Column generation-based stochastic school bell time and bus scheduling optimization

Given a trip plan (a sequence of bus stops), the integrated school bell time and bus scheduling problem simultaneously optimizes the school bell time and bus routes (the assignment of buses to serve trips at a specific time) while maintaining the minimum level-of-service requirements. The objective is to minimize the total number of buses and the total vehicle time. The previous efforts focused on the deterministic problem. However, uncertainty is unavoidable in a real-world problem. Thus, we consider this integrated problem with stochastic travel times. Two Mixed Integer Programming (MIP) models and a Column Generation-based algorithm are proposed. The algorithm incorporates the merits of the Column Generation, Simulated Annealing, Insertion Algorithm, and Greedy Randomized Adaptive Search Procedure and gains the computational power that the existing methods do not have. The experiments on a real-world problem show that, after the bell time optimization, up to 41% of current buses can be saved with even better service in terms of the higher punctuality and shorter student ride time.