## 3 Bipolar pqr-Spherical Fuzzy Sets
_Fermatean fuzzy set (FFS) [26] if \(p=q=3\) and \(\eta_{\tau}^{+}(\rho)=\theta_{\tau}^{-}(\rho)=\eta_{\tau}^{-}(\rho)=\lambda_{ \tau}^{-}(\rho))=0\)._
15. _Pythagorean fuzzy set (PyFS) [3] if \(p=q=2\) and \(\eta_{\tau}^{+}(\rho)=\theta_{\tau}^{-}(\rho)=\eta_{\tau}^{-}(\rho)=\lambda_{ \tau}^{-}(\rho))=0\)._
16. _intuitionistic fuzzy set (IFS) [2] if \(p=q=1\) and \(\eta_{\tau}^{+}(\rho)=\theta_{\tau}^{-}(\rho)=\eta_{\tau}^{-}(\rho)=\lambda_{ \tau}^{-}(\rho))=0\)._
17. _fuzzy set (FS) [1] if \(p=1\) and \(\eta_{\tau}^{+}(\rho)=\lambda_{\tau}^{+}(\rho))=\theta_{\tau}^{-}(\rho)=\eta_{ \tau}^{-}(\rho)=\lambda_{\tau}^{-}(\rho))=0\)._  
Next, Table 1 highlights the exceptional performance of our novel model compared to existing models, excelling in membership degree (MD), neutral degree (NMD), non-membership degree (\(\
eg\)MD), offering greater flexibility in parameter selection, and effectively capturing bipolarity. Now, we present the fundamental operations on B\({}_{\text{pqr}}\)SFNs. Definition 8: Let \(\bar{\tau}=(\theta_{\bar{\tau}}^{+},\eta_{\bar{\tau}}^{+},\lambda_{\bar{\tau} }^{+},\theta_{\bar{\tau}}^{-},\eta_{\bar{\tau}}^{-},\lambda_{\bar{\tau}}^{-})\) and \(\bar{\ell}=(\theta_{\ell}^{+},\eta_{\bar{\ell}}^{+},\lambda_{\bar{\ell}}^{+}, \theta_{\bar{\ell}}^{-},\eta_{\bar{\ell}}^{-},\lambda_{\bar{\ell}}^{-})\) be two B\({}_{pqr}\)SFNs. Then, we have the following:  
1. \(\bar{\tau}\bar{\subseteq}\bar{\ell}\), if \(\theta_{\tau}^{+}\leq\theta_{\ell}^{+}\), \(\eta_{\tau}^{+}\leq\eta_{\bar{\tau}}^{+}\), \(\lambda_{\bar{\tau}}^{+}\geq\lambda_{\bar{\ell}}^{+}\), \(\theta_{\tau}^{-}\geq\theta_{\bar{\tau}}^{-}\), \(\eta_{\tau}^{-}\geq\eta_{\bar{\ell}}^{-}\), and \(\lambda_{\bar{\tau}}^{-}\leq\lambda_{\bar{\ell}}^{-}\). 2. \(\bar{\tau}=\bar{\ell}\) if \(\bar{\tau}\bar{\subseteq}\bar{\ell}\) and \(\bar{\ell}\bar{\subseteq}\bar{\tau}\). 3. \(\bar{\tau}^{\epsilon}=(\lambda_{\bar{\tau}}^{+},\eta_{\bar{\tau}}^{+},\lambda_{ \bar{\tau}}^{-},\eta_{\bar{\tau}}^{-},\theta_{\bar{\tau}}^{-})\), where \(\bar{\tau}^{\epsilon}\) means the complement of \(\bar{\tau}\). \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Model** & **MD** & **NMD** & **\(\
eg\)MD** & **FSDP \({}^{*}\)** & **Bipolarity** \\ \hline FS [1] & Yes & No & No & No & No \\ IFS [2] & Yes & No & Yes & No & No \\ PyFS [3] & Yes & No & Yes & No & No \\ FFS [26] & Yes & No & Yes & No & No \\ q-ROFS [4] & Yes & No & Yes & No & No \\ mn-ROFS [7] & Yes & No & Yes & Yes & No \\ PFS [8] & Yes & Yes & Yes & No & No \\ SFS [9;25] & Yes & Yes & Yes & No & No \\ T-SFS [9] & Yes & Yes & Yes & No & No \\ pqr-SFS [12;13] & Yes & Yes & Yes & Yes & No \\ BIFS [17] & Yes & No & Yes & No & Yes \\ BPyFS [18] & Yes & No & Yes & No & Yes \\ FBFS [19] & Yes & No & Yes & No & Yes \\ Bmn-ROFS [21] & Yes & No & Yes & Yes & Yes \\ BPFS [24] & Yes & Yes & Yes & No & Yes \\ SBFS [22] & Yes & Yes & Yes & No & Yes \\ BT-SFS [23] & Yes & Yes & Yes & No & Yes \\ Proposed & Yes & Yes & Yes & Yes & Yes \\ \hline \hline \end{tabular}  
* FSDP means Flexibility in Selecting Different Parameters.  
\end{table} 
Table 1: Superiority of the proposed model compared to existing ones. \[4.\quad\bar{\tau}\bar{\cap}\bar{\ell}=\left( \begin{array}{c}\min(\theta_{\tau}^{+}, \theta_{\bar{\ell}}^{+}),\min(\eta_{\bar{\tau}}^{+},\eta_{\bar{\ell}}^{+}),\max( \lambda_{\bar{\tau}}^{+},\lambda_{\bar{\ell}}^{+}),\\ \max(\theta_{\tau}^{-},\theta_{\bar{\ell}}^{-}),\max(\eta_{\tau}^{-},\eta_{ \bar{\ell}}^{-}),\min(\lambda_{\tau}^{-},\lambda_{\bar{\ell}}^{-})\\ \max(\theta_{\tau}^{+},\theta_{\bar{\ell}}^{+}),\min(\eta_{\tau}^{+},\eta_{ \bar{\ell}}^{+}),\min(\lambda_{\tau}^{+},\lambda_{\bar{\ell}}^{+}),\\ \min(\theta_{\tau}^{-},\theta_{\bar{\ell}}^{-}),\max(\eta_{\tau}^{-},\eta_{ \bar{\ell}}^{-}),\max(\lambda_{\tau}^{-},\lambda_{\bar{\ell}}^{-})\\ \end{array} \right).\]  
**Proposition 1**.