Facet analysis in data envelopment analysis: some pitfalls of the CRS models

The nonparametric estimator of the Extended Facet reference technology for the Constant Returns to Scale case has attracted some attention, because the associated production frontier by construction does only include strongly efficient faces of maximal dimension, or strongly efficient faces that are part of such strongly efficient faces of maximal dimension. The strongly efficient faces of maximal dimension are denoted Full Dimensional Efficient Facets (FDEFs). The identification of such strongly efficient facets is facilitated by removing all inefficient and all strongly efficient but not extreme efficient DMUs from the estimation procedure of the technology set. Any face that i) is passing through the origin and with (s + m − 1) linear independent extreme efficient observed DMUs positioned on it and ii) with a normal vector with strict positive (strict negative) output (input) components, is a FDEF, where s (m) is the number of outputs (inputs). It is, however, not correct that every face that satisfies only i) is a FDEF. We denote a face (a subface) that satisfies only condition i) but not condition ii) for an AP-face (an AP-subface). It is proved that a radial projection of any output input combination in the estimated EXFA technology set is positioned on the strongly efficient frontier if and only if i) no A P-(sub)faces exist, ii) a regulaty condition RC1 is satisfied and only dual multiplier constraints corresponding to extreme efficient DMUs are included in the estimation. A test for the fulfillment of the condition that no A P-(sub)faces exist is provided.