=141−sg2ϖjt=0.71−1−0.520.41−0.620.31−0.320.11−0.620.2=0.7393.
Now we have(47)GTSFWGT1,T2,…,Tm,Tg=0.6374,0.0,0.7393.


## 8. Conclusion

In this manuscript, it is pointed out that existing geometric aggregation operators fail when the opinion of a senior expert is also involved with moderator’s opinion because all decision makers are not much familiar with alternatives that is why an opinion of expert is necessary. In it, a generalized parameter is defined for TSFSs. Then, by using this, generalized TSF geometric operators are proposed. Then, these operators are extended to group-generalized TSF geometric operators which deal with a group of experts’ opinion. Then, an algorithm is developed to solve MADM problem. The validity of defined operators is checked by a numerical example. A comparative analysis is also constructed in which the defined operators are reduced to other fuzzy structures such as SFSs, PFSs, PyFSs, and IFSs by using some conditions. An example is also solved by using proposed operators in which information is given in the form of PyFS. In future, it would be interesting to extend the concept generalized and group-generalized parameter to other aggregation operators and other structures like soft sets.



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