This study addresses the challenges of the expanding global online retail market by developing an innovative inventory management strategy that promotes sustainable development. We propose an integrated planning model that utilizes a newly enhanced Parallel Chicken Swarm Optimization (PCSO) algorithm alongside the established Economic Order Quantity (EOQ) model and the (s, S) inventory policy. Our model demonstrates its effectiveness through simulation experiments, showcasing its ability to dynamically adjust inventory thresholds based on daily forecasted demand, which significantly optimizes replenishment strategies. Notably, our model achieves a total cost reduction while maintaining a high service level of 0.761467 and reduces computational time to 82.32 s, highlighting its efficiency and adaptability. The innovation of this research lies in the application of the PCSO algorithm to dynamically manage inventory in real-time, aligning with sustainable inventory management practices. This study significantly contributes to the field by enhancing the economic and environmental sustainability of supply chain operations.

The hesitant fuzzy \(\beta \)-covering rough set offers stronger representational capabilities than earlier hesitant fuzzy rough sets. Its flexibility makes it more suitable for hesitant fuzzy multi-attribute decision-making (MADM). As a result, it has become a popular research focus in decision analysis and has drawn significant attention from scholars. However, the existing hesitant fuzzy \(\beta \)-covering rough set based on t-norms cannot handle the overlap and correlation between hesitant information well. Addressing this problem, we propose the hesitant fuzzy overlap function and hesitant fuzzy \(\beta \)-covering \(({\mathcal {I}},\) \({\mathcal {O}})\) rough set (HF\(\beta \)CIORS) models based on the hesitant fuzzy overlap function. First, we establish the definition of the hesitant overlap function and representable hesitant fuzzy overlap function on a partial order relation. Based on proposed definitions, we provide examples of representable and unrepresentable hesitant fuzzy overlap functions and offer a detailed proof to explain the unrepresentable function. Second, we construct four types of HF\(\beta \)CIORS models and prove some of its important properties. Thirdly, we integrate the HF\(\beta \)CIORS models with the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method and apply them to solve MADM problems. The validity of the proposed method is demonstrated through a practical application, and its stability and effectiveness are confirmed via sensitivity and comparative analyses. Based on these validations, our method proves effective in addressing MADM problems, offering reliable decision-making support.

Some homomorphism theorems of neutrosophic extended triplet group (NETG) are proved in the paper [Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 2018, 10(8), 321; doi:10.3390/sym10080321]. These results are revised in this paper. First, several counterexamples are given to show that some results in the above paper are not true. Second, two new notions of normal NT-subgroup and complete normal NT-subgroup in neutrosophic extended triplet groups are introduced, and their properties are investigated. Third, a new concept of perfect neutrosophic extended triplet group is proposed, and the basic homomorphism theorem of perfect neutrosophic extended triplet groups is established.

To accurately predict the remaining useful life (RUL) of rolling bearings under strong noise interference, this paper proposes an RUL prediction method based on adaptive Variational Mode Decomposition (VMD) and a hybrid Temporal Convolutional Network-Gated Recurrent Unit-Self-Attention (TCN-GRU-SA) framework. First, a parameter-optimized VMD algorithm is developed by integrating the Grey Wolf Optimizer (GWO) with VMD to extract effective intrinsic mode components (IMFs) and reconstruct denoised signals, thereby mitigating the impact of strong background noise. Subsequently, time-domain degradation features are extracted from the reconstructed signals to generate more representative feature datasets. These degradation features are then fed into a parallel TCN-GRU-SA prediction model. To enhance the model’s generalization capability and RUL prediction performance, the proposed hybrid architecture combines a Temporal Convolutional Network (TCN), which captures local temporal patterns, a Gated Recurrent Unit (GRU) for modeling long-term dependencies, and a Self-Attention (SA) mechanism to prioritize critical degradation-related features. Experimental validation on the PHM2012 rolling bearing accelerated lifetime dataset demonstrates that the proposed method achieves superior noise robustness and prediction accuracy compared to existing approaches. Specifically, it reduces the root mean square error (RMSE) by 18.7 % and improves the coefficient of determination (R2) by 12.3 % under high-noise conditions, confirming its effectiveness in industrial predictive maintenance applications.

The overlap function is an important class of aggregation function that is closely related to the continuous triangular norm. It has important applications in information fusion, image processing, information classification, intelligent decision-making, etc. The usual multi-attribute decision-making (MADM) is to select the decision object that performs well on all attributes (indicators), which is quite demanding. The MADM based on fuzzy quantifiers is to select the decision object that performs well on a certain proportion or quantification (such as most, many, more than half, etc.) of attributes. Therefore, it is necessary to study how to express and calculate fuzzy quantifiers such as most, many, etc. In this paper, the Sugeno integral based on the overlap function (called the O-Sugeno integral) is used as a new information fusion tool, and some related properties are studied. Then, the truth value of a linguistic quantified proposition can be estimated by using the O-Sugeno integral, and the O-Sugeno integral semantics of fuzzy quantifiers is proposed. Finally, the MADM method based on the O-Sugeno integral semantics of fuzzy quantifiers is proposed and the feasibility of our method is verified by several illustrative examples such as the logistics park location problem.