# Research on Fleet Size of Demand Response Shuttle Bus Based on Minimum Cost Method
## 1 Introduction  
With the acceleration of urbanization and the diversification of travel needs, people are demanding higher quality travel services [1]. Traveling as a service has become a development trend of urban public transport in the future [2]. The traditional fixed-route transit (FRT) mode can meet the travel demand of high-density passenger flow through the reasonable arrangement of the travel plan, but in the area where the travel demand of the residents is small in scale and spatially scattered, the conventional bus mode will produce a series of problems [3]. On the one hand, the line can only stop at fixed stations and the passenger direct access rate is low, resulting in many passenger service blind spots, insufficient passenger flow, and serious vehicle idling. Bus companies, in order to improve economic efficiency, will reduce the frequency of buses and shorten service times. However, this not only fails to solve the problem but also increases passenger waiting times and reduces service levels. Consequently, it becomes difficult to cultivate a stable passengerflow along the route [4]. On the other hand, when residents face public transportation that cannot meet their travel demands, they must choose other modes of travel. This provides conditions and space for the increase in private car ownership and illegal transportation operations [5; 6]. Therefore, in reality, there is an urgent need for flexible and economic modes of transport organization to serve the residents of such areas [7; 8]. Demand-responsive transport is a flexible mode of operation. After years of development, a different form of layout evolved, mainly containing five kinds: point deviation (PD), new request stops (NRS), flexible route segments (FRS), demand-responsive connector (DRC), and zone route (ZR). Demand-responsive connector (DRC) is a common mode serving low-density areas, which requires vehicles to respond to passengers' dynamic requests in real time within an area, but requires that they arrive at pre-determined transfer points (at least one) at a fixed time to ensure interchange with other modes in the bus network [9; 10; 11; 12]. An important feature of a DRC service is the ability to achieve door-to-door service, which provides a high degree of flexibility. However, it increases vehicle diversions distances while meeting the needs of dispersed passenger travel [13; 14; 15]. Therefore, how to reasonably allocate the number of vehicles to meet the demand and ensure the service quality is a key issue that needs to be solved for DRC services [16]. Researchers have proposed various pricing methodologies from diverse perspectives. For instance, [PERSON] et al. [17] analyzed traveler responsiveness to fare adjustments to inform bus pricing strategies. [PERSON] et al. [18] integrated marginal cost pricing into agent-based models to optimize short-distance fares, while [PERSON] [19] explored optimal pricing frameworks under varying service conditions. [PERSON] et al. used the demand elasticity coefficient to design the bus fare setting model [20]. There are also several studies focusing on bus schedule optimization. [PERSON] et al. developed a model with the objective of minimizing passenger transfers and total travel time to optimize bus routes and frequency between suburban and urban areas [21]. [PERSON] et al. proposed an equilibrium bus boarding model to study the optimal pricing and bus departure intervals in the presence of inefficiencies of bus commuting during high peak hours [22]. [PERSON] et al. proposed a model to optimize bus schedules by considering the stations and routes of urban buses in order to optimize the bus scheduling process [23]. Some scholars have considered the uncertainty of passenger demand and optimized the bus schedules accordingly. [PERSON] et al. constructed a bi-objective mathematical model to solve the multi-period urban bus schedules taking into account the uncertainty of passenger demand [24].