## 6 Numerical Experiments
### Test Results
Compared with the CPLEX, our algorithm has apparent advantages regarding its calculation time. Furthermore, the calculation precision of the BC algorithm is verified to be exact, showcasing its ability to find an optimal solution. The total revenue convergence process of Case 6 by applying the branch-and-cut algorithm is shown in Figure 5.  
Figure 5 illustrates the number of containers of various types across multiple voyages on different shipping routes. Specifically, Groups 1(1) to 1(4) represent the container counts for different types across the first to fourth voyages on the first route. Similarly, the final group, which is 3(4), corresponds to the container counts for the fourth voyage on the third route. Each group comprises four columns of data, representing, from left to right, the number of containers for TS cargo, ordinary cargo, postponed cargo, and empty containers. It is important to note that the container numbers for TS and ordinary cargo are measured against the left vertical axis, while the numbers for postponed cargo and empty containers are aligned with the right vertical axis. As indicated in Figure 6, TS cargo has the largest volume of containers, followed by ordinary cargo. The quantity of TS cargo containers is approximately 1.2 to 1.4 times that of ordinary cargo containers. This is  
Figure 5: Solution process of the branch-and-cut algorithm. attributed to the fact that the freight rate for TS cargo is higher, leading liner companies to prioritize containers that offer higher revenue. The results further reveal that the number of delayed containers tends to increase on voyages with higher volumes of TS cargo and empty containers, largely due to the limited capacity of ship resources. Additionally, it is evident that every route requires the repositioning of empty containers. Although repositioning empty containers does not generate direct revenue, it is necessary to meet the demand for empty containers from shippers. Given the high leasing costs for empty containers, repositioning them becomes a strategic choice aimed at balancing storage costs with leasing expenses. In some cases, the cost of repositioning empty containers between certain ports may exceed the leasing cost. Therefore, from a revenue management perspective, it is sometimes more cost-effective not to reposition empty containers between specific ports. Table 4 presents the ratio of empty container transportation to laden container transportation for various voyages and routes. The values indicate that this ratio remains within 6%, underscoring the liner company's prioritization of laden containers with a higher revenue potential. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Route1** & **Route2** & **Route3** \\ \hline Voyage 1 & 0 & 0 & 0 \\ Voyage 2 & 2.358\% & 0.798\% & 4.354\% \\ Voyage 3 & 2.444\% & 1.92\% & 3.324\% \\ Voyage 4 & 0 & 0.207\% & 1.707\% \\ Voyage 5 & 0.093\% & 0.253\% & 1.594\% \\ Voyage 6 & 2.941\% & 2.017\% & 3.748\% \\ Voyage 7 & 2.517\% & 1.627\% & 5.617\% \\ Voyage 8 & 0.062\% & 0.014\% & 1.172\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The ratio of empty container transport number to loaded container transport number. Figure 6: Diagram of number for four types of containers. The numerical variations in Table 4 reflect the periodic pattern of empty container repositioning. Specifically, the ratio is negligible (almost 0) for the first voyage of each route, primarily due to sufficient initial stocks of empty containers at the port, thus obviating the need for repositioning. In contrast, substantial increases in empty container repositioning occur during the second and third voyages, while volumes decrease for the fourth and fifth voyages. This cycle repeats with higher levels observed again during the sixth and seventh voyages. During the initial voyage, the port's stockpile of empty containers meets the demand, eliminating the necessity for repositioning.