Vehicle scheduling under stochastic trip times: An approximate dynamic programming approach

Due to unexpected demand surge and supply disruptions, road traffic conditions could exhibit substantial uncertainty, which often makes bus travelers encounter start delays of service trips and substantially degrades the performance of an urban transit system. Meanwhile, rapid advances of information and communication technologies have presented tremendous opportunities for intelligently scheduling a bus fleet. With the full consideration of delay propagation effects, this paper is devoted to formulating the stochastic dynamic vehicle scheduling problem, which dynamically schedules an urban bus fleet to tackle the trip time stochasticity, reduce the delay and minimize the total costs of a transit system. To address the challenge of “curse of dimensionality”, we adopt an approximate dynamic programming approach (ADP) where the value function is approximated through a three-layer feed-forward neural network so that we are capable of stepping forward to make decisions and solving the Bellman's equation through sequentially solving multiple mixed integer linear programs. Numerical examples based on the realistic operations dataset of bus lines in Beijing have demonstrated that the proposed neural-network-based ADP approach not only exhibits a good learning behavior but also significantly outperforms both myopic and static polices, especially when trip time stochasticity is high.