## 3 Methodology
### Problem Statement
A three-dimensional spatio-temporal state network can provide a clearer description of the problem, and the problem can be modelled and solved by adding the required attributes. This paper studies the coal transport mode and path selection problem under the long-distance transport system of the coal supply chain, so the transport mode is added to the spatio-temporal network as the third dimension, and the coal transport nodes at the fixed departure moments of railway transport and waterway transport are taken as the spatio-temporal points so as to construct a three-dimensional spatio-temporal network of time-space-transport mode and to describe and study the problem. On the basis of the physical network, the time dimension and transport-mode selection dimension are added to construct the time-space-transport-mode three-dimensional spatio-temporal network \(G(N,A,M)\), where \(N\) denotes the set of spatio-temporal points, \(A\) denotes the set of spatio-temporal arcs, and \(M\) denotes the set of transport modes. \(i\), \(j\) denotes the serial number of the physical network node, \(t_{s},t_{A}\) denotes the time serial number, and \(m,n\) denotes the transport-mode serial number. A simple three-bit spatio-temporal network schematic is shown in Figure 3, including three transport modes--road, railway, and water transport. This paper explores the spatio-temporal network and finds that the main nodes are the following two:  
1. Departure node: \(\left(i,t_{ps},m\right)\) indicates that coal starts to be transported from point \(i\) at the moment \(t_{ps}\) by transport mode \(m\). There are also departure points \(\left(i_{1},t_{2},m_{3}\right)\), \(\left(i_{1},t_{2},m_{2}\right)\), etc., in the diagram. 2. Arrival node: \(\left(j,t_{ps},m\right)\) arrives at node \(j\) at the moment \(t_{pa}\) by transport mode \(m\), and there are other departure points in the graph, such as \(\left(i_{3},t_{3},m_{3}\right)\), \(\left(i_{3},t_{3},m_{2}\right)\), and so on. The spatio-temporal arc connecting the spatio-temporal nodes \(\left(i,t_{ps},m\right)\) and \(\left(j,t_{pa},m\right)\) is denoted as \(\left(i,j,t_{ps},t_{pa},m,m\right)\), which indicates that it enters into the spatial arc \(\left(i,j\right)\) at the moment \(t_{ps}\) with mode of transport \(m\) and ends with mode of transport \(m\) at the moment \(t_{pa}\). According to the transport process under the integrated coal transport system, the spatio-temporal arcs of the spatio-temporal state network in this paper mainly include the following kinds:  
1. Transport arc: \(\left(i,j,t_{ps},t_{pa},m,m\right)\) denotes the transport arc, such as the arc from the space-time node \(\left(i_{1},t_{2},m_{2}\right)\) to \(\left(i_{3},t_{3},m_{2}\right)\), i.e., the \(m\) transportation mode is used to depart from \(i_{1}\) at the moment \(t_{2}\) and arrive at node \(i_{3}\) at the moment \(t_{3}\). Figure 3: Time–space–transport network diagram. 2. Transit arc: \((j,j,t_{fs},t_{ff},m,n)\) denotes the transit arc, \((i_{3},i_{3},i_{3},t_{4},m_{2},m_{3})\) denotes that the \(m_{2}\) transport mode arrives at the \(i_{3}\) point at the moment \(t_{3}\) and starts the transport mode transit to \(n\), and the transport mode transit is completed at the moment \(t_{4}\). 3. Waiting arc: \((j,j,t_{ws},t_{wf},n,n)\) represents the waiting arc. Due to the transport mode, it is often necessary to wait for the next train to start or for the next class of routes to start, and this waiting for the start of the state that node \(j\) transit is completed waiting for the departure of the waiting arc. \((i_{3},i_{3},t_{4},t_{5},m_{3},m_{3})\) means that after completing the transit at point \(i_{3}\) at time \(t_{4}\), the departure is waited for at time \(t_{5}\). 4. Storage arc: \((s^{t},s^{t+1})\) denotes a storage arc, which indicates that coal is stored at the same location for a period of time at different time intervals, and \(\left(s^{t},s^{t+1}\right)\) indicates that coal is stored for 1 unit of time at the point \(s\).