# Optimization of the Classification Yard Location Problem Based on Train Service Network

^{*}

## Abstract

**:**

## 1. Introduction

- The location-allocation problem of classification yards in a rail network is investigated combined with the TSNDP. This combination and the interaction between the CYLP and TSNDP are elaborated on the basis of several small-scale artificial cases, so that the significance and necessity of addressing CYLP can be clarified.
- Not only the scenario of improving the scale of an existing yard is considered, but also that of downsizing or demolishing a yard is taken into account. A set of the decision variables are designed in order to determine whether an original existing yard or a new yard to-be-built exists after optimization in the given network.
- Two nonlinear programming models are proposed aiming at striking a balance between economic costs and car-hour consumptions in rail. Moreover, two linearization approaches are presented to transform the nonlinear models into the linear ones.

## 2. The Classification Yard Location Problem

## 3. Mathematical Model for the CYLP

#### 3.1. Notations

#### 3.2. The Nonlinear 0-1 Programming Model for CYLP

#### 3.3. The Linearization Techniques

## 4. Numerical Example

#### 4.1. The Input Data and Computational Results for M3

#### 4.2. The Input Data and Computational Results for M4

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Notation | Description |
---|---|

Sets | |

${V}^{\mathrm{All}}$ | The set of all the yards, including yards to-be-built and reconstructed; |

${V}^{\mathrm{Stay}}$ | The set of the existing yards that stay unchanged, that is, without any reconstructions, in a rail network; |

${V}^{\mathrm{Modified}}$ | The set of the existing yards that might require reconstructions or demolitions in a rail network; |

${V}^{\mathrm{New}}$ | The set of the yards which are possible to be built in a rail network; |

$plan(k)$ | The investment plan of yard $k$; |

$\vartheta (i,j)$ | The set of yards through which a flow from $i$ to $j$ may pass excluding yard $i$ and yard $j$. |

Parameters | |

${c}_{i}$ | The accumulation parameter of yard $i$; (hour) |

${m}_{ij}$ | The size of a train dispatched from yard $i$ to $j$; (cars) |

${\tau}_{k}^{}$ | The classification cost for each railcar at yard $k$; (hours per railcar) |

${\tau}_{k}^{p-\mathrm{New}}$ | The classification cost for each railcar at the new yard $k$ in accordance with the investment plan $p$ if yard $k$ is determined to be built after optimization; (hours per railcar) |

${\tau}_{k}^{p-\mathrm{Modified}}$ | The classification cost for each railcar at the reconstructed yard $k$ in accordance with the investment plan $p$; (hours per railcar) |

${C}_{k}^{Total}$ | The original total reclassification capacity of the existing yard $k$; (cars per day) |

$\beta $ | The proportion of reserved reclassification capacity for the local trains at the existing yard $k$; |

${\rho}_{k}^{p-\mathrm{New}}$ | The reclassification capacity of the new yard $k$ according to the investment plan $p$ if yard $p$ is optimized to be built; (cars per day) |

${\rho}_{k}^{p-\mathrm{Modified}}$ | The changed (increased or reduced) reclassification capacity (excluding for the local trains) of yard $k$ according to the investment plan $p$ if yard $k$ needs to be reconstructed after optimizing; (cars per day) |

${T}_{k}^{\mathrm{Total}}$ | The original number of sorting tracks of the existing yard $k$; (tracks) |

$\theta $ | The proportion of reserved sorting tracks for the local railcars at the existing yard $k$; |

${\sigma}_{k}^{p-\mathrm{New}}$ | The number of sorting tracks of the new yard to be built in line with the investment plan $p$ after optimization; (tracks) |

${\sigma}_{k}^{p-\mathrm{Modified}}$ | The changed (increased or reduced) number of sorting tracks of yard $k$ in line with the investment plan $p$ after optimization; (tracks) |

$M$ | An extremely huge number, which is used to express the logical relations among variables; |

${I}_{k}^{p}$ | The cost of investment plan $p$ of yard $k$; (CNY) |

$\alpha $ | The converting coefficient that unifies conflicting units of costs; (car-hour per CNY) |

$\omega $ | The budget of investment; (CNY) |

${N}_{ij}$ | The original demand which originates at yard $i$ and is destined for yard $j$. |

Decision variables | |

${y}_{k}^{p}$ | Investment variable; it takes value one if plan $p$ is selected for yard k; otherwise, it is zero; |

${\varsigma}_{ij}$ | Train service variable; it takes the value of one if the train service from yard $i$ to j is provided; otherwise, it is zero; |

${r}_{ij}^{k}$ | Car flow variable; it takes value one if railcars destinating for yard j areconsolidated into train service i→k at yard $i$; Otherwise, it is zero. |

${x}_{i}$ | Yard variable; it takes value one if yard $i$ is existing or newly established; otherwise, it is zero. |

Auxiliary variables | |

f_{ij} | The actual traffic flow from yard $i$ to j including the original demand from i to j and the reclassified flows sorted at yard i from other yards (notice that the origins of the reclassified flows could not be i); |

D_{ij} | The service flow from yard $i$ to j, which is used to indicate the traffic volume shipped by train service from i to j. |

Yard | Y1 | Y2 | Y3 | Y4 | Y5 | Y6 |
---|---|---|---|---|---|---|

Y1 | 0.00 | 79.51 | 84.23 | — | 54.72 | 89.52 |

Y2 | 96.03 | 0.00 | 13.06 | — | 84.85 | 46.44 |

Y3 | 18.86 | 44.89 | 0.00 | — | 92.26 | 17.28 |

Y4 | — | — | — | 0.00 | — | — |

Y5 | 62.18 | 118.67 | 95.94 | — | 0.00 | 24.94 |

Y6 | 71.44 | 68.73 | 69.87 | — | 87.29 | 0.00 |

Yard | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{\tau}}_{\mathit{k}}$ | $(1-\mathit{\beta}){\mathit{C}}_{\mathit{k}}^{\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ | $(1-\mathit{\theta}){\mathit{T}}_{\mathit{i}}^{\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ |
---|---|---|---|---|

Y1 | 11.0 | 4.1 | 450 | 15 |

Y2 | 10.9 | 3.8 | 330 | 15 |

Y3 | 11.5 | 4.2 | 640 | 12 |

Y5 | 11.8 | 3.9 | 560 | 7 |

Y6 | 10.5 | 4.4 | 400 | 11 |

Y1 | 11.0 | 4.1 | 450 | 15 |

No. | Origin | Destination | ${\mathit{N}}_{\mathit{i}\mathit{j}}$ | ${\mathit{f}}_{\mathit{i}\mathit{j}}$ | Train Service Sequence |
---|---|---|---|---|---|

1 | Y1 | Y2 | 79.51 | 79.51 | (Y1→Y2) |

2 | Y1 | Y3 | 84.23 | 84.23 | (Y1→Y2) & (Y3→Y2) |

3 | Y1 | Y5 | 54.72 | 54.72 | (Y1→Y2) & (Y2→Y5) |

4 | Y1 | Y6 | 89.52 | 89.52 | (Y1→Y2) & (Y2→Y6) |

5 | Y2 | Y1 | 96.03 | 186.33 | (Y2→Y1) |

6 | Y2 | Y3 | 13.06 | 97.29 | (Y2→Y3) |

7 | Y2 | Y5 | 84.85 | 139.57 | (Y2→Y5) |

8 | Y2 | Y6 | 46.44 | 135.96 | (Y2→Y6) |

9 | Y3 | Y1 | 18.86 | 18.86 | (Y3→Y2) & (Y2→Y1) |

10 | Y3 | Y2 | 44.89 | 44.89 | (Y3→Y2) |

11 | Y3 | Y5 | 92.26 | 92.26 | (Y3→Y5) |

12 | Y3 | Y6 | 17.28 | 17.28 | (Y3→Y6) |

13 | Y5 | Y1 | 62.18 | 62.18 | (Y5→Y1) |

14 | Y5 | Y2 | 118.67 | 118.67 | (Y5→Y2) |

15 | Y5 | Y3 | 95.94 | 165.81 | (Y5→Y3) |

16 | Y5 | Y6 | 24.94 | 24.94 | (Y5→Y6) |

17 | Y6 | Y1 | 71.44 | 71.44 | (Y6→Y2) & (Y2→Y1) |

18 | Y6 | Y2 | 68.73 | 68.73 | (Y6→Y2) |

19 | Y6 | Y3 | 69.87 | 69.87 | (Y6→Y5) & (Y5→Y3) |

20 | Y6 | Y5 | 87.29 | 87.29 | (Y6→Y5) |

Yard | Reclassified Railcars | Capacity Utilization | Tracks Occupied | Remaining Tracks |
---|---|---|---|---|

Y1 | 307.98 | 68.44% | 9 | 6 |

Y2 | 318.77 | 96.60% | 10 | 5 |

Y3 | 173.29 | 27.08% | 4 | 8 |

Y5 | 371.6 | 66.36% | 7 | 0 |

Y6 | 297.33 | 74.33% | 8 | 3 |

Ave. | 293.79 | 66.56% | 8 | 5 |

Y2 | Y3 | Y5 | Y4 | ||
---|---|---|---|---|---|

${\tau}_{k}^{1-\mathrm{Modified}}$ | 3.4 | 3.8 | 3.5 | ${\tau}_{k}^{1-\mathrm{New}}$ | 4.0 |

${\tau}_{k}^{2-\mathrm{Modified}}$ | 3.6 | 4.0 | 3.7 | ${\tau}_{k}^{2-\mathrm{New}}$ | 4.5 |

${\tau}_{k}^{3-\mathrm{Modified}}$ | 4.0 | 4.4 | 4.1 | ${\tau}_{k}^{3-\mathrm{New}}$ | 3.5 |

${\rho}_{k}^{1-\mathrm{Modified}}$ | 230 | 8 | 80 | ${\rho}_{k}^{1-\mathrm{New}}$ | 400 |

${\rho}_{k}^{2-\mathrm{Modified}}$ | 91 | −92 | −20 | ${\rho}_{k}^{2-\mathrm{New}}$ | 300 |

${\rho}_{k}^{3-\mathrm{Modified}}$ | −109 | −292 | −220 | ${\rho}_{k}^{3-\mathrm{New}}$ | 450 |

${\sigma}_{k}^{1-\mathrm{Modified}}$ | 2 | 1 | 6 | ${\sigma}_{k}^{1-\mathrm{New}}$ | 20 |

${\sigma}_{k}^{2-\mathrm{Modified}}$ | 1 | −2 | 2 | ${\sigma}_{k}^{2-\mathrm{New}}$ | 15 |

${\sigma}_{k}^{3-\mathrm{Modified}}$ | −4 | −6 | −2 | ${\sigma}_{k}^{3-\mathrm{New}}$ | 25 |

${I}_{k}^{1}$ | 0.2 | 0.1 | 0.2 | ${I}_{k}^{1}$ | 0.4 |

${I}_{k}^{2}$ | 0.1 | 0.2 | 0.3 | ${I}_{k}^{2}$ | 0.2 |

${I}_{k}^{3}$ | 0.4 | 0.6 | 0.2 | ${I}_{k}^{3}$ | 0.3 |

No. | Ori. | Des. | ${\mathit{N}}_{\mathit{i}\mathit{j}}$ | ${\mathit{f}}_{\mathit{i}\mathit{j}}$ | Train Service Sequence |
---|---|---|---|---|---|

1 | Y1 | Y2 | 79.51 | 79.51 | (Y1→Y2) |

2 | Y1 | Y3 | 84.23 | 84.23 | (Y1→Y2) & (Y2→Y3) |

3 | Y1 | Y4 | 78.54 | 78.54 | (Y1→Y2) & (Y2→Y4) |

4 | Y1 | Y5 | 54.72 | 54.72 | (Y1→Y2) & (Y2→Y5) |

5 | Y1 | Y6 | 89.52 | 89.52 | (Y1→Y6) |

6 | Y2 | Y1 | 96.03 | 111.59 | (Y2→Y1) |

7 | Y2 | Y3 | 13.06 | 97.29 | (Y2→Y3) |

8 | Y2 | Y4 | 78.55 | 157.09 | (Y2→Y4) |

9 | Y2 | Y5 | 84.85 | 139.57 | (Y2→Y5) |

10 | Y2 | Y6 | 46.44 | 46.44 | (Y2→Y5) & (Y5→Y6) |

11 | Y3 | Y1 | 18.86 | 18.86 | (Y3→Y1) |

12 | Y3 | Y2 | 44.89 | 44.89 | (Y3→Y2) |

13 | Y3 | Y4 | 85.45 | 85.45 | (Y3→Y4) |

14 | Y3 | Y5 | 92.26 | 92.26 | (Y3→Y5) |

15 | Y3 | Y6 | 17.28 | 17.28 | (Y3→Y6) |

16 | Y4 | Y1 | 15.56 | 15.56 | (Y4→Y2) & (Y2→Y1) |

17 | Y4 | Y2 | 25.43 | 25.43 | (Y4→Y2) |

18 | Y4 | Y3 | 35.44 | 35.44 | (Y4→Y3) |

19 | Y4 | Y5 | 28.74 | 28.74 | (Y4→Y5) |

20 | Y4 | Y6 | 78.55 | 78.55 | (Y4→Y6) |

21 | Y5 | Y1 | 62.18 | 62.18 | (Y5→Y1) |

22 | Y5 | Y2 | 118.67 | 187.40 | (Y5→Y2) |

23 | Y5 | Y3 | 95.94 | 165.81 | (Y5→Y3) |

24 | Y5 | Y4 | 87.45 | 123.23 | (Y5→Y4) |

25 | Y5 | Y6 | 24.94 | 71.38 | (Y5→Y6) |

26 | Y6 | Y1 | 71.44 | 71.44 | (Y6→Y1) |

27 | Y6 | Y2 | 68.73 | 68.73 | (Y6→Y5) & (Y5→Y2) |

28 | Y6 | Y3 | 69.87 | 69.87 | (Y6→Y5) & (Y5→Y3) |

29 | Y6 | Y4 | 35.78 | 35.78 | (Y6→Y5) & (Y5→Y4) |

30 | Y6 | Y5 | 87.29 | 87.29 | (Y6→Y5) |

Yard | Reclassified Railcars | Capacity Utilization | Tracks Occupied | Remaining Tracks |
---|---|---|---|---|

Y1 | 386.52 | 85.89% | 11 | 4 |

Y2 | 551.98 | 98.57% | 10 | 7 |

Y3 | 258.74 | 40.43% | 5 | 7 |

Y4 | 183.72 | 61.24% | 4 | 11 |

Y5 | 610.00 | 95.31% | 11 | 2 |

Y6 | 333.11 | 83.28% | 9 | 2 |

Ave. | 387.35 | 77.45% | 9 | 6 |

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**MDPI and ACS Style**

Zhao, Y.; Lin, B.
Optimization of the Classification Yard Location Problem Based on Train Service Network. *Symmetry* **2020**, *12*, 872.
https://doi.org/10.3390/sym12060872

**AMA Style**

Zhao Y, Lin B.
Optimization of the Classification Yard Location Problem Based on Train Service Network. *Symmetry*. 2020; 12(6):872.
https://doi.org/10.3390/sym12060872

**Chicago/Turabian Style**

Zhao, Yinan, and Boliang Lin.
2020. "Optimization of the Classification Yard Location Problem Based on Train Service Network" *Symmetry* 12, no. 6: 872.
https://doi.org/10.3390/sym12060872