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 |
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.
Citation: |
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 |
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 |
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 |
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) |
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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Two-stage DEA system
Two-stage R & D value chain
Efficiency evaluation from an optimistic perspective by area
Efficiency evaluation from a pessimistic perspective by area
Efficiency evaluation from double frontier perspective by area