\(p,q,r-\)Fractional fuzzy sets and their aggregation operators and applications

Using \(p,q,r-\) fractional fuzzy sets (\(p,q,r-\) FFS) to demonstrate the stability of cryptocurrencies is considered due to the complex and volatile nature of cryptocurrency markets, where traditional models may fall short in capturing nuances and uncertainties. \(p,q,r-\) FFS provides a flexible framework for modeling cryptocurrency stability by accommodating imprecise data, multidimensional analysis of various market factors, and adaptability to the unique characteristics of the cryptocurrency space, potentially offering a more comprehensive understanding of the factors influencing stability. Existing studies have explored Picture Fuzzy Sets and Spherical Fuzzy Sets, built on membership, neutrality, and non-membership grades. However, these sets can’t reach the maximum value (equal to \(1\)) due to grade constraints. For example, when considering \(\wp =(h,\langle \text{0.9,0.8,1.0}\rangle \left|h\in H\right.)\), these sets fall short. This is obvious when a decision-maker possesses complete confidence in an alternative, they have the option to assign a value of 1 as the assessment score for that alternative. This signifies that they harbor no doubts or uncertainties regarding the chosen option. To address this, \(p,q,r-\) Fractional Fuzzy Sets (\(p,q,r-\) FFSs) are introduced, using new parameters \(p\), \(q\), and \(r\). These parameters abide by \(p\),\(q\ge 1\) and \(r\) as the least common multiple of \(p\) and \(q\). We establish operational laws for \(p,q,r-\) FFSs. Based on these operational laws, we proposed a series of aggregation operators (AOs) to aggregate the information in context of \(p,q,r-\) fractional fuzzy numbers. Furthermore, we constructed a novel multi-criteria group decision-making (MCGDM) method to deal with real-world decision-making problems. A numerical example is provided to demonstrate the proposed approach.