## 5 Conclusions  
In view of the few studies involving quantitative considerations of ship vulnerability and IMO guidelines in ocean route planning models, this study proposes a multi-objective nonlinear programming model which integrates ship vulnerability and meteorological and marine environment factors to obtain a comprehensive risk assessment, and it implements the IMO guidelines as hard constraints to solve the model using the bidirectional A* algorithm. Firstly, this study uses the multi-layer fuzzy comprehensive evaluation model, which comprehensively assesses the risk of sailing ships by evaluating ship vulnerability  
Figure 16: IBA*routeB and WrouteB from 16 June to 7 July 2022.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithms** & Dijkstra & A* & Bidirectional A* \\ \hline
**Running Time (s)** & 31.739 & 8.236 & 4.187 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Running time of the Dijkstra, A*, and bidirectional A* algorithms.  
and meteorological and marine environment factors. Secondly, the multi-objective nonlinear programming model is constructed by adding IMO constraints and comprehensively considering the navigation risk, sailing time, and fuel consumption in this study. This study then uses the bidirectional A* algorithm to solve the route planning model. Finally, an accident case in the Singapore Maritime Investigation Report was studied, and the experimental results show that the model planning route in this study is similar to the original route in the Towing Manual, with an average fit of 98.22%, and the average risk is 11.19% lower than that of the original route. Comparing algorithms, the bidirectional A* algorithm is 86.81% faster than the Dijkstra algorithm and 49.16% faster than the A* algorithm. In summary, our model realizes two goals: a safer ocean route and more efficient computing. In future works, the algorithms for ocean ship route planning can be improved, which will greatly improve search efficiency and further approach an optimal solution. Finally, we will continue studying the influence of ship stability under dynamic constraints from the perspective of ship dynamics, and these constraints will be added into the proposed multi-objective programming model.  
Conceptualization, Y.W. and L.Q.; Methodology, Y.W.; Software, Y.W. and D.L.; Validation, L.Q. and D.L.; Investigation, M.H.; Resources, L.Q.; Data curation, Y.L.; Visualization, M.H.; Supervision, Y.L.; Project administration, L.Q. and M.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Humanities and Social Science Fund of Ministry of Education (Grant No. 23 YAAZH111), the Open Project of Key Laboratory of High Impact Weather(special), China Meteorological Administration, the Open Research Fund of Key Laboratory of Water Safety for Beijing-Tianjin-Hebei Region of Ministry of Water Resources (Grant No. IWHR-KLWS-202301), the Natural Science Foundation of Hunan Province (Grant No. 2023 JJ10054), the National Natural Science Foundation of China (Grant No. 42375016), and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. SJCX24_0278).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.