Improvements to smooth data envelopment analysis
Brandão, Luana C.1 ; Soares de Mello, João Carlos C.B.2
1 Post Graduation in Production Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24.210-240 Niterói/RJ, Brazil;
2 Production Engineering Department, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24.210-240 Niterói/RJ, Brazil;
RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 157-171.

Classic data envelopment analysis (DEA) models do not provide unique solutions for multipliers of extreme efficient units. To overcome this problem, previous works have proposed the smooth DEA technique. However, multidimensional models with variable returns to scale (BCC) present deficiencies, as they do not fully ensure the frontier’s convexity. Therefore, the main contribution of this paper is to correct smooth BCC models, with regard to such property. Moreover, we propose improvements to smooth DEA models, so that all projections for the evaluated units, and their efficiency values, are non-negative. Furthermore, based on the corrected and improved smooth model, we propose a solution to avoid a classic BCC distortion, which may be called efficiency by default. Finally, we evaluate the operational performance of Brazilian airlines in 2010, to show the applicability of our model and to illustrate the practical effect of our contributions.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2016012
Classification : 90B50
Mots-clés : Smooth DEA, DEA properties, BCC distortion, efficiency by default, airlines efficiency
Brandão, Luana C. 1 ; Soares de Mello, João Carlos C.B. 2

1 Post Graduation in Production Engineering, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24.210-240 Niterói/RJ, Brazil;
2 Production Engineering Department, Universidade Federal Fluminense, Rua Passo da Pátria 156, 24.210-240 Niterói/RJ, Brazil; 
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Brandão, Luana C.; Soares de Mello, João Carlos C.B. Improvements to smooth data envelopment analysis. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 157-171. doi : 10.1051/ro/2016012. http://www.numdam.org/articles/10.1051/ro/2016012/

A.I. Ali, Streamlined computation for Data Envelopment Analysis. Eur. J. Oper. Res. 64 (1993) 61–67. | DOI | Zbl

P. Andersen and N.C. Petersen, A procedure for ranking efficient units in Data Envelopment Analysis. Manage. Sci. 39 (1993) 1261–1264. | DOI | Zbl

L. Angulo Meza, L. Biondi Neto, J.C.C.B. Soares De Mello and E.G. Gomes, ISYDS – Integrated System for Decision Support (SIAD): a Software Package for Data Envelopment Analysis Model. Pesq. Operac. 25 (2005) 493–503. | DOI

J.V.G. Avellar, A.Z. Milioni and T.N. Rabello, Spherical frontier DEA model based on a constant sum of inputs. J. Oper. Res. Soc. 58 (2007) 1246–1251. | DOI | Zbl

J.V.G. Avellar, A.Z. Milioni, T.N. Rabello, and H.P. Simão, On the redistribution of existing inputs using the spherical frontier DEA model. Pesquisa Oper. 30 (2010) 1–16. | DOI

R.D. Banker and R.M. Thrall, Estimation of returns-to-scale using Data Envelopment Analysis. Eur. J. Oper. Res. 62 (1992) 74–84. | DOI | Zbl

R.D. Banker, A. Charnes and W.W. Cooper, Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis. Manage. Sci. 30 (1984) 1078–1092. | DOI | Zbl

L.C. Brandao and J.C.C.B. Soares de Mello, Smooth Data Envelopment Analysis – a new model. Annals of XLVI SBPO, Salvador/BA, Brazil (2014) 921–931.

L.C. Brandao, F.V.S. Andrade and J.C.C.B. Soares de Mello, UEFA Euro efficiency evaluation based on market expectations. Proc. of 4th International Conference on Mathematics in Sport. KULeuven, Leuven (2013) 32–37.

A. Charnes, W.W. Cooper and E. Rhodes, Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2 (1978) 429–444. | DOI | MR | Zbl

A. Charnes, W.W. Cooper, B.E. Golany, L. Seiford and J. Stutz, Foundations of Data Envelopment Analysis for Pareto-Koopmans efficient empirical production functions. J. Econom. 30 (1985) 91–107. | DOI | MR | Zbl

T.J. Coelli, D.S. Prasada Rao, G.E. Battese, An Introduction to Efficiency and Productivity Analysis, 1st edition. Kluwer Academic Publishers, London (1998). | Zbl

W.W. Cooper, K.S. Park, J.T.P. Ciurana, Marginal rates and elasticities of substitution with additive models in DEA. J. Prod. Anal. 13 (2000) 105–123. | DOI

W.W. Cooper, J.L. Ruiz, I. Sirvent, Choosing weights from alternative optimal solutions of dual multiplier models in DEA. Eur. J. Oper. Res. 180 (2007) 443–458. | DOI | Zbl

T.C.V.D. Correia, J.C.C.B. Soares De Mello and L. Angulo Meza, Assessment of Brazilian airlines technical efficiency: a study using data envelopment analysis and fuzzy sets. Producao 21 (2011) 676–683.

D. Deprins, L. Simar and H. Tulkens, Measuring labour efficiency in post offices, in: The Performance of Public Enterprises: Concepts and Measurements, edited by M. Marchand, P. Petiseau and H. Tulkens. North-Holland, Amsterdam (1984) 243–267.

J.R. Doyle and R.H. Green, Efficiency and Cross-efficiency in DEA: Derivations, Meanings and Uses. Eur. J. Oper. Res. 45 (1994) 567–578. | Zbl

J.H. Dulá, Computations in DEA. Pesquisa Oper. 22 (2002) 165–182. | DOI

C.H. Edwards and D.E. Penney, Multivariable Calculus, 6th edition. Prentice Hall PTR (2002).

E.G. Gomes and M.P.E. Lins, Modelling undesirable outputs with zero sum gains data envelopment analysis models. J. Oper. Res. Soc. 59 (2008) 616–623. | DOI | Zbl

E.G. Gomes, J.C.C.B. Soares De Mello and M.P.E. Lins, Redistribuição de inputs e outputs em modelos de análise de envoltória de dados com ganhos de soma zero. Pesquisa Oper. 24 (2004) 269–284. | DOI

E.G. Gomes, U.G.P. Abreu, J.C.C.B. Soares de Mello, T.B. de Carvalho and S. de ZEN, DEA performance evaluation of livestock systems in Brazil. In: Data Envelopment Analysis and Its Applications to Management, edited by V. Charles and M. Kumar, 1st edition. Cambridge Scholars Publishing, Newcastle upon Tyne (2012) 224–238.

S.F. Gomes Júnior, A.P.S. Rubem, J.C.C.B. Soares De Mello and L. Angulo Meza, Evaluation of Brazilian airlines nonradial efficiencies and targets using an alternative DEA approach. Intl. Trans. Oper. Res. 00 (2015) 1–21.

M.C. Gonzalez-Araya, Projeções Não Radiais em regiões fortemente eficientes da fronteira DEA– Algoritmos e aplicações, Tese de Doutorado, COPPE/UFRJ, Rio de Janeiro (2003).

E.C. Guedes, A.Z. Milioni, J.V.G. Avellar and R.C. Silva, Adjusted spherical frontier model: allocating input via parametric DEA. J. Oper. Res. Soc. 63 (2012) 406–417. | DOI

L. Liang, J. Wu, W.D. Cook and J. Zhu, Alternative secondary goals in DEA cross-efficiency evaluation. Int. J. Prod. Econ. 113 (2008) 1025–1030. | DOI

L. Liang, J. Wu, W.D. Cook and J. Zhu, The DEA Game Cross-Efficiency Model and Its Nash Equilibrium. Oper. Res. 56 (2008) 1278–1288. | DOI | MR | Zbl

M.P.E. Lins, E.G. Gomes, J.C.C.B. Soares De Mello and A.J.R. Soares De Mello,Olympic ranking based on a Zero Sum Gains DEA model. Eur. J. Oper. Res. 148 (2003) 85–95. | Zbl

A.Z. Milioni and L.B. Alves, Ten years of research on parametric data envelopment analysis. Pesquisa Oper. 33 (2013) 89–104. | DOI

A.Z. Milioni, J.V.G. Avellar, E.G. Gomes and J.C.C.B. Soares De Mello, An ellipsoidal frontier model: allocating input via parametric DEA. Eur. J. Oper. Res. 209 (2011) 113–121. | DOI | Zbl

A.Z. Milioni, J.V.G. Avellar, T.N. Rabello and G.M. Freitas, Hyperbolic frontier model: a parametric DEA approach for the distribution of a total fixed input. J. Oper. Res. Soc. 62 (2011) 1029–1037. | DOI

F.B. Nacif, J.C.C.B. Soares De Mello and L. Angulo Meza, Choosing weights in optimal solutions for DEA-BCC models by means of a n-dimensional smooth frontier. Pesquisa Oper. 29 (2009) 623–642. | DOI

O.P. Olesen and N.C. Petersen, Indicators of ill-conditioned data sets and model misspecifications in Data Envelopment Analysis: an extended facet approach. Manage. Sci. 42 (1996) 205–219. | DOI | Zbl

E.R. Pereira and J.C.C.B. Soares De Mello, Smoothed frontier to determine a single set of weights in CCR models. Production 25 (2015) 585–597.

E.R. Pereira, M.C.C. Chaves and J.C.C.B. Soares de Mello, Evaluation of efficiency of Brazilian airlines using the MCDEA-TRIMAP model. Proc. of the 2nd Intl. Conf. on Oper. Res. and Enterprise Syst. Barcelona, Spain (2013) 213–221.

D. Rosen, C. Schaffnit and J. Paradi, Marginal Rates and Two-dimensional Level Curves in DEA. J. Prod. Anal. 9 (1998) 202–232. | DOI

R.C. Silva and A.Z. Milioni, The adjusted spherical frontier model with weight restrictions. Eur. J. Oper. Res. 220 (2012) 729–735. | DOI | MR | Zbl

J.Q. Silveira, J.C.C.B. Soares De Mello and L. Angulo-Meza, Brazilian airlines efficiency evaluation using a data envelopment analysis (DEA) and multiobjective linear programming hybrid model. Ingeniare 20 (2012) 331–342.

J.C.C.B. Soares De Mello, M.P.E. Lins and E.G. Gomes, Construction of a smoothed DEA Frontier. Pesquisa Oper. 22 (2002) 183–201. | DOI

J.C.C.B. Soares De Mello, E.G. Gomes, L. Biondi Neto and M.P.E. Lins, Suavização da fronteira DEA: o caso BCC tridimensional. Investig. Oper. 24 (2004) 89–107.

J.C.C.B. Soares De Mello, L. Angulo-Meza, E.G. Gomes and L. Biondi Neto, Curso de Análise de Envoltória de Dados. Annals of XXXVII SBPO, Gramado/RS, Brazil (2005) 2520–2547.

J.C.C.B. Soares De Mello, L. Angulo Meza and B.P. Branco Da Silva, A ranking for the Olympic Games with unitary input DEA models. IMA J. Manag. Math. 20 (2009) 201–211. | DOI | Zbl

J.C.C.B. Soares De Mello, L. Angulo Meza, J.Q. Silveira and E.G. Gomes, About negative efficiencies in cross evaluation BCC input oriented models. Eur. J. Oper. Res. 229 (2013) 732–737. | DOI

M. Schefzyck, Operational performance of airlines: an extension of traditional measurement paradigms. Strat. Manage. J. 14 (1993) 301–317. | DOI

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