The importance of convex and non-convex functions in the study of optimization is widely established. The concept of convexity also plays a key part in the subject of inequalities due to the behavior of its definition. The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this study, first, Hermite-Hadamard type inequalities for LR- $p$-convex interval-valued functions (LR- $p$-convex-I-V-F) are constructed in this study. Second, for the product of p-convex various Hermite-Hadamard (HH) type integral inequalities are established. Similarly, we also obtain Hermite-Hadamard-Fejér (HH-Fejér) type integral inequality for LR- $p$-convex-I-V-F. Finally, for LR- $p$-convex-I-V-F, various discrete Schur's and Jensen's type inequalities are presented. Moreover, the results presented in this study are verified by useful nontrivial examples. Some of the results reported here for be LR- $p$-convex-I-V-F are generalizations of prior results for convex and harmonically convex functions, as well as $p$-convex functions.

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.