The Reverse Directional Distance Function

Abstract

The aim of any Data Envelopment Analysis (DEA) inefficiency model is to calculate the efficient projection of each unit belonging to a certain finite sample. The reverse directional distance function (RDDF) is a new tool developed in this chapter that allows us to express any known DEA inefficiency model as a directional distance function (DDF). Hence, given a certain DEA inefficiency model, its RDDF is a specific DDF that truly reproduces the functioning of the considered DEA model. Automatically, all the interesting properties that apply to any DDF are directly transferable to the considered DEA model through its RDDF. Hence, the RDDF enlarges the set of properties exhibited by any DEA model. For instance, given any DEA inefficiency model, its economic inefficiency—in any of its three possible versions—, can be easily defined and decomposed as the sum of technical inefficiency and allocative inefficiency thanks to the RDDF. We further propose to transform any non-strong DDF into a strong DDF, i.e., into a DDF that projects all the units onto the strongly efficient frontier. This constitutes another indication of the transference capacity of the RDDF, because its strong version constitutes in itself a strong version of the original DEA model considered. We further propose to search for alternative projections so as to minimize profit inefficiency, and add an appendix showing how to search for multiple optimal solutions in additive-type models.