Path-based DEA models in multiplier form and returns-to-scale analysis

Data envelopment analysis (DEA) models appear in envelopment and multiplier forms, which are in a primal-dual relationship. In this paper, we derive the general multiplier form of path-based models, encompassing radial, directional distance function, and hyperbolic distance function models as special cases. We investigate the economic interpretation of the multiplier models and uncover the link between shadow profit inefficiency and technical inefficiency provided by path-based models. Using the optimality conditions for the primal-dual pair, we precisely describe the two-way relationship between the optimal solutions of the multiplier model and the supporting hyperplanes of the technology set at the projection. This relationship serves as a mathematical justification for extending one of the early approaches to measuring returns-to-scale (RTS) onto the entire class of path-based models. Moreover, we demonstrate the eligibility of this method by revealing the fact that the set of all strongly efficient benchmarks for the assessed unit in path-based models does not need to belong to a single strongly efficient face of technology set. This finding changes the view on traditional approaches to RTS measurement that rely on supporting hyperplanes encompassing a single strongly efficient face of the technology set. In this new perspective, we propose two methods for RTS measurement. The first is based on the hyperplanes at the projection, and the second method adapts the minimum face method to be suitable for path-based models. Both methods are fully justified and brought to an algorithmic form.