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.

The modern world uses an increasing number of robots, notably service robots. Robots will be able to easily manipulate everyday objects in the future, but only if they are paired with planning and decision-making procedures that allow them to comprehend how to complete a task. This research presents new techniques to handling multi-attribute problem solving with trigonometric Pythagorean normal fuzzy numbers. The sine trigonometric Pythagorean fuzzy sets combine the concept of Pythagorean fuzzy sets with sine trigonometric functions to represent uncertainty in decision-making. It is feasible to combine trigonometric Pythagorean fuzzy numbers and normal fuzzy numbers to get trigonometric Pythagorean fuzzy normal numbers. In addition to the fundamental interaction aggregation operators, we define the trigonometric Pythagorean fuzzy normal numbers. The trigonometric Pythagorean fuzzy normal numbers satisfy the following properties: associative, distributive, idempotent, bounded, commutative and monotonicity. Four novel approaches are introduced such as weighted averaging, weighted geometric, generalized weighted averaging and generalized weighted geometric. These operators can be used in the development of a multi-attribute decision-making algorithm. We demonstrate how improved Euclidean and Hamming distances are used in practical situations. For industrial robots, the two most crucial elements are computer science and machine tool technology. The four criteria of weights, orientations, speeds and accuracy may be used to assess robotic systems. They are also more practical, easier to understand, and more adept at identifying the best answer more quickly. The effectiveness and accuracy of the models we are looking at are demonstrated by comparing many existing models with those that have been developed.

In this study, we introduce the concept of the K-Lexicographic Max Product (K−LMP) of neutrosophic graphs and explore its associated degree structure to enhance decision-making frameworks in food safety applications related to risk assessment, including freshness, contamination, and spoilage. Neutrosophic graphs, capable of handling indeterminacy, inconsistency, and incompleteness, provide a flexible mathematical foundation for modelling complex systems. By incorporating the K−LMP into neutrosophic graphs, we offer a novel approach to comparing and ranking food safety scenarios where multiple attributes and uncertain information coexist. We present example graphs and theorems related to K−LMP and further define the K-Lexicographic degree to quantify node significance within the context of neutrosophic graphs. To validate the practical utility of this approach, a food safety analysis is implemented, demonstrating how the model identifies critical control points and supports more robust, transparent decision-making under uncertainty. This work contributes to the advancement of neutrosophic graph theory and its interdisciplinary application in food quality and safety management.