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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. [2] R. D. Banker, A. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092. [3] M. Carrillo and J. M. Jorge, A multiobjective DEA approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139. [4] A. Charnes, W. 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. [5] Y. Chen, Ranking efficient units in DEA, Omega, 32 (2004), 213-217. [6] Y. Chen, J. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625. [7] Y. Chen, H. 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. [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. [9] W. D. Cook, Y. 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. [10] R. H. Green, J. 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. [11] F. Hosseinzadeh Lotfi, A. A. Noora, G. R. Jahanshahloo and M. Reshadi, One DEA ranking method based on applying aggregate units, Expert Systems with Applications, 38 (2011), 13468-13471. [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. [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. [14] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, H. 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. [15] G. R. Jahanshahloo, A. Memariani, F. 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. [16] G. R. Jahanshahloo, H. V. Junior, F. 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. [17] Y. Li, J. Xie, M. 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. [18] S. Lim, Minimax and maximin formulations of cross-efficiency in DEA, Computers & Industrial Engineering, 62 (2012), 726-731. [19] S. Lim, K. 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. [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. [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. [22] M. Oral, O. Kettani and P. Lang, A methodology for collective evaluation and selection of industrial R & D projects, Management Science, 37 (1991), 871-885. [23] A. Oukil and G. R. Amin, Maximum appreciative cross-efficiency in DEA: A new ranking method, Computers & Industrial Engineering, 81 (2015), 14-21. [24] C. Parkan, J. Wang, D. 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. [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. [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. [27] S. Ramezani-Tarkhorani, M. Khodabakhshi, S. 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. [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. [29] S. J. Sadjadi, H. Omrani, S. Abdollahzadeh, M. 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. [30] L. M. Seiford and J. Zhu, Context-dependent data envelopment analysis -Measuring attractiveness and progress, Omega, 31 (2003), 397-408. [31] T. R. Sexton, R. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, (1986), 73-105. [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. [33] J. Sun, J. 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. [34] R. M. Thrall, Duality, classification and slacks in DEA, Annals of Operations Research, 66 (1996), 109-138. doi: 10.1007/BF02187297. [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. [36] Y. M. Wang, K. 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. [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. [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. [39] M. Wang and Y. Li, Supplier evaluation based on Nash bargaining game model, Expert Systems with Applications, 41 (2014), 4181-4185. [40] J. Wu, L. Liang, F. Yang and H. Yan, Bargaining game model in the evaluation of decision making units, Expert Systems with Applications, 36 (2009), 4357-4362. [41] J. Wu, J. Chu, Q. Zhu, P. 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. [42] J. Wu, J. Chu, Q. Zhu, P. Yin and L. Liang, Extended secondary goal models for weights selection in DEA cross-efficiency evaluation, Computers & Industrial Engineering, 93 (2016), 143-151. [43] M. Zerafat Angiz, A. 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.

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. [2] R. D. Banker, A. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092. [3] M. Carrillo and J. M. Jorge, A multiobjective DEA approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139. [4] A. Charnes, W. 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. [5] Y. Chen, Ranking efficient units in DEA, Omega, 32 (2004), 213-217. [6] Y. Chen, J. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625. [7] Y. Chen, H. 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. [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. [9] W. D. Cook, Y. 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. [10] R. H. Green, J. 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. [11] F. Hosseinzadeh Lotfi, A. A. Noora, G. R. Jahanshahloo and M. Reshadi, One DEA ranking method based on applying aggregate units, Expert Systems with Applications, 38 (2011), 13468-13471. [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. [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. [14] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, H. 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. [15] G. R. Jahanshahloo, A. Memariani, F. 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. [16] G. R. Jahanshahloo, H. V. Junior, F. 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. [17] Y. Li, J. Xie, M. 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. [18] S. Lim, Minimax and maximin formulations of cross-efficiency in DEA, Computers & Industrial Engineering, 62 (2012), 726-731. [19] S. Lim, K. 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. [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. [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. [22] M. Oral, O. Kettani and P. Lang, A methodology for collective evaluation and selection of industrial R & D projects, Management Science, 37 (1991), 871-885. [23] A. Oukil and G. R. Amin, Maximum appreciative cross-efficiency in DEA: A new ranking method, Computers & Industrial Engineering, 81 (2015), 14-21. [24] C. Parkan, J. Wang, D. 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. [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. [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. [27] S. Ramezani-Tarkhorani, M. Khodabakhshi, S. 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. [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. [29] S. J. Sadjadi, H. Omrani, S. Abdollahzadeh, M. 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. [30] L. M. Seiford and J. Zhu, Context-dependent data envelopment analysis -Measuring attractiveness and progress, Omega, 31 (2003), 397-408. [31] T. R. Sexton, R. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, New Directions for Evaluation, (1986), 73-105. [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. [33] J. Sun, J. 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. [34] R. M. Thrall, Duality, classification and slacks in DEA, Annals of Operations Research, 66 (1996), 109-138. doi: 10.1007/BF02187297. [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. [36] Y. M. Wang, K. 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. [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. [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. [39] M. Wang and Y. Li, Supplier evaluation based on Nash bargaining game model, Expert Systems with Applications, 41 (2014), 4181-4185. [40] J. Wu, L. Liang, F. Yang and H. Yan, Bargaining game model in the evaluation of decision making units, Expert Systems with Applications, 36 (2009), 4357-4362. [41] J. Wu, J. Chu, Q. Zhu, P. 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. [42] J. Wu, J. Chu, Q. Zhu, P. Yin and L. Liang, Extended secondary goal models for weights selection in DEA cross-efficiency evaluation, Computers & Industrial Engineering, 93 (2016), 143-151. [43] M. Zerafat Angiz, A. 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.
Graphical representation of the hit or miss Monte Carlo method
Relative efficiency scores of A, B, and C
Cross-efficiency matrix
 $\mathbf{DMU_1}$ $\mathbf{DMU_2}$ $\dots$ $\mathbf{DMU_j}$ $\dots$ $\mathbf{DMU_n}$ $\mathbf{DMU_1}$ $E_{11}$ $E_{12}$ $E_{1j}$ $E_{1n}$ $\mathbf{DMU_2}$ $E_{21}$ $E_{22}$ $E_{2j}$ $E_{2n}$ ⋮ $\mathbf{DMU_j}$ $E_{j1}$ $E_{j2}$ $E_{jj}$ $E_{jn}$ ⋮ $\mathbf{DMU_n}$ $E_{n1}$ $E_{n2}$ $E_{nj}$ $E_{nn}$ $E_j^{CE}$ $\frac{1}{n} \sum_{d=1}^n E_{d1}$ $\frac{1}{n} \sum_{d=1}^n E_{d2}$ $\frac{1}{n} \sum_{d=1}^n E_{dj}$ $\frac{1}{n} \sum_{d=1}^n E_{dn}$
 $\mathbf{DMU_1}$ $\mathbf{DMU_2}$ $\dots$ $\mathbf{DMU_j}$ $\dots$ $\mathbf{DMU_n}$ $\mathbf{DMU_1}$ $E_{11}$ $E_{12}$ $E_{1j}$ $E_{1n}$ $\mathbf{DMU_2}$ $E_{21}$ $E_{22}$ $E_{2j}$ $E_{2n}$ ⋮ $\mathbf{DMU_j}$ $E_{j1}$ $E_{j2}$ $E_{jj}$ $E_{jn}$ ⋮ $\mathbf{DMU_n}$ $E_{n1}$ $E_{n2}$ $E_{nj}$ $E_{nn}$ $E_j^{CE}$ $\frac{1}{n} \sum_{d=1}^n E_{d1}$ $\frac{1}{n} \sum_{d=1}^n E_{d2}$ $\frac{1}{n} \sum_{d=1}^n E_{dj}$ $\frac{1}{n} \sum_{d=1}^n E_{dn}$
Data set for illustrative example
 $\mathbf{DMUs}$ A B C Output $\mathbf{y}$ 1 1 1 Input $\mathbf{x_1}$ 1 4 6 Input $\mathbf{x_2}$ 6 2 2
 $\mathbf{DMUs}$ A B C Output $\mathbf{y}$ 1 1 1 Input $\mathbf{x_1}$ 1 4 6 Input $\mathbf{x_2}$ 6 2 2
Ranking DMUs by Super-Efficiency, Cross-Efficiency and CSW
 DMU Efficiency Super- Rank Cross- Rank CSW Rank score eff. score eff. score A 1 4 1 0.55 3 3.98 1 B 1 1.3 2 0.75 1 1.3 2 C 1 1 3 0.72 2 0.73 3
 DMU Efficiency Super- Rank Cross- Rank CSW Rank score eff. score eff. score A 1 4 1 0.55 3 3.98 1 B 1 1.3 2 0.75 1 1.3 2 C 1 1 3 0.72 2 0.73 3
Ranking by ATE with 1000/2000 trials
 $\mathbf{DMUs}$ A B C $N_H$ 804 891 729 $\Psi_i$ 0.804 0.891 0.729 Rank 2 1 3 $N_H$ 1606 1752 1397 $\Psi_i$ 0.803 0.876 0.699 Rank 2 1 3
 $\mathbf{DMUs}$ A B C $N_H$ 804 891 729 $\Psi_i$ 0.804 0.891 0.729 Rank 2 1 3 $N_H$ 1606 1752 1397 $\Psi_i$ 0.803 0.876 0.699 Rank 2 1 3
Normalized inputs and outputs for 20 DMUs
 $\mathbf{DMU}$ $\mathbf{I_1}$ $\mathbf{I_2}$ $\mathbf{I_3}$ $\mathbf{O_1}$ $\mathbf{O_2}$ $\mathbf{O_3}$ $\mathbf{DMU_{1}}$ 0.950 0.700 0.155 0.190 0.521 0.293 $\mathbf{DMU_{2}}$ 0.796 0.600 1.000 0.227 0.627 0.462 $\mathbf{DMU_{3}}$ 0.798 0.750 0.513 0.228 0.970 0.261 $\mathbf{DMU_{4}}$ 0.865 0.550 0.210 0.193 0.632 1.000 $\mathbf{DMU_{5}}$ 0.815 0.850 0.268 0.233 0.722 0.246 $\mathbf{DMU_{6}}$ 0.842 0.650 0.500 0.207 0.603 0.569 $\mathbf{DMU_{7}}$ 0.719 0.600 0.350 0.182 0.900 0.716 $\mathbf{DMU_{8}}$ 0.785 0.750 0.120 0.125 0.234 0.298 $\mathbf{DMU_{9}}$ 0.476 0.600 0.135 0.080 0.364 0.244 $\mathbf{DMU_{10}}$ 0.678 0.550 0.510 0.082 0.184 0.049 $\mathbf{DMU_{11}}$ 0.711 1.000 0.305 0.212 0.318 0.403 $\mathbf{DMU_{12}}$ 0.811 0.650 0.255 0.123 0.923 0.628 $\mathbf{DMU_{13}}$ 0.659 0.850 0.340 0.176 0.645 0.261 $\mathbf{DMU_{14}}$ 0.976 0.800 0.540 0.144 0.514 0.243 $\mathbf{DMU_{15}}$ 0.685 0.950 0.450 1.000 0.262 0.098 $\mathbf{DMU_{16}}$ 0.613 0.900 0.525 0.115 0.402 0.464 $\mathbf{DMU_{17}}$ 1.000 0.600 0.205 0.090 1.000 0.161 $\mathbf{DMU_{18}}$ 0.634 0.650 0.235 0.059 0.349 0.680 $\mathbf{DMU_{19}}$ 0.327 0.700 0.238 0.039 0.190 0.111 $\mathbf{DMU_{20}}$ 0.583 0.550 0.500 0.110 0.615 0.764
 $\mathbf{DMU}$ $\mathbf{I_1}$ $\mathbf{I_2}$ $\mathbf{I_3}$ $\mathbf{O_1}$ $\mathbf{O_2}$ $\mathbf{O_3}$ $\mathbf{DMU_{1}}$ 0.950 0.700 0.155 0.190 0.521 0.293 $\mathbf{DMU_{2}}$ 0.796 0.600 1.000 0.227 0.627 0.462 $\mathbf{DMU_{3}}$ 0.798 0.750 0.513 0.228 0.970 0.261 $\mathbf{DMU_{4}}$ 0.865 0.550 0.210 0.193 0.632 1.000 $\mathbf{DMU_{5}}$ 0.815 0.850 0.268 0.233 0.722 0.246 $\mathbf{DMU_{6}}$ 0.842 0.650 0.500 0.207 0.603 0.569 $\mathbf{DMU_{7}}$ 0.719 0.600 0.350 0.182 0.900 0.716 $\mathbf{DMU_{8}}$ 0.785 0.750 0.120 0.125 0.234 0.298 $\mathbf{DMU_{9}}$ 0.476 0.600 0.135 0.080 0.364 0.244 $\mathbf{DMU_{10}}$ 0.678 0.550 0.510 0.082 0.184 0.049 $\mathbf{DMU_{11}}$ 0.711 1.000 0.305 0.212 0.318 0.403 $\mathbf{DMU_{12}}$ 0.811 0.650 0.255 0.123 0.923 0.628 $\mathbf{DMU_{13}}$ 0.659 0.850 0.340 0.176 0.645 0.261 $\mathbf{DMU_{14}}$ 0.976 0.800 0.540 0.144 0.514 0.243 $\mathbf{DMU_{15}}$ 0.685 0.950 0.450 1.000 0.262 0.098 $\mathbf{DMU_{16}}$ 0.613 0.900 0.525 0.115 0.402 0.464 $\mathbf{DMU_{17}}$ 1.000 0.600 0.205 0.090 1.000 0.161 $\mathbf{DMU_{18}}$ 0.634 0.650 0.235 0.059 0.349 0.680 $\mathbf{DMU_{19}}$ 0.327 0.700 0.238 0.039 0.190 0.111 $\mathbf{DMU_{20}}$ 0.583 0.550 0.500 0.110 0.615 0.764
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 $\mathbf{DMU_{1}}$ 1.000 448 7 938 7 7 9 13 $\mathbf{DMU_{2}}$ 0.901 9 9 9 12 9 $\mathbf{DMU_{3}}$ 0.991 8 8 8 6 6 $\mathbf{DMU_{4}}$ 1.000 921 1 1847 1 2 2 3 $\mathbf{DMU_{5}}$ 0.897 10 10 10 8 8 $\mathbf{DMU_{6}}$ 0.748 15 15 14 11 7 $\mathbf{DMU_{7}}$ 1.000 875 2 1751 2 5 1 2 $\mathbf{DMU_{8}}$ 0.797 12 12 12 16 16 $\mathbf{DMU_{9}}$ 0.787 13 13 13 13 12 $\mathbf{DMU_{10}}$ 0.289 20 20 20 20 20 $\mathbf{DMU_{11}}$ 0.604 16 16 16 14 14 $\mathbf{DMU_{12}}$ 1.000 793 3 1560 3 6 3 4 $\mathbf{DMU_{13}}$ 0.816 11 11 11 10 11 $\mathbf{DMU_{14}}$ 0.469 18 18 18 17 17 $\mathbf{DMU_{15}}$ 1.000 574 6 1162 6 1 4 1 $\mathbf{DMU_{16}}$ 0.639 14 14 15 15 15 $\mathbf{DMU_{17}}$ 1.000 565 5 1155 5 3 5 10 $\mathbf{DMU_{18}}$ 0.472 17 17 17 18 18 $\mathbf{DMU_{19}}$ 0.408 19 19 19 19 19 $\mathbf{DMU_{20}}$ 1.000 735 4 1472 4 4 7 5
 $\mathbf{DMU}$ 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 $\mathbf{DMU_{1}}$ 1.000 448 7 938 7 7 9 13 $\mathbf{DMU_{2}}$ 0.901 9 9 9 12 9 $\mathbf{DMU_{3}}$ 0.991 8 8 8 6 6 $\mathbf{DMU_{4}}$ 1.000 921 1 1847 1 2 2 3 $\mathbf{DMU_{5}}$ 0.897 10 10 10 8 8 $\mathbf{DMU_{6}}$ 0.748 15 15 14 11 7 $\mathbf{DMU_{7}}$ 1.000 875 2 1751 2 5 1 2 $\mathbf{DMU_{8}}$ 0.797 12 12 12 16 16 $\mathbf{DMU_{9}}$ 0.787 13 13 13 13 12 $\mathbf{DMU_{10}}$ 0.289 20 20 20 20 20 $\mathbf{DMU_{11}}$ 0.604 16 16 16 14 14 $\mathbf{DMU_{12}}$ 1.000 793 3 1560 3 6 3 4 $\mathbf{DMU_{13}}$ 0.816 11 11 11 10 11 $\mathbf{DMU_{14}}$ 0.469 18 18 18 17 17 $\mathbf{DMU_{15}}$ 1.000 574 6 1162 6 1 4 1 $\mathbf{DMU_{16}}$ 0.639 14 14 15 15 15 $\mathbf{DMU_{17}}$ 1.000 565 5 1155 5 3 5 10 $\mathbf{DMU_{18}}$ 0.472 17 17 17 18 18 $\mathbf{DMU_{19}}$ 0.408 19 19 19 19 19 $\mathbf{DMU_{20}}$ 1.000 735 4 1472 4 4 7 5
Spearman rank-order correlation index
 Models ATE Super-eff Cross-eff CSW ATE 1 0.962 0.943 0.859 Super-efficiency 1 0.925 0.865 Cross-efficiency 1 0.937 CSW 1
 Models ATE Super-eff Cross-eff CSW ATE 1 0.962 0.943 0.859 Super-efficiency 1 0.925 0.865 Cross-efficiency 1 0.937 CSW 1
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