The T-Spherical Fuzzy Einstein Interaction Operation Matrix Energy Decision-Making Approach: The Context of Vietnam Offshore Wind Energy Storage Technologies Assessment

Abstract

Fuzzy multi-criteria decision making (FMCDM) is a critical field that addresses the inherent uncertainty and imprecision in complex decision scenarios. This study tackles the significant challenge of evaluating energy storage technologies (ESTs) in Vietnam’s offshore wind sector, where traditional decision-making models often fall short due to their inability to handle fuzzy data and complex criteria interactions effectively. To overcome these limitations, the novel T-spherical fuzzy Einstein interaction operation matrix energy decision-making approach is introduced. This methodology integrates T-spherical fuzzy sets with matrix energy concepts and Einstein interaction operations, thereby eliminating the need for traditional aggregation processes and criteria weight determinations. My approach provides a structured evaluation of ESTs, highlighting that hydrogen storage, among others, demonstrates significant potential for high energy capacity and long-term storage. The findings not only underscore the robustness of this new method in managing the complexities of renewable energy assessment but also offer a comprehensive tool for selecting the most suitable ESTs to support Vietnam’s energy transition strategies. This study recognizes limitations related to data dependency, which could affect the generalizability of the results. Future research is suggested to expand the ESTs considered and integrate extensive real-world operational data, aiming to deepen the exploration of economic impacts and long-term viability of these technologies. This revised approach emphasizes both the challenge of evaluating ESTs under uncertain conditions and my innovative solution, enhancing the relevance and applicability of the findings.

1. Introduction

The evolution of fuzzy set theory initiated with Zadeh’s pioneering introduction of a system in which objects could have varying degrees of membership, transcending the traditional binary categorizations [1,2]. This significant enhancement broadened the scope of set theory, especially in contexts where binary logic was found to be too limiting. Building upon this fundamental idea, Atanassov introduced intuitionistic fuzzy sets (IFS), which incorporated both membership and non-membership functions, enriching the framework for uncertainty representation [3]. Despite their utility, these sets were limited by the requirement that the combined degrees of membership and non-membership should not surpass one, spurring further developments in the domain. The introduction of Pythagorean fuzzy sets (PyFS) by Yager was a crucial evolution that eased these restrictions, allowing the square of the membership and non-membership degrees to remain within the unit interval [0, 1] [4]. This adjustment provided decision-makers with more latitude while preserving the coherence needed for robust decision making. Yager further advanced the field by proposing the q-rung orthopair fuzzy set (q-ROFS), which eliminated the overarching constraints on assigning membership and non-membership values, thus offering a more detailed and nuanced framework for depicting uncertainty [5]. Following this development, Cuong introduced the picture fuzzy set (PFS), adding the dimensions of abstention and refusal [6]. However, like earlier models, it encountered limitations when the sum of these dimensions exceeded one, highlighting the need for continued refinement. Addressing this challenge, Kutlu et al. introduced spherical fuzzy sets (SFS) and later developed T-spherical fuzzy sets (T-SFS) [7,8,9], which allowed for the square sum of membership, abstention, and non-membership degrees to fit within the unit interval, thus providing decision-makers with unparalleled flexibility in quantifying uncertainties.

The literature on the application of T-SFS in multiple criteria decision making (MCDM) is burgeoning, highlighting its versatility and robustness in dealing with ambiguity and imprecise data. Ju et al. focus on multi-criteria group decision making (MCGDM) with incomplete weight information in a T-spherical fuzzy setting [10]. They propose several operation laws and aggregation operators, such as the T-spherical fuzzy weighted averaging interaction (T-SFWAI) and the T-spherical fuzzy weighted geometric interaction (T-SFWGI) operators. The study extends traditional TODIM approaches to accommodate T-spherical fuzzy numbers, offering comprehensive sensitivity and comparative analyses to demonstrate the robustness and adaptability of their proposed methods. In 2022, Fan et al. introduce a novel MCDM method within a T-spherical fuzzy environment, presenting a technique using the correlation coefficient and standard deviation (CCSD) to determine attribute weights when such information is partially or completely unknown [11]. They further explore a T-spherical fuzzy complex proportional assessment (COPRAS) method, providing a numerical example to validate its feasibility and effectiveness in practical scenarios. Later, Khan and his team extend the framework by integrating Archimedean t-conorm (ATCN) and t-norm (ATN), essential for creating expansive operational rules within T-spherical fuzzy structures [12]. Their research develops and examines new weighted averaging and geometric aggregation operators based on these norms, illustrating the application through a numerical example that underscores the superiority of these methods in solving MADM problems. Currently, Farman et al. (2024) introduce T-spherical fuzzy soft rough sets (TSFSRSs) as a model to tackle uncertainties in MCGDM, detailing a variety of aggregation operators and their practical applications in real-life scenarios [13]. Their approach aims to capture the nuances of parameterized conflicting information, showing how these models surpass existing techniques in terms of representation capabilities. Further, Eti et al. explore the efficiency of solar panels in minimizing hospital energy costs using a T-spherical fuzzy decision-making trial and evaluation laboratory (DEMATEL) method. They innovate by integrating aspects of the technique for order preference by similarity to ideal solution (TOPSIS) to enhance the classical DEMATEL approach, demonstrating the reliability and coherence of their findings across different t values [14]. Rani and Mishra develop an integrated decision-making method for selecting electric vehicle charging station (EVCS) locations under the Fermatean fuzzy set (FFS) context, effectively handling uncertainty. Their approach combines multi-objective optimization, maximizing deviation method, and novel Einstein aggregation operators within the FFS environment, demonstrating its effectiveness through an illustrative study [15]. Together, these studies underscore the dynamic development of T-SFS and Einstein aggregation operators applications in MCDM, each contributing unique insights and methodologies that enhance decision making in environments characterized by uncertainty and fuzziness.

The concept of matrix energy, developed by Ivan Gutman in the 1970s, is a significant measure in algebraic graph theory, originally aimed at studying molecular energy levels in chemistry [16]. It is defined as the sum of the absolute values of a matrix’s eigenvalues. This measure has since been widely applied in various fields including network analysis, physics, and graph theory. Matrix energy is particularly valuable for analyzing the structural properties and stability of graphs [17]. The energy provides insights into the graph’s connectivity, symmetry, and irregularity. Its theoretical and practical implications make it a critical tool for both understanding complex systems and solving real-world problems [18]. Recent advancements in the application of matrix energy to fuzzy set theories and MCDM have highlighted its potential to significantly improve decision-making processes by integrating uncertainty and precision management. Notable among these advancements are the studies conducted by Donbosco and Ganesan (2022), Li and Ye (2023), and Rui et al. (2023), each contributing unique perspectives and methodologies to the field [19,20,21]. In 2022, Donbosco and Ganesan introduced the concept of Rough neutrosophic matrix energy in the context of MCDM, exploring its utility in selecting the optimal location for school construction. Their approach combines rough set theory with neutrosophic set theory, enhancing decision making under conditions of indeterminacy [19]. The energy of the rough neutrosophic matrix, as defined in their study, incorporates both the lower and upper energy limits, which facilitates a more structured evaluation of decision-making criteria. Later, Li and Ye (2023) extended matrix energy to intuitionistic fuzzy matrices, addressing a significant gap in the literature by applying matrix energy to intuitionistic fuzzy set information in MCDM. Their methodology not only captures true and false matrix energies but also aligns these with attribute weights, decision maker weights, and attribute values, thus providing a comprehensive framework that enhances decision making in healthcare facility location [20]. Meanwhile, the study of Rui et al. extends the concept of matrix energy to linguistic and fuzzy environments by introducing the energy of a linguistic neutrosophic matrix (LNM), aimed at improving MCGDM processes. It presents a novel MCGDM technique based on LNM energy, covering decision-maker weights, criteria weights, and alternative evaluations. The technique’s effectiveness is demonstrated through a case study on hospital location selection, confirming its validity and usability in complex decision-making scenarios [21]. While T-SFS has been applied in MCDM, their combination with matrix energy remains underexplored as a significant theoretical gap. This presents an opportunity to develop a new framework that merges the robustness of T-SFS with the analytical capabilities of matrix energy, potentially transforming decision-making processes in environments characterized by uncertainty and fuzziness. The T-spherical Einstein interaction operation fuzzy matrix energy decision-making approach developed in this study distinguishes itself from these existing methods by integrating T-spherical fuzzy sets with matrix energy concepts and Einstein interaction operations in a novel framework. Unlike the methods by Donbosco and Ganesan, and Li and Ye, which focus on rough neutrosophic and intuitionistic fuzzy sets, respectively, our approach leverages the flexibility and multidimensional nature of T-spherical fuzzy sets, providing a more detailed representation of uncertainty and membership degrees. Moreover, while Rui et al. focus on linguistic aspects within neutrosophic settings, our method applies a broader spectrum of fuzzy logic, combining it with matrix energy to enhance the decision-making process without the need for traditional aggregation or weight determination processes. This not only simplifies the complexity inherent in other methods but also offers a direct and efficient way to handle multi-criteria decision making under uncertainty.

Offshore wind energy has emerged as a pivotal component of Vietnam’s strategy to diversify its energy portfolio and reduce carbon emissions [22,23]. The geographic and climatic conditions of Vietnam, particularly along its extensive coastline, offer significant potential for offshore wind farms [24]. However, like all renewable energy sources, offshore wind energy is subject to intermittency issues due to the variable nature of wind speeds [25]. This variability poses challenges for grid stability and the consistent delivery of power, necessitating robust solutions for energy storage [26]. The energy storage technologies (ESTs) are critical in mitigating the intermittency of wind energy and ensuring a reliable and continuous energy supply. ESTs provide a means to store excess energy generated during peak wind periods and release it during lulls, thereby balancing energy supply with demand and stabilizing the electrical grid [27,28,29]. The selection of appropriate ESTs for offshore wind applications in Vietnam is a complex decision-making problem that involves multiple criteria and stakeholders. In 2021, Thanh et al. assess the financial viability of grid-tied rooftop solar power systems with and without battery storage in Vietnam’s northeast region, finding both to be financially viable [30]. However, they noted that the addition of battery storage increases flexibility but at the cost of economic feasibility, leading to a longer payback period and significantly lower profits due to the high costs of inverters and batteries. Meanwhile, Wang et al. recently explore alternative metal-ion batteries to lithium-ion in Vietnam, using a Fuzzy MCDM approach to rank sodium-ion, magnesium-ion, and other batteries, identifying sodium-ion batteries as the most promising. Both studies highlight critical economic and sustainability considerations for adopting energy storage technologies in emerging markets [31]. While existing research like Nguyen et al. and Wang et al. explores the viability and alternatives of energy storage technologies for solar power, there is a notable lack of practical, localized research on assessment ESTs with offshore wind energy in Vietnam.

This study sets out to address the above critical theoretical and practical gaps. The main objective is to develop and validate a novel decision-making framework by integrating T-SFS with matrix energy concepts and Einstein interaction operations. This approach aims to leverage the sophisticated decision-making capabilities of T-SFS and the structural analysis strengths of matrix energy, to effectively manage the complexities and uncertainties. Furthermore, this study will conduct a comprehensive assessment of various ESTs, focusing on their economic viability, sustainability, and practical applicability to Vietnam’s offshore wind sector. By accomplishing these objectives, the research will provide robust tools and methodologies that can significantly improve decision-making processes in the context of Vietnam’s renewable energy strategy and carbon emissions reduction goals.

The primary novelty of this approach lies not merely in the application of matrix energy to a dataset but in the integration of three advanced theoretical concepts—T-spherical fuzzy sets, matrix energy concepts, and Einstein interaction operations—into a unified decision-making framework. This integration represents a significant theoretical advancement that enriches the field of MCDM by enhancing its robustness and applicability under conditions of uncertainty and fuzziness. The applicability and practical effectiveness of this framework have been demonstrated specifically within the context of Vietnam’s offshore wind energy sector, addressing a critical need for effective decision-making tools in renewable energy. This practical application underscores not only the theoretical capabilities of the framework but also its real-world efficacy.

This study is organized into five main sections: introduction, methods, case study, discussion, and conclusion. The introduction outlines the background, motivation, and objectives of the research. The methods section describes the approach taken in this study. The case study and discussion analyze the findings from the research. Finally, the conclusion summarizes the key insights and discusses their implications.

2. Materials and Methods

2.1. Preliminaries

Definition 1

([32]). A T-spherical fuzzy set $\tilde{A}$ on the universe of discourse $X$ is defined as

$$\tilde{A}=\left\{\left(x,\alpha \left(x\right), \gamma \left(x\right),\beta \left(x\right)\right)\right|x\in X\}$$

(1)

$$\mathrm{where} \alpha ,\gamma ,\beta : X\to \left[0,1\right] \mathrm{and} 0\le {\alpha }^t\left(x\right)+{\gamma }^t\left(x\right)+{\beta }^t\left(x\right)\le 1 \forall t\in \mathbb{N},x\in X$$

(2)

The notation $\alpha \left(x\right), \gamma \left(x\right),$ and $\beta \left(x\right)$ are the degree of membership, hesitancy, and non-membership of $x$ to $\tilde{A}$, respectively. The refusal degree of $x$ in $\tilde{A}$ is defined as

$$r\left(x\right)=\sqrt[\mathrm{t}]{1-\left({\alpha }^t\left(x\right)+{\gamma }^t\left(x\right)+{\beta }^t\left(x\right)\right)}$$

(3)

 and the $\tilde{A}=\left(\alpha ,\gamma ,\beta \right)$ is known as T-spherical number (T-SFN).

Definition 2

([32]). The score value $\left(SV\right)$ of the T-SFN $\tilde{A}=\left(\alpha ,\gamma ,\beta \right)$ is defined as 

$$SV\left(\tilde{A}\right)={\alpha }^t-{\beta }^t$$

(4)

 and the accuracy value $\left(AV\right)$ of the T-SFN $\tilde{A}=\left(\alpha ,\gamma ,\beta \right)$ is defined as

$$AV\left(\tilde{A}\right)={\alpha }^t+{\gamma }^t+{\beta }^t$$

(5)

Definition 3

([32]). Consider two T-SFN ${\tilde{A}}_1=\left({\alpha }_1,{\gamma }_1,{\beta }_1\right)$ and ${\tilde{A}}_2=\left({\alpha }_2,{\gamma }_2,{\beta }_2\right)$, the comparison is defined as

$$\begin{gathered} {\tilde{A}}_1<{\tilde{A}}_2 if \mathrm{and} \mathrm{only} \mathrm{if} \\ i.SV\left({\tilde{A}}_1\right)<SV\left({\tilde{A}}_2\right) \mathrm{or} \\ ii.SV\left({\tilde{A}}_1\right)=SV\left({\tilde{A}}_2\right) \mathrm{and} AV\left({\tilde{A}}_1\right)<AV\left({\tilde{A}}_2\right) \end{gathered}$$

(6)

Definition 4

([33]). Consider two T-SFN ${\tilde{A}}_1=\left({\alpha }_1,{\gamma }_1,{\beta }_1\right)$ and ${\tilde{A}}_2=\left({\alpha }_2,{\gamma }_2,{\beta }_2\right)$, the Einstein interaction operations are defined as

$${\tilde{A}}_1\oplus {\tilde{A}}_2=\left(\begin{array}{c}\sqrt[t]{{\displaystyle \frac{\left(1+{\alpha }_1^t\right)\left(1+{\alpha }_2^t\right)-\left(1-{\alpha }_1^t\right)\left(1-{\alpha }_2^t\right)}{\left(1+{\alpha }_1^t\right)\left(1+{\alpha }_2^t\right)+\left(1-{\alpha }_1^t\right)\left(1-{\alpha }_2^t\right)}}}, \sqrt[t]{{\displaystyle \frac{\left(1+{\gamma }_1^t\right)\left(1+{\gamma }_2^t\right)-\left(1-{\gamma }_1^t\right)\left(1-{\gamma }_2^t\right)}{\left(1+{\gamma }_1^t\right)\left(1+{\gamma }_2^t\right)+\left(1-{\gamma }_1^t\right)\left(1-{\gamma }_2^t\right)}}},\\ \sqrt[t]{{\displaystyle \frac{2\left(\left(1-{\alpha }_1^t-{\gamma }_1^t\right)\left(1-{\alpha }_2^t-{\gamma }_2^t\right)-\left(1-{\alpha }_1^t-{\gamma }_1^t-{\beta }_1^t\right)\left(1-{\alpha }_2^t-{\gamma }_2^t-{\beta }_2^t\right)\right)}{\left(1+{\alpha }_1^t\right)\left(1+{\alpha }_2^t\right)+\left(1-{\alpha }_1^t\right)\left(1-{\alpha }_2^t\right)}}}\end{array}\right)$$

(7)

$${\tilde{A}}_1\otimes {\tilde{A}}_2=\left(\begin{array}{c}{\left({\displaystyle \frac{2\left(\left(1-{\beta }_1^t-{\gamma }_1^t\right)\left(1-{\beta }_2^t-{\gamma }_2^t\right)-\left(1-{\alpha }_1^t-{\gamma }_1^t-{\beta }_1^t\right)\left(1-{\alpha }_2^t-{\gamma }_2^t-{\beta }_2^t\right)\right)}{\left(1+{\beta }_1^t\right)\left(1+{\beta }_2^t\right)+\left(1-{\beta }_1^t\right)\left(1-{\beta }_2^t\right)}}\right)}^{{\displaystyle \frac{1}{t}}}, \\ {\left({\displaystyle \frac{\left(1+{\gamma }_1^t\right)\left(1+{\gamma }_2^t\right)-\left(1-{\gamma }_1^t\right)\left(1-{\gamma }_2^t\right)}{\left(1+{\gamma }_1^t\right)\left(1+{\gamma }_2^t\right)+\left(1-{\gamma }_1^t\right)\left(1-{\gamma }_2^t\right)}}\right)}^{{\displaystyle \frac{1}{t}}},{\left({\displaystyle \frac{\left(1+{\beta }_1^t\right)\left(1+{\beta }_2^t\right)-\left(1-{\beta }_1^t\right)\left(1-{\beta }_2^t\right)}{\left(1+{\beta }_1^t\right)\left(1+{\beta }_2^t\right)+\left(1-{\beta }_1^t\right)\left(1-{\beta }_2^t\right)}}\right)}^{{\displaystyle \frac{1}{t}}}\end{array}\right)$$

(8)

$$\omega {\tilde{A}}_1=\left(\begin{array}{c}\sqrt[t]{{\displaystyle \frac{{\left(1+{\alpha }_1^t\right)}^{\omega }-{\left(1-{\alpha }_1^t\right)}^{\omega }}{{\left(1+{\alpha }_1^t\right)}^{\omega }+{\left(1-{\alpha }_1^t\right)}^{\omega }}}},\sqrt[t]{{\displaystyle \frac{{\left(1+{\gamma }_1^t\right)}^{\omega }-{\left(1-{\gamma }_1^t\right)}^{\omega }}{{\left(1+{\alpha }_1^t\right)}^{\omega }+{\left(1-{\alpha }_1^t\right)}^{\omega }}}},\\ \sqrt[t]{{\displaystyle \frac{2\left({\left(1-{\alpha }_1^t-{\gamma }_1^t\right)}^{\omega }-{\left(1-{\alpha }_1^t-{\gamma }_1^t-{\beta }_1^t\right)}^{\omega }\right)}{{\left(1+{\alpha }_1^t\right)}^{\omega }+{\left(1-{\alpha }_1^t\right)}^{\omega }}}}\end{array}\right), \omega >0$$

(9)

$${\left({\tilde{A}}_1\right)}^{\omega }=\left(\begin{array}{c}\sqrt[t]{{\displaystyle \frac{2\left({\left(1-{\beta }_1^t-{\gamma }_1^t\right)}^{\omega }-{\left(1-{\alpha }_1^t-{\gamma }_1^t-{\beta }_1^t\right)}^{\omega }\right)}{{\left(1+{\beta }_1^t\right)}^{\omega }+{\left(1-{\beta }_1^t\right)}^{\omega }}}},\\ \sqrt[t]{{\displaystyle \frac{{\left(1+{\gamma }_1^t\right)}^{\omega }-{\left(1-{\gamma }_1^t\right)}^{\omega }}{{\left(1+{\gamma }_1^t\right)}^{\omega }+{\left(1-{\gamma }_1^t\right)}^{\omega }}}},\sqrt[t]{{\displaystyle \frac{{\left(1+{\beta }_1^t\right)}^{\omega }-{\left(1-{\beta }_1^t\right)}^{\omega }}{{\left(1+{\beta }_1^t\right)}^{\omega }+{\left(1-{\beta }_1^t\right)}^{\omega }}}}\end{array}\right), \omega >0$$

(10)

Remark 1.

Definitions 1, 2, 3, and 4 can be reduced for 

(1) 

PFSs if $t=1$;

(2) 

SFSs if $t=2$;

(3) 

q-ROFSs if $\gamma =0$;

(4) 

PyFSs if t = 2 and γ = 0;

(5) 

IFSs if t = 1 and γ = 0;

(6) 

FSs if t = 1, γ = 0, and β = 0.

Definition 5

([34]). Let $Q_k\left(\mathbb{C}\right)$ be the space of $k\times k$ matrices with entries in $\mathbb{C}$ and $L$ be a matrix in $Q_k\left(\mathbb{C}\right)$. The energy of $L$ can be defined as 

$$E\left(L\right)={\displaystyle \sum }_{j=1}^k\left|{\lambda }_j-\mu \right|$$

(11)

where ${\lambda }_j$ and $\mu$ are the eigenvalues of $L$ and the mean of eigenvalues, respectively.

Definition 6.

Let $\tilde{L}$ be the T-spherical fuzzy square matrix with the size $k\times k$. The matrix $\tilde{L}$ can be expressed as three submatrices. Those submatrices contain the membership values $\left(x_{ij}\right)$, the hesitancy values $\left(y_{ij}\right)$, and the non-membership values $\left(z_{ij}\right)$. It can be denoted as 

$$\tilde{L}={\left[L\left({\alpha }_{ij}^t\right),L\left({\gamma }_{ij}^t\right),L\left({\beta }_{ij}^t\right) \right]}_{k\times k}$$

(12)

where $x_{ij}\in L{\left({\alpha }_{ij}^t\right)}_{k\times k}, y_{ij}\in L{\left({\gamma }_{ij}^t\right)}_{k\times k},$ and $z_{ij}\in L{\left({\beta }_{ij}^t\right)}_{k\times k}$.

The energy of the T-spherical fuzzy matrix $L$ can be defined as 

$$E\left(\tilde{L}\right)=\left[L{\left({\alpha }_{ij}^t\right)}_{k\times k},L{\left({\gamma }_{ij}^t\right)}_{k\times k},L{\left({\beta }_{ij}^t\right)}_{k\times k} \right]=\left[{\displaystyle \sum }_{j=1}^k\left|{\lambda }_j-{\mu }_{\lambda }\right|,{\displaystyle \sum }_{j=1}^k\left|{\vartheta }_j-{\mu }_{\vartheta }\right|,{\displaystyle \sum }_{j=1}^k\left|{\varphi }_j-{\mu }_{\varphi }\right|\right]$$

(13)

where ${\lambda }_j$, ${\vartheta }_j$, and ${\varphi }_j$ are the eigenvalues of the membership, hesitancy, and non-membership values, respectively. The ${\mu }_{\lambda }, {\mu }_{\vartheta }$, and ${\mu }_{\varphi }$ are the mean values of ${\lambda }_j$, ${\vartheta }_j$, and ${\varphi }_j$, respectively.

2.2. The T-SF Einstein Interaction Operation Matrix Energy Decision-Making Approach

In this section, the proposed T-SF Einstein interaction operation matrix energy decision-making approach is described according to the following steps as shown in Figure 1.

Step 1. Define the group of $K$ decision-makers ($k=1\dots K)$.

Step 2. Let a given T-SFNs ${\tilde{Q}}_k=\left({\alpha }_k^t,{\gamma }_k^t,{\beta }_k^t\right)$ is the degree of expertise of $k$th decision-maker. The weight $\left({\omega }_k\right)$ of $k$th decision maker is determined as

$${\omega }_k={\displaystyle \frac{1-\sqrt[t]{\frac{\left(\left(1-{\alpha }_k^t\right)+{\beta }_k^t+{\gamma }_k^t\right)}{3}}}{{{\displaystyle \sum }}_{k=1}^K\left(1-\sqrt[t]{\frac{\left(\left(1-{\alpha }_k^t\right)+{\beta }_k^t+{\gamma }_k^t\right)}{3}}\right)}} k=1\dots K$$

(14)

Step 3. Through the Delphi method procedure, the $I$ alternatives $\left(i=1\dots I\right)$ and the $J$ evaluation criteria $\left(j=1\dots J\right)$ are defined by the decision-makers perspectives.

Step 4. Each decision-maker provides linguistic assessment on the influence of criteria. Then, those linguistic assessments are transformed into the T-SFNs to construct the T-SF criteria influence matrix $\left(\widetilde{CRI}\right)$ as below.

$$\widetilde{CRI}={\left[\left({\alpha }_{kj}^t,{\gamma }_{kj}^t,{\beta }_{kj}^t\right)\right]}_{K\times J}$$

(15)

Step 5. Establish the weighted T-SF criteria influence matrix by multiply the T-SF criteria influence matrix and weight of decision makers according to Equation (9) as

$$\widetilde{wCRI}={\omega }_k\widetilde{CRI}={\left[{\omega }_k\left({\alpha }_{kj}^t,{\gamma }_{kj}^t,{\beta }_{kj}^t\right)\right]}_{K\times J}={\left[\left(a_{kj}^t,b_{kj}^t,c_{kj}^t\right)\right]}_{K\times J}$$

(16)

Step 6. The crisp submatrices of weighted criteria influence for the membership $\left(wCR{I}^m\right)$, the hesitancy $\left(wCR{I}^h\right)$, and non-membership $\left(wCR{I}^n\right)$ values are defined as

$$wCR{I}^m={\left[a_{kj}^t\right]}_{K\times J}$$

(17)

$$wCR{I}^h={\left[b_{kj}^t\right]}_{K\times J}$$

(18)

$$wCR{I}^n={\left[c_{kj}^t\right]}_{K\times J}$$

(19)

Step 7. Decision-makers provide linguistic assessment on the performance of each alternative according to criteria. Then, those linguistic assessments are transformed into the T-SFNs to construct the T-SF decision matrix $\left({\widetilde{ATL}}_i\right)$ as below.

$${\widetilde{ATL}}_i={\left[\left({\alpha }_{jk}^t,{\gamma }_{jk}^t,{\beta }_{jk}^t\right)\right]}_{J\times K} i=1\dots I$$

(20)

Step 8. Establish the weighted T-SF decision matrix by multiplying the T-SF decision matrix and weight of decision makers according to Equation (9) as

$${\widetilde{wATL}}_i={\omega }_k{\widetilde{ATL}}_i={\left[{\omega }_k\left({\alpha }_{jk}^t,{\gamma }_{jk}^t,{\beta }_{jk}^t\right)\right]}_{J\times K}={\left[\left(x_{jk}^t,y_{jk}^t,z_{jk}^t\right)\right]}_{J\times K} i=1\dots I$$

(21)

Step 9. The weighted crisp decision submatrices for the membership $\left(wAT{L}_i^m\right)$, the hesitancy $\left(wAT{L}_i^m\right)$, and non-membership $\left(wAT{L}_i^n\right)$ values of each alternative are defined as

$$wAT{L}_i^m={\left[x_{jk}^t\right]}_{J\times K} i=1\dots I$$

(22)

$$wAT{L}_i^h={\left[y_{jk}^t\right]}_{J\times K} i=1\dots I$$

(23)

$$wAT{L}_i^n={\left[z_{jj}^t\right]}_{J\times K} i=1\dots I$$

(24)

Step 10. For each alternative, the performance square matrices of membership $\left(R_i^m\right)$, hesitancy $\left(R_i^h\right)$, and non-membership $\left(R_i^n\right)$ are constructed by multiplying the weighted decision matrices and the weighted criteria influence matrices as below.

$$R_i^m=wCR{I}^m\times wAT{L}_i^m={\left[a_{kj}^t\right]}_{K\times J}\times {\left[x_{jk}^t\right]}_{J\times K}={\left[r_{lk}^m\right]}_{K\times K } i=1\dots I$$

(25)

$$R_i^h=wCR{I}^h\times wAT{L}_i^h={\left[b_{kj}^t\right]}_{K\times J}\times {\left[y_{jk}^t\right]}_{J\times K}={\left[r_{lk}^h\right]}_{K\times K} i=1\dots I$$

(26)

$$R_i^n=wCR{I}^n\times wAT{L}_i^n={\left[c_{kj}^t\right]}_{K\times J}\times {\left[z_{jj}^t\right]}_{J\times K}={\left[r_{lk}^n\right]}_{K\times K} i=1\dots I$$

(27)

Step 11. Determine the energy of the performance square matrices according to Equation (11) as

$$E\left(R_i^m\right) i=1\dots I$$

(28)

$$E\left(R_i^h\right) i=1\dots I$$

(29)

$$E\left(R_i^n\right) i=1\dots I$$

(30)

Step 12. The final score of alternatives can be calculated as follows. In which, the energy of membership values is the driver of the final scores. Thus, the higher the final score, the better alternative.

$$Scor{e}_i=E\left(R_i^m\right)-E\left(R_i^h\right)-E\left(R_i^n\right) i=1\dots I$$

(31)

3. Case Study

This section focuses on the practical application of the novel decision-making framework developed for evaluating ESTs in Vietnam’s offshore wind sector. It delves into the specific criteria and suitability for ESTs in context of Vietnam, addressing the critical need for robust energy storage solutions to manage the intermittency of wind energy. In this study, the linguistic judgements are considered with corresponding T-SFNs as shown in

Table 1. The linguistic judgement and corresponding T-SFNs.

Linguistic Judgments	Notation	T-SFN $\left(\mathit{\alpha },\mathit{\gamma },\mathit{\beta }\right)$
Extremely low	EL	0.040, 0.040, 0.960
Very low	VL	0.155, 0.155, 0.845
Low	L	0.270, 0.270, 0.730
Slightly low	SL	0.385, 0.385, 0.615
Medium	M	0.500, 0.500, 0.500
Slightly high	SH	0.615, 0.385, 0.385
Very high	VH	0.730, 0.270, 0.270
High	H	0.845, 0.155, 0.155
Extremely high	EH	0.960, 0.040, 0.040

with $t=2$. For other applications, the definition of linguistics judgments can freely set up by decision makers as the advantages of T-SFS MCDM methods.

Then, a panel of eight high qualification experts in the field of renewable energy in Vietnam were invited to independently participate in this study’s Delphi process as decision makers (DM). In

Table 2. The decision makers’ qualifications.

No.	Highest Degree	Years of Experience	Position Title	Expertise Linguistics Judgment	Expertise T-SFN Judgment
DM1	PhD in Energy Systems Engineering	20	Chief Technology Officer	Extremely High	(0.960, 0.04, 0.04)
DM2	PhD in Renewable Energy Technologies	15	Director of Innovation	High	(0.845, 0.155, 0.155)
DM3	PhD in Environmental Science	18	Head of Sustainability Research	Very High	(0.730, 0.270, 0.270)
DM4	PhD in Electrical Engineering (Smart Grids)	12	Lead Grid Integration Engineer	Very High	(0.730, 0.270, 0.270)
DM5	Master of Science in Energy Policy	25	Senior Policy Advisor	High	(0.845, 0.155, 0.155)
DM6	Master of Engineering in Mechanical Engineering	20	Renewable Energy Project Manager	Very High	(0.730, 0.270, 0.270)
DM7	PhD in Chemical Engineering	10	Energy Storage Specialist	Slightly High	(0.615, 0.385, 0.385)
DM8	Master of Business Administration (Energy Management)	15	Vice President of Strategic Investments	Very High	(0.730, 0.270, 0.270)

, their expertise and the T-SFN’s corresponding with their expertise are presented. Based on those expertise T-SFN judgment, the decision makers’ weight $\left({\omega }_k\right)$ is calculated according to Equation (14) as shown in Figure 2.

In the next step of the proposed approaches, by in-depth interviews with DMs, ESTs and criteria to evaluate their suitability for the Vietnamese offshore wind energy context are identified as shown in

Table 3. The considered energy storage technologies for the Vietnamese offshore wind energy context.

EST	Notation	Description	Key Attributes
Lithium-Ion Batteries	EST-1	Widely used due to their high energy density and efficiency. They are commonly found in portable electronics, electric vehicles, and increasingly in grid storage.	High energy density, fast charging capabilities, good cycle life, but concerns over cost, resource availability (like lithium and cobalt), and safety issues related to overheating.
Flow Batteries	EST-2	These batteries store chemical energy in external tanks instead of within the battery container. They are well-suited for stationary storage with long discharge times.	Lower energy density but offer longer cycle life and quicker response times. They are scalable and have less environmental impact than lithium-ion batteries.
Compressed Air Energy Storage (CAES)	EST-3	Stores energy by compressing air in underground caverns or containers, releasing it to generate power when needed.	Potential for large-scale energy storage, low cost per cycle, and long lifespan. However, it requires specific geological conditions and can be less efficient due to thermal losses.
Flywheel Energy Storage	EST-4	Uses rotational energy stored in a spinning flywheel to maintain electricity flow. Best suited for short-duration high-quality power applications.	High power density, long lifetime, and minimal environmental impact. Efficiency drops over time due to frictional losses.
Hydrogen Storage	EST-5	Stores energy by converting electricity into hydrogen via electrolysis. The hydrogen can be stored and later used to generate electricity via fuel cells.	Offers a clean energy solution with the potential for high energy capacity and long-term storage. Challenges include high costs, energy losses during conversion processes, and infrastructure requirements.

and

Table 4. The proposed assessment criteria for ESTs.

Criteria	Notation	Description
Technical Performance	CRI-1	Technical Performance assesses how effectively an energy storage technology (EST) operates, focusing on its efficiency in energy conversion, the compactness of energy storage (energy density), and its dependability over time and various conditions (reliability).
Economic Factors	CRI-2	Economic Factors determine the financial viability of ESTs, considering both the cost-effectiveness relative to benefits and performance, and the return on investment, which measures the profitability of the technology compared to its costs.
Environmental Impact	CRI-3	Environmental Impact evaluates the ecological footprint of ESTs, including their sustainability in terms of environmental friendliness and the ease and environmental effects of recycling or disposing of materials at the end of their life cycle.
Scalability and Flexibility	CRI-4	Scalability and Flexibility gauge an EST’s ability to be efficiently scaled up or down to meet different needs and its adaptability to different operational demands or integration with other systems.
Regulatory and Safety Compliance	CRI-5	Regulatory and Safety Compliance involves ensuring that ESTs meet all applicable regulations and safety standards, covering everything from environmental compliance to operational risks.
Market and Social Factors	CRI-6	Market and Social Factors explore the commercial preparedness of ESTs for widespread deployment and their acceptance by the public, which can significantly influence adoption rates.

. Next, DMs provided their opinions, in linguistics assessment form, on the level of influence for each criterion as shown in

Table A1. The linguistic assessments of criteria influence.

DM	CRI-1	CRI-2	CRI-3	CRI-4	CRI-5	CRI-6
DM1	SH	VH	M	M	M	EH
DM2	SH	EH	H	M	SH	SH
DM3	M	M	VH	M	EH	EH
DM4	EH	EH	H	SH	M	VH
DM5	VH	EH	H	VH	VH	EH
DM6	VH	VH	H	SH	H	VH
DM7	VH	M	EH	M	M	SH
DM8	SH	EH	VH	M	SH	EH

in the Appendix A. Those assessments are transformed into T-SFN, according to Table 1, to construct the T-SF criteria influence matrix $\left(\widetilde{CRI}\right)$ as shown in

Table 5. The T-SF criteria influence matrix.

DM	CRI-1	CRI-2	CRI-3	CRI-4	CRI-5	CRI-6
DM1	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)
DM2	(0.615, 0.385, 0.385)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)
DM3	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)
DM4	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500))	(0.730, 0.270, 0.270)
DM5	(0.730, 0.270, 0.270)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.960, 0.040, 0.040)
DM6	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.730, 0.270, 0.270)
DM7	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)
DM8	(0.615, 0.385, 0.385)	(0.960, 0.040, 0.040)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.960, 0.040, 0.040)

.

By applying Einstein interaction operation in Equation (9), the T-SF criteria influence matrix $\left(\widetilde{CRI}\right)$ are multiplied with the corresponding decision maker’s weight $\left({\omega }_k\right)$ to establish the weighted T-SF criteria influence matrix $(\widetilde{wCRI}$) as shown in

Table 6. The weighted T-SF criteria influence matrix.

DM	CRI-1	CRI-2	CRI-3	CRI-4	CRI-5	CRI-6
DM1	(0.263, 0.162, 0.237)	(0.321, 0.114, 0.174)	(0.211, 0.211, 0.318)	(0.211, 0.211, 0.318)	(0.211, 0.211, 0.318)	(0.521, 0.018, 0.052)
DM2	(0.235, 0.144, 0.215)	(0.467, 0.016, 0.048)	(0.351, 0.059, 0.107)	(0.188, 0.188, 0.289)	(0.235, 0.144, 0.215)	(0.235, 0.144, 0.215)
DM3	(0.171, 0.171, 0.265)	(0.171, 0.171, 0.265)	(0.26, 0.092, 0.145)	(0.171, 0.171, 0.265)	(0.425, 0.014, 0.044)	(0.425, 0.014, 0.044)
DM4	(0.425, 0.014, 0.044)	(0.425, 0.014, 0.044)	(0.319, 0.053, 0.098)	(0.213, 0.131, 0.197)	(0.171, 0.171, 0.265)	(0.26, 0.092, 0.145)
DM5	(0.287, 0.102, 0.158)	(0.467, 0.016, 0.048)	(0.351, 0.059, 0.107)	(0.287, 0.102, 0.158)	(0.287, 0.102, 0.158)	(0.467, 0.016, 0.048)
DM6	(0.26, 0.092, 0.145)	(0.26, 0.092, 0.145)	(0.319, 0.053, 0.098)	(0.213, 0.131, 0.197)	(0.319, 0.053, 0.098)	(0.26, 0.092, 0.145)
DM7	(0.235, 0.083, 0.132)	(0.154, 0.154, 0.242)	(0.384, 0.013, 0.041)	(0.154, 0.154, 0.242)	(0.154, 0.154, 0.242)	(0.192, 0.118, 0.179)
DM8	(0.213, 0.131, 0.197)	(0.425, 0.014, 0.044)	(0.26, 0.092, 0.145)	(0.171, 0.171, 0.265)	(0.213, 0.131, 0.197)	(0.425, 0.014, 0.044)

. In the next phase of Delphi process, the decision makers also provide the linguistic assessment on performance (or suitability in this case study context) of ESTs according to criteria (see

Table A2. The linguistic performance assessment of ESTs.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1	CRI-1	VH	M	VH	M	VH	SL	SH	H
	CRI-2	M	M	VH	L	L	L	SH	L
	CRI-3	L	SL	M	SL	M	SH	SH	SL
	CRI-4	VH	L	L	SH	M	SH	SL	SH
	CRI-5	M	EH	H	EH	M	SH	M	M
	CRI-6	VH	SL	L	H	SL	M	SH	SH
EST-2	CRI-1	M	H	EH	L	SL	SH	H	L
	CRI-2	M	H	VH	EH	L	SH	M	SL
	CRI-3	VH	L	L	H	L	L	M	M
	CRI-4	M	M	SL	M	EH	H	EH	SH
	CRI-5	VH	M	SL	SL	VH	VH	L	M
	CRI-6	M	SL	H	L	M	VH	VH	H
EST-3	CRI-1	SL	M	SH	SL	H	EH	M	SH
	CRI-2	SH	H	SH	L	VH	VH	H	EH
	CRI-3	VH	VH	L	SH	VH	VH	SH	L
	CRI-4	SH	EH	EH	SH	L	L	VH	SH
	CRI-5	VH	M	H	EH	M	M	SL	L
	CRI-6	SL	EH	VH	L	SL	L	SL	M
EST-4	CRI-1	M	SL	EH	M	H	L	SL	VH
	CRI-2	L	SH	L	M	EH	VH	SH	H
	CRI-3	H	SH	SH	M	L	SH	H	M
	CRI-4	SL	VH	M	EH	SL	SH	H	SH
	CRI-5	H	L	SL	L	L	EH	H	L
	CRI-6	L	VH	EH	H	VH	M	VH	SL
EST-5	CRI-1	VH	M	SL	L	H	SH	M	EH
	CRI-2	SL	EH	L	EH	SH	H	H	SH
	CRI-3	M	VH	EH	M	SH	M	SH	EH
	CRI-4	H	VH	L	SL	H	L	EH	SH
	CRI-5	SL	L	L	VH	SH	SL	EH	EH
	CRI-6	M	SL	L	M	EH	EH	EH	VH

in the Appendix A). Similarly, those linguistic assessments are transformed into T-SFNs according to Table 1 to establish the T-SF decision matrix $\left({\widetilde{ATL}}_i\right)$ as presented in

Table 7. The T-SF decision matrix.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1 $({\widetilde{ATL}}_1)$	CRI-1	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.385, 0.385, 0.615)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)
	CRI-2	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.270, 0.270, 0.730)
	CRI-3	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	(0.385, 0.385, 0.615)
	CRI-4	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.385, 0.385, 0.615)	(0.615, 0.385, 0.385)
	CRI-5	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)
	CRI-6	(0.730, 0.270, 0.270)	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)	(0.845, 0.155, 0.155)	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)
EST-2 $({\widetilde{ATL}}_2)$	CRI-1	(0.500, 0.500, 0.500)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.27, 0.27, 0.73)
	CRI-2	(0.500, 0.500, 0.500)	(0.845, 0.155, 0.155)	(0.730, 0.270, 0.270)	(0.960, 0.040, 0.040)	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)
	CRI-3	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)
	CRI-4	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)	(0.615, 0.385, 0.385)
	CRI-5	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.385, 0.385, 0.615)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.500, 0.500, 0.500)
	CRI-6	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.845, 0.155, 0.155)
EST-3 $({\widetilde{ATL}}_3)$	CRI-1	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.385, 0.385, 0.615)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)
	CRI-2	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)	(0.270, 0.270, 0.730)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)
	CRI-3	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.730, 0.270, 0.270)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	(0.270, 0.270, 0.730)
	CRI-4	(0.615, 0.385, 0.385)	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)	(0.615, 0.385, 0.385)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)
	CRI-5	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.845, 0.155, 0.155)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)
	CRI-6	(0.385, 0.385, 0.615)	(0.960, 0.040, 0.040)	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.500, 0.500, 0.500)
EST-4 $({\widetilde{ATL}}_4)$	CRI-1	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.730, 0.270, 0.270)
	CRI-2	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.270, 0.270, 0.730)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)
	CRI-3	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.270, 0.270, 0.730)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.500, 0.500, 0.500)
	CRI-4	(0.385, 0.385, 0.615)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.385, 0.385, 0.615)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)
	CRI-5	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)
	CRI-6	(0.270, 0.270, 0.730)	(0.730, 0.270, 0.270)	(0.960, 0.040, 0.040)	(0.845, 0.155, 0.155)	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.385, 0.385, 0.615)
EST-5 $({\widetilde{ATL}}_5)$	CRI-1	(0.730, 0.270, 0.270)	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.27, 0.27, 0.73)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)
	CRI-2	(0.385, 0.385, 0.615)	(0.960, 0.040, 0.040)	(0.270, 0.270, 0.730)	(0.960, 0.040, 0.040)	(0.615, 0.385, 0.385)	(0.845, 0.155, 0.155)	(0.845, 0.155, 0.155)	(0.615, 0.385, 0.385)
	CRI-3	(0.500, 0.500, 0.500)	(0.730, 0.270, 0.270)	(0.960, 0.040, 0.040)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.500, 0.500, 0.500)	(0.615, 0.385, 0.385)	(0.960, 0.040, 0.040)
	CRI-4	(0.845, 0.155, 0.155)	(0.730, 0.270, 0.270)	(0.270, 0.270, 0.730)	(0.385, 0.385, 0.615)	(0.845, 0.155, 0.155)	(0.270, 0.270, 0.730)	(0.960, 0.040, 0.040)	(0.615, 0.385, 0.385)
	CRI-5	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)	(0.270, 0.270, 0.730)	(0.730, 0.270, 0.270)	(0.615, 0.385, 0.385)	(0.385, 0.385, 0.615)	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)
	CRI-6	(0.500, 0.500, 0.500)	(0.385, 0.385, 0.615)	(0.270, 0.270, 0.730)	(0.500, 0.500, 0.500)	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)	(0.960, 0.040, 0.040)	(0.730, 0.270, 0.270)

. The decision maker’s weights are applied to the T-SF decision matrix to construct the weighted T-SF decision matrix $\left({\widetilde{wATL}}_i\right)$ by Einstein interaction operation as shown on

Table 8. The weighted T-SF decision matrix.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1 $({\widetilde{wATL}}_1)$	CRI-1	(0.321, 0.114, 0.174)	(0.188, 0.188, 0.289)	(0.260, 0.092, 0.145)	(0.171, 0.171, 0.265)	(0.287, 0.102, 0.158)	(0.130, 0.130, 0.285)	(0.192, 0.118, 0.179)	(0.319, 0.053, 0.098)
	CRI-2	(0.211, 0.211, 0.318)	(0.188, 0.188, 0.289)	(0.260, 0.092, 0.145)	(0.091, 0.091, 0.322)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.192, 0.118, 0.179)	(0.091, 0.091, 0.322)
	CRI-3	(0.113, 0.113, 0.390)	(0.144, 0.144, 0.311)	(0.171, 0.171, 0.265)	(0.130, 0.130, 0.285)	(0.188, 0.188, 0.289)	(0.213, 0.131, 0.197)	(0.192, 0.118, 0.179)	(0.130, 0.130, 0.285)
	CRI-4	(0.321, 0.114, 0.174)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.213, 0.131, 0.197)	(0.188, 0.188, 0.289)	(0.213, 0.131, 0.197)	(0.118, 0.118, 0.259)	(0.213, 0.131, 0.197)
	CRI-5	(0.211, 0.211, 0.318)	(0.467, 0.016, 0.048)	(0.319, 0.053, 0.098)	(0.425, 0.014, 0.044)	(0.188, 0.188, 0.289)	(0.213, 0.131, 0.197)	(0.154, 0.154, 0.242)	(0.171, 0.171, 0.265)
	CRI-6	(0.321, 0.114, 0.174)	(0.144, 0.144, 0.311)	(0.091, 0.091, 0.322)	(0.319, 0.053, 0.098)	(0.144, 0.144, 0.311)	(0.171, 0.171, 0.265)	(0.192, 0.118, 0.179)	(0.213, 0.131, 0.197)
EST-2 $({\widetilde{wATL}}_2)$	CRI-1	(0.211, 0.211, 0.318)	(0.351, 0.059, 0.107)	(0.425, 0.014, 0.044)	(0.091, 0.091, 0.322)	(0.144, 0.144, 0.311)	(0.213, 0.131, 0.197)	(0.288, 0.048, 0.090)	(0.091, 0.091, 0.322)
	CRI-2	(0.211, 0.211, 0.318)	(0.351, 0.059, 0.107)	(0.260, 0.092, 0.145)	(0.425, 0.014, 0.044)	(0.101, 0.101, 0.352)	(0.213, 0.131, 0.197)	(0.154, 0.154, 0.242)	(0.130, 0.130, 0.285)
	CRI-3	(0.321, 0.114, 0.174)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.319, 0.053, 0.098)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.154, 0.154, 0.242)	(0.171, 0.171, 0.265)
	CRI-4	(0.211, 0.211, 0.318)	(0.188, 0.188, 0.289)	(0.130, 0.130, 0.285)	(0.171, 0.171, 0.265)	(0.467, 0.016, 0.048)	(0.319, 0.053, 0.098)	(0.384, 0.013, 0.041)	(0.213, 0.131, 0.197)
	CRI-5	(0.321, 0.114, 0.174)	(0.188, 0.188, 0.289)	(0.130, 0.130, 0.285)	(0.130, 0.130, 0.285)	(0.287, 0.102, 0.158)	(0.260, 0.092, 0.145)	(0.082, 0.082, 0.293)	(0.171, 0.171, 0.265)
	CRI-6	(0.211, 0.211, 0.318)	(0.144, 0.144, 0.311)	(0.319, 0.053, 0.098)	(0.091, 0.091, 0.322)	(0.188, 0.188, 0.289)	(0.260, 0.092, 0.145)	(0.235, 0.083, 0.132)	(0.319, 0.053, 0.098)
EST-3 $({\widetilde{wATL}}_3)$	CRI-1	(0.161, 0.161, 0.344)	(0.188, 0.188, 0.289)	(0.213, 0.131, 0.197)	(0.130, 0.130, 0.285)	(0.351, 0.059, 0.107)	(0.425, 0.014, 0.044)	(0.154, 0.154, 0.242)	(0.213, 0.131, 0.197)
	CRI-2	(0.263, 0.162, 0.237)	(0.351, 0.059, 0.107)	(0.213, 0.131, 0.197)	(0.091, 0.091, 0.322)	(0.287, 0.102, 0.158)	(0.260, 0.092, 0.145)	(0.288, 0.048, 0.090)	(0.425, 0.014, 0.044)
	CRI-3	(0.321, 0.114, 0.174)	(0.287, 0.102, 0.158)	(0.091, 0.091, 0.322)	(0.213, 0.131, 0.197)	(0.287, 0.102, 0.158)	(0.260, 0.092, 0.145)	(0.192, 0.118, 0.179)	(0.091, 0.091, 0.322)
	CRI-4	(0.263, 0.162, 0.237)	(0.467, 0.016, 0.048)	(0.425, 0.014, 0.044)	(0.213, 0.131, 0.197)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.235, 0.083, 0.132)	(0.213, 0.131, 0.197)
	CRI-5	(0.321, 0.114, 0.174)	(0.188, 0.188, 0.289)	(0.319, 0.053, 0.098)	(0.425, 0.014, 0.044)	(0.188, 0.188, 0.289)	(0.171, 0.171, 0.265)	(0.118, 0.118, 0.259)	(0.091, 0.091, 0.322)
	CRI-6	(0.161, 0.161, 0.344)	(0.467, 0.016, 0.048)	(0.260, 0.092, 0.145)	(0.091, 0.091, 0.322)	(0.144, 0.144, 0.311)	(0.091, 0.091, 0.322)	(0.118, 0.118, 0.259)	(0.171, 0.171, 0.265)
EST-4 $({\widetilde{wATL}}_4)$	CRI-1	(0.211, 0.211, 0.318)	(0.144, 0.144, 0.311)	(0.425, 0.014, 0.044)	(0.171, 0.171, 0.265)	(0.351, 0.059, 0.107)	(0.091, 0.091, 0.322)	(0.118, 0.118, 0.259)	(0.260, 0.092, 0.145)
	CRI-2	(0.113, 0.113, 0.390)	(0.235, 0.144, 0.215)	(0.091, 0.091, 0.322)	(0.171, 0.171, 0.265)	(0.467, 0.016, 0.048)	(0.260, 0.092, 0.145)	(0.192, 0.118, 0.179)	(0.319, 0.053, 0.098)
	CRI-3	(0.393, 0.066, 0.117)	(0.235, 0.144, 0.215)	(0.213, 0.131, 0.197)	(0.171, 0.171, 0.265)	(0.101, 0.101, 0.352)	(0.213, 0.131, 0.197)	(0.288, 0.048, 0.090)	(0.171, 0.171, 0.265)
	CRI-4	(0.161, 0.161, 0.344)	(0.287, 0.102, 0.158)	(0.171, 0.171, 0.265)	(0.425, 0.014, 0.044)	(0.144, 0.144, 0.311)	(0.213, 0.131, 0.197)	(0.288, 0.048, 0.090)	(0.213, 0.131, 0.197)
	CRI-5	(0.393, 0.066, 0.117)	(0.101, 0.101, 0.352)	(0.130, 0.130, 0.285)	(0.091, 0.091, 0.322)	(0.101, 0.101, 0.352)	(0.425, 0.014, 0.044)	(0.288, 0.048, 0.090)	(0.091, 0.091, 0.322)
	CRI-6	(0.113, 0.113, 0.309)	(0.287, 0.102, 0.158)	(0.425, 0.014, 0.044)	(0.319, 0.053, 0.098)	(0.287, 0.102, 0.158)	(0.171, 0.171, 0.265)	(0.235, 0.083, 0.132)	(0.130, 0.130, 0.285)
EST-5 $({\widetilde{wATL}}_5)$	CRI-1	(0.321, 0.114, 0.174)	(0.188, 0.188, 0.289)	(0.130, 0.130, 0.285)	(0.091, 0.091, 0.322)	(0.351, 0.059, 0.107)	(0.213, 0.131, 0.197)	(0.154, 0.154, 0.242)	(0.425, 0.014, 0.044)
	CRI-2	(0.161, 0.161, 0.344)	(0.467, 0.016, 0.048)	(0.091, 0.091, 0.322)	(0.425, 0.014, 0.044)	(0.235, 0.144, 0.215)	(0.319, 0.053, 0.098)	(0.288, 0.048, 0.09)	(0.213, 0.131, 0.197)
	CRI-3	(0.211, 0.211, 0.318)	(0.287, 0.102, 0.158)	(0.425, 0.014, 0.044)	(0.171, 0.171, 0.265)	(0.235, 0.144, 0.215)	(0.171, 0.171, 0.265)	(0.192, 0.118, 0.179)	(0.425, 0.014, 0.044)
	CRI-4	(0.393, 0.066, 0.117)	(0.287, 0.102, 0.158)	(0.091, 0.091, 0.322)	(0.130, 0.130, 0.285)	(0.351, 0.059, 0.107)	(0.091, 0.091, 0.322)	(0.384, 0.013, 0.041)	(0.213, 0.131, 0.197)
	CRI-5	(0.161, 0.161, 0.344)	(0.101, 0.101, 0.352)	(0.091, 0.091, 0.322)	(0.260, 0.092, 0.145)	(0.235, 0.144, 0.215)	(0.130, 0.130, 0.285)	(0.384, 0.013, 0.041)	(0.425, 0.014, 0.044)
	CRI-6	(0.211, 0.211, 0.318)	(0.144, 0.144, 0.311)	(0.091, 0.091, 0.322)	(0.171, 0.171, 0.265)	(0.467, 0.016, 0.048)	(0.425, 0.014, 0.044)	(0.384, 0.013, 0.041)	(0.260, 0.092, 0.145)

.

Then, the weighted T-SF criteria influence matrix can be split into three weighed crisp criteria influence submatrices for membership $\left(wCR{I}^m\right)$, hesitancy $\left(wCR{I}^h\right)$, and non-membership $\left(wCR{I}^n\right)$ values are shown in

Table 9. The weighed crisp criteria influence submatrices.

Crisp Value	DM	CRI-1	CRI-2	CRI-3	CRI-4	CRI-5	CRI-6
Membership $\left(wCR{I}^m\right)$	DM1	0.263	0.321	0.211	0.211	0.211	0.521
	DM2	0.235	0.467	0.351	0.188	0.235	0.235
	DM3	0.171	0.171	0.260	0.171	0.425	0.425
	DM4	0.425	0.425	0.319	0.213	0.171	0.260
	DM5	0.287	0.467	0.351	0.287	0.287	0.467
	DM6	0.260	0.260	0.319	0.213	0.319	0.260
	DM7	0.235	0.154	0.384	0.154	0.154	0.192
	DM8	0.213	0.425	0.260	0.171	0.213	0.425
Hesitancy $\left(wCR{I}^h\right)$	DM1	0.162	0.114	0.211	0.211	0.211	0.018
	DM2	0.144	0.016	0.059	0.188	0.144	0.144
	DM3	0.171	0.171	0.092	0.171	0.014	0.014
	DM4	0.014	0.014	0.053	0.131	0.171	0.092
	DM5	0.102	0.016	0.059	0.102	0.102	0.016
	DM6	0.092	0.092	0.053	0.131	0.053	0.092
	DM7	0.083	0.154	0.013	0.154	0.154	0.118
	DM8	0.131	0.014	0.092	0.171	0.131	0.014
Non-membership $\left(wCR{I}^n\right)$	DM1	0.237	0.174	0.318	0.318	0.318	0.052
	DM2	0.215	0.048	0.107	0.289	0.215	0.215
	DM3	0.265	0.265	0.145	0.265	0.044	0.044
	DM4	0.044	0.044	0.098	0.197	0.265	0.145
	DM5	0.158	0.048	0.107	0.158	0.158	0.048
	DM6	0.145	0.145	0.098	0.197	0.098	0.145
	DM7	0.132	0.242	0.041	0.242	0.242	0.179
	DM8	0.197	0.044	0.145	0.265	0.197	0.044

. Similarly, the crisp weighted decision matrices for membership $\left(wAT{L}_i^m\right)$, hesitancy $\left(wAT{L}_i^h\right)$, and non-membership $\left(wAT{L}_i^n\right)$ values of each ESTs are determined. The crisp weighted decision matrices for membership values are presented in

Table 10. The crisp weighted decision matrix for membership values of ESTs.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1 $\left(wAT{L}_1^m\right)$	CRI-1	0.321	0.188	0.260	0.171	0.287	0.130	0.192	0.319
	CRI-2	0.211	0.188	0.260	0.091	0.101	0.091	0.192	0.091
	CRI-3	0.113	0.144	0.171	0.130	0.188	0.213	0.192	0.130
	CRI-4	0.321	0.101	0.091	0.213	0.188	0.213	0.118	0.213
	CRI-5	0.211	0.467	0.319	0.425	0.188	0.213	0.154	0.171
	CRI-6	0.321	0.144	0.091	0.319	0.144	0.171	0.192	0.213
EST-2 $\left(wAT{L}_2^m\right)$	CRI-1	0.211	0.351	0.425	0.091	0.144	0.213	0.288	0.091
	CRI-2	0.211	0.351	0.260	0.425	0.101	0.213	0.154	0.130
	CRI-3	0.321	0.101	0.091	0.319	0.101	0.091	0.154	0.171
	CRI-4	0.211	0.188	0.130	0.171	0.467	0.319	0.384	0.213
	CRI-5	0.321	0.188	0.130	0.130	0.287	0.260	0.082	0.171
	CRI-6	0.211	0.144	0.319	0.091	0.188	0.260	0.235	0.319
EST-3 $\left(wAT{L}_3^m\right)$	CRI-1	0.161	0.188	0.213	0.130	0.351	0.425	0.154	0.213
	CRI-2	0.263	0.351	0.213	0.091	0.287	0.260	0.288	0.425
	CRI-3	0.321	0.287	0.091	0.213	0.287	0.260	0.192	0.091
	CRI-4	0.263	0.467	0.425	0.213	0.101	0.091	0.235	0.213
	CRI-5	0.321	0.188	0.319	0.425	0.188	0.171	0.118	0.091
	CRI-6	0.161	0.467	0.260	0.091	0.144	0.091	0.118	0.171
EST-4 $\left(wAT{L}_4^m\right)$	CRI-1	0.211	0.144	0.425	0.171	0.351	0.091	0.118	0.260
	CRI-2	0.113	0.235	0.091	0.171	0.467	0.260	0.192	0.319
	CRI-3	0.393	0.235	0.213	0.171	0.101	0.213	0.288	0.171
	CRI-4	0.161	0.287	0.171	0.425	0.144	0.213	0.288	0.213
	CRI-5	0.393	0.101	0.130	0.091	0.101	0.425	0.288	0.091
	CRI-6	0.113	0.287	0.425	0.319	0.287	0.171	0.235	0.130
EST-5 $\left(wAT{L}_5^m\right)$	CRI-1	0.321	0.188	0.130	0.091	0.351	0.213	0.154	0.425
	CRI-2	0.161	0.467	0.091	0.425	0.235	0.319	0.288	0.213
	CRI-3	0.211	0.287	0.425	0.171	0.235	0.171	0.192	0.425
	CRI-4	0.393	0.287	0.091	0.130	0.351	0.091	0.384	0.213
	CRI-5	0.161	0.101	0.091	0.260	0.235	0.130	0.384	0.425
	CRI-6	0.211	0.144	0.091	0.171	0.467	0.425	0.384	0.260

, while the matrices for hesitancy and non-membership values are presented in

Table A3. The crisp weighted decision matrix for hesitancy values of ESTs.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1 $\left(wAT{L}_1^h\right)$	CRI-1	0.114	0.188	0.092	0.171	0.102	0.130	0.118	0.053
	CRI-2	0.211	0.188	0.092	0.091	0.101	0.091	0.118	0.091
	CRI-3	0.113	0.144	0.171	0.130	0.188	0.131	0.118	0.130
	CRI-4	0.114	0.101	0.091	0.131	0.188	0.131	0.118	0.131
	CRI-5	0.211	0.016	0.053	0.014	0.188	0.131	0.154	0.171
	CRI-6	0.114	0.144	0.091	0.053	0.144	0.171	0.118	0.131
EST-2 $\left(wAT{L}_2^h\right)$	CRI-1	0.211	0.059	0.014	0.091	0.144	0.131	0.048	0.091
	CRI-2	0.211	0.059	0.092	0.014	0.101	0.131	0.154	0.130
	CRI-3	0.114	0.101	0.091	0.053	0.101	0.091	0.154	0.171
	CRI-4	0.211	0.188	0.130	0.171	0.016	0.053	0.013	0.131
	CRI-5	0.114	0.188	0.130	0.130	0.102	0.092	0.082	0.171
	CRI-6	0.211	0.144	0.053	0.091	0.188	0.092	0.083	0.053
EST-3 $\left(wAT{L}_3^h\right)$	CRI-1	0.161	0.188	0.131	0.130	0.059	0.014	0.154	0.131
	CRI-2	0.162	0.059	0.131	0.091	0.102	0.092	0.048	0.014
	CRI-3	0.114	0.102	0.091	0.131	0.102	0.092	0.118	0.091
	CRI-4	0.162	0.016	0.014	0.131	0.101	0.091	0.083	0.131
	CRI-5	0.114	0.188	0.053	0.014	0.188	0.171	0.118	0.091
	CRI-6	0.161	0.016	0.092	0.091	0.144	0.091	0.118	0.171
EST-4 $\left(wAT{L}_4^h\right)$	CRI-1	0.211	0.144	0.014	0.171	0.059	0.091	0.118	0.092
	CRI-2	0.113	0.144	0.091	0.171	0.016	0.092	0.118	0.053
	CRI-3	0.066	0.144	0.131	0.171	0.101	0.131	0.048	0.171
	CRI-4	0.161	0.102	0.171	0.014	0.144	0.131	0.048	0.131
	CRI-5	0.066	0.101	0.130	0.091	0.101	0.014	0.048	0.091
	CRI-6	0.113	0.102	0.014	0.053	0.102	0.171	0.083	0.130
EST-5 $\left(wAT{L}_5^h\right)$	CRI-1	0.114	0.188	0.130	0.091	0.059	0.131	0.154	0.014
	CRI-2	0.161	0.016	0.091	0.014	0.144	0.053	0.048	0.131
	CRI-3	0.211	0.102	0.014	0.171	0.144	0.171	0.118	0.014
	CRI-4	0.066	0.102	0.091	0.130	0.059	0.091	0.013	0.131
	CRI-5	0.161	0.101	0.091	0.092	0.144	0.130	0.013	0.014
	CRI-6	0.211	0.144	0.091	0.171	0.016	0.014	0.013	0.092

and

Table A4. The crisp weighted decision matrix for non-membership values of ESTs.

EST	Criteria	DM1	DM2	DM3	DM4	DM5	DM6	DM7	DM8
EST-1 $\left(wAT{L}_1^h\right)$	CRI-1	0.174	0.289	0.145	0.265	0.158	0.285	0.179	0.098
	CRI-2	0.318	0.289	0.145	0.322	0.352	0.322	0.179	0.322
	CRI-3	0.390	0.311	0.265	0.285	0.289	0.197	0.179	0.285
	CRI-4	0.174	0.352	0.322	0.197	0.289	0.197	0.259	0.197
	CRI-5	0.318	0.048	0.098	0.044	0.289	0.197	0.242	0.265
	CRI-6	0.174	0.311	0.322	0.098	0.311	0.265	0.179	0.197
EST-2 $\left(wAT{L}_2^h\right)$	CRI-1	0.318	0.107	0.044	0.322	0.311	0.197	0.090	0.322
	CRI-2	0.318	0.107	0.145	0.044	0.352	0.197	0.242	0.285
	CRI-3	0.174	0.352	0.322	0.098	0.352	0.322	0.242	0.265
	CRI-4	0.318	0.289	0.285	0.265	0.048	0.098	0.041	0.197
	CRI-5	0.174	0.289	0.285	0.285	0.158	0.145	0.293	0.265
	CRI-6	0.318	0.311	0.098	0.322	0.289	0.145	0.132	0.098
EST-3 $\left(wAT{L}_3^h\right)$	CRI-1	0.344	0.289	0.197	0.285	0.107	0.044	0.242	0.197
	CRI-2	0.237	0.107	0.197	0.322	0.158	0.145	0.090	0.044
	CRI-3	0.174	0.158	0.322	0.197	0.158	0.145	0.179	0.322
	CRI-4	0.237	0.048	0.044	0.197	0.352	0.322	0.132	0.197
	CRI-5	0.174	0.289	0.098	0.044	0.289	0.265	0.259	0.322
	CRI-6	0.344	0.048	0.145	0.322	0.311	0.322	0.259	0.265
EST-4 $\left(wAT{L}_4^h\right)$	CRI-1	0.318	0.311	0.044	0.265	0.107	0.322	0.259	0.145
	CRI-2	0.390	0.215	0.322	0.265	0.048	0.145	0.179	0.098
	CRI-3	0.117	0.215	0.197	0.265	0.352	0.197	0.090	0.265
	CRI-4	0.344	0.158	0.265	0.044	0.311	0.197	0.090	0.197
	CRI-5	0.117	0.352	0.285	0.322	0.352	0.044	0.090	0.322
	CRI-6	0.390	0.158	0.044	0.098	0.158	0.265	0.132	0.285
EST-5 $\left(wAT{L}_5^h\right)$	CRI-1	0.174	0.289	0.285	0.322	0.107	0.197	0.242	0.044
	CRI-2	0.344	0.048	0.322	0.044	0.215	0.098	0.090	0.197
	CRI-3	0.318	0.158	0.044	0.265	0.215	0.265	0.179	0.044
	CRI-4	0.117	0.158	0.322	0.285	0.107	0.322	0.041	0.197
	CRI-5	0.344	0.352	0.322	0.145	0.215	0.285	0.041	0.044
	CRI-6	0.318	0.311	0.322	0.265	0.048	0.044	0.041	0.145

in Appendix A.

In order to apply matrix energy concepts, as mentioned in Equations (25)–(27), the performance square matrices are computed by multiplying the weighted crisp decision matrices and the weighted crisp criteria influence matrices for membership, hesitancy, and non-membership values of each EST. For instance, the performance square matrix for membership values of EST-1 $\left(R_1^m\right)$ can be calculated as a product of the weighed crisp criteria influence matrix $\left(wCR{I}^m\right)$ and the crisp weighted decision matrix for membership values of EST-1 $\left(wAT{L}_1^m\right)$. The performance square matrix for membership values $\left(R_i^m\right)$ of ESTs is presented in

Table 11. The performance square matrices for membership values of ESTs.

EST-1 $\left(R_1^m\right)$	0.455	0.399	0.401	0.450	0.533	0.394	0.295	0.424
	0.335	0.345	0.379	0.345	0.423	0.352	0.244	0.335
	0.322	0.356	0.323	0.373	0.416	0.335	0.247	0.333
	0.402	0.343	0.431	0.354	0.469	0.373	0.264	0.371
	0.302	0.294	0.288	0.335	0.371	0.298	0.241	0.286
	0.287	0.278	0.293	0.288	0.357	0.283	0.225	0.276
	0.310	0.306	0.283	0.326	0.380	0.285	0.227	0.307
	0.333	0.293	0.304	0.346	0.389	0.303	0.239	0.304
EST-2 $\left(R_2^m\right)$	0.413	0.425	0.418	0.436	0.523	0.414	0.328	0.412
	0.381	0.395	0.320	0.440	0.475	0.352	0.261	0.384
	0.435	0.383	0.354	0.453	0.499	0.359	0.276	0.410
	0.339	0.416	0.294	0.403	0.465	0.337	0.273	0.379
	0.349	0.316	0.350	0.334	0.428	0.336	0.240	0.321
	0.401	0.364	0.372	0.390	0.480	0.358	0.257	0.380
	0.378	0.340	0.316	0.394	0.452	0.333	0.267	0.350
	0.349	0.297	0.327	0.306	0.406	0.295	0.227	0.328
EST-3 $\left(R_3^m\right)$	0.402	0.436	0.406	0.435	0.524	0.413	0.322	0.411
	0.604	0.551	0.525	0.574	0.725	0.513	0.399	0.582
	0.436	0.397	0.415	0.423	0.527	0.400	0.282	0.411
	0.290	0.309	0.349	0.304	0.380	0.330	0.242	0.287
	0.381	0.414	0.342	0.454	0.486	0.376	0.309	0.390
	0.353	0.391	0.312	0.446	0.452	0.359	0.298	0.359
	0.309	0.338	0.266	0.350	0.403	0.295	0.231	0.321
	0.365	0.382	0.281	0.406	0.459	0.314	0.230	0.378
EST-4 $\left(R_4^m\right)$	0.350	0.389	0.400	0.394	0.463	0.399	0.325	0.354
	0.394	0.371	0.340	0.389	0.479	0.341	0.275	0.384
	0.471	0.380	0.409	0.456	0.524	0.391	0.324	0.422
	0.411	0.356	0.350	0.389	0.486	0.346	0.273	0.381
	0.465	0.454	0.356	0.502	0.559	0.382	0.286	0.468
	0.376	0.398	0.405	0.380	0.485	0.385	0.274	0.385
	0.397	0.396	0.399	0.395	0.500	0.387	0.302	0.392
	0.339	0.362	0.274	0.395	0.432	0.313	0.248	0.346
EST-5 $\left(R_5^m\right)$	0.407	0.386	0.363	0.438	0.499	0.383	0.307	0.383
	0.417	0.475	0.340	0.486	0.551	0.393	0.314	0.445
	0.239	0.282	0.241	0.288	0.324	0.265	0.253	0.250
	0.368	0.406	0.338	0.390	0.476	0.344	0.245	0.395
	0.584	0.506	0.520	0.560	0.679	0.498	0.389	0.544
	0.463	0.407	0.387	0.433	0.532	0.364	0.280	0.449
	0.536	0.491	0.518	0.496	0.646	0.480	0.346	0.516

. By calculating the eigenvalues of each performance square matrix, the energy of that matrix $\left(E\left(R_i^m\right)\right)$ is computed according to Equation (11). This procedure is repeated for hesitancy and non-membership values as shown in

Table A5. The performance square matrices for hesitancy values of ESTs.

EST-1 $\left(R_1^h\right)$	0.137	0.095	0.090	0.072	0.057	0.072	0.107	0.077
	0.110	0.081	0.097	0.042	0.045	0.070	0.081	0.062
	0.093	0.063	0.065	0.041	0.037	0.049	0.057	0.053
	0.097	0.068	0.080	0.035	0.042	0.054	0.058	0.061
	0.150	0.111	0.089	0.083	0.064	0.076	0.101	0.091
	0.117	0.096	0.076	0.065	0.052	0.067	0.087	0.072
	0.117	0.087	0.075	0.062	0.051	0.062	0.085	0.070
	0.112	0.085	0.063	0.067	0.047	0.058	0.082	0.067
EST-2 $\left(R_2^h\right)$	0.155	0.127	0.123	0.078	0.068	0.098	0.126	0.095
	0.120	0.099	0.066	0.077	0.054	0.064	0.090	0.077
	0.088	0.060	0.051	0.050	0.036	0.043	0.063	0.051
	0.093	0.080	0.055	0.057	0.045	0.050	0.067	0.065
	0.084	0.073	0.058	0.046	0.037	0.053	0.069	0.048
	0.088	0.063	0.065	0.040	0.037	0.049	0.065	0.050
	0.079	0.045	0.053	0.034	0.027	0.040	0.054	0.037
	0.130	0.082	0.079	0.063	0.053	0.060	0.083	0.075
EST-3 $\left(R_3^h\right)$	0.130	0.103	0.097	0.066	0.056	0.078	0.101	0.079
	0.102	0.066	0.057	0.045	0.047	0.042	0.059	0.062
	0.071	0.050	0.058	0.028	0.029	0.042	0.053	0.038
	0.091	0.068	0.074	0.038	0.039	0.054	0.060	0.056
	0.106	0.083	0.059	0.066	0.045	0.057	0.083	0.062
	0.089	0.064	0.046	0.056	0.037	0.044	0.068	0.051
	0.100	0.079	0.063	0.051	0.046	0.053	0.067	0.063
	0.092	0.087	0.059	0.055	0.044	0.056	0.069	0.062
EST-4 $\left(R_4^h\right)$	0.111	0.092	0.092	0.051	0.052	0.068	0.084	0.073
	0.115	0.080	0.083	0.052	0.048	0.062	0.079	0.066
	0.104	0.064	0.061	0.054	0.042	0.047	0.065	0.062
	0.106	0.061	0.079	0.036	0.042	0.052	0.065	0.056
	0.086	0.071	0.050	0.052	0.039	0.046	0.058	0.057
	0.087	0.074	0.068	0.045	0.036	0.057	0.066	0.052
	0.064	0.050	0.055	0.028	0.028	0.041	0.053	0.037
	0.106	0.081	0.066	0.056	0.045	0.056	0.068	0.065
EST-5 $\left(R_5^h\right)$	0.133	0.097	0.083	0.071	0.053	0.073	0.097	0.072
	0.099	0.088	0.065	0.052	0.048	0.056	0.068	0.067
	0.074	0.064	0.057	0.040	0.036	0.046	0.064	0.048
	0.102	0.086	0.060	0.059	0.045	0.056	0.066	0.065
	0.099	0.053	0.060	0.044	0.038	0.043	0.062	0.052
	0.110	0.068	0.065	0.047	0.047	0.046	0.057	0.066
	0.061	0.036	0.048	0.014	0.026	0.028	0.027	0.036

and

Table A6. The performance square matrices for non-membership values of ESTs.

EST-1 $\left(R_1^n\right)$	0.367	0.333	0.410	0.068	0.245	0.224	0.204	0.305
	0.379	0.344	0.424	0.070	0.253	0.232	0.211	0.315
	0.307	0.279	0.344	0.057	0.205	0.188	0.171	0.256
	0.287	0.260	0.321	0.053	0.191	0.176	0.160	0.239
	0.400	0.363	0.447	0.074	0.267	0.245	0.223	0.333
	0.347	0.315	0.388	0.064	0.231	0.212	0.193	0.288
	0.288	0.262	0.323	0.054	0.192	0.176	0.161	0.240
	0.323	0.293	0.361	0.060	0.216	0.198	0.180	0.269
EST-2 $\left(R_2^n\right)$	0.384	0.348	0.429	0.071	0.256	0.235	0.214	0.319
	0.345	0.313	0.386	0.064	0.230	0.211	0.192	0.287
	0.279	0.253	0.312	0.052	0.186	0.171	0.156	0.232
	0.317	0.287	0.354	0.059	0.211	0.194	0.176	0.263
	0.358	0.325	0.400	0.066	0.239	0.219	0.199	0.297
	0.262	0.237	0.293	0.049	0.174	0.160	0.146	0.217
	0.246	0.224	0.276	0.046	0.164	0.151	0.137	0.205
	0.339	0.308	0.379	0.063	0.226	0.208	0.189	0.282
EST-3 $\left(R_3^n\right)$	0.358	0.325	0.400	0.066	0.239	0.219	0.199	0.297
	0.223	0.202	0.249	0.041	0.148	0.136	0.124	0.185
	0.238	0.216	0.266	0.044	0.158	0.145	0.132	0.198
	0.324	0.294	0.362	0.060	0.216	0.198	0.180	0.269
	0.326	0.296	0.364	0.061	0.217	0.199	0.182	0.271
	0.295	0.267	0.329	0.055	0.196	0.180	0.164	0.245
	0.275	0.250	0.308	0.051	0.183	0.168	0.153	0.229
	0.319	0.290	0.357	0.059	0.213	0.195	0.178	0.265
EST-4 $\left(R_4^n\right)$	0.397	0.360	0.444	0.074	0.265	0.243	0.221	0.330
	0.334	0.303	0.373	0.062	0.223	0.204	0.186	0.278
	0.274	0.249	0.307	0.051	0.183	0.168	0.153	0.228
	0.298	0.271	0.334	0.055	0.199	0.183	0.166	0.248
	0.315	0.286	0.352	0.058	0.210	0.193	0.175	0.262
	0.277	0.252	0.310	0.051	0.185	0.170	0.154	0.230
	0.199	0.181	0.223	0.037	0.133	0.122	0.111	0.165
	0.311	0.282	0.348	0.058	0.207	0.190	0.173	0.258
EST-5 $\left(R_5^n\right)$	0.383	0.347	0.428	0.071	0.255	0.234	0.213	0.318
	0.312	0.283	0.349	0.058	0.208	0.191	0.174	0.259
	0.383	0.348	0.429	0.071	0.255	0.234	0.213	0.319
	0.314	0.285	0.351	0.058	0.210	0.192	0.175	0.261
	0.215	0.195	0.240	0.040	0.143	0.132	0.120	0.179
	0.287	0.260	0.321	0.053	0.191	0.176	0.160	0.239
	0.150	0.136	0.168	0.028	0.100	0.092	0.084	0.125

in the Appendix A. As the results, the energy of performance square matrices for all ESTs are presented in

Table 12. The matrix energy results.

EST	Energy of the Performance Square Matrix
	Membership $\left(\mathit{E}\left({\mathit{R}}_{\mathit{i}}^{\mathit{m}}\right)\right)$	Hesitancy $\left(\mathit{E}\left({\mathit{R}}_{\mathit{i}}^{\mathit{h}}\right)\right)$	Non-Membership $\left(\mathit{E}\left({\mathit{R}}_{\mathit{i}}^{\mathit{n}}\right)\right)$
EST-1	4.710	1.062	3.528
EST-2	5.125	1.003	3.301
EST-3	5.518	0.922	2.978
EST-4	5.449	0.906	3.169
EST-5	6.111	0.845	2.953

.

Based on the obtained energy of matrix results, the final score of ESTs can be calculated according to Equation (31). As shown in Figure 3, the final scores from the evaluation of ESTs using the T-spherical fuzzy Einstein interaction operation matrix energy decision-making approach present a clear ranking based on comprehensive performance across specified criteria. Hydrogen Storage (EST-5) leads with the highest score of 2.313, indicating its superior adaptability and potential despite higher costs and infrastructural demands, particularly useful in long-term energy storage scenarios. Following it, Compressed Air Energy Storage (EST-3) scores 1.618, highlighted by its low operational costs and significant storage capacity, although its applicability is geographically limited. Flywheel Energy Storage (EST-4) comes next with a score of 1.374, favored for its high power density and minimal environmental impact, suitable for high-quality, short-duration energy applications. Flow Batteries (EST-2) score 0.821, recognized for their scalability and environmental friendliness, though they lag in energy density and economic factors. Lastly, Lithium-Ion Batteries (EST-1) score the lowest at 0.120, where their widespread usage and high efficiency are overshadowed by safety concerns, resource scarcity, and environmental issues. This evaluation underscores the importance of a holistic view in selecting energy storage technologies, where factors like environmental impact, economic viability, and technical performance are crucial in determining the most appropriate technology for specific energy needs.

4. Discussion

4.1. Theorical Implications

The T-spherical fuzzy Einstein interaction operation matrix energy decision-making approach presents significant theoretical advancements within the fields of fuzzy set theory and decision making under uncertainty. This innovative method integrates T-SFS with matrix energy concepts, advancing fuzzy set theory by allowing for more nuanced expressions of uncertainty. The incorporation of matrix energy adds a complex layer, enhancing the robustness of handling ambiguous information and pushing the boundaries of traditional fuzzy applications. A key advantage of your approach is its sophisticated framework for multi-criteria decision making (MCDM), which utilizes Einstein interaction operations combined with matrix energy. This framework eliminates the need for the traditional aggregation process and determining criteria weights, streamlining the decision-making process. By doing so, it provides a direct, structured analysis of alternatives under multiple criteria, facilitating both comparative and holistic evaluations. This is particularly beneficial in scenarios where decision variables are interdependent, and a balanced evaluation of competing criteria is crucial.

Moreover, the operationalization of matrix energy in practical decision-making settings marks a significant theoretical expansion from its traditional confines within graph theory and network analysis. This adaptation introduces new methodologies for fields where matrix properties like eigenvalues and structural characteristics critically influence system dynamics. Applying this framework to the evaluation of energy storage technologies not only showcases its practical implications but also emphasizes its adaptability to specific sector challenges, enhancing its credibility and applicability in real-world settings.

In terms of uncertainty modeling, the use of T-SFS and Einstein operations enhances the capability to capture degrees of hesitancy and membership in decision-making processes more effectively. This enhancement is crucial in sectors like renewable energy, where outcomes and variables often embody inherent uncertainty, impacting decision making. The mathematical rigor involved in this approach also contributes to the fields of applied mathematics and operations research, providing a structured way to address complex mathematical models in strategic decision-making scenarios. Overall, these theoretical implications enrich academic discussions and open new avenues for research and application, particularly in technology assessment and strategic management.

4.2. Practical Implications

This study’s outcomes offer actionable insights that can shape strategic decision making, particularly in enhancing the reliability and efficiency of energy storage solutions critical to supporting the intermittent nature of wind energy.

The top-ranking of hydrogen storage technologies in this study underscores its potential as a highly suitable solution for long-term energy storage needs in the offshore wind sector. Given Vietnam’s strategic goals to increase renewable energy sources, hydrogen storage presents an opportunity to overcome intermittency challenges associated with wind energy. Its ability to store energy for prolonged periods can ensure a stable and continuous power supply, which is crucial for maintaining grid stability. However, the high costs, energy conversion losses, and substantial infrastructure requirements highlighted in this study suggest that while hydrogen storage holds great promise, it also requires significant investment in technology and infrastructure development.

This study also highlights the diversity of ESTs, each suitable for different operational needs and scales. For instance, Compressed Air Energy Storage (CAES) and Flywheel Energy Storage offer benefits in terms of cost-efficiency and response times, making them suitable for large-scale storage and high-quality power applications, respectively. The Vietnamese energy sector could leverage these technologies depending on specific regional needs and the scale of wind energy projects.

The findings regarding the lower scores for lithium-ion batteries due to concerns about resource availability, cost, and environmental impacts emphasize the need for Vietnam to diversify its approach to ESTs. Flow batteries emerge as a less impactful alternative, offering scalability and quicker response times, which are crucial for enhancing the operational flexibility of wind energy systems. This study advocates for a balanced approach, considering not only the technical performance but also economic and environmental sustainability of ESTs.

The evaluation indicates the necessity for supportive policies and regulations to facilitate the adoption of advanced ESTs. Regulatory frameworks that encourage investment in cutting-edge storage technologies and ensure safety and environmental compliance are essential. Such policies could catalyze the deployment of the most effective technologies, as identified by this study, thereby accelerating Vietnam’s progress towards its renewable energy targets. Finally, the aspect of market and social factors addressed in this study points to the importance of public and market acceptance. Technologies that rank well in technical and economic terms must also gain social acceptance to be successfully implemented. Efforts to educate stakeholders and the public about the benefits and potential of various ESTs will be crucial in overcoming resistance and fostering a supportive environment for innovative energy solutions.

5. Conclusions

The conclusion of this study encapsulates the critical exploration of ESTs for Vietnam’s offshore wind sector, driven by the necessity to enhance energy storage solutions that can manage the intermittency and ensure the reliability of wind power. The motivation behind this research stems from Vietnam’s ambitious goals to expand its renewable energy capacity, necessitating robust, efficient, and sustainable energy storage systems to stabilize and support the grid.

Employing a novel T-spherical fuzzy Einstein interaction operation matrix energy decision-making approach, this study developed and validated a comprehensive framework for assessing various ESTs. The methodology integrated T-spherical fuzzy sets with matrix energy concepts and Einstein interaction operations, providing a nuanced tool for decision making that eschews the traditional need for aggregating processes and determining criteria weights. This allowed for a direct, structured evaluation of ESTs against multiple criteria tailored to Vietnam’s specific energy requirements.

The findings reveal significant differences in the suitability of various ESTs, with hydrogen storage emerging as the most favorable option due to its potential for high energy capacity and long-term storage. Other technologies like CAES and flywheel storage also demonstrated valuable attributes for specific applications within the offshore wind sector. This study underscored the importance of considering a broad range of factors, including technical performance, economic viability, environmental impact, and regulatory compliance, in selecting the most appropriate technology.

This research makes multiple significant contributions, enriching both the theoretical framework and practical applications related to decision making in the renewable energy domain. From a theoretical perspective, it advances the utilization of fuzzy set theory and decision making in uncertain environments by bringing matrix energy concepts into practical decision-making scenarios. On a practical level, it equips stakeholders within Vietnam’s renewable energy sector with a sophisticated tool for evaluating and selecting the most appropriate ESTs, thus facilitating the country’s progression towards its renewable energy goals.

Despite its strengths, this study recognizes certain limitations, notably the reliance on expert qualitative judgments for assessing ESTs. These assessments are inherently subjective and may affect the generalizability of the findings. To address this, while the current study employs T-spherical fuzzy sets and Einstein’s interaction operations adeptly capturing the inherent ambiguity in expert opinions, future research should aim to complement these qualitative assessments with quantitative data. Incorporating real-world operational data and broadening the array of ESTs analyzed will enhance the robustness and applicability of the decision-making framework, thereby improving its reliability and reducing the dependency on purely qualitative judgments. Moreover, examining the economic aspects of deploying these technologies on a larger scale and conducting long-term studies to evaluate their ongoing performance would yield more comprehensive insights into their sustainability and effectiveness over time.