# Dynamic Demand-Responsive Feeder Bus Network Design for a Short Headway Trunk Line
## 1 Introduction  
The modal share for ridesharing has been in decline over the past few decades; however, it still has played an important role in mobility as one of the influencing travel modes in the United States [1]. There has been significant consideration given to on-demand ride services in recent years from federal and state governments and private sectors for improving on-demand transit services, especially in areas with limited accessibility to public transit services. Fewer mobility options are available in suburban areas because of weaker production/attraction centers of trips in a relatively large area and unpredictable travel demand based on low population density [2]. The flexible demand-responsive transit system (DRT), as an on-demand ride service, has been considered as an efficient mobility option in urban and suburban areas in many studies; however, the efficiency of these systems is open to discussion because of the different approaches and perspectives toward considering passengers and operator costs [3, 4]. Generally, DRT systems have been proved in past studies as transit systems capable of satisfying even large transit demands and as a complementary transit mode that supports conventional transit systems [5]. Considering the advancement of new communication and computational technologies in transportation and transit systems such as automation and connectivity of vehicles, the demand-responsive transit systems with time windows (DRTTW) can take advantage of these technologies to improve the quality of service and maximize serving demands. The proposed DRT feeder system in the current study performs as a complementary transit system with a rail system. A feeder system should follow routes and timetables to match demand requests to maximize demand coverage while improving the quality of service by maximizing fleet efficiency [6]. However, the perspectives of passengers and operators using a DRT system are different. From users' perspective, their in-vehicletravel time should be minimized; therefore, direct routes are preferable for them because this will decrease their travel time and consequently the total passengers' travel cost. On the other hand, the operator seeks to minimize vehicles' traveled distances because that will decrease vehicle travel time and consequently the total transit operational costs. This problem should consider both user and operator costs at the same time as total costs, and therefore, it has a complex structure. In past studies, it has been introduced as an NP-Hard problem that should be solved by heuristics and metaheuristic methods [7]. Previously, an algorithm for a feeder bus routing problem was developed which accommodated multiple stations and feeder buses while allowing relocations of feeder buses [8]. For that research, it was assumed that the headway of the rail service is long enough for the feeder buses to come back by the next train. Because the maximum feeder service cycle time is determined by the rail headway, matching between feeder buses and the trains is not necessary. However, if the headway of the rail service is not long enough for the feeder buses to return before the next train, then the algorithm should find not only matching between passengers and feeder buses but also matching between feeder buses and trains, which makes the problem much more complicated and distinguishes this research from the previous research. In this research, using the previously developed algorithm and the model network, the headway of trains is decreased from 20 min to 5 min, making the problem more realistic and usable not only for suburban areas but also for the higher-frequency rail lines serving urban areas. Transportation network companies (TNCs) such as Uber and Lyft have actively expanded their services in the U.S. in recent years; more realistic and optimized routing algorithms for these companies can remarkably decrease their operating costs and resources while increasing the satisfaction of time-sensitive origin-to-destination passengers in urban areas.