Diophantine spherical vague sets and their applications for micro-technology robots based on multiple-attribute decision-making

We introduce the concept of Diophantine spherical vague set approach to multiple-attribute decision-making. The Spherical vague set is a novel expansion of the vague set and interval valued spherical fuzzy set. Three new concepts have been introduce such as Diophantine spherical vague weighted averaging operator, Diophantine spherical vague weighted geometric operator, generalized Diophantine spherical vague weighted averaging operator and generalized Diophantine spherical vague weighted geometric operator. We provide a numerical example to show how Euclidean distance and Hamming distance interact. Applications of the Diophantine spherical vague number include idempotency, boundedness, commutativity and monotonicity in algebraic operations. They can determine the optimal option and are more well-known and reasonable. Our goal was to identify the optimal choice by comparing expert opinions with the criteria. As a result, the model's output was more accurate as well as in the range of the natural number (sic). The weighted averaging distance and weighted geometric distance operators are distance measure that is based on aggregating model. By comparing the models under discussion with those suggested in the literature, we hoped to show their worth and reliability. It is possible to find a better solution more quickly, simply, and practically. Our objective was to compare the expert evaluations with the criteria and determine which option was the most suitable. Because they yield more precise solutions, these models are more accurate and more related to models with (sic). To show the superiority and the validity of the proposed aggregation operations, we compared it with the existing method and concluded from the comparison and sensitivity analysis that our proposed technique is more effective and reliable. This investigation yielded some intriguing results.