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The directional distance function and the translation invariance property

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  • Aparicio, Juan
  • Pastor, Jesus T.
  • Vidal, Fernando

Abstract

Recently, in a Data Envelopment Analysis (DEA) framework, Färe and Grosskopf [15] argued that the input directional distance function is invariant to affine data transformations under variable returns to scale (VRS), which includes, as a particular case, the property of translation invariance. In this paper we show that, depending on the directional vector used, the translation invariance may fail. In order to identify the directional distance functions (DDFs) that are translation invariant under VRS, we establish a necessary and sufficient condition that the directional vector must fulfill. As a consequence, we identify the characteristics that the DDFs should verify to be translation invariant. We additionally show some distinguished members that satisfy the aforementioned condition. We finally give several examples of DDFs, including input and output DDFs, which are not translation invariant.

Suggested Citation

  • Aparicio, Juan & Pastor, Jesus T. & Vidal, Fernando, 2016. "The directional distance function and the translation invariance property," Omega, Elsevier, vol. 58(C), pages 1-3.
    • Handle: RePEc:eee:jomega:v:58:y:2016:i:c:p:1-3
    DOI: 10.1016/j.omega.2015.04.012
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    References listed on IDEAS

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    Cited by:

    1. Aparicio, Juan & Pastor, Jesús T. & Vidal, Fernando & Zofío, José L., 2017. "Evaluating productive performance: A new approach based on the product-mix problem consistent with Data Envelopment Analysis," Omega, Elsevier, vol. 67(C), pages 134-144.
    2. Chen, Zhenling & Zhang, Xiaoling & Ni, Guohua, 2020. "Decomposing capacity utilization under carbon dioxide emissions reduction constraints in data envelopment analysis: An application to Chinese regions," Energy Policy, Elsevier, vol. 139(C).
    3. Aparicio, Juan & Kapelko, Magdalena, 2019. "Accounting for slacks to measure dynamic inefficiency in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 278(2), pages 463-471.
    4. Falavigna, G. & Ippoliti, R., 2020. "The socio-economic planning of a community nurses programme in mountain areas: A Directional Distance Function approach," Socio-Economic Planning Sciences, Elsevier, vol. 71(C).
    5. Liang-Han Ma & Jin-Chi Hsieh & Yung-Ho Chiu, 2020. "Comparing regional differences in global energy performance," Energy & Environment, , vol. 31(6), pages 943-960, September.
    6. Kejia Yang & Yalin Lei, 2017. "The carbon dioxide marginal abatement cost calculation of Chinese provinces based on stochastic frontier analysis," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 85(1), pages 505-521, January.
    7. Charles, Vincent & Färe, Rolf & Grosskopf, Shawna, 2016. "A translation invariant pure DEA model," European Journal of Operational Research, Elsevier, vol. 249(1), pages 390-392.
    8. Juan Aparicio & Magdalena Kapelko, 2019. "Enhancing the Measurement of Composite Indicators of Corporate Social Performance," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 144(2), pages 807-826, July.
    9. Fangqing Wei & Junfei Chu & Jiayun Song & Feng Yang, 2019. "A cross-bargaining game approach for direction selection in the directional distance function," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(3), pages 787-807, September.
    10. Bowen Xiao & Dongxiao Niu & Han Wu & Haichao Wang, 2017. "Marginal Abatement Cost of CO 2 in China Based on Directional Distance Function: An Industry Perspective," Sustainability, MDPI, Open Access Journal, vol. 9(1), pages 1-1, January.
    11. Zhang, Yue-Jun & Chen, Ming-Ying, 2018. "Evaluating the dynamic performance of energy portfolios: Empirical evidence from the DEA directional distance function," European Journal of Operational Research, Elsevier, vol. 269(1), pages 64-78.

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