## 3 Proposed Methods
### Container Multimodal Transport Model
Based on the above parameter definitions, this study constructs a container intermodal transport path optimization model for inland ports within a given planning period. Considering practical constraints related to cost, transport time windows, and carbon emissions, the objective function is defined as shown in Equation (9). \[Z=\omega_{1}C+\omega_{2}T+\omega_{3}E=\omega_{1}\big{(}C_{w}+C_{t}+C_{p}+C_{ ct}\big{)}+\omega_{2}(T_{s}+T_{r})+\omega_{3}\Big{(}E_{f}+E_{m}\Big{)}, \tag{9}\]  
where \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) denote the weighting coefficients for total transport costs, transport time, and carbon emissions. To ensure path uniqueness, the model restricts the selection to at most one transport mode between any two nodes, as specified in Equation (10). \[\sum_{k\in K}X_{ij}^{k}\leq 1\hskip 56.905512 pt\forall i,j\in N,\ \forall k\in K. \tag{10}\]  
The number of mode transitions per node is limited to one, as reflected in Equation (11), thereby avoiding multiple conversions between transport modes at the same node. \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]0The coherence of transport modes is maintained by Equation (12), which ensures that if a mode transition occurs at node \(i\), the inbound and outbound transport modes are logically consistent. \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]1  
No transport mode conversion is allowed at the origin or destination node, as specified in Equation (13). \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]2  
The non-negativity and normalization of weight parameters are specified in Equation (14), which ensures that all weights range between 0 and 1 and their total equals one. \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]3  
To control emissions, Equation (15) limits total carbon emissions during the transport process, keeping them within the prescribed upper threshold. \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]4  
The arrival time of goods at node \(j\) is calculated based on the departure time from node \(i\), the transition time, and the travel time, as shown in Equation (16). \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]5  
The total transport time is constrained to remain within a predefined hard time window, as indicated in Equation (17). \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]6  
The binary nature of the decision variables is specified in Equations (18) and (19), where both \(X_{ij}^{k}\) and \(Y_{i}^{kh}\) are restricted to values of 0 or 1. \[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]7  
\[C_{t}=\sum_{i\in N}\sum_{k\in N}\sum_{h\in K}qc_{i}^{kh}Y_{i}^{kh}, \tag{2}\]8  
The balance of cargo flow is maintained by Equation (20), which ensures that the inflow and outflow at each node remain equal throughout the scheduling process. \[\sum_{i,j\in N,k\in K}X_{ij}^{k}-\sum_{i,j\in N,k\in K}X_{ij}^{k}\left\{ \begin{array}{ll}1,&j=d\\ 0,j\in N\backslash\{o,d\}\\ -1,&j=o\end{array} \right. \forall i,j\in N,\forall k\in K. \tag{20}\]  
The proposed model integrates soft time window constraints, alongside time penalty costs and carbon tax considerations, to balance the economic and environmental objectives. It includes a comprehensive set of constraints that guarantee the selection of feasible transport modes, compliance with time constraints, control of carbon emissions, and balance of flow. This approach establishes a strong foundation for optimizing intermodal transport scheduling with a focus on both cost efficiency and low-carbon objectives.