Using the Fermatean vague normal set (FVNS), problems requiring multiple attribute decision making (MADM) have been resolved in this article. This article focuses on the log Fermatean vague normal weighted averaging (log FVNWA), logarithmic Fermatean vague normal weighted geometric (log FVNWG), log generalized Fermatean vague normal weighted averaging (log GFVNWA) and log generalized Fermatean vague normal weighted geometric (log GFVNWG) operators. Described the scoring function, accuracy function and operational laws of the log FVNS. The Euclidean and Humming distance are extended with numerical examples. The features of the log FVNS based on the algebraic operations, including idempotency, boundedness, commutativity and monotonicity are also examined. A field of applied engineering called agricultural robotics has been compared to computer science and machine tool technology. Five distinct agricultural robotics including autonomous mobile robots, articulated robots, humanoid robots, cobot robots, and hybrid robots are randomly chosen. Findings can be compared to established criteria to determine which robotics are the most successful. The results of the models are expressed as a natural number $ \alpha $. We contrast several existing with those that have been developed in order to show the effectiveness and accuracy of the models.

This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies.