$$k^{n}_{m}$$-Rung picture fuzzy information in a modern approach to multi-attribute group decision-making

Abstract

AbstractA useful tool for expressing fuzziness and uncertainty is the recently developed n,m-rung orthopair fuzzy set (n,m-ROFS). Due to their superior ability to manage uncertain situations compared to theories of q-rung orthopair fuzzy sets, the n,m-rung orthopair fuzzy sets have variety of applications in decision-making in daily life. To deal with ambiguity and unreliability in multi-attribute group decision-making, this study introduces a novel tool called $$k^{n}_{m}$$ k m n -rung picture fuzzy set ($$k^{n}_{m}$$ k m n -RPFS). The suggested $$k^{n}_{m}$$ k m n -RPFS incorporates all of the benefits of n,m-ROFS and represents both the quantitative and qualitative analyses of the decision-makers. The presented model is a fruitful advancement of the q-rung picture fuzzy set (q-RPFS). Furthermore, numerous of its key notions, including as complement, intersection, and union are explained and illustrated with instances. In many decision-making situations, the main benefit of $$k^{n}_{m}$$ k m n -rung picture fuzzy sets is the ability to express more uncertainty than q-rung picture fuzzy sets. Then, along with their numerous features, we discover the basic set of operations for the $$k^{n}_{m}$$ k m n -rung picture fuzzy sets. Importantly, we present a novel operator, $$k^{n}_{m}$$ k m n -rung picture fuzzy weighted power average ($$k^{n}_{m}$$ k m n -RPFWPA) over $$k^{n}_{m}$$ k m n -rung picture fuzzy sets, and use it to multi-attribute decision-making issues for evaluating alternatives with $$k^{n}_{m}$$ k m n -rung picture fuzzy information. Additionally, we use this operator to pinpoint the countries with the highest expat living standards and demonstrate how to choose the best option by comparing aggregate findings and applying score values. Finally, we compare the outcomes of the q-RPFEWA, SFWG, PFDWA, SFDWA, and SFWA operators to those of the $$k^{n}_{m}$$ k m n -RPFWPA operator.