# Utilizing Soft Computing Techniques to Estimate the Axial Permanent Deformation of Asphalt Concrete
## 5 Results and Analysis
The relative importance generally quantifies the contribution of each input variable in the prediction of the output in the network. Input variables with higher values matter much in the model's accuracy. On the other hand, the impact analysis indicates how changes in a particular input affect the output by identifying whether increasing an input will increase or decrease the output. Garson algorithm method [23] is used to compute the relative importance of the input variables, as shown in the results in Figure 7. In this method, Equation (4) was used to calculate the contribution of each input variable using the absolute values of the connection weights \(w_{1}\)(\(i\),\(j\)) and \(w_{2}\)(\(i\)), resulting in Table 7.  
\[I_{j}=\frac{\sum_{m=1}^{m=Nh}\left(\frac{w_{jm}^{0h}}{\sum_{k=1}^{Nh}w_{km}^{ 0h}}w_{mn}^{0h}\right)}{\sum_{k=1}^{k-Ni}\left[\sum_{m=1}^{m=-Nh}\left(\frac{w_ {jm}^{0h}}{\sum_{k=1}^{Nh}w_{km}^{0h}}w_{mn}^{0h}\right)\right]} \tag{4}\]  
where  
\(I_{j}\): relative importance (contribution) of \(j\)th input variables  
\(k\): input neurons  
\(m\): hidden neurons  
\(n\): output neuron  
\(i\): input layer  
\(h\): hidden layer  
\(o\): output layer  
\(Ni\): number of neurons in the input layer  
\(Nh\): number of neurons in the hidden layer  
\(w\): connection weights  
The results indicate that the percentage of aggregate passing the 4.75 mm sieve (PPNo.4) and theoretical maximum specific gravity (Gmm) were the most influential variables, contributing 13% and 11.8%, respectively. These findings align with those of [24], who reported that aggregate gradation significantly impacts the rutting potential of asphalt mixtures due to its role in load distribution and shear resistance. The importance of Gmm is also consistent with previous research, which has shown that asphalt mixtures with higher Gmm values tend to have improved stability and resistance to permanent deformation [25]. In contrast, asphalt viscosity (Vis) and normal stress levels (Stress) were found to have the lowest contributions, with values of 1.1% and 3.3%, respectively. While these factors influence the mix behavior, their relatively lower impact can be attributed to their interaction with other variables, such as temperature and binder content, which have a more pronounced effect on deformation characteristics. This observation is consistent with the findings of [25], who reported that stress levels alone do not fully dictate rutting  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Neuron**} & \multicolumn{6}{c}{**w\({}_{1}\)(\(\epsilon\))**} & \multicolumn{6}{c}{**w\({}_{2}\)(\(\epsilon\))**} & \multicolumn{6}{c}{**w\({}_{1}\)(\(\epsilon\))**} \\ \cline{2-13}  & **n** & **Temp. ** & **AC** & **Vis** & **Stress** & **PP12** & **PPNo.4** & **AV** & **VMA** & **Gmm** & **Cmb** & **Gab** & \\ \hline \(1\) & \(-0.063\) & \(1.362\) & \(0.621\) & \(-0.309\) & \(0.565\) & \(0.909\) & \(-1.396\) & \(2.534\) & \(0.386\) & \(-1.  
behavior, as other parameters, including binder stiffness and mix volumetrics, play a more dominant role. The impact of the input variables in predicting the (\(\epsilon\)p) values is analyzed using the connection weight method by [PERSON] and [PERSON] [26]. The connection weight approach of input variables was calculated using Equation (5). The notation (\(Ix\)) represents the impact of input neuron (x), the (w\({}_{xy}\)) is the input to hidden connection weights, and the (w\({}_{yz}\)) is the hidden to output connection weights. \[I_{x}=\sum_{y=1}^{m}(w_{xy}.w_{yz}) \tag{5}\]  
Figure 8 shows the computed impact of all input variables in the model. It is demonstrated that only the normal stress, \(AV\), and \(G_{sb}\) variables negatively impact the permanent strain (\(\epsilon\)p) value. Within the investigated data, the marginal negative impact of normal stress can be attributed to its interaction with other variables, such as temperature and binder content, which have a more pronounced effect on deformation characteristics.