# A Hybrid Machine Learning and Population Knowledge Mining Method to Minimize Makespan and Total Tardiness of Multi-Variety Products

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## Abstract

**:**

## 1. Introduction

- (1)
- A hybrid machine learning and population knowledge mining method is proposed to solve multi-objective job shop scheduling problems.
- (2)
- Five attributes, namely operation feature, processing time, remaining time, due date, and priorities, are selected to mine initial population knowledge.
- (3)
- The ADSM method is designed to reprioritize operations after the population knowledge mining.
- (4)
- Three populations (rules, mixed, and random) with different iterations and population sizes are compared, and three performance metrics are defined to explore the effectiveness of the proposed method.

## 2. Literature Review

## 3. Problem Description and Goal

_{0}, J

_{1}, …, J

_{n}} under multiple objectives processed on m machines {M

_{0}, M

_{1}, …, M

_{m}} with different routes [31]. The processing order of each part and the processing time of each operation o

_{ij}on a machine is determined. To illustrate the problem, the encoding of a benchmark of 10 × 10 job shop (LA18) [32] here can take an example as {9, 8, 5, 4, 2, 1, 3, 6, 7, 0, …,5, 8, 9}. Each element in the operation sequence code represents a job. The ith occurrence of the same value means the ith operation of this job. Here, the MOJSSP aims to find an order to optimize the above two objectives simultaneously. The constraints include each job should be processed on only one machine at a time, each machine can process only one job at a time, and the preemption is not allowed [33].

_{ij}: the jth operation of the job i

_{i}: the completion time of the job i

_{ijk}: the processing time of the jth operation of the job i on machine M

_{k}

_{ijk}the completion time of the jth operation of the job i on machine M

_{k}

_{i}: the due date of the ith job

_{ij}: set of available machines for the jth operation of job i

## 4. The Hybrid Machine Learning and Population Knowledge Mining Method

#### 4.1. Attributes

#### 4.1.1. Priorities

#### 4.1.2. Operation Feature

#### 4.1.3. Processing Time

#### 4.1.4. Remaining Time

#### 4.1.5. Due Date

#### 4.2. Data Preparation

#### 4.3. Rule Mining

#### 4.4. Initial Population Generated Using ADSM

## 5. Experiment

#### 5.1. Selected Algorithm

- Nondominated sorting genetic algorithm II (NSGA-II)Based on NSGA, developed by Deb et al. [36], NSGA-II has the advantage of fast running speed, good robustness, and fast convergence. The encoding method is consistent with the above described. Each gene means an operation. Each chromosome/individual represents a solution or a sequence of all the operations waiting to be scheduled. NSGA-II combines parent and offspring into a new population and then obtains the non-dominated solutions by fast non-dominated sorting. The parent is generated by disrupting genes in the chromosome. The offspring is generated by the crossover and mutation method. The parameters are shown in Table 5.
- Simulated annealing (SA)Simulated annealing is an approximate method based on Monte Carlo design. It was first introduced by Kirkpatrick et al. [37] to solve the optimization problem. SA accepts a solution that is worse than the current solution by the Metropolis criterion, thus it is possible to jump out of this local optimal solution to reach the global optimal solution. Table 6 lists the parameters used in SA.

- Knowledge mining heuristic optimization method (rule)The initial population was generated using the proposed hybrid machine learning and population knowledge mining method.
- Heuristic optimization method (random)The initial population of this method was completely randomized.
- Hybrid population optimization method (mixed)Half of the initial population was generated by knowledge mining and the other half was randomly generated.

#### 5.2. Performance Metrics

- Relative Error (RE)Extending the method of Arroyo and Leung [41], we analyzed the performance of the two acquired objectives using the relative error (RE) metric. The formulation is as follows:$$RE=\frac{\overline{F}-{F}_{\mathrm{best}}}{{F}_{\mathrm{best}}}\times 100$$
- Coverage of Two Sets (Cov)This indicator was used to measure the dominance between two sets of solutions. The definition is as follows:$$Cov(X,Y)=\frac{\left|\right\{y\in Y;\exists x\in X:x\prec =y\left\}\right|}{\left|Y\right|}$$
- SpacingIt measures the standard deviation of the minimum distance from each solution to other solutions. The smaller the Spacing value is, the more uniform the solution set is. The expression is as follows:$$Spacing=\sqrt{\frac{1}{n-1}{\displaystyle {\sum}_{1}^{n}{(\overline{d}-{d}_{i})}^{2}}}$$

#### 5.3. Results and Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Distribution of makespan and total tardiness of 100 initial population sizes under different iteration numbers: (

**a**) makespan under 100 iterations; (

**b**) total tardiness under 100 iterations; (

**c**) makespan under 300 iterations; and (

**d**) total tardiness under 300 iterations.

**Figure 6.**Distribution of makespan and total tardiness of 50 initial population sizes under different iteration numbers: (

**a**) makespan under 100 iterations; (

**b**) total tardiness under 100 iterations; (

**c**) makespan under 300 iterations; and (

**d**) total tardiness under 300 iterations.

**Figure 7.**Distribution of makespan and total tardiness of 25 initial population sizes under different iteration numbers: (

**a**) makespan under 100 iterations; (

**b**) total tardiness under 100 iterations; (

**c**) makespan under 300 iterations; and (

**d**) total tardiness under 300 iterations.

**Figure 8.**Box diagram of Cov of the rule method vs. random method with different sizes of initial population under 100 and 300 iterations: (

**a**) IP = 25 under 100 iterations; (

**b**) IP = 50 under 100 iterations; (

**c**) IP = 100 under 100 iterations; (

**d**) IP = 25 under 300 iterations; (

**e**) IP = 50 under 300 iterations; and (

**f**) IP = 100 under 300 iterations.

Job No. | Due Date | Class |
---|---|---|

0 | 1100 | tight |

1 | 1150 | tight |

2 | 1150 | tight |

3 | 1180 | tight |

4 | 1180 | tight |

5 | 1200 | slack |

6 | 1200 | slack |

7 | 1250 | slack |

8 | 1250 | slack |

9 | 1290 | slack |

Operation | Machine | Processing Time | Remaining Time | Operation Feature | Processing Time | Remaining Time | Due Date |
---|---|---|---|---|---|---|---|

00 | 6 | 54 | 507 | first | middle | long | tight |

01 | 0 | 87 | 420 | secondary | long | long | tight |

02 | 4 | 48 | 372 | secondary | middle | middle | tight |

03 | 3 | 60 | 312 | middle | middle | middle | tight |

04 | 7 | 39 | 273 | middle | middle | middle | tight |

05 | 8 | 35 | 238 | middle | short | middle | tight |

… | … | … | … | … | … | … | … |

97 | 9 | 85 | 105 | later | long | short | slack |

98 | 5 | 46 | 59 | later | middle | short | slack |

99 | 0 | 59 | 0 | last | middle | short | slack |

ID | Operation | Operation Feature | Processing Time | Remaining Time | Due Date | Priority |
---|---|---|---|---|---|---|

0 | 40 | first | long | middle | tight | 0 |

1 | 40 | first | long | middle | tight | 0 |

2 | 90 | first | middle | long | slack | 0 |

3 | 90 | first | middle | long | slack | 0 |

4 | 90 | first | middle | long | slack | 0 |

5 | 40 | first | long | middle | tight | 0 |

6 | 40 | first | long | middle | tight | 0 |

7 | 80 | first | short | long | slack | 0 |

8 | 80 | first | short | long | slack | 0 |

9 | 00 | first | middle | long | tight | 0 |

… | … | … | … | … | … | … |

6594 | 99 | last | middle | short | slack | 9 |

6595 | 79 | last | middle | Short | slack | 9 |

6596 | 99 | last | middle | short | slack | 9 |

6597 | 89 | last | short | short | slack | 9 |

6598 | 69 | last | long | short | slack | 9 |

6599 | 69 | last | long | short | slack | 9 |

Id | Rule Set | Priority | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||

0 | first long long slack | 0.67 | 0.15 | 0.14 | 0.05 | ||||||

1 | first long middle tight | 0.80 | 0.20 | ||||||||

2 | first middle long slack | 0.71 | 0.28 | 0.01 | |||||||

3 | first middle long tight | 0.98 | 0.02 | ||||||||

4 | first middle middle tight | 0.82 | 0.18 | ||||||||

5 | first short long slack | 1.00 | |||||||||

6 | first short long tight | 0.33 | 0.48 | 0.12 | 0.07 | ||||||

7 | last long short slack | 0.02 | 0.01 | 0.09 | 0.11 | 0.78 | |||||

8 | last long short tight | 0.08 | 0.76 | 0.17 | 0.00 | 0.00 | |||||

9 | last middle short slack | 0.07 | 0.02 | 0.17 | 0.75 | ||||||

10 | last middle short tight | 0.08 | 0.21 | 0.39 | 0.32 | ||||||

11 | last short short slack | 0.03 | 0.27 | 0.70 | |||||||

12 | last short short tight | 0.06 | 0.16 | 0.02 | 0.14 | 0.63 | |||||

13 | later long short slack | 0.01 | 0.08 | 0.09 | 0.06 | 0.19 | 0.39 | 0.19 | |||

14 | later long short tight | 0.12 | 0.13 | 0.16 | 0.29 | 0.13 | 0.18 | ||||

15 | later middle middle slack | 0.05 | 0.26 | 0.05 | 0.26 | 0.39 | |||||

16 | later middle short slack | 0.12 | 0.18 | 0.27 | 0.24 | 0.19 | |||||

17 | later middle short tight | 0.01 | 0.06 | 0.08 | 0.17 | 0.13 | 0.43 | 0.12 | |||

18 | later short short slack | 0.05 | 0.13 | 0.12 | 0.18 | 0.37 | 0.15 | ||||

19 | later short short tight | 0.18 | 0.53 | 0.10 | 0.15 | 0.04 | |||||

20 | middle long middle slack | 0.01 | 0.16 | 0.24 | 0.24 | 0.14 | 0.13 | 0.07 | |||

21 | middle long middle tight | 0.14 | 0.37 | 0.04 | 0.08 | 0.36 | 0.02 | ||||

22 | middle long short tight | 0.05 | 0.27 | 0.40 | 0.21 | 0.08 | |||||

23 | middle middle middle slack | 0.01 | 0.19 | 0.22 | 0.13 | 0.17 | 0.14 | 0.11 | 0.02 | ||

24 | middle middle middle tight | 0.02 | 0.24 | 0.39 | 0.26 | 0.09 | |||||

25 | middle middle short tight | 0.30 | 0.36 | 0.15 | 0.18 | ||||||

26 | middle short middle slack | 0.14 | 0.14 | 0.52 | 0.14 | 0.07 | |||||

27 | middle short middle tight | 0.07 | 0.18 | 0.15 | 0.11 | 0.08 | 0.22 | 0.17 | 0.02 | ||

28 | middle short short slack | 0.18 | 0.26 | 0.03 | 0.48 | 0.05 | |||||

29 | middle short short tight | 0.36 | 0.12 | 0.30 | 0.06 | 0.15 | |||||

30 | secondary long long slack | 0.03 | 0.62 | 0.17 | 0.05 | 0.04 | 0.05 | 0.05 | 0.01 | ||

31 | secondary long long tight | 0.03 | 0.82 | 0.15 | |||||||

32 | secondary middle long slack | 0.38 | 0.39 | 0.22 | 0.02 | ||||||

33 | secondary middle middle slack | 0.14 | 0.39 | 0.30 | 0.02 | 0.06 | 0.04 | 0.06 | |||

34 | secondary middle middle tight | 0.11 | 0.23 | 0.34 | 0.21 | 0.08 | 0.03 | 0.01 | |||

35 | secondary short long slack | 0.38 | 0.42 | 0.20 | |||||||

36 | secondary short middle tight | 0.18 | 0.33 | 0.39 | 0.10 |

Parameters | Values |
---|---|

population size | 25, 50, 100 |

mutation rate | 0.002 |

crossover rate | 0.9 |

size of tournament selection | 10 |

number of iterations | 100, 300 |

Parameters | Values |
---|---|

population size | 25, 50, 100 |

initial temperature | 100 |

end temperature | 0.01 |

cooling rate | 0.001 |

number of iterations | 100, 300 |

IP | IT | NSGA-II | NSGA-II + SA | |
---|---|---|---|---|

25 | 100 | best | 933 | 853 |

average | 974 | 909 | ||

300 | best | 895 | 848 | |

average | 947 | 910 | ||

50 | 100 | best | 895 | 854 |

average | 941 | 912 | ||

300 | best | 865 | 848 | |

average | 931 | 915 | ||

100 | 100 | best | 898 | 848 |

average | 938 | 919 | ||

300 | best | 861 | 848 | |

average | 912 | 943 |

IP | IT | NSGA-II | NSGA-II + SA | |
---|---|---|---|---|

25 | 100 | best | −3860 | −4529 |

average | −3350 | −4338 | ||

300 | best | −4001 | −4524 | |

average | −3757 | −4537 | ||

50 | 100 | best | −4241 | −4630 |

average | −3743 | −4400 | ||

300 | best | −4279 | −4815 | |

average | −3982 | −4569 | ||

100 | 100 | best | −4273 | −4698 |

average | −3703 | −4495 | ||

300 | best | −4539 | −4698 | |

average | −4229 | −4587 |

Methods | Iteration | F1 | F2 | RE | |
---|---|---|---|---|---|

rules | 100 | best | 848 | −4696 | 7.5|4 |

average | 912 | −4500 | / | ||

300 | best | 848 | −4717 | 6.7|2 | |

average | 905 | −4622 | / | ||

mixed | 100 | best | 848 | −4618 | 8.1|2.7 |

average | 916.3 | −4492 | / | ||

300 | best | 848 | −4846 | 10.7|4.3 | |

average | 939 | −4639 | / | ||

random | 100 | best | 848 | −4698 | 8.4|4.3 |

average | 919 | −4495 | / | ||

300 | best | 848 | −4698 | 11.2|2.4 | |

average | 940 | −4587 | / |

Methods | Iteration | F1 | F2 | RE | |
---|---|---|---|---|---|

rules | 100 | best | 852 | −4568 | 5.94|3.27 |

average | 902.6 | −4418.4 | / | ||

300 | best | 848 | −4835 | 8.54|4.87 | |

average | 920.4 | −4599.3 | / | ||

mixed | 100 | best | 848 | −4618 | 6.03|3.52 |

average | 899.1 | −4455.6 | / | ||

300 | best | 848 | −4868 | 10.74|6.04 | |

average | 939.1 | −4573.8 | / | ||

random | 100 | best | 854 | −4630 | 6.77|4.97 |

average | 911.8 | −4400 | / | ||

300 | best | 848 | −4815 | 7.88|5.12 | |

average | 914.8 | −4568.5 | / |

Methods | Iteration | F1 | F2 | RE | |
---|---|---|---|---|---|

rules | 100 | best | 852 | −4693 | 8.49|6.58 |

average | 924.3 | −4384.3 | / | ||

300 | best | 848 | −4731 | 5.83|4.34 | |

average | 897.4 | −4525.9 | / | ||

mixed | 100 | best | 855 | −4593 | 5.38|5.11 |

average | 901 | −4358.2 | / | ||

300 | best | 848 | −4810 | 7.63|5.59 | |

average | 912.7 | −4540.9 | / | ||

random | 100 | best | 853 | −4529 | 6.58|4.22 |

average | 909.1 | −4337.8 | / | ||

300 | best | 848 | −4824 | 7.28|5.94 | |

average | 909.7 | −4537.4 | / |

Methods | IP | IT100 | IT300 | |||
---|---|---|---|---|---|---|

Cov | Spacing | Cov | Spacing | |||

rules | 25 | best | 1 | 37 | 1 | 8.8 |

average | 0.54 | 0.4 | ||||

50 | best | 1 | 25 | 1 | 39 | |

average | 0.59 | 0.38 | ||||

100 | best | 1 | 29 | 1 | 22 | |

average | 0.56 | 0.47 | ||||

random | 25 | best | 0.75 | 76 | 1 | 50.5 |

average | 0.19 | 0.35 | ||||

50 | best | 0.9 | 65 | 1 | 45 | |

average | 0.21 | 0.43 | ||||

100 | best | 0.8 | 37 | 0.86 | 18 | |

average | 0.19 | 0.29 |

Methods | IP | IT100 | IT300 | |||
---|---|---|---|---|---|---|

Cov | Spacing | Cov | Spacing | |||

mixed | 25 | best | 1 | 19 | 1 | 48 |

average | 0.63 | 0.31 | ||||

50 | best | 1 | 24 | 0.86 | 37 | |

average | 0.45 | 0.35 | ||||

100 | best | 1 | 16 | 0.92 | 20 | |

average | 0.41 | 0.45 | ||||

random | 25 | best | 0.27 | 76 | 1 | 50.5 |

average | 0.11 | 0.43 | ||||

50 | best | 1 | 65 | 1 | 45 | |

average | 0.33 | 0.4 | ||||

100 | best | 1 | 37 | 1 | 18 | |

average | 0.39 | 0.33 |

Methods | IP | IT100 | IT300 | |||
---|---|---|---|---|---|---|

Cov | Spacing | Cov | Spacing | |||

rules | 25 | best | 1 | 37 | 1 | 8.8 |

average | 0.24 | 0.46 | ||||

50 | best | 1 | 25 | 1 | 39 | |

average | 0.47 | 0.44 | ||||

100 | best | 1 | 29 | 1 | 22 | |

average | 0.53 | 0.41 | ||||

mixed | 25 | best | 1 | 19 | 0.8 | 48 |

average | 0.41 | 0.24 | ||||

50 | best | 1 | 24 | 0.88 | 37 | |

average | 0.31 | 0.41 | ||||

100 | best | 1 | 16 | 1 | 20 | |

average | 0.21 | 0.37 |

Methods | IP | IT | F1 | F2 | RE | |
---|---|---|---|---|---|---|

ADSM | 25 | 100 | best | 852 | −4693 | 8.49|6.58 |

average | 924.3 | −4384.3 | / | |||

300 | best | 848 | −4731 | 5.83|4.34 | ||

average | 897.4 | −4525.9 | / | |||

50 | 100 | best | 852 | −4568 | 5.94|3.27 | |

average | 902.6 | −4418.4 | / | |||

300 | best | 848 | −4835 | 8.54|4.87 | ||

average | 920.4 | −4599.3 | / | |||

100 | 100 | best | 848 | −4696 | 7.5|4 | |

average | 912 | −4500 | / | |||

300 | best | 848 | −4717 | 6.7|2 | ||

average | 905 | −4622 | / | |||

Nasiri’s | 25 | 100 | best | 853 | −4587 | 4|5.8 |

average | 887 | −4319 | / | |||

300 | best | 848 | −4810 | 8.84|5.51 | ||

average | 923 | −4545 | / | |||

50 | 100 | best | 848 | −4558 | 7.8|1.34 | |

average | 914.2 | −4497 | / | |||

300 | best | 848 | −4822 | 7.1|5.4 | ||

average | 908 | −4561.5 | / | |||

100 | 100 | best | 848 | −4642 | 6|3.8 | |

average | 896.3 | −4467 | / | |||

300 | best | 848 | −4799 | 12.7|4.6 | ||

average | 956 | −4576 | / |

Methods | IP | IT100 | IT300 | |||
---|---|---|---|---|---|---|

Cov | Spacing | Cov | Spacing | |||

ADSM | 25 | best | 1 | 37 | 1 | 8.8 |

average | 0.54 | 0.42 | ||||

50 | best | 1 | 25 | 1 | 39 | |

average | 0.59 | 0.47 | ||||

100 | best | 1 | 29 | 1 | 22 | |

average | 0.55 | 0.39 | ||||

Nasiri’s | 25 | best | 0.75 | 41 | 1 | 44 |

average | 0.19 | 0.35 | ||||

50 | best | 0.9 | 46 | 1 | 29 | |

average | 0.21 | 0.33 | ||||

100 | best | 0.8 | 71 | 1 | 64 | |

average | 0.16 | 0.37 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Qiu, Y.; Ji, W.; Zhang, C. A Hybrid Machine Learning and Population Knowledge Mining Method to Minimize Makespan and Total Tardiness of Multi-Variety Products. *Appl. Sci.* **2019**, *9*, 5286.
https://doi.org/10.3390/app9245286

**AMA Style**

Qiu Y, Ji W, Zhang C. A Hybrid Machine Learning and Population Knowledge Mining Method to Minimize Makespan and Total Tardiness of Multi-Variety Products. *Applied Sciences*. 2019; 9(24):5286.
https://doi.org/10.3390/app9245286

**Chicago/Turabian Style**

Qiu, Yongtao, Weixi Ji, and Chaoyang Zhang. 2019. "A Hybrid Machine Learning and Population Knowledge Mining Method to Minimize Makespan and Total Tardiness of Multi-Variety Products" *Applied Sciences* 9, no. 24: 5286.
https://doi.org/10.3390/app9245286