## 3 A Normalized Approach for the Throughput Prediction
Although simpler machine learning methods could potentially be applied given the small errors observed in our test results, the adaptability and scalability of neural networks make them well suited for complex manufacturing scenarios. Before proceeding to the experimental part, it is useful to note that [PERSON] and [PERSON]'s mathematical model already shows a confirmation of what stated, at least in thePWC case. Indeed, by adopting the previously showed PWC law (Table 1), by dividing both members by \(r_{b}\), we obtain:  
\[\frac{TH}{r_{b}}=\frac{w}{W_{0}+w-1}=f(w,W_{0}) \tag{5}\]  
In other words, the left part represents a throughput normalized with respect to the bottleneck rate, whereas the right one is a quantity that is independent of the average processing time but dependent only on the level of WIP in the production line and on \(W_{0}\), the critical WIP that in this balanced case is equal to the number of machines. What we expect is that, when the assumption of exponential variability is relaxed, the quantity on the right may be also dependent on the coefficient of variation, but the independence from the average processing time still remains. Due to the lack of an exact measure for these cases, a simulation tool is here used to experimentally verify if this condition is true. To accomplish this, a simulation tool capable of simulating multiple production lines in a scalable manner has been developed, taking advantages of both discrete-event and agent-based simulation techniques with the use of Anylogic simulation software. The simulation tool is straightforward and consists of two distinct agent types: a first one, named \"Main\" (Figure 3), in which the CONWIP system is modeled and includes all the customization parameters necessary to characterize the various simulation scenarios; and a second one, named \"Machine\" (Figure 3), that consists of a service block capable of simulating both the queue of orders waiting to be processed at the machine and the waiting delay representing the processing time. The reason for developing this agent type is to create a simulator that is not only parametric in terms of average processing time, work in progress, and variability but is also able to effectively scale on the line length, measured in terms of number of machines. Finally, an automatism that can independently simulate multiple sessions (even in parallel) and save all simulation data in an Excel file in terms of performance has been developed. The simulation tool was then validated against the previously mentioned PWC case to determine whether two-year simulation times are sufficient to achieve steady-state values of the simulated production system and whether the model is statistically representative of the theoretical performance known from the [PERSON] et al. [25] model, as reported in Table 1. In this regard, after evaluating the \(TH\) and \(CT\) for various (fixed) WIP values, a \(t\)-test was conducted to validate the results, and it was determined, with a confidence level  
Figure 3: The Main agent and the Machine agent of the simulation tool. of 99% percent, that the simulation tool obtained the same average \(TH\) and \(CT\) values as those calculated using [PERSON] et al. [25]'s PWC model. Coming back to the objective of this section, having fixed the number of machines in the single-product, single-routing line at five (it should be noted that the same conclusion may be found also when considering production lines with different number of machines), an experimental campaign was carried out according to a full factorial plan with factorial levels shown in Table 2. Each simulation was run for a simulated time of two years, collecting the \(TH\) value in terms of ratio between unit produced and the time simulated in hours (i.e., units/hours) and the normalized throughput (\(TH/r_{b}\)) which is a dimensionless number between 0 and 1.