## 3 Methodology  
### Problem Statement  
In this section, we examine the three-level coal supply chain network of coal sources, coal storage and distribution bases, and consumption sites. A coal storage and distribution base serves as a critical hub for centralized coal warehousing, transportation, and deployment, addressing the issue of long transportation distances for coal supplied from a coal source. These bases are strategically located with respect to coal production and distribution areas, consumption concentration, major railroad transportation nodes, and key receiving and discharging ports. The layout and abstraction of this transportation network are illustrated in Figure 1. Large-scale coal enterprises generally have self-built railroad lines connecting coal mines to demand points, with railroad transportation being the primary mode of transport. Coal is typically transported southward through ports, with a small portion handled by short-distance road transportation. At the consumption point, coal consumption can be predicted per unit hour. The total cost of the whole system includes inventory costs, transportation costs, transit costs, and arrival-time penalty costs across the three nodes of the coal supply chain. For different transportation demands, the coal requirement at a demand point can be met either directly from the coal source or from the coal storage and distribution base. Therefore, different transportation schemes and modes, as well as path selections, directly impact the transportation cost and arrival time, resulting in varying inventory costs. The entire coal collection and transportation system needs to consider minimizing overall costs while continually optimizing the transportation scheme. The transportation optimization problem is studied in this section to simplify the complex coal transportation problem into a path integer planning problem within a three-level supply chain system, comprising multiple coal sources, multiple storage and distribution bases, and multiple coal consumption places. The objective is to minimize the total system cost while considering capacity constraints and inventories at each node. Ultimately, an optimized scheme for coal path selection and flow distribution is obtained, taking into account multilevel node inventories. A spatio-temporal network can more clearly and accurately represent the spatio-temporal displacement of coal flow and vehicle flow in a network, as shown in the figure below. The transport spatio-temporal network diagram contains a number of coal sources (\(o\)), coal storage and distribution bases (\(s\)), and coal consumption places (\(d\)). The coal can be transported directly from the source to the coal consumption place, for instance, by transporting the coal through the transport arc \((o_{4},d_{1},T_{1},T_{3})\), which departs from \(o_{4}\) at the moment \(T_{1}\) and arrives at \(d_{1}\) at the moment \(T_{3}\). It can be stored for a certain period of time at the storage and distribution coal base and then transported to the coal consumption place. It can also be stored in the storage and distribution base for a certain period of time and then transported to the coal consumption place, for example, it can first be transported to the storage and distribution base through the transport arc \((o_{3},s_{1},T_{1},T_{2})\), then stored in the storage arc \((s_{1},s_{1},T_{2},T_{3})\), and then transported to the coal consumption place through the transport arc \((s_{1},d_{3},T_{3},T_{4})\). Taking this graph network as a prototype, the coal transfer optimization model is established, as shown in Figure 2.  
Figure 1: Coal multimodal transportation network. Figure 2: Coal transportation network topology. For more complex problems in practical applications, two-dimensional spatio-temporal networks can be further expanded into three-dimensional spatio-temporal state networks by adding state dimensions such as speed, capacity, and mode of transport.