## 3 Methodology
### Construction of Spatio-Temporal-Mode Network for Coal Heavy Rail  
The steps for constructing the spatio-temporal network are as follows:  
1. Set the relevant parameters for each batch of coal \(k\in K\). Add the start point, \(i=\phi^{k}\), and the end arrival point, \(d^{k}\), for \(k\) batches of coal, while each batch of coal is assigned a time window for the earliest dispatch time and the latest arrival time, \([k,t_{l}^{k}]\), and a weight, \(q^{k}\). 2. Add running arc segments \(\big{(}i,j,t_{ps},t_{pa},m,m\big{)}\). An interval running arc represents the process of coal being sent from station \(i\) at the moment \(t_{ps}\), transported by transport mode \(m\), and arriving at station \(j\) at the moment \(t_{pa}\), and the transport arc is determined by the departure and arrival moments of the trains running through the stations, among which the railway transport and shipping are determined by the timetable, and the road transport is relatively more flexible in terms of the departure time, so when generating the transport arc segment, it generates the road transport arc that departs at each time slot, and each time slot is 15 min. 3. Add the transit arc \((j,j,t_{fs},t_{ff},m,n)\). The transit arc represents the process of coal being transported to transit node \(j\) by transport mode \(m\) at the time \(t_{fs}\) and then waiting for the next train of transport mode \(n\) to be sent out from the station at the time \(t_{ff}\) after the completion of the transit operation. Introducing the transit succession time window, \([t_{f}^{lhs},t_{f}^{rhs}]\), which needs to be satisfied for any batch of coal, ensures the minimum duration of the transit operation of different transport modes, \(t_{f}^{lhs}\), and also restricts the maximum residence time of coal in the station, \(t_{f}^{rhs}\). When the transport modes of the front and rear transport arcs are different at transit node \(j\) and the transit time constraint is satisfied, then the transit operation will be carried out at that point and the next train of transport mode n is sent from the station at \(t_{ff}\). Then, a trans-shipment operation is performed at that point. 4. Add storage arc segments \((s,s,t,t+1)\). Due to the long distance of coal transport, the time limit for transport is high. In the coal transport network, there is a coal storage and distribution base to ensure a certain amount of coal stockpiling. The storage arc indicates the coal in the coal storage and distribution base \(m\) at moment \(t\) for a given time period. Coal can be stored for more than one time period in a coal storage and distribution base. A Simultaneous Model for Inventory and Schedule Optimization Considering Production, Transportation, and Sales  
The optimization objective of this model is to minimize the cost generated by the coal logistics system as a whole, which is composed of three parts: the first one is the transportation cost, the second one is the transit cost, and the third one is the inventory cost. Coal is transported from the coal source to the consumer end of deployment. The main modes of coal transport are railroad transport and waterway transport. The end of a railroad line is connected to a factory, or other consumer places can be reached by road, which constitute a small part of the overall transport network. The main routes are as follows: (1) coal source--railway transportation--port--water transportation--consumption area--neighboring power plants; (2) coal source--direct railway transportation--consumption area--neighboring power plants. For transport mode (1), the overall transport cost includes the cost of railroad and water transport, but also includes the cost of transit at the port. The advantage of this mode is that the cost of sea transport is lower, which can greatly save on the cost of transport. For transport mode (2), the organization is relatively simple compared with sea transport; railroad transport is faster, and it can avoid the additional cost of transit due to the rail and sea transit at a port.