Abstract
In this study, we present novel cosine similarity measures designed for \(q\)-rung orthopair fuzzy sets (\(q\)-ROFSs), offering a comprehensive analysis of both direction and magnitude aspects in fuzzy set representations. Unlike traditional cosine similarity measures, which primarily focus on the direction (cosine of the angle) between vectors, our proposed measures address this limitation by incorporating a lengths difference control term. This enhancement becomes crucial, especially when dealing with overlapping vector representations of \(q\)-ROFS components with a height difference, where traditional measures yield a similarity measure of \(1\). We demonstrate the effectiveness of these improved cosine similarity measures, showcasing their superiority not only over traditional counterparts for \(q\)-ROFSs but also in enhancing existing measures for intuitionistic fuzzy sets and Pythagorean fuzzy sets. The proposed measures consist of an average or Choquet integral of two components. The first component quantifies the cosine similarity between two \(q\)-ROFSs at each element, while the second component captures the difference in lengths between the vector representations of these \(q\)-ROFSs at the same element. This innovative length-difference term ensures sensitivity to variations in both direction and magnitude, making the measures well-suited for applications where both aspects are crucial. The Choquet integral-based measure further considers interactions among elements, enhancing sensitivity in diverse applications. In addition to introducing these cosine similarity measures, we extend our contributions to the realm of multi-criteria group decision making (MCGDM) through an extended The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) methodology. The proposed TOPSIS methodology is applied to a real-world problem in green technology selection, providing a comprehensive evaluation framework. Our comparative analysis with some other MCGDM methods further highlights the effectiveness of our proposed approach.