## 3 Mathematical Model
### Model Formulation
#### 3.3.2 Constraints  
1. [label=(0)]
2. Ticket price constraint 1. Since the ticket price of a high-speed railway needs to account for the revenue of railway transport enterprises and social benefits, the ticket price is restricted to float within a certain range, as shown in Equation (9). \[\mathrm{p}_{\mathrm{ij}}\leq\mathrm{p}_{\mathrm{ij}}^{\mathrm{t}}\leq\mathrm{ p}_{\mathrm{ij}}^{\mathrm{T}}\ \forall\mathrm{t}\in\mathrm{T}\] (9) According to the current policy on ticket prices, the published price is the highest limit of the implementation price; that is, the upper limit \(\mathrm{p}_{\mathrm{ij}}\) of section (i,j) is a certain value. The lower limit \(\mathrm{p}_{\mathrm{ij}}\) is the product of the published ticket price and the lowest discount. As shown in Equation (10), \(\mathrm{\beta}_{\mathrm{ij}}\) is the lowest ticket price discount rate in section (i,j). \[\mathrm{p}_{\mathrm{ij}}\mathrm{=p}_{\mathrm{ij}}^{\mathrm{t}}.\mathrm{\beta}_ {\mathrm{ij}}\] (10)
3. To encourage passengers to purchase tickets earlier during the pre-sale period, the ticket price in section (i,j) will only rise as the departure date approaches. \[\mathrm{p}_{\mathrm{ij}}^{\mathrm{t-1}}\leq\mathrm{p}_{\mathrm{ij}}^{\mathrm{t }}\ \forall\mathrm{t}\in\mathrm{T}\] (11)
4. In the first period of the pre-sale period, the ticket price should be at a lower level to provide room for increases in later periods. \[\mathrm{p}_{\mathrm{ij}}^{\mathrm{1}}\leq\mathrm{p}_{\mathrm{ij}}^{\mathrm{a}}\] (12)
5. Ticket allocation constraint 4. The number of tickets during the pre-sale period consists of two parts: the pre-allocation tickets and the standby tickets. Standby tickets will be sold at the last period \(\mathrm{T}\) only when the pre-allocation tickets are sold out in the former T-1 periods. Therefore, the number of tickets allocated in the former T-1 periods should be less than or equal to the number of pre-allocation tickets. \[\sum\limits_{\mathrm{t=0}}^{\mathrm{T}-1}x_{\mathrm{ij}}^{\mathrm{t}}\leq \mathrm{N}_{\mathrm{ij}}\] (13)
6. The number of tickets allocated in each OD section cannot be negative and must be an integer. \[x_{\mathrm{ij}}^{\mathrm{t}}\geq 0,\ x_{\mathrm{ij}}^{\mathrm{t}}\in Z\ \ \forall \mathrm{t}\in\mathrm{T}\] (14)
7. Train seat capacity constraint 5. The number of tickets allocated in each OD section cannot be negative and must be an integer. \[x_{\mathrm{ij}}^{\mathrm{t}}\geq 0,\ x_{\mathrm{ij}}^{\mathrm{t}}\in Z\ \ \forall \mathrm{t}\in\mathrm{T}\] (15)
8. Train seat capacity constraint 6. The number of tickets allocated in each OD section cannot be negative and must be an integer. \[x  
The total number of tickets allocated to each OD section during the pre-sale period should be less than the train seat capacity in the current segment.  
\[\sum_{t=0}^{T}\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}\varphi_{h}\cdot x_{ij}\leq C_{h} \tag{15}\]  
(4) Calculation of standby ticket-purchasing demand  
The standby passenger flows in each section come from the passengers whose ticket-purchasing demands have not been met in the former T-1 periods. Their value is the difference between the passenger demand and the seats allocated and is not negative. \[\delta_{\bar{ij}}\cdot M_{\bar{ij}}^{T-1}=\sum_{t=0}^{T-1}d_{\bar{ij}}^{t}-x_{ ij}^{t}=\sum_{t=0}^{T-1}d_{\bar{ij}}^{t}-\sum_{t=0}^{T-1}\min\left\{d_{ij}^{t} \cdot x_{ij}^{t}\right\} \tag{16}\]  
(5) Average passenger seat utilization constraint  
\[\gamma.\frac{\sum_{t=0}^{T}\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}E\left[X_{ij}\left( \xi_{\bar{ij}}^{k}\right)\right]\cdot l_{ij}}{C\cdot L}\leq\frac{\sum_{t=0}^{ T}\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}min\left\{d_{ij}^{t}\cdot x_{ij}^{t}\right\} \cdot l_{ij}}{C\cdot L},\,\gamma\in(0,1) \tag{17}\]  
The Equation (17) indicates that the average passenger seat utilization rate of the train after the implementation of dynamic pricing shall not be lower than a certain limit, and the value range of the dimensionless parameter \(\gamma\) is (0,1).