## 6 Discussions
### Comparison with Previous Studies
Therefore, the relationship between the passenger demand and ticket price was necessary for the study in this paper, which was used to connect the two decisions of ticket pricing and allocation and realize the joint optimization. As for the expression of passengers' demand for tickets, most previous studies believe that the demand is sensitive to the ticket price. For example, [30] argues that the passengers' demand is a function of the ticket price and assumes that the demand function is known and deterministic. However, it ignores that demand changes over time during the pre-sale period. The authors of [33] consider the difference in demand in different periods during the pre-sale period but describe the demand in every period as the deterministic demand according to the given passenger flow, which ignores the randomness of the ticket-purchasing demand. In reality, passengers' ticket-purchasing process is a simple stochastic process with a continuous time and discrete states. This paper investigated the past ticket-purchasing data of passengers during the pre-sale period and found that the arrival time of passengers' ticket-purchasing requests is not uniformly distributed in the time dimension. Therefore, the compound non-homogeneous Poisson process was used to simulate the ticket-purchasing process of passengers, and the sparse method was used to simulate the dynamic ticket-purchasing demand of passengers during the pre-sale period, so that the demand in different OD sections and different periods of the pre-sale period is expressed as a stochastic function affected by the price. The methods proposed in this study describe the ticket-purchasing process and the dynamic demand of passengers during the pre-sale period more realistically so as to improve the mismatch between supply and demand. In terms of model construction, in previous similar studies on dynamic pricing or ticket allocation for HSRs, the following constraints were generally considered: the number of seats allocated to each train cannot exceed the capacity of any two neighboring stations, upper and lower price limits, and the number of seats allocated to each train must be a non-negative integer, e.g., in [28; 36]. In our opinion, these constraints are necessary to ensure the reliability of the study results, but they are insufficient for practical situations. Unlike previous studies, this paper adds the constraints of the average passenger seat utilization and standby passenger flow calculation. To a certain extent, the average passenger seat utilization constraint ensures that under the goal of maximizing train revenue, it will not have a significant impact on the average passenger seat utilization rate of trains after the implementation of floating pricing, which is more in line with the actual transportation situation of high-speed railways in China. Considering the standby ticketing service in China Railway's passenger-ticketing organization, the demand for standby tickets was calculated, and the ticket purchasing expenses of standby passengers were simultaneously included in the train's total revenue of the objective function for optimization, thereby allowing the model to better fit the actual situation in China.