## 3 Orthogonal Experiments and Grey Relational Analysis
### Orthogonal Experimental Design  
In this experiment, an \(L_{16}(4^{5})\) orthogonal array was used for research and analysis. Blank holder force, stamping speed, friction coefficient, and die clearance were selected as experimental factors, while \"maximum thickening rate\" and \"maximum thinning rate\" were used as quality evaluation indicators. In this case, there was a blank column in theorthogonal test table, and it did not work in the test. The factor level table and orthogonal experimental table are shown in Tables 2 and 3, respectively. By analyzing the quality assessment indicators in Table 3, it is evident that both the \"maximum thickening rate\" and the \"maximum thinning rate\" do not meet the production requirements (with the \"maximum thickening rate\" being less than 5% and the \"maximum thinning rate\" being less than 30%). This indicates that wrinkling and cracking are the main issues affecting the quality of the engine hood. Therefore, it is necessary to conduct a comprehensive study on the \"maximum thickening rate\" and the \"maximum thinning rate\" to analyze the extent to which these parameters impact part quality. By adopting a comprehensive scoring method, multi-objective problems can be converted into single-objective problems, which can effectively solve multi-objective optimization issues. In this study, there were two quality evaluation indicators: \"maximum thickening rate\" and \"maximum thinning rate\". Both of these indicators were converted into their membership values, and these membership values were used to represent the scores of comprehensive scoring. The formula for calculating membership is as follows [15]:  
\[\mu_{i}=\frac{x_{i}-min(x)}{max(x)-min(x)} \tag{5}\]  
where \(\mu_{i}\) is the membership, \(x_{i}\) is the result of the \(i\)-th experimental scheme, and \(\min(x)\) and \(\max(x)\) are the minimum and maximum values among all experimental results, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Factor** & **Blank Holder Force** & **Stamping Speed** & **Friction Coefficient** & **Die Clearance** /**mm** & **Blank Column** \\ /**Level** & **/kN** & **/mm\(\cdot\)s\({}^{-1}\)** & **Friction Coefficient** & **Die Clearance** /**mm** & **Thickening Rate** /**\%** \\ \hline
1 & 150 & 1000 & 0.050 & 0.90 & 1 \\
2 & 150 & 3000 & 0.075 & 1.05 & 2 \\
3 & 150 & 5000 & 0.100 & 1.20 & 3 \\
4 & 150 & 7000 & 0.125 & 1.35 & 4 \\
5 & 300 & 1000 & 0.075 & 1.20 & 4 \\
6 & 300 & 3000 & 0.050 & 1.35 & 5.46 & 32.08 \\
7 & 300 & 5000 & 0.125 & 0.90 & 2 & 6.01 & 28.98 \\
8 & 300 & 7000 & 0.100 & 1.05 & 1 & 5.58 & 38.69 \\
9 & 450 & 1000 & 0.100 & 1.35 & 2 & 5.49 & 39.61 \\
10 & 450 & 3000 & 0.125 & 1.20 & 1 & 4.75 & 32.54 \\
11 & 450 & 5000 & 0.050 & 1.05 & 4 & 4.96 & 30.97 \\
12 & 450 & 7000 & 0.075 & 0.90 & 3 & 5.07 & 38.95 \\
13 & 600 & 1000 & 0.125 & 1.05 & 3 & 4.84 & 32.35 \\
14 & 600 & 3000 & 0.100 & 0.90 & 4 & 5.34 & 33.89 \\
15 & 600 & 5000 & 0.075 & 1.35 & 1 & 5.56 & 31.71 \\
16 & 600 & 7000 & 0.050 & 1.20 & 2 & 5.19 & 30.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Factor level table. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test Program** & **Blank Holder Force** & **Stamping Speed** & **Friction Coefficient** & **Die Clearance** /**mm** & **Maximum** \\
**/**Level** & **/kN** & **/mm\(\cdot\)s\({}^{-1}\)** & **Friction Coefficient** & **Die Clearance** /**mm** & **Thickening Rate** /**\%** \\ \hline
1 & 150 & 1000 & 0.050 & 0.90 & 1 & 5.32 & 35.26 \\
2 & 150 & 3000 & 0.075 & 1.05 & 2 & 4.86 & 29.78 \\
3 & 150 & 5000 & 0.100  
Since wrinkling and cracking are both major quality defects of equal importance, the weights of the \"maximum thickening rate\" and \"maximum thinning rate\" were both set at 0.5.