Skip to main content
SpringerLink
Log in

Discrete and Integer Valued Inputs and Outputs in Data Envelopment Analysis

  • Chapter
  • First Online:
Book cover Data Envelopment Analysis

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 221))

Abstract

Standard axioms of free disposability, convexity and constant returns to scale employed in Data Envelopment Analysis (DEA) implicitly assume continuous, real-valued inputs and outputs. However, the implicit assumption of continuous data will never hold with exact precision in real world data. To address the discrete nature of data explicitly, various formulations of Integer DEA (IDEA) have been suggested. Unfortunately, the axiomatic foundations and the correct mathematical formulation of IDEA technology has caused considerable confusion in the literature. This chapter has three objectives. First, we re-examine the axiomatic foundations of IDEA, demonstrating that some IDEA formulations proposed in the literature fail to satisfy the axioms of free disposability of continuous inputs and outputs, and natural disposability of discrete inputs and outputs. Second, we critically examine alternative efficiency metrics available for IDEA. We complement the IDEA formulations for the radial input measure with the radial output measure and the directional distance function. We then critically discuss the additive efficiency metrics, demonstrating that the optimal slacks are not necessarily unique. Third, we consider estimation of the IDEA technology under stochastic noise, modeling inefficiency and noise as Poisson distributed random variables.

Abbreviations of key concepts referred to in this chapter: DEA = Data Envelopment Analysis, DMU = Decision Making Unit, CNLS = Convex Nonparametric Least Squares, IDEA = Integer DEA, MILP = Mixed Integer Linear Programming, RTS = Returns To Scale, SFA = Stochastic Frontier Analysis, StoNED = Stochastic Nonparametric Envelopment of Data.

Abbreviations of articles frequently cited in this chapter: KJS = Kuosmanen, Johnson and Saastamoinen (in this volume), KKM = Kuosmanen and Kazemi Matin (2009), KMK = Kazemi Matin and Kuosmanen (2009), KSM = Khezrimotlagh, Salleh, and Mohsenpour (2012, (2013a, (2013b), LV = Lozano and Villa (2006, 2007).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We will henceforth use the term “DMU” to refer to any entity that transforms inputs to output, including both non-profit firms and for-profit companies. DMU can refer to a production plant, facility, or sub-division of a company, or to an aggregate entity such as an industry, a region, or a country.

  2. 2.

    Previous studies such as Banker and Morey (1986), Kamakura (1988), and Rousseau and Semple (1993) (among others) consider inputs and outputs measured on the categorical or ordinal scale, which are obviously integer valued. However, input-output variables defined on the interval or ratio scales can be integer valued as well.

  3. 3.

    The minimum extrapolation principle was formally introduced by Banker et al. (1984), but formal minimum extrapolation theorems (and proofs) date back at least to Afriat (1972).

  4. 4.

    For clarity, we denote vectors by bold lower case letters (e.g., x) and matrices by bold capital letters (e.g., X).

  5. 5.

    The correct way of implementing weak disposability in DEA has caused some confusion in the literature: see Kuosmanen (2005), Färe and Grosskof (2009), Kuosmanen and Podinovski (2009), and Podinovski and Kuosmanen (2011) for an interesting debate on this issue.

  6. 6.

    This is another issue that has caused confusion: see Cherchye et al. (2001).

  7. 7.

    KSM (2012) state: ”Now, if it has been supposed that only the integer numbers set is considered, then it should not have been used the real number variable in the integer axioms! In fact, a new axiom must not have any doubts or parallel affects with those previous axioms. In other words, an axiom is an evident premise as to be accepted as true without controversy.” This discussion reveals that KSM do not understand the economic meaning of axioms in DEA. In fact, none of the standard DEA axioms can meet the requirements of KSM.

  8. 8.

    DEA is a nonparametric estimator subject to the curse of dimensionality. This implies that the precision of DEA estimator deteriorates rapidly as the number of input and output variables increases. Also the discriminating power of DEA is affected: when the dimensionality is large, almost all DMUs appear as inefficient.

  9. 9.

    In the single output case, Afriat (1972) proves the similar minimum extrapolation result for the smallest production function satisfies axioms (A1), (A1) + (A2), or (A1) + (A2) + (A5).

  10. 10.

    In their original manuscript, KKM stated their MILP formulation using inequality constraints. They later introduced slacks by request of a reviewer.

  11. 11.

    In most applications we can think of, it would seem more natural to treat integer-valued inputs as quasi-fixed factors, and project DMUs to the frontier in the direction of continuous inputs.

  12. 12.

    An interested reader can easily verify the empirical results reported by KKM and KMK. For transparency, the data and the computational codes for GAMS and LINGO are freely available on the website:

    http://nomepre.net/index.php/integerdea.

  13. 13.

    The Poisson distribution is the most widely used discrete probability distribution in statistics. It can be derived from the probability of a given number of events occurring in a fixed interval of time and/or space when the events occur with a known average rate and independently of the time since the last event. Note that the Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials approaches to infinity and the expected number of successes is fixed.

  14. 14.

    The second set of constraints imposes convexity, applying the Afriat theorem (Afriat 1972). The convexity axiom can be relaxed by replacing CNLS with isotonic regression, see Keshvari and Kuosmanen (2013), for details.

  15. 15.

    See, e.g., Simar and Wilson (2010) for a more detailed discussion about the wrong skewness problem in stochastic frontier estimation.

  16. 16.

    KJS, Sect. 7.4.8, discusses some of this literature in the context of StoNED estimation.

References

  • Afriat SN (1972) Efficiency estimation of production functions. Int Econ Rev 13(3):568–598

    Article  Google Scholar 

  • Alirezaee MR, Sani MRR (2011) An enumeration algorithm for integer-valued data envelopment analysis. Int Trans Oper Res 18(6):729–740

    Article  Google Scholar 

  • Banker RD, Morey RC (1986) The use of categorical variables in data envelopment analysis. Manage Sci 32(12):1613–1627

    Article  Google Scholar 

  • Banker RD, Charnes A, Cooper WW, (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30(9):1078–1092

    Article  Google Scholar 

  • Bogetoft P (1996) DEA on relaxed convexity assumptions. Manage Sci 42(3):457–465

    Article  Google Scholar 

  • Bogetoft P, Tama JM, Tind J (2000) Convex input and output projections of nonconvex production possibility sets. Manage Sci 46(6):858–869

    Article  Google Scholar 

  • Chambers RG, Chung YH, Färe R (1996) Benefit and distance functions. J Econ Theory 70(2):407–419

    Article  Google Scholar 

  • Chambers RG, Chung YH, Färe R (1998) Profit, directional distance function, and Nerlovian efficiency. J Optim Theory Appl 98(2):351–364

    Article  Google Scholar 

  • Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444

    Article  Google Scholar 

  • Charnes A, Cooper WW, Golany B, Seiford LM, Stutz J (1985) Foundations of data envelopment analysis for Pareto-Koopmans efficient empirical production functions. J Econom 30(1):91–107

    Article  Google Scholar 

  • Cherchye L, Kuosmanen T, Post T (2001). Alternative treatments of congestion in DEA. Eur J Oper Res 132:75–80

    Article  Google Scholar 

  • Chen CM., Du J, Huo J, Zhu J (2012) Undesirable factors in integer-valued DEA: evaluating the operational efficiencies of city bus systems considering safety records. Decis Support Syst 54(1):330–335

    Article  Google Scholar 

  • Chen Y, Djamasbi S, Du J, Lim S (2013) Integer-valued DEA super-efficiency based on directional distance function with an application of evaluating mood and its impact on performance. Int J Prod Econ 146(2):550–556

    Article  Google Scholar 

  • Cooper WW, Park KS, Pastor JT (1999) RAM: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. J Prod Anal 11:5–42

    Article  Google Scholar 

  • Deprins D, Simar L, Tulkens H, (1984) Measuring labor-efficiency in post offices. In: Marchand M, Pestieau P, Tulkens H, (eds) The performance of public enterprises, concepts and measurement. Elsevier Science Ltd., North Holland

    Google Scholar 

  • Du J, Chen CM, Chen Y, Cook WD., Zhu J (2012) Additive super-efficiency in integer-valued data envelopment analysis. Eur J Oper Res 218(1):186–192

    Article  Google Scholar 

  • Farrell M (1957) The measurement of productive efficiency. J Royal Stat Soc Ser A 120(3):253–290

    Article  Google Scholar 

  • Färe R, Grosskopf S (2009) A Comment on weak disposability in nonparametric production analysis. Am J Agric Econ 91(2):535–538

    Article  Google Scholar 

  • Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19:233–238

    Article  Google Scholar 

  • Kamakura WA (1988) A note on the use of categorical variables in data envelopment analysis. Manage sci 34(10):1273–1276

    Article  Google Scholar 

  • Kazemi Matin, R., Emrouznejad, A. 2011. An integer-valued data envelopment analysis model with bounded outputs. Int Trans Oper Res 18(6):741–749

    Article  Google Scholar 

  • Kazemi Matin R, Kuosmanen T (2009) Theory of integer-valued data envelopment analysis under alternative returns to scale axioms. Omega 37(5):988–995

    Article  Google Scholar 

  • Keshvari A Kuosmanen T (2013) Stochastic non-convex envelopment of data: applying isotonic regression to frontier estimation. Eur J Oper Res 231(2):481–491

    Article  Google Scholar 

  • Khezrimotlagh D, Salleh S, Mohsenpour Z (2012) A comment on theory of integer-valued data envelopment analysis. Appl Math Sci 6(116):5769–5774

    Google Scholar 

  • Khezrimotlagh D, Salleh S, Mohsenpour Z (2013a) A new robust mixed integer-valued model in DEA. Appl Math Model 37(24):9885–9897

    Google Scholar 

  • Khezrimotlagh D, Salleh S, Mohsenpour Z (2013b) A note on integer-valued radial model in DEA. Comput Ind Eng 66(1):199–200

    Google Scholar 

  • Koopmans TC (1951a) Analysis of production as an efficient combination of activities. In: Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 33–97

    Google Scholar 

  • Koopmans TC (ed) (1951b) Activity analysis of production and allocation, cowles commission for research in economics. Wiley, New York

    Google Scholar 

  • Kuosmanen T (2001) DEA with efficiency classification preserving conditional convexity. Eur J Oper Res 132(2):326–342

    Article  Google Scholar 

  • Kuosmanen T (2003) Duality theory of non-convex technologies. J Prod Anal 20:273–304

    Article  Google Scholar 

  • Kuosmanen T (2005) Weak disposability in nonparametric production analysis with undesirable outputs. Am J Agric Econ 87:1077–1082

    Article  Google Scholar 

  • Kuosmanen T (2008) Representation theorem for convex nonparametric least squares. Econom J 11:308–325

    Article  Google Scholar 

  • Kuosmanen T, Johnson AL (2010) Data envelopment analysis as nonparametric least squares regression. Oper Res 58(1):149–160

    Article  Google Scholar 

  • Kuosmanen T, Kazemi Matin R (2009) Theory of integer-valued data envelopment analysis. Eur J Oper Res 192(2):658–667

    Article  Google Scholar 

  • Kuosmanen T, Kortelainen M (2012) Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints. J Prod Anal 38(1):11–28

    Article  Google Scholar 

  • Kuosmanen T, Podinovski VV (2009) Weak disposability in nonparametric production analysis: reply to Färe and Grosskopf. Am J Agric Econ 91(2):539–545

    Article  Google Scholar 

  • Kuosmanen T, Post GT, Sipiläinen T (2004) Shadow price approach to total factor productivity measurement: with an application to Finnish grass-silage production. J Prod Anal 22(1):95–121

    Article  Google Scholar 

  • Kuosmanen T, Johnson AL, Saastamoinen A (2014) Stochastic nonparametric approach to efficiency analysis: a unified framework. In: Zhu J (ed) data envelopment analysis, Springer

    Google Scholar 

  • Lozano S (2013) Using DEA to find the best partner for a horizontal cooperation. Comput Ind Eng 66(2):286–292

    Article  Google Scholar 

  • Lozano S, Villa G (2006) Data envelopment analysis of integer-valued inputs and outputs. Comput Oper Res 33(10):3004–3014

    Article  Google Scholar 

  • Lozano S, Villa G (2007) Integer DEA models. In Zhu J, Cook WD (eds) Modeling data irregularities and structural complexities in data envelopment analysis. Springer, New York, pp 271–290

    Chapter  Google Scholar 

  • Lozano S, Villa G, Canca D (2011) Application of centralised DEA approach to capital budgeting in Spanish ports. Comput Ind Eng 60(3):455–465

    Article  Google Scholar 

  • Nöhren M, Heinzl A (2012) Measuring the relative efficiency of global delivery models in IT outsourcing. Lect Notes Bus Inf Process 130:61–75

    Article  Google Scholar 

  • Petersen, N. C. (1990) Data Envelopment analysis on a relaxed set of assumptions, Management Science 20(3), 305–314

    Google Scholar 

  • Podinovski VV (2004) Bridging the gap between the constant and variable returns-to-scale models: selective proportionality in data envelopment analysis. J Oper Res Soc 55(3):265–276

    Article  Google Scholar 

  • Post GT (2001) Transconcave data envelopment analysis. Eur J Oper Res 132(2):374–389

    Article  Google Scholar 

  • Rousseau JJ, Semple JH (1993) Notes: categorical outputs data envelopment analysis. Manage Sci 39(3):384–386

    Article  Google Scholar 

  • Shephard R, (1970) Theory of cost and production functions. Princeton University Press, Princeton

    Google Scholar 

  • Simar L, Wilson PW (2010) Inferences from cross-sectional, stochastic frontier models. Econom Rev 29(1):62–98

    Article  Google Scholar 

  • Skellam JG (1946) The frequency distribution of the difference between two Poisson variates belonging to different populations. J Royal Stat Soc Ser A 109(3):296–296

    Article  Google Scholar 

  • Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130:498–509

    Article  Google Scholar 

  • Tulkens H (1993) On FDH efficiency analysis: some methodological issues and applications to retail banking, courts, and urban transit. J Prod Anal 4:183–210

    Article  Google Scholar 

  • Von Neumann J (1945–1946) A model of general economic equilibrium. Rev Econ Stud 13(1):1–9

    Google Scholar 

  • Wu J Zhou Z, Liang L (2009) Measuring the performance of nations at Beijing summer olympics using integer-valued DEA model. J Sports Econ 11(5):549–566

    Google Scholar 

  • Wu J, Liang L, Song H (2010) Measuring hotel performance using the integer DEA model. Tour Econ 16(4):867–882

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aalto University School of Business, Helsinki, Finland

    Timo Kuosmanen & Abolfazl Keshvari

  2. Department of Mathematics, College of Basic Science, Karaj Branch, Islamic Azad University, Alborz, Iran

    Reza Kazemi Matin

Authors
  1. Timo Kuosmanen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abolfazl Keshvari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reza Kazemi Matin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Timo Kuosmanen .

Editor information

Editors and Affiliations

  1. Department of Management, Worcester Polytechnic Institute, Worcester, Massachusetts, USA

    Joe Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this chapter

Cite this chapter

Kuosmanen, T., Keshvari, A., Matin, R. (2015). Discrete and Integer Valued Inputs and Outputs in Data Envelopment Analysis. In: Zhu, J. (eds) Data Envelopment Analysis. International Series in Operations Research & Management Science, vol 221. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7553-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation