## 2 Materials and Methods
### Robust Optimization Approach  
The CQAs of CKD-497 are selected through QFD, and a robust optimization design based on DoE is performed. As a first step, the significance of the effects of IVs on ORVs is statistically evaluated through a full-factorial experiment. Second, based on the response surface method, an optimal combination of IVs that satisfies the target value of ORVs is derived. To this end, we first derive a DS based on the overlaid contour plot. Based on this, the response surface design and analysis are performed. At this time, the optimal solution is derived by applying the weighting scenario that gives priority to the ORVs. Finally, RE is conducted to compare the results of the existing experiments with the optimum predictive value and the RE. Figure 3 is a schematic of the robust optimal approach used here.  
In the case of RSM, which is used for optimally deriving, polynomial regression of the second-order model is mainly used, and it has the advantage of producing curvature effects and precision optimization through repetition and axial points of the center point. The RSM regression formula is as follows  
\[\hat{y}(\mathrm{X})=\hat{\beta}_{0}+\sum\
olimits_{i=1}^{n}\hat{\beta}_{i}x_{ i}+\sum\
olimits_{i=1}^{n}\sum\
olimits_{j=1}^{n}\hat{\beta}_{ij}x_{i}x_{j},\ i<j, \tag{1}\]  
where \(\hat{\beta}\) denotes the estimated coefficient in the regression functions, and \(\mathrm{X}\) is the design matrix of IVs. The coefficient can be estimated by using the least square method as follows [23]  
\[\hat{\beta}=\left(\mathrm{X}^{\mathrm{T}}\mathrm{X}\right)^{-1}\mathrm{X}^{ \mathrm{T}}y. \tag{2}\]  
The main goal of the optimization was to identify the optimal settings of the process variables. Among the various optimization methods available, the desirability function (DF)-based optimization method was utilized to solve the simultaneous dual-objectives [6], including enhancing the significant ingredients of CKD-497 where \"the nominal better (N)-type\" was applied. The N-type can be defined by  
\begin{table}
\begin{tabular}{c c} \hline \hline
**IVs** & **ORVs** \\ \hline X1 (Extraction temperature) & Y1 (TDD) \\ X2 (Circulation velocity) & Y2 (SZD) \\ X3 (Filter paper size) & Y3 (Yield) \\  & Y4 (Solid flour) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input variables (IVs) and output response variables (ORVs) derived from QFD. TDD, 6,12-tetradadiene-8,10-diyne-1,3-diol; SZD, Schizandrin.  
\[d_{i}(\hat{y}_{i})=\left\{\begin{array}{c}0\text{ \ if }\hat{y}_{i}(x)<L_{i}\\ \left(\frac{\hat{y}_{i}(x)-L_{i}}{T_{i}-U_{i}}\right)^{w_{1}}if\ L_{i}\leq\hat{y}_{i}(x) \leq T_{i}\\ \left(\frac{U_{i}-\hat{y}_{i}(x)}{U_{i}-T_{i}}\right)^{w_{2}}if\ T_{i}\leq\hat{y}_{ i}(x)\leq U_{i}\\ 0\text{ \ if }\hat{y}_{i}(x)>U_{i}\end{array}\right.. \tag{3}\]  
where \(\hat{y}_{i}\) denotes the estimated function between ingredients of CKD-497, and \(L_{i}\) and \(T_{i}\) denote the lower and upper values, respectively [23]. The superscripts \(w_{1}\) and \(w_{2}\) represent the importance of the individual ranges. The overall desirability, \(D\), a geometric mean of all the DFs, can be expressed as follows [23]  
\[D=\left[d_{1}\big{(}\hat{y}_{1}\big{)}d_{2}\big{(}\hat{y}_{2}\big{)}\cdots d_ {m}\big{(}\hat{y}_{m}\big{)}\right]^{1/m}. \tag{4}\]  
The optimal combinations of ingredients are obtained when the overall desirability \(D\) is maximized.