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A slacks-based model for dynamic data envelopment analysis

  • * Corresponding author: Mohammad Afzalinejad

    * Corresponding author: Mohammad Afzalinejad 
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  • Dynamic Data Envelopment Analysis (DDEA) deals with efficiency analysis of decision making units in time dependent situations. A finite number of time periods and some carry-over activities between each two consecutive periods are assumed in DDEA. There are many models in DEA for efficiency evaluation of decision making units over time periods. One important class of dynamic models is the class of slacks-based models. By using a numerical example we show that some slacks-based DDEA models, especially ones proposed by Tone and Tsutsui, suffer from efficiency overestimation. A new dynamic slacks-based DEA model is proposed to overcome the deficiencies of the available slacks-based models. The model proposed in this paper is capable of revealing all sources of inefficiencies and providing more discrimination between decision making units. The theoretical and practical examinations demonstrate the merits of the new model.

    Mathematics Subject Classification: Primary: 90C05, 90B50; Secondary: 91B06.

    Citation:

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  • Figure 1.  $DMU_{b}$ is dominated by $DMU_{a}$ and is not efficient.

    Figure 2.  Comparison of rank order (input-oriented).

    Table 1.  Data of 10 bank branches

    DMUs Average monthly salaries Operating expense Total loans Net profit Loan Losses
    t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3 t=1 t=2 t=3
    DMU1 2.828 2.705 3.775 27.55 35.25 50.43 40.01 49.85 54.38 57.95 58.85 66.64 12.41 7.88 7.4
    DMU2 5.667 5.825 7.657 84.5 122 105.5 282.9 297.6 322.5 94.18 87.29 111.6 41.34 34.95 28.64
    DMU3 6.23 6.32 8.899 183.6 159.5 170.8 184.5 191.4 188.4 103.7 120.5 121.6 28.44 22.71 21.41
    DMU4 5.577 5.532 7.552 122.7 94.48 94.97 195.9 200.5 202.2 58.98 58.42 58.25 22.8 25.68 26.69
    DMU5 3.864 4.526 5.72 57.19 38.43 40.27 106.2 102.9 98.36 32.41 42.5 48.91 8.51 6.25 8.93
    DMU6 4.696 4.601 6.196 72.07 2.64 3.41 175.5 176.1 190.7 60.7 58.88 47.68 10.35 11.89 10.22
    DMU7 3.582 3.108 4.221 21.83 21.3 29.76 21.56 24.38 28.28 18.68 19.17 19.42 1.91 1.24 2.02
    DMU8 5.395 5.522 7.139 63.85 56.14 49 133 147.1 156.8 76.77 99.79 100.9 30.49 21.06 18.07
    DMU9 7.761 7.522 10.746 27.93 34.4 31.14 872.9 815.4 803.3 314.7 312.8 31.21 80.96 119.5 115.5
    DMU10 3.748 3.593 5.138 59.99 96.5 60.43 113.7 121.6 122.9 72.64 84.51 81.45 7.33 3.28 13.53
     | Show Table
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    Table 2.  Comparison of the efficiency scores resulted from Tone and Tsutsui's model and the proposed model.

    Overall input-oriented efficiency Overall output-oriented efficiency Non-oriented combined efficiency
    DMUs $ \theta^*_{o}(\text{TT})$ $ \theta^{*}_{o}$ $ \tau^{*}_{o}(\text{TT})$ $ \varphi^{*}_{o}$ Model (16) with TT objective: $ \varphi^{*}_{o}(\text{TT})$ $ \theta^{*}_{o}(\text{TT})\times\tau^{*}_{o}(TT)$ $ \theta^{*}_{o}\times\varphi^{*}_{o}$
    DMU1 0.9194 0.8792 0.6771 0.7587 0.6771 0.6225 0.6671
    DMU2 1 0.7040 1 0.8458 0.7492 1 0.5954
    DMU3 0.6521 0.4735 0.7618 0.7959 0.7230 0.4968 0.3761
    DMU4 0.5133 0.6773 0.5840 0.7117 0.5456 0.2998 0.4821
    DMU5 0.7648 0.7614 0.7916 0.8100 0.7286 0.6054 0.6168
    DMU6 1 1 1 1 1 1 1
    DMU7 0.8854 0.6421 0.9409 0.8979 0.8700 0.8331 0.5766
    DMU8 0.7765 0.7020 0.7482 0.7766 0.6841 0.5810 0.5451
    DMU9 1 1 1 1 1 1 1
    DMU10 1 0.5660 0.1 0.6979 0.9977 1 0.3950
     | Show Table
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    Table 3.  The input-oriented period efficiency scores resulted from the TT model and the proposed model

    DMUs Tone and Tsutsui's model The proposed model
    Period 1 efficiency Period 2 efficiency Period 3 efficiency Period 1 efficiency Period 2 efficiency Period 3 efficiency
    DMU1 0.7574 1 1 0.6364 1 1
    DMU2 1 1 1 0.2769 0.8169 1
    DMU3 0.5217 0.6987 0.7450 0.1962 0.4393 0.7247
    DMU4 0.3887 0.6334 0.5324 0.1802 0.8321 1
    DMU5 0.7047 0.8411 0.7477 0.5053 0.7744 1
    DMU6 1 1 1 1 1 1
    DMU7 0.6562 1 1 0.4236 0.5221 0.9991
    DMU8 0.4524 0.8718 1 0.2319 0.8498 1
    DMU9 1 1 1 1 1 1
    DMU10 1 1 1 0.2602 0.8350 0.5901
     | Show Table
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