American Institute of Mathematical Sciences

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

A slacks-based model for dynamic data envelopment analysis

Mohammad Afzalinejad , and  Zahra Abbasi

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

* Corresponding author: Mohammad Afzalinejad

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.

Keywords: Data envelopment analysis, dynamic efficiency, DDEA, slacks-based model, carry-over activity.
Mathematics Subject Classification: Primary: 90C05, 90B50; Secondary: 91B06.
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
References:
[1]

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.  Google Scholar

[2]

W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software, 2nd edition, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-45283-8.  Google Scholar

[3]

D. K. Despotis, D. Sotiros and G. Koronakos, A network DEA approach for series multi-stage processes, Omega, 61 (2016), 35-48.  doi: 10.1016/j.omega.2015.07.005.  Google Scholar

[4]

A. Emrouznejad and E. Thanassoulis, A mathematical model for dynamic efficiency using data envelopment analysis, Applied Mathematics and Computation, 160 (2005), 363-378.  doi: 10.1016/j.amc.2003.09.026.  Google Scholar

[5]

R. Färe, S. Grosskopf, C. A. K. Lovell and C. Pasurka, Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach, The Review of Economics and Statistics, 71 (1989), 90-98.  doi: 10.2307/1928055.  Google Scholar

[6]

R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Springer-Verlag, Netherlands, 1996. doi: 10.1007/978-94-009-1816-0.  Google Scholar

[7]

R. Färe, S. Grosskopf, M. Norris and Z. Zhang, Productivity growth, technical progress, and efficiency change in industrialized countries, The American Economic Review, 84 (1994), 66-83.   Google Scholar

[8]

H. Fukuyama and W. L. Weber, Measuring Japanese bank performance: A dynamic network DEA approach, Journal of Productivity Analysis, 44 (2015), 249-264.  doi: 10.1007/s11123-014-0403-1.  Google Scholar

[9]

S. Hung, D. He and W. Lu, Evaluating the dynamic performances of business groups from the carry-over perspective: A case study of Taiwan's semiconductor industry, Omega, 46 (2014), 1-10.  doi: 10.1016/j.omega.2014.01.003.  Google Scholar

[10]

C. Kao, Dynamic data envelopment analysis: A relational analysis, European Journal of Operational Research, 227 (2013), 325-330.  doi: 10.1016/j.ejor.2012.12.012.  Google Scholar

[11]

C. Kao and S. Hwang, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429.  doi: 10.1016/j.ejor.2006.11.041.  Google Scholar

[12]

G. A. Klopp, The Analysis of the Efficiency of Production System with Multiple Inputs and Outputs, Ph. D thesis, Industrial and System Engineering College, University of Illinois in Chicago, 1985. Google Scholar

[13]

L. Liang, W. D. Cook and J. Zhu, DEA models for two-stage processes: Game approach and efficiency decomposition, Naval Research Logistics, 55 (2008), 643-653.  doi: 10.1002/nav.20308.  Google Scholar

[14]

P. Moreno and S. Lozano, Super SBI Dynamic Network DEA approach to measuring efficiency in the provision of public services, International Transactions in Operational Research, 25 (2018), 715-735.  doi: 10.1111/itor.12257.  Google Scholar

[15]

J. Nemoto and M. Goto, Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies, Economic Letters, 64 (1999), 51-56.  doi: 10.1016/S0165-1765(99)00070-1.  Google Scholar

[16]

J. Nemoto and M. Goto, Measuring dynamic efficiency in production: an application of data envelopment analysis to Japanese electric utilities, Journal of Productivity Analysis, 19 (2003), 191-210.  doi: 10.1023/A:1022805500570.  Google Scholar

[17]

H. Omrani and E. Soltanzadeh, Dynamic DEA models with network structure: An application for Iranian airlines, Journal of Air Transport Management, 57 (2016), 52-61.  doi: 10.1016/j.jairtraman.2016.07.014.  Google Scholar

[18]

K. S. Park and K. Park, Measurement of multiperiod aggregative efficiency, European Journal of Operational Research, 193 (2009), 567-580.  doi: 10.1016/j.ejor.2007.11.028.  Google Scholar

[19]

H. Scheel, Undesirable outputs in efficiency valuations, European Journal of Operational Research, 132 (2001), 400-410.  doi: 10.1016/S0377-2217(00)00160-0.  Google Scholar

[20]

J. K. Sengupta, A dynamic efficiency model using data envelopment analysis, International Journal of Production Economics, 62 (1999), 209-218.  doi: 10.1016/S0925-5273(98)00244-8.  Google Scholar

[21]

M. Shafiee, M. Sangi and M. Ghaderi, Bank performance evaluation using dynamic DEA: A slacks-based measure approach, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-12.  doi: 10.5899/2013/dea-00026.  Google Scholar

[22]

M. Soleimani-damaneh, An effective computational attempt in DDEA, Applied Mathematical Modelling, 33 (2009), 3943-3948.  doi: 10.1016/j.apm.2009.01.013.  Google Scholar

[23]

T. Sueyoshi and K. Sekitani, Returns to scale in dynamic DEA, European Journal of Operational Research, 161 (2005), 536-544.  doi: 10.1016/j.ejor.2003.08.055.  Google Scholar

[24]

K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509.  doi: 10.1016/S0377-2217(99)00407-5.  Google Scholar

[25]

K. Tone and M. Tsutsui, Dynamic DEA: A slacks-based measure approach, Omega, 38 (2010), 145-156.  doi: 10.1016/j.omega.2009.07.003.  Google Scholar

[26]

K. Tone and M. Tsutsui, Dynamic DEA with network structure: A slacks-based measure approach, Omega, 42 (2014), 124-131.  doi: 10.1016/j.omega.2013.04.002.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software, 2nd edition, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-45283-8.  Google Scholar

[3]

D. K. Despotis, D. Sotiros and G. Koronakos, A network DEA approach for series multi-stage processes, Omega, 61 (2016), 35-48.  doi: 10.1016/j.omega.2015.07.005.  Google Scholar

[4]

A. Emrouznejad and E. Thanassoulis, A mathematical model for dynamic efficiency using data envelopment analysis, Applied Mathematics and Computation, 160 (2005), 363-378.  doi: 10.1016/j.amc.2003.09.026.  Google Scholar

[5]

R. Färe, S. Grosskopf, C. A. K. Lovell and C. Pasurka, Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach, The Review of Economics and Statistics, 71 (1989), 90-98.  doi: 10.2307/1928055.  Google Scholar

[6]

R. Färe and S. Grosskopf, Intertemporal Production Frontiers: With Dynamic DEA, Springer-Verlag, Netherlands, 1996. doi: 10.1007/978-94-009-1816-0.  Google Scholar

[7]

R. Färe, S. Grosskopf, M. Norris and Z. Zhang, Productivity growth, technical progress, and efficiency change in industrialized countries, The American Economic Review, 84 (1994), 66-83.   Google Scholar

[8]

H. Fukuyama and W. L. Weber, Measuring Japanese bank performance: A dynamic network DEA approach, Journal of Productivity Analysis, 44 (2015), 249-264.  doi: 10.1007/s11123-014-0403-1.  Google Scholar

[9]

S. Hung, D. He and W. Lu, Evaluating the dynamic performances of business groups from the carry-over perspective: A case study of Taiwan's semiconductor industry, Omega, 46 (2014), 1-10.  doi: 10.1016/j.omega.2014.01.003.  Google Scholar

[10]

C. Kao, Dynamic data envelopment analysis: A relational analysis, European Journal of Operational Research, 227 (2013), 325-330.  doi: 10.1016/j.ejor.2012.12.012.  Google Scholar

[11]

C. Kao and S. Hwang, Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan, European Journal of Operational Research, 185 (2008), 418-429.  doi: 10.1016/j.ejor.2006.11.041.  Google Scholar

[12]

G. A. Klopp, The Analysis of the Efficiency of Production System with Multiple Inputs and Outputs, Ph. D thesis, Industrial and System Engineering College, University of Illinois in Chicago, 1985. Google Scholar

[13]

L. Liang, W. D. Cook and J. Zhu, DEA models for two-stage processes: Game approach and efficiency decomposition, Naval Research Logistics, 55 (2008), 643-653.  doi: 10.1002/nav.20308.  Google Scholar

[14]

P. Moreno and S. Lozano, Super SBI Dynamic Network DEA approach to measuring efficiency in the provision of public services, International Transactions in Operational Research, 25 (2018), 715-735.  doi: 10.1111/itor.12257.  Google Scholar

[15]

J. Nemoto and M. Goto, Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies, Economic Letters, 64 (1999), 51-56.  doi: 10.1016/S0165-1765(99)00070-1.  Google Scholar

[16]

J. Nemoto and M. Goto, Measuring dynamic efficiency in production: an application of data envelopment analysis to Japanese electric utilities, Journal of Productivity Analysis, 19 (2003), 191-210.  doi: 10.1023/A:1022805500570.  Google Scholar

[17]

H. Omrani and E. Soltanzadeh, Dynamic DEA models with network structure: An application for Iranian airlines, Journal of Air Transport Management, 57 (2016), 52-61.  doi: 10.1016/j.jairtraman.2016.07.014.  Google Scholar

[18]

K. S. Park and K. Park, Measurement of multiperiod aggregative efficiency, European Journal of Operational Research, 193 (2009), 567-580.  doi: 10.1016/j.ejor.2007.11.028.  Google Scholar

[19]

H. Scheel, Undesirable outputs in efficiency valuations, European Journal of Operational Research, 132 (2001), 400-410.  doi: 10.1016/S0377-2217(00)00160-0.  Google Scholar

[20]

J. K. Sengupta, A dynamic efficiency model using data envelopment analysis, International Journal of Production Economics, 62 (1999), 209-218.  doi: 10.1016/S0925-5273(98)00244-8.  Google Scholar

[21]

M. Shafiee, M. Sangi and M. Ghaderi, Bank performance evaluation using dynamic DEA: A slacks-based measure approach, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-12.  doi: 10.5899/2013/dea-00026.  Google Scholar

[22]

M. Soleimani-damaneh, An effective computational attempt in DDEA, Applied Mathematical Modelling, 33 (2009), 3943-3948.  doi: 10.1016/j.apm.2009.01.013.  Google Scholar

[23]

T. Sueyoshi and K. Sekitani, Returns to scale in dynamic DEA, European Journal of Operational Research, 161 (2005), 536-544.  doi: 10.1016/j.ejor.2003.08.055.  Google Scholar

[24]

K. Tone, A slacks-based measure of efficiency in data envelopment analysis, European Journal of Operational Research, 130 (2001), 498-509.  doi: 10.1016/S0377-2217(99)00407-5.  Google Scholar

[25]

K. Tone and M. Tsutsui, Dynamic DEA: A slacks-based measure approach, Omega, 38 (2010), 145-156.  doi: 10.1016/j.omega.2009.07.003.  Google Scholar

[26]

K. Tone and M. Tsutsui, Dynamic DEA with network structure: A slacks-based measure approach, Omega, 42 (2014), 124-131.  doi: 10.1016/j.omega.2013.04.002.  Google Scholar

Figure 1.  $DMU_{b}$ is dominated by $DMU_{a}$ and is not efficient.
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Figure 2.  Comparison of rank order (input-oriented).
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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
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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
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
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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
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
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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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