Relative partial efficiency: network and black box SBM DEA interpretations in multiplier form

  • Fatemeh BolooriEmail author
  • Rashed Khanjani-Shiraz
  • Hirofumi Fukuyama
Original paper


In traditional black-box DEA when the ratio-based multiplier DEA model is estimated to obtain a technical efficiency score, the estimated multipliers (shadow prices) serve as the weights that maximize the ratio of the aggregation of weighted sum of outputs (virtual output) to that of inputs (virtual input) of the assessed DMU in comparison with the other decision making units (DMUs). With respect to the ratio-based multiplier model of non-radial slack-based measure (SBM), however, there does not exist such a nice efficiency interpretation. For the purpose of providing a reasonable efficiency interpretation for both black-box and network SBM models, this paper introduces a concept called relative partial efficiency (RPE). In the black box structure, RPEs are defined for each input–output pair and a multi objective programming is formed in order to maximize RPEs. Then, it is proved that its equivalent single objective programming problem is the same SBM multiplier DEA model. The obtained explicit efficiency interpretation coming from this novel concept is then generalized for the multiplier network SBM DEA model represented by Boloori (Comput Ind Eng 95:83–96, 2016).


Data envelopment analysis Slack based DEA model Network SBM Relative partial efficiency 



The financial support of the University of Tabriz in the context of the research fund (No. 31/27) devoted to the first author is gratefully acknowledged.


  1. Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manage Sci 30:1078–1092CrossRefGoogle Scholar
  2. Banker RD, Cooper WW, Seiford L, Thrall RM, Zhu J (2004) Returns to scale in different DEA models. Eur J Oper Res 154:345–362CrossRefGoogle Scholar
  3. Boloori F (2016) A slack-based network DEA model for generalized structures: an axiomatic approach. Comput Ind Eng 95:83–96CrossRefGoogle Scholar
  4. Boloori F, Pourmahmoud J (2016) A modified SBM-NDEA approach for the efficiency measurement of bank branches. Oper Res Int J 16:301–326CrossRefGoogle Scholar
  5. Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2:429–444CrossRefGoogle Scholar
  6. 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 11:5–42CrossRefGoogle Scholar
  7. Cooper WW, Seiford LM, Tone K (2006) General discussion in “introduction to data envelopment analysis and its uses: with DEA-solver software and references”. Springer, New York, pp 1–2Google Scholar
  8. Cooper WW, Pastor JT, Aparicio J, Borras F (2011) Decomposing profit inefficiency in DEA through the weighted additive model. Eur J Oper Res 212:411–416CrossRefGoogle Scholar
  9. Lozano S (2011) Scale and cost efficiency analysis of networks of processes. Expert Syst Appl 38:6612–6617CrossRefGoogle Scholar
  10. Lozano S (2015) Alternative SBM model for network DEA. Comput Ind Eng 32:33–40CrossRefGoogle Scholar
  11. Mehrabian S, Jahanshahloo GH, Alirezaee MR, Amin GR (2000) An assurance interval for the non-Archimedean epsilon in DEA. Oper Res 48:344–347CrossRefGoogle Scholar
  12. Mirdehghan SM, Fukuyama H (2016) Pareto–Koopmans efficiency and network DEA. Omega 61:78–88CrossRefGoogle Scholar
  13. Pastor JT, Ruiz JL, Sirvant I (1999) An enhanced DEA Russell graph efficiency measure. Eur J Oper Res 115:596–607CrossRefGoogle Scholar
  14. Russell RR (1985) Measures of technical efficiency. J Econ Theory 35:109–126CrossRefGoogle Scholar
  15. Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130:498–509CrossRefGoogle Scholar
  16. Tone K, Tsutsui M (2009) Network DEA: a slacks-based measure approach. Eur J Oper Res 197:243–252CrossRefGoogle Scholar
  17. Zimmermann HJ (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1:45–55CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Fatemeh Boloori
    • 1
    Email author
  • Rashed Khanjani-Shiraz
    • 1
  • Hirofumi Fukuyama
    • 2
  1. 1.School of Mathematical ScienceUniversity of TabrizTabrizIran
  2. 2.Faculty of CommerceFukuoka UniversityFukuokaJapan

Personalised recommendations