# Combining Optimization and Simulation for Designing a Robust Short-Sea Feeder Network
## 5 Computational Study
### Determining Global Estimated Optimal Routes
#### 5.3.3 Analyzing the Best Deterministic Solution  
The best deterministic solution is slightly cheaper than all solutions found by the solution triggered feedback approach. However, these cost savings come at the expense of a lack of robustness for harsh weather conditions. Table 4 provides the performance characteristics for simulating the optimal deterministic solution under weather conditions without any performance-improving strategy. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Strategy**} & \multirow{2}{*}{**Cost [\%]**} & **Dur.** & **Dur.** & **Sync.** & **Sync.** & **Idle** & **Idle** & **Vessel** \\  & & **Viol.** & **Mag.** & **Viol.** & **Mag.** & **Time** & **Use [\%]** & **Size** \\ \hline None & 102.1 & 0 & 0 & 200 & 1958 & 219 & 14.8 & 200 \\ Slack 10\% & 101.4 & 4 & 18.7 & 61 & 450 & 171 & 3.9 & 300 \\ Realistic & 101.4 & 4 & 17.1 & 52 & 319 & 162 & 2.5 & 300 \\ Seasonal & 101.5 & 3 & 4.3 & 204 & 1859 & 180 & 17.3 & 300 \\ Speed-up & 102.8 & 2 & 4.1 & 125 & 927 & 188 & 8.9 & 300 \\ Combined & 101.2 & 1 & 0.7 & 21 & 192 & 160 & 1.8 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance characteristics for the different solutions. In particular, the high number of duration violations renders the best deterministic solution infeasible in practice. While almost all of the planned idle time is used for sailing to mitigate delays, the system is still unable to recover. The poor performance of the optimal deterministic solution clearly shows the importance of taking into account weather uncertainty. All solutions found the using optimization-simulation approach (see Table 3) perform significantly better, while being less than 3% more expensive. The solution from the Combined strategy for example, reduces the number of duration violations from 277 to 1 and the duration magnitude from 45,429 h to 0.7 h, while the costs only increase by 1.2%. Pushing the analysis further, we simulate the best deterministic solution with the Speed-up strategy. This combination results in a cost increase of 2.2%, which is due to the higher fuel consumption from the required speed-ups. However, the number of duration violations is reduced from 277 to 28, and the number of synchronization violation is reduced from 194 to 181. Even though these are significant improvements, the solution still performs much worse than the solutions found by the optimization-simulation method. In another analysis, we consider another criterion of a well-performing solution. It consists of measuring the stability of arrival times, which translates directly into predicting future arrivals and maintaining a given schedule. For the Combined-strategy solution, the arrival times for the summer \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Cost [\%]**} & **Dur.** & **Dur.** & **Sync.** & **Sync.** & **Idle** & **Idle** & **Vessel** \\  & **Viol.** & **Mag.** & **Viol.** & **Mag.** & **Time** & **Use [\%]** & **Size** \\ \hline
100 & 277 & 45,429 & 194 & 14,332 & 192 & 82.5 & 300 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance characteristics for the best deterministic solution. Figure 10: The difference between the summer and winter solutions when using the combined strategy. The main route sails further north in the winter season due to rough weather conditions. and winter routes are presented in Figures 11 and 12. The mother vessels are scheduled to depart from the main continental port at time zero. The arrival times for all vessels refer to this departure time. The small vertical line represents the median arrival time. The width of each box corresponds to the interquartile range (\(IQR\)) defined by the first and third quartile, and accounts for 50% of all port arrivals in the simulation period. The left and right whiskers capture all arrival times within \(1.5\cdot IQR\). Arrival times outside this range are considered outliers and represented with circles.