The traditional predictive method cannot fully reflect the complex nonlinear characteristics and regularities of automobile and parts sales data, so the prediction precision is not high. The purpose of this paper is to propose the gray GM(1,1) nonlinear periodic predictive model by introducing the seasonal variation index to improve predictive accuracy of the single GM(1,1) model. Firstly, the paper analyzes concept of GM(1,1) and then proposes the gray GM(1,1) nonlinear periodic predictive model to forecast automobile parts sales. The model algorithm used gray theory and accumulated technology to generate new data and set up unified differential equations to find the fitting curve of automobile parts sales prediction by the seasonal variation index to remove random elements. Lastly, the gray GM(1,1) nonlinear periodic predictive model is used for empirical analysis; the result of example shows that the model proposed in the paper is feasible. The superiority of the proposed predictive model compared with the single gray GM(1,1) model is demonstrated. The reliability of this model is experienced by the accuracy test, which provides a theoretical guidance for the prediction of automobile part sales. And the average relative error is reduced by 8.52% compared with the single GM(1,1) model.
In recent years, China’s automobile industry has grown rapidly with the development of the automotive market. Profiting from the improvement of the level of information, sales data of the dealer are increasingly abundant. Therefore, it is very important to predict the sales of automobile parts, regardless of the policy makers of the automobile industry or the research of the market strategy of the automobile or parts manufacturers [
Many scholars have studied gray relational analysis theory based on the gray relational analysis model proposed by Professor Deng and have achieved many valuable research results [
The gray predictive model is an important part of gray system theory. The classical GM(1,1) model is suitable for the prediction of single exponential growth, but it has some errors that are difficult to take into account the abnormal sequence data. On this basis, we construct the gray GM(1,1) nonlinear periodic predictive model by introducing seasonal variation index to improve the GM(1,1) model. The following analysis is used to build the gray GM(1,1) nonlinear periodic predictive model.
Setting
Setting
Then, the basic form of the GM(1,1) model is as follows:
Setting as a nonnegative sequence,
The GM(1,1) model is a differential equation that contains only one variable of time. The time series of original data acquisition is shown in Figure
The steps of gray GM(1,1) nonlinear periodictype predictive model are shown in Figure
Data processing program.
Steps of the gray GM(1,1) nonlinear periodic predictive model.
(
(
(
(
(
The accuracy of gray prediction model can be proved by using the residual, the correlation degree, and the posteriori difference testing [
The original sequence of the variable
The relative error
For a given value
Setting the original sequence
According to the calculation, the mean square deviation of
The mean variance ratio
According to the analysis, the fitting accuracy of the model prediction can be determined by the calculation of
Model fitting accuracy class.
Accuracy class  Index critical value  

Relative error 
Correlation degree 
Mean variance ratio 
Small error probability 

The first grade (good)  1  0.90  0.35  0.95 
The second grade (qualified)  5  0.80  0.50  0.80 
The third grade (possibly qualified)  10  0.70  0.65  0.70 
The fourth grade (unqualified)  20  0.60  0.80  0.60 
Data for empirical analysis are from automotive seat sales in a month of the Fcompany that is an automotive parts supplier since 2012–2018, as shown in Table
Automotive seat sales of FCompany in 2012–2018 (unit: ten thousand).
Year  Month sales  

1  2  3  4  5  6  7  8  9  10  11  12  
2012  15.21  15.03  18.23  16.54  20.36  19.63  13.64  19.05  24.36  23.15  20.48  22.36 
2013  16.14  15.96  18.63  16.5  20.15  19.56  13.42  19.52  24.65  23.54  21.35  21.41 
2014  16.89  16.52  19.54  16.34  20.01  20.82  14.76  18.06  25.13  23.68  21.68  21.59 
2015  17.05  16.98  19.15  16.8  20.08  20.93  14.07  18.16  25.52  23.98  21.51  21.65 
2016  17.04  17.06  17.84  16.35  20.6  20.41  15.46  19.42  26.6  27.37  24.28  27.52 
2017  17.11  17.62  19.56  16.52  20.89  21.06  16.58  20.25  26.72  27.69  24.39  27.88 
2018  18.16  18.10  20.45  18.12  22.21  22.14  16.32  20.89  27.85  27.41  24.51  25.91 
In the view of dual character, according to the gray GM(1,1) nonlinear periodic predictive model, firstly the prediction model of automotive seat sales was established, which can reflect the sales growth trend. Secondly, the seasonal variation index was used to fit automotive seat sales of seasonal fluctuations. Finally, we form the gray GM(1,1) nonlinear seasonal automotive seat sales forecasting model. The detailed calculation is as follows.
(
Sorting the original sequence values and eliminating the influence of extreme values in the sequence, the accumulated data are generated by using equations (
According to equation (
Seasonal variation index (SVI).
Month  SVI  Month  SVI  Month  SVI  Month  SVI 

1  0.830981  4  0.827722  7  0.734796  10  1.24856 
2  0.828724  5  1.020258  8  0.956497  11  1.117194 
3  0.943878  6  1.022932  9  1.278393  12  1.190064 
(
By using equation (
By Matlab, it can be known that
(
Therefore, the predicted average of automotive seat sales of the company is 21.8419 thousand in 2018. Taking into account the effect of the seasonal variation, according to equation (
Based on the sales data of automotive seats in the year of 2012–2017, the gray G(1,1) nonlinear periodic predictive model introduced the seasonal variation index to forecast the monthly sales for the year of 2018, as shown in Table
Sales predictive results of automotive seats in 2018 (unit: ten thousand).
Month and year  Actual sales 
Predictive value 

Jan, 2018  18.16  18.15 
Feb, 2018  18.10  18.10 
Mar, 2018  20.45  20.62 
Apr, 2018  18.12  18.08 
May, 2018  22.21  22.28 
Jun, 2018  22.14  22.34 
Jul, 2018  16.32  16.05 
Aug, 2018  20.89  20.89 
Sept, 2018  27.85  27.92 
Oct, 2018  27.41  27.27 
Nov, 2018  24.51  24.40 
Dec, 2018  25.91  25.99 
The residual sequence can be obtained by equation (
By referring to Table
According to equation (
By Equation (
Therefore, the predictive model of the mean variance ratio of the qualified, precision grade is the first grade and the predictive accuracy is very good.
Small error probability is
By referring to Table
In a word, the gray GM(1,1) nonlinear periodic predictive model is feasible to forecast the sales of automotive parts.
In order to show the advantages of the model, compared with the calculation results of the single GM(1,1) predictive model and the gray GM(1,1) nonlinear periodic type predictive model, the results are shown in Table
Comparison of the results of the two models (unit: ten thousand).
Month and year  Actual value  Single GM(1,1) forecasting model predicted value  Gray GM(1,1) nonlinear periodic model predicted value 

Jan, 2018  18.16  17.11  18.15 
Feb, 2018  18.10  16.76  18.10 
Mar, 2018  20.45  17.61  20.62 
Apr, 2018  18.12  18.50  18.08 
May, 2018  22.21  19.44  22.28 
Jun, 2018  22.14  20.42  22.34 
Jul, 2018  16.32  21.46  16.05 
Aug, 2018  20.89  22.54  20.89 
Sept, 2018  27.85  23.69  27.92 
Oct, 2018  27.41  24.89  27.27 
Nov, 2018  24.51  26.15  24.40 
Dec, 2018  25.91  27.47  25.99 
Table
Figure
Comparison of single GM(1,1) and gray GM(1,1) nonlinear periodic predicted model.
The gray GM(1,1) nonlinear periodic predictive model is constructed by using gray theory and applied to the prediction of automotive seat. And the reliability of this model is experienced by the accuracy test, which provides a theoretical guidance for the prediction of automobile part sales. The following conclusions can be obtained:
The gray GM(1,1) theory was studied and the fluctuation caused by the different demand of the automotive seat between offseason and busy season was considered, establishing the gray GM(1,1) model. The predictive model of single GM(1,1) was modified by the introduction of seasonal variation index, and the accuracy of the model was verified by examples.
The accuracy of the model is built on the basis of the analysis of historical data. Although it has been verified by the year of 2018, because of the limitation of the data, the accuracy of the model still needs to be improved. At the same time, the impact of the market, national policy, area, seasonal, and other factors are all influencing the sales. In the next stage of research, we would analyze the intelligent forecasting model to adapt to the changes of the market and demand and then guide the enterprise to make decisions in advance.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This research was funded by the Ministry of Education of Humanities and Social Science Research Program of China (Grant no. 15YJCZH049), Chongqing Research Program of Basic Research and Frontier Technology (Grant nos. cstc2016jcyjA0385, and cstc2017jcyjAX0343), Doctoral Scientific Research Startup Foundation of Hubei University of Technology (Grant no.BSQD2019004), and Open Program of Academic Team of Advanced Manufacturing Technology and Equipment & Inter Disciplinary Research Institute of Product Quality Engineering, HBUT.