March  2022, 18(2): 1339-1363. doi: 10.3934/jimo.2021023

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 Published  March 2022 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 and Management Optimization, 2022, 18 (2) : 1339-1363. 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

Figure 1.  The dynamic framework for a DMU comprising of $ T $ periods with multiple carryovers connecting consecutive periods
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]
[1]

Mohammad Afzalinejad, Zahra Abbasi. A slacks-based model for dynamic data envelopment analysis. Journal of Industrial and Management Optimization, 2019, 15 (1) : 275-291. doi: 10.3934/jimo.2018043

[2]

Pooja Bansal. Sequential Malmquist-Luenberger productivity index for interval data envelopment analysis. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022058

[3]

Mahdi Mahdiloo, Abdollah Noorizadeh, Reza Farzipoor Saen. Developing a new data envelopment analysis model for customer value analysis. Journal of Industrial and Management Optimization, 2011, 7 (3) : 531-558. doi: 10.3934/jimo.2011.7.531

[4]

Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311

[5]

Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial and Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014

[6]

Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046

[7]

Hasan Hosseini-Nasab, Vahid Ettehadi. Development of opened-network data envelopment analysis models under uncertainty. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022027

[8]

Saber Saati, Adel Hatami-Marbini, Per J. Agrell, Madjid Tavana. A common set of weight approach using an ideal decision making unit in data envelopment analysis. Journal of Industrial and Management Optimization, 2012, 8 (3) : 623-637. doi: 10.3934/jimo.2012.8.623

[9]

Ali Hadi, Saeid Mehrabian. A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022006

[10]

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan. Microlocal analysis of borehole seismic data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022026

[11]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[12]

Zheng Dai, I.G. Rosen, Chuming Wang, Nancy Barnett, Susan E. Luczak. Using drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. Mathematical Biosciences & Engineering, 2016, 13 (5) : 911-934. doi: 10.3934/mbe.2016023

[13]

Mostafa Karimi, Noor Akma Ibrahim, Mohd Rizam Abu Bakar, Jayanthi Arasan. Rank-based inference for the accelerated failure time model in the presence of interval censored data. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 107-112. doi: 10.3934/naco.2017007

[14]

George Siopsis. Quantum topological data analysis with continuous variables. Foundations of Data Science, 2019, 1 (4) : 419-431. doi: 10.3934/fods.2019017

[15]

Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001

[16]

Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001

[17]

Pankaj Sharma, David Baglee, Jaime Campos, Erkki Jantunen. Big data collection and analysis for manufacturing organisations. Big Data & Information Analytics, 2017, 2 (2) : 127-139. doi: 10.3934/bdia.2017002

[18]

Runqin Hao, Guanwen Zhang, Dong Li, Jie Zhang. Data modeling analysis on removal efficiency of hexavalent chromium. Mathematical Foundations of Computing, 2019, 2 (3) : 203-213. doi: 10.3934/mfc.2019014

[19]

Erik Carlsson, John Gunnar Carlsson, Shannon Sweitzer. Applying topological data analysis to local search problems. Foundations of Data Science, 2022  doi: 10.3934/fods.2022006

[20]

Débora A. F. Albanez, Maicon J. Benvenutti. Continuous data assimilation algorithm for simplified Bardina model. Evolution Equations and Control Theory, 2018, 7 (1) : 33-52. doi: 10.3934/eect.2018002

2020 Impact Factor: 1.801

Article outline

Figures and Tables

[Back to Top]