# American Institute of Mathematical Sciences

January  2019, 15(1): 275-291. doi: 10.3934/jimo.2018043

## A slacks-based model for dynamic data envelopment analysis

 Department of Mathematics, Tafresh University, Tafresh, 3951879611, Iran

Received  May 2017 Revised  October 2017 Published  April 2018

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.

Citation: Mohammad Afzalinejad, Zahra Abbasi. A slacks-based model for dynamic data envelopment analysis. Journal of Industrial & Management Optimization, 2019, 15 (1) : 275-291. doi: 10.3934/jimo.2018043
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##### References:
$DMU_{b}$ is dominated by $DMU_{a}$ and is not efficient.
Comparison of rank order (input-oriented).
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
 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
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
 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
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
 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
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