Efficiency Measurement in Special Production Stages

  • Chiang Kao
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 240)


The preceding three chapters discuss how to measure the efficiency of a DMU based on the production frontier constructed from the peer DMUs. The production possibility set is assumed to be convex. In the classical production theory, production is separated into three stages, as depicted in Fig. 5.1 (Ferguson and Gould 1986). At the beginning, the output rises at an increasing rate as the input increases, then at a decreasing rate, and finally at a negative rate. The first stage corresponds to use of the variable input X to point b, where the average product (AP) achieves its maximum. At this point the marginal product (MP) equals the average product. Stage II starts from this point to point c, where the marginal product of X drops to zero and the total product (TP) of X culminates. Stage III corresponds to use of the variable input X to the right of this point, where the marginal product is negative.


Efficiency Score Marginal Product Production Possibility Output Efficiency Free Disposal Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Agrell PJ, Tind J (2001) A dual approach to nonconvex frontier models. J Prod Anal 16(2):129–147CrossRefGoogle Scholar
  2. Banker RD, Maindiratta A (1986) Piecewise loglinear estimation of efficient production surfaces. Manag Sci 32:126–135CrossRefGoogle Scholar
  3. Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale efficiencies in data envelopment analysis. Manag Sci 30:1078–1092CrossRefGoogle Scholar
  4. Banker RD, Charnes A, Cooper WW, Schinnar A (1981) A bi-extremal principle for frontier estimation and efficiency evaluation. Manag Sci 27:1370–1382CrossRefGoogle Scholar
  5. Blancard S, Boussemart JP, Leleu H (2011) Measuring potential gains from specialization under non-convex technologies. J Oper Res Soc 62:1871–1880CrossRefGoogle Scholar
  6. Bogetoft P (1996) DEA on relaxed convexity assumptions. Manag Sci 42:457–465CrossRefGoogle Scholar
  7. Bogetoft P, Tama JM, Tind J (2000) Convex input and output projections of nonconvex production possibility sets. Manag Sci 46:858–869CrossRefGoogle Scholar
  8. Briec W, Kerstens K, Leleu H, Vanden Eeckaut P (2000) Returns to scale information on nonparametric deterministic technologies: a simplification of goodness-of-fit methods. J Prod Anal 14(3):267–274CrossRefGoogle Scholar
  9. Briec W, Kerstens K, Vanden Eeckaut P (2004) Non-convex technologies and cost functions: definitions, duality and nonparametric tests of convexity. J Econ 81(2):155–192CrossRefGoogle Scholar
  10. Byrnes P, Färe R, Grosskopf S (1984) Measuring productive efficiency: an application to Illinois strip mines. Manag Sci 30:671–681CrossRefGoogle Scholar
  11. Byrnes P, Färe R, Grosskopf S, Lovell CAK (1988) The effect of union on productivity: US surface mining of coal. Manag Sci 34:1037–1053CrossRefGoogle Scholar
  12. Charnes A, Cooper WW, Seiford L, Stutz J (1982) A multiplicative model for efficiency analysis. Socio Econ Plan Sci 16:223–224CrossRefGoogle Scholar
  13. Charnes A, Cooper WW, Seiford L, Stutz J (1983) Invariant multiplicative efficiency and piecewise Cobb-Douglas envelopments. Oper Res Lett 2(3):1010–1013CrossRefGoogle Scholar
  14. Cherchye L, Kuosmanen T, Post T (2001a) FDH directional distance functions: with an application to European commercial banks. J Prod Anal 15:201–215CrossRefGoogle Scholar
  15. Cherchye L, Kuosmanen T, Post T (2001b) Alternative treatments of congestion in DEA: a rejoinder to Cooper, Gu, and Li. Eur J Oper Res 132:75–80CrossRefGoogle Scholar
  16. Cooper WW, Deng H, Huang ZM, Li SX (2002) A one-model approach to congestion in data envelopment analysis. Socio Econ Plan Sci 36:231–238CrossRefGoogle Scholar
  17. Cooper WW, Seiford LM, Zhu J (2000) A unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA. Socio Econ Plan Sci 34:1–25CrossRefGoogle Scholar
  18. Cooper WW, Thompson RG, Thrall RM (1996) Introduction: extension and new developments in DEA. Ann Oper Res 66:3–45Google Scholar
  19. De Witte K, Marques RC (2011) Big and beautiful? On non-parametrically measuring scale economies in non-convex technologies. J Prod Anal 35:213–226CrossRefGoogle Scholar
  20. Deprins D, Simar L, Tulkens H (1984) Measuring labor efficiency in post offices. In: Marchand M, Pestieu P, Tulkens H (eds) The performance of public enterprises: concepts and measurements. North Holland, AmsterdamGoogle Scholar
  21. Färe R, Grosskopf S (1983) Measuring congestion in production. Nationalökonomie 43:257–271CrossRefGoogle Scholar
  22. Färe R, Grosskopf S (2000) Decomposing technical efficiency with care. Manag Sci 46:167–168CrossRefGoogle Scholar
  23. Färe R, Grosskopf S (2001) When can slacks be used to identify congestion? An answer to W.W. Cooper, L. Seiford and J. Zhu. Socio Econ Plan Sci 35:217–221CrossRefGoogle Scholar
  24. Färe R, Grosskopf S, Lovell CAK (1985) The measurement of efficiency of production. Kluwer-Nijhoff, BostonCrossRefGoogle Scholar
  25. Färe R, Svensson L (1980) Congestion of production factors. Econometrica 48:1745–1753CrossRefGoogle Scholar
  26. Ferguson CE, Gould JP (1986) Microeconomic theory, 6th edn. Irwin, Homewood, ILGoogle Scholar
  27. Henderson JM, Quandt RE (1980) Microeconomic theory, 3rd edn. McGraw-Hill, New YorkGoogle Scholar
  28. Intriligator MD (1971) Mathematical optimization and economic theory. Prentice-Hall, Englewood Cliffs, NJGoogle Scholar
  29. Kao C (2010) Congestion measurement and elimination under the framework of data envelopment analysis. Int J Prod Econ 123:257–265CrossRefGoogle Scholar
  30. Keshvari A, Hardoroudi ND (2008) An extended numeration method for solving free disposal hull models in DEA. Asia Pac J Oper Res 25:689–696CrossRefGoogle Scholar
  31. Kerstens K, Van De Woestyne I (2014) Solution methods for nonconvex free disposal hull models: A review and some critical comments. Asia Pac J Oper Res 31. doi: 10.1142/S0217595914500109
  32. Kerstens K, Vanden Eeckaut P (1999) Estimating returns to scale using non-parametric deterministic technologies: a new method based on goodness-of-fit. Eur J Oper Res 113:206–214CrossRefGoogle Scholar
  33. Khodabakhshi M, Hosseinzadeh F, Aryavash K (2014) Review of input congestion estimating methods in DEA. J Appl Math. doi: 10.1155/2014/963791 Google Scholar
  34. Leleu H (2006) A linear programming framework for free disposal hull technologies and cost functions: primal and dual models. Eur J Oper Res 168:340–344CrossRefGoogle Scholar
  35. Leleu H (2009) Mixing DEA and FDH models together. J Oper Res Soc 60:1730–1737CrossRefGoogle Scholar
  36. McDonald J (1996) A problem with the decomposition of technical inefficiency into scale and congestion components. Manag Sci 42:473–474CrossRefGoogle Scholar
  37. Mehdiloozad M, Sahoo BK, Roshdi I (2014) A generalized multiplicative directional distance function for efficiency measurement in DEA. Eur J Oper Res 232:679–688CrossRefGoogle Scholar
  38. Petersen NC (1990) Data envelopment analysis on a relaxed set of assumptions. Manag Sci 36:305–314CrossRefGoogle Scholar
  39. Podinovski VV (2004a) On the linearisation of reference technologies for testing returns to scale in FDH models. Eur J Oper Res 152:800–802CrossRefGoogle Scholar
  40. Podinovski VV (2004b) Local and global returns to scale in performance measurement. J Oper Res Soc 55:170–178CrossRefGoogle Scholar
  41. Sharma MJ, Yu SJ (2013) Multi-stage data envelopment analysis congestion model. Oper Res 13:399–413Google Scholar
  42. Shephard RW (1970) Theory of cost and production functions. Princeton University Press, Princeton, NJGoogle Scholar
  43. Soleimani-damaneh M, Jahanshahloo GR, Reshadi M (2006) On the estimation of returns to scale in FDH models. Eur J Oper Res 174:1055–1059CrossRefGoogle Scholar
  44. Soleimani-damaneh M, Mostafaee A (2009) Stability of the classification of resturns to scale in FDH models. Eur J Oper Res 196:1223–1228CrossRefGoogle Scholar
  45. Tone K, Sahoo BK (2004) Degree of scale economies and congestion: a unified DEA approach. Eur J Oper Res 158:755–772CrossRefGoogle Scholar
  46. Tulkens H (1993) On FDH efficiency: some methodological issues and application to retail banking, courts, and urban transit. J Prod Anal 4(1):183–210CrossRefGoogle Scholar
  47. Wei QL, Yan H (2004) Congestion and returns to scale in data envelopment analysis. Eur J Oper Res 153:641–660CrossRefGoogle Scholar
  48. Zare-Haghighi H, Rostamy-Malkhalifeh M, Jahanshaloo RG (2014) Measurement of congestion in the simultaneous presence of desirable and undesirable outputs. J Appl Math. doi: 10.1155/2014/512157 Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Chiang Kao
    • 1
  1. 1.Department of Industrial and Information ManagementNational Cheng Kung UniversityTainanTaiwan

Personalised recommendations