## 6 Tuning Up the Parameters
### Pareto Front Calibration Analysis  
Figures 18-20 show the solution space obtained by calibration experiments. The horizontal axis represents the energy \(E_{op}\) consumed by the machines. The vertical axis represents the time completion \(C_{max}\) of jobs. To determine the best individuals of all experiments, the Pareto front is calculated for each case. Each point represents a combination of parameters from 162 experiments explained in Section 6.1. Each front consists of numbered points from lowest to highest according to their DI. The closer to 1, the better the place in the enumeration is (see Section 3.3).  
Figure 19: Pareto front for four machines per stage.  
Figure 17: Means and 95% LSD confidence intervals of mutation probability—six machines per stage.  
Tables 13-15 show the ordering of the front points regarding a combination of \(E_{op}\), \(C_{max}\), and Desirability Index (DI) for each Pareto front. We see, for example, that the parameter combination 156 corresponds to a higher DI, followed by 129, etc.  
Table 16 shows the parameters of these combinations. The crossover PMX has higher values of DI. The mutation operator Displacement is used in each of the top three combinations. The crossover probability is maintained at 0.9 in the two most significant combinations. The mutation probability remains at the highest with 0.2 in all three, as does the population with 50 individuals.  
The most influential parameters correspond to the combination 156, which has greater DI in three cases (see Tables 13-15).  
Our analysis shows that crossover PMX and mutation operator Displacement are the best among those tested. The best probability of crossover is 0.9 and that of mutation is 0.2. The population of 50 individuals is statistically more significant.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Comb.** & \(E_{op}\) & \(C_{max}\) & **DI** \\ \hline
1 & 156 & 6993.41 & 97.03 & 0.843 \\
2 & 129 & 6968.70 & 97.15 & 0.816 \\
3 & 128 & 7014.08 & 97.01 & 0.782 \\
4 & 120 & 7046.87 & 96.94 & 0.677 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Optimum parameter combination selection for bi-objective genetic algorithm (GA) for two machines per stage.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**N** & **Comb.** & \(E_{op}\) & \(C_{max}\) & **DI** \\ \hline
1 & 156 & 17,018.73 & 26.07 & 0.891 \\
2 & 129 & 16,961.46 & 26.12 & 0.883 \\
3 & 120 & 17,108.33 & 26.05 & 0.847 \\
4 & 147 & 16,953.30 & 26.19 & 0.823 \\
5 & 138 & 17,189.52 & 26.01 & 0.816 \\
6 & 75 & 16,925.85 & 26.23 & 0.796 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Optimum parameter combination selection for bi-objective GA for six machines per stage.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**N** & **Comb.** & \(E_{op}\) & \(C_{max}\) & **DI** \\ \hline
1 & 156 & 17,018.73 & 26.07 & 0.891 \\
2 & 129 & 16,961.46 & 26.12 & 0.883 \\
3 & 120 & 17,108.33 & 26.05 & 0.847 \\
4 & 147 & 16,953.30 & 26.19 & 0.823 \\
5 & 138 & 17,189.52 & 26.01 & 0.816 \\
6 & 75 & 16,925.85 & 26.23 & 0.796 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Optimum parameter combination selection for bi-objective GA for six machines per stage.  
Figure 20: Pareto front for six machines per stage.