## 5 Conclusions  
In this study, we expanded the realm of multi-criteria decision making under uncertainty by introducing an innovative application of the VIKOR method, which incorporates Pythagorean hesitant fuzzy sets. This enhanced method was specifically tailored for a holistic safety assessment of quayside container cranes in port environments, effectively gauging the safety risk levels of such critical equipment. Our research systematically \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\hat{k}_{i}^{+}\) & \(\hat{k}_{i}^{-}\) \\ \hline Column 1 & \(\langle\)(0.98,0.98,0.98\(\}\), \(\{0.05,0.05,0.05\}\)) & \(\langle\)(0.05,0.05,0.05\}\), \(\{0.98,0.98,0.98\(\}\)) \\ Column 2 & \(\langle\)(0.98,0.98,0.98\(\}\), \(\{0.05,0.05,0.05\}\)) & \(\langle\)(0.05,0.05,0.05\}\), \(\{0.98,0.98,0.98\(\}\)) \\ Column 3 & \(\langle\)(0.98,0.98,0.98\(\}\), \(\{0.05,0.05,0.05\}\)) & \(\langle\)(0.05,0.05,0.05\}\), \(\{0.98,0.98,0.98\(\}\)) \\ Column 4 & \(\langle\)(0.98,0.98,0.98\(\}\), \(\{0.05,0.05,0.05\}\)) & \(\langle\)(0.05,0.05,0.05\}\), \(\{0.98,0.98,0.98\}\) \\ Column 5 & \(\langle\)(0.98,0.98,0.98\(\}\), \(\{0.05,0.05,0.05\}\)) & \(\langle\)(0.05,0.05,0.05\}\), \(\{0.98,0.98,0.98\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Calculation results of \(\hat{k}_{i}^{+}\) and \(\hat{k}_{i}^{-}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Evaluation Criteria** & **Evaluation Results** & **Evaluation Criteria** & **Evaluation Results** \\ \hline \(C_{1}\) & 0.0068 & \(E_{1}\) & 0.0808 \\ \(C_{2}\) & 0.0180 & \(E_{2}\) & 0.0449 \\ \(C_{3}\) & 0.0031 & \(E_{3}\) & 0.0299 \\ \(C_{4}\) & 0.0052 & \(E_{4}\) & 0.0449 \\ \(C_{5}\) & 0.0052 & \(E_{5}\) & 0.0180 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: _Cont._deconstructed the quayside container crane into five distinct safety evaluation subsystems, pinpointing a suite of pertinent safety risk indicators for each. To accurately weigh these indicators, the Best-Worst Method (BWM) was employed, thereby providing a nuanced understanding of their collective impact on the overarching safety of quayside container crane operations. Given the inherent qualitative nature and the accompanying challenges in quantifying many of the safety risk indicators for quayside container cranes, coupled with the limitations of expert evaluations in yielding precise numerical scores, the proposed Pythagorean hesitant fuzzy set-based VIKOR method emerged as a robust solution. It adeptly bridges the gap between qualitative judgment and quantitative analysis, offering a comprehensive tool for risk level assessment. By applying this method to a real-world scenario, specifically an operational quayside container crane at a port in China, we delved into the practicality of this technique within the project risk evaluation sphere. The case study not only illustrated the approach's applicability but also substantiated its viability, underscoring the potential benefits of integrating Pythagorean hesitant fuzzy sets into the VIKOR framework for enhanced decision making in complex, uncertain environments. The traditional VIKOR method, primarily utilized for selecting optimal solutions, is expanded in this study through the Pythagorean hesitant fuzzy set-based VIKOR approach. This method enhances grading by ranking evaluation targets against standard levels, addressing existing challenges in the safety risk assessment of port quayside container cranes and pioneering future research directions. The method's adaptability suggests potential applications in aviation, transportation, industrial production, and evaluations of efficiency and economy. However, adaptability issues arise when assessment targets change, necessitating re-evaluation of safety indicator weights due to variations in port equipment, work environments, management practices, and operator habits. Future research may concentrate on refining computational algorithms or expanding the model to include real-time data analytics for dynamic safety evaluations. Methodology, J.Y. ; investigation, F.S. and L.Y.