Decision support model for blockchain adoption using EDAS technique in the complex q-rung orthopair fuzzy structure

Abstract

Blockchain technology (BCT) is dramatically altering industries by providing a decentralized, secure, and transparent system to trade without any intermediaries—promoting trust and automation in public and private networks. Managers and decision-makers need to apply multi-attribute group decision-making (MAGDM) processes under uncertainty in order to identify the factors that influence BCT adoption and its possible uses. This study presents a systematic approach to the identification and addressing of barriers preventing the implementation of BCT. First, we introduce a number of intricate aggregation processors based on complex q-rung orthopair fuzzy settings (Cq-ROFSs) for synthesizing the views of specialists. Moreover, we propose a set of interesting complex q-rung orthopair fuzzy Einstein geometric operators such as Cq-ROFEWG, Cq-ROFEOWG, and Cq-ROFEHG operators to enhance aggregation under uncertainty. Then, we describe an advanced EDAS-based decision-making process that meets MAGDM requirements, complete with an explanatory flowchart step-by-step. The proposed technique is used to review the BCT implementation over fourteen critical aspects, thus validating its practical application. Finally, both conventional models and the proposed framework performance comparison reveal the proposed framework can offer enhanced reliability and realistic insights for decision-makers.

Introduction

Blockchain (BC) is a novel innovation that facilitates decentralized networks, altering data storage, advanced computing, cybersecurity, financial transactions, and digital interactions. BC can potentially promote both the transformation and development of business activities¹. BC is a decentralized, peer-to-peer digital ledger that promotes a new era of transactional applications, guaranteeing transparency and trust. It facilitates the documenting of transactions and the oversight of digital assets across business networks². The system enables proper monitoring processes alongside validation activities for complex transactions that require authentication and traceability. BCT operates as a resilient digital ledger that establishes protected, clear, and distributed operations across business systems for data and information sharing. A globally distributed system functions through interconnected blocks or nodes that deliver secure information for all transactions across its worldwide network. Each block can receive or transmit transactions to and from other blocks, so ensuring that local information is synchronized with the comprehensive dataset. The BC network improves and structures itself by removing redundancies and reducing reliance on mediators.

The main benefit of BCT technology is its ability to provide network members equal visibility into shared data, which improves transparency in the network. The common data accessibility function between network users creates minimal conditions for disputes and fraud. Luo et al.^3,4 developed a symbiotic blockchain consensus in wireless blockchain systems to boost consent reliability. The widespread adoption of BCT requires the solution of scalability issues alongside addressing energy use problems, regulatory requirements, interoperability challenges, and privacy protection measures. Ongoing technological advancement produces programs to solve existing problems so different industries can access BCT’s maximum potential. Governments, engineers, and businesses worldwide are interested in BCT because of its potential to promote increased efficiency and trust and open up new business models. Technological progress can transform industries, foster innovation, and offer new opportunities for producing value and collaboration in the digital age.

Zadeh⁵ proposed a fuzzy set (FS), allowing for a more nuanced representation by establishing the membership degree (MD) among 0 and 1. Atanassov⁶ elaborated on this concept through intuitionistic fuzzy sets (IFS), which integrate both the MD and the non-membership degree (NMD), with the range . Khan et al.⁷ developed the dombi aggregation operators in the IFS structure. An IFS operates offers and betters practical solutions for unreliable or uncertain information than the FS does. Atanassov⁸ created interval-valued IFS (IVIFS) through the implementation of interval-valued MD and NMD pairs to address difficult situations in the application. Yager⁹ created Pythagorean fuzzy sets (PyFS) from generalized IFS by using the condition as . Yager¹⁰ established the q-rung orthopair fuzzy set (q-ROFS) by altering the PyFS condition to , while using . Khan et al.¹¹ proposed the aczel-alsina aggregation operators for q-ROFS and extended a MAGDM technique using their presented operators.

Ramot et al.¹² expanded the concept of FS by replacing the unit interval with the unit disc, thereby establishing the Complex FS (CFS). A complex number denotes the MD in CFS , in which the real and imaginary components are restricted to the unit interval. Alkouri and Salleh¹³ created Complex IFS (CIFS), which is made up of Complex-MD and Complex-NMD , with the condition that the sum of the real and imaginary components does not exceed the interval . Ullah et al.¹⁴ created complex PyFS (CPyFS) to refine uncertainty modeling, ensuring that the sum of squared complex-MD and complex-NMD retained within . Liu et al.¹⁵ presented the complex q-ROFS (Cq-ROFS), which is characterized by the following conditions: and . The Cq-ROFS is a more extensive generalization than CFS, CIFS, and CPFS, giving a better framework for dealing with complex decision-making challenges. Khan et al.¹⁶ developed the aczel-alsina AOs for Cq-ROFS and established a decision-making technique to resolve MAGDM problems.

Many scholars have investigated other t-norms and t-conorms; however, the Einstein t-norm and t-conorm have garnered significant attention from researchers owing to their distinctive features and uses. Wang and Liu¹⁷ suggested Einstein aggregation operators (AOs) for IFS to help with the problems that come up with multiple attribute group decision-making (MAGDM) issues. Ghorabaee et al.¹⁸ developed the Evaluation Based on the Distance from Average Solution (EDAS) technique, which delivers a straightforward but efficient way of addressing multi-criteria group decision-making (MCGDM) challenges. An accurate clarification approach for contentment estimation is presented within factor decorrelation, developing a description regularity¹⁹. Also, this paper stated an adaptive learning-based technique for useful mobile communications deployment, implementing high overall consumer contentment and method productivity²⁰. Although BCT has yet to utilize Cq-ROFNs to tackle challenges fully, their application in BCT adoption could enhance decision-making. To the best of the authors’ knowledge, nothing else has examined the use of Einstein geometric operators in the EDAS approach with Cq-ROFNs. With the help of Cq-ROFNs and Einstein geometric operators, our main research goal was to find a new way to solve some MAGDM problems more effectively within the EDAS framework.

The structure of the subsequent sections of this manuscript is as follows: “Literature review” section offers an extensive examination of the literature concerning fuzzy MADGM techniques, as well as the Einstein geometric operators and EDAS technique. “Preliminaries” section offers an in-depth review of the fundamental concepts and principles that form the basis of the study. It establishes the foundation for the next discourse by underscoring the intricacies of addressing information difficulties and accentuating the necessity for a more dependable, precise, and enhanced methodology. “Einstein’s operations” section lays out the Einstein operating rules for Cq-ROFNs. Then, the Cq-ROFWG, Cq-ROFEOWG, and Cq-ROFEHG operators are fully explained, focusing on their most important properties and roles. “Extended EDAS technique based on einstein geometric operators” section presents a novel EDAS technique for addressing MAGDM issues, grounded in the principles of Einstein geometric operators. “Application of the proposed EDAS method for blockchain technology” section contains a numerical study demonstrating the suggested methods’ practical applicability. “Comparative and sensitivity analysis” section offers a thorough comparative analysis to evaluate the feasibility of the proposed strategy. The assessment will compare the methodology to recognized methods, emphasizing criteria of dependability and accuracy. This section outlines the principal benefits of the suggested technique. “Conclusion and future developments” section concludes the study and offers ideas for future research.

Motivation and problem statement

In recent years, some fuzzy set models, such as q-ROFS¹⁰ and complex FS (CFS)¹², have gained significant attention due to their ability to model uncertainty and hesitation in complex decision-making environments. After careful analysis of the existing literature on the topic, it becomes clear that despite the popularity of the existing models, the extension of the Einstein AOs to complex q-rung orthopair fuzzy sets (Cq-ROFS) has never been explored in detail. Until now, no study has used the EDAS method in the Cq-ROFS context to support decision-making about proper BCT choices. In response to this lack of exploration, the study recommends an innovative EDAS-based MAGDM strategy, which is designed explicitly for Cq-ROFS environs. The application of a robust weight allocation approach increases the reliability and consistency of BCT alternatives for decision-making. This research deals with the pressing need for reliable decision-support systems in BCT against the backdrop of increased intricacy and risks in the discipline. The model demonstrates improved evaluation effectiveness and supports the choice of more informed technologies, as described in the subsequent sections.

• The Cq-ROFS model’s extensive applicability enables it to represent intricate data flexibly and comprehensively, making it a valuable tool in decision-making strategies across various industries. Given its potential, further exploration and attention to the decision-making mechanisms within the Cq-ROFS framework are warranted.

• The Einstein operations demonstrate effective implementation in the Cq-ROFS framework through their parameter-dependent flexibility. We establish new operational laws based on Einstein mathematics to work specifically for Cq-ROFSs. The developed laws enable the construction of Einstein-ordered and Einstein-hybrid aggregation operators with specific fundamental characteristics. The improvement that was developed makes Cq-ROFSs suitable for complex decision-making situations and increases their overall applicability.

• The Evaluation based on Distance from Average Solution (EDAS) method is popular for its user-friendliness and reliable decision-making results. EDAS simplifies decision-making and improves dependability by evaluating options based on their distance from an average answer, unlike current methods.

• Multiple obstacles exist throughout the MAGDM process when selecting the most suitable BCT organization. Our proposed model incorporates Cq-ROFS data to address these difficulties while effectively identifying top BCT organizations. ​

Contributions of the study

The key objective of this study is to examine Einstein geometric operators in a Cq-ROFNs framework. This study’s primary contributions include the following:

1. The Cq-ROFNs combine CIFNs and CPFNs to enhance proficiency, reliability, and comprehensiveness in handling uncertain information during decision-making. Moreover, the investigation of complex q-rung orthopair fuzzy numbers involves the use of Einstein geometric operators. Therefore, expanding the Einstein geometric operator to handle MAGDM problems with complex q-rung orthopair fuzzy numbers is highly important.

2. The Einstein geometric operator, which is based on Cq-ROFNs, is an effective method for managing two-dimensional information inside a single set. This article how to construct the Cq-ROF Einstein weighted geometric (Cq-ROFWG) operator, the Cq-ROF Einstein ordered weighted geometric (Cq-ROFEOWG) operator, the Cq-ROF Einstein hybrid geometric (Cq-ROFEHG) operator and examine certain properties of these operators.

3. The proposed Cq-ROFns MAGDM solutions are operated with Einstein geometric operators using the new EDAS technique. The entire operation is divided into two fundamental components:

   • I.
     The procedure begins with collecting. The information collection procedure starts with the implementation of Cq-ROFNs Einstein geometric operators.
   • II.
     The EDAS algorithm forms inside the Cq-ROFNs structure to identify the best choice. This method presents an authentic method to handle complex decision problems by integrating modern practices of decision-making theory.

4. The proposed method illustrates its application through an adoption of BCT case analysis employing Cq-ROFNs information techniques. A sensitivity evaluation method and comparison studies are used to validate that the proposed approach remains robust.

Literature review

This section presents an in-depth research study focusing on BCT adoption and briefly examining Einstein laws and the EDAS technique. The study highlights crucial motivations and substantial contributions to the area.

The collective process of organizational decision-making uses data analytics as a tool to deliver correct answers for predetermined objectives. MAGDM represents a flexible problem-solving approach that handles the resolution of decision problems with numerous alternatives and opposing criteria. In recent years, an expanding number of researchers from a variety of scientific disciplines have embraced it as a solution to a variety of intricate decision-making challenges. Dhumras et al.^21,22 developed a strategic framework for green supply chain management in the energy industry based on a q-rung picture fuzzy AHP-WASPAS and Federated Learning-based q-rung picture fuzzy TOPSIS/VIKOR models strategic planning in electronic marketing. Singh et al.²³ extended the TOPSIS method with an R-norm picture fuzzy discriminant measure and used their stated model for supplier selection. Zhang et al.^24,25 developed the heterogeneous attribute reconstruction and representation model using distance measures by prognostic attribute values. Dhumras et al.²⁶ assessed the consumer electronics market in the T-spherical fuzzy sets.

Görçün et al.²⁷ developed an innovative MAGDM system to choose blockchain technology for logistics by adopting FFS and dombi aggregation. Meskini et al.²⁸ applied IVIF TOPSIS to discuss the barriers facing BCT adoption in insurance companies. Mishra et al.²⁹ used an IVPF entropy-based decision support system in the healthcare supply chain evaluation process for BCT adoption. Li and Wang³⁰ analyzed the dynamics of an HIV infection model uniting logistic mitosis and direct cell-to-cell transmission. Yang et al.³¹ designed a blockchain-based framework, DeepICE, integrating multimodal neural convolution in graph networks that predicts appropriate leaving timings and travel expenditures, which improves the transportation procedure for personal automobile passengers while retaining security.

The aggregation operators (AOs) maintain strong scientific interest because the research subject continues to grow in both importance and excitement. Azeem et al.³² investigated Einstein operations within CIFS through a detailed study that demonstrated their applications for resolving MAGDM according to their findings. Janani et al.³³ created the CPFS Einstein AOs as a powerful decision-making tool to identify the ideal breed of horse gram, demonstrating its efficacy in managing complexity in MAGDM. Ali et al.³⁴ investigated Einstein’s Geometric AOs within a novel CIVPF framework, illustrating their effectiveness in optimizing green supply chain management implementation. Liu et al.³⁵ introduced a variety of related methodologies that employed Cq-ROFNs. Liu et al.³⁶ used the EDAS method in the Pythagorean fuzzy structure and used their developed model for the circular supplier selection in the manufacturing industry. A comprehensive review of the current literature indicated that there has been insufficient study on the utilization of Einstein geometric operations to create new operators employing Cq-ROFNs. It is crucial to investigate how Einstein procedures operate when processing Cq-ROFNs data. The developed concepts for Cq-ROFNs enable more accurate uncertainty and inconsistency evaluation to enhance decision-making performances.

The main purpose of MAGDM analysis is to identify the optimal solution between multiple options by using various evaluation factors. Ghorabaee et al.¹⁸ presented EDAS as a widely recognized concept for assessing MAGDM problems with conflicting criteria. The EDAS method needs to establish both Positive Distance from the Average Solution (PDAS) and Negative Distance from the Average Solution (NDAS) values within the Average Solution framework. The EDAS method for handling MAGDM problems was described by Li et al.³⁷ using the q-rung orthopair fuzzy theory. Mishra et al.³⁸ adopted the IF with EDAS method to evaluate technology for efficiently managing healthcare waste. Ying et al.³⁹ stated an analytical decision technique that utilizes obvious research that combines inadequate interpretation and design concepts using conceptual modeling, mining associations, and fuzzy Dempster-Shafer theory to boost the process of decision-making. Dhumras and Bajaj⁴⁰ made a significant advancement in multi-criteria decision-making (MCDM) for robotic agrifarming by developing a Modified EDAS method, incorporating picture fuzzy soft Dombi aggregation operators. Shi et al.⁴¹ planned an analytical framework implementing graphs and a Fokker–Planck equation to investigate the behavioral patterns of agents in governmental competitions. Du et al.⁴² developed the frank AOs for CqROFS to resolve MADM complications.

This study recognized the limitations and developed the proposed approach accordingly. In the preliminary stage, Cq-ROFN Einstein operators were created to facilitate the review of data to identify dependable vendors. In the next phase, the generated Einstein AOs are utilized within the EDAS method to establish an organized approach for identifying the most suitable use of BCT. Despite the inability of existing studies to address the resulting issues, Which model is most appropriate for determining the key factors influencing the application of BCT across many areas in settings of significant uncertainty? What is the suitable model for identifying the most appropriate area for the application of BCT? This article presents a novel EDAS method that uses Einstein geometric operators within the Cq-ROFSs framework to deal with the problems of adopting BCT.

Preliminaries

In this section, we will revisit the existing concepts and introduce the concept of Cq-ROFSs. We will also thoroughly discuss the operational laws of Cq-ROFSs.

Definition 2.1

¹⁰ The q-rung orthopair fuzzy sets in a universe discourse that is described as

where, and indicate the membership degree (MD) and non-membership degree (NMD), satisfying the conditions: . Furthermore, the hesitancy degree is expressed as .

Definition 2.2

¹⁵ The complex q-rung orthopair fuzzy sets on a non-empty set with the following description:

where, and indicate the complex-valued MD and complex-valued NMD, fulfill conditions: and . The Cq-ROFNs are represented by: .

Definition 2.3

¹⁵ For the Cq-ROFNs, the score and accuracy functions are represented as follows:

1

2

where and . To analyze the relationships among any two Cq-ROFNs, and , the following rules are considered:

1. If then

2. If then .

3. If then

4. If then .

Definition 2.4

⁴² Let and be two Cq-ROFNs. The subsequent operations are specified on Cq-ROFNs:

1. 

2. 

3. ;

4. .

Einstein’s operations

The following section will examine the key ideas of Einstein’s t-norm and t-conorm, which model the intersection and union of fuzzy sets. These operations are then implemented within the Cq-ROFSs environment to improve decision-making under uncertainty.

Definition 3.2

Let and be two Cq-ROFNs and , be any real number, then we have

1. 

2. ;

3. and

4. 

Theorem 1

Let and be two Cq-ROFNs and is the real number, then we have

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Proof

The proof of theorem is provided in the "Appendix-1" (cc supplementary material).

Complex q-rung orthopair fuzzy Einstein geometric aggregation operators

This section presents a few Einstein AOs, specifically the Cq-ROFEWG, Cq-ROFEOWG, and Cq-ROFEHG operators. We will also highlight certain aspects of these operators.

Definition 3.1.1

Let , m be a collection of Cq-ROFNs with linked weight vector (WV) satisfying conditions and . Then, the Cq-ROFEWG operator is a mapping from to , such that:

Theorem 3.1.2

Let where , be the family of Cq-ROFNs. Then Cq-ROFEWG is derived as follows:

3

Proof

Mathematical induction is applied to prove Eq. (3).

Case 1: if then

The values of and are obtained using (4) of Definition 3.2,

Thus, the result holds when

Case 2: Next, check for , and Eq. (3) holds.

Case 3: lastly, check for , i.e.,

Hence, the result holds for , proving that Eq. (3) is true for all

Proposition 3.1.3

Let , be a collection of Cq-ROFNs with linked WV satisfying conditions and . Then,

(i) Idempotency: If then

Proof

(ii) Boundedness: Let and then

Proof

Since and

Therefore, we have

So,

then we have

(iii) Monotonicity: If then

Proof

We omit it because it is comparable to (ii).

Complex q-rung orthopair fuzzy Einstein-ordered weighted geometric operators

The Cq-ROFEOWG operator is now established, and some properties are presented.

Definition 3.2.1

Let , be a set of Cq-ROFNs with linked WV satisfying conditions and . The Cq-ROFEOWG operator is a mapping from to , i.e.,

where is the permutation of such that

Theorem 3.2.2

Let where , be the collection of Cq-ROFNs, then Cq-ROFEOWG is derived as follows:

4

Proof

Similar to theorem 3.1.2.

Proposition 3.2.2

Let , be a family of Cq-ROFNs, and is the WV i.e., and

(i) Idempotency: If then

(ii) Boundedness: Let and then

(iii) Monotonicity: If then

Proof

Similar to proposition 3.1.3.

Complex q-rung orthopair fuzzy Einstein, hybrid geometric operator

Definition 3.3.1

Let , be a family of Cq-ROFNs, and the Cq-ROFEHG operator is mapping such that

where is WV of with and Additionally, with another WV and

Theorem 3.3.2

Let where , be the family of Cq-ROFNs. Then, Cq-ROFEHG is derived as follows:

5

Proof

Similar to theorem 3.1.2.

Proposition 3.3.3

Let , be a family of Cq-ROFNs, and is the WV i.e., and

(i) Idempotency: If then

(ii) Boundedness: Let and then

(iii) Monotonicity: If then

Proof

Similar to proposition 3.1.3.

Extended EDAS technique based on Einstein geometric operators

In this section, we develop an algorithm that implements the EDAS method and the proposed complex q-rung orthopair fuzzy set to resolve the conventional MAGDM problem. Several decision matrices for the Cq-ROF MAGDM problem contain preference values for all alternatives according to criteria set with Cq-ROFNs.

Let represent a family of alternatives, be the collection of attributes and experts . Let represent the attributes WV, be the experts in WV, where and Let expert evaluates the attribute for the alternative using Cq-ROFN

The procedure for implementing the proposed method is outlined below:

Step 1 Gather the assessment data for each alternative against their attribute from qualified experts and construct the resulting decision matrix.

6

Step 2 When the decision matrices display various types of attributes, it is necessary to apply a normalization transformation to ensure consistency, as given below:

7

Step 3 Review the ordered matrices for every alternative.

Step 4 The proposed method consolidates the expert data with their weights to create the aggregated decision matrix, as shown below:

8

Step 5 Compute the average solution according to the criteria shown below:

9

Step 6 Compute the positive distance from average and negative distance from average matrices according to the criteria type (benefit and cost), as illustrated below:

and , where

10

and

11

Step 7 For proposed operators, compute the positive weighted distance and negative weighted distance .

12

13

Step 8 For each , normalize the values of and

14

15

Step 9 Compute the integrative-appraisal score

16

Step 10 Rank the alternatives in descending order by their appraisal score , with the highest indicating the most beneficial option. This ranking provides a basis for categorizing each alternative accordingly.

The flowchart of the proposed algorithm is presented in the following Fig. 1.

Fig. 1.

Framework of developed model.

Application of the proposed EDAS method for blockchain technology

This section will analyze the application of the previously described model to identify the key factors influencing the adoption of BCT.

The description of BCT

BCT, as a tool for secure data storage in blocks using cryptographic methods in a decentralized environment, has become a supply chain transformation tool. Even though the technology gained fame as part of the attention to Bitcoin’s success, its provenance of transparency, traceability, and trust make it a valuable tool in handling supply chain activities⁴³. However, the integration of blockchain technology into real-world supply chains is a challenging decision-making process due to a variety of technological, organizational, behavioral, and regulatory barriers. These challenges make the adoption of BCT a form of an MCDM, in which decision-makers are required to carefully analyze and integrate an array of possibly conflicting variables in order to determine its viability and value.

Several key benchmarks must be applied when viewing an available choice. Technological readiness is also important, and the question is whether the blockchain platforms can be plugged into the existing infrastructure and handle real-time data streams. Evaluating the cost–benefit of the investment to make sure that the improvements are financially feasible and validating organizational readiness to ensure that the company can sustain the deployment through proper skills and infrastructure. It is critical to keep data transparent and secure, protecting the reliability and privacy of information in the supply chain. Both regulatory compliance and stakeholder involvement need to be assessed as both legal risks and the disinclination of others can hamper the use of blockchain technology. Companies can identify vital blocks, strengths, and weaknesses of various blockchain options by using these evaluation standards systematically. With the help of these results, the businesses can optimize their planning, minimize risk, and make wise investment decisions, which enables them to, for example, as is the case with Maersk, IBM, etc.

Alternatives

The following are the alternatives’ descriptions.

• Finance Technology : BCT has revolutionized the financial sector and continues unlocking new finance and banking possibilities. A recent World Economic Forum report predicts that by 2025, blockchain will hold nearly 10% of the global GDP⁴⁴. Blockchain significantly impacts cross-border payments, stock trading, identity verification, efficient syndicated lending, and the standardization of accounting and auditing practices. Additionally, decentralized crypto hedge funds built on blockchain are drawing more investors and participants, further accelerating the technology’s adoption and impact on the financial industry.

• Healthcare ⁴⁵: Smart healthcare systems are organization-focused, with hospitals in charge of all patient information and medical data via hospital-centric interoperability. BCT found that key properties such as data provenance, transparency, tamper-resistance, and robust security could aid in the transition to a patient-centric data model. Integrating BCT into healthcare systems presents several hurdles, particularly regarding secrecy on a public blockchain, along with speed and scalability issues. Experts believe employing blockchain to store merely data hashes, with the actual data stored off-chain, could help alleviate privacy issues. Governments, entrepreneurs, and healthcare organizations are actively incorporating blockchain concepts, with systems such as MedChain pioneering secure healthcare data storage.

• Logistics and supply chain ⁴⁶: BCT is revolutionizing the logistics and supply chain industries by providing secure, tamper-proof audit trails that improve trust and transparency. The modification tackles long-standing challenges of cargo theft, fraud, and inefficient tracking in traditional systems that rely significantly on trust among individual parties. BCT integration ensures transparency, traceability, and authenticity, benefiting all stakeholders, from producers to customers, by bringing various interests together on a trustworthy platform. Numerous organizations are utilizing blockchain concepts to streamline and secure supply chains, lowering errors and enhancing responsibility across the network. For instance, IBM employs blockchain in Africa to combat counterfeit medications, allowing end consumers to make more informed decisions.

• Food and Agriculture Industry ⁴⁷: Tracing products, often maintained by third-party middlemen, is a fundamental challenge in the agriculture industry. Implementing BCT in these operations could result in significant reforms and automation in important areas such as payments, product tracking, and supply chains. This transparency addresses consumer concerns about product quality, authenticity, and sources. Furthermore, eliminating intermediaries via BCT increases profitability, reduces delays, and improves supply chain efficiency.

Criteria description

This study evaluates BCT adoptions using the following fourteen criteria based on a literature review and insights from academic and industry experts.

• Scalability⁴⁸: Scalability refers to the platform’s ability to maintain constant performance as the network size, number of nodes, and transactions per second improve. This criterion demonstrates the blockchain’s ability to scale and manage increased traffic.

• Privacy⁴⁹: This factor involves protecting private information on the blockchain so that only authorized users or organizations may access it. In many industries, privacy is crucial for upholding confidentiality and satisfying legal obligations.

• Transaction Speed⁵⁰: It determines the pace of operational completions and verification processes. Fast transaction speeds represent a critical requirement for applications that need immediate settlement, especially when applied to financial services and supply chain management.

• Scope⁵¹: The prospective applications and business sectors for BCT deployment pertain to the scope aspect. Blockchain possesses the capacity to broaden its applications across other disciplines, hence enhancing its market impact within the healthcare sectors, banking, and supply chain.

• Security⁵⁰: The security systems employed for blockchain data protection against external fraud attempts, unauthorized access, and cyber-attacks. Network sustainability depends heavily on robust security systems to maintain the dependability, trustworthiness, and integrity of its operations.

• Government Policies⁴⁸: The implementation of BCT exists within the boundaries set by government laws and regulations defined as government policies. Eligible policies that support innovation combined with investment attract more adoption and growth, whereas restrictive policies hinder both measures.

• Transparency⁴⁹: All network participants need blockchain activity’s complete visibility regarding transactions and smart contracts for transparency to succeed. The combination of transparency helps stakeholders build confidence because it permits independent examination and audit functions for blockchain activities.

• Blockchain maintenance⁴⁹: Blockchain maintenance consists of permanent responsibilities to repair flaws while enhancing and optimizing the system’s networks to maintain correct operational functionality. A well-maintained blockchain achieves both reliability and efficient long-term functionality.

• Adaptability⁴⁸: A blockchain system can establish connections to various industries apart from different environments and technologies through its adaptability. Flexible blockchains allow easy configuration for diverse uses, which boosts their potential for varied field applications.

• Network Availability⁵⁰: The availability of network resources allows consumers to access services and execute transactions without interruptions continuously.

• Accountability⁴⁹: Accountability represents blockchain’s power to create individual member identification for taking responsibility in network operations through member accounting systems. The system boosts ethical conduct through mechanisms that help users respect established ethical standards while abiding by existing laws.

• Flexibility⁵¹: High levels of adaptability are essential for blockchain systems to handle changes, which include different use case protocols along with fresh features. Blockchain systems with flexible features remain adaptable to market developments and technological advancements.

• Cost⁵²: Financial expenditures for blockchain systems development with subsequent platform configuration and maintenance require both infrastructure investment and transaction fees and ongoing operational funding. Blockchain technology needs lower costs to become accessible for use by businesses and individuals.

• Energy Consumption⁵¹: A blockchain network’s energy requirements reach notable and substantial levels. Reduced energy usage is critical for increasing the environmental sustainability of BCT, particularly resource-intensive consensus methods like proof-of-work.

Alternatives and criteria for the adoption of BCT are given in the following Fig. 2.

Fig. 2.

The alternatives and criteria for BCT adoption.

Numerical example

This section implements the proposed MADGM model in a particular study about adopting blockchain technology across many domains, as explored by Siddiqui and Haroon⁵¹. Through a comprehensive literature review, four critical domains were recognized as potential sectors for BCT adoption as follows: = {: Finance technology (https://fsdhmerchantbank.com/2025/04/17/future-of-tech-finance-digital-economy/), : Healthcare (https://www.gebauer.com/blog/healthcare-technology-for-your-facility), : Logistics and supply chain (https://www.linkedin.com/pulse/digital-transformation-supply-chain-paving-path-efficiency-ali-9abzc), : Food and agriculture industry (https://www.linkedin.com/posts/uc-davis_ourucdavis-infocus-activity-7191194959427870720-JCmF)}. We selected these sectors based on multiple variables in BCT adoption. The factors that have been identified as impacting blockchain adoption are as follows: Consider a team of three experts, represented as with corresponding weights and

be the attribute’s weights. A team of three decision-making experts (DMEs) designated as has been established to guide the BCT adoption process and analyze the choices and variables examined. The DMEs encompass several sectors and have over 15 years of expertise in industrial and computational applications. Figure 2 shows the criterion for factor adoption of BCT. Based on this information, we explain the subsequent decision-making procedures.

Step 1 According to Garg et al.⁴⁴, Table 1 presents the language ratings that reflect the importance of DMEs and the domains investigated for BCT adoption concerning the assessment factors.

Table 1.

The performance value of the option expressed in Cq-ROF.

Linguistic ratings
Extremely-high (EHg)	0.95	0.9	0.02	0.1
Very-high (VHg)	0.9	0.8	0.05	0.15
High (Hg)	0.8	0.75	0.1	0.2
Slightly-high (SHg)	0.7	0.7	0.25	0.1
Medium (Me)	0.65	0.5	0.3	0.4
Slightly-low (SLo)	0.5	0.3	0.45	0.6
Low (Lo)	0.4	0.25	0.5	0.6
Very-low (VLo)	0.25	0.2	0.6	0.65
Extremely-low (ELo)	0.15	0.1	0.65	0.7

Based on the findings of^33,35,42, the expert-provided initial evaluation matrix has been compiled into Cq-ROFS and is displayed in Table 2.

Table 2.

The initial evaluation matrix of BCT.

Criteria
	(VLo, SLo, VLo)	(SLo, Lo, VLo)	(SLo, SHg, Me)	(SHg, SHg, Me)
	(Lo, SLo, Me)	(Lo, VLo, VLo)	(Me, SLo, SHg)	(Hg, SHg, Me)
	(VHg, Hg, Hg)	(VHg, Me, SLo)	(SHg, Me, Me)	(SHg, Me, ELo)
	(Me, Me, SLo)	(Hg, Me, SHg)	(SHg, SLo, Me)	(SHg, Hg, SLo)
	(Hg, SHg, Me)	(SLo, Lo, VLo)	(SHg, SLo, Lo)	(SLo, Lo, VLo)
	(Lo, Lo, VLo)	(VLo, Lo, VLo)	(SHg, Hg, Me)	(SHg, Lo, Me)
	(Lo, SHg, Me)	(SHg, Hg, Lo)	(Me, SHg, Hg)	(SLo, Hg, Me)
	(Lo, Hg, SHg)	(Hg, Lo, VLo)	(SHg, Me, Lo)	(Me, SHg, Lo)
	(Hg, Me, SHg)	(ELo, Hg, Lo)	(Hg, SHg, Me)	(SLo, VLo, Lo)
	(Hg, ELo, SHg)	(Hg, Me, VLo)	(SLo, Slo, Me)	(SLo, ELo, Lo)
	(SLg, Lo, VLo)	(Lo, Lo, SLo)	(Me, SHg, Lo)	(Lo, SLo, Me)
	(Lo, SLo, Me)	(SLo, Lo, VLo)	(Me, Hg, SLo)	(SHg, SLo, Hg)
	(Hg, Me, SHg)	(Lo, SHg, Me)	(Hg, Lo, Me)	(SLo, Me, SHg)
	(Me, Hg, SLo)	(Hg, SHg, Me)	(Hg, Me, Lo)	(Me, SHg, Lo)

Step 2 This example does not require normalization because all criteria are in the same category and measured on a similar scale.

Step 3 Tables 3, 4 and 5 organizes and presents a complete set of normalized decision matrices.

Table 3.

Ordered matrix of .

Table 4.

Ordered matrix of .

Table 5.

Ordered matrix of .

Step 4 The aggregated decision matrix is developed by utilizing suggestions from DMEs and applying attribute weighting via the Cq-ROFEOWG operator (Eq. 3), as shown in Table 6.

Table 6.

Aggregate matrix for Cq-ROFEOWG operator.

Step 5 We Use Eq. 9 to determine the average solution matrix, as follows:

Step 6 Find the PDA and NDA matrices given in Tables 7 and 8, respectively, using the Eqs. 10 and 11.

Table 7.

PDAS matrix.

Table 8.

NDAS matrix.

Step 7 Using Eqs. 12 and 13, construct the positive weighted distance and negative weighted distance .

And

Step 8 The normalized values for and are obtained using Eqs. 14 and 15, respectively.

And

Step 9 The integrative appraisal score can be calculated by utilizing Eq. 16, as demonstrated in the following: .

Step 10 Afterward, the arranged options in decreasing order according to their appraisal scores indicate that the top-ranked option is the most desired choice, while the lowest-ranked option represents the Cq-ROFEOWG (Cq-ROFEWG) least suitable alternative

This proposed method offers significant improvements over the traditional EDAS method, enabling specialists to express their opinions with greater precision and adaptability.

Comparative and sensitivity analysis

This section presents a thorough sensitivity analysis of our proposed model, emphasizing its robustness and efficacy and a detailed comparative assessment against previous research findings.

Sensitivity study the influence of parameter

To investigate the influence of parameter on the results, their values are systematically varied within the framework of Problem 6.2. Table 9 and Fig. 3 present the results corresponding to various -values as determined by the Cq-ROEOWG operator. Simple variation in AC is observed with parameter alterations, indicating that the final result is not substantially affected. This further substantiates the effectiveness of our operators, with the Cq-ROFS demonstrating its advantage in scenarios where CIFS¹³ and CPFS¹⁴ fail. The Cq-ROFS operator enables decision-makers to allocate higher values enhancing flexibility and precision, as demonstrated by the findings in Table 9. The analysis of the provided tables indicates that the ranking of alternatives is stable across different values of , with consistently identified as the optimal option in every case.

Table 9.

Comparison of and ranking for various values in Cq-ROFEOWG.

	Appraisal score	Ranking

Fig. 3.

Impact of parameter on ranking results.

Comparative analysis

We will evaluate our suggested model from three viewpoints in the subsequent subsection. Initially, we will evaluate the novel EDAS approach against current EDAS techniques within several fuzzy frameworks. In the subsequent step, the proposed EDAS approach will be compared to the extant aggregation operators within the Cq-ROFS framework. In the end, we will evaluate the efficacy of our methodology in comparison to the TOPSIS technique within the Cq-ROFS framework.

Comparison with EDAS techniques in different fuzzy structures

Traditional approaches like fuzzy-EDAS¹⁸ usually use MD and NMD intervals to assess alternatives. In contrast, more advanced models, including MD and NMD, throughout the evaluation process include IFS-EDAS³⁸ and PFS-EDAS³⁶. Güneri and Deveci⁵³ expanded the EDAS technique to encompass the q-rung orthopair fuzzy and interval-valued q-rung orthopair fuzzy frameworks. However, these techniques are still limited in adequately handling different parameterizations. Table 10 contains a full theoretical comparison of our new EDAS approach with current EDAS techniques for various fuzzy set systems.

Table 10.

Comparative evaluation using current EDAS systems.

Structure	Appraisal score				Ranking
Fuzzy EDAS18	n/a				n/a
IFS-EDAS38	n/a				n/a
PFS-EDAS36	n/a				n/a
q-ROFS-EDAS53	n/a				n/a
Proposed-Cq-ROF-EDAS	0.99794	0.92923	0.60263	0.02090

The EDAS structure developed proficiently selects the most efficient BCT disposal technique. Table 10 illustrates its advantage over current approaches, establishing its dependability and resilience in optimum decision-making.

Comparison with different existing methods

To highlight the utility, strength, and applicability of the suggested approach, we perform a thorough comparative analysis, comparing the performance of the proposed operators against various established decision-making techniques from previous studies. A comparative analysis is performed to assess the proposed method along Cq-ROF with various operators, i.e., AOs¹⁵, frank AOs⁴², yager AOs⁵⁴, and interaction AOs⁵⁵. To enhance accuracy, the Cq-ROFEOWG operator implemented in Step 2 is substituted with the operators specified in Sect. 6.2, but the other data remains unchanged. The classification results that were produced by employing different operators are displayed in Table 11 and Fig. 4. As indicated in Table 11, the ranking levels for the BCT disposal approach vary depending on the methodology used by different operators to synthesize the assessment information. However, the rankings for the BCT disposal approach remain stable, with being accepted as the most efficient option.

Table 11.

Comparative with different existing methods.

Operator’s	Appraisal score/score values	Ranking
Cq-ROFWA15
Cq-ROFWG15
Cq-ROFYWA54
Cq-ROFYWG54
Cq-ROFFWA42
Cq-ROFFWG42
Cq-ROFIWA55
Cq-ROFIWG55
Proposed Method

Fig. 4.

Comparison with different aggregation operators.

The findings indicate that our research corresponds with the optimal choices identified in prior studies, confirming the reliability and efficacy of the suggested strategy for selecting a disposal approach for BCTs. This highlights the strength and importance of the proposed methodology. However, slight variations in rankings were observed, probably attributable to the Einstein operations employed, which effectively manage complex and ambiguous data while reducing distortion and loss. The current operators demonstrate insufficient capability to fully integrate precise and rapidly changing information, highlighting potential areas for enhancement.

Comparison with different decision-making methods in the same structure

This section evaluates the effectiveness of the proposed Cq-ROF-EDAS method by comparing it with existing approaches. To ensure a fair comparison, we adapt the VIKOR⁵⁶ and TOPSIS⁵⁷ methods to align with the Cq-rung orthopair fuzzy framework and conduct a detailed performance analysis. The third phase involves a comprehensive comparison of options using the three methodologies, VIKOR, TOPSIS, and EDAS, to determine the final ranking. The current research employed VIKOR and TOPSIS approaches for data processing. The alternatives’ computed values are as follows: for TOPSIS, , , , and and for VIKOR, , , , and 0.9705. The methodologies determine possible results for selecting BCT. The ranking features derived through separate approaches appear in Table 12, and Fig. 5 illustrates them.

Table 12.

Decision outcomes of various approaches.

Approach	Appraisal score/score values	Ranking
TOPSIS57
VIKOR56
PROPOSED

Fig. 5.

Comparison with various approaches.

Variations in the ranks of alternatives have been noted based on the outcomes derived from various techniques. The extended EDAS approach is efficient and trustworthy, particularly in complicated assessments. It handles several criteria accurately, which is superior to older methods. The suggested strategy is adaptable and well-suited to solving MAGDM issues involving Cq-ROFNs.

Results and discussion

The research demonstrates an effective method to resolve MAGDM problems, which include contradictory characteristics. Strengthened EDAS methodology demonstrates enhanced effectiveness in its applications since it produces better results than previous research and aligns with existing methodological findings. This approach establishes a benefit by using detailed restriction data while reducing uncertainties and creating an exact depiction of factual information. This characteristic provides a solution for adding attribute-related information, which enhances its usefulness as a decision-support tool. Furthermore, the comparison analysis emphasizes EDAS’s unique capacity to manage higher levels of uncertain data, lowering the probability of errors generated by assigning scores to parameters without fully accounting for their interconnections. The EDAS technique guarantees more dependable, resilient, and risk-averse decision-making by accurately reflecting the perception and similarity between solutions and averting conclusions based on insufficient data, thereby providing substantial advantages over conventional methods.

A group of three experts evaluated four alternative technologies during the assessment phase. Based on assessment scores obtained through Cq-ROF-EDAS, the options received the rankings . The assessment results demonstrate that delivers the best performance while represents the poorest choice between the four examined alternatives. The proposed method demonstrates excellence in resolving problems involving linguistic values and descriptive phrases through evaluative information. The proposed model demonstrates the superior capability to handle information from diverse formats, which extends to linguistic expressions. An excellent outcome occurs when MD and NMD total values stay within the language parameters established by the model, leading to precise and trustworthy results.

The proposed technique is more adaptable and effective for complex decision-making situations than the alternative strategies mentioned in Table 12. An in-depth examination indicates that the ranking order produced by our methodology closely corresponds with the results of Mahmood and Ali⁵⁷ and Garg et al.⁵⁶, highlighting its consistency and reliability. Our Cq-ROF-EDAS strategy covers both operational difficulties and mathematical complexities challenges. This contrasts with other methods requiring more computing expenses and extended processing durations. The proposed method is a viable solution for decision-making challenges due to its efficacy, rapid execution, and user-friendliness.

Advantages and limitations of the proposed method

The suggested approach has numerous advantages, significantly improving its reliability and operational efficacy. Table 13 summarizes the key characteristics of the model studied in this study and other similar methods.

• The proposed approach demonstrates high versatility when working with numerous information types, including language-based data. The approach delivers optimal results when the MD and NMD together do not exceed the scope of linguistic terms. Expert satisfaction comes from MD ratings through the Cq-ROFS framework, while NMD scores represent their dissatisfaction.

• The structure of Cq-ROFS provides a complete capture of uncertainty while simultaneously supporting MD and NMD operations. This facilitates a more precise depiction of the uncertain and unclear data frequently observed in real-world situations.

• The Cq-ROFS framework handles MAGDM situations well by using Einstein AOs that include expert opinions and the weights that go with them. This enables the method to assess alternatives based on various criteria, resulting in more reliable and well-informed decisions.

• The Cq-ROF-EDAS technique that we created significantly reduces the operational and mathematical difficulties of the algorithm that is being implemented. As a result, the method is both user-friendly and practical.

Table 13.

Comparative table of the Cq-ROFS of various fuzzy set theories.

	Set type	Aggregated information in complex form	Attributes	Advantages	Limitations
Zadeh5	FS	×	✓	Deals uncertainty using MD	Unable to handle NMD
Atanassov6	IFS	×	✓	Deals uncertainty using MD and NMD	Unable of complex fuzzy values
Atanassov8	IVIFS	×	✓	Deals uncertainty using MD and NMD intervals	Unable of complex fuzzy values
Yager9	PFS	×	✓	Deals uncertainty using MD and NMD	Unable of complex fuzzy values
Senapati58	FFS	×	✓	Deals uncertainty using MD and NMD	Unable of complex fuzzy values
Yager10	q-ROFS	×	✓	Deals uncertainty using MD and NMD	Unable of complex fuzzy values
Ju et al.19	IVq-ROFS	×	✓	Deals uncertainty using MD and NMD intervals	Unable of complex fuzzy values
Proposed	Cq-ROFS	✓	✓	Deals with amplitude and phase information	Unable to handle more complex computations

The proposed model presents certain limitations, described as follows:

• This investigation neglects the incorporation of subjective weights for factors influencing BCT adoption. The methodology neglects expert assessments in determining criteria weights, a crucial element of subjective weighing techniques.

• The accuracy and comprehensiveness of the input data are critical to the efficacy of the EDAS approach. Variations in BCT adoption frameworks or expert assessments may compromise the reliability of the results.

• Inconsistencies and unpredictability in qualitative criteria can skew quantitative data and lead to poor conclusions.

• The involvement of experts from both global and local perspectives is crucial for attaining more comprehensive and universally applicable decision findings.

The BCT adoption structure outlined in this research, implementing Cq-ROFS and Einstein geometric operators in the EDAS approach, presents important implications for the expected investigation of dynamical systems and computational modeling, particularly in convoluted decision-making circumstances. The investigations carried out by Li et al.^59–66 associated with singular points, bifurcations, and system integrability correlate with the effective use of fuzzy decision models to handle nonlinear dynamics and unpredictability. The implementation of Cq-ROFS can enhance decision assistance in BCT adoption, especially in dynamic mechanisms that have multifaceted, undetermined decision factors. The proposed model can help manage compatibility and linear capacity obstacles by demonstrating a novel approach for blockchain network preference.

Conclusion and future developments

This study extends Einstein’s geometric operations into the Cq-ROFS framework, utilizing the fundamental principles of the Cq-ROF and Einstein’s geometric operations. We commence by presenting a collection of Einstein operational rules designed explicitly for the Cq-ROFS. We provide a set of novel complicated q-rung orthopair fuzzy Einstein geometric operators, namely the Cq-ROFEWG, Cq-ROFEOWG, and Cq-ROFEHG operators, which offer a comprehensive framework for addressing intricate decision-making issues. We investigated their primary results and the fundamental properties of Idempotency, boundedness, and monotonicity. We extend the EDAS method, which uses Einstein AOs to solve MAGDM problems. We have utilized the EDAS method to prioritize the application domains for BCT adoption, and the case study showcases the effectiveness of the proposed approach. To show that the results are stable and reliable, sensitivity analysis was done on different values of the strategy coefficient, and comparisons were made with other models. The results showed that it gave correct and dependable answers well, making it an important tool for decision-making about BCT adoption.

This study’s insights and contributions create a favorable basis for the next studies, thereby providing chances for more investigation and field developments. Future initiatives should involve a broader range of worldwide decision-making experts to fully evaluate application areas and the elements affecting the acceptance of blockchain technology. This study also makes it easier to look into other frameworks, such as complex T-spherical picture fuzzy sets, complex bipolar fuzzy sets, linguistic dual hesitant rough sets, and interval-valued complex q-rung orthopair fuzzy sets. Future studies may investigate integrating deep learning and machine learning techniques with Cq-ROFSs to assess the adoption of blockchain technology, considering essential drivers and constraints across various application industries.

Author contributions

U. A., J. Li. and U. Z. wrote the main manuscript text. K. B. A. S. and R. A. Conceptualization, Methodology, Validation, Formal analysis, Data curation, Writing—review & editing. I. S., I. S. and R. M. Z. Methodology, Validation, Writing—original draft, Formal analysis. I.S. prepared Figs. 1 and 2.

Data availability

All the data used and analyzed is available in the manuscript.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-05971-5.