## 3 Orthogonal Experiments and Grey Relational Analysis
### Grey Relational Analysis  
If the orthogonal experimental method is used to determine the key process parameters affecting the \"maximum thickening rate\" and \"maximum thinning rate\", it may be difficult to effectively distinguish whether the variations in the \"maximum thickening rate\" and \"maximum thinning rate\" caused by different levels of specific process parameters are due to differences in the selected parameter levels or experimental errors. In such cases, the differences in experimental results may not accurately reflect the actual impact of the process parameter levels but may instead be influenced by errors, thereby affecting the reliability of the research conclusions. Therefore, in the research process, the influence of experimental errors must be comprehensively considered. The grey relational analysis method should be used to evaluate the degree of association of blank holder force, stamping speed, friction coefficient, and die clearance with the \"maximum thickening rate\" and \"maximum thinning rate\". A sequence is set as follows [16]:  
\[x=[x(1),x(2),\cdots,x(n)] \tag{7}\]  
Interval transformation is performed [16]:  
\[f[x(k)]=\frac{x(k)-\frac{min}{k}x(k)}{\underset{k}{max}x(k)-\underset{k}{min }x(k)}=y(k) \tag{8}\]  
A reference sequence \(x_{0}=\{x_{0}(k)\ |\ k=1,\ 2,\ \ldots,\ n\}\) is selected, where \(k\) represents the time. If there exists a set of comparison sequences \(x_{i}=\{x_{i}(k)\ |\ k=1,\ 2,\ \ldots,\ n\}\), \(i=1,\ 2,\ \ldots m\), the following is said [16]:  
\[\xi_{i}=\frac{\underset{s}{\overset{min\,min\,}{t}}|x_{0}(t)-x_{s}(t)|+ \underset{s}{\overset{max\,}{t}}|x_{0}(t)-x_{s}(t)|}{|x_{0}(k)-x_{i}|+ \underset{s}{\overset{max\,max\,}{t}}|x_{0}(t)-x_{s}(t)|} \tag{9}\]  
At \(k\) time, \(x_{i}\) is the relational degree of comparison sequence to the reference sequence \(x_{0}\), \(\rho\) is the resolution coefficient, \(\underset{s}{\overset{min\,}{min\,}}|x_{0}(t)-x_{s}(t)|\) is the minimum difference between two levels, and \(\underset{s}{\overset{max\,}{t}}|x_{0}(t)-x_{s}(t)|\) is the maximum difference between two levels. The relational degree defined in Equation (7) describes the correlation of the comparison sequence with the reference sequence at a certain time. However, there are different relational degrees at different times; therefore, a relative degree of the comparison sequence to the reference sequence is given as follows [16]:  
\[r_{i}=\frac{1}{n}\underset{k=1}{\overset{n}{\underset{k=1}{\sum}}}\zeta_{i} (k) \tag{10}\]  
In the formula, \(n\) represents the total number of reference sequences; \(r_{i}\) represents the value of relational degree. The larger the relational degree, the closer the comparison sequence is to the reference sequence, meaning a greater impact. Based on the above, the \"comprehensive scores\" of the 16 sets of process parameter combinations were used as the reference array, and the level values of the 16 sets of process parameters were used as the comparison array. These values were then substituted into Equations (8) to (10) in sequence for computational analysis. The selected data normalization method was min-max normalization to maintain the relative relationships and differences of the original arrays. A discrimination coefficient of 0.5 was chosen, which is a commonly used value that ensures sensitivity to differences in data analysis. MATLAB software ([[https://www2.mathworks.cn/products/matlab.html](https://www2.mathworks.cn/products/matlab.html)]([https://www2.mathworks.cn/products/matlab.html](https://www2.mathworks.cn/products/matlab.html))) was used to solve the correlation degree of each process parameter with respect to the \"maximum thickening rate\" and \"maximum thinning rate\", as shown in Table 6.  
According to Table 6, the blank holder force and die clearance have a significant impact on the \"maximum thickening rate\" and \"maximum thinning rate\" of the part, whereas the stamping speed and friction coefficient have a lesser impact on the \"maximum thickening rate\" and \"maximum thinning rate\" of the part.