# Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line
## 3 Methodology
### Algorithm
Each color corresponds to a distinct bus. As shown, the blue and red buses are relocated to another station. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Train/** \\ **Station** \\ \end{tabular}  } & \multicolumn{3}{c}{**Station A**} & \multicolumn{3}{c}{**Station B**} & \multicolumn{3}{c}{**Station C**} & \multicolumn{3}{c}{**Station D**} \\ \cline{2-13}  & & \multicolumn{3}{c}{**Average Direct**} & \multirow{2}{*}{**B**} & \multirow{2}{*}{**A**} & \multicolumn{2}{c}{**Average Direct Travel**} & \multirow{2}{*}{**B**} & \multirow{2}{*}{**A**} & \multicolumn{2}{c}{**Average**} & \multirow{2}{*}{**Average**} & \multirow{2}{*}{**B**} & \multirow{2}{*}{**A**} & \multicolumn{2}{c}{**Average Direct Travel**} \\  & & & \multicolumn{3}{c}{**Travel Distance (km)**} & & & & & & & & & & **Distance (km)** \\ \hline Bus set 1 & 1 & 1 & 1.01 & 1 & 1 & 1.01 & 1 & 1 & 1.01 & 1 & 1 & 1.01 & 1.01 \\ \hline Bus set 2 & 3 & 1 & 1.01 & 4 & 2 &



Figure 5: Illustration for the feeder bus schedules and movements. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Station**} & **Total Vehicle** & **Total Passenger** & **Average Passenger** & **Average Total** & **Average Total** & **Average Total** \\  & **Traveled** & **Passenger** & **Distance Traveled** & **Passenger Average** & **Passenger** & **Passenger** \\  & **Distance (km)** & **Travel Time** & **to each Station** & **Distance Traveled** & **Travel Cost** & **Bus Operating** \\  & **Distance (km)** & **(km)** & **(km)** & **(km)** & **(USD)** & **Cost (USD)** \\ \hline \#1 & 67.69 & 3.09 & 1.66 & & & \\ \#2 & 48.31 & 2.51 & 1.47 & & & \\ \#3 & 70.92 & 3.08 & 1.90 & & & \\ \#4 & 64.34 & 3.13 & 1.50 & & & \\ \hline \end{tabular}
\end{table} 
Table 3: The results of computations for the model. ## 6 Discussion and Conclusions

This study aimed to create a routing algorithm for a demand-responsive feeder transit system aligned with short headway mass transit. Building on a previous model [8] designed for long headway trunk lines in less dense areas, the new routing accommodates more frequent train services. Developing a similar algorithm for use in more congested urban areas motivated the authors to develop a new specialized algorithm where the headway of metro trains is much shorter. Therefore, a 5 min headway was considered in the hypothetical network of this study. When the headway of the rail service is long enough for the feeder buses to come back by the next train, then the feeder network algorithm is rather easy because the maximum feeder service cycle time is determined by the rail headway, and matching between feeder buses and the trains is not necessary. However, if the headway of the rail service is not long enough for the feeder buses to return before the next train, then the algorithm should find not only matching between passengers and feeder buses but also matching between feeder buses and trains because the bus can deliver passengers any time as long as it is no later than their requested time. This makes fundamental changes in the algorithm. In this study, a simulated annealing (SA) algorithm was developed for flexible feeder bus routing on short trunk lines, taking into account the relocation of buses across multiple stations. The objective function of the model incorporated both operating costs and passenger travel expenses. The proposed algorithm effectively managed bus relocations when optimal routing was infeasible due to bus availability constraints at specific stations. For future research, it is recommended to develop a feeder bus routing algorithm for trains with equivalent short headways, including temporary stops for feeder buses. Furthermore, incorporating composite heuristics for the larger and real networks and exploring the use of more advanced metaheuristics within the algorithm are suggested. Conceptualization, Y.-J.L. ; methodology, Y.-J.L. and A.N. ; software, A.N. ; validation, A.N. and Y.-J.L. ; formal analysis, A.N. ; investigation, A.N. and Y.-J.L.