Aggregation operators based on Einstein averaging under q-spherical fuzzy rough sets and their applications in navigation systems for automatic cars

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.


Introduction
Zadeh [1] introduced the concept of fuzzy sets, which inspired subsequent extensions such as interval-valued fuzzy sets (IVFSs) [2], Atanassov's intuitionistic fuzzy sets (IFSs) [3], and interval-valued intuitionistic fuzzy sets (IVIFSs) [4] as alternative structures within the fuzzy set framework.However, Zadeh's study [5] and other studies [6,7] have demonstrated that Atanassov's intuitionistic fuzzy sets (IFS) and IVFSs are mathematically equal.IFS is closely linked to the representation of intuitionistic fuzzy information, necessitating a thorough understanding of both intuitive fuzzy sets and aggregation methods [8].The concept of IFSs has evolved, allowing non-membership and membership functions to take interval values, leading to the development of interval-valued intuitionistic fuzzy sets (IVIFSs) [9].IVIFS offers a more complicated representation of uncertainty and imprecision in fuzzy set theory.The invention of intuitionistic fuzzy numbers (IFNs) was a significant step towards determining decision outcomes.To elaborate, Xu [10] presented other aggregation operators, including IFWA, IFOWA, and IFHA, and detailed their unique properties.Notably, Xu and Yager [11] introduced three new geometric aggregation operators for IFNs: IFWG, IFOGA, and IFHG.The application of these mathematical principles to a wide range of topics and situations is intriguing.Zeng and Su [12] examined the development of an intuitionistic fuzzy ordered weighted distance (IFOWD) operator and its use in group decision-making for system selection.Their finding throws light on this vital location, paving the way for future discoveries.The IFOWD operator shows fascinating applications throughout a wide spectrum.When decision-makers express preference values like ζ A = 0.6 and ξ A = 0.7, the total of both values exceeds 1.This violates the criterion of an intuitionistic fuzzy set.Yager's Pythagorean fuzzy sets [13] solve these problems by ensuring that ) ≤ 1. Pythagorean fuzzy sets are recognized to manage uncertainty better than intuitionistic fuzzy sets (IFSs), making Pythagorean fuzzy set theory a more popular and exciting research topic.Yager and Abbasov [14] developed many aggregation approaches to address Multiple Criteria Decision Making (MCDM) issues in the Pythagorean fuzzy framework.Neutronosophic cubic sets [15] are derived from the neutrosophic set [16], which represents an astounding step forward beyond conventional fuzzy sets.According to the existing literature, major research efforts have been focused on the study of neutrosophic sets (NSs), neutrosophic cubic sets (NCSs), and the accompanying aggregation operators.Alia et al. [17] investigate the notion of NCSs and their use in pattern recognition.Furthermore, Je [18] developed operations and aggregation methods specifically for NCSs.Ajay et al. [19] used geometric Bonferroni mean operators for multicriteria decision-making (MCDM) using neutrosophic cubic sets (NCSs).Coung et al. [20] introduced the concept of picture fuzzy sets, with the restriction that the sum of all memberships lies inside the interval [0,1].In more recent work, Atta et al. [21] used the notion of neutrosophic sets (NSs) in an upgraded picture steganography system that is based on modification direction.Gundogdu et al. [22] developed the notion of spherical fuzzy sets (SFS) and its associated theory, describing it as a unique extension of fuzzy set theory.This paradigm is differentiated by its triple membership structure, which contains membership, non-membership, and hesitation functions.They look at the positive, neutral, and negative membership functions with total squares equal to or less than one.In dealing with uncertainty, imprecision, and vagueness, the SFS model surpasses Pythagorean fuzzy sets.A thorough examination of the most recent literature demonstrates a growing preference for investigations into SFS.Ashraf et al. [23] developed a set of aggregation strategies, particularly for a spherical fuzzy framework.Ashraf et al. [24] took a unique approach by providing a grey technique (GRA) based on the groundbreaking notion of spherical linguistic fuzzy Choquet integrals.Furthermore, Jin et al. [25] developed and utilized logarithmic operators designed for spherical fuzzy sets (SFSs) in decision support systems.Rafiq et al. [26] introduced a cosine similarity measure tailored particularly to the SFS model, intending to improve decision-making in scenarios including ambiguous and imprecise data.Furthermore, Ashraf et al. [27] proposed a group decision-making technique customized for the spherical fuzzy environment and used it to solve challenges in multi-criteria group decision-making (MCGM).Gundogdu et al. [28] updated the well-known VIKOR technique to include the spherical fuzzy set (SFS) model and used it in a Multi-Criteria Decision Making (MCDM) context within a spherical fuzzy setting.Acharjya and Rathi [29] proposed an integrated decision-making approach that combines fuzzy rough sets and genetic algorithm models.They evaluated its effectiveness in a relevant MCDM scenario about smart agriculture.Sharaff et al. [30,31] investigated a fuzzy-based technique for text summarization extraction and proposed a document categorization strategy based on a fuzzy clustering algorithm.Gou et al. [32] devised exponential operating rules for interval fuzzy sets (IFSs) and introduced novel aggregation operators within the IFS framework.Recent developments in decision-making methodologies have utilized diverse fuzzy set theories and aggregation operators to address intricate problems in various fields.For instance, Seikh and Mandal [33] developed interval-valued Fermatean fuzzy Dombi aggregation operators and applied the SWARA-based PROMETHEE II method for effective biomedical waste management.In another study, Seikh and Chatterjee [34] utilized the SWARA-ARAS method in a confidence level-based interval-valued Fermatean fuzzy environment to determine the best renewable energy sources in India.Mahmood et al. [35] proposed a decision-making and medical diagnosis method using spherical fuzzy sets.In contrast, Wang [36] employed T-spherical fuzzy rough interactive power Heronian mean aggregation operators for multiple attribute group decision-making.Furthermore, Wang et al. [37] created a new CODAS approach using the Heronian Minkowski distance operator for T-spherical fuzzy group decision-making, while Ali et al. [38] introduced Heronian mean operators for multi-attribute decision-making using T-spherical fuzzy information.Using bipolar complex fuzzy soft information, Jaleel [39] extended the WASPAS approach to agricultural robotics systems based on Dombi aggregation operators.For MADM problems, Ali et al. [40] investigated averaging aggregation operators in the context of q-rung orthopair image fuzzy soft sets.Additionally, Azim et al. [41] applied sine trigonometric q-SFR aggregation operators for group decision-making in digital transformation [42], evaluated indoor positioning systems using a q-SFR TOPSIS analysis [43], and prioritized Industry 4.0 projects using this method.

Logics to define Einstein aggregation operators and their advantages
Einstein operations, a subset of fuzzy logic operators, are used to create aggregation operators.These operators are used to aggregate multiple fuzzy sets or fuzzy values into a single fuzzy set or value.The logic behind Einstein's aggregation is derived from Einstein's addition and multiplication operations, which provide a way to combine values while considering their inherent fuzziness.Below are the advantages of Einstein aggregation operators.
A.B. Azim et al. accommodate a wide range of responses, including positive and negative assessments, uncertainty, and even abstention from answering.By incorporating the parameter q, the q-SFS provides decision-makers with greater flexibility to express their preferences for membership, non-membership, and the degree of ambiguity in their judgments.This enhances decision-making by offering a more nuanced approach to handling complex and ambiguous information, thereby increasing the practical value of q-SFS in various application areas.
Rough set theory, introduced by Pawlak [48,49] in the early 1980s, is a mathematical approach designed to handle vagueness and uncertainty in data analysis, especially when information is imprecise or incomplete.At its core, rough set theory revolves around the concept of an information system, which is essentially a table comprising objects and their attributes.Objects are considered indiscernible if they cannot be distinguished based on the available attributes, leading to the formation of equivalence classes.The theory defines two key approximations: the lower approximation, which includes all objects that belong to a target set, and the upper approximation, which includes all objects that possibly belong to the target set.The boundary region, representing the difference between these approximations, signifies the uncertainty in classification.Rough set theory is also concerned with identifying reducts, which are minimal subsets of attributes that retain the classification capability of the entire attribute set, and the core, which is the intersection of all reducts, representing the most essential attributes.These concepts are particularly useful in various applications, such as feature selection, where the theory helps identify the most relevant attributes in a dataset, reducing dimensionality and improving machine learning efficiency.In data mining and knowledge discovery, rough set theory aids in uncovering patterns and relationships within data, providing valuable insights.It is also beneficial in decision support systems by offering a framework to manage imprecise information for making informed decisions, and in classification and clustering tasks, it helps group objects based on their attributes even amidst uncertainty.For instance, in a medical diagnosis system, rough set theory can pinpoint the most relevant symptoms and test results, classify patients into diagnostic categories, and generate diagnostic rules from the data.The primary advantages of rough set theory include its independence from preliminary information such as probability distributions or membership A.B. Azim et al. functions, its effectiveness in handling uncertainty, and its straightforward, intuitive concepts.Overall, rough set theory is a powerful tool for dealing with imprecise and uncertain information, making it invaluable in various domains such as data analysis, machine learning, and decision-making.Several researchers [50,51] have significantly contributed to the development and application of rough set theory since its inception by Zdzisław Pawlak.Their work spans various domains, including data mining, machine learning, and decision support systems.These researchers have explored feature selection, rule generation, and the handling of uncertainty in data.Their contributions have enhanced the theory's robustness and expanded its practical applications.As a result, rough set theory continues to evolve and find new uses in diverse fields.The q-SFS is acknowledged as the most effective way of dealing with uncertainty and ambiguity.This advanced mathematical framework improves the capacity to tackle these difficulties more effectively than traditional fuzzy sets.The q-SFS, which includes an extra parameter, gives better flexibility and accuracy in expressing complicated decision-making scenarios.This method is especially useful when data is confusing or partial, providing a reliable result.As a result, it has earned a reputation for being exceptionally adept in solving complicated issues in a variety of sectors.The integration of fuzzy logic with rough set theory has been significantly advanced by the theories of IFRS [52], PyFRS [53], and q-ROFRS [54,55].By extending the traditional rough set structure, these sophisticated models help data analysts more effectively handle ambiguity and imprecision.By combining intuitionistic fuzzy sets with rough sets, IFRS enhances its capacity to manage ambiguity and hesitation.PyFRS allows for a more comprehensive depiction of uncertainty by incorporating Pythagorean fuzzy sets.Similarly, to provide even more flexibility and precision, q-ROFRS makes use of q-rung orthopair fuzzy sets.These developments highlight how rough set theory is always changing and how it can be applied to intricate decision-making procedures.The examination of voting may be conducted using a theoretical framework known as the PFRS.This approach successfully integrates the ideas of picture fuzzy sets with rough set theory, resulting in a strong technique for dealing with the inherent uncertainty and ambiguity in voting scenarios.Despite having lower and upper approximations expressed as , there are inherent limitations.These limitations arise from the necessity that the combined values of the lower and upper approximations must fall within the [0, 1] range.Immediate and urgent action is needed to fully resolve this issue and ensure the framework's effectiveness and accuracy.However, when decision-makers are presented with information in the form of SFRS, which includes both lower and upper approximations, such as {(0.7,0.8,0.9),(0.9,0.8,0.7)}, the combined values of these approximations frequently exceed the [0, 1] interval.This disparity poses a serious problem because it might cause errors and inconsistencies while making decisions.To solve this issue, a detailed investigation of the theoretical underpinnings is required, as well as the development of improved methodologies to assure the validity and dependability of the approximations within the prescribed period.This project is critical for successfully implementing the PFRS framework in real-world voting and decision-making contexts.The numbers (0⩽̸ 0.7 2 +0.8 2 +0.9 2 ⩽̸ 1) and (0⩽̸ 0.9 2 +0.8 2 +0.7 2 ⩽̸ 1) are not appropriate for the SFRS framework, indicating a restriction in its use.In their 2023 research paper, Azim et al. [56] presented the notion of q-SFRS as a solution to this problem.Information with lower and upper approximations can be represented using the q-SFRS framework as follows: {(0.7,0.8,0.9),(0.9,0.8,0.7)}.This ensures that the total values of these approximations stay within the interval [0,1].This change overcomes the prior limits, increasing the SFRS framework's applicability and accuracy in decision-making processes.This fuzzy set combines the strengths of rough sets and q-SFS.Our study presents a realistic decision-making framework based on q-SFRS, considerably adding to the field's current body of knowledge.The q-SFRS framework employs three distinct parameters for lower and upper approximations, enhancing its flexibility and accuracy.The primary goal of this project is to advance future research by developing new aggregation operators and defuzzification procedures.A detailed examination demonstrates that q-SFRSs hold significant potential as a new concept, opening up several options for future research and practical applications.This breakthrough shows that q-SFRSs might play a critical role in addressing difficult decision-making challenges, enabling the development of more sophisticated and effective solutions across a variety of disciplines.

Research questions behind the combination of q-spherical fuzzy sets and rough sets
The combination of q-SFSs with rough sets may introduce additional complexity, but this complexity is justified by the enhanced ability to model real-world problems involving intricate uncertainties and imprecisions.This relevance is particularly evident in fields requiring high precision and reliability in decision-making under uncertainty.Therefore, the practical necessity of this model lies in its superior capability to handle complex, uncertain, and imprecise data, making it a valuable tool despite its complexity.Traditionally, fuzzy sets allow for partial membership, which is useful for representing uncertainty in a single dimension.The extended q-SFSs introduce a parameter q, which can model more complex uncertainties by incorporating higher-order hesitancy and ambiguity in membership degrees.This makes them more versatile in real-world applications where uncertainty is not merely binary or linear.On the other hand, rough sets are adept at dealing with vagueness by approximating sets with a pair of lower and upper bounds, which is useful when precise boundary information is unavailable.Combining q-SFSs with rough sets leverages the strengths of both approaches.The q-SFS provides a nuanced representation of membership degrees, while the rough set theory deals effectively with boundary vagueness.This hybrid model can thus capture more complex patterns and relationships in data that are not possible with either model alone.For instance, in decision-making systems where data is both ambiguous (fuzzy) and incomplete (rough), this combined approach ensures more accurate and reliable outcomes.Autonomous vehicles rely heavily on sensor data (such as LIDAR, radar, and cameras) to navigate.This data often contains uncertainties and imprecisions due to various factors like weather conditions, lighting, and occlusions.The q-spherical fuzzy sets can model the ambiguous nature of sensor readings more effectively, while rough sets can handle incomplete information, providing a more robust interpretation of the environment.Navigation decisions in autonomous vehicles require precise handling of dynamic and uncertain environments.The combination of q-spherical fuzzy sets and rough sets allows for better modeling of uncertainties related to obstacles, traffic conditions, and road layouts, leading to safer and more A.B. Azim et al. reliable decision-making processes.Autonomous vehicles need to continuously plan and re-plan their paths to avoid obstacles and navigate efficiently.The hybrid model enhances the vehicle's ability to generate optimal paths by accurately assessing the uncertainties in the environment and the vehicle's position, ensuring smooth and safe navigation even in complex and unpredictable scenarios.While the combination does increase the mathematical complexity, the practical benefits often justify this complexity.The nuanced modeling of uncertainty leads to more accurate and reliable decision-making, which is crucial in critical applications like autonomous driving.The hybrid model allows for a more comprehensive analysis of data, capturing subtleties that might be missed with simpler models.Such models can adapt to a wide range of applications, providing flexibility in modeling different types of uncertainties.The combination of q-SFSs with rough sets may introduce additional complexity, but this complexity is justified by the enhanced ability to model real-world problems involving intricate uncertainties and imprecisions.This relevance is particularly evident in autonomous navigation systems, where precise interpretation of sensor data, robust decision-making, and reliable path planning are crucial for safety and efficiency.Therefore, the practical necessity of this model lies in its superior capability to handle complex, uncertain, and imprecise data, making it a valuable tool despite its complexity.

Gap before establishing the proposed operators
The preceding gap in the domain of q-SFR sets may be briefly expressed as follows: Traditional fuzzy sets helped control uncertainty, but they struggled to absorb complicated information and effectively express decision-makers' preferences.This constraint created a gap in decision-making processes, especially when dealing with complex and unexpected data that required more efficient treatment.The introduction of q-SFR sets addressed this gap by improving the ability of conventional fuzzy sets to handle complicated information.However, a critical necessity remained for sophisticated aggregation operators capable of negotiating the intricacies of q-SFR sets while accurately conveying decision-makers' preferences and uncertainties.Einstein's operations and operators, such as q-SFREWA, q-SFREOWA, and q-SFREHWA, were designed to fill the current gap.These operators combine the advantages of q-SFR sets with Einstein aggregation techniques, allowing for more accurate and adaptive decision-making in complex and unpredictable circumstances.They provide powerful and comprehensive ways of gathering information and accurately reflecting decision-makers' preferences.In essence, before the advent of Einstein's operations and operators, there was a lack of sophisticated aggregation approaches capable of processing complicated information and properly representing decision-makers' preferences.Einstein's operators effectively bridge this gap by incorporating powerful and thorough aggregation methods into decision-making procedures.

Motivations for the proposed operators
The motivations behind the introduction of q-SFRSs and their associated aggregation operators, such as q-SFREWA, q-SFREOWA, and q-SFREHWA, can be articulated as follows.
1. q-spherical fuzzy rough sets: These sets are designed to handle and represent complex and ambiguous information more effectively than traditional fuzzy sets.They offer a versatile and precise framework for decision-making in unpredictable and complex scenarios.This enhanced flexibility allows decision-makers to navigate uncertainty with greater accuracy and adaptability, making it invaluable across various contexts and applications 2. q-spherical fuzzy rough Einstein weighted averaging: This operator calculates a weighted average by integrating the benefits of q-SFRSs with Einstein aggregation.The aim is to enhance the handling of complex and ambiguous data, leading to more accurate and adaptable decision-making.3. q-spherical fuzzy rough Einstein ordered weighted averaging: By applying the concept of ordered weighted averages to q-SFRSs, this operator enables decision-makers to account for the importance of criteria and their levels of optimism or pessimism.The goal is to capture decision-makers' preferences and uncertainties comprehensively.4. q-spherical fuzzy rough Einstein weighted averaging This hybrid operator combines both weighted and hybrid averages, considering the weights and hybrid distances between q-spherical fuzzy rough sets.The aim is to offer a robust and precise aggregation method that reflects the influence of both weights and hybrid distances accurately.
Integrating q-SFR sets into various decision support systems has proven to significantly enhance the robustness and efficiency of these systems.The proposed structure fulfills the need for more advanced aggregation operators capable of handling the complexities of q-SFR sets while accurately reflecting decision-makers' preferences and uncertainties.

Significance
1.The proposed study on q-spherical fuzzy rough sets (q-SFRSs) and their associated Einstein aggregation operators is significant due to its innovative approach to handling complex, ambiguous, and uncertain information.This study addresses several critical gaps in the current state of fuzzy set theory and rough set theory, providing a comprehensive framework that enhances decision-making processes across various fields.Below are the key aspects highlighting the significance of this research: 2. Traditional fuzzy sets and rough sets are limited in their ability to fully capture the nuances of complex uncertainty.q-SFRSs introduce an additional parameter, q, which allows for a more nuanced representation of membership, non-membership, and hesitancy degrees, thus offering a richer framework for modeling uncertainty.3.By combining the strengths of fuzzy sets and rough sets, q-SFRSs improve the accuracy of decision-making processes.This hybrid approach provides a more detailed and accurate representation of data, leading to better-informed decisions, especially in environments characterized by high uncertainty and complexity.4. The introduction of Einstein aggregation operators, such as q-SFREWA, q-SFREOWA, and q-SFREHWA, provides sophisticated tools for aggregating q-SFRNs.These operators leverage the unique properties of Einstein operations to offer more precise and reliable aggregation methods, addressing the limitations of traditional aggregation techniques. 5.The proposed framework applies to various domains, including autonomous systems, medical diagnosis, and multi-attribute decision-making (MADM).The ability to handle complex and uncertain data makes q-SFRSs particularly valuable in fields requiring high precision and reliability.
This work lays the groundwork for further investigations into the fields of rough set theory and fuzzy set theory.It opens doors for additional research and development in linked sectors by presenting cutting-edge ideas and techniques.

Objectives
The main objectives of the study are outlined as follows.
1. To formally define and develop the mathematical framework for q-SFRSs.This involves establishing the basic properties, operational laws, and algebraic structures of q-SFRSs, ensuring they effectively combine the features of both fuzzy sets and rough sets.2. To design and define new Einstein aggregation operators specifically tailored for q-SFRNs.This includes developing operators such as q-SFREWA (q-spherical fuzzy rough Einstein weighted averaging), q-SFREOWA (q-spherical fuzzy rough Einstein ordered weighted averaging), and q-SFREHWA (q-spherical fuzzy rough Einstein hybrid weighted averaging).These operators will leverage the properties of Einstein operations to handle the complexities of q-SFRSs.3. To conduct a comparative analysis of the proposed Einstein aggregation operators with existing methods.This involves evaluating the performance of the new operators against traditional aggregation techniques and other advanced methods like Dombi and Archimedean aggregation operators.Metrics for comparison will include accuracy, robustness, and ability to handle extreme values and outliers.4.There will be case studies in domains including autonomous car navigation, medical diagnosis, and multi-attribute group decisionmaking (MAGDM) to illustrate the practical usefulness of q-SFRSs and associated aggregation operators in real-world decisionmaking scenarios.These examples will serve as proof of the suggested framework's efficacy.5. To perform sensitivity and robustness analysis on the proposed methods.This involves assessing how changes in the input data and parameters affect the performance of the q-SFRS-based decision-making framework.The goal is to ensure that the proposed methods are reliable and robust under varying conditions.
The work seeks to make a substantial addition to the domains of fuzzy set theory, rough set theory, and uncertainty-based decisionmaking.The suggested q-SFRSs and Einstein aggregation operators offer enhanced tools for modeling and handling complicated, ambiguous, and uncertain data, opening the way for more accurate and trustworthy decision-making processes.
The paper is structured as follows: Section 2 provides a comprehensive review of several topics, including PFS, SFS, q-SFS, RS, and q-SFRS, laying the foundation for subsequent sections.Section 3 discusses the "main results (aggregation operators based on q-SFRS),"  exploring three distinct aggregation operators based on Einstein's operations.In Section 4, the "multi-criteria group decision-making approach" is introduced, focusing on the development of a multi-criteria group decision-making framework.Section 5 examines the managerial implications of the proposed q-SFRSs and Einstein aggregation operators for decision-making processes, including comparative and sensitivity analyses to highlight strengths and weaknesses.Section 6, titled "Conclusion and Future Directions," summarizes the key findings, underscores the study's significance, and proposes numerous avenues for future research.Fig. 1 illustrates the overall structure of the article.

Preliminaries
In this section, we will explore a diverse array of mathematical concepts, starting with a comprehensive review of PFS, SFS, q-SFS, and RS.Each of these theories plays a crucial role in addressing different facets of uncertainty and imprecision in data analysis and decision-making.Definition 1. [20] Let X be a non-empty set.A picture fuzzy set A in X is defined by a membership function ζ A , a neutral membership function η A , and a non-membership function ξ A .The mathematical characterization of a PFS is as follows: ζ A : X → [0, 1] represents the degree of membership of each element in X to the set A .
η A : X → [0, 1] represents the degree of neutral membership of each element in X to the set A .
ξ A : X → [0, 1] represents the degree of non-membership of each element in X to the set A .These functions must satisfy the following condition for every element x ∈ X : Definition 2. [22] Consider a non-empty set X .A spherical fuzzy set A in X is characterized by a membership function ζ A , a neutral membership function η A , and a non-membership function ξ A .The mathematical formulation of a spherical fuzzy set is provided below: represents the degree of membership of each element in X to the set A .
η A : X → [0, 1] represents the degree of neutral membership of each element in X to the set A .
ξ A : X → [0, 1] represents the degree of non-membership of each element in X to the set A .These functions must satisfy the following condition for every element x ∈ X : Definition 3. [47] Let X be a non-empty set.A q-SFS A in X is characterized by the membership function ζ A , the neutral membership function η A , and the non-membership function ξ A .The mathematical formulation of a q-SFS is provided as follows: Fig. 2. Visual comparison between the picture fuzzy set, spherical fuzzy set, and q-spherical fuzzy set in three dimensions.
A.B. Azim et al. ] represents the degree of membership of each element in X to the set A .
η A : X → [0, 1] represents the degree of neutral membership of each element in X to the set A .
ξ A : X → [0, 1] represents the degree of non-membership of each element in X to the set A .
The following requirement must be met by these functions for any element x in X : 0 ≤ (ζ A (x)) q + (η A (x)) q + (ξ A (x)) q ≤ 1 for all q ≥ 1.The graphical depiction of PFS, SFS, and q-SFS differences in three-dimensional space is shown in Fig. 2. Definition 4. [56] Let G 1 and G 2 are non-empty sets.Typically, G 1 is the universe of discourse, and G 2 is a set of attributes or conditions.Let R ⊆ G 1 × G 2 be a binary relation between elements of G 1 and G 2 .The lower and upper set approximations for a subset X ⊆ G 1 and ⊆ G 2 are as follows: Here, [x] A represents the idea of indiscernibility within the context of these subsets.
The pair (R(A ), R(A )) is sometimes referred to as a rough set.
Definition 5. [56] A q-SF relation R is a q-SF subset of G 1 × G 2 and is defined as follows: Definition 6. [56] Consider the universal set G 1 and the set of attributes G 2 .Let R be a q-SF relation from G 1 to G 2 .The q-SF approximation space is defined as the triplet (G 1 ,G 2 ,R).The lower and upper approximation spaces of a given element p with respect to the approximation space (G 1 , G 2 , R) are defined and elaborated as follows: Fig. 3. Three-dimensional graphical representation of a q-SFRS.
A.B. Azim et al. with the condition that A q-SFRS is characterized by a pair of q-SFSs with distinct lower and upper set approximations.This concept is denoted as A = (A , A ) and referred to as a q-SFR number.The collection of all q-SFR numbers is represented as A i .A q-SFRS comprises a pair of q-SFSs where A differs from A .Fig. 3 illustrates the three-dimensional graphical representation of a q-SFRS.

Definition 7. [56] Consider
) be any three q-SFRNs, and ω > 0, then, ) denote a q-SFRN.A specific formula is employed to compute the score value Sco(A ) for A .This score value numerically represents the q-SFRN, facilitating comparison and analysis.The formula for calculating the score value is provided below: Where 0 ≤ Sco(A ) ≤ 1.This formula allows for the determination of the score value of the q-SFRN.

Definition 9. [56] Let
) ) denote a q-SFRN.A specific formula is employed to determine the accuracy of A , denoted as Acc(A ).To calculate the accuracy of A , use the formula provided below: Where − 1 ≤ Acc(A ) ≤ 1.This formula is used to calculate the accuracy value of the q-SFRN.
Let ω, ω 1 and ω 2 be any positive integers.Then the following properties hold: Definition 12. Using the t-norm Ҭ, t-conorm S , Einstein's operations are as follows:

Proposed operational laws for q-SFRSs
In this section, we develop a comprehensive set of operational laws within the framework of q-SFRSs.This involves defining and exploring various mathematical properties and operations that apply to these sets.Using these established operational laws, we provide a diversified collection of aggregation operators designed specifically for the integration of q-SFR information.This technique greatly increases the flexibility and accuracy of aggregation procedures within the stated framework, resulting in higher decisionmaking efficacy in complex settings.The development of these operational laws and aggregation operators significantly enhances the capacity to process and analyze q-SFR information.By providing a robust framework for the integration and manipulation of such information, these tools improve the flexibility and accuracy of aggregation procedures.Consequently, they offer greater decisionmaking efficacy in complex settings, making them valuable for applications in various fields requiring nuanced handling of uncertainty and vagueness.

Operational laws
) as any three q-SFRNs, where ω > 0, the essential Einstein's operations for q-SFRNs based on t-norm and t-conorm are presented as follows: ( . (

Q-SFREWA operators
) ne a collection of q-SFRNs, the q-spherical fuzzy rough Einstein averaging operator (q-SFREWA) operator is defined as a mapping q − SFREWA : A n ⟶A characterized by ) be a collection of q − SFRNs and (ω 1 , ω 2 , …, ω n ) T signifies the weight vector adhering to the condition ω i > 0 and the constrain Then the aggregated values obtained by the q-SFREWA operator is also a q-SFRN.
Proof: This proof can be easily established using mathematical induction about the natural number n.
Step 1.When n = 1 the value of ω 1 become 1 and upon evaluating the left side of Definition ( 14), we get: .
Also, for the right-hand side of Definition ( 14), we obtain: Thus, the condition n = 1 is true for Definition (14).
Step 2. Assume that Definition (14), is valid for n = k, where k is any real number, given this supposition, Definition (14), can be represented as: Step 3. Now for n = k + 1, we are examining the following equations: )ω k+1 .
Hence for n = k + 1 Definition ( 14) holds.By combining observations from steps (1), (2), and (3), this result applies to all values of n inside the natural numbers.
Step 2: Suppose that Theorem (2) is true for n = k.
Step 3: We will prove that Theorem (2) is true for n = k + 1.

Theorem 3 (Idempotency). Assuming
) be a collection of q − SFRNs and (ω 1 , ω 2 , …, ω n ) T signifies the weight vector adhering to the condition ω i > 0 and the constrain ) be a collection of q − SFRNs and (ω 1 , ω 2 , …, ω n ) T signifies the weight vector adhering to the condition ω i > 0 and the constrain and Proof: From Theorem 2, we have

For lower and upper memberships
We have

For lower and upper neutral memberships
We have

For lower and upper non-memberships
We have

For the lower and upper memberships
As,

For the lower and upper non-memberships
As,

Some specific cases regarding the q − SFREWA operator
From Theorem 2, we have Fig. 4. Specific cases regarding the q − SFREWA operator.
We are facing the following cases. )ωi ) (The spherical fuzzy rough Einstein weighted averaging operator).
(The picture fuzzy rough Einstein weighted averaging operator).

Case 4.
If ξ i = ξ i = 0 and q = 1 then (The intuitionistic fuzzy rough Einstein weighted averaging operator).Fig. 4 represents the specific cases regarding the q − SFREWA operator.

q − SFREOWA operator
) ne a collection of q-SFRNs, the q-spherical fuzzy rough Einstein ordered averaging operator (q-SFREOWA) operator is defined as a mapping q − SFREOWA : A n ⟶A associated with the weight vector (ω 1 , ω 2 , …, ω n ) T adhering the condition ω i > 0 and the constrain ( ) ) ( ) Fig. 5. Specific cases regarding the q − SFREOWA operator.
(The Pythagorean fuzzy rough Einstein ordered weighted averaging operator). ) ( ) (The picture fuzzy rough Einstein ordered weighted averaging operator).

Proof:
The proof follows the same steps as in Theorem 5.

Some specific cases regarding the q − SFREHWA operator
From Theorem 10, we have ) ) (The picture fuzzy rough Einstein hybrid weighted averaging operator).
Step 2: Construct the normalized decision matrices (Tables 5-8).The closer the distance, the better.As a result, a shorter distance is regarded as advantageous, converting distance to destination into a cost-type criterion.Also, light traffic is preferable, hence less traffic is regarded as advantageous.Thus, traffic condition is a cost-type criterion.

Table 12
Ranking of alternatives based on their respective q parameter values.approach is robust and adaptable to various decision-making scenarios, maintaining consistent and accurate rankings.The method's versatility and reliability extend its applicability across a broad spectrum of decision-making scenarios, thereby enhancing its practical utility.These attributes underscore the effectiveness and dependability of the proposed approach in assessing and prioritizing alternatives based on the parameter q.
Table 12 provides a detailed ranking of alternatives based on their respective q-parameter values, illustrating the steadfastness and reliability of the ranking order across a spectrum of q-values.The data shows a clear pattern: as the q values increase from 1 to 10, the alternatives values consistently increase, showcasing a positive relationship between the q parameter and the values of these alternatives.Each alternative value increases steadily, reflecting a positive relationship between q and the values of these alternatives.The minor fluctuations do not significantly impact the overall increasing trend.The graphical representation of different q values is shown in Fig. 10.
The method's versatility and reliability extend its applicability across a broad spectrum of decision-making scenarios, thereby enhancing its practical utility.These attributes underscore the effectiveness and dependability of the proposed approach in assessing and prioritizing alternatives based on the parameter q.

Test of validity
To demonstrate the flexibility and robustness of the recommended approach in a variety of circumstances, we use the assessment procedures proposed by Wang and Triantaphyllou [57], as follows: Step 1: The initial validation stage involves substituting the rating values of suboptimal alternatives with those of even lower quality.This substitution should not affect the determination of the optimal alternative.Specifically, the choice that appears to be the best should not change, as long as the relative weights of the criterion stay stable.This stage guarantees that the model continues to identify the best alternative, independent of changes in the lower-ranked alternatives.
Step 2: The second validation step requires that the procedure follows the principle of transitivity.Transitivity in decision-making means that if alternative A is preferred over alternative B and alternative B is preferred over alternative C, then alternative A should also be preferred over alternative C.This step verifies that the model maintains logical consistency in its ranking process, ensuring that the preference order is coherent and rational.
Step 3: The third validation step tests the model's consistency when a given decision-making problem is divided into smaller subproblems.When applying the same decision-making process to these smaller problems, the initial ranking of the alternatives should remain unchanged.This step ensures that the model's rankings are stable and reliable, even when the problem is broken down into simpler, more manageable parts.
By adhering to these evaluation protocols, the proposed technique demonstrates its validity and reliability in various decisionmaking scenarios.These steps confirm that the model.i. Maintains the selection of the highest-rated alternative even when less-than-ideal ratings are replaced.ii.Follows the transitivity principle to maintain a coherent and rational preference order.iii.Preserves the initial ranking of alternatives, even when the problem is subdivided into smaller segments.
These tests of validity provide strong evidence that the proposed technique is both adaptable and robust, capable of delivering consistent and reliable results across different settings and scenarios.
Test of validity utilizing criteria 1: The alternatives ranked according to our proposed technique are To assess the method's stability under test criterion 1, we replaced the non-optimal alternative V 1 with the lowest-ranked alternative V 1 , with the best alternative remaining the same as in the initial suggested approach.Thus, the findings consistently support test criteria 1, demonstrating that replacing a less-than-ideal alternative with an inferior one does not affect the correspondence of the best alternative.
Test of validity employing criteria 2 and 3: To further assess the validity of our decision-making process based on criteria 2 and 3, we examined fragmented decision-making subcases.These subcases included specific sets of alternatives: Through our suggested procedures, we ranked these subcases as follows: For the subcase {V 1 , V 2 , V 3 }, the ranking order was V For the subcase {V 2 , V 3 , V 4 }, the ranking order was V 2 > V 4 > V 3 .For the subcase {V 1 , V 3 , V 4 }, the ranking order was V By joining all these conclusions, the complete position seems as This mirrors the results obtained from the initial decision-making method.This consistency demonstrates that our proposed approach satisfies requirements 2 and 3, ensuring logical consistency (transitivity) and stability when the problem is decomposed into smaller components.selecting the most suitable maintenance techniques for their equipment or systems.By evaluating various maintenance procedures, managers can enhance operational efficiency and prolong asset lifespan.Furthermore, the approach is useful in evaluating robots employed in industrial situations.It assists managers in evaluating the utility and application of various robotic systems, allowing them to make better decisions that improve production and operational efficiency.The model may also be used to choose material handling equipment.Managers may use the framework to make educated decisions on the most effective and productive equipment for their specific needs, resulting in smooth and efficient material handling operations.While the q-SFR Einstein operators provide a scientific and systematic approach to decision-making, it is essential to recognize that the process within this framework is substantially influenced by the preferences and judgments of the experts and individuals involved.The model offers a structured method, but the final decisions and rankings are shaped by the subjective assessments of the decision-makers.Therefore, the inclusion of experts and stakeholders is crucial to ensuring the accuracy and relevance of the findings.Two critical analyses are performed to validate and strengthen the obtained results.
Comparative analysis is a useful tool for decision-makers to review and compare rankings and results across several options, each evaluated using its own set of criteria.It raises awareness of trade-offs and facilitates better-informed decision-making by stressing the advantages and disadvantages of each option.Sensitivity analysis reveals important information on the stability and sensitivity of the results.Decision-makers can assess the multitude of factors influencing their decisions, thereby enhancing their capacity to make adaptable judgments in evolving contexts.Integrating this study into the decision-making process enables managers to enhance the reliability and confidence in their strategic decisions.The q-SFR Einstein operators, supported by comparative and sensitivity analyses, provide a comprehensive framework that equips managers across various sectors and applications with the tools needed to make informed and resilient choices.

Comparative analysis
Table 13 presents a comparative evaluation of rankings achieved through the utilization of the q-SFR Einstein operators in contrast to four other methods for multi-criteria decision-making (MCDM).A review of recent research publications reveals diverse  contributions to fuzzy set theory and decision-making processes.Khan et al. [58] propose a spherical fuzzy rough EDAS method for cache replacement policies that use Einstein aggregation operators, highlighting the necessity of innovative aggregation methodologies in enhancing caching strategies.On the other hand, Ashraf et al. [59] study multi-attribute decision-making settings, highlighting the flexibility and utility of spherical fuzzy sets.Furthermore, Ashraf, Abdullah, and Mahmood [60] widen the study of group decision-making challenges by presenting spherical fuzzy Dombi aggregation operators as a novel method.An examination of recent research articles indicates diverse contributions to fuzzy set theory and decision-making processes.Khan et al. [58] introduce a spherical fuzzy rough EDAS method for cache replacement policies that employ Einstein aggregation operators, emphasizing the need for novel aggregation approaches in improving caching strategies.In contrast, Ashraf et al. [59] examine multi-attribute decision-making scenarios, emphasizing the adaptability and usefulness of spherical fuzzy sets.Furthermore, Ashraf, Abdullah, and Mahmood [60] broadened their investigation into group decision-making issues by using spherical fuzzy Dombi aggregation operators as a new strategy.
Fig. 11 depicts the ranking comparison between various operators and the proposed operators.Amidst the propositions, calculations, and practical applications outlined above, the unique benefits of adopting q-SFRSs become evident, paving the way for enhanced decision-making.
1. Traditional fuzzy sets and intuitionistic fuzzy sets, while useful, sometimes fail to capture all important information in certain cases.Membership and non-membership degrees may limit decision-makers' ability to explain nuanced perspectives.2. To address these restrictions, Yager developed Pythagorean fuzzy sets, which expanded the extent of representation and enabled a wider range of applications.However, in the context of uncertain information, such as voting systems, the rigidity of picture fuzzy sets may be limiting, particularly in terms of decision-maker flexibility.4. The spherical fuzzy sets, offer a solution that gracefully navigates diverse information sets without exceeding the bounds of unity.This adaptability empowers decision-makers to allocate membership values according to their unique preferences.5.The incorporation of q-spherical fuzzy rough sets, along with associated algorithms, presents a versatile framework with farreaching implications across various decision-making processes.6.Furthermore, the proposed aggregation operators excel in handling imprecise information, offering a level of reliability that surpasses existing methodologies.7. The applicability of q-SFRSs spans a multitude of domains, including stock investment analysis, airline service quality evaluation, investment banking authority selection, and electronic learning factor assessment, underscoring their broad utility and relevance.8.By embracing the advantages inherent in q-spherical fuzzy rough sets, decision-makers are better equipped to traverse the intricate landscapes of decision-making with heightened confidence and precision.9. Regarding the specific concerns about the limitations of picture fuzzy rough sets (PFSRS) and spherical fuzzy rough sets, it's important to acknowledge that they are constrained by specific numerical bounds within their approximations.In contrast, qspherical fuzzy rough sets offer a broader scope of representation, allowing for a more nuanced handling of information sets.This distinction underscores the versatility and potential superiority of q-SFRS in handling complex decision-making scenarios.10.The q-SFRSs are more general than other algebraic structures because they incorporate lower and upper approximations with membership, neutral, and non-membership degrees.The inclusion of a q-parameter further enhances their robustness compared to PFSs and SFSs.This parameter allows for more flexible and nuanced modeling of uncertainty.Additionally, the merging of Einstein operators strengthens their robustness.
Fig. 12 represents some particular cases of q-SFRSs.

Sensitivity analysis
In this study, the developed model undergoes rigorous testing through two distinct sensitivity analyses, one of which examines variations in criteria and decision-making weights and their impact on final rankings.
1.The criteria weight analysis explores the effect of varying the weights of different criteria on the overall ranking of alternatives.It ensures that the model's outcomes are robust and not overly sensitive to changes in the criteria's importance.2. The decision-maker weight analysis examines the impact of different weight distributions assigned to decision-makers on the final rankings.By evaluating various scenarios, it ensures that the model remains consistent and reliable across different decisionweighting configurations.
By incorporating these analyses, managers can have greater confidence in the robustness and reliability of the decision-making process facilitated by the q-SFR Einstein operators, making it a valuable tool in strategic decision-making across various sectors.In conclusion, the q-SFR Einstein operators provide a flexible and reliable decision-making framework that can be applied across numerous industries, from automotive to industrial environments.Their ability to incorporate expert judgments and adapt to varying criteria and decision-maker preferences ensures that the results are both credible and applicable to real-world scenarios.This innovative approach helps managers make informed, strategic decisions that enhance operational efficiency, productivity, and overall success.The initial sensitivity analysis conducted a temporal examination to assess how varying priority levels assigned to reference criteria-high, equal, and low-affect overall ranking outcomes.This strategy requires running the model for each criteria individually and applying appropriate reference weights.Fig. 13 displays the results from twenty different scenarios.Surprisingly, in every example, option V 2 ranks first, whereas option V 1 always ranks last.Despite major changes in criteria weights, the model stays somewhat sensitive to these changes, demonstrating its ability to maintain consistent ranks across priority levels.The second sensitivity analysis adjusts the weights assigned to decision-makers, resulting in four distinct scenarios with varying weight distributions.Fig. 14 illustrates the outcomes of these scenarios.Throughout all configurations, alternative V 2 consistently emerges as the most preferred option, while alternative V 1 remains the least favored.Although the rankings of other alternatives fluctuate with different decision-maker weights, the proposed model demonstrates robustness and consistency across a broad spectrum of weight distributions.These sensitivity analyses underscore the model's reliability and robustness.The first analysis confirms that the model is largely insensitive to variations in criteria weights, consistently identifying the top and bottom alternatives.The second analysis demonstrates that the model remains stable and reliable even when the weights assigned to decision-makers are adjusted, reaffirming the preferred and least favored alternatives.This comprehensive verification process ensures that the model can be confidently applied in various decision-making scenarios, providing consistent and dependable results.

Advantages
The suggested approach has several key advantages.
1.By incorporating the parameter q into the aggregation operators, decision-makers benefit from enhanced flexibility.This adaptability enables customization of settings to suit the specific demands and preferences of each decision-making scenario.The technique's flexibility accommodates varying degrees of membership, neutral membership, and non-membership with lower and upper set approximations, making it suitable for diverse real-world applications.This customization ensures that the decisionmaking process accurately reflects the intricacies and complexities of different scenarios.2. The parametric structure of the proposed operators enables decision-makers to adjust the influence of membership, neutral membership, and non-membership levels along with lower and upper set approximations.This level of control is critical as it allows decision-makers to tailor the aggregation process to their preferences and the specific details of the scenario at hand.By fine-tuning these parameters, decision-makers can ensure that the aggregation process aligns perfectly with their preferred decision-making approach, leading to more precise and dependable results.3. The symmetry of the proposed aggregation operators with respect to the parameter ensures that the ranking orders of alternatives remain consistently stable across different parameter values.This consistency is a significant advantage in decision-making, as it mitigates the potential impact of decision-makers' inherent biases, whether pessimism or optimism.The stability of ranking orders across various parameter settings enhances the reliability of the decision-making process, instilling confidence that the results are robust and dependable, independent of subjective inclinations of decision-makers.
Overall, the proposed technique significantly improves the decision-making process by offering enhanced flexibility, precise control, and consistent reliability.By incorporating the parameter q, decision-makers can adapt the process to a wide variety of scenarios, ensuring accurate and trustworthy outcomes.This adaptability and robustness make the technique a valuable tool for effective decision-making across diverse applications.

Limitations
Every research project has inherent limitations, and the approach detailed in this study is no different.The following points outline these constraints.
1.The proposed technique may be restricted in its applicability to specific domains or decision contexts.While it offers significant flexibility and precision, its effectiveness and relevance might be limited outside the contexts for which it was designed.Understanding these boundaries is crucial for determining when and where the recommended strategy can be optimally utilized.Decision-makers should assess the suitability of this approach to the particularities of their domain to ensure it is the best fit for their needs.2. Like any research method, the proposed approach is built upon certain assumptions and simplifications that facilitate analysis.
These assumptions are necessary for creating a manageable and understandable model, but they may not perfectly align with realworld conditions.This misalignment could potentially limit the broader or practical applicability of the results.Decision-makers should be aware of these assumptions and consider their potential impact on the applicability of the findings in more complex or varied real-world scenarios.3. The validation of the proposed framework is based on a case study involving four alternatives and four criteria.While this case study provides initial evidence of the method's effectiveness, it is important to acknowledge that the framework's scalability and robustness need further testing.Future research should aim to expand the framework to incorporate more alternatives and criteria, thereby evaluating its performance and adaptability in more complex and diverse decision-making environments.4. The current study explores alternative ranking orders for several values of the parameter q.However, the investigation is not exhaustive, and additional research is necessary to fully understand how varying this parameter impacts the hierarchical order of alternatives.Further studies should focus on exploring a wider range of parameter values to provide a more comprehensive understanding of their effects.This would help in identifying the conditions under which the proposed method performs optimally and ensuring its robustness across different parameter settings.

Conclusion and recommendation for future work
In this paper, we explored aggregation operators, focusing on new q-SFREWA, q-SFREOWA, and q-SFREHWA operators for q-SFRSs.Clear operational laws are crucial in decision-making, and Einstein operators effectively incorporate experts' preferences.Our goal was to enhance decision-making using these operators, resulting in smoother and more effective processes.We presented and investigated the Einstein sum and Einstein product for q-SFRNs, resulting in numerous q-SFREWA, q-SFREOWA, and q-SFRHWA operators.These operators establish a framework that incorporates decision-makers' preferences.We also explored the relationships between different aggregation operators to gain a deeper understanding of their interactions.To show its practical utility, we suggested a new approach for multiple attribute group decision-making (MAGDM) that successfully solves MCDM.We evaluated our approach using a practical example of selecting a navigation system for automatic cars, performing a comparative study with existing methods, and a sensitivity analysis to confirm its effectiveness.Looking ahead, we plan to extend our framework to address fuzziness and ambiguity in various decision-making parameters, such as design choices, construction options, site selection, and decision-making in a soft-set environment.We will also explore other extensions of fuzzy sets, like quasirung fuzzy sets, which handle uncertainty and ambiguity effectively.Seikh and Mandal [61,62] work on MAGDM with quasirung orthopair fuzzy sets, particularly for electric vehicle charging station site selection, highlighting their potential in complex decisions.Integrating quasirung fuzzy sets into our framework is a promising direction for future research.Additionally, enhancing our literature review, historical background, and future work sections by incorporating recent advancements and seminal works will provide a more comprehensive context for our research.Hussain et al. [63,64] research on decision algorithms for picture fuzzy sets and Aczel Alsina aggregation operators, along with their innovative approach using Schweizer-Sklar aggregation operators for picture fuzzy sets with unknown weights, will significantly expand our study's scope.This integration ensures our work is well-grounded in existing knowledge while pushing the boundaries of current decision-making frameworks.

Fig. 10 .
Fig. 10.Ranking of alternatives with different q values.
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