## 7 Conclusions  
With the gradual deepening of China Railway's market-oriented reform, based on the revenue management theory, this paper tried to solve the shortcomings of the existing ticketing organization in terms of the joint optimization of ticket pricing and allocation, considering the dynamic demand characteristics of passengers during the pre-sale period. The main conclusions are as follows:  
1. Since the arrival times of passengers' ticket-purchasing requests are unevenly distributed in the time dimension, this paper uses the compound non-homogeneous Poisson process to describe the passengers' ticket-purchasing process, and adopts the sparse method to simulate the passengers' ticket demand during the pre-sale period. Through this method, the ticket-purchasing process of passengers and the dynamic demand of passengers during the pre-sale period are described realistically, so as to propose the stochastic demand function in different OD sections and different periods within the pre-sale period. This allows the joint optimization model of ticket pricing and allocation to better improve the mismatch between supply and demand.
2. Taking the ticket pricing and allocation as the decision variables, a stochastic nonlinear-programming mathematical model was established with the goal of maximizing the single train revenue, considering the constraints of the upper and lower prices, the ticket allocation conditions among each period, the train's seat capacity, the average passenger seat utilization, and the standby ticket-purchasing demand. According to the characteristics of the model, a particle swarm algorithm was designed to solve the problem. This study enriches the application of revenue management theory in the high-speed railway passenger transport market and makes certain academic contributions.
3. This study took the G19 high-speed train as a case study to verify the effectiveness of the model and algorithm. Based on the past data of passengers' ticket purchases, the dynamic passenger demand of each OD section during the pre-sale period was simulated, and the optimal ticket price and allocation of each OD section in each period of the pre-sale period were obtained through the optimization model. The results show that the joint optimization scheme of ticket pricing and allocation considering dynamic demand yields a revenue of CNY 601,881, which increases the revenue by 1.01% compared with the fixed ticket price and pre-allocation scheme. With the increase in train revenue, the operation income of the whole transportation enterprise will be gradually improved, which is conducive to dealing with the challenge of an imbalanced income and expenditure and promoting sustainable development.  
* In this paper, only the single train and the second-class seats are considered when studying the joint optimization of ticket pricing and allocation. In an actual situation, the passenger demand not only transfers among parallel trains, but also among different seat classes. In future research, the joint optimization of ticket pricing and the allocation for multiple parallel trains and multiple seat classes should be further studied to develop more realistic schemes and strategies.  
The authors confirm the contribution to the paper as follows. Conceptualization, X.Y., D.L., W.R. and Z.L.; methodology, X.Y., D.L., W.R. and Z.L.; software, X.Y. and Z.L.; validation, X.Y., D.L., W.R. and Z.L.; writing--original draft preparation, X.Y., D.L., W.R. and Z.L.; writing--review and editing, X.Y. and D.L.; visualization, X.Y. and Z.L.; supervision, W.R.; project administration, D.L. All authors have read and agreed to the published version of the manuscript.  
This research is financially supported by National Railway Administration of the people's Republic of China. The name of the research project is \"Research on the Railway Ticket Internet Purchasing Behavior Regulation\" (TYFK201940).  
Not applicable.  
The authors declare no conflict of interest.