Abstract
This study offers a comprehensive analysis of novel information for linear diophantine multi-fuzzy sets and illustrates its applications in practical scenarios. We introduce innovative similarity metrics tailored for linear diophantine multi-fuzzy sets, including Cosine similarity, Jaccard similarity, and Exponential similarity. Additionally, we propose Entropy, Inclusion, and Distance measures, providing a robust theoretical foundation supported by developed theorems that explain the interactions between these metrics. The practical implications of these theoretical advancements are demonstrated through various case studies. Specifically, we apply the similarity measures to predict preeclampsia, a severe condition affecting pregnant women, showcasing their potential in medical diagnostics. The entropy measure is used to identify the optimal materials manufacturing method for medical surgical robots, underscoring its importance in ensuring patient safety and the effectiveness of medical procedures. Furthermore, the inclusion measure is employed in pattern recognition tasks, highlighting its utility in complex data analysis. The comparative and superiority analysis shows the effectiveness of our research. The novel aspect of this study is the implementation of information metrics for LDMFS. These efforts aim to enhance the impact and practical applicability of linear diophantine multi-fuzzy sets, fostering innovation and improving outcomes across multiple fields.