## 3 Ore Bonding Parameter Determination Method  
Response Surface Methodology (RSM) is a statistical modeling technique that aims to establish relationships between input variables and output variables [26]. This methodology is widely utilized for optimizing experimental design and analysis [27], with the objective of identifying the optimal combination of input parameters that yields the desired output results. In Response Surface Methodology (RSM), a mathematical model is typically constructed to describe the impact of input variables on output variables. This mathematical model is frequently based on a multiple linear regression model. A widely adopted re  
Figure 2: Particle filling diagram. sponse surface model is the second-order polynomial regression model, which has the following general form:  
\[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{11}x_{1}^{2}+\beta_{22}x_{2}^{2 }+\beta_{12}x_{1}x_{2}+\varepsilon \tag{7}\]  
where: \(y\) is the response value; \(x_{1}\) and \(x_{2}\) are the input variables; \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), \(\beta_{11}\), \(\beta_{22}\), \(\beta_{12}\) are the model parameters; \(\varepsilon\) is the error. Let \(x_{1}^{2}=x_{3}\), \(x_{2}^{2}=x_{4}\), \(\beta_{11}=\beta_{3}\), \(\beta_{12}=\beta_{5}\), \(\beta_{22}=\beta_{4}\) and \(x_{1}x_{2}=x_{5}\), be expressed as follows:  
\[y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{1}^{2}+\beta_{4}x_{2}^{ 2}+\beta_{5}x_{1}x_{2}+\varepsilon \tag{8}\]  
For systems involving m input variables, a multivariate linear regression equation can be employed to characterize the general response surface. If \(n\) predictions are needed (\(n>m\)), \(y_{s}\) is the \(s\)th prediction value (\(1\leq s\leq n\)), and \(x_{sh}\) is the regression value \(x_{h}\) at the \(s\)th level. The data derived from the linear regression analysis, meticulously documented in Table 1, forms the cornerstone for the system's model development. In this system, the observation errors conform to a normal distribution with mean 0 and variance \(\sigma^{2}\), and \(\{\epsilon_{s}\}\) are mutually independent random variables. From Table 1, for the predicted value \(y_{s}\), the multiple linear regression model it fits is as follows:  
\[y_{i}=\beta_{0}+\beta_{1}x_{s1}+\beta_{2}x_{s2}+\cdots+\beta_{k}x_{sm}+ \varepsilon_{s}=\beta_{0}+\sum_{h=1}^{m}\beta_{j}x_{sh}+\varepsilon_{s,s}(1 \leq s\leq n) \tag{9}\]  
By rewriting Formula (9) into matrix form, we get the following:  
\[y=X\beta+\varepsilon \tag{10}\]  
Therefore, the matrix representation is as follows:  
\[y= \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{bmatrix} ,X= \begin{bmatrix}1&x_{11}&x_{12}&\cdots&x_{1m}\\ 1&x_{21}&x_{22}&\cdots&x_{2m}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&x_{n1}&x_{n2}&\cdots&x_{nm}\end{bmatrix} ,\ \beta= \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \vdots\\ \beta_{m}\end{bmatrix} ,\varepsilon= \begin{bmatrix}\varepsilon_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{bmatrix} \]  
\(y\) employing the least squares method to solve Equation (10), the minimum value of the squared error \(\varepsilon\) can be determined, thereby identifying the optimal fitting surface. This method serves to approximate the actual surface by minimizing the deviations between the predicted and empirical data points, ensuring an accurate and precise representation of the underlying phenomenon.