## Methodologies  
### Building the ANN Models and the Sensitivity Analysis  
First, experiments on the mechanical properties of RAC with different RAs were carried out in the laboratory, which had 46 concrete mixes and were divided into 3 groups. The RAs were categorized into 3 groups according to their sources: (1) RAs derived from 3 different sources and crushed  
Figure 1: A typical artificial neural networks (ANN) model. using different methods; (2) RAs derived from concrete cubes made in the laboratory with different compressive strengths (35-85 MPa); (3) RAs contained different amounts of masonry added (clay bricks or tiles). As many sources of natural and recycled aggregates were used in these mixes, 8 aggregate characteristics, including fineness modulus of the fine aggregate (FM), residual mortar content (M\({}_{\text{C}}\)), 10% fines value (TFV), aggregate crushing value (ACV), water absorption value (W\({}_{\text{a}}\)), specific gravity (SG\({}_{\text{SSD}}\)), impurity content (\(\delta\)) and masonry content (m) of the coarse aggregate, were comprehensively measured and quantified. These factors, together with the mix proportions and concrete curing time, were selected as the input variables of the ANN for modeling the compressive strength and elastic modulus. To facilitate the analysis, factors including the mix proportions (5 variables) and curing time (1 variable) were designated as \"certainties\", while the other factors (8 variables) were named \"uncertainties\". The experimental results obtained from the above mixes at different ages were divided into three groups, acting as: (i) the training set; (ii) the validation set; and (iii) the testing set, respectively. The corresponding ANN model could be established using the procedures described previously [20]. For each model, the ANN network parameters were determined when the error values reached the minimum. Based on the comparison of the error of integral testing set after a series of trials, the initial network architecture and parameters used in this study were as follows:  
* Number of input layer units = 14
* Number of hidden layers = 1
* Number of hidden layer units = 40
* Number of output layer units = 1
* Momentum rate = 0.9
* Learning rate = 0.01
* Learning cycle = 15,000  
In this study, the mean absolute percentage error (_MAPE_), root-mean-squared error (_RMS_) and absolute fraction of variance (\(R^{2}\)) computed using Equations (2)-(4) were used to access the accuracy of the ANN model developed. \[MAPE=\left(\frac{o_{j}-t_{j}}{o_{j}}\right) \tag{2}\]  
\[RMS=\sqrt{\frac{1}{p}\times\sum_{j}\abs{t_{j}-o_{j}}^{2}} \tag{3}\]  
\[R^{2}=1-\left(\frac{\sum_{j}\abs{t_{j}-o_{j}}^{2}}{\sum_{j}\left(o_{j}\right)^ {2}}\right) \tag{4}\]  
where \(t\): the predicted output of the network; \(o\): the actual output of the network; \(p\): the total number of training and testing patterns; \(t_{j}\): the predicted output of \(j\)th pattern of the network; \(o_{j}\): the actual output of \(j\)th pattern of the network. After the construction of the ANN models, the sensitivity analysis was then conducted according to Figure 2.  
Step 1: A comparison of the performance between the models (\(ANN_{14}\)) with all variables (14 variables) and that (\(ANN_{6}\)) with certainties (only 6 variables) was first made, while keeping the other networks parameters constant. Step 2: Various combinations of the uncertainties (aggregate characteristics) together with the \"certainties\" were used as the inputs of each model to find the best model with the minimum error. However, it would take a huge amount of time if all the combinations were tried out one by one. Considering the interaction and constraints among the aggregate characteristics, a simple method developed to determine the best combination of variables is shown in Figure 3.  
* At Stage a, the resulted error of \(ANN_{6}\) was compared with that of the networks (\(ANN_{7}\)) when each \"uncertainty\" was sequentially added as an input variable.