Measuring energy efficiency incorporating regional heterogeneities: A meta-frontier method with log-linear technology

ANG Sheng; DENG Fang; YANG Feng

doi:10.3969/j.issn.0253-2778.2020.03.001

Open Access JUSTC Original Paper

Measuring energy efficiency incorporating regional heterogeneities: A meta-frontier method with log-linear technology

Funds: Supported by the National Natural Science Foundation of National Science Foundation(71631006,71601173), Special Fund for Basic Scientific Research Expenses of Central Universities.

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https://doi.org/10.3969/j.issn.0253-2778.2020.03.001

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Author Bio:
ANG Sheng,male,born in 1987,PhD cadidate.Research field:Decision method and its application.E-mail:shengang@ustc.edu.cn
Corresponding author: YANG Feng
Received Date: 16 April 2018
Accepted Date: 05 June 2018
Rev Recd Date: 05 June 2018
Publish Date: 31 March 2020

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Abstract

Abstract

China’s provinces vary greatly in the economic development, resource endowments, and science and technology levels, which leads the heterogeneity of production technology. A data envelopment analysis model is proposed based on the log-linear energy technology and meta-frontier method, which considers the heterogeneity of production technology. Energy performances of China’s 30 provinces during the 2006-2015 period were measured and causes of energy inefficiency were analyzed with the model. Empirical results indicate that, most of the provinces in China have great potential for energy efficiency improvement. It is also shown that the eastern region has the highest average annual energy efficiency while that of the western region is the lowest, with the central region somewhere in between. It is suggested to enhance management capabilities for the eastern provinces should be enhanced, improve both technology and management efficiency for central and western provinces, to improve their energy performances.

Abstract

China’s provinces vary greatly in the economic development, resource endowments, and science and technology levels, which leads the heterogeneity of production technology. A data envelopment analysis model is proposed based on the log-linear energy technology and meta-frontier method, which considers the heterogeneity of production technology. Energy performances of China’s 30 provinces during the 2006-2015 period were measured and causes of energy inefficiency were analyzed with the model. Empirical results indicate that, most of the provinces in China have great potential for energy efficiency improvement. It is also shown that the eastern region has the highest average annual energy efficiency while that of the western region is the lowest, with the central region somewhere in between. It is suggested to enhance management capabilities for the eastern provinces should be enhanced, improve both technology and management efficiency for central and western provinces, to improve their energy performances.

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References(48)

References

[1]	BP. Statistical review of world energy[EB/OL]. [2018.03.25], Available online: http://www.bp.com/statisticalreview,2011.
[2]	General Office of the State Council of PRC. Circular of the General Office of the State Council of PRC on Printing and Issuing the 13th FYP for energy conservation and emission reduction[EB/OL]. [2018.03.25],http://www.gov.cn/zhengce/content/2017-01/05/content_5156789.htm.
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[4]	CHARNES A, COOPER W W, RHODES E. Measuring the efficiency of decision making units [J]. European Journal of Operational Research, 1978, 2(6): 429-444.
[5]	HU J L, WANG S C. Total-factor energy efficiency of regions in China [J]. Energy Policy, 2006, 34(17): 3206-3217.
[6]	魏楚,沈满洪.能源效率及其影响因素:基于DEA的实证分析[J].管理世界,2007,167(8):66-76. WEI C, SHEN M H. Energy efficiency and its influencing factors: An empirical analysis based on DEA [J]. Management World, 2007, 167 (8): 66-76.
[7]	曾胜, 黄登仕. 中国能源消费、经济增长与能源效率——基于1980-2007年的实证分析[J]. 数量经济技术经济研究,2009, 26(8): 17-28 ZENG S, HUANG D S. Energy consumption, economic growth and energy efficiency on China [J]. The Journal of Quantitative & Technical Economics, 2009, 26(8): 17-28.
[8]	李国璋, 霍宗杰. 我国全要素能源效率及其收敛性[J].中国人口·资源与环境, 2010, 20(1): 11-16. LI G Z, HUO Z J. China's total factor energy efficiency and its convergence [J]. China Population, Resources and Environment, 2010, 20(1): 11-16.
[9]	HU J L, KAO C H. Efficient energy-saving targets for APEC economies [J]. Energy Policy, 2007, 35(1): 373-382.
[10]	HONMA S, HU J L. Total-factor energy efficiency of regions in Japan [J]. Energy Policy, 2008, 36(2): 821-833.
[11]	ZHAO C, ZHANG H, ZENG Y, et al. Total-factor energy efficiency in BRI countries: An estimation based on three-stage DEA model [J]. Sustainability, 2018, 10(1): No.278(1-15).
[12]	WATANABE M, TANAKA K. Efficiency analysis of Chinese industry: A directional distance function approach [J]. Energy Policy, 2007, 35(12):6323-6331.
[13]	FRE R, GROSSKOPF S. Measuring output efficiency [J]. European Journal of Operational Research, 1983, 13(2): 173-179.
[14]	FRE R, GROSSKOPF S, PASURKA C. Effects on relative efficiency in electric power generation due to environmental controls [J]. Resources and Energy, 1986, 8(2): 167-184.
[15]	袁晓玲,张宝山,杨万平. 基于环境污染的中国全要素能源效率研究[J].中国工业经济,2009, (2): 76-86. YUAN X L, ZHANG B S, YANG W P. Research on China's total factor energy efficiency based on environmental pollution [J]. China Industrial Economics, 2009, (2): 76-86.
[16]	何文强,汪明星.全要素能源效率的DEA模型评价——基于中国1991～2007年数据的实证检验[J]. 上海商学院学报. 2009, 10(5):92-96. HE W Q, WANG M X. Evaluation for the DEA model in calculating total-factor energy efficiency: An empirical test based on China’s data from 1991 to 2007 [J]. Journal of Shanghai Business School, 2009, 10(5):92-96.
[17]	YEH T L, CHEN T Y, LAI P Y. A comparative study of energy utilization efficiency between Taiwan and Mainland of China [J]. Energy Policy, 2010, 38(5):2386-2394.
[18]	徐盈之,管建伟.中国区域能源效率趋同性研究：基于空间经济学视角[J]. 财经研究，2011, 37 (1): 112-123. XU Y Z, GUAN J W. Research on the convergence of regional energy efficiency in China: Based on the perspective of spatial economics [J]. Journal of Finance and Economics, 2011, 37 (1): 112-123.
[19]	RAMLI N A, MUNISAMY S. Modeling undesirable factors in efficiency measurement using data envelopment analysis: A review [J]. Journal of Sustainability Science and Management, 2013, 8(1): 126-135.
[20]	VALADKHANI A, ROSHDI I, SMYTH R. A multiplicative environmental DEA approach to measure efficiency changes in the world's major polluters [J]. Energy Economics, 2016, 54: 363-375.
[21]	ZHOU P, ANG B W. Linear programming models for measuring economy-wide energy efficiency performance [J]. Energy Policy, 2008, 36(8): 2911-2916.
[22]	LIU J P, YANG Q R, HE L. Total-factor energy efficiency (TFEE) evaluation on thermal power industry with DEA, malmquist and multiple regression techniques [J]. Energies, 2017, 10(7): 1039.
[23]	HAYAMI Y. Sources of agricultural productivity gap among selected countries [J]. American Journal of Agricultural Economics, 1969, 51(3): 564-575.
[24]	HAYAMI Y, RUTTAN V W. AgriculturalDevelopment: An International Perspective [M].London: The Johns Hopkins Press, 1971.
[25]	BATTESE G E, RAO D S P. Technology gap, efficiency, and a stochastic metafrontier function [J]. International Journal of Business & Economics, 2002, 1(2):87-93.
[26]	O’DONNELL C J, RAO D S P, BATTESE G E. Metafrontier frameworks for the study of firm-level efficiencies and technology ratios [J]. Empirical Economics, 2008, 34(2): 231-255.
[27]	YU Y, HUANG J, LUO N. Can more environmental information disclosure lead to higher eco-efficiency? Evidence from china [J]. Sustainability, 2018, 10(2):No.528(1-20).
[28]	WANG Q, ZHOU P, ZHAO Z, et al. Energy efficiency and energy saving potential in China&58; A directional meta-frontier DEA approach [J]. Sustainability, 2014, 6(8):5476-5492.
[29]	SUN J, WANG Z, LI G. Measuring emission-reduction and energy-conservation efficiency of Chinese cities considering management and technology heterogeneity [J]. Journal of Cleaner Production, 2018, 175:561-571.
[30]	SUN J, WANG C, JI X, et al. Performance evaluation of heterogeneous bank supply chain systems from the perspective of measurement and decomposition [J]. Computers & Industrial Engineering, 2017: S0360835217302322.
[31]	BANKER R D. Estimating most productive scale size using data envelopment analysis [J]. European Journal of Operational Research, 1984, 17(1): 35-44.
[32]	BANKER R D, MAINDIRATTA A. Piecewise loglinear estimation of efficient production surfaces [J]. Management Science, 1986, 32(1): 126-135.
[33]	MEHDILOOZAD M, SAHOO B K, ROSHDI I. A generalized multiplicative directional distance function for efficiency measurement in DEA [J]. European Journal of Operational Research, 2014, 232(3): 679-688.
[34]	CHARNES A, COOPER W W, SEIFORD L, et al. A multiplicative model for efficiency analysis [J]. Socio-Economic Planning Sciences, 1982, 16(5): 223-224.
[35]	BATTESE G E, RAO D S P, O'DONNELL C J. A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies [J]. Journal of Productivity Analysis, 2004, 21(1): 91-103.
[36]	KUOSMANEN T. Weak disposability in nonparametric production analysis with undesirable outputs [J]. American Journal of Agricultural Economics, 2005, 87(4): 1077-1082.
[37]	ASSAF A, BARROS C P, JOSIASSEN A. Hotel efficiency: A bootstrapped metafrontier approach [J]. International Journal of Hospitality Management, 2010, 29(3): 468-475.
[38]	Chiu C R, Liou J L, Wu P I, et al. Decomposition of the environmental inefficiency of the meta-frontier with undesirable output [J]. Energy Economics, 2012, 34(5): 1392-1399.
[39]	FRE R, GROSSKOPF S, LOVELL C A K, et al. Measuring efficiency when some outputs are undesirable: a nonparametric approach [J]. Review of Economics and Statistics, 1989, 71(1): 90-98.
[40]	WANG Q, ZHAO Z, ZHOU P, et al. Energy efficiency and production technology heterogeneity in China: a meta-frontier DEA approach [J]. Economic Modelling, 2013, 35: 283-289.
[41]	LIN C H, CHIU Y H, HUANG C W. Assessment of technology gaps of tourist hotels in productive and service processes [J]. The Service Industries Journal, 2012, 32(14): 2329-2342.
[42]	ZHANG N, KONG F, YU Y. Measuring ecological total-factor energy efficiency incorporating regional heterogeneities in China [J]. Ecological Indicators, 2015, 51: 165-172.
[43]	Zhang N, Zhou P, Choi Y. Energy efficiency, CO2 emission performance and technology gaps in fossil fuel electricity generation in Korea: A meta-frontier non-radial directional distance function analysis [J]. Energy Policy, 2013, 56: 653-662.
[44]	ZHANG J, WU G Y, ZHANG J P. The Estimation of China's provincial capital stock: 1952~2000 [J]. Economic Research Journal, 2004, 10(1): 35-44.
[45]	单豪杰. 中国资本存量K的再估算：1952～2006年[J].数量经济技术经济研究, 008, 10: 17-31. SHAN H J. Re-estimating the capital stock of China: 1952~2006[J]. The Journal of Quantitative & Technical Economics, 2008, 10: 17-31.
[46]	CHEN Z, SONG S. Efficiency and technology gap in China's agriculture: A regional meta-frontier analysis [J]. China Economic Review, 2008, 19(2):287-296.
[47]	CHUNG Y H,FRE R, GROSSKOPF S. Productivity and Undesirable Outputs: A Directional Distance Function Approach [J]. Microeconomics, 1995, 51(3):229-240.
[48]	ZHOU P, WANG H. Energy and CO emission performance in electricity generation: A non-radial directional distance function approach [J]. European Journal of Operational Research, 2012, 221(3):625-635. Appendix Following Refs.[40,47-48], we introduce the directional-distance meta-frontier method to compare with our log-linear method. The production possibility sets under group frontier and meta-frontier are shown as Eqs.(15) and (16), respectively. T=k,l,e,y,u:∑Njn=1λnkn≤k,∑Njn=1λnln≤l,∑Njn=1λnen≤e,∑Njn=1λnyn≥1+βjyy,∑Njn=1λnun≤(1-βju)u,λn≥0,n=1,…,Nj Tmeta=k,l,e,y,u:∑Jj=1∑Njn=1λnkn≤k,∑Jj=1∑Njn=1λnln≤l,∑Jj=1∑Njn=1λnen≤e,∑Jj=1∑Njn=1λnyn≥(1+βy)y,∑Jj=1∑Njn=1λnun≤(1-βu)u,λn≥0,n=1,…,Nj（16） Based on the production possibility sets, the directional DEA models under group frontier and meta-frontier are shown as models (17) and (18), respectively.  s.t. Dj=k,l,e,y,u;g=Maxωyβjy+ωuβju∑Njn=1λnkn≤k,∑Njn=1λnln≤l,∑Njn=1λnen≤e,∑Njn=1λnyn≥(1+βjy)y,∑Njn=1λnun≤(1-βju)u,λj≥0,βjy≥0,0≤βju<1（17） s.t. Dj=k,l,e,y,u;g=Maxωyβy+ωuβu∑Jj=1∑Njn=1λnkn≤k,∑Jj=1∑Njn=1λnln≤l,∑Jj=1∑Njn=1λnen≤e,∑Jj=1∑Njn=1λnyn≥(1+βy)y,∑Jj=1∑Njn=1λnun≤(1-βu)u,λj≥0,βy≥0,0≤βu<1,（18） Through models (17) and (18), the energy efficiency under group frontier as GEE and meta-frontier as MEE are computed as follows. GEE=1-βju1+βjy（19） MEE=1-βu1+βy（20）)

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[1]	BP. Statistical review of world energy[EB/OL]. [2018.03.25], Available online: http://www.bp.com/statisticalreview,2011.
[2]	General Office of the State Council of PRC. Circular of the General Office of the State Council of PRC on Printing and Issuing the 13th FYP for energy conservation and emission reduction[EB/OL]. [2018.03.25],http://www.gov.cn/zhengce/content/2017-01/05/content_5156789.htm.
[3]	General Office of the State Council of PRC. Circular of the General Office of the State Council of PRC on Printing and Issuing the 13th FYP for controlling greenhouse gas emissions[EB/OL]. [2018.03.25],http://www.gov.cn/zhengce/content/2016-11/04/content_5128619.htm, 2016.
[4]	CHARNES A, COOPER W W, RHODES E. Measuring the efficiency of decision making units [J]. European Journal of Operational Research, 1978, 2(6): 429-444.
[5]	HU J L, WANG S C. Total-factor energy efficiency of regions in China [J]. Energy Policy, 2006, 34(17): 3206-3217.
[6]	魏楚,沈满洪.能源效率及其影响因素:基于DEA的实证分析[J].管理世界,2007,167(8):66-76. WEI C, SHEN M H. Energy efficiency and its influencing factors: An empirical analysis based on DEA [J]. Management World, 2007, 167 (8): 66-76.
[7]	曾胜, 黄登仕. 中国能源消费、经济增长与能源效率——基于1980-2007年的实证分析[J]. 数量经济技术经济研究,2009, 26(8): 17-28 ZENG S, HUANG D S. Energy consumption, economic growth and energy efficiency on China [J]. The Journal of Quantitative & Technical Economics, 2009, 26(8): 17-28.
[8]	李国璋, 霍宗杰. 我国全要素能源效率及其收敛性[J].中国人口·资源与环境, 2010, 20(1): 11-16. LI G Z, HUO Z J. China's total factor energy efficiency and its convergence [J]. China Population, Resources and Environment, 2010, 20(1): 11-16.
[9]	HU J L, KAO C H. Efficient energy-saving targets for APEC economies [J]. Energy Policy, 2007, 35(1): 373-382.
[10]	HONMA S, HU J L. Total-factor energy efficiency of regions in Japan [J]. Energy Policy, 2008, 36(2): 821-833.
[11]	ZHAO C, ZHANG H, ZENG Y, et al. Total-factor energy efficiency in BRI countries: An estimation based on three-stage DEA model [J]. Sustainability, 2018, 10(1): No.278(1-15).
[12]	WATANABE M, TANAKA K. Efficiency analysis of Chinese industry: A directional distance function approach [J]. Energy Policy, 2007, 35(12):6323-6331.
[13]	FRE R, GROSSKOPF S. Measuring output efficiency [J]. European Journal of Operational Research, 1983, 13(2): 173-179.
[14]	FRE R, GROSSKOPF S, PASURKA C. Effects on relative efficiency in electric power generation due to environmental controls [J]. Resources and Energy, 1986, 8(2): 167-184.
[15]	袁晓玲,张宝山,杨万平. 基于环境污染的中国全要素能源效率研究[J].中国工业经济,2009, (2): 76-86. YUAN X L, ZHANG B S, YANG W P. Research on China's total factor energy efficiency based on environmental pollution [J]. China Industrial Economics, 2009, (2): 76-86.
[16]	何文强,汪明星.全要素能源效率的DEA模型评价——基于中国1991～2007年数据的实证检验[J]. 上海商学院学报. 2009, 10(5):92-96. HE W Q, WANG M X. Evaluation for the DEA model in calculating total-factor energy efficiency: An empirical test based on China’s data from 1991 to 2007 [J]. Journal of Shanghai Business School, 2009, 10(5):92-96.
[17]	YEH T L, CHEN T Y, LAI P Y. A comparative study of energy utilization efficiency between Taiwan and Mainland of China [J]. Energy Policy, 2010, 38(5):2386-2394.
[18]	徐盈之,管建伟.中国区域能源效率趋同性研究：基于空间经济学视角[J]. 财经研究，2011, 37 (1): 112-123. XU Y Z, GUAN J W. Research on the convergence of regional energy efficiency in China: Based on the perspective of spatial economics [J]. Journal of Finance and Economics, 2011, 37 (1): 112-123.
[19]	RAMLI N A, MUNISAMY S. Modeling undesirable factors in efficiency measurement using data envelopment analysis: A review [J]. Journal of Sustainability Science and Management, 2013, 8(1): 126-135.
[20]	VALADKHANI A, ROSHDI I, SMYTH R. A multiplicative environmental DEA approach to measure efficiency changes in the world's major polluters [J]. Energy Economics, 2016, 54: 363-375.
[21]	ZHOU P, ANG B W. Linear programming models for measuring economy-wide energy efficiency performance [J]. Energy Policy, 2008, 36(8): 2911-2916.
[22]	LIU J P, YANG Q R, HE L. Total-factor energy efficiency (TFEE) evaluation on thermal power industry with DEA, malmquist and multiple regression techniques [J]. Energies, 2017, 10(7): 1039.
[23]	HAYAMI Y. Sources of agricultural productivity gap among selected countries [J]. American Journal of Agricultural Economics, 1969, 51(3): 564-575.
[24]	HAYAMI Y, RUTTAN V W. AgriculturalDevelopment: An International Perspective [M].London: The Johns Hopkins Press, 1971.
[25]	BATTESE G E, RAO D S P. Technology gap, efficiency, and a stochastic metafrontier function [J]. International Journal of Business & Economics, 2002, 1(2):87-93.
[26]	O’DONNELL C J, RAO D S P, BATTESE G E. Metafrontier frameworks for the study of firm-level efficiencies and technology ratios [J]. Empirical Economics, 2008, 34(2): 231-255.
[27]	YU Y, HUANG J, LUO N. Can more environmental information disclosure lead to higher eco-efficiency? Evidence from china [J]. Sustainability, 2018, 10(2):No.528(1-20).
[28]	WANG Q, ZHOU P, ZHAO Z, et al. Energy efficiency and energy saving potential in China&58; A directional meta-frontier DEA approach [J]. Sustainability, 2014, 6(8):5476-5492.
[29]	SUN J, WANG Z, LI G. Measuring emission-reduction and energy-conservation efficiency of Chinese cities considering management and technology heterogeneity [J]. Journal of Cleaner Production, 2018, 175:561-571.
[30]	SUN J, WANG C, JI X, et al. Performance evaluation of heterogeneous bank supply chain systems from the perspective of measurement and decomposition [J]. Computers & Industrial Engineering, 2017: S0360835217302322.
[31]	BANKER R D. Estimating most productive scale size using data envelopment analysis [J]. European Journal of Operational Research, 1984, 17(1): 35-44.
[32]	BANKER R D, MAINDIRATTA A. Piecewise loglinear estimation of efficient production surfaces [J]. Management Science, 1986, 32(1): 126-135.
[33]	MEHDILOOZAD M, SAHOO B K, ROSHDI I. A generalized multiplicative directional distance function for efficiency measurement in DEA [J]. European Journal of Operational Research, 2014, 232(3): 679-688.
[34]	CHARNES A, COOPER W W, SEIFORD L, et al. A multiplicative model for efficiency analysis [J]. Socio-Economic Planning Sciences, 1982, 16(5): 223-224.
[35]	BATTESE G E, RAO D S P, O'DONNELL C J. A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies [J]. Journal of Productivity Analysis, 2004, 21(1): 91-103.
[36]	KUOSMANEN T. Weak disposability in nonparametric production analysis with undesirable outputs [J]. American Journal of Agricultural Economics, 2005, 87(4): 1077-1082.
[37]	ASSAF A, BARROS C P, JOSIASSEN A. Hotel efficiency: A bootstrapped metafrontier approach [J]. International Journal of Hospitality Management, 2010, 29(3): 468-475.
[38]	Chiu C R, Liou J L, Wu P I, et al. Decomposition of the environmental inefficiency of the meta-frontier with undesirable output [J]. Energy Economics, 2012, 34(5): 1392-1399.
[39]	FRE R, GROSSKOPF S, LOVELL C A K, et al. Measuring efficiency when some outputs are undesirable: a nonparametric approach [J]. Review of Economics and Statistics, 1989, 71(1): 90-98.
[40]	WANG Q, ZHAO Z, ZHOU P, et al. Energy efficiency and production technology heterogeneity in China: a meta-frontier DEA approach [J]. Economic Modelling, 2013, 35: 283-289.
[41]	LIN C H, CHIU Y H, HUANG C W. Assessment of technology gaps of tourist hotels in productive and service processes [J]. The Service Industries Journal, 2012, 32(14): 2329-2342.
[42]	ZHANG N, KONG F, YU Y. Measuring ecological total-factor energy efficiency incorporating regional heterogeneities in China [J]. Ecological Indicators, 2015, 51: 165-172.
[43]	Zhang N, Zhou P, Choi Y. Energy efficiency, CO2 emission performance and technology gaps in fossil fuel electricity generation in Korea: A meta-frontier non-radial directional distance function analysis [J]. Energy Policy, 2013, 56: 653-662.
[44]	ZHANG J, WU G Y, ZHANG J P. The Estimation of China's provincial capital stock: 1952~2000 [J]. Economic Research Journal, 2004, 10(1): 35-44.
[45]	单豪杰. 中国资本存量K的再估算：1952～2006年[J].数量经济技术经济研究, 008, 10: 17-31. SHAN H J. Re-estimating the capital stock of China: 1952~2006[J]. The Journal of Quantitative & Technical Economics, 2008, 10: 17-31.
[46]	CHEN Z, SONG S. Efficiency and technology gap in China's agriculture: A regional meta-frontier analysis [J]. China Economic Review, 2008, 19(2):287-296.
[47]	CHUNG Y H,FRE R, GROSSKOPF S. Productivity and Undesirable Outputs: A Directional Distance Function Approach [J]. Microeconomics, 1995, 51(3):229-240.
[48]	ZHOU P, WANG H. Energy and CO emission performance in electricity generation: A non-radial directional distance function approach [J]. European Journal of Operational Research, 2012, 221(3):625-635. Appendix Following Refs.[40,47-48], we introduce the directional-distance meta-frontier method to compare with our log-linear method. The production possibility sets under group frontier and meta-frontier are shown as Eqs.(15) and (16), respectively. T=k,l,e,y,u:∑Njn=1λnkn≤k,∑Njn=1λnln≤l,∑Njn=1λnen≤e,∑Njn=1λnyn≥1+βjyy,∑Njn=1λnun≤(1-βju)u,λn≥0,n=1,…,Nj Tmeta=k,l,e,y,u:∑Jj=1∑Njn=1λnkn≤k,∑Jj=1∑Njn=1λnln≤l,∑Jj=1∑Njn=1λnen≤e,∑Jj=1∑Njn=1λnyn≥(1+βy)y,∑Jj=1∑Njn=1λnun≤(1-βu)u,λn≥0,n=1,…,Nj（16） Based on the production possibility sets, the directional DEA models under group frontier and meta-frontier are shown as models (17) and (18), respectively.  s.t. Dj=k,l,e,y,u;g=Maxωyβjy+ωuβju∑Njn=1λnkn≤k,∑Njn=1λnln≤l,∑Njn=1λnen≤e,∑Njn=1λnyn≥(1+βjy)y,∑Njn=1λnun≤(1-βju)u,λj≥0,βjy≥0,0≤βju<1（17） s.t. Dj=k,l,e,y,u;g=Maxωyβy+ωuβu∑Jj=1∑Njn=1λnkn≤k,∑Jj=1∑Njn=1λnln≤l,∑Jj=1∑Njn=1λnen≤e,∑Jj=1∑Njn=1λnyn≥(1+βy)y,∑Jj=1∑Njn=1λnun≤(1-βu)u,λj≥0,βy≥0,0≤βu<1,（18） Through models (17) and (18), the energy efficiency under group frontier as GEE and meta-frontier as MEE are computed as follows. GEE=1-βju1+βjy（19） MEE=1-βu1+βy（20）)

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Measuring energy efficiency incorporating regional heterogeneities: A meta-frontier method with log-linear technology

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