## 4 Results
### Numerical Examples  
Model adjustment results for the first case example are presented in Table 8. The average values for parameters \(\lambda\), \(\beta\), and \(\theta\) were employed to optimize the times and intensities of preventive maintenance in the second optimization stage. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(k=1\) & & & \(k=2\) & & & \(k=3\) & \\ \hline
**Time \(t\)** & & & **Time \(t\)** & & **Event** & \(s\) & **Time \(t\)** & **Event** & \(s\) \\ (Days) & **Event** & & & **(Days)** & & **(Days)** & **Event** & \(s\) \\ \hline
0.00 & Start & - & 0.00 & Start & - & 0.00 & Start & - \\
39.68 & Failure & - & 29.56 & Failure & - & 52.13 & Failure & - \\
51.99 & Failure & - & 48.12 & Failure & - & 75.27 & Failure & - \\
64.76 & Failure & - & 60.00 & PM & 0.8 & 89.45 & Failure & - \\
80.00 & PM & 1.0 & 99.78 & Failure & - & 105.69 & Failure & - \\
127.74 & Failure & - & 125.96 & Failure & - & 120.00 & PM & 1.0 \\
144.21 & Failure & - & 140.00 & PM & 0.8 & 174.89 & Failure & - \\
154.88 & Failure & - & 194.21 & Failure & - & 217.62 & Failure & - \\
160.00 & PM & 1.0 & 227.26 & Failure & - & 233.97 & Failure & - \\
234.88 & Failure & - & 233.97 & Failure & - & 240.00 & PM & 0.8 \\
263.97 & Failure & - & 240.00 & PM & 0.8 & 284.37 & Failure & - \\
285.35 & Failure & - & 294.88 & Failure & - & 300.00 & End & - \\
300.00 & End & - & 300.00 & End & - & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Heat exchanger failure data. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Numerical Examples** & **Case Study** \\ \hline
**Parameter** & **Value** & **Value** \\ \hline Number of iterations & 100 & 100 \\ Population size & 100 & 100 \\ Mutation rate & 5\% & 1\% \\ Crossover rate & 75\% & 90\% \\ Crossover operator & Two-point & Two-point \\ Elitism rate & 5\% & 5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Genetic algorithm parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Replication** & \(\lambda\) & \(\beta\) & \(\theta\) \\ \hline
#1 & 117.6538 & 1.9994 & 0.9971 \\
#2 & 128.0954 & 1.9916 & 0.6368 \\
#3 & 109.6726 & 1.9993 & 1.2811 \\
#4 & 100.3383 & 1.9758 & 1.6545 \\
#5 & 103.6600 & 1.9979 & 1.5312 \\
#6 & 111.9697 & 1.9928 & 1.1886 \\
#7 & 103.3317 & 1.9943 & 1.5654 \\
#8 & 113.1235 & 1.9969 & 1.1734 \\
#9 & 124.3926 & 1.9937 & 0.8 The average values of times and severities in ten replications are presented in Table 9, which shows the optimal number of maintenance cycles (\(c\)), the optimal PM times \(T_{j},j=1,\ldots,c\) (in days), the severities \(s_{j},j=1,\ldots,c\), and the respective total cost, \(C_{\text{TOT}}\left(T_{j},s_{j},c\right)\). It is possible to observe that all replications resulted in a single maintenance cycle (\(c=1\)), with time \(T_{1}\approx 35\) days and an intervention level of 20%, which has the lowest cost among all possible maintenance activities (Table 1). Since preventive maintenance costs vary as a function of intervention level, according to Table 1, the result that minimizes the total maintenance cost has the lowest intervention level, even though this intervention level promotes the lower improvement in the system intensity of failure. Therefore, optimization is expected to indicate maintenance with maximum levels if preventive maintenance costs are fixed. The results with the preventive maintenance costs set at USD 4000, regardless of the intervention level, are presented in Table 10. It can be observed that the most common solution becomes an intervention level of 90%, as expected. It is possible to observe the occurrence of solutions with other levels of intervention, such as 50%. It is noteworthy that variations in the solution are due to the stochastic characteristic of the genetic algorithm itself. Additionally, there is a relatively small difference in the improvement factor and, consequently, in the expected number of failures, for nearby intervention level values (see Figure 4). For all executions under these same conditions, however, solutions with an intervention level of 20% were not obtained.