## 5 Optimal Nanomaterial Content
NT- and NA-modified asphalts display improved binder attributes at a 6% nanomaterial dosage, whereas NS-modified asphalt reaches peak performance at a 4% inclusion rate, thereby establishing these as the optimum concentrations for ameliorating asphalt binder characteristics per the parameters analyzed. These findingshighlight the importance of careful nanomaterial dosage selection to maximize the overall performance of asphalt mixes. The OD values serve as a quantitative measure, guiding the selection of the most effective nanomaterial content for the desired asphalt properties. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \multirow{2}{*}{**Parameter**} & \multicolumn{6}{c}{**NT Content, \%**} \\ \cline{3-6}  & & **0** & **2** & **4** & **6** & **8** \\ \hline \multirow{2}{*}{Penetration, 1/10 mm} & \(\chi_{1}^{(0)}\) & 49 & 48 & 45 & 42 & 40 \\  & \(\chi_{1}^{*}\) & 0.000 & 0.111 & 0.444 & 0.778 & 1.000 \\  & \(\chi_{2}^{(0)}\) & 49.7 & 50 & 50.4 & 51.5 & 53.1 \\  & \(\chi_{2}^{*}\) & 0.000 & 0.088 & 0.206 & 0.529 & 1.000 \\  & \(\chi_{3}^{(0)}\) & \(-1.33\) & \(-1.3\) & \(-1.34\) & \(-1.22\) & \(-0.96\) \\  & \(\chi_{3}^{*}\) & 0.026 & 0.105 & 0.300 & 0.316 & 1.000 \\  & \(\chi_{4}^{(0)}\) & 0.266 & 0.234 & 0.226 & 0.238 & 0.259 \\  & \(\chi_{4}^{*}\) & 0.000 & 0.808 & 1.000 & 0.695 & 0.169 \\  & \(\chi_{5}^{(0)}\) & 0.4 & 0.8 & 1.3 & 1.7 & 2 \\  & \(\chi_{5}^{*}\) & 1.000 & 0.750 & 0.438 & 0.188 & 0.000 \\  & \(\chi_{6}^{(0)}\) & 719.7 & 793.1 & 904.4 & 916.8 & 982.5 \\  & \(\chi_{6}^{*}\) & 0.000 & 0.279 & 0.703 & 0.750 & 1.000 \\  & \(\chi_{7}^{(0)}\) & 68.85 & 69.92 & 69.93 & 74.61 & 79.47 \\  & \(\chi_{7}^{*}\) & 0.000 & 0.101 & 0.102 & 0.542 & 1.000 \\  & \(\chi_{8}^{(0)}\) & 68.35 & 69.06 & 70.68 & 74.71 & 76.08 \\  & \(\chi_{8}^{(0)}\) & 0.000 & 0.092 & 0.301 & 0.823 & 1.000 \\  & \(\chi_{9}^{(0)}\) & 25.13 & 25.48 & 25.72 & 26.31 & 26.89 \\  & \(\chi_{9}^{*}\) & 1.000 & 0.801 & 0.665 & 0.330 & 0.000 \\  & & & & & & \\ \hline Overall Desirability (OD) & 0.000 & 0.222 & 0.382 & 0.498 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The original values, normalized values, and OD for NT-modified asphalt. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \multirow{2}{*}{**Parameter**} & \multicolumn{6}{c}{**NA Content, \%**} \\ \cline{3-6}  & & **0** & **2** & **4** & **6** & **8** \\ \hline \multirow{2}{*}{Penetration, 1/10 mm} & \(\chi_{1}^{(0)}\) & 49 & 47 & 44 & 41 & 36 \\  & \(\chi_{1}^{*}\) & 0.000 & 0.154 & 0.385 & 0.615 & 1.000 \\  & \(\chi_{2}^{(0)}\) & 49.7 & 50.1 & 51.1 & 52.9 & 54.6 \\  & \(\chi_{2}^{*}\) & 0.000 & 0.082 & 0.286 & 0.653 & 1.000 \\  & \(\chi_{3}^{(0)}\) & \(-1.33\) & \(-1.32\) & \(-1.22\) & \(-0.95\) & \(-0.85\) \\  & \(\chi_{3}^{*}\) & 0.000 & 0.021 & 0.229 & 0.792 & 1.000 \\  & \(\chi_{4}^{(0)}\) & 0.266 & 0.245 & 0.275 & 0.303 & 0.414 \\  & \(\chi_{4}^{*}\) & 0.876 & 1.000 & 0.822 & 0.657 & 0.000 \\  & \(\chi_{5}^{(0)}\) & 0.4 & 0.6 & 0.9 & 1.3 & 1.6 \\  & \(\chi_{5}^{*}\) & 1.000 & 0.833 & 0.583 & 0.250 & 0.000 \\  & \(\chi_{6}^{(0)}\) & 719.7 & 845.2 & 965.9 & 1063 & 1089 \\  & \(\chi_{6}^{*}\) & 1.000 & 0.660 & 0.333 & 0.070 & 0.000 \\  & \(\chi_{6}^{*}\) & 1.000 & 0.660 & 0.333 & 0.070 & 0.000 \\  & \(\chi_{7}^{(0)}\) & 68.85 & 69.97 & 70.91 & 75.36 & 80.42 \\  & \(\chi_{7}^{*}\) & 0.000 & 0.097 & 0.178 & 0.563 & 1.000 \\  & \(\chi_{8}^{(0)}\) & 68.35 & 69.75 & 71.36 & 76.14 & 78.57 \\  & \(\chi_{8}^{*}\) & 0.000 & 0.137 & 0.295 & 0.762 & 1.000 \\  & \(\chi_{9}^{(0)}\) & 25.13 & 24.88 & 25.02 & 25.08 & 25.53 \\  & \(\chi_{9}^{*}\) & 0.615 & 1.000 & 0.785 & 0.692 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The original values, normalized values, and OD for NA-modified asphalt.