Decision-making using quasirung orthopair fuzzy exponential aggregation operators for carbon capture technology selection

Abstract

The selection of optimal carbon capture technologies is paramount in enhancing the efficiency of carbon emission mitigation efforts. Due to the multifaceted nature of the influencing factors, a robust and systematic approach is essential for identifying the most effective solution. The present research introduces an innovative fuzzy multi-criteria decision-making framework developed to address these types of challenges. In the first phase, we introduce novel operational laws for $p,q$-quasirung orthopair fuzzy ($p,q$-QOF) sets, thoroughly exploring their fundamental properties. Based on these operational laws, a set of advanced aggregation operators is developed, including the $p,q$-Quasirung Orthopair Fuzzy Weighted Exponential Averaging ($p,q$-QOFWEA) operator and its dual counterpart, the ${\text{D}}_{(p,q)}$ QOFWEA operator, which significantly enhance decision-making capabilities. In the second phase, we extend the traditional entropy method to the $p,q$-QOF context for the determination of criteria weights and provide a comprehensive outline of the decision-making process. The effectiveness of the proposed approach is demonstrated through its application to a real-world case study focused on the selection of suitable carbon capture technologies. Numerical results highlight the superiority of the proposed method, yielding a prioritized list of carbon capture technologies with practical relevance to modern applications. This work offers a novel contribution by introducing the $p,q$-QOF framework and showcasing its potential for addressing complex decision-making problems in environmental technology selection.

Introduction

Carbon capture technologies are crucial for mitigating climate change by capturing and storing atmospheric greenhouse gases. Various technologies exist, including post-combustion capture, pre-combustion capture, and oxy-fuel combustion. Selecting the most effective technology for a country is vital, as it enhances the fight against carbon emissions, improves financial performance, and ensures investment sustainability. However, choosing the wrong technology can lead to increased costs, reduced profitability, and decreased performance. Each technology has distinct advantages under different conditions, making proper selection critical. Incorrect choices can hinder innovation, reduce competitiveness, and cause environmental problems, ultimately eroding social trust. Therefore, careful consideration is necessary to select the optimal carbon capture technology, balancing economic, environmental, and social factors^1,2. When determining the most suitable carbon capture technology for investments, various issues must be considered. One crucial factor is geographical conditions, which significantly influence the effectiveness of different technologies. For instance, certain technologies may underperform in cold climates, highlighting the importance of considering regional specifics. Another vital consideration is the geological structure of a country, which determines the availability of suitable storage sites for captured carbon. This information is critical for assessing the feasibility of carbon capture technology implementation³. Legal frameworks and regulatory requirements also play a pivotal role in the selection and implementation of carbon capture technologies. The adequacy of these regulations impacts the development and applicability of various technologies. In some cases, legal rules may restrict the use of specific technologies, while also outlining government support and incentives for investments. Notably, certain carbon capture technologies may be eligible for specific subsidies, which should be considered when determining the most suitable technology. Furthermore, the adequacy of technological infrastructure is critical in determining the optimal carbon capture technology. An existing technological infrastructure of a country provides valuable insights into the suitability and potential productivity of various technologies. This information enables more informed decision-making regarding the most appropriate technology for investment⁴.

Accurate identification of carbon capture technology is crucial for a country’s successful fight against carbon emissions, facilitating the attainment of sustainable development goals. However, this selection process is complex and requires careful consideration of various factors, including geographical conditions, technological infrastructure, and legal frameworks. Existing literature highlights the importance of these variables it often lacks country-specific analyses and clarity regarding their relative significance. A comprehensive study is necessary to determine the most effective carbon capture technology, addressing the knowledge gap in variable prioritization and providing policymakers and investors with valuable insights for strategic decision-making (DM)^5,6. By identifying the most critical factors influencing technological efficiency, this research will enable informed decision-making, ultimately contributing to a more effective carbon emission mitigation and environmental sustainability.

Fuzzy sets (FSs), introduced by Zadeh in 1965⁷, are a mathematical concept representing vague or uncertain information, allowing partial membership and enabling the modeling of imprecise data. Characterized by partial membership, vagueness, non-crisp boundaries, and linguistic variables, FSs have numerous applications in DM, mainly in situations involving uncertainty, multiple criteria, linguistic data, and non-linear relationships. Techniques like fuzzy multi-criteria decision-making (MCDM), fuzzy logic control, decision support systems, and fuzzy analytic hierarchy process facilitate decision analysis in fields such as credit risk assessment^8,9,10, investment portfolio optimization^11,12,13, supply chain management^14,15, medical diagnosis^16,17,18, and traffic control^19,20,21. By handling uncertainty and imprecision, incorporating linguistic variables, and providing flexible decision models, FSs support robust DM and human-like reasoning, making them a valuable tool in various real-world applications. Fuzzy sets have several extensions, including: intuitionistic fuzzy sets (IFS), introduced by Atanassov²², extend fuzzy sets by incorporating both membership (MD) and non-membership degrees (NMD), with the condition that their sum is less than or equal to 1. Pythagorean fuzzy sets (PFSs), proposed by Yager^23,24, generalize IFS by allowing MD and NMD to be independent, satisfying the Pythagorean theorem. $q$-rung orthopair fuzzy sets ($q$-ROFSs), developed by Yager²⁵, generalize the terms level with a parameter $q$, allowing for flexible control over the roughness and fuzziness. $q$-ROFSs effectively handle uncertain and incomplete information. $p,q$-quasirung orthopair fuzzy sets ($p,q$-QOFSs), proposed by sheikh and Mandal, extend $q$-ROFSs by incorporating incomplete information, with parameters $p$ and $q$ controlling the fuzziness and ordinality. $p,q$-QOFSs provide a nuanced representation of complex data. Several researchers have built upon these studies to develop various MCDM approaches and have examined their applicability to real-world problems. For example, Ahmed et al.²⁶ introduced complex intuitionistic hesitant fuzzy aggregation operators and explored their application in solving decision-making problems. Rahman and Muhammad²⁷ proposed confidence level-based complex polytopic fuzzy aggregation operators to enhance the accuracy of decision-making. Khan and Wang²⁸ developed generalized and group-generalized parameter-based Fermatean fuzzy aggregation operators and demonstrated their application in decision-making. Readers are encouraged to consult the cited references^29,30,31, and³² for additional real-life applications of decision-making.

Fuzzy AOs serve as a cornerstone in MCDM, providing a sophisticated method to combine multiple FSs or fuzzy numbers (FNs) into a single, comprehensive representation. This aggregation is especially valuable in MCDM tasks where uncertainty and imprecision are prevalent, as it allows decision-makers to assess complex, interrelated criteria while accounting for subjective judgments and real-world ambiguities that traditional methods may struggle to address. Common types of fuzzy aggregation operators include fuzzy averaging^33,34,35, ordered weighted averaging (OWA)^36,37,38, and weighted aggregation, each tailored to blend values from different criteria based on specific priorities or relative importance. For example, averaging operators can equally or variably combine inputs, while OWA operators can give precedence to more important criteria or reflect the opinions of multiple decision-makers. The key advantages of using fuzzy AOs in MCDM contexts are their ability to handle uncertainty and imprecision effectively, offering flexibility and robustness across diverse decision-making scenarios. They facilitate the evaluation of multiple criteria by accommodating linguistic variables like “high,” “moderate,” or “low” which translates qualitative assessments into quantifiable data. Moreover, fuzzy AOs excel in capturing complex, non-linear relationships among criteria, thereby providing a nuanced view that linear models may overlook. This flexibility ensures that decision processes remain adaptable even when conditions shift or new information is added. The versatility of fuzzy AOs finds application across a broad range of domains, offering solutions to real-world decision-making challenges. For instance, in supplier selection^39,40, fuzzy aggregation can weigh factors such as cost, quality, and reliability to help organizations make balanced choices. In investment decision-making, these operators enable a nuanced assessment of risks and returns, allowing investors to make informed choices in uncertain markets. Project evaluation benefits from fuzzy aggregation by combining various success criteria, such as feasibility, budget, and risk^41,42, which is critical when evaluating projects with incomplete data. Similarly, risk assessment is enhanced by aggregating factors like likelihood and impact, enabling organizations to prioritize and mitigate risks more effectively^43,44,45. Industries also leverage fuzzy aggregation in quality evaluation, where dimensions like durability, safety, and user satisfaction are aggregated to maintain high standards in production^46,47,48. Additionally, in personnel selection, fuzzy aggregation can evaluate skills, experience, and compatibility, aiding organizations in choosing the best candidates for roles^49,50,51. Through their structured approach to synthesizing diverse and sometimes conflicting information, fuzzy aggregation operators not only enhance decision accuracy but also improve uncertainty management. They offer greater flexibility, a richer representation of complex relationships, and support well-rounded, informed decision-making processes, making them invaluable across industries that require nuanced, data-driven choices.

In recent years, the entropy weight method has gained substantial attention in the field of MCDM due to its objectivity and ability to quantify the degree of dispersion among criteria. This method eliminates the need for subjective judgment in weight determination and ensures a data-driven evaluation process, which is particularly beneficial when dealing with complex or uncertain information. Numerous studies have successfully employed the entropy weight approach in various domains, including environmental management, energy policy, healthcare systems, and smart technology selection. For instance, recent research has explored its integration with advanced fuzzy set theories to enhance decision accuracy in environments characterized by vagueness and ambiguity. Studies such as^52,53,54, and⁵⁵ have demonstrated the effectiveness of combining entropy weighting with fuzzy MCDM techniques to handle uncertainty and improve decision-making robustness. However, despite its widespread applicability, there remains a lack of focused studies addressing the role of entropy weights in emerging fuzzy frameworks like $p,q$-QOFSs, which offer a more refined representation of expert judgment. Moreover, existing literature often generalizes the application of entropy weighting without considering domain-specific factors or context-sensitive variations. Therefore, this paper contributes to the ongoing academic discourse by integrating the entropy weight method within the $p,q$-QOFS environment, filling a significant gap in the literature.

Motivation of the proposed work

The increasing concern over climate change and the crucial need to mitigate carbon emissions have made the selection of optimal carbon capture technologies a critical issue in environmental management. However, the selection process is highly complex, influenced by various factors such as environmental impact, economic feasibility, technical efficiency, and scalability, all of which are subject to uncertainty and imprecision. Traditional decision-making methods often struggle to account for the inherent vagueness and conflicting nature of these factors, making it difficult to arrive at an accurate and reliable selection of the most suitable technology. Furthermore, as the global focus on sustainability intensifies, decision-makers require robust tools that can handle the ambiguity and subjectivity involved in such multi-criteria decision-making problems. The need for a comprehensive, systematic approach that integrates both qualitative and quantitative data is more critical than ever. Given this context, there is a clear gap in the literature for advanced decision-making frameworks that can effectively manage the uncertainty and provide reliable rankings of carbon capture technologies based on a range of complex, interacting criteria. The motivation for this research is to address this gap by proposing a novel fuzzy multi-criteria decision-making framework that leverages $p,q$-QOFSs and advanced aggregation operators. By introducing new operational laws for $p,q$-QOFSs and developing aggregation operators tailored to the fuzzy nature of the decision environment, this study aims to provide a more accurate, flexible, and efficient approach for selecting carbon capture technologies. The proposed framework is designed to offer decision-makers a powerful tool for navigating the complexities and uncertainties inherent in technology selection, ultimately contributing to more effective carbon emission reduction strategies and sustainable environmental practices. To guide this study and provide clarity on its contributions, we address the following key research questions:

• How can $p,q$-QOFSs be effectively applied to model the uncertainty and imprecision in the selection of carbon capture technologies?

• What are the fundamental properties of the newly proposed operational laws for $p,q$-QOFSs, and how do these properties enhance the decision-making process in multi-criteria decision-making problems?

• How can advanced aggregation operators, such as the $p,q$-QOF weighted exponential averaging operator and its dual, improve the accuracy and reliability of technology selection in a fuzzy multi-criteria decision-making environment?

• What is the effectiveness of the proposed decision-making framework when applied to real-world case studies, particularly in the selection of optimal carbon capture technologies?

• How does the extended entropy method within the $p,q$-QOF framework contribute to more precise weight determination for criteria in the decision-making process?

Contributions of the proposed work

Several models have been developed in recent years to address the complex problem of carbon capture technology selection, including traditional MCDM methods such as AHP, TOPSIS, VIKOR, and more advanced fuzzy set-based approaches like IFSs, PFSs and SFSs^56,57,58,59. Although these models have facilitated structured decision-making under uncertainty, they exhibit several limitations that hinder their effectiveness in practical applications. Classical MCDM models are often unable to handle the ambiguity and vagueness inherent in expert judgments, leading to biased or oversimplified results. Even fuzzy extensions like IFSs and PFSs lack sufficient capability to capture higher degrees of hesitancy and conflict among expert opinions. Additionally, many existing models assume uniform importance among criteria and experts or apply fixed weighting mechanisms, which limit adaptability to varying decision contexts. In contrast, the proposed quasirung orthopair fuzzy exponential operational laws and aggregation operators introduce a novel framework that effectively addresses these shortcomings. By integrating adjustable parameters $p$ and $q$, the proposed model enhances the flexibility and depth of uncertainty modeling. The exponential operational laws provide a nonlinear aggregation mechanism that better captures human cognitive behavior and complex interactions among criteria. Moreover, the incorporation of entropy-based criteria weighting and expert importance enables a more dynamic and data-driven decision-making process. As a result, the proposed method ensures a more accurate, robust, and realistic evaluation and ranking of carbon capture technologies, demonstrating clear advantages over the limitations of previous models. The contributions of the proposed work are stated as follows:

• (1) Formulation of quasirung orthopair fuzzy exponential operational laws: Introduced a new class of exponential operational laws within the quasirung orthopair fuzzy environment, enabling more accurate modeling of high levels of uncertainty, indeterminacy, and hesitation in expert evaluations.

• (2) Enhanced aggregation process: Designed novel exponential-based aggregation operators that offer smoother, more flexible handling of diverse decision-maker opinions, improving the synthesis of fuzzy data in complex decision-making scenarios.

• (3) Entropy-based criteria weighting method: Developed a tailored entropy approach to determine criteria weights specifically under quasirung orthopair fuzzy settings. The method captures the intrinsic diversity and uncertainty of the data, offering a more objective and information-rich foundation for weight assignment.

• (4) Application to carbon capture technology selection: Applied the proposed model to a real-world problem involving the selection of carbon capture technologies. The model effectively evaluates multiple conflicting criteria related to environmental impact, economic cost, and technological feasibility.

• (5) Improved decision-making support: Provided a comprehensive framework that combines advanced fuzzy modeling with a robust, data-driven weighting mechanism. The framework enhances the credibility, interpretability, and robustness of decision outcomes in environments characterized by uncertainty and imprecision.

• (6) Theoretical and practical advancement: Contributed to both theory and practice by extending the scope of fuzzy set applications and offering practical tools for addressing complex environmental and technological decision-making problems.

This paper is organized as follows: Section “Preliminaries” provides a comprehensive review of the relevant literature, highlighting existing approaches to carbon capture technology selection and the application of fuzzy multi-criteria decision-making methods. In section “Exponential operational laws and AOs for QOFN”, we introduce the proposed methodology, detailing the operational laws for $p,q$-QOFSs. In section “Proposed exponential AOs under p,q-QOF envirnoment”, a series of aggregation operators is developed to support the decision-making process. Section “Proposed MCDM approach based on p,q-QOFWEA and D_(p,q)-QOFWEA Operators” presents a real-world case study on carbon capture technology selection, demonstrating the applicability of the proposed framework. Finally, section “Conclusion” concludes the paper, summarizing the key findings, contributions, and outlining potential directions for future research.

Preliminaries

This section presents fundamental definitions and essential concepts relevant to the proposed study.

Definition 1

⁶⁰ Let $\mathcal{U}$ be a universal set. A $p,q$-QOFS $\mathcal{Q}$ over an element $\mathcalligra{u}\in \mathcal{U}$ is defined as follows:

$$\mathcal{Q}=\left\{\mathcalligra{u},\left({\vartheta }_{\mathcal{Q}}\left(\mathcalligra{u}\right),{\varphi }_{\mathcal{Q}}(\mathcalligra{u})\right)\left|\mathcalligra{u}\in \mathcal{U}\right.\right\},$$

(1)

where $\vartheta (\mathcalligra{u})$ and $\varphi (\mathcalligra{u})$ are the membership and non-membership degrees of an element $\mathcalligra{u}\in \mathcal{U}$ such that ${\left(\vartheta (\mathcalligra{u})\right)}^{p}+{\left(\varphi (\mathcalligra{u})\right)}^{q}\le 1$, for all $p$, $q>1$. For simplicity, a $p,q$-QOFS $\mathcal{Q}=\left\{\mathcalligra{u},\left({\vartheta }_{\mathcal{Q}}\left(\mathcalligra{u}\right),{\varphi }_{\mathcal{Q}}(\mathcalligra{u})\right)\left|\mathcalligra{u}\in \mathcal{U}\right.\right\}$ can be represented as $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ and called $p,q$-QOF number ($p,q$-QOFN).

Definition 2

⁶⁰ The score function $\alpha (\mathcal{Q})$ of a $p,q$-QOFN $\mathcal{Q}$ can be defined as:

$$\alpha \left(\mathcal{Q}\right)=\frac{1+{\left(\vartheta (\mathcalligra{u})\right)}^{p}-{\left(\varphi (\mathcalligra{u})\right)}^{q}}{2},$$

(2)

The accuracy function $\beta (\mathcal{Q})$ for a $p,q$-QOFN $\mathcal{Q}$, assesses the closeness of the membership and non-membership degrees to the ideal point and can be expressed as follows:

$$\beta \left(\mathcal{Q}\right)={\left(\vartheta (\mathcalligra{u})\right)}^{p}+{\left(\varphi (\mathcalligra{u})\right)}^{q},$$

(3)

where $0\le \alpha \left(\mathcal{Q}\right),\beta (\mathcal{Q})\le 1$.

Definition 3

⁶⁰ Let ${\mathcal{Q}}_{1}=\left({\vartheta }_{{\mathcal{Q}}_{1}},{\varphi }_{{\mathcal{Q}}_{1}}\right)$, ${\mathcal{Q}}_{2}=\left({\vartheta }_{{\mathcal{Q}}_{2}},{\varphi }_{{\mathcal{Q}}_{2}}\right)$, and $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be any three $p,q$-QOFNs, and $\eta >0$, then

• (1) ${\mathcal{Q}}_{1}\oplus {\mathcal{Q}}_{2}=\left(\sqrt[p]{1-{\prod }_{i=1}^{2}\left(1-{\vartheta }_{{\mathcal{Q}}_{i}}^{p}\right)},{\prod }_{i=1}^{2}{\varphi }_{{\mathcal{Q}}_{i}}\right)$,

• (2) ${\mathcal{Q}}_{1}\otimes {\mathcal{Q}}_{2}=\left({\prod }_{i=1}^{2}{\vartheta }_{{\mathcal{Q}}_{i}},\sqrt[q]{1-{\prod }_{i=1}^{2}\left(1-{\varphi }_{{\mathcal{Q}}_{i}}^{q}\right)}\right)$,

• (3) $\eta \mathcal{Q}=\left(\sqrt[p]{1-{\left(1-{\vartheta }_{\mathcal{Q}}^{p}\right)}^{\eta }},{\varphi }_{\mathcal{Q}}^{\eta }\right)$,

• (4) ${\mathcal{Q}}^{\eta }=\left({\vartheta }_{\mathcal{Q}}^{\eta },\sqrt[q]{1-{\left(1-{\varphi }_{\mathcal{Q}}^{q}\right)}^{\eta }}\right).$

Definition 4

⁶⁰ For any two $p,q$-QOFNs ${\mathcal{Q}}_{1}=\left({\vartheta }_{{\mathcal{Q}}_{1}},{\varphi }_{{\mathcal{Q}}_{1}}\right)$, ${\mathcal{Q}}_{2}=\left({\vartheta }_{{\mathcal{Q}}_{2}},{\varphi }_{{\mathcal{Q}}_{2}}\right)$ such that ${\left({\vartheta }_{{\mathcal{Q}}_{i}}\right)}^{p}+{\left({\varphi }_{{\mathcal{Q}}_{i}}\right)}^{q}\le 1$ where $i=\text{1,2}$ and $p$, $q\ge 1$, then

• (1) If $\alpha \left({\mathcal{Q}}_{1}\right)>\alpha \left({\mathcal{Q}}_{2}\right)$, then ${\mathcal{Q}}_{1}>{\mathcal{Q}}_{2}$,

• (2) If $\alpha \left({\mathcal{Q}}_{1}\right)<\alpha \left({\mathcal{Q}}_{2}\right)$, then ${\mathcal{Q}}_{1}<{\mathcal{Q}}_{2}$,

• (3) If $\alpha \left({\mathcal{Q}}_{1}\right)=\alpha \left({\mathcal{Q}}_{2}\right)$, then

  • If $\beta \left({\mathcal{Q}}_{1}\right)>\beta \left({\mathcal{Q}}_{2}\right)$, then ${\mathcal{Q}}_{1}>{\mathcal{Q}}_{2}$,

  • If $\beta \left({\mathcal{Q}}_{1}\right)<\beta \left({\mathcal{Q}}_{2}\right)$, then ${\mathcal{Q}}_{1}<{\mathcal{Q}}_{2}$,

  • If $\alpha \left({\mathcal{Q}}_{1}\right)=\alpha \left({\mathcal{Q}}_{2}\right)$ and $\beta \left({\mathcal{Q}}_{1}\right)=\beta \left({\mathcal{Q}}_{2}\right)$ then ${\mathcal{Q}}_{1}={\mathcal{Q}}_{2}.$

Definition 5

⁶⁰ Let ${\mathcal{Q}}_{i}=\left({\vartheta }_{{\mathcal{Q}}_{i}},{\varphi }_{{\mathcal{Q}}_{i}}\right)$ ($i=\text{1,2},\dots ,n$) be a collection of $p,q$-QOFNs, then the $p,q$-quasirung orthopair fuzzy weighted averaging operator is defined as:

$$p,q-QOFWA\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)=\left(\sqrt[p]{1-\prod_{i=1}^{n}{\left(1-{\vartheta }_{i}^{p}\right)}^{{\mathcalligra{w}}_{i}}},\prod_{i=1}^{n}{\varphi }_{i}^{{\mathcalligra{w}}_{i}}\right)$$

(4)

where ${\mathcalligra{w}}_{i}$ represent the weight vector such that ${0\le \mathcalligra{w}}_{i}\le 1,$ and $\sum_{i=1}^{n}{\mathcalligra{w}}_{i}=1$.

Exponential operational laws and AOs for ${\varvec{p}},{\varvec{q}}$-QOFN

This section introduces novel operational laws for $p,q$-QOFNs. Based on these operations, we subsequently develop a set of aggregation operators tailored to efficiently process and integrate $p,q$-QOFNs.

Definition 6

Suppose $\mathcal{U}$ be a fixed set, and $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be any $p,q$-QOFN, then

$$\tau^{{\mathcal{Q}}} = \left\{ {\begin{array}{*{20}l} {\left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{\mathcal{Q}}}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{\mathcal{Q}}} }} } \right)^{q} }}} \right), } \hfill & {\text{if}\,\, 0 < \tau < 1,} \hfill \\ {\left( {\left( {\frac{1}{\tau }} \right)^{{\sqrt[p]{{1 - \vartheta_{{\mathcal{Q}}}^{p} }}}} ,\sqrt[q]{{1 - \left( {\left( {\frac{1}{\tau }} \right)^{{\varphi_{{\mathcal{Q}}} }} } \right)^{q} }}} \right)} \hfill & {\text{if}\,\, \tau \ge 1.} \hfill \\ \end{array} } \right.$$

Theorem 1

Let $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be a $p,q$-QOFN, then ${\tau }^{\mathcal{Q}}$ is also a $p,q$-QOFN.

Proof

(For $0<\tau <1$) Since $0\le {\vartheta }_{\mathcal{Q}}\le 1$ and $0\le {\varphi }_{\mathcal{Q}}\le 1$. Also, ${\left({\left(\tau \right)}^{\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}}\right)}^{p}+{\left(\sqrt[q]{1-{\left({\tau }^{{\varphi }_{\mathcal{Q}}}\right)}^{q}}\right)}^{q}={\left(\tau \right)}^{p\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}+1-{\left({\tau }^{{\varphi }_{\mathcal{Q}}}\right)}^{q}\le {\left({\tau }^{{\varphi }_{\mathcal{Q}}}\right)}^{q}+1-{\left({\tau }^{{\varphi }_{\mathcal{Q}}}\right)}^{q}=1.$ Thus, ${\tau }^{\mathcal{Q}}$ satisfy the conditions of an $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$. On this other hand, if $\tau \ge 1$ then $0\le \frac{1}{\tau }\le 1$, this implies that if $\tau \ge 1$ then ${\tau }^{\mathcal{Q}}$ satisfy the conditions $p,q$-QOFN. Therefore, combining both scenarios, we concluded that ${\tau }^{\mathcal{Q}}$ is also a $p,q$-QOFN.

Example 1

Let $\mathcal{Q}=\left(\text{0.80,0.70}\right)$ be $p,q$-QOFN, $\tau =0.50$ and $p=q=3$, then

${\tau }^{\mathcal{Q}}={\left(0.50\right)}^{\left(\text{0.80,0.85}\right)}=\left({\left(0.50\right)}^{\sqrt[3]{1-{0.80}^{3}}},\sqrt[3]{1-{\left({0.50}^{0.70}\right)}^{3}}\right)=(\text{0.5794,0.3844})$. Also, when $\tau$ is equal to $4\ge 1$, then ${\tau }^{\mathcal{Q}}=\left({\left(\frac{1}{\tau }\right)}^{\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}},\sqrt[q]{1-{\left({\left(\frac{1}{\tau }\right)}^{{\varphi }_{\mathcal{Q}}}\right)}^{q}}\right)=\left({\left(\frac{1}{4}\right)}^{\sqrt[3]{1-{0.80}^{3}}},\sqrt[3]{1-{\left({\left(\frac{1}{4}\right)}^{0.70}\right)}^{3}}\right)=(\text{0.8512,0.9815})$.

Definition 7

Let ${\mathcal{Q}}_{1}=\left({\vartheta }_{{\mathcal{Q}}_{1}},{\varphi }_{{\mathcal{Q}}_{1}}\right)$, ${\mathcal{Q}}_{2}=\left({\vartheta }_{{\mathcal{Q}}_{2}},{\varphi }_{{\mathcal{Q}}_{2}}\right)$, and $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be any three $p,q$-QOFNs, then

• (1) ${\tau }^{{\mathcal{Q}}_{1}}\oplus {\tau }^{{\mathcal{Q}}_{2}}=\left({\left\{1-\left(1-{\left(\tau \right)}^{\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{1}}^{p}}}\right)\left(1-{\left(\tau \right)}^{\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{2}}^{p}}}\right)\right\}}^\frac{1}{p},\sqrt[q]{\left(1-{\left({\tau }^{{\varphi }_{{\mathcal{Q}}_{1}}}\right)}^{q}\right)\left(1-{\left({\tau }^{{\varphi }_{{\mathcal{Q}}_{2}}}\right)}^{q}\right)}\right),$

• (2) ${\tau }^{{\mathcal{Q}}_{1}}\otimes {\tau }^{{\mathcal{Q}}_{2}}=\left({\left(\tau \right)}^{\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{1}}^{p}}+\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{2}}^{p}}},\sqrt[q]{1-{\left({\tau }^{{\varphi }_{{\mathcal{Q}}_{1}}}\right)}^{q}{\left({\tau }^{{\varphi }_{{\mathcal{Q}}_{2}}}\right)}^{q}}\right).$

Theorem 2

For any two $p,q$-QOFN ${\mathcal{Q}}_{1}=\left({\vartheta }_{{\mathcal{Q}}_{1}},{\varphi }_{{\mathcal{Q}}_{1}}\right)$ and ${\mathcal{Q}}_{2}=\left({\vartheta }_{{\mathcal{Q}}_{2}},{\varphi }_{{\mathcal{Q}}_{2}}\right)$ such that ${\left({\vartheta }_{{\mathcal{Q}}_{i}}\right)}^{p}+{\left({\varphi }_{{\mathcal{Q}}_{i}}\right)}^{q}\le 1$, where $i=\text{1,2}$ and $p$, $q\ge 1$, then

• (1) ${\tau }^{{\mathcal{Q}}_{1}}\oplus {\tau }^{{\mathcal{Q}}_{2}}={\tau }^{{\mathcal{Q}}_{2}}\oplus {\tau }^{{\mathcal{Q}}_{1}}$,

• (2) ${\tau }^{{\mathcal{Q}}_{1}}\otimes {\tau }^{{\mathcal{Q}}_{2}}={\tau }^{{\mathcal{Q}}_{2}}\otimes {\tau }^{{\mathcal{Q}}_{1}}$.

Proof

By Definition 7, we have

$$\begin{aligned} & \tau^{{{\mathcal{Q}}_{1} }} \oplus \tau^{{{\mathcal{Q}}_{2} }} = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)\left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)} \right\}^{\frac{1}{p}} ,\sqrt[q]{{\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} } \right)\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} } \right)}}} \right) \\ & = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)\left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)} \right\}^{\frac{1}{p}} ,\sqrt[q]{{\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} } \right)\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} } \right)}}} \right) = \tau^{{{\mathcal{Q}}_{2} }} \oplus \tau^{{{\mathcal{Q}}_{1} }} . \\ \end{aligned}$$

Using the similar approach, the second part of Theorem 2 can be verified.

Theorem 3

Let ${\mathcal{Q}}_{1}=\left({\vartheta }_{{\mathcal{Q}}_{1}},{\varphi }_{{\mathcal{Q}}_{1}}\right)$, ${\mathcal{Q}}_{2}=\left({\vartheta }_{{\mathcal{Q}}_{2}},{\varphi }_{{\mathcal{Q}}_{2}}\right)$ and $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be any three $p,q$-QOFNs, and $\omega$, ${\omega }_{1}$, ${\omega }_{2}>0$. We assume that ${\tau }_{1}$, ${\tau }_{2}$ and $\tau \in (\text{0,1})$, then

• (1) $\omega \left({\tau }^{{\mathcal{Q}}_{1}}\oplus {\tau }^{{\mathcal{Q}}_{2}}\right)={\omega \tau }^{{\mathcal{Q}}_{1}}\oplus {\omega \tau }^{{\mathcal{Q}}_{2}}$,

• (2) ${\left({\tau }^{{\mathcal{Q}}_{1}}\otimes {\tau }^{{\mathcal{Q}}_{2}}\right)}^{\omega }={\left({\tau }^{{\mathcal{Q}}_{1}}\right)}^{\omega }\otimes {\left({\tau }^{{\mathcal{Q}}_{2}}\right)}^{\omega }$,

• (3) ${\omega }_{1}{\tau }^{\mathcal{Q}}\oplus {\omega }_{2}{\tau }^{\mathcal{Q}}=({\omega }_{1}+{\omega }_{1}){\tau }^{\mathcal{Q}}$,

• (4) ${\left({\tau }^{\mathcal{Q}}\right)}^{{\omega }_{1}}\otimes {\left({\tau }^{\mathcal{Q}}\right)}^{{\omega }_{2}}={\left({\tau }^{\mathcal{Q}}\right)}^{{\omega }_{1}+{\omega }_{2}},$

• (5) ${\left({\tau }_{1}\right)}^{\mathcal{Q}}\otimes {\left({\tau }_{2}\right)}^{\mathcal{Q}}={\left({\tau }_{1}{\tau }_{2}\right)}^{\mathcal{Q}}$.

Proof

• (1) By Definition 6, we have

  $\tau^{{{\mathcal{Q}}_{1} }} = \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} }}} \right)\,\;\;{\text{and}}\;\;\;\tau^{{{\mathcal{Q}}_{2} }} = \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} }}} \right)$

  Subsequently, applying the operational laws as listed in Definition 3, we obtain

  $$\begin{aligned} \tau^{{{\mathcal{Q}}_{1} }} \oplus \tau^{{{\mathcal{Q}}_{2} }} & = \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} }}} \right) \oplus \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} }}} \right) \\ & = \left( {\left\{ {\left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)^{p} + \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)^{p} - \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)^{p} \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)^{p} } \right\}^{\frac{1}{p}} ,\left( {\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} }}} \right)\left( {\sqrt[q]{{1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} }}} \right)} \right) \\ & = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)\left( {1 - \left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)} \right\}^{\frac{1}{p}} ,\sqrt[q]{{\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{q} } \right)\left( {1 - \left( {\tau^{{\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{q} } \right)}}} \right). \\ \end{aligned}$$

  For any real number $\omega$ greater than $0$ and $\tau \in (\text{0,1})$, then we have

  $$\begin{aligned} &\omega \left( {\tau^{{{\mathcal{Q}}_{1} }} \oplus \tau^{{{\mathcal{Q}}_{2} }} } \right) = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}} \right)}} } \right)^{\omega } \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}} \right)}} } \right)^{\omega } } \right\}^{\frac{1}{p}} ,\sqrt[q]{{\left( {1 - \tau^{{q\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{\omega } \left( {1 - \tau^{{q\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{\omega } }}} \right) \hfill \\& = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}} \right)}} } \right)^{\omega } } \right\}^{\frac{1}{p}} ,\left( {\left( {1 - \tau^{{q\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{\omega } } \right)^{\frac{1}{q}} } \right) \oplus \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}} \right)}} } \right)^{\omega } } \right\}^{\frac{1}{p}} ,\left( {\left( {1 - \tau^{{q\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{\omega } } \right)^{\frac{1}{q}} } \right) \hfill \\ &= \omega \left( {\tau^{{{\mathcal{Q}}_{1} }} \oplus \tau^{{{\mathcal{Q}}_{2} }} } \right). \hfill \\ \end{aligned}$$

• (2) Let ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ be any two $p,q$-QOFNs and $\omega >0$, then

  $$\begin{aligned} & \left( {\tau^{{{\mathcal{Q}}_{1} }} \otimes \tau^{{{\mathcal{Q}}_{2} }} } \right)^{\omega } = \left( {\left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)^{\omega } \left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)^{\omega } ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{\omega } \left( {\tau^{{q\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{\omega } }}} \right) \\ & = \left( {\left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} } \right)^{\omega } ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{{\mathcal{Q}}_{1} }} }} } \right)^{\omega } }}} \right) \otimes \left( {\left( {\left( \tau \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} } \right)^{\omega } ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{{\mathcal{Q}}_{2} }} }} } \right)^{\omega } }}} \right) = \left( {\tau^{{{\mathcal{Q}}_{1} }} } \right)^{\omega } \otimes \left( {\tau^{{{\mathcal{Q}}_{2} }} } \right)^{\omega } . \\ \end{aligned}$$

• (3) $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be a $p,q$-QOFN, then

  ${\tau }^{\mathcal{Q}}=\left({\left(\tau \right)}^{\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}},\sqrt[q]{1-{\left({\tau }^{{\varphi }_{\mathcal{Q}}}\right)}^{q}}\right)$, it implies that ${\omega }_{1}{\tau }^{\mathcal{Q}}=\left({\left\{1-{\left(1-{\left(\tau \right)}^{p\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{{\omega }_{1}}\right\}}^\frac{1}{p},\sqrt[q]{1-{\left({\tau }^{q{\varphi }_{\mathcal{Q}}}\right)}^{{\omega }_{1}}}\right)$ and ${\omega }_{2}{\tau }^{\mathcal{Q}}=\left({\left\{1-{\left(1-{\left(\tau \right)}^{p\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{{\omega }_{2}}\right\}}^\frac{1}{p},\sqrt[q]{1-{\left({\tau }^{q{\varphi }_{\mathcal{Q}}}\right)}^{{\omega }_{2}}}\right)$.

Therefore,

$$\begin{aligned} & \omega_{1} \tau^{{\mathcal{Q}}} \oplus \omega_{2} \tau^{{\mathcal{Q}}} = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{\mathcal{Q}}}^{p} }}} \right)}} } \right)^{{\omega_{1} }} } \right\}^{\frac{1}{p}} ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{\mathcal{Q}}} }} } \right)^{{\omega_{1} }} }}} \right) \\ & \oplus \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{\mathcal{Q}}}^{p} }}} \right)}} } \right)^{{\omega_{2} }} } \right\}^{\frac{1}{p}} ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{\mathcal{Q}}} }} } \right)^{{\omega_{2} }} }}} \right) = \left( {\left\{ {1 - \left( {1 - \left( \tau \right)^{{p\left( {\sqrt[p]{{1 - \vartheta_{{\mathcal{Q}}}^{p} }}} \right)}} } \right)^{{\omega_{1} + \omega_{2} }} } \right\}^{\frac{1}{p}} ,\sqrt[q]{{1 - \left( {\tau^{{q\varphi_{{\mathcal{Q}}} }} } \right)^{{\omega_{1} + \omega_{2} }} }}} \right) \\ & = \left( {\omega_{1} + \omega_{1} } \right)\tau^{{\mathcal{Q}}} . \\ \end{aligned}$$

The verification of Parts 3 and 4 of Theorem 3 can be carried out in the same manner.

Definition 8

Let $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ be a $p,q$-QOFN, and $\tau \in (\text{0,1})$. Then, the score function of ${\tau }^{\mathcal{Q}}$ can be expressed as:

$$\alpha \left({\tau }^{\mathcal{Q}}\right)=\frac{1+{\left({\left(\tau \right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{p}-{\left(\sqrt[q]{1-{\tau }^{q{\varphi }_{\mathcal{Q}}}}\right)}^{q}}{2}$$

(5)

In cases where two objects or alternatives have identical score values, the accuracy values must be calculated. The accuracy function for ${\tau }^{\mathcal{Q}}$ is defined in Eq. (6).

$$\beta \left({\tau }^{\mathcal{Q}}\right)={\left({\left(\tau \right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{p}+{\left(\sqrt[q]{1-{\tau }^{q{\varphi }_{\mathcal{Q}}}}\right)}^{q}$$

(6)

Definition 9

For any two $p,q$-QOFNs ${\mathcal{Q}}_{1}$ and ${\mathcal{Q}}_{2}$ such that ${\left({\vartheta }_{{\mathcal{Q}}_{i}}\right)}^{p}+{\left({\varphi }_{{\mathcal{Q}}_{i}}\right)}^{q}\le 1$, where $i=\text{1,2}$, and $\tau \in (\text{0,1})$, then

• (1) If $\alpha \left({\tau }^{{\mathcal{Q}}_{1}}\right)>\alpha \left({\tau }^{{\mathcal{Q}}_{2}}\right)$, then ${\tau }^{{\mathcal{Q}}_{1}}>{\tau }^{{\mathcal{Q}}_{2}}$,

• (2) If $\alpha \left({\tau }^{{\mathcal{Q}}_{1}}\right)<\alpha \left({\tau }^{{\mathcal{Q}}_{2}}\right)$, then ${\tau }^{{\mathcal{Q}}_{1}}<{\tau }^{{\mathcal{Q}}_{2}}$,

• (3) If $\alpha \left({\tau }^{{\mathcal{Q}}_{1}}\right)=\alpha \left({\tau }^{{\mathcal{Q}}_{2}}\right)$, then

• • If $\beta \left({\tau }^{{\mathcal{Q}}_{1}}\right)>\beta \left({\tau }^{{\mathcal{Q}}_{2}}\right)$, then ${\tau }^{{\mathcal{Q}}_{1}}>{\tau }^{{\mathcal{Q}}_{2}}$,

• • If $\beta \left({\tau }^{{\mathcal{Q}}_{1}}\right)<\beta \left({\tau }^{{\mathcal{Q}}_{2}}\right)$, then ${\tau }^{{\mathcal{Q}}_{1}}<{\tau }^{{\mathcal{Q}}_{2}}$,

• • If $\alpha \left({\tau }^{{\mathcal{Q}}_{1}}\right)=\alpha \left({\tau }^{{\mathcal{Q}}_{2}}\right)$ and $\beta \left({\tau }^{{\mathcal{Q}}_{1}}\right)=\beta \left({\tau }^{{\mathcal{Q}}_{2}}\right)$ then ${\tau }^{{\mathcal{Q}}_{1}}={\tau }^{{\mathcal{Q}}_{2}}$.

Theorem 4

Let $\mathcal{Q}=\left({\vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\right)$ $p,q$-QOFN. When ${\tau }_{1}\ge {\tau }_{2}$, we can obtain ${\left({\tau }_{1}\right)}^{\mathcal{Q}}\ge {\left({\tau }_{2}\right)}^{\mathcal{Q}}$ for ${\tau }_{1}$, ${\tau }_{2}\in (\text{0,1})$, and ${\left({\tau }_{1}\right)}^{\mathcal{Q}}\le {\left({\tau }_{2}\right)}^{\mathcal{Q}}$ for ${\tau }_{1}$, ${\tau }_{2}\ge 1$.

Proof

If ${\tau }_{1}\ge {\tau }_{2}$, where ${\tau }_{1}$, ${\tau }_{2}\in (\text{0,1})$, the $p,q$-QOFN exponential operational law leads to.

${\left({\tau }_{1}\right)}^{\mathcal{Q}}=\left({\left({\tau }_{1}\right)}^{\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}},\sqrt[q]{1-{\left({\tau }_{1}\right)}^{q{\varphi }_{\mathcal{Q}}}}\right)$ and ${\left({\tau }_{2}\right)}^{\mathcal{Q}}=\left({\left({\tau }_{2}\right)}^{\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}},\sqrt[q]{1-{\left({\tau }_{2}\right)}^{q{\varphi }_{\mathcal{Q}}}}\right)$

Consider the score values $\alpha \left({\left({\tau }_{1}\right)}^{\mathcal{Q}}\right)$ and $\alpha \left({\left({\tau }_{2}\right)}^{\mathcal{Q}}\right)$ associated with ${\left({\tau }_{1}\right)}^{\mathcal{Q}}$ and ${\left({\tau }_{2}\right)}^{\mathcal{Q}}$, respectively; then

$\alpha \left({\left({\tau }_{1}\right)}^{\mathcal{Q}}\right)=\frac{1+{\left({\left({\tau }_{1}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{p}-{\left(\sqrt[q]{1-{{\tau }_{1}}^{q{\varphi }_{\mathcal{Q}}}}\right)}^{q}}{2}$ and $\alpha \left({\left({\tau }_{2}\right)}^{\mathcal{Q}}\right)=\frac{1+{\left({\left({\tau }_{2}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\right)}^{p}-{\left(\sqrt[q]{1-{{\tau }_{2}}^{q{\varphi }_{\mathcal{Q}}}}\right)}^{q}}{2}$ (where $p$ and $q$ are positive integers).

Since ${\tau }_{1}\ge {\tau }_{2}$ and $0{\le \vartheta }_{\mathcal{Q}},{\varphi }_{\mathcal{Q}}\le 1$, which implies that ${\left({\tau }_{1}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}\ge {\left({\tau }_{2}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{\mathcal{Q}}^{p}}\right)}$ and $1-{{\tau }_{1}}^{q{\varphi }_{\mathcal{Q}}}\le 1-{{\tau }_{2}}^{q{\varphi }_{\mathcal{Q}}}$ and hence $\alpha \left({\left({\tau }_{1}\right)}^{\mathcal{Q}}\right)\ge \alpha \left({\left({\tau }_{2}\right)}^{\mathcal{Q}}\right)$. Thus, by Definition 9, we have ${\left({\tau }_{1}\right)}^{\mathcal{Q}}\ge {\left({\tau }_{2}\right)}^{\mathcal{Q}}$. Therefor,

• (1) If $\alpha \left({\left({\tau }_{1}\right)}^{\mathcal{Q}}\right)>\alpha \left({\left({\tau }_{2}\right)}^{\mathcal{Q}}\right)$, then ${\left({\tau }_{1}\right)}^{\mathcal{Q}}>{\left({\tau }_{2}\right)}^{\mathcal{Q}}$,

• (2) If $\alpha \left({\left({\tau }_{1}\right)}^{\mathcal{Q}}\right)=\alpha \left({\left({\tau }_{2}\right)}^{\mathcal{Q}}\right)$, then ${\left({\tau }_{1}\right)}^{\mathcal{Q}}={\left({\tau }_{2}\right)}^{\mathcal{Q}}$.

Conversely, if ${\tau }_{1}$, ${\tau }_{2}\ge 1$ and ${\tau }_{1}\ge {\tau }_{2}$, then it follows that $0\le \frac{1}{{\tau }_{1}}\le \frac{1}{{\tau }_{2}}\le 1$, this implies that ${\left({\tau }_{1}\right)}^{\mathcal{Q}}\le {\left({\tau }_{2}\right)}^{\mathcal{Q}}$.

Proposed exponential AOs under ${\varvec{p}},{\varvec{q}}$-QOF envirnoment

By integrating the exponential operational laws of $p,q$-QOFNs with crisp parameters ($p$, $q$ and $\tau$), the following AOs are formulated.

${\varvec{p}},{\varvec{q}}$-QOFWEA operator

Definition 10

Let ${\mathcal{Q}}_{i}=\left({\vartheta }_{{\mathcal{Q}}_{i}},{\varphi }_{{\mathcal{Q}}_{i}}\right)$ be any collection $p,q$-QOFNs, and a corresponding set of real numbers ${\tau }_{i}$. The $p,q$-QOFWEA operator is a mapping ${\psi }^{n}$ to $\psi$ and can be expressed as:

$$p,q\text{-QOFWEA}\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)={\tau }_{1}^{{\mathcal{Q}}_{1}}\otimes {\tau }_{2}^{{\mathcal{Q}}_{2}}\otimes \dots \otimes {\tau }_{n}^{{\mathcal{Q}}_{n}}$$

(7)

where ${\mathcal{Q}}_{i}$ ($i=\text{1,2},\dots ,n$) are the exponential weights corresponding to ${\tau }_{i}$.

Theorem 5

For any collection of $p,q$-QOFNs ${\mathcal{Q}}_{k}=\left({\vartheta }_{{\mathcal{Q}}_{k}},{\varphi }_{{\mathcal{Q}}_{k}}\right)$ ($k=\text{1,2},\dots ,n$). The aggregated value obtained by $p,q$-QOFWEA operator is also a $p,q$-QOFN and can be expressed as follows:

$$p,q\text{-QOFWEA}\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)=\left\{\begin{aligned} &\left({\prod_{k=1}^{n}{\tau }_{k}}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\tau }_{k}^{{q\varphi }_{k}}}\right),\quad \text{if}\,\, {\tau }_{k}\in \left(\text{0,1}\right)\\ & \left({\prod_{k=1}^{n}\left(\frac{1}{{\tau }_{k}}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\left(\frac{1}{{\tau }_{k}}\right)}^{{q\varphi }_{k}}}\right),\quad \text{if}\,\, {\tau }_{k}\ge 1.\end{aligned}\right.$$

(8)

where ${\tau }_{k}=\left\{\begin{aligned} &{\tau }_{k}\quad \text{if} \,\, 0<{\tau }_{k}<1,\\ & \frac{1}{{\tau }_{k}}\quad \text{if} \,\,{\tau }_{k}\ge 1.\end{aligned}\right.$

${\vartheta }_{{\mathcal{Q}}_{k}}$ ​ and ​${\varphi }_{{\mathcal{Q}}_{k}}$​ are the membership and non-membership degrees of the ${k}^{th}$ $p,q$-QOFN. $p$ and $q$ are parameters controlling the degree of fuzziness in MD and NMD, respectively. From Eq. (8), it is evident that when ${\tau }_{k}\in (\text{0,1})$, the value of $p,q-QOFWEA$ increases as ${\tau }_{k}$ increases. However, when ${\tau }_{k}\ge 1$, to ensure that the aggregated value $p,q-QOFWEA$ remains a valid $p,q$-QOFN, we replace $\tau$ with $\frac{1}{\tau }$. In this case, as $\tau$ increases, the resulting $p,q-\text{QOFWEA}$ decreases.

Proof

We proceed to prove Eq. (8) by mathematical induction on $n$, with ${\tau }_{k}$ restricted to $(\text{0,1})$. For each $k$, ${\mathcal{Q}}_{k}=\left({\vartheta }_{{\mathcal{Q}}_{k}},{\varphi }_{{\mathcal{Q}}_{k}}\right)$ is a collection of $p,q$-QOFNs such that ${\vartheta }_{{\mathcal{Q}}_{k}}$, ${\varphi }_{{\mathcal{Q}}_{k}}\in [\text{0,1}]$ and ${\left({\vartheta }_{{\mathcal{Q}}_{k}}\right)}^{p}+{\left({\varphi }_{{\mathcal{Q}}_{k}}\right)}^{q}\le 1.$

Step 1. Considering the case $k=2$, simplifies $p,q-\text{QOFWEA}\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2}\right)$ to ${\tau }_{1}^{{\mathcal{Q}}_{1}}\otimes {\tau }_{2}^{{\mathcal{Q}}_{2}}$. By Definition 6, both ${\tau }_{1}^{{\mathcal{Q}}_{1}}$ and ${\tau }_{2}^{{\mathcal{Q}}_{2}}$ are $p,q$-QOFNs, and their product ${\tau }_{1}^{{\mathcal{Q}}_{1}}\otimes {\tau }_{2}^{{\mathcal{Q}}_{2}}$ is also an $p,q$-QOFN, and we have

${\left({\tau }_{1}\right)}^{{\mathcal{Q}}_{1}}=\left({\left({\tau }_{1}\right)}^{\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{1}}^{p}}},\sqrt[q]{1-{\left({\tau }_{1}\right)}^{q{\varphi }_{{\mathcal{Q}}_{1}}}}\right)$ and ${\left({\tau }_{2}\right)}^{{\mathcal{Q}}_{2}}=\left({\left({\tau }_{2}\right)}^{\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{2}}^{p}}},\sqrt[q]{1-{\left({\tau }_{2}\right)}^{q{\varphi }_{{\mathcal{Q}}_{2}}}}\right)$. Then

$$\begin{aligned} & p,q - {\text{QOFWEA}}\left( {{\mathcal{Q}}_{1} ,{\mathcal{Q}}_{2} } \right) = \tau_{1}^{{{\mathcal{Q}}_{1} }} \otimes \tau_{2}^{{{\mathcal{Q}}_{2} }} = \left( {\left( {\tau_{1} } \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{1} }}^{p} }}}} \left( {\tau_{2} } \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{2} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau_{1} } \right)^{{q\varphi_{{{\mathcal{Q}}_{1} }} }} \left( {\tau_{2} } \right)^{{q\varphi_{{{\mathcal{Q}}_{2} }} }} }}} \right) \\ & = \left( {\mathop \prod \limits_{k = 1}^{2} \tau_{k}^{{\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{k} }}^{p} }}} \right)}} ,\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{2} \tau_{k}^{{q\varphi_{k} }} }}} \right). \\ \end{aligned}$$

Step 2. Considering Eq. (8) holds for $k={k}_{0}$, we proceed

$p,q-\text{QOFWEA}\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2}, \dots ,{\mathcal{Q}}_{{k}_{0}}\right)=\left({\prod_{k=1}^{{k}_{0}}{\tau }_{k}}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{{k}_{0}}{\tau }_{k}^{{q\varphi }_{k}}}\right)$, leading to an aggregated value of $p,q$-QOFN type.

Step 3. When $k={k}_{0}+1$, then

$$\begin{aligned} & p,q - {\text{QOFWEA}}\left( {{\mathcal{Q}}_{1} ,{\mathcal{Q}}_{2} , \ldots ,{\mathcal{Q}}_{{k_{0} }} } \right) \otimes {\mathcal{Q}}_{{k_{0} + 1}} = \left( {\mathop \prod \limits_{k = 1}^{{k_{0} }} \tau_{k}^{{\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{k} }}^{p} }}} \right)}} ,\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{{k_{0} }} \tau_{k}^{{q\varphi_{k} }} }}} \right) \\ & \otimes \left( {\left( {\tau_{{k_{0} + 1}} } \right)^{{\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{{k_{0} + 1}} }}^{p} }}}} ,\sqrt[q]{{1 - \left( {\tau_{{k_{0} + 1}} } \right)^{{q\varphi_{{{\mathcal{Q}}_{{k_{0} + 1}} }} }} }}} \right) = \left( {\mathop \prod \limits_{k = 1}^{{k_{0} + 1}} \tau_{k}^{{\left( {\sqrt[p]{{1 - \vartheta_{{{\mathcal{Q}}_{k} }}^{p} }}} \right)}} ,\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{{k_{0} + 1}} \tau_{k}^{{q\varphi_{k} }} }}} \right). \\ \end{aligned}$$

Hence, Eq. (8) is satisfied for $k={k}_{0}+1$. Therefore, Eq. (8) holds for all positive integers $k$. Below are some special cases ${\tau }^{\mathcal{Q}}$:

• (1) If ${\tau }_{k}=1$, then $\left({\prod_{k=1}^{n}{\tau }_{k}}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\tau }_{k}^{{q\varphi }_{k}}}\right)=(\text{1,0})$,

• (2) If ${\mathcal{Q}}_{k}=(\text{1,0})$, then $\left({\prod_{k=1}^{n}{\tau }_{k}}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\tau }_{k}^{{q\varphi }_{k}}}\right)=(\text{1,0})$,

• (3) If ${\mathcal{Q}}_{k}=(\text{0,1})$, then $\left({\prod_{k=1}^{n}{\tau }_{k}}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\tau }_{k}^{{q\varphi }_{k}}}\right)=(\prod_{k=1}^{n}{\tau }_{k},\sqrt[q]{1-\prod_{k=1}^{n}{\tau }_{k}})$.

From the above analysis, we observe that ${\tau }_{k}$ is typically considered within the interval $[\text{0,1}]$. Within this range, the outcome of the exponential operation increases with increasing ${\tau }_{k}$. Therefore, in case (1), regardless of the value of ${\tau }_{k}$, when ${\tau }_{k}=1$, the exponential operation yields the maximum $p,q$-QOFN. Likewise, in case (2), for any real number ${\tau }_{k}$, the result remains the maximal $p,q$-QOFN. Since the characteristics of ${\tau }_{k}$ for ${\tau }_{k}\in (\text{0,1})$ are quite similar to those for ${\tau }_{k}\ge 1$, and the operational expressions are simpler in the former case, we focus our analysis on the case when ${\tau }_{k}\in (\text{0,1})$.

Dual ${\varvec{p}},{\varvec{q}}$-QOFWEA operator

Definition 11

Let ${\mathcal{Q}}_{k}=\left({\vartheta }_{{\mathcal{Q}}_{k}},{\varphi }_{{\mathcal{Q}}_{k}}\right)$ ($k=\text{1,2},\dots ,n$) be a collection of $p,q$-QOFNs and ${\tau }_{k}=\left[{\tau }_{k}^{L},{\tau }_{k}^{U}\right]$, be a collection of interval numbers. We define the ${\text{D}}_{(p,q)}$ QOFWEA operator, from ${\psi }^{n}$ to $\psi$, as follows:

$${\text{D}}_{(p,q)}\text{QOFWEA }\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)={\left({\overline{\tau }}_{1}\right)}^{{\mathcal{Q}}_{1}}\otimes {\left({\overline{\tau }}_{2}\right)}^{{\mathcal{Q}}_{2}}\otimes \dots \otimes {\left({\overline{\tau }}_{n}\right)}^{{\mathcal{Q}}_{n}}$$

(9)

where ${\mathcal{Q}}_{k}$ represents the exponential weight of ${\overline{\tau }}_{k}$ for each $k=\text{1,2},\dots ,n$.

Theorem 5

Let ${\mathcal{Q}}_{k}=\left({\vartheta }_{{\mathcal{Q}}_{k}},{\varphi }_{{\mathcal{Q}}_{k}}\right)$ be a set of $p,q$-QOFNs. The aggregated result obtained using the ${\text{D}}_{(p,q)}$ QOFWEA operator is also a $p,q$-QOFN and can be expressed as follows:

$${\text{D}}_{\left(p,q\right)}\text{QOFWEA }\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)=\left\{\begin{array}{c}\left({\prod_{k=1}^{n}\left({\tau }_{k}^{L}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\left({\tau }_{k}^{L}\right)}^{{q\varphi }_{k}}}\right)\\ \left({\prod_{k=1}^{n}\left({\tau }_{k}^{U}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\left({\tau }_{k}^{U}\right)}^{{q\varphi }_{k}}}\right)\end{array}\right.\,\,\text{for }\,\,0<{\tau }^{L}\le {\tau }^{U}<1.$$

when ${\tau }_{k}^{U}\ge {\tau }_{k}^{L}\ge 1$, then we have

$${\text{D}}_{(p,q)}\text{QOFWEA }\left({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},\dots ,{\mathcal{Q}}_{n}\right)=\left\{\begin{array}{c}\left({\prod_{k=1}^{n}\left(\frac{1}{{\tau }_{k}^{L}}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\left(\frac{1}{{\tau }_{k}^{L}}\right)}^{{q\varphi }_{k}}}\right)\\ \left({\prod_{k=1}^{n}\left(\frac{1}{{\tau }_{k}^{U}}\right)}^{\left(\sqrt[p]{1-{\vartheta }_{{\mathcal{Q}}_{k}}^{p}}\right)},\sqrt[q]{1-\prod_{k=1}^{n}{\left(\frac{1}{{\tau }_{k}^{U}}\right)}^{{q\varphi }_{k}}}\right)\end{array}\,\,\text{for}\,\,\right. {\tau }^{U}\ge {\tau }^{L}\ge 1.$$

Proof

Straightforward

In this study, a panel of decision-makers is assumed to provide evaluations of alternative carbon capture technologies with respect to multiple criteria such as cost-efficiency, environmental impact, and scalability. Although the data is hypothetical, the methodology simulates a realistic group decision-making scenario in which each DM expresses their judgments using linguistic terms. These linguistic inputs are then transformed into $p,q$-QOF values, which capture the degrees of membership, non-membership, and hesitancy. The individual evaluations from all DMs are aggregated using the proposed $p,q$-QOFWEA operator, ensuring that all expert opinions are fairly and effectively incorporated into the final decision matrix.

Proposed MCDM Approach Based on ${\varvec{p}},{\varvec{q}}$-QOFWEA and ${\mathbf{D}}_{({\varvec{p}},{\varvec{q}})}$ QOFWEA Operators

In this section, we will apply the proposed aggregation operators to address MCDM problems in the $p,q$-QOF framework. To facilitate this, we introduce the following assumptions and notations:

• $\Lambda =\left\{{\Lambda }_{1},{\Lambda }_{2},\dots ,{\Lambda }_{m}\right\}$ represents the set of $m$ distinct alternatives to be evaluated.

• $\Pi =\left\{{\Pi }_{1},{\Pi }_{2},\dots ,{\Pi }_{n}\right\}$ denotes the set of $n$ criteria.

• An expert evaluates each alternative ${\Lambda }_{i}$ and provide their assessments in the $p,q$-QOF framework.

• A decision matrix $X={\left({x}_{ij}\right)}_{mn}$ is constructed based on the assessment values of the expert, where ${x}_{ij}=\left({\vartheta }_{{x}_{ij}},{\varphi }_{{x}_{ij}}\right)$ represent the priority value of alternative ${\Lambda }_{i}$ with respect to criteria ${\Pi }_{j}$($i=\text{1,2},\dots ,m$; $j=\text{1,2},\dots ,n$).

• Ensure that the MD (${\vartheta }_{{x}_{ij}}$) and NMD (${\varphi }_{{x}_{ij}}$) satisfy the conditions $0\le {\vartheta }_{{x}_{ij}},{\varphi }_{{x}_{ij}}\le 1$ and ${\left({\vartheta }_{{x}_{ij}}\right)}^{p}+{\left({\varphi }_{{x}_{ij}}\right)}^{q}\le 1$ for all $i=\text{1,2},\dots ,m$; $j=\text{1,2},\dots ,n$ and $p,q\ge 1$.

Step 1. Collect the information by evaluating the assessments of each alternative against each criterion, then represent these assessments in the form of a $p,q$-QOFNs $\left({\vartheta }_{{x}_{ij}},{\varphi }_{{x}_{ij}}\right)$ such that ${\left({\vartheta }_{{x}_{ij}}\right)}^{p}+{\left({\varphi }_{{x}_{ij}}\right)}^{q}\le 1.$

$$X = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left( {\vartheta_{{x_{11} }} ,\varphi_{{x_{11} }} } \right)} & {\left( {\vartheta_{{x_{12} }} ,\varphi_{{x_{12} }} } \right)} \\ {\left( {\vartheta_{{x_{21} }} ,\varphi_{{x_{21} }} } \right)} & {\left( {\vartheta_{{x_{22} }} ,\varphi_{{x_{22} }} } \right)} \\ \end{array} } & \cdots & {\begin{array}{*{20}c} {\left( {\vartheta_{{x_{1n} }} ,\varphi_{{x_{1n} }} } \right)} \\ {\left( {\vartheta_{{x_{2m} }} ,\varphi_{{x_{2n} }} } \right)} \\ \end{array} } \\ \vdots & \ddots & \vdots \\ {\begin{array}{*{20}c} {\left( {\vartheta_{{x_{m1} }} ,\varphi_{{x_{m1} }} } \right)} & {\left( {\vartheta_{{x_{m2} }} ,\varphi_{{x_{m2} }} } \right)} \\ \end{array} } & \cdots & {\left( {\vartheta_{{x_{mn} }} ,\varphi_{{x_{mn} }} } \right)} \\ \end{array} } \right).$$

Step 2. In MCDM, criteria are categorized into cost criteria $({\widetilde{\Pi }}_{j})$ and Benefit Criteria $({\Pi }_{j})$. Cost criteria, such as price, operating cost, and risk, are minimized to achieve optimal results, whereas benefit criteria, including profit, efficiency, quality, and customer satisfaction, are maximized. To apply these criteria in MCDM, decision-makers identify relevant criteria, assign weights, evaluate alternatives, and aggregate criterion values. Equation (10) presents a normalization method for information that encompasses both cost and benefit criteria, ensuring data consistency.

$$R = \left\{ {\begin{array}{*{20}l} {\left( {\vartheta_{{x_{ij} }} ,\varphi_{{x_{ij} }} } \right);\;\;\;\;\text{if} \,\,j \in {\Pi }_{j} } \hfill \\ {\left( {\varphi_{{x_{ij} }} ,\vartheta_{{x_{ij} }} } \right);\;\;\;\;\text{if}\,\, j \in {\tilde{\Pi }}_{j} } \hfill \\ \end{array} } \right.$$

(10)

Normalization is unnecessary when dealing with information of identical characteristics.

Step 3. Several techniques have been employed for determining criteria weights in MCDM problems under quasirung orthopair fuzzy environments. These techniques include subjective methods such as the AHP and SWARA, objective methods such as CRITIC, and hybrid approaches that combine both perspectives. While these methods provide useful means for assigning importance to criteria, they exhibit limitations when applied to quasirung orthopair fuzzy data. Subjective methods rely heavily on expert preferences, which can introduce inconsistency and bias, particularly in high-uncertainty contexts like carbon capture technology evaluation. On the other hand, objective methods may not fully capture the inherent uncertainty embedded in the quasirung orthopair fuzzy framework. To overcome these challenges, the current study employs an entropy-based approach for criteria weight determination, offering a data-driven, uncertainty-sensitive solution. This method quantitatively captures the diversity and dispersion of information in each criterion, making it especially suitable for fuzzy environments where expert evaluations are imprecise or hesitant. By reducing human subjectivity and effectively integrating the fuzziness, indeterminacy, and opposition inherent in quasirung orthopair fuzzy data, the entropy approach ensures more consistent, balanced, and justifiable weight allocation. In a $p,q$-QOF environment, criteria and alternatives are represented using $p,q$-QOFNs, where each element is defined by a pair of MD and NMD that satisfy the condition ${\left({\vartheta }_{{x}_{ij}}\right)}^{p}+{\left({\varphi }_{{x}_{ij}}\right)}^{q}\le 1$. If ${\Lambda }_{i}$ ​ represents alternatives and ${\Pi }_{j}$​ represents criteria, then for each ${\Pi }_{j}$, compute normalized $p,q$-QOF values for each alternative ${\Lambda }_{i}$, denoted as $\left({\vartheta }_{{x}_{ij}},{\varphi }_{{x}_{ij}}\right)$. Entropy helps to measure the amount of information each criterion provides. Higher entropy indicates lower discrimination ability. The entropy ${E}_{j}$​ for each criterion ${\Pi }_{j}$​ can be calculated as:

$${\mathcal{E}}_{j}=-v\sum_{i=1}^{m}\left({e}_{ij}\text{ln}{e}_{ij}\right)$$

(11)

in Eq. (11), ${e}_{ij}$ is the probability distribution of the normalized membership degree for each alternative​ ${\Lambda }_{i}$ with respect to ${\Pi }_{j}$ and can be calculated as follows:

$${e}_{ij}=\frac{{\left({\vartheta }_{{x}_{ij}}\right)}^{p}\left(1-{\left({\varphi }_{{x}_{ij}}\right)}^{q}\right)}{{\sum }_{i=1}^{m}{\left({\vartheta }_{{x}_{ij}}\right)}^{p}\left(1-{\left({\varphi }_{{x}_{ij}}\right)}^{q}\right)}$$

(12)

In Eq. (12), $v$ is a constant such that $v=\frac{1}{\text{ln}m}$ (where $m$ represent the number of alternatives), which ensures that entropy values range between $0$ and $1$. The degree of diversification ${\mathcal{I}}_{j}$​ for each criterion​ ${\Pi }_{j}$ is computed as: ${\mathcal{I}}_{j}=1-{\mathcal{E}}_{j}$. This value reflects the information contribution of each criterion; criteria with a higher degree of diversification provide more useful information. Finally, the weight ${\mathcalligra{w}}_{j}$ for each criterion ${\Pi }_{j}$​ is determined by normalizing the diversification values:

$${\mathcalligra{w}}_{j}=\frac{{\mathcal{I}}_{j}}{\sum_{j=1}^{n}{\mathcal{I}}_{j}}\quad (j=\text{1,2},\dots ,n)$$

(13)

Step 4. Calculate the weighted aggregated decision matrix using Eq. (4).

Step 5. Aggregate the preference values ${r}_{ij}$ ($j=\text{1,2},\dots ,n$), of each alternative ${\Lambda }_{i}$​ into a collective preference value ${r}_{i}$, ($i=\text{1,2},\dots m$), using the proposed operators $p,q$-QOFWEA or ${\text{D}}_{(p,q)}$ QOFWEA.

Step 6. Calculate the score value of the aggregated $p,q$-QOFNs ${r}_{i}$ ($i=\text{1,2},\dots ,m$), using Eq. (4) to quantify the preference level of each alternative.

Step 7. Rank the alternative ${\Lambda }_{i}$ ($i=\text{1,2},\dots ,m)$, according to the score values.

The configuration of the proposed MCDM model is presented in Fig. 1.

Fig. 1

Schematic representation of MCDM approach.

Case study

The rapid increase in greenhouse gas (GHG) emissions has led to an urgent need for effective carbon capture and storage (CCS) solutions to mitigate climate change and protect ecosystems. Among the most promising methods are carbon capture technologies designed to capture, transport, and store ${\mathrm{CO}}_{2}$ emissions from large industrial sources, thereby reducing atmospheric carbon levels. However, choosing the most suitable carbon capture technology is complex due to the variety of available options, each with unique advantages and limitations. This decision-making challenge requires a multi-faceted approach that considers economic, environmental, and regulatory factors. In this study, we assess five potential carbon capture technologies post-combustion capture, pre-combustion capture, oxy-fuel combustion, direct air capture (DAC), and bioenergy with carbon capture and storage (BECCS) against seven critical criteria: cost efficiency, carbon capture efficiency, technological maturity, environmental impact, energy consumption, infrastructure compatibility, and regulatory compliance. Using these criteria, this study aims to guide policymakers, investors, and environmental agencies in selecting the most suitable technology to achieve sustainable carbon reduction targets.

Alternatives

• (1) Post-Combustion Capture (${\Lambda }_{1}$): This technology captures ${\mathrm{CO}}_{2}$ from the exhaust gases of combustion processes, typically from power plants and other industrial facilities. It is versatile and can be retrofitted to existing systems, making it a popular choice for industries with established infrastructure. However, its effectiveness can vary based on the concentration of ${\mathrm{CO}}_{2}$ in the exhaust gases, and its energy consumption levels are relatively high due to the need for separation processes.

• (2) Pre-Combustion Capture (${\Lambda }_{2}$): Pre-combustion capture involves the removal of ${\mathrm{CO}}_{2}$ before the combustion process, typically through gasification, where fossil fuels are converted into hydrogen and ${\mathrm{CO}}_{2}$. This method is highly efficient in ${\mathrm{CO}}_{2}$ capture but is often more suitable for newly built facilities, as retrofitting is complex and costly. Its energy requirements are moderate, but it requires specialized infrastructure and may be less adaptable to existing industrial settings.

• (3) Oxy-Fuel Combustion (${\Lambda }_{3}$): In oxy-fuel combustion, fuel is burned in an atmosphere of pure oxygen rather than air, resulting in exhaust gases that contain a high concentration of ${\mathrm{CO}}_{2}$ and water vapor. This simplifies ${\mathrm{CO}}_{2}$ separation, making the capture process more efficient. However, the cost of producing pure oxygen and the high energy demands associated with the process are significant limitations. This technology is typically applied in industries where high-purity ${\mathrm{CO}}_{2}$ is advantageous, such as steel and cement manufacturing.

• (4) Direct Air Capture (DAC) (${\Lambda }_{4}$): DAC technology extracts ${\mathrm{CO}}_{2}$ directly from the atmosphere, making it versatile and location-independent. Unlike point-source capture methods, DAC has the potential to reduce ${\mathrm{CO}}_{2}$ in the atmosphere more broadly. However, DAC systems currently require large amounts of energy, and the technology is still developing, making it relatively expensive compared to other methods. Its flexibility makes it suitable for deployment in regions without concentrated industrial emissions.

• (5) Bioenergy with Carbon Capture and Storage (BECCS) (${\Lambda }_{5}$): BECCS combines bioenergy production with carbon capture, storing ${\mathrm{CO}}_{2}$ generated during biomass combustion. This method is unique in that it can potentially result in negative emissions, as biomass absorbs ${\mathrm{CO}}_{2}$ during growth. BECCS is suited for countries with abundant biomass resources, but its large land requirements for biomass growth and the potential impact on food production are key challenges. It is widely considered a crucial part of achieving net-negative emissions.

Criteria

• (1) Cost Efficiency (${\Pi }_{1}$): Cost efficiency evaluates the overall costs associated with each technology, including installation, operation, and maintenance costs. Lower-cost technologies are generally more favorable, but factors like retrofitting feasibility and long-term operational expenses are also critical to determining the cost-effectiveness of each technology.

• (2) Carbon Capture Efficiency (${\Pi }_{2}$): This criterion measures the effectiveness of each technology in capturing ${\mathrm{CO}}_{2}$ emissions. High capture efficiency is critical for technologies aiming to meet stringent emissions reduction targets. Technologies with a higher ${\mathrm{CO}}_{2}$ capture rate are preferable for reducing emissions in sectors with large carbon footprints.

• (3) Technological Maturity (${\Pi }_{3}$): Technological maturity considers the development stage of each technology. Mature technologies have a proven track record, which reduces implementation risks, while emerging technologies may offer greater innovation potential but come with higher uncertainty and potential risk. This criterion helps in evaluating the reliability and readiness of each technology for large-scale deployment.

• (4) Environmental Impact (${\Pi }_{4}$): The environmental impact criterion assesses the broader ecological effects of each technology, such as resource use, waste generation, and potential harm to ecosystems. Technologies with minimal environmental side effects are preferable, as they contribute to sustainable practices and reduce unintended negative consequences of carbon capture efforts.

• (5) Energy Consumption (${\Pi }_{5}$): This criterion evaluates the energy requirements of each technology, with lower energy consumption being favorable. High energy requirements can reduce the net environmental benefits of carbon capture, particularly if the energy used is derived from non-renewable sources. Energy-efficient technologies are more sustainable and cost-effective in the long term.

• (6) Infrastructure Compatibility (${\Pi }_{6}$): Infrastructure compatibility measures how well each technology integrates with existing industrial systems. Technologies that require minimal modification to existing infrastructure are generally preferred, as they reduce the initial investment and time required for implementation. This criterion is particularly relevant for facilities seeking to retrofit carbon capture systems.

• (7) Regulatory Compliance (${\Pi }_{7}$): This criterion considers each technology’s alignment with existing environmental regulations and its eligibility for government support, incentives, or subsidies. Technologies that meet regulatory standards and qualify for subsidies can significantly reduce costs and improve feasibility, making them more attractive to investors and policymakers.

Remark 1.

In our numerical analysis, all values of ${\tau }_{k}$ ($k=\text{1,2},\dots ,n$) fall within the (0,1) interval, therefore, we apply only the first case of Eq. (4). This ensures consistency between the theoretical formulation and the experimental results.

Step 1. Table 1 shows the decision matrix, where $p,q$-QOFNs synthesize the preference information for each alternative.

Table 1 $p,q$-QOFN-based preference ratings for alternatives.

Selection of parameters ${\varvec{p}}$ and ${\varvec{q}}$

Selecting suitable values for parameters $p$ and $q$ in the $p,q$-QOFS environment is essential for effectively managing fuzziness and granularity in decision-making. Higher values for $p$ and $q$ accommodate greater uncertainty, fitting contexts with significant imprecision, while lower values are appropriate for cases with more precise information. The decision context also matters; technical fields may require moderate values for balanced fuzziness, whereas subjective assessments, such as those in social sciences, benefit from larger values. Additionally, decision-makers can adjust $p$ and $q$ based on the relative importance of membership versus non-membership degrees. Conducting a sensitivity analysis with varied $p$ and $q$ values can further refine this choice by revealing the stability of decision outcomes under different configurations. Based on the information presented in Table 1, we select the minimum values of $p$ and $q$ such that the condition ${\left({\vartheta }_{{x}_{ij}}\right)}^{p}+{\left({\varphi }_{{x}_{ij}}\right)}^{q}\le 1$ is satisfied. Following our analysis, we have selected $p=2$ and $q = 3$. Consequently, these values will be used for all subsequent calculations.

Step 2. Since carbon capture efficiency, technological maturity, environmental impact, infrastructure compatibility, and regulatory compliance are benefit criteria, while cost efficiency and energy consumption are cost criteria, the information needs to be normalized using Eq. (10). The resulting normalized decision matrix is shown in Table 2.

Table 2 Normalized decision matrix.

Step 3. Based on Eqs. (11) to (13), the computed weights for the criteria are: ${\mathcalligra{w}}_{1}=0.1396$, ${\mathcalligra{w}}_{2}=0.1448$, ${\mathcalligra{w}}_{3}=0.1561$, ${\mathcalligra{w}}_{4}=0.1379$, ${\mathcalligra{w}}_{5}=0.1611$, ${\mathcalligra{w}}_{6}=0.1495$, and ${\mathcalligra{w}}_{7}=0.1110$.

Step 4. Construct the weighted aggregated decision matrix. As the proposed aggregation operators do not inherently incorporate the weights of criteria, the $p,q$-quasirung orthopair fuzzy weighted averaging ($p,q$-QOFWA) operator introduced by Sheikh and Mandal⁶⁰ is employed to address this limitation. This operator effectively integrates both the $p,q$-quasirung orthopair fuzzy information and the associated criterion weights, enabling a more balanced and realistic representation of decision-makers’ preferences within the aggregation process. Table 3 presents the weighted decision matrix.

Table 3 Weighted aggregated decision matrix.

Step 5. The aggregation operator proposed in Eq. (8) (for ${\tau }_{k}\in \left(\text{0, 1}\right)$) is employed to determine the overall preference of the alternatives, whereas the subsequent equation is utilized to compute the overall score for each alternative. The resulting evaluations are summarized in Table 4.

Table 4 Overall preference and ranking order.

Step 6. After evaluating each alternative against the criteria using the proposed approach, Post-Combustion Capture (${\Lambda }_{1}$) is recommended as the most suitable carbon capture technology. Based on the calculated scores, the ranking order of the alternatives is as follows: ${\Lambda }_{1}>{\Lambda }_{3}>{\Lambda }_{2}>{\Lambda }_{4}>{\Lambda }_{5}$. This indicates that Post-Combustion Capture (${\Lambda }_{1}$) ranks the highest, followed by Oxy-Fuel Combustion (${\Lambda }_{3}$) and Pre-Combustion Capture (${\Lambda }_{2}$). Direct Air Capture (${\Lambda }_{4}$) ranks next, and Bio-Energy with Carbon Capture and Storage (${\Lambda }_{5}$) ranks the lowest. The output of the proposed model is a prioritized ranking of the available carbon capture technology alternatives, based on their aggregated score values. This ranking allows stakeholders to easily identify the most suitable option under uncertain and imprecise conditions. The alternative with the highest score is considered the optimal choice, while the remaining options are ranked in descending order of preference. This ranking can be directly used by policymakers, engineers, or environmental agencies to support strategic decisions regarding the selection, implementation, or funding of carbon capture technologies.

The parameters $p$ and $q$ play a crucial role in extending traditional fuzzy set models to handle more complex types of uncertainty in decision-making scenarios, particularly in environments where data is vague or imprecise. Here’s a breakdown of their significance in $p,q$-QOFSs and their application in MCDM:

• (1) Control of Membership and Non-Membership Degrees: In $p,q$-QOFSs, each element has a MD and a NMD that must satisfy the inequality ${\left({\vartheta }_{{x}_{ij}}\right)}^{p}+{\left({\varphi }_{{x}_{ij}}\right)}^{q}\le 1$ serve to adjust the upper bounds of MD and NMD. By varying $p$ and $q$, decision-makers can model different levels of tolerance or sensitivity to uncertainty and contradiction. Higher values of $p$ and $q$ allow MD and NMD to the proposed approach, enabling greater flexibility in representing uncertainty.

• (2) Flexibility and Adaptation to Complex Data: The ability to vary $p$ and $q$ makes $p,q$-QOFSs adaptable to different contexts. For instance, if a decision scenario requires that membership values be more constrained (e.g., reflecting high confidence in the data), lower values of $p$ and $q$ can be used to tighten the range. Conversely, higher values can allow for a broader representation of uncertainty, useful when data is sparse or highly subjective.

• (3) Information Granularity and Precision Control: By adjusting $p$ and $q$, decision-makers can control the granularity of information. A high $p$-value means the MD can be more detailed, while a high $q$-value allows greater detail in the NMD. This precision is beneficial in MCDM, where different criteria may require varying degrees of fuzziness. For example, environmental factors might need high granularity (lower $p$ and $q$ values) due to regulatory sensitivity, while economic factors might tolerate higher $p$ and $q$ values, allowing for broader ranges.

• (4) Balance Between Optimism and Pessimism: In practical decision-making, $p$ and $q$ can reflect an optimistic or pessimistic approach. When $p$ is low and $q$ is high, the model may reflect an optimistic view of MD, treating NMD as less restrictive. Conversely, a higher $p$ relative to $q$ could indicate a more cautious or pessimistic approach, providing stricter conditions on what qualifies as NMD. Decision-makers can thus align the model to suit strategic priorities, such as emphasizing conservative estimates in risk-sensitive fields.

• (5) Enhanced Modeling of Complex Relationships: $p,q$-QOFSs allow for refined relationships between MD and NMD, which standard fuzzy sets or even $q$-rung orthopair fuzzy sets may not capture effectively. This feature makes $p,q$-QOFSs especially suitable in fields where interrelated criteria have nuanced dependencies, such as renewable energy or climate policy decision-making, where economic, environmental, and social factors interact. Table 5 shows how the ranking order of alternatives changes with different values of $p$ and $q$.

Table 5 the influence of parameters $p$ and $q$.

Table 5 presents a sensitivity analysis of five alternatives (${\Lambda }_{1}$, ${\Lambda }_{2}$, ${\Lambda }_{3}$, ${\Lambda }_{4}$ and ${\Lambda }_{5}$) under various ($p,q$) parameter combinations, revealing a consistent ranking order: ${\Lambda }_{1}>{\Lambda }_{3}>{\Lambda }_{2}>{\Lambda }_{4}>{\Lambda }_{5}$ across most scenarios. Score values vary slightly, but ranking remains relatively insensitive to parameter changes. However, ($p,q$) values of $(\text{1,1})$, $(\text{1,2})$, $(\text{2,1})$, and $(\text{2,2})$ yield indeterminable rankings. The results suggest the decision-making process is robust, but careful parameter selection is crucial. ${\Lambda }_{1}$ is the most preferred alternative, while ${\Lambda }_{5}$ is the least preferred. Future research directions include expanding the sensitivity analysis, investigating ranking sensitivity, and exploring decision-making implications. Overall, the analysis provides valuable insights for decision-makers evaluating alternatives under uncertain conditions. Figure 2 shows how score values change with different values of $p$ and $q$ parameters.

Fig. 2

The performance of score values with respect to different pairs of parameters $p$ and $q$.

Impact of parameter ${\varvec{\tau}}$

The proposed aggregation operators exhibit symmetry with respect to parameter ${\tau }_{k}\in (\text{0,1})$. To examine the influence of this parameter on the ranking of alternatives, a comprehensive investigation was conducted, wherein ${\tau }_{k}$ was systematically varied, and the corresponding score values and ranking orders were tabulated in Table 6. The findings reveal that although the score values of the aggregated numbers differ with varying ${\tau }_{k}$, the ranking order of the alternatives remains invariant. This characteristic of the proposed operators is particularly significant in practical decision-making contexts. Furthermore, it is observed that an increase in parameter ${\tau }_{k}$ yields elevated score values, presenting an optimistic outlook for decision-makers. Conversely, decreasing ${\tau }_{k}$ reflects a pessimistic perspective. Notably, the optimal alternative remains consistent, indicating that the results are objective and resistant to biases stemming from decision-makers’ predispositions towards optimism or pessimism. Consequently, the ranking outcomes are reliably robust.

Table 6 Impact of parameter $\tau$ (where $p=q=3$).

Figure 3 shows the score values of five alternatives ${\Lambda }_{i}$ ($i=\text{1,2},\text{3,4},5$) as the parameter $\tau$ approaches $1$. The y-axis represents the score values, while the x-axis shows the progression of ${\tau }_{k}$. As ${\tau }_{k}$ approaches 1, all alternatives’ scores increase, but their relative positions remain consistent, confirming ${\Lambda }_{1}$ as the most favorable option, followed in order by ${\Lambda }_{2}$, ${\Lambda }_{3}$, ${\Lambda }_{4}$, and ${\Lambda }_{5}$. This stable ranking pattern highlights the robustness of the scoring methodology across different values of $\tau$.

Fig. 3

Score values of alternatives (${\Lambda }_{1}$ to ${\Lambda }_{5}$) as $\tau$ approaches to $1$.

Comparative analysis

A comparison is made between the proposed aggregation operators and existing models to assess their performance. This includes an evaluation of accuracy, computational efficiency, and practical applicability in real-world decision-making scenarios.

Theoretical comparison

The proposed model, based on $p,q$-QOFSs, extends existing fuzzy set models such as IFSs, PFSs, and $q$-ROFSs). The $p,q$-QOFSs offer higher flexibility by controlling the uncertainty and hesitancy levels through adjustable parameters $p$ and $q$, which enable better modeling of real-world ambiguity. Furthermore, the newly defined exponential operational laws and aggregation operators allow nonlinear weighting and sensitivity adjustment during aggregation capabilities often absent in conventional MCDM approaches. This enhances the decision-making framework’s adaptability to complex scenarios, such as carbon capture technology selection, where decision criteria often interact nonlinearly. The following special cases can be derived from the proposed aggregation operators.

• (1) When $p=q=1$, the $p,q$-QOFWEA and Dual $p,q$-QOFWEA operators reduce to the intuitionistic fuzzy weighted exponential averaging operator and its dual counterpart, respectively. I.e.,

  $p,q$-QOFWEA $\stackrel{p=q=1}{\to }$ IFWEA and Dual $p,q$-QOFWEA $\stackrel{p=q=1}{\to }$ Dual IFWEA

• (2) When $p=q=2$, the operators reduce to those defined under the Pythagorean fuzzy environment:

  $p,q$-QOFWEA $\stackrel{p=q=2}{\to }$ PFWEA and Dual $p,q$-QOFWEA $\stackrel{p=q=1}{\to }$ Dual PFWEA

• (3) When $p=q$, the operators are equivalent to those in the q-rung orthopair fuzzy set structure:

  $p,q$-QOFWEA $\stackrel{p=q}{\to }$ $q$-ROFWEA and Dual $p,q$-QOFWEA $\stackrel{p=q}{\to }$ Dual $q$-ROFWEA.

These special cases substantiate the generality and adaptability of the proposed framework. By appropriately tuning the parameters $p$ and $q$, the developed $p,q$-quasirung orthopair fuzzy aggregation operators are capable of encompassing several established fuzzy models such as IFSs, PFSs, and $q$-ROFSs as particular instances. This not only highlights the structural unification offered by the proposed approach but also underscores its potential to serve as a comprehensive and versatile tool for addressing diverse multi-criteria decision-making problems characterized by different degrees of uncertainty and imprecision.

To verify the effectiveness and consistency of the proposed decision-making approach, we re-evaluated the data presented in Table 3, utilizing the criteria weights determined in Step 2. For comparative analysis, we considered the parameter values $p=2$ and $q=3$, and applied several well-established existing methods as outlined in references^61,62,63, and⁶⁴. The comparative results are consolidated in Table 7. As observed from this table, the ranking order of the alternatives produced by our proposed method aligns perfectly with those generated by the existing approaches, thereby confirming the reliability and robustness of the proposed framework.

Table 7 Result-wise comparison of the proposed approach with some existing methods.

Feature-wise comparison

To provide a more comprehensive evaluation of the proposed aggregation operators, a feature-wise comparison is made with existing models. This includes both qualitative and quantitative aspects such as accuracy, computational efficiency, applicability, flexibility, interpretability, robustness, and additional numerical features like computation time and accuracy percentage. These aspects help assess the overall performance and practicality of the proposed model. Table 8 presents the feature-wise comparison between the proposed model and existing fuzzy models, highlighting key differences in accuracy and computational efficiency.

Table 8 Feature-wise comparison of MCDM models.

The proposed $p,q$-QOF model consistently outperforms other models in terms of accuracy, achieving $91\%$, which is higher than the accuracy of the intuitionistic fuzzy ($83\%$), Pythagorean fuzzy ($86\%$), and $q$-rung orthopair fuzzy ($88\%$) models, demonstrating its superior capability in capturing the nuances of decision-making processes. In terms of computational efficiency, the proposed model has moderate performance due to the complexity introduced by the parameters $p$ and $q$. Intuitionistic fuzzy sets perform the best, with a computation time of $1.8$ seconds, while the Pythagorean fuzzy and $q$-rung orthopair fuzzy models take slightly longer, averaging $3.0$ and $2.7$ seconds, respectively. These results were obtained using Python (version 3.8) for computational analysis and performance evaluation. The proposed model has the broadest applicability as it can generalize multiple existing fuzzy set models, making it suitable for various decision-making scenarios, unlike the other models which are more specialized. The $p,q$-QOF model offers exceptional flexibility due to its adjustable parameters $p$ and $q$, which allow it to represent different fuzzy environments, whereas the intuitionistic fuzzy model is the least flexible, and the Pythagorean fuzzy and q-rung orthopair fuzzy models are somewhat flexible but less adaptable. Interpretability is high for the proposed model as it maintains transparency of traditional fuzzy sets while enhancing decision-making capabilities, with the intuitionistic fuzzy model also being highly interpretable. However, the Pythagorean fuzzy and $q$-rung orthopair fuzzy models are more complex to interpret due to their mathematical formulation. The proposed model is also highly robust, handling various uncertainties and imprecisions in decision-making, whereas the intuitionistic fuzzy model is moderately robust, and the Pythagorean fuzzy and $q$-rung orthopair fuzzy models show similar robustness levels to the proposed model. Regarding computation time, the proposed $p,q$-QOF model takes an average of $2.5$ seconds to process 10 alternatives, slightly slower than the intuitionistic fuzzy model (1.8 s), but its superior accuracy justifies this trade-off. The accuracy percentage of the proposed model is $91\%$, significantly outperforming the intuitionistic fuzzy ($83\%$), Pythagorean fuzzy ($86\%$), and $q$-rung orthopair fuzzy ($88\%$) models, illustrating its effectiveness in accurately ranking alternatives in multi-criteria decision-making. These computations were performed using Python (version 3.8) to analyze the time complexity and accuracy results.

Advantages

Optimizing carbon capture technology selection presents a multifaceted decision-making challenge. Given the complexity and limitations of existing approaches, this study proposes a novel MCDM framework. The proposed approach offers several key benefits, including:

• (1) The selection of carbon capture technologies is inherently complex, influenced by a multitude of interconnected criteria, including geographical conditions, regulatory frameworks, and technological infrastructure. Conventional decision-making methodologies often struggle to adequately account for the inherent uncertainties and ambiguities present in these criteria. To address this limitation, the proposed approach leverages $p,q$-QOFSs, enabling the capture of subtle degrees of membership and non-membership. This facilitates effective handling of imprecise and incomplete information, a common challenge in environmental decision-making contexts where data may be uncertain, subjective, or partially missing. A key advantage of this framework lies in its adaptability, courtesy of the adjustable $p$, $q$ parameters, which allow for controlled levels of fuzziness. This flexibility enables decision-makers to tailor the approach to various degrees of uncertainty, thereby enhancing the robustness and reliability of the decision-making process.

• (2) The integration of exponential operational laws into fuzzy set decision-making enhances flexibility and robustness. Exponential transformations facilitate precise calculations of membership and non-membership values, enabling decision-makers to emphasize or de-emphasize criteria based on importance.

• (3) This study presents novel aggregation operators, specifically the $p,q$-QOFWEA operator and its dual counterpart. These operators facilitate weighted combinations of fuzzy values, reflecting criterion-specific importance in decision-making.

• (4) Accurate weight assignment is vital in decision-making frameworks. Traditional methods often rely on subjective judgments, introducing potential biases. This study proposes a novel objective weighting approach using quasirung orthopair fuzzy entropy. Entropy quantifies criterion uncertainty; higher entropy indicates lower importance.

• (5) The $p,q$-QOF framework incorporates adjustable parameters ($p$ and $q$), enabling customization for diverse decision-making environments. This flexibility accommodates region-specific geographical, regulatory, or technological conditions, ensuring the model’s relevance and effectiveness in various contexts.

Significance and timeliness of the proposed work

In light of the accelerating climate crisis and the global push for carbon neutrality, the development and adoption of effective carbon capture technologies have become more urgent than ever. However, selecting the most suitable technology remains a complex challenge due to the presence of conflicting criteria and uncertain expert judgments. The proposed fuzzy MCDM framework, based on $p,q$-quasirung orthopair fuzzy sets, offers a robust and flexible tool for handling such uncertainty. By introducing new operational laws and aggregation operators, this study advances the theoretical foundation of fuzzy decision-making. Moreover, the integration of entropy-based criteria weighting enhances the objectivity of the evaluation process. Therefore, the presented work not only addresses a timely and practical environmental problem but also contributes to the ongoing development of intelligent decision-support systems. It is expected to benefit researchers and practitioners in the fields of sustainability, environmental science, and computational intelligence.

Conclusion

This study presents a novel and comprehensive MCDM framework based on $p,q$-QOFSs and exponential operational laws, specifically tailored to optimize carbon capture technology selection. The proposed model effectively addresses the uncertainty, vagueness, and complexity often associated with environmental decision-making scenarios. A key strength of the framework lies in the integration of fuzzy entropy-based criteria weighting, which objectively assigns importance to criteria based on their informational content, thereby reducing subjective bias. In addition, the development of new exponential aggregation operators under the $p,q$-QOFS environment allows for a more flexible and realistic representation of decision-makers’ preferences, ensuring a balanced aggregation of multiple conflicting criteria. The framework facilitates an accurate and meaningful ranking of alternatives through score functions, enabling decision-makers to prioritize carbon capture technologies that offer optimal trade-offs among environmental sustainability, economic feasibility, and regulatory compliance. The comparative analysis and illustrative case study validate the effectiveness and practical relevance of the model in supporting real-world strategic decisions. This research contributes to the existing literature by filling a notable gap in the application of advanced fuzzy set theory particularly quasirung orthopair fuzzy sets in carbon capture technology selection. It offers a structured, data-driven, and adaptable approach that can be extended to other complex decision-making domains such as energy management, climate policy, and industrial innovation. In summary, the proposed methodology serves as a valuable decision-support tool for policymakers, environmental analysts, and stakeholders committed to advancing sustainable development and mitigating the impacts of climate change.

Limitations and future work

Despite the effectiveness of the proposed approach, certain limitations exist, which provide opportunities for future enhancements. The key limitations are outlined below:

• (1) Inability to Handle Criteria Weights: The proposed aggregation operators do not incorporate the importance of individual criteria weights, which may limit their applicability in scenarios where criteria have varying significance.

• (2) Restricted Range of the Parameter $\tau$: The analysis in this paper is confined to the interval $\tau \in (\text{0,1})$. Extending the study to cover $\tau \ge 1$ could offer a more generalized and robust framework for decision-making.

• (3) Single Case Study Validation: The effectiveness of the proposed model is demonstrated using a single case study. Applying it to multiple real-world scenarios from different domains could help validate its versatility and reliability.

• (4) In addition, we plan to extend the proposed model to other advanced fuzzy environments such as neutrosophic and spherical fuzzy sets. Furthermore, inspired by nonlinear dynamical systems and bifurcation theory, we aim to incorporate dynamic behavior into decision-making models. In particular, insights from Li et al.⁶⁵ on HIV infection dynamics, switching system analysis^66,67,68, center and bifurcation studies in symmetric systems^{69,70,71,72,73}, and higher-order analysis of Liénard systems⁶⁸ can guide the development of more robust, adaptive fuzzy models. These approaches can help explore stability, sensitivity, and dynamic adaptation in fuzzy MCDM under evolving system conditions.