**2. Preliminaries**

In this section, we present some basic concepts related to IFS, PyFS, PFS, SFS, and T-SFS over the universal set X.

**Definition 1.** [2] *An IFS on X consists of membership and non-membership functions defined as*

$$P = \{ \langle \mathbf{x}, m(\mathbf{x}), n(\mathbf{x}) \rangle \mid \mathbf{x} \in X \}$$

*such that m*, *n* : *X* → [0, 1] *with a condition* 0 ≤ *m*(*x*) + *n*(*x*) ≤ 1 ∀ *x* ∈ *X Further, the degree of refusal of x in P is r*(*x*) = 1 − (*m*(*x*) + *n*(*x*)) *and the pair* (*<sup>m</sup>*, *n*) *is regarded as an IFN.*

**Definition 2.** [4] *A Pythagorean fuzzy set (PyFS) on X consists of membership and non-membership functions defined as*

$$P = \{ \langle \mathfrak{x}, m(\mathfrak{x}), n(\mathfrak{x}) \rangle \mid \mathfrak{x} \in X \}$$

*such that m*, *n* : *X* → [0, 1] *with a condition that* 0 ≤ *m*<sup>2</sup>(*x*) + *n*<sup>2</sup>(*x*) ≤ 1 ∀ *x* ∈ *X. Further, the degree of refusal of x in P is r*(*x*) = L1 − (*m*<sup>2</sup>(*x*) + *n*<sup>2</sup>(*x*)) *and the pair* (*<sup>m</sup>*, *n*) *is regarded as a Pythagorean fuzzy number (PyFN).*

**Definition 3.** [6] *A picture fuzzy set (PFS) on X consists of membership, abstinence, and non-membership functions defined as*

$$P = \{ \langle \mathbf{x}, m(\mathbf{x}), i(\mathbf{x}), n(\mathbf{x}) \rangle \mid \mathbf{x} \in X \}$$

*such that m*, *i*, *n* : *X* → [0, 1] *with a condition that* 0 ≤ *m*(*x*) + *<sup>i</sup>*(*x*) + *n*(*x*) ≤ 1 ∀ *x* ∈ *X Further, the degree of refusal of x in P is r*(*x*) = 1 − (*m*(*x*) + *<sup>i</sup>*(*x*) + *n*(*x*)) *and* (*<sup>m</sup>*, *i*, *n*) *is regarded as a picture fuzzy number (PFN).*

**Definition 4.** [34] *A spherical fuzzy set (SFS) on X consists of membership, abstinence, and non-membership functions defined as*

$$P = \{ \langle \mathbf{x}, m(\mathbf{x}), i(\mathbf{x}), n(\mathbf{x}) \rangle \mid \mathbf{x} \in X \}$$

*such that m*, *i*, *n* : *X* → [0, 1] *with a condition that* 0 ≤ *m*<sup>2</sup>(*x*) + *i* <sup>2</sup>(*x*) + *n*<sup>2</sup>(*x*) ≤ 1 ∀ *x* ∈ *X Further, the degree of refusal of x in P is r*(*x*) = L1 − (*m*<sup>2</sup>(*x*) + *<sup>i</sup>*<sup>2</sup>(*x*) + *n*<sup>2</sup>(*x*)) *and* (*<sup>m</sup>*, *i*, *n*) *is regarded as a spherical fuzzy number (SFN).*

**Definition 5.** [34] *A T-SFS on X consists of membership, abstinence, and non-membership functions defined as*

$$P = \{ \langle \mathbf{x}, m(\mathbf{x}), i(\mathbf{x}), n(\mathbf{x}) \rangle \mid \mathbf{x} \in X \}$$

*such that m*, *i*, *n* : *X* → [0, 1] *with a condition that* 0 ≤ *m<sup>t</sup>*(*x*) + *<sup>i</sup><sup>t</sup>*(*x*) + *n<sup>t</sup>*(*x*) ≤ 1 ∀ *x* ∈ *X t* = 1, 2, ... *k*. *Further, the degree of refusal of x in P is r*(*x*) = L*t* 1 − (*m<sup>t</sup>*(*x*) + *it*(*x*) + *n<sup>t</sup>*(*x*)) *and* (*<sup>m</sup>*, *i*, *n*) *is regarded as a T-spherical fuzzy number (T-SFN).*

**Definition 6.** [34] *Let P* = (*<sup>m</sup>*, *i*, *n*) *be a T-SFS. Then the score value of P is defined as*

$$\mathcal{SC}(P) = m^t - n^t$$

*and accuracy function is defined as*

$$AC(P) = m^t + i^t + n^t$$

*The one which has a greater score is the superior value. If the score of two T-SFNs is equal, then we rank them using the accuracy value, and a number is called superior if it has greater accuracy. If again accuracy values of two T-SFNs become equal, then both numbers are considered as similar.*

**Definition 7.** [39] *Let P* = (*mP*, *nP*) *and P* = (*mP* , *nP* ) *be two IFNs. Then the existing operational laws between them are defined as*


**Definition 8.** *For any collection of T-SFNs Pj* = 8*mj*, *ij*, *nj*9 (*j* = 1, 2, 3, . . . . . . , *k*)*, [34] defined the T-spherical fuzzy weighted geometric aggregation operator (T-SFWGA) as*

$$\mathbf{T}-\text{SFWGA}\_{\mathbf{W}}(\mathbf{P}\_{1},\mathbf{P}\_{2},\ldots,\mathbf{P}\_{\mathbf{k}})=\begin{pmatrix}\sqrt{\prod\_{j=1}^{\mathbf{k}}\left(\mathbf{m}\_{\mathbf{j}}^{\mathbf{t}}+\mathbf{i}\_{\mathbf{j}}^{\mathbf{t}}\right)^{\mathbf{w}\_{\mathbf{j}}^{\mathbf{t}}}-\prod\_{j=1}^{\mathbf{k}}\left(\mathbf{i}\_{\mathbf{j}}^{\mathbf{t}}\right)^{\mathbf{w}\_{\mathbf{j}}^{\mathbf{t}}}\prod\_{j=1}^{\mathbf{k}}\left(\mathbf{i}\_{\mathbf{j}}\right)^{\mathbf{w}\_{\mathbf{j}}}\\\sqrt{1-\prod\_{j=1}^{\mathbf{k}}\left(1-\mathbf{n}\_{\mathbf{j}}^{\mathbf{t}}\right)^{\mathbf{w}\_{\mathbf{j}}^{\mathbf{t}}}}\end{pmatrix}\tag{1}$$

*where w* = (*<sup>w</sup>*1, *w*2,...... *wk*)*<sup>T</sup> be the weighting vector of T-SFNs Pj with wj* ∈ (0, 1] *and* <sup>∑</sup>*kj*=<sup>1</sup> *wj* = 1 *and t* = 1, 2, . . . . . . *k*.

#### **3. Proposed Operational Laws and Aggregation Operators**

This section is divided into two subsections. One presents the improved operations laws for the T-SFSs, while other presents some improved geometric AOs under the T-SFS environment.

#### *3.1. Improved Operational Laws*

In this section, we present some new, improved operations laws by incorporating the features of the degree of refusal into the analysis.

**Definition 9.** *Let P*1 = -*mP*1 , *iP*1 , *nP*1 . *and P*2 = -*mP*2 , *iP*2 , *nP*2 . *be two T-SFNs. Then, the proposed operational laws are defined as*

$$(1)\quad P\_1 \otimes P\_2 = \left(\sqrt[4]{\frac{(1-n\_{P\_1}^t)(1-n\_{P\_2}^t) - (1-m\_{P\_1}^t-i\_{P\_1}^t-n\_{P\_1}^t)(1-m\_{P\_2}^t-i\_{P\_2}^t-n\_{P\_2}^t) - i\_{P\_1}^t i\_{P\_2}^t}{\sqrt[4]{1-(1-i\_{P\_1}^t)(1-n\_{P\_2}^t)}\sqrt[4]{1-(1-n\_{P\_1}^t)(1-n\_{P\_2}^t)}}}\right)$$

$$(2)\quad P^{\lambda} = \left(\sqrt[t]{\left(1 - n\_P^t\right)^{\lambda} - \left(1 - m\_P^t - i\_P^t - n\_P^t\right)^{\lambda} - i\_P^{t\lambda}}, \sqrt[t]{1 - \left(1 - i\_P^t\right)^{\lambda}}, \sqrt[t]{1 - \left(1 - n\_P^t\right)^{\lambda}}\right)$$

For two T-SFNs, *P*1 = -*mP*1 , *iP*1 , *nP*1 . and *P*2 = -*mP*2 , *iP*2 , *nP*2 ., new operations of multiplication can be construed from four aspects, such as between:


These multiplication rules are of the form:

1. *<sup>E</sup>*-*nP*1 , *nP*2 . = *nP*1 .*nP*2 . Therefore, *nP*1⊗*P*2 = *<sup>t</sup>*+-*nPt*1 + *nPt*2 − *nPt*1*nPt*2. is considered as a probability non-membership (PN) function operator, that is,

$$PN(n\_{P\_1\prime}, n\_{P\_2}) = \sqrt[t]{n\_{P\_1}^{t} + n\_{P\_2}^{t} - n\_{P\_1}^{t}n\_{P\_2}^{t}}$$

2. *<sup>E</sup>*-*mP*1,*mP*2. = -*mP*1 <sup>+</sup>*iP*1..-*mP*1 <sup>+</sup>*iP*1.. Therefore, *mP*1⊗*P*2 = *t* 71− 1− *mtP*1 <sup>+</sup>*<sup>i</sup>tP*1<sup>1</sup><sup>−</sup> *mtP*2 <sup>+</sup>*<sup>i</sup>tP*2 is considered as a probability membership (PM) function operator, that is,

$$PM(m\_{P\_1}, m\_{P\_2}) = \sqrt[4]{1 - \left(1 - m\_{P\_1}^t - i\_{P\_1}^t\right)\left(1 - m\_{P\_1}^t - i\_{P\_1}^t\right)}$$

3. *<sup>I</sup>*-*nP*1 , *mP*2 . = *<sup>t</sup>*+-*mPt*2 + *iP<sup>t</sup>*2.*nPt*1.*<sup>I</sup>*-*nP*1 , *mP*2 . is considered as a probability heterogeneous (PH) function operator, that is,

$$PH(n\_{P\_1}, m\_{P\_2}) = \sqrt[t]{m\_P t\_2 n\_{P\_1} t + i\_P t\_2 n\_{P\_1} t}$$

4. *<sup>I</sup>*-*iP*1 , *iP*2 . = *iP*1 .*iP*2 . Therefore, *iP*1⊗*P*2 = *<sup>t</sup>*+-*iP<sup>t</sup>*1 + *iPt*2 − *iP<sup>t</sup>*1*iP<sup>t</sup>*2..*iP*1⊗*P*2 is considered as a probability neutral (PNe) function operator, that is,

$$P\mathcal{N}e\left(i\_{P\_1\prime},i\_{P\_2}\right) = \sqrt[t]{i\_{P\_1}t + i\_{P\_2}t - i\_{P\_1}i\_{P\_2}t}$$

From the proposed laws, it is observed that the several existing laws can be considered as a special case of it. For instance,


Further, it is observed that for the above defined PN, PH satisfies the following properties:

**Theorem 1.** *Let P* = *mP*, *iP*, *nP*, *Q* = 8*mQ*, *iQ*, *nQ*9, *R* = *mR*, *iR*, *nR and D* = *mD*, *iD*, *nD be four T-SFNs. Then, we have:*

