## 3 Solution Method
### Recursive Method for Waiting Time Calculation
Step 5: Generate the subsets of R. Based on the value of \(y_{i,d}\), R is divided into \(\mid\)D\(\mid\) disjoint subsets \(DR_{1},\ DR_{2},\ \ldots,\ DR_{\mid\mid D\mid}\left(R=DR_{1}\cup DR_{2}\cup\ldots\cup DR_{ \mid\mid D\mid}\right)\), where \(DR_{d}\) is the set of cargoes that select \(d\) as a downstream station. Let \(d=1\), and proceed to Step 6. Step 6: Calculate \(t_{i}^{5}\). Step 6.1: If \(d\leq\mid\)D\(\mid\), let \(TS=DR_{d}\), \(r=0\), and proceed to Step 6.2; otherwise, skip to Step 7. Step 6.2: If \(TS=\Phi\), let \(d=d+1\), and return to Step 6.1; otherwise, proceed to Step 6.3. Step 6.3: Search for elements in \(TS\). If \(i\in TS\) has the earliest arrival moment at \(d\), i.e., for \(j\in TS\) satisfies \(ADM_{i}=min\left\{ADM_{j}\right\},i\&j\in TS\), then,  
\[LDSM_{i}=max\{ADM_{i},r\},i\in R. \tag{23}\]  
Substitute \(ADM_{i}\) and \(LDSM_{i}\) into Equation (16) to obtain \(t_{i}^{5}\). Step 6.4: Calculate \(t_{i}^{6}\) by Equation (17). Let \(r=max\{LDSM_{i},\ ADM_{i}\}+t_{i}^{6}\), \(TS=TS-\{i\}\), and return to Step 6.2. Step 7: Calculate \(EM_{i}\). Applying Equation (18) to obtain \(t_{i}^{7}\). Substitute \(t_{i}^{1}\), \(t_{i}^{2}\), \(t_{i}^{3}\), \(t_{i}^{4}\), \(t_{i}^{5}\), \(t_{i}^{6}\), \(t_{i}^{7}\), and \(SM_{i}\) into Equation (11) to obtain \(EM_{i}\) for each \(i\in R\). Step 8: End. Unlike existing multimodal transport studies that mainly calculate average waiting times for cargoes at hubs, WTC, with known hub allocation and transshipment sequence, can recursively derive minimum waiting times at hubs based on arrival moment, thereby providing a more precise estimate of \(EM_{i}\). Steps 1 to 3 are responsible for calculating \(t_{i}^{1}\), \(t_{i}^{2}\), and \(t_{i}^{3}\) given \(x_{i,u}\) and \(z_{i,j}^{u}\): The waiting time \(t_{j}^{2}\) of the cargo that earliest loaded at \(u\) is determined in Step 3.3; the minimum waiting time \(t_{i}^{2}\) of other cargo is obtained in Step 3.5; \(t_{i}^{1}\) and \(t_{i}^{3}\) are, respectively, gained in Step 1 and Step 3.4. Based on Steps 1 to 3, Steps 4 to 6 are primarily responsible for calculating \(t_{i}^{4}\), \(t_{i}^{5}\), and \(t_{i}^{6}\) given \(y_{i,d}\): \(t_{i}^{5}\) is obtained in Step 6.3; \(t_{i}^{4}\) and \(t_{i}^{6}\) are obtained in Step 4 and Step 6.4, respectively. Then, based on \(t_{i}^{7}\) given in Step 7, \(EM_{i}\) can be obtained. It should be noted that unlike the FCFS rule commonly used in the existing literature, in WTC, \(z_{i,j}^{u}\) is independent of \(AUM_{i}\) (see Step 3), while \(h_{i,j}^{d}\) is determined by \(ADM_{i}\) (see Step 6). This means that an arbitrary loading sequence at upstream stations is allowed in WTC, while the unloading sequence at downstream stations is determined by the FCFS rule. It can be easily demonstrated that, in the model shown in Section 2.2, when hub allocation and upstream station transshipment sequence are known, the FCFS rule is the optimal sequencing rule for all downstream stations. If upstream stations also determine the transshipment sequence (\(z_{i,j}^{u}\)) based on the arrival moments of cargoes, it can be simplified to the FCFS rule. Therefore, WTC can be seen as a more general waiting time calculation method that can be adapted to any sequencing rule. Below, a simple example is provided to illustrate the advantage of WTC in waiting time calculation and the necessity of considering transshipment sequence selection. Cargoes from A, B, and C are transported to a destination via upstream stations (1 and 2) and downstream stations (3 and 4). A hub allocation result is shown in Table 3 where A and B share 2, while B and C share 4. Therefore, congestion may occur at 2 and 4, necessitating the determination of the transshipment sequence. According to WTC, the transshipment sequence at 4 is determined by the FCFS rule, so it is only necessary to select transshipment sequences for A and B at 2. From Table 3, it can be seen that there are two feasible sequencing schemes for A\(\rightarrow\)B and B\(\rightarrow\)A.