small target probability of failure, the accuracy of MCS around the optimum is not as
good as in regions with high probability of failure. Furthermore, the probability of failure
in some regions may be so low that it is calculated to be zero by MCS. This flat zero
probability of failure does not provide gradient information to guide the optimization
procedure.
The probabilistic sufficiency factor is readily available from the results of MCS
with little extra computational cost. The noise problems of MCS motivate the use of
response surface approximation (RSA, e.g., Khuri and Cornell 1996). Response surface
approximations typically employ low-order polynomials to approximate the probability
of failure or safety index in terms of design variables in order to filter out noise and
facilitate design optimization. These response surface approximations are called design
response surface approximation (DRS) and are widely used in the RBDO (e.g., Sues et al.
1996).
The probability of failure often changes by several orders of magnitude over
narrow bands in design space, especially when the random variables have small
coefficients of variation. The steep variation of probability of failure requires DRS to use
high-order polynomials for the approximation, such as quintic polynomials (chapter 5),
increasing the required number of probability calculations (Qu et al. 2000). An additional
problem arises when Monte Carlo simulations (MCS) are used for calculating
probabilities. For a given number of simulations, the accuracy of the probability estimates
deteriorates as the probability of failure decreases.
The numerical problems associated with steep variation of probability of failure
led to consideration of alternative measures of safety. The most common one is to use the