School Bus Routing Problem with a Mixed Ride, Mixed Load, and Heterogeneous Fleet

In this dissertation, we solve a school bus routing problem of transporting students including special education (handicapped) students and assigning them in Oldham county education district. The main contribution of this research is that we consider special education students (Type-2) along with other students (Type-1) and design a comprehensive school bus schedule to transport both kinds of students at the same time. Also, a student assignment mathematical model is presented to optimize the number of bus stops in use as well as one important measure of service quality, the total student walking distance. Comparing to the classic clustering methods, heuristic methods, or other methods from previous literatures, a mathematical optimization model is developed to solve a student assignment problem and to obtain the global optimal solution. The modeling constraints include budget limit, travel time limit, equity, school time window, and etc. Especially, the main difference between our model and other models is that it takes Type-2 students into consideration along with critical constraints accordingly, and solves the resulting more complex problem. Moreover, the school bus routing model in this work is one of the most general optimization models representing the school bus routing problem. On the other hand, similar to all existing models, the developed model considers the total system cost as the objective function value to minimize, different bus capacities, and common vehicle routing constraints such as flow conservation on routes and subtour elimination. Furthermore, another main difference is that the bus scheduling and school time window is also considered and solved in the model. With two different types of students, both Type-1 and Type-2, the time restrictions are varying, resulting in more complexity and additional constraints. The results in this work present the difficulties of meeting the requirement of Type-2 student riding time limit and school time window simultaneously. Also, the constraints regarding service equity and quality are provided and they can be used by decision makers if necessary. Either densely populated urban areas or sparsely populated rural areas, the school bus routing problem is difficult to solve due to a large number of students or long travel distance. The school bus routing problem falls under vehicle routing problem (VRP) with additional requirements because each student represents one unit of capacity. In this dissertation, we present a modeling framework that solves a student assignment problem with bus stop selection, and subsequently a school bus routing problem with school

Generally, Kandy city and traffic congestion go hand in hand. This situation becomes worse during school time. The geographical location of the city, the limited number of entrances and the large number of schools located within a small area are contributing to this condition. This study attempts to propose a new model to reduce school time traffic congestion using School Bus Routing Problem (SBRP) method. Network analysis in GIS facilitates the build of this system. SBRP seeks to plan an efficient schedule for school buses to pick up students from a bus stop and drop them to school by satisfying various constraints like; school time window, maximum walking distance and riding time, and the traffic. SBRP selects the nearest bus stops, generates suitable bus routes and visualizes traffic congestion on roads in Kandy city. Vehicle Routing Problem (VRP), closest facility, and location-allocation tools identify GIS-based solutions to SBRP. The proposed model selected suitable bus stops to drop students to a school and to generate the fastest bus route for the school buses. Visualizing a 12-hour daytime traffic in Kandy City, all schools are covered by the 56 fastest bus routes. The maximum riding time and distance limits to 59 minutes and 16 km respectively. Historical traffic data identifies 6.00 am to 8.00 am and 12.00 pm to 2.00 pm as the most traffic-congested time in the city. Finally, all problems of SBRP are solved and a systematic method is introduced to transport school students by optimizing the traveling time of students with existing traffic congestion.

A school bus problem is one of the most important logistical issues faced by governments and school administrations, which is widely depends on route optimization and scheduling. This study presents a mathematical model for the single-load school bus problem for urban areas, which considers bus stop selection, bus route generation, and school bus scheduling simultaneously. This model considers the possibility of demand outsourcing (e.g., students using parallel systems for transportation) and time windows in the school bus routing and scheduling (SBRS) problem. Each vehicle transports students from a bus stop to their school under the given constraints. Our goal is to maximize the transportation firm’s profits and increase the quality of service provided for students. To validate the presented model, the problem is solved by GAMS software, and the results are shown and analyzed. Finally, some sensitivity analyses are performed to show the efficiency of the presented model.

A multi-loading school bus routing problem

This paper proposes a School Bus Stop location and Routing Problem with Walking Accessibility and Mixed Load (SBSLRP-WA-ML), where the individual difference of walking accessibilities among students and the possibility of serving students attending different schools with the same bus simultaneously are considered. We first develop a mixed integer programming model for SBSLRP-WA-ML with the objective of minimizing the total commuting time, including walking time from the residence to school, in-vehicle travel time, and service time at stops. A two-stage solution method is then developed. In stage 1, an iterative clustering method based on k-means and Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is used to locate bus stops aiming at minimizing the number of stops subject to various walking accessibilities. In stage 2, an improved ant colony optimization algorithm (IACO) integrating two local search operators is devised, which is used to generate bus routes with minimal total commuting time. A number of instances of different sizes are generated to verify the solution approach, and the influential factors with respect to total commuting time are analyzed. The model is also compared to the door-to-door school bus services. Comparison to similar methods and sensitivity analysis of parameters are also conducted to analyze the performance and robustness of the proposed approach.

A column generation approach for a school bus routing problem with resource constraints

A post-improvement procedure for the mixed load school bus routing problem

The school bus routing problem (SBRP) is crucial because of its impact on economic and social objectives. A single bus is assigned to each route, picking up students and arriving at the school within a specified time window. SBRP aims to find the fewest buses needed to cover all of the routes while minimizing the total travel distance and meeting required constraints. We propose a mathematical formulation responding to the overbooking policies applied in a real-world school district. According to our empirical studies, the probability of a student riding a bus varies from 22% to 77%, opening the opportunity to overbook the buses to improve utilization of their capacity. However, SBRP with overbooking has not attracted much attention in previous studies. In this work, overbooking is modeled via chance constrained programming. Additionally, to account for the uncertainty of the total travel time of the buses, a constraint limiting the probability of being late to school is also proposed in this paper. As a result of the NP-hard nature of the problem, a cascade simplification algorithm is proposed to partition the multiple stage SBRP problems into multiple multi-depot and one-school subproblems that are solved sequentially, where the results for one are data inputs for the next. Furthermore, we develop column-generation-based algorithms to solve the scheduling problem, and different instances of the problem are examined. Our computational experiments on a real-world school district demonstrate desirable cost savings in terms of the total number of buses used.

This paper describes a case study of the school bus routing problem. It is formulated as a multi-objective combinatorial optimisation problem. The objectives considered include minimising the total number of buses required, the total travel time spent by pupils at all pick-up points, which is what the school and parents are concerned with most, and the total bus travel time. It also aims at balancing the loads and travel times between buses. A heuristic algorithm for its solution is proposed. The algorithm has been programmed and run efficiently on a PC. Numerical results are reported using test data from a kindergarten in Hong Kong. It has shown to be effective with a saving of 29% in total travelling times when comparing to current practice.

In the current system, school bus stops in Howard County, Maryland are manually placed along the school bus routes based on safety, cost-efficiency, and many other variables. With such liberal placement, bus stops are sometimes placed unnecessarily. This issue is prevalent in many school districts and often results in needlessly close bus stop proximity. In this study, the authors implemented a GIS-based heuristic to assist school officials in optimizing their districts bus stop placement. They also estimated the proportion of county-wide bus stops that could be eliminated by this approach. Following the constraints determined by State and local guidelines, the ArcGIS Network Analyst Extension was used to identify unnecessary bus stops across the study area. The initial output was re-evaluated by school officials in order to determine if those bus stops would be eliminated. The results indicate that approximately 30% of the existing bus stops were marked as “candidates for elimination” by the GIS process. After a review of these candidates, it was determined that at least 15% of the total school bus stops could be eliminated. Statistical estimates lent credence to the benefit of a re-evaluation of these bus stops. The method developed in this study can easily be replicated. Hence, it may inspire other school systems to exercise the same approach. Additionally, the results provide a gateway for future studies in examining more efficient school bus routes with less travel time, as well as investigating how much the carbon footprint of school bus fleets can be reduced.

Un Algoritmo de Estimación de Distribuciones copulado con la Distribución Generalizada de Mallows para el Problema de Ruteo de Autobuses Escolares con Selección de Paradas

In our country, school bus routes are determined by experiments of driver without considering the scientific optimum route and location. Traversing additional routes will always result an increase in vehicle movements and fuel consumption and enormous costs. Hence, this paper will study the school bus routing in Tehran considering special students and a model will be presented to minimize traveling distance and to prevent repetitive crossings through the bus stops and to determine the shortest routes by presenting a way to propel several students to a bus stop. The proposed model will solve via GAMS software. Because the model is NP-Hard, the Genetic algorithm is used to solve the large scale problem. The contribution of this paper is to consider gender separation in schools and buses. To solve this problem, an integer linear programming model is developed. The conclusion indicates a decrease in transportation time.

Optimizing school bus routes is essential for improving student travel efficiency and safety, yet it presents significant challenges due to the stochastic nature of student demand. Traditional approaches often struggle to fully utilize bus capacity while adapting to daily fluctuations in demand. In response to these challenges, this paper provides a new algorithm for the school bus routing problem with stochastic demand, incorporating industrial process flexibility. Firstly, an open chain k-overlapped-route network is presented, allowing for route overlaps to fully utilize bus capacity. Secondly, a new recourse algorithm is developed, which introduces partial overlaps in bus fleet routes and employs recourse actions to determine the start and end depots for each bus. This algorithm can adapt to daily fluctuations in student demand, while ensuring that school buses operate normally on familiar routes. Finally, numerical and empirical studies validate the feasibility and applicability of the proposed algorithm, demonstrating its superiority in optimizing both the number of school buses used and the total travel distance.

Feeder buses provide a small but important part of the public transport system by carrying people between residential areas and transport interchanges. A feeder bus to a train station planned in advance will attract new residents of a housing development to use the bus. The bus route can influence the location choice of a buyer concerned about access for children, the elderly or anyone not wishing to drive a car. Our bus route modelling starts with the bus stops – not the route – to be reached from each dwelling by the shortest possible walk. In demand terms, people locating close to bus stops are more likely to use the service than those choosing more distant locations, and the nearby residences have higher values. The stops-first application determines a feeder bus route to enhance an irregular residential plan covering an area of one square kilometre. The planned road and housing lot locations provide the data for calculating the access measure from each dwelling to each potential bus stop, the closest stop being used. A genetic algorithm tests potential bus stops to find demand maximizing locations, the propensity to use the bus being formulated as an exponential (increasing elasticity) function of walking distance. Then a ‘travelling salesman’ genetic algorithm finds the shortest route linking the stops, so that an efficient circuit route is generated for each alternative number of bus stops, ranging from 7 to 11. More stops not only give better access but also increase the route length, so that total accessibility must be assessed against route length. The distribution of walking distances shows most between 150 and 240 metres, with none more than 400 metres. The results indicate that planning policy should require prior design of a bus route to achieve good walking accessibility, so that residents become accustomed to the convenience of using the bus. This study shows that, at the planning stage, estimating a bi-objective model giving a Pareto front between accessibility and route length can reveal a policy compromise that shortens the route with little reduction in expected patronage.

In this paper is a solution to the School Bus Routing Problem by the application of a bio-inspired algorithm in the vertical transfer of genetic material to offspring or the inheritance of genes by subsequent generations. The vertical transfer algorithm or Genetic algorithm uses the clusterization population pre-selection operator, tournament selection, crossover-k operator and an intelligent mutation operator called mutation-S. The use of the bio-inspired algorithm to solve SBRP instances show good results about Total Bus Travel Distance and the Number of Buses with the Routes.KeywordsTransportationCombinatorial OptimizationAlgorithmsSchool Bus Routing ProblemSBRPbio-inspired algorithm