# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem
## 6 Results  
The algorithm successfully generated the school bus routings for both morning (AM) and afternoon (PM) periods. The following Figure 4, as an example, shows the routings in the afternoon period for middle school M1. In these figures, Figure 4a shows the location of students, Figure 4b represents the optimal routing when the routing is the same (identically reversed) for morning and afternoon time windows, Figure 4c represents the optimal routing for the morning time window, and Figure 4d shows the optimal routing for the afternoon time window. The proposed algorithm provided a solution that optimally minimizes the total costs, including both the travel time of students and school bus operations costs. One of the main complaints of parents whose children use school buses in the U.S. is that their children have to remain on the bus for a long time while their homes are relatively close to the school. To address this issue, the school buses should have different routes for morning and afternoon time windows with different picking up and dropping off sequences that can decrease travel distance and the operating costs of the public school bus service. Although the amount of cost savings depends on the distribution of students and the scale of the  
Figure 3: The locations of the schools in the hypothetical network. Figure 2: Simplified conceptual operation of school buses in afternoon trips. problem, theoretically, the total costs of the public school bus operation would be reduced when it comes to choosing different routes for the morning and afternoon time windows. To compare with the developed integrated single-framework algorithm, identically reversed routings with three separated levels were generated, and their costs were computed as shown in Table 3. The computed total cost was USD 2072.05, and it was proved that USD 92.17 could be saved by the identically reversed routings with the integrated single-framework algorithm, which is equivalent to 4.5% savings. Different routings for morning and afternoon with an integrated single-frame algorithm saved USD 256.09 compared to identically reversed routings with a separated framework, which came out to 12.4% savings. This comparison shows that the integrated single-framework algorithm can produce more efficient routings than the separated-framework algorithm. Moreover, in the present example of this study, the total costs of the integrated routing framework for different morning and afternoon time windows are 8.28% less than the same routings (identically reversed) for morning and afternoon time windows, while the bus operating cost is 1.67% higher when we implement the integrated routing framework for the different routings. The results of the analysis also show that although high schools, middle schools, and elementary schools serve the same amount of students (240 students) with the same number of total buses (12 buses), the total cost for high schools is more than that of middle schools, and the total cost for middle schools is more than that of elementary schools, as shown in Table 2, because high schools serve the widest area with more dispersed students than do middle schools, and middle schools serve wider areas with more dispersed students than elementary schools do. Table 3 summarizes comparisons of total costs for different scenarios. Figure 4: The middle school (M1) student distribution and routings.