# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
###### Abstract  
Given the distinct operational features of flex-route transit (FRT) compared to conventional fixed-route systems, this study integrates FRT with urban rail transit and proposes a collaborative optimization approach for feeder flex-route operations and scheduling. The model incorporates path planning, timetable design, and vehicle scheduling to minimize travel costs, enhance operational efficiency, and improve service quality. A mixed-integer nonlinear programming method is applied to optimize scheduling while aligning with rail timetables and passenger transfer behavior. Simulation experiments based on realistic urban transit scenarios are conducted to validate the model, optimizing key operational parameters such as departure intervals, vehicle deployment, and stop selection. The results demonstrate that the proposed approach effectively adjusts schedules and vehicle assignments to accommodate fluctuating passenger demand. This study offers both a theoretical foundation and a practical framework for integrated scheduling of feeder flex-route services, with potential applications for improving coordination and performance in multimodal public transit systems. flex-route transit; scheduling; urban rail transit; transfer analysis; operation coordination 2020 IEEEexample:BSTcontrol  
despite its growing adoption, the optimization of FRT scheduling and operation planning remains underexplored. Existing research primarily focuses on fixed-route timetabling or fully demand-responsive transit, overlooking the unique challenges posed by FRT, such as balancing operational efficiency with passenger convenience and integrating FRT with existing urban transit networks. Addressing these challenges is critical for improving public transit accessibility, service reliability, and network efficiency. This study aims to bridge this gap by proposing a collaborative optimization approach for FRT operation and timetable scheduling, integrating passenger demand dynamics, urban rail connectivity, and real-time scheduling constraints. By developing a comprehensive optimization framework, this research provides theoretical insights and practical solutions to enhance FRT's role in modern urban transport systems. In the absence of flexible travel demand, FRT operates like a conventional bus, adhering to a fixed route and station stops. Upon reaching the interchange station area, FRT can dynamically pick up or drop off passengers from the base route, effectively completing the \"first/last mile\" segment of their journey. As shown in Figure 1, there are two types of passengers in the \(L\times W\) service area. Type I passengers: transfer from the subway station (Origin) to the flex-route transit and then reach the demand point (Destination); Type II passengers: from the demand point (Origin), board the flex-route transit to the subway station (Destination) and then enter the subway station. Therefore, optimizing the coordinated scheduling and route planning of feeder flex-route transit is essential to enhancing operational efficiency, improving service quality, minimizing costs for both the system and passengers, and increasing public transit attractiveness and ridership. To address the limitations of conventional bus-rail coordination in meeting last-mile travel needs, this research introduces a collaborative optimization framework that integrates feeder flex-route transit with urban rail systems. A mixed-integer nonlinear programming model is formulated to jointly optimize routing, scheduling, and vehicle allocation, incorporating passenger transfer behavior and spatiotemporal constraints. The proposed approach contributes both theoretically and practically to the advancement of flexible feeder service scheduling in multimodal urban transport systems. The remainder of this paper is organized as follows: In Section 2, the relevant literature is reviewed and key research gaps are identified. In Section 3, the flex-route transit operation organization model is proposed.