Annals of Operations Research

, Volume 271, Issue 2, pp 1067–1078 | Cite as

Enumeration algorithms for FDH directional distance functions under different returns to scale assumptions

  • Kristiaan KerstensEmail author
  • Ignace Van de Woestyne
Short Note


Computing directional distance functions for a free disposal hull (FDH) technology in general requires solving nonlinear mixed integer programs. Cherchye et al. (J Product Anal 15(3):201–215, 2001) provide an enumeration algorithm for the FDH directional distance function in case of a variable returns to scale technology. In this contribution, we provide fast enumeration algorithms for the FDH directional distance functions under constant, nonincreasing, and nondecreasing returns to scale assumptions. Consequently, enumeration algorithms are now available for all commonly used returns to scale assumptions.


Directional distance function Enumeration Free disposal hull CRS NDRS NIRS 

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  1. Afriat, S. (1972). Efficiency estimation of production functions. International Economic Review, 13(3), 568–598.CrossRefGoogle Scholar
  2. Agrell, P., & Tind, J. (2001). A dual approach to nonconvex frontier models. Journal of Productivity Analysis, 16(2), 129–147.CrossRefGoogle Scholar
  3. Akçay, A., Ertek, G., & Büyüközkan, G. (2012). Analyzing the solutions of DEA through information visualization and data mining techniques: SmartDEA framework. Expert Systems with Applications, 39(9), 7763–7775.CrossRefGoogle Scholar
  4. Alam, I., & Sickles, R. (2000). Time series analysis of deregulatory dynamics and technical efficiency: The case of the US Airline Industry. International Economic Review, 41(1), 203–218.CrossRefGoogle Scholar
  5. Balaguer-Coll, M., Prior, D., & Tortosa-Ausina, E. (2007). On the determinants of local government performance: A two-stage nonparametric approach. European Economic Review, 51(2), 425–451.CrossRefGoogle Scholar
  6. Barr, R. (2004). DEA software tools and technology: A state-of-the-art survey. In W. Cooper, L. Seiford, & J. Zhu (Eds.), Handbook on data envelopment analysis (pp. 539–566). Boston: Kluwer.CrossRefGoogle Scholar
  7. Briec, W., & Kerstens, K. (2006). Input, output and graph technical efficiency measures on non-convex FDH models with various scaling laws: An integrated approach based upon implicit enumeration algorithms. TOP, 14(1), 135–166.CrossRefGoogle Scholar
  8. Briec, W., Kerstens, K., & Vanden Eeckaut, P. (2004). Non-convex technologies and cost functions: Definitions, duality and nonparametric tests of convexity. Journal of Economics, 81(2), 155–192.CrossRefGoogle Scholar
  9. Cesaroni, G. (2011). A complete FDH efficiency analysis of a diffused production network: The case of the Italian Driver and Vehicle Agency. International Transactions in Operational Research, 18(2), 205–229.CrossRefGoogle Scholar
  10. Cesaroni, G., Kerstens, K., & Van de Woestyne, I. (2017). A new input-oriented plant capacity notion: Definition and empirical comparison. Pacific Economic Review, 22(4), 720–739.CrossRefGoogle Scholar
  11. Chambers, R., Chung, Y., & Färe, R. (1998). Profit, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications, 98(2), 351–364.CrossRefGoogle Scholar
  12. Charnes, A., Cooper, W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.CrossRefGoogle Scholar
  13. Cherchye, L., Kuosmanen, T., & Post, T. (2001). FDH directional distance functions with an application to European commercial banks. Journal of Productivity Analysis, 15(3), 201–215.CrossRefGoogle Scholar
  14. Chong, E., & Zak, S. (2001). Introduction to optimization (2nd ed.). New York: Wiley.Google Scholar
  15. Cullinane, K., Song, D.-W., & Wang, T. (2005). The Application of mathematical programming approaches to estimating container port production efficiency. Journal of Productivity Analysis, 24(1), 73–92.CrossRefGoogle Scholar
  16. Cummins, D., & Zi, H. (1998). Comparison of Frontier efficiency methods: An application to the U.S. Life Insurance Industry. Journal of Productivity Analysis, 10(2), 131–152.CrossRefGoogle Scholar
  17. De Borger, B., & Kerstens, K. (1996). Cost efficiency of Belgian Local Governments: A comparative analysis of FDH, DEA, and econometric approaches. Regional Science and Urban Economics, 26(2), 145–170.CrossRefGoogle Scholar
  18. De Witte, K., & Marques, R. (2011). Big and beautiful? On non-parametrically measuring scale economies in non-convex technologies. Journal of Productivity Analysis, 35(3), 213–226.CrossRefGoogle Scholar
  19. Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor efficiency in post offices. In M. Marchand, P. Pestieau, & H. Tulkens (Eds.), The performance of public enterprises: Concepts and measurements (pp. 243–268). Amsterdam: North Holland.Google Scholar
  20. Destefanis, S., & Sena, V. (2005). Public capital and total factor productivity: New evidence from the Italian regions, 1970–98. Regional Studies, 39(5), 603–617.CrossRefGoogle Scholar
  21. Diewert, W., & Parkan, C. (1983). Linear programming test of regularity conditions for production functions. In W. Eichhorn, K. Neumann, & R. Shephard (Eds.), Quantitative studies on production and prices (pp. 131–158). Würzburg: Physica-Verlag.CrossRefGoogle Scholar
  22. Dulá, J. (2008). A computational study of DEA with massive data sets. Computers & Operations Research, 35(4), 1191–1203.CrossRefGoogle Scholar
  23. Eiselt, H., & Sandblom, C.-L. (2007). Linear programming and its applications. Berlin: Springer.Google Scholar
  24. Epstein, M., & Henderson, J. (1989). Data envelopment analysis for managerial control and diagnosis. Decision Sciences, 20(1), 90–119.CrossRefGoogle Scholar
  25. Färe, R., Grosskopf, S., & Zaim, O. (2002). Hyperbolic efficiency and return to the dollar. European Journal of Operational Research, 136(3), 671–679.CrossRefGoogle Scholar
  26. Fried, H., Lovell, C., & Turner, J. (1996). An analysis of the performance of university affiliated credit unions. Computers & Operations Research, 23(4), 375–384.CrossRefGoogle Scholar
  27. Fried, H., Lovell, C., & Vanden Eeckaut, P. (1993). Evaluating the performance of U.S. credit unions. Journal of Banking & Finance, 17(2–3), 251–265.CrossRefGoogle Scholar
  28. Green, R. (1996). DIY DEA: Implementing data envelopment analysis in the mathematical programming language AMPL. Omega, 24(4), 489–494.CrossRefGoogle Scholar
  29. Hackman, S. (2008). Production economics: Integrating the microeconomic and engineering perspectives. Berlin: Springer.Google Scholar
  30. Kerstens, K., & Managi, S. (2012). Total factor productivity growth and convergence in the petroleum industry: Empirical analysis testing for convexity. International Journal of Production Economics, 139(1), 196–206.CrossRefGoogle Scholar
  31. Kerstens, K., & Van de Woestyne, I. (2014a). Comparing Malmquist and Hicks–Moorsteen productivity indices: Exploring the impact of unbalanced vs. balanced panel data. European Journal of Operational Research, 233(3), 749–758.CrossRefGoogle Scholar
  32. Kerstens, K., & Van de Woestyne, I. (2014b). Solution methods for nonconvex free disposal hull models: A review and some critical comments. Asia-Pacific Journal of Operational Research, 31(1), 1450010.CrossRefGoogle Scholar
  33. Kerstens, K., & Vanden Eeckaut, P. (1999). Estimating returns to scale using nonparametric deterministic technologies: A new method based on goodness-of-fit. European Journal of Operational Research, 113(1), 206–214.CrossRefGoogle Scholar
  34. Leleu, H. (2006). A linear programming framework for free disposal hull technologies and cost functions: Primal and dual models. European Journal of Operational Research, 168(2), 340–344.CrossRefGoogle Scholar
  35. Leleu, H. (2009). Mixing DEA and FDH models together. Journal of the Operational Research Society, 60(1), 1730–1737.CrossRefGoogle Scholar
  36. Luenberger, D. (1992a). Benefit function and duality. Journal of Mathematical Economics, 21(5), 461–481.CrossRefGoogle Scholar
  37. Luenberger, D. (1992b). New optimality principles for economic efficiency and equilibrium. Journal of Optimization Theory and Applications, 75(2), 221–264.CrossRefGoogle Scholar
  38. Luenberger, D. (1995). Microeconomic theory. Boston: McGraw-Hill.Google Scholar
  39. Mairesse, F., & Vanden Eeckaut, P. (2002). Museum assessment and FDH technology: Towards a global approach. Journal of Cultural Economics, 26(4), 261–286.CrossRefGoogle Scholar
  40. Mayston, D. (2014). Effectiveness analysis of quality achievements for university departments of economics. Applied Economics, 46(31), 3788–3797.CrossRefGoogle Scholar
  41. Olesen, O., & Petersen, N. (1996). A presentation of GAMS for DEA. Computers & Operations Research, 23(4), 323–339.CrossRefGoogle Scholar
  42. Podinovski, V. (2004). On the linearisation of reference technologies for testing returns to scale in FDH models. European Journal of Operational Research, 152(3), 800–802.CrossRefGoogle Scholar
  43. Sueyoshi, T. (1992). Measuring technical, allocative and overall efficiencies using a DEA algorithm. Journal of the Operational Research Society, 43(2), 141–155.CrossRefGoogle Scholar
  44. Tulkens, H. (1993). On FDH efficiency analysis: Some methodological issues and applications to retail banking, courts, and urban transit. Journal of Productivity Analysis, 4(1–2), 183–210.CrossRefGoogle Scholar
  45. Walden, J., & Tomberlin, D. (2010). Estimating fishing vessel capacity: A comparison of nonparametric frontier approaches. Marine Resource Economics, 25(1), 23–36.CrossRefGoogle Scholar
  46. Zhu, D. (2010). A hybrid approach for efficient ensembles. Decision Support Systems, 48(3), 480–487.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.CNRS-LEM (UMR 9221)IESEG School of ManagementLilleFrance
  2. 2.Research Unit MEESKU LeuvenBrusselBelgium

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