## 9 Conclusions  
In this work, we define a new notion, named bipolar pqr-spherical fuzzy sets (BpqrSFS), which is a combination of bipolar fuzzy sets and pqr-spherical fuzzy sets. Then, we investigate some basic properties of BpqrSFS, including the basic operational laws, complement operation, union operation, and intersection operation. This new framework allows for a more nuanced representation of uncertainty by accounting for different degrees of membership, non-membership, and neutrality, in addition to bipolarity. This adaptability of bipolar pqr-SFS has proven to be immensely valuable in intricate decision-making processes that involve evaluating both affirmative and adverse perceptions. Its utility in multi-criteria group decision-making problems has been demonstrated, and it has been shown that this framework is well-suited for capturing the complexities of bipolar judgmental reasoning during the decision-making process. This has been conducted by using two algorithms--the first one is referred to as the 'technique for order of preference by similarity to ideal solution' (TOPSIS) and the second one involves using the proposed operators themselves in the BpqrSF context, as well as providing a numerical example, which demonstrates the usefulness of the proposed set. Targeted applications were developed in the multi-criteria decision-making field to address problems in the BpqrSF environment using the two aforementioned aggregation operators. Additionally, a comparative study was used to show the applicability and viability of the proposed approach. Future works will be applied with more decision-making methods, like VIKOR. In addition, a few other aggregation operators can be introduced to enhance the existing approaches owing to the generation of a more precise ranking outcome. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Method** & **Parameter(s)** & **Ranking Results** & **Optimal Alternative** \\ \hline \multirow{5}{*}{pqr-SFWAO [12]} & \(p=3,q=4,r=5\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{3}\succ\tau_{1}\) & \(\tau_{4}\) \\  & \(p=5,q=4,r=3\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{3}\succ\tau_{1}\) & \(\tau_{4}\) \\  & \(p=2,q=4,r=6\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{3}\succ\tau_{1}\) & \(\tau_{4}\) \\  & \(p=3,q=5,r=7\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{3}\succ\tau_{1}\) & \(\tau_{4}\) \\  & \(p=7,q=5,r=3\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{1}\succ\tau_{3}\) & \(\tau_{4}\) \\ \hline \multirow{5}{*}{B\({}_{\text{q}}\)-ROFWAO [20]} & \(q=1\) & \(\tilde{\tau}_{4}\succ\tilde{\tau}_{1}\succ\tilde{\tau}_{2}\succ\tilde{\tau}_{3}\) & \(\tilde{\tau}_{4}\) \\  & \(q=2\) & \(\tau_{4}\succ\tau_{1}\succ\tau_{2}\succ\tau_{3}\) & \(\tau_{4}\) \\  & \(q=3\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{1}\succ\tau_{3}\) & \(\tau_{4}\) \\  & \(q=5\) & \(\tilde{\tau}_{4}\succ\tilde{\tau}_{2}\succ\tilde{\tau}_{1}\succ\tilde{\tau}_{3}\) & \(\tilde{\tau}_{4}\) \\  & \(q=15\) & \(\tau_{4}\succ\tau_{2}\succ\tau_{3}\succ\tilde{\  
**Author Contributions:** Conceptualization, Z.A.A. and H.F.M.S. ; methodology, Z.A.A., H.F.M.S. and B.A.A. ; formal analysis, Z.A.A., H.F.M.S., A.I.A. and R.A.M. ; investigation, H.F.M.S., A.I.A., R.A.M. and B.A.A. ; writing--original draft preparation, Z.A.A. and H.F.M.S. ; writing--review and editing, Z.A.A., H.F.M.S., A.I.A. and R.A.M. ; funding acquisition, A.I.A. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. **Acknowledgments:** The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for the financial support (QU-APC-2025). **Conflicts of Interest:** The authors declare no conflicts of interest. **Appendix A. Proof**  
**Proof of proposition 1.