## 4 The Throughput Prediction with a Deep Learning Neural Network
Since the neural network is necessary to estimate the normalized throughput, which is known to range from 0 to 1, the choice of the latter activation function is not coincidental due that a value of normalized throughput larger than 1 should not be predicted. Figure 6: The throughput and normalized throughput when varying the number of machines (in different colors) with alpha value fixed at 1 and mean processing time fixed to 10. Obviously, other experiments with several neural network architectures were conducted before reaching the proposed one, which we omitted for space considerations. For the assessment of each architectures, the mean squared error (MSE) was considered as loss function. Trials with alternative learning rates were also conducted in this scenario, and the results have a satisfactory MSE value also in the test phase (MSE = 6.42 \(\times\) 10\({}^{-6}\)). For the training phase, the random seed for the initial neural net weight was fixed to guarantee an effective comparison of the outcomes, examining the model's sensitivity to differing hyper-parameters while using the ADAM optimizer with a learning rate of 0.001. In terms of epoch number, we found that the proposed structure performed best at roughly 20 epochs, which prevented over-fitting the neural network model. Figure 8 depicts the outcomes of the deep learning neural network model. After training the model, it was subjected to a series of stress tests in which the model's performance was evaluated under extreme or challenging conditions that deviated from the assumptions made during model development. These tests were performed for specific cases by comparing model predictions to those obtained from the simulator and [PERSON]'s iterative model. It should be noted that both models produce predictions with negligible errors when the hypothesis of exponential processing times holds (i.e., when the coefficient of variation is equal to one). However, as one moves away from the exponential \begin{table}
\begin{tabular}{c c c} \hline \hline
**Experimental Factor** & **Level** & **Unit of Measure** \\ \hline Mean processing time & 5 \(\div\) 50, step 5 & [min] \\ Shape factor (\(\alpha\)) & 0.5 \(\div\) 3, step 0.1 & [] \\ WIP level & 1 \(\div\) 30, step 1 & [units] \\ Number of machines (i.e., \(W_{0}\)) & 5 \(\div\) 15, step 1 & [machines] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental factors and levels for the data collection  
Figure 8: Training results of the proposed deep learning model. Figure 7: Layout of the proposed estimation neural network. distribution (both for lower and higher coefficients of variation), [PERSON] and [PERSON]'s iterative model exhibits an increasing amount of drift with respect to true throughput values. In fact, without losing generality, focusing the discussion on a 10-machine scenario with average processing times set at 10 min, in Figure 9 it is possible to observe how the two models under analysis behave in the limit scenarios of the experimental plan considered (i.e., with \(\alpha\) value equal to 3 and 0.5). In the first case, when a coefficient of variation equal to 0.33 is considered, and thus a variability less than the exponential case is represented, the performance of the analyzed production system is on average better than the scenario with exponential variability. In fact, as it is possible to observe in Figure 9, the proposed neural network model (in yellow) captures the different trend completely and perfectly overlaps with the simulated values (in green), whereas the iterative model underestimates the performance (in blue). In the second case, when a coefficient of variation equal to 2 is considered, and thus a variability bigger than the exponential case is represented, the performance of the analyzed production system is on average lower than in the scenario with exponential variability. Again, the proposed model adequately captures this trend, whereas the iterative model overestimates it.