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