## 5 Conclusions  
This paper has described a hybrid algorithm that applies a global search with genetic crossover and mutation operators in a CA-like neighborhood. These operators mainly optimize the sequence of operations. Random-restart hill-climbing performs a local search to allocate the best machine to each critical operation. This GA-RRHC feature makes it suitable for FJSSP instances with high flexibility. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Instance** & **n \(\times\) m** & \(o\) & \multicolumn{2}{c}{**GLNSA**} & \multicolumn{2}{c}{**GA-RRHC**} \\ \hline  & & & **Best** & **Avg.** & **Best** & **Avg.** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Experiment with large instances. Figure 20: Comparison of the RPD difference per algorithm for the Vdata dataset. The CA-like neighborhood allows the concurrent application of genetic operations, which gives the GA-RRHC a better exploration capability. Hill-climbing uses a similar number of iterations compared to other algorithms and is easy to implement, reflecting a satisfactory complexity. Four popular datasets (covering 101 problems) were used for the numerical experimentation of the GA-RRHC. The results expose good performance compared to recent algorithms taken as reference, especially for instances with high flexibility. The GA-RRHC opens a new way to apply CA-like neighborhoods that concurrently apply the exploration and exploitation operators to solve task scheduling problems; for instance, the flowshop, the job shop, or the open shop cases. As a possible future work, it is suggested to use another type of exploitation technique such as simulated annealing, similar to the one proposed in [56], to reduce the complexity of the local search. Other variants of scaling algorithms can be tested for optimizing the sequences of operations to solve FJSSP instances with low flexibility, in addition to extending this methodology for the optimization of multi-objective problems. Conceptualization, N.J.E.-S. and J.C.S.-T.-M.; methodology, N.J.E.-S., J.C.S.-T.-M. and J.M.M. ; software, N.J.E.-S. and J.C.S.-T.-M.; validation, J.M.-M., I.B.-V. and J.R.C.-A. ; formal analysis, N.J.E.-S., J.C.S.-T.-M. and J.M.-M.; investigation, N.J.E.-S., J.C.S.-T.-M. and I.B.-V.; resources, J.M.-M. and J.C.S.-T.-M.; data curation, N.J.E.-S., J.C.S.-T.-M.; writing--original draft preparation, J.C.S.-T.-M., I.B.-V. and J.R.C.-A; writing--review and editing, J.C.S.-T.-M., I.B.-V. and J.R.C.-A. visualization, N.J.E.-S., J.C.S.-T.-M. and J.M.M. ; supervision, J.M.-M. and J.R.C.-A. ; project administration, J.M.-M. and J.C.S.-T.-M.; funding acquisition, J.M.-M. and J.C.S.-T.-M. All authors have read and agreed to the published version of the manuscript. This study was supported by the Autonomous University of Hidalgo (UAEH) and the National Council for Science and Technology (CONACYT) with projects numbers CB-2017-2018-A1-S-43008 and F003/320109. [PERSON] was supported by CONACYT grant number 1013175. Not applicable. The GA-RRHC source code is available on Github [[https://github.com/juanscek/GA-RRHC](https://github.com/juanscek/GA-RRHC)]([https://github.com/juanscek/GA-RRHC](https://github.com/juanscek/GA-RRHC)) (accessed on 5 August 2022). The datasets taken to test the GA-RRHC are available at [[https://people.idsia.ch/~monaldo/fjsp.html](https://people.idsia.ch/~monaldo/fjsp.html)]([https://people.idsia.ch/~monaldo/fjsp.html](https://people.idsia.ch/~monaldo/fjsp.html)) (accessed on 5 August 2022). The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.