# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem
## 2 Literature Review
In the 2000s, more sophisticated constraints were added to the SBRP algorithms. First, [PERSON] and [PERSON] [26] added new constraints on the earliest pick-up time of students while their proposed algorithm tried to minimize the total number of buses required and the total traveling time of students. They used a heuristic method to transform the problem into a Traveling Salesman Problem (TSP). [PERSON] et al. [27] considered the concept of temporary stops in the algorithm, which are potential spots within walking distance for students; therefore, the school bus will not serve as a door-to-door service. The objective function of their proposed model included minimizing students' walking distance and school bus travel time. Second, the school buses serve only one educational stage at a time, while three separate time windows are connected to each other in reality, so the algorithm should consider three separate time windows in a unit framework in order to generate optimal routes for the whole time period. Finally, in reality, and in the case of the United States, the school buses serve two separate time windows--morning and afternoon. This means that school buses leave the depot early in the morning to pick up high school students and deliver them to their designated high schools, then the buses leave the high schools and travel to middle school students' locations to pick them up and deliver them to the middle schools, and, finally, the buses will leave the middle schools for elementary students' locations to pick them up and deliver them to elementary schools. Therefore, the other contribution of this study is to consider all three separate time windows together in each of these frameworks for morning and afternoon time periods as an integrated multi-period scope (IM). The reader is referred to the literature review studies on SBRPs [28; 29] for a more comprehensive discussion of past developed school bus routing algorithms. These studies categorized the SBRP's features based on their practical aspects. Most of the past studies used real (R) network data rather than hypothetical (H); similarly, many studies used a single-period time window (S) scope of the problem, while some recent studies developed a multi-period scope (M) that considered both morning and afternoon time windows. Regarding the object function of SBRPs, most of the studies consider the minimization of fleet size (F), buses' traveled distance or time (BTD), and students' traveled distance or time (STD), while other objects like the minimization of students' time loss (STL) and students' walking distance (SWD) are frequently used. As to the constraints, capacity of buses (C) and school time window (TW) were used predominantly, while maximum riding time (MRT) and maximum students' walking time (MWT) were used frequently. Finally, regarding the approach to solving the problem, initial problems, especially in recent decades, used simple computer-based and heuristics methods; however, in recent years, by considering more complex concepts of the SBRP, most of the studies used metaheuristics and hybrid methods. Upon reviewing the background studies, the authors found limited knowledge available on solving the SBRP within an integrated framework. However, such integrity ensures efficient resource utilization and potentially reduces both operational costs and environmental impacts. Therefore, the current study tries to address this gap by providing a more realistic integrated SBRP, wherein the sequential operation of fleet services across all three tiers is addressed simultaneously within an IM scope. Table 1 summarizes selected studies about multi-school bus routing problems.