# Improved Differential Evolution Algorithm for Flexible Job Shop Scheduling Problems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Flexible Job Shop Scheduling Problems by Using Other Metaheuristic Methods

#### 2.2. Flexible Job Shop Scheduling Problems by Using Differential Evolution Algorithm

#### 2.3. Differential Evolution Algorithm for Solving Other Problems

## 3. Flexible Job Shop Scheduling Problem Pattern and Mathematical Model

#### 3.1. Flexible Job Shop Scheduling Problem (FJSP)

_{max}), the number of tardy jobs, and the maximum lateness. This paper employed the minimized makespan (C

_{max}) for evaluation in accordance with Equation (1). While C1, C2, …, C

_{p}are possible solutions to produce scheduling, the solution resulting in the longest processing time was selected. Moreover, Z refers to the objective of production scheduling—to achieve the lowest value—as described in Equation (1). Figure 2 shows a sample of solutions from the total processing time of entire jobs:

_{1}, C

_{2}, …, C

_{p}).

_{min}is the optimal solution to the algorithm, and BKS is the best known solution.

#### 3.2. Mathematical Model of the Flexible Job Shop Scheduling Problem

#### 3.2.1. Indices

#### 3.2.2. Parameter

#### 3.2.3. Decision Variables

## 4. General Differential Evolution Algorithm

_{m}is the dimension of vector m. Furthermore, ${D}_{{m}_{rand}}$ denotes a random integer number of vector m in the range [1, N], where N is the vector size. Consequently, the value of trial vector m in dimension n in iteration round G equals the mutant vector when there is a random number of vector m at dimension n that is less than CR; in other words, ${D}_{{m}_{rand}}$ equals D

_{m}. (4) The formula for selecting a target vector for the next round described in Equation (19). To solve the equation for the minimum target value (minimization), the less than or equal to sign is used:

#### 4.1. Procedure of FJSP by Using Differential Evolution Algorithm

#### 4.1.1. Calculation Using the General Differential Evolution Algorithm DE/rand/1 and Binomial Crossover

#### 4.1.2. Procedure of FJSP by Using the Improved Differential Evolution Algorithm

Algorithm 1. Pseudo-code of the improved DE for the FJSP |

Setup the initial DE parameterDo while from first iteration to final iterationDo while from first DE to final DESetup the initial parameters: job, operation, machine, processing time, operation sequence, machine assignment. Do while from first task to final taskFind the start/following task where the fitness is the makespan of the data instances Input the scaling factor, crossover rate, NP, job assignment, machine assignment, and local search to data list Produce the four mutation equations:
$${\mathrm{V}}_{\mathrm{m},\mathrm{n},\mathrm{G}}={\mathrm{X}}_{\mathrm{r}1,\mathrm{n},\mathrm{G}}+\mathrm{F}1\left({\mathrm{X}}_{\mathrm{r}2,\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}3,\mathrm{n},\mathrm{G}}\right)+\mathrm{F}2\left({\mathrm{X}}_{\mathrm{best},\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}1,\mathrm{n},\mathrm{G}}\right)$$
$${\mathrm{V}}_{\mathrm{m},\mathrm{n},\mathrm{G}}={\mathrm{X}}_{\mathrm{r}1,\mathrm{n},\mathrm{G}}+\mathrm{F}\left({\mathrm{X}}_{\mathrm{r}2,\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}3,\mathrm{n},\mathrm{G}}+{\mathrm{X}}_{\mathrm{r}4,\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}5,\mathrm{n},\mathrm{G}}\right)$$
$${\mathrm{V}}_{\mathrm{m},\mathrm{n},\mathrm{G}}={\mathrm{X}}_{\mathrm{best},\mathrm{n},\mathrm{G}}+\mathrm{F}\left({\mathrm{X}}_{\mathrm{r}2,\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}3,\mathrm{n},\mathrm{G}}+{\mathrm{X}}_{\mathrm{r}4,\mathrm{n},\mathrm{G}}-{\mathrm{X}}_{\mathrm{r}5,\mathrm{n},\mathrm{G}}\right)$$
$${\mathrm{U}}_{\mathrm{i},\mathrm{j},\mathrm{G}}=\{\begin{array}{c}{\mathrm{V}}_{\mathrm{i},\mathrm{j},\mathrm{G}}\mathrm{when}\mathrm{j}\text{}\le {\mathrm{rand}}_{\mathrm{i},1}\mathrm{and}j\ge {\mathrm{rand}}_{\mathrm{i},2}\\ {\mathrm{X}}_{\mathrm{i},\mathrm{j},\mathrm{G}}\mathrm{when}{\mathrm{rand}}_{\mathrm{i},1}j{\mathrm{rand}}_{\mathrm{i},2}\end{array}.$$
$${\mathrm{X}}_{\mathrm{m},\mathrm{n},\mathrm{G}+1}=\{\begin{array}{c}{\mathrm{U}}_{\mathrm{m},\mathrm{n},\mathrm{G}}\mathrm{i}\mathrm{f}\mathrm{f}\left({\mathrm{U}}_{\mathrm{m},\mathrm{n},\mathrm{G}}\right)\le \mathrm{f}\left({\mathrm{X}}_{\mathrm{m},\mathrm{n},\mathrm{G}}\right)\\ {\mathrm{X}}_{\mathrm{m},\mathrm{n},\mathrm{G}},\mathrm{else}\end{array}.$$
End doEnd doSelect the best solution from all DEs in the iteration End doShow/select the best solution from all DEs in all iterations |

#### 4.1.3. Procedure of FJSP by the Using Local Search with the Jump Search

_{2,1}→ O

_{4,1}→ O

_{4,2}→ O

_{4,3}→ O

_{2,2}→ O

_{2,3}→ O

_{2,4}→ T.

_{2,1}is checked for possible intervention by a predecessor under these circumstances and priorities with a lower processing time and compatibility with an operating machine. Accordingly, operations are checked in the following consecutive order O

_{4,1}→ O

_{4,2}→ O

_{4,3}→ O

_{2,2}→ O

_{2,3}→ O

_{2,4}. Then, the fitness value is calculated in each round until completion of a set number of iterations.

## 5. Analysis of the Results from the Experiment on DE for Solving FJSP

#### 5.1. Results of Solving the Flexible Job Shop Scheduling Problem with Sample Problems from Kacem et al.

#### 5.2. Results of Solving the Flexible Job Shop Scheduling Problem with Sample Problems of Brandimarte

#### 5.3. Results of Solving the Flexible Job Shop Scheduling Problem with Sample Problems of Dauzere-Peres and Paulli

## 6. The Results of the Comparison of the DE Algorithm with Other Metaheuristic Methods

#### 6.1. Results of Solving the Flexible Job Shop Scheduling Problem with Sample Problems of Brandimarte

#### 6.2. Results of Solving the Flexible Job Shop Scheduling Problem with Sample Problems of Dauzere-Peres and Paulli

## 7. Conclusions and Suggestions

## Author Contributions

**.**

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Gantt chart of solutions for flexible job shop scheduling problem Mk1 presented by Brandimarte [30].

Job | Operation | Machines | ||
---|---|---|---|---|

M_{1} | M_{2} | M_{3} | ||

J_{1} | O_{1,1} | 5 | - | 3 |

O_{1,2} | - | 5 | 10 | |

O_{1,3} | 5 | 9 | - | |

J_{2} | O_{2,1} | - | 10 | 7 |

O_{2,2} | 20 | 6 | - | |

O_{3,3} | 2 | - | 11 | |

J_{3} | O_{3,1} | 2 | 5 | 4 |

O_{3,3} | 2 | 5 | 10 |

45. 35. 1 2 2 5 3 4 4 1 5 2 5 1 5 2 4 3 5 4 7 5 5 5 1 4 2 5 3 5 4 4 5 5 35. 1 2 2 5 3 4 4 7 5 8 5 1 5 2 6 3 9 4 8 5 5 5 1 4 2 5 3 4 4 54 5 5 45. 1 9 2 8 3 6 4 7 5 9 5 1 6 2 1 3 2 4 5 5 4 5 1 2 2 5 3 4 4 2 5 4 5 1 4 2 5 3 2 4 1 5 5 25. 1 1 2 5 3 2 4 4 5 12 5 1 5 2 1 3 2 4 1 5 2 |

Jobs | Operations | Machines | ||||
---|---|---|---|---|---|---|

M1 | M2 | M3 | M4 | M5 | ||

J1 | O_{1,1} | 2 | 5 | 4 | 1 | 2 |

O_{1,2} | 5 | 4 | 5 | 7 | 5 | |

O_{1,3} | 4 | 5 | 5 | 4 | 5 | |

J2 | O_{2,1} | 2 | 5 | 4 | 7 | 8 |

O_{2,2} | 5 | 6 | 9 | 8 | 5 | |

O_{2,3} | 4 | 5 | 4 | 54 | 5 | |

J3 | O_{3,1} | 9 | 8 | 6 | 7 | 9 |

O_{3,2} | 6 | 1 | 2 | 5 | 4 | |

O_{3,3} | 2 | 5 | 4 | 2 | 4 | |

O_{3,4} | 4 | 5 | 2 | 1 | 5 | |

J4 | O_{4,1} | 1 | 5 | 2 | 4 | 12 |

O_{4,2} | 5 | 1 | 2 | 1 | 2 |

NP | Dimensions, D | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |

1 | 0.55 | 0.32 | 0.70 | 0.12 | 0.64 | 0.89 | 0.96 | 0.81 | 0.38 | 0.55 | 0.27 | 0.71 |

2 | 0.17 | 0.80 | 0.94 | 0.93 | 0.44 | 0.36 | 0.77 | 0.35 | 0.13 | 0.42 | 0.17 | 0.11 |

3 | 0.42 | 0.35 | 0.15 | 0.61 | 0.10 | 0.34 | 0.93 | 0.51 | 0.08 | 0.59 | 0.63 | 0.50 |

4 | 0.65 | 0.72 | 0.30 | 0.58 | 0.02 | 0.74 | 0.59 | 0.17 | 0.14 | 0.07 | 0.73 | 0.31 |

5 | 0.72 | 0.32 | 0.04 | 0.20 | 0.89 | 0.28 | 0.42 | 0.67 | 0.15 | 0.49 | 0.09 | 0.81 |

Random Vector | r1 | r2 | r3 |
---|---|---|---|

1 | 2 | 5 | 3 |

2 | 3 | 3 | 5 |

3 | 4 | 2 | 2 |

4 | 5 | 1 | 1 |

5 | 1 | 4 | 4 |

Mutation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.77 | 0.74 | 0.42 | 0.11 | 2.02 | 0.24 | −0.25 | 0.67 | 0.27 | 0.22 | −0.91 | 0.73 |

2 | 0.80 | 0.13 | 0.74 | 0.24 | 1.78 | 0.50 | −0.47 | 0.04 | 1.81 | 0.38 | −0.43 | 0.37 |

3 | 0.16 | 0.70 | 0.56 | 0.61 | 1.45 | 0.43 | −0.74 | 0.24 | 1.30 | −0.08 | 0.02 | 0.40 |

4 | 0.81 | 0.43 | 0.24 | −0.36 | 0.03 | 0.09 | 0.97 | 0.69 | −0.70 | 1.04 | 1.19 | 1.04 |

5 | 0.87 | 1.30 | 0.65 | 0.69 | 1.05 | 1.39 | 0.23 | −0.86 | 1.31 | 0.60 | 0.31 | 0.91 |

Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.83 | 0.05 | 0.75 | 0.80 | 0.95 | 0.39 | 0.29 | 0.60 | 0.30 | 0.44 | 0.59 | 0.65 |

2 | 0.68 | 0.36 | 0.48 | 0.47 | 0.70 | 0.96 | 0.04 | 0.76 | 0.64 | 0.42 | 0.16 | 0.44 |

3 | 0.32 | 0.40 | 0.97 | 0.38 | 0.63 | 0.69 | 0.71 | 0.92 | 0.65 | 0.83 | 0.92 | 0.49 |

4 | 0.56 | 0.18 | 0.06 | 0.38 | 0.47 | 0.23 | 0.11 | 0.85 | 0.80 | 0.30 | 0.65 | 0.02 |

5 | 0.81 | 0.35 | 0.70 | 0.50 | 0.89 | 0.89 | 0.84 | 0.29 | 0.01 | 0.21 | 0.41 | 0.83 |

Trial Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.55 | 0.74 | 0.42 | 0.12 | 0.64 | 0.24 | −0.25 | 0.67 | 0.27 | 0.22 | −0.91 | 0.73 |

2 | 0.80 | 0.13 | 0.74 | 0.24 | 1.78 | 0.50 | −0.47 | 0.04 | 1.81 | 0.38 | −0.43 | 0.37 |

3 | 0.16 | 0.70 | 0.56 | 0.61 | 1.45 | 0.43 | −0.74 | 0.24 | 1.30 | −0.08 | 0.02 | 0.40 |

4 | 0.65 | 0.72 | 0.30 | 0.58 | 0.02 | 0.74 | 0.59 | 0.17 | 0.14 | 0.07 | 0.73 | 0.31 |

5 | 0.72 | 0.32 | 0.04 | 0.20 | 0.89 | 0.28 | 0.42 | 0.67 | 0.15 | 0.49 | 0.09 | 0.81 |

Vector | 11 | 7 | 4 | 10 | 6 | 9 | 3 | 1 | 5 | 8 | 12 | 2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | −0.91 | −0.25 | 0.12 | 0.22 | 0.24 | 0.27 | 0.42 | 0.55 | 0.64 | 0.67 | 0.73 | 0.74 |

O_{i,j} | 1, 1 | 1, 2 | 1, 3 | 2, 1 | 2, 2 | 2, 3 | 3, 1 | 3, 2 | 3, 3 | 3, 4 | 4, 1 | 4, 2 |

Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.55 | 0.74 | 0.42 | 0.12 | 0.64 | 0.24 | −0.25 | 0.67 | 0.27 | 0.22 | −0.91 | 0.73 |

O_{i,j} | 3,2 | 4,2 | 3,1 | 1,3 | 3,3 | 2,2 | 1,2 | 3,4 | 2,3 | 2,1 | 1,1 | 4,1 |

M | 2 | 2 | 4 | 1 | 1 | 5 | 2 | 4 | 3 | 1 | 4 | 1 |

PT | 1 | 1 | 6 | 4 | 2 | 5 | 4 | 1 | 4 | 2 | 1 | 1 |

Target Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Target |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.55 | 0.32 | 0.70 | 0.12 | 0.64 | 0.89 | 0.96 | 0.81 | 0.38 | 0.55 | 0.27 | 0.71 | 21 |

2 | 0.17 | 0.80 | 0.94 | 0.93 | 0.44 | 0.36 | 0.77 | 0.35 | 0.13 | 0.42 | 0.17 | 0.11 | 20 |

3 | 0.42 | 0.35 | 0.15 | 0.61 | 0.10 | 0.34 | 0.93 | 0.51 | 0.08 | 0.59 | 0.63 | 0.50 | 22 |

4 | 0.65 | 0.72 | 0.30 | 0.58 | 0.02 | 0.74 | 0.59 | 0.17 | 0.14 | 0.07 | 0.73 | 0.31 | 24 |

5 | 0.72 | 0.32 | 0.04 | 0.20 | 0.89 | 0.28 | 0.42 | 0.67 | 0.15 | 0.49 | 0.09 | 0.81 | 19 |

Trial Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Target |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.55 | 0.74 | 0.42 | 0.12 | 0.64 | 0.24 | −0.25 | 0.67 | 0.27 | 0.22 | −0.91 | 0.73 | 18 |

2 | 0.80 | 0.13 | 0.74 | 0.24 | 1.78 | 0.50 | −0.47 | 0.04 | 1.81 | 0.38 | −0.43 | 0.37 | 25 |

3 | 0.16 | 0.70 | 0.56 | 0.61 | 1.45 | 0.43 | −0.74 | 0.24 | 1.30 | −0.08 | 0.02 | 0.40 | 16 |

4 | 0.65 | 0.72 | 0.30 | 0.58 | 0.02 | 0.74 | 0.59 | 0.17 | 0.14 | 0.07 | 0.73 | 0.31 | 17 |

5 | 0.72 | 0.32 | 0.04 | 0.20 | 0.89 | 0.28 | 0.42 | 0.67 | 0.15 | 0.49 | 0.09 | 0.81 | 26 |

Vector | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Target |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.55 | 0.74 | 0.42 | 0.12 | 0.64 | 0.24 | −0.25 | 0.67 | 0.27 | 0.22 | −0.91 | 0.73 | 18 |

2 | 0.17 | 0.80 | 0.94 | 0.93 | 0.44 | 0.36 | 0.77 | 0.35 | 0.13 | 0.42 | 0.17 | 0.11 | 20 |

3 | 0.16 | 0.70 | 0.56 | 0.61 | 1.45 | 0.43 | −0.74 | 0.24 | 1.30 | −0.08 | 0.02 | 0.40 | 16 |

4 | 0.65 | 0.72 | 0.30 | 0.58 | 0.02 | 0.74 | 0.59 | 0.17 | 0.14 | 0.07 | 0.73 | 0.31 | 17 |

5 | 0.72 | 0.32 | 0.04 | 0.20 | 0.89 | 0.28 | 0.42 | 0.67 | 0.15 | 0.49 | 0.09 | 0.81 | 19 |

Problem | BKS | Mutation Strategy | |||
---|---|---|---|---|---|

DE * | DE ** | DE *** | DE **** | ||

K01 | 11 | 12 (9.09) | 12 (9.09) | 11 (0.00) | 11 (0.00) |

K02 | 14 | 15 (7.14) | 15 (7.14) | 15 (7.14) | 15 (7.14) |

K03 | 11 | 11 (0.00) | 11 (0.00) | 11 (0.00) | 11 (0.00) |

K04 | 7 | 7 (0.00) | 7 (0.00) | 7 (0.00) | 7 (0.00) |

K05 | 11 | 12 (9.09) | 12 (9.09) | 12 (9.09) | 12 (9.09) |

MRE | 5.06 | 5.06 | 3.25 | 3.25 |

**Table 15.**Summary of solving the flexible job shop scheduling problem with sample problems Brandimarte [30].

Problem | BKS | Mutation Strategy | |||
---|---|---|---|---|---|

DE * | DE ** | DE *** | DE **** | ||

Mk1 | 40 | 43 (7.50) | 43 (7.50) | 40 (0.00) | 40 (0.00) |

Mk2 | 27 | 28 (7.69) | 28 (7.69) | 28 (7.69) | 28 (7.69) |

Mk3 | 204 | 204 (0.00) | 204 (0.00) | 204 (0.00) | 204 (0.00) |

Mk4 | 60 | 71 (18.33) | 71 (18.33) | 71 (18.33) | 71 (18.33) |

Mk5 | 174 | 178 (2.30) | 178 (2.30) | 179 (2.87) | 179 (2.87) |

Mk6 | 59 | 73 (23.73) | 73 (23.73) | 73 (23.73) | 73 (23.73) |

Mk7 | 143 | 149 (4.20) | 149 (4.20) | 148 (3.50) | 146 (2.10) |

Mk8 | 523 | 528 (0.96) | 528 (0.96) | 528 (0.96) | 528 (0.96) |

Mk9 | 307 | 324 (5.54) | 321 (4.56) | 323 (5.21) | 321 (4.56) |

Mk10 | 212 | 234 (10.38) | 233 (9.90) | 236 (11.32) | 235 (10.85) |

MRE | 8.06 | 7.92 | 7.36 | 7.11 |

**Table 16.**Summary of solving the flexible job shop scheduling problem with sample problems Dauzere-Peres and Paulli.

Problem | BKS | Mutation Strategy | |||
---|---|---|---|---|---|

DE * | DE ** | DE *** | DE **** | ||

01a | 2530 | 2895 (14.42) | 2750 (8.70) | 2615 (3.36) | 2645 (4.55) |

04a | 2555 | 2859 (11.90) | 2770 (8.41) | 2650 (3.72) | 2610 (2.15) |

07a | 2396 | 2759 (15.15) | 2650 (10.60) | 2650 (10.60) | 2510 (4.76) |

09a | 2074 | 2281 (9.98) | 2269 (9.40) | 2210 (6.56) | 2150 (3.66) |

11a | 2078 | 2378 (14.44) | 2366 (13.86) | 2221 (6.88) | 2200 (5.87) |

MRE | 13.18 | 10.19 | 6.22 | 4.20 |

**Table 17.**Summary of comparing the differential evolution algorithm with other metaheuristic algorithms.

Problem | n × m × k * | BKS ** | Chen et al. (GA) [32] | Girish and Jawahar (PSO) [33] | DE-FJSP |
---|---|---|---|---|---|

C_{max} | C_{max} | C_{max} | C_{max} | ||

Mk01 | 10 × 6 × 55 | 40 | 40 (0.00) | 40 (0.00) | 40 (0.00) |

Mk02 | 10 × 6 × 58 | 27 | 29 (6.89) | 27 (0.00) | 28 (7.69) |

Mk03 | 15 × 8 × 150 | 204 | 204 (0.00) | 204 (0.00) | 204 (0.00) |

Mk04 | 15 × 8 × 90 | 60 | 63 (4.76) | 62 (3.22) | 71 (18.33) |

Mk05 | 15 × 4 × 106 | 174 | 181 (3.86) | 178 (2.24) | 179 (2.87) |

Mk06 | 10 × 15 × 150 | 59 | 60 (1.66) | 78 (24.35) | 73 (23.73) |

Mk07 | 20 × 5 × 100 | 143 | 148 (3.38) | 147 (2.72) | 146 (2.10) |

Mk08 | 20 × 10 × 225 | 523 | 523 (0.00) | 523 (0.00) | 528 (0.96) |

Mk09 | 20 × 10 × 240 | 307 | 308 (0.32) | 341 (9.97) | 321 (4.56) |

Mk10 | 20 × 15 × 240 | 212 | 212 (0.00) | 252 (15.07) | 235 (10.85) |

MRE | 2.08 | 7.75 | 7.11 |

**Table 18.**Summary of comparing the differential evolution algorithm with other metaheuristic algorithms.

Problem | n × m × k * | BKS ** | Wisittipanich (1ST-DE) [12] | DE-FJSP |
---|---|---|---|---|

C_{max} | C_{max} | C_{max} | ||

01a | 10 × 5 × 196 | 2530 | 2645 (4.55) | 2645 (4.55) |

04a | 10 × 5 × 196 | 2555 | 2616 (2.39) | 2610 (2.15) |

07a | 15 × 8 × 293 | 2396 | 2582 (7.76) | 2510 (4.76) |

09a | 15 × 8 × 293 | 2074 | 2153 (3.81) | 2150 (3.66) |

11a | 15 × 8 × 293 | 2078 | 2221 (6.88) | 2200 (5.87) |

MRE | 5.08 | 4.20 |

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**MDPI and ACS Style**

Sriboonchandr, P.; Kriengkorakot, N.; Kriengkorakot, P. Improved Differential Evolution Algorithm for Flexible Job Shop Scheduling Problems. *Math. Comput. Appl.* **2019**, *24*, 80.
https://doi.org/10.3390/mca24030080

**AMA Style**

Sriboonchandr P, Kriengkorakot N, Kriengkorakot P. Improved Differential Evolution Algorithm for Flexible Job Shop Scheduling Problems. *Mathematical and Computational Applications*. 2019; 24(3):80.
https://doi.org/10.3390/mca24030080

**Chicago/Turabian Style**

Sriboonchandr, Prasert, Nuchsara Kriengkorakot, and Preecha Kriengkorakot. 2019. "Improved Differential Evolution Algorithm for Flexible Job Shop Scheduling Problems" *Mathematical and Computational Applications* 24, no. 3: 80.
https://doi.org/10.3390/mca24030080