A Meta-heuristic Algorithm to Solve the Large-Scale School Bus Scheduling Problem

Abstract

Given school bus trips for each school in a school district, if a school bus can serve multiple trips, the efficiency of school bus service can be improved in terms of the number of buses needed and the total travel cost. The school bus scheduling problem(SBSP), a class of school bus routing problem(SBRP), is concerned with assigning a fleet of buses to serve all the given trips and aims to get optimal bus schedules. Each school has its xed time window within which school bus must arrive at the destination school of the trip. In existing literatures, SBSP is usually formulated as a transportation problem(TP) or an assignment problem(AP). However, many existing algorithms for vehicle routing problem(VRP) have not been fully utilized to solve the problem effectively. This paper proposes a meta-heuristic algorithm for large-scale SBSP. Treating a trip as a virtual stop with time window, the problem can be converted to a vehicle routing problem with time windows(VRPTW). Therefore, the SBSP can be solved in a VRP algorithm framework. After a set of feasible solutions are generated using construction heuristic algorithm, a simulated annealing(SA) algorithm is designed to improve the initial solutions iteratively. Four general operators for VRP, one-point move, two point move, two-opt move and cross-exchange move, are used in the neighborhood search. In addition to the SBSP objectives of minimizing the number of the routes and the total travel distance, the sum of squared number of route stops is added as a new objective. This will guide the neighborhood search toward the situation that deleting some routes more easily. For avoiding local optimum, some worsening neighborhood solutions can be accepted with a certain probability. Computational tests on 15 instances with a homogeneous fleet show the effectiveness of the proposed approaches. Compared with the existing SBSP solutions, the proposed algorithm can solve large-scale SBSP in a reasonable time and find better solutions using fewer buses. In addition, the algorithm can be easily integrated with GIS for solving real world school bus scheduling.