Two-Stage Stochastic Program for Scheduling Bus Service Under Demand Uncertainty

Abstract

To improve the reliability of public transport (PT) service, this paper develops a PT operating mode considering both regular and flexible services. Flexible bus services are introduced to accommodate demand that cannot be served by regular operations under demand uncertainty. A two-stage stochastic program is proposed to jointly optimize the service design decisions (i.e., stop allocation and service headway) and vehicle scheduling decisions. To solve the problem efficiently, the concept of service reliability (SR) is introduced to reformulate the coupled two-stage problem into two independently tractable subproblems. An SR-based gradient solution framework integrating fixed-point structure and branch-and-price is developed to efficiently obtain a high-quality solution. The proposed model and the SR-based approach are evaluated on both a toy service line and a real bus line in Shenzhen, China. The effects of demand variability on different costs of the transit system are further analyzed. Numerical results show that the proposed SR-based approach can effectively balance service design and operational adjustments to reduce the expected total system cost under demand uncertainty.