# A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling

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## Abstract

**:**

## 1. Introduction

- Develop GNDO using the LRV rule for PFSSP.
- Improve GNDO using the swap mutation operator to avoid being stuck into local minima.
- Enhance the local search strategy using the scramble mutation operator for accelerating the convergence speed toward the near-optimal solution.
- Integrate the improved local search strategy and the standard one with the improved GNDO and GNDO for tackling the PFSSP.
- The experimental findings show that IHGNDO and HIGNDO are better in terms of standard deviation and computational cost and final accuracy.

## 2. Description of the Permutation Flow Shop Scheduling Problem

## 3. Standard Algorithm: Generalized Normal Distribution Optimization

#### 3.1. Exploitation Operator

#### 3.2. Exploration Operator

## 4. The Proposed Work

#### 4.1. Initialization

#### 4.2. Swap Mutation Operator

#### 4.3. Scramble Mutation Operator

#### 4.4. Improved Local Search Strategy (ILSS)

Algorithm 1 Improved LSS (ILSS). |

Input: ${X}^{*}$**For**I = 1: n- $X={X}^{*}$
**For**j = 1: n- $r:$ create a random number between 0 and 1.
**If**(r < LSP)- ${X}_{j}={X}_{i}^{*}$
- Applying scramble mutation operator on $X$
- Calculate the fitness of $X$.
- Update ${X}^{*}$ if $X$ is better.
**End if****End for****End for**
Return ${X}^{*}$ |

Algorithm 2 HIGNDO. |

Input: N, ${t}_{max}$- $t=0$
- Initialization phase.
**While**$t<{t}_{max}$**For**$i=1:N$- Create a number $\alpha $ randomly between [0, 1].
- Create a number ${\alpha}_{1}$ randomly between [0, 1].
**If**$\alpha >{\alpha}_{1}$- Calculate
**M**using Equation (8) - Calculate ${\mu}_{i},{\delta}_{i},and\eta $
- Calculate ${T}_{i}{}^{t}$ using Equation (6).
**If**$f({T}_{i}{}^{t})<f\left({X}_{i}{}^{t}\right)$- ${X}_{i}{}^{t}={T}_{i}{}^{t}$
**End If****Else**- Compute ${T}_{i}{}^{t}$ according to Equation (11).
**If**$f({T}_{i}{}^{t})<f\left({X}_{i}{}^{t}\right)$- ${X}_{i}{}^{t}={T}_{i}{}^{t}$
**End If****End If****For**$j=1:0.1*n$- $T$: Applying the swap mutation on the best-so-far solution.
**If**$f\left(T\right)<f\left({X}^{*}\right)$- ${X}^{*}=T$
**End If****End for****Applying algorithm 1 without Line 7.****End For**- $t++$;
**End while**
Output: return ${X}^{*}$ |

## 5. Results and Comparisons

#### 5.1. Comparison under Carlier

#### 5.2. Comparison under Reeves

#### 5.3. Comparison under Heller

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Position, Job | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

Position, ${T}_{i}{}^{t}$ | 0.1 | 0.5 | 0.8 | 0.2 | 0.6 | 0.7 | 0.9 |

Job, $T{T}_{i}{}^{t}$ | 6 | 4 | 1 | 5 | 3 | 2 | 0 |

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

Name | n | m | ${\mathit{Z}}^{*}$ | Name | n | m | ${\mathit{Z}}^{*}$ | Name | N | m | ${\mathit{Z}}^{*}$ | Name | n | m | ${\mathit{Z}}^{*}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Hel1 | 20 | 10 | 516 | Car07 | 7 | 7 | 6590 | Rec13 | 20 | 15 | 1930 | Rec29 | 23 | 15 | 2287 |

Hel2 | 100 | 10 | 136 | Car08 | 8 | 8 | 8366 | Rec15 | 20 | 15 | 1950 | Rec31 | 50 | 10 | 3045 |

Car01 | 11 | 5 | 7038 | Rec01 | 20 | 5 | 1247 | Rec17 | 20 | 15 | 1902 | Rec33 | 50 | 10 | 3114 |

Car02 | 13 | 4 | 7166 | Rec03 | 20 | 5 | 1109 | Rec19 | 30 | 10 | 2017 | Rec35 | 50 | 10 | 3277 |

Car03 | 12 | 5 | 7312 | Rec05 | 20 | 5 | 1242 | Rec21 | 30 | 10 | 2011 | Rec37 | 75 | 20 | 4951 |

Car04 | 14 | 4 | 8003 | Rec07 | 20 | 10 | 1566 | Rec23 | 30 | 10 | 2011 | Rec39 | 75 | 20 | 5087 |

Car05 | 10 | 6 | 7720 | Rec09 | 20 | 10 | 1537 | Rec25 | 30 | 15 | 2513 | Rec41 | 75 | 20 | 4960 |

Car06 | 8 | 9 | 8505 | Rec011 | 20 | 10 | 1431 | Rec27 | 30 | 15 | 2373 |

Instances | Algorithm | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Car01 | IHGNDO | 7038 | 0.0000 | 0.0000 | 0.0000 | 7038.0000 | 0.0077 | 0.0000 | Car04 | 7720 | 0.0000 | 0.0000 | 0.0000 | 7720.0000 | 0.0558 | 0.0000 |

HIGNDO | 0.0000 | 0.0000 | 0.0000 | 7038.0000 | 0.0078 | 0.0000 | 0.0000 | 0.0131 | 0.0014 | 7731.1667 | 0.0636 | 30.1575 | ||||

HGNDO | 0.0000 | 0.0000 | 0.0000 | 7038.0000 | 0.0079 | 0.0000 | 0.0000 | 0.0131 | 0.0028 | 7741.9000 | 0.1949 | 36.3129 | ||||

HMPA | 0.0000 | 0.0000 | 0.0000 | 7038.0000 | 0.0192 | 0.0000 | 0.0000 | 0.0039 | 0.0011 | 7728.4000 | 1.0569 | 10.1114 | ||||

HWOA | 0.0000 | 0.0000 | 0.0000 | 7038.0000 | 0.0052 | 0.0000 | 0.0000 | 0.0039 | 0.0015 | 7731.4333 | 0.1961 | 11.6152 | ||||

HEO | 0.0000 | 0.0195 | 0.0011 | 7045.9000 | 0.0091 | 29.9426 | 0.0000 | 0.0135 | 0.0048 | 7756.7000 | 0.2393 | 37.5501 | ||||

HSCA | 0.0000 | 0.0456 | 0.0043 | 7068.2667 | 0.0830 | 76.6159 | 0.0000 | 0.0486 | 0.0076 | 7778.6667 | 0.4299 | 104.3537 | ||||

HSSA | 0.0000 | 0.0617 | 0.0048 | 7072.1000 | 0.0079 | 107.6947 | 0.0000 | 0.0486 | 0.0078 | 7780.0000 | 0.0668 | 72.7736 | ||||

HTSA | 0.0000 | 0.0997 | 0.0395 | 7315.6667 | 0.3338 | 286.3707 | 0.0000 | 0.1124 | 0.0334 | 7977.8333 | 0.6940 | 255.2532 | ||||

HGA | 0.0000 | 0.0169 | 0.0018 | 7050.6000 | 0.0105 | 29.1692 | 0.0000 | 0.0153 | 0.0099 | 7796.1000 | 0.6052 | 40.0161 | ||||

Car02 | IHGNDO | 7166 | 0.0000 | 0.0000 | 0.0000 | 7166.0000 | 0.0200 | 0.0000 | Car05 | 8505 | 0.0000 | 0.0000 | 0.0000 | 8505.0000 | 0.0134 | 0.0000 |

HIGNDO | 0.0000 | 0.0000 | 0.0000 | 7166.0000 | 0.0151 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 8505.0000 | 0.0192 | 0.0000 | ||||

HGNDO | 0.0000 | 0.0000 | 0.0000 | 7166.0000 | 0.0152 | 0.0000 | 0.0000 | 0.0076 | 0.0008 | 8511.5000 | 0.0713 | 19.5000 | ||||

HMPA | 0.0000 | 0.0293 | 0.0010 | 7173.0000 | 0.3175 | 37.6962 | 0.0000 | 0.0540 | 0.0070 | 8564.4333 | 0.5601 | 101.8279 | ||||

HWOA | 0.0000 | 0.0000 | 0.0000 | 7166.0000 | 0.0224 | 0.0000 | 0.0000 | 0.0076 | 0.0003 | 8507.1667 | 0.0552 | 11.6679 | ||||

HEO | 0.0000 | 0.0293 | 0.0078 | 7222.0000 | 0.0980 | 92.8655 | 0.0000 | 0.0396 | 0.0076 | 8569.7000 | 0.1632 | 78.3812 | ||||

HSCA | 0.0000 | 0.1136 | 0.0347 | 7414.6000 | 0.2283 | 344.7938 | 0.0000 | 0.0770 | 0.0223 | 8694.8667 | 0.5217 | 190.9799 | ||||

HSSA | 0.0000 | 0.1231 | 0.0183 | 7297.4333 | 0.0313 | 262.8529 | 0.0000 | 0.0366 | 0.0109 | 8597.9667 | 0.0492 | 95.4919 | ||||

HTSA | 0.0000 | 0.1749 | 0.0788 | 7730.5333 | 0.4812 | 420.7919 | 0.0000 | 0.1250 | 0.0461 | 8897.4000 | 0.8095 | 345.5301 | ||||

HGA | 0.0000 | 0.0293 | 0.0063 | 7211.1000 | 0.1966 | 84.1088 | 0.0000 | 0.0582 | 0.0084 | 8576.1667 | 0.4249 | 115.9747 | ||||

Car03 | IHGNDO | 7312 | 0.0000 | 0.0000 | 0.0000 | 7312.0000 | 0.0953 | 0.0000 | Car06 | 6590 | 0.0000 | 0.0000 | 0.0000 | 6590.0000 | 0.0067 | 0.0000 |

HIGNDO | 0.0000 | 0.0000 | 0.0000 | 7312.0000 | 0.0455 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 6590.0000 | 0.0051 | 0.0000 | ||||

HGNDO | 0.0000 | 0.0074 | 0.0027 | 7331.8000 | 0.2018 | 26.0223 | 0.0000 | 0.0000 | 0.0000 | 6590.0000 | 0.0067 | 0.0000 | ||||

HMPA | 0.0000 | 0.0254 | 0.0073 | 7365.2000 | 1.3745 | 42.6375 | 0.0000 | 0.0478 | 0.0089 | 6648.5333 | 0.1175 | 81.3858 | ||||

HWOA | 0.0000 | 0.0074 | 0.0042 | 7342.6000 | 0.2496 | 26.7589 | 0.0000 | 0.0000 | 0.0000 | 6590.0000 | 0.0206 | 0.0000 | ||||

HEO | 0.0000 | 0.0150 | 0.0090 | 7378.0667 | 0.2009 | 37.8919 | 0.0000 | 0.0347 | 0.0067 | 6634.3333 | 0.0380 | 63.7377 | ||||

HSCA | 0.0000 | 0.1002 | 0.0146 | 7418.7333 | 0.4578 | 174.9874 | 0.0000 | 0.0347 | 0.0125 | 6672.4667 | 0.4517 | 77.5735 | ||||

HSSA | 0.0000 | 0.1265 | 0.0180 | 7443.3000 | 0.0630 | 184.0828 | 0.0000 | 0.0247 | 0.0062 | 6631.1667 | 0.0329 | 53.6452 | ||||

HTSA | 0.0000 | 0.1570 | 0.0631 | 7773.0667 | 0.6972 | 401.4574 | 0.0000 | 0.0900 | 0.0313 | 6796.0333 | 0.7878 | 180.7279 | ||||

HGA | 0.0000 | 0.0150 | 0.0084 | 7373.6667 | 0.6146 | 38.4598 | 0.0000 | 0.0437 | 0.0098 | 6654.2667 | 0.6137 | 67.0020 | ||||

Car04 | IHGNDO | 8003 | 0.0000 | 0.0000 | 0.0000 | 8003.0000 | 0.0139 | 0.0000 | Car07 | 8366 | 0.0000 | 0.0000 | 0.0000 | 8366.0000 | 0.0111 | 0.0000 |

HIGNDO | 0.0000 | 0.0000 | 0.0000 | 8003.0000 | 0.0082 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 8366.0000 | 0.0063 | 0.0000 | ||||

HGNDO | 0.0000 | 0.0000 | 0.0000 | 8003.0000 | 0.0146 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 8366.0000 | 0.0070 | 0.0000 | ||||

HMPA | 0.0000 | 0.0014 | 0.0000 | 8003.3667 | 0.1111 | 1.9746 | 0.0000 | 0.0225 | 0.0009 | 8373.7000 | 0.1041 | 34.3581 | ||||

HWOA | 0.0000 | 0.0000 | 0.0000 | 8003.0000 | 0.0205 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 8366.0000 | 0.0085 | 0.0000 | ||||

HEO | 0.0000 | 0.0112 | 0.0004 | 8006.0000 | 0.0479 | 16.1555 | 0.0000 | 0.0135 | 0.0008 | 8372.8000 | 0.0484 | 25.6013 | ||||

HSCA | 0.0000 | 0.0659 | 0.0068 | 8057.3667 | 0.1228 | 151.0125 | 0.0000 | 0.0634 | 0.0092 | 8443.0333 | 0.2099 | 146.8188 | ||||

HSSA | 0.0000 | 0.0947 | 0.0115 | 8095.0000 | 0.0291 | 212.0660 | 0.0000 | 0.0000 | 0.0000 | 8366.0000 | 0.0167 | 0.0000 | ||||

HTSA | 0.0000 | 0.1369 | 0.0485 | 8390.7667 | 0.4935 | 366.4514 | 0.0000 | 0.0865 | 0.0233 | 8560.8333 | 0.4741 | 228.3998 | ||||

HGA | 0.0000 | 0.0000 | 0.0000 | 8003.0000 | 0.1037 | 0.0000 | 0.0000 | 0.0069 | 0.0008 | 8372.9667 | 0.1663 | 17.8783 |

Inst | Algorithm | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD | Inst | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

reC01 | IHGNDO | 1247 | 0.0000 | 0.0016 | 0.0016 | 1248.9333 | 0.6132 | 0.3590 | reC13 | 1930 | 0.0031 | 0.0249 | 0.0116 | 1952.3333 | 0.8099 | 12.3216 |

HIGNDO | 0.0000 | 0.0032 | 0.0015 | 1248.8667 | 0.6007 | 0.7180 | 0.0026 | 0.0212 | 0.0122 | 1953.5000 | 0.8224 | 10.1415 | ||||

HGNDO | 0.0000 | 0.0144 | 0.0026 | 1250.2667 | 0.9136 | 3.3559 | 0.0052 | 0.0430 | 0.0196 | 1967.8000 | 0.9858 | 17.7696 | ||||

HMPA | 0.0016 | 0.0265 | 0.0065 | 1255.1000 | 2.4415 | 8.9976 | 0.0093 | 0.0425 | 0.0191 | 1966.7667 | 2.6874 | 14.5709 | ||||

HWOA | 0.0016 | 0.0465 | 0.0047 | 1252.8333 | 1.1080 | 10.3765 | 0.0026 | 0.0415 | 0.0166 | 1962.0000 | 1.4368 | 17.8419 | ||||

HEO | 0.0016 | 0.0634 | 0.0133 | 1263.6333 | 0.3845 | 19.3503 | 0.0067 | 0.0974 | 0.0307 | 1989.1667 | 0.3930 | 31.0656 | ||||

HSCA | 0.0000 | 0.1291 | 0.0112 | 1260.9667 | 0.8572 | 35.4847 | 0.0016 | 0.1528 | 0.0450 | 2016.8000 | 0.9266 | 93.9260 | ||||

HSSA | 0.0016 | 0.1588 | 0.0401 | 1296.9667 | 0.1210 | 63.0611 | 0.0083 | 0.0383 | 0.0232 | 1974.7667 | 0.1342 | 15.1738 | ||||

HTSA | 0.0016 | 0.1764 | 0.0640 | 1326.8000 | 1.0840 | 89.2007 | 0.0026 | 0.1741 | 0.0594 | 2044.6333 | 1.1585 | 125.2979 | ||||

HGA | 0.0016 | 0.0634 | 0.0117 | 1261.5333 | 0.9869 | 19.5699 | 0.0088 | 0.0440 | 0.0231 | 1974.5667 | 1.2510 | 18.9520 | ||||

reC03 | IHGNDO | 1109 | 0.0000 | 0.0018 | 0.0011 | 1110.2000 | 0.4757 | 0.9798 | reC15 | 1950 | 0.0056 | 0.0200 | 0.0118 | 1973.0667 | 0.8296 | 6.4028 |

HIGNDO | 0.0000 | 0.0027 | 0.0013 | 1110.4667 | 0.5128 | 1.0873 | 0.0067 | 0.0308 | 0.0125 | 1974.4667 | 0.8251 | 9.7151 | ||||

HGNDO | 0.0000 | 0.0036 | 0.0013 | 1110.4000 | 0.7720 | 1.1431 | 0.0026 | 0.0400 | 0.0172 | 1983.6000 | 0.9996 | 18.6683 | ||||

HMPA | 0.0000 | 0.0216 | 0.0035 | 1112.8333 | 2.4883 | 5.8085 | 0.0082 | 0.0426 | 0.0230 | 1994.9333 | 2.6970 | 23.7079 | ||||

HWOA | 0.0000 | 0.0090 | 0.0013 | 1110.4333 | 0.7880 | 1.9093 | 0.0036 | 0.0426 | 0.0195 | 1988.1000 | 1.4496 | 21.4357 | ||||

HEO | 0.0000 | 0.0911 | 0.0124 | 1122.7000 | 0.3519 | 19.8899 | 0.0118 | 0.0923 | 0.0298 | 2008.1667 | 0.3997 | 32.4860 | ||||

HSCA | 0.0000 | 0.1623 | 0.0361 | 1149.0667 | 0.8191 | 57.8653 | 0.0051 | 0.1369 | 0.0295 | 2007.5000 | 0.9629 | 49.8108 | ||||

HSSA | 0.0018 | 0.1587 | 0.0314 | 1143.8000 | 0.1230 | 55.4403 | 0.0108 | 0.1246 | 0.0314 | 2011.3000 | 0.1352 | 40.7015 | ||||

HTSA | 0.0000 | 0.1659 | 0.0768 | 1194.2000 | 1.0001 | 67.7655 | 0.0056 | 0.1441 | 0.0634 | 2073.6667 | 1.1775 | 103.7662 | ||||

HGA | 0.0000 | 0.0379 | 0.0091 | 1119.1000 | 0.8489 | 11.2141 | 0.0082 | 0.0508 | 0.0266 | 2001.8000 | 1.2887 | 25.2645 | ||||

reC05 | IHGNDO | 1242 | 0.0024 | 0.0024 | 0.0024 | 1245.0000 | 0.6343 | 1.9746 | reC17 | 1902 | 0.0000 | 0.0484 | 0.0225 | 1944.7000 | 0.8176 | 17.6921 |

HIGNDO | 0.0024 | 0.0113 | 0.0027 | 1245.3667 | 0.6424 | 1.9746 | 0.0000 | 0.0389 | 0.0245 | 1948.5333 | 0.8119 | 15.7623 | ||||

HGNDO | 0.0024 | 0.0113 | 0.0044 | 1247.5000 | 0.8638 | 3.8536 | 0.0079 | 0.0705 | 0.0319 | 1962.7000 | 1.0376 | 25.6621 | ||||

HMPA | 0.0024 | 0.0217 | 0.0060 | 1249.4000 | 2.2647 | 6.5605 | 0.0105 | 0.0715 | 0.0337 | 1966.0667 | 2.9044 | 26.5806 | ||||

HWOA | 0.0024 | 0.0113 | 0.0063 | 1249.8333 | 0.9428 | 4.3134 | 0.0000 | 0.0436 | 0.0302 | 1959.4333 | 1.3958 | 16.1734 | ||||

HEO | 0.0024 | 0.0217 | 0.0102 | 1254.7000 | 0.3134 | 8.7527 | 0.0131 | 0.0615 | 0.0364 | 1971.2000 | 0.3743 | 20.4163 | ||||

HSCA | 0.0024 | 0.0902 | 0.0178 | 1264.1333 | 0.7352 | 33.1831 | 0.0047 | 0.1456 | 0.0397 | 1977.4333 | 0.8843 | 57.9555 | ||||

HSSA | 0.0024 | 0.1272 | 0.0241 | 1271.9000 | 0.1013 | 45.6350 | 0.0110 | 0.0589 | 0.0341 | 1966.9000 | 0.1288 | 22.1696 | ||||

HTSA | 0.0024 | 0.1449 | 0.0401 | 1291.8000 | 0.9273 | 54.0724 | 0.0131 | 0.1887 | 0.0838 | 2061.4667 | 1.0997 | 108.3872 | ||||

HGA | 0.0024 | 0.0250 | 0.0061 | 1249.6000 | 0.8465 | 7.5745 | 0.0179 | 0.0657 | 0.0369 | 1972.1333 | 1.1925 | 22.5961 | ||||

reC07 | IHGNDO | 1566 | 0.0000 | 0.0115 | 0.0070 | 1576.9333 | 0.6078 | 8.4929 | reC19 | 2017 | 0.0436 | 0.0645 | 0.0514 | 2120.7000 | 1.6120 | 9.2273 |

HIGNDO | 0.0000 | 0.0115 | 0.0039 | 1572.0667 | 0.5225 | 8.4456 | 0.0407 | 0.0649 | 0.0515 | 2120.8000 | 1.5874 | 10.9891 | ||||

HGNDO | 0.0000 | 0.0115 | 0.0053 | 1574.3667 | 0.6376 | 8.7349 | 0.0446 | 0.0709 | 0.0516 | 2121.0667 | 1.9258 | 11.9022 | ||||

HMPA | 0.0000 | 0.0383 | 0.0112 | 1583.4667 | 2.3519 | 10.6356 | 0.0471 | 0.1715 | 0.0613 | 2140.7000 | 3.5589 | 43.3083 | ||||

HWOA | 0.0000 | 0.0115 | 0.0053 | 1574.2333 | 0.9048 | 8.4565 | 0.0436 | 0.0704 | 0.0538 | 2125.4333 | 2.4811 | 14.5113 | ||||

HEO | 0.0013 | 0.0383 | 0.0160 | 1591.1000 | 0.3607 | 15.4561 | 0.0471 | 0.0788 | 0.0655 | 2149.0333 | 0.5302 | 14.6708 | ||||

HSCA | 0.0000 | 0.1277 | 0.0272 | 1608.5333 | 0.8516 | 60.5286 | 0.0456 | 0.2152 | 0.0820 | 2182.3000 | 1.2905 | 112.9936 | ||||

HSSA | 0.0000 | 0.0383 | 0.0153 | 1590.0000 | 0.1187 | 13.4313 | 0.0545 | 0.2181 | 0.0767 | 2171.7000 | 0.2010 | 77.7321 | ||||

HTSA | 0.0000 | 0.1750 | 0.0684 | 1673.1000 | 1.1179 | 97.7628 | 0.0491 | 0.2583 | 0.1413 | 2302.0667 | 1.5909 | 159.0463 | ||||

HGA | 0.0000 | 0.0230 | 0.0110 | 1583.3000 | 1.1020 | 7.7981 | 0.0530 | 0.0912 | 0.0680 | 2154.2333 | 1.5769 | 19.2036 | ||||

reC09 | IHGNDO | 1537 | 0.0000 | 0.0241 | 0.0068 | 1547.4000 | 0.5749 | 11.7774 | reC21 | 2011 | 0.0174 | 0.0224 | 0.0189 | 2049.0000 | 1.6123 | 2.2361 |

HIGNDO | 0.0000 | 0.0325 | 0.0065 | 1547.0667 | 0.5424 | 13.5153 | 0.0174 | 0.0194 | 0.0187 | 2048.5333 | 1.5865 | 1.9276 | ||||

HGNDO | 0.0000 | 0.0390 | 0.0085 | 1550.1000 | 0.7766 | 14.3256 | 0.0174 | 0.0214 | 0.0185 | 2048.1333 | 2.0050 | 2.2470 | ||||

HMPA | 0.0000 | 0.0410 | 0.0201 | 1567.9000 | 2.4736 | 16.1211 | 0.0174 | 0.0254 | 0.0192 | 2049.7000 | 3.7467 | 2.7221 | ||||

HWOA | 0.0000 | 0.0416 | 0.0176 | 1564.0000 | 1.1298 | 16.4033 | 0.0104 | 0.0194 | 0.0186 | 2048.3333 | 2.5842 | 3.5056 | ||||

HEO | 0.0085 | 0.0885 | 0.0251 | 1575.5667 | 0.3463 | 21.1624 | 0.0174 | 0.0363 | 0.0238 | 2058.7667 | 0.5330 | 10.2524 | ||||

HSCA | 0.0065 | 0.1516 | 0.0387 | 1596.4333 | 0.8261 | 66.8265 | 0.0174 | 0.1914 | 0.0344 | 2080.1667 | 1.2679 | 92.3263 | ||||

HSSA | 0.0072 | 0.1314 | 0.0308 | 1584.2667 | 0.1162 | 40.1355 | 0.0194 | 0.1875 | 0.0395 | 2090.3667 | 0.2040 | 91.4033 | ||||

HTSA | 0.0000 | 0.1913 | 0.0501 | 1614.0333 | 1.0061 | 89.8719 | 0.0174 | 0.1994 | 0.0864 | 2184.6667 | 1.5538 | 156.4899 | ||||

HGA | 0.0007 | 0.0416 | 0.0222 | 1571.1667 | 1.0296 | 15.5844 | 0.0174 | 0.0537 | 0.0266 | 2064.5333 | 1.5350 | 17.1692 | ||||

reC11 | IHGNDO | 1431 | 0.0000 | 0.0210 | 0.0070 | 1441.0000 | 0.6339 | 8.4735 | reC23 | 2011 | 0.0050 | 0.0234 | 0.0120 | 2035.0333 | 1.6254 | 12.0706 |

HIGNDO | 0.0000 | 0.0356 | 0.0091 | 1444.0667 | 0.5742 | 13.0024 | 0.0045 | 0.0338 | 0.0131 | 2037.2667 | 1.5891 | 14.7827 | ||||

HGNDO | 0.0000 | 0.0314 | 0.0153 | 1452.8333 | 0.8737 | 11.5126 | 0.0050 | 0.0264 | 0.0141 | 2039.4333 | 1.8821 | 14.7912 | ||||

HMPA | 0.0000 | 0.0894 | 0.0175 | 1456.1000 | 2.1720 | 23.8039 | 0.0060 | 0.0363 | 0.0220 | 2055.2333 | 3.4993 | 14.7098 | ||||

HWOA | 0.0000 | 0.0594 | 0.0176 | 1456.2333 | 1.1155 | 18.7664 | 0.0035 | 0.0318 | 0.0165 | 2044.1333 | 2.4370 | 16.3477 | ||||

HEO | 0.0049 | 0.0587 | 0.0240 | 1465.3000 | 0.3542 | 21.4043 | 0.0154 | 0.0467 | 0.0298 | 2070.9667 | 0.5241 | 16.7381 | ||||

HSCA | 0.0000 | 0.1600 | 0.0378 | 1485.0667 | 0.7505 | 72.6443 | 0.0050 | 0.1611 | 0.0278 | 2066.8667 | 1.2726 | 70.7327 | ||||

HSSA | 0.0000 | 0.1593 | 0.0273 | 1470.0333 | 0.1148 | 49.8233 | 0.0104 | 0.1785 | 0.0418 | 2095.1333 | 0.1994 | 77.4277 | ||||

HTSA | 0.0000 | 0.1824 | 0.0733 | 1535.9333 | 0.9973 | 108.4789 | 0.0080 | 0.2004 | 0.0755 | 2162.8667 | 1.5456 | 149.6422 | ||||

HGA | 0.0000 | 0.0496 | 0.0249 | 1466.6000 | 0.9958 | 18.7183 | 0.0050 | 0.0383 | 0.0266 | 2064.5667 | 1.5332 | 16.6046 |

Inst | Algorithm | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD | Inst | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

reC25 | IHGNDO | 2513 | 0.0092 | 0.0342 | 0.0220 | 2568.2667 | 1.8688 | 16.9232 | reC35 | 3277 | 0.0000 | 0.0000 | 0.0000 | 3277.0000 | 1.1526 | 0.0000 |

HIGNDO | 0.0131 | 0.0390 | 0.0259 | 2577.9667 | 1.8352 | 17.1065 | 0.0000 | 0.0000 | 0.0000 | 3277.0000 | 0.9999 | 0.0000 | ||||

HGNDO | 0.0123 | 0.0390 | 0.0240 | 2573.2667 | 2.0970 | 19.5276 | 0.0000 | 0.0000 | 0.0000 | 3277.0000 | 0.6110 | 0.0000 | ||||

HMPA | 0.0139 | 0.0493 | 0.0319 | 2593.2000 | 3.8974 | 22.4179 | 0.0000 | 0.0034 | 0.0005 | 3278.7667 | 3.6242 | 3.7299 | ||||

HWOA | 0.0064 | 0.0458 | 0.0270 | 2580.8667 | 3.0302 | 21.6791 | 0.0000 | 0.0000 | 0.0000 | 3277.0000 | 1.2182 | 0.0000 | ||||

HEO | 0.0163 | 0.0505 | 0.0373 | 2606.6333 | 0.5912 | 21.3690 | 0.0000 | 0.0275 | 0.0026 | 3285.6333 | 0.9912 | 16.8750 | ||||

HSCA | 0.0107 | 0.1675 | 0.0493 | 2637.0000 | 1.4448 | 118.6139 | 0.0000 | 0.1202 | 0.0047 | 3292.5333 | 1.6490 | 70.4134 | ||||

HSSA | 0.0111 | 0.0517 | 0.0362 | 2604.0667 | 0.2289 | 23.8005 | 0.0000 | 0.1428 | 0.0148 | 3325.4000 | 0.4445 | 125.6247 | ||||

HTSA | 0.0147 | 0.1823 | 0.0729 | 2696.1000 | 1.7504 | 155.5443 | 0.0000 | 0.1385 | 0.0607 | 3476.0333 | 2.6465 | 199.8649 | ||||

HGA | 0.0171 | 0.0505 | 0.0348 | 2600.3333 | 1.8883 | 19.7017 | 0.0000 | 0.0284 | 0.0029 | 3286.6667 | 2.6794 | 16.4100 | ||||

reC27 | IHGNDO | 2373 | 0.0088 | 0.0388 | 0.0220 | 2425.1667 | 1.8698 | 17.2937 | reC37 | 4951 | 0.0297 | 0.0562 | 0.0448 | 5172.7667 | 17.3840 | 31.6698 |

HIGNDO | 0.0097 | 0.0367 | 0.0191 | 2418.3333 | 1.8362 | 18.8279 | 0.0250 | 0.0578 | 0.0410 | 5154.2333 | 17.8260 | 34.6763 | ||||

HGNDO | 0.0105 | 0.0641 | 0.0222 | 2425.6000 | 2.1352 | 28.8335 | 0.0218 | 0.0523 | 0.0384 | 5141.0000 | 20.0077 | 41.1671 | ||||

HMPA | 0.0093 | 0.0426 | 0.0226 | 2426.5333 | 3.9037 | 17.9327 | 0.0382 | 0.0673 | 0.0495 | 5196.1333 | 13.1585 | 34.4710 | ||||

HWOA | 0.0088 | 0.0396 | 0.0197 | 2419.6667 | 3.0528 | 19.2238 | 0.0315 | 0.0529 | 0.0426 | 5161.7333 | 27.3170 | 31.0987 | ||||

HEO | 0.0147 | 0.0615 | 0.0303 | 2444.8667 | 0.5939 | 25.0941 | 0.0458 | 0.0766 | 0.0562 | 5229.4333 | 2.4868 | 35.6779 | ||||

HSCA | 0.0097 | 0.1774 | 0.0279 | 2439.1333 | 1.4628 | 67.6307 | 0.0307 | 0.2135 | 0.0681 | 5287.9667 | 6.5152 | 262.3587 | ||||

HSSA | 0.0139 | 0.1909 | 0.0364 | 2459.4667 | 0.2311 | 71.7378 | 0.0400 | 0.0755 | 0.0563 | 5229.8000 | 1.4355 | 42.7398 | ||||

HTSA | 0.0122 | 0.2174 | 0.0778 | 2557.7333 | 1.7549 | 192.4243 | 0.0372 | 0.2127 | 0.0757 | 5325.9000 | 7.3935 | 288.5887 | ||||

HGA | 0.0122 | 0.0590 | 0.0300 | 2444.2333 | 1.8881 | 24.9475 | 0.0404 | 0.0689 | 0.0562 | 5229.4667 | 7.4647 | 38.0059 | ||||

reC29 | IHGNDO | 2287 | 0.0087 | 0.0468 | 0.0237 | 2341.1667 | 1.8542 | 22.1105 | reC39 | 5087 | 0.0179 | 0.0352 | 0.0255 | 5216.5333 | 17.2536 | 20.5065 |

HIGNDO | 0.0031 | 0.0704 | 0.0259 | 2346.3333 | 1.8314 | 33.3320 | 0.0090 | 0.0271 | 0.0188 | 5182.6667 | 18.4729 | 22.1891 | ||||

HGNDO | 0.0057 | 0.0503 | 0.0241 | 2342.1333 | 2.1269 | 25.1869 | 0.0094 | 0.0297 | 0.0208 | 5192.7333 | 20.9996 | 26.3438 | ||||

HMPA | 0.0144 | 0.0647 | 0.0319 | 2359.8667 | 3.8849 | 27.4733 | 0.0173 | 0.0472 | 0.0293 | 5235.8667 | 13.1493 | 39.1550 | ||||

HWOA | 0.0092 | 0.0582 | 0.0260 | 2346.5667 | 3.0756 | 28.2756 | 0.0132 | 0.0299 | 0.0202 | 5189.8667 | 27.4727 | 17.3661 | ||||

HEO | 0.0240 | 0.1552 | 0.0470 | 2394.4667 | 0.6039 | 54.5422 | 0.0283 | 0.0554 | 0.0406 | 5293.4000 | 2.5226 | 38.4106 | ||||

HSCA | 0.0153 | 0.2239 | 0.0772 | 2463.5000 | 1.4498 | 174.4799 | 0.0155 | 0.1928 | 0.0554 | 5368.9667 | 6.6178 | 299.1470 | ||||

HSSA | 0.0210 | 0.2147 | 0.0600 | 2424.1667 | 0.2285 | 125.7885 | 0.0348 | 0.2048 | 0.0575 | 5379.3667 | 1.4566 | 230.4488 | ||||

HTSA | 0.0149 | 0.2317 | 0.0726 | 2453.1000 | 1.7457 | 185.6707 | 0.0161 | 0.2058 | 0.0817 | 5502.6333 | 7.2613 | 387.9788 | ||||

HGA | 0.0162 | 0.0700 | 0.0359 | 2369.1000 | 1.8481 | 28.7557 | 0.0261 | 0.0499 | 0.0368 | 5274.1000 | 7.0018 | 28.4843 | ||||

reC31 | IHGNDO | 3045 | 0.0085 | 0.0276 | 0.0207 | 3108.0000 | 4.8699 | 18.6744 | reC41 | 4960 | 0.0302 | 0.0569 | 0.0422 | 5169.2000 | 17.2578 | 31.5546 |

HIGNDO | 0.0039 | 0.0276 | 0.0131 | 3084.9667 | 4.7237 | 19.3278 | 0.0252 | 0.0466 | 0.0368 | 5142.5667 | 18.9444 | 30.6884 | ||||

HGNDO | 0.0033 | 0.0276 | 0.0145 | 3089.2000 | 5.7295 | 24.3604 | 0.0204 | 0.0546 | 0.0359 | 5137.9333 | 20.3596 | 38.5650 | ||||

HMPA | 0.0151 | 0.0309 | 0.0245 | 3119.5000 | 6.4491 | 12.0796 | 0.0310 | 0.0575 | 0.0425 | 5170.6667 | 13.1488 | 34.8715 | ||||

HWOA | 0.0026 | 0.0348 | 0.0177 | 3098.9333 | 7.4365 | 26.6632 | 0.0236 | 0.0514 | 0.0373 | 5144.8667 | 27.8736 | 29.6533 | ||||

HEO | 0.0197 | 0.0525 | 0.0334 | 3146.8000 | 1.0823 | 20.8477 | 0.0369 | 0.0722 | 0.0527 | 5221.5333 | 2.4905 | 36.3187 | ||||

HSCA | 0.0154 | 0.1846 | 0.0547 | 3211.6000 | 2.6588 | 187.0270 | 0.0349 | 0.2119 | 0.0516 | 5216.0667 | 6.6015 | 151.4369 | ||||

HSSA | 0.0187 | 0.2125 | 0.0695 | 3256.5333 | 0.5026 | 203.7153 | 0.0399 | 0.0704 | 0.0549 | 5232.0667 | 1.4595 | 36.8953 | ||||

HTSA | 0.0138 | 0.1941 | 0.0663 | 3247.0000 | 3.1755 | 214.0903 | 0.0331 | 0.2407 | 0.0648 | 5281.2000 | 7.3321 | 276.2412 | ||||

HGA | 0.0223 | 0.0693 | 0.0342 | 3149.2000 | 3.0078 | 27.9552 | 0.0411 | 0.0774 | 0.0547 | 5231.3333 | 7.0161 | 35.3802 | ||||

reC33 | IHGNDO | 3114 | 0.0058 | 0.0109 | 0.0084 | 3140.1333 | 4.8495 | 2.1868 | ||||||||

HIGNDO | 0.0000 | 0.0202 | 0.0085 | 3140.5000 | 4.7668 | 8.2735 | ||||||||||

HGNDO | 0.0013 | 0.0202 | 0.0078 | 3138.3333 | 5.6585 | 11.2497 | ||||||||||

HMPA | 0.0083 | 0.0202 | 0.0109 | 3147.9667 | 6.3048 | 13.6808 | ||||||||||

HWOA | 0.0083 | 0.0083 | 0.0083 | 3140.0000 | 7.0735 | 0.0000 | ||||||||||

HEO | 0.0071 | 0.0369 | 0.0160 | 3163.9000 | 1.0421 | 20.2227 | ||||||||||

HSCA | 0.0013 | 0.1532 | 0.0190 | 3173.2000 | 2.6069 | 98.5341 | ||||||||||

HSSA | 0.0039 | 0.1689 | 0.0250 | 3191.8667 | 0.4736 | 118.6611 | ||||||||||

HTSA | 0.0022 | 0.1811 | 0.0923 | 3401.2667 | 3.0636 | 240.2365 | ||||||||||

HGA | 0.0080 | 0.0466 | 0.0148 | 3160.0000 | 2.8183 | 24.0680 |

Inst | Algorithm | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD | Inst | ${\mathit{Z}}^{*}$ | BRE | WRE | ARE | ${\mathit{Z}}_{\mathit{A}\mathit{v}\mathit{g}}$ | Time(MS) | SD |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Hel1 | IHGNDO | 516 | −0.0019 | 0.0000 | −0.0005 | 515.7667 | 6.3275 | 0.4230 | He12 | 136 | 0.0000 | 0.0074 | 0.0040 | 136.5333 | 0.6642 | 0.4819 |

HIGNDO | −0.0019 | 0.0019 | −0.0001 | 515.9333 | 7.0816 | 0.3590 | 0.0000 | 0.0074 | 0.0040 | 136.5333 | 0.5195 | 0.4819 | ||||

HGNDO | −0.0019 | 0.0058 | −0.0001 | 515.9667 | 13.8456 | 0.7520 | 0.0000 | 0.0147 | 0.0059 | 136.8000 | 0.7227 | 0.5416 | ||||

HMPA | 0.0000 | 0.0058 | 0.0016 | 516.8333 | 12.3427 | 1.0355 | 0.0000 | 0.0294 | 0.0098 | 137.3333 | 2.3587 | 0.9428 | ||||

HWOA | −0.0019 | 0.0000 | −0.0002 | 515.9000 | 7.5461 | 0.3000 | 0.0000 | 0.0147 | 0.0044 | 136.6000 | 0.7650 | 0.6110 | ||||

HEO | −0.0019 | 0.0174 | 0.0045 | 518.3333 | 3.2206 | 2.0221 | 0.0000 | 0.0368 | 0.0154 | 138.1000 | 0.3749 | 1.3503 | ||||

HSCA | −0.0019 | 0.1105 | 0.0180 | 525.2667 | 6.0083 | 20.0382 | 0.0000 | 0.1176 | 0.0137 | 137.8667 | 0.8138 | 3.5659 | ||||

HSSA | 0.0000 | 0.1105 | 0.0155 | 524.0000 | 2.0489 | 15.3188 | 0.0000 | 0.1397 | 0.0213 | 138.9000 | 0.1299 | 3.3101 | ||||

HTSA | −0.0019 | 0.1202 | 0.0470 | 540.2333 | 7.4086 | 27.0502 | 0.0000 | 0.1618 | 0.0551 | 143.5000 | 1.0492 | 8.4370 | ||||

HGA | −0.0019 | 0.0078 | 0.0028 | 517.4667 | 7.9189 | 1.4314 | 0.0000 | 0.0515 | 0.0162 | 138.2000 | 1.0993 | 1.4697 |

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## Share and Cite

**MDPI and ACS Style**

Abdel-Basset, M.; Mohamed, R.; Abouhawwash, M.; Chang, V.; Askar, S.S.
A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling. *Appl. Sci.* **2021**, *11*, 4837.
https://doi.org/10.3390/app11114837

**AMA Style**

Abdel-Basset M, Mohamed R, Abouhawwash M, Chang V, Askar SS.
A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling. *Applied Sciences*. 2021; 11(11):4837.
https://doi.org/10.3390/app11114837

**Chicago/Turabian Style**

Abdel-Basset, Mohamed, Reda Mohamed, Mohamed Abouhawwash, Victor Chang, and S. S. Askar.
2021. "A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling" *Applied Sciences* 11, no. 11: 4837.
https://doi.org/10.3390/app11114837