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Estimating the hyperbolic distance function: A directional distance function approach

Author

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  • Färe, Rolf
  • Margaritis, Dimitris
  • Rouse, Paul
  • Roshdi, Israfil

Abstract

Färe, Grosskopf, and Lovell (1985) merged Farrell’s input and output oriented technical efficiency measures into a new graph-type approach known as hyperbolic distance function (HDF). In spite of its appealing special structure in allowing for the simultaneous and equiproportionate reduction in inputs and increase in outputs, HDF is a non-linear optimization and it is hard to solve particularly when dealing with technologies operating under variable returns to scale. By connecting HDF to the directional distance function, we propose a linear programming based procedure for estimating the exact value of HDF within the non-parametric framework of data envelopment analysis. We illustrate the computational effectiveness of the algorithm on several real-world and simulated data sets, generating the optimal value of HDF through generally solving at most two linear programs. Moreover, our approach has several desirable properties such as: (1) introducing a computational dual formulation for the HDF and providing an economic interpretation in terms of shadow prices; (2) being readily adaptable to measure hyperbolic-oriented super-efficiency; and (3) being flexible to deal with HDF-based efficiency measures on environmental technologies.

Suggested Citation

  • Färe, Rolf & Margaritis, Dimitris & Rouse, Paul & Roshdi, Israfil, 2016. "Estimating the hyperbolic distance function: A directional distance function approach," European Journal of Operational Research, Elsevier, vol. 254(1), pages 312-319.
    • Handle: RePEc:eee:ejores:v:254:y:2016:i:1:p:312-319
    DOI: 10.1016/j.ejor.2016.03.045
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    Cited by:

    1. Minviel, Jean Joseph & De Witte, Kristof, 2017. "The influence of public subsidies on farm technical efficiency: A robust conditional nonparametric approach," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1112-1120.
    2. Jun Xu & Yuchen Jiang & Xin Guo & Li Jiang, 2021. "Environmental Efficiency Assessment of Heavy Pollution Industry by Data Envelopment Analysis and Malmquist Index Analysis: Empirical Evidence from China," IJERPH, MDPI, vol. 18(11), pages 1-17, May.
    3. Kristiaan Kerstens & Jafar Sadeghi & Ignace Van de Woestyne, 2020. "Plant capacity notions in a non-parametric framework: a brief review and new graph or non-oriented plant capacities," Annals of Operations Research, Springer, vol. 288(2), pages 837-860, May.
    4. Masayuki Shimizu, 2020. "The relationship between pollution abatement costs and environmental regulation: Evidence from the Chinese industrial sector," Review of Development Economics, Wiley Blackwell, vol. 24(2), pages 668-690, May.
    5. Briec, Walter & Fukuyama, Hirofumi & Ravelojaona, Paola, 2021. "Exponential distance function and duality theory," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1002-1014.
    6. Mustapha D. Ibrahim & Sahand Daneshvar & Mevhibe B. Hocaoğlu & Olasehinde-Williams G. Oluseye, 2019. "An Estimation of the Efficiency and Productivity of Healthcare Systems in Sub-Saharan Africa: Health-Centred Millennium Development Goal-Based Evidence," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 143(1), pages 371-389, May.
    7. Juan Aparicio & Jose Manuel Cordero & Carlos Díaz-Caro, 2020. "Efficiency and productivity change of regional tax offices in Spain: an empirical study using Malmquist–Luenberger and Luenberger indices," Empirical Economics, Springer, vol. 59(3), pages 1403-1434, September.
    8. Halická, Margaréta & Trnovská, Mária, 2019. "Duality and profit efficiency for the hyperbolic measure model," European Journal of Operational Research, Elsevier, vol. 278(2), pages 410-421.
    9. Mushtaq Taleb & Ruzelan Khalid & Ali Emrouznejad & Razamin Ramli, 2023. "Environmental efficiency under weak disposability: an improved super efficiency data envelopment analysis model with application for assessment of port operations considering NetZero," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 25(7), pages 6627-6656, July.
    10. Simar, Léopold & Wilson, Paul, 2022. "Modern Tools for Evaluating the Performance of Health-Care Providers," LIDAM Discussion Papers ISBA 2022006, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Imane Bounadi & Khalil Allali & Aziz Fadlaoui & Mohammed Dehhaoui, 2023. "Water Pollution Abatement in Olive Oil Industry in Morocco: Cost Estimates and Policy Implications," Sustainability, MDPI, vol. 15(5), pages 1-19, February.
    12. Yao Hu & Tai-Hua Yan & Feng-Wen Chen, 2020. "Energy and Environment Performance of Resource-Based Cities in China: A Non-Parametric Approach for Estimating Hyperbolic Distance Function," IJERPH, MDPI, vol. 17(13), pages 1-23, July.
    13. Roshdi, Israfil & Hasannasab, Maryam & Margaritis, Dimitris & Rouse, Paul, 2018. "Generalised weak disposability and efficiency measurement in environmental technologies," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1000-1012.
    14. Halická, Margaréta & Trnovská, Mária & Černý, Aleš, 2024. "A unified approach to radial, hyperbolic, and directional efficiency measurement in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 312(1), pages 298-314.
    15. Sebastián Lozano & Narges Soltani, 2020. "A modified discrete Raiffa approach for efficiency assessment and target setting," Annals of Operations Research, Springer, vol. 292(1), pages 71-95, September.
    16. Léopold Simar & Paul W. Wilson, 2023. "Another look at productivity growth in industrialized countries," Journal of Productivity Analysis, Springer, vol. 60(3), pages 257-272, December.
    17. Hasannasab, Maryam & Margaritis, Dimitris & Roshdi, Israfil & Rouse, Paul, 2019. "Hyperbolic efficiency measurement: A conic programming approach," European Journal of Operational Research, Elsevier, vol. 278(2), pages 401-409.
    18. Margaréta Halická & Mária Trnovská, 2018. "Negative features of hyperbolic and directional distance models for technologies with undesirable outputs," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(4), pages 887-907, December.

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