## 3 A Normalized Approach for the Throughput Prediction  
As said before, the purpose of this work is to develop a model based on recent machine learning techniques able to estimate the performance (in terms of throughput and cycle time) of a CONWIP single-product, single-routing production system given the operating parameters of the production line and the WIP released in it. We consider the system at the steady-state, so it is sufficient to estimate only one of either TH or CT given that the other can be computed by [PERSON]'s law (i.e., \(WIP=TH\cdot CT\)), the WIP being a controlled parameter. As a result, the discussion will now focus on throughput estimation, while the cycle time will be computed as \(CT=WIP/TH\). Given these premises, the purpose of this paragraph is to determine which inputs are strictly necessary for the performance estimation model in order to minimize the size of the experimental campaign required to feed the model. In particular, focusing on the hypothesized production line, which is still assumed to be balanced in this phase, the operating parameters of the line are the following:  
* The average processing time at the stations;
* A measure of the processing time's variability (e.g., the coefficient of variation);
* The production line critical WIP (i.e., the previously defined \(W_{0}\));
* The level of WIP released in the production line. With regards to the processing time variability, it is worth noting that a dimensionless measure of processing time variability can be represented by the coefficient of variation, which is defined as the ratio of the processing time standard deviation over its mean. Since we are modeling processing times distribution by means of a gamma distribution function, and given that the scale parameter \(\beta\) is computed as a function of the average processing time and the shape parameter \(\alpha\), the coefficient of variation can be simply computed as \(1/\sqrt{\alpha}\), confirming the fact that the variability depends only on \(\alpha\). With regard to the critical WIP \(W_{0}\), it can be shown that, in the case of balanced production lines, it exactly equals the number of machines (remember that single-machine stations are considered). As such, the number of machines is then a metric that unambiguously characterizes the production line. Finally, the last parameter is the level of WIP in the production line, which, as previously discussed, has a direct effect on the performances since it hinders the propagation of the disruptive effects of variability. The more WIP in the production line, the higher the throughput achieved or, in other words, the closer this is to the limiting value represented by the best case and, conversely, the longer the average cycle time required for the WIP to cross the production line. Finally, the average processing time relates to the bottleneck rate (i.e., the inverse of \(r_{b}\)) since the case of the balanced line is under consideration. While it is widely recognized that training a machine learning algorithm, particularly neural networks, often requires a large dataset, they can still be effectively employed with moderate datasets, albeit at the cost of potentially reduced performance. In this study, we explore the possibility of removing the \(r_{b}\) parameter from both data generation and prediction, given its multiplicative effect on the throughput value and the hope that it does not alter its dynamic behavior. Our aim is to derive models of throughput performance that are normalized with respect to the bottleneck rate \(r_{b}\), thus only dependent on the variability of processing times and completely independent of the average value. This approach offers two key advantages: first, it reduces the need for large datasets to represent any different average processing time; second, the machine learning model, specifically the neural network, can be trained once (and deeply) on normalized data and then used for various values of average processing time while maintaining high accuracy.