The Journal of Grey System

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1. 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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2. Modeling and Predicting the Socio-Economic Performance of Countries Using Grey Relational Analysis and K-NN Algorithm

Hande Hakan, Ecem Coşar Canlıer, Çiğdem Özarı, Esin Nesrin Can

The Journal of Grey System 2024, 36 (1): 4-15.

摘要（196）

The main purpose of this study is to forecast the countries’ socio-economic performance with the fewest possible parameters. To do this, we propose a model consisting of methods from Multi-Criteria Decision Making and Machine Learning. Since the existence of different classifications of countries and several socioeconomic parameters, it becomes difficult to make a prediction of their belonging group and compare countries based on these parameters. Using the Grey Relational Analysis and the Critic method, we classify the countries into four different subgroups based on several socio-economic dimensions. K-Nearest Neighbor (K-NN) algorithm with basic macro-economic parameters is implemented to predict the countries' socioeconomic groups. The results rank the countries according to their socio-economic performance and predict the countries’ development levels for the future. The main findings indicated that the proposed approach can be used for similar research questions. The highest prediction percentages are accurate for small values of k. This study provides a convenient and effective method for grouping countries at different levels of development using basic economic parameters and provides a simple and practical method to predict the belonging group.

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3. Supplier Risk Assessment Research Based on Improved QFD-GC Method with Choquet Integral

Daao Wang, Huan Wang, Zhigeng Fang

The Journal of Grey System 2024, 36 (1): 16-21.

摘要（143）

In the realm of production and manufacturing, a symbiotic relationship with suppliers underscores the significance of rigorous supplier risk assessment. To address this growing need, our approach comprises four essential stages. Firstly, we have pioneered the development of an interval grey number QFD platform, a pioneering tool designed to discern and prioritize pivotal quality risk factors. Secondly, the Choquet integral is skillfully employed to navigate the intricate web of risk events' interrelations, thereby deriving the essential weightings of these risk factors. Thirdly, a cutting-edge grey clustering evaluation model is meticulously crafted, integrating the weightings of the risk factors. This model, a cornerstone of our methodology, is instrumental in classifying suppliers based on their respective risk profiles, optimizing risk management strategies. Lastly, our method's practicality and effectiveness are unequivocally validated through a real-world numerical example, conclusively showcasing its value in the context of supplier risk management.

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4. Evaluation of Care Service Quality for Disabled Elderly Individuals in the Community Based on the Prospect-Grey Target House Model

Yun Fan, Jun Liu, Shuli Yan, Na Zhang, Xiaojun Guo, Zhigeng Fang, Sifeng Liu

The Journal of Grey System 2024, 36 (1): 22-31.

摘要（186）

Considering the varying levels of disability among elderly people, the capability layer → demand layer → service layer structure was decomposed step by step. This enabled us to establish a quality house model that forges an association matrix between care needs and care service attributes for disabled elderly individuals residing in the community. Considering the decision maker's expected grey target and irrational risk attitude towards care service attributes, the prospect-grey target house care service quality evaluation model for disabled elderly individuals in the community was constructed based on prospect theory and grey target decision-making. By evaluating the quality of care services for disabled elderly individuals in the community, the care service levels of different types of disabled elderly individuals and groups that require improvements in care services were identified.

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5. On Grey Real Number and Its Operation Rules

Xican Li, Li Li

The Journal of Grey System 2024, 36 (1): 32-44.

摘要（157）

In order to reveal the mathematical mechanism of generating grey numbers and the law of grey number evolution, the extension principle of grey set is proposed in this paper firstly, which provides a theoretical basis for grey number operation. Secondly, based on the possibility function of grey set, the concept and properties of grey convex set are given. According to grey convex set, the definitions of grey real number, universal grey real number and interval grey number are put forward. Thirdly, the basic operation rules of grey real number are given based on the extension principle, and the operations of data block expression are further given to meet the reversibility of grey real number operation. Finally, the sufficient and necessary condition for the reversibility of universal grey real number operation is given, and it is proved that the grey real number operations using data block expression are reversible. Examples show that the grey real number operations using data block expression proposed in this paper are feasible and effective. The research results enrich the grey system theory and provide a theoretical basis for grey algebra research.

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6. A study on the grey relational weighted evaluation model for the selection of leading industries in the airport economic zone

Hongqing Liao, Zhigeng Fang, Qin Zhang, Xiaqing Liu, Chuanhui Wang, Ding Chen, Xiaochao Qian

The Journal of Grey System 2024, 36 (1): 45-55.

摘要（127）

The evaluation of leading industries involves the use of evaluation index data that exhibit multi-source uncertainty. This implies that the evaluation index values encompass various types of numbers, such as grey numbers, white numbers, fuzzy numbers, interval value fuzzy numbers, and more. The evaluation index system possesses a multi-level structure with interconnections among the indicators. In this study, we propose a generalized grey relational weighted evaluation model based on multi-source uncertainty for assessing leading industries. To begin with, the generalized grey number is used to represent all uncertain data, and the weights of multi-level indicators are calculated by combining the generalized grey number with the information entropy model. Subsequently, the correlation between the indexes is examined utilizing the theorem of the generalized grey absolute relational degree model. This analysis leads to the construction of a generalized grey absolute relational degree matrix, from which eigenvalues and eigenvectors are derived. Based on the generalized grey weight of the multi-level indexes, the eigenvalue and eigenvector of the generalized grey absolute relational matrix, and the grey relational weighted evaluation model for leading industry evaluation are established. Finally, the effectiveness and feasibility of the proposed model are validated through a case study involving the evaluation of the leading industry in the airport economic zone.

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7. Interaction-based nonlinear INDGM(1,N) model and its application

Ye Li, Dongyu Liu, Junjuan Liu, Meidan Xiao

The Journal of Grey System 2024, 36 (1): 56-62.

摘要（152）

Multivariate grey models are commonly used to evaluate the independent effects of related factors, but they may fail to account for any nonlinear interactions that could exist between them. To address this limitation, this study introduces the INDGM(1,N) model, which considers the nonlinear interactions between related factors. Simultaneously, to depict the nonlinear impact of both the system behavior sequence and corresponding factor sequences more accurately, varying power parameters have been incorporated into the model proposed in this paper. Moreover, by adjusting the parameter values of the INDGM(1,N) model, it can be converted into several other models, such as the DGM(1,N) model, GM(1,N) model, DGM(1,1) model, or GM(1,1) model. This study uses a genetic algorithm to obtain the time response of the INDGM(1,N) model by solving its nonlinear characteristic parameters. We then use the model to simulate and forecast China's CO2 emissions and compare its performance with that of other models. The results show that the INDGM(1,N) model provides more accurate simulation and prediction accuracy than other models, highlighting its effectiveness.

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8. Time-Delay TLDBGM(1,N) model with dynamic background value and its application

Dang Luo, Xinqing Qiao

The Journal of Grey System 2024, 36 (1): 63-78.

摘要（171）

Since the traditional multivariable grey prediction model has insufficient consideration of the coupling effect of background value and time-delay accumulative term, which leads to the low prediction accuracy of the model. Based on this, we propose a new multivariable time-delay grey prediction model with dynamic background value. The model adds dynamic background value coefficient, time-delay parameters, linear correction term, and grey action quantity term to the traditional GM(1,N) model. First, the delay periods of driver factors are determined by using the grey time-delay correlation analysis method. Second, the parameter estimation method of the model is discussed and the direct solution of the TLDBGM(1,N) time response function is given by defining the derived form of the TLDBGM(1,N) model. Finally, the model background value coefficient and time-delay parameters are identified and optimized based on the differential evolutionary algorithm. The model is applied to the problem of grain yield prediction in Henan Province. Result shows that the simulation and prediction accuracy of TLDBGM(1,N) are better than other multivariable grey prediction models. The model is theoretically more generalized. And it is shown that GM(1,1), GM(1,N), and TLGM(1,N) models are all special forms of the model for different parameter values.

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9. 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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10. Grey Target Decision Extended Model Based on Prospect Theory and Dual Hesitant Fuzzy Set

Sen Zheng , Lirong Jian, Sifeng Liu, Jie Zhang

The Journal of Grey System 2024, 36 (2): 13-26.

摘要（161）

Grey target decision-making, as an effective method for multi-attribute decision problems, has been applied to evaluations in various domains. However, existing grey target decision-making method fails to mitigate the inherent subjectivity of decision-makers’ preferences and cannot address the dependency issues in raw data. To address these limitations, this paper proposes an extended model of grey target decision-making based on prospect theory in a dual hesitant fuzzy environment. Firstly, by referencing the positive, negative, and median expected points of each dual hesitant fuzzy decision matrix and employing the prospect theory value function for numerical transformation, a dual hesitant fuzzy prospect matrix is formed. Secondly, utilizing the projection distance and the rewardpenalty principle, the comprehensive target center distance of the decision matrix is calculated to identify the superiority or inferiority of different alternatives. Furthermore, a nonlinear model with minimal comprehensive target center distance is established based on the principle of maximum entropy to optimize attribute weights. Finally, the effectiveness and rationality of the proposed decision-making method are validated through a case study of ecological assessment in the aviation industry cluster.

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11. GN.Gompertz Model under Information Progressive Coverage for Reliability Growth Based on Complex Equipment Collaborative Network

Yangyang Du , Yadong Zhang, Sifeng Liu , Zhigeng Fang

The Journal of Grey System 2024, 36 (2): 27-36.

摘要（118）

The development of complex equipment has the feature of "main manufacturer-supplier", with a large amount of underlying data and a small amount of top-level data for reliability growth. This paper firstly introduces the basic model of reliability growth. For small samples and uncertain of complex equipment, Bayesian methods and grey system theory have also been investigated. It has been found that existing research cannot describe this distributed collaborative network and the inherent mechanism of reliability growth, and the information data of reliability growth had not been fully utilized. This paper proposes a new model that combines Bayesian and GERT networks. This model is suitable for evaluating the reliability growth process in collaborative development networks, and can make full use of supplier information, historical information, and experimental information to comprehensively evaluate the reliability growth. Through case analysis of the flight control system, it is proven that the new model performs well in comprehensively utilizing various types of information.

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12. Forecasting PM_2.5Concentration with a Novel Seasonal Discrete Multivariable Grey Model Incorporating Spatial Influencing Factors

Yuanping Ding, Yaoguo Dang, Junjie Wang, Qingyuan Xue

The Journal of Grey System 2024, 36 (2): 37-53.

摘要（190）

Given that PM2.5 concentration is not only related to local pollutants, but also affected by long-distance transmission of PM2.5 in adjacent areas, the key to improving the prediction accuracy of PM2.5 concentration is to comprehensively consider the effect of local influencing factors and the transmission effect of PM2.5 in adjacent areas. For this purpose, a novel seasonal discrete multivariable grey prediction model, encompassing spatial influencing factors, has been established. Firstly, we analyze the mechanism of spatial influencing factors and the reasonableness of using PM2.5 concentration in adjacent areas as the spatial influencing factors. Secondly, based on the spatial agglomeration characteristics of PM2.5 concentration, the K − means clustering algorithm is used to cluster adjacent cities with similar PM2.5 concentration, then the comprehensive value of PM2.5 concentration in each city cluster is calculated by weighted average combination. On this basis, a driving term of spatial influencing factors and a cosine trigonometric function term are introduced into the novel model to characterize the effect of spatial influencing factors on PM2.5 concentration and the seasonal fluctuation of itself, respectively. More importantly, the Genetic Algorithm Toolbox is employed to optimally determine the emerging parameters of this model, and the time response function of the novel model is calculated by mathematical induction method. Lastly, the new model is deemed valid through testing its PM2.5 concentration predictions for the cities of Beijing, Tianjin, and Baoding in the Beijing-TianjinHebei region. Based on the original observations from 2018Q1 to 2023Q2, the novel model is built for PM2.5 concentration prediction in 2023Q3 to 2024Q2 for the three cities. The findings imply that the newly developed model outperforms its competitors significantly and has the potential to serve as a robust tool for predicting PM2.5 concentration.

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13. Reinforcement Model for Unmanned Combat System of Systems Based on Multi-Layer Grey Target

Xueting Hao, Zhigeng Fang, Jingru Zhang, Fei Deng, Ankang Jiang, Shuyu Xiao

The Journal of Grey System 2024, 36 (2): 54-66.

摘要（179）

In the future battlefield, unmanned combat mode will be crucial. Its attack strategy formulation is a significant and complex task. To address the issue of decisionmaking in unmanned combat system of systems(UCSoS) striking strategy, this paper begins by analyzing the characteristics of UCSoS. By utilizing the GERT concept, an unmanned combat A-GERT network is developed to provide parameter support for evaluating its effectiveness. Secondly, for effectiveness evaluation, a multi-layer gray target model built on the A-GERT network is proposed. This model is utilized to formulate the optimal striking scheme for the UCSoS. And Agent technology is employed to solve the intelligent learning decision-making problem for UCSoS based on multi-layer grey target model. Finally, a case study illustrates the efficiency and effectiveness of the reinforcement model for UCSoS based on multi-layer grey target.

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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. Parameter Estimation of Integro-differential Equation-based Grey Predator-prey Model From Noisy Data

Zhaoya Zhang, Naiming Xie, Lu Yang, Xiaolei Wang

The Journal of Grey System 2024, 36 (2): 79-89.

摘要（99）

The grey predator-prey model is widely applied in many fields for its effectiveness. While most of the research on the grey predator-prey model focuses on the different model forms, less attention has been paid to identifying parameters and initial value from noisy data. In this work, considering the measurement noise in practice, we propose a two-stage approach to estimate the structural parameters and initial value of the grey predator-prey model with integro-differential equations(IDEs). First, by introducing an integral operator, we propose another form of the grey predator-prey model, namely the IDE-based grey predator-prey model. Next, we develop a parameter estimation approach based on the idea of a two-stage in the presence of measurement noise, where in the first stage we smooth time series using cubic B-Spline and in the second stage estimate structural parameters and initial conditions by the separable nonlinear least squares. Then, the finite sample performance of the IDE-based model is investigated with Monte Carlo numerical experiments. Results demonstrate that the IDE-based model has better performance in terms of accuracy than the Cusum-based model. Finally, we use a case to further verify the accuracy and stability of the proposed method.

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16. Discovering the mechanism of grey forecasting models from the perspective of dynamic system modelling

Xiaolei Wang, Naiming Xie

The Journal of Grey System 2024, 36 (2): 90-99.

摘要（144）

Grey forecasting models have found extensive applications across various domains, but the connection between their theory and practice has not yet been fully revealed. This paper seeks to discuss the modelling mechanism of grey forecasting models from the perspective of dynamic system modelling and illustrate how to establish grey forecasting models to address real-world challenges. Firstly, we outline the grey forecasting models under the traditional and direct frameworks. Then, the problem description and model assumptions of grey forecasting models are discussed by incorporating model characteristics and Prof. Deng's original concepts. The complete process of model establishment as well as the purpose and tasks of each step is elaborated in detail. Ultimately, taking the inventory of perishable products as a case study, this article discusses the utilization of grey forecasting models in inventory management and elucidates the application process using citrus as a specific example.

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17. Ordinal Multivariate Grey Incidence Model and Its Application on Early Warning of Construction Quality Risk

Ke Zhang, Min Ma, Feizhen Zhang, Yuxin Zhou, Chunyong She, Zheng Zhang

The Journal of Grey System 2024, 36 (3): 1-10.

摘要（185）

Government supervision is the highest level of construction quality management system. Due to a large number of constructions in progress, timely and accurate risk early warning is imperative for improving the efficiency of supervision. Aiming at the small-scale, ordinal, and unequal length multivariate time series of government supervision data, this paper proposes a construction quality risk early warning method based on ordinal multivariate grey incidence analysis. Firstly, to measure the dynamic similarity between risk indicators of projects, the proximity grey incidence model based on ordinal dynamic time warping (DTW) and the similarity grey incidence model based on ordinal L1 norm DTW are constructed respectively. Then, the two models are integrated to construct a comprehensive similarity model for construction quality risk warning. Combining the comprehensive similarity and k-nearest neighbour (k-NN) algorithm, a method of construction quality risk level classification and early warning is constructed. Finally, the method is applied to the quality supervision of water conservancy and hydropower projects in Zhejiang Province, and the results show that the proposed method can effectively solve the problem of construction quality risk early warning based on small-scale and ordinal data.

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18. Multi-steps Carbon Emission Forecasts Using a Novel Grey Multivariable Convolution Model

Song Ding, Juntao Ye, Zhijian Cai, Xing'ao Shen, Huahan Zhang

The Journal of Grey System 2024, 36 (3): 11-24.

摘要（131）

The accurate forecasting of provincial carbon emissions is pivotal for China as it strives to meet its carbon neutrality goals. To this end, an improved grey multivariable convolution model has been developed, employing a unified new-information-based method for the preliminary accumulation of data. The particle swarm optimization (PSO) algorithm is then applied to determine the optimal parameters within this sophisticated model. Moreover, to identify the relevant factors for provincial carbon emissions, a comprehensive determination of these factors was conducted from two aspects: literature research and grey relational analysis. For validation, carbon emission data from two provinces are analyzed, and the model’s efficacy is thoroughly compared with five competitors across three different predictive horizons. The empirical results indicate that the proposed model has distinct advantages over the competing models. Additionally, the model’s robustness and comprehensive forecasting abilities for provincial carbon emissions are confirmed through detailed Monte Carlo simulations and parameter sensitivity analyses across various forecasting horizons.

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19. Some Properties of Generalized Whiteness of Interval Grey Number

Li Li, Xican Li

The Journal of Grey System 2024, 36 (3): 25-36.

摘要（192）

In order to mine the intrinsic information of interval grey number, the concept of the generalized whiteness of interval grey number is first given in this paper based on the generalized greyness of interval grey number. Then the static and dynamic properties of generalized whiteness on bounded background domain, infinite background domain and infinitesimal background domain are analyzed, and the concepts of the extreme white system and extreme white spot are given. Finally, the conservation law of the generalized whiteness of interval grey numbers is given, and the generalized whiteness is applied to the ranking of interval grey numbers. The results show that the generalized whiteness of interval grey number on the bounded background domain has the static properties of the relativity, normality, unity, opposition, connectivity, justice and graduality; while on the infinite background domain and the infinitesimal background domain, the generalized whiteness of interval grey number has the above static properties except the graduality. On the bounded background domain, the generalized whiteness changes with the expansion of the background domain and the value domain, while on the infinite background domain and the infinitesimal background domain, the static and dynamic properties of the generalized whiteness of interval grey number are the same, it is not affected by the expansion change of the background domain and value domain, and the generalized whiteness of interval grey numbers is conserved. The research results not only enrich the grey system theory, but also provide a theoretical basis for the analysis and utilization of interval grey numbers.

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20. Grey Generalized Stochastic Petri-Bayesian Network Testability Model for High-reliability Complex Systems

Cuiping Niu, Zhigeng Fang, Shuyu Xiao, Youpeng Liu

The Journal of Grey System 2024, 36 (3): 37-50.

摘要（167）

Aiming at the issues of paucity of fault information, complexity of functional logic relationships between fault modes, and uncertainty of fault information and its propagation path in the testability analysis of high-reliability complex systems, a grey generalized stochastic Petri-Bayesian network (Grey-GSPBN) testability model is proposed in this study. Firstly, typical failure modes and their severity are obtained through the failure modes, effects and analysis (FMECA) study, and the failure modes are coded and coloured accordingly to construct the generalized stochastic Petri network (GSPN) model. Then, the correlation matrix between failure modes and test points is established by using the reachability algorithm, based on which the equivalent isomorphic grey Bayesian network (GBN) model is established, and grey number theory is introduced to integrate multi-source grey information to determine the grey prior and posterior distribution matrix of testability indexes. Finally, the grey probabilistic testability evaluation matrix is calculated using GreyGSPBN model, and the testability indicators are analyzed. A certain liquid rocket engine system is taken as a case to verify the scientificity and superiority of the proposed model in the testability modelling of high-reliability complex systems, and the model can provide a valuable reference for engineering applications

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21. 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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22. Risk-transmission Mechanism of Industry Chain under a Multi-parameter Grey-GERT Network

Lan Xu, Yingying Shang

The Journal of Grey System 2024, 36 (3): 63-73.

摘要（133）

Aiming at the risk of local obstruction or rupture in the operation process of the industry chain, a Grey-GERT network model of industry chain risk transmission is constructed based on the effect of input resources of each link of the industry chain, and the key links and their degree of risk in the process of industry chain network value transmission are identified and analysed to reveal the risk transmission mechanism of the industry chain. Finally, an empirical study is conducted on China’s integrated circuit industry chain to verify the feasibility and effectiveness of the proposed model and to propose targeted control measures for the key links and their value transmission risks. The results show that the proposed model can effectively solve the problem of incomplete information on multiple transmission parameters in industry chain network activities, thoroughly analyse the risk transmission mechanism of the industry chain, and provide theoretical support for strengthening the risk control of the industry chain.

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23. 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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24. Predicting Solar Array Power Output on a Spacecraft Using a Fractional-Order Grey Model and Particle Swarm Optimization

Liang Ren, Yuanhe Gao, Feng Yang, Yongcong He

The Journal of Grey System 2024, 36 (3): 86-97.

摘要（90）

During eclipse periods, the spacecraft relies on electricity, which its solar arrays produce and store in batteries. Forecasting a solar array’s power output employed during space missions is of significant importance. Varying space environments and satellite loads, which are characterized by significant randomness and uncertainty, affect the generated power of the spacecraft’s solar array. These challenges pose difficulties in power prediction. To address these issues and achieve a more accurate estimation of the solar array’s generated power during a space mission, this study develops a metabolic model termed TDGM(1, 1, r) that incorporates an enhanced accumulating fractional-order, optimizing it within the discrete grey TDGM(1,1) model’s framework with three parameters. The optimization model’s objective function is defined as the mean absolute percentage error (MAPE) within the modeling context. In order to minimize MAPE, the differential equation’s order and accumulation number are determined using a particle swarm approach. The TDGM(1, 1, r) demonstrates superior forecasting performance in comparison to the classical GM(1,1) and Grey–Markov models. These findings indi-cate the superiority of TDGM(1,1,r) over GM(1,1) and Grey–Markov, with improvements of 84.2% and 81.2% for MAPE (from 1.83% to 0.29% and from 1.54% to 0.29%). The metabolic TDGM(1,1,r) employing the particle swarm algorithm (PSO) is better suited for short-term predictions. Finally, relevant suggestions for future development of the prediction model are proposed.

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25. 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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26. Maintenance Modeling and Grey Lease Pricing for a Series Manufacturing System with the Machines from Multiple Suppliers

Yaping Li, Yuhong Cheng, Tangbin Xia, Zhen Chen, Naiming Xie, Ershun Pan

The Journal of Grey System 2024, 36 (4): 14-25.

摘要（151）

Leasing is an important mode of service-oriented manufacturing where a manufacturer may lease machines from multiple suppliers to form a manufacturing system. While the suppliers provide machine maintenance, their individual interests may not be necessarily optimal to the system. Therefore, we propose a maintenance modeling and lease pricing (MMLP) framework to find the optimal maintenance policy for the system, and make a lease pricing scheme as an incentive mechanism to promote the realization of the cooperative maintenance among the suppliers. An optimization model is established to obtain a long-term maintenance policy by minimizing a cost rate function. And then, the grey lease pricing scheme based on the fair allocation of the benefits from the cooperation is suggested, with the marginal contribution of each supplier in the cooperation measured by the grey Shapley value method. Finally, a case study is used to show the application of the MMLP framework, presenting that the joining of the suppliers can make maintenance cost reduced and that the lease prices determined by using the grey Shapley value demonstrate the emergence and risk of the cooperation.

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27. Hydrogen Load Demand Prediction in Unified Energy System Based on Grey Ridgelet Neural Network

Dou Qin, Bin Zhao

The Journal of Grey System 2024, 36 (4): 26-32.

摘要（144）

Hydrogen will play critical pole in industrial field, heating field and transportation field, which can achieve mutual conversion of different energies. Hydrogen load prediction demand is important for establishing unified energy system, a novel prediction model is established based on particle swarm algorithm (PSA) and grey Ridgelet neural network (GRNN) to improve medium and long term hydrogen load demand prediction accuracy. Firstly hydrogen load demand prediction model in unified energy system is established, which concludes hydrogen load demand prediction models in industrial field, heating field and transportation field, and then total hydrogen demand model is deduced. Secondly, model of GRNN is constructed based on grey system theory and Ridgelet neural network, analysis procedure of GRNN is established. Structure of GRNN is confirmed, and mathematical model is constructed. To enhance prediction effectiveness of GRNN, PSA is used to optimize parameters of GRNN. Finally hydrogen load demand data in a province is selected to carry out prediction simulation, results show that prediction error of proposed PSA-GRNN ranges from 1.88% to 3.02%, which is less than that of other three prediction models, and fit goodness of proposed PSA-GRNN ranges from 0.958 to 0.985, which is also less than that of other three prediction models. Therefore proposed PSA-GRNN has better prediction precision and efficiency, which can obtain better precision effect and applicability. Hydrogen load demand prediction results in heating field based on PSAGRNN are closer to real value than that based on other three prediction models, results show that proposed PSA-GRNN has better prediction accuracy that other three prediction models.

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28. Construction of Symbiosis System for Rural Industry Revitalization Based on Lotka-Volterra Model and Stability Strategy Study

Na Zhang, Shuting Shi, Zihao Li

The Journal of Grey System 2024, 36 (4): 33-54.

摘要（164）

The symbiotic system of rural industrial revitalization comprises farming households, grassroots governments, and new rural collective economies as symbiotic units. The optimal symbiosis mode within this symbiotic system is reciprocal symbiosis. This study focuses on Yuhang District in Zhejiang Province, Yining County in Xinjiang Uygur Autonomous Region, and Wangcheng District in Hunan Province as research areas. Firstly, the grey Lotka-Volterra (GLV) model is employed to analyze the interaction within the current symbiotic system of rural industrial revitalization across these regions using data from 2019-2023. Secondly, this paper utilizes the GM(1,1) model to predict future data for these regions and analyze their future symbiosis. Subsequently, this paper examines the equilibrium point and stability of the symbiosis systems within these regions. Finally, based on an evolutionary game model approach, key factors influencing the evolution of a stable symbiosis system when satisfying and balancing interest demands among symbiotic units are explored. The findings reveal distinct characteristics within each subject's rural industrial revitalization symbiotic system. Government subsidies' intensity and cooperation benefits and costs primarily influence its evolution towards stability.

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29. 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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30. Prediction of Digital Economy Development Levels in Urban Cities Based on the GCSA-GM(1,N) Model

Chengxuan Wu, Cheng Tian, Fang Wang, Wenxin Cheng

The Journal of Grey System 2024, 36 (4): 69-77.

摘要（190）

Based on the digital economy index (DEI) and Technological Innovation, Industrial Structure, GDP and Openness to the Development Index data of 15 sub-provincial cities from 2017 to 2021, we construct a framework to predict the development potential of the urban digital economy and analyse the spatial evolution trend under the ‘small data’ scenario using geometric causal strength analysis GM(1,N) and the gravity center model. The empirical analysis reveals that,15 sub-provincial cities, at least one of the influencing factors has a causal relationship with the urban DEI that is greater than 0.5. The average forecast error of the GM(1,N) model based on causality strength in 15 sub-provincial cities is less than 1% in 2022. This reflects that four influencing factors can be used as an effective indicator to measure the level of digital economic development. The forecast results also indicate that the digital economy center of China’s sub-provincial cities will evolve from north to south and from east to west in 2022-2025. Finally, this study presents suggestions from three aspects: Strengthening technological innovation, promoting industrial digital transformation and upgrading, and strengthening cross-regional cooperation and exchanges.

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31. Residual Life Prediction of High-pressure Pipeline Erosion Based on the Grey Markov Model

Liu Xiong, Mo Li

The Journal of Grey System 2024, 36 (4): 78-89.

摘要（92）

Based on the digital economy index (DEI) and Technological Innovation, Industrial Structure, GDP and Openness to the Development Index data of 15 sub-provincial cities from 2017 to 2021, we construct a framework to predict the development potential of the urban digital economy and analyse the spatial evolution trend under the ‘small data’ scenario using geometric causal strength analysis GM(1,N) and the gravity center model. The empirical analysis reveals that,15 sub-provincial cities, at least one of the influencing factors has a causal relationship with the urban DEI that is greater than 0.5. The average forecast error of the GM(1,N) model based on causality strength in 15 sub-provincial cities is less than 1% in 2022. This reflects that four influencing factors can be used as an effective indicator to measure the level of digital economic development. The forecast results also indicate that the digital economy center of China’s sub-provincial cities will evolve from north to south and from east to west in 2022-2025. Finally, this study presents suggestions from three aspects: Strengthening technological innovation, promoting industrial digital transformation and upgrading, and strengthening cross-regional cooperation and exchanges.

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32. Assessing the Service Quality of Fisherman's Homestays in China: a Hybrid MADM Approach Consisting of DANP and Grey Clustering Evaluation

Peng Jiang, Rui Chi, Hui Xia, Longyun Zhang, Chuandong Ju

The Journal of Grey System 2024, 36 (4): 90-110.

摘要（155）

Nowadays, a new emerging tourism industry, named fisherman's homestay, has become a very representative marine leisure tourism project, widely promoted in coastal areas in China. However, the development of China's fisherman's homestay industry is not yet mature and there are many problems, such as incomplete laws and regulations, lack of infrastructure, and severe homogenization. Therefore, it is very important to help the owners of fisherman's homestay improve the service quality. Based on the Service Quality Gap Model, this paper establishes an evaluation index that affects the service quality of fishermen's homestay. We apply decisionmaking trial and evaluation laboratory-based analytic network process (DANP) to identify the critical factors and the causal relationship between them. And the grey clustering model is used to evaluate the service quality of Liaoning province and other five places. Experimental results reveal those six factors, including surrounding facilities, online marketing, room infrastructure, personalized service, community co-prosperity, and attention to harmonization with the environment, contribute to improving the service quality of fisherman's homestays. The fisherman's homestays in Shandong Province, Liaoning Province, and Zhejiang Province are in "excellent" level, and the fisherman's homestays in Fujian Province and Guangxi Zhuang Autonomous Region are in "good" level.

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33. Evaluation of Barriers to Disabled Elderly’s Access to eHealth in China Using Grey Relational Analysis

Muhammad Nawaz, Sifeng Liu, Naiming Xie, Mohammed Atef, Muhammad Wasif Hanif

The Journal of Grey System 2024, 36 (5): 1-14.

摘要（192）

This study aimed to identify and rank the barriers faced by disabled elderly in China while accessing eHealth primary care services. Primary data were collected from the disabled elderly based on technological, individual, relational, environmental, and organizational constructs. The Dynamic Grey Relational Analysis (DGRA) and Multiple-criteria Decision-making (MCDM) based TOPSIS techniques were used to identify and rank the barriers. We found that the most significant barrier was “aging limitation (reduction in hearing, sight, memory, and fine motor control)” in both (DGRA and MCDM) cases. The Kruskal-Wallis test was used to investigate the significance of this barrier in different age groups of disabled elderly. We found no significant differences among the three age groups of disabled elderly, which shows that the barrier “aging limitation (reduction in hearing, sight, memory, and fine motor control)” is the most significant barrier at each age group (when age ≥ 60) of disabled elderly. The average value of Grey Relational Grades (GRGaverage) and the sorting outcomes of the MCDM of the construct individual were higher than those of all other constructs. This study is the first of its kind to apply the DGRA, MCDM and KWT to expose the barriers while accessing eHealth services for the disabled elderly in China.

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34. 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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35. Seasonal Grey Forecasting Model Based on Damping Accumulation and Its Application

Ye Li, Chengyun Wang, Qiwen Wei, Shi Yao

The Journal of Grey System 2024, 36 (5): 27-42.

摘要（148）

A new damping nonlinear grey multivariate seasonal forecasting power model DAFGM(1,N, , ) is proposed to solve the problem of small sample forecasting with seasonal, nonlinear, and uncertain system behavior characteristic sequence. Firstly, the seasonal moving filter is used to eliminate the seasonal characteristics of the original series. Then, according to the principle of "new information priority ", the damping accumulation coefficient is introduced, the unknown factors which are difficult to collect are simulated by introducing dummy variables, and a new seasonal forecasting model is constructed. Finally, the model is used to forecast the quarterly wind power generation in China. The results show that the model has good practicability and effectiveness.

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36. Novel Grey SIRS Model Forecasts Credit Risk with Nonlinear Infection

Qian Lv, Xinping Xiao, Mingyun Gao

The Journal of Grey System 2024, 36 (5): 43-57.

摘要（106）

Epidemic models are widely used in financial risk prediction. The problems of nonlinear changes in infection rates and limited data samples in financial risk remain to be addressed. To this end, this paper proposes a nonlinear grey SIRS (abbreviated as GSIRS) model based on short-term data. This model employs a time-varying function to capture the nonlinear dynamics of infection rates, and integrates the system grey prediction model to analyze short-term data. Parameter optimization is achieved through the least square method and the whale optimization algorithm. The GSIRS model shows good prediction accuracy across three financial crisis datasets, with MAPE ranging from 3.379% to 4.981% for training sets and 2.913% to 3.212% for test sets. These values are significantly better than those of competition models. In addition, the CWC values of the interval prediction under the 95% confidence level of the model are 0.13, 0.14 and 0.33, respectively. The combination of excellent RMSE and STD metrics further proves the stable forecasting ability. Meanwhile, the sensitivity analysis shows that changes of infection rate have a 1-2 period lagged effect on the infected individual density.

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37. A Temperature Error Correction Method with the ARIMA–GM(1,1) Model

Xin Feng, Juncheng Jiang, Ni Lei, Li Lei, Haibing Feng, Zhiquan Chen, Shu Li

The Journal of Grey System 2024, 36 (5): 58-69.

摘要（141）

To address the problem of temperature errors in secondary instruments operating in high- and low-temperature environments, this paper proposed a temperature correction method based on the ARIMA–GM(1,1) model. First, a standard source was connected to a temperature secondary instrument placed in a high- and low-temperature circulation box. The errors between the measurements of the standard source and the secondary instrument could be calculated and obtained a set of error sequences. Second, the error sequences were used to establish an ARIMA model and obtained a set of predicted values. And the residual between the errors and the predicted values could be calculated. To improve the accuracy of the ARIMA model, a GM(1,1) residual correction model was established based on the residual sequences. Lastly, the ARIMA and the GM(1,1) models were combined to formulate an ARIMA–GM model that could perform error self-correction for the temperature secondary instrument. In application experiments, the model achieved smaller average relative errors than a traditional ARIMA and hybrid models. Finally, we developed the ARIMA–GM(1,1) model into a software and applied it to cases of actual detection.

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38. A New Grey Forecasting Model with Fractional Order Accumulation Generation Operation and Its Application in GDP Forecasting

Qifeng Xu, Yongjun Guan , Yunbao Xu, Ran Wang

The Journal of Grey System 2024, 36 (5): 70-79.

摘要（119）

In this paper, a new fractional-order grey forecasting model with a temporal power term that can handle both annual and quarterly forecasting tasks for GDP is presented. The model's characteristic is that it has a dynamic simulation parameter, which can automatically adjust the structure of the model according to the need of the prediction task to achieve the purpose of accurate prediction. In addition, the fractional order parameter and power term parameter of the model play an important role in enhancing the adaptive performance of the model. In particular, an excellent intelligent optimization algorithm, the Ant lion optimizer, is used to solve the model's programming model to obtain the hyperparameters for modeling quickly. In this study, China's annual GDP and quarterly GDP are used as research objects to verify the validity of the new model. The experimental results show that all evaluation indicators of the proposed method are better than those of its competitors. Therefore, the model has some application value.

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39. An Improved Grey Time Power Model for Forecasting the Ecological Environmental Water Consumption In the Upper Yangtze River Basin

Rui Duan, Shuliang Li, Weizhe Sun, Wei Meng, Dajin Zeng, Kui Yu

The Journal of Grey System 2024, 36 (5): 80-95.

摘要（150）

Scientific and accurate forecast of ecological environmental water consumption (EEWC) in the upper Yangtze River basin is of major prominence to the sustainable development of the basin and the formulation of eco-environmental protection policies. Firstly, a two parameter variable weight buffer operator is used to pre-processing the system shock behavior sequence. Then, an improved grey model IGM4(λ,γ,t^a) with four background values is established, introducing power exponential terms and linear correction terms to characterize data series with mixed linear and nonlinear relationships. The particle swarm optimization (PSO) algorithm is employed to find optimal parameters. Additionally, the model’s effectiveness is evaluated by comparing the fitting values of models with other grey models. The final results demonstrate that the IGM4( λ,γ,t^a) performs best with mean absolute percentage error only 0.0199%. Finally, model IGM4( λ,γ,t^a) is utilized to predict the EEWC in the upper Yangtze River basin from 2023 to 2028. The reasonableness of the predicted results is analyzed, and related policy measures are put forward.

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40. An Optimization Scheme for Enhancing the Performance of Fractional-order Grey Prediction Models in Seasonal Forecasting Tasks: the Case of the Fractional-order GM(1,1) Model

Yanan Li, Liang Zeng

The Journal of Grey System 2024, 36 (5): 96-105.

摘要（107）

Fractional-order grey prediction models have gained wide recognition for their computational efficiency and straightforward modeling mechanisms. However, their performance in seasonal forecasting tasks still needs improvement. To address this, this paper designs a novel optimization scheme and applies it to the representative fractional-order grey GM(1,1) model (FGM(r,1)) to advance research in this area. In this optimization scheme, the dummy variable is used to enable the model to directly handle seasonal time series, the discretization technique is employed to simplify the computational steps, and the Bernoulli parameter and the linearly weighted hybrid fractional-order accumulation strategy are used to enhance the model's fitting capability. To verify the effectiveness of the proposed method, the optimized model and some benchmark algorithms are used to model three quarterly data sets. The experimental results show that the optimized model can produce better performance, which verifies the effectiveness of this optimization scheme.

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