## 4 Numerical Experiment
### Feasibility Verification of Mixed Time Window  
Currently, there are no studies that employ an MTW in the GIMRP under the uncertainty of both capacity and carbon tax rate. Therefore, there is no existing evidence verifying the feasibility of the MTW under these conditions. With reference to [PERSON] et al. (2016), this study compares the MTW with the HTW and STW to show its feasibility.  
Under the HTW, the delivery of containers must be accomplished within \([\pi,\pi^{\prime}]\). In this case, the objective of model formulating the GIMRP with a HTW under capacity and carbon tax rate uncertainty is shown in Equation (31), and its constraints are shown in Equations (2)-(7), (14)-(18), (21), (27)-(29), (32), and (33).  
\[\min\sum_{(i,j)\in A}\sum_{r\in S_{ij}}c_{ijr}\cdot q\cdot x_{ijr}+\sum_{j\in N }\sum_{r\in S_{j}}c_{j}^{rs}\cdot q\cdot y_{j}^{rs}+\xi \tag{31}\]  
\[z\geq\pi \tag{32}\]\[z\leq\pi^{\prime} \tag{33}\]  
The comparisons between the HTW and MTW under different optimization levels are illustrated in Figures 2 and 3. According to Figures 2 and 3, the two types of time window yield the same results under lower optimization levels of 0.5 and 0.6. However, when the optimization level is improved to satisfy the receiver's requirement of a higher RT, the use of MTW can reduce both the TCs and CEs of the PIMR when compared to the HTW. Consequently, the MTW is more feasible than the HTW in the GIMRP under capacity and carbon tax rate uncertainty.  
Figure 3: CEs of the PIMR under MTW and HTW.  
Figure 2: TCs of the PIMR under MTW and HTW.  
When using the STW, the GIMRP takes Equation (19) as the objective and its constraints cover Equations (2)-(9), (14)-(18), (21), and (27)-(29). The comparisons between the STW and MTW under different optimization levels are illustrated in Figures 4 and 5.  
As shown in Figures 4 and 5, when the receiver pays less attention to the RT and determines lower optimization levels of 0.5, 0.6, and 0.7, the STW helps to reduce the TCs and CEs of the PIMR when compared to the MTW. When the receiver demands higher optimization levels to ensure reliable transportation, the two time windows give identical results. Furthermore, we calculate the delivery time of containers under the STW, which is given in Figure 6.  
As shown in Figure 6, when the STW obtains better optimization results than the MTW, the delivery times of containers under STW are earlier than 40 (i.e., 16:00 on day 2), which  
Figure 4: TCs of the PIMR under MTW and STW.  
Figure 5: CEs of the PIMR under MTW and STW.  
is too early and violates the receiver's timeliness requirement indicated by the tolerable time window. Consequently, the MTW enables the receiver's requirements to be satisfied in all the decision-making scenarios. Therefore, the MTW is more feasible than the STW in the GIMRP investigated in this study.  
Above all, the HTW could lead to higher TCs and CEs, and the STW could fail to avoid overly early delivery in some decision-making scenarios. However, the MTW is able to avoid these issues. Consequently, we have verified that the MTW is better than the HTW and STW when employed in a GIMRP that considers uncertainty in the capacity and carbon tax rate.