For example, if the required probability P,- is 10-4 and the sample size of Monte Carlo
simulation M~ is 106, Ps, is equal to the highest safety factor among the 100 samples
(n=PM with the lowest safety factors. The calculation of Ps, requires only sorting the
lowest safety factors in the Monte Carlo samples. While the probability of failure changes
by several orders of magnitude the probabilistic sufficiency factor usually varies by less
than one order of magnitude in a given design space.
For problems with k reliability constraints, the most critical safety factor is
calculated first for each Monte Carlo sample,
s(xi) = nu (6-16)
Then the sorting of the Ilth minimum safety factor can be proceeded as in (6-14). When n
is small, it may be more accurate to calculate Ps, as the average between the Ilth and
(n+1)th l0WeSt safety factor in the Monte Carlo samples.
The probabilistic sufficiency factor provides more information than probability of
failure or safety index. Even in the regions where the probability of failure is so small
that it cannot be estimated accurately by the MCS with given sample size M, the accuracy
ofPs,is maintained. Using the probabilistic sufficiency factor also gives designers useful
insights on how to change the design to satisfy safety requirements as shown in section
2.1. The estimate is not readily available from the probability of failure or the safety
index. The probabilistic sufficiency factor is based on the ratio of allowable to response,
which exhibits much less variation than the probability of failure or safety index.
Therefore, approximating probabilistic sufficiency factor in design optimization is easier
than approximating probability of failure or safety index as discussed in the next section.