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corresponding variable so that the resulting variable has all its values
positive and then perform the analysis of the translated data set by means of
the (weighted) additive model. What the translation invariance tells us is that
the obtained results are exactly the same as if the original data set were
analyzed. In fact, by means of a linear programming code we can solve the
model resorting to the original data set. The reason for performing the
translation so as to achieve non-negative data is that the DEA software
packages typically require this condition. Moreover, in the presence of non-
discretionary inputs or outputs (Banker and Morey, 1986) the same result
holds, since the model formulation only skips the corresponding slack or
excess variables from the objective function.
As already pointed out by Ali and Seiford (1990), the translation
invariance property of the additive model may be very useful for dealing
with the multiplicative model (Charnes et al., 1983), which is simply the
additive model applied to the logarithms of the original data, in order to
achieve a piecewise Cobb-Douglas envelopment. Let us now consider the
case of the two basic radial models, starting with the VRS case.
Proposition 2. The envelopment form of the input (output)-oriented BCC
model is translation invariant with respect only to outputs
(inputs) and to non-discretionary inputs (outputs).
This result means that, for example, we can deal with any output variable
in the input-oriented BCC model, even if all its data are negative, provided
all the input variables have non-negative values. If non-discretionary inputs
are present, they may also contain negative data. In that case we may
consider each non-discretionary input as an output of the model, just by
reversing its sign, and then the model can be solved (see Lozano-Vivas et al.
(2002) for an application where all the environmental variables are located
on the output side of an input-oriented BCC model, although some of them
are originally non-discretionary inputs).
The presence of the efficiency score as a variable at, say, the input
restrictions of an input-oriented BCC model prevents the equivalence
between the set of original restrictions and the set of restrictions obtained
after translating the inputs. Moreover, this equivalence holds if, and only if,
the efficiency score of the unit under assessment equals 1, which means that
the translated model identifies correctly the weak-efficient units. In other
words, if we translate the inputs in an input-oriented BCC model, the
classification of the units as weak-efficient or inefficient remains, but the
efficiency score of each inefficient unit is distorted.
It should also be noted that both the BCC and the additive models are
variable returns to scale (VRS) models, in contrast to the CCR model, which
exhibits constant returns to scale (CRS). In fact, being a VRS model is the
key for satisfying translation invariance or, in other words, the convexity
Pastor & Ruiz, Variables with Negative Values in DEA