## 4 Numerical Experiment  
### Design of the Numerical Experiment and Sensitivity Analysis  
This study considers the 35-node water-rail-road intermodal network proposed by [PERSON] and [PERSON] [24] in the design of the numerical experiment and to verify the proposed methods. Such an intermodal network has been commonly used in related studies to verify and evaluate the IMRP [7; 10; 25; 26; 27].  
In the numerical experiment, the in-transit distances and the upper bounds of the interval fuzzy capacities of the transportation modes and their transfers at the transfer nodes are set according to [PERSON] and [PERSON] [24], and are presented in Table S1 in the Supplementary Materials. We assume that the lower bounds of the interval fuzzy capacities are 0.7 of their corresponding upper bounds. The in-transit costs (in CNY/TEU) of rail, road, and water transportation are calculated using Equation (30) [1]. The speeds and CE factors of the three transportation modes and the times, costs, and CE factors to transfer the containers between different transportation modes are all set according to [PERSON] et al. [10], and can be seen in Tables S2-S4 in the Supplementary Materials.  
\[c_{ijp}=\begin{cases}500+2.03\cdot d_{ijr}&r=\text{Rail}\\ 15+8\cdot d_{ijr}&r=\text{Road}\ \ \forall(i,j)\in A\ \forall r\in S_{ij}\\ 950&r=\text{Water}\end{cases} \tag{30}\]  
Referring to [PERSON] et al. [28], the interval fuzzy carbon tax rate is set to CNY (0.22, 2.20)/kg. In this study, we assume that the 35 TEU containers of a transportation order need to be transported from node 1 to node 35, and their release time at node 1 is 9:00 on day 1. The receiver's MTW yields an expected time window from 23:00 on day 2 to 2:00 on day 3, and a tolerable time window from 16:00 on day 2 to 8:00 on day 3. The penalty costs caused by early and late delivery concerning the expected time window are CNY 15/(TEU\(\cdot\)h) and CNY 30/(TEU\(\cdot\)h), respectively.  
The optimization level \(\lambda\) is determined by the receiver according to its attitude towards the RT, and might influence the optimization result. We considered that the receiver prefers a high optimization level that falls in the interval [0.5, 1.0] to ensure reliable transportation. We accordingly increased the optimization level from 0.5 to 1.0 with a step of 0.1 to determine the TCs and CEs of the PIMRs under different optimization levels. The sensitivity of the GIMRPMTWIFCCTR concerning the optimization level is illustrated in Figure 1.  
As we can see from Figure 1, with the improvement in the optimization level, the TCs and CEs of the PIMR both increase, which means that addressing the capacity and carbon tax rate uncertainty significantly influences routing optimization. It is clear that enhancing the RT through improving the optimization level worsens both the economic and environmental sustainability of the GIMRPMTWIFCCTR. Based on Figure 1, the receiver can determine a suitable optimization level that satisfies its requirements to balance the objectives of lowering CTs, reducing CEs, and enhancing the RT. The IMO can then make a plan based on the given optimization level and accordingly organize the IMT.  
Figure 1: Sensitivity of the TCs and CEs of the PIMR concerning the optimization level.