## 3 Results
### MOGAs Optimization  
The regression models obtained in Equation (6) are substituted into Equation (2) to define the three objectives to minimize according to the multi-objective optimization problem described in Equation (5). Defined below are the models for each objective to be minimized using the multi-objective genetic algorithm.  
\[\begin{split}\text{F}(\text{Y}_{1}(\text{X}))&= \exp(\text{Y}_{1}(\text{X}))/1\ -\ \exp(\text{Y}_{1}(\text{X}))\\ \text{F}(\text{Y}_{2}(\text{X}))&=\exp(\text{Y}_{2}( \text{X}))/1\ -\ \exp(\text{Y}_{2}(\text{X}))\\ \text{F}(\text{Y}_{3}(\text{X}))&=\exp(\text{Y}_{3}( \text{X}))/1\ -\ \exp(\text{Y}_{3}(\text{X}))\end{split} \tag{7}\]  
The first model is a function of the category in which less growth exists; therefore, it can be taken as a priority when selecting the best solution of the Pareto Front obtained by the multi-objective genetic algorithm shown in Table 6.  
The set of solutions of the Pareto Front (Figure 9) meets the requirements of the optimization problem. The solution closest to zero was selected for the first objective, related to the lesser fungal growth probability. In Figure 9, the set of solutions between the first and second objectives is shown, where the point corresponding to the solution closest to zero for the first objective can be determined, and it corresponds to the solution in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pareto Front** & **X1** & **X2** & **X3** & **F(Y1(X))** & **F(Y2(X))** & **F(Y3(X))** \\ \hline
28 & 0.98357 & 0.98818 & \(-0.66040\) & 0.03183 & 0.20663 & 0.06601 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimal solution selected from the Pareto Front.  
\begin{table}
\begin{tabular}{l r r r r} \hline \hline
**Predictor** & **Coef** & **SE Coef** & **Z** & **_p_ \\ \hline \(\alpha\)1 & \(-0.844483\) & 0.368023 & \(-2.29\) & 0.022 \\ \(\alpha\)2 & 1.41937 & 0.403828 & 3.51 & 0.000 \\ \(\alpha\)3 & 3.20650 & 0.748000 & 4.29 & 0.000 \\ \(\text{X}_{1}\) & \(-0.115466\) & 0.304130 & \(-0.38\) & 0.704 \\ \(\text{X}_{2}\) & \(-0.103887\) & 0.304001 & \(-0.34\) & 0.733 \\ \(\text{X}_{3}\) & \(-0.503327\) & 0.310272 & \(-1.62\) & 0.105 \\ \(\text{X}_{1}\)\(\times\)\(\text{X}_{2}\) & 0.328445 & 0.305680 & 1.07 & 0.283 \\ \(\text{X}_{1}\)\(\times\text{X}_{3}\) & \(-0.159936\) & 0.304776 & \(-0.52\) & 0.600 \\ \(\text{X}_{2}\)\(\times\text{X}_{3}\) & \(-0.0594171\) & 0.303632 & \(-0.20\) & 0.845 \\ \(\text{X}_{1}\)\(\times\text{X}_{2}\) & \(-0.517629\) & 0.310828 & \(-1.67\) & 0.096 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Coefficients for the ordinal logistic regression model.  
The optimization result indicates that the optimal fungal growth inhibition can be obtained using large size nanoparticles with a high concentration, using an ultrasonic treatment for 2 and 2.5 s. The three experimental factors showed an effect on fungal growth inhibition; therefore, they are factors to consider while designing products based on composite materials from PUR and AgNPs.  
The ordinal logistic regression model is more accurate for categoric-type responses and can slightly modify the levels of the graphics for the main observed effects (see Figure 7).