Although China’s high-speed railway (HSR) is maturing after more than ten years of construction and development, the load factor and revenue of HSR could still be improved by optimizing the ticket fare structure. Different from the present unitary and changeless fare structure, this paper explores the application of multigrade fares to China’s HSR. On the premise that only one fare grade can be offered for each origin-destination (O-D) at the same time, this paper addresses the questions of how to adjust ticket price over time to maximize the revenue. First, on the basis of piecewise pricing strategy, a ticket fare optimization model is built, which could be transformed to convex program to be solved. Then, based on the analysis of passenger arrival regularity using historical ticket data of Beijing-Shanghai HSR line, several experiments are performed using the method proposed in the paper to explore the properties of the optimal multigrade fare scheme.
The competition between railway passenger transportation and other modes of passenger transportation is increasingly fierce. In this situation, the present unitary and changeless fare structure gradually becomes the prevention of railway revenue increase and railway system development. Statistical data provided for 2014 by the Beijing-Shanghai High-Speed Railway (HSR) company shows that 13 percent of the trains had a load factor less than 65 percent. It has been confirmed theoretically and in practice that an advance-purchase discount based on the uncertainty and valuation of travel demand can assist in attaining efficient allocation of capacity [
A multigrade fare structure for China’s high-speed railway is proposed in this paper. On the premise that only one fare grade can be offered for each origin-destination (O-D) pair at the same time, this paper examines how to adjust ticket price over time based on passenger choice behavior for each O-D.
A number of previous papers have examined aspects of the problem of dynamic pricing. Kincaid and Darling (1963) [
Pricing belongs to a tactical problem. On the practical level, methods are needed to dynamically decide the optimal timing of price changes. For a unique price change allowed to be either higher or lower, Feng and Gallego (1995) [
The research on dynamic pricing started late in China, where ticket price has traditionally remained changeless over decades. From the practical point of view, Shi [
In our study, each train service is represented as a linear network with stations as nodes and arcs that connect O-D station pairs served. The multigrade fare strategy is generated to meet the demand of each O-D. The optimal ticket fare for each O-D over time is driven by passenger demand, fluctuating within a certain range of the standard fare, either upward or downward. The remainder of the paper is organized as follows. In the next section we present a ticket fare optimization model based on piecewise pricing policy. In Section
The notations used in this paper are as follows.
The problem we address in the paper is described as follows. The railway company, which is in a market with imperfect competition, sells tickets for a train service to passengers having different O-D itineraries in a limited time horizon (ticket-selling period). Before the ticket-selling period, the number of seats is predetermined and unchangeable, and afterwards unsold tickets have zero salvage value. One of the tactical level decisions the company has to make is determining the number of fare grades and the price of each grade for each O-D to maximize the total revenue. Gallego and van Ryzin (1994) [
The transport service between a pair of consecutive stations is defined as a resource
Customers are usually divided into two categories, myopic and strategic [
Passenger demand is usually characterized as a Poisson process, for which demand within a certain time period is a stochastic variable. In a network formulation, the stochastic element vastly increases the complexity and difficulty of the problem. Following a frequently employed method, we use a deterministic model as an approximation of the stochastic problem. We denote The demand function is continuously differentiable and strictly decreasing with The revenue rate is
Examples include the commonly used linear demand function
Although we seek optimal prices, it is convenient to employ the demand density function
The objective function (
According to property 2 of the demand function, the objective function of the model is concave, and the constraints are concave, too. By introducing
Train G205, which has a low load factor, was the case to verify our multigrade fare strategy. The route map of G205 (Figure
Route map of train G205.
Train G205 has a passenger capacity of 1005, thus the initial capacity of each resource,
We employ the log-linear demand function commonly used in economics to describe the relationship between demand and price:
In (
The price elasticity of demand is a key parameter of the demand function, indicating the response degree of demand to a change in price. We estimated the price elasticity of demand of the O-Ds of train G205 using ticket data for the summer of 2013, when discounts were offered. The price elasticity of demand, which is the ratio of the percentage change in demand and the percentage change in price, could be calculated using the following equation, and the resulting estimates are shown in Table
Estimates for price elasticity of demand.
O-D | Mileage (km) | Price elasticity of demand |
---|---|---|
Jinan West-Xuzhou East | 286 | |
Xuzhou East-Nanjing South | 331 | |
Beijing South-Jinan West | 406 | |
Jinan West-Nanjing South | 617 | |
Beijing South-Xuzhou East | 692 | |
Beijing South-Nanjing South | 1023 | |
The resulting estimates of price elasticity of demand are in conformity with the existing research results, from the aspect of competition between the high-speed railway and other transport modes [
We apply MATLAB to solve the ticket fare optimization problem proposed above. Two experiments are performed to examine the characteristics of the optimal fare scheme. The first explores the relationship between the optimal fare scheme and price elasticity of demand.
We analyze the passenger arrival regularity as the basis of the first experiment, using historical ticket data for the Beijing-Shanghai High-Speed Railway during the period March 1, 2013, to March 13, 2014. For a certain train,
The demand density ratio of each day by O-D.
As Figure
Input parameters.
| | | | ||
---|---|---|---|---|---|
| | | | ||
1 | 399 | 0.051 | 68 | 1.481 | 184.5 |
2 | 399 | 0.051 | 68 | 0.982 | 309 |
3 | 375 | 0.081 | 92 | 0.849 | 443.5 |
4 | 423 | 0.004 | 45.5 | 0.376 | 129.5 |
5 | 399 | 0.008 | 69.6 | 0.254 | 279 |
6 | 399 | 0.037 | 70.7 | 0.943 | 149.5 |
It is commonly assumed that leisure travelers who are sensitive to price tend to arrive early, whereas business travelers who are not so sensitive to price arrive late. Under this hypothesis, we set the price elasticity of demand of the first subperiod as 10% higher than that of the second subperiod. The price elasticity of demand calculated from historical ticket data (see Table
Relation between optimal fare scheme and price elasticity of demand.
We can see that the train revenue and load factor increase with the price elasticity of demand, which indicates that the more sensitive the demand is to price, the better the application effect of a multigrade fare strategy is. This is important in a market with a variety of competitive transportation modes.
The second experiment studies the relationship between optimal fare scheme and the number of fare grades. We determine the optimal fare schemes for two, three, and four fare grades, along with the train revenue and load. Table
The relation between optimal fare scheme and number of fare grades.
Revenue (Yuan) | Revenue increase pct. | Load factor | Load factor increase pct. | |
---|---|---|---|---|
Basic fare scheme | 112462.63 | 24.67% | ||
Two fare grades | 119647.30 | 6.39% | 26.88% | 8.94% |
Three fare grades | 119772.47 | 6.50% | 25.87% | 4.90% |
Four fare grades | 120326.56 | 6.99% | 25.14% | 1.92% |
Table
Parameters and optimal solution of the three-grade fare strategy.
| | | | | | |
---|---|---|---|---|---|---|
| 1.07 | 0.029 | 184.5 | 303 | 0.031 | 172 |
| 0.98 | 0.122 | 184.5 | 96 | 0.120 | 188 |
| 0.89 | 1.481 | 184.5 | 68 | 1.327 | 207 |
| 1.15 | 0.033 | 309 | 231 | 0.038 | 269 |
| 1.05 | 0.077 | 309 | 168 | 0.081 | 294 |
| 0.95 | 0.982 | 309 | 68 | 0.934 | 325 |
| 1.33 | 0.06 | 443.5 | 183 | 0.084 | 333 |
| 1.21 | 0.101 | 443.5 | 192 | 0.125 | 366 |
| 1.10 | 0.849 | 443.5 | 92 | 0.938 | 403 |
| 0.81 | 0.002 | 129.5 | 375 | 0.002 | 130 |
| 0.74 | 0.017 | 129.5 | 48 | 0.014 | 164 |
| 0.67 | 0.376 | 129.5 | 45.5 | 0.307 | 168 |
| 1.07 | 0.005 | 279 | 327 | 0.006 | 259 |
| 0.98 | 0.022 | 279 | 72 | 0.022 | 284 |
| 0.89 | 0.254 | 279 | 69.5 | 0.228 | 314 |
| 0.73 | 0.022 | 149.5 | 279 | 0.018 | 190 |
| 0.67 | 0.075 | 149.5 | 120 | 0.061 | 192 |
| 0.61 | 0.943 | 149.5 | 70.6 | 0.786 | 194 |
The experiments that make use of the actual data show that the multigrade fare strategy can not only increase revenue but also stimulate the potential demand, compared to the present unitary fare structure. This indicates the multigrade fare strategy is suitable for China’s high-speed railway, especially in the market with fierce competition with other transportation modes. The second experiment tells that the amount of fare grades should not be too much, which was also proved in practice by the experience of Britain and France railway [
This paper explores the application of multigrade fare structure in China’s HSR, where the single-grade fare strategy has always been used. Our goal is to maximize the expected revenue of a train by setting multiple ticket prices according with different valuations of passengers.
Based on a piecewise pricing strategy, we first build a ticket fare optimization model that can be transformed to a convex program to solve. Taking train G205 as a case study, we conducted two experiments to examine the properties of the optimal fare scheme. The experiment results show that the more sensitive the demand is to price, the better the application effect of the multigrade fare strategy is. By comparing the optimal solutions of different fare schemes with the basic fare strategy, we find that multigrade fare strategy can increase both the train revenue and load factor. This indicates that multigrade fare is suitable for China’s HSR. At last we give a suggestion about determining the most suitable multigrade fare scheme. Since China’s railway is a seminonprofit organization that should pursue benefit on the premise of meeting passengers’ travel demand as much as possible, we recommend that a three-grade fare scheme that balances train revenue and load factor is the most suitable.
There are certainly many possible directions for future research in this area. In this paper, the passenger behavior is assumed to be deterministic, and the demand quantity is assumed to be fixed. Being more realistic, in the future research we should consider the passengers’ response to ticket fare optimization, and the demand quantity should be regarded as elastic. Another direction is to expand our study to the system with multiple trains. Because the ticket fare adjustment of some train would cause the transfer of passengers between the trains that provide the same O-D transportation service, we should further research how to collaboratively optimize the ticket fare of these trains.
Symbol of source,
Symbol of product,
Initial capacity of resource
Demand density of product
Time duration of subperiod
Price of product
Revenue rate of product
Basis price of product
The number of ticket-selling subperiods
Resource-product incidence matrix
Maximum downward fluctuation ratio of price
Maximum upward fluctuation ratio of price.
The authors declare that they have no competing interests.
This research was supported by China Railway Corporation Technology Research and Development Plan Project (2016X005-E).