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Negative features of hyperbolic and directional distance models for technologies with undesirable outputs

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Abstract

In data envelopment analysis for environmental performance measurement the undesirable outputs are taken into account. Ones of the standard approaches for dealing with the undesirable outputs are the hyperbolic and the directional distance measures. They both allow a simultaneous expansion of desirable outputs and a contraction of undesirable outputs by means of a single parameter. To meet environmental requirements, a technology with no disposability of undesirable outputs is often considered and the outputs are assumed to be only weakly disposable. We show that the combination of this type of technology with the hyperbolic measure, (or with its linearization, which is a special type of the directional distance model) may lead to a misleading efficiency score of the unit under evaluation. We derive the dual of the hyperbolic model under the environmental technology and describe some of its properties. Then, we use the hyperbolic and directional distance dual models for developing a second-phase method. This enables to detect the misleading scores of the decision making units. We illustrate the results on a real world data set.

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Notes

  1. Even in this journal a number of interesting papers has been published recently, where some modifications of the basic models are used in various applications. For example, the two-stage output oriented SBM model with restriction on weights is used in evaluation of countries in olympic games by Jablonsky (2018), an input oriented BCC model is modified to select most efficient information system projects in Toloo et al. (2018), and a modification of the output oriented BCC model is employed to assess impacts of environmental regulations on Indian cement industry in Bandyopadhyay (2010).

  2. We use an extension of the notion of Pareto-Koopmans efficiency to technologies with undesirable outputs described e.g. in Scheel (2001).

  3. An output correspondence \(P(\mathbf{x})\) of the technology set is and alternative way of representation and it is joined with the technology \({{\mathcal {T}}}\) with the equivalence relation \((\mathbf{x},\mathbf{b},\mathbf{y})\in {{\mathcal {T}}} \ \Leftrightarrow \ (\mathbf{b},\mathbf{y})\in P(\mathbf{x})\).

  4. Note that this assumption is not compatible with the axiom of free disposability of outputs, which was nevertheless imposed in the models in a number of papers (see e.g. Färe et al. 1989; Boyd and Mc Clelland 1999; Boyd et al. 2002; Zaim and Taskin 2000a).

  5. We formulate programs equivalent to the maximization programs introduced in Färe et al. (1989).

  6. Färe et al. (1989) use a different linear approximation \(\frac{1}{\theta }\approx 2-\theta \).

  7. Wu et al. (2014) use the second order cone reformulation of the model, following the approach of Sueyoshi and Sekitani (2007) for Russell measure model. However, as shown in Halická and Trnovská (2018), the semidefinite programming is simpler, and allows to formulate a well interpretable dual.

  8. The trace \(tr(\mathbf{P}{} \mathbf{Z})\) represents an inner product in the vector space of symmetric \(2\times 2\) matrices.

  9. See the duality theory for semidefinite programming presented for example in Todd (2001).

  10. The proof of this assertion follows the same pattern as the proof of similar result for the Russell measure model in Halická and Trnovská (2018).

  11. The MIP and the weighted additive models provide the technical inefficiency measure IM being equal zero for an efficient DMU. By the associated efficiency measure we understand \(1- IM\).

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Acknowledgements

The authors are thankful to Israfil Roshdi for sharing the dataset for 92 coal-fired power plants, and three anonymous reviewers for comments. Research of the authors was supported by VEGA Grant 01/0062/18.

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  1. Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, 842 48, Mlynská dolina, Bratislava, Slovakia

    Margaréta Halická & Mária Trnovská

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  1. Margaréta Halická
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  2. Mária Trnovská
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Correspondence to Mária Trnovská.

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Halická, M., Trnovská, M. Negative features of hyperbolic and directional distance models for technologies with undesirable outputs. Cent Eur J Oper Res 26, 887–907 (2018). https://doi.org/10.1007/s10100-018-0567-2

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