## 6 Numerical Example
### Sensitivity Analysis
#### 6.4.2 State Railway Fare per Kilometer \(p_{1}\)  
State railway fare policy affects the fare cost of the travel path, thus affecting passengers' choice behavior. This part mainly analyzes the changes of passengers on the state train and the probability of passengers choosing the state train under the conditions of different state railway fares and values of \(\theta\).  
Figure 11: The iteration times by the proposed algorithm under different \(\theta\) values.  
Passengers on the state train.  
Figure 12 shows the changes of passengers on the state train (No. 8) in different sections under the conditions of various state railway fare policies and \(\theta\) values. The passenger assignment results in Figure 12b are less than they are in Figure 12a because the section from station No. 8 to No. 11 is the last operation section of the state train, and the total number of passengers to the terminal station is small.  
Overall, the passengers in both sections show a downward trend with the increase of state railway fare. This is because passengers choose the relatively cheap suburban trains when fare costs rise with more travel time. When the fare is less than 0.5 CNY/km, it has little impact on the results. However, when it is greater than or equal to 0.5, the effect of fare cost becomes greater. And when \(\theta\) = 0.1, passenger's sensitivity to state railway fares is low, more passengers are assigned to the state train. When \(\theta\) takes other values and the fare is more than 0.7 CNY/km, the results vary widely for different values of \(\theta\). The larger value of \(\theta\), the more passengers are allocated.  
2. State train choice probability  
Figure 13a,b represents the probability of passengers choosing the state train (No. 8) at station No. 5 while the current arriving train is No. 7 and No. 8, respectively. We can see that the state railway fares increase, and the choice probability of the state train decreases gradually. Meanwhile, the probability of Figure 13b is higher than that of Figure 13a because the additional waiting time will be added when the current arriving train is train No. 7.  
Separately, in Figure 13a, when the fare is less than 0.5 CNY/km, the greater the \(\theta\), the higher the probability of choosing the state train and vice versa. Therefore, it can be concluded that when the state railway fare is below 0.5 CNY/km, passengers are willing to wait for the next state train even with a longer waiting time. When the fare is greater than 0.5 CNY/km, the disadvantage of the additional waiting time will not be compensated. At this moment, the higher the \(\theta\) value, the more sensitive passengers are to fares and the lower the probability of choosing the state train. In Figure 13b, the larger the \(\theta\) value is, the greater the choice probability is, regardless of the fare. These passengers do not need additional waiting time, and the state train has a very high attraction.  
The overall trend of Figure 14 is similar to Figure 13b, and the probability is relatively low and less than 26% because fewer passengers arrive at the terminal station among all waiting passengers at station No. 8.  
Figure 12: Sensitivity analysis of state railway fares to state train passengers. (**a**) Section between st.5 and st.8. (**b**) Section between st.8 and st.11.