New cosine similarity and distance measures for Fermatean fuzzy sets and TOPSIS approach.

The most straightforward approaches to checking the degrees of similarity and differentiation between two sets are to use distance and cosine similarity metrics. The cosine of the angle between two n-dimensional vectors in n-dimensional space is called cosine similarity. Even though the two sides are dissimilar in size, cosine similarity may readily find commonalities since it deals with the angle in between. Cosine similarity is widely used because it is simple, ideal for usage with sparse data, and deals with the angle between two vectors rather than their magnitude. The distance function is an elegant and canonical quantitative tool to measure the similarity or difference between two sets. This work presents new metrics of distance and cosine similarity amongst Fermatean fuzzy sets. Initially, the definitions of the new measures based on Fermatean fuzzy sets were presented, and their properties were explored. Considering that the cosine measure does not satisfy the axiom of similarity measure, then we propose a method to construct other similarity measures between Fermatean fuzzy sets based on the proposed cosine similarity and Euclidean distance measures and it satisfies the axiom of the similarity measure. Furthermore, we obtain a cosine distance measure between Fermatean fuzzy sets by using the relationship between the similarity and distance measures, then we extend the technique for order of preference by similarity to the ideal solution method to the proposed cosine distance measure, which can deal with the related decision-making problems not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods.