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.

It is a well-known fact that ${\mathcal{L}}^p$-spaces provide a robust and flexible framework for analyzing functions with different types of behavior, uncertainty, and regularity. They are widely applicable in many areas of mathematics, science, and engineering. In this study, we introduced a novel generalization that combines interval intuitionistic fuzzy sets ( $IFS$s), as proposed by Atanassov ^[8], with circular intuitionistic fuzzy sets ( $C$- $IFS$), introduced by Atanassov ^[9], because these classical sets restrict us. This new concept is known as the ${\mathcal{L}}^p$-intuitionistic fuzzy set (value) ( ${\mathcal{L}}^p$- $IFS$( $V$)). The degrees of membership and non-membership in a ${\mathcal{L}}^p$- $IFS$ are depicted by a diamond shape, circle, star shape, and square with its center defined by non-negative real numbers $"\kappa "$ and $"𝓈"$, ensuring that $\kappa +𝓈\le 1$. The structure of a ${\mathcal{L}}^p$- $IFS$ facilitates the representation of information through points on different shapes with respect to $pth$-norm with a designated center and norm $"\mathrm{\aleph }"$, thereby enabling a more precise characterization of the fuzziness inherent in uncertain data. As a result, a ${\mathcal{L}}^p$- $IFS$ empowers decision-makers to evaluate options within a broader and more flexible framework, leading to the possibility of making more nuanced decisions. After establishing the concept of ${\mathcal{L}}^p$- $IFS$, some fundamental operations involving ${\mathcal{L}}^p$- $IFSs$ were outlined. To establish a novel scoring function and an accuracy function that incorporates the decision-makers' attitude ( $\lambda$), the set's optimistic and pessimistic points were defined. When the decision-maker's viewpoint ( $\lambda$) approached 1, the defuzzification of ${\mathcal{L}}^p$- $IFS$ occurred near its optimistic point, while it occurred near its pessimistic point as ( $\lambda$) approached 0. Moreover, a technique for converting a collection of intuitionistic fuzzy values into a ${\mathcal{L}}^p$-intuitionistic fuzzy values ( ${\mathcal{L}}^p$- $IFVs$) was formulated. Additionally, several algebraic operations between ${\mathcal{L}}^p$- $IFV$ using general triangular $𝓉$-norms and triangular $𝓉$-conorms were proposed. To transform input values represented by ${\mathcal{L}}^p$- $IFVs$ into a single output value, specific weighted aggregation operators based on these algebraic methods were introduced. The proposed methodology was applied to a problem concerning the selection of the optimal artificial intelligence (AI) agricultural field robots multi-attribute decision-making ( $MADM$) framework. Finally, a framework was also presented for addressing $MADM$ challenges within a ${\mathcal{L}}^p$-intuitionistic fuzzy context. It is interesting to note that the time complexity of the proposed method and a comparative analysis were evaluated.