## 4 Optimization  
The experimental data were analyzed statistically to develop regression equations that could explain the relationship between the input variables and the responses. The study utilized the response optimizer option, available within the DOE module of the statistical software package Design Expert v17, to optimize the input parameter combinations that yielded the most favorable tradeoff among the different responses. The optimization was carried out using the desirability function. The criteria for optimization, as outlined in Table 7, included achieving a minimum top kerr width, a minimum bottom kerr width, a minimum cut kerr angle, and a ratio of the top kerr width to the bottom kerr width equal to one. The optimization process in Table 8 assigned weight and importance values to the responses, and the optimal input parameter combinations were determined using the desirability function. The experiments were conducted at the optimal settings to compare the actual responses with those obtained from the optimization. The results are shown in Table 9. The errors were low and indicated a good suitability for engineering applications. The desirability plots of the input parameters are shown in Figure 11. The desirability plot for the laser power and the cutting speed showed that the optimal amount, with the highest desirability value, was 0.892. This indicated that the highest laser power values were preferred in the process to achieve the desired responses. Similarly, the desirability \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & & & & & & & \\ \hline \multirow{2}{*}{**Parameters/ Responses**} & \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Goal**} & \multirow{2}{*}{**Lower Limit**} & \multirow{2}{*}{**Upper Limit**} & \multirow{2}{*}{**Lower Weight**} & \multirow{2}{*}{**Upper Weight**} & \multirow{2}{*}{**Importance**} \\  & & A: LP & is in range & 40 & 60 & 1 & 1 & 3 \\  & B: CS & is in range & 4 & 16 & 1 & 1 & 3 \\  & C: FPP & is in range & \(-2\) & 2 & 1 & 1 & 3 \\ \hline \multirow{4}{*}{**Parameters/ Responses**} & Top Average Width & minimize & 4.436 & 13.453 & 1 & 1 & 3 \\  & Bottom Average Width & minimize & 0 & 9.653 & 1 & 1 & 3 \\  & Ratio & is target = 1 & 0.459 & 3.004 & 1 & 1 & 3 \\  & Cut Kerr Angle & minimize & 0.025 & 1.924 & 1 & 1 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Constraints and criteria of the input parameters and responses. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**\#**} & \multirow{2}{*}{**LP**} & \multirow{2}{*}{**CS**} & \multirow{2}{*}{**FPP**} & \multirow{2}{*}{**Desirability**} & \multirow{2}{*}{**Top Kerr Width**} & \multirow{2}{*}{**Bottom Kerr Width**} & \multirow{2}{*}{**Ratio**} & \multirow{2}{*}{**Cut Kerr Angle**} \\  & & **(W)** & & & & & & **(Degree)** \\ \hline
1 & 50.661 & 4.000 & \(-1.016\) & 0.896 & 4.012 & 2.201 & 1.000 & 0.341 \\
2 & 51.025 & 4.017 & \(-0.950\) & 0.894 & 4.269 & 1.966 & 1.000 & 0.400 \\
3 & 49.966 & 4.000 & \(-1.155\) & 0.892 & 3.430 & 2.751 & 0.998 & 0.230 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Output responses’ solutions for finding the optimum laser cutting parameters ordered by desirability. plot for the cutting speed and the focal plane position showed that the optimal range for the cutting speed was between 4 and 7 mm/s, with the highest desirability values located at the higher end of this range. This indicated that higher cutting speeds were preferred in the process to achieve the desired responses. On the other hand, the desirability plot for the focal plane position showed that the optimal range for the FPP was between \(-1\) and 1 mm, with the highest desirability values located at the center of this range. This indicated that a neutral or central focal plane position was preferred in the process to achieve the desired responses. To investigate the correlation between these input parameters, the overlay plots in Figure 12 were generated.