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presenting for that criterion values more distant to a j . So, the variance must be large
enough to make that probability not so small that the information given by the other
criteria may become irrelevant, giving veto power to a single criterion.
This principle can be satis
ed, in practice, if we estimate the variance of the
evaluations according to each criterion by the variance of the set of values by this
criterion in the various pro
les offered as representative of the classes. This may
lead to overestimation, because the variance in such a sample combines the inner
variance in each class with the variance between classes and only the
first of these
components must be attributed to the uncertainty inherent in each register. In fact,
each alternative should belong to a proper class and differences between classes are
not of a random nature. But a possible overestimation is welcome to avoid the
above mentioned risk of observations being deemed as too far away from some of
the reference pro
les suggested.
Once replaced the exact measurements a j and C ihj by distributions of random
variables X k and Y ihj centered on these measurements, probabilities of outranking
can be calculated. Let us denote by A i + and A i the probabilities of the alternative A
presenting a value for X j respectively above and below those associated to the jth
criterion in the ith class pro
les. By independence between disturbances affecting
the evaluations by a given criterion of different alternatives and different pro
les,
Y
A ij ¼
PX j [
Y ihj
h
and
A ij ¼
Y
:
PX j \
Y ihj
h
To make sure that the probability of an alternative being above the representative
pro
les decreases as increases the order of the class as well as, conversely, the
probability of being below increases, it is necessary that the number of pro
les be
the same in all classes, i.e., that, for all i, n(i) = k, for a constant k.
If the decision maker, when modeling the classes, offers different numbers of
pro
les, additional pro
les may be assigned to the classes with number of pro
les
different of the maximum. To form these complementing pro
les, median or mean
values may be employed.
8.3 Composition Rules
The Choquet integral with respect to a capacity that reflects an assessment of the
importance of the criteria can be used to compose global evaluations of the alter-
native being above and being below each class. Assuming linearity, this can be
reduced to a weighted average.
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