DEA cross-efficiency evaluation under variable returns to scale

Cross‐efficiency evaluation in data envelopment analysis (DEA) has been developed under the assumption of constant returns to scale (CRS), and no valid attempts have been made to apply the cross‐efficiency concept to the variable returns to scale (VRS) condition. This is due to the fact that negative VRS cross‐efficiency arises for some decision‐making units (DMUs). Since there exist many instances that require the use of the VRS DEA model, it is imperative to develop cross‐efficiency measures under VRS. We show that negative VRS cross‐efficiency is related to free production of outputs. We offer a geometric interpretation of the relationship between the CRS and VRS DEA models. We show that each DMU, via solving the VRS model, seeks an optimal bundle of weights with which its CRS‐efficiency score, measured under a translated Cartesian coordinate system, is maximized. We propose that VRS cross‐efficiency evaluation should be done via a series of CRS models under translated Cartesian coordinate systems. The current study offers a valid cross‐efficiency approach under the assumption of VRS–one of the most common assumptions in performance evaluation done by DEA.