Utilizing Monte Carlo Method for Ranking Extreme Efficient Units in Data Envelopment Analysis

Subject Areas : Statistics

Gh-R  .Jahanshahloo ¹ (Department of Mathematics, Kharazmi University, Tehran, Iran)
M.  Zahedi-Seresht ² (Department of Mathematics, Kharazmi University, Tehran, Iran Corresponding author)

Received: 2015-05-15 Accepted : 2015-12-16 Published : 2015-03-21

Keywords: تحلیل پوششی داده‌ها, بهره‌وری, رتبه‌بندی, شبیه‌سازی مونت کارلو, روش Cross-efficiency, روش Super-efficiency,

Abstract :

Data envelopment analysis (DEA) is a mathematical programming method for calculatingefficiency of decision making units (DMU). In calculating the efficiency score of unitsthrough DEA we may come up with some efficient units. But the question is among theseefficient units which of them is better. As we know, it is possible to rank inefficient unitsthrough efficiency score; however, for ranking efficient units it is not helpful and othermethods should be developed in these regards. To obviate this problem there have been somany attempts in the literature which have their pros and cons. Cross-efficiency method wasfirst introduced by Sexon et al. for ranking efficient units. The major problem of this methodis alternative optimal solutions in each model which must be solved for each DMU. Anotherproblem of this method is dependency of obtained solutions on the solution obtained by otherunits. Another method which has widely been used is super efficiency, presented byAnderson and Petersen. There are several flaws in their suggested method. Infeasibility,instability, dependency of the model on the input and output orientation and non-zero slackvariables are the weaknesses of this method which may occur in specific problems. Thisarticle is an attempt to present a method which does not have the aforementioned problemsand can be utilized to calculate the rank of extreme efficient units through using the Hit orMiss Monte Carlo method. At the end of the article some examples are made in order to showthe efficiency of the presented method.

References:

[1] Charnes A., Cooper W.W. , Rhodes E. , Measuring the efficiency of decision making units. European Journal of Operational Research, 2(1978), 429–441

[2] Banker R.D., Charnes A., Cooper W.W., Some models for estimating technical and scale efficiencies in data envelopment analysis, Management Science ,30(1984), 1078–1092.

[3] Abello J.M., Pardalos P.M., Resende M.G.C. , Handbook of massive data sets. Kluwer, Dordrecht(2002).

[4] Sexton T.R., Silkman R.H , Hogan A.J, Data Envelopment Analysis: Critique and Extensions. in Measuring Efficiency: An Assessment of Data Envelopment Analysis. San Francisco: JosseyBass,(1986),73-105.

[5] Sungmook Lim,Minimax and maximin formulations of cross-efficiency in DEA, Computers & Industrial Engineering 62(2012), 726-731.

[6] Andersen P., Petersen N.C., A procedure for ranking efficient units in data envelopment analysis, Management Science,39(10)(1993), 1261-1294.

[7] Mehrabian S. , Alirezaee M.R., Jahanshahloo G.R.,Acomplete efficiency ranking of decisionmaking units in data envelopment analysis. Comput. Optim. Appl. ,14(1999), 261–266.

[8] Saati M.S., Zarafat Angiz M., Jahanshahloo G.R.,A model for ranking decision making units in data envelopment analysis. Ricerca Operativa 31(97) (2001), 47–59.

[9] Jahanshahloo G.R., Hosseinzadeh Lotfi F., Zhiani Rezai H., F. Rezai Balf, UsingMont Carlo method for ranking efficient DMUs. Appl. Math. Comput. 162(1) (2005), 371–379.

[10] Jahanshahloo,G.R., Afzalinejad,M, A ranking method based on a full-inefficient frontier, Applied Mathematical Modelling,30(2005), 248–260.

[11] Jahanshahloo G.R., Hosseinzadeh Lotfi F.,, Shoja. N., Tohidi G., Razavian S.,Ranking using L1 norm in data envelopment analysis. Appl. Math. Comput. 153(1) (2004), 215–224 .

[12] Amirteimoori. A., Jahanshahloo G.R., Kordrostami S.,Ranking of decision making units in data envelopment analysis. A distance-based approach. Appl. Math. Comput. 171(2005), 122–135 .

[13] Jahanshahloo G.R, Sanei M., Hosseinzadeh Lotfi F., Shoja N.,Using the gradient line for ranking DMUs in DEA. Appl.Math. Comput. 151(1) (2004), 209–219 .