School Bus Routing with Stochastic Demand and Duration Constraints

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

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.

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.

Special education students are usually routed on different buses than non-disabled general education students. To make the routes more efficient, this study proposes serving general students and special education students on the same bus at the same time (mixed ride) while allowing heterogeneous fleets and mixed loads. ESRI Location-Allocation tools and Google OR-Tools are used for bus stop selection, route generation, and bus stop optimization. Parallel cheapest insertion heuristic and metaheuristic, simulated annealing, are adopted to generate school bus routes. The effectiveness of the mixed ride approach is tested for three schools with 178 synthetic students’ locations data (including 12 with wheelchair) in New York City using a fleet of 14 buses spread over four types. The results show the mixed ride approach achieved 14.32% reduction in total travel distance and 10.46% reduction in total travel time. The mixed ride approach tends to return solutions with fewer vehicles and fewer bus stops, less average travel distance, and shorter average travel time.

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

We consider the classic School Bus Routing Problem (SBRP), where a fleet of school buses is used to transport students from their residences and to a common destination (school) subject to certain constraints. The constraints that are typically imposed include a maximum fleet size, a maximum walking distance to a pickup point and a maximum commute time for each student. The SBRP is a special case of the Vehicle Routing Problem (VRP) with a common destination. We propose a new approach for solving the SBRP based on the existing notion of a shareability network, which has been used recently in the context of dynamic ridepooling problems. To adapt this method to large-scale SBRP instances, we propose i) a node compression of the shareability network method with optimality guarantees, and ii) a heuristic-based edge compression technique that works well in practice. We show that the compressed problem leads to an Integer Linear Programming (ILP) of reduced dimensionality that can be solved very efficiently using off-the-shelf ILP solvers. Numerical experiments on small-scale, large-scale and benchmark networks are used to evaluate the performance of our approach and compare it to an existing large-scale SBRP solver based on a metaheuristic approach.

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.

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

This paper presents the implementation of simulated annealing (SA) method, an artificial intelligence technique, to solve the optimization problem known as the school bus routing problem (SBRP). A specific challenge in all artificial intelligence optimization techniques is the selection of appropriate value parameters. One contribution of this paper is the implementation of a design of experiments technique to provide statistical support for parameter selection. The SBRP is formulated as a 0–1 integer linear programming model, where the objective function is to minimize the total cost. Because this problem is combinatorial in nature, it is not possible to find exact solutions in an adequate time, calling for the use of an artificial intelligence optimization technique. The proposed technique is SA due to its modeling flexibility and processing speed. To demonstrate the performance of the proposed algorithm, several experiments with real instances were carried out, showing that the metaheuristic algorithm performs better in quality and time than the classic routing method.

The uncertainty in travel time for water through the unsaturated and saturated zones of Yucca Mountain and vicinity was determined by considering uncertainty associated with input parameters to the hydrologic models of these zones. A first-order analysis was used to investigate uncertainty in water travel time through the unsaturated zone at Yucca Mountain, based on an analytic solution for water flow. Results of the investigation of uncertainty for the unsaturated zone indicated that uncertainty in the percolation estimate contributed significantly more to uncertainty in travel time than uncertainty in estimates of hydraulic conductivity. Monte Carlo and first-order approaches were used to investigate uncertainty in ground-water travel time for different cases that varied in the treatment of the input parameters to the hydrologic model of the unsaturated zone.

The School Bus Routing Problem (SBRP) is a combinatorial optimization problem focused on designing efficient routes for transporting students between their homes and school while satisfying constraints such as vehicle capacity and maximum walking distance. This study proposes a hybrid GRASP-ILS metaheuristic applied to a real case study in Fortaleza, Brazil, considering a homogeneous fleet and a single-school setting. Two mathematical models were developed: the first minimizes total travel distance to reduce operational costs, and the second, inspired by the p-centers problem, minimizes the maximum walking distance to improve accessibility and equity. An initial feasible solution is constructed using GRASP and refined through Iterated Local Search. The algorithm was tested on 51 benchmark instances and real data. Results indicate that GRASP-ILS outperforms a Variable Neighborhood Search approach, achieving cost reductions of up to 35% while satisfying all constraints.

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.

A multi-loading school bus routing problem

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

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