Abstract
Considering the effective service life of products, this study initially defined a generalized fractional-order accumulation generation matrix covering the effective accumulation percentage. We suggested a generalized fractional-order accumulation grey power model (GFAGMP(1,1) model) using this matrix, along with its parameter estimate, error analysis, and time response function solution. We studied transformation and the link from the generalized GM(1,1) model to GFAGMP(1,1) model on the basis of integral and power function transformation and deduced three derivation forms of this model and their application range via the class ratio analysis. Finally, different forecasting models were compared with the actual sales data of Chinese refrigerators. The results of comparison demonstrated the feasibility and effectiveness of the GFAGMP(1,1) model in forecasting the home-appliance supply chain demand in China.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Chen CI (2008) Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate. Chaos Solitons Fractals 37(1):278–287. https://doi.org/10.1016/j.chaos.2006.08.024
Chen CI, Hsin PH, Wu CS (2010) Forecasting Taiwan’s major stock indices by the Nash nonlinear grey Bernoulli model. Expert Syst Appl 37(12):7557–7562. https://doi.org/10.1016/j.eswa.2010.04.088
Chen YY, Liu HT, Hsieh HL (2017) Time series interval forecast using GM(1,1) and NGBM(1,1) models. Soft Comput 23:1–15. https://doi.org/10.1007/s00500-017-2876-0
Chen X, Wang XJ, Wang L, Shi QH, Li YL (2018) Uncertainty quantification of multi-dimensional parameters for composite laminates based on grey mathematical theory. Appl Math Model 55:299–313. https://doi.org/10.1016/j.apm.2017.10.038
Chen Y-Y, Liu H-T, Hsieh H-L (2019) Time series interval forecast using GM(1,1) and NGBM(1,1) models. Soft Comput 23(5):1541–1555. https://doi.org/10.1007/s00500-017-2876-0
Chen L, Nan G, Li M, Feng B, Liu Q (2022) Manufacturers online selling strategies under spillovers from online to offline sales. J Oper Res Soc (forcoming). https://doi.org/10.1080/01605682.2022.2032426
Deng JL (1982) Control problems of grey systems. Syst Control Lett 1(5):288–294. https://doi.org/10.1016/S0167-6911(82)80025-X
Deng JL (2002) The basis of grey theory. Huazhong University of Science & Technology Press, Wuhan
Evans M (2014) An alternative approach to estimating the parameters of a generalised grey Verhulst model: an application to steel intensity of use in the UK. Expert Syst Appl 41(4):1236–1244. https://doi.org/10.1016/j.eswa.2013.08.006
Gao MY, Mao SH, Yan XP, Wen JH (2015) Estimation of Chinese \(\text{ CO}_{2}\) emission based on a discrete fractional accumulation grey model. J Grey Syst 27(4):114–130
Gao MY, Yang HL, Xiao QZ, Goh M (2021) A novel fractional grey Riccati model for carbon emission prediction. J Clean Prod 282:124471. https://doi.org/10.1016/j.jclepro.2020.124471
Gao MY, Yang HL, Xiao QZ, Goh M (2022a) A novel method for carbon emission forecasting based on Gompertz’s law and fractional grey model: evidence from American industrial sector. Renew Energy 181:803–819. https://doi.org/10.1016/j.renene.2021.09.072
Gao MY, Yang HL, Xiao QZ, Goh M (2022b) COVID-19 lockdowns and air quality: evidence from grey spatiotemporal forecasts. Socio-Econ Plan Sci 83:101228. https://doi.org/10.1016/j.seps.2022.101228
Guo XJ, Liu SF, Yang YJ (2019) A prediction method for plasma concentration by using a nonlinear grey Bernoulli combined model based on a self-memory algorithm. Comput Biol Med 105:81–91. https://doi.org/10.1016/j.compbiomed.2018.12.004
Hsin PH, Chen CI (2015) Application of game theory on parameter optimization of the novel two-stage Nash nonlinear grey Bernoulli model. Commun Nonlinear Sci Numer Simul 27(1):168–174. https://doi.org/10.1016/j.cnsns.2015.03.006
Hsu LC (2010) A genetic algorithm based nonlinear grey Bernoulli model for output forecasting in integrated circuit industry. Expert Syst Appl 37(6):4318–4323. https://doi.org/10.1016/j.eswa.2009.11.068
Hu YC (2020) A multivariate grey prediction model with grey relational analysis for bankruptcy prediction problems. Soft Comput 24(6):4259–4268. https://doi.org/10.1007/s00500-019-04191-0
Kang Y, Mao S, Zhang Y (2022) Fractional time-varying grey traffic flow model based on viscoelastic fluid and its application. Transp Res Part B Methodol 157:149–174. https://doi.org/10.1016/j.trb.2022.01.007
Li DC, Chang CJ, Chen CC, Chen WC (2012) A grey-based fitting coefficient to build a hybrid forecasting model for small data sets. Appl Math Model 36(10):5101–5108. https://doi.org/10.1016/j.apm.2011.12.050
Liu SF, Dang YG, Fang ZG, Xie NM (2010) Grey system theory and its application. Science Press, Beijing
Ma X, Liu ZB (2018) The kernel-based nonlinear multivariate grey model. Appl Math Model 56:217–238. https://doi.org/10.1016/j.apm.2017.12.010
Ma X, Mei X, Wu W, Wu X, Zeng B (2019) A novel fractional time delayed grey model with grey wolf optimizer and its applications in forecasting the natural gas and coal consumption in Chongqing China. Energy 178(1):487–507. https://doi.org/10.1016/j.energy.2019.04.096
Mao SH, Gao MY, Xiao XP, Zhu M (2016) A novel fractional grey system model and its application. Appl Math Model 40(7–8):5063–5076. https://doi.org/10.1016/j.apm.2015.12.014
Mao SH, Zhu M, Wang XP, Xiao XP (2020) Grey-Lotka-Volterra model for the competition and cooperation between third-party online payment systems and online banking in china. Appl Soft Comput 95:106501. https://doi.org/10.1016/j.asoc.2020.106501
Meng QL, Dong ZJS, He L, Dong J (2018) Forecasting service quality dynamics based on integrated method with GM(1,1) model and Markov chain. Teh Vjesn Tech Gaz 25(5):1437–1446. https://doi.org/10.17559/TV-20180624094340
Ministry of Finance, Ministry of Commerce, and Ministry of Industry and Information Technology of China, Guanyu quanguo tuiguang jia dian xia xiang gongzuo de tongzhi (notice from three government departments about the national implementtiona of ’home appliances to the countryside’ program). http://www.gov.cn/gzdt/2008-12/05/content_1169347.htm. Accessed 10 Mar 2019
Mustaffa AS, Shabri A (2020) An improved rolling NGBM(1,1) forecasting model with GRG nonlinear method of optimization for fossil carbon dioxide emissions in Malaysia and Singapore. In: 2020 11th IEEE control and system graduate research colloquium (ICSGRC), pp 32–37. https://doi.org/10.1109/ICSGRC49013.2020.9232665
Rajesh R (2019) Social and environmental risk management in resilient supply chains: a periodical study by the grey-Verhulst model. Int J Prod Res 57(11):3748–3765. https://doi.org/10.1080/00207543.2019.1566656
Rao C, Gao Y (2022) Evaluation mechanism design for the development level of urban-rural integration based on an improved Topsis method. Mathematics 10(3):380. https://doi.org/10.3390/math10030380
Rao C, Gao M, Wen J, Goh M (2022) Multi-attribute group decision making method with dual comprehensive clouds under information environment of dual uncertain z-numbers. Inf Sci 602:106–127. https://doi.org/10.1016/j.ins.2022.04.031
Utkucan S (2020) Projections of Turkey’s electricity generation and installed capacity from total renewable and hydro energy using fractional nonlinear grey Bernoulli model and its reduced forms. Sustain Prod Consum 23:52–62. https://doi.org/10.1016/j.spc.2020.04.004
Wang ZX (2017) A weighted non-linear grey Bernoulli model for forecasting non-linear economic time series with small data sets. Econ Comput Econ Cybern Stud Res 51(1):169–186
Wang ZX, Hipel Keith W, Wang Q, Whale S (2011) He, An optimized \(\text{ NGBM(1,1) }\) model for forecasting the qualified discharge rate of industrial wastewater in \(\text{ China }\). Appl Math Model 35(12):5524–5532. https://doi.org/10.1016/j.apm.2011.05.022
Wang ZX, Li Q, Pei LL (2017) Grey forecasting method of quarterly hydropower production in China based on a data grouping approach. Appl Math Model 51:302–316. https://doi.org/10.1016/j.apm.2017.07.003
Wu LF, Liu SF, Yao LG, Yan SL, Liu DL (2013) Grey system model with the fractional order accumulation. Commun Nonlinear Sci Numer Simul 18(7):1775–1785. https://doi.org/10.1016/j.cnsns.2012.11.017
Wu LF, Liu SF, Fang ZG, Xu HY (2015) Properties of the \(\text{ GM(1,1) }\) with fractional order accumulation. Appl Math Comput 252:287–293. https://doi.org/10.1016/j.amc.2014.12.014
Wu WQ, Ma X, Zeng B, Wang Y, Cai W (2019) Forecasting short-term renewable energy consumption of China using a novel fractional nonlinear grey Bernoulli model. Renew Energy 140:70–87. https://doi.org/10.1016/j.renene.2019.03.006
Xiao XP, Mao SH (2013) The method of grey prediction and decision. Science Press, Beijing
Xiao XP, Guo H, Mao SH (2014) The modeling mechanism, extension and optimization of grey \(\text{ GM(1,1) }\) model. Appl Math Model 38(5–6):1896–1910. https://doi.org/10.1016/j.apm.2013.10.004
Xiao XP, Yang JW, Mao SH, Wen JH (2017) An improved seasonal rolling grey forecasting model using a cycle truncation accumulated generating operation for traffic flow. Appl Math Model 51:386–404. https://doi.org/10.1016/j.apm.2017.07.010
Yang Y, Xue D (2016) Continuous fractional-order grey model and electricity prediction research based on the observation error feedback. Energy 115:722–733. https://doi.org/10.1016/j.energy.2016.08.097
Yang L, Xie N, Wei B, Wang X (2022) On unified framework for nonlinear grey system models: an integro-differential equation perspective. Commun Nonlinear Sci Numer Simul 108:106250. https://doi.org/10.1016/j.cnsns.2022.106250
Ying KC, Liao CK, Hsu YT (2000) Generalized admissible region of class ratio for \(\text{ GM(1,1) }\). J Grey Syst 12(2):153–156
Yousuf MU, Al-Bahadly I, Avci E (2021) A modified GM(1,1) model to accurately predict wind speed. Sustain Energy Technol Assess 43:100905. https://doi.org/10.1016/j.seta.2020.100905
Zeng XY, Shu L, Yan SL, Shi YC, He FL (2019) A novel multivariate grey model for forecasting the sequence of ternary interval numbers. Appl Math Model 69:273–286. https://doi.org/10.1016/j.apm.2018.12.020
Zhou JZ, Fang RC, Li YH, Zhang YC, Peng B (2009) Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization. Appl Math Comput 207(2):292–299. https://doi.org/10.1016/j.amc.2008.10.045
Acknowledgements
The authors are grateful to the editor and the reviewers for their valuable comments.
Funding
This work is supported by the China Scholarship Council (CSC) (No. 201906130025, 201906130049), and the National Natural Science Foundation of China (Grant No. 71790593, 72071072).
Author information
Authors and Affiliations
Contributions
HY Conceptualization of the presented idea, Supervised the findings of this work. MG Software, Validation, Visualization, Writing-Original draft preparation, Formal analysis. QX Data curation, Writing- Original draft preparation.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yang, H., Gao, M. & Xiao, Q. A novel fractional-order accumulation grey power model and its application. Soft Comput 27, 1347–1365 (2023). https://doi.org/10.1007/s00500-022-07634-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-022-07634-3