Case

---

This section describes a case study for the Windsor School District (WSD), a semirural school district located approximately 25 miles south of St. Louis, Missouri, to demonstrate the applicability of the models and some of the real-world complications in school bus routing. The school district has approximately 3,000 students, 2,300 of whom are transported to and from school by school buses. WSD has five schools; Windsor High School (HS) for grades 9–12,Windsor Middle School (MS) for grades 6–8, Windsor Intermediate School (IS) for grades 3–5, Windsor Elementary School (WE) for grades PreK-2, and Freer Elementary School (FE) for grades PreK-3.Note that this includes four levels of schools and an overlap in grade 3 between IS and FE. The school district is approximately a rectangle of 17.1 square miles, measuring 5.25 miles north-to-south and 3.25 miles east-to-west. For this case study, the buses and the student population being served by special-needs buses are excluded. One complication for WSD concerns the students in grade 3. Elementary school students in the southwest quadrant of the district attend nearby school FE for grades Pre K-3, but all other elementary students attend school WE for grades Pre K-2, and then move to the intermediate school IS for grade 3. Another interesting feature of WSD is that schools HS, MS and IS are very close together in the east-central part of the district, with WE and the bus depot also located nearby. Here are the basic data on the bus riders for each school. Bus rider population for HS, MS, IS, WE, and FE are 484, 806, 405, 387, and 219, respectively, for a total of 2,301. The mix of schools and overlapping assignments of grades reflects the complexity of real-world school planning.

Currently WSD utilizes mixed routing with 22 buses, with each bus completing two trips in the morning and two trips in the afternoon. We concentrate on the morning trips in the case study. Currently, the first trip for all 22 buses is a mixed trip for students attending MS and HS, with each trip starting at the bus depot, making an average 8.4 stops picking up approximately 7 students per stop, and then transporting the students to MS first, and then to HS. All second trips for students attending IS or an elementary school begin at HS, with eight of the mixed trips serving the students for IS and FE (with an average of 6.4 students per stop) and the remaining fourteen mixed trips serving students for IS and WE(with an average of 5.2 students per stop).

The current policy in WSD can be analyzed using the continuous approximation travel distance equations developed in this research. To calculate the total expected distance for the current WSD mixed bus trips, we use the actual locations of the bus depot and the schools, but assume that the bus stops are randomly located over the district. The current mixed routing policy for WSD has a total expected travel distance from the continuous approximation models for both sets of 22 trips of 296.6miles (see [9] for details). In addition to analyzing the current policy for WSD, we also analyzed serving all students with non-mixed trips, where the first set of 22 trips consists of 8 non-mixed HS trips and 14 non-mixed MS trips, and then the second set of 22 trips consists of 10 non-mixed trips for IS, 8 non-mixed trips for WE, and 4 non-mixed trips for FE. This produced a total expected travel distance only 1% greater than with mixed trips. A closer examination of the travel distance for each school revealed a long distance travel between the schools in the mixed trips serving far apart schools IS and FE, due to the start time for IS preceding that for FE. Thus, FE students in the southwest were being transported across the district to IS to meet its earlier start time, before returning back to FE (with the later start time). To eliminate this long travel distance for the FE students, we considered a hybrid strategy where the first 22 trips are mixed trips for HS and MS, but the second trips are split into 14 mixed trips for nearby schools IS and WE, and then 4 non-mixed trips for FE and 4 non-mixed trips for IS. This hybrid strategy produced savings in travel distance of 8.5%, which shows the benefits of tailoring school bus transportation to the specific details in the school district, especially school locations. These results suggest that replacing the current transportation policy with a non-mixed policy throughout the district will result in a negligible change in the total distance travelled, but using a hybrid policy could result in a noticeable reduction in the total distance travelled (about 8.5%).

We also applied the three-phase heuristic to develop discrete mixed trip bus routes for WSD. We first geocoded the bus stop and school locations and calculated the road travel distance using the shortest path through the road network between pairs of locations. Then, we analyzed the current bus routes in WSD by modeling each of the routes for the 22 buses to determine the total travel distance was 434.9miles. As expected, this is considerably larger than the distance from the continuous approximation models as that used straight-line distances and the heuristic uses shortest paths on the actual road network. Lengthy travel was observed again for back-and-forth travel between FE and IS due to the start time conflict for these schools. We then used the three-phase heuristic to determine the bus routes, and it produced much shorter routes totaling 329.2miles with only 17 buses. Interestingly, the heuristic produced non-mixed trips routes serving FE and mixed trips serving the other schools, which reflects the hybrid strategy that was suggested to be beneficial by the continuous approximation modeling. These hybrid routes from the discrete heuristic used fewer buses than the current routes in WSD (17 vs. 22), and they respected the 45min ride time restriction and school start times employed by WSD. However, the routes from the heuristic were of longer duration and carried more students than the routes currently used by WSD, as the heuristic tended to better fill the buses. We acknowledge though that there may be good practical reasons to design somewhat shorter routes, so that some unexpected delays en route (e.g., traffic congestion, longer than expected loading) can be accommodated without making the students arrive late at school. We also used the three-phase heuristic with an additional restriction that all trips should be non-mixed, and this produced routes requiring 21 buses and a total distance of 354.8miles, an increase of 7.8% compared to the mixed trips. So in summary, the results for the discrete model with the three-phase heuristic for the WSD case study are in general agreement with the findings from the continuous approximation modeling in showing the benefits of mixed trips and the value of a hybrid strategy. However, we do underline that caution is needed when comparing the travel distances from the continuous approximation model and the heuristic due to the different ways of measuring distance. Note also that the shortest path distances used in the heuristic may understate actual bus travel distances, because the large buses may not always follow shortest paths on the roads for safety reasons.