The Journal of Grey System

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1. Improved Fractional Order Single Optimization Parameter Grey Model

Jiangtao Wei, Yonghong Wu

The Journal of Grey System 2023, 35 (4): 154-171.

摘要（360）

Some grey prediction models suffer from outliers and overfitting, and their prediction performance can be improved. Based on the fractional grey model (FGM) and the fractional time delayed grey model (FTDGM), an improved fractional single optimization parameter grey model (IFSGM) is proposed in this paper. A timetranslation power term is used to reduce model overfitting. The data is pre-processed to reduce the influence of outliers based on data transformation in cumulants and summations. The convolutional computational formulation is used to perform the prediction and improve the prediction effect. The genetic optimization algorithm is used to optimize the order of the fractional cumulants, find the optimal value of the order, and improve the model fit and prediction effect. In 6 data sets, the IFSGM and the 7 grey models are compared and tested. The experimental results show that IFSGM achieves excellent prediction performance.

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2. The Greyness and Applications of Grey Set

Xican Li, Li Li

The Journal of Grey System 2024, 36 (6): 42-53.

摘要（302）

In order to quantitatively describe the grey properties of grey set, based on the possibility function of grey set, this paper discusses the expression method of the greyness of grey set and its applications. Firstly, the axiomatic definition and a method of calculating the greyness of grey set are given, and its rationality is analyzed through examples. Then, according to the principle of unity of opposites, the concept of whiteness of grey set is given, and the applications of greyness of grey set are analyzed. Finally, some examples are given to verify the validity of greyness of grey set and its application model. The results show that the axiomatic definition of greyness of grey set not only conforms to the grey immortal axiom, but also can quantitatively describes the dynamic evolution state of grey hazy set (grey set). The application examples show that the grey relational degree model and the grey decision model based on the greyness of grey set are feasible and effective. The research results not only enrich the theory of grey mathematics and grey system, but also provide a new method for grey relational analysis and grey decision.

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3. Entropy-weighted TOPSIS Multi-attribute Decision-making Model and Its Applications Based on Generalized Greyness

Li Zhang, Xican Li

The Journal of Grey System 2024, 36 (5): 15-26.

摘要（300）

In order to solve the decision-making problem that the attributive values are internal grey numbers and the attributive weights are unknown, this paper try to construct an entropy-weighted TOPSIS model based on the generalized greyness of interval grey number from the perspectives of proximity and equilibrium. Firstly, the properties of greyness distance are analyzed and the simplified formula for computing greyness distance is given. Then, a method to determine entropy weight based on greyness distance is given, and an entropy weighted TOPSIS decision-making model is established. Finally, the constructed model is applied to selecting brackish water irrigation pattern of winter wheat in North China Plain, China, so to verify its feasibility and effectiveness. The results show that the model proposed in this paper not only fully utilizes the measurement information of interval grey numbers, but also overcomes the influence of subjective factors on weights, and provide a new method for decision-making of unknown attributive weights and attributive value with interval grey number, and the interval grey numbers coexist with the real numbers. The application examples show that the model proposed in this paper is feasible and valid.

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4. Data-driven Dynamic Grey-Verhulst SEIRD Model for Public Health Emergencies Forecasting

Shuhua Zhang, Ming Liu, Bingjun Li

The Journal of Grey System 2025, 37 (1): 33-46.

摘要（283）

Determining parameters in infectious disease dynamics models is crucial for simulating and predicting the development trends of public health emergencies. Utilizing real-time epidemic data and grey systems theory, our innovative approach bridges the Dynamic Grey Verhulst model and the SEIRD model, which respectively have advantages in short-term and long-term forecasting. The new model features a dynamically adjusting decision cycle to accommodate evolving epidemic data. We constructed a dynamic grey Verhulst model using the principle of metabolism, enabling it to dynamically update and iterate important parameters of infectious disease models. This results in accurate simulation and prediction of epidemic dynamics. Taking the SARS-CoV-2 Omicron outbreak in Shanghai, China, in the spring of 2022 as an example, the proposed Dynamic Grey-Verhulst SEIRD model (DGVM-SEIRD) provides a data-driven, high-sensitivity and high-precision method for predicting public health emergencies. Sensitivity tests also confirm the superiority of our model. Furthermore, validation with H1N1 influenza data from Beijing, the COVID-19 outbreak in Wuhan and SARSCoV-2 emergencies in the UK reinforces our model’s accuracy. This methodology provides a highly flexible and responsive analytical tool for public health emergency management, offering scientific support for formulating more effective epidemic prevention and control strategies and emergency responses.

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5. A New Optimized Grey Forecasting Model with Polynomial Term and Its Application

An Wang, Yaoguo Dang, Junjie Wang

The Journal of Grey System 2024, 36 (3): 74-85.

摘要（272）

China’s total energy consumption and production rank first in the world. However, China’s energy structure is not reasonable. Therefore, accurate prediction of future energy trends is of great significance for the Chinese government to adjust the energy structure. In this paper, we propose an optimized Grey Euler model with polynomial term, which is abbreviated as OSGEM(1,1,N), to forecast the total energy consumption and production of China in comparison with the commonly used prediction models. The data from 2002 to 2018 are used to simulate the parameters in the proposed model, and the data from 2019 to 2021 are used to test the improved approach. The results show that the OSGEM(1,1,N) model outperforms the other models. Finally, the OSGEM(1,1,N) model is used to forecast the total energy consumption of China from 2022 to 2025 and different results from the previous research results have been obtained.

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6. A Novel Logistic Multivariate Grey Prediction Model for Energy Consumption: A case study of China Coal

Sihao Chen, Yongshan Liu, Huiming Duan

The Journal of Grey System 2023, 35 (4): 132-153.

摘要（263）

Coal consumption plays a pivotal role in the national economic growth, and the coal-based energy supply system as the main guarantee of China's energy is impossible to change in the short term. In order to ensure the security of China's energy supply, accurate coal consumption forecast can provide important theoretical basis for the development of scientific and effective energy planning and decision-making. Starting from the classical Logistic model, this paper introduces a variety of related factors such as energy consumption, economic growth rate and carbon dioxide emission growth rate to expand the modeling objects of the Logistic model and improve the performance of the model by relying on its ability to capture the historical trend of the data model and accurately predict the future value. At the same time, a new logistic multivariate grey prediction model of energy consumption is established by introducing the principle of grey variance information organically combined with the logistic model. The modeling steps of the model are obtained by using mathematical methods such as least squares estimation of parameters and number multiplication transformation. Finally, the new model is applied to the prediction of Chinese coal consumption, and the validity of the model is verified from different perspectives of three cases, showing that the fitted and predicted data of the new model have good consistency with the actual results. The new model has a high accuracy for China's coal short-term forecast, and uses simulation and prediction effects of 1.68220% and 1.29866%, respectively, to effectively forecast China's coal consumption in 2022-2026, and points out the development trend of Chinese coal consumption, and provides a basis for China to make scientific and effective energy planning and decision-making.

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7. Exploring Death Population Prediction and Cemetery Planning in Chongqing amid an Aging Population: A Grey Forecast Model Based on Interval Grey Number

Huaan Wu, Yuhua Jin, Yinhe Xue, Bo Zeng, Hui Wang

The Journal of Grey System 2024, 36 (3): 51-62.

摘要（255）

China’s population is steadily aging, contributing to the increase in the number of deceased people and the growing disparity between supply and demand for cemeteries. To provide theoretical support and data reference for cemetery planning, this study considers Chongqing, a city with a high rate of aging population, as an example to apply the interval grey number model to model and predict the size of the death population in Chongqing. The following conclusions are drawn: (1) The accuracy of the interval grey number prediction model in simulating the size of the death population in Chongqing exceeds 98%, indicating that the model employed in the study is suitable for medium- to long-term prediction; (2) The prediction results show that the annual death scale of registered population in Chongqing will range between 220 and 330 thousand from 2022 to 2030, with a fluctuating upward trend; (3) According to the size of the death population predicted, the cemetery market in Chongqing will experience a shortage of supply within 10 years. Therefore, in order to ensure a balance between the supply and demand of cemeteries in Chongqing, the government should actively promote the concept of green funerals and reduce the demand for cemeteries. Alternatively, it is also necessary to accelerate the planning and construction of cemeteries to avoid the predicament of people wanting to be buried without a tomb.

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8. Forecasting the Two-Stage Regional Population Ageing Structure by Employing Grey Compositional Model

Hui Li, Naiming Xie, Rafał Mierzwiak

The Journal of Grey System 2025, 37 (1): 1-15.

摘要（255）

Population ageing is a significant and global concern, particularly pronounced in China, where rapid ageing growth has been observed. This growth is uneven across regions, presenting urgent challenges for local governments. Accurate forecast of regional ageing structure is vital for developing and adjusting population, social, and economic policies. To address this, based on the compositional data, population ageing is firstly delineated into two stages: the structure of the elderly and that of the disabled elderly, and a data collection and pre-processing framework based on this division is constructed. Then, a novel non-linear dynamic grey Markov compositional model is developed to tackle this two-stage issue. Finally, using this model, the ageing structure is predicted and studied in Jiangsu Province, China, as an illustrative case. Experimental results show that the ageing structure will be further “aged” and “disabled”, and moderate disability is the core component of the rise in the disabled elderly. These forecasts align with current trends in ageing and provide a quantitative basis for governmental policy-making and adjustments.

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9. Research on China's GDP Growth Forecast Based on Grey Machine Learning Model

Tianxiang Yao, Xichun Liu

The Journal of Grey System 2024, 36 (4): 1-13.

摘要（245）

Based on Keynesian macroeconomic theory, this paper introduces economic indicators with Chinese characteristics, and constructs a multivariate grey machine learning forecasting model (IGM (1, N, X1 (0) )-IPSO-LSTM) to predict China's GDP growth. Firstly, IGM (1, N) model is constructed by changing the background value construction method of GM (1, N) model and introducing grey action constant A which reflects the change from the grey differential equation to the difference equation. Secondly, due to the low frequency and small amount of GDP data, constructing a two-layer LSTM model to increase the model complexity, so that the data can be fully trained. In addition, this paper uses nonlinear descending function instead of w to construct Improved Particle Swarm Optimization algorithm (IPSO), and adds Genetic Algorithm (GA) to IPSO to reduce the risk of particles falling into the local optimal solution. Finally, using IPSO to find the optimal parameters of LSTM model to predict China's GDP growth. By comparing the prediction accuracy of IGM (1, N, X1 (0) )-IPSO-LSTM model with other benchmark models, the prediction result of IGM (1, N, X1 (0) )-IPSO-LSTM model is the best. It is predicted that China's GDP growth rate in 2024 is 5.18% and in 2025 is 5.12%. By analyzing the trend development of China's economic, it is found that the forecast results are consistent with the expected trend of macro economy, which increases the credibility of the forecast results.

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10. Comparative Analysis of Grey Forecasting Models for Population Aging Prediction: A Case Study of Egypt's Demographic Evolution

Islam Mahmoud Sharafeldin, Naiming Xie

The Journal of Grey System 2025, 37 (1): 108-117.

摘要（230）

Population aging in developing nations presents complex demographic challenges that conventional forecasting approaches often struggle to address effectively, particularly when confronted with endogenous volatility in demographic structures and limited data availability. This study introduces an enhanced hybrid grey forecasting framework to predict population aging patterns in Egypt, incorporating advanced grey models to improve prediction accuracy and capture regional demographic variations. Using comprehensive demographic data from 2011-2023, we evaluate multiple grey forecasting models to identify optimal prediction methodologies for different population segments. Our findings reveal that the Grey Optimization Model with Interval Analysis (GOM_IA (1,1)) demonstrates superior predictive performance, achieving the lowest Mean Absolute Percentage Error for urban populations, rural and aged populations during the testing period. While, Unbiased GOM (1,1) model give the best performance for the total population prediction over the other grey models. The model projects significant regional variations in aging patterns, with urban areas experiencing accelerated aging rates compared to rural regions. This study makes several key contributions by it establishing a robust methodological framework for demographic forecasting in developing nations with limited data availability. As well as providing quantitative evidence of regional disparities in aging patterns across Egypt. Finally, offering a data-driven insights for policy formulation in healthcare infrastructure development and social service delivery. The findings have significant implications for resource allocation and policy planning in Egypt and other developing nations experiencing similar demographic transitions. Furthermore, our research demonstrated the effectiveness of grey forecasting models in capturing complex demographic patterns and supports evidence-based decision-making in addressing the challenges of population aging.

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11. Grey Information Relational Estimation Model of Soil Organic Matter Content Based on Hyperspectral data

Hong Che, Xican Li, Guozhi Xu

The Journal of Grey System 2024, 36 (4): 56-68.

摘要（225）

In order to overcome the uncertainty in hyperspectral estimation of soil organic matter content, this paper aim to establish a grey information relational estimation model of soil organic matter content based on hyperspectral data and grey information theory. Based on 76 samples in Zhangqiu District of Jinan City, Shandong province of China, the spectral data are first transformed by the nine methods such as square root, first order differentiation of the logarithm reciprocal, and so on, the correlation coefficient is calculated, and the estimation factors are selected by using the principle of great maximum correlation. Then, according to the principle of increasing information and taking maximum method, the spectral estimation factors of each sample are sorted from small to large, and the grey information sequence is formed, and the grey relational estimation model of soil organic matter content is constructed based on the information chain. Finally, the estimation results based on different information chains are fused twice, and compared with the commonly used estimation methods. The results of the method in this paper show that the average relative error of the 12 test samples is 5.576%, and the determination coefficient R2 is 0.934, and the estimation accuracy is higher than that of commonly used methods such as multiple linear regression, BP neural network and support vector machine. The results show that the grey information relational estimation model using hyperspectral data proposed in this paper is feasible and effective, and it provides a new way for hyperspectral estimation of soil organic matter and other soil properties.

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12. A Novel Time-varying Non-homogeneous Discrete Grey Model and Its Application in Forecasting Solar Energy Generation in Total North America

Lin Xia, Yuhong Wang, Yuxuan Han, Ke Zhou, Youyang Ren, Yiyang Fu

The Journal of Grey System 2025, 37 (1): 96-107.

摘要（223）

Accurate forecasting of solar energy generation in total North America is crucial for effective energy planning and environmental protection. However, challenges arise from the limited and complex nature of the data. This paper introduces a novel Time-Varying Non-Homogeneous Discrete Grey Model (TVNDGM(1,1)) to address these challenges. The model introduces an anti-forgetting accumulated generating operator as the weight accumulation function to effectively prioritize new information. Additionally, it extends the homogeneous discrete grey model into a non-homogeneous format, enhancing model adaptability to various samples. Applying the Whale Optimization Algorithm in selecting non-structural parameters further improves accuracy. Case study results demonstrate that the model achieves fitting and test errors of 2.28% and 1.65%, respectively, outperforming seven other methods, thus indicating superior predictive accuracy and stability. Forecasts suggest that from 2024 to 2030, solar energy generation in total North America will continue to rise, with an average annual growth rate of 20.31%. This study enriches the theory of new information prioritization within grey forecasting methods and provides technological support for global energy planning and development.

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13. Research on Grey Prediction of Regional Dual Energy Consumption Under Carbon Emission Constraints

Yuhan Xie, Chuanmin Mi

The Journal of Grey System 2025, 37 (1): 79-95.

摘要（217）

To enhance the modeling capability of the grey prediction model in the spatiotemporal domain, the paper proposes a novel spatiotemporal grey prediction model integrated with heterogeneous adjacency accumulation. Initially, an improved economic geographic gravity matrix is employed to characterize the spatial flow patterns of regional energy consumption, vividly illustrating the spatial interplay between non-adjacent provinces. Subsequently, a heterogeneous adjacent accumulation operator is incorporated to mirror regional discrepancies in energy consumption and bolster the robustness of the spatiotemporal prediction model. Ultimately, the novel prediction model is utilized to forecast the evolution of regional dual energy consumption within the constraints of carbon emissions. The findings of this research reveal the following: (1) By 2030, the total energy demand is projected to surge to 6.839 billion tons of standard coal, surpassing the predefined threshold of 6 billion tons. The prompt implementation of energy-saving strategies is paramount to expedite the attainment of carbon peaking. (2) Energy consumption intensity exhibits notable regional variability, with a spatially positive correlation in energy consumption intensity among regions. By 2030, it is anticipated that only 12 provinces, including Beijing, Guangdong, Shanghai, and Jiangsu, will attain the energy efficiency benchmarks of advanced developed countries.

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14. An Optimized Multivariate Grey Bernoulli Model for Forecasting Fossil Energy Consumption in China

Ye Li , Dongyu Liu, Meidan Xiao, Bin Liu

The Journal of Grey System 2024, 36 (2): 67-78.

摘要（213）

Given the increasing severity of energy shortages, the exploration of effective strategies to optimize energy structures has become imperative. This requires careful consideration of energy consumption patterns, especially since these data are fundamental inputs for policy formulation. Given the uncertainty in the rate of change in energy consumption, this paper proposes an optimized multivariable grey Bernoulli model that is rooted in the grey Bernoulli model and incorporates background values and genetic algorithms. The grey Bernoulli model effectively linearizes nonlinear problems, thus simplifying computational procedures. In addition, to account for random fluctuations of relevant factors that may affect the model's predictions, this model introduces nonlinear correction terms that allow simulation and prediction values to adhere to the grey index law. The incorporation of background values enhances the model's ability to process information, providing it with a superior grasp of real data. Genetic algorithms can be used to refine the model's parameters, increasing its adaptability and precision. Finally, this paper applies the refined model to examine China's energy consumption patterns, validating its efficacy and versatility. Furthermore, energy consumption patterns for the next four years are forecast, with the analysis revealing that the growth rate of energy consumption from 2021 to 2024 shows a downward trend, particularly notable in 2024, where the growth rate is 1.64%.

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15. Equipment Maintenance Reliability Based On Grey Relational Decision Optimization Model

Qiang Li, Shupin Chen, Shumiao Fang, Ailing Yan, Wenjie Dong

The Journal of Grey System 2025, 37 (1): 133-144.

摘要（211）

Aiming at the selection of maintenance strategy for equipment reliability, This article first proposed a evaluation index system for equipment maintenance reliability from three perspectives: equipment operation guarantee, equipment maintenance, and equipment daily management and gives the modeling steps and flow chart of the grey correlation decision model, Delphi method and analytic hierarchy process are used to combine qualitative and quantitative analysis methods to determine and optimize the weight of qualitative indicators. Then, combined with the actual data of coating equipment operation and maintenance in a semiconductor panel manufacturing industry, a grey correlation decision optimization model is constructed to calculate the effect vector, the ideal optimal effect vector and the grey comprehensive correlation degree of the decision scheme of each index. Finally, through the grey correlation analysis, the optimal strategy selection of Coating equipment maintenance reliability is realized. The research in this paper has practical guiding significance for coating equipment maintenance decision-making, improving coating equipment maintenance reliability and reducing equipment maintenance cost.

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16. A Conformable Fractional Non-homogeneous Grey Forecasting Model with Adjustable Parameters CFNGMA(1,1,k,c) and its Application

Wenqing Wu , Xin Ma , Bo Zeng , Peng Zhang

The Journal of Grey System 2024, 36 (2): 1-12.

摘要（206）

The inconsistency between the whitening differential equation and the grey basic form of the non-homogeneous continuous grey model CFNGM(1,1,k,c) will result in internal errors. Thus this paper proposes a CFNGMA(1,1,k,c) model with adjustable parameters, which improves the accuracy of the CFNGM(1,1,k,c). This paper first elucidates reasons for the internal errors generated by the continuous grey model CFNGM(1,1,k,c), and explains the classic method, the discrete grey forecasting model, of eliminating internal errors. On the basis of an in-depth analysis of the modeling mechanism of CFNGM(1,1,k,c) model, a new parameter adjustable grey forecasting model is proposed by introducing parameter adjustment factors to modify model’s parameters. Finally, the new model is applied to explore the gross regional product of Chengdu and Deyang in the Chengdu metropolitan area. The calculation results indicate that the newly proposed model can obtain more accurate results.

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17. Forecasting New Energy Vehicle Sales in China Based on a Novel Grey Lotka-Volterra Model and Assessing Its Environmental Impact

Wuyong Qian, Tingting Zou, Yuhong Wang, Chunyi Ji, Minghao Ran

The Journal of Grey System 2023, 35 (3): 82-99.

摘要（205）

With China's vigorous implementation of the two-carbon policy, the automotive industry is transitioning from traditional to alternative energy sources. Accurate forecasting of new energy vehicle sales can provide statistical support for policy planning. This paper proposed a grey Lotka-Volterra model based on an HP filter to forecast new energy vehicle sales trends. The proposed model overcomes the limitations of the original model in forecasting cyclical data and improves its applicability in small sample data systems. And the competitive forecasting model allows us to explore the market competition or cooperation between various vehicle types. Compared to the two existing models, the new model performs better in forecasting vehicle sales. The study further applies the LCA method to assess the scale of energy consumption and GHG emissions in the automotive sector. The prediction and evaluation results show that new energy vehicle sales will surpass traditional fuel vehicles by 2035 and dominate the vehicle market before 2050. Unfortunately, vehicle electrification has not significantly reduced the transportation industry's environmental concerns. To fully utilize the potential for energy-savings and emission-reduction of new energy vehicles, it is urgent to pursue technological advancements in battery production and power-generating structure.

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18. A Novel Power-sum Time-varying Grey Prediction Model and Its Applications

Kai Cai, Lianyi Liu, Sifeng Liu

The Journal of Grey System 2025, 37 (1): 64-78.

摘要（203）

The purpose of this paper is to propose an improved power-sum accumulation time-varying grey model (PATGM) to enhance the ability to mine the heterogeneity of sparse data. Firstly, a novel power-sum accumulation grey generating operator is introduced to smooth the observed values according to data fluctuations, mitigating the model's ill-conditioned property. Secondly, a time-varying function is introduced as a parameter structure to the traditional model, providing the model with flexibility in complex systems modeling. Finally, based on the Dingo Optimization Algorithm, a hyperparameter calibration strategy for PATGM is provided. The power-sum accumulation grey generating operator can amplify or minimize the nonlinear characteristics of the observations, thus significantly improving the adaptivity of the grey modeling approach to fluctuating sequences. Meanwhile, the elastic-net regression method is employed to obtain a more reasonable and stable parameter structure. The hyperparameters are calculated using the Dingo optimization algorithm, which effectively controls the noise resistance and nonlinearity in the prediction system. PATGM solves the data smoothing processing and model structure selection problems of the traditional grey model. This new model is suitable for processing data prediction tasks with complex characteristics, especially provides an effective prediction method for complex engineering and system modeling.

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19. Memorabilia of the Establishment and Development of Grey System Theory

Sifeng Liu, Liangyan Tao, Wei Tang

The Journal of Grey System 2024, 36 (1): 1-3.

摘要（201）

This article summarizes and records important historical events in the establishment and 40 year development process of continuing innovation and dissemination of grey system theory, providing reference for scholars who pay attention to the evolution laws of grey system theory, a new branch of uncertainty system research, as well as colleagues engaged in grey system theory research. If there are any important omissions, we sincerely welcome readers to supplement and improve.

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20. Research on Image Recognition of Wood Defects Using TGARG Based on Edge Detection and Characteristic Combination

Yanping Qin, Jun Zhang, Huaqiong Duo

The Journal of Grey System 2025, 37 (1): 118-132.

摘要（197）

Wood defects affect the use value and commodity value of wood, so the research on effective recognition of wood defects has important practical significance. In this study, three types of wood defect images (live knots, cracks, and dead knots) were used as research objects. To investigate the impact of edge detection and characteristics combination on recognition rate, the recognition method based on threedimensional grey absolute relational grade (TGARG) is constructed and the recognition rates of different types of wood defects were compared under different edge detection and characteristics combinations. The results show that, based on TGARG, the recognition rate of live knots is the highest (0.76) under edge detection by Canny operator as well as characteristics combination of energy and homogeneity. The recognition rate of cracks is the highest (1.00) under edge detection by Roberts or Sobel operator as well as characteristics combination of contrast and correlation. The recognition rate of dead knots is the highest (0.90) under without edge detection as well as characteristics combination of correlation, energy and homogeneity. The research method and conclusion proposed in this study on the selection of wood defect recognition from a new perspective will contribute to the development of wood defect recognition.

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