# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
## 2 Literature Review  
When optimizing transit operations, while some studies focus solely on headway adjustments, a greater number explore the combined optimization of headway alongside other factors such as load factor, vehicle type, stop selection, and fare policies [3, 4, 5, 6, 7]. The urgent need for flexibility and personalization of travel has prompted continuous research and innovation related to the bus bridging service. [PERSON] and [PERSON] proposed a non-uniform departure timetable optimization model based on the bus connecting high-speed rail stations as the research object. The results show that, compared to a uniform departure schedule, a non-uniform departure interval can effectively reduce passenger transfer waiting times [8]. [PERSON] and [PERSON] [9] developed an optimization model to determine the optimal vehicle departure times and type ratios, focusing on railway station-bus connectivity. Their approach reduced operating costs by adjusting the vehicle types used in feeder bus services. [PERSON] et al. [10] examined the collaborative optimization of community shuttle bus and subway services and employed a hybrid approach integrating genetic and Frank-Wolfe algorithms to derive an optimal synchronous scheduling scheme. [PERSON] and [PERSON] [11] addressed the issue of large numbers of passengers stranded due to rail transit network interruptions. They developed a timetable optimization model for bus-rail connections and proposed corresponding strategies to enhance connectivity. To address rail transit interruptions, [PERSON] and [PERSON] [12] proposed a comprehensive optimization framework for bus bridging routes and timetables under time-varying demand, incorporating a merging strategy for fast and short-distance routes to improve service efficiency. [PERSON] et al. [13] introduced a column-generation algorithm within a two-step framework to generate candidate bus bridge routes. A multi-network flow model was established, and a detailed headway design was implemented. Accounting for bus travel time uncertainties, [PERSON] et al. [14] proposed a robust optimization method to determine bus crossing routes and frequencies. [PERSON] and [PERSON] [15] proposed an integrated model for segmenting bus routes and scheduling periodic timetables. [PERSON] and [PERSON] [16] introduced a comprehensive route planning and cyclic timetable model, incorporating a cross-entropy heuristic algorithm for optimization. [PERSON] et al. [17] optimized bus route frequencies by adjusting route structures to minimize passenger delays. [PERSON] and [PERSON] [18] proposed a nonlinear model and genetic algorithm integrating shuttle bus timetables with vehicle scheduling while accounting for skip-stop operations. While existing studies have explored timetable optimization, headway adjustments, and bus-rail coordination, limited research has focused on the integrated optimization of flex-route transit operation planning and timetable synchronization with urban rail systems. Prior research primarily addresses either fixed-route bus connections or fully demand-responsive transit models, yet a significant gap remains in optimizing the hybrid characteristics of flex-route transit, particularly in balancing cost efficiency with service quality and ensuring seamless connectivity between flex-route services and urban rail networks. Furthermore, most studies focus on static scheduling models, whereas real-world transit operations require dynamic and adaptable scheduling frameworks that respond to fluctuating passenger demand and transit network conditions. Passengers take urban rail transit schedules into account when planning their short-distance travel to optimize their overall trip efficiency. To ensure seamless connectivity, it is essential to align the operational strategies of flex-route transit with urban rail transit services. Bus schedule formulation comprises two critical aspects: vehicle scheduling design and timetable preparation, both of which are fundamental to enhancing the efficiency and service quality of the bus system.