## 3 Results and Discussion
### Multi-Objective Optimization Based on Grey Relational Analysis
\begin{table}
\begin{tabular}{c c c c} \hline
**Run** & **GRG** & **S/N of GRG** & **Rank** \\ \hline
1 & 0.352256 & \(-\)9.062842 & 15 \\
2 & 0.389364 & \(-\)8.192876 & 12 \\
3 & 0.497816 & \(-\)6.058616 & 8 \\
4 & 0.587804 & \(-\)4.615348 & 6 \\
5 & 0.568405 & \(-\)4.906848 & 7 \\
6 & 0.931034 & \(-\)6.620685 & 1 \\
7 & 0.483295 & \(-\)6.315751 & 9 \\
8 & 0.833333 & \(-\)1.386325 & 2 \\
9 & 0.358985 & \(-\)8.898844 & 13 \\
10 & 0.621265 & \(-\)4.134458 & 5 \\
11 & 0.403068 & \(-\)7.929424 & 10 \\
12 & 0.355488 & \(-\)8.983491 & 14 \\
13 & 0.833241 & \(-\)1.584587 & 3 \\
14 & 0.337255 & \(-\)9.440835 & 16 \\
15 & 0.801387 & \(-\)1.923150 & 4 \\
16 & 0.398681 & \(-\)7.987490 & 11 \\ \hline \end{tabular}
\end{table} 
Table 10: Grey relational grade and the corresponding S/N conversion. \begin{table}
\begin{tabular}{c c c c c} \hline
**Run** & \multicolumn{2}{c}{**Normalization**} & \multicolumn{2}{c}{**Grey Relational Coefficient**} \\ \cline{2-5}
**Run** & **S/N of** & **S/N of Wear** & **S/N of** & **S/N of Wear** \\  & **Micro-Hardness** & **Volume** & **Micro-Hardness** & **Volume** \\ \hline
1 & 0.068966 & 0.092000 & 0.349398 & 0.355114 \\
2 & 0.267241 & 0.160000 & 0.405594 & 0.373134 \\
3 & 0.612069 & 0.344000 & 0.563107 & 0.432526 \\
4 & 0.801724 & 0.412000 & 0.716049 & 0.459559 \\  
Figure 5 shows the Anderson-Darling test for normality on the S/N of GRG, its p-value is larger than 0.05. Thus, the normal distribution hypothesis is accepted. The ANOVA result is showing in Table 11 below. From the _p_-value of each processing parameters, it can conclude that laser power, scanning speed and SiC powder ratio had significant influence on the grey relational grade, when larger micro-hardness and less wear volume were desired concurrently. Gas flow displayed statistically insignificant based on its _p_-value. The average of S/N GRG for each of the four processing parameters in their four different levels were shown in Table 12. Delta stands for the absolute difference between maximum and minimum for each factor. From the sorting rank (largest to smallest), it can conclude that the influence on GRG was following the order of SiC powder ratio, scanning speed, laser power, then gas flow as the last. The maximum value in each category provided the possible optimal set point to achieve the desired GRG, which was LP\({}_{\text{level}}\) 4, SS\({}_{\text{level}}\) 1, GF\({}_{\text{level}}\) 2, PR\({}_{\text{level}}\) 4, referring to 1800 W laser power, 6 mm/s scanning speed, 1100 L/h gas flow, and 30 % SiC powder ratio. Since this set of processing parameters is not in the original L16 Taguchi orthogonal design setup, experimental validation is necessary to verify this prediction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Source** & **DF** & **Seq SS** & **Adj SS** & **Adj MS** & **F Value** & **Contribute** & _p_**Value** \\ \hline LP & 3 & 7.458 & 7.458 & 2.4861 & 22.65 & 5.28\% & 0.015 \\ SS & 3 & 9.900 & 9.900 & 3.2999 & 30.06 & 7.01\% & 0.010 \\ GF & 3 & 2.279 & 2.279 & 0.7597 & 6.92 & 1.61\% & 0.073 \\ PR & 3 & 121.248 & 121.248 & 40.4161 & 368.19 & 85.86\% & \textless{}0.001 \\ Error & 3 & 0.329 & 0.329 & 0.1098 & & 0.23\% & \\ Total & 15 & 141.215 & & & & 100\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 11: ANOVA analysis on S/N of GRG. Figure 5: Test for normality on S/N of grey relational grade (GRG).