## 4 Results and Discussion
### Determine the Optimal Content of Ingredients  
The multilinear regression equations show that the mechanical properties of mortar depend on time, GO, FA, and the interaction between these factors. It is necessary to determine the appropriate content of GO and FA to ensure the best strength on time with 95% reliability to serve the production of concrete in the following experiment. The regression Equations (1) and (2) indicate that the relationship functions have extreme values because the coefficients of the second-order terms are less than 0. We can easily determine the extremum of the function. Assuming the considered range, 0 \(\leq\) GO \(\leq\) 0.05 wt%, 0 \(\leq\) FA \(\leq\) 30 wt%, 3 \(\leq\) time \(\leq\) 28 days, we can find the extreme values of the functions Y1 and Y2 as follows:  
Determine the optimal values of the variables to achieve the maximum compressive strength from Equation (1):  
\[\left\{ \begin{array}{l}\frac{\partial\mathrm{Y1}}{\partial\mathrm{GO}}=351.3 -10638\mathrm{GO}-0.765\mathrm{FA}+2.206\mathrm{t}=0\\ \frac{\partial\mathrm{Y1}}{\partial\mathrm{FA}}=0.5128-0.041676\mathrm{FA}-0.7 65\mathrm{GO}-0.002487\mathrm{t}=0\\ \frac{\partial\mathrm{Y1}}{\partial\mathrm{t}}=5.0089-0.24256\mathrm{t}+2.206 \mathrm{GO}-0.002487\mathrm{FA}=0\\ \end{array} \right. \tag{3}\]  
\[=>\left\{ \begin{array}{l}\mathrm{GO}=0.036246\%\\ \mathrm{FA}=10.39512\%\\ \mathrm{t}=20.84631\;\;\mathrm{days}\end{array} \right.\]  
Determine the optimal values of the variables to achieve the maximum flexural strength from Equation (2):  
\[\left\{ \begin{array}{l}\frac{\partial\mathrm{Y2}}{\partial\mathrm{GO}}=52.3 4-1576.2\mathrm{GO}-0.1296\mathrm{FA}+0.3299\mathrm{t}=0\\ \frac{\partial\mathrm{Y2}}{\partial\mathrm{FA}}=0.06898-0.005384\mathrm{FA}-0. 1296\mathrm{GO}-0.000276\mathrm{t}=0\\ \frac{\partial\mathrm{Y2}}{\partial\mathrm{t}}=0.6836-0.033236\mathrm{t}+0.3299 \mathrm{GO}-0.000276\mathrm{FA}=0\\ \end{array} \right. \tag{4}\]  
\[=>\left\{ \begin{array}{l}\mathrm{GO}=0.036746\%\\ \mathrm{FA}=10.02801\%\\ \mathrm{t}=20.84952\;\;\mathrm{days}\end{array} \right.\]  
Thus, from the obtained results (3) and (4), it was shown that both equations had quite similar optimal variables. The optimal values of the variables are as follows:  
\[\mathrm{GO}=0.036\;\mathrm{wt\%}\mathrm{;}\mathrm{FA}=10\;\mathrm{wt\%}\mathrm{; }\mathrm{t}=20\;\mathrm{days}\]  
The values of each of the response for these settings are as follows:  
\[\mathrm{Y1}=67.1681\;\mathrm{MPa}\mathrm{ and}\;\mathrm{Y2}=10.14661\;\mathrm{MPa}\]  
In many cases, the experimenter has to optimize a number of responses at the same time. The challenge when handling multiple responses is that there may be conflicting objectives due to the different requirements of each response. The experimenter needs to find a solution that satisfies each requirement as much as possible without significantly affecting any other requirement. To address multi-response optimization, desirability functions [26] are used. Each \(\mathrm{i_{th}}\) response is assigned a desirability function, \(\mathrm{di}\), where the value of \(\mathrm{d_{i}}\) varies between 0 and 1. The function, \(\mathrm{d_{i}}\), is defined differently based on the objective of the response. If the response is to be maximized, then \(\mathrm{d_{i}}\) is defined as follows:  
\[\mathrm{d_{i}}=\left\{ \begin{array}{cc}0&\mathrm{y_{i}}<\mathrm{L}\\ \left(\frac{\mathrm{y_{i}}-\mathrm{L}}{\mathrm{T-L}}\right)^{\omega}&\mathrm{ L}\leq\mathrm{y_{i}}\leq\mathrm{T}\\ 1&\mathrm{y_{i}}>\mathrm{T}\end{array} \right. \tag{5}\]  
where T represents the target value of the \(\mathrm{i_{th}}\) response, \(\mathrm{y_{i}}\), L represents the acceptable lower limit value for this response, and \(\omega\) represents the weight. When \(\omega=1\), the function \(\mathrm{di}\) is linear. If \(\omega>1\), then more importance is placed on achieving the target for the response, \(\mathrm{y_{i}}\). When \(\omega<1\), less weight is assigned to achieving the target for the response, \(\mathrm{y_{i}}\).