Probabilistic Sensitivity Analysis With Respect to Bounds of Truncated Distributions

Bounds on variables are often implemented as a part of a quality control program to ensure a sufficient pedigree of a product component, and these bounds may significantly affect the product’s design through constraints such as cost, manufacturability, and reliability. Thus, it is useful to determine the sensitivity of the product reliability to the imposed bounds. In this work, a method to compute the partial derivatives of the probability-of-failure and the response moments, such as mean and the standard deviation, with respect to the bounds of truncated distributions are derived for rectangular truncation. The sensitivities with respect to the bounds are computed using a supplemental “flux” integral that can be combined with the probability-of-failure or response moment information. The formulation is exact in the sense that the accuracy depends only upon the numerical algorithms employed. The flux integral is formulated as a special case of the probability integral for which the sensitivities are being computed. As a result, the methodology can be implemented with any probabilistic method, such as sampling, first order reliability method, conditional expectation, etc. Moreover, the maximum and minimum values of the sensitivities can be obtained without any additional computational cost. The methodology is quite general and can be applied to both component and system reliability. Several numerical examples are presented to demonstrate the advantages of the proposed method. In comparison, the examples using Monte Carlo sampling demonstrated that the flux-based methodology achieved the same accuracy as a standard finite difference approach using approximately 4 orders of magnitude fewer samples. This is largely due to the fact that this method does not rely upon subtraction of two near-equal numbers.