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