# Ticket Allocation Optimization of Fuxing Train Based on Overcrowding Control: An Empirical Study from China

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## Abstract

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## 1. Introduction

#### Literature Review and Innovation of This Paper

- (1)
- Single-train ticket allocation is the most basic research entry. Ciancimino et al. [4] first proposed treating different classes of carriages as different train products, so the problem of train ticket allocation can be transformed into a seat control problem in single trains, with multiple sections and a single ticket price. Some studies [5] assumed that the passenger demand obeys an independent normal distribution and used the particle swarm optimization algorithm (PSO) to solve the final optimization model. Shan et al. [6] predicted the passenger flow by adopting time series firstly. Then, they obtained a ticket allocation method through the formulation of rules such as long distance before short distance, seat before no seat, and allocating tickets in advance by quantity and then by proportion. Gopalakrishnan et al. [7] mainly considered the long-distance passenger demand of Indian railways and studied the utilization method of single-train seat capacity.
- (2)
- Ticket allocation involving multiple trains is further complicated on the basis of single-train ticket allocation. Jiang et al. [8] integrated the short-term passenger flow demand forecasting method into the railway multi-train ticket allocation model, which solved the situation that some stations were short of tickets and other stations were rich in tickets. Yan et al. [9] developed a seat allocation model for multiple HSR trains with flexible train formation. Jiang et al. [10] proposed a dynamic adjustment method for ticket allocation. Zhao et al. [11] proposed a probabilistic nonlinear programming model for the problem of railway passenger ticket allocation. Deng et al. [12] focused on the joint pricing and ticket allocation problem for multiple HSR trains with different stop patterns. Luo et al. [13] developed a multi-train seat inventory control model based on revenue management theory. The above was discussed with respect to the complexity of ticket allocation.
- (3)
- Customer choice behavior plays an important role in estimating customer demand, which has also been more discussed in ticket allocation in recent years. It is an obvious research conclusion that there are many factors influencing customer choice behavior, and these factors can be divided mainly into two aspects, personal attributes and trip attributes, in a great deal of the literature [14,15,16,17]. As for the method describing passenger choice behavior, Wang et al. [18] described different passenger needs through the Logit model and established a multi-stage random ticket allocation model under passenger selection behavior.
- (4)
- In the two most important sub-studies of revenue management, ticket allocation is always constrained by ticket prices. However, it is unfortunate that in the past few years, pricing and ticket allocation issues have always been treated separately, and there is a gap in the research regarding joint pricing and ticket allocation models [2,19]. Weatherford [20] first stressed the importance of considering prices and suggested them as decision variables for the ticket allocation problem. After Weatherford’s contribution, we can see more and more research that has optimized ticket allocation and ticket prices at the same time. Hetrakul et al. [19] put forward a comprehensive optimization model of dynamic ticket pricing and ticket allocation based on the impact of passenger heterogeneity on ticket allocation. Some of the literature regards ticket allocation as the basis of dynamic pricing and gives priority to solving the problem of ticket allocation while solving dynamic pricing [21,22,23,24].

## 2. Research Methodology and Process

#### 2.1. Model Assumptions and Notations

- (1)
- Passengers whose requests are fulfilled will not cancel their reservations or change their ticket.
- (2)
- Take the second-class seat of HSR as the research object.
- (3)
- If passengers need to extend their travel section after getting on the train, they only make up the tickets to the terminal. The travel section that extends from the boarding station to the terminal is called the “target travel section (OD pair)”. In order to get on the train first, passengers buy a short OD pair ticket, and the travel section corresponding to these tickets is called the “short travel section (OD pair)”. A target travel section corresponds to several short travel sections.

#### 2.2. Model Formulation

- (1)
- Train capacity constraints: the ticket number allocated for each train between $(r,s)$ cannot exceed their capacity constraint. ${C}_{k}$ is the capacity of train $k$$$\sum _{r\text{}=\text{}1}^{h}{\displaystyle \sum _{s\text{}=\text{}h\text{}+\text{}1}^{l}{b}_{rs}^{\left(k\right)}}}\le {C}_{k},h=1,2\dots l-1$$
- (2)
- Train service constraints:$$\{\begin{array}{l}{\eta}_{rs}^{\left(k\right)}=1,{b}_{rs}^{\left(k\right)}>0,{b}_{rs}^{\left(k\right)}\in {Z}^{+}\\ {\eta}_{rs}^{\left(k\right)}=0,{b}_{rs}^{\left(k\right)}=0\end{array}$$

#### 2.3. Train Overcrowding Control

## 3. Solution Algorithm

## 4. Results and Analysis

#### 4.1. Basic Data

#### 4.2. Computational Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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References | Revenue Maximization | Overcrowding Control | ||
---|---|---|---|---|

Single Train | Multiple Trains | Pricing | ||

[4,5,6,7] | √ | × | × | × |

[8,9,10,11,12,13] | × | √ | × | × |

[2,19,20,21,22,23,24] | × | × | √ | × |

This paper | × | √ | × | √ |

Symbols | Definition | Unit |
---|---|---|

Variable or Parameters | ||

$(r,s)$ | $\mathrm{Origin}\u2013\mathrm{destination}\text{}\mathrm{pair}\text{}\mathrm{in}\text{}\mathrm{the}\text{}\mathrm{transportation}\text{}\mathrm{service},\text{}\mathrm{which}\text{}\mathrm{is}\text{}\mathrm{composed}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{train}\text{}\mathrm{stops},\text{}\mathrm{from}\text{}\mathrm{station}\text{}r\text{}\mathrm{to}\text{}\mathrm{station}\text{}s$ | / |

$t$ | $\mathrm{Station}\text{}\mathrm{serial}\text{}\mathrm{number},\text{}\mathrm{the}\text{}\mathrm{train}\text{}\mathrm{arrives}\text{}\mathrm{at}\text{}\mathrm{the}\text{}t\mathrm{th}\text{}\mathrm{station}$ | / |

$h$ | $\mathrm{Section}\text{}\mathrm{serial}\text{}\mathrm{number},\text{}h=1,2\dots l-1$ | / |

$k$ | $\mathrm{Train}\text{}\mathrm{serial}\text{}\mathrm{number},\text{}k=1,2\dots K$ | / |

${p}_{rs}^{(k)}$ | $\mathrm{Price}\text{}\mathrm{of}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{between}\text{}(r,s)$ | CNY |

${\eta}_{rs}^{\left(k\right)}$ | $\mathrm{The}\text{}0\u20131\text{}\mathrm{binary}\text{}\mathrm{parameter}\text{}\mathrm{denoting}\text{}\mathrm{whether}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{can}\text{}\mathrm{provide}\text{}\mathrm{services}\text{}\mathrm{between}\text{}(r,s)$ | / |

${t}_{rs}^{\left(k\right)}$ | $\mathrm{Travel}\text{}\mathrm{time}\text{}\mathrm{of}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{between}\text{}(r,s)$ | min |

${\alpha}_{1},{\alpha}_{2}$ | Parameters of utility function | / |

${U}_{rs}^{\left(k\right)}$ | $\mathrm{Total}\text{}\mathrm{utility}\text{}\mathrm{of}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{for}\text{}\mathrm{passengers}\text{}\mathrm{between}\text{}(r,s)$ | / |

${w}_{rs}^{\left(k\right)}$ | $\mathrm{Probability}\text{}\mathrm{of}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{chosen}\text{}\mathrm{by}\text{}\mathrm{passengers}\text{}\mathrm{between}\text{}(r,s)$ | % |

${Q}_{rs}$ | $\mathrm{Expected}\text{}\mathrm{sales}\text{}\mathrm{volume}\text{}\mathrm{between}\text{}(r,s)$ | / |

${C}_{k}$ | $\mathrm{Capacity}\text{}\mathrm{of}\text{}\mathrm{train}\text{}k$ | passengers |

${q}_{t}^{\left(k\right)}$ | $\mathrm{Total}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{passengers}\text{}\mathrm{on}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{arriving}\text{}\mathrm{at}\text{}\mathrm{station}\text{}t$ | passengers |

${q}_{t\xb7}^{\left(k\right)}$ | $\mathrm{The}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{passengers}\text{}\mathrm{boarding}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{when}\text{}\mathrm{arriving}\text{}\mathrm{at}\text{}\mathrm{station}\text{}t$ | passengers |

${\widehat{q}}_{t}^{\left(k\right)}$ | $\mathrm{The}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{passengers}\text{}\mathrm{who}\text{}\mathrm{get}\text{}\mathrm{on}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{before}\text{}\mathrm{station}\text{}t$$\text{}\mathrm{and}\text{}\mathrm{get}\text{}\mathrm{off}\text{}\mathrm{the}\text{}\mathrm{train}\text{}\mathrm{after}\text{}\mathrm{station}\text{}t$ | passengers |

${q}_{\xb7t}^{\left(k\right)}$ | $\mathrm{For}\text{}\mathrm{train}\text{}k$$,\text{}\mathrm{the}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{passengers}\text{}\mathrm{who}\text{}\mathrm{should}\text{}\mathrm{get}\text{}\mathrm{off}\text{}\mathrm{before}\text{}\mathrm{station}\text{}t,\text{}\mathrm{but}\text{}\mathrm{are}\text{}\mathrm{still}\text{}\mathrm{on}\text{}\mathrm{the}\text{}\mathrm{train}\text{}\mathrm{after}\text{}\mathrm{extending}\text{}\mathrm{their}\text{}\mathrm{travel}\text{}\mathrm{section}$ | passengers |

${g}_{rs}$ | $\mathrm{The}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{passengers}\text{}\mathrm{who}\text{}\mathrm{buy}\text{}\mathrm{tickets}\text{}\mathrm{between}\text{}(r,s),\text{}\mathrm{but}\text{}\mathrm{extend}\text{}\mathrm{the}\text{}\mathrm{travel}\text{}\mathrm{section}\text{}\mathrm{to}\text{}\mathrm{the}\text{}\mathrm{terminal}$ | passengers |

${\beta}_{rs}$ | Parameter, travel extension coefficient | / |

$\gamma $ | Parameter, risk coefficient | / |

${x}_{rl}$ | $\mathrm{Passenger}\text{}\mathrm{demand}\text{}\mathrm{under}\text{}\gamma $$\text{}\mathrm{for}\text{}\mathrm{target}\text{}\mathrm{travel}\text{}\mathrm{sec}\mathrm{tion}\text{}(r,l)$ | / |

${F}_{rl}^{-1}(x)$ | $\mathrm{Inverse}\text{}\mathrm{function}\text{}\mathrm{of}\text{}\mathrm{cumulative}\text{}\mathrm{probability}\text{}\mathrm{function}\text{}\mathrm{of}\text{}\mathrm{passenger}\text{}\mathrm{demand}\text{}\mathrm{for}\text{}\mathrm{target}\text{}\mathrm{travel}\text{}\mathrm{sec}\mathrm{tion}\text{}(r,l)$ | / |

Decision Variable | ||

${b}_{rs}^{\left(k\right)}$ | $\mathrm{The}\text{}\mathrm{number}\text{}\mathrm{of}\text{}\mathrm{tickets}\text{}\mathrm{allocated}\text{}\mathrm{to}\text{}\mathrm{train}\text{}k$$\text{}\mathrm{between}\text{}(r,s)$ | / |

Train | Factors of Passengers Choice | OD | |||||
---|---|---|---|---|---|---|---|

(SH, BS) | (SH, NS) | (SH, JW) | (NS, BS) | (NS, JW) | (JW, BS) | ||

G2 | Price (CNY) | 553 | 134.5 | 398.5 | 443.5 | 279 | 184.5 |

Travel time (min) | 268 | 60 | 179 | 206 | 117 | 87 | |

G22 | Price (CNY) | 553 | 134.5 | / | 443.5 | / | / |

Travel time (min) | 258 | 60 | / | 196 | / | / |

Parameter | OD | |||||
---|---|---|---|---|---|---|

(SH, BS) | (SH, NS) | (SH, JW) | (NS, BS) | (NS, JW) | (JW, BS) | |

${\mu}_{rs}$ | 1701.4 | 170 | 89.6 | 277 | 13.4 | 96.6 |

${\sigma}_{rs}$ | 222.5 | 92.4 | 68.7 | 142.4 | 14 | 70.3 |

Arrive at the Station t | NS | JW | ||
---|---|---|---|---|

Target travel section | (SH, BS) | (NS, BS) | (SH, BS) | |

Short travel section | (SH, BS) | (NS, JW) | (SH, NS) | (SH, JW) |

$\mathrm{Travel}\text{}\mathrm{extension}\text{}\mathrm{coefficient}\text{}{\beta}_{rs}$ | 0.065 | 0.1 | 0.065 | 0.035 |

Train | OD | Number of Tickets Finally Allocated |
---|---|---|

G2 | (SH, BS) | 833 |

G2 | (SH, NS) | 110 |

G2 | (SH, JW) | 117 |

G2 | (NS, BS) | 153 |

G2 | (NS, JW) | 43 |

G2 | (JW, BS) | 160 |

G22 | (SH, BS) | 919 |

G22 | (SH, NS) | 184 |

G22 | (NS, BS) | 184 |

Train | OD | Number of Tickets Finally Allocated |
---|---|---|

G2 | (SH, BS) | 833 |

G2 | (SH, NS) | 75 |

G2 | (SH, JW) | 131 |

G2 | (NS, BS) | 139 |

G2 | (NS, JW) | 64 |

G2 | (JW, BS) | 195 |

G22 | (SH, BS) | 888 |

G22 | (SH, NS) | 225 |

G22 | (NS, BS) | 225 |

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**MDPI and ACS Style**

Wang, Y.; Shan, X.; Wang, H.; Zhang, J.; Lv, X.; Wu, J.
Ticket Allocation Optimization of Fuxing Train Based on Overcrowding Control: An Empirical Study from China. *Sustainability* **2022**, *14*, 7055.
https://doi.org/10.3390/su14127055

**AMA Style**

Wang Y, Shan X, Wang H, Zhang J, Lv X, Wu J.
Ticket Allocation Optimization of Fuxing Train Based on Overcrowding Control: An Empirical Study from China. *Sustainability*. 2022; 14(12):7055.
https://doi.org/10.3390/su14127055

**Chicago/Turabian Style**

Wang, Yu, Xinghua Shan, Hongye Wang, Junfeng Zhang, Xiaoyan Lv, and Jinfei Wu.
2022. "Ticket Allocation Optimization of Fuxing Train Based on Overcrowding Control: An Empirical Study from China" *Sustainability* 14, no. 12: 7055.
https://doi.org/10.3390/su14127055