response surface approximation is more accurate than the probability design response
surface approximation.
Besides the average errors over the design space, it is instructive to compare errors
measured in probability of failure in the important region of the design space. For
optimization problems, the important region is defined as the region containing the
optimum. Here it is the curve of target reliability according to each design response
surface, on which the reliability constraint is satisfied critically, and the probability of
failure should be 0.00135 if design response surface approximation does not have errors.
For each design response surface approximation, 11 test points were selected along a
curve of target reliability and given in the Appendix. The average percentage errors at
these test points, shown in Table 6-4, demonstrate the accuracy advantage of the
probabilistic sumfciency factor approach. For the target reliability, the standard error due
to Monte Carlo simulation of 100,000 samples is 8.6%, which is comparable to the
response surface error for the Pyf. For the other two response surfaces, the errors are
apparently dominated by the modeling errors due to the cubic polynomial approximation.
Table 6-4. Averaged errors in cubic design response surface approximations of
probabilistic sumfciency factor, safety index and probability of failure at 1 1
points on the curves of target reliability
Design Response Probability of Probabilistic
Safety Index (Pof)
Surface of failure L suffciency factor
Average Percentage
Error in Probability 213.86% 92.38% 10.32%
of Failure
The optima found by using the design response surface approximations of Table 6-
3 are compared in Table 6-5. The probabilistic sufficiency factor design response surface
clearly led to a better design, which has a safety index of 3.02 according to Monte Carlo
simulation. It is seen that the design from probabilistic sumfciency factor design response