Skip to main content
SpringerLink
Log in

Bounded directional distance function models

  • Original Paper
  • Published:
Central European Journal of Operations Research Aims and scope Submit manuscript

Abstract

Bounded additive models in data envelopment analysis (DEA) under the assumption of constant returns to scale (CRS) were recently introduced in the literature (Cooper et al. in J Product Anal 35(2):85–94, 2011; Pastor et al. in J Product Anal 40:285–292, 2013; Pastor et al. in Omega 56:16–24, 2015). In this paper, we propose to extend the so far generated knowledge about bounded additive models to the family of directional distance function (DDF) models in DEA, giving rise to a completely new subfamily of bounded or partially-bounded CRS-DDF models. We finally check the new approach on a real agricultural panel data set estimating efficiency and productivity change over time, resorting to the Luenberger indicator in a context where at least one variable is naturally bounded.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Luenberger (1992) introduced the concept of benefit function as a representation of the amount that an individual is willing to trade, in terms of a specific reference commodity bundle g, for the opportunity to move from a consumption bundle to a utility threshold. Luenberger also defined a so-called shortage function (1992, p. 242, Definition 4.1), which basically measures the amount by which a specific plan is short of reaching the frontier of the technology. Later, Chambers et al. (1996a, 1998) redefined the benefit function and the shortage function as the directional distance function.

  2. Since we are interesting in measuring productivity, we will assume from now on CRS. The definition of a Bounded DDF model under any other returns to scale assumption is straightforward.

  3. In our empirical context, the bottlers are also marketers.

  4. Since olive oil is a perishable product and must be consumed during its first year of life, all the yearly production must be sold.

  5. This is the part of the production of olive oil with the highest quality, either extra virgin olive oil or virgin olive oil.

  6. Our empirical application and that published in Vidal et al. (2014) are different regarding the data because Vidal et al. (2014) used the available information of the years 2008, 2009 and 2010 for 17 DMUs, while we utilize data from 2012 and 2013 for 22 DMUs. The inputs are similar but the outputs are more different. Vidal et al. (2014) did not use Revenue as output. They also did not utilize the ratio VOO/POO, which measures the ‘quality’ of the produced outputs. Instead, Vidal et al. (2014) directly used tons of olive oil. In Vidal et al. (2014), none of the variables is bounded in a natural way and were really bounded by the minimum observed input and maximum observed output. Additionally, from a methodological point of view, these approaches are so different since (1) Vidal et al. (2014) resorted to the ‘Graph’ version of a BAM, which are based upon the Weighted Additive Model, bounding all inputs and outputs, whereas our approach is based on an output-oriented (Bounded) DDF, where also only one dimension is bounded; and (2) the objective in Vidal et al. (2014) was to measure technical efficiency and its evolution, while we also determined productivity change over time and its drivers.

References

  • Ali AI, Seiford LM (1993) The mathematical programming approach to efficiency analysis. In: Fried H, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency: techniques and applications. Oxford University Press, Inc, Oxford

    Google Scholar 

  • Balk B (2001) Scale efficiency and productivity change. J Prod Anal 15:159–183

    Article  Google Scholar 

  • Briec W, Kerstens K (2009a) Infeasibility and directional distance functions with application of determinateness of the Luenberger productivity indicator. J Optim Theory Appl 141:55–73

    Article  Google Scholar 

  • Briec W, Kerstens K (2009b) The Luenberger productivity indicator: an economic specification leading to infeasibilities. Econ Model 26:597–600

    Article  Google Scholar 

  • Caves DW, Christensen LR, Diewert WE (1982) The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50:1393–1414

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Chambers RG, Färe R, Grosskopf S (1996b) Productivity growth in APEC countries. Pac Econ Rev 1:181–190

    Article  Google Scholar 

  • Chambers RG, Chung Y, Färe R (1998) Profit, directional distance functions, 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 2:429–444

    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 others models and measures in DEA. J Prod Anal 11:5–42

    Article  Google Scholar 

  • Cooper WW, Pastor JT, Borras F, Aparicio J, Pastor D (2011) BAM: a bounded adjusted measure of efficiency for use with bounded additive models. J Prod Anal 35(2):85–94

    Article  Google Scholar 

  • Färe R, Grosskopf S (2010) Directional distance functions and slacks-based measures of efficiency. Eur J Oper Res 200:320–322

    Article  Google Scholar 

  • Färe R, Grosskopf S, Lindgren B, Roos P (1992) Productivity changes in swedish pharmacies 1980–1989: a non-parametric malmquist approach. J Prod Anal 3:85–101

    Article  Google Scholar 

  • Grifell-Tatjé E, Lovell CAK (1995) A note on the malmquist productivity index. Econ Lett 47:169–175

    Article  Google Scholar 

  • Hollingsworth B, Smith P (2003) Use of ratios in data envelopment analysis. Appl Econ Lett 10:733–735

    Article  Google Scholar 

  • Kapelko M, Horta IM, Camanho AS, Oude Lansink A (2015) Measurement of input specific productivity growth with an application to the construction industry in Spain and Portugal. Int J Prod Econ 166:64–71

    Article  Google Scholar 

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

    Google Scholar 

  • Liu JS, Lu LYY, Lu W-M, Lin BJY (2013) Data envelopment analysis 1978–2010: a citation-based literature survey. Omega 41:3–15

    Article  Google Scholar 

  • Lovell CAK (2003) The decomposition of Malmquist productivity indexes. J Prod Anal 20:437–458

    Article  Google Scholar 

  • Luenberger DG (1992) New optimality principles for economic efficiency and equilibrium. J Optim Theory Appl 75(2):221–264

    Article  Google Scholar 

  • MAGRAMA (2016) Estadisticas. Producciones Agricolas. Ministerio de Agricultura, Alimentacion y Medio Ambiente. http://www.magrama.gob.es/es/agricultura/estadisticas/. Accessed 28 Dec 2016 (in Spanish)

  • Olesen OB, Petersen NC, Podinovski VV (2015) Efficiency analysis with ratio measures. Eur J Oper Res 245(2):446–462

    Article  Google Scholar 

  • Pastor JT, Aparicio J, Monge JF, Pastor D (2013) Modeling CRS bounded additive DEA models and characterizing their Pareto-efficient points. J Prod Anal 40:285–292

    Article  Google Scholar 

  • Pastor JT, Del Campo FJ, Vidal F, Pastor D (2014) Efficiency in attracting tourists via the Web—an application to the Mediterranean countries. Tour Econ 20(1):195–202

    Article  Google Scholar 

  • Pastor JT, Aparicio J, Alcaraz J, Vidal F, Pastor D (2015) An enhanced BAM for unbounded or partially bounded CRS additive models. Omega 56:16–24

    Article  Google Scholar 

  • Rashidi K, Saen RF (2015) Measuring eco-efficiency based on green indicators and potentials in energy saving and undesirable output abatement. Energy Econ 50:18–26

    Article  Google Scholar 

  • Ray SC, Desli E (1997) Productivity growth, technical progress, and efficiency change in industrialized countries. Am Econ Rev 87:1033–1039

    Google Scholar 

  • Shephard RW (1953) Cost and production functions. Princeton University Press, Princeton

    Google Scholar 

  • Toloo M, Tavana M, Santos-Arteaga FJ (2017) An integrated data envelopment analysis and mixed integer non-linear programming model for linearizing the common set of weights. CEJOR. https://doi.org/10.1007/s10100-017-0510-y

    Article  Google Scholar 

  • Toloo M, Nalchigar S, Sohrabi B (2018) Selecting most efficient information system projects in presence of user subjective opinions: a DEA approach. CEJOR. https://doi.org/10.1007/s10100-018-0549-4

    Article  Google Scholar 

  • Vidal F, Pastor JT, Borras F, Pastor D (2013) Efficiency analysis of the designations of origin in the Spanish wine sector. Span J Agri Res 11(2):294–304

    Article  Google Scholar 

  • Vidal F, Aparicio J, Pastor JT, Pastor D (2014) Las Denominaciones de Origen de aceite de oliva virgen en España. Un analisis de su eficiencia tecnica. ITEA 2:208–222 (in Spanish)

    Google Scholar 

Download references

Acknowledgements

We thank the guest editors of the special issue DEA 2017 and two anonymous referees for providing constructive comments and help in improving the contents and presentation of this paper. Additionally, J.T. Pastor, J. Aparicio, J. Alcaraz and F. Vidal thank the financial support from the Spanish Ministry for Economy and Competitiveness (Ministerio de Economía, Industria y Competitividad), the State Research Agency (Agencia Estatal de Investigacion) and the European Regional Development Fund (Fondo Europeo de DEsarrollo Regional) under Grant MTM2016-79765-P (AEI/FEDER, UE).

Author information

Authors and Affiliations

  1. Center of Operations Research (CIO), University Miguel Hernandez of Elche (UMH), 03202, Elche, Alicante, Spain

    Jesus T. Pastor, Juan Aparicio & Javier Alcaraz

  2. Environmental Economics Department, University Miguel Hernandez of Elche (UMH), 03212, Orihuela (Alicante), Spain

    Fernando Vidal

  3. Physical and Sports Education, University Miguel Hernandez of Elche (UMH), 03202, Elche, Alicante, Spain

    Diego Pastor

Authors
  1. Jesus T. Pastor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Aparicio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Javier Alcaraz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fernando Vidal
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Diego Pastor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Aparicio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pastor, J.T., Aparicio, J., Alcaraz, J. et al. Bounded directional distance function models. Cent Eur J Oper Res 26, 985–1004 (2018). https://doi.org/10.1007/s10100-018-0562-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10100-018-0562-7

Keywords

Search

Navigation