## 3 Case Study
### Results and Analysis
#### 3.2.3 Comparison and Verification of Routes  
When there is little difference between the two planned routes visually, the similarity between TMroute and IBA*route can be compared using the goodness of fit \(R_{NL}\)[38], and the calculation formula of \(R_{NL}\) is shown in Equation (39).  
\[R_{NL}=1-\sqrt{\frac{\Sigma(y_{i}-\hat{y}_{i})^{2}}{\sum y_{i}{}^{2}}} \tag{39}\]  
The average relative error \(R_{E}\) of TMroute and IBA*route is calculated as follows:  
\[R_{E}=\frac{\sum\lvert y_{i}-\hat{y}_{i}\rvert}{n(y_{\text{max}}-y_{\text{min} })} \tag{40}\]  
where \(y_{i}\) and \(\hat{y}_{i}\) are the latitude grid values corresponding to the \(i\)th longitude grid value of TMroute and IBA*route, respectively, and \(y_{\text{max}}\) and \(y_{\text{min}}\) are the maximum latitude grid value and the minimum latitude grid value of TMroute, respectively.  
TMrouteA and TMrouteB are compared with IBA*routeA and IBA*routeB, respectively. The goodness of fit and the average relative error are calculated, as shown in Table 6.  
In Table 6, the goodness-of-fit values of IBA*routeA and IBA*routeB separately compared with TMrouteA and TMrouteB are both higher than 95%, and their average relative errors are lower than 5%, indicating that IBA*route and TMroute are highly similar.  
According to the investigation report, the ship deviated from the TMroute between 19 June and 2 July and entered an area of increasingly severe weather conditions, as shown in Figure 11.  
In Figure 11, the route that the CB actually sailed, which is marked as Realroute in this manuscript, entered the high-risk area around 2 July after deviating from TMroute, while according to the ship's planned arrival time in the investigation report, it can be calculated that TMrouteB had already left the open sea area on 30 June. IBA*routeB bypassed the high-risk area on 28 June and also left the open sea on 30 June.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **From 1 to 12 May 2022** & **From 16 June to 7 July 2022** \\ \hline \(R_{NL}\) & 0.9969 & 0.9675 \\ \(R_{E}\) & 0.0224 & 0.0302 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The comparison of \(R_{NL}\) and \(R_{E}\) between TMroute and IBA*route.  
Figure 11: The routes of Realroute, TMrouteB, and IBA*routeB from 26 June to 2 July 2022.  
To study the risks of Realroute, TMrouteB, and IBA*routeB during the period of the accident, the average risks of the three routes during the period from Colombo, Sri Lanka to Salalah were calculated, respectively, as shown in Table 7, and the box plots of the three routes are shown in Figure 12.  
In Table 7, Realroute is larger than TMrouteB and IBA*routeB in terms of the average risk, maximum risk, and minimum risk values. Moreover, the average risk value of IBA*routeB is only 0.3929, the maximum risk value is only 0.6592, and the minimum risk value is only 0.1252, all of which are the minimum values in the three routes, so it can be seen that IBA*routeB exhibits the minimum overall risk.  
In Figure 12, IBA*routeB has the smallest maximum, lower quartile, and upper quartile values of risk, which indicates that our planning model has a strong risk avoidance ability.