To wait or to transfer? A three-level optimization framework for intermodal transfer coordination in first train timetabling and bus bridging services management

This study addresses the integrated optimization of the first train timetabling and bus bridging service design (FTT-BBSD) for morning transfer challenges, two critical but interdependent passenger services in the public transit system. In contrast to most existing studies and conventional approaches, this study explicitly models the influence of passenger path choices and transfer mode selections on FTT-BBSD. Through a novel dual-level network representation that integrates subway and bus systems, we formulate the FTT-BBSD problem as a mixed-integer nonlinear programming model. The model simultaneously determines subway and bus timetables and bridging line deployment to minimize total travel time for all first train passengers. Subsequently, a three-level optimization framework based on an improved Benders decomposition algorithm is developed to solve the model. The first level updates travel paths and transfer modes to achieve a minimum travel time. The second level enhances the Benders decomposition to optimize first train and bus departure times, where a master problem optimizes passenger riding plans and a subproblem determines subway and bus departure times. The final level focuses on bridging line generation, performing a rolling station search to find the most efficient bridging services scheme. Ultimately, based on the Beijing subway case study, we reduce the total passenger travel time by 52,316 min and the transfer waiting time by 23.34% by operating 76 bridging buses across the dual-level network.