RETRACTED ARTICLE: Confidence level based complex polytopic fuzzy Einstein aggregation operators and their application to decision-making process

Abstract

A complex Polytopic fuzzy set (CPoFS) extends a Polytopic fuzzy set (PoFS) by handling vagueness with degrees that range from real numbers to complex numbers within the unit disc. This extension allows for a more nuanced representation of uncertainty. In this research, we develop Complex Polytopic Fuzzy Sets (CPoFS) and establish basic operational laws of CPoFS. Leveraging these laws, we introduce new operators under a confidence level, including the confidence complex Polytopic fuzzy Einstein weighted geometric aggregation (CCPoFEWGA) operator, the confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (CCPoFEOWGA) operator, the confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (CCPoFEHGA) operator, the induced confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (I-CCPoFEOWGA) operator and the induced confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (I-CCPoFEHGA) operator, enhancing decision-making precision in uncertain environments. We also investigate key properties of these operators, including monotonicity, boundedness, and idempotency. With these operators, we create an algorithm designed to solve multiattribute decision-making problems in a Polytopic fuzzy environment. To demonstrate the effectiveness of our proposed method, we apply it to a numerical example and compare its flexibility with existing methods. This comparison will underscore the advantages and enhancements of our approach, showing its efficiency in managing complex decision-making scenarios. Through this, we aim to demonstrate how our method provides superior performance and adaptability across different situations.

Introduction

Decision-making across various fields like manufacturing, finance, and medicine often involves choosing among options based on multiple criteria, which complicates the process. When each option is evaluated on several criteria, an aggregation method is needed to derive a single value per option. Multi-attributes group decision-making (MAGDM) helps manage this complexity. The challenge is heightened by the inherent ambiguity in information, especially when it's binary (yes-or-no) rather than a spectrum. Traditional logic's binary nature does not accommodate the nuances found in real-world decision-making.

Zadeh¹ developed the notion of fuzzy sets (FSs) theory. Fuzzy sets assign degrees of membership to better handle imprecision and uncertainty in decision-making. This approach enhances analysis in applications like decision support and control systems. A key feature is the membership degree (MD), but it does not include a non-membership degree (NMD). For instance, if MD is 0.2, then its NMD would be calculated as 1 − 0.2 = 0.8. Thus, it offers a crucial means to manage uncertainty in real-world scenarios, enhancing decision support systems. Yet, its limitation arises when confronted with data containing both satisfactory and unsatisfactory information, posing challenges for effective processing and decision-making in such contexts. Atanassov² proposed intuitionistic fuzzy sets (IFSs) to overcome the limitations of traditional FSs in handling imprecise or uncertain information. IFSs offer a more comprehensive framework for managing such complexities, providing a broader scope for representing ambiguity and hesitation. In IFSs, each element is represented as a pair $\left( {\mu ,\hbar } \right)$, where $\mu$ and $\hbar$ are values indicating MD and NMD, respectively, with $\mu + \hbar \le 1$. Wang and Liu^3,4 and Tesic and Bozanic⁵, proposed the utilization of IFNs to introduce various operators and enhancing decision-making frameworks by effectively managing uncertainty. Yager⁵ expanded IFS to Pythagorean fuzzy sets (PyFSs), where each element is represented as a pair $\left( {\mu ,\hbar } \right)$ with $\mu^{2} + \hbar^{2} \le 1$. These extensions are designed to better handle uncertainty and hesitation with a more nuanced approach, enhancing the capabilities of IFS theory. Through PyFSs, the representation of uncertain and hesitant information becomes more adept, providing a richer framework for analysis and decision-making. Several researchers, like Garg^6,7, Chohan et al.⁸ and Rahman and Ali⁹ developed various operators for Pythagorean fuzzy numbers, enhancing their application in decision-making processes. These operators help in better handling uncertainty and imprecision in complex decision scenarios. Their work has significantly contributed to the field by providing more robust tools for evaluating and aggregating fuzzy information. Senapati and Yager¹⁰ introduced Fermatean fuzzy sets (FFSs) to overcome the limitations of PyFSs, specifically addressing the constraints like $\mu^{2} + \hbar^{2} \le 1$ to $\mu^{3} + \hbar^{3} \le 1$. They offer a more flexible approach for representing uncertainty, enhancing the capability to handle complex decision-making scenarios effectively. Khan and Wang¹¹ introduced several operators for Fermatean fuzzy numbers, significantly improving their utility in decision-making processes. These operators enhance the ability to handle uncertainty and imprecision more effectively. Their work broadens the application scope of Fermatean fuzzy numbers in complex decision scenarios. Yager¹² introduced q-rung orthopair fuzzy sets (q-ROFSs) to address limitations in FFSs, particularly the constraints like $\mu^{3} + \hbar^{3} \le 1$ to $\mu^{q} + \hbar^{q} \le 1$. These sets provide a flexible way to express varying degrees of membership and non-membership, offering a nuanced representation of uncertainty. The developed operators for q-ROFSs are essential for integrating information and understanding complex scenarios effectively.

In real-world voting scenarios, traditional models often fall short in accurately representing human opinions, especially when multiple response options like ‘yes’, ‘abstain’, ‘no’, and ‘refusal’ are involved. Consider an election where a candidate receives 1000 votes, divided into 'vote for' (502), 'abstain' (198), 'vote against' (240), and 'refusal' (60). Although the candidate wins with a majority, the decision-making complexity is evident as abstentions and refusals significantly impact the outcome. Traditional fuzzy set theories may struggle to handle such complexities, failing to account for the nuances of voter behavior, such as last-minute decisions influenced by perceived majority support. This highlights the need for more adaptive models that can better capture the intricacies of human decision-making in voting processes.

Cuong et al.¹³ introduced the concept of picture fuzzy sets (PcFSs) as an innovative approach to address existing challenges. Within the framework of PcFSs, each element is represented mathematically as $\left( {\mu ,\eta ,\hbar } \right)$ with $\mu + \eta + \hbar \le 1$ offering a distinct way to encapsulate information. Jana and Nunic¹⁴ created a linear programming model that operates within a picture fuzzy environment, integrating more nuanced uncertainty and ambiguity into decision-making processes. Ashraf et al.¹⁵ introduced spherical fuzzy sets (SpFSs) to overcome the limitations of PcFSs, specifically addressing the constraints like $\mu + \eta + \hbar \le 1$ to $\mu^{2} + \eta^{2} + \hbar^{2} \le 1$. They are used in decision-making processes and data analysis, providing a more nuanced approach by incorporating degrees of membership, non-membership, and hesitancy. This allows for more accurate modeling of uncertainty and ambiguity in complex systems. Sarfraz¹⁶ presented innovative techniques utilizing spherical fuzzy information. These methods aim to enhance decision-making processes and improve computational efficiency. Later on, Beg et al.¹⁷ introduced Polytopic fuzzy sets (PoFSs). In PoFSs each element can be presented mathematically under conditions as $\left( {\mu ,\eta ,\hbar } \right)$ with $0 \prec \mu^{q} + \eta^{q} + \hbar^{q} \le 1$. PoFSs emerge as a potent tool for navigating intricate decision-making scenarios characterized by the simultaneous consideration of multiple factors. Their inherent flexibility renders them invaluable across diverse applications, including decision support, control systems, and pattern recognition. These sets offer a robust framework that accommodates the complexities inherent in real-world decision contexts, allowing for a more nuanced and adaptable representation of information.

While the fuzzy models mentioned earlier excel in addressing uncertainty and vagueness within data, they do encounter limitations, notably in explicitly representing partial ignorance and tracking its evolution over time. This drawback poses a challenge, especially when dealing with periodic information, as none of the previously discussed fuzzy set theories and their extensions are adept at handling this temporal aspect. The intricacies of complex datasets, encompassing ambiguity, vagueness, and variations in periodicity, are prevalent in domains such as, image analysis, health assessment, biometric database analysis, and audio processing. These datasets, though rich in information, also introduce a significant challenge due to their vastness, emphasizing the need for advanced models that can effectively navigate and extract meaningful insights from such complex and dynamic data sources. Ramot et al.¹⁸ have pioneered a noteworthy development within fuzzy set theory known as complex fuzzy sets (CFSs). Unlike traditional fuzzy sets, CFSs offer a more intricate framework for managing uncertainty and ambiguity. This advancement provides a sophisticated means of representing complex relationships, enabling a more nuanced portrayal of uncertainty across different applications. In essence, CFSs enhance the capacity to handle uncertainty by offering a more advanced approach within the realm of fuzzy set theory. Alkouri and Salleh¹⁹ introduced complex intuitionistic fuzzy sets (CIFs) as an extension of intuitionistic fuzzy sets (IFSs) by integrating complex numbers. This innovation provides a more nuanced way to represent uncertainty in decision-making. CIFs enhance the traditional membership and non-membership degrees with complex numbers, offering a richer depiction of truth and falsity. This approach allows for capturing additional dimensions of uncertainty in various scenarios. Ahmed et al.²⁰, Rani and Garg²¹ and Ma et al.²² have significantly advanced our knowledge in this field with their distinct and valuable research contributions. Their work has provided unique insights and deepened our understanding of the subject. Ullah et al.²³ presented complex Pythagorean fuzzy sets (CPyFSs), advancing the concept of CIFSs. CPyFSs address and overcome several restrictions inherent in CIFSs. This innovation enhances the flexibility and applicability of fuzzy set theory. Researchers Rahman and Iqbal²⁴, Iqbal and Khan²⁵ and Rahman et al.²⁶ have developed a set of operators designed for CPyFNs. These operators improve the way information expressed by CPyFNs is combined, providing a more sophisticated approach for integrating uncertain and imprecise data. Liu et al.²⁷ introduced complex q-rung orthopair fuzzy sets (Cq-ROFSs), which offer a more flexible and adaptable framework for handling uncertainty and imprecision compared to traditional CPyFSs. Akram et al.²⁸ introduced the concept of complex picture fuzzy sets (CPcFSs), offering a more advanced and effective tool than previous models. This new approach enhances the ability to handle uncertainty and imprecision in data analysis. Naeem et al.²⁹ introduced complex spherical fuzzy sets (CSpFSs) to overcome the limitations found in CPcFSs. This new concept enhances the ability to handle uncertainty and vagueness in data more effectively. Choudhary et al.³⁰ developed some new operators utilizing CSpFNs and applied these operators to enhance decision-making processes. Their approach aims to improve the precision and effectiveness of decision-making in complex scenarios. Rahman et al.³¹ introduced complex Polytopic fuzzy sets (CPoFSs) and developed innovative methods utilizing algebraic operational laws. These advancements offer new approaches for handling and analyzing fuzzy data. Rahman and Jan^32,33 developed several algebraic aggregation operators tailored for complex Polytopic fuzzy information. These operators are designed to efficiently handle and process the intricacies of fuzzy data within Polytopic structures. Their work contributes to enhancing decision-making frameworks in environments characterized by uncertainty and complexity. No one introduce Einstein operators based on complex Polytopic fuzzy information under confidence level. This paper introduces Einstein norms, a novel concept based on complex Polytopic fuzzy information, as alternatives to traditional algebraic norms. It develops new operational laws for these norms and formulates innovative methods for data aggregation at different confidence levels. These methods aim to improve the accuracy and reliability of data aggregation.

Motivation of the study

Motivated by previous research^31,32,33 on algebraic aggregation operators, this paper introduces the Einstein t-norm and t-conorm, which offer greater flexibility and reliability than traditional norms. We develop several aggregation operators based on the Einstein t-norm and t-conorm under varying confidence levels. These new operators enhance decision-making processes by providing more precise adjustments and improved robustness.

Contributions of the study

This paper makes significant strides in CPoFSs by introducing innovative techniques, enhancing both the precision and the practical application of fuzzy decision-making methods. Each contribution enriches the field, advancing its methodologies.

• 1. Einstein aggregation operators under confidence level: The paper introduces a set of innovative techniques under confidence level, including CCPoFEWGA, CCPoFEOWGA, and CCPoFEHGA operators. Each operator is specifically designed with unique structural properties to enhance its adaptability and effectiveness in aggregating information.

• 2. Induced Einstein aggregation operators under confidence level: The paper introduces induced techniques under confidence level, namely I-CCPoFEOWGA and I-CCPoFEHGA operators. These operators are groundbreaking contributions to the field of aggregation theory, offering a novel approach to decision-making processes.

• 3. Development of decision-making algorithm: The key contribution of the paper is the development of algorithm. This algorithm provides a systematic and structured framework for applying the novel operators to real-world decision scenarios.

• 4. Demonstration of visibility and reliability in decision-making: The paper highlights the practicality of the new operators through extensive real-world applications, validating their effectiveness. This real-world usage provides valuable insights into their practical application, showcasing their reliability.

• 5. Illustrative example: The paper supports its contributions by showcasing the visibility of the recently introduced operators using a practical example. The example not only validates the proposed operators but also offers valuable insights into their practical use, thereby contributing significant knowledge to the field of decision science.

Organization of the study

The following sections of the paper will be structured as follows:

Section 2—This segment provides a detailed review of existing models related to our research, aiming to map out the current landscape and pinpoint areas needing enhancement.

Section 3—This section introduces CPoFSs and establishes fundamental operational laws for CPoFNs. Additionally, it presents several key results pertinent to CPoFSs.

Section 4—This section presents Einstein aggregation operators, which are tailored according to confidence levels. These operators are examined for fundamental properties such as monotonicity, boundedness, and idempotency. The focus is on understanding how these properties influence the behavior and application of the operators.

Section 5—This section introduces induced Einstein aggregation operators customized for specific confidence levels. It analyzes their essential traits like monotonicity, boundedness, and idempotency to grasp their operational nuances and applications effectively.

Section 6—This section introduces a model tailored for rapid emergency decision-making. It outlines its structure and components, offering clarity on its application in urgent scenarios demanding quick and efficient decisions.

Section 7—This section offers a concrete example to illustrate the practical application of the concepts discussed earlier, providing clarity and enhancing understanding. The example serves to demonstrate the relevance and applicability of the discussed methodologies in solving real-world problems, enriching the readers' comprehension and facilitating their ability to apply these concepts in similar contexts.

Section 8—This section provides a thorough examination of various models, highlighting their advantages and limitations. By pinpointing areas for enhancement, it offers valuable insights essential for furthering knowledge and practical applications in the field.

Section 9—This section emphasizes the advantages of newly introduced techniques, showcasing their improved effectiveness and precision in achieving the study's objectives. These innovations hold significant practical implications and open avenues for future research.

Section 10—This section succinctly presents the research findings, offering insights and conclusions derived from the study. It emphasizes the key outcomes and their implications, encapsulating the significance of the results in a concise manner.

Preliminaries

In this segment, we introduce fundamental concepts rooted in complex fuzzy information. Complex fuzzy set theory is an extension of the conventional fuzzy set theory, but it incorporates complex numbers. This expansion offers a more sophisticated means of representing and managing uncertainty and vagueness across diverse domains. The adaptability of complex fuzzy logic enables a more authentic portrayal of uncertain information, ultimately enhancing decision-making processes and system performance.

Definition 2.1

(Ramot et al.¹⁸) Let $\Re$ is a CFS, then mathematically, it can be written on a universal set ${\mathbb{R}}$ as:$\Re = \left\{ {\tau ,\mu_{\Re } (\tau )e^{{i\varsigma_{\Re } (\tau )}} \left| {\tau \in {\mathbb{R}}} \right.} \right\}$, where $\mu_{\Re } (\tau )$ is a function of complex number associated with each $\tau$ for all $\tau \in {\mathbb{R}}$.

Definition 2.2

(Alkouri and Salleh¹⁹) Let $L$ is a CIFS, then it can be expressed on a universal set ${\mathbb{R}}$ as: $L = \left\{ {\left. {\left\langle {\tau ,\mu_{L} (\tau )e^{{i\varsigma_{L} (\tau )}} ,\hbar_{L} (\tau )e^{{i\chi_{L} (\tau )}} } \right\rangle } \right|\tau \in {\mathbb{R}}} \right\}$, where, $\mu_{L} (\tau ):{\mathbb{R}} \to [0,1]$, $\hbar_{L} (\tau ):{\mathbb{R}} \to [0,1][0,1]$ $\varsigma_{L} (\tau ) \in [0,2\pi ]$, $\chi_{L} (u) \in [0,2\pi ]$ with $\mu_{L} \left( \tau \right)$, $\hbar_{L} \left( \tau \right)$ are complex valued functions associated with each element $\tau$ under conditions:$0 \prec \mu_{L} (\tau ) + \hbar_{L} (\tau ) \le 1$ and ${0} \prec \frac{{\varsigma_{L} \left( \tau \right)}}{2\pi }{ + }\frac{{\chi_{L} \left( \tau \right)}}{2\pi } \le 1$.

Definition 2.3

(Ullah et al.²³) Let ${\rm P}$ is a CPyFS, then mathematically, it can be expressed on a universal set ${\mathbb{R}}$ as: ${\rm P} = \left\{ {\left. {\left\langle {\tau ,\mu_{\rm P} (\tau )e^{{i\varsigma_{\rm P} (\tau )}} ,\hbar_{\rm P} (\tau )e^{{i\chi_{\rm P} (\tau )}} } \right\rangle } \right|\tau \in {\mathbb{R}}} \right\}$, where $\mu_{\rm P} (\tau ):{\mathbb{R}} \to [0,1]$, $\hbar_{\rm P} (\tau ):{\mathbb{R}} \to [0,1]$, $\varsigma_{\rm P} (\tau ) \in [0,2\pi ]$, $\chi_{\rm P} (\tau ) \in [0,2\pi ]$ with $\mu_{\rm P} \left( \tau \right)$, $\hbar_{\rm P} \left( \tau \right)$ are complex valued functions associated with each element $\tau$ in the universal set under conditions: $0 \prec \left( {\mu_{\rm P} (\tau )} \right)^{2} + \left( {\hbar_{\rm P} (\tau )} \right)^{2} \le 1$ and ${ 0} \prec \, \left( {\frac{{\varsigma_{\rm P} \left( \tau \right)}}{2\pi }} \right)^{2} { + }\left( {\frac{{\chi_{\rm P} \left( \tau \right)}}{2\pi }} \right)^{2} \le 1$.

Definition 2.4

(Liu et al.²⁷) Let $A$ is a Cq-OFS, then it can be expressed on a universal set ${\mathbb{R}}$ as: $A = \left\{ {\left. {\left\langle {\tau ,\mu_{A} (\tau )e^{{i\varsigma_{A} (\tau )}} ,\hbar_{A} (\tau )e^{{i\chi_{A} (\tau )}} } \right\rangle } \right|\tau \in {\mathbb{R}}} \right\}$, where $\mu_{A} \left( \tau \right):{\mathbb{R}} \to [0,1]$, $\hbar_{A} \left( \tau \right):{\mathbb{R}} \to [0,1]$$\varsigma_{A} (\tau ) \in [0,2\pi ]$, $\chi_{A} (\tau ) \in [0,2\pi ]$ with $\mu_{A} \left( \tau \right)$, $\hbar_{A} \left( \tau \right)$ are complex valued functions under conditions: $0 \prec \left( {\mu_{A} (\tau )} \right)^{q} + \left( {\hbar_{A} (\tau )} \right)^{q} \le 1$ and ${0} \prec \, \left( {\frac{{\varsigma_{A} \left( \tau \right)}}{2\pi }} \right)^{q} { + }\left( {\frac{{\chi_{A} \left( \tau \right)}}{2\pi }} \right)^{q} \le$ and $q \ge 1$.

Complex polytopic fuzzy model and their operations

In this segment, we delved into the concept of complex Polytopic fuzzy sets, exploring the score function and its fundamental operational laws. CPoFSs provide a versatile framework for handling complex, multivariate, and uncertain information, making them valuable in various fields where these characteristics are present simultaneously. Score functions play a crucial role in evaluating the performance and efficacy of fuzzy systems or models. These functions provide a quantitative measure of how well a fuzzy system or model is performing, helping to assess its effectiveness in handling the inherent complexities of uncertain or vague information. Essentially, score functions offer a means to gauge the success and reliability of fuzzy systems in scenarios where clear-cut distinctions may not be feasible. Operational laws play a pivotal role in providing a robust and flexible framework to navigate uncertainty and address complex systems. These laws contribute to a more nuanced comprehension of uncertainty, facilitating the modeling of intricate and dynamic real-world scenarios.

Definition 3.1

(Rahman³¹) Let $F$ is a CPoFS, then it can be expressed on a universal set ${\mathbb{R}}$ as: $F = \left\{ {\left. {\left\langle {\tau ,\mu_{F} (\tau )e^{{i\varsigma_{F} (\tau )}} ,\eta_{F} (\tau )e^{{i\xi_{F} (\tau )}} ,\hbar_{F} (\tau )e^{{i\chi_{F} (\tau )}} } \right\rangle } \right|\tau \in {\mathbb{R}}} \right\}$, where $\mu_{F} \left( \tau \right):{\mathbb{R}} \to [0,1]$ $\eta_{F} \left( \tau \right):{\mathbb{R}} \to [0,1]$,$\hbar_{F} \left( \tau \right):{\mathbb{R}} \to [0,1]$, $\varsigma_{{_{F} }} \left( \tau \right) \in [0,2\pi ]$, $\xi_{F} \left( \tau \right) \in [0,2\pi ]$,$\chi_{{_{F} }} \left( \tau \right) \in [0,2\pi ]$ with $\mu_{F} \left( \tau \right)$, $\eta_{F} \left( \tau \right)$, $\hbar_{F} \left( \tau \right)$ are complex valued functions associated with each element $\tau$ in the universal set with conditions, that is $0 \prec \left( {\mu_{F} \left( \tau \right)} \right)^{q} + \left( {\eta_{F} \left( \tau \right)} \right)^{q} + \left( {\hbar_{F} \left( \tau \right)} \right)^{q} \le 1 \, \left( {1 \le q} \right)$ and ${ 0} \prec \, \left( {\frac{{\varsigma_{F} \left( \tau \right)}}{2\pi }} \right)^{q} { + }\left( {\frac{{\xi_{F} \left( \tau \right)}}{2\pi }} \right)^{q} + \left( {\frac{{\chi_{F} \left( \tau \right)}}{2\pi }} \right)^{q} \le 1$. Furthermore, its hesitancy can be stated as

$\pi_{F} = \left( {1 - \left( {\left( {\mu_{F} \left( \tau \right)} \right)^{q} + \left( {\eta_{F} \left( \tau \right)} \right)^{q} + \left( {\hbar_{F} \left( \tau \right)} \right)^{q} } \right)} \right)^{\frac{1}{q}} e^{{\left( {1 - \left( {\left( {\varsigma_{F} \left( \tau \right)} \right)^{q} + \left( {\xi_{F} \left( \tau \right)} \right)^{q} + \left( {\chi_{F} \left( \tau \right)} \right)^{q} } \right)} \right)^{\frac{1}{q}} }}$ $\forall \tau \in {\mathbb{R}}$.

Definition 3.2

(Rahman³¹) Let $\rho = \left( {\mu e^{i\varsigma } ,\eta e^{i\xi } ,\hbar e^{i\chi } } \right)$ be a CPoFN, then its score and accuracy can be expressed as:$S\left( \rho \right) = \frac{1}{3}\left[ {\left( {1 + \mu^{q} + \eta^{q} - \hbar^{q} } \right) + \left( {1 + \varsigma^{q} + \xi^{q} - \chi^{q} } \right)} \right]$ with $S\left( \rho \right) \in [ - 2,2]$ and $A\left( \rho \right) = \frac{1}{2}\left[ {\left( {1 + \max \left( {\mu^{q} ,\eta^{q} } \right) - \hbar^{q} } \right) + \left( {1 + \max \left( {\varsigma^{q} ,\xi^{q} } \right) - \chi^{q} } \right)} \right]$ with $S\left( \rho \right) \in [ - 2,2]$ respectively.

Definition 3.3

(Rahman³¹) Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le 2} \right)$ are CPoFNs, then

• 1. If, $S\left( {\rho_{1} } \right) \succ S\left( {\rho_{2} } \right)$, this implies that $\rho_{1} \succ \rho_{2}$

• 2. If, $S\left( {\rho_{1} } \right) \prec S\left( {\rho_{2} } \right)$, this implies that $\rho_{1} \prec \rho_{2}$

• 3. If, $S\left( {\rho_{1} } \right) = S\left( {\rho_{2} } \right)$, then accuracy function can be used:

• i. If, $A\left( {\rho_{1} } \right) \succ A\left( {\rho_{2} } \right)$, this implies that $\rho_{1} \succ \rho_{2}$

• ii. If, $A\left( {\rho_{1} } \right) \prec A\left( {\rho_{2} } \right)$, this implies that $\rho_{1} \prec \rho_{2}$

Definition 3.4

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le 2} \right)$ be group of CPoFNs, and $\delta \succ 0$, then

• i. $\rho_{1} \oplus \rho_{2} = \left[ \begin{gathered} \tfrac{{\left( {\mu_{1}^{q} + \mu_{2}^{q} } \right)^{\frac{1}{q}} }}{{\left( {1 + \mu_{1}^{q} + \mu_{2}^{q} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\left( {\frac{{\varsigma_{1} }}{2\pi }} \right)^{q} + \left( {\frac{{\varsigma_{2} }}{2\pi }} \right)^{q} } \right)^{\frac{1}{q}} }}{{\left( {1 + \left( {\frac{{\varsigma_{1} }}{2\pi }} \right)^{q} \left( {\frac{{\varsigma_{2} }}{2\pi }} \right)^{q} } \right)^{\frac{1}{q}} }}} \right)}} ,\tfrac{{\eta_{1} \eta_{2} }}{{\left( {1 + \left( {1 - \eta_{1}^{q} } \right)\left( {1 - \eta_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\frac{{\xi_{1} }}{2\pi }} \right)\left( {\frac{{\xi_{2} }}{2\pi }} \right)}}{{\left( {1 + \left( {1 - \xi_{1}^{q} } \right)\left( {1 - \xi_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\hbar_{1} \hbar_{2} }}{{\left( {1 + \left( {1 - \hbar_{1}^{q} } \right)\left( {1 - \hbar_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\frac{{\chi_{1} }}{2\pi }} \right)\left( {\frac{{\chi_{2} }}{2\pi }} \right)}}{{\left( {1 + \left( {1 - \chi_{1}^{q} } \right)\left( {1 - \chi_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right]$

• ii. $\rho_{1} \otimes \rho_{2} = \left[ \begin{gathered} \tfrac{{\mu_{1} \mu_{2} }}{{\left( {1 + \left( {1 - \mu_{1}^{q} } \right)\left( {1 - \mu_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\frac{{\varsigma_{1} }}{2\pi }} \right)\left( {\frac{{\varsigma_{2} }}{2\pi }} \right)}}{{\left( {1 + \left( {1 - \varsigma_{1}^{q} } \right)\left( {1 - \varsigma_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\eta_{1} \eta_{2} }}{{\left( {1 + \left( {1 - \eta_{1}^{q} } \right)\left( {1 - \eta_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\frac{{\xi_{1} }}{2\pi }} \right)\left( {\frac{{\xi_{2} }}{2\pi }} \right)}}{{\left( {1 + \left( {1 - \xi_{1}^{q} } \right)\left( {1 - \xi_{2}^{q} } \right)} \right)^{\frac{1}{q}} }}} \right)}} ,\tfrac{{\left( {\hbar_{1}^{q} + \hbar_{2}^{q} } \right)^{\frac{1}{q}} }}{{\left( {1 + \hbar_{1}^{q} + \hbar_{2}^{q} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\left( {\frac{{\chi_{1} }}{2\pi }} \right)^{q} + \left( {\frac{{\chi_{2} }}{2\pi }} \right)^{q} } \right)^{\frac{1}{q}} }}{{\left( {1 + \left( {\frac{{\chi_{1} }}{2\pi }} \right)^{q} \left( {\frac{{\chi_{2} }}{2\pi }} \right)^{q} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right]$

• iii. $\left( \rho \right)^{\delta } = \left[ \begin{gathered} \tfrac{{\left( {2\left( {\mu^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \mu^{q} } \right)^{\delta } + \left( {\mu^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {\left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {\left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} ,\tfrac{{\left( {2\left( {\eta^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \eta^{q} } \right)^{\delta } + \left( {\eta^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {\left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {\left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\left( {1 + \hbar^{q} } \right)^{\delta } - \left( {1 - \hbar^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {1 + \hbar^{q} } \right)^{\delta } + \left( {1 - \hbar^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\left( {1 + \left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } - \left( {1 - \left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {1 + \left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {1 - \left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right]$

• iv. $\delta \left( \rho \right) = \left[ \begin{gathered} \tfrac{{\left( {\left( {1 + \mu^{q} } \right)^{\delta } - \left( {1 - \mu^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {1 + \mu^{q} } \right)^{\gamma } + \left( {1 - \mu^{q} } \right)^{\gamma } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\left( {1 + \left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } - \left( {1 - \left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {1 + \left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {1 - \left( {\frac{\varsigma }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\left( {\eta^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \eta^{q} } \right)^{\delta } + \left( {\eta^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {\left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {\left( {\frac{\xi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} ,\tfrac{{\left( {2\left( {\hbar^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \hbar^{q} } \right)^{\delta } + \left( {\hbar^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {\left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}{{\left( {\left( {2 - \left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } + \left( {\left( {\frac{\chi }{2\pi }} \right)^{q} } \right)^{\delta } } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right]$

Theorem 3.5

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le 3} \right)$ are CPoFNs, then

• i. $\rho_{1} \otimes \rho_{2} = \rho_{2} \otimes \rho_{1}$

• ii. $\rho_{1} \oplus \rho_{2} = \rho_{2} \oplus \rho_{1}$

• iii. $\left( {\rho_{1} \otimes \rho_{2} } \right) \otimes \rho_{3} = \rho_{1} \otimes \left( {\rho_{2} \otimes \rho_{3} } \right)$

• iv. $\left( {\rho_{1} \oplus \rho_{2} } \right) \oplus \rho_{3} = \rho_{1} \oplus \left( {\rho_{2} \oplus \rho_{3} } \right)$

• v. $\rho_{1} \otimes \left( {\rho_{2} \oplus \rho_{3} } \right) = \left( {\rho_{1} \otimes \rho_{2} } \right) \oplus \left( {\rho_{1} \otimes \rho_{3} } \right)$

• vi. $\left( {\rho_{1} \oplus \rho_{2} } \right) \otimes \rho_{3} = \left( {\rho_{1} \otimes \rho_{3} } \right) \oplus \left( {\rho_{2} \otimes \rho_{3} } \right)$

Proof

Given in the Supplementary Material A.

Theorem 3.6

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le 3} \right)$ be a group of CPoFNs, then

• i. $\rho_{1} \cup \rho_{2} = \rho_{2} \cup \rho_{1}$

• ii. $\rho_{1} \cap \rho_{2} = \rho_{2} \cap \rho_{1}$

• iii. $\rho_{1} \cup \rho_{2} \cup \rho_{3} = \rho_{1} \cup \rho_{3} \cup \rho_{2}$

• iv. $\rho_{1} \cap \rho_{2} \cap \rho_{3} = \rho_{1} \cap \rho_{3} \cap \rho_{2}$

Proof

We need to prove only (i, ii) and (iii, iv) can be proved by the same process.

• (i) As $\rho_{1} = \left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right)$ and $\rho_{2} = \left( {\mu_{2} e^{{i\varsigma_{2} }} ,\eta_{2} e^{{i\xi_{2} }} ,\hbar_{2} e^{{i\chi_{2} }} } \right)$ are CPoFNs, then

  $$ \begin{gathered} \rho_{1} \cup \rho_{2} \hfill \\ = \left( {\max \left\{ {\mu_{1} ,\mu_{2} } \right\}e^{{i\left( {\max \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\}} \right)}} ,\max \left\{ {\eta_{1} ,\eta_{2} } \right\}e^{{i\left( {\max \left\{ {\xi_{1} ,\xi_{2} } \right\}} \right)}} ,\min \left\{ {\hbar_{1} ,\hbar_{2} } \right\}e^{{i\left( {\min \left\{ {\chi_{1} ,\chi_{2} } \right\}} \right)}} } \right) \hfill \\ = \left( {\max \left\{ {\mu_{2} ,\mu_{1} } \right\}e^{{i\left( {\max \left\{ {\varsigma_{2} ,\varsigma_{1} } \right\}} \right)}} ,\max \left\{ {\eta_{2} ,\eta_{1} } \right\}e^{{i\left( {\max \left\{ {\xi_{2} ,\xi_{1} } \right\}} \right)}} ,\min \left\{ {\hbar_{2} ,\hbar_{1} } \right\}e^{{i\left( {\min \left\{ {\chi_{2} ,\chi_{1} } \right\}} \right)}} } \right) \hfill \\ { = }\rho_{2} \cup \rho_{1} \hfill \\ \end{gathered} $$

• (ii) Again, $\rho_{1} = \left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right)$ and $\rho_{2} = \left( {\mu_{2} e^{{i\varsigma_{2} }} ,\eta_{2} e^{{i\xi_{2} }} ,\hbar_{2} e^{{i\chi_{2} }} } \right)$ are CPoFNs, then

  $$ \begin{gathered} \rho_{1} \cap \rho_{2} \hfill \\ = \left( {\min \left\{ {\mu_{1} ,\mu_{2} } \right\}e^{{i\left( {\min \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\}} \right)}} ,\min \left\{ {\eta_{1} ,\eta_{2} } \right\}e^{{i\left( {\min \left\{ {\xi_{1} ,\xi_{2} } \right\}} \right)}} ,\max \left\{ {\hbar_{1} ,\hbar_{2} } \right\}e^{{i\left( {\max \left\{ {\chi_{1} ,\chi_{2} } \right\}} \right)}} } \right) \hfill \\ = \left( {\min \left\{ {\mu_{2} ,\mu_{1} } \right\}e^{{i\left( {\min \left\{ {\varsigma_{2} ,\varsigma_{1} } \right\}} \right)}} ,\min \left\{ {\eta_{2} ,\eta_{1} } \right\}e^{{i\left( {\min \left\{ {\xi_{2} ,\xi_{1} } \right\}} \right)}} ,\max \left\{ {\hbar_{2} ,\hbar_{1} } \right\}e^{{i\left( {\max \left\{ {\chi_{2} ,\chi_{1} } \right\}} \right)}} } \right) \hfill \\ { = }\rho_{2} \cap \rho_{1} \hfill \\ \end{gathered} $$

Thus, the proof is completed.

Theorem 3.7

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le 3} \right)$ be a group of CPoFNs, then

• i. $\left( {\rho_{1} \cap \rho_{2} } \right) \cup \rho_{1} = \rho_{1}$

• ii. $\left( {\rho_{1} \cup \rho_{2} } \right) \cap \rho_{1} = \rho_{1}$

• iii. $\left( {\rho_{1} \cap \rho_{2} } \right) \cup \rho_{2} = \rho_{2}$

• iv. $\left( {\rho_{1} \cup \rho_{2} } \right) \cap \rho_{2} = \rho_{2}$

Proof

We need to prove parts (i, ii), and (iii, iv) can be showed using the similar procedure.

• (i) As $\rho_{1} = \left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right)$ and $\rho_{2} = \left( {\mu_{2} e^{{i\varsigma_{2} }} ,\eta_{2} e^{{i\xi_{2} }} ,\hbar_{2} e^{{i\chi_{2} }} } \right)$ are CPoFNs, then

  $$ \begin{gathered} \left( {\rho_{1} \cap \rho_{2} } \right) \cup \rho_{1} \hfill \\ = \left( \begin{gathered} \min \left\{ {\mu_{1} ,\mu_{2} } \right\}e^{{i\left( {\min \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\}} \right)}} ,\min \left\{ {\eta_{1} ,\eta_{2} } \right\}e^{{i\left( {\min \left\{ {\xi_{1} ,\xi_{2} } \right\}} \right)}} ,\max \left\{ {\hbar_{1} ,\hbar_{2} } \right\}e^{{i\left( {\max \left\{ {\chi_{1} ,\chi_{2} } \right\}} \right)}} \hfill \\ \cup \left( {\mu_{1} e^{{i\tau_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right) \hfill \\ \end{gathered} \right) \hfill \\ { = }\left( \begin{gathered} \max \left\{ {\min \left\{ {\mu_{1} ,\mu_{2} } \right\},\mu_{1} } \right\}e^{{i\left( {\max \left\{ {\min \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\},\varsigma_{1} } \right\}} \right)}} ,\max \left\{ {\min \left\{ {\eta_{1} ,\eta_{2} } \right\},\eta_{1} } \right\}e^{{i\left( {\max \left\{ {\min \left\{ {\xi_{1} ,\xi_{2} } \right\},\xi_{1} } \right\}} \right)}} , \hfill \\ \min \left\{ {\max \left\{ {\hbar_{1} ,\hbar_{2} } \right\},\hbar_{1} } \right\}e^{{i\left( {\min \left\{ {\max \left\{ {\chi_{1} ,\chi_{2} } \right\},\chi_{1} } \right\}} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ { = }\left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right) = \rho_{1} \hfill \\ \end{gathered} $$

• (ii) Again, $\alpha_{1} = \left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\nu_{1} e^{{i\tau_{1} }} } \right)$ and $\alpha_{2} = \left( {\mu_{2} e^{{i\varsigma_{2} }} ,\eta_{2} e^{{i\xi_{2} }} ,\nu_{2} e^{{i\tau_{2} }} } \right)$ are CPoFNs, then

  $$ \begin{gathered} \left( {\rho_{1} \cup \rho_{2} } \right) \cap \rho_{1} \hfill \\ = \left( \begin{gathered} \max \left\{ {\mu_{1} ,\mu_{2} } \right\}e^{{i\left( {\max \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\}} \right)}} ,\max \left\{ {\eta_{1} ,\eta_{2} } \right\}e^{{i\left( {\max \left\{ {\xi_{1} ,\xi_{2} } \right\}} \right)}} ,\min \left\{ {\hbar_{1} ,\hbar_{2} } \right\}e^{{i\left( {\min \left\{ {\chi_{1} ,\chi_{2} } \right\}} \right)}} \hfill \\ \cap \left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\nu_{1} e^{{i\tau_{1} }} } \right) \hfill \\ \end{gathered} \right) \hfill \\ { = }\left( \begin{gathered} \min \left\{ {\max \left\{ {\mu_{1} ,\mu_{2} } \right\},\mu_{1} } \right\}e^{{i\left( {\min \left\{ {\max \left\{ {\varsigma_{1} ,\varsigma_{2} } \right\},\varsigma_{1} } \right\}} \right)}} ,\min \left\{ {\max \left\{ {\eta_{1} ,\eta_{2} } \right\},\eta_{1} } \right\}e^{{i\left( {\min \left\{ {\max \left\{ {\xi_{1} ,\xi_{2} } \right\},\xi_{1} } \right\}} \right)}} , \hfill \\ \max \left\{ {\min \left\{ {\hbar_{1} ,\hbar_{2} } \right\},\hbar_{1} } \right\}e^{{i\left( {\max \left\{ {\min \left\{ {\chi_{1} ,\chi_{2} } \right\},\chi_{1} } \right\}} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ { = }\left( {\mu_{1} e^{{i\varsigma_{1} }} ,\eta_{1} e^{{i\xi_{1} }} ,\hbar_{1} e^{{i\chi_{1} }} } \right) \hfill \\ = \rho_{1} \hfill \\ \end{gathered} $$

Einstein aggregation operators under confidence level

In this segment, we introduce a series of sophisticated Polytopic fuzzy Einstein operators designed around varying confidence levels. These operators, including the CCPoFEWGA, CCPoFEOWGA, and CCPoFEHGA, serve as essential mathematical tools for consolidating diverse data points, values, or information into a single representative value. Each operator is meticulously crafted with unique properties to cater to a broad spectrum of applications. We delve into several key characteristics of these innovative aggregation operators. Monotonicity ensures that the operators preserve the order of input data, maintaining consistency and reliability in their outputs. Boundedness defines the constraints within which the output values are confined, ensuring that results remain within specified limits and thereby enhancing the reliability of the data fusion process. Idempotency is another critical property, highlighting the operators' ability to produce consistent outcomes even when applied multiple times to the same data set, which is essential for iterative processes. These traits collectively enhance the efficacy and versatility of our proposed aggregation methods. The CCPoFEWGA, CCPoFEOWGA, and CCPoFEHGA operators are particularly valuable in scenarios demanding comprehensive data fusion and synthesis, such as multiattribute decision-making, where integrating various information sources is crucial. By employing these operators, we can achieve more accurate and reliable results, ultimately improving decision-making processes in complex and uncertain environments. Their robustness and adaptability make them indispensable tools in the realm of advanced fuzzy logic and decision analysis.

Definition 4.1

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ are CPoFNs, along with their weights $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)^{T}$ with $\kappa_{j} \in \left[ {0,1} \right]$ and $\sum\limits_{j = 1}^{n} {\kappa_{j} = 1}$. Let $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$, then the CCPoFEWGA operator mathematically, can be expressed as:

$$ \begin{gathered} {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) \hfill \\ = \left[ \begin{gathered} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$

Example 4.2

Let's construct an example with four CPoFNs, such as

$$ \begin{aligned} \rho_{1} & = \left\langle {\left( {0.70e^{{i2\pi \left( {0.50} \right)}} ,0.50e^{{i2\pi \left( {0.60} \right)}} ,0.80e^{{i2\pi \left( {0.70} \right)}} } \right),0.50} \right\rangle \\ \rho_{2} & = \left\langle {\left( {0.50e^{{i2\pi \left( {0.70} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.90e^{{i2\pi \left( {0.60} \right)}} } \right),0.60} \right\rangle \\ \rho_{3} & = \left\langle {\left( {0.40e^{{i2\pi \left( {0.40} \right)}} ,0.40e^{{i2\pi \left( {0.80} \right)}} ,0.60e^{{i2\pi \left( {0.70} \right)}} } \right),0.80} \right\rangle \\ \rho_{4} & = \left\langle {\left( {0.60e^{{i2\pi \left( {0.60} \right)}} ,0.50e^{{i2\pi \left( {0.70} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} } \right),0.70} \right\rangle \\ \end{aligned} $$

with weighted vector $\kappa = \left( {0.1,0.2,0.3,0.4} \right)$ and $q = 5$. First we need to compute the values below:

$$ \prod\limits_{j = 1}^{4} {\left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = } \left( {\left( {0.70} \right)^{5} } \right)^{0.1 \times 0.50} \left( {\left( {0.50} \right)^{5} } \right)^{0.2 \times 0.60} \left( {\left( {0.40} \right)^{5} } \right)^{0.3 \times 0.80} \left( {\left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} = 0.098 $$

$$ \begin{aligned} & \prod\limits_{j = 1}^{4} {\left( {2 - \mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \\ & = \left( {2 - \left( {0.70} \right)^{5} } \right)^{0.1 \times 0.50} \left( {2 - \left( {0.50} \right)^{5} } \right)^{0.2 \times 0.60} \left( {2 - \left( {0.40} \right)^{5} } \right)^{0.3 \times 0.80} \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.583}} \\ \end{aligned} $$

$$ \mathop \prod \limits_{j = 1}^{4} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {\left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {\left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.60} \left( {\left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.80} \left( {\left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.110}} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {2 - \left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} \hfill \\ = \left( {2 - \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {2 - \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.60} \left( {2 - \left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.80} \left( {2 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.575}} \hfill \\ \end{gathered} $$

$$ \prod\limits_{j = 1}^{4} {\left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = } \left( {\left( {0.50} \right)^{5} } \right)^{0.1 \times 0.50} \left( {\left( {0.60} \right)^{5} } \right)^{0.2 \times 0.60} \left( {\left( {0.40} \right)^{5} } \right)^{0.3 \times 0.80} \left( {\left( {0.50} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.078}} $$

$$ \begin{gathered} \prod\limits_{j = 1}^{4} {\left( {2 - \eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \hfill \\ = \left( {2 - \left( {0.50} \right)^{5} } \right)^{0.1 \times 0.50} \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.1 \times 0.60} \left( {2 - \left( {0.40} \right)^{5} } \right)^{0.1 \times 0.80} \left( {2 - \left( {0.50} \right)^{5} } \right)^{0.1 \times 0.70} = {1}{\text{.595}} \hfill \\ \end{gathered} $$

$$ \mathop \prod \limits_{j = 1}^{4} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {\left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {\left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {\left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.80} \left( {\left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.70} = {0}{\text{.300}} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {2 - \left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {2 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {2 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.60} \left( {2 - \left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.80} \left( {2 - \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ \, = {1}{\text{.497}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} \hfill \\ \left( {1 + \left( {0.80} \right)^{5} } \right)^{0.1 \times 0.50} \left( {1 + \left( {0.90} \right)^{5} } \right)^{0.2 \times 0.60} \left( {1 + \left( {0.60} \right)^{5} } \right)^{0.3 \times 0.80} \left( {1 + \left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.114}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {1 - \left( {0.80} \right)^{5} } \right)^{0.1 \times 0.50} \left( {1 - \left( {0.90} \right)^{5} } \right)^{0.2 \times 0.60} \left( {1 - \left( {0.60} \right)^{5} } \right)^{0.3 \times 0.80} \left( {1 - \left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ \, = {0}{\text{.844}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} \hfill \\ \left( {1 + \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {1 + \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.50} \left( {1 + \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {1 + \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.50} = {1}{\text{.064}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} \hfill \\ \left( {1 - \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.50} \left( {1 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.60} \left( {1 - \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.80} \left( {1 - \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.930}} \hfill \\ \end{gathered} $$

Next by using the CCPoFEWGA operator, we have

$$ \begin{gathered} {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,\rho_{4} } \right) \hfill \\ = \left[ \begin{subarray}{l} \tfrac{{\left( {2\left( {0.098} \right)} \right)^{\frac{1}{5}} }}{{\left( {1.583 + 0.098} \right)^{\frac{1}{5}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {0.110} \right)} \right)^{\frac{1}{5}} }}{{\left( {1.575 + 0.110} \right)^{\frac{1}{5}} }}} \right)}} ,\tfrac{{\left( {2\left( {0.078} \right)} \right)^{\frac{1}{5}} }}{{\left( {1.595 + 0.078} \right)^{\frac{1}{5}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {0.300} \right)} \right)^{\frac{1}{5}} }}{{\left( {1.597 + 0.300} \right)^{\frac{1}{5}} }}} \right)}} , \\ \tfrac{{\left( {1.114 - 0.844} \right)^{\frac{1}{5}} }}{{\left( {1.114 + 0.844} \right)^{\frac{1}{5}} }}e^{{i2\pi \left( {\tfrac{{\left( {1.064 - 0.930} \right)^{\frac{1}{5}} }}{{\left( {1.064 + 0.930} \right)^{\frac{1}{5}} }}} \right)}} \end{subarray} \right] \hfill \\ = \left[ {0.650e^{{i2\pi \left( {0.665} \right)}} ,0.622e^{{i2\pi \left( {0.794} \right)}} ,0.670e^{{i2\pi \left( {0.582} \right)}} } \right] \hfill \\ \end{gathered} $$

Theorem 4.3

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ are CPoFNs, and $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)^{T}$ be their weights. Let $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$, then their aggregated value under CCPoFEWGA operator remains still CPoFN, such that

$$ \begin{gathered} {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) = \hfill \\ \left[ \begin{gathered} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$

(1)

Proof

Given in the Supplementary Material B.

Property 4.4

(Idempotency) Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ be a group of CPoFNs, such that $\rho_{j} = \rho$ for all $j,$ and $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)^{T}$ be their weights, and $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level, then

$$ {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,...,\rho_{n} } \right) = \left( {\rho_{1} } \right)^{{\kappa_{1} \ell_{1} }} \otimes \left( {\rho_{2} } \right)^{{\kappa_{2} \ell_{2} }} \otimes ... \otimes \left( {\rho_{n} } \right)^{{\kappa_{n} \ell_{n} }} { = }\rho $$

(2)

Property 4.5

(Boundedness) Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ be a finite collection of CPoFNs, and $\ell_{j}$ be their confidence level. Let $\rho_{\max } = \left( {\mu_{\max } e^{{i\varsigma_{\max } }} ,\eta_{\max } e^{{i\xi_{\max } }} ,\hbar_{\max } e^{{i\chi_{\max } }} } \right).$ Additionally,$\mu_{\max } = \mathop {\max }\limits_{j} \left\{ {\mu_{j} } \right\}$$\varsigma_{\max } = \mathop {\max }\limits_{j} \left\{ {\varsigma_{j} } \right\},$ $\eta_{\max } = \mathop {\max }\limits_{j} \left\{ {\eta_{j} } \right\},$ $\xi_{\max } = \mathop {\max }\limits_{j} \left\{ {\xi_{j} } \right\}.$ $\hbar_{\max } = \mathop {\max }\limits_{j} \left\{ {\hbar_{j} } \right\},$ $\chi_{\max } = \mathop {\max }\limits_{j} \left\{ {\chi_{j} } \right\}.$ Again, $\rho_{\min } = \left( {\mu_{\min } e^{{i\varsigma_{\min } }} ,\eta_{\min } e^{{i\xi_{\min } }} ,\hbar_{\min } e^{{i\chi_{\min } }} } \right).$ Additionally, we have $\mu_{\min } = \mathop {\min }\limits_{j} \left\{ {\mu_{j} } \right\}$$\varsigma_{\min } = \mathop {\min }\limits_{j} \left\{ {\varsigma_{j} } \right\}$$\eta_{\min } = \mathop {\min }\limits_{j} \left\{ {\eta_{j} } \right\},$ $\xi_{\min } = \mathop {\min }\limits_{j} \left\{ {\xi_{j} } \right\},$ $\hbar_{\min } = \mathop {\min }\limits_{j} \left\{ {\hbar_{j} } \right\}$ and $\chi_{\min } = \mathop {\min }\limits_{j} \left\{ {\chi_{j} } \right\},$ then we have

$$ \rho_{\min } \le {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) \le \rho_{\max } $$

(3)

Property 4.6

(Monotonicity) Let there are two families of CPoFNs, such that $\left( {\rho_{j} ,\ell_{j} } \right)\left( {1 \le j \le n} \right)$ and $\left( {\rho_{j}^{ * } ,\ell_{j}^{ * } } \right) = \left( {1 \le j \le n} \right)$ satisfying the conditions, such as: $\mu_{j} \le \mu_{j}^{ * },$ $\varsigma_{j} \le \varsigma_{j}^{ * },$ $\eta_{j} \le \eta_{j}^{ * },$ $\xi_{j} \le \xi_{j}^{ * },$ $\hbar_{j} \ge \hbar_{j}^{ * }$ and $\chi_{j} \ge \chi_{j}^{ * },$ then

$$ {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) \le {\text{CCPoFEWGA}}_{\kappa } \left( {\rho_{1}^{ * } ,\rho_{2}^{ * } ,\rho_{3}^{ * } ,...,\rho_{n}^{ * } } \right) $$

(4)

Definition 4.7

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ is a group of CPoFNs, along with their weights $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)^{T}$ with $\kappa_{j} \in \left[ {0,1} \right]$ and $\sum\limits_{j = 1}^{n} {\kappa_{j} = 1}$. Let $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$. Moreover $\left( {\sigma \left( 1 \right),\sigma \left( 2 \right),...,\sigma \left( n \right)} \right)$ is any rearrangement of $\left( {1,2,...,n} \right)$ under inequality $\rho_{\sigma \left( j \right)} \le \rho_{{\sigma \left( {j - 1} \right)}}$, then CCPoFEOWGA operator, can be expressed as follows:

$$ \begin{gathered} {\text{CCPoFEOWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) = \hfill \\ \left[ \begin{subarray}{l} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \end{subarray} \right] \hfill \\ \end{gathered} $$

The CCPoFEHGA operator represents a versatile and sophisticated approach that blends the strengths of the CCPoFEWGA and CCPoFEOWGA operators. By integrating weighted geometric and ordered weighted geometric methodologies within the context of complex Polytopic fuzzy Einstein operations, this operator offers a unique and flexible aggregation mechanism. The CCPoFEHGA operator stands out for its ability to adapt to diverse decision-making scenarios, effectively handling varying preferences and requirements. This hybrid operator leverages geometric principles, ensuring a robust aggregation of complex Polytopic fuzzy information. It incorporates both weighted factors, which emphasize the importance of individual elements, and ordered factors, which account for the relative ranking of these elements. Its adaptability makes it an invaluable tool for decision-makers who need to synthesize complex, multi-faceted information. By ensuring that all relevant factors are considered and appropriately weighted, the CCPoFEHGA operator facilitates more accurate and informed decision-making processes. Ultimately, the CCPoFEHGA operator's ability to merge geometric principles with flexible weighting schemes results in a powerful aggregation tool. This operator not only enhances the precision and reliability of the aggregation process but also ensures that the resulting decisions are well-rounded and reflective of all pertinent information. As such, the CCPoFEHGA operator is a critical advancement in the field of complex Polytopic fuzzy decision-making, offering significant benefits for various applications and research endeavors.

Definition 4.8

Let $\rho_{j} = \left( {\mu_{j} e^{{i\varsigma_{j} }} ,\eta_{j} e^{{i\xi_{j} }} ,\hbar_{j} e^{{i\chi_{j} }} } \right)\left( {1 \le j \le n} \right)$ be a family of CPoFNs. Let their associated vector and weighted vectors are respectively presented by $\varphi = \left( {\varphi_{1} ,\varphi_{2} ,...,\varphi_{n} } \right)$, $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)$, and both vectors satisfy the conditions of belonging to the closed interval [0,1] and having a sum equal to one. Furthermore,$\mathop \rho \limits^{.}_{\sigma \left( j \right)} = n\varphi_{j} \rho_{j}$, where $\mathop \rho \limits^{.}_{\sigma \left( j \right)}$ be the maximum value in the equation is determined by the parameter n, which represents a positive number known as the balancing coefficient. This coefficient plays a crucial role in ensuring a balanced and effective equation by influencing the overall outcome and magnitude of the maximum value. If $\varphi = \left( {\varphi_{1} ,\varphi_{2} ,...,\varphi_{n} } \right)$ tends to $\left( {\frac{1}{n},\frac{1}{n},...,\frac{1}{n}} \right)$, then $\left( {n\varphi_{1} \rho_{1} ,n\varphi_{2} \rho_{2} ,...,n\varphi_{n} \rho_{n} } \right)$ tends to $\rho = \left( {\rho_{1} ,\rho_{2} ,...,\rho_{n} } \right)$. Let $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$, then mathematically, CCPoFEHGA operator can be expressed as:

$$ \begin{gathered} {\text{CCPoFEHGA}}_{\varphi ,\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,...,\rho_{n} } \right) = \hfill \\ \left[ \begin{gathered} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\upsilon_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$

Induced aggregation operators under confidence level

In this section, we introduce two novel induced aggregation operators, I-CCPoFEOWGA and I-CCPoFEHGA. These operators are designed as mathematical tools to integrate multiple inputs or criteria into a unified outcome, specifically tailored to capture and incorporate the unique characteristics of the given context. By utilizing these operators, analysts can effectively merge diverse data sets or criteria, thus enhancing the decision-making process with a more holistic perspective. The I-CCPoFEOWGA operator excels in weighted geometric averaging, providing a balanced approach to aggregating inputs while maintaining sensitivity to the importance of each criterion. Similarly, the I-CCPoFEHGA operator leverages harmonic averaging techniques, ensuring that the resulting aggregation reflects the intrinsic properties and relationships among the data points. These operators offer a structured methodology for synthesizing information, enabling a comprehensive and nuanced understanding of the factors involved. Their application ensures that the final decisions are well-informed and consider all relevant aspects, thus improving the precision and reliability of the outcomes. By integrating these advanced aggregation techniques, decision-makers can achieve a more accurate and insightful analysis, paving the way for superior decision-making in complex scenarios.

Definition 5.1

Let $\left( {\left\langle {u_{j} ,\rho_{j} } \right\rangle ,\ell_{j} } \right)\left( {1 \le j \le n} \right)$ be a group of 2-tuple of CPoFNs, $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$ and $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)$ be their corresponding weights such that $\kappa_{j} \in \left[ {0,1} \right]$ and $\sum\limits_{j = 1}^{n} {\rho_{j} = 1}$. Moreover, $u_{j} \in \left\langle {u_{j} ,\rho_{j} } \right\rangle$ be the ordered pair of CPoFOWGA having the j-th largest value is known as the order inducing variable, such as $u_{j} \in \left\langle {u_{j} ,\rho_{j} } \right\rangle$ and $\rho_{j}$ as the CPoF argument. Then the I-CCPoFEOWGA operator mathematically, can be expressed as:

$$ \begin{gathered} {\text{I - CCPoFEOWGA}}_{\kappa } \left( {\left( {\left\langle {u_{1} ,\rho_{1} } \right\rangle ,\ell_{1} } \right),\left( {\left\langle {u_{2} ,\rho_{2} } \right\rangle ,\ell_{2} } \right),...,\left( {\left\langle {u_{n} ,\rho_{n} } \right\rangle ,\ell_{n} } \right)} \right) \hfill \\ = \left[ \begin{gathered} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\mu_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\varsigma_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\eta_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\xi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \hbar_{\sigma \left( j \right)}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\chi_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$

Example 5.2

To enhance the above definition, consider the following example based on CPoFNs. Here, we illustrate this with specific CPoFNs, ensuring clarity and practical application in the context.

$$ \left( {\left\langle {u_{1} ,\rho_{1} } \right\rangle ,\ell_{1} } \right) = \left\langle {0.70,\left( {0.50e^{{i2\pi \left( {0.70} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.80e^{{i2\pi \left( {0.60} \right)}} } \right),0.50} \right\rangle $$

$$ \left( {\left\langle {u_{2} ,\rho_{2} } \right\rangle ,\ell_{2} } \right) = \left\langle {0.90,\left( {0.60e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.70e^{{i2\pi \left( {0.60} \right)}} } \right),0.60} \right\rangle $$

$$ \left( {\left\langle {u_{3} ,\rho_{3} } \right\rangle ,\ell_{3} } \right) = \left\langle {0.60,\left( {0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} ,0.40e^{{i2\pi \left( {0.80} \right)}} } \right),0.70} \right\rangle $$

$$ \left( {\left\langle {u_{4} ,\rho_{4} } \right\rangle ,\ell_{4} } \right) = \left\langle {0.80,\left( {0.40e^{{i2\pi \left( {0.80} \right)}} ,0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} } \right),0.80} \right\rangle $$

Let $\kappa = \left( {0.1,0.2,0.3,0.4} \right)$ be their weights and $q = 5$.

Based on the inducing variable, the values are ordered as follows. This arrangement reflects the hierarchy determined by the variable's influence, ensuring clarity and precision in the dataset's representation.$\left( {\left\langle {u_{2} ,\rho_{2} } \right\rangle ,\ell_{2} } \right) = \left\langle {0.90,\left( {0.60e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.70e^{{i2\pi \left( {0.60} \right)}} } \right),0.60} \right\rangle$

$\left( {\left\langle {u_{4} ,\rho_{4} } \right\rangle ,\ell_{4} } \right) = \left\langle {0.80,\left( {0.40e^{{i2\pi \left( {0.80} \right)}} ,0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} } \right),0.80} \right\rangle$

$\left( {\left\langle {u_{1} ,\rho_{1} } \right\rangle ,\ell_{1} } \right) = \left\langle {0.70,\left( {0.50e^{{i2\pi \left( {0.70} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.80e^{{i2\pi \left( {0.60} \right)}} } \right),0.50} \right\rangle$

$\left( {\left\langle {u_{3} ,\rho_{3} } \right\rangle ,\ell_{3} } \right) = \left\langle {0.60,\left( {0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} ,0.40e^{{i2\pi \left( {0.80} \right)}} } \right),0.70} \right\rangle$.

Next, we have the following ordering to consider.

$$ \left( {\left\langle {u_{\sigma \left( 1 \right)} ,\rho_{\sigma \left( 1 \right)} } \right\rangle ,\ell_{\sigma \left( 1 \right)} } \right) = \left\langle {0.90,\left( {0.60e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.70e^{{i2\pi \left( {0.60} \right)}} } \right),0.60} \right\rangle $$

$$ \left( {\left\langle {u_{\sigma \left( 2 \right)} ,\rho_{\sigma \left( 2 \right)} } \right\rangle ,\ell_{\sigma \left( 2 \right)} } \right) = \left\langle {0.80,\left( {0.40e^{{i2\pi \left( {0.80} \right)}} ,0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} } \right),0.80} \right\rangle $$

$$ \left( {\left\langle {u_{\sigma \left( 3 \right)} ,\rho_{\sigma \left( 3 \right)} } \right\rangle ,\ell_{\sigma \left( 3 \right)} } \right) = \left\langle {0.70,\left( {0.50e^{{i2\pi \left( {0.70} \right)}} ,0.60e^{{i2\pi \left( {0.60} \right)}} ,0.80e^{{i2\pi \left( {0.60} \right)}} } \right),0.50} \right\rangle $$

$$ \left( {\left\langle {u_{\sigma \left( 4 \right)} ,\rho_{\sigma \left( 4 \right)} } \right\rangle ,\ell_{\sigma \left( 4 \right)} } \right) = \left\langle {0.60,\left( {0.80e^{{i2\pi \left( {0.40} \right)}} ,0.60e^{{i2\pi \left( {0.50} \right)}} ,0.40e^{{i2\pi \left( {0.80} \right)}} } \right),0.70} \right\rangle $$

To proceed with further calculations, we need to first determine the following values:

$$ \prod\limits_{j = 1}^{4} {\left( {\mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = } \left( {\left( {0.60} \right)^{5} } \right)^{0.1 \times 0.60} \left( {\left( {0.40} \right)^{5} } \right)^{0.2 \times 0.80} \left( {\left( {0.50} \right)^{5} } \right)^{0.3 \times 0.50} \left( {\left( {0.80} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.179}} $$

$$ \begin{gathered} \prod\limits_{j = 1}^{4} {\left( {2 - \mu_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} } = \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.1 \times 0.60} \left( {2 - \left( {0.40} \right)^{5} } \right)^{0.2 \times 0.80} \left( {2 - \left( {0.50} \right)^{5} } \right)^{0.3 \times 0.50} \left( {2 - \left( {0.80} \right)^{5} } \right)^{0.4 \times 0.70} \, \hfill \\ \, = {1}{\text{.484}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {\left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {\left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {\left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80} \left( {\left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {\left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ { = 0}{\text{.134}} \hfill \\ \end{gathered} $$

$$ \begin{aligned} & \mathop \prod \limits_{j = 1}^{4} \left( {2 - \left( {\frac{{\varsigma_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {2 - \left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {2 - \left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80}\\ & \quad \left( {2 - \left( {\frac{0.70}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {2 - \left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.502}} \hfill \\ \end{aligned} $$

$$ \prod\limits_{j = 1}^{4} {\left( {\eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = } \left( {\left( {0.60} \right)^{5} } \right)^{0.1 \times 0.60} \left( {\left( {0.80} \right)^{5} } \right)^{0.2 \times 0.80} \left( {\left( {0.60} \right)^{5} } \right)^{0.3 \times 0.50} \left( {\left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.239}} $$

$$ \begin{gathered} \prod\limits_{j = 1}^{4} {\left( {2 - \eta_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = } \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.1 \times 0.60} \left( {2 - \left( {0.80} \right)^{5} } \right)^{0.2 \times 0.80} \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.3 \times 0.50} \left( {2 - \left( {0.60} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ \, = {1}{\text{.495}} \hfill \\ \end{gathered} $$

$$ \mathop \prod \limits_{j = 1}^{4} \left( {\left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {\left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {\left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80} \left( {\left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {\left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.106}} $$

$$ \begin{aligned} & \mathop \prod \limits_{j = 1}^{4} \left( {2 - \left( {\frac{{\xi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {2 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {2 - \left( {\frac{0.40}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80} \\ & \quad \left( {2 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {2 - \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.548}} \hfill \\ \end{aligned} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 + \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {1 + \left( {0.70} \right)^{5} } \right)^{0.1 \times 0.60} \left( {1 + \left( {0.60} \right)^{5} } \right)^{0.2 \times 0.80} \left( {1 + \left( {0.80} \right)^{5} } \right)^{0.3 \times 0.50} \left( {1 + \left( {0.40} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ \, = {1}{\text{.091}} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \mathop \prod \limits_{j = 1}^{4} \left( {1 - \hbar_{j}^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {1 - \left( {0.70} \right)^{5} } \right)^{0.1 \times 0.60} \left( {1 - \left( {0.60} \right)^{5} } \right)^{0.2 \times 0.80} \left( {1 - \left( {0.80} \right)^{5} } \right)^{0.3 \times 0.50} \left( {1 - \left( {0.40} \right)^{5} } \right)^{0.4 \times 0.70} \hfill \\ \, = {0}{\text{.889}} \hfill \\ \end{gathered} $$

$$ \begin{aligned} & \mathop \prod \limits_{j = 1}^{4} \left( {1 + \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {1 + \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {1 + \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80} \\ & \quad \left( {1 + \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {1 + \left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {1}{\text{.110}} \hfill \\ \end{aligned} $$

$$ \begin{aligned} & \mathop \prod \limits_{j = 1}^{4} \left( {1 - \left( {\frac{{\chi_{j} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} = \left( {1 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.1 \times 0.60} \left( {1 - \left( {\frac{0.50}{{2\pi }}} \right)^{5} } \right)^{0.2 \times 0.80} \\ & \quad \left( {1 - \left( {\frac{0.60}{{2\pi }}} \right)^{5} } \right)^{0.3 \times 0.50} \left( {1 - \left( {\frac{0.80}{{2\pi }}} \right)^{5} } \right)^{0.4 \times 0.70} = {0}{\text{.870}} \end{aligned} $$

Next, by using the I-CCPoFEWGA operator, we have

$$ \begin{gathered} {\text{I - CCPoFEWGA}}_{\kappa } \left( {\rho_{1} ,\rho_{2} ,\rho_{3} ,\rho_{4} } \right) \hfill \\ = \left[ \begin{gathered} \tfrac{{\left( {2\left( {0.179} \right)} \right)^{\frac{1}{4}} }}{{\left( {1.484 + 0.179} \right)^{\frac{1}{4}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {0.134} \right)} \right)^{\frac{1}{4}} }}{{\left( {1.502 + 0.134} \right)^{\frac{1}{4}} }}} \right)}} ,\tfrac{{\left( {2\left( {0.239} \right)} \right)^{\frac{1}{4}} }}{{\left( {1.495 + 0.239} \right)^{\frac{1}{4}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\left( {0.106} \right)} \right)^{\frac{1}{4}} }}{{\left( {1.548 + 0.106} \right)^{\frac{1}{4}} }}} \right)}} , \hfill \\ \tfrac{{\left( {1.091 - 0.889} \right)^{\frac{1}{4}} }}{{\left( {1.091 + 0.889} \right)^{\frac{1}{4}} }}e^{{i2\pi \left( {\tfrac{{\left( {1.110 - 0.870} \right)^{\frac{1}{4}} }}{{\left( {1.110 + 0.870} \right)^{\frac{1}{4}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ = \left[ {{0}{\text{.735}}e^{{i2\pi \left( {{0}{\text{.696}}} \right)}} ,{0}{\text{.701}}e^{{i2\pi \left( {{0}{\text{.663}}} \right)}} ,{0}{\text{.633}}e^{{i2\pi \left( {{0}{\text{.655}}} \right)}} } \right] \hfill \\ \end{gathered} $$

Definition 5.3

Let $\left( {\left\langle {u_{j} ,\rho_{j} } \right\rangle ,\ell_{j} } \right)\left( {1 \le j \le n} \right)$ be a group of 2-tuple of CPoFNs, and $\ell_{j} \left( {1 \le j \le n} \right)$ be their confidence level with $\ell \in \left[ {0,1} \right]$. Let their associated vector and weighted vectors are respectively presented by $\varphi = \left( {\varphi_{1} ,\varphi_{2} ,...,\varphi_{n} } \right)$, $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)$, and both vectors satisfy the conditions of belonging to the closed interval [0,1] and having a sum equal to one. Furthermore,$\mathop \rho \limits^{.}_{\sigma \left( j \right)} = n\varphi_{j} \rho_{j}$, where $\mathop \rho \limits^{.}_{\sigma \left( j \right)}$ be the maximum value in the equation is determined by the parameter n, which represents a positive number known as the balancing coefficient. This coefficient plays a crucial role in ensuring a balanced and effective equation by influencing the overall outcome and magnitude of the maximum value. If $\varphi = \left( {\varphi_{1} ,\varphi_{2} ,...,\varphi_{n} } \right)$ tends to $\left( {\frac{1}{n},\frac{1}{n},...,\frac{1}{n}} \right)$, then $\left( {n\varphi_{1} \rho_{1} ,n\varphi_{2} \rho_{2} ,...,n\varphi_{n} \rho_{n} } \right)$ tends to $\rho = \left( {\rho_{1} ,\rho_{2} ,...,\rho_{n} } \right)$, In the CPoFOWG system, the element in the ordered pair $u_{j} \in \left\langle {u_{j} ,\rho_{j} } \right\rangle$ that holds the jth largest value is identified as the order-inducing variable. This variable influences the overall ordering or prioritization within the system, contributing to the determination of significance or weight in the decision-making process. Then mathematically, I-CCPoFEHGA operator can be expressed as:

$$ \begin{gathered} {\text{I - CCPoFEHGA}}_{\varphi ,\kappa } \left( {\left\langle {u_{1} ,\rho_{1} } \right\rangle ,\left\langle {u_{2} ,\rho_{2} } \right\rangle ,...,\left\langle {u_{n} ,\rho_{n} } \right\rangle } \right) \hfill \\ = \left[ \begin{gathered} \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \mu \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \varsigma \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\mathop \eta \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {2\mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {2 - \left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {\left( {\frac{{\mathop \xi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} , \hfill \\ \tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\mathop \hbar \limits^{.}_{\sigma \left( j \right)} } \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}e^{{i2\pi \left( {\tfrac{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} - \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}{{\left( {\mathop \prod \limits_{j = 1}^{n} \left( {1 + \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} + \mathop \prod \limits_{j = 1}^{n} \left( {1 - \left( {\frac{{\mathop \chi \limits^{.}_{\sigma \left( j \right)} }}{2\pi }} \right)^{q} } \right)^{{\kappa_{j} \ell_{j} }} } \right)^{\frac{1}{q}} }}} \right)}} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$

An application of the proposed approaches

The decision-making process is a cognitive function central to human behavior, encompassing the act of choosing one course of action from multiple alternatives. Found in various life domains, from personal to professional spheres, it represents a pivotal moment of opting for one option over others. Trusting one's judgment is paramount in this process, yet it is equally crucial to remain open to seeking advice or input from others when needed. This fundamental aspect of human life requires a harmonious blend of rational analysis and intuitive judgment for effective decision-making. Adaptability and continuous improvement, informed by feedback and experience, are integral components that contribute to successful decision-making in both personal and professional contexts. In this research, the focus revolves around decision-making processes involving complex Polytopic fuzzy information. The study introduces various geometric techniques under confidence level, including the CCPoFEWGA operator, CCPoFEOWGA operator, CCPoFEHGA operator, I-CCPoFEOWGA operator, and I-CCPoFEHGA operator. These operators serve the purpose of aggregating results derived from the complex Polytopic fuzzy information, aiding in the identification and selection of the most optimal alternative. The investigation delves into the application of these operators to streamline decision-making and enhance the efficiency of the selection process. In this process, we consider a fixed group of m alternative,$A = \left\{ {A_{1} ,A_{2} ,...,A_{m} } \right\}$, a fixed group of n attributes/options $T = \left\{ {T_{1} ,T_{2} ,...,T_{n} } \right\}$ whose weighted vector is $\kappa = \left( {\kappa_{1} ,\kappa_{2} ,...,\kappa_{n} } \right)$ and a fixed group of k decision-makers $D = \left\{ {D_{1} ,D_{2} ,...,D_{k} } \right\}$ whose weighted vector is $\omega = \left( {\omega_{1} ,\omega_{2} ,...,\omega_{k} } \right)$.

The decision-making process typically involves the following steps.

• Step 1: During this phase, we articulate the insights provided by experts using a matrix that encompasses both attributes and alternatives. This representation can be outlined as follows:

  $$ \begin{gathered} \begin{array}{*{20}l} { \, T_{1} } \hfill & { \, T_{2} } \hfill & {\begin{array}{*{20}c} . & . & . \\ \end{array} } \hfill & { \, T_{n} } \hfill \\ \end{array} \hfill \\ D_{l} = \begin{array}{*{20}c} {A_{1} } \\ \begin{gathered} \hfill \\ A_{2} \hfill \\ \end{gathered} \\ {\begin{array}{*{20}c} . \\ . \\ . \\ \end{array} } \\ {A_{m} } \\ \end{array} \left( {\begin{array}{*{20}c} {\rho_{11}^{\left( k \right)} } & {\rho_{12}^{\left( k \right)} } & {\begin{array}{*{20}c} . & . & . \\ \end{array} } & {\rho_{1n}^{\left( k \right)} } \\ {\rho_{21}^{\left( k \right)} } & {\rho_{22}^{\left( k \right)} } & {\begin{array}{*{20}c} . & . & . \\ \end{array} } & {\rho_{2n}^{\left( k \right)} } \\ {\begin{array}{*{20}c} . \\ . \\ . \\ \end{array} } & {\begin{array}{*{20}c} . \\ . \\ . \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} . & . & . \\ \end{array} } \\ {\begin{array}{*{20}c} . & . & . \\ \end{array} } \\ {\begin{array}{*{20}c} . & . & . \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} . \\ . \\ . \\ \end{array} } \\ {\rho_{m1}^{\left( k \right)} } & {\rho_{m2}^{\left( k \right)} } & {\begin{array}{*{20}c} . & . & . \\ \end{array} } & {\rho_{mn}^{\left( k \right)} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

• Step 2: To create a unified decision matrix, aggregate all individual matrices by averaging the corresponding criteria scores from each matrix. This consolidated approach ensures a comprehensive and balanced evaluation, reflecting the collective input of all decision-makers.

• Step 3: Utilize all the suggested methods for computing preference values. By utilizing the collective decision matrix, you can systematically compute preference values, ensuring a comprehensive evaluation that incorporates diverse factors and considerations.

• Step 4: Calculating the score functions for all the preference values involves determining the numerical assessments or scores associated with each preference in a given context.

• Step 5: When ranking based on scores, the process involves assessing and ordering items according to their respective score values. The item with the highest score is then selected.

Illustrative example

In many developing nations, agriculture plays a crucial role in sustaining the economy. It provides employment for a significant portion of the population and generates essential income. As a backbone industry, it supports food security and drives economic growth. The agricultural sector plays a crucial role in the world for several reasons, contributing significantly to economic development, food security, employment, and overall well-being. Pakistan is recognized as one of the countries with a significant reliance on agriculture. The agricultural sector holds a pivotal position in Pakistan's economic landscape, serving as a major force in driving the country's development. Over the years, it has consistently stood out as one of the primary contributors to Pakistan's Gross Domestic Product (GDP) and has been a substantial source of employment, establishing itself as a vital pillar of the nation's economy. The significance of agriculture is evident in its substantial contribution to Pakistan's overall economic output, playing a crucial role in fostering and sustaining economic growth. The agricultural sector in Pakistan confronts several hurdles, including issues like water scarcity, reliance on outdated farming methods, and susceptibility to the impacts of climate change. In order to unlock the full economic potential of this sector, it is imperative for Pakistan to tackle these challenges head-on by implementing policy reforms, making substantial investments in infrastructure, and providing support to farmers for the adoption of contemporary farming techniques and technologies. Despite being an agrarian nation, Pakistan needs comprehensive measures to enhance the efficiency and sustainability of its agricultural practices, considering the diverse array of crops cultivated throughout the year. Hence, Pakistan aims to address the challenges outlined above and enhance the progress of its agricultural sector. To achieve this objective, the Pakistani government has entrusted a committee comprising four decision-makers, such as $D = \left\{ {D_{1} ,D_{2} ,D_{3} ,D_{4} } \right\}$ with the responsibility, whose weighted vector is $\omega = \left( {0.2,0.3,0.4,0.1} \right)$. Among numerous crops being evaluated, experts have narrowed their focus to four options: wheat, rice, barley, and sugarcane. By prioritizing these crops, experts aim to maximize agricultural productivity and economic viability.

• A₁: Wheat: Wheat holds significant importance in Pakistan's agricultural landscape and economic structure, serving as a key staple crop. The country stands among the leading global producers of wheat, emphasizing its pivotal role in sustaining both the agricultural sector and the overall economy.

• A₂: Rice: Rice plays a crucial role in Pakistan's agriculture and economy. It's a staple food for the majority of the population, and Pakistan is one of the world's largest producers and exporters of rice. Basmati rice, known for its aroma and long grains, is particularly famous and has a significant market globally.

• A₃: Sugarcane: Sugarcane holds considerable importance in Pakistan as a prominent crop, serving as a crucial component in the nation's agricultural sector. Not only is it a major source of income, functioning as a key cash crop, but it also plays a pivotal role in shaping the overall landscape of agriculture in the country.

• A₄: Barley: Cotton holds significant economic value for Pakistan, serving as a crucial cash crop and playing a vital role in the nation's textile sector. Renowned for its high-quality cotton production, Pakistan relies on this crop to drive its textile exports. The country's ability to cultivate and produce top-notch cotton has become a cornerstone of its textile industry, further bolstering its position in the global market.

The decision-makers evaluate the four alternatives based on four distinct criteria, whose weighted vector is given by $\kappa = \left( {0.2,0.2,0.2,0.4} \right)$.

• T₁: Market demand: Market demand assesses the level of consumer interest or need for a particular product or service. It measures how many people are willing to buy it. Understanding market demand helps businesses gauge potential sales and market viability.

• T₂: Investment cost: Investment cost encompasses both the initial setup expenses and ongoing operational costs required to initiate and sustain a project, serving as a crucial financial commitment essential for its launch and longevity.

• T₃: Economic importance: Economic importance looks at how a project or industry can affect the local or national economy. This includes evaluating how many jobs it can create and how much revenue it can generate. It assesses the overall contribution to economic growth.

• T₄: Water management system: The water management system evaluates how efficiently and sustainably water is used in the project. It aims to reduce environmental impact and conserve resources. The focus is on optimizing water usage to ensure long-term sustainability. The water management system evaluates how efficiently and sustainably water is used in the project. It aims to reduce environmental impact and conserve resources. The focus is on optimizing water usage to ensure long-term sustainability. By weighing these factors, decision-makers aim to choose the most viable and beneficial alternative.

By Einstein techniques

• Step 1: In Step 1, the data of various experts is organized into matrices, specifically Tables 1, 2, 3 and 4. These tables likely present information such as expertise areas, qualifications, experience, and possibly ratings or scores assigned to each expert's proficiency in certain domains. By structuring the data in this way, it becomes easier to analyze and compare the expertise of different individuals efficiently.

  Table 1 Assessment of expert $D_{1}$.

  Table 2 Assessment of expert $D_{2}$.

  Table 3 Assessment of expert $D_{3}$.

  Table 4 Assessment of expert $D_{4}$.

• Step 2: In this step, the process involved combining all individual matrices into a unified matrix using CCPoFEWGA technique, where $\omega = \left( {0.2,0.3,0.4,0.1} \right)$ and $q = 4$, shown on Table 5.

  Table 5 Collective assessment of all experts.

• Step 3: Next, again using the CCPoFEWGA operator, with $\kappa = \left( {0.2,0.2,0.2,0.4} \right)$, and get the following preference values:

  $$ y_{1} = \left( {0.74e^{{i2\pi \left( {0.62} \right)}} ,0.85e^{{i2\pi \left( {0.64} \right)}} ,0.70e^{{i2\pi \left( {0.61} \right)}} } \right),y_{2} = \left( {0.69e^{{i2\pi \left( {0.58} \right)}} ,0.86e^{{i2\pi \left( {0.66} \right)}} ,0.64e^{{i2\pi \left( {0.70} \right)}} } \right) $$
  $$ y_{3} = \left( {0.75e^{{i2\pi \left( {0.66} \right)}} ,0.70e^{{i2\pi \left( {0.77} \right)}} ,0.78e^{{i2\pi \left( {0.71} \right)}} } \right),y_{4} = \left( {0.74e^{{i2\pi \left( {0.51} \right)}} ,0.78e^{{i2\pi \left( {0.66} \right)}} ,0.81e^{{i2\pi \left( {0.58} \right)}} } \right) $$

• Step 4: Calculating the score functions of all preference values.

  $$ S\left( {y_{1} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.74} \right)^{4} + \left( {0.85} \right)^{4} - \left( {0.70} \right)^{4} } \right) + \left( {1 + \left( {0.62} \right)^{4} + \left( {0.64} \right)^{4} - \left( {0.61} \right)^{4} } \right)} \right] = 0.92 $$
  $$ S\left( {y_{2} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.69} \right)^{4} + \left( {0.86} \right)^{4} - \left( {0.64} \right)^{4} } \right) + \left( {1 + \left( {0.58} \right)^{4} + \left( {0.66} \right)^{4} - \left( {0.70} \right)^{4} } \right)} \right] = 0.88 $$
  $$ S\left( {y_{3} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.75} \right)^{4} + \left( {0.70} \right)^{4} - \left( {0.78} \right)^{4} } \right) + \left( {1 + \left( {0.66} \right)^{4} + \left( {0.77} \right)^{4} - \left( {0.71} \right)^{4} } \right)} \right] = 0.86 $$
  $$ S\left( {y_{4} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.74} \right)^{4} + \left( {0.78} \right)^{4} - \left( {0.81} \right)^{4} } \right) + \left( {1 + \left( {0.51} \right)^{4} + \left( {0.67} \right)^{4} - \left( {0.58} \right)^{4} } \right)} \right] = 0.80 $$

• Step 5: Thus, the best option is $A_{1}$.

By induced Einstein techniques

Next, we consider the above data and using the induced aggregation operators, which provide flexibility in aggregating diverse types of information.

• Step 1: Again make matrices based on the inducing variable, shown on Tables 6, 7, 8 and 9.

  Table 6 Assessment of expert $D_{1}$ under inducing variable.

  Table 7 Assessment of expert $D_{2}$ under inducing variable.

  Table 8 Assessment of expert $D_{3}$ under inducing variable.

  Table 9 Assessment of expert $D_{4}$ under inducing variable.

• Step 2: Combine all the individual matrices into a unified matrix by employing the I-CCPoFEOWGA operator, where $\omega = \left( {0.2,0.3,0.4,0.1} \right)$ and $\, q = 4$, shown on Table 10.

  Table 10 Collective assessment of all experts under inducing variable.

• Step 3: Next, again by using the I-CCPoFEOWGA operator, with $\kappa = \left( {0.2,0.2,0.2,0.4} \right)$ and get the following preference values:

  $$ y_{1} = \left( {0.74e^{{i2\pi \left( {0.67} \right)}} ,0.72e^{{i2\pi \left( {0.49} \right)}} ,0.73e^{{i2\pi \left( {0.58} \right)}} } \right),y_{2} = \left( {0.73e^{{i2\pi \left( {0.67} \right)}} ,0.72e^{{i2\pi \left( {0.68} \right)}} ,0.70e^{{i2\pi \left( {0.69} \right)}} } \right) $$
  $$ y_{3} = \left( {0.82e^{{i2\pi \left( {0.64} \right)}} ,0.69e^{{i2\pi \left( {0.51} \right)}} ,0.75e^{{i2\pi \left( {0.67} \right)}} } \right),y_{4} = \left( {0.85e^{{i2\pi \left( {0.68} \right)}} ,0.68e^{{i2\pi \left( {0.49} \right)}} ,0.80e^{{i2\pi \left( {0.79} \right)}} } \right) $$

• Step 4: Computing the score functions are as follows.

  $$ Sc\left( {r_{1} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.74} \right)^{4} + \left( {0.72} \right)^{4} - \left( {0.73} \right)^{4} } \right) + \left( {1 + \left( {0.67} \right)^{4} + \left( {0.49} \right)^{4} - \left( {0.58} \right)^{4} } \right)} \right] = 0.83 $$
  $$ Sc\left( {r_{2} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.73} \right)^{4} + \left( {0.72} \right)^{4} - \left( {0.70} \right)^{4} } \right) + \left( {1 + \left( {0.67} \right)^{4} + \left( {0.68} \right)^{4} - \left( {0.69} \right)^{4} } \right)} \right] = 0.81 $$
  $$ Sc\left( {r_{3} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.82} \right)^{4} + \left( {0.69} \right)^{4} - \left( {0.75} \right)^{4} } \right) + \left( {1 + \left( {0.64} \right)^{4} + \left( {0.51} \right)^{4} - \left( {0.67} \right)^{4} } \right)} \right] = 0.79 $$
  $$ Sc\left( {r_{4} } \right) = \frac{1}{3}\left[ {\left( {1 + \left( {0.85} \right)^{4} + \left( {0.68} \right)^{4} - \left( {0.80} \right)^{4} } \right) + \left( {1 + \left( {0.68} \right)^{4} + \left( {0.49} \right)^{4} - \left( {0.79} \right)^{4} } \right)} \right] = 0.73 $$

• Step 5: Thus, the best alternative is $A_{1}$.

The following Tables 11 and 12, show the score functions of various methods and Fig. 1 show their ranking based on the score function.

Table 11 Scores functions.

Table 12 Score functions of various approaches.

Figure 1

Ranking of all methods.

Comparative sensitivity analysis

Complex Polytopic fuzzy sets are more successful extension of fuzzy sets that allow for the modeling of complex, higher-dimensional fuzzy information. These sets are particularly useful in applications where decision-making, control, and pattern recognition require the handling of complex and multidimensional data. They focus on complex numbers, which can represent both magnitude and phase of uncertainty. Therefore, in this section, we compared the proposed model with some of their existing models, namely FSs, IFSs, PyFSs, CFSs, CIFSs, and CPyFSs. FSs represent degree of membership for elements in a set, typically ranging between 0 and 1. They do not inherently handle non-membership degrees. IFSs extend traditional FSs by introducing additional degrees called non-membership and hesitation. They are based on real numbers and do not handle complex numbers. PyFSs extend IFSs by introducing both membership and non-membership degrees, allowing for a more nuanced representation of uncertainty. They are based on real numbers and do not inherently handle complex numbers. CFSs extend traditional FSs by allowing the use of complex numbers for membership degrees, which can represent both magnitude and phase. They can represent uncertainties with complex values. They do not inherently handle non-membership degrees. CIFSs combine the concepts of complex numbers and hesitation degrees found in IFSs. They can represent complex uncertainties while accounting for hesitation. They are useful in scenarios with complex and hesitant information. CPyFSs can represent complex membership and non-membership degrees, providing a more comprehensive representation of uncertainty compared to traditional sets. Complex Polytopic fuzzy sets may be used for Polytopic fuzzy data by taking the phase terms zero. Moreover, it can be used for q-Rung orthopair fuzzy data by setting their neutral and phase terms zero. Thus, our proposed model is more elastic and flyable as compared to their existing models (Table 13).

Table 13 Comparative and sensitivity analysis.

Advantages and benefit of novel methods

• 1) Our proposed model, such as CPoFSs generalizes multiple fuzzy set models, including IFSs, PyFSs, FFSs, Q-ROFSs, PcFSs, SpFSs, PoFSs, CIFSs, CPyFSs, and CQ-ROFSs. This comprehensive framework offers enhanced flexibility and precision in representing uncertainty and ambiguity.

• 2) The existing methods under IFSs, PyFSs, FFSs, and q-ROFSs can be viewed as special cases of our novel proposed methods by setting both the neutral term and the phase term to zero. This simplification demonstrates the versatility of our novel operators.

• 3) The existing methods under SpFSs and PoFSs become special cases of our novel methods when the phase term is set to zero. This simplification highlights the versatility and generality of our proposed operators, demonstrating their broader applicability across various fuzzy set models.

• 4) The existing methods under CIFSs and CPyFSs are special cases of our proposed methods, achievable by setting the neutral term to zero. This generalization demonstrates the versatility and broader applicability of our new aggregation methods.

• 5) Based on the comparisons and analysis, our methods outperform existing techniques for aggregating CPoFNs. Consequently, they are better suited for addressing MPDA problems, offering enhanced accuracy and effectiveness

Conclusions and implications

Confidence levels quantify how certain we can be about statistical results, ensuring their reliability and accuracy. This article leverages complex Polytopic fuzzy information within a confidence framework, introducing innovative operations to create complex Polytopic fuzzy Einstein operators. These new operators, including CCPoFEWGA, CCPoFEOWGA, CCPoFEHGA, I-CCPoFEOWGA, and I-CCPoFEHGA, enhance the flexibility and precision of data aggregation. The distinctive features and benefits of these operators are discussed, demonstrating their advanced capabilities in managing complex data sets. These operators significantly improve decision-making processes in uncertain and fuzzy environments under confidence. We then developed an algorithm to tackle MADM problems within a complex Polytopic fuzzy framework under confidence, employing the CCPoFEWGA and I-CCPoFEOWGA operators. Our results highlight the algorithm's exceptional capacity to handle complex and uncertain data with confidence, greatly enhancing decision-making accuracy and effectiveness. This improvement indicates the algorithm's potential for wider use in fields that need precise and efficient decision-making tools.

Implications of the study

CPoFSs provide a nuanced representation for systems with multiple and overlapping characteristics, enhancing decision-making in complex environments.

Vital points of the study

The research highlights the critical importance of interdisciplinary collaboration for solving complex problems and underscores the necessity of continuous adaptation to emerging technologies and methodologies.

Limitations of the study

CPoFSs can be computationally intensive and difficult to interpret due to their intricate structure and high-dimensional data representation.

Further research direction of the study

In the future, our aim is to implement the suggested complex Polytopic fuzzy Einstein operators in various practical applications, expanding beyond their current scope. This includes applying these approaches to real-world scenarios such as medical diagnosis, pattern recognition, machine learning, and the detection of brain hemorrhages. These applications will involve adapting the techniques to operate effectively in environments that incorporate fuzzy logic extensions, enhancing their versatility and performance in diverse settings.