## 3 Solution Method  
### Recursive Method for Waiting Time Calculation  
It can be seen from Equation (11), the multimodal transport time of \(i\) consists of transport time, transshipment time, and waiting time. From assumptions (4) to (8), it is evident that the first two times depend only on \(x_{i,u}\) and \(y_{i,d}\). Therefore,\(t_{i}^{1}\), \(t_{i}^{3}\), \(t_{i}^{4}\), \(t_{i}^{6}\), and \(t_{i}^{7}\) can be directly calculated using Equations (12), (14), (15), (17) and (18) when given hub allocation. Nevertheless, calculating minimum waiting times at hubs (\(t_{i}^{2}\) and \(t_{i}^{5}\)) is more complex. From Equation (13) (or Equation (16)), \(t_{i}^{2}\) (or \(t_{i}^{5}\)) is the difference between \(LUSM_{i}\) (or \(LDSM_{i}\)) and \(AUIM_{i}\) (or \(ADM_{i}\)). According to assumptions (9) to (11), \(LUSM_{i}\), \(ADM_{i}\), and \(LDSM_{i}\), except for \(AUIM_{i}\), depend not only on \(x_{i,u}\) and \(y_{i,d}\), but also on \(x_{j,u}\) and \(y_{j,d}\) (\(j\
eq i\)) as well as the position of \(i\) in the transshipment sequence (\(z_{ij}^{u}\) and \(h_{i,j}^{d}\), \(j\
eq i\)). Consequently, calculating \(t_{i}^{2}\) and \(t_{i}^{5}\) under the premise of hub allocation and transshipment sequence becomes a key issue in solving \(EM_{i}\). With growing interest in simulation models for scheduling [44], this section proposes a recursive method (after this referred to as WTC) to more accurately calculate \(t_{i}^{2}\), \(t_{i}^{5}\), and \(EM_{i}\) at hubs by simulating actual cargo flow, replacing the prevalently used average waiting time. The detailed process is as follows. Step 1: Calculate \(AUIM_{i}\). According to assumptions (1) and (5), \(AUIM_{i}\) equals the sum of the multimodal transport start moment of \(i\) and the time consumed by operations 1\(\!\):  
\[AUM_{i}=t_{i}^{1}+SM_{i},i\in R. \tag{19}\]  
Given the value of \(x_{i,u}\), \(t_{i}^{1}\) can be calculated by Equation (12). Using Equation (19) and repeating Step 1 for each \(i\in R\), the arrival moments at upstream stations can be obtained. Step 2: Generate the subsets of R. Based on the values of \(x_{i,u}\), R is divided into 1\(\!\) U 1 disjoint subsets \(UR_{1},~{}UR_{2},~{}\ldots,UR_{|U|}\left(R=UR_{1}\cup UR_{2}\cup\ldots\cup UR_ {|U|}\right)\), where \(UR_{u}\) is the set of cargoes that select \(u\) as an upstream station. Let \(u=1\), and proceed to Step 3. Step 3: Calculate \(t_{i}^{2}\). Step 3.1: If \(u\leq\!\) U 1, proceed to Step 3.2; otherwise, skip to Step 4. Step 3.2: If \(UR_{u}=\Phi\) (i.e., no cargo selects \(u\)), let \(u=u+1\), and return to Step 3.1; otherwise, proceed to Step 3.3. Step 3.3: Search for elements in \(UR_{u}\). If there is no immediate predecessor activity for \(j\in UR_{u}\), i.e., it satisfies:  
\[\sum_{i\in UR_{u}}z_{ij}^{u}=0,j\in UR_{u}, \tag{20}\]  
then \(LUSM_{j}=AUM_{j}\). According to Equation (13), \(t_{i}^{2}=0\); that is, \(j\) will start operation 2\(\!\) 2 immediately upon arrival at \(u\) without waiting. Let \(c=j\), and proceed to Step 3.4. Step 3.4: Calculate \(t_{c}^{3}\) by Equation (14). Step 3.5: If no \(i\in UR_{u}\) satisfies \(z_{ij}^{u}=1\), then let \(u=u+1\), and return to Step 3.1; otherwise,  
\[LUSM_{i}=max\left\{AUM_{i},\sum_{k=1}^{3}t_{c}^{k}+SM_{c}\right\},i\in R \tag{21}\]  
Substitute \(AUM_{i}\) and \(LUSM_{i}\) into Equation (13) to obtain \(t_{i}^{2}\). Let \(c=i\), and return to Step 3.4. Step 4: Calculate \(ADM_{i}\). According to assumptions (1) to (5), \(ADM_{i}\) equals the sum of the multimodal transport start moment of \(i\) and the time consumed by operations 1\(\!\), 2\(\!\) 1\(\!\) 2\(\!\) 2\(\!\) 2\(\!\), and 3\(\!\), i.e.,  
\[ADM_{i}=\sum_{k=1}^{4}t_{i}^{k}+SM_{i},i\in R. \tag{22}\]Given the values of \(x_{i,u}\) and \(y_{i,d}\), \(t_{i}^{4}\) can be calculated by Equation (15). Using Equation (22) and repeating Step 4 for each \(i\in R\), the arrival moments at downstream stations can be obtained.