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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
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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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