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December  2017, 7(4): 403-416. doi: 10.3934/naco.2017025

A new Monte Carlo based procedure for complete ranking efficient units in DEA models

1. 

Department of Mathematics, Kharazmi University, Tehran, Iran

2. 

Department of Econometrics, University of Economics, Prague, Prague, Czech Republic

* Corresponding author: jablon@vse.cz

Received  February 2017 Revised  July 2017 Published  October 2017

Fund Project: This paper was prepared at the occasion of The 12th International Conference on Industrial Engineering (ICIE 2016), Tehran, Iran, January 25-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Assoc. Prof. A. (Nima) Mirzazadeh, Kharazmi University, Tehran, Iran, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

Traditional data envelopment analysis (DEA) models split DMUs into two classes – namely efficient and inefficient. Due to the identical maximum efficiency scores of the efficient units, they cannot be ranked directly. That is why various models allowing the complete ranking of DMUs have been proposed in the past. Those models are based on different principles and have various advantages and disadvantages (infeasibility, alternative optimum, computational aspects, etc.). The method proposed in this paper uses the magnitude of the area under the efficient curve. In order to estimate this magnitude we suggest to use Monte Carlo simulation for the complete ranking originally efficient DMUs so as to overcome the problems arisen from other ranking methods and it is very simple, computationally. This method generates random weights for the inputs and outputs in the feasible region and finally derives probability the DMUs are efficient. The procedure proposed is illustrated by a numerical example and its results are compared with three of most important and popular methods for ranking efficient units (i.e. cross-efficiency evaluation, Andersen and Petersen super-efficiency model, and common set of weights method).

Citation: Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025
References:
[1]

P. Andersen and N. C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science, 39 (1993), 1261-1264.   Google Scholar

[2]

R. D. BankerA. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092.   Google Scholar

[3]

M. Carrillo and J. M. Jorge, A multiobjective DEA approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139.   Google Scholar

[4]

A. CharnesW. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[5]

Y. Chen, Ranking efficient units in DEA, Omega, 32 (2004), 213-217.   Google Scholar

[6]

Y. ChenJ. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625.   Google Scholar

[7]

Y. ChenH. Morita and J. Zhu, Context-dependent DEA with an application to Tokyo public libraries, International Journal of Information Technology & Decision Making, 4 (2005), 385-394.   Google Scholar

[8]

Y. Chen and L. Liang, Super-efficiency DEA in the presence of infeasibility: One model approach, European Journal of Operational Research, 212 (2011), 141-147.  doi: 10.1016/j.ejor.2011.01.022.  Google Scholar

[9]

W. D. CookY. Roll and A. Kazakov, A DEA model for measuring the relative efficiency of highway maintenance patrols, INFOR: Information Systems and Operational Research, 28 (1990), 113-124.   Google Scholar

[10]

R. H. GreenJ. R. Doyle and W. D. Cook, Preference voting and project ranking using DEA and cross-evaluation, European Journal of Operational Research, 90 (1996), 461-472.   Google Scholar

[11]

F. Hosseinzadeh LotfiA. A. NooraG. R. Jahanshahloo and M. Reshadi, One DEA ranking method based on applying aggregate units, Expert Systems with Applications, 38 (2011), 13468-13471.   Google Scholar

[12]

M. Izadikhah and R. Farzipoor Saen, A new data envelopment analysis method for ranking decision making units: an application in industrial parks, Expert Systems, 32 (2015), 596-608.   Google Scholar

[13]

J. Jablonsky, Multicriteria approaches for ranking of efficient units in DEA models, Central European Journal of Operations Research, 20 (2012), 435-449.  doi: 10.1007/s10100-011-0223-6.  Google Scholar

[14]

G. R. JahanshahlooF. Hosseinzadeh LotfiH. Zhiani Rezai and F. Rezai Balf, Using Monte Carlo method for ranking efficient DMUs, Applied Mathematics and Computation, 162 (2005), 371-379.  doi: 10.1016/j.amc.2003.12.139.  Google Scholar

[15]

G. R. JahanshahlooA. MemarianiF. H. Lotfi and H. Z. Rezai, A note on some of DEA models and finding efficiency and complete ranking using common set of weights, Applied Mathematics and Computation, 166 (2005), 265-281.  doi: 10.1016/j.amc.2004.04.088.  Google Scholar

[16]

G. R. JahanshahlooH. V. JuniorF. H. Lotfi and D. Akbarian, A new DEA ranking system based on changing the reference set, European Journal of Operational Research, 181 (2007), 331-337.   Google Scholar

[17]

Y. LiJ. XieM. Wang and L. Liang, Super efficiency evaluation using a common platform on a cooperative game, European Journal of Operational Research, 255 (2016), 884-892.  doi: 10.1016/j.ejor.2016.06.001.  Google Scholar

[18]

S. Lim, Minimax and maximin formulations of cross-efficiency in DEA, Computers & Industrial Engineering, 62 (2012), 726-731.   Google Scholar

[19]

S. LimK. W. Oh and J. Zhu, Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market, European Journal of Operational Research, 236 (2014), 361-368.  doi: 10.1016/j.ejor.2013.12.002.  Google Scholar

[20]

F. H. Liu and H. Hsuan Peng, Ranking of units on the DEA frontier with common weights, Computers & Operations Research, 35 (2008), 1624-1637.   Google Scholar

[21]

W.-M. Lu and S.-F. Lo, An interactive benchmark model ranking performers -Application to financial holding companies, Mathematical and Computer Modelling, 49 (2009), 172-179.  doi: 10.1016/j.mcm.2008.06.008.  Google Scholar

[22]

M. OralO. Kettani and P. Lang, A methodology for collective evaluation and selection of industrial R & D projects, Management Science, 37 (1991), 871-885.   Google Scholar

[23]

A. Oukil and G. R. Amin, Maximum appreciative cross-efficiency in DEA: A new ranking method, Computers & Industrial Engineering, 81 (2015), 14-21.   Google Scholar

[24]

C. ParkanJ. WangD. Wu and G. Wei, Data envelopment analysis based on maximin relative efficiency criterion, Computers & Operations Research, 39 (2012), 2478-2487.  doi: 10.1016/j.cor.2011.12.015.  Google Scholar

[25]

V. V. Podinovski, DEA models for the explicit maximisation of relative efficiency, European Journal of Operational Research, 131 (2001), 572-586.  doi: 10.1016/S0377-2217(00)00099-0.  Google Scholar

[26]

V. V. Podinovski and A. D. Athanassopoulos, Assessing the relative efficiency of decision making units using DEA models with weight restrictions, Journal of the Operational Research Society, 49 (1998), 500. doi: 10.1016/j.ejor.2016.04.035.  Google Scholar

[27]

S. Ramezani-TarkhoraniM. KhodabakhshiS. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in DEA, Applied Mathematical Modelling, 38 (2014), 3890-3890.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar

[28]

J. L. Ruiz and I. Sirvent, On the DEA total weight flexibility and the aggregation in cross-efficiency evaluations, European Journal of Operational Research, 223 (2012), 732-738.  doi: 10.1016/j.ejor.2012.06.011.  Google Scholar

[29]

S. J. SadjadiH. OmraniS. AbdollahzadehM. Alinaghian and H. Mohammadi, A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran, Expert Systems with Applications, 38 (2011), 10875-10881.   Google Scholar

[30]

L. M. Seiford and J. Zhu, Context-dependent data envelopment analysis -Measuring attractiveness and progress, Omega, 31 (2003), 397-408.   Google Scholar

[31]

T. R. SextonR. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, (1986), 73-105.   Google Scholar

[32]

M. Soltanifar and F. Hosseinzadeh Lotfi, The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis, Computers & Industrial Engineering, 60 (2011), 585-592.   Google Scholar

[33]

J. SunJ. Wu and D. Guo, Performance ranking of units considering ideal and anti-ideal DMU with common weights, Applied Mathematical Modelling, 37 (2013), 6301-6310.  doi: 10.1016/j.apm.2013.01.010.  Google Scholar

[34]

R. M. Thrall, Duality, classification and slacks in DEA, Annals of Operations Research, 66 (1996), 109-138.  doi: 10.1007/BF02187297.  Google Scholar

[35]

K. Tone, A slacks-based measure of super-efficiency in data envelopment analysis, European Journal of Operational Research, 143 (2002), 32-41.  doi: 10.1016/S0377-2217(01)00324-1.  Google Scholar

[36]

Y. M. WangK. S. Chin and J. B. Yang, Measuring the performances of decision-making units using geometric average efficiency, Journal of the Operational Research Society, 58 (2007), 929-937.  doi: 10.1016/j.cam.2005.12.025.  Google Scholar

[37]

Y.-M. Wang and K.-S. Chin, Some alternative models for DEA cross-efficiency evaluation, International Journal of Production Economics, 128 (2010), 332-338.   Google Scholar

[38]

Y.-M. Wang and P. Jiang, Alternative mixed integer linear programming models for identifying the most efficient decision making unit in data envelopment analysis, Computers & Industrial Engineering, 62 (2012), 546-553.   Google Scholar

[39]

M. Wang and Y. Li, Supplier evaluation based on Nash bargaining game model, Expert Systems with Applications, 41 (2014), 4181-4185.   Google Scholar

[40]

J. WuL. LiangF. Yang and H. Yan, Bargaining game model in the evaluation of decision making units, Expert Systems with Applications, 36 (2009), 4357-4362.   Google Scholar

[41]

J. WuJ. ChuQ. ZhuP. Yin and L. Liang, DEA cross-efficiency evaluation based on satisfaction degree: an application to technology selection, International Journal of Production Research, 54 (2016), 5990-6007.   Google Scholar

[42]

J. WuJ. ChuQ. ZhuP. Yin and L. Liang, Extended secondary goal models for weights selection in DEA cross-efficiency evaluation, Computers & Industrial Engineering, 93 (2016), 143-151.   Google Scholar

[43]

M. Zerafat AngizA. Mustafa and M. J. Kamali, Cross-ranking of decision making units in data envelopment analysis, Applied Mathematical Modelling, 37 (2013), 398-405.  doi: 10.1016/j.apm.2012.02.038.  Google Scholar

show all references

References:
[1]

P. Andersen and N. C. Petersen, A procedure for ranking efficient units in data envelopment analysis, Management Science, 39 (1993), 1261-1264.   Google Scholar

[2]

R. D. BankerA. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092.   Google Scholar

[3]

M. Carrillo and J. M. Jorge, A multiobjective DEA approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139.   Google Scholar

[4]

A. CharnesW. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[5]

Y. Chen, Ranking efficient units in DEA, Omega, 32 (2004), 213-217.   Google Scholar

[6]

Y. ChenJ. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625.   Google Scholar

[7]

Y. ChenH. Morita and J. Zhu, Context-dependent DEA with an application to Tokyo public libraries, International Journal of Information Technology & Decision Making, 4 (2005), 385-394.   Google Scholar

[8]

Y. Chen and L. Liang, Super-efficiency DEA in the presence of infeasibility: One model approach, European Journal of Operational Research, 212 (2011), 141-147.  doi: 10.1016/j.ejor.2011.01.022.  Google Scholar

[9]

W. D. CookY. Roll and A. Kazakov, A DEA model for measuring the relative efficiency of highway maintenance patrols, INFOR: Information Systems and Operational Research, 28 (1990), 113-124.   Google Scholar

[10]

R. H. GreenJ. R. Doyle and W. D. Cook, Preference voting and project ranking using DEA and cross-evaluation, European Journal of Operational Research, 90 (1996), 461-472.   Google Scholar

[11]

F. Hosseinzadeh LotfiA. A. NooraG. R. Jahanshahloo and M. Reshadi, One DEA ranking method based on applying aggregate units, Expert Systems with Applications, 38 (2011), 13468-13471.   Google Scholar

[12]

M. Izadikhah and R. Farzipoor Saen, A new data envelopment analysis method for ranking decision making units: an application in industrial parks, Expert Systems, 32 (2015), 596-608.   Google Scholar

[13]

J. Jablonsky, Multicriteria approaches for ranking of efficient units in DEA models, Central European Journal of Operations Research, 20 (2012), 435-449.  doi: 10.1007/s10100-011-0223-6.  Google Scholar

[14]

G. R. JahanshahlooF. Hosseinzadeh LotfiH. Zhiani Rezai and F. Rezai Balf, Using Monte Carlo method for ranking efficient DMUs, Applied Mathematics and Computation, 162 (2005), 371-379.  doi: 10.1016/j.amc.2003.12.139.  Google Scholar

[15]

G. R. JahanshahlooA. MemarianiF. H. Lotfi and H. Z. Rezai, A note on some of DEA models and finding efficiency and complete ranking using common set of weights, Applied Mathematics and Computation, 166 (2005), 265-281.  doi: 10.1016/j.amc.2004.04.088.  Google Scholar

[16]

G. R. JahanshahlooH. V. JuniorF. H. Lotfi and D. Akbarian, A new DEA ranking system based on changing the reference set, European Journal of Operational Research, 181 (2007), 331-337.   Google Scholar

[17]

Y. LiJ. XieM. Wang and L. Liang, Super efficiency evaluation using a common platform on a cooperative game, European Journal of Operational Research, 255 (2016), 884-892.  doi: 10.1016/j.ejor.2016.06.001.  Google Scholar

[18]

S. Lim, Minimax and maximin formulations of cross-efficiency in DEA, Computers & Industrial Engineering, 62 (2012), 726-731.   Google Scholar

[19]

S. LimK. W. Oh and J. Zhu, Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market, European Journal of Operational Research, 236 (2014), 361-368.  doi: 10.1016/j.ejor.2013.12.002.  Google Scholar

[20]

F. H. Liu and H. Hsuan Peng, Ranking of units on the DEA frontier with common weights, Computers & Operations Research, 35 (2008), 1624-1637.   Google Scholar

[21]

W.-M. Lu and S.-F. Lo, An interactive benchmark model ranking performers -Application to financial holding companies, Mathematical and Computer Modelling, 49 (2009), 172-179.  doi: 10.1016/j.mcm.2008.06.008.  Google Scholar

[22]

M. OralO. Kettani and P. Lang, A methodology for collective evaluation and selection of industrial R & D projects, Management Science, 37 (1991), 871-885.   Google Scholar

[23]

A. Oukil and G. R. Amin, Maximum appreciative cross-efficiency in DEA: A new ranking method, Computers & Industrial Engineering, 81 (2015), 14-21.   Google Scholar

[24]

C. ParkanJ. WangD. Wu and G. Wei, Data envelopment analysis based on maximin relative efficiency criterion, Computers & Operations Research, 39 (2012), 2478-2487.  doi: 10.1016/j.cor.2011.12.015.  Google Scholar

[25]

V. V. Podinovski, DEA models for the explicit maximisation of relative efficiency, European Journal of Operational Research, 131 (2001), 572-586.  doi: 10.1016/S0377-2217(00)00099-0.  Google Scholar

[26]

V. V. Podinovski and A. D. Athanassopoulos, Assessing the relative efficiency of decision making units using DEA models with weight restrictions, Journal of the Operational Research Society, 49 (1998), 500. doi: 10.1016/j.ejor.2016.04.035.  Google Scholar

[27]

S. Ramezani-TarkhoraniM. KhodabakhshiS. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in DEA, Applied Mathematical Modelling, 38 (2014), 3890-3890.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar

[28]

J. L. Ruiz and I. Sirvent, On the DEA total weight flexibility and the aggregation in cross-efficiency evaluations, European Journal of Operational Research, 223 (2012), 732-738.  doi: 10.1016/j.ejor.2012.06.011.  Google Scholar

[29]

S. J. SadjadiH. OmraniS. AbdollahzadehM. Alinaghian and H. Mohammadi, A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran, Expert Systems with Applications, 38 (2011), 10875-10881.   Google Scholar

[30]

L. M. Seiford and J. Zhu, Context-dependent data envelopment analysis -Measuring attractiveness and progress, Omega, 31 (2003), 397-408.   Google Scholar

[31]

T. R. SextonR. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, (1986), 73-105.   Google Scholar

[32]

M. Soltanifar and F. Hosseinzadeh Lotfi, The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis, Computers & Industrial Engineering, 60 (2011), 585-592.   Google Scholar

[33]

J. SunJ. Wu and D. Guo, Performance ranking of units considering ideal and anti-ideal DMU with common weights, Applied Mathematical Modelling, 37 (2013), 6301-6310.  doi: 10.1016/j.apm.2013.01.010.  Google Scholar

[34]

R. M. Thrall, Duality, classification and slacks in DEA, Annals of Operations Research, 66 (1996), 109-138.  doi: 10.1007/BF02187297.  Google Scholar

[35]

K. Tone, A slacks-based measure of super-efficiency in data envelopment analysis, European Journal of Operational Research, 143 (2002), 32-41.  doi: 10.1016/S0377-2217(01)00324-1.  Google Scholar

[36]

Y. M. WangK. S. Chin and J. B. Yang, Measuring the performances of decision-making units using geometric average efficiency, Journal of the Operational Research Society, 58 (2007), 929-937.  doi: 10.1016/j.cam.2005.12.025.  Google Scholar

[37]

Y.-M. Wang and K.-S. Chin, Some alternative models for DEA cross-efficiency evaluation, International Journal of Production Economics, 128 (2010), 332-338.   Google Scholar

[38]

Y.-M. Wang and P. Jiang, Alternative mixed integer linear programming models for identifying the most efficient decision making unit in data envelopment analysis, Computers & Industrial Engineering, 62 (2012), 546-553.   Google Scholar

[39]

M. Wang and Y. Li, Supplier evaluation based on Nash bargaining game model, Expert Systems with Applications, 41 (2014), 4181-4185.   Google Scholar

[40]

J. WuL. LiangF. Yang and H. Yan, Bargaining game model in the evaluation of decision making units, Expert Systems with Applications, 36 (2009), 4357-4362.   Google Scholar

[41]

J. WuJ. ChuQ. ZhuP. Yin and L. Liang, DEA cross-efficiency evaluation based on satisfaction degree: an application to technology selection, International Journal of Production Research, 54 (2016), 5990-6007.   Google Scholar

[42]

J. WuJ. ChuQ. ZhuP. Yin and L. Liang, Extended secondary goal models for weights selection in DEA cross-efficiency evaluation, Computers & Industrial Engineering, 93 (2016), 143-151.   Google Scholar

[43]

M. Zerafat AngizA. Mustafa and M. J. Kamali, Cross-ranking of decision making units in data envelopment analysis, Applied Mathematical Modelling, 37 (2013), 398-405.  doi: 10.1016/j.apm.2012.02.038.  Google Scholar

Figure 1.  Graphical representation of the hit or miss Monte Carlo method
Figure 2.  Relative efficiency scores of A, B, and C
Table 1.  Cross-efficiency matrix
$\mathbf{DMU_1}$ $\mathbf{DMU_2}$ $\dots$ $\mathbf{DMU_j}$$\dots$ $\mathbf{DMU_n}$
$\mathbf{DMU_1}$ $E_{11}$ $E_{12}$ $E_{1j}$$E_{1n}$
$\mathbf{DMU_2}$ $E_{21}$ $E_{22}$ $E_{2j}$ $E_{2n}$
$\mathbf{DMU_j}$ $E_{j1}$ $E_{j2}$ $E_{jj}$ $E_{jn}$
$\mathbf{DMU_n}$ $E_{n1}$ $E_{n2}$ $E_{nj}$ $E_{nn}$
$E_j^{CE}$ $\frac{1}{n} \sum_{d=1}^n E_{d1}$ $\frac{1}{n} \sum_{d=1}^n E_{d2}$ $\frac{1}{n} \sum_{d=1}^n E_{dj}$ $\frac{1}{n} \sum_{d=1}^n E_{dn}$
$\mathbf{DMU_1}$ $\mathbf{DMU_2}$ $\dots$ $\mathbf{DMU_j}$$\dots$ $\mathbf{DMU_n}$
$\mathbf{DMU_1}$ $E_{11}$ $E_{12}$ $E_{1j}$$E_{1n}$
$\mathbf{DMU_2}$ $E_{21}$ $E_{22}$ $E_{2j}$ $E_{2n}$
$\mathbf{DMU_j}$ $E_{j1}$ $E_{j2}$ $E_{jj}$ $E_{jn}$
$\mathbf{DMU_n}$ $E_{n1}$ $E_{n2}$ $E_{nj}$ $E_{nn}$
$E_j^{CE}$ $\frac{1}{n} \sum_{d=1}^n E_{d1}$ $\frac{1}{n} \sum_{d=1}^n E_{d2}$ $\frac{1}{n} \sum_{d=1}^n E_{dj}$ $\frac{1}{n} \sum_{d=1}^n E_{dn}$
Table 2.  Data set for illustrative example
$\mathbf{DMUs}$ A B C
Output $\mathbf{y}$111
Input $\mathbf{x_1}$146
Input $\mathbf{x_2}$622
$\mathbf{DMUs}$ A B C
Output $\mathbf{y}$111
Input $\mathbf{x_1}$146
Input $\mathbf{x_2}$622
Table 3.  Ranking DMUs by Super-Efficiency, Cross-Efficiency and CSW
DMU Efficiency Super- Rank Cross- Rank CSW Rank
score eff. score eff. score
A1410.5533.981
B11.320.7511.32
C1130.7220.733
DMU Efficiency Super- Rank Cross- Rank CSW Rank
score eff. score eff. score
A1410.5533.981
B11.320.7511.32
C1130.7220.733
Table 4.  Ranking by ATE with 1000/2000 trials
$\mathbf{DMUs}$ A B C
$N_H$804891729
$\Psi_i$0.8040.8910.729
Rank213
$N_H$160617521397
$\Psi_i$0.8030.8760.699
Rank213
$\mathbf{DMUs}$ A B C
$N_H$804891729
$\Psi_i$0.8040.8910.729
Rank213
$N_H$160617521397
$\Psi_i$0.8030.8760.699
Rank213
Table 5.  Normalized inputs and outputs for 20 DMUs
$\mathbf{DMU}$ $\mathbf{I_1}$ $\mathbf{I_2}$ $\mathbf{I_3}$ $\mathbf{O_1}$ $\mathbf{O_2}$ $\mathbf{O_3}$
$\mathbf{DMU_{1}}$0.9500.7000.1550.1900.5210.293
$\mathbf{DMU_{2}}$0.7960.6001.0000.2270.6270.462
$\mathbf{DMU_{3}}$0.7980.7500.5130.2280.9700.261
$\mathbf{DMU_{4}}$0.8650.5500.2100.1930.6321.000
$\mathbf{DMU_{5}}$0.8150.8500.2680.2330.7220.246
$\mathbf{DMU_{6}}$0.8420.6500.5000.2070.6030.569
$\mathbf{DMU_{7}}$0.7190.6000.3500.1820.9000.716
$\mathbf{DMU_{8}}$0.7850.7500.1200.1250.2340.298
$\mathbf{DMU_{9}}$0.4760.6000.1350.0800.3640.244
$\mathbf{DMU_{10}}$0.6780.5500.5100.0820.1840.049
$\mathbf{DMU_{11}}$0.7111.0000.3050.2120.3180.403
$\mathbf{DMU_{12}}$0.8110.6500.2550.1230.9230.628
$\mathbf{DMU_{13}}$0.6590.8500.3400.1760.6450.261
$\mathbf{DMU_{14}}$0.9760.8000.5400.1440.5140.243
$\mathbf{DMU_{15}}$0.6850.9500.4501.0000.2620.098
$\mathbf{DMU_{16}}$0.6130.9000.5250.1150.4020.464
$\mathbf{DMU_{17}}$1.0000.6000.2050.0901.0000.161
$\mathbf{DMU_{18}}$0.6340.6500.2350.0590.3490.680
$\mathbf{DMU_{19}}$0.3270.7000.2380.0390.1900.111
$\mathbf{DMU_{20}}$0.5830.5500.5000.1100.6150.764
$\mathbf{DMU}$ $\mathbf{I_1}$ $\mathbf{I_2}$ $\mathbf{I_3}$ $\mathbf{O_1}$ $\mathbf{O_2}$ $\mathbf{O_3}$
$\mathbf{DMU_{1}}$0.9500.7000.1550.1900.5210.293
$\mathbf{DMU_{2}}$0.7960.6001.0000.2270.6270.462
$\mathbf{DMU_{3}}$0.7980.7500.5130.2280.9700.261
$\mathbf{DMU_{4}}$0.8650.5500.2100.1930.6321.000
$\mathbf{DMU_{5}}$0.8150.8500.2680.2330.7220.246
$\mathbf{DMU_{6}}$0.8420.6500.5000.2070.6030.569
$\mathbf{DMU_{7}}$0.7190.6000.3500.1820.9000.716
$\mathbf{DMU_{8}}$0.7850.7500.1200.1250.2340.298
$\mathbf{DMU_{9}}$0.4760.6000.1350.0800.3640.244
$\mathbf{DMU_{10}}$0.6780.5500.5100.0820.1840.049
$\mathbf{DMU_{11}}$0.7111.0000.3050.2120.3180.403
$\mathbf{DMU_{12}}$0.8110.6500.2550.1230.9230.628
$\mathbf{DMU_{13}}$0.6590.8500.3400.1760.6450.261
$\mathbf{DMU_{14}}$0.9760.8000.5400.1440.5140.243
$\mathbf{DMU_{15}}$0.6850.9500.4501.0000.2620.098
$\mathbf{DMU_{16}}$0.6130.9000.5250.1150.4020.464
$\mathbf{DMU_{17}}$1.0000.6000.2050.0901.0000.161
$\mathbf{DMU_{18}}$0.6340.6500.2350.0590.3490.680
$\mathbf{DMU_{19}}$0.3270.7000.2380.0390.1900.111
$\mathbf{DMU_{20}}$0.5830.5500.5000.1100.6150.764
Table 6.  Computational results
$\mathbf{DMU}$ Eff. $N_H$ ATE $N_H$ ATE Super- Cross- CSW
score rank. rank. eff. eff. rank.
1000 1000 2000 2000 rank. rank.
trials trials trials trials
$\mathbf{DMU_{1}}$1.000448793877913
$\mathbf{DMU_{2}}$0.901999129
$\mathbf{DMU_{3}}$0.99188866
$\mathbf{DMU_{4}}$1.000921118471223
$\mathbf{DMU_{5}}$0.89710101088
$\mathbf{DMU_{6}}$0.748151514117
$\mathbf{DMU_{7}}$1.000875217512512
$\mathbf{DMU_{8}}$0.7971212121616
$\mathbf{DMU_{9}}$0.7871313131312
$\mathbf{DMU_{10}}$0.2892020202020
$\mathbf{DMU_{11}}$0.6041616161414
$\mathbf{DMU_{12}}$1.000793315603634
$\mathbf{DMU_{13}}$0.8161111111011
$\mathbf{DMU_{14}}$0.4691818181717
$\mathbf{DMU_{15}}$1.000574611626141
$\mathbf{DMU_{16}}$0.6391414151515
$\mathbf{DMU_{17}}$1.0005655115553510
$\mathbf{DMU_{18}}$0.4721717171818
$\mathbf{DMU_{19}}$0.4081919191919
$\mathbf{DMU_{20}}$1.000735414724475
$\mathbf{DMU}$ Eff. $N_H$ ATE $N_H$ ATE Super- Cross- CSW
score rank. rank. eff. eff. rank.
1000 1000 2000 2000 rank. rank.
trials trials trials trials
$\mathbf{DMU_{1}}$1.000448793877913
$\mathbf{DMU_{2}}$0.901999129
$\mathbf{DMU_{3}}$0.99188866
$\mathbf{DMU_{4}}$1.000921118471223
$\mathbf{DMU_{5}}$0.89710101088
$\mathbf{DMU_{6}}$0.748151514117
$\mathbf{DMU_{7}}$1.000875217512512
$\mathbf{DMU_{8}}$0.7971212121616
$\mathbf{DMU_{9}}$0.7871313131312
$\mathbf{DMU_{10}}$0.2892020202020
$\mathbf{DMU_{11}}$0.6041616161414
$\mathbf{DMU_{12}}$1.000793315603634
$\mathbf{DMU_{13}}$0.8161111111011
$\mathbf{DMU_{14}}$0.4691818181717
$\mathbf{DMU_{15}}$1.000574611626141
$\mathbf{DMU_{16}}$0.6391414151515
$\mathbf{DMU_{17}}$1.0005655115553510
$\mathbf{DMU_{18}}$0.4721717171818
$\mathbf{DMU_{19}}$0.4081919191919
$\mathbf{DMU_{20}}$1.000735414724475
Table 7.  Spearman rank-order correlation index
Models ATE Super-eff Cross-eff CSW
ATE10.9620.9430.859
Super-efficiency10.9250.865
Cross-efficiency10.937
CSW1
Models ATE Super-eff Cross-eff CSW
ATE10.9620.9430.859
Super-efficiency10.9250.865
Cross-efficiency10.937
CSW1
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