Wasserstein Distance-Based Distributionally Robust Modeling of Fuzzy Numbers for Fuzzy Fault Tree Analysis
Abstract
Fuzzy fault tree analysis commonly uses interval, triangular, trapezoidal, or Gaussian fuzzy numbers to describe the uncertainty of basic-event failure probabilities. However, the support intervals and parameters of conventional fuzzy numbers are usually determined by expert judgment and limited historical data, which may be insufficient to capture tail risks caused by environmental changes, cognitive bias, and data scarcity. To address this issue, this paper proposes a Wasserstein distance-based distributionally robust modeling method for fuzzy numbers. The membership function of a fuzzy number is first normalized into a probability density function, and the induced probability distribution is taken as the nominal distribution. A Wasserstein ambiguity set is then constructed to expand the fuzzy probability within a bounded distributional deviation. Two modeling schemes are developed: a structure-preserving ambiguity set, which expands interval, triangular, trapezoidal, and Gaussian fuzzy numbers in their parameter spaces, and a general ambiguity set, which approximates membership functions using step-function discretization. Numerical experiments show that the proposed method effectively expands $\alpha$-cuts under a given Wasserstein radius. An application example of fuzzy fault tree analysis further shows that expanding the $\alpha$-cuts of fuzzy failure probabilities of basic events can increase the membership degrees of true failure probabilities under marginal operating conditions in both basic-event and top-event fuzzy probabilities, thereby improving the coverage of marginal scenarios and tail risks.