Improved cosine similarity measures for q-Rung orthopair fuzzy sets

In this short correspondence, we introduce some novel cosine similarity measures tailored for \(q\)-rung orthopair fuzzy sets (\(q\)-ROFSs), which capture both the direction and magnitude aspects of fuzzy set representations. Traditional cosine similarity measures focus solely on the direction (cosine of the angle) between vectors, neglecting the crucial information embedded in the lengths of these vectors. To address this limitation, we propose some improved cosine similarity measures, which extend the conventional cosine similarity by incorporating a length difference control term. These measures not only outperform traditional cosine similarity measures for \(q\)-ROFSs but also improve the existing cosine similarity measures for intuitionistic fuzzy sets and Pythagorean fuzzy sets, making them a valuable addition to the fuzzy set cosine similarity measures. These similarity measures are defined as the average and Choquet integral of two components: the first component, \(\text{Cos}_{A,B}(x_{j})\), quantifies the cosine similarity between \(q\)-ROFS \(A\) and \(B\) at each element \(x_{j}\). The second component, \(L_{A,B}(x_{j})\), represents the difference in lengths between the vector representations of \(A\) and \(B\) at the same element \(x_{j}\). This length-difference term ensures that the measures are sensitive to variations in both direction and magnitude, making them particularly suitable for applications where both aspects are significant. The measure derived through the Choquet integral also takes into account the interaction among the elements, thereby enhancing the sensitivity of solutions in various applications.