This article's purpose is to investigate and generalize the concepts of rough set, in addition to the q-spherical fuzzy set, and to introduce a novel concept that is called q-spherical fuzzy rough set (q-SFRS). This novel approach avoids the complications of more recent ideas like the intuitionistic fuzzy rough set, Pythagorean fuzzy rough set, and q-rung orthopair fuzzy rough set. Since mathematical operations known as "aggregation operators" are used to bring together sets of data. Popular aggregation operations include the arithmetic mean and the weighted mean. The key distinction between the weighted mean and the arithmetic mean is that the latter allows us to weight the various values based on their importance. Various aggregation operators make different assumptions about the input (data kinds) and the kind of information that may be included in the model. Because of this, some new q-spherical fuzzy rough weighted arithmetic mean operator and q-spherical fuzzy rough weighted geometric mean operator have been introduced. The developed operators are more general. Because the picture fuzzy rough weighted arithmetic mean (PFRWAM) operator, picture fuzzy rough weighted geometric mean (PFRWGM) operator, spherical fuzzy rough weighted arithmetic mean (SFRWAM) operator and spherical fuzzy rough weighted geometric mean (SFRWGM) operator are all the special cases of the q-SFRWAM and q-SFRWGM operators. When parameter q = 1, the q-SFRWAM operator reduces the PFRWAM operator, and the q-SFRWGM operator reduces the PFRWGM operator. When parameter q = 2, the q-SFRWAM operator reduces the SFRWAM operator, and the q-SFRWGM operator reduces the SFRWGM operator. Besides, our approach is more flexible, and decision-makers can choose different values of parameter q according to the different risk attitudes. In addition, the basic properties of these newly presented operators have been analyzed in great depth and expounded upon. Additionally, a technique called multi-criteria decision-making (MCDM) has been established, and a detailed example has been supplied to back up the recently introduced work. An evaluation of the offered methodology is established at the article's conclusion. The results of this research show that, compared to the q-spherical fuzzy set, our method is better and more effective.

The Fourth Industrial Revolution, also known as Industry 4.0, is attracting a significant amount of attention because it has the potential to revolutionize a variety of industries by developing a production system that is fully automated and digitally integrated. The implementation of this transformation, however, calls for a significant investment of resources and may present difficulties in the process of adapting existing technology to new endeavors. Researchers have proposed integrating the Analytic Hierarchy Process (AHP) with extensions of fuzzy rough sets, such as the three-dimensional q-spherical fuzzy rough set (q-SFRS), which is effective in handling uncertainty and quantifying expert judgments, to prioritize projects related to Industry 4.0. This would allow the projects to be ranked in order of importance. In this article, a novel framework is presented that combines AHP with q-SFRS. To calculate aggregated values, the new framework uses a new formula called the q-spherical fuzzy rough arithmetic mean, when applied to a problem involving the selection of a project with five criteria for evaluation and four possible alternatives, the suggested framework produces results that are robust and competitive in comparison to those produced by other multi-criteria decision-making approaches.

The combination of q-spherical fuzzy sets and rough sets has emerged as a useful paradigm for fuzzy mathematics and decision-making. The hybrid structure of q-spherical fuzzy sets and rough sets has shown to be useful in the fields of fuzzy mathematics and decision-making. The primary goal of this research is to present Einstein’s operational principles for q-spherical rough numbers (q-SFRNs). The fundamental goal of this research is to develop geometric aggregation operators (AOs), such as q-spherical fuzzy rough Einstein weighted geometric (q-SFREWG) and q-spherical fuzzy rough Einstein ordered weighted geometric (q-SFREOWG) operators. We will look at the idempotency, boundedness, and other theorems linked with the suggested AOs. Recognizing the importance of multi-criteria decision-making (MCDM) in dealing with real-world difficulties, it is important to note that traditional MCDM procedures sometimes provide contradicting outcomes. Using the proposed AOs, this study presents a robust MCDM approach designed to address picture understanding and interpretation issues inside the q-SFRS framework. In addition, a complete comparative study is carried out to assess the suggested method’s efficacy and value in comparison to existing procedures. The findings from these comparison investigations show that our developed technique outperforms current approaches. The study emphasizes the expanded capabilities of the suggested technique in resolving the complexities of picture perception and interpretation within the q-SFRS environment, bringing a potential addition to the field of decision-making and fuzzy mathematics.

This study investigates advanced data collection methodologies and their implications for understanding employee and customer behavior within specific locations. Employing a comprehensive multi-criteria decision-making framework, we evaluate various technologies based on four distinct criteria and four technological alternatives. To identify the most effective technological solution, we employ the q-spherical fuzzy rough TOPSIS method, integrating three key parameters: lower set approximation, upper set approximation, and parameter q (where q ≥ 1). Our novel approach combines the TOPSIS method with q-spherical fuzzy rough set theory, providing deeper insights into data-driven decision-making processes in corporate environments. By comparing our proposed framework with existing multi-criteria decision-making methodologies, we demonstrate its strength and competitiveness. This research contributes to enhancing decision-making capabilities in corporate settings and beyond.

Multi-criteria decision-making (MCDM) approaches prove to be effective and reliable in addressing problems under uncertain conditions. The q-spherical fuzzy rough set (q-SFRS) represents the latest advancement in fuzzy set theory. This article aims to introduce a novel approach, q-spherical fuzzy rough Combinative Distance-based Assessment (q-SFR-CODAS), by integrating CODAS with q-SFR set to address MCDM problems. The method utilizes the Hamming distance as the primary measure and the Euclidean distance as the secondary measure to assess the desirability of alternatives, calculated concerning the negative-ideal solution. Additionally, an illustrative example is presented to demonstrate the applicability of the proposed methodology. A comprehensive sensitivity analysis is conducted to validate the results of q-SFR-CODAS, comparing them with existing MCDM methods.

This study introduces innovative operational laws, Einstein operations, and novel aggregation algorithms tailored for handling q-spherical fuzzy rough data. The research article presents three newly designed arithmetic averaging operators: q-spherical fuzzy rough Einstein weighted averaging, q-spherical fuzzy rough Einstein ordered weighted averaging, and q-spherical fuzzy rough Einstein hybrid weighted averaging. These operators are meticulously crafted to enhance precision and accuracy in arithmetic averaging. By thoroughly examining their characteristics and interrelations with existing aggregate operators, the article uncovers their distinct advantages and innovative contributions to the field. Furthermore, the study illustrates the actual implementation of these newly constructed operators in a variety of attribute decision-making scenarios employing q-SFR data, yielding useful insights. Our suite of decision-making tools, including these operators, is specifically designed to address complex and uncertain data. To validate our approach, this study offers a numerical example showcasing the real-world applicability of the proposed operators. The results not only corroborate the efficacy of the proposed method but also underscore its potential significance in practical decision-making processes dealing with intricate and ambiguous data. Additionally, comparative and sensitivity analyses are presented to assess the effectiveness and robustness of our proposed work relative to other approaches. The acquired knowledge enriches the current understanding and opens new avenues for future research.