# A Hybrid Search Using Genetic Algorithms and Random-Restart Hill-Climbing for Flexible Job Shop Scheduling Instances with High Flexibility

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Description of the FJSSP

## 3. Genetic Algorithm and Random-Restart Hill-Climbing (GA-RRHC)

#### 3.1. Encoding and Decoding Solutions

#### 3.2. Qualitative Description of the GA

#### 3.2.1. Population Selection

#### 3.2.2. Crossover Operators

#### 3.2.3. Mutation Operators

#### 3.2.4. CA-Type Neighborhood to Apply Genetic Operators

#### 3.3. Random-Restart Hill-Climbing (RRHC)

#### 3.4. Integration of the GA-RRHC Algorithm

Algorithm 1: Pseudocode of the GA-RRHC |

## 4. Results of Experiments

#### 4.1. GA-RRHC Parameter Tuning

#### 4.2. Comparison with Other Methods

#### 4.3. Kacem Dataset

#### 4.4. Brandimarte Dataset

#### 4.5. Rdata Dataset

#### 4.6. Vdata Dataset

#### 4.7. Generated Large Dataset

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FJSSP | Flexible job shop scheduling problem |

GA | Genetic algorithm |

RRHC | Random-restart hill-climbing algorithm |

CA | Cellular automata |

POX | Precedence operation crossover |

JBX | Job-based crossover |

OS | Operation sequence |

MS | Machine sequence |

## References

- Chen, H.; Ihlow, J.; Lehmann, C. A genetic algorithm for flexible job-shop scheduling. IEEE Int. Conf. Robot. Autom.
**1999**, 2, 1120–1125. [Google Scholar] [CrossRef] - Pinedo, M.L. Scheduling Theory, Algorithms, and Systems, 5th ed.; Springer: Berlin/Heidelberg, Germany, 2016; pp. 1–670. [Google Scholar] [CrossRef]
- Amjad, M.K.; Butt, S.I.; Kousar, R.; Ahmad, R.; Agha, M.H.; Faping, Z.; Anjum, N.; Asgher, U. Recent research trends in genetic algorithm based flexible job shop scheduling problems. Math. Probl. Eng.
**2018**, 2018, 9270802. [Google Scholar] [CrossRef] - Pezzellaa, F.; Morgantia, G.; Ciaschettib, G. A genetic algorithm for the flexible job-shop scheduling problem. Comput. Oper. Res.
**2008**, 35, 3202–3212. [Google Scholar] [CrossRef] - Garey, M.; Johnson, D.; Sethi, R. The complexity of flowshop and jobshop scheduling. Math. Oper. Res.
**1976**, 1, 117–129. [Google Scholar] [CrossRef] - Qing-dao-er ji, R.; Wang, Y. A new hybrid genetic algorithm for job shop scheduling problem. Comput. Ind. Eng.
**2012**, 1, 2291–2299. [Google Scholar] [CrossRef] - Yu, Y. A research review on job shop scheduling problem. E3S Web Conf.
**2021**, 253, 02024. [Google Scholar] [CrossRef] - Morrison David, R.; Jacobson, S.H.; Sauppe, J.J.; Sewell, E.C. Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning. Discret. Optim.
**2016**, 19, 79–102. [Google Scholar] [CrossRef] - Mallia, B.; Das, M.; Das, C. Fundamentals of transportation problem. Int. J. Eng. Adv. Technol. (IJEAT)
**2021**, 10, 90–103. [Google Scholar] [CrossRef] - Che, P.; Tang, Z.; Gong, H.; Zhao, X. An improved Lagrangian relaxation algorithm for the robust generation self-scheduling problem. Math. Probl. Eng.
**2018**, 2018, 6303596. [Google Scholar] [CrossRef] - Holland, J.H. Adaptation in Natural and Artificial Systems; MIT Press Ltd.: Cambridge, MA, USA, 1975; p. 232. [Google Scholar]
- Goldberg, D.E. Genetic Algorithms in Search, Optimization and Machine Learning; Addison-Wesley: Reading, MA, USA, 1989. [Google Scholar]
- Dorigo, M. Optimization, Learning and Natural Algorithms. Ph.D. Thesis, Politecnico di Milano, Milano, Italy, 1992. [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm pptimization. In Proceedings of the ICNN’95—International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar] [CrossRef]
- Glover, F. Tabu seach—Part I. ORSA J. Comput. Summer
**1989**, 1, 190–206. [Google Scholar] [CrossRef] - Brucker, P.; Schlie, R. Job-shop scheduling with multi-purpose machines. Computing
**1990**, 45, 369–375. [Google Scholar] [CrossRef] - Brandimarte, P. Routing and scheduling in a flexible job shop by tabu search. Ann. Oper. Res.
**1993**, 41, 157–183. [Google Scholar] [CrossRef] - Mastrolilli, M.; Gambardella, L.M. Effective neighbourhood functions for the flexible job shop problem. J. Sched.
**2000**, 3, 3–20. [Google Scholar] [CrossRef] - Li, X.; Gao, L. An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem. J. Prod. Econ.
**2016**, 174, 93–110. [Google Scholar] [CrossRef] - Deng, Q.; Gong, G.; Gong, X.; Zhang, L.; Liu, W.; Ren, Q. A bee evolutionary guiding nondominated sorting genetic algorithm II for multiobjetive flexible job shop scheduling. Comput. Intell. Neurosci.
**2017**, 27, 5232518. [Google Scholar] [CrossRef] - Gao, J.; Sun, L.; Gen, M. A hybrid genetic and variable neighborhood descent algorithm for flexible job shop scheduling problems. Comput. Oper. Res.
**2008**, 35, 2892–2907. [Google Scholar] [CrossRef] - Zhang, G.; Shao, X.; Li, P.; Gao, L. An effective hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problem. Comput. Ind. Eng.
**2009**, 56, 1309–1318. [Google Scholar] [CrossRef] - Amiri, M.; Zandieh, M.; Yazdani, M.; Bagheri, A. A variable neighbourhood search algorithm for the flexible job-shop scheduling problem. Int. J. Prod. Res.
**2010**, 48, 5671–5689. [Google Scholar] [CrossRef] - Li, J.q.; Pan, Q.k.; Liang, Y.C. An effective hybrid tabu search algorithm for multi-objective flexible job-shop scheduling problems. Comput. Ind. Eng.
**2010**, 59, 647–662. [Google Scholar] [CrossRef] - Dalfarda, V.M.; Mohammadi, G. Two meta-heuristic algorithms for solving multi-objective flexible job-shop scheduling with parallel machine and maintenance constraints. Comput. Math. Appl.
**2012**, 64, 2111–2117. [Google Scholar] [CrossRef] - Yuan, Y.; Xu, H.; Yang, J. A hybrid harmony search algorithm for the flexible job shop scheduling problem. Appl. Soft Comput.
**2013**, 13, 3259–3272. [Google Scholar] [CrossRef] - Li, J.Q.; Pan, Q.K.; Tasgetiren, M.F. A discrete artificial bee colony algorithm for the multi-objective flexible job-shop scheduling problem with maintenance activities. Appl. Math. Model.
**2014**, 38, 1111–1132. [Google Scholar] [CrossRef] - Gao, K.Z.; Suganthan, P.N.; Chua, T.J.; Chong, C.S.; Cai, T.X.; Pan, Q.K. A two-stage artificial bee colony algorithm scheduling flexible job-shop scheduling problem with new job insertion. Expert Syst. Appl.
**2015**, 42, 7652–7663. [Google Scholar] [CrossRef] - Li, X.; Peng, Z.; Du, B.; Guo, J.; Xu, W.; Zhuang, K. Hybrid artificial bee colony algorithm with a rescheduling strategy for solving flexible job shop scheduling problems. Comput. Ind. Eng.
**2017**, 113, 10–26. [Google Scholar] [CrossRef] - Rodriguez Kato, E.R.; de Aguiar Aranha, G.D.; Tsunaki, R.H. A new approach to solve the flexible job shop problem based on a hybrid particle swarm optimization and random-restart hill climbing. Comput. Ind. Eng.
**2018**, 125, 178–189. [Google Scholar] [CrossRef] - Gong, G.; Deng, Q.; Gong, X.; Liu, W.; Ren, Q. A new double flexible job-shop scheduling problem integrating processing time, green production, and human factor indicators. J. Clean. Prod.
**2018**, 174, 560–576. [Google Scholar] [CrossRef] - Sreekara Reddy, M.; Ratnam, C.; Rajyalakshmi, G.; Manupati, V. An effective hybrid multi objective evolutionary algorithm for solving real time event in flexible job shop scheduling problem. Measurement
**2018**, 114, 78–90. [Google Scholar] [CrossRef] - Meng, T.; Pan, Q.K.; Sang, H.Y. A hybrid artificial bee colony algorithm for a flexible job shop scheduling problem with overlapping in operations. Int. J. Prod. Res.
**2018**, 56, 5278–5292. [Google Scholar] [CrossRef] - Wu, X.; Shen, X.; Li, C. The flexible job-shop scheduling problem considering deterioration effect. Comput. Ind. Eng.
**2019**, 135, 1004–1024. [Google Scholar] [CrossRef] - Lin, J.; Zhu, L.; Wang, Z.J. A hybrid multi-verse optimization for the fuzzy flexible job-shop scheduling problem. Comput. Ind. Eng.
**2019**, 127, 1089–1100. [Google Scholar] [CrossRef] - Goerler, A.; Lalla-Ruiz, E.; Voß, S. Late acceptance hill-climbing matheuristic for the general lot sizing and scheduling problem with rich constraints. Algorithms
**2020**, 13, 138. [Google Scholar] [CrossRef] - Defersha, F.M.; Rooyani, D. An efficient two-stage genetic algorithm for a flexible job-shop scheduling problem with sequence dependent attached/detached setup, machine release date and lag-time. Comput. Ind. Eng.
**2020**, 147, 106605. [Google Scholar] [CrossRef] - Alzaqebah, M.; Jawarneh, S.; Alwohaibi, M.; Alsmadi, M.K.; Almarashdeh, I.; Mohammad, R.M.A. Hybrid brain storm optimization algorithm and late acceptance hill climbing to solve the flexible job-shop scheduling problem. J. King Saud Univ. Comput. Inf. Sci.
**2020**, 34, 2926–2937. [Google Scholar] [CrossRef] - Ding, H.; Gu, X. Hybrid of human learning optimization algorithm and particle swarm optimization algorithm with scheduling strategies for the flexible job-shop scheduling problem. Neurocomputing
**2020**, 414, 313–332. [Google Scholar] [CrossRef] - Escamilla-Serna, N.; Seck-Tuoh-Mora, J.C.; Medina-Marin, J.; Hernandez-Romero, N.; Barragan-Vite, I.; Corona Armenta, J.R. A global-local neighborhood search algorithm and tabu search for flexible job shop scheduling problem. PeerJ Comput. Sci.
**2021**, 7, e574. [Google Scholar] [CrossRef] [PubMed] - Jacobson, S.H.; Yücesan, E. Analyzing the performance of generalized hill climbing algorithms. J. Heuristics.
**2004**, 10, 387–405. [Google Scholar] [CrossRef] - McIntosh, H.V. One Dimensional Cellular Automata; Luniver Press: Bristol, UK, 2009. [Google Scholar]
- Eiben, A.E.; Smith, J.E. Introduction to evolutionary computing. Nat. Comput. Ser.
**2015**, 2, 287. [Google Scholar] [CrossRef] - Kari, J. Theory of cellular automata: A survey. Theor. Comput. Sci.
**2005**, 334, 3–33. [Google Scholar] [CrossRef] - Shi, Y.; Liu, H.; Gao, L.; Zhang, G. Cellular particle swarm optimization. Inf. Sci.
**2011**, 181, 4460–4493. [Google Scholar] [CrossRef] - Lagos-Eulogio, P.; Seck-Tuoh-Mora, J.C.; Hernandez-Romero, N.; Medina-Marin, J. A new design method for adaptive IIR system identification using hybrid CPSO and DE. Nonlinear Dyn.
**2017**, 88, 2371–2389. [Google Scholar] [CrossRef] - Seck-Tuoh-Mora, J.C.; Medina-Marin, J.; Martinez-Gomez, E.S.; Hernandez-Gress, E.S.; Hernandez-Romero, N.; Volpi-Leon, V. Cellular particle swarm optimization with a simple adaptive local search strategy for the permutation flow shop scheduling problem. Arch. Control Sci.
**2019**, 29, 205–226. [Google Scholar] - Hernández-Gress, E.S.; Seck-Tuoh-Mora, J.C.; Hernández-Romero, N.; Medina-Marín, J.; Lagos-Eulogio, P.; Ortíz-Perea, J. The solution of the concurrent layout scheduling problem in the job-shop environment through a local neighborhood search algorithm. Expert Syst. Appl.
**2020**, 144, 113096. [Google Scholar] [CrossRef] - Kacem, I.; Hammadi, S.; Borne, P. Approach by localization and multiobjective evolutionary optimization for flexible job-shop scheduling problems. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
**2002**, 32, 1–13. [Google Scholar] [CrossRef] - Hurink, J.; Jurisch, B.; Thole, M. Tabu search for the job-shop scheduling problem with multi-purpose machines. Oper. Res. Spektrum
**1994**, 15, 205–215. [Google Scholar] [CrossRef] - Baykasoğlu, A.; Madenoğlu, F.S.; Hamzadayı, A. Greedy randomized adaptive search for dynamic flexible job-shop scheduling. J. Manuf. Syst.
**2020**, 56, 425–451. [Google Scholar] [CrossRef] - Caldeira, R.H.; Gnanavelbabu, A. Solving the flexible job shop scheduling problem using an improved Jaya algorithm. Comput. Ind. Eng.
**2019**, 137, 106064. [Google Scholar] [CrossRef] - Zarrouk, R.; Bennour, I.E.; Jemai, A. A two-level particle swarm optimization algorithm for the flexible job shop scheduling problem. Swarm Intell.
**2019**, 13, 145–168. [Google Scholar] [CrossRef] - Derrac, J.; García, S.; Molina, D.; Herrera, F. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol. Comput.
**2011**, 1, 3–18. [Google Scholar] [CrossRef] - Sun, L.; Lin, L.; Li, H.; Gen, M. Large scale flexible scheduling optimization by a distributed evolutionary algorithm. Comput. Ind. Eng.
**2019**, 128, 894–904. [Google Scholar] [CrossRef] - Sajid, M.; Jafar, A.; Sharma, S. Hybrid Genetic and Simulated Annealing Algorithm for Capacitated Vehicle Routing Problem. In Proceedings of the 2020 Sixth International Conference on Parallel, Distributed and Grid Computing (PDGC), Waknaghat, India, 6–8 November 2020; pp. 131–136. [Google Scholar]

**Figure 1.**Gantt diagram of one possible solution for the FJSSP in Table 1.

**Figure 9.**CA-type neighborhood used in the GA-RRHC. Colors represent the selection and modification of different smart-cells.

**Figure 10.**Critical path of the solution described in Figure 1.

Job | Op. | ${\mathit{M}}_{1}$ | ${\mathit{M}}_{2}$ | ${\mathit{M}}_{3}$ |
---|---|---|---|---|

${J}_{1}$ | ${O}_{1,1}$ | 3 | 4 | 4 |

${O}_{1,2}$ | 1 | 2 | 1 | |

${J}_{2}$ | ${O}_{2,1}$ | 2 | 3 | 3 |

${O}_{2,2}$ | 3 | 3 | 2 | |

${J}_{3}$ | ${O}_{3,1}$ | 3 | 3 | 3 |

${O}_{3,2}$ | 2 | 2 | 1 |

${\mathit{G}}_{\mathit{n}}$ | total iterations of the algorithm | 250 |

${G}_{b}$ | limit of stagnation iterations | 50 |

${S}_{n}$ | number of smart-cells | 100 |

${E}_{p}$ | proportion of elite smart-cells | 0.02 |

l | neighbors of each smart-cell | 3 |

${\alpha}_{m}$ | mutation probability | 0.1 |

${H}_{n}$ | iterations of the RRHC | 100 |

${H}_{r}$ | iterations to restart the RRHC | 30 |

${\alpha}_{c}$ | probability of moving a critical operation in the RRHC | 0.05 |

Method | Complexity | Rank |
---|---|---|

TlPSO | $\mathcal{O}\left(o\left({G}_{n}(2X+{G}_{n}2X)\right)\right)$ | 1 |

GA-RRHC | $\mathcal{O}\left(o\left({G}_{n}({S}_{n}+X+{S}_{n}{H}_{n}m)\right)\right)$ | 2 |

GLNSA | $\mathcal{O}\left(o\left({G}_{n}({S}_{n}+X+{S}_{n}{H}_{n}m)\right)\right)$ | 2 |

IJA | $\mathcal{O}(o\left({G}_{n}(3X+X{H}_{n}m)\right)$ | 3 |

HA | $\mathcal{O}\left(o\left({G}_{n}(4X+X{H}_{n}m)\right)\right)$ | 4 |

HBSO-LAHC | $\mathcal{O}\left(o\left({G}_{n}(4X+X{H}_{n}m)\right)\right)$ | 4 |

GRASP | $\mathcal{O}\left({o}^{2}({G}_{n}\ast {H}_{n})\right)$ | 5 |

Instance | n × m | $\mathit{\beta}$ | GA-RRHC | GLNSA | GRASP | HA | IJA | TlPSO |
---|---|---|---|---|---|---|---|---|

K1 | 4 × 5 | 1. | 11 | 11 | − | − | 11 | 11 |

K2 | 8 × 8 | 0.81 | 14 | 14 | 14 | 14 | 14 | 14 |

K3 | 10 × 7 | 1 | 11 | 11 | − | − | 11 | − |

K4 | 10 × 10 | 1 | 7 | 7 | 7 | 7 | 7 | 7 |

K5 | 15 × 10 | 1 | 11 | 11 | 11 | 11 | 11 | − |

Instance | n × m | $\mathit{\beta}$ | BKV | BSO-LAHC | GA-RRHC | GLNSA | GRASP | HA | IJA | TlPSO |
---|---|---|---|---|---|---|---|---|---|---|

MK01 | 10 × 6 | 0.2 | 36 | 40 * | 40 * | 40 * | 40 * | 40 * | 40 * | 40 * |

MK02 | 10 × 6 | 0.35 | 24 | 26 * | 26 * | 26 * | 26 * | 26 * | 27 | 26 * |

MK03 | 15 × 8 | 0.3 | 204 | 204 * | 204 * | 204 * | 204 * | 204 * | 204 * | 204 * |

MK04 | 15 × 8 | 0.2 | 48 | 60 * | 60 * | 60 * | 60 * | 60 * | 60 * | 60 * |

MK05 | 15 × 4 | 0.15 | 168 | 173 | 172 * | 173 | 172 * | 172 * | 172 * | 173 |

MK06 | 10 × 15 | 0.3 | 33 | 61 | 58 | 58 | 64 | 57 * | 57 * | 60 |

MK07 | 20 × 5 | 0.3 | 133 | 141 | 139 * | 139 * | 139 * | 139 * | 139 * | 139 * |

MK08 | 20 × 10 | 0.15 | 523 | 523 * | 523 * | 523 * | 523 * | 523 * | 523 * | 523 * |

MK09 | 20 × 10 | 0.3 | 299 | 307 * | 307 * | 307 * | 307 * | 307 * | 307 * | 307 * |

MK10 | 20 × 15 | 0.2 | 165 | 204 | 198 | 205 | 205 | 197 * | 197 * | 205 |

Algorithm: | BSO-LAHC | GA-RRHC | GLNSA | GRASP | HA | IJA | TlPSO |
---|---|---|---|---|---|---|---|

Average $RPD$: | 11.3881 | 10.6710 | 11.0121 | 11.4890 | 10.5289 | 11.2665 | 11.2017 |

Rank: | 6 | 2 | 3 | 7 | 1 | 5 | 4 |

p-value: | 0.0455 | ~ | 0.1573 | 0.0435 | 0.1643 | 0.0803 | 0.0833 |

Instance | n × m | $\mathit{\beta}$ | BKV | GA-RRHC | GLNSA | HA | IJA |
---|---|---|---|---|---|---|---|

mt06 | 6 × 6 | $0.33$ | 47 | 47 * | 47 * | 47 * | 47 * |

mt10 | 10 × 10 | $0.2$ | 686 | 686 * | 686 * | 686 * | 686 * |

mt20 | 20 × 5 | $0.4$ | 1022 | 1022 * | 1022 * | 1024 | 1024 |

la01 | 10 × 5 | $0.4$ | 570 | 571 | 571 | 570 * | 571 |

la02 | 10 × 5 | $0.4$ | 529 | 530 * | 530 * | 530 * | 530 * |

la03 | 10 × 5 | $0.4$ | 477 | 477 * | 477 * | 477 * | 477 * |

la04 | 10 × 5 | $0.4$ | 502 | 502 * | 502 * | 502 * | 502 * |

la05 | 10 × 5 | $0.4$ | 457 | 457 * | 457 * | 457 * | 457 * |

la06 | 15 × 5 | $0.4$ | 799 | 799 * | 799 * | 799 * | 799 * |

la07 | 15 × 5 | $0.4$ | 749 | 749 * | 749 * | 749 * | 749 * |

la08 | 15 × 5 | $0.4$ | 765 | 765 * | 765 * | 765 * | 765 * |

la09 | 15 × 5 | $0.4$ | 853 | 853 * | 853 * | 853 * | 853 * |

la10 | 15 × 5 | $0.4$ | 804 | 804 * | 804 * | 804 * | 804 * |

la11 | 20 × 5 | $0.4$ | 1071 | 1071 * | 1071 * | 1071 * | 1071 * |

la12 | 20 × 5 | $0.4$ | 936 | 936 * | 936 * | 936 * | 936 * |

la13 | 20 × 5 | $0.4$ | 1038 | 1038 * | 1038 * | 1038 * | 1038 * |

la14 | 20 × 5 | $0.4$ | 1070 | 1070 * | 1070 * | 1070 * | 1070 * |

la15 | 20 × 5 | $0.4$ | 1089 | 1089 * | 1089 * | 1090 | 1090 |

la16 | 10 × 10 | $0.2$ | 717 | 717 * | 717 * | 717 * | 717 * |

la17 | 10 × 10 | $0.2$ | 646 | 646 * | 646 * | 646 * | 646 * |

la18 | 10 × 10 | $0.2$ | 666 | 666 * | 666 * | 666 * | 666 * |

la19 | 10 × 10 | $0.2$ | 647 | 700 * | 700 * | 700 | 702 |

la20 | 10 × 10 | $0.2$ | 756 | 756 * | 756 * | 756 * | 760 |

la21 | 15 × 10 | $0.2$ | 808 | 850 | 852 | 835 * | 854 |

la22 | 15 × 10 | $0.2$ | 737 | 770 | 774 | 760 * | 760 * |

la23 | 15 × 10 | $0.2$ | 816 | 850 | 854 | 840 * | 852 |

la24 | 15 × 10 | $0.2$ | 775 | 810 | 826 | 806 * | 806 * |

la25 | 15 × 10 | $0.2$ | 752 | 800 | 803 | 789 * | 803 |

la26 | 20 × 10 | $0.2$ | 1056 | 1070 | 1075 | 1061 * | 1061 * |

la27 | 20 × 10 | $0.2$ | 1085 | 1100 | 1109 | 1089 * | 1109 |

la28 | 20 × 10 | $0.2$ | 1075 | 1090 | 1096 | 1079 * | 1081 |

la29 | 20 × 10 | $0.2$ | 993 | 999 | 1008 | 997 * | 997 * |

la30 | 20 × 10 | $0.2$ | 1068 | 1088 | 1096 | 1078 * | 1078 * |

la31 | 30 × 10 | $0.2$ | 1520 | 1521 * | 1527 | 1521 * | 1521 * |

la32 | 30 × 10 | $0.2$ | 1657 | 1667 | 1667 | 1659 * | 1659 * |

la33 | 30 × 10 | $0.2$ | 1497 | 1500 | 1504 | 1499 * | 1499 * |

la34 | 30 × 10 | $0.2$ | 1535 | 1539 | 1540 | 1536 * | 1536 * |

la35 | 30 × 10 | $0.2$ | 1549 | 1553 | 1555 | 1550 * | 1555 |

la36 | 15 × 15 | $0.13$ | 1016 | 1050 | 1053 | 1028 * | 1050 |

la37 | 15 × 15 | $0.13$ | 989 | 1092 | 1093 | 1074 * | 1092 |

la38 | 15 × 15 | $0.13$ | 943 | 995 | 999 | 960 * | 995 |

la39 | 15 × 15 | $0.13$ | 966 | 1030 | 1034 | 1024 * | 1031 |

la40 | 15 × 15 | $0.13$ | 955 | 998 | 997 | 970 * | 993 |

Algorithm: | GA-RRHC | GLNSA | HA | IJA |
---|---|---|---|---|

Average $RPD$: | $1.5761$ | $1.7768$ | $1.0872$ | $1.5155$ |

Rank: | 3 | 4 | 1 | 2 |

p-value: | ~ | $0.0001$ | $0.0001$ | $0.9195$ |

Instance | n × m | $\mathit{\beta}$ | BKV | GA-RRHC | GLNSA | HA | IJA |
---|---|---|---|---|---|---|---|

mt06 | 6 × 6 | $0.5$ | 47 | 47 * | 47 * | 47 * | 47 * |

mt10 | 10 × 10 | $0.5$ | 655 | 655 * | 655 * | 655 * | 655 * |

mt20 | 20 × 5 | $0.5$ | 1022 | 1022 * | 1022 * | 1022 * | 1024 |

la01 | 10 × 5 | $0.5$ | 570 | 570 * | 570 * | 570 * | 571 |

la02 | 10 × 5 | $0.5$ | 529 | 529 * | 529 * | 529 * | 529 * |

la03 | 10 × 5 | $0.5$ | 477 | 477 * | 477 * | 477 * | 477 * |

la04 | 10 × 5 | $0.5$ | 502 | 502 * | 502 * | 502 * | 502 * |

la05 | 10 × 5 | $0.5$ | 457 | 457 * | 457 * | 457 * | 457 * |

la06 | 15 × 5 | $0.5$ | 799 | 799 * | 799 * | 799 * | 799 * |

la07 | 15 × 5 | $0.5$ | 749 | 749 * | 749 * | 749 * | 749 * |

la08 | 15 × 5 | $0.5$ | 765 | 765 * | 765 * | 765 * | 765 * |

la09 | 15 × 5 | $0.5$ | 853 | 853 * | 853 * | 853 * | 853 * |

la10 | 15 × 5 | $0.5$ | 804 | 804 * | 804 * | 804 * | 804 * |

la11 | 20 × 5 | $0.5$ | 1071 | 1071 * | 1071 * | 1071 * | 1071 * |

la12 | 20 × 5 | $0.5$ | 936 | 936 * | 936 * | 936 * | 936 * |

la13 | 20 × 5 | $0.5$ | 1038 | 1038 * | 1038 * | 1038 * | 1038 * |

la14 | 20 × 5 | $0.5$ | 1070 | 1070 * | 1070 * | 1070 * | 1070 * |

la15 | 20 × 5 | $0.5$ | 1089 | 1089 * | 1089 * | 1089 * | 1089 * |

la16 | 10 × 10 | $0.5$ | 717 | 717 * | 717 * | 717 * | 717 * |

la17 | 10 × 10 | $0.5$ | 646 | 646 * | 646 * | 646 * | 646 * |

la18 | 10 × 10 | $0.5$ | 663 | 663 * | 663 * | 663 * | 665 |

la19 | 10 × 10 | $0.5$ | 617 | 617 * | 617 * | 617 * | 618 |

la20 | 10 × 10 | $0.5$ | 756 | 756 * | 756 * | 756 * | 758 |

la21 | 15 × 10 | $0.5$ | 800 | 804 * | 806 | 804 * | 806 |

la22 | 15 × 10 | $0.5$ | 733 | 737 * | 737 * | 738 | 738 |

la23 | 15 × 10 | $0.5$ | 809 | 813 * | 813 * | 813 * | 813 * |

la24 | 15 × 10 | $0.5$ | 773 | 777 * | 777 | 777 * | 778 |

la25 | 15 × 10 | $0.5$ | 751 | 754 * | 754 | 754 * | 754 * |

la26 | 20 × 10 | $0.5$ | 1052 | 1053 * | 1054 | 1053 * | 1054 |

la27 | 20 × 10 | $0.5$ | 1084 | 1085 * | 1085 * | 1085 * | 1085 * |

la28 | 20 × 10 | $0.5$ | 1069 | 1070 * | 1070 * | 1070 * | 1070 * |

la29 | 20 × 10 | $0.5$ | 993 | 994 * | 994 * | 994 * | 994 * |

la30 | 20 × 10 | $0.5$ | 1068 | 1069 * | 1069 * | 1069 * | 1069 * |

la31 | 30 × 10 | $0.5$ | 1520 | 1520 * | 1520 * | 1520 * | 1521 |

la32 | 30 × 10 | $0.5$ | 1657 | 1658 * | 1658 * | 1658 * | $1658*$ |

la33 | 30 × 10 | $0.5$ | 1497 | 1497 * | 1497 * | 1497 * | 1497 * |

la34 | 30 × 10 | $0.5$ | 1535 | 1535 * | 1535 * | 1535 * | 1535 * |

la35 | 30 × 10 | $0.5$ | 1549 | 1549 * | 1549 * | 1549 * | 1549 * |

la36 | 15 × 15 | $0.5$ | 948 | 948 * | 948 * | 948 * | 950 |

la37 | 15 × 15 | $0.5$ | 986 | 986 * | 986 * | 986 * | 986 * |

la38 | 15 × 15 | $0.5$ | 943 | 943 * | 943 * | 943 * | 943 * |

la39 | 15 × 15 | $0.5$ | 922 | 922 * | 922 * | 922 * | 922 * |

la40 | 15 × 15 | $0.5$ | 955 | 955 * | 955 * | 955 * | 956 |

Algorithm: | GA-RRHC | GLNSA | HA | IJA |
---|---|---|---|---|

Average $RPD$: | $0.0693$ | $0.0772$ | $0.0724$ | $0.1177$ |

Rank: | 1 | 3 | 2 | 4 |

p-value: | ~ | $0.1573$ | $0.3173$ | $0.0005$ |

Instance | $\mathbf{n}\times \mathbf{m}$ | o | GLNSA | GA-RRHC | ||
---|---|---|---|---|---|---|

Best | Avg. | Best | Avg. | |||

VL01 | $50\times 20$ | 704 | 592 | $617.3$ | 551 | $570.9$ |

VL02 | $60\times 30$ | 1246 | 759 | $781.3$ | 705 | $717.8$ |

VL03 | $80\times 50$ | 2773 | 1155 | $1183.1$ | 1041 | $1058.4$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Escamilla-Serna, N.J.; Seck-Tuoh-Mora, J.C.; Medina-Marin, J.; Barragan-Vite, I.; Corona-Armenta, J.R.
A Hybrid Search Using Genetic Algorithms and Random-Restart Hill-Climbing for Flexible Job Shop Scheduling Instances with High Flexibility. *Appl. Sci.* **2022**, *12*, 8050.
https://doi.org/10.3390/app12168050

**AMA Style**

Escamilla-Serna NJ, Seck-Tuoh-Mora JC, Medina-Marin J, Barragan-Vite I, Corona-Armenta JR.
A Hybrid Search Using Genetic Algorithms and Random-Restart Hill-Climbing for Flexible Job Shop Scheduling Instances with High Flexibility. *Applied Sciences*. 2022; 12(16):8050.
https://doi.org/10.3390/app12168050

**Chicago/Turabian Style**

Escamilla-Serna, Nayeli Jazmin, Juan Carlos Seck-Tuoh-Mora, Joselito Medina-Marin, Irving Barragan-Vite, and José Ramón Corona-Armenta.
2022. "A Hybrid Search Using Genetic Algorithms and Random-Restart Hill-Climbing for Flexible Job Shop Scheduling Instances with High Flexibility" *Applied Sciences* 12, no. 16: 8050.
https://doi.org/10.3390/app12168050