## 3 The P2 TLWBM Operator and P2 TLWGBM Operator
### P2 TLWGBM Operator  
Thereafter, we extend WGBM to P2 TLN and introduced the Pythagorean 2-tuple linguistic WGBM (P2 TLWGBM) operator. **Definition 7**. : _Let \(t,r>0\) and \(p_{i}=((s_{i},\rho_{i}),\mu_{i},v_{i})(i=1,2,\cdots,n)\) be a set of P2 TLNs with weights \(\omega=(\omega_{1},\omega_{2},\cdots,\omega_{n})^{T},\omega_{i}\in[0,1]\), and \(\sum\limits_{i=1}^{n}\omega_{i}=1\). If_  
\[\text{P2 TLWGBM}_{\omega}^{t,r}(p_{1},p_{2},\cdots,p_{n})=\frac{1}{t+r}\sum \limits_{i,j=1}^{n}\big{(}tp_{i}\oplus rp_{j}\big{)}^{\omega_{i}\omega_{j}}, \tag{35}\]_then the P2 TLWGBM operator has four properties._  
**Property 5**. : _Let \(t,r>0\) and \(p_{i}=((s_{i},\rho_{i}),\mu_{i},v_{i})(i=1,2,\cdots,n)\) be a set of P2 TLNs. The aggregated result of P2 TLWGBM is a P2 TLN._  
\[ \begin{array}{l}\text{P2 TLWGBM}_{\omega}^{t,r}(p_{1},p_{2},\cdots p_{n})\\ =\frac{1}{t+r}\underset{i,j=1}{\overset{n}{\otimes}}\left(tp_{i}\oplus rp_{j} \right)^{\omega_{i}\omega_{j}}\\ =\left(\Delta\left(\frac{1}{t+r},\underset{i,j=1}{\overset{n}{\prod}}\left(t \Delta^{-1}(S_{i},\rho_{i})+r\Delta^{-1}\big{(}S_{j},\rho_{j}\big{)}\right)^{ \omega_{i}\omega_{j}}\right),\\ \left(\left(\sqrt{1-\left(1-\underset{i,j=1}{\overset{n}{\prod}}\left(1-\mu_{ i}^{2}\right)^{t}\left(1-\mu_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}}} \right)^{1/(t+r)}\right.\\ \left.\left(\sqrt{1-\underset{i,j=1}{\overset{n}{\prod}}\left(1-v_{i}^{2t}v_{ j}^{2r}\right)^{\omega_{i}\omega_{j}}}\right)^{1/(t+r)}\right)\right)\end{array}  \tag{36}\]  
**Proof. ** Through Definition 3, we can obtain  
\[tp_{i}=\bigg{(}\Delta\Big{(}t\Delta^{-1}(S_{i},\rho_{i})\Big{)},\bigg{(}\sqrt {1-\left(1-\mu_{i}^{2}\right)^{t}},v_{i}^{t}\bigg{)}\bigg{)}, \tag{37}\]  
\[rp_{j}=\bigg{(}\Delta\Big{(}r\Delta^{-1}\big{(}S_{j},\rho_{j}\big{)}\Big{)}, \bigg{(}\sqrt{1-\left(1-\mu_{j}^{2}\right)^{r}},v_{i}^{r}\bigg{)}\bigg{)}. \tag{38}\]  
Thereafter,  
\[ \begin{array}{l}\left(tp_{i}\oplus rp_{j}\right)^{\omega_{i}\omega_{j}}\\ =\bigg{(}\Delta\Big{(}\left(t\Delta^{-1}(S_{i},\rho_{i})+r\Delta^{-1}\big{(}S_ {j},\rho_{j}\big{)}\right)^{\omega_{i}\omega_{j}}\Big{)},\\ \bigg{(}\bigg{(}\sqrt{1-\left(1-\mu_{i}^{2}\right)^{t}\left(1-\mu_{j}^{2} \right)^{r}}\bigg{)}^{\omega_{i}\omega_{j}}\bigg{)},\sqrt{1-\left(1-v_{i}^{2t} v_{j}^{2r}\right)^{\omega_{i}\omega_{j}}}\bigg{)}\bigg{)}\end{array}  \tag{39}\]  
Therefore,  
\[ \begin{array}{l}\underset{i,j=1}{\overset{n}{\otimes}}\left(tp_{i}\oplus rp _{j}\right)^{\omega_{i}\omega_{j}}\\ =\bigg{(}\Delta\bigg{(}\overset{n}{\underset{i,j=1}{\overset{n}{\prod}}\left( t\Delta^{-1}(S_{i},\rho_{i})+r\Delta^{-1}\big{(}S_{j},\rho_{j}\big{)}\right)^{ \omega_{i}\omega_{j}}}\bigg{)},\\ \bigg{(}\bigg{(}\sqrt{\underset{i,j=1}{\overset{n}{\prod}}\left(1-\mu_{i}^{2} \right)^{t}\left(1-\mu_{j}^{2}\right)^{r}}\bigg{)}^{\omega_{i}\omega_{j}}, \sqrt{1-\underset{i,j=1}{\overset{n}{\prod}}\left(1-v_{i}^{2t}v_{j}^{2r} \right)^{\omega_{i}\omega_{j}}}\bigg{)}\bigg{)}\end{array}  \tag{41}\]Thus,  
\[ \begin{array}{ll}&\text{P2 TLWGBM}_{\omega}^{tr}(p_{1},p_{2},\cdots p_{n})\\ &=\frac{1}{t+r}\mathop{\otimes}\limits_{i,j=1}^{n}\left(tp_{i}\oplus rp_{j} \right)^{\omega_{i}\omega_{j}}\\ &=\left(\Delta\left(\frac{1}{t+r}\mathop{\prod}\limits_{i,j=1}^{n}\left(t \Delta^{-1}(S_{i},\rho_{i})+r\Delta^{-1}\big{(}S_{j},\rho_{j}\big{)}\right)^{ \omega_{i}\omega_{j}}\right),\\ &\left(\sqrt{1-\left(1-\mathop{\prod}\limits_{i,j=1}^{n}\left(1-\left(1-\mu_{i }^{2}\right)^{t}\left(1-\mu_{j}^{2}\right)^{r}\right)^{\omega_{i}\omega_{j}} \right)^{1/\left(t+r\right)}}\right.\\ &\left(\sqrt{1-\mathop{\prod}\limits_{i,j=1}^{n}\left(1-v_{i}^{2t}v_{j}^{2r} \right)^{\omega_{i}\omega_{j}}}\right)^{1/\left(t+r\right)}\right)\end{array}  \tag{42}\]  
Hence, (36) is maintained. Thereafter,  
\[\left(t\Delta^{-1}(S_{i},\rho_{i})+r\Delta^{-1}\big{(}S_{j},\rho_{j}\big{)} \right)^{\omega_{i}\omega_{j}}\leq\left((t+r)\Delta^{-1}(S_{\max},\rho_{\max })\right)^{\omega_{j}\omega_{j}}, \tag{43}\]  
\[