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doi: 10.3934/jimo.2022081
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A novel simultaneous grey model parameter optimization method and its application to predicting private car ownership and transportation economy

1. 

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, China

2. 

College of Automobile and Traffic Engineering, Nanjing Forestry University, Nanjing, 210037, China

*Corresponding author: Maolin Cheng

Received  October 2021 Revised  February 2022 Early access May 2022

Fund Project: This work is supported in part by the National Natural Science Foundation of China(11401418)

Common models used for grey system predictions include the GM (1, 1), GM (1, N), and GM (N, 1). Their whitening equations are ordinary differential equations. However, the objects and factors are generally characterized by mutual restrictions and variable interactions. Thus, the relationship cannot be adequately described using a single differential equation. Therefore, this paper proposes a novel simultaneous grey model. A parameter optimization method is developed, and the time response equation of a simultaneous grey model with 2 interactive variables is derived. The model has high simulation and prediction precision because the parameter optimization's objective function minimizes the average relative errors of the simulation and prediction. The proposed model with two mutually influencing variables is used to predict the private car ownership and the added value of the transportation industry in China. The results indicate that the simultaneous grey model has significantly higher precision than the conventional single grey model. The novel model and method improve the grey modeling system and are significant for in-depth studies, popularization, and application of grey models.

Citation: Maolin Cheng, Zhun Cheng. A novel simultaneous grey model parameter optimization method and its application to predicting private car ownership and transportation economy. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022081
References:
[1]

M. L. Cheng and B. Liu, An extended grey GM(1, 1) power model and its application, J. Statistics and Information, 36 (2021), 3-11. 

[2]

J. L. Deng, Grey Prediction and Grey Decision, Huazhong University of Science and Technology Press, Wuhan, (2002).

[3]

S. Ding, A novel self-adapting intelligent grey model for forecasting China's natural-gas demand, Energy, 162 (2018), 393-407.  doi: 10.1016/j.energy.2018.08.040.

[4]

S. Ding, A novel discrete grey multivariable model and its application in forecasting the output value of China's high-tech industries, Computers & Industrial Engineering, 127 (2019), 749-760.  doi: 10.1016/j.cie.2018.11.016.

[5]

Y. X. Jiang and Q. S. Zhang, Background-value optimization of model GM(1, 1), Chinese J. Management Science, 23 (2015), 146-151. 

[6]

Y. Q. JiaoJ. FengL. L. YangG. S. Zhao and H. J. Fan, Prediction study of grey model GM(1, 1) for growth volume of pine, Math. Practice and Theory, 50 (2020), 83-88. 

[7]

B. Li and Y. Wei, Optimizes grey derivative of GM(1, 1), Systems Engineering-Theory & Practice, 29 (2009), 100-105. 

[8]

X. LiL. L. Zhao and X. Z. Jin, Prediction on dynamic tendency of the number of COVID-19 patients in Wuhan –-Based on grey prediction model, Soft Science of Health, 34 (2020), 85-87. 

[9]

S. F. Liu, T. B. Guo and Y. G. Dang, Grey system theory and its application, Science Press, Beijing, (1999).

[10]

Y. M. Ma and S. C. Wang, Construction and application of improved GM(1, 1) Power model, J. Quantitative Economics, 36 (2019), 84-88. 

[11]

J. Ruan, Difference equations and ordinary differential, Fudan University Press, Shanghai, (2002).

[12]

Q. Tong, Weighted non-equal interval gray GM(1, 1) model based on function $\cot ({x^\alpha })$ transformation and its application, Mathematics in Practice and Theory, 51 (2021), 209-215. 

[13]

Z. X. Wang and Q. Li, Modelling the nonlinear relationship between CO2 emissions and economic growth using a PSO algorithm-based grey Verhulst model, J. Cleaner Production, 207 (2019), 214-224.  doi: 10.1016/j.jclepro.2018.10.010.

[14]

Z. X. Wang and Y. F. Zhao, GM (1, 1) model with seasonal dummy variables and its application, Systems Engineering - Theory & Practice, 40 (2020), 2981-2990. 

[15]

M. Xie and L. F. Wu, Short-term traffic flow prediction based on GM (1, N) power model optimized by rough set algorithm, Mathematics In Practice and Theory, 51 (2021), 241-249. 

[16]

N. Xu and Y. G. Dang, An optimized grey GM(2, 1) model and forecasting of highway subgrade settlement, Mathematical Problems in Engineering, 2015 (2015), 1-6.  doi: 10.1155/2015/606707.

[17]

N. XuY. G. Dang and S. Ding, Optimization method of background value in GM(1, 1) model based on least error, Control and Decision, 30 (2015), 283-288. 

[18]

N. XuY. G. Dang and Y. D. Gong, Novel grey prediction model with nonlinear optimized time response method for forecasting of electricity consumption in China, Energy, 118 (2017), 473-480.  doi: 10.1016/j.energy.2016.10.003.

[19]

X. Q. Yan, Research on forecast of total freight volume in Guangdong province based on grey forecasting model, Math. Practice and Theory, 50 (2020), 294-302. 

[20]

B. Zeng and C. Li, Forecasting the natural gas demand in China using a self-adapting intelligent grey model, Energy, 112 (2016), 810-825.  doi: 10.1016/j.energy.2016.06.090.

[21]

Y. H. ZhaiR. D. Ren and D. D. Ren, Application of grey prediction model in fatigue life prediction under compressive stress, Mechanical Research & Application, 33 (2020), 36-43. 

[22]

L. Zhang and D. H. Zhou, Uncertainty evaluation of pulverized coal fineness of boiler based on grey model, J. Anhui University of Technology (Natural Science), 37 (2020), 351-355. 

[23]

Y. Zhong, L. Q. Peng and B. W. Liu, Ordinary differential equations and their solution with Maple and Matlab, Tsinghua University Press, Beijing, (2007).

show all references

References:
[1]

M. L. Cheng and B. Liu, An extended grey GM(1, 1) power model and its application, J. Statistics and Information, 36 (2021), 3-11. 

[2]

J. L. Deng, Grey Prediction and Grey Decision, Huazhong University of Science and Technology Press, Wuhan, (2002).

[3]

S. Ding, A novel self-adapting intelligent grey model for forecasting China's natural-gas demand, Energy, 162 (2018), 393-407.  doi: 10.1016/j.energy.2018.08.040.

[4]

S. Ding, A novel discrete grey multivariable model and its application in forecasting the output value of China's high-tech industries, Computers & Industrial Engineering, 127 (2019), 749-760.  doi: 10.1016/j.cie.2018.11.016.

[5]

Y. X. Jiang and Q. S. Zhang, Background-value optimization of model GM(1, 1), Chinese J. Management Science, 23 (2015), 146-151. 

[6]

Y. Q. JiaoJ. FengL. L. YangG. S. Zhao and H. J. Fan, Prediction study of grey model GM(1, 1) for growth volume of pine, Math. Practice and Theory, 50 (2020), 83-88. 

[7]

B. Li and Y. Wei, Optimizes grey derivative of GM(1, 1), Systems Engineering-Theory & Practice, 29 (2009), 100-105. 

[8]

X. LiL. L. Zhao and X. Z. Jin, Prediction on dynamic tendency of the number of COVID-19 patients in Wuhan –-Based on grey prediction model, Soft Science of Health, 34 (2020), 85-87. 

[9]

S. F. Liu, T. B. Guo and Y. G. Dang, Grey system theory and its application, Science Press, Beijing, (1999).

[10]

Y. M. Ma and S. C. Wang, Construction and application of improved GM(1, 1) Power model, J. Quantitative Economics, 36 (2019), 84-88. 

[11]

J. Ruan, Difference equations and ordinary differential, Fudan University Press, Shanghai, (2002).

[12]

Q. Tong, Weighted non-equal interval gray GM(1, 1) model based on function $\cot ({x^\alpha })$ transformation and its application, Mathematics in Practice and Theory, 51 (2021), 209-215. 

[13]

Z. X. Wang and Q. Li, Modelling the nonlinear relationship between CO2 emissions and economic growth using a PSO algorithm-based grey Verhulst model, J. Cleaner Production, 207 (2019), 214-224.  doi: 10.1016/j.jclepro.2018.10.010.

[14]

Z. X. Wang and Y. F. Zhao, GM (1, 1) model with seasonal dummy variables and its application, Systems Engineering - Theory & Practice, 40 (2020), 2981-2990. 

[15]

M. Xie and L. F. Wu, Short-term traffic flow prediction based on GM (1, N) power model optimized by rough set algorithm, Mathematics In Practice and Theory, 51 (2021), 241-249. 

[16]

N. Xu and Y. G. Dang, An optimized grey GM(2, 1) model and forecasting of highway subgrade settlement, Mathematical Problems in Engineering, 2015 (2015), 1-6.  doi: 10.1155/2015/606707.

[17]

N. XuY. G. Dang and S. Ding, Optimization method of background value in GM(1, 1) model based on least error, Control and Decision, 30 (2015), 283-288. 

[18]

N. XuY. G. Dang and Y. D. Gong, Novel grey prediction model with nonlinear optimized time response method for forecasting of electricity consumption in China, Energy, 118 (2017), 473-480.  doi: 10.1016/j.energy.2016.10.003.

[19]

X. Q. Yan, Research on forecast of total freight volume in Guangdong province based on grey forecasting model, Math. Practice and Theory, 50 (2020), 294-302. 

[20]

B. Zeng and C. Li, Forecasting the natural gas demand in China using a self-adapting intelligent grey model, Energy, 112 (2016), 810-825.  doi: 10.1016/j.energy.2016.06.090.

[21]

Y. H. ZhaiR. D. Ren and D. D. Ren, Application of grey prediction model in fatigue life prediction under compressive stress, Mechanical Research & Application, 33 (2020), 36-43. 

[22]

L. Zhang and D. H. Zhou, Uncertainty evaluation of pulverized coal fineness of boiler based on grey model, J. Anhui University of Technology (Natural Science), 37 (2020), 351-355. 

[23]

Y. Zhong, L. Q. Peng and B. W. Liu, Ordinary differential equations and their solution with Maple and Matlab, Tsinghua University Press, Beijing, (2007).

Table 1.  Modeling results of the grey model GM (1, 1) of China's private car ownership and the added value of the transportation industry
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ GM (1, 1) model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation Value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2931.301 25.6 13152.26 7.93
2007 3 2876.22 14605.1 3504.56 21.8 14455.08 1.03
2008 4 3501.39 16367.6 4189.929 19.7 15886.95 2.94
2009 5 4574.91 16522.4 5009.331 9.5 17460.66 5.68
2010 6 5938.71 18783.6 5988.98 0.846 19190.26 2.16
2011 7 7326.79 21842.0 7160.214 2.27 21091.18 3.44
2012 8 8838.6 23763.2 8560.499 3.15 23180.4 2.45
2013 9 10501.68 26042.7 10234.63 2.54 25476.57 2.17
2014 10 12339.36 28534.4 12236.17 0.836 28000.2 1.87
2015 11 14099.1 30519.5 14629.13 3.76 30773.8 0.833
2016 12 16330.2 33028.7 17490.07 7.1 33822.15 2.4
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 20910.51 12.9 37172.46 0.136
2018 14 20574.93 40337.2 24999.87 21.5 40854.64 1.28
2019 15 22508.99 42466.3 29888.97 32.8 44901.56 5.73
Average relative error of the simulation (2005-2016) - 8.83 - 2.99
Average relative error of the prediction (2017-2019) - 22.41 - 2.38
Average relative error (2005-2019) - 11.74 - 2.86
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ GM (1, 1) model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation Value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2931.301 25.6 13152.26 7.93
2007 3 2876.22 14605.1 3504.56 21.8 14455.08 1.03
2008 4 3501.39 16367.6 4189.929 19.7 15886.95 2.94
2009 5 4574.91 16522.4 5009.331 9.5 17460.66 5.68
2010 6 5938.71 18783.6 5988.98 0.846 19190.26 2.16
2011 7 7326.79 21842.0 7160.214 2.27 21091.18 3.44
2012 8 8838.6 23763.2 8560.499 3.15 23180.4 2.45
2013 9 10501.68 26042.7 10234.63 2.54 25476.57 2.17
2014 10 12339.36 28534.4 12236.17 0.836 28000.2 1.87
2015 11 14099.1 30519.5 14629.13 3.76 30773.8 0.833
2016 12 16330.2 33028.7 17490.07 7.1 33822.15 2.4
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 20910.51 12.9 37172.46 0.136
2018 14 20574.93 40337.2 24999.87 21.5 40854.64 1.28
2019 15 22508.99 42466.3 29888.97 32.8 44901.56 5.73
Average relative error of the simulation (2005-2016) - 8.83 - 2.99
Average relative error of the prediction (2017-2019) - 22.41 - 2.38
Average relative error (2005-2019) - 11.74 - 2.86
Table 2.  Modeling results of the simultaneous grey model of China's private car ownership and the added value of the transportation industry
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ Simultaneous grey model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2179.25 6.6 12463.53 2.27
2007 3 2876.22 14605.1 2733.22 4.97 14011.13 4.07
2008 4 3501.39 16367.6 3517.1 0.45 15692.37 4.13
2009 5 4574.91 16522.4 4513.1 1.35 17508.01 5.97
2010 6 5938.71 18783.6 5702.81 3.97 19454.64 3.57
2011 7 7326.79 21842.0 7067.92 3.53 21526.11 1.45
2012 8 8838.6 23763.2 8590.72 2.8 23714.56 0.205
2013 9 10501.68 26042.7 10254.4 2.35 26011.18 0.121
2014 10 12339.36 28534.4 12043.4 2.4 28406.75 0.447
2015 11 14099.1 30519.5 13943.1 1.11 30892.05 1.22
2016 12 16330.2 33028.7 15940.5 2.39 33458.13 1.3
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 18023.7 2.65 36096.45 2.76
2018 14 20574.93 40337.2 20182.0 1.91 38799.06 3.81
2019 15 22508.99 42466.3 22405.7 0.46 41558.58 2.14
Average relative error of the simulation (2005-2016) - 2.90 - 2.25
Average relative error of the prediction (2017-2019) - 1.67 - 2.90
Average relative error (2005-2019) - 2.64 - 2.39
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ Simultaneous grey model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2179.25 6.6 12463.53 2.27
2007 3 2876.22 14605.1 2733.22 4.97 14011.13 4.07
2008 4 3501.39 16367.6 3517.1 0.45 15692.37 4.13
2009 5 4574.91 16522.4 4513.1 1.35 17508.01 5.97
2010 6 5938.71 18783.6 5702.81 3.97 19454.64 3.57
2011 7 7326.79 21842.0 7067.92 3.53 21526.11 1.45
2012 8 8838.6 23763.2 8590.72 2.8 23714.56 0.205
2013 9 10501.68 26042.7 10254.4 2.35 26011.18 0.121
2014 10 12339.36 28534.4 12043.4 2.4 28406.75 0.447
2015 11 14099.1 30519.5 13943.1 1.11 30892.05 1.22
2016 12 16330.2 33028.7 15940.5 2.39 33458.13 1.3
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 18023.7 2.65 36096.45 2.76
2018 14 20574.93 40337.2 20182.0 1.91 38799.06 3.81
2019 15 22508.99 42466.3 22405.7 0.46 41558.58 2.14
Average relative error of the simulation (2005-2016) - 2.90 - 2.25
Average relative error of the prediction (2017-2019) - 1.67 - 2.90
Average relative error (2005-2019) - 2.64 - 2.39
Table 3.  Modeling results of the grey Bernoulli model proposed by reference [10]
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ Grey Bernoulli model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2154.17 7.68 11774.58 3.38
2007 3 2876.22 14605.1 2979.14 3.58 13832.31 5.29
2008 4 3501.39 16367.6 3879.82 10.8 15743.02 3.82
2009 5 4574.91 16522.4 4880.55 6.68 17628.32 6.69
2010 6 5938.71 18783.6 6002.34 1.07 19543.92 4.05
2011 7 7326.79 21842.0 7265.91 0.831 21522.63 1.46
2012 8 8838.6 23763.2 8693.06 1.65 23587.32 0.74
2013 9 10501.68 26042.7 10307.4 1.85 25756.02 1.1
2014 10 12339.36 28534.4 12135.1 1.66 28044.29 1.72
2015 11 14099.1 30519.5 14205.1 0.752 30466.44 0.174
2016 12 16330.2 33028.7 16549.9 1.35 33036.24 0.0228
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 19205.67 3.73 35767.38 3.65
2018 14 20574.93 40337.2 22213.37 7.96 38673.75 4.12
2019 15 22508.99 42466.3 25618.87 13.8 41769.67 1.64
Average relative error of the simulation (2005-2016) - 3.45 - 2.59
Average relative error of the prediction (2017-2019) - 8.50 - 3.14
Average relative error (2005-2019) - 4.53 - 2.71
Year No. $ x_{1}^{(0)} $ $ x_{2}^{(0)} $ Grey Bernoulli model
Simulation value of $ x_{1}^{(0)} $ Relative error (%) Simulation value of $ x_{2}^{(0)} $ Relative error (%)
2005 1 1848.07 10668.8 - - - -
2006 2 2333.32 12186.3 2154.17 7.68 11774.58 3.38
2007 3 2876.22 14605.1 2979.14 3.58 13832.31 5.29
2008 4 3501.39 16367.6 3879.82 10.8 15743.02 3.82
2009 5 4574.91 16522.4 4880.55 6.68 17628.32 6.69
2010 6 5938.71 18783.6 6002.34 1.07 19543.92 4.05
2011 7 7326.79 21842.0 7265.91 0.831 21522.63 1.46
2012 8 8838.6 23763.2 8693.06 1.65 23587.32 0.74
2013 9 10501.68 26042.7 10307.4 1.85 25756.02 1.1
2014 10 12339.36 28534.4 12135.1 1.66 28044.29 1.72
2015 11 14099.1 30519.5 14205.1 0.752 30466.44 0.174
2016 12 16330.2 33028.7 16549.9 1.35 33036.24 0.0228
Prediction value Relative error (%) Prediction value Relative error (%)
2017 13 18515.1 37121.9 19205.67 3.73 35767.38 3.65
2018 14 20574.93 40337.2 22213.37 7.96 38673.75 4.12
2019 15 22508.99 42466.3 25618.87 13.8 41769.67 1.64
Average relative error of the simulation (2005-2016) - 3.45 - 2.59
Average relative error of the prediction (2017-2019) - 8.50 - 3.14
Average relative error (2005-2019) - 4.53 - 2.71
Table 4.  Prediction results of China's private car ownership and the added value of the transportation industry
Year China's private car ownership Added value of the transportation industry
2020 24686.5 44368.31
2021 27016.9 47222.15
2022 29390.2 50114.61
2023 31800.6 53040.82
2024 34243.0 55996.41
Year China's private car ownership Added value of the transportation industry
2020 24686.5 44368.31
2021 27016.9 47222.15
2022 29390.2 50114.61
2023 31800.6 53040.82
2024 34243.0 55996.41
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