## 8 Application for Optimum Inspection Planning
### Optimum Inspection Planning with Extended Service Life-Based State Function  
The optimum inspection application times and associated values of \(p_{f}\) are shown in Figures 12 and 13. The objective of the optimization problem is to minimize the probability of failure (\(p_{f}\)) formulated with the extended service life-based state function (see Equation (13)). The number of inspections (\(N_{i}\)) is one and two in Figures 12 and 13, respectively. The 100 solutions obtained using MCS, KDE, and the combination of KDE and GPD are illustrated in parallel coordinates in Figure 12a-c (or Figure 13a-c), respectively.  
When MCS was used to compute the probability of failure (\(p_{f}\)) considering one inspection, the optimum inspection application time (\(t_{\textit{insp},1}\)) was equal to 10 years (see Figure 12a). If the probability of failure (\(p_{f}\)) is estimated using KDE, the inspection application time of \(t_{\textit{insp},1}=10\) years should be applied to minimize \(p_{f}\) (see Figure 12b). Figure 12c indicates that one of the optimum solutions was \(t_{\textit{insp},1}=9.93\) years. As shown in Figure 12a-c, (a) the solutions were not dispersed, and (b) the constraint for the optimization problem (i.e., 1 year \(\leq t_{\textit{insp},i}-t_{\textit{insp},i-1}\leq 10\) years) was active because the 100 solutions shown in Figure 12a,b had the same optimum inspection application time of 10 years.  
Figure 13 shows the optimum inspection application times and corresponding probability of failure (\(p_{f}\)) when two inspections are considered to formulate the optimization problem. The optimum inspection application times (\(t_{\textit{insp},1}\) and \(t_{\textit{insp},2}\)) obtained using MCS were 7.033 and 13.714 years, respectively (see Figure 13a). When KDE was used to compute \(p_{f}\), \(t_{\textit{insp},1}\) was 7.033 years, and \(t_{\textit{insp},2}\) was 13.718 years (see Figure 13b). With the use of the combination of KDE and GPD, \(t_{\textit{insp},1}\) was 8.606 years, and \(t_{\textit{insp},2}\) was 17.171 years (see Figure 13c). As shown in Figure 13, (a) the optimum inspection application times obtained using MCS are similar to those obtained using KDE; (b) the optimum inspection application times (\(t_{\textit{insp},1}\) and \(t_{\textit{insp},2}\)) computed using MCS and KDE were less dispersed than those computed using the combination of KDE and GPD; and (c) the objective values based on MCS are more dispersed than those based on KDE.  
Figure 11: Optimum inspection planning to minimize the probability of failure based on damage detection time-based state function with one-time inspection: (**a**) MCS; (**b**) KDE; and (**c**) the combination of KDE and GPD.  
Figure 12: Optimum inspection application time to minimize the probability of failure based on extended service life-based state function with one-time inspection: (**a**) MCS; (**b**) KDE; and (**c**) the combination of KDE and GPD.  
Figure 13: Optimum inspection application time to minimize the probability of failure based on extended service life-based state function with two-time inspection: (**a**) MCS; (**b**) KDE; and (**c**) the combination of KDE and GPD.  
The dispersions of the optimum inspection times are affected by the probability of failure based on the extended service life-based state function (see Equation (13)). MCS computes the probability of failure directly (see Equation (2)). However, the probability of failure using KDE and the combination of KDE and GPD is obtained through extrapolation with the sample data generated from MCS. Therefore, KDE and GPD can provide a more stable probability of failure than MCS during the optimization process using GA.