The influence of Acrylate Styrene Acrylonitrile (ASA) and ASA/nanosilica (ASA/Si) additives was investigated by using a dynamic shear rheometer (DSR). Firstly, an ASA polymer was blended with the virgin asphalt binder at two different concentrations (3% ASA and 5% ASA). After observing that 5% ASA was the optimum concentration for modification, nanosilica particles were further incorporated into the 5% ASA-modified asphalt binder with two different percentages (5% ASA 3%Si; 5% ASA 5%Si). Frequency sweep tests were conducted across various frequencies at elevated temperatures. The experimental outcomes were analyzed using master curves, rutting, and fatigue resistance parameter plots. Additionally, to provide a more holistic analysis, two different multicriteria decision analysis (MCDA) techniques, namely the Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) and the Technique for the Order of Preference by a Similarity to Ideal Solution (TOPSIS), were conducted to identify the best-performing asphalt binder by considering three different parameters: workability, performance under different conditions, and cost. The frequency sweep tests showed that the 5% ASA 5%Si asphalt worked best in terms of resistance to rutting. On the other hand, the virgin binder performed better than all modified binders when it failed to resist fatigue. On the other hand, the PROMETHEE analysis identified the 5% ASA-modified asphalt binder as the optimal choice, while the TOPSIS analysis determined that the 5% ASA 3%Si-modified binder provided the best performance. The differences between the experimental results and the MCDA were due to using more than one evaluation parameter and looking at how well the asphalt binder worked at different temperature ranges at the same time.