## 2 Preliminaries
Where \(\hat{s}_{\mu(r)},\hat{s}_{\eta(r)},\hat{s}_{\
u(r)}\in\hat{S}\), with the condition \(3\leq\mu^{2}(r)+\eta^{2}(r)+\
u^{2}(r)\leq\left(\kappa+1\right)^{2},\,\forall \,\,r\in\mathbb{R}\). The term \(\hat{s}_{\pi(r)}\) is called the refusal grade of the element \(r\) to the set \(\Re\), and defined as \(\hat{s}_{\pi(r)}=\delta\sqrt{\frac{(\kappa+1)^{2}-\mu^{2}-\mu^{2}-\mu^{2}}{2}}\)._  
_For convenience, we say \(\widetilde{\Re}=\left\langle\left(\hat{s}_{\mu(r)},\hat{u}\right),\left(\hat{s }_{\eta(r)},\hat{e}\right),\left(\hat{s}_{\
u(r)},\hat{u}\right)\right\rangle\), a 2-tuple spherical fuzzy linguistic number (2 TSFLN), where \(\hat{s}_{\mu(r)},\hat{s}_{\eta(r)},\hat{s}_{\
u(r)}\in\hat{S}\),_  
\(3\leq\Xi^{-1}\left(\hat{s}_{\mu^{2}(r)},\hat{u}\right)+\Xi^{-1}\left(\hat{s}_{ \eta^{2}(r)},\hat{e}\right)+\Xi^{-1}\left(\hat{s}_{\
u^{2}(r)},\hat{u}\right) \leq\left(\kappa+1\right)^{2},\) _and \(\hat{u},\hat{e},\hat{u}\in[-0.5,0.5).\)_  
**Definition 9**. : _Let \(\alpha,\beta,\gamma\) be the results of the aggregation of the indices of a set of labels assessed in a linguistic term set \(\hat{S}\), for example, the result of a symbolic aggregation operation, \(\alpha,\beta,\gamma\in[1,\kappa],3\leq\alpha+\beta+\gamma\leq\kappa+1,\) (where \(\kappa\) be the cardinality of \(\hat{S}\)). Assume that \(\mu=round(\alpha),\eta=round(\beta),\
u=round(\gamma)\) and \(\tilde{a}=\alpha-\mu,\tilde{e}=\beta-\eta,\tilde{u}=\gamma-\
u\) are the six values, such that \(\alpha,\beta,\gamma\in[1,\kappa]\), and \(\tilde{a},\tilde{e},\tilde{u}\in[-0.5,0.5)\), then \(\tilde{a},\tilde{e},\tilde{u}\) are called symbolic translations._  
**Definition 10**. : _Let \(\hat{S}=(\hat{s}_{1},...,\hat{s}_{\kappa})\) be the linguistic set and \(\alpha,\beta,\gamma\in[1,\kappa]\) be the three number value, denoting the aggregation result of linguistic symbolic. Then, a mapping \(\Xi\) is utilized to get the 2-tuple linguistic information equivalent to \(\alpha,\beta,\gamma\) which is defined as:_  
\[\Xi:[1,\kappa]\rightarrow\hat{S}\times[-0.5,0.5), \tag{9}\]  
\[\Xi(\alpha)=\left\{ \begin{array}{c}\hat{s}_{\mu},\mu=round(\alpha)\\ \tilde{u}=\alpha-\mu,\tilde{u}\in[-0.5,0.5),\end{array} \right. \tag{10}\]\[\Xi(\beta)=\left\{ \begin{array}{c}\xi_{\eta},\eta=round(\beta)\\ \bar{e}=\beta-\eta,\bar{e}\in[-0.5,0.5),\end{array} \right. \tag{11}\]  
\[\Xi(\bar{u})=\left\{ \begin{array}{c}\xi_{\
u},\
u=round(\gamma)\\ \bar{u}=\gamma-\
u,\bar{u}\in[-0.5,0.5),\end{array} \right. \tag{12}\]  
where round \((. ),\xi_{\mu},\xi_{\eta},\xi_{\
u}\) and \(\bar{u},\bar{e},\bar{u}\) denoted the usual round operation, closest index label to \(\alpha,\beta,\gamma\) and the value of the symbolic translation, respectively. **Definition 11**. : _Let \(\hat{S}=(\xi_{1},...,\xi_{\kappa})\) be the finite linguistic term set and \(\widetilde{\Re}=\left\langle\left(\xi_{\mu},\bar{u}\right),\left(\xi_{\eta}, \bar{e}\right),\left(\xi_{\
u},\bar{u}\right)\right\rangle\) be a 2-tuple spherical fuzzy linguistic number (2 TPFLN). Then, there exists a mapping \(\Xi^{-1}\), such that from a 2-tuple spherical fuzzy numbers \(\left\langle\left(\xi_{\mu},\bar{u}\right),\left(\xi_{\eta},\bar{e}\right), \left(\xi_{\
u},\bar{u}\right)\right\rangle\) it returns its equivalent numerical value \(\alpha,\beta,\gamma\in[1,\kappa]\subset R\), which is_  
\[\Xi^{-1}:\hat{S}\times[-0.5,0.5)\rightarrow[1,\kappa], \tag{13}\]  
\[\Xi^{-1}(\hat{s}_{\mu},\bar{u})=\mu+\bar{a}=\alpha, \tag{14}\]  
\[\Xi^{-1}(\hat{s}_{\eta},\bar{e})=\eta+\bar{e}=\beta, \tag{15}\]  
\[\Xi^{-1}(\xi_{\
u},\bar{u})=\
u+\bar{u}=\gamma. \tag{16}\]  
From Definitions 2 and 3, we observe that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation:  
\[\Xi\left\langle\xi_{\mu(r)},\xi_{\eta(r)},\xi_{\
u(r)}\right\rangle=\left\langle \left(\xi_{\mu(r)},0\right),\left(\xi_{\eta(r)},0\right),\left(\xi_{\
u(r)},0 \right)\right\rangle. \tag{17}\]  
**Definition 12**.