## Methodology  
### Research Framework  
In this paper, a DRT is proposed that provides services to conveniently transport passengers from demand points to the rail station [1; 40; 41]. Using a cellphone app and an open geo-information system(GIS)tool, we can obtain the traveling information of some passengers and the traffic network in the study area. Each passenger has one or several preferred boarding time windows as well as minimum and maximum expected ride times. During the process of designing the route, each vehicle starts at the dispatch center, visits the demand points to pick up passengers in the specified time periods, and arrives at the rail transit station. This is designed to meet realistic constraints, such as time windows and expected ride time, etc. To reveal the optimal relationship between passenger satisfaction and total mileage to maximize the efficiency in the feeder bus route design, a mixed-integer programming model was formulated to design routes from selected demand points to the rail station, which includes the demand points of passengers in the specified time period from several preferred time windows. These key components are illustrated by a research framework graph, shown in Figure 1.  
Figure 2 also provides another explanation of the principle and scope of the proposed model. Figure 2 includes one rail station (M), three dispatch centers (D1-D3), and five demand points (C1-C5) in the DRT. The numbers below each demand point (Ci, where \(i=1\), \(2\),..., \(5\)) denotes the number of passengers in the location. Two numbers in brackets to the left of each demand point (Ci, where \(i=1\), \(2\),..., \(5\)) denote the minimum and maximum values of preferred boarding time windows and expected ride time. For instance, the value below demand point C1 of Figure 2 is 4, which means that four people will board the bus at this location. They have two preferred boarding time windows of [6:50, 7:00] and [7:05, 7:10], while their minimum and maximum expected ride times are 5 min and 10 min. In this example, the optimization process yields three routes as follows. Route 1 is illustrated by a solid line [D1(7:10)-C5(7:15)-M(7:19)], Route 2 by a dashed line [D3(7:00)-C4(7:06)-C3(7:08)-M(7:13)], and Route 3 by a dotted line [D2(7:02)-C2(7:04)-C1(7:07)-M(7:11)]. For example, Vehicle 3 departs from D2 at 7:02, arrives at demand points C2 and C1 at the times of 7:04 and 7:07 to pick up seven and four people, respectively, before finally returning back to M at 7:11. Thus, the ride times of the two customers are 7 min and 4 min. We also define the passenger satisfaction related to demand points as the amount of ride time minus the minimum ride time divided by the amount of the maximum ride time minus the minimum ride time. In this case, passengers at C2 and C1 would get on the bus in the time periods of [7:00, 7:10] and [7:05, 7:10], respectively, with their passenger satisfactions being \((10-7)/(10-5)=0.6\) and \(1\). In this case, the loading schemes at each stop of Route 1 can be described as {C5(5)}, while this is described for Routes 2 and 3 as {C4(5), C3(11)} and {C2(7), C1(11)}. The satisfaction at each stop of Route 1 can be described as {C5(1)}, while this is {C4(0.6), C3(1)} and {C2(0.6), C1(1)} for Routes 2 and 3. It is important to note that Route 3 may be described as [D2-C1-C2-M] with more mileage and longer ride times, if passengers at the C1 point can only be picked up in the time period of [6:50, 7:00]. Our objective is to find a subgraph that simultaneously maximizes passenger satisfaction and minimizes the total mileage cost of designed feeder routes. To ensure that the proposed DRT model fits well with the real-world situations, this study considers the following assumptions:  
1. Passengers at the demand point are allowed to travel in one or several preferred boarding time windows. It is possible to investigate the number of passengers at each demand point around the rail station and ignore the passenger flow between them. 2.