*3.2. Aggregation Operators*

In this section, based on the above proposed operational laws, we have proposed some series of geometric interactive improved AOs, namely, T-SFWGIA, T-SFOWGIA, and T-SFHGIA, under the T-SFS environment.

**Definition 10.** *For any collection, Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*) *of T-SFNs. If the mapping*

$$T - SFWIGA\_w(P\_1, P\_2, \dots, \dots, P\_k) = \otimes\_{j=1}^k P\_j^{w\_j} \tag{2}$$

*then T* − *SFWGIAw is called a T-Spherical fuzzy weighted geometric interactive averaging (T-SFWGIA) operator, where w* = (*<sup>w</sup>*1, *w*2,...... *wk*)*<sup>T</sup> is the weighting vector of Pj with wj* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *wj* = 1*.*

**Theorem 5.** *For any collection of T-SFNs, Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*)*, the aggregated values obtained by using Definition 10 is still T-SFNs and is given by:*

$$(T - SFWGIA\_{\mathbf{w}}(P\_1, P\_2, \dots, \mathbf{1}\_k) = \begin{pmatrix} \sqrt[l]{\prod\_{j=1}^k (1 - n\_j^t)^{w\_j} - \prod\_{j=1}^k (1 - \mathbf{m}\_j^t - \mathbf{i}\_j^t - n\_j^t)^{w\_j} - \prod\_{j=1}^k (i\_j^t)^{w\_j} \\\\ \sqrt[l]{1 - \prod\_{j=1}^k (1 - i\_j^t)^{w\_j}} \sqrt[l]{1 - \prod\_{j=1}^k (1 - n\_j^t)^{w\_j}} \end{pmatrix}$$

**Proof.** For any collection of T-SFNs, *Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*), we shall proof the result by induction on *k*.

For *k* = 1, we have:

$$T - SFWGIA\_{\mathfrak{w}}(P\_1) = P\_1^{\mathfrak{w}\_1} = \left(m\_1, i\_1, n\_1\right)$$

$$= \left(\sqrt[t]{\left(1 - n\_1^t\right)^1 - \left(1 - \left(m\_1^t + i\_1^t + n\_1^t\right)\right)^1 - \left(i\_1^t\right)^1}, \sqrt[t]{1 - 1 + \left(i\_1^t\right)^1}, \sqrt[t]{1 - 1 + \left(i\_1^t\right)^1}\right)$$

Thus, hold for *k* = 1*.* Now, the result holds for *n* = *m*:

$$(T - SFWGIA\_{\mathfrak{w}}\left(P^1, P^2, \dots, \dots, P\_{\mathfrak{m}}\right) = \begin{pmatrix} \sqrt[f]{\prod\_{j=1}^{\mathfrak{m}} \left(1 - n\_j^{\mathfrak{t}}\right)^{w\_j} - \prod\_{j=1}^{\mathfrak{m}} \left(1 - \mathbf{m}\_{\mathfrak{j}}^{\mathfrak{t}} - \mathbf{i}\_{\mathfrak{j}}^{\mathfrak{t}} - n\_j^{\mathfrak{t}}\right)^{w\_j} - \prod\_{j=1}^{\mathfrak{m}} \left(\mathbf{i}\_j^{\mathfrak{t}}\right)^{w\_j} \\\sqrt[f]{1 - \prod\_{j=1}^{\mathfrak{m}} \left(1 - \mathbf{i}\_j^{\mathfrak{t}}\right)^{w\_j}}, \sqrt[f]{1 - \prod\_{j=1}^{\mathfrak{m}} \left(1 - n\_j^{\mathfrak{t}}\right)^{w\_j}} \end{pmatrix}$$

Then for *k* = *m* + 1, we have:

*T* − *SFWGIAw*(*<sup>P</sup>*1, *P*2,........., *Pm*+<sup>1</sup>) = ⊗*<sup>m</sup>*+<sup>1</sup> *j*=1 *Pwj j* = *T* − *SFWGIAw*(*<sup>P</sup>*1, *P*2,........., *Pm*) ⊗ *Pwm*+<sup>1</sup> *m*+1 = ⎛⎝ *<sup>t</sup>*+∏*mj*=<sup>1</sup> (1 − *ntj*)*wj* − ∏*mj*=<sup>1</sup> (1 − mtj − itj − *ntj*)wj − ∏*mj*=<sup>1</sup> (*itj*)*wj* , *t*+1 − ∏*mj*=<sup>1</sup> (1 − *itj*)*wj* , *t*+1 − ∏*mj*=<sup>1</sup> (1 − *ntj*)*wj* ⎞⎠ ⊗⎛⎝ *t*+(1 − *ntj*)*wj* − (1 − mtj − itj − *ntj*)wj − (*itj*)*wj* , *t*+1 − (1 − *itj*)*wj* , *t*+1 − (1 − *ntj*)*wj* ⎞⎠ = ⎛⎜⎜⎜⎜⎝ *t*K*m*+1 ∏*j*=1 (1 − *ntj*)*wj* − *m*+1 ∏*j*=1 (1 − mtj − itj − *ntj*)wj − *m*+1 ∏*j*=1 (*itj*)*wj* , *t*K1 − *m*+1 ∏*j*=1 (1 − *itj*)*wj* , *t*K1 − *m*+1 ∏*j*=1 (1 − *ntj*)*wj* ⎞⎟⎟⎟⎟⎠

So, the result holds for *k* = *m* + 1*.* Therefore, by the principle of mathematical induction, the result holds for all *k* ∈ *Z*+. -

**Theorem 6.** *If Pj* = -*mj*, *ij*, *nj*.*, j* = 1, ... , *k are T-SFNs. Then the aggregated value using the T-SFWGIA operator is also T-SFN.*

**Proof.** Since *Pj* = -*mj*, *ij*, *nj*. is a T-SFN, *j* = 1, ... , *k*, we have 0 ≤ *mj*, *ij*, *nj* ≤ 1. So 0 ≤ *mtj*, *itj*, *ntj* ≤ 1 and 0 ≤ *mtj* + *itj* + *ntj* ≤ 1. Then:

$$\begin{aligned} \leq \prod\_{j=1}^{k} \left( 1 - n\_j^t \right)^{w\_j} - \prod\_{j=1}^{k} \left( 1 - \mathbf{m}\_j^t - \mathbf{i}\_j^t - n\_j^t \right)^{w\_j} - \prod\_{j=1}^{k} \left( i\_j^t \right)^{w\_j} \leq 1, \\ \quad 0 \leq 1 - \prod\_{j=1}^{k} \left( 1 - i\_j^t \right)^{w\_j} \leq 1 \\ \quad 0 \leq 1 - \prod\_{j=1}^{k} \left( 1 - n\_j^t \right)^{w\_j} \leq 1 \end{aligned}$$

Now:

$$\begin{split} & \sqrt{2\frac{\mathop{\rm l\hbox{ $\!\_{1}$ }}\limits^{k}(1-n\_{j}^{t})^{w\_{j}}-\prod\_{j=1}^{k}(1-(\mathbf{m}\_{j}^{t}+\mathbf{i}\_{j}^{t}+n\_{j}^{t}))^{w\_{j}}-\prod\_{j=1}^{k}(i\_{j}^{t})^{w\_{j}}+\\ & \sqrt{\frac{\mathop{\rm l\hbox{ $\!\_{1}$ }}\limits^{k}(1-\mathbf{i}\_{j}^{t})^{w\_{j}}+1-\prod\_{j=1}^{k}(1-n\_{j}^{t})^{w\_{j}}}{1-\prod\_{j=1}^{k}(1-(\mathbf{m}\_{j}^{t}+\mathbf{i}\_{j}^{t}+n\_{j}^{t}))^{w\_{j}}-\prod\_{j=1}^{k}(i\_{j}^{t})^{w\_{j}}-\prod\_{j=1}^{k}(1-i\_{j}^{t})^{w\_{j}}}\in[0,1] \end{split}$$

Thus, *T* − *SFWGIAw*(*<sup>P</sup>*1,........., *Pk*) is T-SFN.

Further, it is observed that the proposed operator satisfies certain properties, which are listed as follows: -

**Theorem 7.** *If all T-SFNs, Pj*(*j* = 1, 2, . . . , *k*)*, are equal to P*0*, where P*0 *is another T-SFN, then*

$$(T - SFWGIA\_w(P\_1, \ldots, \ldots, \ldots, P\_k) = P\_0)$$

**Proof.** Assume that *Pj* = *P*0 = (*<sup>m</sup>*0, *i*0, *<sup>n</sup>*0) is a T-SFN ∀*j*. Then, by definition of T-SFWGIA operator, we have:

$$\begin{split} T-SFWGIA\_{w}(P\_{1},P\_{2},\ldots,P\_{k}) &= \left( \begin{array}{l} \sqrt[l]{\prod\_{j=1}^{k} \left(1-n\_{j}^{t}\right)^{w\_{j}}}-\prod\_{j=1}^{k} \left(1-\left(\mathfrak{m}\_{j}^{t}+\mathfrak{i}\_{j}^{t}+n\_{j}^{t}\right)^{w\_{j}}-\prod\_{j=1}^{k} \left(\mathfrak{i}\_{j}^{t}\right)^{w\_{j}}\right) \\ & \sqrt[l]{1-\prod\_{j=1}^{k} \left(1-\mathfrak{i}\_{j}^{t}\right)^{w\_{j}}},\sqrt[l]{1-\prod\_{j=1}^{k} \left(1-n\_{j}^{t}\right)^{w\_{j}}} \end{array} \right) \\ &= \left( \begin{array}{l} \sqrt[l]{(1-n\_{j}^{t})^{\sum\_{i=1}^{k}w\_{j}}-\left(1-\left(\mathfrak{m}\_{j}^{t}+\mathfrak{i}\_{i}^{t}+n\_{j}^{t}\right)\right)^{\sum\_{j=1}^{k}w\_{j}}-\left(\mathfrak{i}\_{j}^{t}\right)^{\sum\_{j=1}^{k}w\_{j}} \\ & \sqrt[l]{1-\left(1-\mathfrak{i}\_{j}^{t}\right)^{\sum\_{j=1}^{k}w\_{j}}},\sqrt[l]{1-\left(1-n\_{j}^{t}\right)^{\sum\_{j=1}^{k}w\_{j}}} \\ & = (m\_{0},i\_{0},m\_{0}) \\ & = P\_{0} \end{array} \right) \end{split}$$

**Theorem 8.** *If Pj* = -*mj*, *ij*, *nj*. *is a T-SFN and*

$$\begin{aligned} P^L &= \left( \max\{0, \left( \min\{ m\_j + i\_j + n\_j \} - \min i\_j - \max n\_j \} \right), \min i\_j, \max n\_j \} \right) \\ P^{II} &= \left( \max\{ m\_j + i\_j + n\_j \} - \max i\_j - \min n\_j, \max i\_j, \min n\_j \} . \text{Then, we have } \\ P^L &\le T - SFWGIA\_w(P\_1, \dots, \dots, P\_k) \le P^{II} \end{aligned}$$

Proof is straightforward.

**Theorem 9.** *For a collection of two different T-SFNs, Aj* = (*mAj* , *iAj* , *nAj*), (*j* = 1, 2, ... , *k*) *and Bj* = (*mBj* , *iBj* , *nBj*), (*j* = 1, 2, ... , *k*)*, which satisfy the following inequalities if nAj* ≥ *nBj* , *iAj* ≥ *iBj and mtAj* + *itAj* + *ntAj* ≤ *mtBj* + *itBj* + *ntBj* ∀*j, then we have*

$$T - SFWGL\_w \text{ ( $A\_1, A\_2, \dots, A\_k$ )} \le T - SFWGL\_w \text{ ( $B\_1, B\_2, \dots, B\_k$ )}$$

**Proof.** Since *nAj* ≥ *nBj*, we have:

$$\sqrt[k]{1 - \prod\_{j=1}^{k} \left(1 - n\_{A\_j}^t \right)^{w\_j}} \ge \sqrt[k]{1 - \prod\_{j=1}^{k} \left(1 - n\_{B\_j}^t \right)^{w\_j}}$$

and *iAj* ≥ *iBj*

$$\sqrt[k]{1 - \prod\_{j=1}^{k} \left(1 - i\_{A\_j}^t \right)^{w\_j}} \ge \sqrt[k]{1 - \prod\_{j=1}^{k} \left(1 - i\_{B\_j}^t \right)^{w\_j}}$$

$$\begin{split} \text{As, } n\_{A\_{j}} &\geq n\_{\mathcal{B}\_{j}}, \ m^{t}\_{A\_{j}} + i^{t}\_{A\_{j}} + n^{t}\_{A\_{j}} \leq m^{t}\_{\mathcal{B}\_{j}} + i^{t}\_{\mathcal{B}\_{j}} + n^{t}\_{\mathcal{B}\_{j}}, \forall j \text{ we have:} \\ & \left( \sqrt[3]{\prod\_{j=1}^{k} \left(1 - n^{t}\_{A\_{j}}\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(1 - \left(\mathbf{m}^{t}\_{\mathcal{A}\_{j}} + \mathbf{i}^{t}\_{\mathcal{A}\_{j}} + n^{t}\_{A\_{j}}\right)\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(i^{t}\_{A\_{j}}\right)^{w\_{j}}}{\sqrt[3]{1 - \prod\_{j=1}^{k} \left(1 - i^{t}\_{A\_{j}}\right)^{w\_{j}}}} \right) \\ & \leq \left( \sqrt[3]{\prod\_{j=1}^{k} \left(1 - n^{t}\_{\mathcal{B}\_{j}}\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(1 - \left(\mathbf{m}^{t}\_{\mathcal{B}\_{j}} + \mathbf{i}^{t}\_{\mathcal{B}\_{j}} + n^{t}\_{\mathcal{B}\_{j}}\right)\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(i^{t}\_{\mathcal{B}\_{j}}\right)^{w\_{j}}}{\sqrt[3]{1 - \prod\_{j=1}^{k} \left(1 - i^{t}\_{\mathcal{B}\_{j}}\right)^{w\_{j}}}} \right) \end{split}$$

Therefore, we have:

$$T - SFWGIA\_w(A\_1, A\_2, \dots, A\_k) \le T - SFWGIA\_w(B\_1, B\_2, \dots, B\_k)$$

**Definition 11.** [34] *For any collection, Pj* = -*mj*, *ij*, *nj*. (*j* = 1, 2, . . . , *k*) *of T-SFNs. The T* − *SFOWGAw* : Ω*n* → Ω *is a mapping defined as*

$$(T - SFOWGA\_{\mathbf{w}}(P\_1, P\_2, \dots, P\_k) = \begin{pmatrix} \sqrt[l]{\prod\_{j=1}^{k} \left( \mathbf{m}\_{\sigma(j)}^{\mathbf{t}} + \mathbf{i}\_{\sigma(j)}^{\mathbf{t}} \right)^{w\_{\parallel}} - \prod\_{j=1}^{k} \left( \mathbf{i}\_{\sigma(j)}^{\mathbf{t}} \right)^{w\_{\parallel}} \prod\_{j=1}^{k} \left( \mathbf{i}\_{\sigma(j)} \right)^{w\_{\parallel}}\\ \sqrt[l]{1 - \prod\_{j=1}^{k} \left( 1 - n\_{\sigma(j)}^{\mathbf{t}} \right)^{w\_{\parallel}}} \end{pmatrix} \tag{3}$$

*where* Ω *is the collection of all T-SFNs, then T* − *SFOWGAw is called a T-SFOWGA operator with weighting vector w* = (*<sup>w</sup>*1, *w*2,...... *wk*)*<sup>T</sup> of Pj with wj* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *wj* = 1*.*

**Definition 12.** *For any collection, Pj* = -*mj*, *ij*, *nj*., (*j* = 1, 2, . . . , *k*) *of T-SFNs. The T* − *SFOWGIAw* : Ω*n* → Ω *is a mapping defined as:*

$$T - SFOWGIA\_w(P\_1, P\_2, \dots, \dots, P\_k) = \bigotimes\_{j=1}^k P\_{\sigma(j)}^{w\_j} \tag{4}$$

*then T* − *SFOWGIAw is called T-SFOWGIA operator, where w* = (*<sup>w</sup>*1, *w*2,......... *wk*)*<sup>T</sup> is the weighting vector of Pj with wj* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *wj* = 1 *and σ is the permutation of* {1, 2, ... , *k*}*, such that σ*(*j* − 1) ≥ *<sup>σ</sup>*(*j*)*.*

**Theorem 10.** *For any collection Pj* = -*mj*, *ij*, *nj*., (*j* = 1, 2, . . . , *k*) *of T-SFNs. Then*

$$(T - SFOWGIA\_w(P\_1, P\_2, \dots, P\_k) = \begin{pmatrix} \sqrt[N]{\prod\_{j=1}^k \left(1 - n\_{\sigma(j)}^t\right)^{w\_j} - \prod\_{j=1}^k \left(1 - \left(n\_{\sigma(j)}^t + l\_{\sigma(j)}^t + n\_{\sigma(j)}^t\right)\right)^{w\_j} - \prod\_{j=1}^k \left(l\_{\sigma(j)}^t\right)^{w\_j} \\ \sqrt[N]{1 - \prod\_{j=1}^k \left(1 - l\_{\sigma(j)}^t\right)^{w\_j}}, \sqrt[N]{1 - \prod\_{j=1}^k \left(1 - n\_{\sigma(j)}^t\right)^{w\_j}} \end{pmatrix}$$

Proof is similar to Theorem 5.

**Theorem 11.** *If Pj* = -*mj*, *ij*, *nj*. *is a T-SFN, j* = 1, . . . , *k. Then the aggregated value using the T-SFOWGIA operator is also T-SFN.*

**Proof.** Since *<sup>P</sup>σ*(*j*) = *<sup>m</sup>σ*(*j*), *<sup>i</sup>σ*(*j*), *<sup>n</sup>σ*(*j*)is a T-SFN, *j* = 1, ... , *k*, we have 0 ≤ *<sup>m</sup>σ*(*j*), *<sup>i</sup>σ*(*j*), *<sup>n</sup>σ*(*j*) ≤ 1. So 0 ≤ *m<sup>t</sup>σ*(*j*), *itσ*(*j*), *ntσ*(*j*) ≤ 1 and 0 ≤ *m<sup>t</sup>σ*(*j*) + *itσ*(*j*) + *ntσ*(*j*) ≤ 1. Then:

$$\begin{aligned} 0 \le \frac{k}{j-1} \left( 1 - n\_{\sigma(j)}^t \right)^{w\_j} - \prod\_{j=1}^k \left( 1 - \left( \mathbf{m}\_{\sigma(j)}^t + \mathbf{i}\_{\sigma(j)}^t + n\_{\sigma(j)}^t \right) \right)^{w\_j} - \prod\_{j=1}^k \left( i\_{\sigma(j)}^t \right)^{w\_j} \le 1 \\\ 0 \le 1 - \prod\_{j=1}^k \left( 1 - i\_{\sigma(j)}^t \right)^{w\_j} \le 1 \\\ 0 \le 1 - \prod\_{j=1}^k \left( 1 - n\_{\sigma(j)}^t \right)^{w\_j} \le 1 \end{aligned}$$

Now:

$$\begin{cases} \begin{aligned} \frac{1}{N} \left( \prod\_{j=1}^{k} \left( 1 - n\_{\sigma(j)}^{t} \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( 1 - \left( \mathbf{m}\_{\sigma(j)}^{t} + \mathbf{i}\_{\sigma(j)}^{t} + n\_{\sigma(j)}^{t} \right) \right) \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( \mathbf{i}\_{\sigma(j)}^{t} \right)^{w\_{j}} + 1 - \prod\_{j=1}^{k} \left( 1 - \mathbf{i}\_{\sigma(j)}^{t} \right)^{w\_{j}} \\ \end{aligned} \\\ \begin{aligned} \frac{1}{N} \left( \prod\_{j=1}^{k} \left( 1 - n\_{\sigma(j)}^{t} \right)^{w\_{j}} \right)^{w\_{j}} \\ = \sqrt{2 - \prod\_{j=1}^{k} \left( 1 - \left( \mathbf{m}\_{\sigma(j)}^{t} + \mathbf{i}\_{\sigma(j)}^{t} + n\_{\sigma(j)}^{t} \right) \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( \mathbf{i}\_{\sigma(j)}^{t} \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( 1 - \mathbf{i}\_{\sigma(j)}^{t} \right)^{w\_{j}} \in [0, 1] \end{aligned} \\\ \begin{aligned} \text{Thus, } T - SFWWGIG\_{w}(P\_{1}, \dots, \dots, \dots, P\_{k}) \text{ is T-SFN.} \ \end{aligned} \end{cases}$$

**Theorem 12.** *T* − *SFOWGIAw*(*<sup>P</sup>*1,........., *Pk*) = *P*0 if *Pj* = *P*0 = -*mj*, *ij*, *nj*. is a T-SFN ∀*j*.

**Proof.** We have:

$$\begin{pmatrix} \frac{T - SFOWIGA\_{\sigma}(P\_{1}, \ldots, P\_{k})}{\sqrt{\prod\_{j=1}^{k} \left(1 - n\_{\sigma(j)}^{t}\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(1 - \left(\mathbf{m}\_{\sigma(j)}^{t} + \mathbf{i}\_{\sigma(j)}^{t} + n\_{\sigma(j)}^{t}\right)\right)^{w\_{j}} - \prod\_{j=1}^{k} \left(\mathbf{i}\_{\sigma(j)}^{t}\right)^{w\_{j}}} \\ \sqrt{1 - \prod\_{j=1}^{k} \left(1 - \mathbf{i}\_{\sigma(j)}^{t}\right)^{w\_{j}}}, \sqrt{1 - \prod\_{j=1}^{k} \left(1 - n\_{\sigma(j)}^{t}\right)^{w\_{j}}} \\ = \left(\sqrt[3]{\left(1 - n\_{\sigma(j)}^{t}\right)^{\sum\_{j=1}^{k} w\_{j}} - \left(1 - \left(\mathbf{m}\_{\sigma(j)}^{t} + \mathbf{i}\_{\sigma(j)}^{t} + n\_{\sigma(j)}^{t}\right)\right)^{\sum\_{j=1}^{k} w\_{j}} - \left(\mathbf{i}\_{\sigma(j)}^{t}\right)^{\sum\_{j=1}^{k} w\_{j}}}\right) \\ \qquad \sqrt[3]{1 - \left(1 - \mathbf{i}\_{\sigma(j)}^{t}\right)^{\sum\_{j=1}^{k} w\_{j}}}, \sqrt[3]{1 - \left(1 - n\_{\sigma(j)}^{t}\right)^{\sum\_{j=1}^{k} w\_{j}}} \\ = \left(m\_{\sigma(0)}, i\_{\sigma(0)}, n\_{\sigma(0)}\right) \\ = P\_{0} \end{pmatrix}$$


> **Theorem 13.** *If Pj* = -*mj*, *ij*, *nj*. *is a T-SFN and*

$$\begin{array}{l} P^{L} = \left( \max\{0, \left( \min\{ m\_{\bar{j}} + i\_{\bar{j}} + n\_{\bar{j}} \} - \min i\_{\bar{j}} - \max n\_{\bar{j}} \} \right) \right), \min i\_{\bar{j}}, \max n\_{\bar{j}} \right), \\\ P^{L} = \left( \max\left( m\_{\bar{j}} + i\_{\bar{j}} + n\_{\bar{j}} \right) - \max i\_{\bar{j}} - \min n\_{\bar{j}} \right) \}, \max i\_{\bar{j}}, \min n\_{\bar{j}}, \text{Then } \\\ P^{L} \leq T - SFOWGIA(P\_{1}, \dots, \dots, P\_{k}) \leq P^{L} \end{array}$$

Proof is straightforward.

**Theorem 14.** *T* − *SFOWGIAw*(*<sup>B</sup>*1*B*2,......, *Bk*) = *T* − *SFOWGIAw*(*<sup>A</sup>*1,......, *Ak*) *if Bj* = *mBj* , *iBj* , *nBj is any permutation of Aj* = *mAj* , *iAj* , *nAj where j* = 1, . . . . . . , *k*.

**Proof.**

$$\begin{pmatrix} T - SFOWIGA\_{\mathcal{w}}(B\_1, B\_2, \dots, B\_k) \\ \sqrt{\prod\_{j=1}^k \left(1 - n\_{B\_{c(j)}}^t\right)^{w\_j} - \prod\_{j=1}^k \left(1 - \left(\mathbf{m}\_{B\_{c(j)}}^t + \mathbf{i}\_{B\_{c(j)}}^t + \mathbf{n}\_{B\_{c(j)}}^t\right)\right)^{w\_j} - \prod\_{j=1}^k \left(\mathbf{i}\_{B\_{c(j)}}^t\right)^{w\_j}}{\sqrt{1 - \prod\_{j=1}^k \left(1 - \mathbf{i}\_{B\_{c(j)}}^t\right)^{w\_j}}} \sqrt{1 - \prod\_{j=1}^k \left(1 - \mathbf{n}\_{B\_{c(j)}}^t\right)^{w\_j}} \\ \sqrt{1 - \prod\_{j=1}^k \left(1 - \mathbf{n}\_{A\_{c(j)}}^t\right)^{w\_j} - \prod\_{j=1}^k \left(1 - \left(\mathbf{m}\_{A\_{c(j)}}^t + \mathbf{i}\_{A\_{c(j)}}^t + \mathbf{n}\_{A\_{c(j)}}^t\right)\right)^{w\_j} - \prod\_{j=1}^k \left(\mathbf{i}\_{A\_{c(j)}}^t\right)^{w\_j}} \\ \sqrt{1 - \prod\_{j=1}^k \left(1 - \mathbf{i}\_{A\_{c(j)}}^t\right)^{w\_j}} \sqrt{1 - \prod\_{j=1}^k \left(1 - \mathbf{n}\_{A\_{c(j)}}^t\right)^{w\_j}} \end{pmatrix}$$

If *Bj* = *mBj* , *iBj* , *nBj* is any permutation of *Aj* = *mAj* , *iAj* , *nAj* then we have *<sup>B</sup>σ*(*j*) = *<sup>A</sup>σ*(*j*). Thus, *T* − *SFOWGIAw*(*<sup>B</sup>*1,......, *Bk*) = *T* − *SFOWGIAw*-*A*1,......, *Ak*.. -

**Definition 13.** *For any collection, Pj* = 8*mj*, *ij*, *nj*9 *of T-SFNs* (*j* = 1, 2, 3, . . . . . . , *k*)*. If the mapping*

$$T - SFHG A\_{\omega, w}(P\_1, P\_2, \dots, \dots, P\_k) = \odot\_{j=1}^k \left(\widetilde{P}\_{\sigma(j)}\right)^{w\_i} \tag{5}$$

*then T* − *SFHGA<sup>ω</sup>*,*<sup>w</sup> is called a T-SFHGA operator, where P j* = -*Pj*.*<sup>n</sup><sup>ω</sup><sup>j</sup> and ω* = (*<sup>ω</sup>*1,...... *<sup>ω</sup>k*)*<sup>T</sup> is the weighting vector of Pj with <sup>ω</sup>j* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *<sup>ω</sup>j* = 1*.*

**Theorem 15.** [34] *For any collection, Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*) *of T-SFNs. If*

$$(T - SFHGA\_{\omega, \mathfrak{w}}(P\_1, P\_2, \dots, \dots, P\_k) = \left( \begin{array}{l} \sqrt[l]{\prod\limits\_{j=1}^k \left( \mathbf{m}\_{\bar{P}\_{\sigma(j)}}^{\mathbf{t}} + \mathbf{i}\_{\bar{P}\_{\sigma(j)}}^{\mathbf{t}} \right)^{\mathbf{w}\_{\bar{j}}} - \prod\limits\_{j=1}^k \left( \mathbf{i}\_{\bar{P}\_{\sigma(j)}}^{\mathbf{t}} \right)^{\mathbf{w}\_{\bar{j}}} \\ \prod\limits\_{j=1}^k \left( \mathbf{i}\_{\bar{P}\_{\sigma(j)}}^{\mathbf{t}} \right)^{\mathbf{w}\_{\bar{j}}}, \sqrt[l]{1 - \prod\limits\_{j=1}^k \left( 1 - n\_{\bar{P}\_{\sigma(j)}}^{\mathbf{t}} \right)^{\mathbf{w}\_{\bar{j}}}}^{\mathbf{w}\_{\bar{j}}} \right) \end{array} \right)$$

*then T* − *SFHGA<sup>ω</sup>*,*<sup>w</sup> is called a T-SFHGA operator with weighting vector ω* = (*<sup>ω</sup>*1, *ω*2,...... *<sup>ω</sup>k*)*<sup>T</sup> of Pj with <sup>ω</sup>j* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *<sup>ω</sup>j* = 1*.*

**Definition 14.** *For any collection, Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*) *of T-SFNs. If the mapping*

$$T - SFHGIA\_{\omega, w}(P\_1, P\_2, \dots, \dots, P\_k) = \odot\_{j=1}^k \hat{P}\_{\sigma(j)}^{w\_j} \tag{6}$$

*then T* − *SFHGIA<sup>ω</sup>*,*<sup>w</sup> is called a T-SFHGIA operator, where ω* = (*<sup>ω</sup>*1, *ω*2,...... *<sup>ω</sup>k*)*<sup>T</sup> is the weighting vector of Pj with <sup>ω</sup>j* ∈ [0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *wj* = 1*.*

**Theorem 16.** *For any collection, Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*) *of T-SFNs. Then*

$$= \left( \sqrt[t]{\prod\_{j=1}^{k} \left( 1 - \mathsf{n}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( 1 - \left( \mathsf{n}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} + \mathsf{i}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} + \mathsf{n}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} \right) \right)^{w\_{j}} - \prod\_{j=1}^{k} \left( \mathsf{i}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} \right)^{w\_{j}}}{\sqrt{1 - \prod\_{j=1}^{k} \left( 1 - \mathsf{i}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} \right)^{w\_{j}}}, \sqrt[t]{1 - \prod\_{j=1}^{k} \left( 1 - \mathsf{n}\_{\tilde{P}\_{\mathcal{C}(j)}}^{t} \right)^{w\_{j}}}} \right)$$

The following example demonstrates these aggregation operators:

**Example 1.** *Let P*1 = (0.3, 0.8, 0.1)*, P*2 = (0.4, 0.3, 0.6)*, P*3 = (0.7, 0.1, 0.5)*, P*4 = (0.9, 0.4, 0.1) *and P*5 = (0.2, 0.6, 0.7) *are T-SFN. The weight vector for Pi* (*i* = 1, 2, . . . , 5) *is ω* = (0.18, 0.22, 0.16, 0.21, 0.23)*T. With loss of generality, we use t* = 2 *for all calculations.*

Firstly, we utilized T-SFHGIA operators on this data to aggregate it.

*P*1 = ⎛⎝ +(1 − 0.1<sup>2</sup>)<sup>5</sup>×0.18 − (1 − (0.3<sup>2</sup> + 0.8<sup>2</sup> + 0.1<sup>2</sup>))<sup>5</sup>×0.18 − (0.8<sup>2</sup>)<sup>5</sup>×0.18, +1 − (1 − 0.8<sup>2</sup>)<sup>5</sup>×0.18, +1 − (1 − 0.1<sup>2</sup>)<sup>5</sup>×0.18 ⎞⎠ = (0.1559, 0.7754, 0.0949) *P*2 = ⎛⎝ +(1 − 0.6<sup>2</sup>)<sup>5</sup>×0.22 − (1 − (0.4<sup>2</sup> + 0.3<sup>2</sup> + 0.6<sup>2</sup>))<sup>5</sup>×0.22 − (0.3<sup>2</sup>)<sup>5</sup>×0.22, +1 − (1 − 0.3<sup>2</sup>)<sup>5</sup>×0.22, +1 − (1 − 0.6<sup>2</sup>)<sup>5</sup>×0.22 ⎞⎠ = (0.4317, 0.3139, 0.6228) *P*3 = ⎛⎝ +(1 − 0.5<sup>2</sup>)<sup>5</sup>×0.16 − (1 − (0.7<sup>2</sup> + 0.1<sup>2</sup> + 0.5<sup>2</sup>))<sup>5</sup>×0.16 − (0.1<sup>2</sup>)<sup>5</sup>×0.16, +1 − (1 − 0.1<sup>2</sup>)<sup>5</sup>×0.16, +1 − (1 − 0.5<sup>2</sup>)<sup>5</sup>×0.16 ⎞⎠ = (0.6629, 0.0895, 0.4534) *P*4 = ⎛⎝ +(1 − 0.1<sup>2</sup>)<sup>5</sup>×0.21 − (1 − (0.9<sup>2</sup> + 0.4<sup>2</sup> + 0.1<sup>2</sup>))<sup>5</sup>×0.21 − (0.4<sup>2</sup>)<sup>5</sup>×0.21, +1 − (1 − 0.4<sup>2</sup>)<sup>5</sup>×0.21, +1 − (1 − 0.1<sup>2</sup>)<sup>5</sup>×0.21 ⎞⎠ = (0.9094, 0.4090, 0.1024) *P*5 = ⎛⎝ +(1 − 0.7<sup>2</sup>)<sup>5</sup>×0.23 − (1 − (0.2<sup>2</sup> + 0.6<sup>2</sup> + 0.7<sup>2</sup>))<sup>5</sup>×0.23 − (0.6<sup>2</sup>)<sup>5</sup>×0.23, +1 − (1 − 0.6<sup>2</sup>)<sup>5</sup>×0.23, +1 − (1 − 0.7<sup>2</sup>)<sup>5</sup>×0.23 ⎞⎠ = (0.2705, 0.6336, 0.7342)

The score values corresponding to these aggregated numbers were obtained as *SC*(*<sup>P</sup>*1) = 0.0153, *SC*(*<sup>P</sup>*2) = −0.2016, *SC*(*<sup>P</sup>*3) = 0.2338, *SC*(*<sup>P</sup>*4) = 0.8166, *SC*(*<sup>P</sup>*5) = −0.4658. Based on the score values, we had the following arrangemen<sup>t</sup> of data:

*<sup>P</sup>σ*(1) = (0.9094, 0.4090, 0.1024), *<sup>P</sup>σ*(2) = (0.6629, 0.0895, 0.4534), *<sup>P</sup>σ*(3) = (0.1559, 0.7754, 0.0949), *<sup>P</sup>σ*(4) = (0.4317, 0.3139, 0.6228), *<sup>P</sup>σ*(5) = (0.2705, 0.6336, 0.7342)

By using the normal distribution-based method, we found *w* = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117)*<sup>T</sup>* and by the definition of T-SFHGIA operator we had

$$(T - SFHGIA\_{\omega, w}(P\_1, P\_2, P\_3, P\_4, P\_5) = (0.4688, 0.5643, 0.4792)$$

**Theorem 17.** *If Pj* = -*mj*, *ij*, *nj*. *is a T-SFN* , *j* = 1, ... , *k, then the aggregated value using the T-SFHGIA operator is also T-SFN.*

Proof is similar as in Theorem 11.

**Theorem 18.** *T* − *SFHGIA<sup>ω</sup>*, *w*(*<sup>P</sup>*1, *P*2,......, *Pk*) = *P*0 *if Pj* = *P*0 = -*mj*, *ij*, *nj*. *is a T-SFN* ∀*j.* Proof is similar as in Theorem 12.

**Theorem 19.** *If Pj* = -*mj*, *ij*, *nj*. *is a T-SFN and*

$$\begin{array}{l} P^{L} = \left( \max\{0, \left( \min\{ m\_{\bar{j}} + i\_{\bar{j}} + n\_{\bar{j}} \} - \min i\_{\bar{j}} - \max n\_{\bar{j}} \right) \}, \min i\_{\bar{j}}, \max n\_{\bar{j}} \right), \\\ P^{LI} = \left( \max\{ m\_{\bar{j}} + i\_{\bar{j}} + n\_{\bar{j}} \} - \max i\_{\bar{j}} - \min n\_{\bar{j}}, \max i\_{\bar{j}}, \min n\_{\bar{j}} \right). \text{ Then} \\\ P^{L} \leq T - SFHIGA\_{\omega, \text{w}}(P\_{1}, \dots, \dots, P\_{k}) \leq P^{L} \end{array}$$

> Proof is straightforward.

**Theorem 20.** *T* − *SFHGIA<sup>ω</sup>*,*<sup>w</sup>*(*<sup>B</sup>*1,......, *Bk*) = *T* − *SFHGIA<sup>ω</sup>*,*<sup>w</sup>*(*<sup>A</sup>*1,......, *Ak*) *if Bj* = *mBj* , *iBj* , *nBj is any permutation of Aj* = *mAj* , *iAj* , *nAj where j* = 1, . . . . . . , *k.* Proof is similar as Theorem 14.

Whenever membership and neutral number of one T-SFN become zero then the membership and abstinence value is not accounted for in the aggregation [34]. However, the geometric interaction averaging operators that are developed in our manuscript overcome this problem. The example below will describe this more clearly.

**Example 2.** *Let P*1 = (0.7, 0.5, 0.6), *P*2 = (0.9, 0.5, 0.4), *P*3 = (0, 0, 0.1), *P*4 = (0.5, 0.3, 0.4) *and P*5 = (0.6, 0.4, 0.5) *are T-SFN. The weight vector for Pi* (*i* = 1, 2, . . . , 5) *is ω* = (0.18, 0.22, 0.16, 0.21, 0.23)*T.*

For the solution, first we will find the T-SFHGA operator. As, 0.7+0.5+0.6 = 1.8 ∈/ [0, 1], 0.7<sup>2</sup> +0.5<sup>2</sup> +0.6<sup>2</sup> = 1.1 ∈/ [0, 1] but 0.7<sup>3</sup> +0.5<sup>3</sup> +0.6<sup>3</sup> = 0.684 ∈ [0, 1] Similarly, *P*2 and *P*4 satisfy the condition for *t* = 3.

*P* 1 = 5 <sup>3</sup>+(0.7<sup>3</sup> + 0.5<sup>3</sup>)<sup>5</sup>×0.18 − (0.5<sup>3</sup>)<sup>5</sup>×0.18, 0.55×0.18, 3+1 − (1 − 0.6<sup>3</sup>)<sup>5</sup>×0.186 = (0.7054, 0.5359, 0.5816) *P* 2 = 5 <sup>3</sup>+(0.9<sup>3</sup> + 0.5<sup>3</sup>)<sup>5</sup>×0.22 − (0.5<sup>3</sup>)<sup>5</sup>×0.22, 0.55×0.22, 3+1 − (1 − 0.4<sup>3</sup>)<sup>5</sup>×0.226 = (0.9041, 0.4665, 0.4125) *P* 3 = 5 <sup>3</sup>+(0<sup>3</sup> + 0<sup>3</sup>)<sup>5</sup>×0.16 − (0<sup>3</sup>)<sup>5</sup>×0.16, 05×0.16, 3+1 − (1 − 0.1<sup>3</sup>)<sup>5</sup>×0.166 = (0, 0, 0.0928) *P* 4 = 5 <sup>3</sup>+(0.5<sup>3</sup> + 0.3<sup>3</sup>)<sup>5</sup>×0.21 − (0.3<sup>3</sup>)<sup>5</sup>×0.21, 0.35×0.21, 3+1 − (1 − 0.4<sup>3</sup>)<sup>5</sup>×0.216 = (0.4874, 0.2885, 0.4063) *P* 5 = 5 <sup>3</sup>+(0.6<sup>3</sup> + 0.4<sup>3</sup>)<sup>5</sup>×0.23 − (0.4<sup>3</sup>)<sup>5</sup>×0.23, 0.45×0.23, 3+1 − (1 − 0.5<sup>3</sup>)<sup>5</sup>×0.236 = (0.5738, 0.3486, 0.5221)

Scores values for these aggregated numbers were obtained as *SC*(*P* 1) = 0.1543, *SC*(*P* 2) = 0.6689, *SC*(*P* 3) = −0.0008, *SC*(*P* 4) = 0.0487, *SC*(*P* 5) = 0.0466, and, based on these score values, we had

*P σ*(1) = (0.9041, 0.4665, 0.4125), *P σ*(2) = (0.7054, 0.5359, 0.5816), *P σ*(3) = (0.4874, 0.2885, 0.4063), *P σ*(4) = (0.5738, 0.3486, 0.5221), *P σ*(5) = (0, 0, 0.0928)

By using the normal distribution-based method, we found *w* = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117)*<sup>T</sup>*, and, by the definition of T-SFHGA operator, we found

$$\text{T}-\text{SFHGA}\_{\text{4}\prime,\text{W}}(\text{P}\_1, \text{P}\_2, \text{P}\_3, \text{P}\_4, \text{P}\_5) = (0, 0, 0.4803) \tag{7}$$

This type of aggregated value seems meaningless, as whenever the membership and abstinence value is zero in any one of the T-SFN it will make the value of the membership and non-membership as zero in the whole aggregated value. This shows that the geometric aggregation operator of T-SFSs [34] does not possess the ability to aggregate such types of information effectively.

On the other hand, the proposed new geometric interactive aggregation operators can process any type of information effectively. Now, the Example 2 was solved using the proposed new aggregation operators in order to justify its effectiveness. For it, we aggregated the data using the T-SFHGIA operator:

*P*1 = ⎛⎝ 3+(1 − 0.6<sup>3</sup>)<sup>5</sup>×0.18 − (1 − (0.7<sup>3</sup> + 0.5<sup>3</sup> + 0.6<sup>3</sup>))<sup>5</sup>×0.18 − (0.5<sup>3</sup>)<sup>5</sup>×0.18, 3+1 − (1 − 0.5<sup>3</sup>)<sup>5</sup>×0.18, 3+1 − (1 − 0.6<sup>3</sup>)<sup>5</sup>×0.18 ⎞⎠ = (0.6656, 0.5359, 0.5816) *P*2 = ⎛⎝ 3+(1 − 0.4<sup>3</sup>)<sup>5</sup>×0.22 − (1 − (0.9<sup>3</sup> + 0.5<sup>3</sup> + 0.4<sup>3</sup>))<sup>5</sup>×0.22 − (0.5<sup>3</sup>)<sup>5</sup>×0.22, 3+1 − (1 − 0.5<sup>3</sup>)<sup>5</sup>×0.22, 3+1 − (1 − 0.4<sup>3</sup>)<sup>5</sup>×0.22 ⎞⎠ = (0.9144, 0.4665, 0.4125) *P*3 = ⎛⎝ 3+(1 − 0.1<sup>3</sup>)<sup>5</sup>×0.16 − (1 − (0<sup>3</sup> + 03 + 0.1<sup>3</sup>))<sup>5</sup>×0.16 − (0<sup>3</sup>)<sup>5</sup>×0.16, 3+1 − (1 − 0<sup>3</sup>)<sup>5</sup>×0.16, 3+1 − (1 − 0.1<sup>3</sup>)<sup>5</sup>×0.16 ⎞⎠ = (0, 0, 0.0928) *P*4 = ⎛⎝ 3+(1 − 0.4<sup>3</sup>)<sup>5</sup>×0.21 − (1 − (0.5<sup>3</sup> + 0.3<sup>3</sup> + 0.4<sup>3</sup>))<sup>5</sup>×0.21 − (0.3<sup>3</sup>)<sup>5</sup>×0.21, 3+1 − (1 − 0.3<sup>3</sup>)<sup>5</sup>×0.21, 3+1 − (1 − 0.4<sup>3</sup>)<sup>5</sup>×0.21 ⎞⎠ = (0.5141, 0.2885, 0.4063) *P*5 = ⎛⎝ 3+(1 − 0.5<sup>3</sup>)<sup>5</sup>×0.23 − (1 − (0.6<sup>3</sup> + 0.4<sup>3</sup> + 0.5<sup>3</sup>))<sup>5</sup>×0.23 − (0.4<sup>3</sup>)<sup>5</sup>×0.23, 3+1 − (1 − 0.4<sup>3</sup>)<sup>5</sup>×0.23, 3+1 − (1 − 0.5<sup>3</sup>)<sup>5</sup>×0.23 ⎞⎠ (0.6422, 0.3486, 0.5221)

The score values of these numbers were obtained as *SC*(*<sup>P</sup>*1) = 0.0981, *SC*(*<sup>P</sup>*2) = 0.6943, *SC*(*<sup>P</sup>*3) = −0.0008, *SC*(*<sup>P</sup>*4) = 0.0688, *SC*(*<sup>P</sup>*5) = 0.1225, and, based on score values, we had the following arrangement:

$$\begin{aligned} P\_{\sigma(1)} &= (0.9144, 0.4665, 0.4125), \\ P\_{\sigma(2)} &= (0.6422, 0.3486, 0.5221), \\ P\_{\sigma(4)} &= (0.5141, 0.2885, 0.4063), P\_{\sigma(5)} = (0, 0, 0.0928) \end{aligned}$$

Now, by using the definition of the T-SFHGIA operator, we found

$$\text{T}-\text{SFHGIA}\_{\omega,\text{W}}(\text{P}\_1, \text{P}\_2, \text{P}\_3, \text{P}\_4, \text{P}\_5) = (0.8375, 0.4223, 0.4928) \tag{8}$$

Clearly, the aggregated value obtained in Equation (8) was an improvement of the one obtained in Equation (7), as it incorporated the zero values occurring in the membership and abstinence of T-SFNs efficiently. The analysis of Equations (7) and (8) proved the significance of proposed aggregation operators.

#### **4. MADM Approach Based on Proposed Operators**

Consider a decision-making problem which consists of a set of alternatives (*Y* = {*y*1, *y*2,......, *yl*}) and set of attributes (*Z* = ,*<sup>z</sup>*1, *z*2,......, *zq*/) associated with weighted vector (*w* = -*<sup>w</sup>*1, *w*2,......, *wq*.*<sup>T</sup>*), where *wk* ∈ (0, 1] and <sup>Σ</sup>*qk*=1*wk* = 1. Suppose every alternative (*yj*) is represented by T-SFNs (*Pjk* = *mjk*, *ijk*, *njk*), which show by which degree alternatives satisfy, neutral, and not satisfy the given attribute. Then, the following steps of the MADM approach, based on the proposed operators, are summarized as follows:

**Step 1** Find the value of *t* for which the information of the decision matrix lies in the T-spherical fuzzy environment.


$$P\_{\sigma(j1)\prime}P\_{\sigma(j2)\prime}\dots\dots\dots\prime P\_{\sigma(jk)\prime}$$


## **5. Numerical Example**

The above-mentioned approach has been illustrated with a real-life decision-making problem under the T-SFS environment, and obtained results have been compared with the other existing results.
