## 3 Modal Construction
### Problem Description and Model Assumptions
#### 3.2.1 Transport Time Analysis  
Within containerized multimodal transportation systems, the total transit duration for cargo movement from origin to destination comprises three operationally linked temporal dimensions: inter-nodal transportation time across network segments, transshipment duration at intermediate nodes, and scheduled waiting periods during modal transfers. These interdependent temporal components collectively determine the overall logistical efficiency through their systematic influence on shipment delivery timelines. The inter-nodal transit duration between transportation nodes is mathematically determined by the average operational velocity of the selected transport mode and the actual geographical distance between consecutive nodes, with this fundamental relationship explicitly formalized in Equation (1). \[t_{ij}^{m}=\frac{d_{ij}^{m}}{y^{m}},\;\forall m\in M,\;\forall i,j\in A \tag{1}\] \begin{table}
\begin{tabular}{c l l} \hline  & **Notation** & **Description** \\ \hline \multirow{4}{*}{Sets} & \multirow{2}{*}{\(c_{i}^{mk}\)} & The unit transshipment cost from mode \(m\) to \(k\) at node \(i\), \\  & & CNY/TEU \\  & & The transportation distances between nodes \(i\) and \(j\) with mode \(m\), km \\  & & Average travel speed of transportation mode m, km/h \\  & & The unit transportation carbon emissions between nodes \(i\) and \(j\) with mode \(m\), kg/(TEU\(\cdot\)km) \\  & & The unit carbon emissions from transportation from mode \(m\) to \(k\) at node \(i\), kg/TEU; \\  & & The container reloading efficiency from mode \(m\) to \(k\) at node \(i\), \\  & & h/TEU \\  & & The transfer transportation distance from mode \(m\) to \(k\) at node \(i\), \\  & & km \\  & & \(v_{i}\) & The average driving speed of transfer vehicles at node \(i\), km/h \\  & & The departure time of the s-th scheduled service for mode \(m\) at \\  & & node \(i\) \\  & & The transportation time between nodes \(i\) and \(j\) with mode \(m\), h \\  & & \(t_{ij}^{80}\) & The departure time of the goods from the starting node \\  & & \(t_{i}^{arrive}\) & The time when the goods arrive at node \(i\) \\  & & \(t_{i}^{trans}\) & The transit time of the goods at node \(i\), h \\  & & \(t_{i}^{wait}\) & The waiting time of the goods at node \(i\), h \\  & & \(t_{i}^{leave}\) & The time when the goods leave node \(i\) \\  & & \(P\) & Carbon price, CNY/kg \\  & & \(U\) & Carbon quota, kg \\  & & \(LT\) & The customer can accept the latest arrival time of the goods \\  & & \(\alpha\) & Confidence level \\  & & \(p_{h}\) & Probability of occurrence of transportation scenario \(h\) \\  & & \(\varphi\) & Regret coefficient \\  & & \(Z_{r}\) & The minimum expected function value corresponding to the \\  & & robust optimal solution \\  & & \(C_{h}(x)\) & The total cost corresponding to x in transportation scenario \(h\) \\  & & \(C_{h*}\) & Transportation scenario \(h\) \\  & & Whether transportation mode m was selected between nodes i \\ Decision & & \(x_{ij}^{m}\) & and j. If yes, then \(x_{ij}^{m}=1\), otherwise \(x_{ij}^{m}=0\). \\ variables & & Whether transportation mode m was converted to l at nodes i. If \\  & & yes, then \(y_{t}^{mk}=1\), otherwise \(y_{t}^{mk}=0\). \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._The intra-nodal transfer duration is analytically decomposed into two constituent temporal components: cargo transloading operation time for modal shifts and internal nodal transportation duration within the node's infrastructure, with this operational bifurcation formally formulated through the mathematical expression presented in Equation (2). \[t_{i}^{trans}=\sum_{m\in M}\sum_{k\in M}y_{i}^{mk}\cdot(2\frac{q}{u_{i}^{mk}}+ \frac{l_{i}^{mk}}{v_{i}}),\forall i\in A \tag{2}\]  
The computational process for determining cargo waiting duration within transfer nodes is formally articulated through Equations (3)-(5). The nodal arrival timestamp is mathematically defined as the cumulative sum of departure time from the preceding node and inter-nodal transportation duration.