# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem
## 7 Further Discussions  
This research developed an optimal integrated school bus routing algorithm that serves three different levels of schools in a single framework. In most U.S. cities, the department of public school education covers all three levels of schools--high schools, middle schools, and elementary schools--and the school buses are delivering all three levels of students with three different time windows. In the morning, the school buses depart from the depot, first collecting and then delivering high school students. The buses depart from the high school to pick up middle school students, and after they are taken to school, the buses gather elementary school students and ferry them to school before returning, empty, to the depot. In the afternoon, the buses depart from the depot and go to high schools to pick up students. They deliver all high school students to their homes, then head to middle schools next, and then elementary schools until all students are delivered. After all deliveries, the buses return to the depot. This research considers three levels of schools in three separate time windows in a single framework to optimize the entire routing for two different morning and afternoon time windows. This study has provided three main findings. First, the results of the comparison between the developed integrated single-framework algorithm and the identically reversed routing model with three separated levels showed that 4.5% of the total cost could be saved by using the identically reversed routings with the integrated single-framework algorithm. Different routings for morning and afternoon with the integrated single frame algorithm saved 12.4% compared to the identically reversed routings with the separated framework. This comparison showed that the integrated single-framework algorithm can produce more efficient routings than the separated-framework algorithm. Second, arranging a different order of serving students for two time windows, morning and afternoon, may noticeably change the total costs of the system. In the proposed example of this study, two algorithms were applied; one algorithm generated two routing sets for morning and afternoon time windows, and the other algorithm generated the same (identically reversed) routing sets for morning and afternoon time windows. Obviously, two different routing sets for morning and afternoon time windows minimized total costs compared to the same (identically reversed) routing set since two different routing sets can optimize each time window even though picking up and dropping off locations are identical because buses' origins and destinations (depot and schools) are different for morning and afternoon time windows. For this particular example, total costs with two different routing sets were 8.28% less compared to the single routing set. Lastly, the total cost of a school bus transit service can be highly dependent on the geographic distribution of students. The results of our developed algorithm for the proposed example clearly show that although high schools, middle schools, and elementary schools serve the same number of students (240 students) with the same number of total buses (12 buses), total costs for serving high schools were the highest due to the larger area involved to cover more students per school. Likewise, middle school service (per school) costs more than that of elementary schools because of the larger areas covered with more widely dispersed students. Also, the results show a 1% increase in traveling distance results in approximately a 0.99% increase in the total operating cost. \begin{table}
\begin{tabular}{c c c} \hline
**Total Cost (USD)** & **Separated Models** & **Integrated Framework** \\ \hline The same routing for morning and afternoon (identically reversed) & 2072.05 & 1979.88 \\ \hline Different routing for morning and afternoon & 2001.64 & 1815.96 \\ \hline \end{tabular}
\begin{tabular}{c} 125.96 \\ (Saves about \%3.5) \\ \end{tabular} \\ \hline \end{tabular}
\end{table}