# Parameters Adjustment Optimization for Prepreg of Copper Clad Laminate Based on Virtual Metrology
## 5 Experimental Results
### Optimizing Adjustment of Input Variables  
Once the VM models were developed, the mathematical model proposed in Section 4 could be adopted to optimize adjustment of input variables. This research tested the mathematical model on the 33.3% of test data using the software Excel Solver. The average normalized deviation of output variables for ratio of resin and gel time 2 with the test data were 0.227 and 0.266, respectively. Based on the adjustment using the results of the mathematical model, the average normalized deviation of output variables for ratio of resin and gel time 2 in the test data were \(5.642\times 10^{-6}\) and 0.117, respectively. The results are summarized in Figure 6 by box plots.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Ratio of Resin**} & \multicolumn{2}{c}{**Gel Time of Resin in Prepregs**} \\ \cline{3-6}  & & **Training** & **Testing** & **Training** & **Testing** \\ \hline
1 & LR & 2.01\% & 1.89\% & 3.60\% & 3.94\% \\
2 & SVR & 2.04\% & 1.74\% & 3.42\% & 3.31\% \\
3 & LR+SVR & 1.85\% & 1.73\% & 3.07\% & 3.12\% \\
4 & 2 nd order+LR & 2.07\% & 1.98\% & 3.51\% & 2.93\% \\
5 & 2 nd order+SVR & 2.13\% & 1.72\% & 3.30\% & 2.73\% \\
6 & 2 nd order+LR+SVR & 1.54\% & 1.41\% & 2.72\% & 2.59\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: The training and testing results.  
Figure 5: Validation results for LR+SVR with different \(\lambda\) and C.  
Figure 6: Performance of proposed mathematical model.  
On average, the proposed mathematical model provided better input variables which can produce output variables that are closer to target values. However, as shown in Figure 6, the proposed mathematical model produced results for output variables of gel time 2 that were far from the target values in some test data. This research then modified the mathematical model proposed in Section 4. Firstly, in order to minimize the deviation of output variables, Equation (15) was added to ensure that the resulting output variables were equal to the target variables. Thus, Equation (12) is redundant in the modified model.  
\[y_{k}=y_{k}^{l}\hskip 28.452756 pt\forall k \tag{15}\]  
Then the objective function was replaced by Equation (16), in order to minimize the normalization of adjustment of all input variables. As the scales of input variables were quite different, this research normalized the adjustment of input variables by adding Equation (17).  
\[minimize\sum_{j=1}^{J}x_{j}^{n} \tag{16}\]  
\[x_{j}^{n}=\frac{\left|x_{j}^{c}-x_{j}\right|}{\underset{\forall i}{max}\big{(} x_{ij}\big{)}-\underset{\forall i}{min}\big{(}x_{ij}\big{)}}\hskip 28.452756 pt \forall j \tag{17}\]  
From Equation (16), the deviation of output variables of the modified mathematical model was reduced to zero. The results for the adjustment of input variables were illustrated in Figure 7. In Figure 7, the mathematical model proposed in Section 4 is called mathematical model 1, and the mathematical model modified in this section is called mathematical model 2. In Figure 7, input variables \(X_{2}\), \(X_{3}\), \(X_{4}\), and \(X_{12}\) were hard to adjust, and their corresponding adjustment for all instances was very close to zero. Moreover, the average adjustment of input variables produced by mathematical model 2 were smaller than those of mathematical model 1; although, in rare instances, the adjustments of input variables \(X_{1}\) and \(X_{11}\) produced by mathematical model 2 were bigger. Mathematical model 2 was more robust in providing values of input variables with less adjustment.