# Optimizing the Paths of Trains Formed at the Loading Area in a Multi-loop Rail Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- Combining with the multi-loop network, a new route optimization strategy for direct train service is proposed. For direct train service, when the origin and destination of the goods are determined, it is not necessary to find the best path in the whole rail network. On the contrary, we only need to choose the best path within the range of the freight channel that may include a multi-loop structure.
- Two integer linear programming models for optimizing the paths of trains formed at the loading area in a multi-loop rail network are proposed. Unlike the models in previous studies, which often choose the best path from a path set (e.g., Lin et al., 1997), the model in this paper does not need to determine the path set in advance but only needs to select the arc for each loop and finally form a path by connecting each arc.
- A set of numerical experiments with various rail loops and train flows are conducted to evaluate the performance of the proposed methods. We use the Lingo solver to solve the proposed models. For small-scale case studies, the results demonstrate the feasibility of the proposed models.
- For large-scale train flows and rail loops, which Lingo may not solve in a short time, a genetic algorithm is designed. This algorithm performs well in optimizing the paths of large-scale train flows formed at the loading area in a multi-loop rail network.

## 3. The Path Problem of Trains Formed at Loading Area in a Multi-Loop Rail Network

## 4. Mathematical Models

#### 4.1. Variables and Parameters

#### 4.2. Mathematical Models under Situation 1 and Situation 2

- The capacity of a network can meet the demands of all train flows;
- A single train flow cannot be split during the itinerary.

**Model I:**

#### 4.3. Mathematical Mode under Situation 3

**Model II:**

## 5. Computational Experiments

^{4}Tons/Year.

#### 5.1. The Results of the Two Models under Situation 1 and Situation 2

#### 5.2. Comparison the Solution Time between Model I and Model II under Situation 1 and Situation 2

#### 5.3. Analysis of the Solution Efficiency of Model II under Situation 3

#### 5.4. A Genetic Algorithm for Solving this Problem

## 6. Conclusions and Future Work

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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No. | Author | Model | Solution Technique | ||
---|---|---|---|---|---|

Model Structure | Decision Variables | Constraints | |||

1 | Haghani (1989) | MINLP | Integer variables | a) Capacity constraint | A heuristic decomposition technique |

2 | Lin et al. (1997) | ILP | 0–1 variables | a) Operation principle b) Capacity constraint | Simulated annealing algorithm |

3 | Borndörfer et al. (2016) | MINLP | 0–1 variables | a) Capacity constraint b) Time constraint | Linearization of the objective function |

4 | Sadykov et al. (2013) | LP | 0–1 and integer variables | a) Operation principle b) Capacity constraint | A column generation approach |

5 | Fügenschuh et al. (2013) | MINLP | 0–1 and integer variables | a) Operation principle b) Capacity constraint | Tree-based reformulation and heuristic cuts |

6 | Samà et al. (2016) | ILP | 0–1 variables | a) Operation principle | Ant colony optimization |

No. | Name | Distance (up) (Km) | Distance (down) (Km) | Capacity (up) (10 ^{4} Tons/Year) | Capacity (down) (10 ^{4} Tons/Year) |
---|---|---|---|---|---|

1 | ${K}_{1}$ | 141 | 111 | 5925 | 4405 |

2 | ${K}_{2}$ | 130 | 95 | 5033 | 6219 |

3 | ${K}_{3}$ | 149 | 158 | 5163 | 4307 |

4 | ${K}_{4}$ | 90 | 78 | 6682 | 5151 |

5 | ${K}_{5}$ | 100 | 125 | 5947 | 4601 |

6 | ${K}_{6}$ | 155 | 138 | 4563 | 5929 |

7 | ${K}_{7}$ | 72 | 103 | 5342 | 4041 |

8 | ${K}_{8}$ | 118 | 144 | 4903 | 5000 |

No. | Name | Volume (10^{4} Tons/Year) | Freight rate No. 1 (¥/Ton) | Freight rate No. 2 (¥/Ton-km) |
---|---|---|---|---|

1 | ${f}^{1}$ | 241 | 5.7 | 0.0336 |

2 | ${f}^{2}$ | 381 | 6.4 | 0.0378 |

3 | ${f}^{3}$ | 111 | 7.6 | 0.0435 |

4 | ${f}^{4}$ | 375 | 9.6 | 0.0484 |

5 | ${f}^{5}$ | 285 | 10.4 | 0.0549 |

6 | ${f}^{6}$ | 186 | 14.8 | 0.0765 |

7 | ${f}^{7}$ | 251 | 6.4 | 0.0378 |

8 | ${f}^{8}$ | 213 | 7.6 | 0.0435 |

9 | ${f}^{9}$ | 216 | 9.6 | 0.0484 |

10 | ${f}^{10}$ | 481 | 7.6 | 0.0435 |

11 | ${f}^{11}$ | 137 | 9.6 | 0.0484 |

12 | ${f}^{12}$ | 462 | 6.4 | 0.0378 |

13 | ${f}^{13}$ | 182 | 7.6 | 0.0435 |

14 | ${f}^{14}$ | 326 | 9.6 | 0.0484 |

15 | ${f}^{15}$ | 128 | 5.7 | 0.0336 |

16 | ${f}^{16}$ | 192 | 6.4 | 0.0378 |

17 | ${f}^{17}$ | 194 | 7.6 | 0.0435 |

18 | ${f}^{18}$ | 399 | 9.6 | 0.0484 |

19 | ${f}^{19}$ | 483 | 10.4 | 0.0549 |

20 | ${f}^{20}$ | 342 | 9.6 | 0.0484 |

21 | ${f}^{21}$ | 477 | 7.6 | 0.0435 |

22 | ${f}^{22}$ | 475 | 9.6 | 0.0484 |

23 | ${f}^{23}$ | 226 | 9.6 | 0.0484 |

24 | ${f}^{24}$ | 427 | 14.8 | 0.0765 |

25 | ${f}^{25}$ | 336 | 6.4 | 0.0378 |

26 | ${f}^{26}$ | 402 | 7.6 | 0.0435 |

27 | ${f}^{27}$ | 205 | 9.6 | 0.0484 |

28 | ${f}^{28}$ | 497 | 7.6 | 0.0435 |

29 | ${f}^{29}$ | 377 | 9.6 | 0.0484 |

30 | ${f}^{30}$ | 162 | 10.4 | 0.0549 |

No. | Name | Paths |
---|---|---|

1 | ${f}^{1}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

2 | ${f}^{2}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

3 | ${f}^{3}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

4 | ${f}^{4}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

5 | ${f}^{5}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

6 | ${f}^{6}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

7 | ${f}^{7}$ | |

8 | ${f}^{8}$ | |

9 | ${f}^{9}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

10 | ${f}^{10}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{down}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

11 | ${f}^{11}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

12 | ${f}^{12}$ | |

13 | ${f}^{13}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

14 | ${f}^{14}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

15 | ${f}^{15}$ | |

16 | ${f}^{16}$ | |

17 | ${f}^{17}$ | |

18 | ${f}^{18}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

19 | ${f}^{19}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

20 | ${f}^{20}$ | |

21 | ${f}^{21}$ | |

22 | ${f}^{22}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

23 | ${f}^{23}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

24 | ${f}^{24}$ | |

25 | ${f}^{25}$ | |

26 | ${f}^{26}$ | |

27 | ${f}^{27}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

28 | ${f}^{28}$ | |

29 | ${f}^{29}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

30 | ${f}^{30}$ |

No. | Parameters | Model I | Model II | No. | Parameters | Model I | Model II |
---|---|---|---|---|---|---|---|

1 | (10,8) | < 1 s | < 1 s | 8 | (30,4) | 1 s | < 1 s |

2 | (20,8) | < 1 s | < 1 s | 9 | (30,6) | 1 s | < 1 s |

3 | (30,8) | 1 s | 3 s | 10 | (30,10) | 1 s | 3 s |

4 | (40,8) | 1 s | 24 s | 11 | (30,12) | 1 s | 3 s |

5 | (50,8) | 1 s | 28 s | 12 | (30,14) | 1 s | 6 s |

6 | (60,8) | 1 s | 154 s | 13 | (30,16) | 1 s | 16 s |

7 | (70,8) | 1 s | 275 s |

**Table 6.**Parameters of the rail network based on Table 2.

No. | Name | Distance (up) (Km) | Distance (down) (Km) | Capacity (up) (10 ^{4} Tons/Year) | Capacity (down) (10 ^{4} Tons/Year) |
---|---|---|---|---|---|

1 | ${K}_{1}$ | 141 | 111 | 5925 | 4405 |

2 | ${K}_{2}$ | 130 | 95 | 5033 | 6219 |

3 | ${K}_{3}$ | 149 | 158 | 4163 | 4307 |

4 | ${K}_{4}$ | 90 | 78 | 6682 | 5151 |

5 | ${K}_{5}$ | 100 | 125 | 5947 | 4601 |

6 | ${K}_{6}$ | 155 | 138 | 4563 | 5929 |

7 | ${K}_{7}$ | 72 | 103 | 5342 | 4041 |

8 | ${K}_{8}$ | 118 | 144 | 4903 | 5000 |

No. | Name | Paths |
---|---|---|

1 | ${f}^{1}$ | — |

2 | ${f}^{2}$ | |

3 | ${f}^{3}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

4 | ${f}^{4}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

5 | ${f}^{5}$ | |

6 | ${f}^{6}$ | |

7 | ${f}^{7}$ | |

8 | ${f}^{8}$ | |

9 | ${f}^{9}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

10 | ${f}^{10}$ | |

11 | ${f}^{11}$ | $s$→${K}_{1}^{\mathrm{Down}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

12 | ${f}^{12}$ | |

13 | ${f}^{13}$ | |

14 | ${f}^{14}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

15 | ${f}^{15}$ | - |

16 | ${f}^{16}$ | |

17 | ${f}^{17}$ | |

18 | ${f}^{18}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

19 | ${f}^{19}$ | |

20 | ${f}^{20}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Up}}$→$t$ |

21 | ${f}^{21}$ | |

22 | ${f}^{22}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

23 | ${f}^{23}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Down}}$→${K}_{3}^{\mathrm{Up}}$→${K}_{4}^{\mathrm{Down}}$→${K}_{5}^{\mathrm{Down}}$→${K}_{6}^{\mathrm{Down}}$→${K}_{7}^{\mathrm{Down}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

24 | ${f}^{24}$ | |

25 | ${f}^{25}$ | - |

26 | ${f}^{26}$ | |

27 | ${f}^{27}$ | |

28 | ${f}^{28}$ | |

29 | ${f}^{29}$ | $s$→${K}_{1}^{\mathrm{Up}}$→${K}_{2}^{\mathrm{Up}}$→${K}_{3}^{\mathrm{Down}}$→${K}_{4}^{\mathrm{Up}}$→${K}_{5}^{\mathrm{Up}}$→${K}_{6}^{\mathrm{Up}}$→${K}_{7}^{\mathrm{Up}}$→${K}_{8}^{\mathrm{Down}}$→$t$ |

30 | ${f}^{30}$ |

No. | Parameters | Solution Times | No. | Parameters | Solution Times |
---|---|---|---|---|---|

1 | (10,8) | < 1 s | 8 | (30,4) | 3 s |

2 | (20,8) | < 1 s | 9 | (30,6) | 4 s |

3 | (30,8) | 5 s | 10 | (30,10) | 5 s |

4 | (40,8) | > 24 h | 11 | (30,12) | 21 s |

5 | (50,8) | > 24 h | 12 | (30,14) | 35 s |

6 | (60,8) | > 24 h | 13 | (30,16) | 44 s |

7 | (70,8) | > 24 h |

No. | Parameters | Solution Times | No. | Parameters | Solution Times |
---|---|---|---|---|---|

1 | (10,8) | 41.5 s | 8 | (30,4) | 129 s |

2 | (20,8) | 107.1 s | 9 | (30,6) | 134.6 s |

3 | (30,8) | 135.9 s | 10 | (30,10) | 153.1 s |

4 | (40,8) | 216.2 s | 11 | (30,12) | 159.3 s |

5 | (50,8) | 254.4 s | 12 | (30,14) | 169.8 s |

6 | (60,8) | 326.1 s | 13 | (30,16) | 182.1 s |

7 | (70,8) | 380.9 s |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Lin, B.; Zhao, Y.
Optimizing the Paths of Trains Formed at the Loading Area in a Multi-loop Rail Network. *Symmetry* **2019**, *11*, 844.
https://doi.org/10.3390/sym11070844

**AMA Style**

Li X, Lin B, Zhao Y.
Optimizing the Paths of Trains Formed at the Loading Area in a Multi-loop Rail Network. *Symmetry*. 2019; 11(7):844.
https://doi.org/10.3390/sym11070844

**Chicago/Turabian Style**

Li, Xingkui, Boliang Lin, and Yinan Zhao.
2019. "Optimizing the Paths of Trains Formed at the Loading Area in a Multi-loop Rail Network" *Symmetry* 11, no. 7: 844.
https://doi.org/10.3390/sym11070844