DEA cross-efficiency evaluation under variable returns to scale

Abstract

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.

Keywords:

• data envelopment analysis
• cross-efficiency
• variable returns to scale (VRS)

Acknowledgements

The authors are grateful to the two anonymous referees for their constructive comments and suggestions on a previous version of this paper.

Notes

1 Type II free production of outputs can be interpreted as consumption (opposite to production) of outputs without any inputs, and it can be considered as extended free disposability of outputs, which may not be unacceptable. However, we pursue our development assuming it is unacceptable in this paper, which provides a more general framework. In case type II free production of outputs is acceptable; the developed framework of VRS cross-efficiency can be easily simplified to suit the case.

2 We should note that O″ is not the only choice for O* along that does not give rise to the negative-output problem, and resulting cross-efficiencies depend on the choice of O*. However, the choice of an adjusted origin on the x-axis makes it possible to derive a general formula (that does not depend on coefficients β_k) for VRS cross-efficiency as shown in the subsequent paragraphs.