A novel fuzzy multi criteria decision making approach based on the energy of the matrix

MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF ECONOMICS HO CHI MINH CITY COLLEGE OF TECHNOLOGY AND DESIGN Technology "land UNIVERSITY-LEVEL RESEARCH PROJECT A novel fuzzy multi-crite

INTRODUCTION

The quest for sustainable and renewable energy sources has led to a growing interest in wave energy, recognized for its immense potential and significance (Clemente, Rosa-Santos,

Wave energy presents a consistent and powerful renewable energy source, largely untapped and capable of exceeding global energy demands (Taveira-Pinto, 2021; Gallutia et al., 2022) Its development could revolutionize the energy sector by providing a clean and inexhaustible alternative, significantly reducing our dependence on fossil fuels (IEA, 2022) Additionally, wave energy offers substantial environmental benefits, serving as a greener option that lowers carbon footprints and minimizes ecological disruption, thus playing a crucial role in the transition to sustainable energy solutions (Quadrelli & Peterson, 2007) As technology advances, benchmarking Wave Energy Converter (WEC) systems becomes essential for evaluating their effectiveness and cost-efficiency, facilitating ongoing improvement and innovation in the wave energy sector (Choupin et al., 2021).

The article offers crucial insights for developers, investors, and policymakers, guiding future advancements, optimizing resource allocation, and setting industry standards (Gao et al., 2020) By showcasing successful technologies and practices, effective benchmarking can expedite the adoption of wave energy, fostering broader acceptance and implementation in the sector.

Benchmarking Wave Energy Converter (WEC) technologies requires sophisticated analysis, where Multiple Criteria Decision-Making (MCDM) methods prove invaluable These methods assess a variety of factors, including technical performance, economic viability, environmental impact, and social acceptance, ensuring a balanced evaluation of complex decision-making scenarios Among the prominent MCDM techniques, the Method based on the Removal Effects of Criteria (MEREC) objectively determines the significance of various criteria by analyzing the effects of their removal This leads to equitable decision-making by accurately reflecting each criterion's contribution Additionally, the Spherical Fuzzy Combine Compromise Solution (SF-C0C0S0) method enhances decision analysis through the use of spherical fuzzy sets, which capture uncertainty and vagueness more effectively than traditional fuzzy numbers By incorporating three-dimensional membership, non-membership, and hesitancy degrees, SFNs provide a richer representation of human judgment complexities This innovative approach improves the robustness and precision of MCDM analyses, making it particularly beneficial in uncertain decision-making environments The integration of MEREC and SF-C0C0S0 in evaluating WEC technologies promises a comprehensive and accurate assessment, facilitating the identification of the most efficient wave energy converters and marking a significant advancement in the field.

7 robust tools for tackling the complexities of technology assessment in renewable energy systems.

Despite extensive research on Wave Energy Converters (WECs) and Multi-Criteria Decision Making (MCDM), there is a notable gap in integrating advanced fuzzy logic with objective weighting methods for WEC benchmarking Specifically, the application of spherical fuzzy sets in conjunction with the MEREC method is limited, highlighting an opportunity to enhance objectivity and precision in WEC assessments by addressing uncertainty and subjectivity Additionally, while methods like CoCoSo effectively balance competing criteria, their use in WEC benchmarking, particularly with spherical fuzzy logic, has not been thoroughly explored This study seeks to bridge these gaps by developing a fuzzy-based, objectively weighted decision-making approach, thereby refining the methodology for sustainable energy decisions.

This study employs an integrated Multi-Criteria Decision-Making (MCDM) approach to effectively address the complex challenges of benchmarking Wave Energy Converter (WEC) technologies By combining various MCDM methodologies, the approach evaluates a diverse set of criteria, refining the decision-making process for enhanced accuracy, comprehensiveness, and reliability This innovative strategy advances the benchmarking efforts in the wave energy sector, facilitating more informed and effective decisions.

This study aims to enhance the benchmarking process of Wave Energy Converter (WEC) technologies by integrating the objective weighting capabilities of the MEREC method with the detailed decision analysis provided by the SF-C0C0S0 approach By combining these methodologies, the research seeks to deliver a thorough and balanced evaluation of WEC technologies, highlighting their crucial role in promoting wave energy as a sustainable energy source This benchmarking not only identifies leading technologies but also informs policy decisions, guides research and development, and fosters industry-wide standards and best practices.

This study introduces an innovative, integrated Multi-Criteria Decision-Making (MCDM) approach for benchmarking Wave Energy Converter (WEC) technologies, which enhances the clarity, accuracy, and effectiveness of technology assessments By employing this groundbreaking methodology, the research aims to support the strategic development and deployment of wave energy converters, representing a significant advancement in the sustainable utilization of wave energy.

LITERATURE REVIEW

Studies of MCDM approaches

In Multi-Criteria Decision Making (MCDM), determining the weights of criteria is crucial, as it greatly impacts the final decision Two primary methods for weight assignment are identified: subjective and objective Subjective methods, such as the Analytic Hierarchy Process (AHP) and the Delphi method, rely on the decision-maker's judgment and expert opinions, but may introduce bias due to variability in expertise and consistency (Le & Nhicu, 2022c; Y Liu, Eckert, & Earl, 2020; Wang, Nguyen, Nhieu, & Hsueh, 2023) Conversely, objective methods assign weights based on the inherent structure of the decision matrix, minimizing personal bias.

External judgments play a crucial role in decision-making processes, as highlighted by Diakoulaki, Mavrotas, and Papayannakis (1995) Techniques such as the Entropy method and the CRITIC method are well-regarded for minimizing subjectivity by deriving weights from variations in criteria data (Diakoulaki et al., 1995; Peng, Zhang, & Luo, 2020; Shannon, 1948) Additionally, the MEREC method offers an objective approach to weighting criteria by evaluating how the removal of each criterion affects the decision-making process (Kcshavarz-Ghorabacc et al., 2021) Notably, MEREC is distinguished in the literature for its innovative perspective on the interdependencies and influence of each criterion within the overall decision-making framework.

Compromise solution-based methods focus on identifying solutions that are closest to the ideal while being furthest from the anti-ideal, emphasizing satisfactory decision-making rather than merely maximizing or minimizing criteria (Tavana et al., 2021) The literature highlights the VIKOR method, which ranks and selects solutions based on their proximity to the ideal solution, making it effective for balancing competing and non-commensurable criteria (Kutlu Giindogdu & Kahraman, 2019a) A recent advancement in this area is the Combined Compromise Solution (CoCoSo) method, which integrates results from three different compromise ranking methods to provide a comprehensive solution (Yazdani et al., 2019) CoCoSo has garnered attention for its robustness and capacity to produce stable and reliable rankings by addressing the limitations of individual compromise methods (Wang et al., 2023).

Fuzzy extensions of traditional multi-criteria decision-making (MCDM) methods, such as fuzzy AHP and fuzzy TOPSIS, have been widely researched and utilized in various domains due to their effectiveness in capturing the uncertainty inherent in subjective evaluations These methods enable a more sophisticated approach to decision-making (Wang, Pham, & Nhieu, 2021) Additionally, spherical fuzzy sets (SFS) enhance this framework by introducing a three-dimensional model that represents membership, non-membership, and hesitancy degrees, allowing for an even more precise representation of uncertainty (Kutlu Giindogdu).

Recent research has increasingly focused on the role of Spherical Fuzzy Sets (SFS) in decision-making, particularly in managing uncertainty and vagueness Kaushik D and Sankar K.R (2023) explored T-spherical fuzzy sets (T-SFS) and developed hybrid operators to reduce bias in Multi-Attribute Decision-Making (MADM) problems, as noted by Debnalh & Roy (2023) Their findings emphasize the effectiveness of weighted power partitioned neutral average and geometric operators within the T-SFS framework, particularly in applications like hydrogen refueling station site selection Additionally, Muhammad Saad and Ayesha Rafiq (2023) enhanced the applicability of T-SFS by introducing correlation coefficients, which proved useful in pattern recognition and decision-making, especially for selecting appropriate COVID-19 related solutions.

Arun Sarkar et al (2023) introduced the T-spherical fuzzy hypersoft set (T-SFHSS), which enhances the accuracy of fuzzy set calculations and proposes novel aggregation operators, demonstrating its effectiveness in managing imprecise data, particularly in natural agribusiness applications Additionally, D Ajay et al (2023) defined new exponential and Einstein exponential operational laws for spherical fuzzy sets (SFS) to improve decision-making in psychotherapy evaluations The integration of spherical fuzzy sets into multi-criteria decision-making (MCDM) methods significantly increases flexibility and expressiveness in handling uncertainty, as highlighted by various researchers (Gul, 2020; Kutlu Gundogdu & Kahraman, 2019; Le & Nhieu, 2022a; Wang, Pham, et al., 2023).

The literature on Multi-Criteria Decision-Making (MCDM) has evolved from traditional subjective methods to advanced data-driven techniques aimed at minimizing bias and effectively managing uncertainty Objective approaches such as MEREC, compromise solution methods like CoCoSo, and the implementation of fuzzy theory, especially spherical fuzzy sets, exemplify this progression.

13 significant advancements in the field, offering nuanced and robust frameworks for decision making in complex, multi-criteria environments.

MCDM Applications for WECs

The integration of Multi-Criteria Decision-Making (MCDM) approaches in assessing Wave Energy Converters (WECs) marks a significant leap in optimizing renewable energy technologies MCDM methods enable thorough evaluations by considering a range of technical, economic, social, and environmental factors critical for determining the feasibility and sustainability of WEC technologies Recent studies highlight the growing use of MCDM methodologies in WEC evaluations For instance, Sadaf Nasrollahi et al (2023) utilized the fuzzy Delphi method and PROMETHEE to identify the optimal WEC technologies for the Caspian Sea, favoring Pelamis based on comprehensive criteria Similarly, Shadmani et al (2023) crafted a novel MCDM strategy that integrates wave energy storage and production metrics to pinpoint ideal WEC deployment sites along Oman’s coast Additionally, Meng Shao et al (2024) combined GIS, MCDM, and ANN techniques to improve site selection and wave power forecasting for WPPs in Hainan Island, identifying suitable deployment areas Daekook Kang et al (2024) proposed an innovative hybrid MCDM methodology using fuzzy SWARA and ELECTRE, emphasizing point absorber technology as the most suitable WEC Moreover, Shabnam et al (2023) highlighted the potential of combining offshore wind and wave energy through MCDM methods for optimal site identification for integrated farms, while also noting the need for assessments of seabed conditions and climate change impacts.

Recent studies demonstrate the effectiveness of Multi-Criteria Decision-Making (MCDM) approaches in addressing the complexities of Wave Energy Converter (WEC) technology assessment and site selection These methodologies provide a structured framework for integrating various criteria, facilitating a comprehensive evaluation of potential technologies and locations The incorporation of fuzzy logic and advanced techniques enhances the decision-making process by accommodating uncertainty and subjectivity However, a significant gap persists in fully leveraging fuzzy-based objective weighting methods, especially in integrating spherical fuzzy sets with the MEREC method Most research continues to rely on traditional MCDM approaches, often overlooking the advancements in fuzzy logic that could better address uncertainty and hesitancy.

The existing research on Wave Energy Converters (WECs) and Multi-Criteria Decision-Making (MCDM) approaches provides valuable insights into the technical, environmental, and economic dimensions of WEC technologies However, there is a significant gap in integrating advanced fuzzy logic with objective weighting methods for evaluating these technologies Specifically, studies utilizing spherical fuzzy sets in conjunction with the MEREC method to enhance the weighting and evaluation process are scarce This gap presents an opportunity to develop a novel approach that improves the objectivity and precision of WEC technology assessments by addressing inherent uncertainties in decision-making Furthermore, while robust compromise solution-based methods like CoCoSo have been acknowledged for their effectiveness in balancing competing criteria, their application in WEC technology benchmarking, particularly when combined with spherical fuzzy logic, remains largely unexplored This study seeks to fill these gaps by creating and implementing a fuzzy-based, objectively weighted integrated decision-making approach, thereby offering a refined methodology for benchmarking WEC technologies that accurately reflects the complex, multidimensional nature of sustainable energy decisions.

METHODOLOGY

Preliminary

Fuzzy theory, established by Lotfi Zadeh in the 1960s, provides a mathematical framework for addressing uncertainty and imprecision, moving beyond traditional binary logic to incorporate degrees of truth This approach enables more realistic modeling of complex systems that lack clear boundaries A recent advancement in this field, Spherical Fuzzy Sets (SFS), enhances the handling of uncertainty by introducing a third parameter, thus improving the representation and processing of vagueness in data This three-dimensional model offers a comprehensive tool for managing uncertain information, proving valuable in areas such as artificial intelligence, decision-making, and complex system analysis.

2007 Spherical fuzzy sets Nonstationary fuzzy sets (Fatma & Cengiz) (Garibaldi & Ozen) \

Type-2 Intuitionist ic fuzzy sc

(Sambuc, Jahn, Grattan-Guiness ,Zadeh) (Atanassov)

Fig 1 Development of fuzzy sets (Le & Nhieu, 2022a)

Definition 1 In the universe of discourse s, the spherical fuzzy set N is defined by

N = {{s,(ớs(s),Mjv(s),7rs(s))|seS} (1) where

^m, fJa/,Tĩịỉí:s-^ [0,1] and 0 < ữ~(x) + ụ~(x) + Tĩ~(x) < 1 Vs e s (2)

The parameters ứ/ự(s),^(s)/and 71 fits) are the membership degree, non-membership degree, and hesitancy degree of each s to N respectively.

Definition 2 In the universe of discourse S ị and s2 two spherical fuzzy numbers (SFN) /V = ($N>U n ’K n ) nnd AỈ = (^M'^M'^ m ) have the basic operators are defined as Equation

Definition 3 Consider the weight vector (p = ((P2> —> where 0 < (Pi 0 (Kutlu Gundogdu & Kahraman, 2019b).

Definition 5 The defuzzied value (DV) of spherical fuzzy number N = (^N> P n ^ n ) is defined as Equation (15) (Mathew, Chakrabortly, & Ryan 2020).-

The proposed spherical fuzzy objectively weighting integrated decision-making approach

To take advantage of the advantages of MEREC and SF-C0C0S0, this study introduces an integrated approach which is performed according to the following procedure:

Step 1 Experts (k = 1 K) (or decision makers), who have expertise and experience in the field, are identified Based on their expertise, the weights of the experts are determined With the given SFN Ek = ($ Ẽk.ỊẤị^TTpk) representing the expertise of the kth expert, the weight (o)k) of the fcth expert is determined as Equation (16) (F Liu, Aiwu, Lukovac, & Vukic, 2018).

S?=1 ự - ((1 - ớ?,) + 4,+4)/3)2 J where Zk=i W j = 1 and 0 < dịk + ụịk + Tĩịk < ĩ

Table 1 illustrates the SFN Ẽk, which reflects the expertise of analysts or higher-level decision-makers in linguistic terms, based on attributes such as years of experience and qualifications.

Table 1 Linguistic terms and corresponding SFN for experts ’ expertise

Linguistic term Spherical fuzzy number

Step 2: The benchmarking criteria (/ = 1 /), and alternatives (i = 1 /) are defined based on literature review and experts’ opinions.

Step 3 Experts provide linguistic assessments of alternatives according to the criteria These linguistic assessments are then transformed into the corresponding SFNs as shown in Table 2, which is provided by experts or decision makers, to form SF decision matrices SF decision matrices are represented as Equation (17) In other applications, the linguistic scale can be defined by decision makers or experts The SFN values should be symmetrically distributed around the neutral point, designated as "Medium" (0.500, 0.500, 0.500), ensuring a balanced and consistent progression in the evaluation scale This symmetry implies that as judgments move from neutral to extremely positive or negative, the membership and non membership degrees adjust inversely, maintaining logical coherence (Farman et al., 2024; Le

Table 2 Linguistic terms and corresponding SFN for alternative assessments

Low (L) Slightly low (SL) Moderate (M) Slightly high (SH)

Step 4 The aggregated decision matrix is constructed using the SWAM based on experts’ weights as Equations (18)-( 19). where fin = swAM^nh.nl n^) = w^ + w2n)( + •■• + a>Kn^,

(18) I=1-i',“™ step 5 The crisp decision matrix is constructed according to Equation (15).

Step 6 The crisp decision matrix is normalized, according to Equations (21)-(22), to transform all criteria into the non-beneficial criteria. m = M, X ; where mij = min nợ nu

Step 7 The overall performance of the alternatives (Si) is calculated using a logarithmic measure with equal criteria weights according to Equation (23) It is based on the non-linear function as shown in Fig 2 Therefore, the smaller value of rtiij yield higher value of Sị.

Step 8 The removal overall performance of the alternatives (S'ij) calculated as the Step 7 but one-by-one criterion is removed from the matrix as shown on Equation (24) The removal overall performance of the alternatives (S'ij) is denoted for the overall performance of ith alternatives with the removal of jth criterion.

Step 9 The weights of criteria (Wj) are computed based on the removal effect (Ej) of the jth criterion according to Equation (25).

Step 10 Based on the criteria weight (W/) the weighted sequences of alternatives are determined using the SWAM and the SWGM according to Equation (26)-(27) They are denoted as SWAi and SWGiy respectively.

Step 11 The defuzzied value of SWAị and SWGi are determined and denoted as SWAị and SWGi according to Equation (15).

Step 12 The additive normalized importance (ộf) and the relative importance (C^) of the SWAM and the SWGM are calculated as Equation (28) and (29), respectively In addition, the trade-off importance (0f) of the alternatives is determined as Equation (30) with the stability and flexibility represented by the coefficient s, which is selected by the decision makers.

= - J - i min(Sl4Mi) min(SVEGj)' • '

Step 13 The final evaluation score (4>J of alternatives is determined as Equation (31) The final rank of alternatives is ranked in descending order of the value of Oj In other words, the best alternative has the largest value of 4\.

NUMERICAL RESULTS

WECs benchmarking by the proposed approach

The benchmarking of Wave Energy Conversion (WEC) technologies includes Oscillating Water Columns (OWC), Point Absorbers (PAB), Attenuators (ATE), and Overtopping Devices (OTD), along with Salinity Gradient Power (SGP) and Tidal Stream Turbines (TST) OWCs harness water movement in a chamber to generate electricity via an air turbine, making them ideal for areas with strong wave action Point absorbers convert wave motion into electricity using hydraulic or mechanical systems, offering adaptability across varying wave conditions Attenuators absorb wave energy, converting it to heat or sound while also protecting coastlines SGP devices, still under development, utilize salinity differences between seawater and freshwater as a promising energy source Tidal stream turbines, akin to wind turbines but for tidal currents, are effective in both shallow and deep waters, representing a well-established technology Overtopping devices efficiently harness water from wave barriers to drive turbines or pumps, particularly in high wave height regions.

To benchmark WEC technologies, a panel of six experts utilized the Delphi method to conduct a survey, as outlined in Table 3, which recommends corresponding SFNs based on their expertise The experts' weights were calculated using Equation (16) and are also presented in Table 3 Subsequently, during the Delphi interview process, the experts proposed benchmarking criteria (BC) and provided linguistic judgments for WEC technologies relevant to each BC Table 4 displays the benchmarking assessments and linguistic judgments from the first expert Utilizing the SFNs from Table 2, each expert's linguistic judgments were transformed into SFNs, resulting in the creation of individual SF benchmarking matrices based on the calculated weights.

21 the experts, which were obtained above, the individual benchmarking matrices are aggregated according to equations (18)-(I9) The aggregated SF benchmarking matrix is shown in Table

5 To start the procedure to determine the objective weights of the BCs, the defuzzification and normalization process is performed according to Equation (15) and Equation (22), respectively

As the results, the obtained the crisp benchmarking matrix and the normalized benchmarking matrix are presented in Table Al and Table A2 in Appendix As described in Equations (23)

(25), the removal effects of the benchmarking criteria are calculated as shown in Table 6.

Table 3 Experts ’ qualifications and weights

1 12 Wave energy research Ph.D Very high (0.85, 0.15,

2 13 Renewable energy manufacturer Ph.D Very high (0.85,0.15,

3 10 Renewable energy manufacturing Master High (0.60, 0.20,

4 8 Wave energy research Ph.D High (0.60, 0.20,

5 11 Renewable energy investment Master High (0.60, 0.20,

6 5 Wave energy research Master Medium (0.35, 0.25,

Table 4 Benchmarking criteria and linguistics judgments by the 1st expert

& Guercio, 2021; Xu Wang Xi, L SL SL SH SL L

Wang & Xu 2022) (Aderinto & Li 2019; Ahamed

McKee & Howard 2022; Curio el al

SL SH SL L SH SH

Sheng De Silva & Aggidis 2023) (Ahamed et al., 2022; Curio el al

2021: B Guo el al 2022; Xu cl al„ L SL SH M SL L

Power output (BC7) Operating range (BC8)

Ease of installation and maintenance (BC10)

(Curto et al., 2021; Rehman Alhcms.

Alam, Wang, & Toor, 2023) SL H SH SH H SH (Ahamed et al 2022: Curto et al

2021: B Guo el al 2022: c Guo el L SL SH H SL SH al., 2023; Xu et al., 2022)

(Cuno et al., 2021: Xu el al 2022) H H SH SL SL SH (Aderinto & Li 2019: Curto et al

(Cuno et al., 2021; Xu el al 2022) L SH SH H SH H (Curto et al 2021: Rehman el al

(Curto el al 2021; c Guo et al 2023;

Xu el al 2022) SH H SL SL M L

(Cuno et al., 2021; Xu el al 2022) H M M H SL L (Curto et al., 2021; Rehman et al

Table 5 The aggregated SF benchmarking matrix

BC owe PAB ATE SGP TST OTD

Table 6 The removal effect matrix

The objective weight of the benchmarking criteria for Wave Energy Converters (WECs) reveals that Grid Connection is the most critical factor, assigned a weight of 0.18, highlighting its importance for integration with existing power grids Required Wave Conditions closely follow with a weight of 0.15, emphasizing the need for efficient operation in diverse marine environments Cost is also significant at 0.11, underscoring the economic viability necessary for market penetration Efficiency and Ease of Installation and Maintenance each receive a weight of 0.09, reflecting the importance of effective energy conversion and practical deployment Moderately weighted factors like Robustness and Environmental Impact, both at 0.07, suggest they are important but less critical than the top criteria Survivability is weighted at 0.06, indicating resilience to extreme conditions is essential Power Output and Operating Range are assigned a weight of 0.03, viewed as baseline expectations rather than primary selection drivers The Technology Readiness Level, at 0.08, signifies the importance of technology maturity in decision-making, while Social Acceptance, with the lowest weight of 0.02, indicates that societal factors are less influential in the technical and economic evaluations of WEC technologies.

In the next stage, the aggregated SF benchmarking matrix is used to determine the Spherical

Equations (7)-(8) Then, their crisp values are obtained by Equation (15) and shown in Table

7 According to Equations (28)-(30), the importance parameters of WEC technologies are defined as shown in Table 8 Ultimately, the final benchmark score is calculated according to Equation (31).

Fig 3 The objective weight of benchmarking criteria

Table 7 The weighted sequences of WEC technologies

WEC Technology SWAi SWGi Crisp SWAM Crisp SWGM owe

PAB ATE SGP TST OTD

Table 8 SF-CoCoSo importance parameters (Ổ = 0 5)

WEC Technology Additive normalized importance

Relative importance Trade-off importance

Sensitivity analysis

This section presents a sensitivity analysis of the stability and flexibility coefficient (Ổ) in relation to the benchmarking results of Wave Energy Converter (WEC) technologies, uncovering significant trends and implications The analysis covers a range of Ô values from 0.1 to 0.9, demonstrating how changes in this coefficient influence the ranking and performance assessment of WEC technologies, as illustrated in Figures 4 and 5.

Owe technology demonstrates exceptional stability with consistent performance scores of 1.8 across various Ổ values, only slightly increasing to 1.9 at 5=0.9, indicating its robustness and reliability In contrast, PAB and SGP technologies are highly sensitive to changes in the stability and flexibility coefficient, with PAB scores rising significantly from 2.1 to 5.0 and SGP scores from 1.9 to 5.4 as the coefficient increases from 0.1 to 0.9 This variability highlights how the perceived effectiveness of these technologies for wave energy conversion can greatly depend on specific evaluation criteria Conversely, ATE technology shows a declining performance trend, with scores dropping from 1.8 to 1.4 as the 5 value increases, suggesting it may be less favorable when stability and flexibility are prioritized in decision-making.

Fig 4 The final benchmarking score of WECs according to the stability and flexibility coefficients

THE STABILITY AND FLEXIBILITY COEFFICIENT (Ổ)

- owe -PAB - ATE -SGP - TST -OTD

Fig 5 The ranking results of WECs according to the stability and flexibility coefficients

At higher Ố values, technologies such as PAB, SGP, TST, and OTD exhibit significant score increases, highlighting a preference for stability and flexibility in their selection This trend underscores the need to consider the operational environment and specific project requirements when choosing WEC technologies The notable rise in scores for these technologies at high 5 values, especially at 0.9, suggests that the SF-C0C0S0 method may overestimate their benefits under high stability and flexibility conditions Therefore, it is crucial to carefully calibrate the Ổ value to align with realistic operational expectations and project needs.

The benchmarking results for wave energy conversion (WEC) technologies reveal that PAB technology leads with a score of 2.0286, indicating its strong performance in efficiency, cost-effectiveness, grid compatibility, and adaptability to various wave conditions This high score suggests that PAB aligns well with industry priorities, including environmental impact and social acceptance, positioning it as a frontrunner in the sector Following PAB, the OWE technology scored 1.8745 and TST technology scored 1.7432, both demonstrating competitive strengths such as robust design, high power output, and operational reliability The slightly lower scores of OWE and TST compared to PAB may highlight areas for improvement or reflect strategic design trade-offs that influence their overall performance.

Fig 6 WEC technology final benchmarking score

ATE and SGP technologies are closely ranked with scores of 1.6079 and 1.6061, indicating they may share similar capabilities or challenges in meeting benchmarking criteria Their lower performance scores could be due to factors like higher costs, lower technology readiness levels, or less favorable environmental impacts However, these technologies might still provide specific advantages in the wave energy sector Meanwhile, OTD technology, with a score of 1.6580, suggests a balanced suite of attributes, placing it respectably in the evaluation This score indicates potential for targeted improvements and suitability for niche applications where its strengths are most beneficial.

This benchmarking study on Wave Energy Converter (WEC) technologies using the MEREC and SF-C0C0S0 methods provides critical insights for decision-making in the wave energy sector It serves as a strategic guide for industry leaders and investors to direct investments toward the most efficient and reliable WEC technologies, facilitating informed resource allocation in research and development Policymakers can leverage these findings to create regulations and incentives that promote the adoption of superior technologies Manufacturers can enhance their competitive positioning by emphasizing strengths identified through the benchmarking process in their marketing strategies Additionally, aligning supply chain decisions with the production needs of promising WEC technologies can lead to improved efficiency and cost-effectiveness Understanding the diverse risk profiles of each technology enables tailored risk mitigation strategies, while a balanced portfolio of WEC technologies can help spread risk and enhance resilience, especially for companies entering new markets Sustainability considerations are crucial, as managers can use environmental impact data to guide their organizations toward sustainable practices, fulfilling corporate social responsibility objectives Finally, insights on social acceptance are vital for public relations and stakeholder engagement, aiding in project approvals and community support Overall, the managerial implications of this study significantly influence investment, strategic planning, and operations, providing a roadmap for competitive advantage and progress toward sustainable energy solutions.

The study focused on the growing field of wave energy, highlighting the significant untapped potential of ocean waves as a renewable energy source It aimed to evaluate and benchmark Wave Energy Converter (WEC) technologies to identify the most efficient solutions To achieve this, an integrated approach was employed, utilizing the MEREC method to objectively assess key evaluation criteria, while the SF-C0C0S0 method was used to analyze and aggregate complex decision-making data, ultimately providing a final evaluation score for each WEC technology.

This study offers a comprehensive framework for benchmarking wave energy converter (WEC) technologies, aiding stakeholders in making informed decisions It enhances the use of integrated multi-criteria decision-making (MCDM) approaches in the renewable energy sector, effectively combining MEREC and SF-C0C0S0 to navigate complex decision-making scenarios Our findings establish a clear hierarchy of WEC technologies based on performance metrics such as efficiency, cost, environmental impact, and grid connectivity The PAB technology emerges as the leading option due to its superior performance, followed by the owe and TST technologies as viable alternatives Furthermore, the study underscores the significance of grid connectivity and adaptability to varying wave conditions as essential factors in the benchmarking process.

This study provides valuable insights into wave energy converter (WEC) technologies but acknowledges several limitations that warrant further research The evaluation criteria, while comprehensive, may not encompass all performance-influencing factors, suggesting a need to expand these criteria to include emerging elements as the renewable energy landscape evolves Additionally, the objectivity of the criteria weightings, though a strength, could be impacted by the dynamic nature of the wave energy market and technological advancements, indicating a need for adaptive methodologies that can respond to real-time changes Future research should explore more flexible weighting mechanisms and investigate alternative multi-criteria decision-making (MCDM) methods to capture the complexities of WEC technology assessment more effectively Continuous improvement in assessment methodologies is essential to align with technological progress and market shifts, encouraging validation against real-world performance data to ensure robustness and relevance Furthermore, integrating advancements in data analytics and artificial intelligence, such as machine learning for predictive analysis, could significantly enhance the benchmarking process in the wave energy sector.

Table Al The crisp benchmarking matrix

Table A2 The normalized benchmarking matrix

Aderinto, T., & Li, H (2019) Review on power performance and efficiency of wave energy converters Energies, /2(22), 4329.

Ahamed, R., McKee, K., & Howard, I (2022) A review of the linear generator type of wave energy converters’ power take-off systems Sustainability, I4(Ỉ6), 9936.

Ajay, D., Selvachandran, G., Aidring, J., Thong, p H., Son, L H., & Cuong, B c (2023)

Einstein exponential operation laws of spherical fuzzy sets and aggregation operators in decision making Multimedia tools and applications, 82(27), 41767-41790.

Ashraf, s., Abdullah, s., Mahmood T., Ghani, F., & Mahmood, T (2019) Spherical fuzzy sets and their applications in multi-attribute decision making problems Journal of intelligent & fuzzy systems, 36(3), 2829-2844 doi: 10.3233/jifs-172009

Bellman, R E., & Zadeh, L A (1970) Decision-Making in a Fuzzy Environment

Bertram, D V., Tarighaleslami, A H., Walmsley, M R w., Atkins, M J., & Glasgow, G D

E (2020) A systematic approach for selecting suitable wave energy converters for potential wave energy farm sites Renewable and Sustainable Energy Reviews, 132, 110011.

Chang, G., Jones, c A., Roberts, J D., & Neary, V s (2018) A comprehensive evaluation of factors affecting the levelized cost of wave energy conversion projects Renewable energy, Ỉ2 7, 344-354.

Choupin, O., Pinheiro Andutta, F., Etemad-Shahidi, A., & Tomlinson, R (2021) A decision making process for wave energy converter and location pairing Renewable and

Sustainable Energy Reviews, 147, 111225 doi: 10.1016/j.rser.2021.111225

Clemente, D., Rosa-Santos, p., & Taveira-Pinto, F (2021) On the potential synergies and applications of wave energy converters: A review Renewable and Sustainable Energy

Curio, D., Franzitta, V., & Guercio, A (2021) Sea wave energy A review of the current technologies and perspectives Energies, 14(20), 6604.

Debnath, K., & Roy, s K (2023) Power partitioned neutral aggregation operators for T- spherical fuzzy sets: An application to H2 refuelling site selection Expert Systems with

Diakoulaki, D., Mavrotas, G., & Papayannakis, L (1995) Determining objective weights in multiple criteria problems: The critic method Computers á Operations Research, 22(7), 763-770.

Farman, s., Khan F M., & Bibi, N (2024) T-Spherical fuzzy soft rough aggregation operators and their applications in multi-criteria group decision-making Granular Computing, 9(1), 6.

Folley, M., & Whittaker, T (2010) Spectral modelling of wave energy converters Coastal

Foteinis, s., & Tsoutsos, T (2017) Strategies to improve sustainability and offset the initial high capital expenditure of wave energy converters (WECs) Renewable and

Gallutia, D., Fard, M T., Soto, M G., & He, J (2022) Recent advances in wave energy conversion systems: From wave theory to devices and control strategies Ocean

Gao, Q., Ertugrul, N., Ding, B., & Negnevitsky, M (2020) Offshore Wind, Wave and

Integrated Energy Conversion Systems: A Review and Future Paper presented at the

Australasian Universities Power Engineering Conference, Hobart TAS, Australia.

Gul, s (2020) Spherical fuzzy extension of DEMATEL (SF-DEMATEL) International

Journal of Intelligent Systems, 35(9), 1329-1353 doi: 10.1002/int.22255

Recent studies have focused on the advancements in wave energy technology, highlighting key reviews in the field Guo et al (2022) provided an in-depth analysis of point absorber wave energy converters, published in the Journal of Marine Science and Engineering, emphasizing their potential in harnessing ocean energy Additionally, Guo et al (2023) explored the levelized cost of wave energy through a techno-economic model in their article in Energies, offering insights into the economic viability of wave energy solutions These reviews underscore the growing importance of innovative approaches to wave energy conversion and cost assessment in the pursuit of sustainable energy sources.

Harris, R E., Johanning, L., & Wolfram, J (2004) Mooring systems for wave energy converters: A review of design issues and choices Marec2004 180-189.

Hosseinzadeh, s., Etemad-Shahidi, A., & Stewart, R A (2023) Site selection of combined offshore wind and wave energy farms: a systematic review Energies, 16(4), 2074.

IEA (2022) World Energy Investment 2022 Retrieved from Paris: https://www.iea.org/rcports/world-cncrgy-investmcnt-2022

Kang, D., Suvitha, K Narayanamoorthy, s., Sandra, M., & Pamucar, D (2024) Evaluation of wave energy converters based on integrated ELECTRE approach Expert Systems with

Katsidoniotaki, E., Psarommatis, F., & Goteman, M (2022) Digital Twin for the Prediction of

Extreme Loads on a Wave Energy Conversion System Energies, /5(15), 5464.

Keshavarz-Ghorabaee, M., Amiri, M., Zavadskas, E K., Turskis, z., & Antucheviciene, J

(2021) Determination of Objective Weights Using a New Method Based on the Removal Effects of Criteria (MEREC) Symmetry, 13(4) doi:10.3390/sym 13040525 Kutlu Gundogdu, F., & Kahraman, c (2019) Extension of WASPAS with Spherical Fuzzy

Kutlu Giindogdu, F., & Kahraman, c (2019a) A novel VIKOR method using spherical fuzzy sets and its application to warehouse site selection Journal of intelligent & fuzzy systems, 37(ỉ), 1197-1211 doi: 10.3233/jifs-182651

Kutlu Gundogdu, F., & Kahraman, c (2019b) Spherical fuzzy sets and spherical fuzzy

TOPSIS method Journal of intelligent & fuzzy systems, 36(1), 337-352 doi:10.3233/jifs-181401

Langhamer, o (2010) Effects of wave energy converters on the surrounding soft-bottom macrofauna (west coast of Sweden) Marine environmental research, 69(5), 374-381.

Le, M.-T., & Nhieu, N.-L (2022a) A Behavior-Simulated Spherical Fuzzy Extension of the

Integrated Multi-Criteria Decision-Making Approach Symmetry, 14(6), 1136 doi:10.3390/sym 14061136

Le, M.-T., & Nhieu, N.-L (2022b) A Novel Multi-Criteria Assessment Approach for Post-

COVID-19 Production Strategies in Vietnam Manufacturing Industry: OPA-Fuzzy EDAS Model Sustainability, 14(S), 4732 doi: 10.3390/su 14084732

Le, M.-T., & Nhieu, N.-L (2022c) An Offshore Wind-Wave Energy Station Location

Analysis by a Novel Behavioral Dual-Side Spherical Fuzzy Approach: The Case Study of Vietnam Applied Sciences, Ỉ2( 10), 5201 doi: 10.3390/app 12105201

Liu et al (2018) developed a multicriteria model utilizing a single valued neutrosophic DEMATEL approach for selecting transport service providers, highlighting its effectiveness in decision-making processes within management and engineering contexts In a related study, Liu, Eckert, and Earl (2020) reviewed fuzzy Analytic Hierarchy Process (AHP) methods, emphasizing their application in decision-making scenarios that involve subjective judgments, thereby contributing valuable insights to the field of expert systems.

Magagna, D., & Uihlein, A (2015) Ocean energy development in Europe: Current status and future perspectives International Journal of Marine Energy, 11, 84-104.

Majidi, A G., Bingốlbali, B., Akpinar, A., & Rusu, E (2021) Wave power performance of wave energy converters at high-energy areas of a semi-enclosed sea Energy, 220, 119705.

Mathew, M., Chakrabortty, R K., & Ryan, M J (2020) A novel approach integrating AHP and TOPSIS under spherical fuzzy sets for advanced manufacturing system selection

Engineering Applications of Artificial Intelligence, 96, 103988. doi: 10.1016/j.engappai.2020.103988

Nasrollahi, s., Kazemi, A., Jahangir, M.-H., & Aryaee, S (2023) Selecting suitable wave energy technology for sustainable development, an MCDM approach Renewable energy, 202, 756-772.

Peng, X., Zhang, X., & Luo, z (2020) Pythagorean fuzzy MCDM method based on CoCoSo and CRITIC with score function for 5G industry evaluation Artificial Intelligence

Quadrelli, R., & Peterson, s (2007) The energy-climate challenge: Recent trends in CO2 emissions from fuel combustion Energy Policy, 35(11), 5938-5952. doi:https://doi.org/10.1016/j.enpol.2007.07.001

Rehman, s., Alhems, L M., Alam, M M., Wang, L., & Toor, z (2023) A review of energy extraction from wind and ocean: Technologies, merits, efficiencies, and cost Ocean

Saad, M., & Rafiq, A (2023) Correlation coefficients for T-spherical fuzzy sets and their applications in pattern analysis and multi-attribute decision-making Granular

Sahoo, s K., & Goswami, s s (2023) A comprehensive review of multiple criteria decision making (MCDM) Methods: advancements, applications, and future directions

Sarkar, A., Senapati, T., Jin, L., Mesiar, R., Biswas, A., & Yager, R R (2023) Sugeno-Weber triangular norm-based aggregation operators under T-spherical fuzzy hypersoft context

Shadmani, A., Nikoo, M R., Gandomi, A H., & Al-Rawas, G (2023) A Multi-Criteria

Decision-Making Approach for Selection of Wave Energy Converter Optimal Site.

In 1948, Shannon introduced the concept of entropy in his influential paper published in the Bell System Technical Journal, which has since laid the groundwork for information theory More recently, Shao et al (2024) developed a novel framework for site selection of wave power plants and wave forecasting, integrating Geographic Information Systems (GIS), Multi-Criteria Decision Making (MCDM), and Artificial Neural Networks (ANN) Their case study focused on Hainan Island in Southern China, highlighting innovative approaches to harnessing wave energy effectively.

Stojcic, M., Zavadskas, E K., Pamucar, D., Stevie, z., & Mardani, A (2019) Application of

MCDM methods in sustainability engineering: A literature review 2008-2018

In their 2021 study, Tavana et al present an innovative approach for supplier selection in reverse supply chains by integrating a group fuzzy best-worst method with a combined compromise solution utilizing Bonferroni functions This research, published in Cleaner Logistics and Supply Chain, offers a robust framework aimed at enhancing decision-making processes in sustainable supply chain management.

100009 doi: 10.1016/j.clscn.2O21.100009 Wang, C.-N., Nguyen, T.-D., Nhieu, N L., & Hsueh M.-H (2023) A Novel Psychological

Decision-Making Approach for Healthcare Digital Transformation Benchmarking in ASEAN Applied Sciences, 13(6) doi: 10.3390/app 13063711

MANAGERIAL IMPLICATIONS

This benchmarking study on Wave Energy Converter (WEC) technologies, utilizing MEREC and SF-C0C0S0 methods, delivers crucial insights for decision-making in the wave energy sector It offers industry leaders and investors a strategic ranking of WEC technologies, guiding investments toward the most efficient options and enabling informed allocation of research and development resources Policymakers can leverage these findings to create supportive regulations and incentives, while manufacturers can enhance their competitive positioning by highlighting strengths identified in the benchmarking process The study's outcomes also optimize supply chain decisions, aligning strategies with the production needs of promising WEC technologies for greater efficiency Understanding diverse risk profiles allows for tailored risk mitigation strategies, promoting investment diversification and resilience Sustainability considerations are vital, as managers can utilize environmental impact data to drive sustainable practices and enhance corporate reputation Additionally, the findings on social acceptance provide a framework for addressing public concerns, essential for project approvals and community support Overall, the managerial implications of this study significantly influence investment, strategic planning, and operations, offering a roadmap for competitive advantage and advancing sustainable energy solutions.

The study focused on the promising field of wave energy, highlighting the significant untapped potential of ocean waves as a renewable energy source It aimed to evaluate and benchmark Wave Energy Converter (WEC) technologies to identify the most efficient solutions for converting wave power into electricity To accomplish this, the research utilized an integrated approach, combining the MEREC and SF-C0C0S0 methods MEREC was employed to objectively assess the criteria essential for evaluating WEC technologies, while SF-C0C0S0 facilitated the aggregation and analysis of complex decision-making data, ultimately providing a final evaluation score for each technology.

This study offers a comprehensive framework for benchmarking wave energy converter (WEC) technologies, enabling stakeholders to make informed decisions By integrating multi-criteria decision-making (MCDM) approaches, it showcases the effectiveness of combining MEREC and SF-C0C0S0 in navigating complex decisions within the renewable energy sector Our research establishes a clear hierarchy of WEC technologies based on key performance criteria such as efficiency, cost, environmental impact, and grid connectivity The findings position PAB technology as the leading option due to its overall superior performance, with owe and TST technologies emerging as viable alternatives Furthermore, the study underscores the significance of grid connection and adaptability to varying wave conditions as essential factors in the benchmarking process.

This study highlights valuable insights into wave energy converter (WEC) technologies while acknowledging limitations that suggest areas for future research The current evaluative criteria, though comprehensive, may not encompass all factors affecting WEC performance, necessitating a broader scope that includes emerging elements as the renewable energy landscape evolves Additionally, the objectivity of criteria weightings, while a strength, may need to adapt to the dynamic wave energy market and technological advancements, pointing to the need for flexible methodologies that can adjust in real-time Future studies should explore alternative multi-criteria decision-making (MCDM) methods that could better capture the complexities of WEC technology assessment, potentially leading to more efficient benchmarking approaches Continuous improvement in assessment methodologies is essential to align with ongoing technological and market developments, encouraging research that expands criteria, investigates new MCDM methods, and validates benchmarking processes against real-world data Furthermore, integrating advancements in data analytics and artificial intelligence, such as machine learning for predictive analysis, could significantly enhance the benchmarking process in the wave energy sector.