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

Abstract

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.

Keywords: Cosine distance measure; Cosine similarity measure; Decision making; Ideal solutions; p,q -Quasirung orthopair fuzzy sets.

Fig. 1

$p,q-$ QOFSs and their specific instances.

Fig. 2

Paper layout.

Fig. 3

Schematic depiction of the proposed MCDM.

Fig. 4

Flowchart of the proposed model.

Fig. 5

Relative closeness of the alternatives.

Fig. 6

Closeness index obtained by existing approaches.

Fig. 7

The graphical view of Closeness index for different pairs of parameters $p$ and $q$.