## 6 Numerical Experiments
### Sensitivity Analysis
The variation in satisfaction ratios for these routes ranges from \(-10.71\%\) to \(2.02\%\), from \(-6.61\%\) to \(5.14\%\), and from \(-5.51\%\) to \(6.62\%\). Figure 7: Sensitivity analysis of unit freight on the profit. To explore the trade off between the allocation of laden and empty containers in the CSAECRP, this section analyzes the \(\alpha\) indicator. Figure 10 investigates the impact of unit rates and unit costs on \(\alpha\). Figure 10 examines how changes in unit rates (including TS cargo and ordinary cargo rates) affect \(\alpha\), while Figure 10 illustrates the influence of changes in unit costs (covering the transportation costs for laden and empty containers, leasing costs, and stockpiling costs) on \(\alpha\). Figure 8: Sensitivity analysis of unit cost on the profit. Figure 9: Sensitivity analysis of leasing cost on the empty container reposition satisfaction. As shown in Figure 10a, as unit revenues increase, the value of \(\alpha\) decreases rapidly at first and then stabilizes. When unit rates are low (indicating a depressed shipping market), liner shipping companies tend to allocate more slots to empty containers, resulting in a faster trade off between empty and laden containers. Notably, \(\alpha\) is more sensitive to changes in the unit rates for TS cargoes. As the unit rate for TS cargo increases, \(\alpha\) declines sharply from 13.55% to 2.63%. For ordinary cargo rates, \(\alpha\) also decreases, though less dramatically, from 10.92% to 2.63%. However, when unit rates reach a certain threshold (indicating a stable or flourishing shipping market), the number of slots allocated for empty containers decreases gradually, and the trade off between laden and empty containers becomes more stable. At this stage, the sensitivity to changes in unit rates for both TS cargo and ordinary cargo is similar, and it reduces slowly from 2.63% to 2%. The impact of each unit cost on the value of varies, as shown in Figure 10b, leading to the following four conclusions:  
Firstly, as the unit transportation and storage costs for laden containers rise, the value of \(\alpha\) shows a noticeable upward trend, with greater sensitivity to transportation costs. Since all other factors remain constant, an increase in the unit transportation cost of laden containers reduces the revenue gained from transporting them. Consequently, the allocation of slots for empty containers increases, while slots for laden containers decrease, causing the value of \(\alpha\) to rise sharply from 1.65% to 10.83%. Secondly, as the unit storage cost rises, liner companies reduce the number of empty containers stored at the port and allocate more slots to empty containers to maximize revenue. As a result, the value of \(\alpha\) increases from 0.49% to 4.99%. Thirdly, with the rise in unit leasing costs, the value of \(\alpha\) grows slowly from 2.03% to 3.01%. When leasing costs rise, liner companies prefer to reposition empty containers rather than lease them. However, since the transportation volume for empty containers is relatively small compared to laden containers, the change in \(\alpha\) is modest. Lastly, as the unit transportation cost for empty containers increases, the value of \(\alpha\) declines rapidly from 7.93% to 1.54%. When the transportation cost for empty containers is low, liner companies are more inclined to reposition empty containers to save on leasing and storage costs, resulting in more slots being allocated for empty containers. However, as transportation costs for empty containers rise, companies shift to leasing, leading to a rapid reduction in the number of slots allocated for empty containers. Figure 10: Sensitivity analysis on the empty-laden ratio.