# A Novel Method of Calibrating Micro-Scale Parameters of PFC Model and Experimental Validation
## 3 BB-DFO Experimental Design Method
### Desirability Function Optimization Method  
Actually, the response surface methodology is only applicable to optimizing single response variable. If multiple responses are required to optimize at the same time, the global solution of the independent variables cannot be derived directly using this method. Consequently, it is necessary to develop a multi-objective optimization approach. For this purpose, a desirability function method proposed by [PERSON] et al. [39] was used to optimize the micro-scale parameters of PFC model. The individual desirability function of a response variable can be expressed as  
\[d_{j}\big{(}\hat{y}_{j}(x)\big{)}=\left\{\begin{array}{ll}0&\hat{y}_{j}(x)<L_ {j}\\ \left(\frac{\hat{y}_{j}(x)-L_{j}}{U_{j}-L_{j}}\right)^{8}&L_{j}\leq\hat{y}_{j} (x)\leq U_{j}\\ 1&\hat{y}_{j}(x)>U_{j}\end{array}\right. \tag{6}\]  
\[d_{j}\big{(}\hat{y}_{j}(x)\big{)}=\left\{\begin{array}{ll}1&\hat{y}_{j}(x)<L _{j}\\ \left(\frac{U_{j}-\hat{y}_{j}(x)}{U_{j}-L_{j}}\right)^{t}&L_{j}\leq\hat{y}_{j} (x)\leq U_{j}\\ 0&\hat{y}_{j}(x)>U_{j}\end{array}\right. \tag{7}\]  
\[d_{j}\big{(}\hat{y}_{j}(x)\big{)}=\left\{\begin{array}{ll}0&\hat{y}_{j}(x)<L _{j}\\ \left(\frac{\hat{y}_{j}(x)-L_{j}}{T_{j}-L_{j}}\right)^{8}&L_{j}\leq\hat{y}_{j} (x)<T_{j}\\ 1&\hat{y}_{j}(x)=T_{j}\\ \left(\frac{U_{j}-\hat{y}_{j}(x)}{U_{j}-T_{j}}\right)^{t}&T_{j}<\hat{y}_{j}(x) \leq U_{j}\\ 0&\hat{y}_{j}(x)>U_{j}\end{array}\right. \tag{8}\]  
where \(d_{j}(\hat{y}_{j})\) (\(j=1\), \(2\), \(3\), \(\ldots\), \(K\)) denotes the desirability of the response variable \(\hat{y}_{j}\) whose value is between \(0\) to \(1\); \(d_{j}(\hat{y}_{j})=0\) means poor response and \(d_{j}(\hat{y}_{j})=1\) stands for ideal response. \(L_{j}\), \(U_{j}\), and \(T_{j}\) represent the lower limit, upper limit, and optimal target value of the response variable, respectively. \(s\) and \(t\) are the weights set by the analyst to determine how important they are for \(\hat{y}_{i}\) to be close to the maximum and the minimum, respectively. In this work, their values were both set to \(1\).  
Note that Equation (6) is suitable for the larger-the-better typed response variable, which indicates that the larger the response variable is, the higher the desirability will be. For Equation (7), the situation is opposite. By contrast, Equation (8) is applicable to the situation where the specific target value is required; that is, the closer the response variable to the set target value, the higher the desirability. Thus, the overall desirability of the multiple response variables can be calculated using the following equation:  
\[D=\left(\prod_{j=1}^{K}d_{j}^{\tau}\right)^{\frac{1}{\tau_{1}+\tau_{2}+\tau_{K }}} \tag{9}\]where \(D\) is the overall desirability of the response variables. Clearly, the closer \(D\) is to 1, the more satisfactory is the result. \(\tau_{j}\) (\(j=1\), \(2\), \(3\),..., \(K\)) is the importance of each response variable relative to the others, with a value ranging from 1 to 5. The larger the value, the more significant the response variable.  
In this study, there are three independent variables (\(\sigma_{c}\), \(E_{c}\), and \(k_{\text{n}}\)/\(k_{\text{s}}\), denoted by \(A\), \(B\), and \(C\), respectively) and three response variables (uniaxial compressive strength \(y_{1}\), [PERSON]'s modulus \(y_{2}\), and [PERSON]'s ratio \(y_{3}\)). In other words, we need to determine the values of \(A\), \(B\), and \(C\) by the proposed BB-DFO experimental design method based on the laboratory test results of \(y_{1}\), \(y_{2}\), and \(y_{3}\). Figure 2 illustrates the technical route of multi-objective optimization of these micro-scale parameters, which is composed of four parts: design of experiment, modeling, analysis of variance, and multiple optimization.