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On selecting directions for directional distance functions in a non-parametric framework: a review

  • Ke Wang
  • Yujiao Xian
  • Chia-Yen Lee
  • Yi-Ming Wei
  • Zhimin Huang
DEA in Data Analytics

Abstract

Directional distance function (DDF) has been a commonly used technique for estimating efficiency and productivity over the past two decades, and the directional vector is usually predetermined in the applications of DDF. The most critical issue of using DDF remains that how to appropriately project the inefficient decision-making unit onto the production frontier along with a justified direction. This paper provides a comprehensive literature review on the techniques for selecting directional vector of the directional distance function. It begins with a brief introduction of the existing methods around the inclusion of the exogenous direction techniques and the endogenous direction techniques. The former commonly includes arbitrary direction and conditional direction techniques, while the latter involves the techniques for seeking theoretically optimized directions (i.e., direction towards the closest benchmark or indicating the largest efficiency improvement potential) and market-oriented directions (i.e., directions towards cost minimization, profit maximization, or marginal profit maximization benchmarks). The main advantages and disadvantages of these techniques are summarized, and the limitations inherent in the exogenous direction-selecting techniques are discussed. It also analytically argues the mechanism of each endogenous direction technique. The literature review is end up with a numerical example of efficiency estimation for power plants, in which most of the reviewed directions for DDF are demonstrated and their evaluation performance are compared.

Keywords

Data envelopment analysis (DEA) Least distance Endogenous mechanism Cost efficiency Profit efficiency Marginal profit maximization 

Notes

Acknowledgements

We gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 71471018, 71521002, and 71642004), the Joint Development Program of Beijing Municipal Commission of Education, the Social Science Foundation of Beijing (Grant No. 16JDGLB013), the National Key R&D Program (Grant No. 2016YFA0602603), and the Ministry of Science and Technology of Taiwan (MOST103-2221-E-006-122-MY3).

References

  1. Adler, N., & Voltab, N. (2016). Accounting for externalities and disposability: A directional economic environmental distance function. European Journal of Operational Research, 250, 314–327.CrossRefGoogle Scholar
  2. Agee, M. D., Atkinson, S. E., & Crocker, T. D. (2012). Child maturation, time-invariant, and time-varying inputs: Their interaction in the production of child human capital. Journal of Productivity Analysis, 35, 29–44.CrossRefGoogle Scholar
  3. Aparicio, J., Ruiz, J. L., & Sirvent, I. (2007). Closest targets and minimum distance to the Pareto-efficient frontier in DEA. Journal of Productivity Analysis, 28, 209–218.CrossRefGoogle Scholar
  4. Arabi, B., Munisamy, S., Emrouznejad, A., & Shadman, F. (2014). Power industry restructuring and eco-efficiency changes: A new slacks-based model in Malmquist–Luenberger Index measurement. Energy Policy, 68, 132–145.CrossRefGoogle Scholar
  5. Baek, C., & Lee, J. (2009). The relevance of DEA benchmarking information and the least-distance measure. Mathematical and Computer Modelling, 49, 265–275.CrossRefGoogle Scholar
  6. Ball, E., Färe, R., Grosskopf, S., & Zaim, O. (2005). Accounting for externalities in the measurement of productivity growth: The Malmquist cost productivity measure. Structural Change and Economic Dynamics, 16(3), 374–394.CrossRefGoogle Scholar
  7. Bellenger, M. J., & Herlihy, A. T. (2009). An economic approach to environmental indices. Ecological Economics, 68, 2216–2223.CrossRefGoogle Scholar
  8. Chambers, R. G., Chung, Y., & Färe, R. (1996a). Benefit and distance functions. Journal of Economic Theory, 70, 407–419.CrossRefGoogle Scholar
  9. Chambers, R., Färe, R., & Grosskopf, S. (1996b). Productivity growth in APEC countries. Pacific Economic Review, 1, 181–190.CrossRefGoogle Scholar
  10. Chen, C.-M., & Delmas, M. A. (2012). Measuring eco-inefficiency: A new frontier approach. Operations Research, 60(5), 1064–1079.CrossRefGoogle Scholar
  11. Chen, Y., Du, J., & Huo, J. Z. (2013). Super-efficiency based on a modified directional distance function. Omega, 41, 621–625.CrossRefGoogle Scholar
  12. Chung, Y., Färe, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: A directional distance function approach. Journal of Environmental Management, 51, 229–240.CrossRefGoogle Scholar
  13. Coggins, J. S., & Swinton, J. R. (1996). The price of pollution: A dual approach to valuing \(\text{ SO }_2\) allowances. Journal of Environmental Economics and Management, 30, 58–72.CrossRefGoogle Scholar
  14. Cong, R. G., & Wei, Y. M. (2010). Potential impact of (CET) carbon emissions trading on China’s power sector: A perspective from different allowance allocation options. Energy, 35(9), 3921–3931.CrossRefGoogle Scholar
  15. Cong, R. G., & Wei, Y. M. (2012). Experimental comparison of impact of auction format on carbon allowance market. Renewable & Sustainable Energy Reviews, 16(6), 4148–4156.CrossRefGoogle Scholar
  16. Cooper, W. W., Park, K. S., & Pastor, J. T. (1999). RAM: A range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. Journal of Productivity Analysis, 11, 5–42.CrossRefGoogle Scholar
  17. Dervaux, B., Leleu, H., Minvielle, E., Valdmanis, V., Aegerter, P., & Guidet, B. (2009). Performance of French intensive care units: A directional distance function approach at the patient level. International Journal of Production Economics, 120, 585–594.CrossRefGoogle Scholar
  18. Du, L. M., Hanley, A., & Zhang, N. (2016). Environmental technical efficiency, technology gap and shadow price of coal-fuelled power plants in China: A parametric meta-frontier analysis. Resource and Energy Economics, 43, 14–32.CrossRefGoogle Scholar
  19. Farrell, M. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society Series A (General) Part O, 120(3), 253–281.CrossRefGoogle Scholar
  20. Färe, R., & Grosskopf, S. (2004). New directions: Efficiency and productivity. Boston: Kluwer Academic Publishers.Google Scholar
  21. Färe, R., & Grosskopf, S. (2010). Directional distance functions and slacks-based measures of efficiency. European Journal of Operational Research, 200, 320–322.CrossRefGoogle Scholar
  22. Färe, R., Grosskopf, S., Lovell, C. A. K., & Pasurka, C. (1989). Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach. Review of Economics and Statistics, 71(1), 90–98.CrossRefGoogle Scholar
  23. Färe, R., Grosskopf, S., Noh, D. W., & Weber, W. (2005). Characteristics of a polluting technology: Theory and practice. Journal of Econometrics, 126, 469–492.CrossRefGoogle Scholar
  24. Färe, R., Grosskopf, S., & Pasurka, C. (1986). Effects on relative efficiency in electric power generation due to environmental controls. Resources and Energy, 8, 167–184.CrossRefGoogle Scholar
  25. Färe, R., Grosskopf, S., & Pasurka, C. (2016). Technical change and pollution abatement cost. European Journal of Operational Research, 248, 715–724.CrossRefGoogle Scholar
  26. Färe, R., Grosskopf, S., & Pasurka, C. A. (2007). Environmental production functions and environmental directional distance functions. Energy, 32, 1055–1066.CrossRefGoogle Scholar
  27. Färe, R., Grosskopf, S., Pasurka, C. A., & Weber, W. (2012). Substitutability among undesirable outputs. Applied Economics, 44, 39–47.CrossRefGoogle Scholar
  28. Färe, R., Grosskopf, S., & Weber, W. L. (2006). Shadow prices and pollution costs in U.S. agriculture. Ecological Economics, 56, 89–103.CrossRefGoogle Scholar
  29. Färe, R., Grosskopf, S., & Wittaker, G. (2013). Directional output distance functions: Endogenous constraints based on exogenous normalization constraints. Journal of Productivity Analysis, 40, 267–269.CrossRefGoogle Scholar
  30. Färe, R., & Logan, J. (1992). The rate of return regulated version of Farrell efficiency. International Journal of Production Economics, 27(2), 161–165.CrossRefGoogle Scholar
  31. Färe, R., & Primont, D. (1995). Multi-output production and duality: Theory and applications. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
  32. Frei, F. X., & Harker, P. T. (1999). Projections onto efficient frontiers: Theoretical and computational extensions to DEA. Journal of Productivity Analysis, 11, 275–300.CrossRefGoogle Scholar
  33. Fukuyama, H., & Weber, W. L. (2009). A directional slacks-based measure of technical inefficiency. Socio-Economic Planning Sciences, 43, 274–287.CrossRefGoogle Scholar
  34. Granderson, G., & Prior, D. (2013). Environmental externalities and regulation constrained cost productivity growth in the US electric utility industry. Journal of Productivity Analysis, 39, 243–257.CrossRefGoogle Scholar
  35. Halkos, G. E., & Tzeremes, N. G. (2013). A conditional directional distance function approach for measuring regional environmental efficiency: Evidence from UK regions. European Journal of Operational Research, 227, 182–189.CrossRefGoogle Scholar
  36. Hampf, B., & Krüger, J. J. (2014a). Optimal directions for directional distance functions: An exploration of potential reductions of greenhouse gases. American Journal of Agricultural Economics, 97, 920–938.CrossRefGoogle Scholar
  37. Hampf, B., & Kruger, J. J. (2014b). Technical efficiency of automobiles—A nonparametric approach incorporating carbon dioxide emissions. Transportation Research Part D, 33, 47–62.CrossRefGoogle Scholar
  38. Hailu, A., & Chambers, R. G. (2012). A Luenberger soil quality indicator. Journal of Productivity Analysis, 38(2), 145–154.CrossRefGoogle Scholar
  39. Kumar, S. (2006). Environmentally sensitive productivity growth: A global analysis using Malmquist-Luenberger index. Ecological Economics, 56, 280–293.CrossRefGoogle Scholar
  40. Kuosmanen, T. (2005). Weak disposability in nonparametric productivity analysis with undesirable outputs. American Journal of Agricultural Economics, 87(4), 1077–1082.CrossRefGoogle Scholar
  41. Lee, C.-Y. (2014). Meta-data envelopment analysis: Finding a direction towards marginal profit maximization. European Journal of Operational Research, 237, 207–216.CrossRefGoogle Scholar
  42. Lee, C.-Y. (2016). Nash-profit efficiency: A measure of changes in market structures. European Journal of Operational Research, 255, 659–663.CrossRefGoogle Scholar
  43. Lee, C.-Y., & Johnson, A. L. (2015). Measuring efficiency in imperfectly competitive markets: An example of rational inefficiency. Journal of Optimization Theory and Applications, 164(2), 702–722.CrossRefGoogle Scholar
  44. Lee, J., Park, J., & Kim, T. (2002). Estimation of the shadow prices of pollutants with production/environment inefficiency taken into account: A nonparametric directional distance function approach. Journal of Environmental Management, 64, 365–375.CrossRefGoogle Scholar
  45. Leleu, H. (2013). Shadow pricing of undesirable outputs in nonparametric analysis. European Journal of Operational Research, 231, 474–480.CrossRefGoogle Scholar
  46. Luenberger, D. G. (1992). New optimality principle for economic efficiency and equilibrium. Journal of Optimization Theory and Applications, 75(2), 221–264.CrossRefGoogle Scholar
  47. Macpherson, A. J., Principe, P. P., & Smith, E. R. (2010). A directional distance function approach to regional environmental-economic assessments. Ecological Economics, 69, 1918–1925.CrossRefGoogle Scholar
  48. Matsushita, K., & Yamane, F. (2012). Pollution from the electric power sector in Japan and efficient pollution reduction. Energy Economics, 34, 1124–1130.CrossRefGoogle Scholar
  49. Murty, S., Russell, R. R., & Levkoff, S. B. (2012). On modeling pollution-generating technologies. Journal of Environmental Economics and Management, 64, 117–135.CrossRefGoogle Scholar
  50. Njuki, E., & Bravo-Ureta, B. E. (2015). The economic costs of environmental regulation in U.S. dairy farming: A directional distance function approach. American Journal of Agricultural Economics, 97(4), 1087–1106.Google Scholar
  51. Oh, D. (2010). A global Malmquist–Luenberger productivity index. Journal of Productivity Analysis, 34, 183–197.CrossRefGoogle Scholar
  52. Oum, T. H., Pathomsiri, S., & Yoshida, S. (2013). Limitations of DEA-based approach and alternative methods in the measurement and comparison of social efficiency across firms in different transport modes: An empirical study in Japan. Transportation Research Part E, 57, 16–26.CrossRefGoogle Scholar
  53. Pathomsiri, S., Haghani, A., Dresner, M., & Windle, R. J. (2008). Impact of undesirable outputs on the productivity of US airports. Transportation Research Part E, 44, 235–259.CrossRefGoogle Scholar
  54. Picazo-Tadeo, A. J., Beltrán-Esteve, M., & Gómez-Limón, J. A. (2012). Assessing eco-efficiency with directional distance functions. European Journal of Operational Research, 220, 798–809.CrossRefGoogle Scholar
  55. Picazo-Tadeo, A. J., & Prior, D. (2009). Environmental externalities and efficiency measurement. Journal of Environmental Management, 90, 3332–3339.CrossRefGoogle Scholar
  56. Picazo-Tadeo, A. J., Reig-Martínez, E., & Hernández-Sancho, F. (2005). Directional distance functions and environmental regulation. Resource Energy Economics, 27, 131–142.CrossRefGoogle Scholar
  57. Podinovski, V. V., & Førsund, F. R. (2010). Differential characteristics of efficient frontiers in data envelopment analysis. Operations Research, 58(6), 1743–1754.CrossRefGoogle Scholar
  58. Portela, M., Borges, P. C., & Thanassoulis, E. (2003). Finding closest targets in non-oriented DEA models: The case of convex and non-convex technologies. Journal of Productivity Analysis, 19, 251–269.CrossRefGoogle Scholar
  59. Ray, S. C., Chen, L., & Mukherjee, K. (2008). Input price variation across locations and a generalized measure of cost efficiency. International Journal of Production Economics, 116, 208–218.CrossRefGoogle Scholar
  60. Ray, S. C., & Mukherjee, K. (2000). Decomposition of cost competitiveness in US manufacturing: Some state by state comparisons. Indian Economic Review, 35, 133–153.Google Scholar
  61. Rezek, J. P., & Campbell, R. C. (2007). Cost estimates for multiple pollutants: A maximum entropy approach. Energy Economics, 29, 503–519.CrossRefGoogle Scholar
  62. Shephard, R. W. (1970). Theory of cost and production functions. Princeton, NJ: Princeton University Press.Google Scholar
  63. Simar, L., Vanhems, A., & Wilson, P. W. (2012). Statistical inference for DEA estimators of directional distances. European Journal of Operational Research, 220, 853–864.CrossRefGoogle Scholar
  64. Vardanyan, M., & Noh, D. W. (2006). Approximating pollution abatement costs via alternative specifications of a multi-output production technology: A case of the U.S. electric utility industry. Journal of Environmental Management, 80, 177–190.CrossRefGoogle Scholar
  65. Wang, K., & Wei, Y. M. (2016). Sources of energy productivity change in China during 1997–2012: A decomposition analysis based on the Luenberger productivity indicator. Energy Economics, 54, 50–59.CrossRefGoogle Scholar
  66. Wang, K., Wei, Y. M., & Zhang, X. (2012). A comparative analysis of China’s regional energy and emission performance: Which is the better way to deal with undesirable outputs? Energy Policy, 46, 574–584.CrossRefGoogle Scholar
  67. Wang, K., Wei, Y. M., & Zhang, X. (2013). Energy and emissions efficiency patterns of Chinese regions: A multi-directional efficiency analysis. Applied Energy, 104, 105–116.CrossRefGoogle Scholar
  68. Wang, K., Xian, Y., Wei, Y. M., & Huang, Z. (2016). Sources of carbon productivity change: A decomposition and disaggregation analysis based on global Luenberger productivity indicator and endogenous directional distance function. Ecological Indicators, 66, 545–555.CrossRefGoogle Scholar
  69. Watanabe, M., & Tanaka, K. (2007). Efficiency analysis of Chinese industry: A directional distance function approach. Energy Policy, 35, 6323–6331.CrossRefGoogle Scholar
  70. Zhou, P., Ang, B. W., & Wang, H. (2012). Energy and CO\(_2\) emission performance in electricity generation: A non-radial directional distance function approach. European Journal of Operational Research, 221, 625–635.CrossRefGoogle Scholar
  71. Zofio, J. L., Paster, J. T., & Aparicio, J. (2013). The directional profit efficiency measure: On why profit efficiency is either technical or allocative. Journal of Productivity Analysis, 40(3), 257–266.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Ke Wang
    • 1
    • 2
    • 3
  • Yujiao Xian
    • 1
  • Chia-Yen Lee
    • 4
  • Yi-Ming Wei
    • 1
    • 2
    • 3
  • Zhimin Huang
    • 1
    • 5
  1. 1.Center for Energy and Environmental Policy Research & School of Management and EconomicsBeijing Institute of TechnologyBeijingChina
  2. 2.Collaborative Innovation Center of Electric Vehicles in BeijingBeijingChina
  3. 3.Sustainable Development Research Institute for Economy and Society of BeijingBeijingChina
  4. 4.Institute of Manufacturing Information and SystemsNational Cheng Kung UniversityTainanTaiwan
  5. 5.Robert B. Willumstad School of BusinessAdelphi UniversityGarden CityUSA

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