## 3 Results
### System Modeling  
Once the manifest variables were grouped into a small number of latent variables, the parameters of the ANN that better fit the relationship between the input and output variables were found. In order to do so, we adopted the Root Mean Square (RMS). Thus, in Figure 6a,b, we present the RMS of the errors of both the training and test sets in terms of the hidden layer size, i.e., the number of neurons in the hidden layer, and the training set size for the environmental performance of the product; in Figure 6c,d we show the RMS of the errors in terms of the environmental performance of the product; in Figure 6e,f we present the same metric, but for the economic performance. Due to the relatively low number of available data, we adopted a k-fold approach to reuse data for test and training. Since this process highly depends on the test-train splitting ratio, we decided to sweep the training set from 100 to 180 elements. In this way, we assessed the performance of configurations with different combinations of neuron numbers and test-train split ratios, which are the two main hyperparameters of our model. In order to reduce the sensitivity to the training and test set partition, for each case, we performed the training and test evaluation of 10 random partitions, and computed the ensemble average of the RMS error values.  
It is worth mentioning that even if the RMS error of the test set is a priori more relevant than that of the training set, it is important to show the error of both sets, to test whether biasing or overfitting is affecting the modeling. Observing the different plots, it is possible to observe that, in most cases, the RMS error presents high values for low neuron numbers, where the model suffers from biasing. That is, the ANN is too simple to accurately model the relationship between the input and output latent variables. As the number of neurons in the hidden layer increases, the error yields a relatively flat level. Indeed, looking at the  
Figure 6: RMS error, in terms of the hidden layer, and training set sizes for the training and the test sets, for (**a,b**) the environmental performance of the product, (**c,d**) the environmental performance of the process, and (**e,f**) the economic performance.  
test error, it is impossible to observe a sensitive increase even for a number of neurons in the hidden layer as high as 32, which means no overfitting is present. On the other hand, looking at the dependency of the performance in terms of the training set size, it is possible to observe that the predicted performance tends to be better for larger training sets. This can be explained by the fact that, as the number of elements in the training set increases and the number of elements in the test set decreases, the number of outliers in the latter reduces. Indeed, the presence of outliers in the test set can also explain the non-monotonic behavior in terms of the training set size. The RMS values presented in Figure 6b,d,f reveal that a hidden layer with less than eight neurons incurs biasing, whereas a larger number does not significantly improve the model's performance. Therefore, a configuration with eight neurons represents a good trade-off between performance and computational complexity.