## 4 Results
### Results of the Case Study  
First, we present the results of model adjustment of the three heat exchangers discussed in this case study. Table 15 displays the average values of parameters \(\lambda\), \(\beta\), and \(\theta\) from ten  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\lambda\) & \(\beta\) & \(\theta\) & \(c\) & \(\hat{p}\) & \(T_{j}\) & \(\mathbf{s}_{j}\) & \(\mathbf{C_{TOT}(T_{j}\mathbf{s}_{j}\mathbf{c})}\) \\ \hline
111.3703 & 2.5 & 1.2348 & 1 & 1.00 & 33 & 0.2 & 2461.28 \(\times\)\(N\) \\ \hline
107.6006 & 3.0 & 1.3936 & 4 & 0.50 & 34, 74, 139, 353 & 0.9, 0.9, 0.2, 0.2 & 5604.27 \(\times\)\(N\) \\ \hline
109.7674 & 3.5 & 1.2675 & 5 & 0.50 & 32, 70, 145, 270, 443 & 0.9, 0.9, 0.9, 0.2, 0.2 & 9790.07 \(\times\)\(N\) \\ \hline
110.6717 & 4.0 & 1.2472 & 6 & 0.55 & 47, 91, 138, 224, 338, 493 & 0.9, 0.9, 0.9, 0.9, 0.2, 0.2 & 16,371.01 \(\times\)\(N\) \\ \hline
103.8392 & 4.5 & 1.5262 & 8 & 0.35 & 29, 60, 125, 236, 339, 454, 519, 608 & 0.9, 0.9, 0.9, 0.9, 0.9, 0.2, 0.2 & 21,828.61 \(\times\)\(N\) \\ \hline
113.7472 & 5.0 & 1.1006 & 7 & 0.30 & 24, 53, 109, 199, 292, 401, 538 & 0.9, 0.9, 0.9, 0.9, 0.9, 0.4, 0.4 & 54,680.71 \(\times\)\(N\) \\ \hline \end{tabular}
\end{table}
Table 14: Preventive maintenance optimization regarding the different expected numbers of failures over time. Illustration of first case example.  
Figure 7: Variable improvement factor as a function of the intervention level.  
runs. These values were later utilized for optimizing the predictive maintenance times and intervention levels in the second stage of the optimization process. The model adjustment outcomes revealed similarities in the values of \(\lambda\) and \(\beta\) across the three equipment types, but there was considerable variation in parameter \(\theta\). It is essential to note that this parameter directly influenced the preventive maintenance improvement factor, indicating a higher effectiveness of preventive maintenance in this scenario. A more in-depth analysis of this result can be based on the optimization outcomes for the time and intervention levels of preventive maintenance, as presented in Table 16. The table displays the best results achieved in ten model replications, along with the mean and standard deviation of the total cost.  
As the parameter \(\theta\) increased, the recommended number of maintenances also rose, leading to values of 4, 5, and 6 for \(k\) = 1, 2, and 3, respectively (with \(\theta\) values of 0.9690, 1.1862, and 1.6559). This effect was also noticeable in the intervention levels allowed for each equipment. Heat exchangers 1 and 2 typically required a 100% intervention, while equipment \(k\) = 3 allowed interventions of 70%, for example. Interestingly, for \(k\) = 3, lower intervention levels were suggested at the beginning of the planning horizon, possibly due to the higher reliability during the initial analysis period and the increased effectiveness of preventive maintenance in this scenario (as indicated by the higher value of \(\theta\)).  
Furthermore, it is crucial to discuss the results concerning the total maintenance costs in each case. It became apparent that a higher number of preventive maintenance actions led to a reduction in the overall cost, which was directly influenced by the effectiveness of these actions. This behavior can be attributed to the decrease in the expected number of failures following preventive maintenance, which also holds true for maintenance actions with lower intervention levels, even though the effect may be less pronounced in these instances. Clearly, there exists a threshold beyond which reducing the total cost per unit of time by increasing the number of maintenance actions is no longer beneficial.