Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5073053, 8 pages

http://dx.doi.org/10.1155/2016/5073053

## The Research on Ticket Fare Optimization for China’s High-Speed Train

School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China

Received 6 June 2016; Accepted 28 July 2016

Academic Editor: Paolo Crippa

Copyright © 2016 Jinzi Zheng and Jun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the constant deepening of the railway reform in China, the competition between railway passenger transportation and other modes of passenger transportation has been getting fiercer. In this situation, the existing unitary and changeless fare structure gradually becomes the prevention of railway revenue increase and railway system development. This paper examines a new method for ticket fare optimization strategy based on the revenue management theory. On the premise that only one fare grade can be offered for each OD at the same time, this paper addresses the questions of how to determine the number of fare grades and the price of each fare grade. First, on the basis of piecewise pricing strategy, we build a ticket fare optimization model. After transforming this model to a convex program, we solve the original problem by finding the Kuhn-Tucker (K-T) point of the convex program. Finally, we verify the proposed method by real data of Beijing-Shanghai HSR line. The calculating result shows that our pricing strategy can not only increase the revenue, but also play a part in regulating the existing demand and stimulating potential demand.

#### 1. Introduction

With the constant deepening of the railway reform in China, railway passenger transportation has been involved in fiercer competition with airline and highway transportation. In the environment of market economy, the primary goal of railway sector is not only to realize passengers’ transportation demand, but also to get as much revenue as possible. However, in the past several decades, what we have been applying is the unitary and changeless fare structure, of which the ticket fare rate is the same all over the country and for all kinds of passengers. Statistical data provided for 2014 by the Beijing-Shanghai High-Speed Railway Company shows that 13 percent of the trains had a load factor less than 65 percent, which means there exists large potential of revenue in railway passenger transportation. It is hard for the railway to sustain the position as before if we do not make any transition to adapt to the new situation. Since 2012, the Shanghai-Nanjing, Shanghai-Hangzhou, Beijing-Shanghai, and Beijing-Guangzhou high-speed railways have tried to discount off-season ticket fares. One characteristic of these preference strategies is that the timing and discount level are the same for all the ODs of a specific train. This unitary discount strategy is easy to be implemented but cannot accommodate to diversified demand characteristics (such as demand density and demand valuation) of various ODs in different time.

Revenue management, which has been applied to air transportation for years, has successfully increased the revenue of air transportation by providing proper customers with proper prices and effective capacity control strategies. General and detailed discussion of revenue management can be seen in Weatherford and Bodily [1], McGill and Van Ryzin [2], Bitran and Caldentey [3], and Elmaghraby and Keskinocak [4]. Since railway passenger transportation has many characters in common with air transportation, we are confident that revenue management can also bring benefit to railway if being utilized properly. Armstrong and Meissner [5] analyzed the similarities and differences between airline and railway passenger transportation and gave an overview of the existing research results on railway revenue management.

A number of previous works have examined the aspects of the problem of price-based revenue management, which is aimed at increasing profits by dynamically setting optimal pricing strategies [6]. Dynamic pricing has already been successfully applied in many industries such as airline, hotel, and car rental. Kincaid and Darling [7] laid the foundation for the study of multigrade price, building two optimization models to generate the optimal prices of each time from the aspects of whether the price is announced. Chatwin [8] researched how to choose the optimal prices from a given price set according to the current inventory. Gallego and Ryzin [9] analyzed the structural characteristics of optimal price for a single product, proving that the optimal solution of the deterministic problem is the upper bound of the stochastic problem. On this basis, Gallego and Van Ryzin [10] found the upper bound of the optimal expected revenue by analyzing the deterministic version of the problem for multiple products. 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 [11] put forward a control strategy based on a time threshold that depends on the number of unsold items.

In China, research on dynamic pricing started late because the ticket price has traditionally remained changeless. From the practical point of view, Shi [12] translated the optimal solution to a feasible approximate optimal solution. Zhang [13] applied maximum concave envelope theory to determine the optimal fare discount for each OD of a train. With the application of dynamic game theory, Zhang et al. [14] established a dynamic pricing model between HSR and air transportation. Yao et al. [15] analyzed the pricing strategy for HSR in Wuhan-Guangzhou corridor with the consideration of the competition between passenger rail transportation and other transport modes. Bingyi [16] developed dynamic programming model to deal with the HSR revenue optimization problem with multitrain, multisegment, and multiclass, by considering passengers choice behavior.

Our paper studies the ticket fare optimization method for single high-speed train, aiming at optimizing the revenue of the train. A train service is represented as a linear network with stations as nodes and arcs that connect OD station pairs served. We propose a method to determine the number of fare grades according to the regularity of passenger arrival and the optimal prices that meet the demand of each OD. A demand driven fare is allowed to fluctuate within a certain range, either upward or downward, around the standard fare.

This paper proceeds as follows. Section 2.1 is a fine description of optimization model. Section 2.2 makes use of Lagrange Multiplier and Karush-Kuhn-Tucker (KKT) conditions to solve the previous model. Section 3 shows an experimental study for verifying the proposed method.

#### 2. Model and Algorithm

We will firstly provide in this section the basic terminologies and notations necessary for modeling. After that, the model and algorithm of ticket fare optimization problem are put forward.

##### 2.1. The Notations and Model

We define the “product” in our study as origin-destination (OD) transportation service offered by a passenger train, which contains stops including the starting and terminal stations and sections between each two stations. The product in our model is denoted by , referring to the OD transportation service from station to station . All the products produced by a passenger train are stored in a sequence according to the incremental order of and , . After numbering all the products in from left to right, we can get a new product sequence indexed by the serial number, , and . The section, which is a basic element to make up OD, is deemed as the resource of product, denoted as .

By analyzing the historical ticket-sale data, we found the demand density for a certain OD product varies with time regularly in ticket-sale period. As we deem, the passenger ticket-purchasing behavior is influenced by various factors, the details on which will be discussed in Section 3.1. For a certain product, the external factors such as travel distance, alternative product, and departure time are fixed; thus the difference of ticket-purchasing pattern by time is mainly determined by the passengers’ valuations. As is usually done in airline revenue management, we assume that passengers with low valuation arrive earlier and those with high valuation arrive later. Based on the third-degree price discrimination, profit could be increased by providing different customers with corresponding prices. Therefore, the idea of floating pricing strategy proposed here is to set prices separately by product, and for a certain product the price is adjusted according to the change of passenger arrival pattern.

We firstly divide the ticket-sale period into several subperiods according to the statistical regularity of passenger arrival process of the product we concern and then set the price in each subperiod. For an OD product, the ticket-sale period is divided into subperiods; the one is denoted by. The time duration of subperiod of productis. In this research, we make the following hypotheses: () the passenger transportation market is an imperfect competition market and the demand of a certain product changes with its price; () we only consider myopic passengers who buy tickets as soon as the offered price is less than their willingness to pay, in contrast to the strategic passengers who will optimize their purchase behavior by predicting the price strategies; () for simplification, we assume there is only one class of seats on the train. Thus, the demand density of product in subperiod can be expressed as the function of and current price , denoted as . We control demand density instead of price in this model for the convenient description of capacity constraint. The price of product is denoted as the inverse function of demand function, . To ensure the price is feasible, we define a feasible price set . The decision variables in the model are demand densities of each product in each subperiod, where ().

In our problem, the initial inventory is expressed as a vector is the set of positive integers), where is the initial inventory of resource , representing the number of available seats of this section. In addition, we define a 0-1 variable to denote the relationship between product and resource ; that is, if product occupies resource , ; else . The preallocation plan is a peculiarity of railway passenger organization in China, which is to set the range of application for all seats ahead of the ticket-sale period. In our model, is used to denote the maximum capacity of product produced by the preallocation plan, which is another important capacity constraint.

The total revenue of the train we concern is as below:

The optimal price strategy is the one ( in this model) that will maximize the total revenue .

As stated above, we take into consideration two kinds of capacity constraints: one is the “resource capacity constraint”, which ensures the capacity of any resource cannot be exceeded; the other is the “product capacity constraint” produced by preallocation plan. These constraints are expressed as (2) and (3), respectively:

##### 2.2. The Algorithm

The revenue rate function is continuous, bounded, and concave [10]. As the product of real number and concave function and the sum of several concave functions are both concave, it is easy to deduce that the objective function is concave too. Therefore, the nonlinear programming problem we propose can be solved using Lagrange Multiplier and KKT conditions. According to the requirement of KKT condition, we introduce a new objective function which is the opposite of . Then the initial problem can be rewritten as follows:

The gradient of the objective function and constraint functions can be gained by taking the derivation:

Supposing that is the optimal solution of the initial nonlinear programming problem (see (1)–(3)), and the derivation and are linearly independent, by introducing Lagrange multipliers and , we can have

By solving the above equations set we can get the optimal solution and also the optimal price of each product in each subperiod.

#### 3. Case Study

In this case study, we select two high-speed trains of Beijing-Shanghai High-Speed Railway G1 and G3, which have many things in common with each other in operation properties. Both of G1 and G3 trains depart from Beijing South station, stop once at Nanjing South station, and terminate at Shanghai Hongqiao station. The distance of Beijing-Nanjing product is 1023 km, Beijing-Shanghai product 1318 km, and Nanjing-Shanghai product 295 km. The single difference between these two trains is that G1 departs at 9:00 am and G3 at 2:00 pm.

##### 3.1. Passenger Arrival Regularity

Passenger arrival regularity is the basis of segmenting the ticket-selling period. We apply the ticket-sale data in 2012 (the data of December is not included for the lack of data in this month) as the sample to analyze these two trains’ passenger arrival regularity by product and show the statistical results in Figure 1, of which each point represents the percentage of tickets sold in that day to the total amount of tickets sold in the entire period for the product.