# Joint Decision Model of Group Ticket Booking Limits and Individual Passenger Dynamic Pricing for the High-Speed Railway

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- (1)
- We proposed a comprehensive model of seat inventory control and dynamic pricing considering all-or-none group reservations, which extends the joint decision of capacity control and dynamic pricing.
- (2)
- We investigated how group discounts affect total revenue when dynamic pricing is applied for individuals. The way of discounting group tickets can effectively stimulate group demand, and the booking of group discount tickets needs to be limited to maximize total revenue. Numerical experiments showed that decision making is superior to simple dynamic pricing when demand is weak.
- (3)
- We investigated the effects of time and the sales volume of tickets on total revenue, and the effects of group demand and total demand on the total expected revenue. These rules will help rail operators to apply the comprehensive policy appropriately.

## 3. Methodology

#### 3.1. Problem Description

- (1)
- The minimum price of dynamic pricing for individual orders is not less than the discount price for group orders, so as to ensure that groups will be attracted by the discount.
- (2)
- The order arrival rate is time homogeneous. In fact, although rail operators cannot predict demand accurately, they can estimate the average order arrival probability and the average proportion of group purchase orders in total orders based on historical data, so that we can consider the arrival probability of each type of order as time homogeneous.
- (3)
- The reserve price distribution information is symmetrical for the rail operator and passengers.
- (4)
- Overbooking, no-shows, refunds, and standing-room-only tickets were not considered.

#### 3.2. Dynamic Programming Model for Joint Decision

#### 3.2.1. Notation

$t$ | Time periods after sale begins,$t\in \left[0,T\right]$; ticket presale begins at $t=0$ and ends after $t=T$ |

$\alpha $ | Average order arrival probability within one period, $0<\alpha <1$ |

$\beta $ | Probability of the arriving order belonging to group order, $0<\beta <1$${\alpha}_{1}$ Average arrival probability of group order within one period, ${\alpha}_{1}=\alpha \cdot \beta $ |

${\alpha}_{2}$ | Average arrival probability of individual order within one period, ${\alpha}_{2}=\alpha \cdot (1-\beta )$ |

$\theta $ | Discounted price for group |

$x$ | Reserve price for individuals |

${F}_{t}\left(x\right)$ | Cumulative distribution function of $x$ |

${f}_{t}\left(x\right)$ | Density function of $x$ |

$y$ | Number of passengers included in a group order |

${P}_{j}$ | Probability of a group order including $j$ passengers |

$s$ | Number of tickets sold at the initiation of any interval |

$C$ | Total seats or tickets |

$V\left(t,s\right)$ | Value function, indicating expected revenue |

$r\left(t,s\right)$ | Dynamic fare for individuals (abbreviated as $r$ later) |

#### 3.2.2. Value Functions

#### 3.2.3. Boundary Conditions

- $V\left(T+1,s\right)=0$, $\forall s$ indicates that the ticket’s residual value should be zero at the departure time of the train; and
- $V\left(t,C\right)=0$, $\forall t$ indicates that when tickets are sold out at the beginning of time period $t$, the expected revenue from time period t to T should be zero.

#### 3.3. Optimal Policy

**Theorem**

**1.**

**Proof.**

#### 3.4. Inverse Recursive Algorithm

Algorithm 1 Inverse recursive algorithm |

1: Initialize $\forall s,V\left(T+1,s\right)\text{}=\text{}0$, and $\forall t,V\left(t,C\right)\text{}=\text{}0$ |

2: for $t\text{}=\text{}T$ to 1 do |

3: for $s=C-1$ to 1 do |

4: calculate $r\left(t,\text{}s\right)$ according to (7) |

5: If $\left(C-\text{}s\right)20$, then |

6: calculate $V\left(t,s\right)$ according to (6) |

7: else |

8: calculate $V\left(t,s\right)$ according to (2) |

9: end if |

10: end for |

11: end for |

#### 3.5. Application Process for the Rail Operator

## 4. Numerical Experiments

#### 4.1. Data

#### 4.2. Results and Discussion

#### 4.2.1. Comparison between CD and SD

#### 4.2.2. Comparison between CD and DF

#### 4.2.3. Impact of Proportion of Group Orders on Total Expected Revenue

#### 4.2.4. Law of Change of Value Function

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Yuan, W.; Nie, L.; Wu, X.; Fu, H. A Dynamic Bid Price Approach for the Seat Inventory Control Problem in Railway Networks with Consideration of Passenger Transfer. PLoS ONE
**2018**, 13, e0201718. [Google Scholar] [CrossRef] - Zhang, X.; Ma, L.; Zhang, J. Dynamic Pricing for Passenger Groups of High-Speed Rail Transportation. J. Rail Transp. Plan. Manag.
**2017**, 6, 346–356. [Google Scholar] - Talluri, K.T.; Van Ryzin, G. The Theory and Practice of Revenue Managemen; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Mcgill, J.I.; Van Ryzin, G. Revenue Management: Research Overview and Prospects. Transp. Sci.
**1999**, 33, 233–256. [Google Scholar] [CrossRef] - Bitran, G.; Caldentey, R. An Overview of Pricing Models for Revenue Management. IEEE Eng. Manag. Rev.
**2016**, 44, 134. [Google Scholar] [CrossRef] - Gallego, G.; Van Ryzin, G. Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons. Manag. Sci.
**1994**, 40, 999–1020. [Google Scholar] [CrossRef] [Green Version] - You, P.S. Dynamic Pricing in Airline Seat Management for Flights with Multiple Flight Legs. Transp. Sci.
**1999**, 33, 192–206. [Google Scholar] [CrossRef] [Green Version] - Chatwin, R.E. Optimal Dynamic Pricing of Perishable Products with Stochastic Demand and a Finite Set of Prices. Eur. J. Oper. Res.
**2000**, 125, 149–174. [Google Scholar] [CrossRef] - Chen, M.; Chen, Z.L. Recent Developments in Dynamic Pricing Research: Multiple Products, Competition, and Limited Demand Information. Prod. Oper. Manag.
**2015**, 24, 704–731. [Google Scholar] [CrossRef] - Tekin, P.; Erol, R. A New Dynamic Pricing Model for the Effective Sustainability of Perishable Product Life Cycle. Sustainability
**2017**, 9, 1330. [Google Scholar] [CrossRef] - Weatherford, L.R. Using Prices More Realistically as Decision Variables in Perishable-Asset Revenue Management Problems. J. Comb. Optim.
**1997**, 1, 277–304. [Google Scholar] [CrossRef] - Feng, Y.Y.; Xiao, B.C. Integration of Pricing and Capacity Allocation for Perishable Products. Eur. J. Oper. Res.
**2006**, 168, 17–34. [Google Scholar] [CrossRef] - Chew, E.P.; Lee, C.L.; Liu, R.J. Joint Inventory Allocation and Pricing Decisions for Perishable Products. Int. J. Prod. Econ.
**2009**, 120, 139–150. [Google Scholar] [CrossRef] - Gans, N.; Savin, S. Pricing and Capacity Rationing for Rentals with Uncertain Durations. Manag. Sci.
**2007**, 53, 390–407. [Google Scholar] [CrossRef] [Green Version] - Li, H.; Xiong, Z.K.; Qu, W.D.; Xiong, Y. Comprehensive Model of Air Passenger Seat Control and Dynamic Pricing Based on Passenger Classification. Theory Pract. Syst. Eng.
**2011**, 31, 1062–1170. [Google Scholar] - Kuyumcu, A.; Garcia-Diaz, A. A Polyhedral Graph Theory Approach to Revenue Management in the Airline Industry. Comput. Ind. Eng.
**2000**, 38, 375–395. [Google Scholar] [CrossRef] - Zhao, X.; Atkins, D.; Hu, M.; Zhang, W.S. Revenue Management under Joint Pricing and Capacity Allocation Competition. Eur. J. Oper. Res.
**2017**, 257, 957–970. [Google Scholar] [CrossRef] - Armstrong, A.; Meissner, J. Railway Revenue Management: Overview and Models; Working Paper; Lancaster University Management School: Lancaster, UK, 2010. [Google Scholar]
- Wang, Y.; Meng, Q.; Du, Y. Liner container seasonal shipping revenue management. Transp. Res. Part B
**2015**, 82, 141–161. [Google Scholar] [CrossRef] - Ciancimino, A.; Inzerillo, G.; Lucidi, S.; Palagi, L. A Mathematical Programming Approach for the Solution of the Railway Yield Management Problem. Transp. Sci.
**1999**, 33, 168–181. [Google Scholar] [CrossRef] - You, P.S. An efficient computational approach for railway booking problems. Eur. J. Oper. Res.
**2008**, 185, 811–824. [Google Scholar] [CrossRef] - Jiang, X.; Chen, X.; Zhang, L.; Zhang, R. Dynamic Demand Forecasting and Ticket Assignment for High Speed Rail Revenue Management in China. Transp. Res. Rec. J. Transp. Res. Board
**2015**, 2475, 37–45. [Google Scholar] [CrossRef] - Wang, X.; Wang, H.; Zhang, X. Stochastic Seat Allocation Models for Passenger Rail Transportation under Customer Choice. Transp. Res. Part E Logist. Transp. Rev.
**2016**, 96, 95–112. [Google Scholar] [CrossRef] - Nuzzolo, A.; Crisalli, U.; Gangemi, F. A Behavioral Choice Model for the Evaluation of Railway Supply and Pricing Policies. Transp. Res. Part A Policy Pract.
**2000**, 34, 395–404. [Google Scholar] [CrossRef] - Zheng, J.Z.; Liu, J. The Research on Ticket Fare Optimization for China’s High-Speed Train. Math. Probl. Eng.
**2016**, 2016, 1–9. [Google Scholar] [CrossRef] - Zheng, J.Z.; Liu, J.; Clarke, D.B. Ticket Fare Optimization for China’s High-Speed Railway Based on Passenger Choice Behavior. Discret. Dyn. Nat. Soc.
**2017**, 2017, 1–6. [Google Scholar] [CrossRef] - Hetrakul, P.; Cirillo, C. Accommodating Taste Heterogeneity in Railway Passenger Choice Models Based on Internet Booking Data. J. Choice Model.
**2013**, 6, 1–16. [Google Scholar] [CrossRef] - Hetrakul, P.; Cirillo, C. A Latent Class Choice Based Model System for Railway Optimal Pricing and Seat Allocation. Transp. Res. Part E Logist. Transp. Rev.
**2014**, 61, 68–83. [Google Scholar] [CrossRef] - Han, P.; Nie, L.; Fu, H.; Gong, Y.; Wang, G. A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network. Symmetry
**2019**, 11, 670. [Google Scholar] [CrossRef] - Qu, Z.; He, S. A Time-Space Network Model Based on a Train Diagram for Predicting and Controlling the Train Congestion in a Station Caused by an Emergency. Symmetry
**2019**, 11, 780. [Google Scholar] [CrossRef] - Wang, J.; Zhou, L.; Yue, Y. Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem. Symmetry
**2019**, 11, 983. [Google Scholar] [CrossRef]

**Figure 3.**Comparison between comprehensive model of group booking limits and dynamic pricing for individual passengers (CD) and simple dynamic pricing model without group discounts (SD) for the total expected revenue.

**Figure 5.**Comparison of total expected revenue between CD and dynamic pricing with fixed inventory for groups (DF).

$\mathit{\alpha}$ | C = 100 | C = 200 | C = 300 | |||
---|---|---|---|---|---|---|

CD | SD | CD | SD | CD | SD | |

0.1 | 58.89 | 104.02 | 58.89 | 184.02 | 58.89 | 264.02 |

0.2 | 114.89 | 130.45 | 117.78 | 210.50 | 117.78 | 290.50 |

0.3 | 154.61 | 156.58 | 176.64 | 236.98 | 176.67 | 316.98 |

0.4 | 183.12 | 182.27 | 231.07 | 263.45 | 235.56 | 343.45 |

0.5 | 205.31 | 204.22 | 275.10 | 289.93 | 294.08 | 369.93 |

0.6 | 223.47 | 222.20 | 311.31 | 316.35 | 347.36 | 396.41 |

0.7 | 238.84 | 237.42 | 342.00 | 342.36 | 393.15 | 422.89 |

0.8 | 252.16 | 250.60 | 368.61 | 367.81 | 432.98 | 449.37 |

0.9 | 263.91 | 262.24 | 392.10 | 391.13 | 468.17 | 475.83 |

1.0 | 274.43 | 272.65 | 413.13 | 412.04 | 499.68 | 502.20 |

π | SD | CD |
---|---|---|

0.6 | 474.95 | 123.72 |

1.2 | 556.32 | 247.44 |

1.8 | 611.82 | 370.94 |

2.4 | 667.41 | 494.66 |

3.0 | 722.91 | 613.77 |

3.6 | 778.50 | 715.16 |

4.2 | 833.99 | 801.04 |

4.8 | 889.58 | 875.65 |

5.4 | 945.05 | 941.39 |

6.0 | 1000.15 | 1000.32 |

6.6 | 1052.72 | 1053.55 |

7.2 | 1101.27 | 1102.24 |

β | α = 0.4 | α = 0.6 | α = 0.8 | ||||||
---|---|---|---|---|---|---|---|---|---|

SD | CD-SD | % | SD | CD-SD | % | SD | CD-SD | % | |

0.01 | 291.51 | 181.80 | 62.37 | 437.26 | 185.86 | 42.50 | 581.55 | 123.99 | 21.32 |

0.02 | 288.56 | 271.45 | 94.07 | 432.84 | 206.33 | 47.67 | 575.97 | 128.24 | 22.26 |

0.03 | 285.62 | 286.90 | 100.45 | 428.43 | 209.04 | 48.79 | 570.35 | 131.35 | 23.03 |

0.04 | 282.67 | 289.32 | 102.35 | 424.01 | 211.58 | 49.90 | 564.68 | 134.47 | 23.81 |

0.05 | 279.73 | 291.05 | 104.05 | 419.59 | 214.09 | 51.02 | 558.98 | 137.61 | 24.62 |

0.06 | 276.78 | 292.74 | 105.77 | 415.18 | 216.59 | 52.17 | 553.23 | 140.77 | 25.44 |

0.07 | 273.84 | 294.42 | 107.52 | 410.76 | 219.07 | 53.33 | 547.45 | 143.95 | 26.29 |

0.08 | 270.89 | 296.09 | 109.30 | 406.34 | 221.55 | 54.52 | 541.65 | 147.16 | 27.17 |

0.09 | 267.95 | 297.76 | 111.12 | 401.93 | 224.03 | 55.74 | 535.81 | 150.39 | 28.07 |

0.10 | 265.01 | 299.42 | 112.99 | 397.51 | 226.50 | 56.98 | 529.96 | 153.63 | 28.99 |

β | α = 0.4 | α = 0.6 | α = 0.8 |
---|---|---|---|

0.05 | 570.78 | 633.68 | 696.58 |

0.06 | 569.53 | 631.76 | 694.00 |

0.07 | 568.26 | 629.83 | 691.41 |

0.08 | 566.99 | 627.90 | 688.81 |

0.09 | 565.71 | 625.95 | 686.20 |

0.10 | 564.42 | 624.01 | 683.59 |

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## Share and Cite

**MDPI and ACS Style**

Yan, Z.; Zhang, P.; Zhang, Y.; Liu, H.; Feng, C.; Li, X.
Joint Decision Model of Group Ticket Booking Limits and Individual Passenger Dynamic Pricing for the High-Speed Railway. *Symmetry* **2019**, *11*, 1128.
https://doi.org/10.3390/sym11091128

**AMA Style**

Yan Z, Zhang P, Zhang Y, Liu H, Feng C, Li X.
Joint Decision Model of Group Ticket Booking Limits and Individual Passenger Dynamic Pricing for the High-Speed Railway. *Symmetry*. 2019; 11(9):1128.
https://doi.org/10.3390/sym11091128

**Chicago/Turabian Style**

Yan, Zhenying, Pingting Zhang, Yujia Zhang, Hui Liu, Chenxi Feng, and Xiaojuan Li.
2019. "Joint Decision Model of Group Ticket Booking Limits and Individual Passenger Dynamic Pricing for the High-Speed Railway" *Symmetry* 11, no. 9: 1128.
https://doi.org/10.3390/sym11091128