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

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.
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  • 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).

    Mathematics Subject Classification: Primary: 90C05; Secondary: 90C90.

    Citation:

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  • 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}$
    $\mathbf{DMU_2}$ $E_{21}$ $E_{22}$ $E_{2j}$ $E_{2n}$
    $\mathbf{DMU_j}$ $E_{j1}$ $E_{j2}$ $E_{jj}$ $E_{jn}$
    $\mathbf{DMU_n}$ $E_{n1}$ $E_{n2}$ $E_{nj}$ $E_{nn}$
    $E_j^{CE}$ $\frac{1}{n} \sum_{d=1}^n E_{d1}$ $\frac{1}{n} \sum_{d=1}^n E_{d2}$ $\frac{1}{n} \sum_{d=1}^n E_{dj}$ $\frac{1}{n} \sum_{d=1}^n E_{dn}$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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