# Efficient numerical simulation methods for estimating fuzzy failure probability based importance measure indices

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## Abstract

For the problem with a fuzzy failure state which commonly exists in degradation structures and systems, the fuzzy failure probability based importance measure indices can be used to measure the effect of the input variables on the fuzzy failure probability effectively. However, the computational cost is unaffordable for estimating the indices directly. For efficiently evaluating the fuzzy failure probability based importance measure indices, this paper proposed two numerical simulation methods, i.e., the direct Monte Carlo simulation method based on the Bayesian formula (B-DMCS) and the adaptive radial-based importance sampling method based on the Bayesian formula (B-ARBIS). The two proposed methods employ the Bayesian formula to eliminate the dependence of the computational cost on the dimensionality of the input variables. Compared with the B-DMCS method, the B-ARBIS method can enhance the computational efficiency significantly due to repeatedly utilizing the same group of samples of the input variables and the strategy to adaptively search the optimal hypersphere in the safety domain. After giving the principles and implementations of the two methods, three examples are employed to validate the effectiveness of the two proposed numerical simulation methods. The results of the examples demonstrate that the effectiveness of the two proposed methods is higher than the direct Monte Carlo method, and the B-ARBIS method can improve the efficiency obviously in contrast with the B-DMCS method.

## Keywords

Fuzzy state Fuzzy failure probability Importance measure indices Monte Carlo simulation Adaptive radial-based importance sampling Bayesian formula## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. NSFC 51775439).

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