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.

The p, q -quasirung orthopair fuzzy ( p, q -QOF) set, an extension of the q -rung orthopair fuzzy set ( q -ROF) set, offers a more comprehensive approach to information representation, adept at managing data uncertainties. Unlike the restrictive conditions of q -ROF set, which require that the sum of q^th power of membership (η) and non-membership function (ϑ) must not exceed one (η ^q + ϑ ^q ≼ 1), p, q -QOFS relaxes these limitations. Here, the combined value of the p^th power of membership and q^th power of non-membership is confined within one i.e., η ^p + ϑ ^q ≼ 1, under the conditions p, q ≽ 1 and various relationships between p and q ( p = q , p ≻ q or p ≺ q ). This study explores leveraging confidence levels tied to each p, q -quasirung orthopair fuzzy number ( p, q –QOFN) to devise a set of averaging and geometric aggregation operators (AOs). These operators effectively combine rating values from distinct criteria, as presented by decision-makers. By harnessing these operators, a novel approach for multi-criteria group decision-making (MCGDM) is formulated, well-suited to resolving real-life decision-making (DM) challenges. An illustrative example underscores the method’s efficacy and validity. Finally, a comparative assessment against existing methods highlights the superior performance of the proposed approach.

In the last few decades, there has been a significant increase in the importance of assessing social and ecological implications within industrial product supply chains. This tendency has given rise to the notion of supplier sustainability, which entails meeting the economic, environmental, and social demands of all suppliers. The supplier selections are typically based on experts’ opinion, which are then reflected in supplier ratings. When examining sustainability indicators, experts may not be fully aware of all aspects of suppliers’ economic, social, and environmental qualities. In the Decision Making (DM) approaches such as Pythagorean Fuzzy (PF) and q-Rung Orthopair Fuzzy (q-ROF) sets, experts are unable to use different powers for membership and non-membership grades simultaneously. In real-life DM problems, experts may require different power levels for membership and non-membership grades. In this paper, we propose a new extension of the COPRAS method under p, q-Quasirung orthopair fuzzy ( $p,q-$ QOF) sets, which allow decision-makers to use different power levels for membership and non-membership grades by incorporating parameters p and q. In the proposed approach, it is assumed that in addition to compiling the expert scores for suppliers, the issue analyst evaluates each expert’s degree of expertise for each criterion. The best choice is then determined by combining the data using the COPRAS method. The Inter-Criteria Correlation (CRITIC) method is employed to determine the unknown criteria weights. To demonstrate the effectiveness of our proposed approach, we apply it to a Multi-Criteria Group Decision-Making (MCGDM) problem that is focused on green supplier selection. Finally, we conduct a comparative study with existing approaches to demonstrate the practicality and applicability of the proposed DM method.