Purpose Accurate energy data prediction holds significant practical importance for optimizing energy structures and informing governmental energy decision making. Due to the uncertainty inherent in energy sequences, this paper proposes a hybrid model combining seasonal–trend decomposition and a nonlinear time-delay grey model. Design/methodology/approach The trend variations inherent in the data are obtained through signal decomposition techniques, and higher weights are assigned to new information within the model. Prediction results are restored via seasonal factors, and the optimal parameter values are obtained using the PSO algorithm. Findings An analysis of the model’s various characteristics is conducted. By fitting and forecasting China’s natural gas production data and comparing the results with those of four other benchmark models, the validity of the model is verified. Originality/value This study extends trend decomposition and seasonal factor restoration by incorporating nonlinear terms, lagged terms and new information priority, enabling the model to better adapt to dynamic trend changes and significantly improve prediction accuracy for time series with prominent dynamic trends and seasonal variations.

Purpose The purpose of this paper is to show Industry 4.0 technologies the potential of Industry 4.0 technologies to reduce the environmental impact of manufacturing processes and transform business models towards sustainability. In this study, the ERP selection process is examined with sustainability and Industry 4.0 criteria under fuzzy and grey uncertainty methodologies. Design/methodology/approach A case study employed the intuitionistic fuzzy sets-based grey ordinal priority approach (OPA-G) methodology to evaluate sustainability criteria that are in line with the 17 Sustainable Development Goals (SDGs) by the United Nations and seven generic framework applications of Industry 4.0 compatibility. This approach was particularly chosen to address the uncertainties involved in selecting appropriate and reliable ERP software. Findings In this study, the enterprises often prioritize economic performance over environmental and social criteria. However, there is also recognition of the importance of integrating Industry 4.0 technologies such as cloud and edge computing, artificial intelligence, the Internet of Things, and big data in ERP selection processes to foster sustainable practices. Practical implications Industry 4.0 technologies enhance sustainability by providing real-time data analytics, increasing supply chain transparency, and optimizing resources to minimize waste and environmental impact. These innovations help businesses align with global sustainability standards, balancing economic, environmental, and social goals considering the circular economy (CE) while appealing to conscious consumers and investors. Originality/value This study introduces a novel use of the intuitionistic fuzzy set-based grey OPA methodology to evaluate sustainability criteria in ERP software selection, focusing on incorporating sustainability uncertainties and highlighting the importance of Industry 4.0 technologies for sustainable development under CE.

Purpose This study proposes a hybrid forecasting framework integrating grey system models (GM) with Gaussian process regression (GPR) to enhance prediction accuracy under small-sample conditions. While GM is effective in capturing trend components under small-sample conditions, it is less suited for modeling nonlinear fluctuations and providing probabilistic uncertainty quantification. To complement this, the proposed framework employs GM to model the trend sub-series, while GPR is applied to the residuals in a probabilistic manner, thereby enabling both accurate forecasts and robust nonlinear interval estimation. Design/methodology/approach The proposed framework first smooths the trend using the moving average method and then extracts seasonal and residual components. The GM model is employed to capture the trend, while the GPR model generates point forecasts and interval estimates for residuals. Particle swarm optimization is applied to optimize GM’s nonlinear parameters, improving overall accuracy and robustness. Findings Experiments show that the nonlinear grey Bernoulli model with fractional-order accumulation-GPR framework outperforms the benchmarks in predictive accuracy. Case studies confirm that combining time series decomposition, grey system modeling and GPR enhances both accuracy and robustness in energy consumption forecasting. Originality/value The primary innovation of this study is a hybrid framework combining trend forecasting with probabilistic residual modeling, using GPR for nonlinear interval estimation. Comparisons with six typical hybrid grey models demonstrate superior predictive performance, offering a novel approach for complex time series modeling and energy forecasting.

Purpose Given the same mixed attribute alternatives, the decision-making result determined by different decision-making basis (DMB)- based generalized grey target decision method (GGTDM) for mixed attributes may be different owing to the different mechanisms of each DMB. To obtain the desired decision-making result that individual GGTDM could not have, a combination GGTDM for mixed attributes based on cooperative game theory is proposed to fulfill this task. Design/methodology/approach Different DMB-based GGTDMs for mixed attributes are regarded as different players in a cooperative game. The characteristic values of different game alliances can be calculated through the method of weighted reciprocal of variance, depending on the error information matrices that originated from the differences between the combination decision-making results and the referenced decision-making results. Then the model for Shapley values of game alliances is built. The weights of GGTDMs can be achieved by normalizing the Shapley values of all players (GGTDMs). Next, the combination decision-making values can be calculated relying on integrating the decision-making values obtained by different GGTDMs with the cooperative game-based weights. Finally, the decision-making is based on the combination decision-making value of each alternative, with which the smaller the better. Findings The proposed combination GGTDM based on cooperative game theory proves to be feasible and effective in its employment. Originality/value The originality of the proposed approach lies in the Shapley value is adopted to be as a tool to obtain the weight of each GGTDM to take part in the combination decision-making.

Purpose This study aims to construct a new grey model to assess the COVID-19's impact on China's domestic tourism revenue. Design/methodology/approach Firstly, the Hausdorff accumulative generation operator and adaptive nonlinear correction term are introduced to the new model, with parameter optimization using PSO. Then, the new model is applied to assess the epidemic's impact on China's domestic tourism revenue. Findings The performance result of the Hausdorff accumulative generation operator and adaptive nonlinear correction show that they can improve the model's prediction stability and simulation accuracy. The model result shows from 2020 to 2022 the average contribution of the epidemic to revenue is −64.51%, with a loss of 132,027.6 million yuan. Originality/value This study has positive implications for enriching the application method of grey prediction model.

Purpose It is necessary but difficult to accurately predict the water levels in front of sluice gates of an open channel water transfer project due to the complex interactions among hydraulic structures. The existing methods have certain shortcomings. For example, although one-dimensional hydrodynamic simulation is technically feasible, little is known about hydrodynamic models for prediction. Another example is that, neural networks can hardly predict the information of nonmonitoring sections. To these problems, this paper presents a novel GRA-NARX and hydrodynamic coupled prediction model (H-GRA-NARX-HPM) that is based on the GRA-NARX (gray relation analysis-nonlinear auto-regressive exogenous) neural network with automatic hyperparameter calibration. Design/methodology/approach Firstly, the GRA is used to determine the correlations of influencing factors and find the optimal influencing factors. Secondly, the selected factors are taken as the input variables of the NARX neural network. Finally, the GRA-NARX neural network with automatic hyperparameter calibration (H-GRA-NARX model) provides accurate 24-h water level prediction to be used as the boundary condition of the hydrodynamic model, and then the H-GRA-NARX-HPM is constructed. Findings The section from the inlet sluice gate of Tang River aqueduct to the outlet sluice gate of Zhang River inverted siphon in the Middle Route of the South-to-North Water Transfer Project, China, is taken as the study area. The water levels before the outlet sluice gate of Anyang River inverted siphon on February 22, 2018 and February 26, 2018, are predicted by the H-GRA-NARX-HPM and then compared with those of the prediction models (GRA-BP-HPM, GRA-NARX-HPM) that use GRA-BP(gray relation analysis-back-propagation) neural network and GRA-NARX neural network prediction information as boundary conditions. The results show that the H-GRA-NARX-HPM has the highest accuracy with MAE values of 0.0028 m and 0.0141 m and MSE values of 1.636 × 10-5 and 2.658 × 10-4 on February 22 and February 26, respectively. In order to verify the universality and applicability of the model, the section from the inlet sluice gate of Ming River aqueduct to the outlet sluice gate of Qili River inverted siphon is taken as another study area. The water levels before the outlet sluice gate of Nansha River inverted siphon on March 18, 2018 and March 19, 2018, are predicted by the H-GRA-NARX-HPM and then also compared with GRA-BP-HPM and GRA-NARX-HPM. The results show that the H-GRA-NARX-HPM has the highest accuracy as well. Originality/value The main contribution of this paper is to propose a novel GRA-NARX and hydrodynamic coupled prediction model (H-GRA-NARX-HPM) which can overcome the main limitations of the individual modelling approaches.

Purpose Accurate multi-city carbon emission forecasts are essential for climate policy. Currently, rapid urbanisation has woven cities into coupled economic and infrastructural networks. Thus, this study aims to model evolving inter-city spillovers to enhance spatiotemporal carbon emission predictions for urban agglomeration. Design/methodology/approach This work develops a spatiotemporal discrete grey multivariate model, denoted as STDGM (1, N, M), in which a composite spatiotemporal-proximity coefficient dynamically fuses geographic distance with time-varying economic distance. Consequently, the proposed mode can track inter-city spillover effects through geographic and economic proximity. Additionally, a particle-swarm algorithm is applied to calibrate the weights between the two distances' parameters. Findings Empirically, the model is tested on annual data for thirteen Jiangsu cities (2010–2022) and outperforms the classical grey models, neural networks and statistical benchmarks, achieving an average mean absolute percentage error of 1.97% and maintaining the narrowest error range under extensive Monte Carlo robustness checks. Research limitations/implications The findings show that embedding adaptive spatiotemporal interactions in a grey prediction model lifts forecast accuracy and yields interpretable results, providing planners with a reliable tool for designing collaborative and region-specific mitigation pathways. Originality/value This work initially embeds adaptive geographic–economic proximity within a grey multivariate model. It combines small-sample efficiency, dynamic spatial realism and interpretability in a unified framework, offering researchers and policymakers a novel tool for coordinated urban-agglomeration decarbonisation.

Purpose China’s accelerating population aging places heavy fiscal pressure on the pension insurance system, challenging its long-term sustainability. Reliable forecasting is therefore essential for informed policy design and fiscal risk management. Design/methodology/approach This paper develops a recursive nonlinear grey Bernoulli model (RNGBM) that integrates the Bernoulli nonlinear differential structure with a recursive updating mechanism and embeds a memory factor to enhance adaptability to new information and sudden policy shifts. Using China’s pension fund data (2010–2024), the RNGBM is evaluated against other grey models (GM(1,1), RGM, NGBM) and exponential smoothing based on mean absolute percentage error (MAPE), and further applied to short-term forecasting and fiscal risk assessment under aging scenarios. Findings The RNGBM consistently outperforms other models. Forecasts suggest that while the fund size will grow over the next three years, expenditure growth will exceed revenue growth, reflecting structural risks driven by demographic aging and policy adjustments. This imbalance could accelerate balance depletion, threatening long-term solvency without timely interventions. Practical implications The RNGBM provides policymakers with a more responsive and robust tool for forecasting pension fund dynamics, supporting evidence-based fiscal governance, optimized resource allocation and proactive reforms to maintain system solvency. Originality/value This paper is the first to combine the Bernoulli differential structure with a recursive updating and memory-enhanced mechanism, forming a nonlinear recursive grey framework. The RNGBM not only improves methodological accuracy but also demonstrates practical utility in addressing fiscal sustainability challenges in aging societies.

Purpose The natural gas production exhibits seasonal oscillations, nonlinear increments and uncertain volatility. Three-parameter interval grey numbers not only contain the mean information but also reflect the uncertainty fluctuation range of the index. To forecast natural gas production, a matrixed Fourier grey Bernoulli model (MFGBM(1,1)) is proposed. Design/methodology/approach To reduce the seasonal volatility of the series, seasonal factors are incorporated into the weighted accumulative generation operator. Additionally, the Fourier correction term and the regular term are introduced into the grey Bernoulli model to further enhance the model’s adaptability to seasonally oscillating series. The grey wolf optimization algorithm is improved based on the convergence factor of the sigmoid function and the Gaussian stochastic wandering strategy. This improved algorithm is used to optimize the model parameters. Findings The accuracy of MFGBM(1,1) is verified by two natural gas-related cases and the model comparison experiment. The natural gas production in China is predicted and analyzed for the four quarters of 2025. The prediction shows that it will grow steadily over the next four seasons. Originality/value The natural gas production has obvious nonlinear characteristics and seasonal oscillations. Therefore, for the data characteristics of natural gas, a three-parameter interval grey number prediction model for natural gas forecasting is proposed.

Purpose The purpose of this paper is to propose a novel model, to forecast demand for a third-party service by using the Grey Systems Theory (GST) and Markov Chains, where its forecast error performance is evaluated through mean percentage error, mean absolute percentage error, where it exceed other models' performance accuracy such as autoregressive integrated moving average. Design/methodology/approach The model performs data characterization to qualify the GM (1,1) model and then applies a Markov Chain transition probability matrix and the GM (1,1) to forecast a time series with high degree of vagueness and imprecision and by providing a forecast kernel range ⊗Aˆ. Findings The MCGM (1,1) model integrates the GST GM (1,1) and Markov Chains in a novel hybrid model, that reduces the mathematical calculation complexity while provides practical forecast performance that exceed or it is equally good as other traditional methods. Research limitations/implications The model outperforms other non-stationary models but does not incorporate multiple variables and requires additional mathematical treatment or combined methods, where its data is stationary, seasonal or negative. Practical implications This model can provide an accurate forecast projection of supply chain demand, for instance the space required in a third-party logistics services provider in Tijuana Mexico, it can be used to forecast complex supply chain systems with minimum, incomplete or poor data, to solve several practical application problems to forecast demand and resources. Social implications The novel MCGM (1,1) hybrid forecasting model combines multiple predictive approaches, allowing for greater accuracy and adaptability. Its implementation enhances decision-making in key sectors such as health, energy and manufacturing, optimizing resources and reducing costs. This drives economic growth, increases sustainability and improves the quality of life in society. Originality/value There are no MCGM (1,1) works applied in supply chain and current works have not established the model characterization criteria. The result of this investigation represents a novel proposal to solve uncertain models with poor information and small amounts of data (>4 records), with higher forecast accuracy.