Lakhasly

2.Ji-yang et al. (2020) investigate the ''many-to-many'' pattern, focusing on the key challenge of determining multi-target station allocation.To ensure sufficient buffer time for gathering transit demand (Ma et al., 2013), the time resolution can vary from days to seconds, depending on the system's computational efficiency in re- optimizing bus scheduling.For instance, Liu et al. (2019) propose a bus ride-sharing service where customers use an online hailing app to upload transit demand, and buses are assigned to customers after sufficient matched requests are integrated.The pioneering work by Flusberg (1976) saw the successful operation of a point deviation bus system in Merrill, Wisconsin, proving the feasibility of on-demand 3

W. Zhang et al. Transportation Research Part E 181 (2024) 103372 service in lower-density areas.In conventional operating modes, urban bus system operators rely on historical and survey data to simulate future demand (Zhang et al., 2011).Subsequently, public transit services with similar concepts, such as customized buses (Cao and Wang, 2017) and personal rapid transit systems, have demonstrated steady growth.The time resolution refers to the length of each time interval in the scheduling plan, indicating the time horizon for collecting passenger demand in on-demand bus services.In this context, customers specify their desired origin and destination stations along with their alighting deadlines beforehand.A comprehensive survey on on-demand bus services conducted by Vansteenwegen et al. (2022) explores systems with different time resolutions.Literature review 2.1.

2. Literature review
   2.1. On-demand bus service
   The concept of on-demand bus service traces back to its initial implementation during the 1970s. The pioneering work by Flusberg (1976) saw the successful operation of a point deviation bus system in Merrill, Wisconsin, proving the feasibility of on-demand
   3

W. Zhang et al. Transportation Research Part E 181 (2024) 103372
service in lower-density areas. Subsequently, public transit services with similar concepts, such as customized buses (Cao and Wang, 2017) and personal rapid transit systems, have demonstrated steady growth. In conventional operating modes, urban bus system operators rely on historical and survey data to simulate future demand (Zhang et al., 2011). This approach leads to the design of services based on predicted demands within the same time horizon. However, the transition from conventional to on-demand bus services has dramatically changed the relationship between passenger demand and bus supply (Wu et al., 2019). Unlike long-term predetermined scheduling in conventional systems, on-demand bus scheduling can be dynamically re-optimized and updated in response to real-time passenger demand during each practical planning horizon.
A comprehensive survey on on-demand bus services conducted by Vansteenwegen et al. (2022) explores systems with different time resolutions. The time resolution refers to the length of each time interval in the scheduling plan, indicating the time horizon for collecting passenger demand in on-demand bus services. To ensure sufficient buffer time for gathering transit demand (Ma et al., 2013), the time resolution can vary from days to seconds, depending on the system’s computational efficiency in re- optimizing bus scheduling. According to the survey, passenger demand typically needs to be stated at least 15 to 45 min before the operation begins (Vansteenwegen et al., 2022). For instance, Liu et al. (2019) propose a bus ride-sharing service where customers use an online hailing app to upload transit demand, and buses are assigned to customers after sufficient matched requests are integrated. Huang et al. (2020) develop a customized bus service, where customers make reservations for transit at least one hour before the operation. Ji-yang et al. (2020) investigate the ‘‘many-to-many’’ pattern, focusing on the key challenge of determining multi-target station allocation. In this context, customers specify their desired origin and destination stations along with their alighting deadlines beforehand. To design an efficient on-demand bus service, Huang et al. (2020) decompose the problem into two phases: the dynamic phase and the static phase. In the dynamic phase, passengers specify their requests, and the decision on whether to provide on-demand service is made. Subsequently, in the static phase, bus scheduling is solved based on the accepted requests. Their model is tested with passengers generated using a uniform distribution. In line with these advancements, the goal of this paper is to equip on-demand bus systems with the capability to efficiently respond to inhomogeneously distributed passenger requests, making them a practical and effective solution for real-world operations.