## 2 Methods
### Design of Experiments (DOE)  
The design of experiments (DOE), or factorial design, is a regression analysis extensively employed in industry to identify and control the critical parameters for enhancing product quality [32]. Experimental studies are normally time-consuming and require examining the effects of two or more factors. Factorial design allows for the investigation of all possible combinations of the levels of the specified factors (see Figure 3). Due to the time and expense of the experiments, the selection of the factors to be analyzed must be made with careful consideration. One of the main reasons to apply the DOE is to minimize the number of experimental trials and to derive a transfer function. Another benefit of using the DOE is the seamless integration of experimental data with numerical methods, such as the finite element method (FEM), to simulate alterations in physical parameters (factors). A complete factorial design incorporates all possible \"levels\" of a given number of \"factors\". These levels are categorized as \"low\" (in red) and \"high\" (in blue) and are represented by the symbols \"\(-\) \" and \"\(+\)\", respectively (see Figure 3). The effect of a factor is defined as the change in the response caused by altering the level of that factor. This is referred to as the main effect because it represents the primary influence in the experiment. The main effect of a factor can be thought of as the difference between the average response at the low level minus the average response at the high level [33]. Furthermore, the difference in response in some experiments between levels of one factor may vary across the levels of the other factors. When this variation occurs, it indicates an interaction between the factors. Analyzing the main effects and interactions between factors results in the achievement of one or more of the following objectives:  
* Determining the optimal condition within a specified range;
* Assessing the contribution of individual factors and their interactions;
* Predicting the response under optimal conditions. The estimation of the contribution of each factor and their interactions will be determined within a specified range or limit. In this study, the parameter range is set to derive a response function (\(\hat{y}\)), which is defined by the parameters or factors (\(x^{\prime}s\)), the coefficients (\(b_{i}\)), and the error term (\(e_{error}\)); see Equations (7)-(9):  
\[\hat{y}=b_{0}+\sum_{i=0}^{k}b_{1}x_{1}+\sum_{i=0}^{k}b_{2}x_{2}+\sum_{i=0}^{k }b_{12}x_{1}x_{2}+\ldots+e_{error} \tag{7}\]  
where  
\[Y=Xb+e_{error} \tag{8}\]  
Figure 3: DOE geometric representation. In each case, high levels are highlighted in blue, low levels in red [32]. \[Y= \begin{bmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{bmatrix} ,\quad X= \begin{bmatrix}a_{11}&a_{12}&\ldots&a_{1k}\\ a_{21}&a_{22}&\ldots&a_{2k}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nk}\end{bmatrix} ,b\quad= \begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{n}\end{bmatrix} ,e\quad= \begin{bmatrix}e_{error_{1}}\\ e_{error_{2}}\\ \vdots\\ e_{error_{n}}\end{bmatrix}  \tag{9}\]  
The response of the \(COP\) is represented by \([Y]\), where \([X]\) is an \(n\times p\) matrix of independent variables (defined as A,B,C,D, and E in Table 2). The vector \(b\) contains the regression coefficients, and \(e_{error}\) is a vector of random errors. The system's \(COP\) is calculated in the following section, using the dimensionless parameters expressed in Equations (3) and (4). Additionally, once the parameters are established, the DOE is primarily analyzed in three stages, encompassing all the theoretical calculation. These stages are as follows:  
* Planning stage;
* Analysis stage;
* Results stage. The most crucial part of the DOE is the planning stage. The development of an orthogonal matrix is vital, to account for the effects of various factors influencing the \(COP\) response.