## 1 Introduction  
Bridges are inspected periodically to ensure their safety, as their performance continuously deteriorates owing to various environmental stressors and loading effects [1, 2, 3]. Depending on the accuracy and reliability of inspection outcomes, effective decisions can be made with respect to inspection schedules, in-depth inspections, type of maintenance, and replacement [4]. To increase the accuracy and reliability of inspection outcomes, the quality of inspection methods and the number of inspections should be improved and increased, respectively. However, such improvements and increases incur additional costs. Therefore, inspection planning is essential to optimize the inspection application times and methods [5, 6]. Over the last few decades, approaches have been investigated for the development of optimum inspection planning with various probabilistic objectives [2, 5, 7]. These probabilistic objectives are generally based on the damage detection time, probability of failure after inspection and maintenance, value of information obtained from inspections, extended service life with inspection and maintenance, life-cycle cost, and risk, among other factors [8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. Several difficulties are encountered in probabilistic optimum inspection planning. The objectives of optimum inspection planning should be formulated based on the uncertainties associated with damage propagation, damage detection, structural performance assessment and prediction, and estimations of the life-cycle cost and risk. Accordingly, the objectives can be expressed in complex forms using various probabilistic parameters and conditions. Solving a single- or multiobjective optimization problem consisting of probabilistic parameters and conditions is time-consuming [15; 18]. The computational time required for optimum inspection planning can be reduced by applying an efficient optimization algorithm and efficient computation of objective values in the optimization algorithm. Genetic algorithms (GAs) are generally adopted when various conditions are considered in a single- or multiobjective optimization problem and the objectives are expressed in non-closed forms, as GAs can address discrete, continuous, and non-differentiable objectives [19]. An increase in the number of objectives increases the computational cost and reduces the accuracy and reliability of solutions. Continuous efforts have been made over the last few decades to develop more advanced GAs that can improve the accuracy, reliability, and computational efficiency of solutions [20; 21; 22; 23; 24; 25]. Formulating the objectives for optimum inspection planning requires the computation of time-dependent probability of failure (or reliability). When the state function used to compute the probability of failure is formulated based on various probabilistic conditions, is not in a closed form, and the computed probability of failure is relatively low, so a long computational time is required to obtain an accurate probability of failure. In this paper, we present extrapolation approaches to efficiently address the difficulties associated with computing the probability of failure (or reliability) for optimum inspection planning. The optimum inspection planning method proposed in this study produces the optimum inspection application times for a given inspection method. The use of reliability for optimum inspection planning was also reviewed. The representative difficulties associated with computing the probability of failure for optimum inspection planning are described. Direct Monte Carlo simulation (MCS), subset simulation, and two extrapolation approaches (kernel density estimation (KDE) and a combination KDE and generalized Pareto distribution (GPD)) were considered candidate approaches to compute the probability of failure for optimum inspection planning.