Special Types of Data

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


In the production process physical inputs are consumed to produce physical outputs. All the data are therefore positive real numbers, and the conventional DEA models are based on this requirement. However, in many applications the inputs and outputs may not be physical factors, and thus the data are not necessarily positive real numbers. For example, if profit is used as an output then it can be negative, although the physical quantity of output is always positive. Another situation is that the physical quantities of the inputs consumed by the DMUs are ordered in ranks, or the abilities of different persons (considered as DMUs) can only be ranked, without real term measures. In these cases the data only reflects the relative differences among the DMUs, instead of the absolute differences represented by the conventional measures. A similar case is that the service level provided by a DMU cannot be evaluated by any measures, and is only subjectively expressed by linguistic terms, such as excellent, good, acceptable, and unacceptable. Data of these three types are referred to as negative, ordinal, and qualitative, and Sects. 7.17.3 will introduce models for handling these.


Fuzzy Number Efficiency Score Linguistic Term Ordinal Data Membership Grade 
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. Azizi H, Kordrostami S, Amirteimoori A (2015) Slacks-based measures of efficiency in imprecise data envelopment analysis: an approach based on data envelopment analysis with double frontiers. Comput Ind Eng 79:42–51CrossRefGoogle Scholar
  2. Banker RD, Morey R (1986) The use of categorical variables in data envelopment analysis. Manag Sci 32:1613–1627CrossRefGoogle Scholar
  3. Branda M (2015) Diversification-consistent data envelopment analysis based on directional-distance measures. Omega 52:65–76CrossRefGoogle Scholar
  4. Charnes A, Cooper WW (1959) Chance-constrained programming. Manag Sci 6:73–79CrossRefGoogle Scholar
  5. Charnes A, Cooper WW (1963) Deterministic equivalences for optimizing and satisfying under chance constraints. Manag Sci 11:18–39Google Scholar
  6. Cheng G, Zervopoulos P, Qian Z (2013) A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis. Eur J Oper Res 225:100–105CrossRefGoogle Scholar
  7. Cook WD, Kress M, Seiford LM (1993) On the use of ordinal data in data envelopment analysis. J Oper Res Soc 44:133–140CrossRefGoogle Scholar
  8. Cook WD, Zhu J (2006) Rank order data in DEA: a general framework. Eur J Oper Res 174:1021–1038CrossRefGoogle Scholar
  9. Cooper WW, Deng H, Huang ZM, Li SX (2002) Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis. J Oper Res Soc 53:1347–1356CrossRefGoogle Scholar
  10. Cooper WW, Huang ZM, Li SX (1996) Satisficing DEA models under chance constraints. Ann Oper Res 66:279–295CrossRefGoogle Scholar
  11. Cooper WW, Park KS, Yu G (1999) IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Manag Sci 45:597–607CrossRefGoogle Scholar
  12. Despotis DK, Smirlis YG (2002) Data envelopment analysis with imprecise data. Eur J Oper Res 140:24–36CrossRefGoogle Scholar
  13. Diabat A, Shetty U, Pakkala TPM (2015) Improved efficiency measure through directional distance formulation of data envelopment analysis. Ann Oper Res 229:325–346CrossRefGoogle Scholar
  14. Emrouznejad A, Anouze AL, Thanassoulis E (2010) A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA. Eur J Oper Res 200:97–304CrossRefGoogle Scholar
  15. Entani T, Maeda Y, Tanaka H (2002) Dual models of interval DEA and its extension to interval data. Eur J Oper Res 136:32–45CrossRefGoogle Scholar
  16. Hatami-Marbini A, Emrouznejad A, Agrell PJ (2014) Interval data with sign restrictions in DEA. Appl Math Model 38:2028–2036CrossRefGoogle Scholar
  17. Hatami-Marbini A, Emrouznejad A, Tavana M (2011) A taxonomy and review of the fuzzy data envelopment analysis literature: two decades in the making. Eur J Oper Res 214:457–472CrossRefGoogle Scholar
  18. Hougaard JL, Balezentis T (2014) Fuzzy efficiency without convexity. Fuzzy Sets Syst 255:17–29CrossRefGoogle Scholar
  19. Huang WT, Chen YW (2013) Qualitative data envelopment analysis by affinity set: a survey of subjective opinions for NPOs. Qual Quant 47:3079–3093CrossRefGoogle Scholar
  20. Inuiguchi M, Mizoshita F (2012) Qualitative and quantitative data envelopment analysis with interval data. Ann Oper Res 195:189–220CrossRefGoogle Scholar
  21. Jahanshahloo GR, Lotfi FH, Balf FR, Rezai HZ (2007) Discriminant analysis of interval data using Monte Carlo method in assessment of overlap. Appl Math Comput 191:521–532Google Scholar
  22. Jahanshahloo GR, Lotfi FH, Malkhalifeh MR, Namin MA (2009) A generalized model for data envelopment analysis with interval data. Appl Math Model 33:3237–3244CrossRefGoogle Scholar
  23. Kao C (2006) Interval efficiency measures in data envelopment analysis with imprecise data. Eur J Oper Res 174:1087–1099CrossRefGoogle Scholar
  24. Kao C, Lin PH (2011) Qualitative factors in data envelopment analysis: a fuzzy number approach. Eur J Oper Res 211:586–593CrossRefGoogle Scholar
  25. Kao C, Liu ST (2000) Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst 113:427–437CrossRefGoogle Scholar
  26. Kao C, Liu ST (2009) Stochastic data envelopment analysis in measuring the efficiency of Taiwan commercial banks. Eur J Oper Res 196:312–322CrossRefGoogle Scholar
  27. Keshvari A, Kuosmanen T (2013) Stochastic non-convex envelopment of data: applying isotonic regression to frontier estimation. Eur J Oper Res 231:481–491CrossRefGoogle Scholar
  28. Kerstens K, Van de Woestyne I (2011) Negative data in DEA: a simple proportional distance function approach. J Oper Res Soc 62:1413–1419CrossRefGoogle Scholar
  29. Kerstens K, Van de Woestyne I (2014) A note on a variant of radial measure capable of dealing with negative inputs and outputs in DEA. Eur J Oper Res 234:341–342CrossRefGoogle Scholar
  30. Khodabakhshi M, Gholami Y, Kheirollahi H (2010) An additive model approach for estimating returns to scale in imprecise data envelopment analysis. Appl Math Model 34:1247–1257CrossRefGoogle Scholar
  31. Kumbhaker SC, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  32. Land K, Lovell CAK, Thore S (1994) Chance-constrained data envelopment analysis. Manag Decis Econ 14:541–554CrossRefGoogle Scholar
  33. Olesen OB, Petersen NC (1995) Chance constrained efficiency evaluation. Manag Sci 41:442–457CrossRefGoogle Scholar
  34. Park KS (2007) Efficiency bounds and efficiency classifications in DEA with imprecise data. J Oper Res Soc 58:533–540CrossRefGoogle Scholar
  35. Pastor J, Ruiz J (2007) Variables with negative values in DEA. In: Zhu J, Cook WD (eds) Modeling data irregularities and structural complexities in data envelopment analysis. Springer, New York, pp 63–84CrossRefGoogle Scholar
  36. Portela MCAS, Thanassoulis E, Simpson G (2004) Negative data in DEA: a directional distance approach applied to bank branches. J Oper Res Soc 55:1111–1121CrossRefGoogle Scholar
  37. Saen RF (2006) Technologies ranking in the presence of cardinal and ordinal data. Appl Math Comput 176:476–487Google Scholar
  38. Sharp JA, Meng W, Liu W (2007) A modified slacks-based measure model for data envelopment analysis with ‘natural’ negative outputs and inputs. J Oper Res Soc 58:1672–1677CrossRefGoogle Scholar
  39. Shokouhi AH, Hatami-Marbini A, Tavana M, Saati S (2010) A robust optimization approach for imprecise data envelopment analysis. Comput Ind Eng 59:387–397CrossRefGoogle Scholar
  40. Tavana M, Shiraz RK, Hatami-Marbini A (2014) A new chance-constrained DEA model with birandom input and output data. J Oper Res Soc 65:1824–1839CrossRefGoogle Scholar
  41. Tsang SS, Chen YF, Lu YH, Chiu CR (2014) Assessing productivity in the presence of negative data and undesirable outputs. Serv Ind J 34:162–174CrossRefGoogle Scholar
  42. Wei Q, Chang TS, Han S (2014) Quantile-DEA classifiers with interval data. Ann Oper Res 217:535–563CrossRefGoogle Scholar
  43. Yager A (1986) A characterization of the extension principle. Fuzzy Sets Syst 18:205–217CrossRefGoogle Scholar
  44. Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28CrossRefGoogle Scholar
  45. Zhu J (2003a) Imprecise data envelopment analysis (IDEA): a review and improvement with an application. Eur J Oper Res 144:513–529CrossRefGoogle Scholar
  46. Zhu J (2003b) Efficiency evaluation with strong ordinal input and output measures. Eur J Oper Res 146:477–485CrossRefGoogle Scholar
  47. Zhu J (2004) Imprecise DEA via standard linear DEA models with a revisit to a Korean mobile telecommunication company. Oper Res 52:323–329CrossRefGoogle Scholar
  48. Zimmermann HJ (1996) Fuzzy set theory and its applications, 3rd edn. Kluwer-Nijhoff, BostonCrossRefGoogle 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

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