$$p,q,r-$$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.