## 3 The Establishment and Analysis of BBD Experimental Design and Prediction Model
### Establishment and Analysis of Grey Relational Degree Prediction Model  
The optimization scope of GRD analysis based on entropy weight is limited, only capable of point-to-point analysis and evaluation of experimental samples, and unable to conduct continuous analysis within a specific range [30]. Therefore, we introduce the RSM to establish the GRD mathematical framework, identifying connections between different parameters and their corresponding responses, and refine the response variables to increase the prediction accuracy. The RSM may be fitted using a single-degree or multi-degree polynomial. Given the few design variables in this paper, a second-order polynomial is selected as the RSM [31; 32]. The friction factor, initial speed, and downward pressing speed are taken as model input variables, and the GRD is taken as the objective function to fit their functional relationship as follows:  
\[Y=a_{0}+\sum_{i=1}^{n}a_{i}X_{i}+\sum_{i=1}^{n}a_{ii}X_{i}^{2}+\sum_{ij(i<j)}^{ n}a_{ij}X_{i}X_{j}+\varepsilon \tag{16}\]  
where \(a_{0}\) denotes the invariant component; \(a_{i}\), \(a_{ii}\), and \(a_{ij}\) represent the coefficients of the first-order, second-order, and interaction terms, respectively; \(X_{i}\) and \(X_{j}\) stand for the fitted equation's variable; and \(\varepsilon\) is the error in the GRD model, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Trial No.** & **GCC of Y** & **GCC of S** & **GCC of Z** & **GRD** \\ \hline
1 & 0.547312642 & 0.409090909 & 0.538461538 & 0.491580957 \\
2 & 0.356333169 & 0.333333333 & 0.497461929 & 0.382923816 \\
3 & 0.853600945 & 0.689189189 & 0.541436464 & 0.711447787 \\
4 & 0.517537581 & 0.583969466 & 0.593939394 & 0.562468419 \\
5 & 1 & 1 & 0.378378378 & 0.843662162 \\
6 & 0.813273341 & 0.448680352 & 0.549019608 & 0.605679581 \\
7 & 0.569739953 & 0.472718447 & 0.662162162 & 0.659944113 \\
8 & 0.405951713 & 0.653846154 & 0.3333333333 & 0.483648128 \\
9 & 0.749222798 & 0.392307692 & 0.717948718 & 0.603195529 \\
10 & 0.989056088 & 0.6375 & 0.592145015 & 0.753145591 \\
11 & 0.3333333333 & 0.379652605 & 0.66440678 & 0.434528495 \\
12 & 0.492171545 & 0.732057416 & 1 & 0.712750222 \\
13 & 0.53674833 & 0.496753247 & 0.568115942 & 0.529155188 \\
14 & 0.541573034 & 0.46223565 & 0.538461538 & 0.510078991 \\
15 & 0.53200883 & 0.448680352 & 0.549019608 & 0.504030587 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Grey Correlation Coefficients and GRD. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & \(\mathbf{e_{k}}\) & \(\mathbf{D_{k}}\) & \(\mathbf{\omega}\) \\ \hline Y & 0.940058 & 0.059942 & 0.3614 \\ S & 0.935803 & 0.064197 & 0.3871 \\ Z & 0.95829 & 0.04171 & 0.2515 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of entropy weight method for calculating weights. Based on the data in Table 5, a second-order polynomial RSM for GRD was fitted in Design-Expert 11 as follows:  
\[ \begin{array}{l}R_{1}=3.15078-0.08137\times F-0.00016\times T-0.70519\times V-0.0 01\times F\times T\\ +0.03084\times F\times V+0.00026\times T\times V+0.56265\times F^{2}+0.0178\times V ^{2}\end{array}  \tag{17}\]  
where \(F\) is the friction coefficient, \(T\) is the initial temperature, and \(V\) is the downward pressure velocity. The response surface plot of the second-order polynomial regression equation was drawn, illustrated in Figure 6. The interaction among variables and their impact on GRD can be reflected in the shape of the response surface and contour plots [33]. The steep and large slopes in the response surface plot indicate a strong interaction impact on two values. The elliptical shape of the contour plot indicates a notable interaction between the variables. From Figure 6a, it is apparent that with a smaller friction factor and higher temperature, the GRD value is larger. At the same friction coefficient, the GRD response value gradually increases with temperature. From Figure 6b, it can be seen that the GRD values are higher under conditions of lower pressing speeds and lower friction coefficients.