## 3 Results
### Multifactorial Optimal Settings with Quadratic Effects  
Upon completing the solver performance testing on the trained models in the previous subsection, it was decided that the quadratic regression model would be used to proceed to the factorial analysis phase. In Table 4, the regression coefficients are tabulated for each of the three quality characteristics. The multiple adjusted R\({}^{2}\) values for the combined replicated datasets were computed, and their values were found to be higher than 0.94 for all three characteristic responses. Using a Bonferroni family-wise error rate of 0.05, the linear and quadratic part of factor A (TMP) is retained for the average permeate flux, as well as the linear term only of factor D (MWCO). Regarding the COD rejection rate \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Characteristic response:** average permeate flux (\(\mathrm{f_{p}}\)) & & & & & & & & \\
**Model type** & Status & RMSE & MSE & R\({}^{2}\) & MAE & MAPE \% & Hyperparameters \\ \hline
**Linear regression** & Trained & 6.29 & 39.58 & 0.91 & 4.90 & 12.80 & Terms—quadratic \\
**SVM** & Trained & 8.07 & 65.11 & 0.86 & 6.73 & 17.20 & Kernel function—quadratic \\  & & & & & & & Standardize data—yes \\ \hline  & & & & & & & & \\  & & & & & & & & \\
**SVM** & Trained & 6.43 & 41.31 & 0.91 & 5.03 & 12.69 & Box constraint—8.2907 \\  & & & & & & & & \\  & & & & & & & & \\  & & & & & & & & \\ \hline
**Characteristic response:** COD rejection rate (COD) & & & & & & & & \\
**Model type** & Status & RMSE & MSE & R\({}^{2}\) & MAE & MAPE \% & Hyperparameters \\
**Linear regression** & Trained & 1.90 & 3.61 & 0.91 & 1.61 & 3.76 & Terms—quadratic \\ \hline
**SVM** & Trained & 3.42 & 11.70 & 0.71 & 2.65 & 6.45 & Kernel function—quadratic \\  & & & & & & & Standardize data—yes \\ \hline
**Characteristic response:** cumulative flux decline (SFD) & & & & & & & & \\ \hline
**Model type** & Status & RMSE & MSE & R\({}^{2}\) & MAE & MAPE \% & Hyperparameters \\
**Linear regression** & Trained & 0.63 & 0.40 & 0.91 & 0.48 & 9.24 & Terms—quadratic \\ \hline
**SVM** & Trained & 0.73 & 0.53 & 0.88 & 0.61 & 11.35 & Kernel function—quadratic \\  & & & & & & & & Standardize data—yes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of the modeling based on linear regression (quadratic), quadratic support vector machine learning, and optimizable support vector machine learning for (1) average permeate flux (\(\mathrm{f_{p}}\)), (2) COD rejection rate (COD), and (3) cumulative flux decline (SFD) characteristic responses in combined replicate datasets (MATLAB R2024a). response, the statistically contributing effects are identified to (1) the linear part of factor A (TMP), (2) the linear term of factor B (CFV), (3) the quadratic part of factor C (temperature), and (4) the linear and quadratic terms of factor D (MWCO). Finally, the cumulative flux decline response is modulated by (1) the linear term of factor A (TMP), (2) the linear term of factor B (CFV), (3) the linear term of factor C (temperature), and (4) the linear term of factor D (MWCO). The resulting response functions for the individual structural modeling of the three quality characteristics are  
\[\rm{J_{P}=57.37+18.04\cdot A-8.83\cdot A^{2}+15.81\cdot D} \tag{1}\]  
\[\rm{COD=38.71+2.01\cdot A+3.29\cdot B+3.58\cdot C^{2}-5.32\cdot D+4.07\cdot D^{2}} \tag{2}\]  
\[\rm{SFD=5.53+1.58\cdot A-0.51\cdot B+0.46\cdot C+1.70\cdot D} \tag{3}\]  
The maximization of the average permeate flux response is attained by maximizing factor D (MWCO) and setting it at 100 kDa. Moreover, the optimal setting for factor A is simply estimated at  
\[\frac{\partial J_{P}}{\partial A}=0\Rightarrow A=\frac{-18.04}{2\cdot(-8.83)} \approx 1.0\] (the third setting in the \[(-1,\ 0,\ 1)\] standardization of the original levels)  
Hence, the optimal setting for factor A (TMP) is 3.0 bar. For optimal COD rejection, factors A (TMP), B (CFV), and C (temperature) should be set at their highest values: 3.0 bar, 1.041 m/s, and 30.0 \({}^{\circ}\)C, respectively.