## 3 Methodology
### Construction of Spatio-Temporal-Mode Network for Coal Heavy Rail
A coal demand time window \((t_{E}^{k},t_{L}^{k})\) is also set, so that if the coal arrives earlier than the time window, a storage cost is incurred, and if it arrives later than the time window, a late penalty cost is incurred:  
\[Z_{3}=\sum\
olimits_{k\in K}c_{h}\cdot q^{k}\cdot Op^{k}\cdot max\{t_{E}^{K}- t_{ar},0\}+\sum\
olimits_{k\in K}c_{p}\cdot q^{k}\cdot Op^{k}\cdot max\{t_{ar}-t_{L}^{K},0\} \tag{5}\]  
(6) The total coal transport system costs are written as follows:  
\[Z_{5}=Z_{1}+Z_{2}+Z_{3}+Z_{4} \tag{6}\]  
The constraints of the model include the following:(1) Transport service selection: The \(k\)-th batch of coal from node \(i\) to \(j\) can be transported by only one transport mode in one time zone or not, i.e., at most one transport arc can be selected in each transport arc. \[\sum_{(i,j,t_{ps},t_{pu},m,m)\in A_{tp}}x_{(i,j,t_{ps},t_{pu},m,m)}^{k}\leq 1, \forall k\in K,\forall(i,j)\in A_{pn} \tag{7}\]  
(2) Limit constraint on the number of transits: This indicates that node \(j\) performs at most one transit, i.e., it selects at most one transit arc. \[\sum \begin{array}{c}(j,j,t_{fs},t_{ff},m,n)\in A_{tf}\end{array} y_{(j,i,t_{ fs},t_{ff},m,n)}^{k}\leq 1,\forall k\in K,\forall f\in F \tag{8}\]  
(3) Transit and transport variable coupling constraint: If the mode of transport of coal imported and exported from point \(j\) changes, then transit has taken place at point \(j\). \[1+M\cdot\left(\sum_{(i,j,t_{ps},m)\in V_{j}^{-}}x_{(i,j,t_{ps},t_{pu},m,m)}^{k} +\sum_{(i^{\prime},t_{ps},t_{ps},t_{ps},t_{ps},n,n)}-2\right)\leq y_{(j,j,t_{ fs},t_{ff},m,n)}^{k},\forall k\in K,\forall j\in F,m\
eq n \tag{9}\]  
(4) Coupling constraints obtain between transport variables and actual coal issued variables:  
\[Op^{k}\leq\sum(i,j,t_{ps},t_{pa},m,m)\in A_{tp}\;x_{(i,j,t_{ps},t_{pa},m,m)}^{k},\forall k\in K \tag{10}\]  
(5) The demand for coal at the place of coal consumption fluctuates within a certain range, and the amount of coal that actually arrives by the specified time is within this range:  
\[\sum_{k\in K}\sum \begin{array}{c}(i,j,t_{ps},t_{pa},m,m)\in A_{tp}\end{array} x_ {(i,j,t_{ps},t_{pa},m,m)}^{k}\cdot q^{k}\cdot Op^{k}\leq q_{dmax} \tag{11}\]  
\[\sum_{k\in K}\sum \begin{array}{c}(i,j,t_{ps},t_{pa},m,m)\in A_{tp}\end{array} x_ {(i,j,t_{ps},t_{pa},m,m)}^{k}\cdot q^{k}\cdot Op^{k}\geq q_{dmin} \tag{12}\]  
(6) Network arc capacity constraints: For any arc in an integrated transport network, there are certain capacity constraints on the amount of throughput allocated to a particular arc, especially for railways:  
\[\sum_{k\in K}x_{(i,j,t_{ps},t_{pa},m,m)}^{k}\cdot Op^{k}\leq u_{(i,j,t_{ps},t_ {pa},m,m)},\forall\big{(}i,j,t_{ps},t_{pa},m,m\big{)}\in A_{tp} \tag{13}\]  
(7) According to the multicommodity flow model, it is necessary to realize the flow balance of the origin node, the intermediate node, and the final destination node, which are the coal source, the intermediate node, and the coal consumption place in this model, and it is also necessary to consider the flow balance of the coal storage and distribution base due to the consideration of the storage of coal by the coal storage and distribution base. The coal source place flow balance is written as follows:  
\[\sum_{(i,j,t_{ps},t_{pa},m,m)}\in A_{tp}\;x_{(i,j,t_{ps},t_{pa},m,m)}^{k}- \sum_{(j^{\prime},i,t^{\prime}_{pa},t_{ps},m,m)}\in A_{tp}\;x_{(j^{\prime},i,t^ {\prime}_{pa},t_{ps},m,m)}^{k}=1,\forall k\in K \tag{14}\]  
\(|i=o^{k},(j,t_{pa},m)\in V_{i}^{+}\)\(|i=o^{k},(j^{\prime},t^{\prime}_{pa},m)\in V_{i}^{-}\)  
(8) Coal storage and distribution base flow balance: The inflow and outflow of coal from the coal storage and distribution base is both the transfer of coal from the coal source and the export of coal to the downstream consumption place, with the remaining coal being saved in stockpiles:\[