## 5 Optimal Nanomaterial Content  
Overall Desirability (OD) is a multi-criteria decision analysis method employed to ascertain optimal conditions that satisfy multiple objectives. This research utilizes an OD analysis to determine the most effective nanomaterial dosage for modifying asphalt binders. Figure 11: Results of DSR after PAV. The method assesses a range of investigated properties, including penetration (X1), the softening point (X2), Penetration Index (X3), mass loss due to RTFO aging (X4), storage stability (X5), viscosity at 135 \({}^{\circ}\)C before aging (X6), the temperature at which the rutting index (G*/sin 8) equals 1 kPa for the original binder (X7), and 2.2 kPa for the RTFO-aged binder (X8), along with the temperature corresponding to a fatigue index (G* sin 8) of 5000 kPa (X9). The OD calculation involves normalizing each of these values on a scale from 0 to 1, where 1 represents the most desirable outcome. The normalization approach can be carried out by using either the maximum value of the properties (the larger the better) or the minimum (the smaller the better) value of the properties in terms of Equation (2) or (3), respectively. For Equation (2), the larger the normalization data, the better the OD, but for Equation (3), it is the opposite. \[x_{i}^{*}(k)=\frac{x_{i}^{(0)}(k)-minx_{i}^{(0)}}{maxx_{i}^{(0)}-minx_{i}^{(0)}} \tag{2}\]  
\[x_{i}^{*}(k)=\frac{maxx_{i}^{(0)}-x_{i}^{(0)}(k)}{maxx_{i}^{(0)}-minx_{i}^{(0)}} \tag{3}\]  
where \(i\) = 1, 2,.., m (m is the index number); \(k\) = 1, 2,.., n (n is the data number); \(x_{i}^{(0)}(k)\) is the \(k\)th value of the ith original sequence; \(x_{i}^{*}(k)\) is the \(k\)-th value of the \(i\)-th normalized sequence; \(maxx_{i}^{(0)}\)\(i\) is the maximum value of the \(i\)-th original sequence; and \(minx_{i}^{(0)}\) is the minimum value of the \(i\)-th original sequence. The normalized values are subsequently combined using a geometric mean to compute the OD score for each nanomaterial dosage level, aiding in the identification of the optimal dosage that maximizes the desired characteristics of the modified asphalt binder as per Equation (4). \[\gamma\ (k)=[x_{1}^{*}\ (k)x_{1}^{*}(k)\ldots x_{m}^{*}\ (k)]^{1/m} \tag{4}\]  
Higher OD scores denote a more favorable combination of properties, thus proving to be essential tools for researchers and engineers to pinpoint the optimal nanomaterial content that enhances asphalt binder performance. Upon examining Tables 7-9, which present the normalized indices and Overall Desirability (OD) for NT-, NA-, and NS-modified asphalt, it is evident that both NT and NA achieve a peak OD at a 6% nanomaterial content, indicating an optimal improvement in asphalt binder properties. This dosage notably improves resistance to deformation, as evidenced by the temperature at which the rutting index equals 1 kPa, which is vital for good performance under high temperatures. For NA-modified asphalt, the 6% inclusion level is distinguished by its enhanced rutting resistance while still offering satisfactory fatigue resistance, as denoted by the fatigue index at 5000 kPa. Although it does not provide the highest fatigue resistance, this concentration delivers a beneficial equilibrium among critical properties such as the softening point, penetration, and mass loss due to aging. In the case of NS-modified asphalt, the OD analysis indicates that a 4% inclusion rate is the most favorable. At this concentration, properties such as penetration, the softening point, and resistance to rutting and fatigue are significantly improved without incurring excessive mass loss due to aging. This suggests that a 4% NS content yields the best balance between enhancing the binder properties and maintaining stability, making it the optimal dosage for NS-modified asphalt within the scope of this study. In summary, the OD method effectively consolidates diverse rheological and performance measures into a single metric, enabling the identification of the most advantageous nanomaterial concentration.