## 3 The Establishment and Analysis of BBD Experimental Design and Prediction Model
### BBD Experimental Design and Grey Relation Analysis
#### 3.1.2 Grey Relational Analysis  
GRA serves as a quantitative method employed to evaluate the interrelation among components within grey systems. The method is applicable to determining the correlation between various factors and various objectives and has found successful applications in hot forging [21], structural design [22; 23], laser welding [24; 25; 26], and other fields. By introducing GRA, multi-response problems can be transformed into corresponding GRDs. The magnitude of the relational degree can serve as a model output to evaluate the superiority or inferiority of the responses. However, the traditional grey relational theory fails to comprehensively consider the true contribution rate of each objective, leading to an inability to optimize the objectives. Therefore, it is necessary to calculate the response weighting based on weighting theory to determine the weights of each optimization objective and to calculate a set of GRDs, transforming the MPO to a GRD optimization task [27]. The flowchart for calculating the GRD is shown in Figure 5.  
Data Preprocessing  
Before conducting GRD analysis, data pre-processing is needed to reduce variation and ensure that the data are equivalent and in order. The forming load, equivalent stress, and damage value are normalized using Equation (9) as follows:  
\[x_{i}^{*}(k)=\frac{\max x_{i}^{0}(k)-x_{i}^{0}(k)}{\max x_{i}^{0}(k)-\min x_{i} ^{0}(k)} \tag{9}\]  
where \(x_{i}^{0}(k)\) and \(x_{i}^{*}(k)\) represent the original sequence and the comparison sequence; \(k=1\), \(2\), \(\ldots\), \(n\); \(i=1\), \(2\), \(\ldots\), \(m\). \(n\) represents the response count; and \(m\) represents the experiment frequency. In the experimental design, \(n=3\) and \(m=15\). The data are depicted in Table 4.  
Figure 5: Flow chart for computing the GRD. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Trial No.** & **F** & **T (\({}^{\circ}\)C)** & **V (mm/s)** & **Y (MN)** & **S (MPa)** & **Z** \\ \hline
9 & 0.5 & 1150 & 7.5 & 568 & 56.1 & 0.503 \\
10 & 0.5 & 1250 & 7.5 & 451 & 41.1 & 0.561 \\
11 & 0.5 & 1150 & 12.5 & 1170 & 57.4 & 0.525 \\
12 & 0.5 & 1250 & 12.5 & 820 & 38 & 0.426 \\
13 & 0.5 & 1200 & 10 & 759 & 47.9 & 0.575 \\
14 & 0.5 & 1200 & 10 & 753 & 50.2 & 0.594 \\
15 & 0.5 & 1200 & 10 & 765 & 51.2 & 0.587 \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._2. Calculation of Grey Correlation Coefficients  
The Grey Correlation Coefficients (GCC) serves as a measure of the link between ideal and actual normalized experimental outcomes, with its calculation proceeding as follows:  
\[\xi(x_{0}^{*}(k),x_{i}^{*}(k))=\frac{\Delta\text{min}+\rho\Delta\text{max}}{ \Delta_{0i}(k)+\rho\Delta\text{max}} \tag{10}\]  
where \(\xi(x_{0}^{*}(k),x_{i}^{*}(k))\) represents the coefficient of correlation, and \(\rho\in[0,1]\) represents the discrimination coefficient, which is generally set to 0.5. \(\Delta_{0i}(k)=\left|x_{0}^{*}(k)-x_{i}^{*}(k)\right|\) is the deviation sequence, \(\Delta\text{min}=\text{min}_{i}\text{min}_{k}\Delta_{0i}(k)\), and \(\Delta\text{max}=\text{max}_{i}\text{max}_{k}\Delta_{0i}(k)\). 3. Weight Calculation  
In GRD analysis, it is necessary to calculate the weights for each response optimization objective. Since each response optimization objective has a different impact on the quality of the forgings, the weight coefficients for each optimization goal are calculated using the entropy weight method. The entropy method is an objective weighting method. In practical calculations, if the entropy value for a certain indicator is smaller, differences in the values of an indicator's samples are larger, the indicator plays a greater role in comprehensive evaluation, and the corresponding index weight is also larger [28].