Super-efficiency based on the directional distance function in the presence of negative data

Abstract

The problem of how to construct an appropriate direction for the directional distance function (DDF) based super-efficiency model arises in the data envelopment analysis field. By exploring the relationships between the optimal value of the DDF-based super-efficiency model and directions, this paper provides the conditions satisfied by the directions, with which the DDF-based super-efficiency model is capable of dealing with negative data and generates bounded super-efficiency scores. Based upon these conditions, two types of directions are put forward. No matter whether negative data exist or not, DDF-based super-efficiency models with these directions are feasible and generate bounded super-efficiency scores. They successfully address the infeasibility issue of the conventional radial super-efficiency model under the assumption of variable returns to scale. Super-efficiency models with the proposed directions are monotonous and unit-invariant. More importantly, compared with the current DDF-based super-efficiency model capable of dealing with negative data, the super-efficiency model with one type of direction is translation invariant for both inputs and outputs. Numerical examples demonstrate the validity and practicality of the proposed conditions and directions.