## 4 The Throughput Prediction with a Deep Learning Neural Network  
The goal of this paper is to take advantage of the machine learning techniques to create a deep learning neural network that, once trained, is able to identify the relationship between WIP and throughput, regardless of the degree of variability in the processing times of the jobs entering the production system. The improvement over the reference scientific literature is that, here, the model is generalized for what concerns the variability of the processing times, while maintaining the assumptions of a balanced line with controlled WIP (i.e., CONWIP), single-product and single-routing. Moreover, given what has been shown in Section 3, it will not be necessary for this neural network model to be trained over different values of the average processing times because, in the case of a balanced line, it is simply a scaling factor of the normalized throughput. It is worth considering whether, similar to what was observed in Section 3, it might be desirable to normalize the throughput value not only in relation to the average value of the processing times but also in relation to the number of machines of the production line. Additionally, for this case, a simulation campaign has been carried out using the previously mentioned simulation tool. Differently from the previous experimental plan, here the number of machines has been varied between 5 and 15 in steps of one based on the levels of the experimental factors listed in Table 2. Figure 5 depicts the traditional throughput and the normalized ones for a fixed value of processing times mean and coefficient of variation (i.e., fixing the alpha value of the gamma distribution). As expected, as the number of machines considered within the production line increases, both the traditionally calculated throughput and the normalized one show decreasing values (Figure 6). The bad news is that, unlike for the processing time means, this link is not as simple linear and thus cannot be easily simplified as in the first case. Then, the proposed deep neural network will have as inputs (i) the value of WIP released in the production line, (ii) the coefficient of variation of processing times, (iii) and the value of \(W_{0}\) which, in the case of a balanced line, directly represents the number of machines. The output value, hence, will be the normalized throughput of the production line (see Figure 7). An intensive simulation campaign has been conducted, with the goal of providing the machine learning model with a complete dataset of the various factors under consideration. The factors and levels of experimentation involved, in particular, are those shown in Table 4 and the experimental plan considered is full factorial. Each parameter combination was reproduced 20 times, with a simulation duration of four years of continuous manufacturing. At the end of each simulation, the absolute and normalized throughput numbers were gathered and saved in an Excel spreadsheet. More than \(1,560,000\) trials were conducted, yielding similar data rows that served as the dataset for the neural network training phase. After the architecture of such a model has been determined, hyper-parameters such as the number of neurons and the structure of the hidden layer, activation functions, and loss function must be specified. However, there is no impartial way to choose these settings [37]. Consequently, these was selected with a \"trial and error\" technique taking advantages of Google Colaboratory and Tensor Flow Keras library for the model construction. After experimenting with various configurations, the ones that best fitted the given problem were composed of: (i) an input layer with the WIP values in the production system, the quadratic coefficient of variation and the critical WIP; (ii) a hidden normalization layer followed by 2 condensed layers of 24 neurons (12 each) with the _elu_ and the _sigmoid_ as activation function; (iii) a final output layer composed of a single neuron with the _tanh_ activation function.