Linguistic interval-valued intuitionistic fuzzy sets, as an extension of interval-valued intuitionistic fuzzy sets, have strong practical value in the management of complex uncertainty system with qualitative evaluation information. This study focuses on the development of several linguistic interval-valued intuitionistic fuzzy Hamacher (LIVIFH) aggregation operators based on the extended Hamacher t-norm and s-norm. First, the extended Hamacher t-norm and s-norm, which are applicable to linguistic information environment, are applied to define the linguistic interval-valued intuitionistic fuzzy Hamacher operational laws. Second, based on the proposed operational laws, this study defines the linguistic interval-valued intuitionistic fuzzy Hamacher weighted average (LIVIFHWA) operator and the linguistic interval-valued intuitionistic fuzzy Hamacher weighted geometric (LIVIFHWG) operator, and then investigates their properties. Furthermore, the degeneracy and monotonicity of the proposed operators with respect to the adjustable parameter are explored. Finally, a multiple attribute group decision-making (MAGDM) approach is developed based on the proposed LIVIFH aggregation operators, and then this approach is applied to a supplier selection problem. Parameter analysis indicates that the adjustable parameter in the proposed LIVIFH aggregation operators could reflect the attitudes of decision makers. The LIVIFHWA operator would be more appropriate to optimistic decision makers, and the LIVIFHWG operator to pessimistic decision makers. In addition, as the adjustable parameter increasing, both attitudes tend to be neutral. The proposed method is also compared with two other approaches to show its feasibility and efficiency.

Background: Hepatocellular carcinoma (HCC), with discouraging morbidity and mortality, ranks as one of the most prevalent tumors worldwide. Pyrimidine metabolism is a critical process that regulates DNA and RNA synthesis in cells. It is imperative to investigate the significance of pyrimidine metabolism in liver cancer. Methods: Transcriptome and clinical data were downloaded from the TCGA database and the GEO database. The genes related to pyrimidine metabolism were sourced from the MSigDB. The pyrimidine metabolism-related signature (PMRS) was constructed through Cox regression and Lasso regression and then verified in the external validation set from the ICGC database. Functional enrichment, immune infiltration analysis, drug sensitivity, and Immunophenoscore (IPS) were further implemented to predict the response to immunotherapy. The role of PMRS in the malignant phenotype of hepatocellular carcinoma was explored by conducting a series of in vitro experiments. Results: Our study developed a four-genes PMRS which demonstrates a substantial correlation with the prognosis of HCC patients, serving as an independent predictor in clinical practice. The result of risk-stratified analysis yielded evidence that low-risk patients experienced more favorable clinical outcomes. The nomogram exhibited remarkable prognostic predictive value. The subsequent results revealed that low-risk patients manifested a more promising response to immunotherapy. Moreover, the results of cell experiments demonstrated that the downregulation of DCK markedly inhibited the malignant phenotype of hepatocellular carcinoma. Conclusions: Our pyrimidine metabolism-centered prognostic signature accurately predicts overall survival, immune status, and treatment response in hepatocellular carcinoma (HCC) patients, offering innovative insights for precise diagnosis, personalized treatment, and improved prognosis.