Geoscience Reference
In-Depth Information
subject to
X
l
r
O
ra
1
for every alternative a
from 1 to n
;
;
;
r
ยต
r
> 0 for all r, r varying from 1 to m, the number of criteria, O
ra
denoting the
evaluation of the a-th alternative by the r-th criterion and a
0
denoting the alternative
being evaluated.
6.2 Use of DEA Scores
The composition of probabilities of preference according to multiple criteria may
employ the DEA multidimensional distances to the frontiers of best or worst per-
formance. The scores of ef
ciency generated by the algorithms of DEA take the role
of global scores of preference. DEA models are based on ef
ciency ratios between
input and output variables, but they can be applied to any situation by treating all
criteria as outputs generated by an input of
identical values along all
the
alternatives.
This approach may be applied even in the problem of comparing production
units. In that case, the output/input productivity ratios may be treated as outputs of
constant input.
Changing signs if necessary, the evaluations according to every criterion can be
oriented in such a way that higher evaluations correspond to better alternatives.
With this, a DEA model with a constant input and with the probabilities of max-
imizing preference according to each criterion as outputs can always be employed.
DEA scores do not depend on the scales of measurement and do not involve
weighting variables. Nevertheless, they are heavily affected by outliers. Any
alternative with an extreme low value in only one input or an extreme high value in
only one output will be necessarily evaluated as fully ef
cient. Its extreme value
will also strongly affect the score of inef
cient units.
By taking into account distances to other alternatives besides those in the
frontier, the probabilistic transformation into probabilities of being the best
increases the resistance of the scores of the alternatives outside the frontier against
the in
uence of such outliers. However, if the DEA algorithm is used, even with the
probabilistic transformation, care must be taken to detect the possibility of an
extreme value in some variable leading to an ef
ciency score equal to 1.
In addition to the model with constant input, the probabilistic composition can
also use DEA algorithms to evaluate ef
ciency of production units employing
combinations of inputs to generate sets of outputs. The probabilistic approach to
this problem can take as criteria the output/input ratios for the different pairs of
input and output or take each input and each output as individual criteria.
In this latter form of modeling, the individual probabilities of preference will be
the probabilities of maximizing some measure of revenue from the sale of each
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