Cosine similarity and distance measures for p,q− 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. p,q− quasirung orthopair fuzzy (p,q− 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 p,q− QOF sets. In this paper, we investigate different cosine similarity and distance measures between to p,q− quasirung orthopair fuzzy sets (p,q− ROFSs). Firstly, the cosine similarity measure and the Euclidean distance measure for p,q− 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 p,q− 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 p,q− 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.