## 4 Results
### Individual Desirability and Global Functions  
The controllability of the process is considered sufficient if the six responses--\(\gamma_{con}\)-d, \(\gamma_{switch}\)-d, \(\gamma_{pour}\)-d, \(\gamma_{con}\)-n, \(\gamma_{switch}\)-n, y \(\gamma_{pour}\)-n--stay below certain values and insufficient if one or various values exceed certain preset limits. In particular, the two responses \(\gamma_{con}\) are considered valid (desirability 1) if they are less than 0.03 and non-acceptable (desirability 0) if they exceed the value of 0.07. The desirability function has a linear variability between both values (\(\gamma_{con}\) is for \(\gamma_{con}\)-d and \(\gamma_{con}\)-n). Thus, the function of individual desirability is:  
\[d_{\gamma con}(\hat{y}_{\gamma con})= \begin{cases}1&if\ \ \hat{y}_{\gamma con}<0.03 \\ \left(\frac{0.07-\hat{y}_{\gamma con}}{0.07-0.03}\right)&if\ \ 0.03<\hat{y}_{\gamma con}<0.07\\ 0&if\ \ 0.07<\hat{y}_{\gamma con}.\end{cases}  \tag{22}\]  
The desirability function is considered for both \(\gamma_{switch}\) (\(\gamma_{switch}\) is for \(\gamma_{switch}\)-d and \(\gamma_{switch}\)-n):  
\[d_{\gamma switch}(\hat{y}_{\gamma switch})= \begin{cases}1&if\ \ \hat{y}_{\gamma switch }<0.03\\ \left(\frac{0.10-\hat{y}_{\gamma switch}}{0.10-0.03}\right)&if\ \ 0.03<\hat{y}_{\gamma switch }<0.10\\ 0&if\ \ 0.10<\hat{y}_{\gamma switch}.\end{cases}  \tag{23}\]  
In the case of \(\gamma_{pour}\)-d, it yields:  
\[d_{\gamma pow}-d\left(\hat{y}_{\gamma pow}-\mathrm{d}\right)= \begin{cases}1& if\ \ \hat{y}_{\gamma pow}-\mathrm{d}<14,150\\ \left(\frac{14,333-\hat{y}_{\gamma pow}-\mathrm{d}}{14,333-14,150}\right)&if \ \ 14,150<\hat{y}_{\gamma pow}-\mathrm{d}<14,333\\ 0&if\ \ 14,333<\hat{y}_{\gamma pow}-\mathrm{d},\end{cases}  \tag{24}\]  
Figure 7: Vertical stripes highlight the acceptable region (values lower than 0.03, in green) and the non-acceptable regions in horizontal stripes (values greater than 0.10, in red). (**a**) Level curves for the response \(\gamma_{switch}\)-d. (**b**) Level curves for the response \(\gamma_{switch}\)-n.  
where as for \(\gamma_{pow}\)-n, we have: \[d_{\gamma pow}-n\big{(}\hat{y}_{\gamma pow}-n\big{)}= \begin{cases}1&if\ \ \hat{y}_{\gamma pow }-n<1327\\ \big{(}\frac{1398-\hat{y}_{\gamma pow}-n}{1398-1327}\big{)}&if\ \ 1327<\hat{y}_{\gamma pow }-n<1398\\ 0&if\ \ 1398<\hat{y}_{\gamma pow}-n.\end{cases} \] (25) The search for configurations that maximize desirability is performed in two stages: 1. Joint desirability, \(D(\mathbf{x})\), is constructed for the four empirically modelled variables as the geometric mean of the four individual desirabilities described above: \[D(\mathbf{x})=\big{(}d_{\gamma com}-d\times d_{\gamma switch}-d\times d_{ \gamma pow}-d\times d_{\gamma switch}-n\big{)}^{1/4}.\] (26)
2. Having identified the region of maximum joint desirability, the individual desirabilities of the other two responses, \(\gamma\)con-n and \(\gamma\)pow-n, are calculated by applying Equations (22) and (25) to the experimental values of the points belonging to the area of maximum desirability. The result of stage (1) is shown in Figure 8. The horizontally striped zone (in red) has a joint desirability of 0; in other words, it fails to comply with one or more of the four acceptable threshold values for the responses: \(\gamma_{cont}\)-d, \(\gamma_{switch}\)-d, \(\gamma_{pow}\)-d, and \(\gamma_{switch}\)-n. The vertically striped zone (in green) has a global desirability of between 0.6 and 0.7. Only three points of the experimental domain are within it: the aforementioned experiment 12 from Table 1, 17, and 18.