## 4 Industrial Applicability  
In the following, a fictitious test case is discussed, in which the complete print head is checked by the final ANN and the number of functioning nozzles is estimated. First of all, the time needed to record the acoustic data is calculated in advance. As already described, this depends directly on the number of samples and can be calculated \(T_{30,000}=\) 480 s for \(n_{sample}=\) 30,000 and \(T_{10,000}=\) 160 s for \(n_{sample}=\) 10,000. The extent to which the reduction in recording time (number of samples) and the associated success rate reduction affect the estimate of the number of functional nozzles can also be subsequently determined by the test case. For the fictitious test case, it is assumed that the acoustic signals of all 768 nozzles are successively tested by the ANN and that the network evaluates an exemplary 454 nozzles as functional and, accordingly, 314 as clogged. By determining a confidence interval as well as a backward calculation, the number of estimated functional nozzles with a confidence level of 95% is to be determined. The following boundary conditions are known for this purpose. \[n_{functional}+n_{clogged}=768 \tag{5}\]  
\[n_{functional}\cdot TQ_{functional}+n_{clogged}\cdot(100-TQ_{clogged})=454 \tag{6}\]  
As can be seen from the above Equations, the boundary conditions represent a system of two Equations, which has the unknowns \(n_{functional}\) and \(n_{clogged}\). \(TQ_{functional}\) and \(TQ_{clogged}\) are the success rates for a functional and for a clogged nozzle, respectively. This distinction is necessary because a functional nozzle is detected with a lower probability than a stuck nozzle. Equations (5) and (6) give an overview of the boundary conditions for determining the estimated functional nozzles \(n_{functional}\) of the print head. First, the sum of estimated functional nozzles \(n_{functional}\) and estimated clogged nozzles \(n_{clogged}\) must be equal to the total number of all nozzles of the print head. On the other hand, the given number of 454 functional nozzles results from the sum of all estimated functional nozzles which were correctly classified as functional nozzles and the number of estimated stuck nozzles which were incorrectly classified as functional by the algorithm. As the two success rates are subject to certain uncertainties, they are to be described as confidence intervals, which corresponds to an estimate of the mean with a given confidence level. Due to the fact that the nozzle state has exactly two possible characteristics (nozzle functional or nozzle clogged), the present case is a Bernoulli process, which leads to a binomial distribution of the success rates. For this, the respective confidence interval can be determined according to Clopper-Pearson [23]. For a number of 1000 test values and a confidence level of 95%, the calculated confidence interval depending on the number of samples can be taken from Table 9. \begin{table}
\begin{tabular}{l l l} \hline
**No. of Samples** & **Success Rate 30 k Samples** & **Success Rate 10 k Samples** \\ \hline Date 1 & 64.1\% & 57.2\% \\ Date 2 & 89.6\% & 74.2\% \\ Date 3 & 61.7\% & 75.8\% \\ Date 4 & 91.8\% & 83.5\% \\
**Average** & **76.8\%** & **72.7\%** \\ \hline \end{tabular}
\end{table} 
Table 8: Success rates for the optimized network architecture for a sample number of 30,000 and 10,000. By solving the system of Equations given by Equations (5) and (6) with the aid of the mapped intervals for the success rates from Table 9, a range can now be determined in which the number of estimated functional nozzles of the print head is located with a confidence level of 95%. This range is listed for the sample numbers 30,000 and 10,000 in Table 10.  
Table 10 shows that for a sample number of 30,000, the interval for the number of estimated functional nozzles is 530-608. With a probability of 95%, the number of functional nozzles of the print head is in this range. In comparison, the number of estimated functional nozzles for a sample number of 10,000 is in a range of 577-681.