Purpose This study aims to address the issues of overfitting and underutilization of new information in traditional grey models for multi-frequency traffic flow forecasting. It proposes the Recursive Grey Multi-frequency Fourier Model (RGMFM) to enhance the extraction of multi-frequency periodic features and enable dynamic updating, thereby improving the accuracy and stability of short-term predictions for small-sample, multi-frequency traffic flow data. Design/methodology/approach The RGMFM integrates Fast Fourier Transform (FFT) and a recursive regression algorithm into the grey modeling framework. FFT extracts primary and secondary periodic components, while recursive regression prioritizes parameter updates with new data. The model incorporates two controllable parameters—an energy threshold and a memory factor—to simultaneously optimize its structure and parameter updating mechanism. Findings Simulation and experimental results demonstrate that RGMFM significantly outperforms benchmarks (SARIMA, LSTM, Transformer), reducing average RMSE by approximately 17% on 50 subsets of the PEMS08 dataset. It accurately captures daily and peak-hour traffic patterns and maintains robust performance under varying noise levels, validating its effectiveness and stability for multi-frequency forecasting. Originality/value The main originality lies in the novel hybrid framework that embeds signal processing (FFT) and recursive learning into grey system theory. The introduction of dual controllable parameters for integrated optimization provides a new, tunable, and interpretable modeling paradigm. This work extends grey models' applicability to complex, dynamic multi-frequency forecasting tasks, offering significant practical value for intelligent transportation systems and similar domains.
Purpose Accurate and reliable energy forecasting is fundamental to strategic decision making for the global transition toward decarbonization and digitalization. This paper proposes a novel nonlinear time-varying fractional grey multivariable model to address the limitations of conventional forecasting methods. Design/methodology/approach The proposed model offers a novel methodological improvement by integrating time-varying coefficients for dynamic parameterization, logarithmic adjustment terms for enhanced nonlinear modeling and a particle swarm algorithm for systematic optimization. Findings Empirical validation across municipal, provincial, and national case studies consistently confirms the superior predictive performance of the proposed model. Comparative analyses with benchmark models show that our model achieves the lowest prediction errors across all evaluation metrics. This outperformance remains robust in both training and testing phases. Practical implications The proposed approach provides a robust methodological framework for energy consumption forecasting, offering substantial improvements in accuracy and reliability for supporting energy strategy formulation and sustainable development planning. Originality/value Its originality stems from the integration of dynamic coefficients and logarithmic adjustment terms. This integrated framework effectively addresses the limitations of static parameters and fixed structures in conventional grey models.
Purpose Aiming at the issue that traditional multivariable grey prediction models cannot fully identify nonlinear trends among sequences, this paper introduces power-exponential and time-power-logarithmic correction terms, combines intelligent algorithms to flexibly optimize the model's unstructured parameters and proposes a new logarithmic flexible discrete grey LFDGM (1, N) model to deeply mine data sequence patterns and adapt to data development trends. Design/methodology/approach Firstly, nonlinear parameters are introduced into system behavior and relevant factor sequences, and a logarithmic time-power correction term is adopted to identify system nonlinear characteristics. Secondly, four mainstream intelligent optimization algorithms are compared to select the optimal one for hyperparameter global optimization with minimum average relative error. Then, the LFDGM (1, N) model is applied to fossil energy production prediction and compared with benchmark models. Finally, model stability is verified via Monte Carlo simulation and varying sample set experiments. Findings The experimental results show that the logarithmic flexible discrete grey LFDGM (1, N) model has better accuracy than other models in the global MAPE, verifying the validity and practicability of the model and indicating that this model has the optimal modeling effect on nonlinear complex systems. In addition, this paper verifies that the LFDGM (1, N) model has good robustness through Monte Carlo simulation and experiments with different sample sets. Practical implications Scientific and accurate prediction of fossil energy output is of great reference significance for China's formulation of energy industry plans and provides solid support for the long-term stable development of its energy industry. Originality/value To tackle complex, variable and highly uncertain system data sequences, this paper develops a logarithmic flexible discrete grey LFDGM (1, N) model. Embedding nonlinear power parameters in the system feature sequence and the related factor sequence, and introducing a logarithmic time-power correction term enhances nonlinear correlation depiction, captures early system abrupt changes and restricts late prediction divergence. Hyperparameters are optimized via preferred intelligent algorithms to improve prediction accuracy.
Purpose This study addresses the grey dynamic flexible job shop scheduling problem (GDFJSP), in which jobs with uncertain grey-number processing times arrive stochastically and must be dispatched in real time. It aims to develop a genetic programming algorithm that evolves interpretable heuristic dispatching rules while handling stochastic arrivals and iterative grey-time updating efficiently. Design/methodology/approach A memory-guided adaptive feature genetic programming (MGAFGP) algorithm is proposed with a dual-tree encoding for routing and sequencing decisions. The algorithm combines parallel simulation for concurrent fitness evaluation, an elite-memory-guided strategy with separate feature probability vectors for routing and sequencing trees, and a generation-dependent parent selection function. Its performance is evaluated across multiple scenarios defined by different objectives, utilization levels, and due-date tightness conditions. Findings MGAFGP reaches high-quality rules substantially faster than standard GP under the tested scenarios, showing corrected significant advantages during early evolution and reaching GP's full-budget mean performance with a substantially smaller iteration budget. No corrected full-budget comparison favours standard GP. The evolved rules outperform classical heuristic combinations after independent test re-evaluation, while feature-use patterns, symbolic expressions, and tree-complexity statistics show that the resulting dispatching logic remains inspectable. Practical implications The approach provides a computationally tractable way to discover interpretable dispatching rules for dynamic manufacturing environments with uncertain processing times and limited historical data. By reaching strong rules earlier, MGAFGP can reduce the simulation budget needed for rule evolution, support managerial inspection of scheduling logic and reduce reliance on expert-designed heuristics. Originality/value The study integrates generalized grey-number processing times into the DFJSP and develops a GP algorithm with separate feature probability adaptation for routing and sequencing. The elite-guided strategy with generation-dependent parent selection provides a mechanism for accelerating convergence in simulation-based GP under grey processing-time uncertainty.
Purpose In the era of big data, the understanding of the complex and uncertain nature of reliability growth data has deepened. Beyond the well-known characteristic that failure data are random variables following specific probability distributions, expert judgments expressed linguistically constitute fuzzy data. Allowable values for critical parameters are often confined to specific ranges, representing typical grey data. Moreover, knowledge regarding specific components, materials and processes frequently manifests as rough data. Effectively utilizing reliability growth data characterized by multiple uncertainties – randomness, fuzziness, greyness and roughness – is therefore key to solving the modeling challenges for reliability growth of high-end intelligent equipment. This paper proposes a novel model and associated new concepts for uncertainty representation and integration to address this gap. Design/methodology/approach Guided by the core principles of big data – which emphasize utilizing all available data beyond random sampling, eliminating confounding factors to discern general trends and prioritizing correlation over strict causality – this research adopts a “full data utilization” perspective. It begins with the collection, identification, and analysis of reliability growth data. Through an in-depth examination of the characteristics and commonalities of data embodying various uncertainties (random, fuzzy, grey and rough), the concept of a standard uncertainty number is defined. The representation of SUNs, conversion rules for transforming diverse uncertainty data into SUNs and a comprehensive operational framework for SUNs are developed. Subsequently, analytical and data mining models based on SUNs are established. These models facilitate multi-dimensional, multi-stage and multi-level exploration of key factors influencing the reliability growth of high-end intelligent equipment, leading to the construction of a reliability growth evaluation index system. To overcome existing modeling bottlenecks, holographic reliability growth evaluation and prediction models are constructed by integrating big data technologies, complex uncertainty data analysis methods, sequence operators, spectrum analysis and intelligent algorithms. Findings The proposed novel concepts and framework demonstrate the feasibility of integrating diverse uncertainties to achieve high reliability for complex equipment. Research limitations/implications The limitation of this research is its coverage of various uncertainties. It is not possible to cover fully all uncertainties due to their unknown status, and the proposed model provides only a method rather than a completed solution to the challenge. Practical implications Manufacturers employ reliability growth tests to iteratively enhance equipment reliability and performance through cycles of “exposing defects – analyzing causes – implementing improvements.” However, when data fail to meet the assumptions of traditional reliability growth models, practitioners often resort to ad hoc measures – such as using simulated data or borrowing data from similar equipment – which may compromise reliability. Traditional models constrained by random sampling are evidently inadequate for the development of complex equipment, necessitating new approaches. The concepts and models proposed in this paper have the potential to significantly improve the quality and reliability of complex products in smart manufacturing. Originality/value While numerous uncertainty models exist, effective frameworks for their integration remain scarce. The definitions, operational systems of SUNs, the various SUN-based data mining models and the holographic reliability growth evaluation and prediction models presented here are original contributions of the authors.
