## 6 Tuning Up the Parameters
### Variance Analysis  
Variance analysis is applied to evaluate the statistical difference between the experimental results and to observe the effect of the parameters on the quality of the results. It is used to determine factors that have a significant effect and to discover the most important factors. The parameters of the problem are considered as factors and their values as levels. We assume that there is no interaction between factors.  
The \(F\)-Ratio is the ratio between Factor Mean Square and Mean Square residue. A high \(F\)-Ratio means that this factor significantly affects the response. The \(p\)-value shows the statistical significance of the factors: \(p\)-values that are less than 0.05 have a statistically significant effect on the response variable (RIBS) with a 95% confidence level. According to the \(F\)-Ratio and \(p\)-value, the most important factors in the case of two machines per stage are the mutation operator and the crossover operator variable. DF shows the number of degrees of freedom (Table 10).  
In the case of four machines per stage, the most important factors are the mutation operator and crossover probability (Table 11). For six machines per stage, the most important factors are the mutation operator and mutation probability (Table 12).  
Figures 14-17 show means and 95% Least Significant Difference (LSD) confidence intervals. Figure 14 shows the results obtained for mutation operators for two, four, and six machines per stage. We can see that operator Displacement is the best mutation among the three tested operators. Figure 15 shows the results for the crossover operators, where we observe that the PMX operator is the best. The results for four and six stages are similar.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **Sum of Squares** & **DF** & **Mean Square** & **\(F\)-Ratio** & \(p\)**-Value** \\ \hline A: Crossover & 1747 & 1 & 1747 & 9.755 & 0.00213 \\ B: Mutation & 11,967 & 1 & 11,967 & 66.833 & \(9.58\times 10^{-14}\) \\ C: Crossover probability & 1519 & 1 & 1519 & 8.481 & 0.00411 \\ D: Mutation probability & 654 & 1 & 654 & 3.652 & 0.05782 \\ E: Population & 1732 & 1 & 1732 & 9.673 & 0.00222 \\ Residuals & 27,934 & 156 & 179 & & \\ \hline \hline \end{tabular}
\end{table}
Table 10: Relative increment of the best solution (RIBS) analysis of variance for two machines per stage.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **Sum of Squares** & **DF** & **Mean Square** & **\(F\)-Ratio** & \(p\)**-Value** \\ \hline A: Crossover & 760 & 1 & 760 & 4.602 & 0.03349 \\ B: Mutation & 13,471 & 1 & 13,471 & 81.565 & \(6.09\times 10^{-16}\) \\ C: Crossover probability & 1540 & 1 & 1540 & 31.223 & \(1.00\times 10^{-7}\) \\ D: Mutation probability & 5157 & 1 & 5157 & 9.326 & 0.00266 \\ E: Population & 3395 & 1 & 3395 & 20.557 & \(1.15\times 10^{-5}\) \\ Residuals & 25,765 & 156 & 165 & & \\ \hline \hline \end{tabular}
\end{table}
Table 11: RIBS analysis of variance for four machines per stage.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Source** & **Sum of Squares** & **DF** & **Mean Square** & **\(F\)-Ratio** & \(p\)**-Value** \\ \hline A: Crossover & 2587 & 1 & 2587 & 18.39 & \(3.14\times 10^{-5}\) \\ B: Mutation & 17,785 & 1 & 17,785 & 126.47 & \(2\times 10^{-16}\) \\ C: Crossover probability & 2886 & 1 & 2886 & 20.52 & \(1.16\times 10^{-5}\) \\ D: Mutation probability & 4840 & 1 & 4840 & 34.42 & \(2.58\times 10^{-8}\) \\ E: Population & 2881 & 1 & 2881 & 20.49 & \(1.18\times 10^{-5}\) \\ Residuals & 21,937 & 156 & 165 & & \\ \hline \hline \end{tabular}
\end{table}
Table 12: RIBS analysis of variance for six machines per stage.