## 3 Proposed Methods
### Container Multimodal Transport Model  
In intermodal container scheduling at inland ports, containers are typically transported directly from the yard to the railway yard or inland waterway terminal by road single-vehicle transport, without intermediate transshipment. To simplify cost calculation, this paper combines pickup and yard-to-yard operations into a single transshipment operation. Based on the actual scheduling process, the model divides the operation costs into three components, which include direct transport cost \(C_{w}\), nodal transshipment cost \(C_{t}\), and time penalty cost \(C_{p}\). This categorization simplifies cost calculation, enhances the model's practicality, and incorporates constraints related to transportation costs, time efficiency, and carbon emissions, thereby aligning more closely with the practical needs of multimodal transport scheduling optimization. \[C_{w}=\sum_{i\in N}\sum_{j\in N}\sum_{k\in K}qd_{ij}^{k}c_{ij}^{k}x_{ij}^{k}, \tag{1}\]  
\[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]  
\[C_{p}=qs_{1}\sum_{i\in N}max(E_{i}-T_{i},0)+qs_{2}\sum_{i\in N}max(T_{i}-L_{i},0). \tag{3}\]  
In this context, \(N\) represents the set of nodes, and \(K\) denotes the set of transport modes. The parameter \(q\) indicates the container demand, while \(d_{ij}^{k}\) and \(c_{ij}^{k}\) refer to the distance and per-unit cost between nodes \(i\) and \(j\) under mode \(k\), respectively. The binary variable \(X_{ij}^{k}\) indicates whether mode \(k\) is selected between nodes \(i\) and \(j\). The term \(c_{i}^{kh}\) denotes the per-unit transition cost from mode \(k\) to mode \(h\) at node \(i\), with \(Y_{i}^{kh}\) representing the corresponding binary transition variable. The pair \((E_{i},L_{i})\) defines the soft time window bounds, \(T_{i}\) is the actual arrival time at node \(i\), and \(s_{1}\) and \(s_{2}\) denote the unit penalty costs for early and late arrivals, respectively. Carbon emissions \(E\) generated during intermodal transport arise from two primary sources. The first is fuel consumption by transport modes in operation, represented as \(E_{f}\). The second source is the use of mechanized equipment in transshipment processes, represented as \(E_{m}\). \[E_{f}=\sum_{i\in N}\sum_{j\in N}\sum_{k\in K}qd_{ij}^{k}e_{k}X_{ij}^{k}, \tag{4}\]  
\[E_{m}=\sum_{i\in N}\sum_{k\in K}\sum_{h\in K}qf_{i}^{kh}Y_{i}^{kh}, \tag{5}\]  
where \(e_{k}\) represents the unit carbon emission factor of mode \(k\), and \(f_{i}^{kh}\) denotes the unit carbon emission factor associated with transitioning from mode \(k\) to \(h\) at node \(i\). In addition to mandatory carbon emission policies, a carbon tax is a key tool for promoting emission reductions by enterprises. By translating carbon emissions into economic costs through carbon tax pricing, the transition to low-carbon transportation can be effectively supported. \[C_{ct}=\rho E=\rho q\left(\sum_{i\in N}\sum_{j\in N}\sum_{k\in K}d_{ij}^{k}e_{ k}X_{ij}^{k}+\sum_{i\in N}\sum_{k\in K}\sum_{h\in K}f_{i}^{kh}Y_{i}^{kh} \right), \tag{6}\]  
where \(\rho\) denotes the carbon tax rate. The expression incorporates both operational and transshipment-related emissions, converting them into monetary terms to reflect their economic impact within the scheduling model. The total transport time \(T\) consists of two components. The first is the travel time of the transport vehicle \(T_{s}\), which represents the duration of goods in transit. The second is the transit time at the transport node \(T_{r}\), which reflects the efficiency of node operations. \[T_{s}=\sum_{i\in N}\sum_{j\in N}\sum_{k\in K}\frac{d_{ij}^{k}}{v_{k}}X_{ij}^{ k}, \tag{7}\]  
\[T_{r}=\sum_{i\in N}\sum_{k\in K}\sum_{h\in K}q\tau_{i}^{kh}Y_{i}^{kh}, \tag{8}\]  
where \(v_{k}\) denotes the average speed of travel using the \(k\)th transport mode, and \(t_{i}^{kh}\) represents the transit time between nodes \(i\) and \(j\) for the selected \(k\)th mode.