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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.   Abstract （152）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （278）            Knowledge map    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.  Related Articles | Metrics

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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.   Abstract （259）            Knowledge map    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. Related Articles | Metrics

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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.   Abstract （211）            Knowledge map    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.  Related Articles | Metrics

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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.   Abstract （237）            Knowledge map    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.   Related Articles | Metrics

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On Grey Real Number and Its Operation Rules Xican Li, Li Li The Journal of Grey System    2024, 36 (1): 32-44.   Abstract （206）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （182）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （255）            Knowledge map    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.  Related Articles | Metrics

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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.   Abstract （254）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （193）            Knowledge map    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.  Related Articles | Metrics

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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.   Abstract （211）            Knowledge map    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. Related Articles | Metrics

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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.   Abstract （204）            Knowledge map    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.  Related Articles | Metrics

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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.   Abstract （237）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （182）            Knowledge map    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. Related Articles | Metrics

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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.   Abstract （212）            Knowledge map    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 Related Articles | Metrics

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Forecasting PM2.5 Concentration 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.   Abstract （241）            Knowledge map    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. Related Articles | Metrics

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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.   Abstract （221）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （163）            Knowledge map    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.   Related Articles | Metrics

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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.   Abstract （243）            Knowledge map    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. Related Articles | Metrics

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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.   Abstract （305）            Knowledge map    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.   Related Articles | Metrics