## 3 Methodology
### Notations
\begin{tabular}{c c} \hline \hline \(G(N,A,M)\) & Three-dimensional space-time network \\ \hline \multirow{5}{*}{\(N\)} & A collection of spatio-temporal network nodes, \(\left(i,t_{ps},m\right)\), \(\left(j,t_{pu},m\right)\), \(\left(j,t_{fs},m\right)\), \\  & \(\left(j,t_{ff},m\right)\), \(\left(j,t_{us},n\right)\), \(\left(j,t_{us},n\right)\)\(\in N\) \\  & The set of arcs \(\left(i,j\right)\in A_{pn}\), \(\left(i,j,t_{ps},t_{pa},m,m\right)\in A_{tp}\), \(\left(j,j,t_{us},t_{us},t_{wf},n,n\right)\in A_{w}\), \\  & \(\left(j,j,t_{fs},t_{ff},m,n\right)\in A_{tf}\) denote the physical network transport arcs, the transport arcs in spatio-temporal networks, the waiting arcs, and the transit arcs, respectively. A collection of modes of transport in spatio-temporal networks, \(m_{1},m_{2},m_{3}\in M\), \\  & \(M\) & \(m_{1}\)—railway transport, \(m_{2}\)—waterway transport, \(m_{3}\)—road transport. \(m,n\) \\ Sets & & indicate different modes of transport \\  & \(T\) & Moments in the transport space-time network, \(t_{ps},t_{pa},t_{fs},t_{fa},t_{us},t_{wf}\in T\). The time slot is 15 min \\ \hline \multirow{5}{*}{Arcs} & \(TT\) & Moment where coal is consumed, time period 1 h \\  & \(T\) & The set of all coal batches, \(k\in K\). Each coal batch contains the following \\  & \(K\) & information: OD pairs, the amount of coal shipped in that batch, and the arrival time window \\  & \(V_{j}^{-}\) & The set of successor nodes of node \(j,j\in N\) \\  & \(V_{j}^{+}\) & The set of successor nodes of node \(i,j\in N\) \\  & \(F\) & Collection of transit nodes \\  & \(S\) & Aggregation of coal storage and distribution bases \\ \hline \multirow{5}{*}{Arcs} & \(\left(i,j\right)\) & Physical network transport arc \\  & \(\left(i,j,t_{ps},t_{pa},m,m\right)\) & Transport arc \\  & \(\left(j,j,t_{fs},t_{ff},m,n\right)\) & Intermediate arc \\  & & Waiting arcs, due to the transit of transport modes, often need to wait for the next \\  & \(\left(j,j,t_{us},t_{wf},n,n\right)\) & train to start or the next route to start, i.e., waiting arcs for node \(j\) wait for \\  & \(\left(s,s,t,t+1\right)\) & Storage arc \\ \hline \multirow{5}{*}{Parametrics} & \(h_{maxd}\) & Maximum stockpile capacity (in tons) where coal is consumed \\  & \(\alpha^{k}\) & Source of coal for lot k coal \\  & \(\delta^{k}\) & Consumption of coal from lot k \\  & \(q^{k}\) & Weight of coal from lot k, in tons \\  & \(\left[t_{L}^{k},t_{L}^{k}\right]\) & Time window in which the k-th instalment of coal must be delivered to the place \\  & \(c_{sh}\) & Time unit and coal storage unit and distribution base storage costs (in \\  & \(\text{CNY/hour-ton}\)) \\  & \(c_{th}\) & Cost of storage per unit of time per unit of coal consumed (in \(\text{CNY/hour-ton}\)) \\  & \(\text{Transport arc }\left(i,j,t_{ps},t_{pa},m,m\right)\) & cost per unit weight per unit distance (in \\  & \(c_{\left(i,j,t_{ps},t_{pa},m,m\right)}\) & \(\text{CNY/ton-km}\)) \\  & \(E_{\left(i,j,t_{s},m,m\right)}\) & Transport arc \(\left(i,j,t_{ps},t_{pa},m,m\right)\) capacity, (in tons) \\  & \(T_{\left(i,j,t_{s},m,m\right)}\) & Transport arc \(\left(i,j,t_{ps},t_{pa},m,m\right)\) transport time (in units of one time slot, i.e., 15 min) \\  & \(c_{\left(j,j,t_{s},t_{ff},m,n\right)}\) & Transit arc \(\left(j,j,t_{fs},t_{ff},m,n\right)\) transit cost per unit weight (in \(\text{CNY/hour-ton}\)) \\  & \(T_{\left(j,j,t_{ps},t_{ff},m,n\right)}\) & Transit arc \(\left(j,j,t_{fs},t_{ff},m,n\right)\) transit time (in time slots) from transport mode m to n \\  & \(t_{s}\left(i,t_{ps},m\right)\) & Moments corresponding to spatio-temporal nodes \(\left(i,t_{ps},m\right)\) from node \(i\) \\  & \(t_{a\left(i,t_{pa},m\right)}\) & The moment corresponding to the spatio-temporal node \(\left(j,t_{pa},m\right)\) arriving at node \(j\) \\  & \(t_{f\left(j,t_{ff},n\right)}\) & The spatio-temporal node \(\left(j,t_{ff},n\right)\) corresponding to the moment when the \\  & \(t_{f\left(j,t_{ff},n\right)}\) & transit is completed at node \(j\) \\ \hline \hline \end{tabular}