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May  2021, 17(3): 1357-1382. doi: 10.3934/jimo.2020025

Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach

Saeed Assani 1,2,, , Jianlin Jiang 1, , Ahmad Assani 3, and  Feng Yang 4,

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2. 

Department of Mathematical Statistics, Faculty of Science, Tishreen University, Latakia, Syria
3. 

Faculty of Computer Science and Business Computer Systems, Karlsruhe University of Applied Science, Karlsruhe, 76133, Germany
4. 

School of Management, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Saeed Assani

Received  December 2018 Revised  October 2019 Published  May 2021 Early access  February 2020

Data envelopment analysis (DEA) is one of the vastly available literature on efficiency analysis. In general, the efficiency of decision making units (DMUs) can be measured from two perspectives, optimistic and pessimistic. Two different perspectives lead to two different conflicting and biased scale efficiency measurements. To deal with the problem, in this paper, we introduce a double frontier approach to integrate both optimistic and pessimistic scale efficiencies' viewpoints in one single scale efficiency term, which will be more realistic and has benchmarking preferences. We first investigate the scale efficiency concept from double frontier perspective in black-box DEA and then extend it to the two-stage DEA framework. Mathematical analysis proved that the double frontier scale efficiency of a two-stage system could be decomposed into the internal stages' double frontier scale efficiencies. Finally, we elaborate applicability and merits of the proposed approach using a case of China's regional R & D value chain in terms of its profitability and marketability.

Keywords: Data envelopment analysis, scale efficiency, optimistic-pessimistic efficiency, double frontier approach, R & D.
Mathematics Subject Classification: Primary: 90B10, 90B30; Secondary: 90B50.
Citation: Saeed Assani, Jianlin Jiang, Ahmad Assani, Feng Yang. Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1357-1382. doi: 10.3934/jimo.2020025
References:
[1]

Q. An, F. Meng, B. Xiong, Z. Wang and X. Chen, Assessing the relative efficiency of Chinese high-tech industries: A dynamic network data envelopment analysis approach, Annals of Operations Research, (2018), 1–23. doi: 10.1007/s10479-018-2883-2.

[2]

S. Assani, J. Jiang, A. Assani and F. Yang, Estimating and decomposing most productive scale size in parallel DEA networks with shared inputs: A case of China's Five-Year Plans, preprint, arXiv: 1910.03421.

[3]

S. Assani, J. Jiang, A. Assani and F. Yang, Most productive scale size of China's regional R & D value chain: A mixed structure network, preprint, arXiv: 1910.03805.

[4]

S. Assani, J. Jiang, R. Cao and F. Yang, Most productive scale size decomposition for multi-stage systems in data envelopment analysis, Computers and Industrial Engineering, 120 (2018), 279-287.  doi: 10.1016/j.cie.2018.04.043.

[5]

H. Azizi, S. Kordrostami and A. Amirteimoori, Slacks-based measures of efficiency in imprecise data envelopment analysis: An approach based on data envelopment analysis with double frontiers, Computers and Industrial Engineering, 79 (2015), 42-51.  doi: 10.1016/j.cie.2014.10.019.

[6]

T. Badiezadeh, R. F. Saen and T. Samavati, Assessing sustainability of supply chains by double frontier network DEA: A big data approach, Computers and Operations Research, 98 (2018), 284-290.  doi: 10.1016/j.cor.2017.06.003.

[7]

R. D. Banker, A. Charnes and W. W. Cooper, Some Models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092.  doi: 10.1287/mnsc.30.9.1078.

[8]

R. D. Banker and R. M. Thrall, Estimation of returns to scale using data envelopment analysis, European Journal of Operational Research, 62 (1992), 74-84.  doi: 10.1016/0377-2217(92)90178-C.

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N. Becheikh, R. Landry and N. Amara, Lessons from innovation empirical studies in the manufacturing sector: A systematic review of the literature from 1993-2003, Technovation, 26 (2006), 644-664.  doi: 10.1016/j.technovation.2005.06.016.

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R. Blundell, R. Griffith and J. V. Reenen, Market share, market value and innovation in a panel of British manufacturing firms, Review of Economic Studies, 66 (1999), 529-554.  doi: 10.1111/1467-937X.00097.

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N. Capon, J. U. Farley and S. Hoenig, Determinants of financial performance, Management Science, 36 (2011), 1143-1159.  doi: 10.1287/mnsc.36.10.1143.

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A. Charnes and W. W. Cooper, The non-archimedean CCR ratio for efficiency analysis: A rejoinder to Boyd and F$\ddot{a}$re, European Journal of Operational Research, 15 (1984), 333-334.  doi: 10.1016/0377-2217(84)90102-4.

[13]

A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.

[14]

K. Chen, M. Kou and X. Fu, Evaluation of multi-period regional R & D efficiency: An application of dynamic DEA to China's regional R & D systems, Omega, 74 (2018), 103-114.  doi: 10.1016/j.omega.2017.01.010.

[15]

X. Chen, Z. Liu and Q. Zhu, Performance evaluation of China's high-tech innovation process: Analysis based on the innovation value chain, Technovation, 74-75 (2018), 42-53.  doi: 10.1016/j.technovation.2018.02.009.

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P. Coto-Millán, V. Inglada, X. L. Fernández, L. Inglada-Pérez and M. Á. Pesquera, The "effect procargo" on technical and scale efficiency at airports: The case of Spanish airports (2009-2011), Utilities Policy, 39 (2016), 29-35.  doi: 10.1016/j.jup.2016.01.004.

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D. M. Decarolis and D. L. Deeds, The impact of stocks and flows of organizational knowledge on firm performance: An empirical investigation of the biotechnology industry, Strategic Management Journal, 20 (1999), 953-968.  doi: 10.1002/(SICI)1097-0266(199910)20:10<953::AID-SMJ59>3.0.CO;2-3.

[19]

A. Emrouznejad and G. Yang, A survey and analysis of the first 40 years of scholarly literature in DEA: 1978–2016, Socio-Economic Planning Sciences, 61 (2018), 4-8.  doi: 10.1016/j.seps.2017.01.008.

[20]

S. B. Graves and N. S. Langowitz, R & D productivity: A global multi-industry comparison, Technological Forecasting and Social Change, 53 (1996), 125-137.  doi: 10.1016/S0040-1625(96)00068-6.

[21]

J. Guan and K. Chen, Measuring the innovation production process: A cross-region empirical study of China's high-tech innovations, Technovation, 30 (2010), 348-358.  doi: 10.1016/j.technovation.2010.02.001.

[22]

B. H. Hall and R. H. Ziedonis, The patent paradox revisited: An empirical study of patenting in the U.S. semiconductor industry, 1979-1995, The RAND Journal of Economics, 32 (2001), 101-101.  doi: 10.2307/2696400.

[23]

M. A. Hitt, R. E. Hoskisson and H. Kim, International diversification: Effects on innovation and firm performance in product-diversified firms, Academy of Management Journal, 40 (1997), 767-798.  doi: 10.2307/256948.

[24]

K. Hosseini and A. Stefaniec, Efficiency assessment of Iran's petroleum refining industry in the presence of unprofitable output: A dynamic two-stage slacks-based measure, Energy, 189 (2019), 116112. doi: 10.1016/j.energy.2019.116112.

[25]

J. L. Jiang, E. P. Chew, L. H. Lee and Z. Sun, DEA based on strongly efficient and inefficient frontiers and its application on port efficiency measurement, OR Spectrum, 34 (2012), 943-969.  doi: 10.1007/s00291-011-0263-2.

[26]

C. Kao and S. N. Hwang, Decomposition of technical and scale efficiencies in two-stage production systems, European Journal of Operational Research, 211 (2011), 515-519.  doi: 10.1016/j.ejor.2011.01.010.

[27]

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

[28]

C. Kao and S. N. Hwang, Scale efficiency measurement in two-stage production systems, International Series in Operations Research and Management Science, 208 (2014), 119-135.  doi: 10.1007/978-1-4899-8068-7_6.

[29]

P. Khoshnevis and P. Teirlinck, Performance evaluation of R & D active firms, Socio-Economic Planning Sciences, 61 (2018), 16-28.  doi: 10.1016/j.seps.2017.01.005.

[30]

S. Lee and H. Lee, Measuring and comparing the R & D performance of government research institutes: A bottom-up data envelopment analysis approach, Journal of Informetrics, 9 (2015), 942-953.  doi: 10.1016/j.joi.2015.10.001.

[31]

J. S. Liu, L. Y. Y. Lu, W. M. Lu and B. J. Y. Lin, A survey of DEA applications, Omega, 41 (2013), 893-902.  doi: 10.1016/j.omega.2012.11.004.

[32]

J. S. Liu and W. M. Lu, DEA and ranking with the network-based approach: A case of R & D performance, Omega, 38 (2010), 453-464.  doi: 10.1016/j.omega.2009.12.002.

[33]

K. Lv, D. Wang and Y. Cheng, Measuring the dynamic performances of innovation production process from the carry-over perspective: An empirical study of China's high-tech industry, Transformations in Business and Economics, 16 (2017), 345-361. 

[34]

M. M. Mousavi, J. Ouenniche and K. Tone, A comparative analysis of two-stage distress prediction models, Expert Systems with Applications, 119 (2019), 322-341.  doi: 10.1016/j.eswa.2018.10.053.

[35]

G. P$\acute{e}$rez-L$\acute{o}$pez, D. Prior and J. L. Zafra-G$\acute{o}$mez, Temporal scale efficiency in DEA panel data estimations. An application to the solid waste disposal service in Spain, Omega, 76 (2018), 18-27.  doi: 10.1016/j.omega.2017.03.005.

[36]

$\ddot{U}$. Sa$\check{g}$lam, Assessment of the productive efficiency of large wind farms in the United States: An application of two-stage data envelopment analysis, Energy Conversion and Management, 153 (2017), 188-214.  doi: 10.1016/j.enconman.2017.09.062.

[37]

B. K. Sahoo, J. Zhu, K. Tone and B. M. Klemen, Decomposing technical efficiency and scale elasticity in two-stage network DEA, European Journal of Operational Research, 233 (2014), 584-594.  doi: 10.1016/j.ejor.2013.09.046.

[38]

L. M. Seiford and J. Zhu, Profitability and marketability of the top 55 U.S. commercial banks, Management Science, 45 (1999), 1270-1288.  doi: 10.1287/mnsc.45.9.1270.

[39]

S. R. Seyedalizadeh Ganji, A. Rassafi and D. L. Xu, A double frontier DEA cross efficiency method aggregated by evidential reasoning approach for measuring road safety performance, Measurement, 136 (2019), 668-688.  doi: 10.1016/j.measurement.2018.12.098.

[40]

A. Sterlacchini, Do innovative activities matter to small firms in non R & D intensive industries? An application to export performance, Research Policy, 28 (1999), 819-832.  doi: 10.1016/S0048-7333(99)00023-2.

[41]

T. Sueyoshi and D. Wang, Measuring scale efficiency and returns to scale on large commercial rooftop photovoltaic systems in California, Energy Economics, 65 (2017), 389-398.  doi: 10.1016/j.eneco.2017.04.019.

[42]

S. Thornhill, Knowledge, innovation and firm performance in high and low technology regimes, Journal of Business Venturing, 21 (2006), 687-703.  doi: 10.1016/j.jbusvent.2005.06.001.

[43]

B. Walheer, Scale efficiency for multi-output cost minimizing producers: The case of the US electricity plants, Energy Economics, 70 (2018), 26-36.  doi: 10.1016/j.eneco.2017.12.017.

[44]

C. H. Wang, Y. H. Lu, C. W. Huang and J. Y. Lee, R & D, productivity, and market value: An empirical study from high-technology firms, Omega, 41 (2013), 143-155.  doi: 10.1016/j.omega.2011.12.011.

[45]

Y. M. Wang, K. S. Chin and J. B. Yang, Measuring the performances of decision-making units using geometric average efficiency, Journal of the Operational Research Society, 58 (2007), 929-937.  doi: 10.1057/palgrave.jors.2602205.

[46]

Y. M. Wang and Y. X. Lan, Estimating most productive scale size with double frontiers data envelopment analysis, Economic Modelling, 33 (2013), 182-186.  doi: 10.1016/j.econmod.2013.04.021.

[47]

Y. M. Wang and J. B. Yang, Measuring the performances of decision-making units using interval efficiencies, Journal of Computational and Applied Mathematics, 198 (2007), 253-267.  doi: 10.1016/j.cam.2005.12.025.

[48]

P. F. Wanke and C. P. Barros, Public-private partnerships and scale efficiency in Brazilian ports: Evidence from two-stage DEA analysis, Socio-Economic Planning Sciences, 51 (2015), 13-22.  doi: 10.1016/j.seps.2015.06.002.

[49]

Z. G. Xin and W. Zhen, The technical efficiency of China's wind power list enterprises: An estimation based on DEA method and micro-data, Renewable Energy, 133 (2019), 470-479.  doi: 10.1016/j.renene.2018.10.049.

[50]

V. Zelenyuk, Aggregation of scale efficiency, European Journal of Operational Research, 240 (2015), 269-277.  doi: 10.1016/j.ejor.2014.06.038.

[51]

B. Zhang, Y. Luo and Y. H. Chiu, Efficiency evaluation of China's high-tech industry with a multi-activity network data envelopment analysis approach, Socio-Economic Planning Sciences, 66 (2019), 2-9.  doi: 10.1016/j.seps.2018.07.013.

[52]

W. Zhong, W. Yuan, S. X. Li and Z. Huang, The performance evaluation of regional R & D investments in China: An application of DEA based on the first official China economic census data, Omega, 39 (2011), 447-455.  doi: 10.1016/j.omega.2010.09.004.

show all references

References:
[1]

Q. An, F. Meng, B. Xiong, Z. Wang and X. Chen, Assessing the relative efficiency of Chinese high-tech industries: A dynamic network data envelopment analysis approach, Annals of Operations Research, (2018), 1–23. doi: 10.1007/s10479-018-2883-2.

[2]

S. Assani, J. Jiang, A. Assani and F. Yang, Estimating and decomposing most productive scale size in parallel DEA networks with shared inputs: A case of China's Five-Year Plans, preprint, arXiv: 1910.03421.

[3]

S. Assani, J. Jiang, A. Assani and F. Yang, Most productive scale size of China's regional R & D value chain: A mixed structure network, preprint, arXiv: 1910.03805.

[4]

S. Assani, J. Jiang, R. Cao and F. Yang, Most productive scale size decomposition for multi-stage systems in data envelopment analysis, Computers and Industrial Engineering, 120 (2018), 279-287.  doi: 10.1016/j.cie.2018.04.043.

[5]

H. Azizi, S. Kordrostami and A. Amirteimoori, Slacks-based measures of efficiency in imprecise data envelopment analysis: An approach based on data envelopment analysis with double frontiers, Computers and Industrial Engineering, 79 (2015), 42-51.  doi: 10.1016/j.cie.2014.10.019.

[6]

T. Badiezadeh, R. F. Saen and T. Samavati, Assessing sustainability of supply chains by double frontier network DEA: A big data approach, Computers and Operations Research, 98 (2018), 284-290.  doi: 10.1016/j.cor.2017.06.003.

[7]

R. D. Banker, A. Charnes and W. W. Cooper, Some Models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30 (1984), 1078-1092.  doi: 10.1287/mnsc.30.9.1078.

[8]

R. D. Banker and R. M. Thrall, Estimation of returns to scale using data envelopment analysis, European Journal of Operational Research, 62 (1992), 74-84.  doi: 10.1016/0377-2217(92)90178-C.

[9]

N. Becheikh, R. Landry and N. Amara, Lessons from innovation empirical studies in the manufacturing sector: A systematic review of the literature from 1993-2003, Technovation, 26 (2006), 644-664.  doi: 10.1016/j.technovation.2005.06.016.

[10]

R. Blundell, R. Griffith and J. V. Reenen, Market share, market value and innovation in a panel of British manufacturing firms, Review of Economic Studies, 66 (1999), 529-554.  doi: 10.1111/1467-937X.00097.

[11]

N. Capon, J. U. Farley and S. Hoenig, Determinants of financial performance, Management Science, 36 (2011), 1143-1159.  doi: 10.1287/mnsc.36.10.1143.

[12]

A. Charnes and W. W. Cooper, The non-archimedean CCR ratio for efficiency analysis: A rejoinder to Boyd and F$\ddot{a}$re, European Journal of Operational Research, 15 (1984), 333-334.  doi: 10.1016/0377-2217(84)90102-4.

[13]

A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.

[14]

K. Chen, M. Kou and X. Fu, Evaluation of multi-period regional R & D efficiency: An application of dynamic DEA to China's regional R & D systems, Omega, 74 (2018), 103-114.  doi: 10.1016/j.omega.2017.01.010.

[15]

X. Chen, Z. Liu and Q. Zhu, Performance evaluation of China's high-tech innovation process: Analysis based on the innovation value chain, Technovation, 74-75 (2018), 42-53.  doi: 10.1016/j.technovation.2018.02.009.

[16]

W. W. Cooper, L. M. Seiford and K. Tone, Introduction to Data Envelopment Analysis and Its Uses. In Introduction to Data Envelopment Analysis and Its Uses: With DEA-Solver Software and References, Springer, Boston, 2006. doi: 10.1007/0-387-29122-9.

[17]

P. Coto-Millán, V. Inglada, X. L. Fernández, L. Inglada-Pérez and M. Á. Pesquera, The "effect procargo" on technical and scale efficiency at airports: The case of Spanish airports (2009-2011), Utilities Policy, 39 (2016), 29-35.  doi: 10.1016/j.jup.2016.01.004.

[18]

D. M. Decarolis and D. L. Deeds, The impact of stocks and flows of organizational knowledge on firm performance: An empirical investigation of the biotechnology industry, Strategic Management Journal, 20 (1999), 953-968.  doi: 10.1002/(SICI)1097-0266(199910)20:10<953::AID-SMJ59>3.0.CO;2-3.

[19]

A. Emrouznejad and G. Yang, A survey and analysis of the first 40 years of scholarly literature in DEA: 1978–2016, Socio-Economic Planning Sciences, 61 (2018), 4-8.  doi: 10.1016/j.seps.2017.01.008.

[20]

S. B. Graves and N. S. Langowitz, R & D productivity: A global multi-industry comparison, Technological Forecasting and Social Change, 53 (1996), 125-137.  doi: 10.1016/S0040-1625(96)00068-6.

[21]

J. Guan and K. Chen, Measuring the innovation production process: A cross-region empirical study of China's high-tech innovations, Technovation, 30 (2010), 348-358.  doi: 10.1016/j.technovation.2010.02.001.

[22]

B. H. Hall and R. H. Ziedonis, The patent paradox revisited: An empirical study of patenting in the U.S. semiconductor industry, 1979-1995, The RAND Journal of Economics, 32 (2001), 101-101.  doi: 10.2307/2696400.

[23]

M. A. Hitt, R. E. Hoskisson and H. Kim, International diversification: Effects on innovation and firm performance in product-diversified firms, Academy of Management Journal, 40 (1997), 767-798.  doi: 10.2307/256948.

[24]

K. Hosseini and A. Stefaniec, Efficiency assessment of Iran's petroleum refining industry in the presence of unprofitable output: A dynamic two-stage slacks-based measure, Energy, 189 (2019), 116112. doi: 10.1016/j.energy.2019.116112.

[25]

J. L. Jiang, E. P. Chew, L. H. Lee and Z. Sun, DEA based on strongly efficient and inefficient frontiers and its application on port efficiency measurement, OR Spectrum, 34 (2012), 943-969.  doi: 10.1007/s00291-011-0263-2.

[26]

C. Kao and S. N. Hwang, Decomposition of technical and scale efficiencies in two-stage production systems, European Journal of Operational Research, 211 (2011), 515-519.  doi: 10.1016/j.ejor.2011.01.010.

[27]

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

[28]

C. Kao and S. N. Hwang, Scale efficiency measurement in two-stage production systems, International Series in Operations Research and Management Science, 208 (2014), 119-135.  doi: 10.1007/978-1-4899-8068-7_6.

[29]

P. Khoshnevis and P. Teirlinck, Performance evaluation of R & D active firms, Socio-Economic Planning Sciences, 61 (2018), 16-28.  doi: 10.1016/j.seps.2017.01.005.

[30]

S. Lee and H. Lee, Measuring and comparing the R & D performance of government research institutes: A bottom-up data envelopment analysis approach, Journal of Informetrics, 9 (2015), 942-953.  doi: 10.1016/j.joi.2015.10.001.

[31]

J. S. Liu, L. Y. Y. Lu, W. M. Lu and B. J. Y. Lin, A survey of DEA applications, Omega, 41 (2013), 893-902.  doi: 10.1016/j.omega.2012.11.004.

[32]

J. S. Liu and W. M. Lu, DEA and ranking with the network-based approach: A case of R & D performance, Omega, 38 (2010), 453-464.  doi: 10.1016/j.omega.2009.12.002.

[33]

K. Lv, D. Wang and Y. Cheng, Measuring the dynamic performances of innovation production process from the carry-over perspective: An empirical study of China's high-tech industry, Transformations in Business and Economics, 16 (2017), 345-361. 

[34]

M. M. Mousavi, J. Ouenniche and K. Tone, A comparative analysis of two-stage distress prediction models, Expert Systems with Applications, 119 (2019), 322-341.  doi: 10.1016/j.eswa.2018.10.053.

[35]

G. P$\acute{e}$rez-L$\acute{o}$pez, D. Prior and J. L. Zafra-G$\acute{o}$mez, Temporal scale efficiency in DEA panel data estimations. An application to the solid waste disposal service in Spain, Omega, 76 (2018), 18-27.  doi: 10.1016/j.omega.2017.03.005.

[36]

$\ddot{U}$. Sa$\check{g}$lam, Assessment of the productive efficiency of large wind farms in the United States: An application of two-stage data envelopment analysis, Energy Conversion and Management, 153 (2017), 188-214.  doi: 10.1016/j.enconman.2017.09.062.

[37]

B. K. Sahoo, J. Zhu, K. Tone and B. M. Klemen, Decomposing technical efficiency and scale elasticity in two-stage network DEA, European Journal of Operational Research, 233 (2014), 584-594.  doi: 10.1016/j.ejor.2013.09.046.

[38]

L. M. Seiford and J. Zhu, Profitability and marketability of the top 55 U.S. commercial banks, Management Science, 45 (1999), 1270-1288.  doi: 10.1287/mnsc.45.9.1270.

[39]

S. R. Seyedalizadeh Ganji, A. Rassafi and D. L. Xu, A double frontier DEA cross efficiency method aggregated by evidential reasoning approach for measuring road safety performance, Measurement, 136 (2019), 668-688.  doi: 10.1016/j.measurement.2018.12.098.

[40]

A. Sterlacchini, Do innovative activities matter to small firms in non R & D intensive industries? An application to export performance, Research Policy, 28 (1999), 819-832.  doi: 10.1016/S0048-7333(99)00023-2.

[41]

T. Sueyoshi and D. Wang, Measuring scale efficiency and returns to scale on large commercial rooftop photovoltaic systems in California, Energy Economics, 65 (2017), 389-398.  doi: 10.1016/j.eneco.2017.04.019.

[42]

S. Thornhill, Knowledge, innovation and firm performance in high and low technology regimes, Journal of Business Venturing, 21 (2006), 687-703.  doi: 10.1016/j.jbusvent.2005.06.001.

[43]

B. Walheer, Scale efficiency for multi-output cost minimizing producers: The case of the US electricity plants, Energy Economics, 70 (2018), 26-36.  doi: 10.1016/j.eneco.2017.12.017.

[44]

C. H. Wang, Y. H. Lu, C. W. Huang and J. Y. Lee, R & D, productivity, and market value: An empirical study from high-technology firms, Omega, 41 (2013), 143-155.  doi: 10.1016/j.omega.2011.12.011.

[45]

Y. M. Wang, K. S. Chin and J. B. Yang, Measuring the performances of decision-making units using geometric average efficiency, Journal of the Operational Research Society, 58 (2007), 929-937.  doi: 10.1057/palgrave.jors.2602205.

[46]

Y. M. Wang and Y. X. Lan, Estimating most productive scale size with double frontiers data envelopment analysis, Economic Modelling, 33 (2013), 182-186.  doi: 10.1016/j.econmod.2013.04.021.

[47]

Y. M. Wang and J. B. Yang, Measuring the performances of decision-making units using interval efficiencies, Journal of Computational and Applied Mathematics, 198 (2007), 253-267.  doi: 10.1016/j.cam.2005.12.025.

[48]

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W. Zhong, W. Yuan, S. X. Li and Z. Huang, The performance evaluation of regional R & D investments in China: An application of DEA based on the first official China economic census data, Omega, 39 (2011), 447-455.  doi: 10.1016/j.omega.2010.09.004.

Figure 1.  Two-stage DEA system
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Figure 2.  Two-stage R & D value chain
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Figure 3.  Efficiency evaluation from an optimistic perspective by area
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Figure 4.  Efficiency evaluation from a pessimistic perspective by area
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Figure 5.  Efficiency evaluation from double frontier perspective by area
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Table 1.  Dataset
DMU Inputs Output
$ X_1 $ $ X_2 $ $ Y $
A 40 7 210
B 32 12 105
C 52 20 304
D 35 13 200
E 32 8 150
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DMU Inputs Output
$ X_1 $ $ X_2 $ $ Y $
A 40 7 210
B 32 12 105
C 52 20 304
D 35 13 200
E 32 8 150
Table 2.  Scale efficiencies and DMUs rankings under optimistic, pessimistic, and double frontier
DMU Optimistic perspective Pessimistic perspective Double frontier
$ E_{opt}^{\ast} $ $ T_{opt}^{\ast} $ $ SE_{opt} $ $ R $ $ E_{pess}^{\ast} $ $ T_{pess}^{\ast} $ $ SE_{pess} $ $ R $ $ SE_{df} $ $ R $
A 1.0000 1.0000 1.0000 1 1.6000 1.0638 1.5040 2 1.2264 2
B 0.5639 1.0000 0.5639 5 1.0000 1.0000 1.0000 5 0.7509 5
C 1.0000 1.0000 1.0000 1 1.7371 1.0000 1.7371 1 1.3180 1
D 0.9838 1.0000 0.9838 3 1.7415 1.1871 1.4670 3 1.2013 3
E 0.8580 1.0000 0.8580 4 1.4286 1.1413 1.2517 4 1.0363 4
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DMU Optimistic perspective Pessimistic perspective Double frontier
$ E_{opt}^{\ast} $ $ T_{opt}^{\ast} $ $ SE_{opt} $ $ R $ $ E_{pess}^{\ast} $ $ T_{pess}^{\ast} $ $ SE_{pess} $ $ R $ $ SE_{df} $ $ R $
A 1.0000 1.0000 1.0000 1 1.6000 1.0638 1.5040 2 1.2264 2
B 0.5639 1.0000 0.5639 5 1.0000 1.0000 1.0000 5 0.7509 5
C 1.0000 1.0000 1.0000 1 1.7371 1.0000 1.7371 1 1.3180 1
D 0.9838 1.0000 0.9838 3 1.7415 1.1871 1.4670 3 1.2013 3
E 0.8580 1.0000 0.8580 4 1.4286 1.1413 1.2517 4 1.0363 4
Table 3.  Summary of inputs and outputs descriptive statistics of China's regional R & D activities
Mean S.D. Minimum Mximum
Employees 1356.5 1111.2 141 5980
Investment in fixed assets (100 million Yuan) 14014.7 16100.7 688 69113
R & D personnel 88048.3 111505.4 2068 424872
R & D projects 11415.9 13971.5 156 53117
R & D expenditure (10000 Yuan) 3084654.9 3771958.6 92528 13765378
Sales volumes (100 million Yuan) 36403 36819.4 1901 141194
Number of patents 26320.2 32272.3 660 146660
Market value (100 million Yuan) 26911.2 57669.3 65 313719
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Mean S.D. Minimum Mximum
Employees 1356.5 1111.2 141 5980
Investment in fixed assets (100 million Yuan) 14014.7 16100.7 688 69113
R & D personnel 88048.3 111505.4 2068 424872
R & D projects 11415.9 13971.5 156 53117
R & D expenditure (10000 Yuan) 3084654.9 3771958.6 92528 13765378
Sales volumes (100 million Yuan) 36403 36819.4 1901 141194
Number of patents 26320.2 32272.3 660 146660
Market value (100 million Yuan) 26911.2 57669.3 65 313719
Table Appendix.1.  Data of R&D activities of 30 provincial-level regions in mainland China for 2014
Region $ X_1 $ $ X_2 $ $ X_3 $ $ X_4 $ $ X_5 $ $ Z_1 $ $ Z_2 $ $ Y $
1 Beijing 5980 11582 57761 2335010 9010 18228 78129 313719
2 Tianjin 1069 16877 79014 3228057 15055 27391 23391 38856
3 Hebei 1448 20762 75142 2606711 8714 46685 8332 2922
4 Shanxi 733 4581 35775 1247027 2726 15214 6107 4846
5 Inner Mongolia 622 14709 27068 1080287 2265 19517 1924 1394
6 Liaoning 1690 26103 63374 3242303 8608 48764 18417 21746
7 Jilin 788 12383 24395 789431 2264 22964 5288 2858
8 Heilongjiang 1154 9229 37509 955820 4324 13139 13468 12028
9 Shanghai 2254 5467 93868 4492192 13821 32458 39133 59245
10 Jiangsu 2151 60663 422865 13765378 53117 141194 146660 54316
11 Zhejiang 1610 9149 290339 7681473 45679 64914 52406 8725
12 Anhui 958 21948 95287 2847303 14648 36505 49960 16983
13 Fujian 848 5269 110892 3153831 10949 37373 12529 3919
14 Jiangxi 554 5440 28803 1284642 4385 28727 4688 5076
15 Shandong 1842 69113 230800 11755482 34353 139627 77298 24929
16 Henan 1639 13262 134256 3372310 12635 67149 19646 4079
17 Hubei 1517 9427 91456 3629506 9955 42012 22536 58068
18 Hunan 1292 20605 77428 3100446 9393 34394 14474 9793
19 Guangdong 3193 16477 424872 13752869 42941 116336 75147 41325
20 Guangxi 977 7054 22793 848808 3260 19629 22237 1158
21 Hainan 219 1184 3484 111010 934 1901 969 65
22 Chongqing 766 2975 43797 1664720 7879 18439 19418 15620
23 Sichuan 2106 8311 62145 1960112 11027 37400 29926 19905
24 Guizhou 766 1515 15659 410132 1682 9053 8203 2004
25 Yunnan 1011 3215 12980 516572 2102 10022 4732 4792
26 Shaanxi 1777 33830 50753 1606946 6668 19947 24399 64002
27 Gansu 704 6623 14380 464410 1894 7886 4986 11452
28 Qinghai 229 688 2068 92528 156 2475 660 2910
29 Ningxia 141 702 5799 186518 1136 3584 2183 318
30 Xinjiang 657 1297 6688 357812 897 9161 2360 282
$X_1$:Employees, $X_2$:Investment in fixed assets (100 million Yuan) $X_3$:R&D personnel, $X_4$:R&D projects, $X_5$:R&D expenditure (10000 Yuan), $Z_1$:Sales volumes (100 million Yuan, ) $Z_2$:Number of patents, $Y$: Market value (100 million Yuan)
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Region $ X_1 $ $ X_2 $ $ X_3 $ $ X_4 $ $ X_5 $ $ Z_1 $ $ Z_2 $ $ Y $
1 Beijing 5980 11582 57761 2335010 9010 18228 78129 313719
2 Tianjin 1069 16877 79014 3228057 15055 27391 23391 38856
3 Hebei 1448 20762 75142 2606711 8714 46685 8332 2922
4 Shanxi 733 4581 35775 1247027 2726 15214 6107 4846
5 Inner Mongolia 622 14709 27068 1080287 2265 19517 1924 1394
6 Liaoning 1690 26103 63374 3242303 8608 48764 18417 21746
7 Jilin 788 12383 24395 789431 2264 22964 5288 2858
8 Heilongjiang 1154 9229 37509 955820 4324 13139 13468 12028
9 Shanghai 2254 5467 93868 4492192 13821 32458 39133 59245
10 Jiangsu 2151 60663 422865 13765378 53117 141194 146660 54316
11 Zhejiang 1610 9149 290339 7681473 45679 64914 52406 8725
12 Anhui 958 21948 95287 2847303 14648 36505 49960 16983
13 Fujian 848 5269 110892 3153831 10949 37373 12529 3919
14 Jiangxi 554 5440 28803 1284642 4385 28727 4688 5076
15 Shandong 1842 69113 230800 11755482 34353 139627 77298 24929
16 Henan 1639 13262 134256 3372310 12635 67149 19646 4079
17 Hubei 1517 9427 91456 3629506 9955 42012 22536 58068
18 Hunan 1292 20605 77428 3100446 9393 34394 14474 9793
19 Guangdong 3193 16477 424872 13752869 42941 116336 75147 41325
20 Guangxi 977 7054 22793 848808 3260 19629 22237 1158
21 Hainan 219 1184 3484 111010 934 1901 969 65
22 Chongqing 766 2975 43797 1664720 7879 18439 19418 15620
23 Sichuan 2106 8311 62145 1960112 11027 37400 29926 19905
24 Guizhou 766 1515 15659 410132 1682 9053 8203 2004
25 Yunnan 1011 3215 12980 516572 2102 10022 4732 4792
26 Shaanxi 1777 33830 50753 1606946 6668 19947 24399 64002
27 Gansu 704 6623 14380 464410 1894 7886 4986 11452
28 Qinghai 229 688 2068 92528 156 2475 660 2910
29 Ningxia 141 702 5799 186518 1136 3584 2183 318
30 Xinjiang 657 1297 6688 357812 897 9161 2360 282
$X_1$:Employees, $X_2$:Investment in fixed assets (100 million Yuan) $X_3$:R&D personnel, $X_4$:R&D projects, $X_5$:R&D expenditure (10000 Yuan), $Z_1$:Sales volumes (100 million Yuan, ) $Z_2$:Number of patents, $Y$: Market value (100 million Yuan)
Table Appendix.2.  Optimistic, pessimistic, and double frontier's CCR, BCC, and scale efficiencies of the two-stage China's R&D value chain
Profitability stage Marketability stage R&D value chain
No Model CCR BCC SE R CCR BCC SE R CCR BCC SE R
1 Opt 1.000 1.000 1.000 7 1.000 1.000 1.000 1 1.000 1.000 1.000 1
Pess 1.000 1.000 1.000 26 176.7 1.000 176.7 1 176.7 1.000 176.7 1
DF 1.000 1.000 1.000 10 13.29 1.000 13.29 1 13.29 1.000 13.29 1
2 Opt 0.547 0.620 0.883 11 0.403 0.412 0.976 10 0.220 0.256 0.862 9
Pess 1.000 1.000 1.000 28 32.14 20.82 1.544 20 32.14 20.82 1.544 20
DF 0.739 0.787 0.939 14 3.600 2.932 1.227 20 2.663 2.308 1.153 12
3 Opt 0.221 0.946 0.234 29 0.076 0.086 0.875 24 0.016 0.082 0.205 28
Pess 1.181 1.000 1.181 8 2.335 1.000 2.335 12 2.758 1.000 2.758 13
DF 0.511 0.972 0.526 29 0.421 0.294 1.430 13 0.215 0.286 0.752 27
4 Opt 0.400 0.711 0.562 21 0.185 0.195 0.950 18 0.074 0.139 0.534 19
Pess 1.000 1.000 1.000 22 10.31 6.918 1.491 21 10.31 6.918 1.491 22
DF 0.632 0.843 0.750 23 1.385 1.163 1.190 21 0.876 0.981 0.893 23
5 Opt 0.165 0.924 0.178 30 0.141 0.174 0.809 26 0.023 0.161 0.144 29
Pess 1.000 1.000 1.000 21 2.721 1.000 2.721 11 2.721 1.000 2.721 14
DF 0.406 0.961 0.422 30 0.620 0.417 1.484 11 0.252 0.401 0.627 29
6 Opt 0.422 0.781 0.540 22 0.275 0.293 0.939 20 0.116 0.229 0.508 21
Pess 1.239 1.000 1.239 6 13.95 7.438 1.876 18 17.29 7.438 2.325 16
DF 0.723 0.883 0.818 19 1.960 1.476 1.327 18 1.418 1.305 1.087 17
7 Opt 0.362 1.000 0.362 26 0.120 0.133 0.907 23 0.043 0.133 0.329 25
Pess 1.441 1.000 1.441 1 4.535 2.328 1.948 15 6.539 2.328 2.808 11
DF 0.723 1.000 0.723 26 0.740 0.556 1.329 17 0.535 0.556 0.961 21
8 Opt 0.533 0.584 0.912 10 0.217 0.221 0.983 7 0.116 0.129 0.897 8
Pess 1.134 1.000 1.134 10 19.62 16.54 1.186 23 22.26 16.54 1.345 25
DF 0.778 0.764 1.017 7 2.068 1.914 1.080 23 1.609 1.464 1.099 14
9 Opt 1.000 1.000 1.000 4 0.370 0.376 0.984 5 0.370 0.376 0.984 4
Pess 1.082 1.000 1.082 16 33.52 6.038 5.552 5 36.28 6.038 6.009 5
DF 1.040 1.000 1.040 6 3.526 1.508 2.338 4 3.668 1.508 2.432 4
10 Opt 1.000 1.000 1.000 2 0.090 0.173 0.522 28 0.090 0.173 0.522 20
Pess 1.349 1.000 1.349 2 8.236 1.000 8.236 3 11.11 1.000 11.11 3
DF 1.161 1.000 1.161 1 0.862 0.416 2.073 7 1.002 0.416 2.408 5
11 Opt 1.000 1.000 1.000 3 0.040 0.041 0.973 13 0.040 0.041 0.973 7
Pess 1.000 1.000 1.000 30 2.919 1.000 2.919 9 2.919 1.000 2.919 10
DF 1.000 1.000 1.000 9 0.343 0.203 1.685 9 0.343 0.203 1.685 9
12 Opt 1.000 1.000 1.000 5 0.083 0.084 0.986 3 0.083 0.084 0.986 2
Pess 1.657 1.280 1.294 4 8.224 1.000 8.224 4 13.63 1.280 10.65 4
DF 1.287 1.131 1.138 2 0.828 0.290 2.848 3 1.066 0.329 3.241 3
13 Opt 0.523 1.000 0.523 23 0.072 0.077 0.932 21 0.037 0.077 0.488 22
Pess 1.000 1.000 1.000 25 3.542 1.823 1.942 16 3.542 1.823 1.942 17
DF 0.723 1.000 0.723 25 0.506 0.376 1.346 15 0.366 0.376 0.974 20
14 Opt 0.321 1.000 0.321 27 0.231 0.266 0.870 25 0.074 0.266 0.279 27
Pess 1.433 1.327 1.080 17 7.806 2.815 2.773 10 11.19 3.737 2.994 9
DF 0.679 1.152 0.589 27 1.344 0.865 1.553 10 0.912 0.997 0.915 22
15 Opt 0.764 1.000 0.764 15 0.076 0.0803 0.958 17 0.058 0.080 0.732 13
Pess 1.178 1.000 1.178 9 5.093 1.000 5.093 6 6.004 1.000 6.004 6
DF 0.949 1.000 0.949 12 0.626 0.283 2.208 5 0.594 0.283 2.096 6
16 Opt 0.458 1.000 0.458 25 0.047 0.051 0.920 22 0.021 0.051 0.421 24
Pess 1.294 1.000 1.294 5 2.159 1.000 2.159 14 2.794 1.000 2.794 12
DF 0.770 1.000 0.770 21 0.320 0.227 1.410 14 0.246 0.227 1.085 18
17 Opt 0.619 0.868 0.713 16 0.613 0.640 0.958 16 0.380 0.555 0.683 15
Pess 1.153 1.135 1.016 18 39.52 21.06 1.876 17 45.60 23.91 1.907 18
DF 0.845 0.992 0.851 17 4.925 3.672 1.341 16 4.163 3.646 1.141 13
18 Opt 0.360 0.629 0.572 20 0.159 0.167 0.947 19 0.057 0.105 0.541 17
Pess 1.000 1.000 1.000 23 8.711 4.929 1.767 19 8.711 4.929 1.767 19
DF 0.600 0.793 0.756 22 1.176 0.909 1.293 19 0.706 0.722 0.978 19
19 Opt 0.848 1.000 0.848 12 0.132 0.136 0.964 14 0.112 0.136 0.818 10
Pess 1.000 1.000 1.000 20 9.085 1.857 4.890 7 9.085 1.857 4.890 7
DF 0.920 1.000 0.920 15 1.095 0.504 2.172 6 1.008 0.504 2.000 8
20 Opt 1.000 1.000 1.000 8 0.012 0.012 0.984 6 0.012 0.012 0.984 5
Pess 2.240 2.033 1.102 12 1.306 1.000 1.306 22 2.928 2.033 1.440 23
DF 1.497 1.426 1.049 4 0.129 0.113 1.134 22 0.193 0.162 1.190 11
21 Opt 0.298 0.981 0.304 28 0.015 1.000 0.015 30 0.004 0.981 0.004 30
Pess 1.000 1.000 1.000 29 1.000 1.000 1.000 30 1.000 1.000 1.000 30
DF 0.546 0.990 0.551 28 0.126 1.000 0.126 30 0.069 0.990 0.069 30
22 Opt 1.000 1.000 1.000 6 0.196 0.199 0.982 9 0.196 0.199 0.982 6
Pess 1.535 1.372 1.118 11 16.23 14.84 1.093 28 24.92 20.38 1.222 28
DF 1.239 1.171 1.057 3 1.785 1.722 1.036 27 2.212 2.018 1.096 16
23 Opt 0.771 0.958 0.805 14 0.161 0.165 0.973 12 0.124 0.158 0.784 12
Pess 1.832 1.664 1.100 14 12.24 5.656 2.164 13 22.43 9.414 2.383 15
DF 1.189 1.263 0.941 13 1.404 0.967 1.451 12 1.670 1.221 1.366 10
24 Opt 0.815 1.000 0.815 13 0.059 0.060 0.983 8 0.048 0.060 0.801 11
Pess 1.597 1.197 1.333 3 4.875 4.323 1.127 26 7.786 5.177 1.503 21
DF 1.140 1.094 1.042 5 0.538 0.511 1.053 24 0.614 0.559 1.097 15
25 Opt 0.319 0.640 0.499 24 0.239 0.249 0.962 15 0.076 0.159 0.480 23
Pess 1.101 1.000 1.101 13 14.44 12.70 1.137 25 15.90 12.70 1.252 26
DF 0.593 0.800 0.741 24 1.860 1.778 1.046 26 1.103 1.423 0.775 26
26 Opt 0.589 0.590 0.998 9 0.642 0.652 0.985 4 0.378 0.384 0.984 3
Pess 1.000 1.000 1.000 27 59.28 13.62 4.352 8 59.28 13.62 4.352 8
DF 0.767 0.768 0.999 11 6.173 2.980 2.071 8 4.739 2.289 2.070 7
27 Opt 0.391 0.646 0.606 19 0.551 0.565 0.975 11 0.216 0.365 0.591 16
Pess 1.085 1.000 1.085 15 38.60 34.25 1.127 27 41.92 34.25 1.224 27
DF 0.652 0.804 0.811 20 4.613 4.399 1.048 25 3.009 3.537 0.850 25
28 Opt 1.000 1.000 1.000 1 0.535 1.000 0.535 27 0.535 1.000 0.535 18
Pess 1.000 1.000 1.000 19 42.81 1.000 42.81 2 42.81 1.000 42.81 2
DF 1.000 1.000 1.000 8 4.789 1.000 4.789 2 4.789 1.000 4.789 2
29 Opt 0.712 1.000 0.712 17 0.034 0.035 0.990 2 0.024 0.035 0.705 14
Pess 1.695 1.695 1.000 24 2.325 2.217 1.048 29 3.944 3.760 1.048 29
DF 1.099 1.302 0.843 18 0.284 0.279 1.019 28 0.313 0.364 0.860 24
30 Opt 0.633 0.956 0.662 18 0.014 0.029 0.491 29 0.009 0.027 0.325 26
Pess 1.217 1.000 1.217 7 1.146 1.000 1.146 24 1.395 1.000 1.395 24
DF 0.878 0.978 0.898 16 0.127 0.170 0.750 29 0.112 0.166 0.674 28
Opt 0.636 0.894 0.709 0.226 0.286 0.878 0.152 0.248 0.636
Pess 1.248 1.123 1.111 19.51 6.406 9.787 21.53 6.965 10.14
DF 0.869 0.996 0.867 2.049 1.131 1.938 1.792 1.041 1.789
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Profitability stage Marketability stage R&D value chain
No Model CCR BCC SE R CCR BCC SE R CCR BCC SE R
1 Opt 1.000 1.000 1.000 7 1.000 1.000 1.000 1 1.000 1.000 1.000 1
Pess 1.000 1.000 1.000 26 176.7 1.000 176.7 1 176.7 1.000 176.7 1
DF 1.000 1.000 1.000 10 13.29 1.000 13.29 1 13.29 1.000 13.29 1
2 Opt 0.547 0.620 0.883 11 0.403 0.412 0.976 10 0.220 0.256 0.862 9
Pess 1.000 1.000 1.000 28 32.14 20.82 1.544 20 32.14 20.82 1.544 20
DF 0.739 0.787 0.939 14 3.600 2.932 1.227 20 2.663 2.308 1.153 12
3 Opt 0.221 0.946 0.234 29 0.076 0.086 0.875 24 0.016 0.082 0.205 28
Pess 1.181 1.000 1.181 8 2.335 1.000 2.335 12 2.758 1.000 2.758 13
DF 0.511 0.972 0.526 29 0.421 0.294 1.430 13 0.215 0.286 0.752 27
4 Opt 0.400 0.711 0.562 21 0.185 0.195 0.950 18 0.074 0.139 0.534 19
Pess 1.000 1.000 1.000 22 10.31 6.918 1.491 21 10.31 6.918 1.491 22
DF 0.632 0.843 0.750 23 1.385 1.163 1.190 21 0.876 0.981 0.893 23
5 Opt 0.165 0.924 0.178 30 0.141 0.174 0.809 26 0.023 0.161 0.144 29
Pess 1.000 1.000 1.000 21 2.721 1.000 2.721 11 2.721 1.000 2.721 14
DF 0.406 0.961 0.422 30 0.620 0.417 1.484 11 0.252 0.401 0.627 29
6 Opt 0.422 0.781 0.540 22 0.275 0.293 0.939 20 0.116 0.229 0.508 21
Pess 1.239 1.000 1.239 6 13.95 7.438 1.876 18 17.29 7.438 2.325 16
DF 0.723 0.883 0.818 19 1.960 1.476 1.327 18 1.418 1.305 1.087 17
7 Opt 0.362 1.000 0.362 26 0.120 0.133 0.907 23 0.043 0.133 0.329 25
Pess 1.441 1.000 1.441 1 4.535 2.328 1.948 15 6.539 2.328 2.808 11
DF 0.723 1.000 0.723 26 0.740 0.556 1.329 17 0.535 0.556 0.961 21
8 Opt 0.533 0.584 0.912 10 0.217 0.221 0.983 7 0.116 0.129 0.897 8
Pess 1.134 1.000 1.134 10 19.62 16.54 1.186 23 22.26 16.54 1.345 25
DF 0.778 0.764 1.017 7 2.068 1.914 1.080 23 1.609 1.464 1.099 14
9 Opt 1.000 1.000 1.000 4 0.370 0.376 0.984 5 0.370 0.376 0.984 4
Pess 1.082 1.000 1.082 16 33.52 6.038 5.552 5 36.28 6.038 6.009 5
DF 1.040 1.000 1.040 6 3.526 1.508 2.338 4 3.668 1.508 2.432 4
10 Opt 1.000 1.000 1.000 2 0.090 0.173 0.522 28 0.090 0.173 0.522 20
Pess 1.349 1.000 1.349 2 8.236 1.000 8.236 3 11.11 1.000 11.11 3
DF 1.161 1.000 1.161 1 0.862 0.416 2.073 7 1.002 0.416 2.408 5
11 Opt 1.000 1.000 1.000 3 0.040 0.041 0.973 13 0.040 0.041 0.973 7
Pess 1.000 1.000 1.000 30 2.919 1.000 2.919 9 2.919 1.000 2.919 10
DF 1.000 1.000 1.000 9 0.343 0.203 1.685 9 0.343 0.203 1.685 9
12 Opt 1.000 1.000 1.000 5 0.083 0.084 0.986 3 0.083 0.084 0.986 2
Pess 1.657 1.280 1.294 4 8.224 1.000 8.224 4 13.63 1.280 10.65 4
DF 1.287 1.131 1.138 2 0.828 0.290 2.848 3 1.066 0.329 3.241 3
13 Opt 0.523 1.000 0.523 23 0.072 0.077 0.932 21 0.037 0.077 0.488 22
Pess 1.000 1.000 1.000 25 3.542 1.823 1.942 16 3.542 1.823 1.942 17
DF 0.723 1.000 0.723 25 0.506 0.376 1.346 15 0.366 0.376 0.974 20
14 Opt 0.321 1.000 0.321 27 0.231 0.266 0.870 25 0.074 0.266 0.279 27
Pess 1.433 1.327 1.080 17 7.806 2.815 2.773 10 11.19 3.737 2.994 9
DF 0.679 1.152 0.589 27 1.344 0.865 1.553 10 0.912 0.997 0.915 22
15 Opt 0.764 1.000 0.764 15 0.076 0.0803 0.958 17 0.058 0.080 0.732 13
Pess 1.178 1.000 1.178 9 5.093 1.000 5.093 6 6.004 1.000 6.004 6
DF 0.949 1.000 0.949 12 0.626 0.283 2.208 5 0.594 0.283 2.096 6
16 Opt 0.458 1.000 0.458 25 0.047 0.051 0.920 22 0.021 0.051 0.421 24
Pess 1.294 1.000 1.294 5 2.159 1.000 2.159 14 2.794 1.000 2.794 12
DF 0.770 1.000 0.770 21 0.320 0.227 1.410 14 0.246 0.227 1.085 18
17 Opt 0.619 0.868 0.713 16 0.613 0.640 0.958 16 0.380 0.555 0.683 15
Pess 1.153 1.135 1.016 18 39.52 21.06 1.876 17 45.60 23.91 1.907 18
DF 0.845 0.992 0.851 17 4.925 3.672 1.341 16 4.163 3.646 1.141 13
18 Opt 0.360 0.629 0.572 20 0.159 0.167 0.947 19 0.057 0.105 0.541 17
Pess 1.000 1.000 1.000 23 8.711 4.929 1.767 19 8.711 4.929 1.767 19
DF 0.600 0.793 0.756 22 1.176 0.909 1.293 19 0.706 0.722 0.978 19
19 Opt 0.848 1.000 0.848 12 0.132 0.136 0.964 14 0.112 0.136 0.818 10
Pess 1.000 1.000 1.000 20 9.085 1.857 4.890 7 9.085 1.857 4.890 7
DF 0.920 1.000 0.920 15 1.095 0.504 2.172 6 1.008 0.504 2.000 8
20 Opt 1.000 1.000 1.000 8 0.012 0.012 0.984 6 0.012 0.012 0.984 5
Pess 2.240 2.033 1.102 12 1.306 1.000 1.306 22 2.928 2.033 1.440 23
DF 1.497 1.426 1.049 4 0.129 0.113 1.134 22 0.193 0.162 1.190 11
21 Opt 0.298 0.981 0.304 28 0.015 1.000 0.015 30 0.004 0.981 0.004 30
Pess 1.000 1.000 1.000 29 1.000 1.000 1.000 30 1.000 1.000 1.000 30
DF 0.546 0.990 0.551 28 0.126 1.000 0.126 30 0.069 0.990 0.069 30
22 Opt 1.000 1.000 1.000 6 0.196 0.199 0.982 9 0.196 0.199 0.982 6
Pess 1.535 1.372 1.118 11 16.23 14.84 1.093 28 24.92 20.38 1.222 28
DF 1.239 1.171 1.057 3 1.785 1.722 1.036 27 2.212 2.018 1.096 16
23 Opt 0.771 0.958 0.805 14 0.161 0.165 0.973 12 0.124 0.158 0.784 12
Pess 1.832 1.664 1.100 14 12.24 5.656 2.164 13 22.43 9.414 2.383 15
DF 1.189 1.263 0.941 13 1.404 0.967 1.451 12 1.670 1.221 1.366 10
24 Opt 0.815 1.000 0.815 13 0.059 0.060 0.983 8 0.048 0.060 0.801 11
Pess 1.597 1.197 1.333 3 4.875 4.323 1.127 26 7.786 5.177 1.503 21
DF 1.140 1.094 1.042 5 0.538 0.511 1.053 24 0.614 0.559 1.097 15
25 Opt 0.319 0.640 0.499 24 0.239 0.249 0.962 15 0.076 0.159 0.480 23
Pess 1.101 1.000 1.101 13 14.44 12.70 1.137 25 15.90 12.70 1.252 26
DF 0.593 0.800 0.741 24 1.860 1.778 1.046 26 1.103 1.423 0.775 26
26 Opt 0.589 0.590 0.998 9 0.642 0.652 0.985 4 0.378 0.384 0.984 3
Pess 1.000 1.000 1.000 27 59.28 13.62 4.352 8 59.28 13.62 4.352 8
DF 0.767 0.768 0.999 11 6.173 2.980 2.071 8 4.739 2.289 2.070 7
27 Opt 0.391 0.646 0.606 19 0.551 0.565 0.975 11 0.216 0.365 0.591 16
Pess 1.085 1.000 1.085 15 38.60 34.25 1.127 27 41.92 34.25 1.224 27
DF 0.652 0.804 0.811 20 4.613 4.399 1.048 25 3.009 3.537 0.850 25
28 Opt 1.000 1.000 1.000 1 0.535 1.000 0.535 27 0.535 1.000 0.535 18
Pess 1.000 1.000 1.000 19 42.81 1.000 42.81 2 42.81 1.000 42.81 2
DF 1.000 1.000 1.000 8 4.789 1.000 4.789 2 4.789 1.000 4.789 2
29 Opt 0.712 1.000 0.712 17 0.034 0.035 0.990 2 0.024 0.035 0.705 14
Pess 1.695 1.695 1.000 24 2.325 2.217 1.048 29 3.944 3.760 1.048 29
DF 1.099 1.302 0.843 18 0.284 0.279 1.019 28 0.313 0.364 0.860 24
30 Opt 0.633 0.956 0.662 18 0.014 0.029 0.491 29 0.009 0.027 0.325 26
Pess 1.217 1.000 1.217 7 1.146 1.000 1.146 24 1.395 1.000 1.395 24
DF 0.878 0.978 0.898 16 0.127 0.170 0.750 29 0.112 0.166 0.674 28
Opt 0.636 0.894 0.709 0.226 0.286 0.878 0.152 0.248 0.636
Pess 1.248 1.123 1.111 19.51 6.406 9.787 21.53 6.965 10.14
DF 0.869 0.996 0.867 2.049 1.131 1.938 1.792 1.041 1.789
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