Infeasibility conditions and resolution strategies for super-efficiency models under weak disposability and null-jointness: A directional distance function approach with endogenous directions

Existing studies have not focused on the infeasibility of super-efficiency models under the weak disposability and null-jointness (WDJ) assumptions, despite the wide adoption of these two conditions in fields where undesirable outputs exist, like the evaluation of energy and environmental efficiency. This paper employs a directional distance function (DDF) approach to investigate super-efficiency feasibility under these assumptions. We note that DDF-based super-efficiency models using frequently-used exogenous directions may encounter infeasibility under the WDJ assumptions, even when constant returns to scale (CRS) are assumed. We present the specific conditions that lead to this infeasibility. By utilizing endogenous directions, we construct a feasible DDF-based CRS super-efficiency model under the WDJ assumptions, ensuring that the DDF super-efficiency scores remain within a maximum of 1. We also find that the DDF-based super-efficiency model under the WDJ assumptions is infeasible in certain cases when variable returns to scale (VRS) are assumed, regardless of whether directions are endogenous or exogenous. To address this issue, we propose a modified DDF-based VRS super-efficiency model that aims to maintain the WDJ assumptions as much as possible. This VRS model ensures feasibility and generates DDF super-efficiency scores below 1. Some properties of the models and their relationships are discussed. Finally, several examples and a real case from the literature validate the feasibility and practical applicability of the proposed models.