Abstract
In this chapter we present an overview of the different approaches that have considered translation invariant Data Envelopment Analysis (DEA) models. Translation invariance is a relevant property for dealing with non-positive input and/or non-positive output values. We start by considering the classical approach and continue revising recent contributions. We also consider non-translation invariant DEA models that are able to deal with negative data at the expense of modifying the model itself. Finally, we propose to study translation invariance in a general framework through a recently introduced distance function: the linear loss distance function.
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Notes
- 1.
Each DEA linear program has a linear dual. Usually the primal or envelopment form evaluates each unit by measuring its “distance” with respect to the frontier, while the dual or multiplier form determines the supporting hyperplane where the unit under evaluation is projected (see Ali and Seiford 1993). Moreover, linear programming theory shows us that the objective function values of both dual programs are the same if they are finite.
- 2.
As mentioned before, Seiford and Zhu (2002) is another previous approach where a method based on the extension of facets is proposed in order to achieve the same goal but resorting to radial models.
- 3.
Each linear restriction is either an equality or a non-strict inequality. On the other hand, the generation of a DEA loss function program under different returns to scale is straightforward (Pastor et al. 2012). For instance, in order to get a non-increasing returns to scale program we just add the restriction \(\alpha \ge 0\); if we want a constant returns to scale program we just delete \(\alpha \) in model (8.13).
- 4.
The last model of Table 8.1 is defined as follows (see Aparicio et al. 2013):\(\max \left\{{{\beta }^{-}}+{{\beta }^{+}}:\sum_{j=1}^{n}{{{\lambda }_{j}}{{x}_{j}}}\le {{x}_{0}}-{{\beta }^{-}}{{g}^{-}},\sum_{j=1}^{n}{{{\lambda }_{j}}{{y}_{j}}}\ge {{y}_{0}}+{{\beta }^{+}}{{g}^{+}},\sum_{j=1}^{n}{{{\lambda }_{j}}}=1,{{\beta }^{-}},{{\beta }^{+}},{{\lambda }_{j}}\ge 0 \right\}\).
- 5.
The known relationship between the values of the Shephard input and output distance functions (Shephard 1953) under CRS suggests the last introduced efficiency score.
- 6.
If we were translating the first output instead of the first input the proof is completely similar. The new translated input would be \(y_{_{1j}}^{'}={{y}_{1j}}+{{k}_{1}},\,\forall j\), which generates a derived feasible solution defined as \(({{c}^{*}},{{p}^{*}},{\alpha }' )\) where \({\alpha }'={{\alpha }^{*}}+p_{1}^{*}{{k}_{1}}\).
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Acknowledgements
We would like to thank Prof. Joe Zhu for kindly inviting us to contribute a chapter to the edition of this book. Additionally, Juan Aparicio is grateful to the Generalitat Valenciana for supporting this research with grant GV/2013/112.
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Pastor, J., Aparicio, J. (2015). Translation Invariance in Data Envelopment Analysis. In: Zhu, J. (eds) Data Envelopment Analysis. International Series in Operations Research & Management Science, vol 221. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7553-9_8
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