# Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. HMP Problem at CR

## 3. Mathematical Model of the HMP Problem

#### 3.1. Notations

#### 3.2. Optimization Objective

#### 3.3. Constraints Analysis

#### 3.4. Model Construction

## 4. Modified Particle Swarm Optimization Algorithm

#### 4.1. Processing of Model Constraints

#### 4.2. General Particle Swarm Optimization Algorithm

#### 4.3. MPSO and Solution Strategy

#### 4.3.1. Inertia Weight

#### 4.3.2. Learning Factor

#### 4.3.3. Update Equations

#### 4.3.4. Encoding Rules and Initial Solution

#### 4.4. Algorithm Steps

## 5. Case Study

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The position of high-level maintenance plan (HMP) in the operation and maintenance management system for the electric multiple unit (EMU) trains.

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

$e$ | The index of EMU trains; |

$E$ | The set of all EMU trains; |

${E}^{\prime}$ | An empty set; |

$et{a}_{e}$ | The eta of the HM for the EMU train $e$; |

$e(m)$ | The type of the EMU train $e$; |

$e(g)$ | The maintenance level of the EMU train $e$; |

${R}_{mg}^{+}$ | The maximum value of the difference between the actual operating mileage before the HM and the $et{a}_{e}$ for the train of which the type is $m$ and maintenance level is $g$; |

${R}_{mg}^{-}$ | The maximum value of the difference between the $et{a}_{e}$ and the actual operating mileage before the HM for the train of which the type is $m$ and the maintenance level is $g$; |

${l}_{e}$ | The average daily operating mileage for the EMU train $e$ before HM. |

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

Indices | |

$e$ | The index of EMU trains; |

$m$ | The index of types for the EMU trains; |

$t$ | The index of dates during the planning horizon; |

$g$ | The index of the maintenance level; |

Set | |

$E$ | The set of all EMU trains; $e\in E$ |

$M$ | The set of all types; $m\in M$ |

$T$ | The set of all dates during the planning horizon; $t\in T$ |

$G$ | The set of all maintenance levels; $g\in G$ |

Input parameters | |

$e(g)$ | The maintenance level of the EMU train $e$; |

$e(m)$ | The type of the EMU train $e$; |

$c$ | The unit penalty fee for the unused mileage before the HM; |

$et{a}_{e}$ | The eta for the EMU train $e$; |

${l}_{e}$ | The average daily operating mileage for the EMU train $e$ before the HM; |

$W{S}_{e}$ | The first date of the time window for the EMU train $e$, $W{S}_{e}=et{a}_{e}-{R}_{e(m),e(g)}^{-}/{l}_{e}$ (See Section 2); |

$W{E}_{e}$ | The last date of the time window for the EMU train $e$, $W{E}_{e}=et{a}_{e}+{R}_{e(m),e(g)}^{+}/{l}_{e}$ (See Section 2); |

${\alpha}_{m}$ | The conversion coefficient for train of which the type is $m$, indicates whether the train includes sixteen cars or not, if yes, then ${\alpha}_{m}$ = 2; otherwise, ${\alpha}_{m}$ = 1; |

${\theta}_{t}$ | The maximum HM rate on the t-th day; |

$Inv$ | The fleet size (the standard set); |

${d}_{m}^{t}$ | The maximum number of an EMU train in the HM state for the $m$ type on the t-th day; |

${b}_{g}$ | The maximum number of an EMU train in the $g$-th level maintenance state; |

${N}_{g}$ | The maximum number of an acceptable EMU train in the $g$-th level maintenance state at the same time; |

$Q$ | A sufficiently large positive number; |

${H}_{m}^{g}$ | The maintenance service time for an EMU train with the $m$-th type and the $g$-th level; |

${J}_{m}^{g}$ | The minimum interval time for delivering another train after an EMU train with the $m$-th type and the $g$-th level enters the workshop; |

$\left|T\right|$ | The length of the planning horizon; |

$\left|E\right|$ | The number of the EMU trains which need to be maintained during the planning horizon; |

Decision Variables | |

${x}_{e}^{t}$ | Binary variable, indicates whether the EMU train $e$ selects the $t$-th day to start the HM procedures during the planning horizon, ${x}_{e}^{t}$ = 1 if yes, ${x}_{e}^{t}$ = 0 otherwise; |

${y}_{e}^{t}$ | Binary variable, indicates whether the EMU train $e$ in the $e(g)$-th level maintenance state on the $t$-th day during the planning horizon, ${y}_{e}^{t}$ = 1 if yes, ${y}_{e}^{t}$ = 0 otherwise; |

${z}_{e}^{t}$ | Binary variable, indicate whether the EMU train $e$ in the $e(g)$-th level delivery interval on the $t$-th day during the planning horizon, ${z}_{e}^{t}$ = 1 if yes, ${z}_{e}^{t}$ = 0 otherwise; |

ID | Type | ${\mathit{l}}_{\mathit{e}}\text{}\left(\mathbf{km}\right)$ | Time Window | Level | ${\mathbf{H}}_{\mathbf{e}(\mathbf{m})}^{\mathbf{e}(\mathbf{g})}$ (Day) |
---|---|---|---|---|---|

1 | m1 | 1600 | [64,164] | 3 | 50 |

2 | m1 | 1600 | [64,164] | 3 | 50 |

3 | m1 | 1600 | [90,190] | 3 | 50 |

4 | m1 | 1600 | [124,224] | 3 | 50 |

5 | m1 | 1600 | [144,244] | 3 | 50 |

6 | m1 | 1600 | [188,288] | 3 | 50 |

7 | m1 | 1600 | [200,300] | 3 | 50 |

8 | m1 | 1600 | [274,374] | 4 | 55 |

9 | m1 | 1600 | [370,470] | 4 | 55 |

10 | m1 | 1600 | [379,479] | 4 | 55 |

11 | m1 | 1600 | [429,479] | 4 | 55 |

12 | m2 | 1600 | [72,172] | 3 | 50 |

13 | m2 | 1600 | [158,258] | 3 | 50 |

14 | m2 | 1600 | [216,316] | 3 | 50 |

15 | m2 | 1600 | [264,354] | 3 | 50 |

16 | m2 | 1600 | [387,484] | 3 | 50 |

17 | m2 | 1600 | [396,484] | 3 | 50 |

18 | m2 | 1600 | [409,484] | 3 | 50 |

19 | m2 | 1600 | [443,484] | 4 | 55 |

21 | m3 | 2000 | [134,234] | 5 | 60 |

21 | m3 | 2000 | [149,249] | 3 | 40 |

22 | m3 | 2000 | [150,250] | 3 | 40 |

23 | m3 | 2000 | [153,253] | 3 | 40 |

24 | m3 | 2000 | [158,258] | 5 | 60 |

25 | m3 | 2000 | [158,258] | 3 | 40 |

26 | m3 | 2000 | [159,259] | 3 | 40 |

27 | m3 | 2000 | [167,267] | 3 | 40 |

28 | m3 | 2000 | [172,272] | 4 | 55 |

29 | m3 | 2000 | [172,272] | 3 | 40 |

30 | m3 | 2000 | [182,282] | 3 | 40 |

31 | m3 | 2000 | [184,284] | 3 | 40 |

32 | m3 | 2000 | [185,285] | 3 | 40 |

33 | m3 | 2000 | [190,290] | 3 | 40 |

34 | m3 | 2000 | [190,290] | 3 | 40 |

35 | m3 | 2000 | [191,291] | 3 | 40 |

36 | m3 | 2000 | [192,292] | 3 | 40 |

37 | m3 | 2000 | [193,293] | 3 | 40 |

38 | m3 | 2000 | [206,306] | 3 | 40 |

39 | m3 | 2000 | [209,309] | 3 | 40 |

40 | m3 | 2000 | [211,311] | 3 | 40 |

41 | m3 | 2000 | [211,311] | 3 | 40 |

42 | m3 | 2000 | [249,349] | 3 | 40 |

43 | m3 | 2000 | [276,376] | 3 | 40 |

44 | m3 | 2000 | [280,380] | 3 | 40 |

45 | m3 | 2000 | [281,381] | 3 | 40 |

46 | m3 | 2000 | [289,389] | 3 | 40 |

47 | m3 | 2000 | [290,390] | 3 | 40 |

48 | m3 | 2000 | [299,399] | 3 | 40 |

49 | m3 | 2000 | [300,400] | 3 | 40 |

50 | m3 | 2000 | [309,409] | 3 | 40 |

51 | m3 | 2000 | [309,409] | 3 | 40 |

52 | m3 | 2000 | [320,420] | 3 | 40 |

53 | m3 | 2000 | [321,421] | 3 | 40 |

54 | m3 | 2000 | [325,425] | 3 | 40 |

55 | m3 | 2000 | [395,494] | 3 | 40 |

56 | m3 | 2000 | [398,494] | 3 | 40 |

57 | m3 | 2000 | [425,494] | 3 | 40 |

58 | m3 | 2000 | [429,494] | 3 | 40 |

59 | m3 | 2000 | [448,494] | 3 | 40 |

60 | m3 | 2000 | [459,494] | 3 | 40 |

e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}$ | e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}\text{}$ | e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}$ | e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}\text{}$ | e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}$ | e | ${{\displaystyle \mathit{t}}}_{\mathit{e}}\text{}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 108 | 11 | 481 | 21 | 182 | 31 | 280 | 41 | 310 | 51 | 409 |

2 | 103 | 12 | 142 | 22 | 220 | 32 | 252 | 42 | 349 | 52 | 419 |

3 | 153 | 13 | 212 | 23 | 250 | 33 | 278 | 43 | 376 | 53 | 421 |

4 | 193 | 14 | 333 | 24 | 240 | 34 | 273 | 44 | 379 | 54 | 425 |

5 | 198 | 15 | 360 | 25 | 248 | 35 | 282 | 45 | 381 | 55 | 472 |

6 | 300 | 16 | 460 | 26 | 188 | 36 | 276 | 46 | 388 | 56 | 498 |

7 | 312 | 17 | 492 | 27 | 218 | 37 | 238 | 47 | 390 | 57 | 500 |

8 | 387 | 18 | 458 | 28 | 268 | 38 | 303 | 48 | 398 | 58 | 502 |

9 | 482 | 19 | 490 | 29 | 270 | 39 | 243 | 49 | 400 | 59 | 496 |

10 | 480 | 20 | 190 | 30 | 246 | 40 | 308 | 50 | 407 | 60 | 494 |

Method | z (km) | Time Consumption (s) |
---|---|---|

Gurobi | 3,213,121 | 3186 s |

MPSO | 3,213,121 | 500 s |

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**MDPI and ACS Style**

Wu, J.; Lin, B.; Wang, H.; Zhang, X.; Wang, Z.; Wang, J.
Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm. *Symmetry* **2018**, *10*, 349.
https://doi.org/10.3390/sym10080349

**AMA Style**

Wu J, Lin B, Wang H, Zhang X, Wang Z, Wang J.
Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm. *Symmetry*. 2018; 10(8):349.
https://doi.org/10.3390/sym10080349

**Chicago/Turabian Style**

Wu, Jianping, Boliang Lin, Hui Wang, Xuhui Zhang, Zhongkai Wang, and Jiaxi Wang.
2018. "Optimizing the High-Level Maintenance Planning Problem of the Electric Multiple Unit Train Using a Modified Particle Swarm Optimization Algorithm" *Symmetry* 10, no. 8: 349.
https://doi.org/10.3390/sym10080349