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Multivariate Fractional Grey Model for Port Throughput Prediction  Xinyu Wang, Xinquan Liu, Yingyi Huang, Che-Jung Chang, Jianhong Guo The Journal of Grey System    2025, 37 (2): 23-32.   Abstract （328）            Knowledge map    Accurate prediction of port throughput is critical for enhancing the precision of port construction decisions. However, identifying the factors influencing port throughput and constructing a robust predictive model remain significant challenges. To address this issue, this paper develops a modeling framework based on grey systems theory. First, grey relational analysis is employed to identify the relevant factors impacting changes in port throughput. These influencing factors are then utilized to construct a multivariate weighted fractional grey model for throughput prediction. To enhance the predictive capability of the grey model, this study establishes an error-based objective function to determine the model's fractional order, followed by corrections to the residual sequence using Long Short-Term Memory (LSTM) neural networks to improve prediction accuracy. To validate the effectiveness of the proposed method, empirical analysis is conducted using port throughput data from Guangxi Province, with comparisons made against four commonly used models. The experimental results demonstrate that the grey modeling framework proposed in this paper achieves high accuracy, with a 0.67% reduction in mean absolute percentage error (MAPE) outperforming alternative models. Furthermore, the model predicts a rapid growth trend in port throughput in Guangxi over the next three years, suggesting that expanding the utilization rate of existing port infrastructure and accelerating investments in capacity expansion are critical for meeting future demand. These results highlight the model's potential as a decision-support tool for strategic planning in the port and logistics sectors.  Related Articles | Metrics

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An Unbiased Grey Model Based on Euler Polynomial and Its Application in China's Primary Energy Production Shuliang Li, Ying Wang, Ruifeng Zheng, Wei Meng, Dajin Zeng The Journal of Grey System    2025, 37 (2): 1-15.   Abstract （237）            Knowledge map    A key property of grey prediction models is their unbiased nature, which focuses on eliminating discontinuity between differencing and differentiation during their construction. Meanwhile, an unbiased proof is required to ensure this property. In this paper, an unbiased EDGM(1,1) model incorporating Euler polynomials is constructed. Firstly, this model integrates Euler polynomials and a time disturbance parameter into the classical GM(1,1) model, enabling this EDGM(1,1) model to flexibly handle sequences with various data characteristics. Subsequently, the difference and its discrete forms are derived, and the latter is then solved using the least squares method and mathematical induction. Then the Particle Swarm Optimization (PSO) algorithm is implemented to improve the model's parameters. Secondly, the compatibility of the EDGM(1,1) model is demonstrated, and its unbiasedness towards three characteristic sequences is proven based on Cramer's rule. Finally, the performance of the EDGM(1,1) model is evaluated comprehensively against five competing models using three metrics: MAPE, RMSE, and R². The comparative analysis shows that the EDGM(1,1) model outshines other models in robustness and accuracy. Furthermore, the novel model is designed to forecast China's primary energy outputs, aiming to provide references for energy policies and decision-making.  Related Articles | Metrics

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GreyShot: Zeroshot and Privacy-preserving Recommender System by GM(1,1) Model  Hao Wang The Journal of Grey System    2025, 37 (2): 16-22.   Abstract （218）            Knowledge map    Every recommendation engineer needs to face the cold start problem when building his system. During the past decades, most scientists adopted transfer learning and meta learning to solve the problem. Although notable exceptions such as ZeroMat etc. have been invented in recent years, cold-start problem remains a challenging problem for many researchers. In this paper, we build a zeroshot and privacypreserving recommender system algorithm GreyShot using GM(1,1) model by taking advantage of the Poisson-Pareto property of the online rating data. Our approach relies on no input data and is effective in generating both accurate and fair results. In conclusion, zeroshot problem of recommender systems could be effectively solved by grey system methods such as GM(1,1).  Related Articles | Metrics

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Model Validation and Visualization Techniques of Grey Relational Analysis  Honghua Wu, Yafang Li, Xue Han, Aqin Hu The Journal of Grey System    2025, 37 (2): 63-76.   Abstract （215）            Knowledge map    To address issues in the validation and analysis methods of panel data-based grey relational analysis models (PD-GRA), this paper introduces various validation methods for the PD-GRA model, including Collision Testing Analysis (CTA), Stability Testing Analysis (STA), Permutation Testing Analysis (PTA), Rolling Window Time Analysis (RWTA), along with corresponding visualization techniques. These approaches are designed to analyze the robustness of grey relational analysis model (GRA). First, the concepts of strong and weak collisions for the GRA model are defined, along with their occurrences at different levels. Conclusions on strong and weak collisions are drawn for different GRA formulas. Second, STA, PTA, and RWTA are systematically presented to evaluate the stability of PD-GRA model from different perspectives. Meanwhile, visualizations for these three tests are also provided. Finally, a simulation analysis is conducted to examine the collision behavior of traditional GRA models. A case study is then used to apply STA, PTA, and RWTA to the PD-GRA model, accompanied by visualizations, offering deeper insights into the model's performance.  Related Articles | Metrics

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Research on Construction of a Novel Grey Clustering Model Based on Possibility Functions Considering Dynamic Contribution Degree and Its Application  Shuaishuai Geng, Zhaohan Hu, Jing Jia, Xiao Xu, Sandang Guo The Journal of Grey System    2025, 37 (2): 50-62.   Abstract （214）            Knowledge map    This paper evaluates the high-quality development of national central cities by proposing a novel grey clustering method for the comprehensive assessment of their developmental levels. This approach facilitates dynamic analysis of urban development status and contributes to the theoretical framework of grey evaluation methods, thereby providing a robust foundation for urban development decision-making. The proposed method addresses the limitations inherent in traditional grey possibility function clustering models, particularly their inadequacy for panel data analysis and their lack of dynamic measurement capabilities in evaluating indicator systems. The novel model incorporates time weighting by optimizing the weights of time point index attributes to derive time domain attribute weights. For panel data, observed values are integrated into a comprehensive evaluation framework, facilitating dimension reduction and the transformation of panel data into cross-sectional data. Subsequently, grey clustering analysis is applied to the cross-sectional data. The validity and feasibility of the proposed model in handling panel data are verified through an evaluation and analysis of the high-quality development of national central cities.  Related Articles | Metrics

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A Novel Grey MCDM Model Assessing Macroeconomic Performance of G7 Countries Alptekin Ulutaş, Ayse Topal, Darjan Karabasevic, Edmundas Kazimeras Zavadskas, Muzaffer Demirbaş, Salim Üre The Journal of Grey System    2025, 37 (3): 61-72.   Abstract （204）            Knowledge map    Macroeconomic indicators offer critical insights into the economic performance of nations. The potential variability of these factors necessitates formulating policies and implementing actions to counteract any adverse situations that may arise. This research aims to evaluate the macroeconomic performances of the seven developed nations, known as the G7 nations. The research identified imports of goods and services, exports of goods and services, gross fixed capital formation, gross domestic savings, unemployment, population, current account balance, inflation, consumer prices gross domestic product as criteria for performance assessment. An integrated framework integrating the LOPCOW-G and RAWEC-G methodologies is presented to assess the macroeconomic performance of G7 nations within the study's framework. The weight values derived from the LOPCOW-G technique indicate that the current account balance is the most significant factor influencing macroeconomic success. The RAWEC-G technique findings indicate that Japan had the highest economic performance, while the USA demonstrated the lowest economic performance. Related Articles | Metrics

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Prediction of China’s Fossil Energy Consumption Using GRNN-Based Grey Multivariable Model Jiayi Liu, Jun Zhang The Journal of Grey System    2025, 37 (3): 1-10.   Abstract （194）            Knowledge map    In view of the dominant position of fossil energy in global energy consumption and the environmental problems caused by excessive use of fossil energy, accurate prediction of fossil energy consumption is of great significance for formulating scientific energy policies and optimizing energy structure. Traditional forecasting methods have limitations when dealing with small samples, nonlinear and multi-factor problems, while grey system theory and neural network model are good at dealing with uncertainty and nonlinear mapping respectively. Therefore, this study hybridizes the generalized regression neural network (GRNN) model on the basis of the dynamic nonlinear grey delay multivariable Logistic model, i.e. NGDM(1, N), and constructs grey NGDM(1, N)-GRNN hybrid model to further optimize the prediction results. Particle swarm optimization (PSO) algorithm was used to optimize grey model parameters, and the optimal smoothing factor of GRNN model was found through cross-validation, which improved the prediction accuracy of the model. The empirical results show that compared with the single NGDM(1, N) model and GRNN model, the proposed hybrid model has smaller errors in the short-term prediction of fossil energy consumption, and has better forecasting effect. Related Articles | Metrics

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Grey BP Neural Network Combinatorial Model with Time-delay Causal Term and its Application Jing Ye, Luolan Zhang The Journal of Grey System    2025, 37 (4): 1-12.   Abstract （175）            Knowledge map    In order to enhance the precisions of grey prediction models, it is essential to address the univariate constraints of the traditional GM(1,1) model and consider the influences of data’s time delay and nonlinear mapping on system behavior. In this paper, based on the grey model with background value optimization, we propose a time-delay optimized grey BP neural network combinatorial model (BP-TDOGM(1,1)) by introducing a time-delay causal term and combining with the BP neural network. The mechanisms of time delay, nonlinear mapping and related factor sequences on system behaviour are discussed in detail. Furthermore, the modelling framework, parameter estimation methods and model resolution techniques are investigated with a view to enhancing the capture of data correlations. These endeavours are designed to extend the scope of application of neural networks in scenarios characterized by limited information and to markedly optimize the prediction accuracy of the grey forecast model. Ultimately, the proposed model's efficacy is validated through an example of forecasting China's power production. This example offers a novel approach to address the practical challenges posed by limited time-delay information. It also serves as a decision-making reference for China's power sector, ensuring the harmonious development of power supply and the economy. Related Articles | Metrics

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A Multi-attribute Quantum Decision Model Based on Grey Relational Analysis Shuli Yan, Yang Hu, Yizhao Xu, Xiangyan Zeng The Journal of Grey System    2025, 37 (4): 43-53.   Abstract （166）            Knowledge map    Aiming at the interference problem between decision-makers in uncertain group decision making, this paper proposes a new interference angle measure method based on quantum-like Bayesian network (QLBN). Firstly, the grey relational degree is extended to the positive and negative range, and then the grey relational degree is calculated according to the sequential growth trend, and it is transformed into the interference value. In this framework, the weights of the decision-makers are set as the first layer of QLBN, where they are determined based on the similarity between the personal opinion and the leading decision-maker’s opinion, and the conditional probabilities are calculated by combining the attributes’ weights and evaluation values. Finally, the quantum probabilities of alternatives are obtained. The alternatives are ranked according to the quantum probabilities, and the robustness and validity of the model are verified by numerical example. The result shows that the interference angle calculation method based on grey relational degree has a good performance in decision-making scenarios with incomplete information, and significantly improves the robustness and applicability of the quantum decision model. Related Articles | Metrics

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A Novel Discrete Grey Model for China’s Carbon Emissions Forecasting  Xinyu Zhang, Jun Zhang, Siqi Dong The Journal of Grey System    2025, 37 (2): 33-49.   Abstract （165）            Knowledge map    Carbon emission projections are pivotal in addressing global climate change and advancing green, low-carbon development. Accurate forecasts of China's carbon emissions provide critical insights for policymakers to understand future emission trends and potential peak levels, thereby enabling the formulation of scientifically grounded and practical mitigation strategies. Discretization has proven to be an effective approach for enhancing the accuracy of grey prediction models. To further refine the performance of discrete grey prediction models, this study integrates integer-order polynomials with time fractional power terms to develop an optimized discrete grey prediction model, DGMPT(1,1,N, ), the particle swarm optimization (PSO) algorithm is utilized to optimize hyperparameters. To validate the model's superiority and predictive accuracy, this study applies it to the carbon emission data of Xinjiang, Shaanxi, and Gansu provinces in China. Empirical results demonstrate that, based on the mean absolute percentage error (MAPE) criterion, the proposed model achieves the lowest MAPE values in both the training and prediction datasets across all three case studies. Finally, the proposed model is employed to forecast China's carbon emissions over the next decade. The results indicate that under current conditions, China is unlikely to achieve its peak carbon emissions by 2030. This underscores the urgency of implementing effective and comprehensive policy measures to improve carbon emission systems and foster a sustainable emission environment.  Related Articles | Metrics

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Forecasting the Amount of Urban Domestic Waste Clearance in China With an Optimized Grey Convolution Model Sandang Guo, Xu Han, Jing Jia, Shuaishuai Geng The Journal of Grey System    2025, 37 (3): 11-23.   Abstract （143）            Knowledge map    Accurate forecasting of urban domestic waste clearance in China is essential for advancing the sustainable development of urbanization. To this end, this paper proposes a novel nonlinear grey convolution model that incorporates background value optimization. Firstly, this model introduces power exponents to enhance its ability to capture the nonlinear characteristics of the system. And then, it also analyzes the sources of errors in the background value calculation of traditional models and effectively addresses this issue by incorporating dynamic interpolation coefficients. Besides, the optimal hyperparameters of the model are determined using particle swarm optimization (PSO). In two distinct case studies, the proposed model was rigorously compared with five forecasting models across three distinct domains, consistently demonstrating its superior performance. Related Articles | Metrics

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An Improved Conformable Fractional Grey Multivariate Model and Its Application Qinqin Shen, Linyun Yang, Yang Cao The Journal of Grey System    2025, 37 (4): 79-90.   Abstract （133）            Knowledge map    The conformable fractional accumulation generating operator (CF-AGO) can effectively handle information differences and deeply explore the laws of information development. Nevertheless, the CF-AGO fails to satisfy the highly crucial new information priority (NIP) principle. In this paper, a novel conformable fractional accumulation generating operator (NCF-AGO), which meets the NIP principle under certain conditions, is introduced firstly. Then an improved conformable fractional grey multivariate model with variable NCF-AGO is constructed. Both linear and nonlinear correction terms are considered in the model structure to fit data sequences with different features. The quantum particle swarm optimization algorithm is adopted to obtain the optimal accumulation orders and the optimal power exponent of the nonlinear correction term. In order to avoid the situation where overfitting of the model leads to poor prediction results, the Tikhonov regularization method, which includes the conventional least squares method as a special case, is proposed solve the involved model parameters. Finally, a case study from bending strength of concrete is given to show the effectiveness of the proposed model and its advantages over the well-known GM(1,N) model and several existing grey multivariate models. Related Articles | Metrics

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A Variable-order Nonlinear Discrete Grey Multivariate Model with New Information Priority Accumulation and Its Applications Yang Cao, Min Sun, Qinqin Shen, Xiaofei Liu The Journal of Grey System    2025, 37 (3): 37-49.   Abstract （130）            Knowledge map    In order to overcome some defects of the existing grey multivariate convolution model with new information priority accumulation (GMCN(1,N)), such as the neglect of variable heterogeneity analysis, the weak capability in nonlinear feature extraction, and the mismatch between model parameter estimation and time response function, based on the ideas of variable-order accumulation and discrete grey models and by introducing an additional nonlinear correction term, a variable-order nonlinear discrete grey multivariate model with new information priority accumulation is proposed. Basic properties of the new model are analyzed. Solution structure as well as model parameters are derived. In addition, the quantum particle swarm optimization algorithm is adopted to seek for the optimal accumulation orders. Finally, the proposed model is applied to two practical cases for multidimensional evaluation. The results indicate that the new model outperforms the classic GM(1,N) model, the existing GMCN(1,N) model, and several recently proposed grey multivariate models in terms of both fitting and prediction accuracy, demonstrating better stability and generalization capability. Related Articles | Metrics

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Grey Double-layer Particle Swarm Optimization Algorithm of Testability Allocation for Complex Systems in the Context of Grey Information  Youpeng Liu, Zhigeng Fang, Shiyun Zhang, Cuiping Niu, Jingru Zhang The Journal of Grey System    2025, 37 (2): 87-99.   Abstract （126）            Knowledge map    The conventional series system and parallel system are no longer able to adequately address the current system condition due to the rapid advancement of science. If faults are not identified and isolated, any system failure can lead to the inability to perform a single task or even multiple tasks. In order to promptly identify and isolate faults, the systems must have high testability. However, problems like omitting structural elements in testability influencing factors and ambiguous testability-related data make standard approaches useless when building and resolving testability allocation models. The grey optimization approach will lose a lot of grey information if a planning model is employed for solution, which will lead to large inaccuracies. Therefore, this paper proposes a testability allocation model for complex systems in the setting of grey information, and proposes the grey double-layer particle swarm optimization algorithm to solve the model. First, the particular factor that influence the testability allocation process is identified. Second, this paper proposes the TOPSIS method based on the improvement of grey entropy weight and determines the weights of the subsystems. Then, this paper proposes the grey nonlinear planning testability allocation model, and proposes the grey double-layer particle swarm optimization algorithm to solve the model. Finally, the viability and efficacy of the model are demonstrated by strength testing and comparison with other algorithms.  Related Articles | Metrics

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Complex Decision Making in Counties Socio-Economic and Security Contexts through Grey Clustering  Camelia Delcea, Constantin Marius Profiroiu, Alina Georgiana Profiroiu, Bianca Raluca Cibu, Ionuț Nica, Liviu-Adrian Cotfas The Journal of Grey System    2025, 37 (2): 115-135.   Abstract （125）            Knowledge map    This paper examines the impact of varying the significance of socio-economic and security indicators on the satisfaction levels of citizens across different counties in Romania and Republic of Moldova. Using grey clustering analysis, a powerful tool offered by the grey systems theory, the counties are grouped into three clusters based on citizen satisfaction. Five different scenarios are explored, each assigning distinct weights to socio-economic and security indexes to evaluate their influence on the clustering outcomes. The findings indicate that increasing the emphasis on socio-economic factors leads to more counties experiencing higher levels of citizen satisfaction, particularly in Romania, where socio-economic stability is more robust. On the other hand, placing greater importance on security factors exposes governance-related challenges, as a clear division between the counties in Romania and Republic of Moldova can be observed. While when considering the security indexes, more counties are pushed in the higher levels of satisfaction clusters for the counties in Republic of Moldova, an opposite trend can be encountered for the counties in Romania.  Related Articles | Metrics

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EEG-GRA Cross-Sequence Feature Extraction Method for Operator Cognitive Fatigue Lidan Chen, Xi Liu, Ying Lin The Journal of Grey System    2025, 37 (3): 83-95.   Abstract （123）            Knowledge map    This paper introduces an advanced EEG-GRA cross-sequence feature extraction method for operator cognitive fatigue detection in industrial settings. Our research addresses key limitations in conventional approaches through three technical innovations: (1) an intelligent adaptive time-varying weight function system that continuously calibrates to operator cognitive states, (2) an advanced multi-scale analysis framework incorporating state-of-the-art wavelet decomposition, and (3) a sophisticated cross-sequence feature fusion mechanism that leverages spatial correlations across EEG channels. Comprehensive performance evaluation reveals significant quantifiable improvements: the system achieves a 45% reduction in processing time (from 100ms to 55ms), enabling genuine real-time monitoring capabilities; detection accuracy shows a remarkable increase of 17.5 percentage points (from 76% to 93.5%); and signal quality demonstrates a substantial improvement of 5.3dB (from 15dB to 20.3dB). These advances are achieved while simultaneously reducing computational demands, with algorithmic optimization decreasing complexity from O(n²) to O(n log n) and memory requirements reduced by 38%. Field implementation in a nuclear power plant control room involving 30 operators under rigorous operational conditions validated the system's exceptional reliability, maintaining 99.99% uptime during 12-hour continuous monitoring shifts. Statistical analysis confirms the significance of these improvements (p < 0.01), establishing a new benchmark for industrial safety systems across high-risk sectors. Related Articles | Metrics

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Risk Assessment of Human Resource Crises in Offshore Engineering Design Enterprises Using a Grey Evaluation Model #br# Tao Li, Luping Zhang, Haoyu Zhang The Journal of Grey System    2025, 37 (2): 100-114.   Abstract （122）            Knowledge map    This study investigates human resource crisis management in Offshore Engineering Design (OED) enterprises, focusing on the specific challenges faced by Chinese firms. The research develops a comprehensive early warning indicator system, integrating 15 qualitative and 21 quantitative indicators across five key dimensions. Given the limitations posed by insufficient data and inherent uncertainties in the evaluation process, this study also presented a grey evaluation method to construct a risk assessment model for OED human resource crises. A case study of AOED reveals critical issues, including inefficiencies in performance assessment and high staff turnover, underscoring the need for enhanced management strategies. The findings contribute to improving OED human resource management and provide a foundation for future research on predictive models in OED contexts. Related Articles | Metrics

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Optimization of Petrochemical Energy Management System Based on Grey-Improved Parallel Moth Flame Optimization  Hongxia Chen, Bin Zhao The Journal of Grey System    2025, 37 (2): 77-86.   Abstract （117）            Knowledge map    In order to improve the profit, reduce the shutdown loss and enhance the environment benefit, an optimization model of petrochemical energy manage system is established based on improved parallel moth flame optimization algorithm (IPMFOA). Firstly, the petrochemical energy management system optimization model is constructed, and the object function and constraint condition are confirmed. Secondly, an Grey-IPMFOA is established. Finally, an energy management system of refinery in a petrochemical company is selected as research object to carry out optimization analysis based on proposed Grey-IPMFOA and other three optimization algorithms, results show that proposed optimization can get best optimal effect and highest optimal efficiency. In addition, proposed Grey-IPMFOA has better convergence performance and computation complexity. The energy management system after optimization has high profit, low shutdown loss and high environment benefit.  Related Articles | Metrics

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Using Grey System Analysis Methods and Models to Achieve the Design and Optimization of Aviation Tire Rubber Formulations Sifeng Liu, Jize Wang, Shaolin Ma, Wei Tang The Journal of Grey System    2025, 37 (5): 1-10.   Abstract （116）            Knowledge map    This study analyzes the critical role of aircraft tire performance in ensuring flight safety and stability. Given that the formulation design of natural rubber composites for aircraft tires still heavily relies on long-term empirical accumulation—a process with considerable randomness—this paper proposes, for the first time, the use of China's original grey system analysis and modeling methods to bring scientific rigor to the formulation design and optimization process. The author also outlines a practical and executable workflow for the formulation design and optimization of natural rubber composites for aircraft tires, which can serve as a reference for relevant industries and rubber formulation researchers. Related Articles | Metrics

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Identification of Key Elements for High-Quality Development of Manufacturing Enterprises Based on Grey Superior Analysis Ruyun Zhang, Zixuan Li, Yimin Huang The Journal of Grey System    2025, 37 (5): 96-112.   Abstract （113）            Knowledge map    Manufacturing enterprises serve as the main implementers for the high-quality development of China's manufacturing industry. Accurately identifying the key elements that evaluate and influence high-quality development is a prerequisite for exploring the path of high-quality enterprise development. This paper defines the connotation of high-quality development of manufacturing enterprises from two dimensions, "state" and "process", and constructs the characteristic behavior sequence and related factor behavior sequence for high-quality development of manufacturing enterprises accordingly. Using the grey superiority analysis model, an empirical study was conducted on 1017 manufacturing enterprises. The study found that: (1) The evaluation elements for high-quality development in different sub-sectors of the manufacturing industry exhibit significant industry-specific characteristics and internal laws. The overall evaluation indicators for the manufacturing industry cover four dimensions: management capability, innovation effectiveness, social effects, and operational benefits. Capital-intensive manufacturing industries focus on operational benefits and management capability; technology-intensive manufacturing industries take innovation effectiveness and social effects as their core; while labor-intensive manufacturing industries emphasize management capability and social effects. (2) The distribution of key relevant factors in different manufacturing sub-sectors presents significant industry heterogeneity and certain regularity. The high-quality development of the overall manufacturing industry and labor-intensive industries is jointly driven by three-dimensional factors of resources, environment and benefits. The key factors of capital-intensive manufacturing are concentrated in resources and environment, while the factors of technology-intensive manufacturing focus on benefits and environment. In summary, theoretically, the indicator system constructed in this paper and the key elements of each sub-sector derived from empirical research provide a clear indicator framework for subsequent scholars to study specific sub-sectors. At the practical level, the research findings offer useful management insights for managers and policymakers of manufacturing enterprises in China. Related Articles | Metrics