The impact of airborne pollutants present in the environment, entering the body through breathing, can cause significant risks of respiratory and heart-related health problems for individuals. For this, different air purifiers are commonly used to eliminate delicate particulate matter PM_2.5, and various studies have examined their effectiveness. This paper aims to analyze airborne pollutants and, by considering them, assess the performance of air purifiers in reducing the concentration of air pollutants in the environment. The aggregation operator (AO) plays a significant role in aggregating the multiple criteria and gives us a result in a singleton set, which assists us in decision-making (DM). So, considering this, the Heronian mean (HM) operator and its special cases such as averaging and geometric operators have been used in this paper. Moreover, circular intuitionistic fuzzy (C-IF) theory is more efficient and comprehensive than the intuitionistic fuzzy set (IFS), as the standard IFS cannot cope with the problems within a circular environment. So, the HM operator under the C-IF environment, such as circular intuitionistic fuzzy Heronian mean (C-IFHM) and it’s averaging and geometric operator, has been presented. Further, some prevalent and crucial properties and theorems have been defined. Then, an algorithm was developed to solve a multicriteria decision-making (MCDM) problem by using the proposed operators within the circular environment. To validate the effectiveness and priority, this paper presents a numerical example of MCDM, and a comparison analysis was conducted to verify the practicality of the proposed approach.

This manuscript presents a new weaker version of supra septarian axioms based on supra $\epsilon$-open sets, along with its essential features, which are called supra-$\epsilon$-$T_{j}$-space, $j=0,1,2$, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra $\epsilon$-symmetric property and supra difference property and study their effects on these version of supra-$\epsilon$-septarian axioms. In especial, we show that the two concepts of supra-$\epsilon$-$T_{0}$-space and supra-$\epsilon$-$T_{1}$-space are the same for any STS that fulfills supra $\epsilon$-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-$\epsilon$-$T_{j}$-space, where $j=0,1,2$, is a supra-hereditary (topological) property.

In this article, as an extension of the concepts of supra-E-T2-space, supra-E-T1-space, and supra-E-T0-space, we present the notion of supra-E completely space. Furthermore, we demonstrate that for any STS (γ; θ), the concepts of supra-E-T2-space and supra-E- completely-space are the same if γ less than or equal 4. We also explore the behavior of this notion with respect to specific forms of supra functions. We demonstrate that, under a bijective supra E-open function, the image of any supra-E- completely -space is a supra-E- completely. Additionally, we demonstrate that every supra subspace of supra-E- completely -space is supra-E-completely. Moreover, four new versions of separation axioms that utilize supra E-open sets are introduced namely: supra-E -regular-space, supra-E-normal-space, supra-E-T3-space, and supra-E-T4-space. We also give a general illustration of their key traits and look at the prerequisites for a number of similar links between them. We also propose a figure 1 graphic that shows these linkages. Furthermore, we demonstrate that every supra-E-R-space (γ; θ) is supra-E-N-space if γ less than or equal 4. This implies that the approaches of supra-E-T3-space and supra-E-T4-space are the same in this case. The necessary counterexamples that validate our findings are finally presented

In this particular piece of work, the new hybrid concept of vague soft rough set theory is introduced. It is a combination of rough set theory, soft set theory, and vague set theory. Based on this new concept, some definitions and operations are introduced. Furthermore, lower and upper vague soft approximations, vague soft rough positive, vague soft negative, and vague soft boundaries are discussed with examples. In addition, several theorems are presented in terms of lower and upper vague soft approximations, supported by examples for better understanding. Finally, an entirely new mathematical structure known as the vague soft rough topological structure is introduced. The related definitions of open sets, closed sets, closure, and interior, as well as their relationships in vague soft rough topological spaces, are addressed. To enhance understanding of this study, numerous examples are provided.