## 5 Experiments
### Sensitivity Analysis
#### 5.3.1 Sensitivity Analysis on the Freight Rate Ratio  
In this section, we further analyze the sensitivity of our bi-objective MIP model to the changes in the freight rate per transported TEU. Generally, we conduct 6 experiments, aiming to illustrate the performance of the bi-objective MIP model in solving instances with different freight rate ratio. The ratio is denoted as \(\alpha\), where the original freight rate is \(p_{od}\), the actual freight rate is \(p_{od}^{\text{actual}}\), and \(\alpha=p_{od}^{\text{actual}}/p_{od}\). To investigate the impact of varying the freight rate ratio of all OD pairs on the model's effectiveness, we design instances by altering the freight rate ratio from 0.7 to 1.2 with a step size of 0.1 while keeping other parameters unchanged. Figure 6 shows the Pareto frontiers of all instances. The results of our bi-objective optimization model are presented as a series of Pareto frontiers. The concept of Pareto frontiers is derived from the work of economist [PERSON] and is a powerful tool for visualizing the trade-offs between two or more conflicting objectives [76]. A solution is considered to be Pareto optimal if it is not possible to improve one objective without making at least one other objective worse. The Pareto frontier is a graph that plots all these optimal, non-dominated solutions. In the context of our study, as shown in Figure 5(a), the Pareto frontier illustrates the inherent trade-off between maximizing profit and maximizing transport volume. Here is how to interpret the graph:  
1. Each point is an optimal strategy: every blue point on the curve represents a complete, viable, and optimal operational plan (i.e., a specific set of decisions about which routes to open, which ships to deploy, and which transport demands to accept). 2. The curve shows the trade-off: The downward slope of the frontier demonstrates the conflict. To make more profit (moving to the right along the curve), the shipping company must accept less transport volume (moving down). Conversely, to increase market share by accepting more volume (moving up), the company must sacrifice some profit (moving to the left). 3. The \"frontier\" is the limit of achievable performance: Any point below and to the left of the curve represents a suboptimal, inefficient strategy. Any point above and to the right of the curve is an unattainable goal given the current model constraints and resources. The frontier itself represents the best possible outcomes. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{k}\) & **Total Volume of Accepted Demand (TEU)** & **Average of \(\mathbf{p_{od}}\) (USD)** \\ \hline
1 & 315,250 & 918 \\
11 & 289,120 & 945 \\
20 & 232,016 & 1015 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Total volume of accepted transport demand and the average freight rate of the transported cargo. As the freight rate ratio increases from 0.7 to 1.2, reflecting higher revenue per transported TEU, the Pareto frontier consistently moves upwards and towards the right, indicating improved profitability for the same or greater transport volumes. At lower freight rates (\(\alpha=0.7\)), the Pareto frontier indicates limited profitability, constraining operators to accept relatively lower transport volumes as many OD pairs become economically unviable. As the freight rate ratio rises, higher profitability allows operators greater flexibility, enabling them to accept more transport demand selectively, thereby significantly increasing total profit and volume simultaneously. The progressive outward shift of the Pareto frontier as the freight rate ratio \(\alpha\) increases illustrates a critical change in the operational landscape: diminishing marginal trade-offs between profit and transport volume. At lower freight rates, increasing Obj 1 often necessitated significant reductions in Obj 2, as lower freight rates for specific OD pairs frequently failed to cover transport costs along particular routes.