Negative features of hyperbolic and directional distance models for technologies with undesirable outputs

In data envelopment analysis for environmental performance measurement the undesirable outputs are taken into account. Ones of the standard approaches for dealing with the undesirable outputs are the hyperbolic and the directional distance measures. They both allow a simultaneous expansion of desirable outputs and a contraction of undesirable outputs by means of a single parameter. To meet environmental requirements, a technology with no disposability of undesirable outputs is often considered and the outputs are assumed to be only weakly disposable. We show that the combination of this type of technology with the hyperbolic measure, (or with its linearization, which is a special type of the directional distance model) may lead to a misleading efficiency score of the unit under evaluation. We derive the dual of the hyperbolic model under the environmental technology and describe some of its properties. Then, we use the hyperbolic and directional distance dual models for developing a second-phase method. This enables to detect the misleading scores of the decision making units. We illustrate the results on a real world data set.