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doi: 10.3934/jimo.2021023
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Integrated dynamic interval data envelopment analysis in the presence of integer and negative data

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

* Corresponding author: Pooja Bansal

Received  October 2019 Revised  September 2020 Early access February 2021

Fund Project: The first author is supported by Council of Scientific Research (CSIR), India

The conventional data envelopment analysis (DEA) models presume that the values of input-output variables of the decision-making units (DMUs) are precisely known. However, some real-life situations can authoritatively mandate the data to vary in concrete fine-tuned ranges, which can include negative values and measures that are allowed to take integer values only. Our study proposes an integrated dynamic DEA model to accommodate interval-valued and integer-valued features that can take negative values. The proposed one-step model follows the directional distance function approach to determine the efficiency of DMUs over time in the presence of carryovers connecting the consecutive periods. We use the pessimistic and optimistic standpoints to evaluate the respective lower and upper bounds of the interval efficiency scores of the DMUs. We compare our proposed approach with a few relevant studies in the literature. We also validate our model on a synthetically generated dataset. Furthermore, we showcase the proposed procedure's applicability on a real dataset from 2014 to 2018 of airlines operating in India.

Citation: Pooja Bansal, Aparna Mehra. Integrated dynamic interval data envelopment analysis in the presence of integer and negative data. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021023
References:
[1]

M. Allahyar and M. Rostamy-Malkhalifeh, Negative data in data envelopment analysis: Efficiency analysis and estimating returns to scale, Computers & Industrial Engineering, 82 (2015), 78-81.  doi: 10.1016/j.cie.2015.01.022.  Google Scholar

[2]

H. AziziA. Amirteimoori and S. Kordrostami, A note on dual models of interval DEA and its extension to interval data, International Journal of Industrial Mathematics, 10 (2018), 111-126.   Google Scholar

[3]

H. Azizi and Y.-M. Wang, Improved DEA models for measuring interval efficiencies of decision-making units, Measurement, 46 (2013), 1325-1332.  doi: 10.1016/j.measurement.2012.11.050.  Google Scholar

[4]

Y. ChenJ. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625.  doi: 10.1016/j.omega.2012.06.006.  Google Scholar

[5]

G. ChengP. Zervopoulos and Z. Qian, A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis, European Journal of Operational Research, 225 (2013), 100-105.  doi: 10.1016/j.ejor.2012.09.031.  Google Scholar

[6]

W. W. CooperK. S. Park and G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management Science, 45 (1999), 597-607.  doi: 10.1287/mnsc.45.4.597.  Google Scholar

[7]

B. EbrahimiM. TavanaM. Rahmani and F. J. Santos-Arteaga, Efficiency measurement in data envelopment analysis in the presence of ordinal and interval data, Neural Computing and Applications, 30 (2018), 1971-1982.  doi: 10.1007/s00521-016-2826-2.  Google Scholar

[8]

A. EmrouznejadA. L. Anouze and E. Thanassoulis, A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA, European Journal of Operational Research, 200 (2010), 297-304.  doi: 10.1007/s10479-009-0639-8.  Google Scholar

[9]

A. EmrouznejadM. Rostamy-MalkhalifehA. Hatami-Marbini and M. Tavana, General and multiplicative non-parametric corporate performance models with interval ratio data, Applied Mathematical Modelling, 36 (2012), 5506-5514.  doi: 10.1016/j.apm.2011.12.040.  Google Scholar

[10]

T. EntaniY. Maeda and H. Tanaka, Dual models of interval DEA and its extension to interval data, European Journal of Operational Research, 136 (2002), 32-45.  doi: 10.1016/S0377-2217(01)00055-8.  Google Scholar

[11]

A. Esmaeilzadeh and A. Hadi-Vencheh, A super-efficiency model for measuring aggregative efficiency of multi-period production systems, Measurement, 46 (2013), 3988-3993.  doi: 10.1016/j.measurement.2013.07.023.  Google Scholar

[12]

R. Färe and S. Grosskopf, Network DEA, Socio-Economic Planning Sciences, 34 (2000), 35-49.   Google Scholar

[13]

R. Färe, S. Grosskopf and P. Roos, Malmquist productivity indexes: A survey of theory and practice, in Index Numbers: Essays in Honour of Sten Malmquist, Kluwer Academic Publishers, Boston, 1998,127–190. Google Scholar

[14]

R. Färe and S. Grosskopf, Intertemporal production frontiers: With dynamic dea, Journal of the Operational Research Society, 48 (1997), 656-656.   Google Scholar

[15]

I.-L. GuoH.-S. Lee and D. Lee, An integrated model for slack-based measure of super-efficiency in additive DEA, Omega, 67 (2017), 160-167.  doi: 10.1016/j.omega.2016.05.002.  Google Scholar

[16]

P. Guo, Fuzzy data envelopment analysis and its application to location problems, Information Sciences, 179 (2009), 820-829.  doi: 10.1016/j.ins.2008.11.003.  Google Scholar

[17]

A. Hatami-MarbiniA. Emrouznejad and P. J. Agrell, Interval data without sign restrictions in DEA, Applied Mathematical Modelling, 38 (2014), 2028-2036.  doi: 10.1016/j.apm.2013.10.027.  Google Scholar

[18]

A. Hatami-MarbiniA. Emrouznejad and M. Tavana, A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research, 214 (2011), 457-472.  doi: 10.1016/j.ejor.2011.02.001.  Google Scholar

[19]

G. R. Jahanshahloo and M. Piri, Data Envelopment Analysis (DEA) with integer and negative inputs and outputs, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-15.   Google Scholar

[20]

C. Kao and S.-T. Liu, Stochastic data envelopment analysis in measuring the efficiency of Taiwan commercial banks, European Journal of Operational Research, 196 (2009), 312-322.  doi: 10.1016/j.ejor.2008.02.023.  Google Scholar

[21]

R. Kazemi Matin and A. Emrouznejad, An integer-valued data envelopment analysis model with bounded outputs, International Transactions in Operational Research, 18 (2011), 741-749.  doi: 10.1111/j.1475-3995.2011.00828.x.  Google Scholar

[22]

K. Kerstens and I. Van de Woestyne, A note on a variant of radial measure capable of dealing with negative inputs and outputs in DEA, European Journal of Operational Research, 234 (2014), 341-342.  doi: 10.1016/j.ejor.2013.10.067.  Google Scholar

[23]

K. Khalili-DamghaniM. Tavana and E. Haji-Saami, A data envelopment analysis model with interval data and undesirable output for combined cycle power plant performance assessment, Expert Systems with Applications, 42 (2015), 760-773.  doi: 10.1016/j.eswa.2014.08.028.  Google Scholar

[24]

T. Kuosmanen and R. K. Matin, Theory of integer-valued data envelopment analysis, European Journal of Operational Research, 192 (2009), 658-667.  doi: 10.1016/j.ejor.2007.09.040.  Google Scholar

[25]

K. Li and M. Song, Green development performance in China: A metafrontier non-radial approach, Sustainability, 8 (2016), 219. doi: 10.3390/su8030219.  Google Scholar

[26]

L. Li, X. Lv, W. Xu, Z. Zhang and X. Rong, Dynamic super-efficiency interval data envelopment analysis, in Computer Science & Education (ICCSE), 10th International Conference, IEEE, 2015. Google Scholar

[27]

R. Lin and Z. Chen, Super-efficiency measurement under variable return to scale: An approach based on a new directional distance function, Journal of the Operational Research Society, 66 (2015), 1506-1510.  doi: 10.1057/jors.2014.118.  Google Scholar

[28]

R. Lin and Z. Chen, A directional distance based super-efficiency DEA model handling negative data, Journal of the Operational Research Society, 68 (2017), 1312-1322.  doi: 10.1057/s41274-016-0137-8.  Google Scholar

[29]

S. Lozano and G. Villa, Centralized DEA models with the possibility of downsizing, Journal of the Operational Research Society, 56 (2005), 357-364.  doi: 10.1057/palgrave.jors.2601838.  Google Scholar

[30]

S. Lozano and G. Villa, Data envelopment analysis of integer-valued inputs and outputs, Computers & Operations Research, 33 (2006), 3004-3014.  doi: 10.1016/j.cor.2005.02.031.  Google Scholar

[31]

F. B. MarizM. R. Almeida and D. Aloise, A review of dynamic data envelopment analysis: State of the art and applications, International Transactions in Operational Research, 25 (2018), 469-505.  doi: 10.1111/itor.12468.  Google Scholar

[32]

O. B. Olesen and N. C. Petersen, Stochastic data envelopment analysis–A review, European Journal of Operational Research, 251 (2016), 2-21.  doi: 10.1016/j.ejor.2015.07.058.  Google Scholar

[33]

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

[34]

M. C. A. S. PortelaE. Thanassoulis and G. Simpson, Negative data in DEA: A directional distance approach applied to bank branches, Journal of the Operational Research Society, 55 (2004), 1111-1121.  doi: 10.1057/palgrave.jors.2601768.  Google Scholar

[35]

R. R. Russell and W. Schworm, Technological inefficiency indexes: A binary taxonomy and a generic theorem, Journal of Productivity Analysis, 49 (2018), 17-23.   Google Scholar

[36]

L. M. Seiford and J. Zhu, Infeasibility of super-efficiency data envelopment analysis models, INFOR: Information Systems and Operational Research, 37 (1999), 174-187.  doi: 10.1080/03155986.1999.11732379.  Google Scholar

[37]

L. M. Seiford and J. Zhu, Modeling undesirable factors in efficiency evaluation, European Journal of Operational Research, 142 (2002), 16-20.  doi: 10.1016/S0377-2217(01)00293-4.  Google Scholar

[38]

J. K. Sengupta, Stochastic data envelopment analysis: A new approach, Applied Economics Letters, 5 (1998), 287-290.  doi: 10.1002/(SICI)1099-0747(199603)12:1<1::AID-ASM274>3.0.CO;2-Y.  Google Scholar

[39]

Y. G. SmirlisE. K. Maragos and D. K. Despotis, Data envelopment analysis with missing values: An interval DEA approach, Applied Mathematics and Computation, 177 (2006), 1-10.  doi: 10.1016/j.amc.2005.10.028.  Google Scholar

[40]

J. SunY. MiaoJ. WuL. Cui and R. Zhong, Improved interval DEA models with common weight, Kybernetika, 50 (2014), 774-785.  doi: 10.14736/kyb-2014-5-0774.  Google Scholar

[41]

Y. TanU. ShettyA. Diabat and T. P. M. Pakkala, Aggregate directional distance formulation of DEA with integer variables, Annals of Operations Research, 235 (2015), 741-756.  doi: 10.1007/s10479-015-1891-8.  Google Scholar

[42]

M. TolooN. Aghayi and M. Rostamy-Malkhalifeh, Measuring overall profit efficiency with interval data, Applied Mathematics and Computation, 201 (2008), 640-649.  doi: 10.1016/j.amc.2007.12.061.  Google Scholar

[43]

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

[44]

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

[45]

T. H. TranY. MaoP. NathanailP. O. Siebers and D. Robinson, Integrating slacks-based measure of efficiency and super-efficiency in data envelopment analysis, Omega, 85 (2019), 156-165.  doi: 10.1016/j.omega.2018.06.008.  Google Scholar

[46]

Y.-M. WangR. Greatbanks and J.-B. Yang, Interval efficiency assessment using data envelopment analysis, Fuzzy Sets and Systems, 153 (2005), 347-370.  doi: 10.1016/j.fss.2004.12.011.  Google Scholar

[47]

Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70.   Google Scholar

[48]

Z. YangD. K. J. Lin and A. Zhang, Interval-valued data prediction via regularized artificial neural network, Neurocomputing, 331 (2019), 336-345.  doi: 10.1016/j.neucom.2018.11.063.  Google Scholar

[49]

S.-H. Yu and C.-W. Hsu, A unified extension of super-efficiency in additive data envelopment analysis with integer-valued inputs and outputs: An application to a municipal bus system, Annals of Operations Research, 287 (2020), 515-535.  doi: 10.1007/s10479-019-03448-z.  Google Scholar

[50]

A. ZanellaA. S. Camanho and T. G. Dias, Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis, European Journal of Operational Research, 245 (2015), 517-530.  doi: 10.1016/j.ejor.2015.03.036.  Google Scholar

show all references

References:
[1]

M. Allahyar and M. Rostamy-Malkhalifeh, Negative data in data envelopment analysis: Efficiency analysis and estimating returns to scale, Computers & Industrial Engineering, 82 (2015), 78-81.  doi: 10.1016/j.cie.2015.01.022.  Google Scholar

[2]

H. AziziA. Amirteimoori and S. Kordrostami, A note on dual models of interval DEA and its extension to interval data, International Journal of Industrial Mathematics, 10 (2018), 111-126.   Google Scholar

[3]

H. Azizi and Y.-M. Wang, Improved DEA models for measuring interval efficiencies of decision-making units, Measurement, 46 (2013), 1325-1332.  doi: 10.1016/j.measurement.2012.11.050.  Google Scholar

[4]

Y. ChenJ. Du and J. Huo, Super-efficiency based on a modified directional distance function, Omega, 41 (2013), 621-625.  doi: 10.1016/j.omega.2012.06.006.  Google Scholar

[5]

G. ChengP. Zervopoulos and Z. Qian, A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis, European Journal of Operational Research, 225 (2013), 100-105.  doi: 10.1016/j.ejor.2012.09.031.  Google Scholar

[6]

W. W. CooperK. S. Park and G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management Science, 45 (1999), 597-607.  doi: 10.1287/mnsc.45.4.597.  Google Scholar

[7]

B. EbrahimiM. TavanaM. Rahmani and F. J. Santos-Arteaga, Efficiency measurement in data envelopment analysis in the presence of ordinal and interval data, Neural Computing and Applications, 30 (2018), 1971-1982.  doi: 10.1007/s00521-016-2826-2.  Google Scholar

[8]

A. EmrouznejadA. L. Anouze and E. Thanassoulis, A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA, European Journal of Operational Research, 200 (2010), 297-304.  doi: 10.1007/s10479-009-0639-8.  Google Scholar

[9]

A. EmrouznejadM. Rostamy-MalkhalifehA. Hatami-Marbini and M. Tavana, General and multiplicative non-parametric corporate performance models with interval ratio data, Applied Mathematical Modelling, 36 (2012), 5506-5514.  doi: 10.1016/j.apm.2011.12.040.  Google Scholar

[10]

T. EntaniY. Maeda and H. Tanaka, Dual models of interval DEA and its extension to interval data, European Journal of Operational Research, 136 (2002), 32-45.  doi: 10.1016/S0377-2217(01)00055-8.  Google Scholar

[11]

A. Esmaeilzadeh and A. Hadi-Vencheh, A super-efficiency model for measuring aggregative efficiency of multi-period production systems, Measurement, 46 (2013), 3988-3993.  doi: 10.1016/j.measurement.2013.07.023.  Google Scholar

[12]

R. Färe and S. Grosskopf, Network DEA, Socio-Economic Planning Sciences, 34 (2000), 35-49.   Google Scholar

[13]

R. Färe, S. Grosskopf and P. Roos, Malmquist productivity indexes: A survey of theory and practice, in Index Numbers: Essays in Honour of Sten Malmquist, Kluwer Academic Publishers, Boston, 1998,127–190. Google Scholar

[14]

R. Färe and S. Grosskopf, Intertemporal production frontiers: With dynamic dea, Journal of the Operational Research Society, 48 (1997), 656-656.   Google Scholar

[15]

I.-L. GuoH.-S. Lee and D. Lee, An integrated model for slack-based measure of super-efficiency in additive DEA, Omega, 67 (2017), 160-167.  doi: 10.1016/j.omega.2016.05.002.  Google Scholar

[16]

P. Guo, Fuzzy data envelopment analysis and its application to location problems, Information Sciences, 179 (2009), 820-829.  doi: 10.1016/j.ins.2008.11.003.  Google Scholar

[17]

A. Hatami-MarbiniA. Emrouznejad and P. J. Agrell, Interval data without sign restrictions in DEA, Applied Mathematical Modelling, 38 (2014), 2028-2036.  doi: 10.1016/j.apm.2013.10.027.  Google Scholar

[18]

A. Hatami-MarbiniA. Emrouznejad and M. Tavana, A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research, 214 (2011), 457-472.  doi: 10.1016/j.ejor.2011.02.001.  Google Scholar

[19]

G. R. Jahanshahloo and M. Piri, Data Envelopment Analysis (DEA) with integer and negative inputs and outputs, Journal of Data Envelopment Analysis and Decision Science, 2013 (2013), 1-15.   Google Scholar

[20]

C. Kao and S.-T. Liu, Stochastic data envelopment analysis in measuring the efficiency of Taiwan commercial banks, European Journal of Operational Research, 196 (2009), 312-322.  doi: 10.1016/j.ejor.2008.02.023.  Google Scholar

[21]

R. Kazemi Matin and A. Emrouznejad, An integer-valued data envelopment analysis model with bounded outputs, International Transactions in Operational Research, 18 (2011), 741-749.  doi: 10.1111/j.1475-3995.2011.00828.x.  Google Scholar

[22]

K. Kerstens and I. Van de Woestyne, A note on a variant of radial measure capable of dealing with negative inputs and outputs in DEA, European Journal of Operational Research, 234 (2014), 341-342.  doi: 10.1016/j.ejor.2013.10.067.  Google Scholar

[23]

K. Khalili-DamghaniM. Tavana and E. Haji-Saami, A data envelopment analysis model with interval data and undesirable output for combined cycle power plant performance assessment, Expert Systems with Applications, 42 (2015), 760-773.  doi: 10.1016/j.eswa.2014.08.028.  Google Scholar

[24]

T. Kuosmanen and R. K. Matin, Theory of integer-valued data envelopment analysis, European Journal of Operational Research, 192 (2009), 658-667.  doi: 10.1016/j.ejor.2007.09.040.  Google Scholar

[25]

K. Li and M. Song, Green development performance in China: A metafrontier non-radial approach, Sustainability, 8 (2016), 219. doi: 10.3390/su8030219.  Google Scholar

[26]

L. Li, X. Lv, W. Xu, Z. Zhang and X. Rong, Dynamic super-efficiency interval data envelopment analysis, in Computer Science & Education (ICCSE), 10th International Conference, IEEE, 2015. Google Scholar

[27]

R. Lin and Z. Chen, Super-efficiency measurement under variable return to scale: An approach based on a new directional distance function, Journal of the Operational Research Society, 66 (2015), 1506-1510.  doi: 10.1057/jors.2014.118.  Google Scholar

[28]

R. Lin and Z. Chen, A directional distance based super-efficiency DEA model handling negative data, Journal of the Operational Research Society, 68 (2017), 1312-1322.  doi: 10.1057/s41274-016-0137-8.  Google Scholar

[29]

S. Lozano and G. Villa, Centralized DEA models with the possibility of downsizing, Journal of the Operational Research Society, 56 (2005), 357-364.  doi: 10.1057/palgrave.jors.2601838.  Google Scholar

[30]

S. Lozano and G. Villa, Data envelopment analysis of integer-valued inputs and outputs, Computers & Operations Research, 33 (2006), 3004-3014.  doi: 10.1016/j.cor.2005.02.031.  Google Scholar

[31]

F. B. MarizM. R. Almeida and D. Aloise, A review of dynamic data envelopment analysis: State of the art and applications, International Transactions in Operational Research, 25 (2018), 469-505.  doi: 10.1111/itor.12468.  Google Scholar

[32]

O. B. Olesen and N. C. Petersen, Stochastic data envelopment analysis–A review, European Journal of Operational Research, 251 (2016), 2-21.  doi: 10.1016/j.ejor.2015.07.058.  Google Scholar

[33]

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

[34]

M. C. A. S. PortelaE. Thanassoulis and G. Simpson, Negative data in DEA: A directional distance approach applied to bank branches, Journal of the Operational Research Society, 55 (2004), 1111-1121.  doi: 10.1057/palgrave.jors.2601768.  Google Scholar

[35]

R. R. Russell and W. Schworm, Technological inefficiency indexes: A binary taxonomy and a generic theorem, Journal of Productivity Analysis, 49 (2018), 17-23.   Google Scholar

[36]

L. M. Seiford and J. Zhu, Infeasibility of super-efficiency data envelopment analysis models, INFOR: Information Systems and Operational Research, 37 (1999), 174-187.  doi: 10.1080/03155986.1999.11732379.  Google Scholar

[37]

L. M. Seiford and J. Zhu, Modeling undesirable factors in efficiency evaluation, European Journal of Operational Research, 142 (2002), 16-20.  doi: 10.1016/S0377-2217(01)00293-4.  Google Scholar

[38]

J. K. Sengupta, Stochastic data envelopment analysis: A new approach, Applied Economics Letters, 5 (1998), 287-290.  doi: 10.1002/(SICI)1099-0747(199603)12:1<1::AID-ASM274>3.0.CO;2-Y.  Google Scholar

[39]

Y. G. SmirlisE. K. Maragos and D. K. Despotis, Data envelopment analysis with missing values: An interval DEA approach, Applied Mathematics and Computation, 177 (2006), 1-10.  doi: 10.1016/j.amc.2005.10.028.  Google Scholar

[40]

J. SunY. MiaoJ. WuL. Cui and R. Zhong, Improved interval DEA models with common weight, Kybernetika, 50 (2014), 774-785.  doi: 10.14736/kyb-2014-5-0774.  Google Scholar

[41]

Y. TanU. ShettyA. Diabat and T. P. M. Pakkala, Aggregate directional distance formulation of DEA with integer variables, Annals of Operations Research, 235 (2015), 741-756.  doi: 10.1007/s10479-015-1891-8.  Google Scholar

[42]

M. TolooN. Aghayi and M. Rostamy-Malkhalifeh, Measuring overall profit efficiency with interval data, Applied Mathematics and Computation, 201 (2008), 640-649.  doi: 10.1016/j.amc.2007.12.061.  Google Scholar

[43]

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

[44]

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

[45]

T. H. TranY. MaoP. NathanailP. O. Siebers and D. Robinson, Integrating slacks-based measure of efficiency and super-efficiency in data envelopment analysis, Omega, 85 (2019), 156-165.  doi: 10.1016/j.omega.2018.06.008.  Google Scholar

[46]

Y.-M. WangR. Greatbanks and J.-B. Yang, Interval efficiency assessment using data envelopment analysis, Fuzzy Sets and Systems, 153 (2005), 347-370.  doi: 10.1016/j.fss.2004.12.011.  Google Scholar

[47]

Z. S. Xu and Q. L. Da, Possibility degree method for ranking interval numbers and its application, Journal of Systems Engineering, 18 (2003), 67-70.   Google Scholar

[48]

Z. YangD. K. J. Lin and A. Zhang, Interval-valued data prediction via regularized artificial neural network, Neurocomputing, 331 (2019), 336-345.  doi: 10.1016/j.neucom.2018.11.063.  Google Scholar

[49]

S.-H. Yu and C.-W. Hsu, A unified extension of super-efficiency in additive data envelopment analysis with integer-valued inputs and outputs: An application to a municipal bus system, Annals of Operations Research, 287 (2020), 515-535.  doi: 10.1007/s10479-019-03448-z.  Google Scholar

[50]

A. ZanellaA. S. Camanho and T. G. Dias, Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis, European Journal of Operational Research, 245 (2015), 517-530.  doi: 10.1016/j.ejor.2015.03.036.  Google Scholar

Figure 1.  The dynamic framework for a DMU comprising of $ T $ periods with multiple carryovers connecting consecutive periods
2a and 2b describe the upper and lower bounds respectively of efficient frontiers for DMUs $ 1 $ (red) and $ 3 $ (blue); the output is allowed to take integer values in the interval">Figure 2.  2a and 2b describe the upper and lower bounds respectively of efficient frontiers for DMUs $ 1 $ (red) and $ 3 $ (blue); the output is allowed to take integer values in the interval
Figure 3.  The dynamic structure of an airline in two consecutive periods
Table 1.  Results from models $ (M1) $ and $ (M2) $ on the data of five DMUs when $ \ell = \ell' = 1 $ and $ k = k' = 3 $; the output is restricted to take only integer values in the intervals
DMU$ _j $ $ [x_{1j}^L, \, x_{1j}^U] $ $ [y_{1j}^L, \, y_{1j}^U] $ $ E_j^{U} $ $ E_j^{L} $
1 [1,2] [2,3] 1.0909 1.025
2 [2.5, 3] [3,4] 1.0545 0.9405
3 [4,5] [5,7] 1.5714 1.0435
4 [6,7] [2,5] 1 0.76
5 [4, 4.5] [1,3] 0.9643 0.8444
DMU$ _j $ $ [x_{1j}^L, \, x_{1j}^U] $ $ [y_{1j}^L, \, y_{1j}^U] $ $ E_j^{U} $ $ E_j^{L} $
1 [1,2] [2,3] 1.0909 1.025
2 [2.5, 3] [3,4] 1.0545 0.9405
3 [4,5] [5,7] 1.5714 1.0435
4 [6,7] [2,5] 1 0.76
5 [4, 4.5] [1,3] 0.9643 0.8444
Table 2.  Results from models $ (M1) $ and $ (M2) $, $ \ell = \ell' = 1 $, $ k = k' = 3 $, in columns 2 and 3; and the interval SORM model [17] in columns 5 and 6 for the dataset of 20 banks described in [17]. Column 4 represents the classification of DMUs into three classes by our approach, while column 7 presents the classification into strictly efficient class ($ E^{++} $), weekly efficient class ($ E^{+} $), and inefficient class ($ E^{-} $) defined in [17]
DMU$ _j $ $ E_j^{L} $ $ E_j^{U} $ class (our approach) $ \overline{E}_j $ $ \underline{E}_j $ class (by [17])
1 1.0063 1.0063 S 1 1 $ E^{++} $
2 0.9858 1.0116 E 0.813 1 $ E^{+} $
3 1.0006 1.0103 S 1 1 $ E^{++} $
4 0.9765 0.9889 IE 0.52 0.698 $ E^{-} $
5 1.0036 1.0084 S 1 1 $ E^{++} $
6 0.9678 0.9818 IE 0.427 0.614 $ E^{-} $
7 1.5072 1.5760 S 1 1 $ E^{++} $
8 0.9858 0.9896 IE 0.253 0.513 $ E^{-} $
9 1.0433 1.1944 S 1 1 $ E^{++} $
10 0.9616 0.9940 IE 0.718 0.901 $ E^{-} $
11 0.9984 1.0002 E 0.777 1 $ E^{+} $
12 0.9970 0.9990 IE 0.667 1 $ E^{+} $
13 0.9897 0.9939 IE 0.49 0.663 $ E^{-} $
14 0.9990 1.0042 E 0.929 1 $ E^{+} $
15 0.9940 1.0127 E 1 1 $ E^{++} $
16 0.9823 0.9928 IE 0.616 0.834 $ E^{-} $
17 0.9635 0.9823 IE 0.47 0.665 $ E^{-} $
18 0.9972 0.9990 IE 0.614 0.846 $ E^{-} $
19 0.9913 1.0051 E 0.866 1 $ E^{+} $
20 1.0066 1.0099 S 1 1 $ E^{++} $
DMU$ _j $ $ E_j^{L} $ $ E_j^{U} $ class (our approach) $ \overline{E}_j $ $ \underline{E}_j $ class (by [17])
1 1.0063 1.0063 S 1 1 $ E^{++} $
2 0.9858 1.0116 E 0.813 1 $ E^{+} $
3 1.0006 1.0103 S 1 1 $ E^{++} $
4 0.9765 0.9889 IE 0.52 0.698 $ E^{-} $
5 1.0036 1.0084 S 1 1 $ E^{++} $
6 0.9678 0.9818 IE 0.427 0.614 $ E^{-} $
7 1.5072 1.5760 S 1 1 $ E^{++} $
8 0.9858 0.9896 IE 0.253 0.513 $ E^{-} $
9 1.0433 1.1944 S 1 1 $ E^{++} $
10 0.9616 0.9940 IE 0.718 0.901 $ E^{-} $
11 0.9984 1.0002 E 0.777 1 $ E^{+} $
12 0.9970 0.9990 IE 0.667 1 $ E^{+} $
13 0.9897 0.9939 IE 0.49 0.663 $ E^{-} $
14 0.9990 1.0042 E 0.929 1 $ E^{+} $
15 0.9940 1.0127 E 1 1 $ E^{++} $
16 0.9823 0.9928 IE 0.616 0.834 $ E^{-} $
17 0.9635 0.9823 IE 0.47 0.665 $ E^{-} $
18 0.9972 0.9990 IE 0.614 0.846 $ E^{-} $
19 0.9913 1.0051 E 0.866 1 $ E^{+} $
20 1.0066 1.0099 S 1 1 $ E^{++} $
Table 3.  Results from models $ (M1) $ and $ (M2) $ with necessary modifications, and the InDEA model in [26] on the data described in [26]. Here, $ Et^{L} $ and $ Et^{U} $ denote the lower and the upper bounds of the efficiency interval obtained by our models at time $ t $, while these bounds are represented by $ q^{Lt} $ and $ q^{Ut}, $ respectively, for the models in [26]. The symbols $ Q, \;Q^{+} $ and $ Q^{++} $ indicate the inefficient, efficient but not super-efficient, and super-efficient DMUs, respectively, by the classification defined in [26]. The ranking of units within each class is carried out by the PD method and recorded for both our proposed models and te one in [26]
DMU$ _j $ ($ j\to $) 1 2 3 4 5
$ E1^{L} $ 1.0591 1.0275 0.9515 1.0108 0.9548
$ E1^{U} $ 1.1077 1.0833 1.0065 1.0108 0.9730
Class S S E S IE
Rank 1 2 4 3 5
$ q^{L1} $ 1.03 1.06 0.74 0.70 0.62
$ q^{U1} $ 2.02 2.04 1.04 0.87 0.85
Class $ Q^{++} $ $ Q^{++} $ $ Q^{+} $ $ Q $ $ Q $
Rank 1 2 3 4 5
$ E2^{L} $ 0.9527 1.0260 1.0559 1.0143 0.9547
$ E2^{U} $ 1.0065 1.0773 1.1046 1.0143 0.9766
Class E S S S IE
Rank 4 2 1 3 5
$ q^{L2} $ 0.75 1.05 1.03 0.71 0.75
$ q^{U2} $ 1.03 1.75 1.88 0.79 0.85
Class $ Q^{+} $ $ Q^{++} $ $ Q^{++} $ $ Q $ $ Q $
Rank 3 2 1 5 4
$ E3^{L} $ 1.0117 1.0291 0.9653 1.0675 0.9652
$ E3^{U} $ 1.0682 1.1144 1.0151 1.1123 1.0220
Class S S E S E
Rank 3 2 5 1 4
$ q^{L3} $ 0.70 1.10 0.73 1.04 0.76
$ q^{U3} $ 0.87 2.04 0.96 2.21 0.83
Class $ Q $ $ Q^{++} $ $ Q $ $ Q^{++} $ $ Q $
Rank 5 2 3 1 4
$ E4^{L} $ 1.0891 1.0818 0.9646 1.0225 0.9886
$ E4^{U} $ 1.1149 1.0818 1.0236 1.0685 1.0330
Class S S E S E
Rank 1 3 5 2 4
$ q^{L4} $ 0.89 0.68 0.73 1.05 0.81
$ q^{U4} $ 1.87 0.95 0.88 1.79 1.33
Class $ Q^{+} $ $ Q $ $ Q $ $ Q^{++} $ $ Q^{+} $
Rank 2 4 1 5 3
$ E5^{L} $ 0.9759 1.0442 0.9765 0.9424 0.9528
$ E5^{U} $ 1.0569 1.0745 1.0185 0.9867 0.9839
Class E S E IE IE
Rank 2 1 3 5 4
$ q^{L5} $ 0.95 1.12 0.78 0.71 0.72
$ q^{U5} $ 1.66 1.76 1.13 0.98 0.94
Class $ Q^{+} $ $ Q^{++} $ $ Q^{+} $ $ Q $ $ Q $
Rank 2 1 3 4 5
DMU$ _j $ ($ j\to $) 1 2 3 4 5
$ E1^{L} $ 1.0591 1.0275 0.9515 1.0108 0.9548
$ E1^{U} $ 1.1077 1.0833 1.0065 1.0108 0.9730
Class S S E S IE
Rank 1 2 4 3 5
$ q^{L1} $ 1.03 1.06 0.74 0.70 0.62
$ q^{U1} $ 2.02 2.04 1.04 0.87 0.85
Class $ Q^{++} $ $ Q^{++} $ $ Q^{+} $ $ Q $ $ Q $
Rank 1 2 3 4 5
$ E2^{L} $ 0.9527 1.0260 1.0559 1.0143 0.9547
$ E2^{U} $ 1.0065 1.0773 1.1046 1.0143 0.9766
Class E S S S IE
Rank 4 2 1 3 5
$ q^{L2} $ 0.75 1.05 1.03 0.71 0.75
$ q^{U2} $ 1.03 1.75 1.88 0.79 0.85
Class $ Q^{+} $ $ Q^{++} $ $ Q^{++} $ $ Q $ $ Q $
Rank 3 2 1 5 4
$ E3^{L} $ 1.0117 1.0291 0.9653 1.0675 0.9652
$ E3^{U} $ 1.0682 1.1144 1.0151 1.1123 1.0220
Class S S E S E
Rank 3 2 5 1 4
$ q^{L3} $ 0.70 1.10 0.73 1.04 0.76
$ q^{U3} $ 0.87 2.04 0.96 2.21 0.83
Class $ Q $ $ Q^{++} $ $ Q $ $ Q^{++} $ $ Q $
Rank 5 2 3 1 4
$ E4^{L} $ 1.0891 1.0818 0.9646 1.0225 0.9886
$ E4^{U} $ 1.1149 1.0818 1.0236 1.0685 1.0330
Class S S E S E
Rank 1 3 5 2 4
$ q^{L4} $ 0.89 0.68 0.73 1.05 0.81
$ q^{U4} $ 1.87 0.95 0.88 1.79 1.33
Class $ Q^{+} $ $ Q $ $ Q $ $ Q^{++} $ $ Q^{+} $
Rank 2 4 1 5 3
$ E5^{L} $ 0.9759 1.0442 0.9765 0.9424 0.9528
$ E5^{U} $ 1.0569 1.0745 1.0185 0.9867 0.9839
Class E S E IE IE
Rank 2 1 3 5 4
$ q^{L5} $ 0.95 1.12 0.78 0.71 0.72
$ q^{U5} $ 1.66 1.76 1.13 0.98 0.94
Class $ Q^{+} $ $ Q^{++} $ $ Q^{+} $ $ Q $ $ Q $
Rank 2 1 3 4 5
Table 4.  Result from models $ (M1) $ and $ (M2) $ with $ \ell = \, \ell' = \, 1 $ and $ k = \, k' = \, 3 $, on the two periods synthetic dataset of 30 DMUs given in Appendix B
DMU$ _j $ interval of efficiency class rank
1 [0.7897, 1.2083] E 28
2 [0.8055, 1.1376] E 25
3 [0.8545, 1.3422] E 6
4 [0.8891, 1.2386] E 7
5 [0.8299, 1.2265] E 21
6 [0.8856, 1.2316] E 9
7 [0.8397, 1.2817] E 15
8 [0.8396, 1.2332] E 16
9 [0.8852, 1.224] E 10
10 [0.7816, 1.2137] E 29
11 [0.8582, 1.2269] E 12
12 [0.7638, 1.1997] E 30
13 [0.8442, 1.2267] E 18
14 [0.8514, 1.1736] E 13
15 [0.8054, 1.1402] E 26
16 [0.8265, 1.2899] E 17
17 [0.842, 1.1696] E 19
18 [0.9258, 1.2041] E 5
19 [0.8057, 1.2573] E 23
20 [0.8351, 1.1974] E 14
21 [0.8808, 1.2 759] E 8
22 [0.8023, 1.2583] E 24
23 [0.9288, 1.2615] E 4
24 [0.8297, 1.2457] E 22
25 [0.7959, 1.1546] E 27
26 [0.875, 1.1831] E 11
27 [0.9845, 1.2724] E 3
28 [1.0122, 1.3216] S 1
29 [0.9875, 1.1374] E 2
30 [0.8106, 1.1771] E 20
DMU$ _j $ interval of efficiency class rank
1 [0.7897, 1.2083] E 28
2 [0.8055, 1.1376] E 25
3 [0.8545, 1.3422] E 6
4 [0.8891, 1.2386] E 7
5 [0.8299, 1.2265] E 21
6 [0.8856, 1.2316] E 9
7 [0.8397, 1.2817] E 15
8 [0.8396, 1.2332] E 16
9 [0.8852, 1.224] E 10
10 [0.7816, 1.2137] E 29
11 [0.8582, 1.2269] E 12
12 [0.7638, 1.1997] E 30
13 [0.8442, 1.2267] E 18
14 [0.8514, 1.1736] E 13
15 [0.8054, 1.1402] E 26
16 [0.8265, 1.2899] E 17
17 [0.842, 1.1696] E 19
18 [0.9258, 1.2041] E 5
19 [0.8057, 1.2573] E 23
20 [0.8351, 1.1974] E 14
21 [0.8808, 1.2 759] E 8
22 [0.8023, 1.2583] E 24
23 [0.9288, 1.2615] E 4
24 [0.8297, 1.2457] E 22
25 [0.7959, 1.1546] E 27
26 [0.875, 1.1831] E 11
27 [0.9845, 1.2724] E 3
28 [1.0122, 1.3216] S 1
29 [0.9875, 1.1374] E 2
30 [0.8106, 1.1771] E 20
Table 5.  Inputs, outputs, desirable and undesirable carryovers applied in the empirical analysis
Inputs Operating expenses (in millions INR)
Desirable carryover Fleet size (in number)
Undesirable carryover Losses carried forward after tax (in millions INR)
Outputs Operating revenue (in millions INR)
Pax load factor per month (in $ \% $)
Weight load factor per month (in $ \% $)
Passengers carried per month (in number)
Cargo carried per month (in tonne)
Inputs Operating expenses (in millions INR)
Desirable carryover Fleet size (in number)
Undesirable carryover Losses carried forward after tax (in millions INR)
Outputs Operating revenue (in millions INR)
Pax load factor per month (in $ \% $)
Weight load factor per month (in $ \% $)
Passengers carried per month (in number)
Cargo carried per month (in tonne)
Table 6.  Data of 11 Indian airlines for the period 2014-15
airline fleet size losses carried operating expenses operating revenue pax load factor weight load factor passengers carried cargo carried
1 101 59058.4 226854.4 206131.6 [73.7, 86.9] [66.7, 77.6] [990986,1217671] [7940,10098]
2 17 611.7 19597.6 22948.2 [56.3, 93.8] [45.4, 98.6] [8430,18674] [0, 32.8]
3 10 1789.2 3034 2279.5 [60.7, 72.2] [55.3, 65.6] [19612,38835] [13,19]
4 3 1333.1 2885 1551.9 [71.7, 84] [40.6, 53.2] [68790,181483] [450,1432]
5 5 -43.9 6310.4 6592 [0, 0] [66.1, 71.9] [0, 0] [9542,11858]
6 19 -363.7 28715.8 30664.3 [75.6, 89.4] [70.6, 82.4] [534795,639264] [3932,5485]
7 94 -16590.3 123578.6 139253.4 [76.8, 91.9] [68.1, 86.4] [2230645,2769283] [10300, 12303.8]
8 107 18137.1 215030.1 195606.1 [77.1, 89.5] [69.7, 78.6] [1186492,1393452] [6355,9124]
9 9 2876.5 16775.2 14229.4 [74.9, 89.7] [68.8, 79.8] [171551,294766] [915,1385]
10 34 6870.5 60885 52015.3 [80, 93.4] [75.3, 87.7] [552726,976517] [3202,6208]
11 6 1990.7 2681.9 691.3 [45.4, 77.6] [34.9, 66.6] [14999,158348] [0, 2151]
airline fleet size losses carried operating expenses operating revenue pax load factor weight load factor passengers carried cargo carried
1 101 59058.4 226854.4 206131.6 [73.7, 86.9] [66.7, 77.6] [990986,1217671] [7940,10098]
2 17 611.7 19597.6 22948.2 [56.3, 93.8] [45.4, 98.6] [8430,18674] [0, 32.8]
3 10 1789.2 3034 2279.5 [60.7, 72.2] [55.3, 65.6] [19612,38835] [13,19]
4 3 1333.1 2885 1551.9 [71.7, 84] [40.6, 53.2] [68790,181483] [450,1432]
5 5 -43.9 6310.4 6592 [0, 0] [66.1, 71.9] [0, 0] [9542,11858]
6 19 -363.7 28715.8 30664.3 [75.6, 89.4] [70.6, 82.4] [534795,639264] [3932,5485]
7 94 -16590.3 123578.6 139253.4 [76.8, 91.9] [68.1, 86.4] [2230645,2769283] [10300, 12303.8]
8 107 18137.1 215030.1 195606.1 [77.1, 89.5] [69.7, 78.6] [1186492,1393452] [6355,9124]
9 9 2876.5 16775.2 14229.4 [74.9, 89.7] [68.8, 79.8] [171551,294766] [915,1385]
10 34 6870.5 60885 52015.3 [80, 93.4] [75.3, 87.7] [552726,976517] [3202,6208]
11 6 1990.7 2681.9 691.3 [45.4, 77.6] [34.9, 66.6] [14999,158348] [0, 2151]
Table 7.  Results of the dynamic integrated interval efficiency models $ (M1) $ and $ (M2) $ with $ \ell = \ell' = 1 $ and $ k = k' = 3, $ and ranking of super-efficient and efficient DMUs
Super-efficient DMUs Efficient DMUs
airline efficiency interval class ranking vector rank ranking vector rank
1 [1.3333, 1.3938] S 7.5 2
2 [1.0026, 1.1066] S 2.459 7
3 [1.0156, 1.0369] S 1.356 9
4 [1.0052, 1.2314] S 4.053 6
5 [1.1249, 1.2199] S 5.373 3
6 [0.9994, 1.0895] E 1.051 10
7 [1.4685, 1.6575] S 8.5 1
8 [1.0950, 1.1924] S 4.847 4
9 [0.9986, 1.0734] E 0.949 11
10 [1.0210, 1.2842] S 4.630 5
11 [1.0081, 1.0591] S 1.781 8
Super-efficient DMUs Efficient DMUs
airline efficiency interval class ranking vector rank ranking vector rank
1 [1.3333, 1.3938] S 7.5 2
2 [1.0026, 1.1066] S 2.459 7
3 [1.0156, 1.0369] S 1.356 9
4 [1.0052, 1.2314] S 4.053 6
5 [1.1249, 1.2199] S 5.373 3
6 [0.9994, 1.0895] E 1.051 10
7 [1.4685, 1.6575] S 8.5 1
8 [1.0950, 1.1924] S 4.847 4
9 [0.9986, 1.0734] E 0.949 11
10 [1.0210, 1.2842] S 4.630 5
11 [1.0081, 1.0591] S 1.781 8
Table 8.  Synthetic dataset of 30 DMUs for $ t = 1 $ with two inputs, two outputs, and one desirable carryover linking the two periods
DMU$ _j $ $ X_{1} $ $ X_{2} $ $ Y_{1} $ $ Y_{2} $ $ C_{1} $
1 [0, 3] [-0.1615, 3.0767] [2,10] [3.6768, 7.233] [-0.9007, 0.3911]
2 [-5, 0] [-1.182, 7.4288] [1,3] [5.0074, 7.5827] [-3.833, -1.0627]
3 [-2, -1] [2.863, 6.6322] [-2, 6] [6.5486, 11.8582] [0.7328, 7.2357]
4 [-11, -5] [5.5777, 9.1795] [-4, -3] [7.2685, 14.1149] [3.8294, -2.4358]
5 [-3, 1] [0.6484, 5.7141] [1,8] [6.5704, 9.7772] [3.5951, 3.4671]
6 [3,6] [-0.2489, 4.7478] [10,10] [5.5064, 9.1181] [5.1994, 0.0173]
7 [-6, -2] [-5.4287, -2.6282] [0, 3] [-1.6806, 3.1973] [-6.5508, -0.0203]
8 [-2, 2] [3.4445, 5.0916] [1,8] [6.8983, 9.4379] [-0.7025, -3.7166]
9 [2,5] [-5.9021, -2.2483] [7,12] [-1.7842, 0.9635] [3.2851, -1.2266]
10 [-2, 3] [-0.7791, 2.458] [2,7] [3.2602, 5.8106] [-3.2601, -0.8168]
11 [-9, -6] [4.5724, 8.613] [-2, -1] [10.5083, 15.6547] [-2.0706, 1.0867]
12 [-1, 2] [1.582, 4.8292] [2,4] [4.7475, 7.3799] [-5.0713, -2.4425]
13 [1,3] [5.5856, 5.9474] [5,8] [9.8552, 9.9694] [-5.5407, 0.0637]
14 [-6, -4] [-1.6635, 1.3701] [2,2] [1.7472, 2.1861] [-2.9329, -1.6078]
15 [-3, 1] [-3.975, -1.1701] [3,3] [-0.7801, 0.2799] [-2.3205, -1.5946]
16 [-7, 0] [-6.6192, -0.3315] [-4, 0] [-1.9427, -1.3506] [-6.3538, -0.7493]
17 [-4, 0] [1.1301, 5.0351] [0, 0] [6.6786, 8.0126] [-1.7038, 1.2584]
18 [6,8] [-3.3653, 2.1456] [12,13] [1.6537, 3.4448] [-1.2121, 5.5107]
19 [-5, 0] [6.2833, 7.7517] [-2, 0] [8.1914, 12.7517] [0.4046, 3.4051]
20 [-3, -1] [1.1149, 1.1633] [-2, 3] [5.6438, 8.1306] [-1.0965, 5.2077]
21 [-5, 4] [8.9549, 9.3198] [5,8] [11.0651, 17.3472] [-0.981, -5.7919]
22 [3,5] [-3.7758, -0.397] [4,7] [2.2879, 3.0219] [-1.1107, -3.3136]
23 [-8, -1] [-5.5734, -1.605] [-1, 3] [-2.0838, 4.5846] [4.0279, -0.1544]
24 [-2, 4] [-2.359, 1.1816] [2,5] [0.7777, 3.9289] [-0.2559, -3.4568]
25 [-1, 5] [2.6094, 2.7898] [5,6] [3.9727, 6.8607] [-0.5585, 0.7262]
26 [-1, -1] [2.9899, 6.6723] [5,6] [8.6948, 11.5419] [-1.5396, -3.3929]
27 [0, 5] [-3.0471, 3.6269] [4,9] [1.2691, 4.5094] [5.9166, 1.5114]
28 [0, 5] [-2.8085, 4.2057] [2,11] [4.4506, 5.2129] [2.597, 9.379]
29 [-1, 3] [1.7709, 3.0374] [2,6] [5.6242, 6.2396] [7.127, -3.5507]
30 [-5, -2] [-1.5041, 1.8533] [-2, 4] [2.1227, 5.3259] [-1.9647, -4.8179]
DMU$ _j $ $ X_{1} $ $ X_{2} $ $ Y_{1} $ $ Y_{2} $ $ C_{1} $
1 [0, 3] [-0.1615, 3.0767] [2,10] [3.6768, 7.233] [-0.9007, 0.3911]
2 [-5, 0] [-1.182, 7.4288] [1,3] [5.0074, 7.5827] [-3.833, -1.0627]
3 [-2, -1] [2.863, 6.6322] [-2, 6] [6.5486, 11.8582] [0.7328, 7.2357]
4 [-11, -5] [5.5777, 9.1795] [-4, -3] [7.2685, 14.1149] [3.8294, -2.4358]
5 [-3, 1] [0.6484, 5.7141] [1,8] [6.5704, 9.7772] [3.5951, 3.4671]
6 [3,6] [-0.2489, 4.7478] [10,10] [5.5064, 9.1181] [5.1994, 0.0173]
7 [-6, -2] [-5.4287, -2.6282] [0, 3] [-1.6806, 3.1973] [-6.5508, -0.0203]
8 [-2, 2] [3.4445, 5.0916] [1,8] [6.8983, 9.4379] [-0.7025, -3.7166]
9 [2,5] [-5.9021, -2.2483] [7,12] [-1.7842, 0.9635] [3.2851, -1.2266]
10 [-2, 3] [-0.7791, 2.458] [2,7] [3.2602, 5.8106] [-3.2601, -0.8168]
11 [-9, -6] [4.5724, 8.613] [-2, -1] [10.5083, 15.6547] [-2.0706, 1.0867]
12 [-1, 2] [1.582, 4.8292] [2,4] [4.7475, 7.3799] [-5.0713, -2.4425]
13 [1,3] [5.5856, 5.9474] [5,8] [9.8552, 9.9694] [-5.5407, 0.0637]
14 [-6, -4] [-1.6635, 1.3701] [2,2] [1.7472, 2.1861] [-2.9329, -1.6078]
15 [-3, 1] [-3.975, -1.1701] [3,3] [-0.7801, 0.2799] [-2.3205, -1.5946]
16 [-7, 0] [-6.6192, -0.3315] [-4, 0] [-1.9427, -1.3506] [-6.3538, -0.7493]
17 [-4, 0] [1.1301, 5.0351] [0, 0] [6.6786, 8.0126] [-1.7038, 1.2584]
18 [6,8] [-3.3653, 2.1456] [12,13] [1.6537, 3.4448] [-1.2121, 5.5107]
19 [-5, 0] [6.2833, 7.7517] [-2, 0] [8.1914, 12.7517] [0.4046, 3.4051]
20 [-3, -1] [1.1149, 1.1633] [-2, 3] [5.6438, 8.1306] [-1.0965, 5.2077]
21 [-5, 4] [8.9549, 9.3198] [5,8] [11.0651, 17.3472] [-0.981, -5.7919]
22 [3,5] [-3.7758, -0.397] [4,7] [2.2879, 3.0219] [-1.1107, -3.3136]
23 [-8, -1] [-5.5734, -1.605] [-1, 3] [-2.0838, 4.5846] [4.0279, -0.1544]
24 [-2, 4] [-2.359, 1.1816] [2,5] [0.7777, 3.9289] [-0.2559, -3.4568]
25 [-1, 5] [2.6094, 2.7898] [5,6] [3.9727, 6.8607] [-0.5585, 0.7262]
26 [-1, -1] [2.9899, 6.6723] [5,6] [8.6948, 11.5419] [-1.5396, -3.3929]
27 [0, 5] [-3.0471, 3.6269] [4,9] [1.2691, 4.5094] [5.9166, 1.5114]
28 [0, 5] [-2.8085, 4.2057] [2,11] [4.4506, 5.2129] [2.597, 9.379]
29 [-1, 3] [1.7709, 3.0374] [2,6] [5.6242, 6.2396] [7.127, -3.5507]
30 [-5, -2] [-1.5041, 1.8533] [-2, 4] [2.1227, 5.3259] [-1.9647, -4.8179]
Table 9.  Synthetic dataset of 30 DMUs for $ t = 2 $ with two inputs, two outputs
DMU$ _j $ $ X_{1} $ $ X_{2} $ $ Y_{1} $ $ Y_{2} $
1 [-3, 1] [1,3] [-11.36, -3.2555] [-3.3936, -1.5096]
2 [1,1] [2,5] [-0.9634, 1.7725] [1.7725, 4.0171]
3 [-7, -1] [0, 2] [1.8694, 3.4901] [3.4901, 8.0933]
4 [-3, 7] [5,7] [-0.4754, 4.3661] [4.3661, 2.8741]
5 [-3, 2] [0, 4] [-2.6313, 4.1259] [4.1259, 2.8374]
6 [-5, 0] [1,3] [0.6498, 3.7561] [3.7561, 2.7529]
7 [-1, 6] [4,9] [-9.4604, -4.8163] [-4.8163, -5.7451]
8 [-8, -4] [-4, -1] [-1.7383, -0.5501] [-0.5501, 1.5019]
9 [2,5] [6,10] [6.2155, 9.9875] [9.9875, 11.2043]
10 [-4, 4] [6,6] [0.3967, 3.3566] [3.3566, 4.4534]
11 [-3, -1] [-2, -2] [-3.638, -0.2749] [-0.2749, 2.2454]
12 [-1, 8] [4,11] [-6.8705, -3.8813] [-3.8813, -2.2912]
13 [-6, 2] [1,4] [1.3005, 3.6518] [3.6518, 5.8577]
14 [0, 1] [1,5] [4.6112, 7.6644] [7.6644, 7.6551]
15 [3,5] [3,8] [-3.1301, -0.6347] [-0.6347, 0.9748]
16 [-6, -3] [-1, 2] [-1.3322, 0.3764] [0.3764, 0.6187]
17 [-3, -2] [0, 4] [-8.6088, -6.2055] [-6.2055, -2.6939]
18 [1,5] [5,8] [-1.0369, 5.3176] [5.3176, 5.2215]
19 [1,4] [6,12] [-3.0645, 2.224] [2.224, 0.3195]
20 [-4, 0] [1,5] [-5.066, -1.4253] [-1.4253, -1.7729]
21 [-2, 3] [3,3] [4.9521, 8.3288] [8.3288, 7.6476]
22 [2,4] [3,15] [-1.5163, 3.1201] [3.1201, 4.2655]
23 [-5, -2] [-2, 1] [-5.5681, -1.3893] [-1.3893, -1.5713]
24 [-10, -4] [-4, -2] [-4.6455, 1.4948] [1.4948, 1.8267]
25 [-3, 2] [0, 4] [-3.4395, 6.5853] [6.5853, 7.1234]
26 [2,6] [8,9] [7.6897, 9.6555] [9.6555, 10.8196]
27 [2,3] [6,12] [-7.7017, -1.5092] [-1.5092, -1.0977]
28 [2,8] [8,10] [8.3769, 11.9711] [11.9711, 13.669]
29 [-1, 5] [0, 7] [-6.1897, 0.5822] [0.5822, 0.6056]
30 [-3, 3] [3,4] [-1.7567, 0.7063] [0.7063, 2.1542]
DMU$ _j $ $ X_{1} $ $ X_{2} $ $ Y_{1} $ $ Y_{2} $
1 [-3, 1] [1,3] [-11.36, -3.2555] [-3.3936, -1.5096]
2 [1,1] [2,5] [-0.9634, 1.7725] [1.7725, 4.0171]
3 [-7, -1] [0, 2] [1.8694, 3.4901] [3.4901, 8.0933]
4 [-3, 7] [5,7] [-0.4754, 4.3661] [4.3661, 2.8741]
5 [-3, 2] [0, 4] [-2.6313, 4.1259] [4.1259, 2.8374]
6 [-5, 0] [1,3] [0.6498, 3.7561] [3.7561, 2.7529]
7 [-1, 6] [4,9] [-9.4604, -4.8163] [-4.8163, -5.7451]
8 [-8, -4] [-4, -1] [-1.7383, -0.5501] [-0.5501, 1.5019]
9 [2,5] [6,10] [6.2155, 9.9875] [9.9875, 11.2043]
10 [-4, 4] [6,6] [0.3967, 3.3566] [3.3566, 4.4534]
11 [-3, -1] [-2, -2] [-3.638, -0.2749] [-0.2749, 2.2454]
12 [-1, 8] [4,11] [-6.8705, -3.8813] [-3.8813, -2.2912]
13 [-6, 2] [1,4] [1.3005, 3.6518] [3.6518, 5.8577]
14 [0, 1] [1,5] [4.6112, 7.6644] [7.6644, 7.6551]
15 [3,5] [3,8] [-3.1301, -0.6347] [-0.6347, 0.9748]
16 [-6, -3] [-1, 2] [-1.3322, 0.3764] [0.3764, 0.6187]
17 [-3, -2] [0, 4] [-8.6088, -6.2055] [-6.2055, -2.6939]
18 [1,5] [5,8] [-1.0369, 5.3176] [5.3176, 5.2215]
19 [1,4] [6,12] [-3.0645, 2.224] [2.224, 0.3195]
20 [-4, 0] [1,5] [-5.066, -1.4253] [-1.4253, -1.7729]
21 [-2, 3] [3,3] [4.9521, 8.3288] [8.3288, 7.6476]
22 [2,4] [3,15] [-1.5163, 3.1201] [3.1201, 4.2655]
23 [-5, -2] [-2, 1] [-5.5681, -1.3893] [-1.3893, -1.5713]
24 [-10, -4] [-4, -2] [-4.6455, 1.4948] [1.4948, 1.8267]
25 [-3, 2] [0, 4] [-3.4395, 6.5853] [6.5853, 7.1234]
26 [2,6] [8,9] [7.6897, 9.6555] [9.6555, 10.8196]
27 [2,3] [6,12] [-7.7017, -1.5092] [-1.5092, -1.0977]
28 [2,8] [8,10] [8.3769, 11.9711] [11.9711, 13.669]
29 [-1, 5] [0, 7] [-6.1897, 0.5822] [0.5822, 0.6056]
30 [-3, 3] [3,4] [-1.7567, 0.7063] [0.7063, 2.1542]
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