# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem
## 1 Introduction  
The school bus system is an essential transportation mode for public school students in most states in the U.S. Since each system covers quite a wide area, contains multiple depots for the buses, and serves many high schools, middle schools, and elementary schools, optimizing the operation is a complicated problem. Among the many aspects of school bus operations, one of the fundamental problems is optimizing the school bus routings. Optimal routing of school buses not only minimizes operating costs but also improves the level of service for students by minimizing their in-vehicle travel times. However, as mentioned, school bus routing is a complex problem because, in most cases, a bus leaves a depot and covers a high school, a middle school, and an elementary school in one trip.  
Optimizing school bus routing is crucial for ensuring efficient resource allocation and cost savings for educational institutions. Additionally, effective routing enhances student wellness, reduces environmental impact, and improves overall logistical efficiency. The benefits of routing optimization for transit service operators have been broadly considered in past studies [1; 2; 3], and, similarly, optimization of school bus routes not only benefits the service provider by decreasing the route length and fleet size but also can benefit students by decreasing their in-vehicle traveling time. [PERSON] and [PERSON] [4] believed optimization of school bus routings could save USD 5 million annually in Boston. Also, optimization of scheduling and start times of schools may save even more costs [5; 6].  
One of the main complaints of parents of students who use school buses in the U.S. is that their children spend a long time on the bus, even if their home is close to the school. To address this issue, the school buses should have different routes for morning and afternoon time windows with different stopping sequences. According to past studies, about 19% of students in the U.S. live within one mile of the school and use school bus services [7].  
Therefore, providing a better school bus routing algorithm that minimizes the in-vehicle travel time of students is important for students and their parents.  
To solve the multi-level school bus routing problem more effectively, rather than solving three independent separate routing problems for each school level, this research develops an integrated single-framework optimization algorithm for multi-level school bus routings in order to consider the nature of the school bus operation, which contains multiple levels of high schools, middle schools, and elementary schools in a single trip. Also, this research compares two routing sets created by the developed algorithm--one for the same routings (identically reversed) for morning and afternoon time windows and the other for the different routing sets for morning and afternoon time windows that optimize routings for individual time windows--to show cost savings achieved by individually optimized routings for each time window and to discuss the policy and planning aspects of such routings.