Discrete Dynamics in Nature and Society

Volume 2017, Article ID 6237642, 6 pages

https://doi.org/10.1155/2017/6237642

## Ticket Fare Optimization for China’s High-Speed Railway Based on Passenger Choice Behavior

^{1}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China^{2}University of Tennessee, Knoxville, TN, USA

Correspondence should be addressed to Jinzi Zheng; nc.ude.utjb@41241101

Received 28 October 2016; Accepted 5 February 2017; Published 21 February 2017

Academic Editor: Lu Zhen

Copyright © 2017 Jinzi Zheng et al. 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

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.

#### 1. Introduction

##### 1.1. Research Background

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 [1]. Therefore, compared to the unitary and changeless fare structure, multigrade fare structure could increase the resource utilization rate, as a result of which the revenue of railway company will be improved.

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.

##### 1.2. Literature Review

A number of previous papers have examined aspects of the problem of dynamic pricing. Kincaid and Darling (1963) [2] 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. Gallego and van Ryzin (1994) [3] 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 (1997) [4] 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 (1995) [5] put forward a control strategy based on a time threshold that depends on the number of unsold items. On the basis of this work, Feng and Xiao (2000) [6] expanded the approach to allow multiple price changes. In further work, Feng and Xiao (2000) [7] allowed the predetermined prices to change reversely.

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 [8] translated the optimal solution to a feasible approximate optimal solution. Zhang [9] applied maximum concave envelope theory to determine the optimal fare discount for each O-D of a train. With the application of dynamic game theory, Xu et al. [10] established a dynamic pricing model between HSR and air transportation. Yao et al. [11] 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 [12] developed a dynamic programming model to deal with the HSR revenue optimization problem with multitrain, multisegment, and multiclass, by considering passengers choice behavior. Wei et al. [13] constructed a bilevel programming model for the high-speed rail fare optimization problem, considering the benefit of both railway unit and passengers. Based on the operation data of high-speed railway, Li and Fu [14] studied the elasticity of demand, which is a key problem when optimizing ticket fare. Zheng and Liu [15] explored the application of multigrade fare in China’s high-speed railway, which set prices separately by O-D and by time period.

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 3, we present several experiments and discuss properties of the optimal fare scheme based on computational results. The last section provides conclusions.

#### 2. Methodology

##### 2.1. Problem Definition and Notation

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) [3] showed that the optimal price changes continuously with the time to departure and the number of seats remaining. Since it is unrealistic to adjust price all through the time, for practical purposes we apply a piecewise pricing policy, where in any interval of the ticket sale period for an O-D there is only one fare grade opened.

The transport service between a pair of consecutive stations is defined as a resource , . The transport service between any pair of stations is defined as a product composed of one or more resource units , . The resource-product incidence matrix is a 0-1 matrix, with when resource is used by product , and otherwise. The initial capacity of resource is , the number of seats on the train. The ticket-selling period is divided into several subperiods, with denoting the number of subperiods of product and denoting the time duration of subperiod of product . For a given product , the sum of the time span of all the subperiods is the duration of ticket-selling period of , which ends as the train departs from the origin of .

Customers are usually divided into two categories, myopic and strategic [16]. Strategic customers optimize their purchase behavior according to the company pricing strategy, while myopic customers buy a product as soon as its price is less than their reservation price. The more expensive the product is, the more necessary it becomes to model strategic customers. However, there is not too much information and time for customers to make strategic decisions when purchasing railway tickets. Therefore, we assume myopic behavior in this paper.

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 as demand density (amount of demand in unit time) of product in subperiod . On the hypothesis of imperfect market competition, demand changes with price. Thus, the demand density is described as a function of time and price, . This function is assumed to have the following properties.(1)The demand function is continuously differentiable and strictly decreasing with , that is, on the feasible price set. The demand function has a unique inverse function .(2)The revenue rate is , which is assumed to be continuously bounded concave.

Examples include the commonly used linear demand function and the exponential demand function .

##### 2.2. Modeling

Although we seek optimal prices, it is convenient to employ the demand density function , which is the unique inverse function of price, as the model decision variable. The ticket fare optimization model is then

The objective function (1) is to maximize the total revenue produced by all the products in the whole ticket-selling period. In this paper we assume that tickets are sold on a first-come, first-served basis. Thus the number of booking requests accepted is restrained only by the capacity of each resource, as shown in inequation (2). Inequation (3) restricts the range that the price fluctuates around the basis price, with the upper bound based on some customer psychological threshold and the lower bound the railway’s transportation cost. We denote the basis price of product as , the maximum downward fluctuation ratio as , and the maximum upward fluctuation ratio as .

According to property 2 of the demand function, the objective function of the model is concave, and the constraints are concave, too. By introducing , the model above can be transformed into a convex program with the objective of minimizing , for which the K-T point is the optimal solution of the original problem. The ticket fare optimization model ((1)–(3)) could be solved by finding the K-T point of the new convex program.

#### 3. Case Study

##### 3.1. Case Introduction

Train G205, which has a low load factor, was the case to verify our multigrade fare strategy. The route map of G205 (Figure 1) shows that there are 3 resources in our case, South-Jinan West, Jinan West-Xuzhou East, Xuzhou East-Nanjing and 6 products, South-Jinan West, Beijing South-Xuzhou East, Beijing South-Nanjing South, Jinan West-Xuzhou East, Jinan West-Nanjing South, Xuzhou East-Nanjing . The resource-product incidence matrix is