## 5 Experiments
### Sensitivity Analysis
#### 5.3.2 Sensitivity Analysis on the Surcharge Intensity  
In this section, we conduct 6 experiments, aiming to illustrate the performance of our proposed model in solving instances with different surcharge intensities. We design instances by altering the unit penalty surcharge \(\mathrm{c_{p}}\) from 0 to 100 with a step size of 20 while keeping other parameters unchanged. Figure 7 shows the Pareto frontiers of all instances.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model Type**} & \multirow{2}{*}{\(\alpha\)} & \multicolumn{2}{c}{**CPU Time (s)**} & \multicolumn{2}{c}{**CPU Time (s)**} \\  & & **(Augmented \(\epsilon\)-Constraint)** & **(Basic \(\epsilon\)-Constraint)** \\ \hline \multirow{4}{*}{MIP} & 0.7 & 9.7 & 161.7 \\  & 0.8 & 9.8 & 188.5 \\  & 0.9 & 9.1 & 149.2 \\  & 1.0 & 8.5 & 136.4 \\  & 1.1 & 9.0 & 142.6 \\  & 1.2 & 9.1 & 154.2 \\ \hline \multirow{4}{*}{PRMIP} & 0.7 & 3.7 & 59.5 \\  & 0.8 & 3.9 & 62.4 \\ \cline{1-1}  & 0.9 & 3.7 & 71.2 \\ \cline{1-1}  & 1.0 & 3.6 & 50.0 \\ \cline{1-1}  & 1.1 & 3.6 & 50.7 \\ \cline{1-1}  & 1.2 & 3.4 & 57.6 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Comparison of computational efficiency for MIP and PRMIP models using two different \(\epsilon\)-constraint methods.  
As the unit penalty surcharge \(c_{p}\) increases from 0 to 100, the Pareto frontier remains largely consistent, indicating that the shipping line can maintain a similar level of performance in terms of both profit and transport volume. This stability indicates that the model is effective at finding potent mitigation strategies. The model achieves this by effectively navigating the complex trade-off between maximizing profit, increasing transport volume, and mitigating the extra penalty. It strategically reconfigures the fleet, for instance by systematically chartering out penalized China-built ships from U.S. routes and chartering in compliant ships to take their place, thereby preserving the core trade-off between Obj 1 and Obj 2.  
To quantitatively dissect this mitigation strategy and reveal the underlying economic trade-offs, we analyze the total penalty paid across all Pareto-optimal solutions for each surcharge level. The results are summarized in Table 12.  
Figure 7: The Pareto frontiers with different freight surcharge intensities.  
It is observed that as \(c_{p}\) rises from 20 to 60, the total penalty incurred generally increases. This indicates that at these lower levels, it is often optimal to accept the surcharge for certain highly profitable routes where the additional cost of chartering ships to avoid using a penalized ship would exceed the penalty itself. A significant strategic shift becomes evident at \(c_{p}=80\). At this level, some of the Pareto-optimal solutions show a total penalty of zero, indicating that for a wide range of strategies, complete avoidance has become the more profitable choice. This trend culminates at \(c_{p}=100\), a level at which the unit penalty cost becomes so high that the total penalty across all Pareto-optimal solutions drops to zero. This illustrates the model's sophisticated decision making, which involves calculating the most profitable strategy: paying a manageable fee on certain routes or altering fleet and route assignments to completely avoid the penalty once the cost becomes prohibitive.