# Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints
## 2 Modular Bus Scheduling Model  
### Preliminaries  
All notations involved in the model are listed in Abbreviations. The modular bus unit scheduling problem under range and charging constraints can be formally defined as follows. Given a directed graph \(G=(N,A)\), the set of nodes \(N=\{o\}\cup D\cup K\cup F\cup\{s\}\) is partitioned into the source node \(o\) (depot departure), the set \(D\) of dummy depot nodes providing temporary stay for buses, the set \(K\) of trips, the set \(F\) of time-expanded charging stations, and the sink node \(s\) (depot arrival). Each node \(i\in N\) is equipped with the start time \(v_{i}\), end time \(w_{i}\), node demand \(d_{i}\), node distance \(l_{i}\) and node energy consumption \(p_{i}\). Without loss of generality, we assume that the demand, distance, and energy consumption of node \(i\in N\backslash K\) are equal to zero, i.e., \(d_{i}=0,l_{i}=0\), \(p_{i}=0,i\in N\backslash K\). The arc set \(A\) contains feasible arcs linking different nodes \(i\) and \(j\), denoted as \((i,j)\in A\). Each arc (\(i\),\(j\)) has an idle time \(t_{ij}\) and an arc energy consumption \(p_{ij}\). The connection rules of feasible arcs are shown in Table 1, distinguished between different nodes. The preprocessing of arcs can effectively reduce unnecessary connections and improve the model's efficiency. The feasibility condition of \(\arc(i,j)\in A,i,j\in D,\ i\
eq j\) between virtual depot nodes is \(w_{i}=v_{j}\) since all virtual depot nodes targeting the same depot do not have spatial location transfer, the dwell time for MB should be continuous. The arc leading from the origin \(o\) only makes sense for the case of going directly to the trip set \(K\) with the feasible condition \(w_{o}+t_{oj}\leq v_{j}\). Similar connection rules exist for the arc to the endpoint \(s\). For the feasibility condition \(v_{j}-t_{b}\leq w_{i}+t_{ij}\leq v_{j}-t_{a}\), in addition to being governed by the end time \(w_{i}\) of the previous node \(i\), the start time \(v_{j}\) of the subsequent node \(j\) and the idle time \(t_{ij}\), it also comprises the waiting threshold \(t_{b}\) introduced to prevent the nodes from being visited prematurely, and the time \(t_{a}\) to ensure the smooth coupling/decoupling action between the MB units. An example consisting of five node types \(o,D,K,F,s\), and some feasible arcs, is used to illustrate the process of constructing the underlying network, as depicted in Figure 2. The travel time along each arc, the duration of nodes, and the sampling interval for charging nodes and virtual charging nodes are all set to one unit. A maximum waiting time of 2 units is allowed outside the depot, while the coupling and decoupling time for vehicles is set to 0.5 units. Starting from the source node \(o\), two sets of trip sequences connected by feasible arcs are generated according to the connectivity rules. In the sequence \(o-k_{1}-k_{3}-d_{6}-d_{7}-k_{6}-s\), trips \(k_{3}\) and \(k_{6}\) are not directly connected because \(11-2\geq 5+1\), which would result in prolonged waiting times for MBs outside the depot. In contrast, modular buses can briefly pause at the virtual depot node after completing trip \(k_{3}\) and then proceed to complete trip \(k_{6}\). It is important to note that the virtual nodes \(d_{6}\) and \(d_{7}\) are both associated with the same depot, thus satisfying condition \(8=8\). \begin{table}
\begin{tabular}{l c c c c c} \hline \(i\)\(j\) & \(o\) & \(D\) & \(K\) & \(F\) & \(s\) \\ \hline \(o\) & \(\times\) & \(\times\) & \(w_{o}+t_{oj}\leq v_{j}\) & \(\times\) & \(\times\) \\ \hline \(D\) & \(\times\) & \(w_{i}=v_{j}\) & \(v_{j}-t_{b}\leq w_{i}+t_{ij}\leq v_{j}-t_{a}\) & \(\times\) \\ \hline \(K\) & \(\times\) & \(v_{j}-t_{b}\leq w_{i}+t_{ij}\leq v_{j}-t_{a}\) & & \(w_{i}+t_{is}\leq v_{s}\) \\ \hline \(F\) & \(\times\) & \(v_{j}-t_{b}\leq w_{i}+t_{ij}\leq v_{j}-t_{a}\) & \(\times\) & \\ \hline \(s\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) \\ \hline \end{tabular}
\end{table} 
Table 1: The connection rules of feasible arcs.