# Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- We construct a flexible scheme that improves the overall utilization of all stations. Different from the current fixed scheme, this scheme allows all trains to stop at any station of the multi-station hub, which balances the operation of stations and improves the whole efficiency of the hub.
- (2)
- Based on the proposed flexible scheme, we formulate a mixed-integer programming node-arc model to route trains at multi-station high-speed railway hubs. In the formulation, the individual train, rather the train flow, is taken as the basic unit, which benefits the determination of various train routes. Each train route is divided into the route between stations and the track allocation inside stations, which is a combination of macroscopic and microscopic TRP.
- (3)
- Considering the passenger demand, we set the passenger traffic zones and define passenger decision variables. Then, the passenger cost is introduced into the objective function to optimize the passenger routes.
- (4)
- A case study of the Zhengzhou railway hub in China is carried out. We analyze the optimized results and compare them with the results of two other scenarios. Scenario 1 is a fixed scheme, and Scenario 2 is the sequential optimization of train routes and passenger routes. The comparison results verify the effectiveness and the benefits of the optimized results produced by the proposed method.

## 2. Literature Review

## 3. Problem Description

- (1)
- Set of nodes

- (2)
- Set of arcs

- (3)
- Arc capacity

- (4)
- Track capacity

## 4. Mathematical Model

#### 4.1. Assumptions

- (1)
- For each connecting direction, the number of trains is predetermined, including departure, arrival, and passing trains. As the capacity occupied by the non-stopping trains is relatively small, we ignore the number of non-stopping trains.
- (2)
- Based on the flexible scheme, trains do not have preferred routes, and they can stop at any station in the hub.
- (3)
- The capacity of arcs and tracks is calculated after deducting the occupation of freight trains, and the occupation of high-speed and normal-speed trains is the same.
- (4)
- The track capacities are different, but their operational cost is the same, which ensures that the trains have no preferred tracks.
- (5)
- For traffic zones, only the passengers departing from the stations are considered. Since the arriving and transit passengers are coming from outside the cities and their amounts are less than that of departing passengers, these parts of passenger demand are ignored.
- (6)
- The scale of each station is enough to accommodate passengers boarding at this station.
- (7)
- The capacity of other hub facilities is sufficient, such as the train maintenance depot. It is because the train operation volume of these facilities is relatively small and does not affect the train routes.

#### 4.2. Variable Definition

#### 4.3. Formulation

## 5. Case Study

#### 5.1. Data Source

#### 5.1.1. Description of Zhengzhou High-Speed Railway Hub

#### 5.1.2. Related Parameters

#### 5.2. Optimized Results

^{®}Core™ [email protected] GHz CPU and 16.00 GB RAM, we substituted the above real data into the proposed model to obtain the optimized results. The computational time is 369.76 s, which proves that the model can be solved efficiently within a short time.

_{1a}, b

_{1b}, b

_{2}, b

_{3a}and b

_{3b}. The routes of other directions can be seen in Appendix B. In Table 6, B refers to the direction, track allocation refers to the occupied tracks, and N refers to the number of trains with the same route. The origin traffic zone (O), the destination direction (B), and the number of passengers boarding at each station are presented in Table 7.

- (1)
- Most trains select the shortest route of all possible routes. For example, all the departure trains in the Chongqing (b6) direction depart from the Zhengzhou South Station (a4 and a5), and most trains to Taiyuan (b8) depart from the Zhengzhou Station (a1). This result proves the effectiveness of the train cost in the objective function, which makes trains prioritize the shortest routes.
- (2)
- The trains with the same destinations take various routes, and their track allocations are also different. For instance, the arrival trains to Beijing (b1b) have five routes, and the track allocation of each train route is also different. This is because the individual train is taken as the basic unit, which diversifies the routes between the same pair of origin and destination. This result illustrates that the proposed method can avoid centralized train routes and reduce the pressure on trunk lines.
- (3)
- All of the passengers board at their nearest stations, which means the passenger cost in the objective function works and the passenger route lengths are minimized. This result illustrates that the stops of the trains are convenient for the passengers, which benefits service quality.

#### 5.3. Discussion

#### 5.3.1. Scenario 1: Fixed Scheme

- (1)
- The total cost of the flexible scheme is 29.35% lower than that of the fixed scheme, where the train and passenger costs are 1.13% and 50% lower, respectively. These results indicate that the optimization model identifies shorter train and passenger routes because the flexible scheme assigns most trains to the shortest routes without considering fixed directions. Additionally, it allows passengers to board at the nearest stations.
- (2)
- The line capacity utilization rates of these two schemes are almost similar. Notably, the partial objective of the optimization model is to minimize the train cost, which usually concentrates trains on the shortest routes and leads to an imbalance among lines. However, for the Zhengzhou railway hub, as the difference in route lengths between routes are not too large, the imbalance in the results produced by the flexible scheme is minor.
- (3)
- Figure 5 shows that the average track capacity utilization rate of the Zhengzhou East Station is much larger than the Zhengzhou South Station in the fixed scheme, and the track capacity utilization between stations is imbalanced. However, this problem does not occur in the flexible scheme, where the track capacity utilization of the stations is more balanced.

#### 5.3.2. Scenario 2: Sequentially Optimize Train and Passenger Routes

- (1)
- The total cost of Scenario 2 is 22.58% higher than that of the optimized results. Although the train cost is slightly lower in the former, the passenger cost is higher by 73.73% because the train cost is considered first in the objective and passengers are passively assigned. This sequential optimization leads to centralized train routes and to passengers in some traffic zones traveling to distant stations, resulting in an increase in passenger cost.
- (2)
- The line capacity utilization in Scenario 2 is more imbalanced. Moreover, Figure 7 shows that, in Scenario 2, many tracks have utilization rates of more than 80%, 18 tracks have utilization rates of more than 90%, and 8 tracks have utilization rates of 100%. Due to the centralized train routes, the operation pressure on some tracks is too high to operate new trains, so this scheme cannot be applied in practice. The above results illustrate that passenger demand should be directly introduced into the objective function rather than the sequential optimization.

## 6. Conclusions and Further Works

- (1)
- For railway operators, when formulating the operation scheme, all stations should be considered together and regarded as a whole. This not only takes advantage of the scale effect of multi-station hubs but also balances the operations of the different stations.
- (2)
- Multiple routes should be planned for each origin–destination pair of trains to avoid congestion on railway lines, and the routes should be detailed in terms of track allocation inside stations.
- (3)
- With the intensification of competition in the transportation market, passenger demand should be considered in the planning of train routes to improve service quality.

- (1)
- The railway hub that we study is mostly relevant to an already constructed railway hub. For hubs still in the planning stage, the location of railway passenger stations can be combined with the optimization scheme to achieve collaborative optimization.
- (2)
- We only study the departure passenger flow and do not consider the transfer passenger flow. In the next stage, we would introduce transfer passenger flow into the model to construct a transfer scheme, which could increase the suitability of the optimization scheme for the transfer passengers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Input Data of the Case Study

D | b_{1a} | b_{1b} | b_{2} | b_{3a} | b_{3b} | b_{4} | b_{5a} | b_{5b} | b_{6} | b_{7a} | b_{7b} | b_{8} | Z | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

O | ||||||||||||||

b_{1a} | 0 | 0 | 0 | 3 | 0 | 0 | 26 | 0 | 0 | 6 | 0 | 0 | 8 | |

b_{1b} | 0 | 0 | 0 | 0 | 8 | 4 | 0 | 64 | 20 | 0 | 20 | 0 | 19 | |

b_{2} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 19 | 12 | 0 | 17 | 0 | 29 | |

b_{3a} | 3 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 24 | 0 | 0 | 10 | |

b_{3b} | 0 | 8 | 0 | 0 | 0 | 0 | 0 | 20 | 13 | 0 | 36 | 4 | 24 | |

b_{4} | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 14 | 22 | |

b_{5a} | 26 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 7 | |

b_{5b} | 0 | 64 | 19 | 0 | 20 | 0 | 0 | 0 | 0 | 0 | 15 | 10 | 25 | |

b_{6} | 0 | 20 | 12 | 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 | 20 | 31 | |

b_{7a} | 6 | 0 | 0 | 24 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 6 | |

b_{7b} | 0 | 20 | 17 | 0 | 36 | 12 | 0 | 15 | 0 | 0 | 0 | 0 | 26 | |

b_{8} | 0 | 0 | 0 | 0 | 4 | 14 | 0 | 10 | 20 | 0 | 0 | 0 | 19 | |

Z | 8 | 19 | 29 | 10 | 24 | 22 | 7 | 25 | 31 | 6 | 26 | 19 |

D | b_{1a} | b_{1b} | b_{2} | b_{3a} | b_{3b} | b_{4} | b_{5a} | b_{5b} | b_{6} | b_{7a} | b_{7b} | b_{8} | Total | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

O | ||||||||||||||

o_{1} | 10 | 30 | 12 | 10 | 20 | 7 | 10 | 30 | 15 | 10 | 30 | 17 | 201 | |

o_{2} | 10 | 30 | 12 | 10 | 20 | 7 | 10 | 30 | 15 | 10 | 30 | 17 | 201 | |

o_{3} | 10 | 30 | 12 | 10 | 20 | 7 | 10 | 30 | 15 | 10 | 30 | 17 | 201 | |

Total | 30 | 90 | 36 | 30 | 60 | 21 | 30 | 90 | 45 | 30 | 90 | 51 |

## Appendix B. Detailed Departure and Arrival Routes of Trains

_{4}, b

_{5a}, b

_{5b}, b

_{6}, b

_{7a}, b

_{7b}and b

_{8}.

**Table A3.**Detailed departure and arrival routes of trains in directions b

_{4}, b

_{5a}, b

_{5b}, b

_{6}, b

_{7a}, b

_{7b}and b

_{8}.

B | Arrival Trains | Track Allocation | N | Departure Trains | Track Allocation | N |
---|---|---|---|---|---|---|

b_{4} | b_{4}-a_{4} | a_{4} [1, 2, 3, 3, 3, 4, 4, 4, 7, 7] | 10 | a_{4}-b_{4} | a_{4} [5, 6, 6, 7, 9, 9, 9, 9] | 8 |

b_{4}-a_{5} | a_{5} [1, 1, 2, 2, 3, 4, 4, 4, 5, 7, 7, 7] | 12 | a_{5}-b_{4} | a_{5} [2, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8] | 14 | |

b_{5a} | b_{5a}-q_{3}-a_{2} | a_{2} [5, 7, 7, 11, 11, 11, 13] | 7 | a_{4}-a_{2}-q_{3}-b_{5a} | a_{4} [1, 2, 2, 4, 7, 7] | 6 |

a_{5}-a_{2}-q_{3}-b_{5a} | a_{5} [1] | 1 | ||||

b_{5b} | b_{5b}-q_{3}-a_{2} | a_{2} [4, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 11, 11, 13, 13] | 25 | a_{4}-a_{2}-q_{3}-b_{5b} | a_{4} [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 8] | 19 |

a_{5}-a_{2}-q_{3}-b_{5b} | a_{5} [1, 1, 1, 1, 3, 8] | 6 | ||||

b_{6} | b_{6}-a_{4} | a_{4} [1, 1, 1, 3, 3, 3, 8] | 7 | a_{4}-b_{6} | a4 [1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 7, 9, 9, 9] | 21 |

b_{6}-a_{5} | a_{5} [1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8] | 24 | a_{5}-b_{6} | a5 [1, 1, 1, 2, 3, 3, 5, 7, 7, 8] | 10 | |

b_{7a} | b_{7a}-q_{4}-a_{2} | a_{2} [5] | 1 | a_{1}-a_{2}-q_{4}-b_{7a} | a_{1} [1, 2, 4, 8, 12] | 5 |

b_{7a}-q_{4}-a_{3} | a_{3} [2, 2, 9, 10, 10] | 5 | a_{4}-a_{3}-q_{4}-b_{7a} | a_{4} [1] | 1 | |

b_{7b} | b_{7b}-q_{4}-a_{2} | a_{2} [4] | 1 | a_{1}-a_{2}-q_{4}-b_{7b} | a_{1} [1, 2, 2, 3, 5, 6, 7, 7, 8, 9, 9, 9, 9, 9, 12, 12, 12, 13] | 18 |

a_{4}-a_{2}-q_{4}-b_{7b} | a_{4} [1, 2, 5, 8, 9] | 5 | ||||

b_{7b}-q_{4}-a_{3} | a_{3} [1, 2, 2, 2, 4, 4, 4, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11] | 25 | a_{4}-a_{3}-q_{4}-b_{7b} | a_{4} [3, 8] | 2 | |

a_{5}-a_{3}-q_{4}-b_{7b} | a_{5} [6] | 1 | ||||

b_{8} | b_{8}-a_{1} | a1 [2, 3, 3, 4, 6, 7, 8, 8, 8, 8, 9, 9, 9, 9, 11, 11, 12, 12, 13] | 19 | a_{1}-b_{8} | a_{1} [2, 3, 3, 4, 4, 5, 6, 6, 6, 9, 10, 12, 12, 13, 13, 13, 13, 13] | 18 |

a_{2}-a_{1}-b_{8} | a_{2} [10] | 1 |

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Parameters | Definition |
---|---|

$V$ | Set of nodes, indexed by $v$ |

$A$ | Set of stations in the hub, indexed by $a$ |

$B$ | Set of connecting directions, indexed by $b$ |

$Bh,Bu,Bi$ | $Bh$ refers to high-speed directions; $Bu$ refers to normal-speed directions; $Bi$ refers to intercity directions |

$Q$ | Set of branch nodes, indexed by $q$ |

$K$ | Set of trains, indexed by $k$, $(b,k)$ indicates the $k$th train of direction $b$ |

${K}_{bd},{K}_{ba},{K}_{mn}$ | ${K}_{bd}$ is the set of departure trains in connecting direction $b$, ${K}_{ba}$ is the set of arrival trains in connecting direction $b$, ${K}_{mn}$ is the set of passing trains from direction $m$ to direction $n$ |

$G$ | Set of tracks, indexed by $g$ |

${G}_{a}$ | ${G}_{a}$ is the set of tracks of station $a$ |

$O$ | Set of traffic zones, indexed by $o$ |

$El$ | Set of arcs, including ${E}_{2},{E}_{3},{E}_{4},{E}_{5}$, indexed by $(i,j)$ |

$Et$ | Set of arcs ${E}_{1}$, indexed by $(o,a)$ |

$n{f}_{b}$ | Number of departure trains moving from the stations in the hub to direction $b$ (trains/day) |

$n{d}_{b}$ | Number of arrival trains moving from direction $b$ to the stations in the hub (trains/day) |

${n}_{mn}$ | Number of passing trains moving from direction $m$ to direction $n$ (trains/day) |

${c}_{ij}$ | Capacity of arc $(i,j)$ (trains/day) |

${d}_{ij}$ | Length of arc $(i,j)$ (km) |

${w}_{ag}$ | Capacity of track $g$ in station $a$ (trains/day) |

${p}_{ob}$ | Number of passengers moving from traffic zone $o$ to direction $b$ (persons/day) |

${d}_{oa}$ | Distance from traffic zone $o$ to station $a$ (km) |

${u}_{l}$ | Per cost of train running on lines (thousand RMB/train·km) |

${u}_{g}$ | Cost of train operation on tracks (thousand RMB/train) |

${u}_{p}$ | Per travel cost of passengers (thousand RMB/person·km) |

${p}_{h}$ | Passenger capacity of high-speed train (persons/train) |

${p}_{u}$ | Passenger capacity of normal-speed train (persons/train) |

${p}_{c}$ | Passenger capacity of intercity train (persons/train) |

Notations | Definition |
---|---|

${x}_{ij}^{bk}$ | Binary {0,1}, equals 1 if the kth departure train of direction $b$ occupies arc $(i,j)$, and 0 otherwise, $\forall b\in B,\forall k\in {K}_{bd},\forall (i,j)\in El$ |

${y}_{ij}^{bk}$ | Binary {0,1}, equals 1 if the $k$th arrival train of direction $b$ occupies arc $(i,j)$, and 0 otherwise, $\forall b\in B,\forall k\in {K}_{ba},\forall (i,j)\in El$ |

${z}_{ij}^{mnk}$ | Binary {0,1}, equals 1 if the $k$th passing train from direction $m$ to direction $n$ occupies arc $(i,j)$, and 0 otherwise, $\forall m,n\in B,\forall k\in {K}_{mn}$, $\forall (i,j)\in El$ |

${r}_{ag}^{bk}$ | Binary {0,1}, equals 1 if the $k$th departure train of direction $b$ stops at track $(a,g)$, and 0 otherwise, $\forall b\in B,\forall k\in {K}_{bd},\forall a\in A,\forall g\in {G}_{a}$ |

${s}_{ag}^{bk}$ | Binary {0,1}, equals 1 if the $k$th arrival train of direction $b$ stops at track $(a,g)$, and 0 otherwise, $\forall b\in B,\forall k\in {K}_{ba},\forall a\in A,\forall g\in {G}_{a}$ |

${t}_{ag}^{mnk}$ | Binary {0,1}, equals 1 if the $k$th passing train from direction $m$ to direction $n$ stops at track (a, g), and 0 otherwise, $\forall m,n\in B,\forall k\in {K}_{mn}$, $\forall a\in A,\forall g\in {G}_{a}$ |

${p}_{oba}$ | Integer representing the passenger number of traffic zone $o$ boarding at station $a$ to travel in direction $b$, $\forall o\in O,\forall b\in B,\forall a\in A$ |

Arcs | Length (km) | Capacity (Trains/Day) | Arcs | Length (km) | Capacity (Trains/Day) |
---|---|---|---|---|---|

Railway lines | |||||

(q_{1}, a_{2}) | 179 | 263 | (q_{4}, a_{1}) | 331 | 150 |

(q_{1}, a_{3}) | 179 | 263 | (q_{4}, a_{2}) | 201 | 120 |

(b_{2}, a_{2}) | 237 | 263 | (q_{4}, a_{3}) | 201 | 221 |

(b_{2}, a_{3}) | 237 | 263 | (b_{8}, a_{1}) | 111 | 287 |

(q_{2}, a_{2}) | 250 | 164 | (b_{1a}, q_{1}) | 0 | 263 |

(q_{2}, a_{3}) | 250 | 164 | (b_{1b}, q_{1}) | 0 | 263 |

(b_{4}, a_{4}) | 212 | 287 | (b_{3a}, q_{2}) | 0 | 164 |

(b_{4}, a_{5}) | 212 | 287 | (b_{3b}, q_{2}) | 0 | 164 |

(q_{3}, a_{1}) | 102 | 150 | (b_{5a}, q_{3}) | 0 | 263 |

(q_{3}, a_{2}) | 91 | 263 | (b_{5b}, q_{3}) | 0 | 150 |

(b_{6}, a_{4}) | 282 | 273 | (b_{7a}, q_{4}) | 0 | 221 |

(b_{6}, a_{5}) | 282 | 273 | (b_{7b}, q_{4}) | 0 | 200 |

Connecting lines | |||||

(a_{2}, a_{4}) | 36 | 331 | (a_{1}, a_{4}) | 43 | 331 |

(a_{2}, a_{5}) | 36 | 331 | (a_{1}, a_{5}) | 43 | 331 |

(a_{3}, a_{4}) | 36 | 331 | (a_{1}, a_{2}) | 15 | 331 |

(a_{3}, a_{5}) | 36 | 331 | |||

Travel routes | |||||

(o_{1}, a_{1}) | 5 | - | (o_{2}, a_{4}) | 10 | - |

(o_{1}, a_{2}) | 10 | - | (o_{2}, a_{5}) | 10 | - |

(o_{1}, a_{3}) | 10 | - | (o_{3}, a_{1}) | 10 | - |

(o_{1}, a_{4}) | 15 | - | (o_{3}, a_{2}) | 15 | - |

(o_{1}, a_{5}) | 15 | - | (o_{3}, a_{3}) | 15 | - |

(o_{2}, a_{1}) | 15 | - | (o_{3}, a_{4}) | 5 | - |

(o_{2}, a_{2}) | 5 | - | (o_{3}, a_{5}) | 5 | - |

(o_{2}, a_{3}) | 5 | - |

Track Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a_{1} | 36 | 38 | 40 | 45 | 40 | 38 | 38 | 38 | 36 | 36 | 36 | 30 | 30 |

a_{2} | 30 | 36 | 40 | 42 | 45 | 38 | 38 | 45 | 45 | 40 | 36 | 30 | 30 |

a_{3} | 30 | 45 | 34 | 36 | 45 | 45 | 38 | 36 | 44 | 42 | 40 | ||

a_{4} | 30 | 36 | 38 | 45 | 44 | 40 | 36 | 38 | 34 | ||||

a_{5} | 30 | 38 | 40 | 45 | 36 | 38 | 38 | 34 |

Parameters | Definition | Value |
---|---|---|

${u}_{l}$ | Per cost of train running on lines | 0.3 (thousand RMB/train·km) |

${u}_{g}$ | Cost of train operation on tracks | 50 (thousand RMB/train) |

${u}_{p}$ | Per travel cost of passengers | 0.04 (thousand RMB/person·km) |

${p}_{h}$ | Passenger capacity of the high-speed train | 1000 (persons/train) |

${p}_{u}$ | Passenger capacity of the normal-speed train | 1460 (persons/train) |

${p}_{p}$ | Passenger capacity of the passing train | 800 (persons/train) |

**Table 6.**Detailed departure and arrival routes of trains in directions b

_{1a}, b

_{1b}, b

_{2}, b

_{3a}and b

_{3b}.

B | Arrival Trains | Track Allocation | N | Departure Trains | Track Allocation | N |
---|---|---|---|---|---|---|

b_{1a} | b_{1a}-q_{1}-a_{2} | a_{2} [5, 9, 10] | 3 | a_{1}-a_{2}-q_{1}-b_{1a} | a_{1} [5] | 1 |

a_{4}-a_{2}-q_{1}-b_{1a} | a_{4} [1, 2] | 2 | ||||

b_{1a}-q_{1}-a_{3} | a_{3} [5, 7, 7, 9,11] | 5 | a_{4}-a_{3}-q_{1}-b_{1a} | a4 [1, 2, 9] | 3 | |

a_{5}-a_{2}-q_{1}-b_{1a} | a5 [3, 5] | 2 | ||||

b_{1b} | b_{1b}-q_{1}-a_{2} | a_{2} [4, 4, 5, 5, 8, 9, 9, 10, 11] | 9 | a_{1}-a_{2}-q_{1}-b_{1b} | a_{1} [1, 4, 5, 5, 5, 10, 11, 12] | 8 |

a_{4}-a_{2}-q_{1}-b_{1b} | a_{4} [2, 3, 4, 5, 5, 7, 9] | 7 | ||||

b_{1b}-q_{1}-a_{3} | a_{3} [6, 8, 8, 8, 9, 9, 10, 10, 10, 11] | 10 | a_{4}-a_{3}-q_{1}-b_{1b} | a_{4} [9] | 1 | |

a_{5}-a_{2}-q_{1}-b_{1b} | a_{5} [1, 1] | 2 | ||||

a_{5}-a_{3}-q_{1}-b_{1b} | a_{5} [1] | 1 | ||||

b_{2} | b_{2}-a_{2} | a_{2} [4, 6, 6, 7, 8, 8, 9] | 7 | a_{1}-a_{2}-b_{2} | a_{1} [3, 3, 4, 5, 6, 9, 10, 10, 10, 11, 12, 12] | 12 |

a_{2}-b_{2} | a_{2} [4, 5, 5, 10, 10, 13] | 6 | ||||

b_{2}-a_{3} | a_{3} [1, 1, 2, 3, 4, 4, 4, 5, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11,11] | 22 | a_{3}-b_{2} | a_{3} [2, 5, 8, 9, 9, 9, 10, 11] | 8 | |

a_{4}-a_{3}-b_{2} | a_{4} [4, 4] | 2 | ||||

a_{5}-a_{3}-b_{2} | a_{5} [5] | 1 | ||||

b_{3a} | b_{3a}-q_{2}-a_{2} | a_{2} [5, 7, 10] | 3 | a_{1}-a_{2}-q_{2}-b_{3a} | a_{1} [1, 1, 2, 2, 2, 4, 4] | 6 |

a_{4}-a_{2}-q_{2}-b_{3a} | a_{4} [5, 6] | 2 | ||||

b_{3a}-q_{2}-a_{3} | _{a3} [3, 7, 8, 8, 10, 11] | 7 | a_{5}-a_{2}-q_{2}-b_{3a} | a_{5} [8] | 1 | |

a_{5}-a_{3}-q_{2}-b_{3a} | a_{5} [8] | 1 | ||||

b_{3b} | b_{3b}-q_{2}-a_{2} | a_{2} [4, 4, 5, 5, 5, 6, 6, 8, 8, 8, 9, 13] | 12 | a_{1}-a_{2}-q_{2}-b_{3b} | a_{1} [3, 3, 3, 4, 4, 5, 6, 9, 10, 11, 11, 12, 12, 13] | 14 |

a_{4}-a_{2}-q_{2}-b_{3b} | a_{4} [2, 2, 5, 5, 9, 9] | 6 | ||||

b_{3b}-q_{2}-a_{3} | a_{3} [2, 6, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11] | 12 | a_{4}-a_{3}-q_{2}-b_{3b} | a_{4} [4, 4, 5] | 3 | |

a_{5}-a_{2}-q_{2}-b_{3b} | a_{5} [2] | 1 |

O/B | b_{1a} | b_{1b} | b_{2} | b_{3a} | b_{3b} | b_{4} | b_{5a} | b_{5b} | b_{6} | b_{7a} | b_{7b} | b_{8} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

o_{1} | a_{1}, 10 | a_{1}, 30 | a_{1}, 12 | a_{1}, 10 | a_{1}, 20 | a_{1}, 7 | a_{1}, 10 | a_{1}, 30 | a_{1}, 15 | a_{1}, 10 | a_{1}, 30 | a_{1}, 17 |

o_{2} | a_{2}, 7.6a _{3}, 2.4 | a_{2}, 21.2a _{3}, 8.8 | a_{2}, 7.6a _{3}, 2.4 | a_{2}, 4a _{3}, 8 | a_{3}, 20 | a_{2}, 6.2a _{3}, 0.8 | a_{2}, 10 | a_{2}, 30 | a_{2}, 15 | a_{2}, 7.6a _{3}, 2.4 | a_{2}, 30 | a_{2}, 17 |

o_{3} | a_{4}, 7.08a _{5}, 2.92 | a_{4}, 26.2a _{5}, 3.8 | a_{4}, 8.6a _{5}, 3.4 | a_{4}, 5.48a _{5}, 4.52 | a_{4}, 19a _{5}, 1 | a_{4}, 0.2a _{5}, 6.8 | a_{4}, 8.54a _{5}, 1.46 | a_{4}, 19.8a _{5}, 10.2 | a_{4}, 7.4a _{5}, 7.6 | a_{4}, 9.2a _{5}, 0.8 | a_{4}, 22.6a _{5}, 7.4 | a_{4}, 8.2a _{5}, 8.8 |

_{1}, 10, there are 10 thousand persons boarding at station a

_{1}.

Station | Yard | Directions |
---|---|---|

Zhengzhou | a_{1} | b_{1a}, b_{3a}, b_{5a}, b_{7a}, b_{8} |

Zhengzhou East | a_{2} | b_{1b}, b_{5b} |

a_{3} | b_{2}, b_{3b}, b_{7b} | |

Zhengzhou South | a_{4} | b_{4} |

a_{5} | b_{6} |

Scheme | Total Cost (Thousand RMB) | Train Cost (Thousand RMB) | Passenger Cost (Thousand RMB) | Line Capacity Utilization (%) | Track Capacity Utilization (%) |
---|---|---|---|---|---|

Scenario 1 | 417,709 | 176,509 | 241,200 | 28.54 | 58.74 |

OP | 295,109 | 174,509 | 120,600 | 28.73 | 58.89 |

Scheme | Total Cost (Thousand RMB) | Train Cost (Thousand RMB) | Passenger Cost (Thousand RMB) | Line Capacity Utilization (%) | Track Capacity Utilization (%) |
---|---|---|---|---|---|

Scenario 2 | 381,206 | 171,686 | 209,520 | 26.01 | 58.56 |

OP | 295,109 | 174,509 | 120,600 | 28.73 | 58.89 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Song, R.; He, S.; Song, Z.
Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub. *Sustainability* **2022**, *14*, 7292.
https://doi.org/10.3390/su14127292

**AMA Style**

Wang Y, Song R, He S, Song Z.
Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub. *Sustainability*. 2022; 14(12):7292.
https://doi.org/10.3390/su14127292

**Chicago/Turabian Style**

Wang, Yidong, Rui Song, Shiwei He, and Zilong Song.
2022. "Train Routing and Track Allocation Optimization Model of Multi-Station High-Speed Railway Hub" *Sustainability* 14, no. 12: 7292.
https://doi.org/10.3390/su14127292