## 1 Introduction  
Increasing sharing of buses can relieve traffic congestion and reduce the emission of pollution. The strategy for designing a bus affects the efficiency of the bus and the design of bus stops, which play a pivotal role in the level of bus service. To make full use of bus resources, fixed bus routes naturally become many types of bus routes, based on whether they are stop services. These types include local, limited-stop, express, semi-restricted zonal service, restricted zonal service, short-turning, and deadheading [1]. Bogota, Los Angeles [2], Santiago [3], Jinan [4], Guangzhou [5], Montreal [6], and many other cities have applied various types of bus routes to make full use of public transportation. A local service provides services at all stops along the route, with accessibility being more important. Vehicles used for limited-stop services only stop at high-demand stops; thus, speed and accessibility are relatively balanced. Limited-stop service can shorten the in-vehicle travel time of the serviced passengers [6] and improve the utilization of bus resources [7], whereas the access or egress time and waiting time for limited-stop service for some passengers will increase. Limited-stop services usually operate in a long bus corridor with a long bus route. The passenger demand along the bus corridor is usually high, and the distribution is unbalanced. Limited-stop services can be beneficial to both bus agencies and passengers under appropriate conditions. For bus agencies, the shorter operating cycle of limited-stop service increases bus vehicle utilization, improving the capacity and operational efficiency. In addition, operating costs are reduced due to lower passenger boarding and alighting times. Limited-stop service is also attractive to passengers, by providing faster trips for some origin-destination (OD) pairs, especially for passengerswith long commuting distances. Mixed service of local and limited stops can balance the advantages of both types, which are shown in Figure 1.  
At present, some bus stop design models are used as express services [8; 9; 10]. Several researchers have studied the design of limited-stop services in different situations. [PERSON] et al. [11] and [PERSON] [12] focused on the dynamics of demand, and the results showed that flexible passenger trips would make better use of the limited-stop service. Compared with the previous literature, [PERSON] [13] considered the constraints of the fleet and redistributed the existing vehicles instead of adding vehicles to the limited-stop service. To expand a limited-stop service to a whole bus network, [PERSON] et al. [14] considered transfers of the limited-stop design. Evaluation of the limited-stop service was also a focus. Customer satisfaction is one of the main ways to measure the performance of limited-stop services [15; 16; 17]. To overcome the limitations of the importance of performance analysis, [PERSON] et al. [18] proposed a three-factor theory based on existing factors, important factors, and basic factors to analyze a limited-stop service's performance. Different from the satisfaction survey, [PERSON] et al. [19] evaluated the newly implemented limited-stop service on Montreal Island, Canada. They collected vehicle location and passenger trip data to compare the on-time rate of the vehicles before and after the implementation of the service. The fare is an essential factor for both bus agencies and passengers in a limited-stop service. If the subsidy from the government remains unchanged and the fare for the limited-stop service is also the same as that for the local service, then the bus agency will have a deficit in their mixed-service operations. Differentiated fares are a gainful way to compensate for the reduction in fare revenue for bus agencies caused by the decreasing number of passengers due to the implementation of limited-stop services. In the current literature, differentiated fares for a limited-stop service and a local service have been insufficiently considered.