New applications of various distance techniques to multi-criteria decision-making challenges for ranking vague sets

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.