## 2 Preliminaries  
In this section, we present basic knowledge about the 2-tuple linguistic term set, spherical fuzzy set, and 2-tuple spherical fuzzy linguistic set. **Definition 1** ([35]). : _Let \(\mathcal{S}=(\xi_{1},...,\kappa)\) be the collection of linguistic term, and \(\kappa\) denotes the odd cardinality, such that \(\xi_{\kappa}\), and \(\kappa\) represent the possible value of the linguistic variable and positive integer, respectively. Generally, \(\kappa\) is taken as \(3,5\), etc. For example, when \(\kappa=5\), then the linguistic term set \(\mathcal{S}\) is defined as \(\{\xi_{1}=\text{poor},\,\xi_{2}=\text{slightly poor},\,\xi_{3}=\text{fair},\,\xi_{4}= \text{slightly good}\,\,\xi_{5}=\text{good}\}\)._  
_If \(\xi_{\kappa},\xi_{t}\in\mathcal{S}\), then the following characteristic must be satisfied;_  
\((1)\) _The ordered of set as:_ \(\xi_{\kappa}\prec\xi_{t},\Leftrightarrow\kappa\prec t\)_;_ \((2)\) _The operator of negation as:_ \(\text{Neg}\,\left(\xi_{\kappa}\right)=\xi_{\kappa-\kappa}\)_;_ \((3)\) _Maximum_ \(\left(\xi_{\kappa},\xi_{t}\right)=\xi_{\kappa}\)_, iff_ \(\xi_{\kappa}\geq\xi_{t}\)_;_ \((4)\) _Minimum_ \(\left(\xi_{\kappa},\xi_{t}\right)=\xi_{\kappa}\)_, iff_ \(\xi_{\kappa}\leq\xi_{t}\)_._  
_Using the concept of symbolic translation, [PERSON] and [PERSON] [50; 51] introduced the 2-tuple fuzzy linguistic model. This model is used to denote the linguistic assessment information by means of a 2-tuple \((\xi_{t},\bar{a}_{t})\) and \(\xi_{t}\) and \(\bar{a}_{t}\) denote the linguistic label and symbolic translation respectively, from the linguistic term set \(\mathcal{S}\) and \(\bar{a}\in[-0.5,0.5]\)._  
_(3) Intersection_  
\[\Re_{1}\cap\Re_{2}=\left( \begin{array}{c}(r,\min\left(\mu_{\Re_{1}}^{2}(r),\mu_ {\Re_{2}}^{2}(r)\right),\ \min\left(\eta_{\Re_{1}}^{2}(r),\eta_{\Re_{2}}^{2}(r)\right),\\ \\ \max\left(\
u_{\Re_{1}}^{2}(r),\
u_{\Re_{2}}^{2}(r)\right)|r\in\mathbb{R}\end{array}  \right);\]  
(4) _Compliment_  
\[\Re_{1}^{c}=\left\{(r,\
u_{\Re_{1}}^{2}(r),\eta_{\Re_{1}}^{2}(r),\mu_{\Re_{1}}^ {2}(r)|r\in\mathbb{R}\right\}.\]  
**Definition 7** ([46]). : _Let \(\Re_{1}=\left\langle\mu_{\Re_{1}},\eta_{\Re_{1}},\
u_{\Re_{1}}\right\rangle\) and \(\Re_{2}=\left\langle\mu_{\Re_{2}},\eta_{\Re_{2}},\
u_{\Re_{2}}\right\rangle\) be two SFNs defined on the universe of discourse \(\mathbb{R}\
eq\alpha\), some operations on SFNs are defined as follows with \(\lambda\geq 0\)._  
\[ \begin{array}{l}1.\quad\Re_{1}\oplus\Re_{2}=\left\{\sqrt{\mu_{\Re_{1}}^{2}+ \mu_{\Re_{2}}^{2}-\mu_{\Re_{1}}^{2}\mu_{\Re_{2}}^{2}},\,\eta_{\Re_{1}}\eta_{ \Re_{2}},\,\
u_{\Re_{1}}\cdot\eta_{\Re_{2}}\right\};\\ 2.\quad\Re_{1}\otimes\Re_{2}=\left\{\,\mu_{\Re_{1}}\cdot\mu_{\Re_{2}}\,,\sqrt{ \mu_{\Re_{1}}^{2}+\mu_{\Re_{2}}^{2}-\mu_{\Re_{1}}^{2}\cdot\eta_{\Re_{2}}^{2}} \,,\sqrt{\
u_{\Re_{1}}^{2}+\
u_{\Re_{2}}^{2}-\
u_{\Re_{1}}^{2}\cdot\
u_{\Re_{2} }^{2}}\right\};\\ 3.\quad\lambda\otimes\Re_{1}=\left\{\sqrt{1-(1-\mu_{\Re_{1}}^{2})^{\lambda}}, \,\left(\eta_{\Re_{1}}\right)^{\lambda},\,\left(\eta_{\Re_{1}}\right)^{\lambda} \right\};\\ 4.\quad\Re_{1}^{\lambda}=\left\{(\mu_{\Re_{1}})^{\lambda},\sqrt{1-(1-\eta_{ \Re_{1}}^{2})^{\lambda}},\sqrt{1-(1-v_{\Re_{1}}^{2})^{\lambda}}\right\};\\ 5.\quad\Re_{1}^{c}=\left\langle\
u_{\Re_{1}},\eta_{\Re_{1}},\mu_{\Re_{1}} \right\rangle;\end{array} \]  
**3. 2-Tuple Spherical Fuzzy Linguistic Sets**  
In the following, we introduced the concept of 2-tuple spherical fuzzy linguistic sets and their basic operations based on the spherical fuzzy set and 2-tuple linguistic information. **Definition 8**. : _A 2-tuple spherical fuzzy linguistic set \(\Re\) in \(\mathbb{R}\
eq\phi\), is defined as;_  
\[\Re=\left\{\left\langle\hat{s}_{\mu(r)},\hat{s}_{\eta(r)},\hat{s}_{\
u(r)}\,| \,\,r\in\mathbb{R}\right\rangle\right\}. \tag{8}\]  
_The numbers \(\hat{s}_{\mu(r)},\hat{s}_{\eta(r)}\) and \(\hat{s}_{\
u(r)}\) are denoted the positive, neutral, and negative membership grades of the number \(r\) to linguistic variable \(\Re\).