# Optimal Integrated Single-Framework Algorithm for the Multi-Level School Bus Network Problem
## 2 Literature Review  
The iconic yellow color for public school buses was established in the late 1930s as part of a nationwide effort to standardize and enhance the safety of school bus transportation. Since then, the U.S. public school bus system has evolved, influenced by various factors such as geographic and demographic shifts, educational policy reforms, and continuous improvements in vehicle design and safety standards [8]. While Canada's school bus system closely mirrors that of the United States, boasting a similar structure and safety protocols, other countries often have distinct arrangements for their public school transportation systems, reflecting their unique infrastructural and educational needs. In the UK, local education authorities determine school transport arrangements, which may not always follow a set-tier system like in North America. Instead, buses may serve multiple age groups simultaneously, depending on the region and school times. Morning and afternoon transport aligns with the start and end times of the schools, which are less staggered than in the U.S., resulting in a more synchronized bus schedule [9]. Australian school bus services cater to various school levels, with some areas offering more dedicated services than others. The timing of these services typically follows the school day, with morning services before school starts and afternoon services following school dismissal [10]. In the 1970s, many scholars considered the school bus routing problem (SBRP) to improve the efficiency and economic performance of school bus services [11; 12; 13; 14]. Basically, the SBRPs, by nature, are like vehicle routing problems (VRPs), as they all seek the efficient size of a fleet on an optimal route. However, the objective functions and constraints may be varied in SBRPs. Since the VRPs are designed to serve passengers (or goods) by alighting/boarding (delivering/picking up) at the same time, the time windows and the vehicles' capacity constraints are emphasized more in these types of problems. The SBRPs, however, highlight serving all students within allowed time windows and total vehicle travel time constraints. For a better insight into various types of VRPs, the reader is referred to the literature review studies [15; 16; 17; 18; 19; 20; 21; 22]. The first attempts at developing SBRPs, conducted between the early 1970s and the 1990s, mostly focused on considering single/multiple school(s) and a limited number of constraints. [PERSON] and [PERSON] [12] introduced the SBRP with a single school. They proposed a basic SBRP model to minimize total student traveling time, and bus capacity was the model's main constraint. At the same time, [PERSON] et al. [11] proposed a school bus scheduling algorithm to minimize the number of routes, vehicle traveled distance, bus load, and traveling time for each route; the main constraints were bus capacity and traveling time limitations for some routes. Later, [PERSON] and [PERSON] [14] completed this algorithm by adding the routing capability to it and considering minimization of total bus travel as the goal of the algorithm with a traveling time window for students. Continuing that work, [PERSON] and [PERSON] [13] solved an SBRP by considering multiple schools and minimizing the total bus travel time and number of routes, with student traveling time and bus capacity as the main constraints of their model. [PERSON] al. [23] developed a single school SBRP model by considering the number of stops, length of routes, and fleet capacity; the model aimed to minimize the total number of bus routes and corresponding distances. Later, [PERSON] et al. [24] developed a single school multi-objective SBRP aimed at minimizing the number of routes, total bus length, bus loads, and walking distance of students while considering bus capacity and the total travel time. Finally, [PERSON] et al. [25] solved a multi-school, multi-objective SBRP for a New York City study to find the optimal fleet size by considering bus capacity, student traveling distance, and school time window.