## 10 Conclusions  
In this study, we proposed an efficient reliability-based optimal inspection planning method for deteriorating bridges using extrapolation approaches, whereby the optimum inspection application times are determined by minimizing the probability of failure. Extrapolation approaches are used to compute the probability of failure for optimum inspection planning. Applications of MCS, subset simulation, and two extrapolation approaches (i.e., KDE and the combination of KDE and GPD) were presented with three general examples. The effects of the approaches (in computing failure probabilities, inspection application times, and the number of inspections) on the probability of failure were investigated using an existing structure. Furthermore, the applications of MCS and two extrapolation approaches for optimum inspection planning of an existing fatigue-sensitive steel girder bridge were presented. Based on this investigation, the following conclusions can be drawn.  
* Probabilistic optimum inspection planning for deteriorating bridges generally requires the estimation of the probability of failure. When a state function is formulated considering multiple probabilistic conditions, it is crucial to select an appropriate approach to compute the probability of failure. In this study, MCS, KDE, and a combination of KDE and GPD were considered as candidate approaches.
* When the number of design variables (i.e., inspection application times) in the inspection planning optimization problem is equal to one and the probability of failure can be reliably obtained using MCS, the optimum inspection application time obtained using MCS is almost the same as that computed using KDE. However, if the number of design variables is two and the probability of failure is low, the solutions after 100 computations with MCS are more dispersed than those with KDE.
* When MCS is used, the accuracy of the computed failure probability considerably depends on the number of samples. A large number of samples is required to obtain an accurate and low probability of failure. Because the computational cost increases with the number of samples, MCS may be inefficient. Although KDE and the combination of KDE and GPD are performed with the samples generated using MCS, the associated computational time is negligible, and these two extrapolation approaches can efficiently compute a low probability of failure.
* The most appropriate approach for reliability-based optimum inspection planning should be selected based on the accuracy and reliability of the solution, the computational efficiency, and the applicability of the probabilistic optimization process. KDE and the combination of KDE and GPD can be easily applied using various program ming languages and existing codes. The presented applications demonstrate that the probability of failure computed using KDE is more accurate and reliable than that estimated using the combination of KDE and GPD.  
K.K.: conceptualization, methodology, computations, and writing--original draft; K.-T.P.: conceptualization and methodology; K.-S.J.: conceptualization and methodology; S.K.: conceptualization, methodology, validation, review, and editing. All authors have read and agreed to the published version of the manuscript.  
This study was funded by the KICT Research Program (project no. 20220217-001, Development of DNA-based smart maintenance platform and application technologies for aging bridges) of the Ministry of Science and ICT of South Korea.  
Not Applicable.  
All data, models, and code generated or used during the study appear in the published article.  
Research for this paper was carried out under the KICT Research Program (project no. 20220217-001, Development of DNA-based smart maintenance platform and application technologies for aging bridges) funded by the Ministry of Science and ICT.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.