## 5 Numerical Example
### Results  
As explained before, the proposed model can solve two dimensions, including the assignment of the number of passengers picked up by the vehicle and the routes of the vehicle. Table 3 summarizes assignment results, which include boarding time, ride time, and satisfaction for each demand point. Taking the vehicle visiting the demand point of C5 as an example, the vehicle leaves the dispatch center of D1 at 8:15 and arrives at the rail station M at 8:27. The ride time of passengers at the customer point C5 is about 10.2 min when they are picked up at 8:17. Due to their expected ride from 5 to 10 min, the satisfaction is calculated by \((20-10.2)/(20-5)=0.65\). Table 3 also guides passengers to travel in the specified time periods from several preferred time windows.  
Table 4 shows the routing plan of each vehicle, in which the two depots of D1 and D4 are selected for these three vehicles. The vehicles have total mileage of 3.1 km, 3.1 km, and 3.0 km, while they have total travel times of 12.2 min, 12.5 min, and 12.1 min, respectively. Tables 2-4 show that the total mileage and time taken in the routes were 9.2 km and 36.8 min, while the total passenger satisfaction was 96.4. The changes in vehicle loading capacity at each point can also be obtained. Taking Route 1 as an example, the vehicle will leave the dispatch center of D1, visit the customer points C5, C6, C7, C8, and C1, and terminate at the rail station M. An empty vehicle departs from D1. This vehicle first  
Figure 3: Spatial distribution of vehicle visiting points (Map resource: Baidu).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Demand Point** & **Boarding Time** & **Ride Time** & **Satisfaction** & **Vehicle** \\ \hline C5 & 8:17 & 10.2 & 0.65 & \\ C6 & 8:18 & 9.7 & 1 & \\ C7 & 8:20 & 7.3 & 1 & V1 \\ C8 & 8:21 & 6.2 & 1 & \\ C15 & 8:25 & 2.7 & 1 & \\ \hline C10 & 8:27 & 2.3 & 1 & \\ C11 & 8:22 & 7.3 & 0.77 & V2 \\ C12 & 8:25 & 3.9 & 1 & \\ C14 & 8:20 & 9.6 & 0.69 & \\ \hline C1 & 8:13 & 2.4 & 1 & \\ C2 & 8:10 & 5.7 & 1 & \\ C3 & 8:12 & 3.4 & 1 & V3 \\ C4 & 8:11 & 4.8 & 1 & \\ C9 & 8:15 & 1.1 & 1 & \\ C13 & 8:05 & 10.5 & 0.63 & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Assignment result of passengers picked up by vehicles.  
stops at C5 to pick up two passengers, thus giving the number of 2 between C5 and C6. After this, the vehicle visits C6 in which three passengers is loaded, thus giving the number of 5 between C6 and C7. Following this, the vehicle visits C7, C8, and C15 to pick up four passengers, one passenger and two passengers, thus giving the number of 9 between C7 and C8, 10 between C8 and C15, and 12 between C15 and M. Finally, this vehicle visits M to deliver its 12 passengers. In this case, the vehicle loading capacity at each point of Route 1 can be described as [D1(0), C5(2), C6(5), C7(9), C8(10), C15(13), M(0)]. Similarly, the capacities for Routes 2 and 3 are described as [D4(0), C14(1), C11(5), C12(6), C10(8), M(0)] and [D4(0), C13(3), C2(4), C4(6), C3(7), C1(10), C9(12), M(0)].  
Furthermore, our proposed method has unique features compared with the traditional DRTs. Figure 4 reveals the difference between the proposed model and DRT with single time window (DRTSTW). When compared with DRT and DRTSTW, the mileage and satisfaction of the proposed model will be reduced by 1.4 km and increased by 7.1%, respectively. This is due to the vehicle arriving at the demand point at any time within one of the multiple time windows, which will reduce invalid travel mileage and time. Therefore, it is very important for passengers to be allowed to choose multiple travel time windows in practical applications. As shown above, it is apparent that the proposed model is valid since all aspects of the model are obtained through this illustrative example.