Cosine similarity and distance measures for quasirung orthopair fuzzy sets: Applications in investment decision-making.

Similarity measures and distance measures are used in a variety of domains, such as data clustering, image processing, retrieval of information, and recognizing patterns, in order to measure the degree of similarity or divergence between elements or datasets. quasirung orthopair fuzzy ( QOF) sets are a novel improvement in fuzzy set theory that aims to properly manage data uncertainties. Unfortunately, there is a lack of research on similarity and distance measure between QOF sets. In this paper, we investigate different cosine similarity and distance measures between to quasirung orthopair fuzzy sets ( ROFSs). Firstly, the cosine similarity measure and the Euclidean distance measure for QOFSs are defined, followed by an exploration of their respective properties. Given that the cosine measure does not satisfy the similarity measure axiom, a method is presented for constructing alternative similarity measures for QOFSs. The structure is based on the suggested cosine similarity and Euclidean distance measures, which ensure adherence to the similarity measure axiom. Furthermore, we develop a cosine distance measure for QOFSs that connects similarity and distance measurements. We then apply this technique to decision-making, taking into account both geometric and algebraic perspectives. Finally, we present a practical example that demonstrates the proposed justification and efficacy of the proposed method, and we conclude with a comparison to existing approaches.