# Energy-Saving Scheduling for Flexible Job Shop Problem with AGV Transportation Considering Emergencies

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

#### 2.1. Introduction of FJSP-AMBRO

- (1)
- Each machine can only process one job at the same time.
- (2)
- Each job can only be processed on one machine at a time.
- (3)
- All AGVs, jobs and machines are ready at time 0.
- (4)
- The AGV can proceed to the next task only after completing the current task.
- (5)
- An AGV can only carry one job at one time.
- (6)
- Ignore the power of AGVs; all AGVs have the same speed.
- (7)
- The collision of AGVs are ignored.
- (8)
- The maintenance time of the broken machine is known.
- (9)
- The arrival time of rush orders, the job number in rush orders are random and the rush orders have processing priority. The machine breakdown and the rush order occur only once, respectively.
- (10)
- AGVs and raw materials are in the warehouse at time 0, and AGVs will drive to the finished product warehouse after completing the transportation tasks.

#### 2.2. Symbol Definitions

#### 2.3. Mathematical Model

## 3. Algorithm Design

#### 3.1. Emergency Handling Process for FJSP-AMBRO

#### 3.2. Improved NSGA-II

#### 3.2.1. Three-Layer Encoding and Decoding

#### 3.2.2. Selection Operator with Improved Crowding Distance Calculation

#### 3.2.3. POX Crossover Operation

Algorithm 1: Procedure of POX crossover operation |

1: Select two parent individuals. 2: classify the jobs into sets S1 and S2. 3: The genes in set S1 of Parent1 are inherited to the same position of Progeny1. 4: The genes in set S2 of Parent2 are inherited to the same position of Progeny2. 5: The rest positions of genes in progenies are temporarily vacant. 6: The remaining genes in Parent1 are filled into the empty positions of Progeny2 in order. 7: The remaining genes in Parent2 are filled into the empty positions of Progeny1 in order. |

#### 3.2.4. Three-Layer Mutation Operator

Algorithm 2: Procedure of three layer mutation operator |

1: Determine the number of mutation individuals mutate_num.2: for i in range(mutate_num):3: Generate a random decimal mutate_num_1 ∈ [0,1].4: Randomly select an individual Xi.5: if rand_num_1 <1/3:6: ( Perform OS mutation.)7: Generate a random decimal mutate_num_2 ∈ [0,1].8: if rand_num_2 <1/3:9: Perform swap mutation on Xi.10: elif rand_num_2 <2/3:11: Perform inverse mutation on Xi.12: else: 13: Perform heuristic mutation on Xi.14: Regenerated MS to ensure the feasibility of solutions 15: elif rand_num_1 <2/3:16: ( Perform MS mutation.)17: Generate a random decimal mutate_num_3 ∈ [0,1].18: if rand_num_3 <1/2:19: (Perform directional mutation on Xi.)20: Randomly select a position in MS of Xi.21: Reselect an available machine with the shortest processing time. 22: else: 23: (Perform random mutation on Xi.)24: Randomly select a position in MS of Xi and randomly reselect an available machine. 25: else: 26: ( Perform AS mutation.)27: Randomly select a different AGV to replace the original one of Xi. |

#### 3.2.5. Local Search Based on Critical Path

## 4. Experimental Analysis

#### 4.1. Example and Performance Metrics

#### 4.2. Parametric Analysis

#### 4.3. Validity Analysis of Improvement Strategies and the Comparison Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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

n | Number of jobs for initial order |

n^{*} | Number of jobs for initial order and rush orders |

m | Number of machines |

g | Number of AGVs |

i | Job index |

j | Operation index |

k | Machine index |

s | AGV index |

${n}_{i}$ | Operation number of job i |

${O}_{ij}$ | The operation j of job i |

${t}_{ijk}$ | The processing time of ${O}_{ij}$ on machine k |

${S}_{ijk}$ | The start time of ${O}_{ij}$ on machine k |

${E}_{ijk}$ | The completion time of ${O}_{ij}$ on machine k |

$A{S}_{ijs}$ | The earliest transportable time for transportation task from processing machine of ${O}_{i(j-1)}$ to processing machine of ${O}_{ij}$ by AGV s |

$A{E}_{ijs}$ | The arrival time of the transportation task from the processing machine of ${O}_{i(j-1)}$ to the processing machine of ${O}_{ij}$ by AGV s |

${T}_{ij}$ | The time for the transportation task from the processing machine of ${O}_{i(j-1)}$ to the processing machine of ${O}_{ij}$ |

${T}_{b}$ | The start time of machine breakdown |

${T}_{r}$ | The available time after machine breakdown |

${C}_{i}$ | The completion time of job i |

${P}_{k}$ | Working power of machine k |

TE | Total energy consumption of all machines |

M | A positive number large enough |

${x}_{ij}^{k}$ | Machine selection decision variable. If ${O}_{ij}$ is processed on machine k, then ${x}_{ij}^{k}=1$; otherwise ${x}_{ij}^{k}=0$ |

${x}_{ij}^{{}^{\prime}k}$ | Reschedule machine selection decision variable. If ${O}_{ij}$ is processed on machine k after rescheduling, then ${x}_{ij}^{{}^{\prime}k}=1$; otherwise ${x}_{ij}^{{}^{\prime}k}=0$ |

${y}_{ij{i}^{\prime}{j}^{\prime}}^{k}$ | Machine condition variables. If ${O}_{ij}$ is the immediately preceding operation of ${O}_{{i}^{\prime}{j}^{\prime}}$ on machine k, then ${y}_{ij{i}^{\prime}{j}^{\prime}}^{k}=1$; otherwis ${y}_{ij{i}^{\prime}{j}^{\prime}}^{k}=0$ |

${z}_{ij}^{s}$ | AGV selection decision variable. If the transportation task from processing machine of ${O}_{i(j-1)}$ to processing machine of ${O}_{ij}$ is carried out by the AGV s, then ${z}_{ij}^{s}=1$; otherwise ${z}_{ij}^{s}=0$ |

${r}_{ij{i}^{\prime}{j}^{\prime}}^{s}$ | AGV condition variable. If the transportation task of AGV s from processing machine of ${O}_{i(j-1)}$ to processing machine of ${O}_{ij}$ is earlier than that from ${O}_{{i}^{\prime}({j}^{\prime}-1)}$ to ${O}_{{i}^{\prime}{j}^{\prime}}$, then ${r}_{ij{i}^{\prime}{j}^{\prime}}^{s}=1$; otherwise ${r}_{ij{i}^{\prime}{j}^{\prime}}^{s}=0$ |

${B}_{k}$ | Machine state variable. If there is a breakdown on machine k, ${B}_{k}$ = 1; Otherwise, ${B}_{k}$ = 0 |

${e}_{ijk}$ | Job status variable. If ${S}_{ijk}$ is less than ${T}_{b}$ and ${E}_{ijk}$ is more than ${T}_{b}$, ${e}_{ijk}$=1; Otherwise, ${e}_{ijk}$= 0 |

Factor | Levels | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

MaxIt1 | 200 | 300 | 400 | 500 | 600 |

PC1 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 |

PM1 | 0.01 | 0.05 | 0.10 | 0.15 | 0.20 |

MaxIt2 | 200 | 300 | 400 | 500 | 600 |

PC2 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 |

PM2 | 0.01 | 0.05 | 0.10 | 0.15 | 0.20 |

Parameter Combinations | Level | IGD | |||||
---|---|---|---|---|---|---|---|

MaxIt1 | PC1 | PM1 | MaxIt2 | PC2 | PM2 | ||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1166.25 |

2 | 1 | 2 | 2 | 2 | 2 | 2 | 1211.99 |

3 | 1 | 3 | 3 | 3 | 3 | 3 | 1042.18 |

4 | 1 | 4 | 4 | 4 | 4 | 4 | 1035.20 |

5 | 1 | 5 | 5 | 5 | 5 | 5 | 1019.37 |

6 | 2 | 1 | 2 | 3 | 4 | 5 | 1122.31 |

7 | 2 | 2 | 3 | 4 | 5 | 1 | 1071.06 |

8 | 2 | 3 | 4 | 5 | 1 | 2 | 1108.89 |

9 | 2 | 4 | 5 | 1 | 2 | 3 | 1396.53 |

10 | 2 | 5 | 1 | 2 | 3 | 4 | 1083.94 |

11 | 3 | 1 | 3 | 5 | 2 | 4 | 976.71 |

12 | 3 | 2 | 4 | 1 | 3 | 5 | 1150.72 |

13 | 3 | 3 | 5 | 2 | 4 | 1 | 1256.44 |

14 | 3 | 4 | 1 | 3 | 5 | 2 | 1277.04 |

15 | 3 | 5 | 2 | 4 | 1 | 3 | 1141.59 |

16 | 4 | 1 | 4 | 2 | 5 | 3 | 1120.62 |

17 | 4 | 2 | 5 | 3 | 1 | 4 | 938.70 |

18 | 4 | 3 | 1 | 4 | 2 | 5 | 972.39 |

19 | 4 | 4 | 2 | 5 | 3 | 1 | 1129.60 |

20 | 4 | 5 | 3 | 1 | 4 | 2 | 1404.60 |

21 | 5 | 1 | 5 | 4 | 3 | 2 | 1084.39 |

22 | 5 | 2 | 1 | 5 | 4 | 3 | 1157.46 |

23 | 5 | 3 | 2 | 1 | 5 | 4 | 1372.36 |

24 | 5 | 4 | 3 | 2 | 1 | 5 | 1013.42 |

25 | 5 | 5 | 4 | 3 | 2 | 1 | 1324.13 |

Problem | IGD | C | |||||
---|---|---|---|---|---|---|---|

NSGA-II | ${\mathit{N}}_{\mathit{l}}$ | INSGA-II | C(NSGA-II, ${\mathit{N}}_{\mathit{l}}$) | C(${\mathit{N}}_{\mathit{l}}$, NSGA-II) | C(INSGA-II, ${\mathit{N}}_{\mathit{l}}$) | C(${\mathit{N}}_{\mathit{l}}$, INSGA-II) | |

Mk-01 | 10.69 | 10.56 | 10.52↓ | 0.10 | 0.50 | 0.13 | 0.48 |

Mk-02 | 11.39 | 11.25 | 11.17↓ | 0.03 | 0.63 | 0.22 | 0.23 |

Mk-03 | 64.75 | 63.37 | 63.10↓ | 0.00 | 0.60 | 0.17 | 0.29 |

Mk-04 | 14.27 | 14.05 | 13.04↓ | 0.25 | 0.29 | 0.27 | 0.29 |

Mk-05 | 48.01 | 47.07 | 46.80↓ | 0.20 | 0.33 | 0.25 | 0.28 |

Mk-06 | 31.09 | 29.74 | 29.09↓ | 0.00 | 0.10 | 0.16 | 0.30 |

Mk-07 | 26.68 | 26.55 | 26.45↓ | 0.00 | 0.57 | 0.19 | 0.29 |

Mk-08 | 67.48 | 67.26 | 64.73↓ | 0.27 | 0.31 | 0.03 | 0.43 |

Mk-09 | 141.63 | 140.25 | 139.42↓ | 0.20 | 0.57 | 0.16 | 0.50 |

Mk-10 | 164.34 | 161.98 | 161.61↓ | 0.12 | 0.51 | 0.29 | 0.34 |

Mk-11 | 207.63 | 205.44 | 204.66↓ | 0.15 | 0.29 | 0.26 | 0.36 |

Mk-12 | 216.12 | 214.10 | 213.51↓ | 0.22 | 0.35 | 0.18 | 0.43 |

Mk-13 | 226.51 | 221.10 | 219.06↓ | 0.15 | 0.36 | 0.26 | 0.42 |

Mk-14 | 365.57 | 364.74 | 361.60↓ | 0.21 | 0.45 | 0.27 | 0.32 |

Mk-15 | 336.60 | 331.41 | 327.76↓ | 0.33 | 0.39 | 0.13 | 0.43 |

Problem | n/m | INSGA-II | NSGA | NSGA-II | SPEA2 |
---|---|---|---|---|---|

Mk-01 | 10/6 | 10.18 | 10.72 | 11.02 | 11.79 |

Mk-02 | 10/6 | 12.49 | 12.59 | 13.27 | 11.96 |

Mk-03 | 15/8 | 63.43 | 66.67 | 63.78 | 67.11 |

Mk-04 | 15/8 | 11.48 | 13.08 | 12.03 | 12.42 |

Mk-05 | 15/4 | 44.35 | 45.13 | 44.87 | 45.01 |

Mk-06 | 10/15 | 28.88 | 30.88 | 29.64 | 33.50 |

Mk-07 | 20/5 | 27.06 | 30.13 | 27.33 | 33.04 |

Mk-08 | 20/10 | 64.70 | 65.33 | 64.29 | 64.83 |

Mk-09 | 20/10 | 141.43 | 145.03 | 142.64 | 146.16 |

Mk-10 | 20/15 | 162.46 | 167.04 | 163.34 | 165.02 |

Mk-11 | 30/5 | 202.23 | 206.45 | 206.01 | 203.44 |

Mk-12 | 30/10 | 209.62 | 217.41 | 215.39 | 211.60 |

Mk-13 | 30/10 | 221.19 | 227.16 | 223.85 | 231.92 |

Mk-14 | 30/15 | 363.32 | 368.55 | 365.66 | 368.33 |

Mk-15 | 30/15 | 337.39 | 346.04 | 341.23 | 344.24 |

Problem | C(INSGA-II, NSGA) | C(NSGA, INSGA-II) | C(INSGA-II, NSGA-II) | C(NSGA-II, INSGA-II) | C(INSGA-II, SPEA2) | C(SPEA2, INSGA-II) |
---|---|---|---|---|---|---|

Mk-01 | 0.48 | 0.11 | 0.30 | 0.21 | 0.34 | 0.20 |

Mk-02 | 0.51 | 0.14 | 0.34 | 0.20 | 0.92 | 0.00 |

Mk-03 | 0.42 | 0.12 | 0.35 | 0.17 | 0.58 | 0.04 |

Mk-04 | 0.48 | 0.08 | 0.29 | 0.18 | 0.72 | 0.08 |

Mk-05 | 0.30 | 0.08 | 0.31 | 0.18 | 0.44 | 0.03 |

Mk-06 | 0.34 | 0.10 | 0.14 | 0.12 | 0.99 | 0.00 |

Mk-07 | 0.54 | 0.07 | 0.35 | 0.26 | 0.84 | 0.00 |

Mk-08 | 0.55 | 0.10 | 0.28 | 0.19 | 0.57 | 0.00 |

Mk-09 | 0.65 | 0.05 | 0.25 | 0.19 | 0.83 | 0.01 |

Mk-10 | 0.59 | 0.03 | 0.38 | 0.09 | 0.77 | 0.01 |

Mk-11 | 0.41 | 0.18 | 0.17 | 0.17 | 0.32 | 0.02 |

Mk-12 | 0.50 | 0.06 | 0.32 | 0.17 | 0.40 | 0.02 |

Mk-13 | 0.70 | 0.05 | 0.25 | 0.16 | 0.89 | 0.00 |

Mk-14 | 0.63 | 0.06 | 0.37 | 0.17 | 0.44 | 0.05 |

Mk-15 | 0.63 | 0.09 | 0.30 | 0.07 | 0.80 | 0.00 |

Algorithm | Mean | Std. Deviation | Std. Error Mean | t | Sig. (2-tailed) |
---|---|---|---|---|---|

INSGA-II-NSGA | −3.46733 | 2.64179 | 0.68211 | −5.083 | 0.000 |

INSGA-II-NSGA-II | −1.61067 | 1.71294 | 0.44228 | −3.642 | 0.003 |

INSGA-II-SPEA2 | −3.34533 | 3.02869 | 0.78200 | −4.278 | 0.001 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, H.; Qin, C.; Zhang, W.; Xu, Z.; Xu, G.; Gao, Z.
Energy-Saving Scheduling for Flexible Job Shop Problem with AGV Transportation Considering Emergencies. *Systems* **2023**, *11*, 103.
https://doi.org/10.3390/systems11020103

**AMA Style**

Zhang H, Qin C, Zhang W, Xu Z, Xu G, Gao Z.
Energy-Saving Scheduling for Flexible Job Shop Problem with AGV Transportation Considering Emergencies. *Systems*. 2023; 11(2):103.
https://doi.org/10.3390/systems11020103

**Chicago/Turabian Style**

Zhang, Hongliang, Chaoqun Qin, Wenhui Zhang, Zhenxing Xu, Gongjie Xu, and Zhenhua Gao.
2023. "Energy-Saving Scheduling for Flexible Job Shop Problem with AGV Transportation Considering Emergencies" *Systems* 11, no. 2: 103.
https://doi.org/10.3390/systems11020103