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doi: 10.3934/jimo.2021026

## CCR model-based evaluation on the effectiveness and maturity of technological innovation

 1 The Department of Finance, School of Business, China University of Political Science and Law, Beijing, 100088, China 2 The Physics Department, School of Arts and Sciences, Boston University, Boston, MA 02215, USA

* Corresponding author: Liling Lin and Linfeng Zhao

Received  June 2020 Revised  December 2020 Published  February 2021

As there are many indexes for evaluating technological innovation in enterprises, it is hard to quantify all those indexes. Therefore, common evaluation methods cannot be applied to solve the absolute value of the evaluation indexes. Therefore, this study used the nonparametric CCR model based on input-output to estimate the relative value of evaluation index, and took dual programming tool to obtain the judgment basis for the most effective and optimal solution. Based on the software evaluation criteria, this paper proposed the concept of "maturity in technological innovation, " its four levels, and an evaluation standard for maturity. Based on the homogeneity, the paper selected four Beijing enterprises as evaluation samples. After comparing and analyzing the efficiency, scale return, production surface projection and maturity, we found that the evaluation results conform to the reality of sampling enterprises. CCR model was used to evaluate decision-making units with multiple inputs and outputs. The results show that this method can help accurately obtain the relative order and the enterprises' ability to make technological innovation. Thus, CCR model is able to help enterprises formulate policies on technological innovation.

Citation: Liling Lin, Linfeng Zhao. CCR model-based evaluation on the effectiveness and maturity of technological innovation. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021026
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Input and output of evaluation departments
 No. Weight Decision-making unit 1 2 ... j ... n Input data 1 $v_{1}$ $x_{11}$ $x_{12}$ ... $x_{1j}$ ... $x_{1n}$ 2 $v_{2}$ $x_{21}$ $x_{22}$ ... $x_{2j}$ ... $x_{2n}$ ... ... ... ... $m$ $v_{m}$ $x_{m1}$ $x_{m2}$ ... $x_{mj}$ ... $x_{mn}$ Output data 1 $u_{1}$ $y_{11}$ $y_{12}$ ... $y_{1j}$ ... $y_{1n}$ 2 $u_{2}$ $y_{21}$ $y_{22}$ ... $y_{2j}$ ... $y_{2n}$ ... ... ... ... $s$ $u_{s}$ $y_{s1}$ $y_{s2}$ ... $y_{sj}$ ... $y_{sn}$
 No. Weight Decision-making unit 1 2 ... j ... n Input data 1 $v_{1}$ $x_{11}$ $x_{12}$ ... $x_{1j}$ ... $x_{1n}$ 2 $v_{2}$ $x_{21}$ $x_{22}$ ... $x_{2j}$ ... $x_{2n}$ ... ... ... ... $m$ $v_{m}$ $x_{m1}$ $x_{m2}$ ... $x_{mj}$ ... $x_{mn}$ Output data 1 $u_{1}$ $y_{11}$ $y_{12}$ ... $y_{1j}$ ... $y_{1n}$ 2 $u_{2}$ $y_{21}$ $y_{22}$ ... $y_{2j}$ ... $y_{2n}$ ... ... ... ... $s$ $u_{s}$ $y_{s1}$ $y_{s2}$ ... $y_{sj}$ ... $y_{sn}$
Data of sample enterprises
 Enter-prises Years R & D funds/product sales revenue (A13) (%)x1 Number of full-time R&D personnel /number of employees (A22) (%)x2 Technology introduction and transformation cost/ product sales revenue (A31) (%)x3 Annual per capita income of R&D personnel /annual per capita income of enterprises (B21) X4 Innovation strategy (B11) X5 Technical level (C11) x6 Number of patents and proprietary technology (C41) x7 Equipment level (D11) x8 Marketing costs for new products /new product sales revenue (E32) (%) x9 Number of full -time sales personnel/ number of employees (E42) x10 New product sales revenue /total product sales revenue (F12) (%) Y1 Enterprise I 01 3.14 13.85 5.38 1.01 75 1 15 0.8 3.81 0.24 24.94 02 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 03 3.65 15.98 6.43 1.11 80 1 15 0.8 6.34 0.26 35.51 04 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 05 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 Enterprise II 01 3.47 14.33 19.69 1.21 70 0.4 9 0.6 20 0.19 7.3 02 4.24 14.51 26.63 1.32 70 0.4 9 0.6 27.91 0.2 8.12 03 4.55 14.63 35.06 1.05 70 0.4 9 0.6 34.12 0.22 11.17 04 3.37 15.66 4.84 1.12 70 0.4 10 0.6 32.12 0.22 13.36 05 3.13 16.05 3.2 1.19 70 0.6 10 0.6 29.66 0.22 13 Enterprise III 01 9.1 7.11 3.17 1.51 80 0.6 6 0.4 13.55 0.1 2.16 02 10.94 7.23 4.66 1.62 80 0.6 6 0.4 14.21 0.1 2.22 03 29.66 7.39 2.65 1.68 80 0.6 6 0.4 15.33 0.1 3.27 04 20.91 7.43 3.05 1.65 80 0.6 6 0.4 17.17 0.1 4.68 05 15.66 7.51 0.74 1.69 80 0.6 6 0.4 15.61 0.1 4.95 Enterprise III 01 2.96 22 21.45 1.22 70 0.4 5 0.4 13.23 0.3 8.085 02 3.36 25 0.48 1.52 70 0.4 5 0.4 8.07 0.3 10.01 03 3.52 21 0.76 1.4 75 0.4 5 0.4 7.13 0.3 15.575 04 4.36 29 3.35 1.29 80 0.4 5 0.4 3.9 0.3 22.14 05 7.47 27 2.35 1.06 80 0.4 5 0.4 2.38 0.3 25.69
 Enter-prises Years R & D funds/product sales revenue (A13) (%)x1 Number of full-time R&D personnel /number of employees (A22) (%)x2 Technology introduction and transformation cost/ product sales revenue (A31) (%)x3 Annual per capita income of R&D personnel /annual per capita income of enterprises (B21) X4 Innovation strategy (B11) X5 Technical level (C11) x6 Number of patents and proprietary technology (C41) x7 Equipment level (D11) x8 Marketing costs for new products /new product sales revenue (E32) (%) x9 Number of full -time sales personnel/ number of employees (E42) x10 New product sales revenue /total product sales revenue (F12) (%) Y1 Enterprise I 01 3.14 13.85 5.38 1.01 75 1 15 0.8 3.81 0.24 24.94 02 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 03 3.65 15.98 6.43 1.11 80 1 15 0.8 6.34 0.26 35.51 04 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 05 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 Enterprise II 01 3.47 14.33 19.69 1.21 70 0.4 9 0.6 20 0.19 7.3 02 4.24 14.51 26.63 1.32 70 0.4 9 0.6 27.91 0.2 8.12 03 4.55 14.63 35.06 1.05 70 0.4 9 0.6 34.12 0.22 11.17 04 3.37 15.66 4.84 1.12 70 0.4 10 0.6 32.12 0.22 13.36 05 3.13 16.05 3.2 1.19 70 0.6 10 0.6 29.66 0.22 13 Enterprise III 01 9.1 7.11 3.17 1.51 80 0.6 6 0.4 13.55 0.1 2.16 02 10.94 7.23 4.66 1.62 80 0.6 6 0.4 14.21 0.1 2.22 03 29.66 7.39 2.65 1.68 80 0.6 6 0.4 15.33 0.1 3.27 04 20.91 7.43 3.05 1.65 80 0.6 6 0.4 17.17 0.1 4.68 05 15.66 7.51 0.74 1.69 80 0.6 6 0.4 15.61 0.1 4.95 Enterprise III 01 2.96 22 21.45 1.22 70 0.4 5 0.4 13.23 0.3 8.085 02 3.36 25 0.48 1.52 70 0.4 5 0.4 8.07 0.3 10.01 03 3.52 21 0.76 1.4 75 0.4 5 0.4 7.13 0.3 15.575 04 4.36 29 3.35 1.29 80 0.4 5 0.4 3.9 0.3 22.14 05 7.47 27 2.35 1.06 80 0.4 5 0.4 2.38 0.3 25.69
Definition of input and output indicators
 Types Names Code Definition Unit Enterprise III X1 A13 R & D funds / product sales revenue % X2 A22 Number of full-time R & D personnel / number of employees % X3 A31 Technology introduction and transformation cost / product sales revenue % X4 B21 Annual per capita income of R & D personnel/ annual per capita income of enterprises % X5 B11 Innovation strategy Point X6 C11 Technical level= 1 × international level + 0.6 × domestic level + 0.3 × enterprise level X7 C41 Number of patents and proprietary technology Piece X8 D11 Equipment level = 1 × international advanced level (%) + 0.8 × international general level (%) + 0.6 × domestic advanced level (%) + 0.4 × domestic general level (%) + 0.2 × others X9 E32 Marketing costs for new products/new product sales revenue % X10 E42 Number of full-time sales personnel/ number of employees Output variable Y1 F12 New product sales revenue/total product sales revenue %
 Types Names Code Definition Unit Enterprise III X1 A13 R & D funds / product sales revenue % X2 A22 Number of full-time R & D personnel / number of employees % X3 A31 Technology introduction and transformation cost / product sales revenue % X4 B21 Annual per capita income of R & D personnel/ annual per capita income of enterprises % X5 B11 Innovation strategy Point X6 C11 Technical level= 1 × international level + 0.6 × domestic level + 0.3 × enterprise level X7 C41 Number of patents and proprietary technology Piece X8 D11 Equipment level = 1 × international advanced level (%) + 0.8 × international general level (%) + 0.6 × domestic advanced level (%) + 0.4 × domestic general level (%) + 0.2 × others X9 E32 Marketing costs for new products/new product sales revenue % X10 E42 Number of full-time sales personnel/ number of employees Output variable Y1 F12 New product sales revenue/total product sales revenue %
Optimal solution and relaxation variables
 Enterprises Year $\theta$ $S1^{-}$ $S2^{-}$ $S3^{-}$ $S4^{-}$ $S5^{-}$ $S6^{-}$ $S7^{-}$ $S8^{-}$ $S9^{-}$ $S10^{-}$ $S1^{+}$ Enterprise 1 01 0.8696334 0 1.06315 2.51657 0.16280 7.44845 0.17362 2.60427 0.13889 0 0.02634 0 02 1.000000 0 0 0 0 0 0 0 0 0 0 0 03 0.9852941 0.27588 1.64544 2.55191 0.09853 0 0 0 0 0.85721 0.00985 0 04 1.000000 0 0 0 0 0 0 0 0 0 0 0 05 1.000000 0 0 0 0 0 0 0 0 0 0 0 Enterprise 2 01 0.3938900 0 0 6.98602 0.20658 7.49104 0 1.33744 0.09783 6.92850 0.00430 0 02 0.4257631 0.15493 0 10.54098 0.27466 7.73962 0 1.45311 0.10515 10.91772 0.00741 0 03 0.5837402 0.37323 0 19.37040 0.21680 10.45333 0 1.99557 0.14390 18.59308 0.02118 0 04 0.7169654 0 0 1.94594 0.26629 11.90772 0 3.18738 0.17519 21.15136 0.02213 0 05 0.5354496 0 0 0.18736 0.16405 3.44490 0 0.70910 0.05036 13.86115 0.00275 0 Enterprise 3 01 0.1498335 1.16151 0.20767 0.24483 0.16572 7.19201 0.02997 0 0.01199 1.70241 0 0 02 0.1539956 1.47713 0.23192 0.48108 0.18726 7.39179 0.03080 0 0.01232 1.85134 0 0 03 0.2268313 6.42205 0.37790 0.25269 0.28944 10.88790 0.04537 0 0.01815 2.98102 0 0 04 0.3246393 6.35059 0.55383 0.49150 0.40450 15.58269 0.06493 0 0.02597 4.86375 0 0 05 0.3677099 5.12719 0.46604 0 0.47199 17.78455 0.08123 0.11527 0.03556 5.09078 0 0 Enterprise 4 01 0.4552423 0 1.58449 8.64749 0.15586 5.94956 0.01313 0 0.02625 4.65633 0.04266 0 02 0.7785724 0.89338 11.53132 0 0.66614 21.97158 0.03695 0 0.07390 3.89102 0.11639 0 03 1.000000 0 0 0 0 0 0 0 0 0 0 0 04 1.000000 0 0 0 0 0 0 0 0 0 0 0 05 1.000000 0 0 0 0 0 0 0 0 0 0 0
 Enterprises Year $\theta$ $S1^{-}$ $S2^{-}$ $S3^{-}$ $S4^{-}$ $S5^{-}$ $S6^{-}$ $S7^{-}$ $S8^{-}$ $S9^{-}$ $S10^{-}$ $S1^{+}$ Enterprise 1 01 0.8696334 0 1.06315 2.51657 0.16280 7.44845 0.17362 2.60427 0.13889 0 0.02634 0 02 1.000000 0 0 0 0 0 0 0 0 0 0 0 03 0.9852941 0.27588 1.64544 2.55191 0.09853 0 0 0 0 0.85721 0.00985 0 04 1.000000 0 0 0 0 0 0 0 0 0 0 0 05 1.000000 0 0 0 0 0 0 0 0 0 0 0 Enterprise 2 01 0.3938900 0 0 6.98602 0.20658 7.49104 0 1.33744 0.09783 6.92850 0.00430 0 02 0.4257631 0.15493 0 10.54098 0.27466 7.73962 0 1.45311 0.10515 10.91772 0.00741 0 03 0.5837402 0.37323 0 19.37040 0.21680 10.45333 0 1.99557 0.14390 18.59308 0.02118 0 04 0.7169654 0 0 1.94594 0.26629 11.90772 0 3.18738 0.17519 21.15136 0.02213 0 05 0.5354496 0 0 0.18736 0.16405 3.44490 0 0.70910 0.05036 13.86115 0.00275 0 Enterprise 3 01 0.1498335 1.16151 0.20767 0.24483 0.16572 7.19201 0.02997 0 0.01199 1.70241 0 0 02 0.1539956 1.47713 0.23192 0.48108 0.18726 7.39179 0.03080 0 0.01232 1.85134 0 0 03 0.2268313 6.42205 0.37790 0.25269 0.28944 10.88790 0.04537 0 0.01815 2.98102 0 0 04 0.3246393 6.35059 0.55383 0.49150 0.40450 15.58269 0.06493 0 0.02597 4.86375 0 0 05 0.3677099 5.12719 0.46604 0 0.47199 17.78455 0.08123 0.11527 0.03556 5.09078 0 0 Enterprise 4 01 0.4552423 0 1.58449 8.64749 0.15586 5.94956 0.01313 0 0.02625 4.65633 0.04266 0 02 0.7785724 0.89338 11.53132 0 0.66614 21.97158 0.03695 0 0.07390 3.89102 0.11639 0 03 1.000000 0 0 0 0 0 0 0 0 0 0 0 04 1.000000 0 0 0 0 0 0 0 0 0 0 0 05 1.000000 0 0 0 0 0 0 0 0 0 0 0
Maturity of technological innovation of sample enterprises
 Enterprises Year DEA optimal solution Technological innovation maturity Enterprise 1 2014 0.8696334 Less maturity 2015 1.000000 Full maturity 2016 0.9852941 Less maturity 2017 1.000000 Full maturity 2018 1.000000 Full maturity Enterprise 2 2014 0.39389 Immaturity 2015 0.425763 Immaturity 2016 0.58374 Less maturity 2017 0.716965 Less maturity 2018 0.53545 Less maturity Enterprise 3 2014 0.149834 Immaturity 2015 0.153996 Immaturity 2016 0.226831 Immaturity 2017 0.324639 Immaturity 2018 0.36771 Immaturity Enterprise 4 2014 0.455242 Immaturity 2015 0.778572 Less maturity 2016 1.000000 Full maturity 2017 1.000000 Full maturity 2018 1.000000 Full maturity
 Enterprises Year DEA optimal solution Technological innovation maturity Enterprise 1 2014 0.8696334 Less maturity 2015 1.000000 Full maturity 2016 0.9852941 Less maturity 2017 1.000000 Full maturity 2018 1.000000 Full maturity Enterprise 2 2014 0.39389 Immaturity 2015 0.425763 Immaturity 2016 0.58374 Less maturity 2017 0.716965 Less maturity 2018 0.53545 Less maturity Enterprise 3 2014 0.149834 Immaturity 2015 0.153996 Immaturity 2016 0.226831 Immaturity 2017 0.324639 Immaturity 2018 0.36771 Immaturity Enterprise 4 2014 0.455242 Immaturity 2015 0.778572 Less maturity 2016 1.000000 Full maturity 2017 1.000000 Full maturity 2018 1.000000 Full maturity
Adjustment range of evaluation indexes of enterprise 1
 Types Names Definition Change in value (unit) Adjusted values (unit) Input variables X1 R & D funds/product sales revenue -0.409351124 (%) 2.730648876 (%) X2 Number of full-time R & D personnel/ number of employees -2.86872741 (%) 10.98127259 (%) X3 Technology introduction and transformation cost / product sales revenue -3.217942308 (%) 2.162057692 (%) X4 Annual per capita income of R & D personnel / annual per capita income of enterprises -0.294470266 (%) 0.715529734 (%) X5 Innovation strategy -17.225945 (Score) 57.774055 (Score) X6 Technical level -0.3039866 0.6960134 X7 Number of patents and proprietary technology -4.559769 10.440231 X8 Equipment level -0.24318328 0.55681672 X9 Marketing costs for new products / new product sales revenue -0.496696746 (%) 3.313303254 (%) X10 Number of full-time sales personnel / number of employees -0.057627984 0.182372016 Output variable Y1 New product sales revenue / total product sales revenue 0 (%) 24.94
 Types Names Definition Change in value (unit) Adjusted values (unit) Input variables X1 R & D funds/product sales revenue -0.409351124 (%) 2.730648876 (%) X2 Number of full-time R & D personnel/ number of employees -2.86872741 (%) 10.98127259 (%) X3 Technology introduction and transformation cost / product sales revenue -3.217942308 (%) 2.162057692 (%) X4 Annual per capita income of R & D personnel / annual per capita income of enterprises -0.294470266 (%) 0.715529734 (%) X5 Innovation strategy -17.225945 (Score) 57.774055 (Score) X6 Technical level -0.3039866 0.6960134 X7 Number of patents and proprietary technology -4.559769 10.440231 X8 Equipment level -0.24318328 0.55681672 X9 Marketing costs for new products / new product sales revenue -0.496696746 (%) 3.313303254 (%) X10 Number of full-time sales personnel / number of employees -0.057627984 0.182372016 Output variable Y1 New product sales revenue / total product sales revenue 0 (%) 24.94
 Enterprise Year Value X Y X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Y1 I 01 S 0 1.06315 2.51657 0.1628 7.44845 0.17362 2.60427 0.13889 0 0.02634 0 T1 3.14 13.85 5.38 1.01 75 1 15 0.8 3.81 0.24 24.94 T2 2.730648876 10.98127259 2.162057692 0.715529734 57.774055 0.6960134 10.440231 0.55681672 3.313303254 0.182372016 24.94 C 0.409351124 2.86872741 3.217942308 0.294470266 17.225945 0.3039866 4.559769 0.24318328 0.496696746 0.057627984 0 02 S 0 0 0 0 0 0 0 0 0 0 0 T1 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 T2 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 C 0 0 0 0 0 0 0 0 0 0 0 03 S 0.27588 1.64544 2.55191 0.09853 0 0 0 0 0.85721 0.00985 0 T1 3.65 15.98 6.43 1.11 80 1 15 0.8 6.34 0.26 35.51 T2 3.320443465 14.09955972 3.783531063 0.995146451 78.823528 0.9852941 14.7794115 0.78823528 5.389554594 0.246326466 35.51 C 0.329556535 1.880440282 2.646468937 0.114853549 1.176472 0.0147059 0.2205885 0.01176472 0.950445406 0.013673534 0 04 S 0 0 0 0 0 0 0 0 0 0 0 T1 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 T2 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 C 0 0 0 0 0 0 0 0 0 0 0 05 S 0 0 0 0 0 0 0 0 0 0 0 T1 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 T2 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 C 0 0 0 0 0 0 0 0 0 0 0 Note: in the table, S represents slack variable; T1 refers to variable value before adjustment; T2 means variable value after adjustment; and C represents difference.
 Enterprise Year Value X Y X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Y1 I 01 S 0 1.06315 2.51657 0.1628 7.44845 0.17362 2.60427 0.13889 0 0.02634 0 T1 3.14 13.85 5.38 1.01 75 1 15 0.8 3.81 0.24 24.94 T2 2.730648876 10.98127259 2.162057692 0.715529734 57.774055 0.6960134 10.440231 0.55681672 3.313303254 0.182372016 24.94 C 0.409351124 2.86872741 3.217942308 0.294470266 17.225945 0.3039866 4.559769 0.24318328 0.496696746 0.057627984 0 02 S 0 0 0 0 0 0 0 0 0 0 0 T1 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 T2 3.37 14.31 3.84 1.01 80 1 15 0.8 5.47 0.25 36.04 C 0 0 0 0 0 0 0 0 0 0 0 03 S 0.27588 1.64544 2.55191 0.09853 0 0 0 0 0.85721 0.00985 0 T1 3.65 15.98 6.43 1.11 80 1 15 0.8 6.34 0.26 35.51 T2 3.320443465 14.09955972 3.783531063 0.995146451 78.823528 0.9852941 14.7794115 0.78823528 5.389554594 0.246326466 35.51 C 0.329556535 1.880440282 2.646468937 0.114853549 1.176472 0.0147059 0.2205885 0.01176472 0.950445406 0.013673534 0 04 S 0 0 0 0 0 0 0 0 0 0 0 T1 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 T2 5.05 17.44 1.1 1.1 85 1 15 0.8 4.29 0.27 35.27 C 0 0 0 0 0 0 0 0 0 0 0 05 S 0 0 0 0 0 0 0 0 0 0 0 T1 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 T2 5.21 19.19 1.4 1.07 90 1 15 0.8 3.11 0.29 35.35 C 0 0 0 0 0 0 0 0 0 0 0 Note: in the table, S represents slack variable; T1 refers to variable value before adjustment; T2 means variable value after adjustment; and C represents difference.
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