The determination of the least distance to the strongly efficient frontier in Data Envelopment Analysis oriented models: modelling and computational aspects

Aparicio, Juan and Cordero, Jose M. and Pastor, Jesús (2016): The determination of the least distance to the strongly efficient frontier in Data Envelopment Analysis oriented models: modelling and computational aspects.

Abstract

Determining the least distance to the efficient frontier for estimating technical inefficiency, with the consequent determination of closest targets, has been one of the relevant issues in recent Data Envelopment Analysis literature. This new paradigm contrasts with traditional approaches, which yield furthest targets. In this respect, some techniques have been proposed in order to implement the new paradigm. A group of these techniques is based on identifying all the efficient faces of the polyhedral production possibility set and, therefore, is associated with the resolution of a NP-hard problem. In contrast, a second group proposes different models and particular algorithms to solve the problem avoiding the explicit identification of all these faces. These techniques have been applied more or less successfully. Nonetheless, the new paradigm is still unsatisfactory and incomplete to a certain extent. One of these challenges is that related to measuring technical inefficiency in the context of oriented models, i.e., models that aim at changing inputs or outputs but not both. In this paper, we show that existing specific techniques for determining the least distance without identifying explicitly the frontier structure for graph measures, which change inputs and outputs at the same time, do not work for oriented models. Consequently, a new methodology for satisfactorily implementing these situations is proposed. Finally, the new approach is empirically checked by using a recent PISA database consisting of 902 schools.

• Item Type: MPRA Paper
• Language: English
• Keywords: Data Envelopment Analysis, least distance, oriented models
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; H - Public Economics > H5 - National Government Expenditures and Related Policies > H52 - Government Expenditures and Education

References

Agasisti, T. and Zoido, P. (2015). The Efficiency of Secondary Schools in an International Perspective: Preliminary Results from PISA 2012 (No. 117). OECD Publishing.

Amirteimoori, A. and Kordrostami, S. (2010). A Euclidean distance-based measure of efficiency in data envelopment analysis. Optimization 59: 985–996.

Ando, K., Kai, A., Maeda, Y. and Sekitani, K. (2012). Least distance based inefficiency measures on the Pareto-efficient frontier in DEA. Journal of the Operations Research Society of Japan 55: 73–91.

Aparicio, J., Ruiz, J.L. and Sirvent, I. (2007). Closest targets and minimum distance to the Pareto-efficient frontier in DEA. Journal of Productivity Analysis 28: 209–218.

Aparicio, J. and Pastor, J.T. (2013). A well-defined efficiency measure for dealing with closest targets in DEA. Applied Mathematics and Computation 219: 9142–9154.

Aparicio, J. and Pastor, J.T. (2014a). On how to properly calculate the Euclidean distance-based measure in DEA. Optimization 63(3): 421–432.

Aparicio, J. and Pastor, J.T. (2014b). Closest targets and strong monotonicity on the strongly efficient frontier in DEA. Omega 44: 51–57.

Baek, C. and Lee, J. (2009). The relevance of DEA benchmarking information and the least-distance measure. Mathematical and Computer Modelling 49: 265–275.

Banker, R.D., Charnes, A. and Cooper, W.W. (1984), Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis, Management Science 30: 1078–1092.

Beale, E.M.L. and Tomlin, J.A. (1970). Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In Proceedings of the Fifth International Conference on Operational Research, edited by Lawrence, J., pp 447-454. London: Tavistock Publications.

Briec, W. (1997) Minimum distance to the complement of a convex set: duality result. Journal of Optimization Theory and Applications 93: 301–319.

Briec, W. (1998). Hölder distance function and measurement of technical efficiency. Journal of Productivity Analysis 11: 111–131.

Briec, W and Lemaire, B. (1999). Technical efficiency and distance to a reverse convex set. European Journal of Operational Research 114:178–187.

Briec, W. and Lesourd, J.B. (1999) Metric distance function and profit: some duality results, Journal of Optimization Theory and Applications 101(1): 15-33.

Briec, W. and Leleu, H. (2003). Dual representations of non-parametric technologies and measurement of technical efficiency. Journal of Productivity Analysis 20: 71–96.

Chambers, R.G. and Mitchell, T. (2001) Homotheticity and Non-Radial Changes. Journal of Productivity Analysis 15: 31–19.

Charnes, A., Cooper, W.W. and Thrall, R.M. (1991) A structure for classifying and characterizing efficiency and inefficiency in data envelopment analysis. Journal of Productivity Analysis 2: 197–237.

Cherchye, L. and Van Puyenbroeck, T. (2001). A comment on multi-stage DEA methodology. Operations Research Letters 28: 93–98.

Coelli, T. (1998) A multi-stage methodology for the solution of orientated DEA models. Operations Research Letters 23: 143–149.

Cooper, W.W., Park, K.S. and Pastor, J.T. (1999) RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models, and Relations to Others Models and Measures in DEA, Journal of Productivity Analysis 11: 5-42.

Cooper, W.W., Seiford, L.M. and Tone, K. (2007) Data envelopment analysis: a comprehensive text with models, applications, references and DEA-solver software. Springer.

Crespo-Cebada, E., Pedraja-Chaparro, F. and Santín, D. (2014). Does school ownership matter? An unbiased efficiency comparison for regions of Spain. Journal of Productivity Analysis, 41(1), 153-172.

De Witte, K. and López-Torres, L. (2015). Efficiency in education. A review of literature and a way forward, Journal of Operational Research Society, forthcoming.

Färe, R., Grosskopf, S. and Lovell, C.A.K. (1985) The Measurement of Efficiency of Production, Kluwer Nijhof Publishing.

Frei, F.X. and Harker, P.T. (1999). Projections onto efficient frontiers: theoretical and computational extensions to DEA. Journal of Productivity Analysis 11: 275–300.

Fukuyama, H., Maeda, Y., Sekitani, K. and Shi, J. (2014a). Input-output substitutability and strongly monotonic p-norm least-distance DEA measures. European Journal of Operational Research 237: 997-1007.

Fukuyama, H., Masaki, H., Sekitani, K. and Shi, J. (2014b). Distance optimization approach to ratio-form efficiency measures in data envelopment analysis. Journal of Productivity Analysis 42: 175-186.

Fukuyama, H., Hougaard, J.L., Sekitani, K. and Shi, J. (2016). Efficiency measurement with a nonconvex free disposal hull technology. Journal of the Operational Research Society, 67: 9–19.

Gonzalez, E. and Alvarez, A. (2001). From efficiency measurement to efficiency improvement: the choice of a relevant benchmark. European Journal of Operational Research 133: 512–520.

Grifell-Tatje, E., Lovell, C.A.K. and Pastor, J.T. (1998) A Quasi-Malmquist Productivity Index. Journal of Productivity Analysis 10: 7–20.

Koopmans, T.C. (1951) Analysis of Production as an Efficient Combination of Activities, in T.C. Koopmans (Ed.), Activity Analysis of Production and Allocation, New York, John Wiley.

Lovell, C.A.K. and Pastor, J.T. (1995) Units Invariant and Translation Invariant DEA Models. Operations Research Letters 18: 147–151.

Lozano, S. and Villa, G. (2005). Determining a sequence of targets in DEA. Journal of Operational Research Society 56: 1439–1447.

Mislevy, R.J., Beaton, A.E., Kaplan, B. and Sheehan, K.M. (1992). Estimating population characteristics from sparse matrix samples of item responses. Journal of Educational Measurement 29(2): 133–161.

OECD (2012), PISA 2009 Technical Report, PISA, OECD Publishing, Paris.

Pastor, J.T., Ruiz, J.L. and Sirvent, I. (1999) An Enhanced DEA Russell Graph Efficiency Measure, European Journal of Operational Research 115: 596–607.

Pastor, J.T. and Aparicio, J. (2010) The relevance of DEA benchmarking information and the Least-Distance Measure: Comment. Mathematical and Computer Modelling 52: 397–399.

Portela, M.C.A.S., Castro, P. and Thanassoulis, E. (2003). Finding closest targets in non-oriented DEA models: The case of convex and non-convex technologies. Journal of Productivity Analysis 19: 251–269.

Prieto, A.M. and Zofio, J.L. (2001) Evaluating Effectiveness in Public Provision of Infrastructure and Equipment: The Case of Spanish Municipalities. Journal of Productivity Analysis 15: 41–58.

Ruiz, J. L and Sirvent, I. (2010) A DEA approach to derive individual lower and upper bounds for the technical and allocative components of the overall profit efficiency. Journal of the Operational Research Society 62: 1907–1916.

Ruiz, J.L. and Sirvent, I. (2015) Common benchmarking and ranking of units with DEA. Omega (in press).

Ruiz, J.L., Segura, J.V. and Sirvent, I. (2015) Benchmarking and target setting with expert preferences: An application to the evaluation of educational performance of Spanish universities. European Journal of Operational Research 242: 594–605.

Santín, D. and Sicilia, G. (2015). Measuring the efficiency of public schools in Uruguay: main drivers and policy implications. Latin American Economic Review, 24(1), 1-28.

Shi, C., Zhang, G., Lu, J. and Zhou, H. (2006) An extended branch and bound algorithm for linear bilevel programming. Applied Mathematics and Computation 180: 529–537.

Tone, K. (2001) A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research 130: 498–509.

Wu, D.D. (2010) Bilevel programming Data Envelopment Analysis with constrained resource, European Journal of Operational Research 207: 856–864.

Wu, M. (2005). The role of plausible values in large-scale surveys. Studies in Educational Evaluation 31(2): 114–128.

Wu, J., Chu, J.-F. and Liang, L. (2015) Target setting and allocation of carbon emissions abatement based on DEA and closest target: an application to 20 APEC economies. Nat Hazards (in press).