## 5 Experiments
### Basic Results
#### 5.2.1 Bi-Objective Optimization Results Analysis  
Figure 5 shows the optimal results of the bi-objective MIP model. Figure 5a shows the Pareto frontier, with the \(x\)-axis representing the total weekly profit (Obj 1) and the \(y\)-axis representing the total weekly transport volume (Obj 2). In Figure 5b-d, the \(x\)-axis represents \(k\), tracking increasing values of Obj 1. The \(y\)-axis of Figure 5b-d corresponds to the values of indicators, including the number of operated routes (ORN) and the average freight rate (AFR) of accepted transport demand, the total fuel cost (TFC) and the total berthing cost (TBC) of all ships, and the revenue from chartering out idle ships minus the cost of chartering in ships (COIR). \begin{table}
\begin{tabular}{c c c} \hline \(\mathbf{Q_{s}}\) (TEUs) & \(\mathbf{a_{sr}}\) & \(\mathbf{b_{sr}}\) \\ \hline
12,000 & 0.01210 & 2.947 \\
15,000 & 0.00433 & 3.314 \\
20,000 & 0.02420 & 2.914 \\ \hline \end{tabular}
\end{table} 
Table 6: The fuel cost parameters for ships of different capacities. The Pareto frontier depicted in Figure 5a demonstrates the inherent trade-off between Obj 1 and Obj 2. The Pareto frontier provides a powerful decision-making reference for shipping managers by revealing two distinct layers of choice. The first layer concerns major strategic decisions. For a shipping company, these correspond to discrete, high-impact actions, for example, activating an entire service route or reallocating a specific ship type. This structure indicates that when the company wants to pursue one objective significantly, such as substantially increasing profit, it must make these major adjustments to its route operations and fleet deployment. This is fundamentally different from making small-scale adjustments to accepted cargo while the network structure remains fixed. In situations where only minor fine-tuning between profit and transport volume is desired, keeping the activated routes and deployed ships constant while adjusting cargo acceptance is the more effective strategy. This clear distinction between major structural changes and minor operational adjustments provides a practical validation of Theorem 5, which establishes that the frontier's overall structure is fundamentally determined by the integer decision variables governing the network design. At the operational level, within each of these fixed strategic configurations, the convex shape of the frontier illustrates the law of diminishing returns. For a shipping manager, this signifies that pushing for an extreme on one objective comes at a disproportionately high cost to the other. This phenomenon of a progressively steepening trade-off is formally established in Theorem 6. Therefore, the frontier serves as a practical guide for finding the optimal trade-off based on current corporate priorities. For instance, if the primary goal is to expand market share, a decision maker would select a solution from the frontier that prioritizes higher transport demand acceptance. Conversely, if the immediate need is to improve profitability and strengthen cash flow, a solution  
Figure 5: Pareto-optimal solutions of the bi-objective MIP model. (**a**) Obj 1 and Obj 2 of instance. (**b**) ORN and AFR of instance. (**c**) TFC and TBC of instance. (**d**) COIR of instance. yielding higher profit would be chosen, even at the expense of some volume. The frontier quantifies these choices, enabling managers to make informed decisions that align their network strategy with their most pressing business objectives. Figure 5b shows that ORN decreases as Obj 1 increases. This reduction in ORN enables the company to concentrate resources on the most profitable transport services rather than dispersing them across marginally profitable ones. The strategic elimination of less profitable routes directly contributes to cost reduction and operational efficiency. This cost reduction is clearly demonstrated in Figure 5c, where both TFC and TBC decrease as Obj 1 increases.