## 6 Numerical Experiments
### Sensitivity Analysis  
To validate the effectiveness and general applicability of the model and method proposed in this paper, a sensitivity analysis was conducted by considering several factors. Figure 6 illustrates the impact of changes in the unit freight rates for TS cargo and ordinary cargo on the total revenue of liner companies. The horizontal axis represents the percentage change in the unit freight rate, ranging from \(-0.9\) (indicating a 90% reduction in the original unit rate) to 1 (indicating a 100% increase). The vertical axis represents the percentage change in total revenue, ranging from \(-57.91\)% (a 57.91% decline in profit compared to the original) to 120.50% (a 120.50% increase in revenue). Three key insights can be drawn from Figure 7:  
1. Total revenue increases with unit rate growth, and this upward trend becomes more pronounced as the unit rate rises. 2. When the unit freight rate decreases to 10-50% of the original, the change in total revenue stabilizes, indicating that revenue fluctuations remain relatively steady in this range. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Profit**} & \multicolumn{2}{c}{**Average Space Utilization**} & \multicolumn{2}{c}{**Average Delivery-Delay Rate**} \\ \cline{2-7}  & **Co.** & **Basic** & **Co.** & **Basic** & **Co.** & **Basic** \\ \hline  & 779,142,116 & 743,739,199 & 87.096\% & 76.944\% & 87.096\% & 76.944\% \\ Enhancing rate & 4.76\% & - & 10.151\% & - & 2.122\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The solutions of different cases. 3. The total revenue is more sensitive to changes in the unit rate of TS cargo compared to ordinary cargo, suggesting that variations in TS cargo rates have a more significant impact on overall profitability. Additionally, this paper explores the impact of changes in unit costs on total revenue, including the unit transportation cost for laden containers, unit transportation cost for empty containers, unit leasing cost for empty containers, and unit stacking cost. Figure 8, consistent with Figure 7, shows the effect of these cost changes. The horizontal axis represents the percentage increase in unit costs, with the same variable range as in Figure 7.  
It is important to note that the total profit change curve for the unit transportation cost of laden containers is based on the right vertical axis, with a range from \(-78.16\%\) to \(79.96\%\). The total revenue change curves for the other costs--empty container transportation, leasing, and stacking--are plotted on the left vertical axis. The ranges for these values are as follows: \(-0.26\%\) to \(1.28\%\) for unit empty container transportation cost, \(-2.65\%\) to \(2.44\%\) for unit empty container leasing cost, and \(-7.52\%\) to \(7.97\%\) for unit stacking cost. Two key insights can be drawn from Figure 7:  
1. Total profit decreases as unit costs increase, and the sensitivity of total profit to each cost varies significantly. 2. Among the different costs, the total revenue is most sensitive to the unit transportation cost of laden containers, while it is least sensitive to the cost of empty container repositioning. This is because the number of slots allocated to empty containers is relatively small compared to those allocated to laden containers. This paper also examines the effect of changes in unit leasing costs on the sourcing of empty containers. Figure 9 illustrates the satisfaction ratio of empty container repositioning volume to the total volume of empty containers (i.e., the sum of repositioning volume and leasing volume). As depicted in Figure 9, with increasing unit leasing costs, the associated rental costs rise as well. Consequently, at the source of empty containers, the proportion of repositioned containers increases, while the proportion of leased containers decreases, which is a logical outcome. The sensitivity of the three routes to changes in empty container leasing costs is roughly similar.