Optimal design of flexible bus service integrated with trunk transit under demand uncertainty

This paper develops a two-stage stochastic optimization framework for designing flexible bus service integrated with trunk transit under uncertain spatial demand. The proposed system jointly supports direct door-to-door service, flexible-to-trunk transfers, and flexible-to-flexible transfers. Passenger willingness to transfer (WTT) is modeled as a demand-partition mechanism, allowing both fixed and service-quality-dependent transfer preferences. To address the strong NP-hardness of the second-stage passenger assignment problem, we develop a reliability-based decomposition method with a tailored trip assignment heuristic for large-scale instances. Numerical experiments based on a Hong Kong case study show that the integrated design reduces total system cost by up to 19.0% relative to a conventional feeder-only flexible-bus-trunk system and by 5.1% relative to a flexible-bus system with ad hoc service. The model also quantifies the value of passenger transfer flexibility, showing that full willingness to transfer can reduce composite system cost by 10.0%. Overall, the results demonstrate the value of jointly coordinating flexible bus deployment, trunk transit integration, and passenger transfer behavior when planning multimodal transit systems under demand uncertainty.