Keywords: imprecise probability, importance sampling, re-analysis.

In reliability design, data for constructing probabilistic models is often scarce. Models whose distribution parameters (e.g. the mean value and standard deviation) vary in known intervals could be more suitable than Bayesian models because the former models do not require assumptions to be made that are not supported by the available evidence. If we use models whose parameters vary in intervals we need to calculate upper and lower bounds of the failure probability (or reliability) of a system in order to make design decisions. Monte Carlo simulation can be used for this purpose, but it is too expensive for all but very simple systems. An efficient Monte Carlo simulation approach for estimation of upper and lower probabilities is proposed. This approach is based on two ideas: a) use an efficient approach for reliability re-analysis of a system, and b) approximate probability distributions of the minimum and maximum failure probabilities using extreme value statistics.

The efficient re-analysis method performs a single reliability analysis of a design by Monte-Carlo simulation and reuses the results of this analysis to estimate the reliability of the same design for different probability distributions of the random variables. A large number of probabilities of failure are calculated using this re-analysis. Then one can estimate directly the minimum and maximum values of these probabilities. An alternative approach is to divide the computed failure probabilities into groups and fit two extreme probability distributions to the maximum and minimum values of these groups. The maximum and minimum values can be found from the parameters of these extreme distributions.

The approach has been demonstrated on an example of a system with a dynamic vibration absorber where the natural frequencies are random and the distribution parameters are specified through bounding intervals. The proposed approach has been found to be as accurate as a Monte Carlo simulation approach, and it takes about one sixtieth of the CPU time of the Monte Carlo simulation approach to estimate the upper and lower bounds of the system failure probability. On the other hand, the method requires a very large number of simulations to find the maximum and minimum failure probabilities because it uses Monte Carlo simulation to explore the space of the distribution parameters.

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