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1. (2008) is to use the software program Qhull. However, this program is not developed exclusively for DEA and it is not always easy to use in DEA applications (see Aparicio et al., 2007; Zhu et al., 2022).
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2. Allen, R., Athanassopoulos, A., Dyson, R.G., & Thanassoulis, E. (1997). Weights restrictions and value judgements in data envelopment analysis: evolution, development and future directions. Annals of Operation Research, 73, 13-34.

3. An alternative advocated by Olesen and Petersen (2003) and Thanassoulis et al.
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4. Banker, R.D., Charnes, A., & Cooper, W.W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30, 1078-1092. https://doi.org/10.1287/mnsc.30.9.1078.

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6. Berger, A., & Humphrey, D. (1992). Measurement and efficiency issues in commercial banking. In Z. Griliches (Ed.): Output measurement in service sectors: National Bureau of Economic Research, studies in income and wealth, 56, University of Chicago Press. https://www.nber.org/system/files/chapters/c7237/c7237.pdf Bogetoft, P., & Otto, L. (2011). Benchmarking with DEA, SFA, and R. International Series in Operations Research and Management Science 157, Springer. https://doi.org/10.1007/978-1-4419-7961-2.

7. Brockett, P.L., Charnes, A., Cooper, W.W., Huang, Z.M., & Sun, D.B. (1997). Data transformations in DEA cone ratio envelopment approaches for monitoring bank performances. European Journal of Operational Research, 98, 250-268.

8. Characterizing and finding full dimensional efficient facets in DEA: a variable returns to scale specification. Journal of Operational Research Society, 65, 14531464. https://doi.org/10.1057/jors.2013.97.
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9. Charnes et al. (1990) used weight vectors with strictly positive components to construct the cone of feasible weights in their application. On the other hand, Brockett et al. (1997) included weight vectors containing zero components as well. We limit our discussion to input-oriented DEA models. The extension of the results to output oriented models is straightforward.
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10. Charnes, A., Cooper, W.W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444. https://doi.org/10.1016/0377-2217(78)90138-8.

11. Charnes, A., Cooper, W.W., & Thrall, R.M. (1991a). A structure for classifying and characterizing efficiency and inefficiency in Data Envelopment Analysis. Journal of Productivity Analysis 2, 197-237. https://doi.org/10.1007/BF00159732.
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12. Charnes, A., Cooper, W.W., Huang, Z.M., & Sun, D.B. (1990). Polyhedral cone- ratio DEA models with an illustrative application to large commercial banks. Journal of Econometrics 46, 73-91. https://doi.org/10.1016/0304-4076(90)90048-X.

13. Charnes, A., Cooper, W.W., Huang, Z.M., & Sun, D.B. (1991b). Relations between half-space and finitely generated cones in polyhedral cone-ratio DEA models.
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14. Charnes, A., Cooper, W.W., Wei, Q.L., & Huang, Z.M. (1989). Cone ratio data envelopment analysis and multi-objective programming. International Journal of Systems Science, 20, 1099-1118. https://doi.org/10.1080/00207728908910197.
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15. Chilingerian, J.A., & Sherman, H. (1997). DEA and primary care physician report cards: Deriving preferred practice cones from managed care service concepts and operating strategies. Annals of Operations Research, 73, 35-66.

16. Ding, J., Feng, C., Bi, G., Liang, L., & Khan, R. (2015). Cone ratio models with shared resources and nontransparent allocation parameters in network DEA. Journal of Productivity Analysis, 44, 137-155. https://doi.org/10.1007/s11123-014-0420-0.

17. Drake, L., Hall, M., & Simper, R. (2009). Bank modelling methodologies: A comparative non-parametric analysis of efficiency in the Japanese banking sector.

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19. Endnotes As model DMUs are considered DEA-efficient DMUs that are viewed as exceptional performers by the DM (Charnes et al., 1990). Agreement on the set of model banks may be unanimous or not, as DMs might have diverging views on what constitutes good performance, or the chosen banks are viewed as different types of examples to follow. From now on, when a quotation is used, the words in brackets and the underlying are our own additions.
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20. Eskelinen, J., Halme, M., & Kallio, M. (2014). Bank branch sales evaluation using extended value efficiency analysis. European Journal of Operational Research, 232, 654-663. https://doi.org/10.1016/j.ejor.2013.08.005.

21. Fethi, M.D., & Pasiouras, F. (2010). Assessing bank efficiency and performance with operational research and artificial intelligence techniques: A survey. European Journal of Operational Research, 204, 189-198. https://doi.org/10.1016/j.ejor.2009.08.003.

22. Fukuyama, H. (1993). Technical and scale efficiency of Japanese commercial banks: A non-parametric approach. Applied Economics, 25, 1101-1112. https://doi.org/10.1080/00036849300000090.
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23. Fukuyama, H., & Sekitani, K. (2012). Decomposing the efficient frontier of the DEA production possibility set into a smallest number of convex polyhedrons by mixed integer programming. European Journal of Operational Research, 221, 165-174. https://doi.org/10.1016/j.ejor.2012.02.035.

24. Fukuyama, H., & Weber, W.L. (2002). Estimating output allocative efficiency and productivity change: Application to Japanese banks. European Journal of Operational Research, 137, 177-190. https://doi.org/10.1016/S03772217 (01)00054-6.

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27. Hoshi, T., & Kashyap, A.K. (2010). Will the U.S. bank recapitalization succeed? Eight lessons from Japan. Journal of Financial Economics, 18, 3-26. https://doi.org/10.1016/j.jfineco.2010.02.005.

28. In: Zhu J (eds.) Data Envelopment Analysis: A Handbook of Models and Methods International Series in Operations Research & Management Science, 221, Springer. https://doi.org/10.1007/978-1-4899-7553-9_6.
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29. Japanese Banking Association (2019). List of members, Japanese Bankers Association, Japan (2019) https://www.zenginkyo.or.jp/en/outline/list-of-members/.
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30. Korhonen, P., Soismaa, M., & Siljamaki, A. (2002). On the use of value efficiency analysis and some further developments. Journal of Productivity Analysis 17:4965.

31. Korhonen, P., Tainio, R., & Wallenius, J. (2001). Value efficiency analysis of academic research. European Journal of Operational Research, 130, 121-132.

32. Lovell, C. A. K., & Pastor, J. T. (1999). Radial DEA models without inputs or without outputs. European Journal of Operational Research, 118, 46-51.

33. Marshall, E., & Shortle, J. (2005). Using DEA and VEA to evaluate quality of life in the Mid-Atlantic States. Agricultural and Resource Economics Review, 34, 185203.

34. Olesen, O.B., & Petersen, N.C. (1996). Indicators of ill-conditioned data sets and model misspecification in Data Envelopment Analysis: An Extended Facet approach.

35. Olesen, O.B., & Petersen, N.C. (2003). Identification and use of efficient faces and facets in DEA. Journal of Productivity Analysis, 20, 323-360.

36. Olesen, O.B., & Petersen, N.C. (2015). Facet Analysis in Data Envelopment Analysis.

37. Oral, M., & Yolalan, R. (1990). An empirical study on measuring operating efficiency and profitability of bank branches. European Journal of Operational Research, 46, 282-294. https://doi.org/10.1016/0377-2217(90)90002-S.

38. Podinovski, V.V. (2004) Production trade-offs and weight restrictions in data envelopment analysis. Journal of the Operational Research Society, 55(12), 1311-1322. https://doi.org/10.1057/palgrave.jors.2601794.

39. Podinovski, V.V. (2005) The explicit role of weight bounds in models of data envelopment analysis. Journal of the Operational Research Society 56(12), 14081418. https://doi.org/10.1057/palgrave.jors.2601969.

40. Portela, M.C.A.S, & Thanassoulis, E. (2006). Zero weights and non-zero slacks: Different solutions to the same problem. Annals of Operations Research, 145, 129-147. https://doi.org/10.1007/s10479-006-0029-4.

41. Räty, T. (2002). Efficient facet based efficiency index: A variable returns to scale specification. Journal of Productivity Analysis 17, 65-82.
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42. Ravanos, P., & Karagiannis, G. (2021). A VEA Benefit-of-the-Doubt Model for the HDI. Social Indicators Research, 155, 27-46. https://doi.org/10.1007/s11205020 -02589-0 .

43. Ravanos, P., & Karagiannis, G. (2022a). On VEA, production trade-offs and weight restrictions. Journal of the Operational Research Society, (in press). https://doi.org/10.1080/01605682.2022.2128909.

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45. See Olesen and Petersen (1996) for a formal definition of FDEFs and non-FDEFs and Olesen and Petersen (2015) for an interesting discussion on alternative definitions of efficient facets. Linear models that test for the existence of at least one FDEF in the CRS or VRS DEA models are given in Olesen and Petersen (2015). For a set of ℛ extreme-efficient DMUs, one may investigate whether these jointly generate at least one FDEF by identifying all the FDEFs of the DEA frontier. The CRS counterpart of model (6) is obtained by removing the free variable ζ.
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46. Talluri, S., & Yoon, K.P. (2000). A cone-ratio DEA approach for AMT justification. International Journal of Production Economics, 66, 119-129.

47. Thanassoulis, E., Portela, M.C.S, & Despíc, O. (2008). Data envelopment analysis: the mathematical programming approach to efficiency analysis. In: H.O. Fried, C.A.K. Lovell, & S.S. Schmidt (Eds.) The measurement of productive efficiency and productivity growth (pp. 251-420). Oxford University Press, New York. https://doi.org/10.1093/acprof:oso/9780195183528.003.0003.
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48. The CRS counterpart of model (15) is provided in Olesen and Petersen (2015, p. 168) and can be obtained by removing the free variable ?? and changing the right-hand side in the fourth constraint to ? − (? + ? − 1). As in most cases not all the DMUs in ? are jointly efficient, it frequently holds that ℛ? ⊂ ?. However, in rare cases there is a facet jointly generated by all the DMUs in ?, in which case ℛ? ≡ ? can be a valid choice.
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49. The CRS counterpart of model (8) is given in Olesen and Petersen (2003) and can be obtained by removing the free variable z. See their footnote 7 in which they noted that weights normal to facets that are not of full dimension could also be included in W.
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50. This CR-DEA variant is related to the Extended Facet (EXFA) efficiency model developed by Olesen and Petersen (1996, 2015), for evaluating efficiency relative to a technology spanned by FDEFs and the extensions of these facets; see their relations (6.27) and (6.40), and footnote 21 in Olesen and Petersen (2015, pp. 167-170).
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51. Thompson, R.G., Dharmapala, P.S., & Thrall, R.M. (1995). Linked-cone DEA profit ratios and technical efficiency with application to Illinois coal mines. International Journal of Production Economics, 39, 99-115.

52. Thompson, R.G., Dharmapala, P.S., Humphrey, D.B., Taylor, W.L. & Thrall, R.M. (1996). Computing DEA/AR efficiency and profit ratio measures with an illustrative bank application. Annals of Operations Research, 68, 301-327.
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53. Thompson, R.G., Dharmapala, P.S., Rothenberg, L.J., & Thrall, R.M. (1994). DEA ARs and CRs applied to worldwide major oil companies. Journal of Productivity Analysis, 5, 181-203. https://doi.org/10.1007/BF01073854.
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54. Zhu, J. (1996). DEA/AR analysis of the 1988-1989 performance of the Nanjing textiles corporation. Annals of Operations Research, 66, 311-335.
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