# Modular Bus Unit Scheduling for an Autonomous Transit System under Range and Charging Constraints
## 4 Computational Results
### Results and Analyses  
Table 3 illustrates the optimal scheduling scheme for MB units in instance (2), where a total of 27 modular buses operates on 18 trip sequences to fulfill the demand of 30 trips. Note that in the \"Trip Sequence\" column, the numbers 1-30, 31-60, and 61-90 represent the 30 trip nodes, 30 temporary depot nodes, and 30 time-extended charging nodes, respectively. The symbols \(o\) and \(s\) are the source and sink, referring to the same depot. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Instance**} & \multicolumn{3}{c}{**GUROBI**} & \multicolumn{3}{c}{**CGBH**} \\ \hline
**Number** & **Trip** & **Node** & **LB (\$)** & **Obj (\$)** & **Time (s)** & **Gap (\%)** & **LB (\$)** & **Obj (\$)** & **Time (s)** & **Gap (\%)** \\ \hline (1) & 10 & 52 & 375.45 & 375.45 & 43.13 & 0.00 & 375.45 & 375.45 & 0.89 & 0.00 \\ (2) & 30 & 92 & 1106.22 & 1147.16 & \textgreater{}7200 & 3.57 & 1144.26 & 1144.26 & 18.93 & 0.00 \\ (3) & 60 & 184 & 2166.27 & 2438.63 & \textgreater{}7200 & 11.17 & 2213.68 & 2217.35 & 464.87 & 0.17 \\ (4) & 90 & 214 & 3094.25 & 4113.42 & \textgreater{}7200 & 24.78 & 3207.08 & 3215.24 & 824.40 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison results of the CGBH algorithm and GUROBI. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multirow{2}{*}{**Trip Sequence**} & \multicolumn{2}{c}{**Number of Units Equipped**} & \multicolumn{2}{c}{**Cumulative Energy**} & \multirow{2}{*}{**Cost (\$)**} \\  & & & **with MB Platoon** & & **Consumption (kWh)** & \\ \hline
1 & _o_-1-7-14-84-21-27-\(s\) & 3 & 45.37 & 155.23 \\
2 & _o_-3-9-80-18-26-30-\(s\) & 1 & 15.76 & 53.25 \\
3 & _o_-2-7-48-51-54-88-25-29-\(s\) & 1 & 15.01 & 48.18 \\
4 & _o_-4-70-10-17-59-26-30-\(s\) & 1 & 18.13 & 50.89 \\
5 & _o_-6-11-18-24-\(s\) & 1 & 18.34 & 39.57 \\
6 & _o_-3-41-13-19-\(s\) & 1 & 17.37 & 34.6 \\
7 & _o_-3-70-10-17-23-28-\(s\) & 3 & 52.26 & 152.08 \\
8 & _o_-5-11-52-55-22-\(s\) & 1 & 15.87 & 32.79 \\
9 & _o_-6-12-54-22-\(s\) & 2 & 33.6 & 67.38 \\
10 & _o_-5-11-18-\(s\) & 1 & 15.76 & 32.91 \\
11 & _o_-15-20-26-30-\(s\) & 3 & 57.16 & 116.52 \\
12 & _o_-9-16-22-\(s\) & 1 & 16.46 & 32.02 \\
13 & _o_-2-8-14-84-57-25-29-\(s\) & 1 & 15.01 & 53.34 \\
14 & _o_-5-11-18-24-\(s\) & 1 & 19.05 & 40.02 \\
15 & _o_-6-12-54-57-60-26-30-\(s\) & 1 & 19.05 & 41.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal modular bus scheduling scheme. First, compared to a traditional electric bus scheduling plan where one bus covers a single trip, the MB scheduling scheme reduces costs from $1534.31 to $1144.26, a decrease of approximately 25%, calculated by \(\left(1534.31-1144.26\right)/1534.31\times 100\%=25\%\), effectively cutting operating expenses for the transit system. Second, as elucidated in \"Trip sequence\", all trips can be categorized into two types. The first type encompasses trips that are covered by multiple sequences, exemplified by Trip 1 being served by both Sequence 1 and Sequence 17. This underscores the advantage of the coupling/decoupling mechanism of modular buses, wherein the consolidation of three MBs and one standalone MB results in four MB units effectively executing Trip 1, followed by their separation to serve subsequent trips. The other type of trip can be fully covered by a single MB platoon. For example, both Trip 21 with Demand 35 and Trip 27 with Demand 42 are only present in Trip Sequence 1, being covered directly by three modular buses with Capacity 45. Additionally, from the usage of charging nodes, the maximum number of MBs served in the occupied charging nodes 68, 70, 77, 80, 84, and 88 is 4, which does not exceed the maximum capacity of 8. Finally, we further verify the effectiveness of the trip sequence. Taking Sequence 1 as an example, a modular platoon with three MB units departs from node \(o\), executes trips 1, 7, and 14 in order, replenishes power at charging node 84, and subsequently executes Trips 21 and 27, finally returning to depot \(s\).