## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **DF** & **Adj SS** & **Adj MS** & **F-Value** & \(p\)**-Value** \\ \hline Mutation Probability & 5 & 1,418,299,280 & 283,659,856 & 93.7 & 0.000 \\ Crossover Probability & 5 & 91,944,634 & 18,388,927 & 6.03 & 0.000 \\ Mutation Probability\({}^{\star}\)Crossover Probability & 25 & 49,082,335 & 1,963,293 & 0.64 & 0.901 \\ Error & 144 & 438,889,368 & 3,047,843 & & \\ Total & 179 & 1,998,215,617 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Output of the analysis of variance for Problem-4. Figure 19: The main effect and residual plots of the analysis of variance. the maximum workload difference), can result in a better workload balance when considered along with the minimization of the maximum and/or the total workload. * The solution representation and the corresponding decoding of the first stage of the two-stage genetic algorithm can generate initial solutions that are highly improved in all the ten objective function terms. * The two-stage genetic algorithm can jointly optimize all the ten objective function terms of the multi-objective FJSP lot streaming considered in this paper and greatly outperform the regular genetic algorithm. * Parallel computation can bring performance improvements in both the two-stage GA and the regular GA. However, the crucial finding is that the sequential two-stage GA using a single CPU outperformed the parallel regular genetic algorithm that uses many CPUs in solving the proposed multi-objective FJSP lot streaming problem. * The performance of the proportional selection method can be significantly improved by the appropriate choice of the fitness transformation function. * Both proportional, linear ranking and tournament selection can result in comparable performance. However, tournament selection with smaller size of tournament slightly outperformed the other two. * Analysis of variance shows the lack of interaction between mutation and crossover probabilities. Thus, the two probabilities can be tuned independently. ### Future Research

The two-stage genetic algorithm may not be directly applicable in scheduling problems with the objective of minimizing earliness-tardiness. In particular, the greedy nature of the first stage is based on finding a schedule that finishes the jobs as early as possible, which is against minimizing earliness. For instance, finishing jobs too early may represent excess work-in-process in a JIT environment. Hence, our future research includes the development of a modified two-stage genetic algorithm to incorporate the minimization of earliness. We also plan to expand the application of the two-stage genetic algorithm for a multi-resource constrained FJSP as flexible job shops are often constrained by many kinds of resources in addition to machines. D.R. : Methodology, Investigation, Validation, Formal analysis, Writing--Original Draft, Data Curation; F.D. : Conceptualization, Methodology, Formal analysis, Software, Validation, Visualization, Writing--Review & Editing, Supervision. All authors have read and agreed to the published version of the manuscript. This research was funded by the Discovery Grant from NSERC, the Natural Science and Engineering Research Counsel of Canada ([[https://www.nserc-crsng.gc.ca/index_eng.asp](https://www.nserc-crsng.gc.ca/index_eng.asp)]([https://www.nserc-crsng.gc.ca/index_eng.asp](https://www.nserc-crsng.gc.ca/index_eng.asp))) (accessed on 10 June 2022). Not applicable. We would like to thank Compute Canada ([[https://www.computecanada.ca/](https://www.computecanada.ca/)]([https://www.computecanada.ca/](https://www.computecanada.ca/)) (accessed on 10 June 2022)) for providing access to high performance parallel computation facility through which the majority the numerical studies were conducted. We also would like to thank all three anonymous reviewers for their suggestions that helped us to improve the paper. The authors declare no conflict of interest.