On $\left(\mathit{p},\mathit{q}\right)$-fractional linear Diophantine fuzzy sets and their applications via MADM approach

The integration of internationally sustainable practices into supply chain management methodologies is known as "green supply chain management". Reducing the supply chain's overall environmental impact is the main objective in order to improve corporate connections and the social, ecological, and economic ties with other nations. To accomplish appropriate and accurate measures to address the issue of emergency decision-making, the paper is divided into three major sections. First, the $\left(p,q\right)$-fractional linear Diophantine fuzzy set represents a new generalization of several fuzzy set theories, including the Pythagorean fuzzy set, $q$-rung orthopair fuzzy set, linear Diophantine fuzzy set, and $q$-rung linear Diophantine fuzzy set, with its key features thoroughly discussed. Additionally, aggregation operators are crucial for handling uncertainty in decision-making scenarios. Consequently, algebraic norms for $\left(p,q\right)$-fractional linear Diophantine fuzzy sets were established based on operational principles. In the second part of the study, we introduced a range of geometric aggregation operators and a series of averaging operators under the $\left(p,q\right)$-fractional linear Diophantine fuzzy set, all grounded in established operational rules. We also explained some flexible aspects for the invented operators. Furthermore, using the newly developed operators for $\left(p,q\right)$-fractional linear Diophantine fuzzy information, we constructed the multi-attribute decision-making ( $MADM$) technique to assess the green supply chain management challenge. Last, we compared the ranking results of the produced approaches with the obtained ranking results of the techniques using several numerical instances to demonstrate the validity and superiority of the developed techniques. Finally, a few comparisons between the findings were made.