Multiplicative and additive distance functions: Efficiency measures and duality

Abstract

In this survey, we present, for the first time, a classification scheme for distance functions, considering two broad groups: the multiplicative and the additive distance functions. Guiding empirical work is one of the objectives of this paper; for this reason, we consider only linear distance functions within a data envelopment analysis (DEA) framework. This also constitutes an easy way of connecting distance functions and efficiency measures. Further, we analyze two classes of distance functions: the ratio-directional distance function and the loss distance function. The former opens the possibility of evaluating productivity change combining directional distance functions, additive in nature, with Malmquist indexes, multiplicative in nature. The latter unifies all the known linear distance functions under a common structure, allowing the numerical evaluation of any linear distance function, as shown by a numerical example. We end up with a revision of duality results so as to highlight the economic relevance of distance functions.