Application to FMEA Priority Assessments - Probabilistic Composition of Preferences, Theory and Applications - page 110

In each of these steps, the comparative assessment in terms of probability of
being the most severe, the least detectable and the most frequent can be applied
under the conditional framework. Thus, a natural measure of risk is the product of
the probabilities of maximizing frequency by the conditional probability of maxi-
mizing undetectability given occurrence and the conditional probability of maxi-
mizing damage given undetectability and occurrence. Such multiplication
corresponds to the pessimist and progressive probabilistic composition of the three
criteria of FMEA.
Besides this theoretical reason, there is a practical reason to replace RPN by
RPP. In the classical calculation of RPN, a change from 1 to 2 in the evaluation of
frequency, for instance, corresponding to a change in the evaluation of the cause
from extremely rare to a little less frequent, doubles the product of the three factors,
so increasing the RPN much more than it would be reduced by a change from 10 to
9, from extremely common to a little less frequent. If the priority is determined by
the probability of being at the same time the least detectable, most frequent and
most serious, this distortion is limited, as the probability according to any criterion
that the risk is the highest varies among alternatives of high risks much more than
among alternatives of low risks.
To be able to perform the probabilistic transformation, one must model the
distributions of the random disturbances. In Sant

'

Anna ( 2012 ) is set for FMEA with
scores ranging from 1 to 10 a triangular distribution with bounds at 0 and 11. With
the limits on the exact possible ends of 1 and 10, as well as with limits more distant,
spaced by a small fraction of these lower and upper ends, for instance to 0.9 and 11,
very close ranks have been obtained in practice.
On the other hand, by setting the ends of the distributions near the largest and the
smallest observed values, instead of the maximum and minimum possible values of
1 and 10, different results are obtained. Approaching the bounds to the observed
values corresponds to discarding the equal weighting of criteria implicit in the
multiplication formula and adopting the weighting implicit in the variability of the
evaluations given by experts.
A basic feature of FMEA is to let the experts the possibility of giving more
weight to one or another criterion by placing a greater distance between the scores
for the modes of failure according to the criterion that they believe to be more
important than the others. It may happen, on the other hand, that it is the greater
dif culty in assessing the importance of the modes of failure according to a criterion

that leads the evaluators to shorten distances in the evaluations according to such
criterion. If this is the case, setting the ends of the distribution next to the observed
values, by increasing the importance of the factors where there is less variability,
will unduly distort the composition.
Finally, instead of just ranking the modes of failure, experts on FMEA some-
times present the results of their evaluation in terms of classes of risk. The prob-
abilistic approach allows for directly classifying the modes of failure by the
trichotomic probabilistic measurements described in Chap. 8 . For that, absolute
pro

les for the classes may be derived. Basic pro les for each class may be

established with equal values for all the criteria.