## 3 Results and Discussion
### Cutting Parameter Optimization  
Once the surface roughness prediction model was built and validated as stated in Section 3.1, the inverse model was utilized for the optimization of cutting parameters. The PSO algorithm [24] was applied to search for the optimal combinations of cutting parameters for different schemes. In this research, the cutting parameter optimization was based on the scenario that the feed rate is maximized to accelerate the milling process subjected to different Ra requirements. Therefore, the fitness function of the optimization process \(J(x)\) is formulated as  
\[J(x)\ =\ f(x)-\alpha|R(x)-R_{\text{exp}}| \tag{3}\]  
where \(f(x)\) represents the feed rate, \(R(x)\) represents the Ra prediction value, \(R_{exp}\) is the expected Ra value, and \(\alpha\) represents the adjusting parameter for weighting the difference between the predicted and expected Ra values.  
The cutting parameters' range (values of lower and upper bounds) was set according to the tool manufacturer's specifications as shown in Table 6 to constrain the searching boundary in the optimization process. As the aforementioned in [1], the surface roughness of work-piece is affected mainly by the cutting speed, feed rate and cutting depth. Furthermore, the clamping force amount of fixture changes the boundary conditions between the vise and work-piece and thus influences the vibration behaviors as well as hereafter the surface roughness of work-piece. In this research, the Ra values of Inconel 718 work-piece were classified to four levels (0.4, 0.6, 0.8 and 1.0 \(\upmu\)m) to represent the different milling quality requirements. With the constraints of Ra values, the feed rate is maximized to decrease the consuming cutting time. Three sets of cutting parameter combinations were obtained as the optimization results of PSO algorithm as shown in Table 7. The selected optimal combinations of cutting parameters were then utilized for the experiment validation. Table 8 shows the experiment results through using the selected optimal combinations of cutting parameters. A reasonable result can be observed that the maximized feed rate has to be decreased apparently for achieving the requirements of fine surface conditions (Ra = 0.6 and 0.4 \(\upmu\)m), as compared with the milling requirements of rough surface conditions (Ra = 1 and 0.8 \(\upmu\)m). For the imprecise requirement of rougher surface quality (Ra = 1.0 and 0.8 \(\upmu\)m), the factor of cutting depth does not affect the Ra value of Inconel 718 work-piece, and thus the optimal values of cutting speed and feed per tooth are set to the values of upper bound since the feed rate needs to be maximized. However, the cutting depth becomes one of the crucial factors for the strict requirement of finer surface quality (Ra = 0.4 and 0.6 \(\upmu\)m), and therefore optimization process tried to search the parameters combination for achieving the maximal feed rate as well as the fine Ra restriction. Through the cutting parameter optimization, it is noted that the predicted Ra values of the Elman neural network are closed to the measured Ra values of actual milling process with the overall accurate MAPE value of 10.38%. Furthermore, the examination was carried out to verify that the feed rate is maximized by using the computed optimal combinations of cutting parameters. The verification result is shown in Table 9. The result indicates that the feed rates can be maximized by using the optimized cutting parameters for the required Ra values of 1, 0.8 and 0.6 \(\upmu\)m since the feed rate errors are less than 1%, while the feed rate could be further accelerated by 24% for the required Ra value of 0.4 \(\upmu\)m.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cutting Parameter** & **Lower Bound** & **Upper Bound** \\ \hline Cutting speed (m/min) & 30 & 45 \\ Feed per tooth (mm/tooth) & 0.05 & 0.25 \\ Cutting depth (mm) & 0.5 & 1.5 \\ Clamping force of vise (N) & 9807 & 62,765 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Cutting parameters range.