# Integrated Optimization of Rolling Stock Scheduling and Flexible Train Formation Based on Passenger Demand for an Intercity High-Speed Railway

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- (1)
- The passenger demand, train formation and rolling stock scheduling of IHSR with fixed segment rolling stock scheduling mode are comprehensively considered, and an integrated optimization model of rolling stock scheduling and flexible train formation based on passenger demand is constructed. The model takes into account the deadhead factor, so that the turnaround and connection between trains in the same direction and different stations can be realized.
- (2)
- A concept of “flexible train formation by time period” is proposed to clarify the relationship between passenger demand and train formation. In each time period, the total formations of the trains in each route are fixed, but the formations of the individual trains are variable. Additionally, through this concept, the relationship between passenger demand and rolling stock scheduling is also established.
- (3)
- In addition to considering the purchase cost of rolling stock, the model also considers the coupling/decoupling cost and the deadhead cost; through the linearization of the model, the commercial Gurobi solver is used to solve it.
- (4)
- A real-world case study of the Beijing–Tianjin IHSR in China is introduced to verify the feasibility of the proposed model. Additionally, compared with the fixed formation mode, this shows that the flexible formation mode can effectively save cost and improve the utilization rate of the rolling stock.

## 3. Problem Description

#### 3.1. Problem Description

#### 3.2. Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

#### 3.3. Illustration Example

## 4. Mathematical Model

#### 4.1. Notations

#### 4.2. Optimization Objectives

#### 4.3. Constraints

- (1)
- Constraints on different train connection types.

- ➀
- Direct turnaround and connection at the station.

- ➁
- Turnaround and connection after entering the EMU depot with coupling/decoupling operations.

- (2)
- Constraints on the origin and destination of rolling stocks.

- a.
- The train turning around directly at the station.
- b.
- The train turning around after entering the EMU depot with coupling/decoupling operations.
- c.
- The EMU depot.

- a.
- Turning around directly with other EMU trains at the station.
- b.
- Turning around with other EMU trains after entering the EMU depot with coupling/decoupling operations.
- c.
- Going to the EMU depot for repair and maintenance.

- (3)
- Constraints on the balance of rolling stocks.

- (4)
- Constraints on flexible train formation by time period based on passenger demand.

#### 4.4. Linearization Process

- ➀
- Formula (5).

- ➁
- Formula (7).

## 5. A Real-World Instance: Beijing-Tianjin IHSR

#### 5.1. Instance Description

#### 5.2. Results and Analysis

#### 5.3. Comparison and Analysis of Results

- (1)
- Comparison of rolling stock operation cost.

- (2)
- Comparison of rolling stock operation efficiency.

## 6. Conclusions and Further Research

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Rolling Stock | Fixed Train Formation Mode | Flexible Train Formation Mode |
---|---|---|

1 | C2552, C2561, C2562, C2029, C2036, C2579, C2058, C2061, C2226, C2591, C2596, C2669 | C2552, C2563, C2564, C2031, C2038, C2579, C2582, C2649, C2674, C2089 |

2 | C2551, C2554, C2019, C2024, C2617, C2638, C2039, C2042, C2049, C2050, C2059, C2224, C2071, C2076, C2671 | C2551, C2554, C2015, C2020, C2569, C2576, C2631, C2652, C2643, C2666, C2651 |

3 | C2555, C2556, C2565, C2566, C2621, C2642, C2629, C2652, C2641, C2588, C2653 | C2201, C2556, C2565, C2055, C2587 |

4 | C2558, C2583 | C2556, C2565, C2055, C2587 |

5 | C2201, C2558, C2583 | C2555, C2558, C2613, C2632, C2619, C2640, C2583, C2584 |

6 | C2610, C2005, C2560, C2213, C2587 | C2630, C2209, C2034, C2621, C2642, C2583, C2584 |

7 | C2614, C2009, C2560, C2213, C2587 | C2553, C2560, C2051, C2059, C2224 |

8 | C2616, C2559, C2564, C2031, C2040, C2053, C2056, C2649, C2668, C2657 | C2614, C2009, C2560, C2051, C2059, C2224 |

9 | C2008, C2015, C2020, C2569, C2568, C2627, C2648, C2635, C2660, C2647, C2674, C2661 | C2202, C2557, C2562, C2617, C2638, C2041, C2044, C2213, C2052, C2639, C2664, C2223, C2080, C2087, C2090 |

10 | C2010, C2205, C2022, C2571, C2570, C2625, C2646, C2060, C2589, C2592, C2595 | C2004, C2203, C2014, C2207, C2566, C2037, C2218, C2581, C2058, C2061, C2226, C2591, C2596, C2669 |

11 | C2622, C2567, C2572, C2045, C2048, C2057, C2222 | C2002, C2003, C2008, C2561, C2568, C2623, C2644, C2629, C2650, C2641, C2064, C2593, C2594, C2661 |

12 | C2004, C2605, C2624, C2017, C2210, C2027, C2034, C2577, C2578, C2222 | C2610, C2005, C2208, C2567, C2570, C2045, C2048, C2633, C2588, C2655 |

13 | C2202, C2557, C2626, C2025, C2032, C2037, C2576, C2631, C2650, C2639, C2062, C2221, C2074, C2597 | C2626, C2571, C2572, C2625, C2646, C2057, C2222, C2071, C2076, C2083, C2086 |

14 | C2204, C2203, C2628, C2613, C2632, C2619, C2640, C2047, C2220, C2637, C2586, C2593, C2594, C2089 | C2616, C2559, C2210, C2027, C2032, C2577, C2578, C2637, C2660, C2647, C2668, C2077, C2230, C2093 |

15 | C2206, C2013, C2018, C2023, C2030, C2033, C2214, C2623, C2644, C2055, C2054, C2643, C2664, C2651, C2678, C2091 | C2010, C2019, C2026, C2573, C2574, C2053, C2056, C2217, C2062, C2221, C2074, C2671 |

16 | C2014, C2207, C2212, C2211, C2216, C2043, C2046, C2633, C2654, C2063, C2066, C2075, C2078, C2085, C2088 | C2010, C2019, C2050, C2585, C2586, C2075, C2078, C2085, C2088 |

17 | C2014, C2207, C2038, C2041, C2044, C2051, C2052, C2065, C2590, C2077, C2230, C2093 | C2624, C2017, C2024, C2029, C2036, C2039, C2042, C2049, C2054, C2065, C2590, C2657 |

18 | C2208, C2026, C2573, C2574, C2585, C2584 | C2622, C2205, C2022, C2615, C2634, C2035, C2040, C2047, C2220, C2589, C2592, C2595 |

19 | C2208, C2630, C2209, C2585, C2584 | C2204, C2605, C2628, C2025, C2030, C2033, C2214, C2043, C2046, C2215, C2060, C2063, C2068, C2653, C2678, C2091 |

20 | C2615, C2634, C2035, C2218, C2581, C2582, C2067, C2068, C2655 | C2206, C2013, C2018, C2023, C2212, C2211, C2216, C2627, C2648, C2635, C2654, C2067, C2066, C2073, C2228, C2597 |

21 | C2553, C2563, C2215, C2217, C2666, C2223, C2080, C2087, C2090 | |

22 | C2002, C2003, C2563, C2215, C2217, C2064, C2073, C2228, C2083, C2086 |

## References

- Zhao, P.; Hu, A. Research on the Circulating Optimization in the Condition of Using High·Speed Passenger Trains in Uncertain Railroad Region. J. Beijing Jiaotong Univ.
**1997**, 21, 621–624. [Google Scholar] - Nie, L.; Zhao, P.; Yang, H.; Hu, A.Z. Study on motor trainset operation in high-speed railway. J. China Railw. Soc.
**2001**, 23, 1–7. [Google Scholar] - Zhao, P.; Tomii, N. Train set scheduling and an algorithm. J. China Railw. Soc.
**2003**, 25, 1–7. [Google Scholar] - Zhao, P.; Tomii, N. An Algorithm for Train-set Scheduling on Weekday Based on Probabilistic Local Search. Syst. Eng.-Theory Pract.
**2004**, 2, 123–129. [Google Scholar] - Zhao, P.; Tomii, N. An Algorithm for Multiple-Bases Train-set Scheduling Based on Path-exchange. J. China Railw. Soc.
**2004**, 1, 7–11. [Google Scholar] - Zhang, C.C.; Chen, J.; Hua, W. Research on the EMU Operation Plan Based on Different Maintenance Capacity. China Railw. Sci.
**2010**, 31, 130–133. [Google Scholar] - Tong, L.; Nie, L.; Zhao, P. Application of Ant Colony Algorithm in Train-Set Scheduling Problem. J. Transp. Syst. Eng. Inf. Technol.
**2009**, 9, 161–167. [Google Scholar] - Alfieri, A.; Groot, R.; Kroon, L.; Schrijver, A. Efficient Circulation of Railway Rolling Stock. Transp. Sci.
**2006**, 40, 378–391. [Google Scholar] [CrossRef] [Green Version] - Peeters, M.; Kroon, L. Circulation of railway rolling stock: A branch-and-price approach. Comput. Oper. Res.
**2008**, 35, 538–556. [Google Scholar] [CrossRef] [Green Version] - Haahr, J.T.; Wagenaar, J.C.; Veelenturf, L.P.; Kroon, L.G. A Comparison of Two Exact Methods for Passenger Railway Rolling Stock (Re)Scheduling. Transp. Res. Part E Logist. Transp. Rev.
**2016**, 91, 15–32. [Google Scholar] [CrossRef] [Green Version] - Ziarati, K.; Soumis, F.; Desrosiers, J.; Gélinas, S.; Saintonge, A. Locomotive assignment with heterogeneous consists at CN North America. Eur. J. Oper. Res.
**1997**, 97, 281–292. [Google Scholar] [CrossRef] - Cordeau, J.F.; Soumis, F.; Desrosiers, J. Simultaneous Assignment of Locomotives and Cars to Passenger Trains. Oper. Res.
**2001**, 49, 531–548. [Google Scholar] [CrossRef] - Cordeau, J.F.; Soumis, F.; Desrosiers, J. A Benders Decomposition Approach for the Locomotive and Car Assignment Problem. Transp. Sci.
**2000**, 34, 133–149. [Google Scholar] [CrossRef] - Giovanni, L.G.; Andrea, D.; Dario, P. Rolling Stock Rostering Optimization under Maintenance Constraints. J. Intell. Transp. Syst. Technol. Plan. Oper.
**2014**, 18, 95–105. [Google Scholar] - Gao, Y.; Schmidt, M.; Yang, L.; Gao, Z. A branch-and-price approach for trip sequence planning of high-speed train units. Omega
**2020**, 92, 102150. [Google Scholar] [CrossRef] - Zhong, Q.; Lusby, R.M.; Larsen, J.; Zhang, Y.; Peng, Q. Rolling stock scheduling with maintenance requirements at the Chinese High-Speed Railway. Transp. Res. Part B Methodol.
**2019**, 126, 24–44. [Google Scholar] [CrossRef] [Green Version] - Lai, Y.C.; Fan, D.C.; Huang, K.L. Optimizing rolling stock assignment and maintenance plan for passenger railway operations. Comput. Ind. Eng.
**2015**, 85, 284–295. [Google Scholar] [CrossRef] - Lai, Y.C.; Wang, S.W.; Huang, K.L. Optimized Train-Set Rostering Plan for Taiwan High-Speed Rail. IEEE Trans. Autom. Sci. Eng.
**2016**, 14, 286–298. [Google Scholar] [CrossRef] - Tong, S.; Chen, S.K.; Liu, G.H. Optimum Circulation Model for Electrical Multiple Units Considering Reliability of Their Operational Connections. J. China Railw. Soc.
**2020**, 42, 27–34. [Google Scholar] - Kang, L.; Xiao, Y.; Sun, H.; Wu, J.; Luo, S.; Buhigiro, N. Decisions on train rescheduling and locomotive assignment during the COVID-19 outbreak: A case of the Beijing-Tianjin intercity railway. Decis. Support Syst.
**2021**, 113600. [Google Scholar] [CrossRef] - Wang, Y.; Tang, T.; Ning, B.; Meng, L. Integrated optimization of regular train schedule and train circulation plan for urban rail transit lines. Transp. Res. Part E Logist. Transp. Rev.
**2017**, 105, 83–104. [Google Scholar] [CrossRef] - Wang, Y.; D’Ariano, A.; Yin, J.; Meng, L.; Tang, T.; Ning, B. Passenger demand-oriented train scheduling and rolling stock circulation planning for an urban rail transit line. Transp. Res. Part B Methodol.
**2018**, 118, 193–227. [Google Scholar] [CrossRef] - Zhao, S.; Yang, H.; Wu, Y. An integrated approach of train scheduling and rolling stock circulation with skip-stopping pattern for urban rail transit lines. Transp. Res. Part C Emerg. Technol.
**2021**, 128, 103170. [Google Scholar] [CrossRef] - Li, S.; Xu, R.; Han, K. Demand-oriented train services optimization for a congested urban rail line: Integrating short turning and heterogeneous headways. Transp. A Transp. Sci.
**2019**, 15, 1459–1486. [Google Scholar] [CrossRef] - Zhao, Y.Q.; Li, D.W.; Yin, Y.; Dong, X.L.; Zhang, S.L. Integrated Optimization of Train Formation Plan and Rolling Stock Scheduling with Multiple Turnaround Operations under Uneven Demand in an Urban Rail Transit Line. In Proceedings of the 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC), Rhodes, Greece, 20–23 September 2020; IEEE: Piscataway, NJ, USA, 2020. [Google Scholar]

Authors | Model Type | Solving Method | Rolling Stock Scheduling Mode | Deadhead | Passenger Demand | Applications |
---|---|---|---|---|---|---|

Zhao P. et al. | IP | RS/AC | Unfixed-segment | N | N | HSR |

Zhang C.C. et al. | IP | AC | Unfixed-segment | N | N | HSR |

Tong L. et al. | IP | AC | Unfixed-segment | N | N | HSR |

Nie L. | IP | AC | Unfixed-segment | Y | N | HSR |

Haahr J.T. | MILP | Cplex/BP | Unfixed-segment | N | N | HSR |

Alfieri A. | MILP | Cplex | Unfixed-segment | N | N | HSR |

Giovanni L.G. | MILP | Cplex | Unfixed-segment | N | N | HSR |

Gao Y. | ILP | Gurobi | Unfixed-segment | Y | N | HSR |

Zhong Q. | MIP | IA | Unfixed-segment | Y | N | HSR |

Wang Y. | MINLP | Cplex/ISA | Fixed-segment | N | Y | URT |

Zhao S. | MINLP | ISA | Fixed-segment | N | Y | URT |

Li S. | MINLP | GA | Fixed-segment | N | Y | URT |

Zhao Y. | MINLP | Cplex | Fixed-segment | N | Y | URT |

This study | MINLP | Gurobi | Fixed-segment | Y | Y | HSR |

Trains | Departure Time | Departure Station | Arrival Time | Arrival Station | Train Formation |
---|---|---|---|---|---|

G1 | 9:10 | 1 | 9:40 | 2 | 3 |

G3 | 10:00 | 1 | 10:30 | 2 | |

G5 | 11:20 | 1 | 12:20 | 3 | 1 |

G2 | 9:00 | 2 | 9:30 | 1 | 1 |

G4 | 9:50 | 3 | 10:50 | 1 | 3 |

G6 | 10:40 | 3 | 11:40 | 1 |

Sets | Description |
---|---|

$I$ | Set of all trains. |

$S$ | Set of all stations. |

$R$ | Set of all route sections. |

$T$ | Set of all time periods. |

$K$ | Set of directions. When $k=0$, it represents the up direction. When $k=1$, it represents the down direction. |

${I}_{t}^{k}$ | Set of trains in the $t$th time period in $k$ direction. |

Parameters | Description |

${t}^{\mathrm{min}}$ | The minimum time for the train to turn around directly at the station. |

${\gamma}^{\mathrm{min}}$ | The minimum time for the train to turn around after coupling/decoupling in the EMU depot. |

${d}_{i}$ | The departure time of train i. |

${a}_{i}$ | The arrival time of train i. |

${t}^{ia\to jd}$ | The travel time from the arrival station of train i to the departure station of train j. |

${t}^{ia\to D}$ | The travel time from the arrival station of train i to the EMU depot. |

${t}^{D\to jd}$ | The travel time from the EMU depot to the departure station of train j. |

${\phi}_{1}$ | The weight of rolling stock cost. |

${\phi}_{2}$ | The weight of rolling stock coupling/decoupling cost. |

${\phi}_{3}$ | The weight of rolling stock deadhead cost. |

${p}_{t,r}^{k}$ | The total passenger demand of $t$th time period in the $r$ route section of $k$ direction. |

${\delta}_{i,r}^{k}$ | The occupancy rate of train $i$ in $r$ route section of $k$ direction. |

$c$ | The number of passengers that can be carried by a single formation. |

Variables | Description |

$Z$ | Objective function value. |

$NR$ | The number of rolling stocks. |

$NC$ | The number of coupling/decoupling times. |

$ND$ | Total deadhead time. |

${x}_{i}$ | The formation of train $i$, 1 indicates single formation, and 2 indicates double formations. |

${x}_{i,r}^{k}$ | The formation of train $i$ in $r$ route of $k$ direction, 1 indicates single formation, and 2 indicates double formations. |

${y}_{D,i}$ | The number of rolling stocks of train $i$ directly provided by the EMU depot. |

${y}_{i,D}$ | The number of rolling stocks that train $i$ directly returns to the EMU depot for repair and maintenance. |

${\alpha}_{i,j}$ | Binary variable, if train $j$ connect with train $i$ in the station directly, ${\alpha}_{i,j}=\hspace{0.17em}1$, otherwise, ${\alpha}_{i,j}=\hspace{0.17em}0$. |

${\beta}_{i,j}$ | Binary variable, if train $j$ connect with train $i$ after coupling/decoupling operations in the EMU depot, ${\beta}_{i,j}=\hspace{0.17em}1$, otherwise, ${\beta}_{i,j}=\hspace{0.17em}0$. |

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

${t}^{\mathrm{min}}$ | 15 min | ${\phi}_{1}$ | 200 | ${\delta}_{i,r}^{k}$ | Actual value |

${\gamma}^{\mathrm{min}}$ | 30 min | ${\phi}_{2}$ | 30 | ${d}_{i}$ | Actual value by timetable |

$c$ | 576 | ${\phi}_{3}$ | 1 | ${a}_{i}$ | Actual value by timetable |

O/D | Beijing South | Tianjin | Tianjin West | Binhai | EMU Depot |
---|---|---|---|---|---|

Beijing South | 0 | 32 min | 35 min | 70 min | 35 min |

Tianjin | 32 min | 0 | 3 min | 25 min | 5 min |

Tianjin West | 35 min | 3 min | 0 | 28 min | 5 min |

Binhai | 70 min | 25 min | 28 min | 0 | 30 min |

EMU depot | 35 min | 5 min | 5 min | 30 min | 0min |

Formation Mode | Direct Cost | Indirect Cost | Objective Value | |
---|---|---|---|---|

Number of Rolling Stocks | Number of Coupling/Decoupling | Deadhead Driving Time | ||

Fixed formation | 22 | 15 | 851 min | 12,301 |

Flexible formation | 20 | 7 | 550 min | 10,760 |

Symbol | Meaning | Symbol | Meaning | Symbol | Meaning |
---|---|---|---|---|---|

${u}^{\mathrm{total}}$ | Total utilization of rolling stocks | ${t}_{i}^{\mathrm{in}}$ | The time when the rolling stock $i$ returns to the depot | ${e}^{\mathrm{total}}$ | Total number of rolling stocks |

${u}_{i}$ | Operating time of rolling stock $i$ | ${o}^{\mathrm{avg}}$ | Average number of tasks performed per rolling stock | $\delta $ | Balance degree of rolling stock scheduling |

${t}_{i}^{\mathrm{out}}$ | The time when the rolling stock $i$ leaves the depot | ${I}^{\mathrm{total}}$ | Total number of train tasks | ${o}_{i}$ | Number of train tasks performed by rolling stock i |

Formation Schemes | Rolling Stock Utilization | Average Number of Tasks Performed by Rolling Stocks | Balance Degree of Rolling Stock Scheduling |
---|---|---|---|

Fixed formation | 65.30% | 9.05 | 4.14 |

Flexible formation | 68.12% | 9.95 | 3.72 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, P.; Li, Y.; Han, B.; Yang, R.; Liu, Z.
Integrated Optimization of Rolling Stock Scheduling and Flexible Train Formation Based on Passenger Demand for an Intercity High-Speed Railway. *Sustainability* **2022**, *14*, 5650.
https://doi.org/10.3390/su14095650

**AMA Style**

Zhao P, Li Y, Han B, Yang R, Liu Z.
Integrated Optimization of Rolling Stock Scheduling and Flexible Train Formation Based on Passenger Demand for an Intercity High-Speed Railway. *Sustainability*. 2022; 14(9):5650.
https://doi.org/10.3390/su14095650

**Chicago/Turabian Style**

Zhao, Peng, Yawei Li, Baoming Han, Ruixia Yang, and Zhiping Liu.
2022. "Integrated Optimization of Rolling Stock Scheduling and Flexible Train Formation Based on Passenger Demand for an Intercity High-Speed Railway" *Sustainability* 14, no. 9: 5650.
https://doi.org/10.3390/su14095650