A novel decision-making system for offshore wind turbine selection integrating cubic bipolar fuzzy set with improved ELECTRE-I

Abstract

Offshore wind turbine (OWT) selection is crucial for the economic viability and stability of offshore wind power projects. However, it is complicated by expert linguistic ambiguity and conflicting criteria, which introduce uncertainties. This paper proposes a novel multi-criteria decision-making (MCDM) system combining principal component analysis (PCA) and cubic bipolar fuzzy extension of improved ELECTRE-I (CBF-ELECTRE-I+). First, twelve secondary indicators are identified through literature review, and their weights are determined by PCA to reduce expert subjectivity. Next, expert evaluations of alternatives are quantified as CBF sets and then aggregated using the weighted geometric aggregation operator, thereby capturing incomplete expert information and integrating divergent opinions. Finally, the optimal scheme is determined by computing the significance of each OWT via CBF-ELECTRE-I+. Compared with the traditional CBF-ELECTRE-I, the proposed CBF-ELECTRE-I+ enhances ranking accuracy and simplifies the computational process. A case study demonstrates the effectiveness of the proposed system for OWT selection.

Keywords

offshore wind turbine selection multi-criteria decision-making cubic bipolar fuzzy set CBF-ELECTRE-I+weighted geometric aggregation operator

Introduction

Background and motivation

To address climate change, accelerate decarbonization, and meet the energy requirements of economic growth, the focus of energy has gradually shifted from fossil fuels to green and low-carbon renewable energy sources. At the 28th conference of the parties of the UNFCCC (COP 28), about 200 countries reached a consensus to limit global temperature rise to 1.5°C by tripling global renewable energy capacity and doubling energy efficiency by 2030 [Williams and Zhao (2024)]. Offshore wind energy is vital for supporting the global energy system and replacing fossil fuels to meet climate goals, due to its scale and reliability. COP 28 laid the foundation for a new phase of growth in offshore wind power over the next decade.

The newly installed offshore wind power capacity in 2023 and 2024, as well as the market forecast for the period 2025–2034, are presented in Figure 1. According to the Global Wind Report 2024, 10.8 GW of new offshore wind capacity was installed globally in 2023, making it the second-highest year for new offshore capacity additions [Lee and Zhao (2024)]. According to the Global Offshore Wind Report 2025, with a compound average annual growth rate (CAGR) of 28% until 2029 and 15% up to 2034, global offshore wind annual capacity additions are expected to sail past the milestones of 30 GW in 2030 and 50 GW by 2033 [Williams et al. (2025)]. By 2034, annual offshore wind installations are expected to reach 55 GW, raising the share of offshore wind in new wind power installations from the current 7% to about 25%. The governments of the Global Offshore Wind Alliance have pledged to collaborate towards installing 380 GW of offshore wind capacity by 2030 and 2000 GW by 2050 [Williams and Zhao (2024)].

Figure 1.

New offshore installations and outlook 2025–2034 (MW) [Williams et al. (2025)].

In the process of offshore wind power development, reducing development costs and improving power generation efficiency are key considerations for developers. Enterprises and research institutions continuously conduct research and innovation to reduce costs in various aspects, such as the installation of offshore wind turbines (OWTs) and foundations, the construction of substations, and the laying of cables. It is worth noting that capital expenditures for OWTs account for approximately 48% of the entire offshore wind power development cycle, making up the largest proportion, as shown in Figure 2 [Lee and Zhao (2024)]. To reduce the cost of offshore wind development and improve power generation efficiency, many wind turbine manufacturers have launched larger rotor diameters with higher rated power, particularly Envision’s 16-18 MW OWT and Mingyang’s recent record-breaking 22 MW wind turbine model [Lee and Zhao (2024)]. This shows that the type of OWT is crucial for offshore wind farm development. In offshore wind power planning, wind farm selection [Dimitriou et al. (2025); Yildiz (2024); Bao et al. (2024); Gil-García et al. (2022); Díaz et al. (2023)] is an important link and belongs to multi-criteria decision-making (MCDM) problem. Besides, the selection of OWTs is a key step in offshore wind power planning. This process requires consideration of factors such as technology, economy, environment, suppliers, and performance, with experts making decisions based on these factors to select the optimal OWT. Therefore, OWT selection is a MCDM problem. Effective selection of OWTs has become a key issue in both research and practical applications [Xue et al. (2025)].

Figure 2.

General cost composition of an offshore wind farm.

However, there are limitations in applying current MCDM models to OWT selection, such as insufficient consideration of uncertain environments, expert linguistic ambiguity, and lack of more nuanced and intuitive ranking. To address these issues, our motivation is to develop an improved MCDM model that can assist operators or investors in selecting the most suitable OWTs. The following section briefly reviews the current research status.

Overview of research status and starting point

We conducted a survey on the research status of OWT selection based on MCDM methods and reviewed some representative articles, as shown in Table 1.

Table 1.

Application of wind turbine selection method.

Author(s)/year	Criteria/Sub-criteria	Alternatives	MCDM methods	Uncertainty method
[Shirgholami et al. (2016)]/2016	4/11	4 wind turbines	AHP	—
[Şağbanşua and Balo (2017)]/2017	4/13	4 wind turbines	AHP	—
[Ţişcă et al. (2017)]/2017	3/-	4 wind turbines	ELECTRE	—
[Bagočius et al. (2014)]/2014	5/-	4 wind turbines	WASPAS	—
[Rehman and Khan (2017)]/2017	5/-	18 wind turbines	WSM	—
[Lee et al. (2012)]/2012	4/14	4 wind turbines	ISM, FANP, VIKOR	Trapezoidal fuzzy set
[Beskese et al. (2020)]/2020	3/9	3 wind turbines	HFAHP, TOPSIS	Hesitant fuzzy set
[Supciller and Toprak (2020)]/2020	4/22	6 wind turbines	SWARA, EDAS, TOPSIS, Borda	Single-valued neutrosophic set
[Narayanamoorthy et al. (2021)]/2021	5/-	5 wind turbines	NWHF-CRITIC, NWHF-MAUT	Hesitant fuzzy set
[Xue et al. (2021)]/2021	4/11	6 wind turbines	Fuzzy Bayesian network, PCA	Triangular fuzzy numbers
[Zhao (2022)]/2022	4/20	—	MCGDM	Linguistic q-rung orthopair fuzzy set
[Wang et al. (2022)]/2022	5/22	4 wind turbines	SWARA, TOPSIS	D-S evidence theory
[Xu et al. (2022)]/2022	5/12	9 wind turbines	PCA, TOPSIS	D numbers
[Yu et al. (2022)]/2022	3/9	4 wind turbines	SWARA II, MEREC, CoCoSo	Interval 2-Tuple linguistic
[Tüysüz and Kahraman (2023)]/2023	6/-	5 wind turbines	Z-Fuzzy AHP&EDAS	Z-Fuzzy numbers

During decision-making, the first step is to establish an indicator system. According to the current research status, we focus on five criteria, including technology, compatibility with wind resources, economy, historical performance of wind turbines, and after-sales service of manufacturers. For example, the factors identified by Şağbanşua and Balo (2017) include technical, economic, environmental, and customer. Beskese et al. (2020) synthesized three main criteria from expert opinions: technical, economic, and vendor-related factors. Wang et al. (2022) and Xu et al. (2022) established five main criteria based on previous research, including technicality, adaptability to wind farms, economics, historical performance, and supplier after-sales service. Yu et al. (2022) constructed a standard system consisting of three main standards: technical, economic, and supplier-related criteria, based on expert opinions. Tüysüz and Kahraman (2023) identified six criteria, including reliability, technical characteristics, performance, cost factors, availability, and maintenance.

The next step is to build a proper MCDM method to evaluate alternatives under the determined indicators. Since the MCDM technique adopted for selecting OWTs affects the selection results and decision-making efficiency, it has been constantly developed and improved. Actually, it is inevitable to rely on empirical rules or the subjective experience of experts [Yazdi and Zarei (2022)]. Most experts tend to express their opinions qualitatively, which generates a lot of uncertain information in the selection process. As a result, many researchers have introduced a combination of uncertainty methods and MCDM methods to handle uncertainty during the selection process of OWTs. Uncertainty methods, including fuzzy set extensions [Lee et al. (2012); Beskese et al. (2020); Narayanamoorthy et al. (2021); Supciller and Toprak (2020); Xue et al. (2021); Zhao (2022); Tüysüz and Kahraman (2023)], D-S evidence theory [Wang et al. (2022)], D numbers [Xu et al. (2022)], and linguistic representation tools [Yu et al. (2022)], have been employed in the OWT selection. Among these, fuzzy sets are the most popular uncertainty theory for expressing linguistic judgments and handling fuzziness [Zadeh (1965)]. Accordingly, this study adopts fuzzy set theory to express expert linguistic judgments in the decision-making process.

Real-life decision analysis often involves large-scale bipolar or dual-sided subjective thoughts [Riaz et al. (2021)]. As shown in Table 1, various fuzzy extensions utilized in wind turbine selection studies, including hesitant fuzzy sets [Beskese et al. (2020)], neutrosophic [Supciller and Toprak (2020)], and q-rung orthopair fuzzy sets [Zhao (2022)], can effectively quantify expert linguistic evaluations yet only interpret expert cognition from a one-sided perspective. In the specific context of OWT selection, decision-makers do not merely focus on unilateral satisfaction or hesitant judgment. Instead, they need to balance positive engineering advantages with adverse technical risks and environmental liabilities. Consequently, current selection models overlook this dual-sided cognitive polarity. To address such ambiguous and uncertain decision scenarios, researchers have developed a variety of bipolar fuzzy theories, including bipolar fuzzy set (BFS) [Zhang (1994)], bipolar neutrosophic set (BNS) [Deli et al. (2015)], and interval-valued bipolar fuzzy set (IVBFS) [Wei et al. (2018)]. However, due to the limited information provided by these theories, they cannot provide sufficient information about ratings or grades, nor can they efficiently describe expert opinions that rely on scheme attributes. Cubic sets [Bae et al. (2012)] containing sufficient information with specific types of data have been successfully used in the decision-making process, and thus [Riaz and Tehrim (2020)] combined BFS and IVBFS into cubic sets to create cubic bipolar fuzzy sets (CBFS) for describing fuzzy information. CBFS provides more dimensional information as compared to the existing bipolar models due to its cubic property, enabling it to handle more complex ambiguities and uncertainties. By providing complete information on scoring occurrence, uncertainty, and polarization, CBFS has been successfully applied to describe uncertain information in decision-making problems [Jamil and Riaz (2022); Riaz and Tehrim (2019); Jamil and Riaz (2024)]. CBFS effectively transform expert evaluations into cubic bipolar fuzzy numbers (CBFNs) containing positive and negative information to reduce information loss, so we apply CBFS to expert evaluation in our work.

In decision-making processes, the critical step is to identify and select the optimal wind turbine from a set of alternatives. As shown in Table 1, techniques such as technique for order preference by similarity to ideal solution (TOPSIS), elimination and choice translating reality (ELECTRE-I), VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), and combined compromise solution (CoCoSo) have been applied to support such decisions. Among them, ELECTRE-I uses outranking relations to assess relative superiority, making it especially suitable for complex trade-offs. However, the traditional ELECTRE-I often yields poorly discriminated aggregated dominance matrices when handling many alternatives. To improve accuracy and simplify computation, the improved ELECTRE-I method [Govindan and Jepsen (2016)] is adopted. This approach removes threshold calculations, ensuring a more straightforward and precise ranking of alternatives. To further enhance the ranking accuracy, this paper develops a new significance formula and incorporates it into an enhanced ELECTRE-I framework, called cubic bipolar fuzzy extension of improved ELECTRE-I (CBF-ELECTRE-I+), which is proven to be effective in the selection result.

Innovation and contribution

In contrast to previous OWT selection models that rely on single-type fuzzy representations and standard distance-based ranking, this study establishes an effective decision-making system that integrates CBFS, the P-order cubic bipolar fuzzy ordered weighted geometric (P-CBFWG) operator, and the improved ELECTRE-I algorithm. In order to effectively reduce the impact of vagueness and directly express the incomplete information from experts, the linguistic assessments are quantified in the form of CBFS, where each entry contains both bipolar fuzzy and interval-valued bipolar fuzzy information, providing richer information than traditional methods. Next, the CBFSs of all experts are aggregated into a new quantified CBFS by using the P-CBFWG operator. This new CBFS contains scoring suggestions from different experts simultaneously. At last, this paper integrates the improved ELECTRE-I algorithm with CBFS to create the new CBF-ELECTRE-I+ model for ranking alternative OWTs. It confirms that CBF-ELECTRE-I+ effectively addresses the existing problem of the traditional cubic bipolar fuzzy ELECTRE-I (CBF-ELECTRE-I) model, including unclear differences between different alternatives and complicated calculation. Specifically, the CBF-ELECTRE-I+ model enhances alternative discrimination and simplifies the overall execution without threshold calculations. In short, the highlights of this article are as follows:

(1) CBFS and the P-CBFWG operator establish a dual-sided representation and aggregation framework, reducing input-side information loss and capturing cognitive polarity.

(2) The CBF-ELECTRE-I+ model eliminates threshold calculations, addressing poor discrimination and high computational complexity found in conventional methods.

(3) MCDM fused with CBFS handles uncertainty caused by fuzziness in linguistic assessments and subjectivity in expert judgments, delivering a regulated and reproducible tool for OWT selection.

The rest of this article is structured as follows. In Section 2, the theoretical basis is introduced. Section 3 elaborates on the decision-making model. Section 4 presents the application of the proposed models and conducts the comparison. Finally, we summarize the research in Section 5.

Preliminaries

Cubic bipolar fuzzy set

CBFS [Riaz and Tehrim (2019); Jamil and Riaz (2024)], including BFS and IVBFS, provides complete information about the occurrence of ratings, uncertainty, and bipolarity in the decision process. In order to better integrate CBFS into the modeling process, we review the basic concepts and operation rules of CBFS in this section, including sum, product, complement, and scalar product under P-order.

Basic concepts

Let V be a universal set. A CBFS is defined on V as

Φ = {〈 χ, {P = [P_{l}, P_{u}], N = [N_{l}, N_{u}]}, {λ, μ} 〉 | χ \in V}

(1)

where

I = {P, N}

is IVBFS,

J = {λ, μ}

is BFS and

[P_{l}, P_{u}] \subseteq [0, 1]

[N_{l}, N_{u}] \subseteq [‒ 1, 0]

λ : V \to [0, 1]

μ : V \to [‒ 1, 0]

Algebraic operation

Let $Φ_{1}$ , $Φ_{2}$ be two CBFNs on the universe $V$ , where $Φ_{1} = 〈 χ, {P_{1}, N_{1}}, {λ_{1}, μ_{1}} 〉$ and $Φ_{2} = 〈 χ, {P_{2}, N_{2}}, {λ_{2}, μ_{2}} 〉$ .

1. The algebraic operations for P-order.

\begin{array}{l} (i) Φ_{1} \oplus Φ_{2} = {⟨ χ, {[P_{1 l} + P_{2 l} ‒ P_{1 l} * P_{2 l}, P_{1 u} + P_{2 u} ‒ P_{1 u} * P_{2 u}], \\ [‒ N_{1 l} * N_{2 l}, ‒ N_{1 u} * N_{2 u}]}, \\ {λ_{1} + λ_{2} ‒ λ_{1} * λ_{2}, ‒ μ_{1} * μ_{2}} ⟩ | χ \in V} . \end{array},

(2)

\begin{array}{l} (i i) Φ_{1} \oplus Φ_{2} = {⟨ χ, {[P_{1 l} * P_{2 l}, P_{1 u} * P_{2 u}], \\ [‒ (‒ N_{1 l} ‒ N_{2 l} ‒ N_{1 l} * N_{2 l}), ‒ (‒ N_{1 u} ‒ N_{2 u} ‒ N_{1 u} * N_{2 u})]}, \\ {λ_{1} * λ_{2}, ‒ (‒ μ_{1} ‒ μ_{2} ‒ μ_{1} * μ_{2})} ⟩ | χ \in V} . \end{array}

(3)

(i i i) α * Φ = \begin{array}{l} {〈 χ, {[1 ‒ {(1 ‒ P_{l})}^{α}, 1 ‒ {(1 ‒ P_{u})}^{α}], [‒ {(‒ N_{l})}^{α}, ‒ {(‒ N_{u})}^{α}]}, \\ {1 ‒ {(1 ‒ λ)}^{α}, ‒ {(‒ μ)}^{α}} 〉 | χ \in V} . \end{array}

(4)

2. The complement of $Φ$ can be defined as

Φ^{c} = {〈 χ, [1 - P_{u}, 1 - P_{l}] [- 1 - N_{u}, - 1 - N_{l}]}, {1 - λ, - 1 - μ} 〉 | χ \in V}

(5)

The normalized Euclidean distance

Let $Φ_{1}$ , $Φ_{2}$ be two CBFNs on the universe $V$ , where $Φ_{1} = 〈 χ, {P_{1}, N_{1}}, {λ_{1}, μ_{1}} 〉$ and $Φ_{2} = 〈 χ, {P_{2}, N_{2}}, {λ_{2}, μ_{2}} 〉$ . The normalized Euclidean distance between $Φ_{1}$ and $Φ_{2}$ is defined as

d (Φ_{1}, Φ_{2}) = \sqrt{\begin{array}{l} \frac{1}{6} ({(P_{1 l} ‒ P_{2 l})}^{2} + {(P_{1 u} ‒ P_{2 u})}^{2} + {(N_{1 l} ‒ N_{2 l})}^{2} \\ + {(N_{1 u} ‒ N_{2 u})}^{2} + {(λ_{1} ‒ λ_{2})}^{2} + {(μ_{1} ‒ μ_{2})}^{2}) \end{array}} .

(6)

P-order cubic bipolar fuzzy ordered weighted geometric operator

Aggregation operators are highly effective functions for combining information in the decision process, especially for correctly fusing large amounts of complex information at the beginning of model construction. Among them, the P-CBFWG operator is presented to aggregate cubic bipolar fuzzy data [Riaz and Tehrim (2020)] while considering the indicator weights. Suppose $Φ_{k} = {〈 {[P_{l k}, P_{u k}], [N_{l k}, N_{u k}]}, {λ_{k}, μ_{k}} 〉}$ , for ${k = 1, 2, . . ., n},$ be a family of cubic bipolar fuzzy elements (CBFEs). Consider a weighted vector $W = {[w_{1}, w_{2}, \dots, w_{n}]}^{T}$ , where $\sum_{k = 1}^{n} ‍ w_{k} = 1$ and $0 \leq w_{k} \leq 1 .$ Then P-CBFWG operator is a mapping: $Φ^{n} \to Φ$ which can be computed as follows:

\begin{array}{l} P ‒ C B F W G (Φ_{1}, Φ_{2}, \dots, Φ_{n}) \\ = {\begin{cases} n \\ \otimes \\ k = 1 \end{cases}}_{P} (w_{k} (Φ_{k})) \\ = {{[\prod_{k = 1}^{n} ‍ {(P_{l_{k}})}^{w_{k}}, \prod_{k = 1}^{n} ‍ {(P_{u_{k}})}^{w_{k}}], \\ [‒ (1 ‒ \prod_{k = 1}^{n} ‍ {(1 ‒ (‒ N_{l_{k}}))}^{w_{k}}), ‒ (1 ‒ \prod_{k = 1}^{n} ‍ {(1 ‒ (‒ N_{u_{k}}))}^{w_{k}})]}, \\ {\prod_{k = 1}^{n} ‍ {(λ_{k})}^{w_{k}}, ‒ (1 ‒ \prod_{k = 1}^{n} ‍ {(1 ‒ (‒ μ_{k}))}^{w_{k}})}} . \end{array}

(7)

Principal components analysis

Principal components analysis (PCA) is used to compute the relative weight of each indicator, minimizing expert subjectivity. Firstly, the indicators are classified into two types: cost indicator and benefit indicator. Then, the standardized matrix $R$ is computed using the range method as follows:

R = {(r_{i j})}_{m \times n} = {\begin{cases} \frac{d_{j}^{i} ‒ \min (d_{j}^{i})}{\max (d_{j}^{i}) ‒ \min (d_{j}^{i})}, j \in τ_{b}, \\ \frac{\max (d_{j}^{i}) ‒ d_{j}^{i}}{\max (d_{j}^{i}) ‒ \min (d_{j}^{i})}, j \in τ_{c} . \end{cases}

(8)

The steps for calculating weights using PCA [Xu et al. (2022)] are as follows:

Step 1. Calculate the contribution rate of each principal component:

u_{i} = \frac{λ_{i}}{\sum_{k = 1}^{m} ‍ λ_{k}} .

(9)

The cumulative contribution rate of the first $m$ principal components must exceed 90% to preserve most of the information in the indicators.

Step 2. Determine the coefficients of the principal components in each linear combination:

l_{i j} = \frac{e_{i j}}{\sqrt{λ_{i}}} (i = 1, 2, \dots, m; j = 1, 2, \dots, p),

(10)

where

e_{i j}

is the load of the ith principal component on the original index variable

d_{j}

Step 3. Determine the coefficients of each factor in the comprehensive scoring model:

d_{j} = \sum_{i = 1}^{m} ‍ u_{i} l_{i j} .

(11)

Step 4. Compute the weights of each factor:

w_{j} = \frac{d_{j}}{\sum_{k = 1}^{p} ‍ d_{k}} .

(12)

Improved ELECTRE-I method

The improved ELECTRE-I method [Govindan and Jepsen (2016)] simplifies the traditional process by eliminating the need for threshold calculations and the generation of separate concordance and discordance dominance matrices. Instead, it directly calculates the aggregated dominance matrix using the concordance matrix and a modified discordance matrix. This modification not only streamlines the computation but also enhances the ranking accuracy. The detailed steps of the improved ELECTRE-I method are outlined below.

Step 1. Standardize the decision matrix $D$ to obtain a standardized decision matrix $D^{'} = {[u_{i j}]}_{m \times n} .$

u_{i j} = \frac{d_{i j}}{\sqrt{Σ_{i = 1}^{n} {(d_{i j})}^{2}}},

(13)

Step 2. Construct a weighted normalization matrix $V = [v_{i j}] m \times n = [u_{i j}] \times w j$ .

Step 3. Determine the concordance set $C_{α β}$ and discordance set $D_{α β}$ . Here, $C_{α β}$ represents attributes where target $α$ is not inferior to target $β$ .

Step 4. Calculate the concordance matrix and discordance matrix to represent satisfaction and dissatisfaction levels, respectively.

Define the concordance indices $x (α, β)$ by

x (α, β) = \sum_{j \in C_{α β}} ‍ w_{j} .

(14)

x (α, β)

reflects the degree to which the performance of target

α

is not inferior to that of target

β

. Compare the concordance indices of all target pairs to form a concordance matrix

X

Define the discordance indices $y (α, β)$ by

y (α, β) = \frac{\max_{j \in D_{α β}} | g_{α j} ‒ g_{β j} |}{\max_{j \in J, s, t \in I} | g_{s j} ‒ g_{t j} |} .

(15)

y (α, β)

represents the degree of dissatisfaction with selecting

α

compared to the target

β

. Compare the discordance indices of all target pairs to form a discordance matrix

Y

Step 5. Determine the corrected discordance matrix $Y^{'} = [y^{'} (α, β)]$ , where

y^{'} (α, β) = 1 - y (α, β) .

(16)

Each element represents the degree of satisfaction or reliability of selecting alternative $α$ relative to target $β$ .

Step 6. Calculate the aggregate dominance matrix $Z = [z (α, β)]$ , where

z (α, β) = x (α, β) \times y^{'} (α, β) .

(17)

z (α, β)

reflects the overall dominance of alternative

α

over

β

. Consequently, a lower

z (α, β)

value indicates that

β

faces weaker competitive pressure from

α

, signifying a superior standing for

β

Step 7. Compute the significance of each alternative. Traditionally, the significance of an alternative is determined by summing the column elements of the dominance matrix $Z$ . However, this approach can lead to information losses due to its multiplicative nature. Specifically, if $y^{'} (j, i)$ is zero, the significant superiority information contained in $x (j, i)$ is completely nullified, regardless of its magnitude. To prevent such information loss and enhance the robustness of the ranking, this study improves the formula as follows:

A S_{i} = \sum_{j = 1}^{m} ‍ x (j, i) [y^{'} (j, i) + 1] .

(18)

The addition of unity ensures that the baseline information in $x (j, i)$ is preserved when $y^{'} (j, i)$ vanishes. This mechanism prevents a single localized veto from overshadowing the comprehensive strengths of an alternative, thereby yielding a more precise and robust ranking. Alternatives are then ranked in ascending order of $A S_{i}$ , where the minimum value represents the optimal choice.

Proposed decision-making model

As shown in Figure 3, the decision-making system introduced for OWT selection includes establishing the indicator system, quantifying expert assessments, integrating expert opinions, and ranking the alternatives using CBF-ELECTRE-I+.

Figure 3.

Flowsheet of the proposed decision-making system.

Establishment of the indicator system

Determine evaluation indicators

According to the literature research, we have identified five main criteria for OWT selection, including technology, matching with the wind resources, economy, historical performance, and after-sales service of the manufacturer. These main criteria are further decomposed into twelve secondary indicators. The detailed information is shown in Table 2.

Table 2.

The evaluation criteria system for offshore wind turbine selection.

Main criteria	Indicator	Type	Explanation
Technology ( $B_{1}$ )	Generator and gearbox ( $C_{1}$ ) [Ma et al. (2021)]	Benefit	Convert impeller mechanical energy to electrical energy and boost speed.
	Professional safety design of wind turbine ( $C_{2}$ ) [Xu et al. (2022); Ederer (2014)]	Benefit	Professional safety design is crucial for efficient power system operation.
	Swept area ( $C_{3}$ ) [Xu et al. (2022); Li et al. (2024); Emeksiz and Yüksel (2022); Sagbansua and Balo (2017)]	Benefit	Larger swept wind area increases wind energy conversion efficiency.
Matching with the wind resources ( $B_{2}$ )	Rated wind speed ( $C_{4}$ ) [Xu et al. (2022); Li et al. (2024); Zhao (2022); Sagbansua and Balo (2017)]	Cost	When wind speed is below rated speed, turbine power output drops.
Matching with the wind resources ( $B_{2}$ )	Wind class (IEC) ( $C_{5}$ ) [Xu et al. (2022); Li et al. (2024); Huang et al. (2020)]	Benefit	A wind turbine’s IEC rating must meet minimum wind speed and turbulence intensity requirements.
Economy ( $B_{3}$ )	Annual power generation ( $C_{6}$ ) [Yu et al. (2022); Ma et al. (2021); Sedaghat et al.(2019)]	Benefit	The annual electricity output expected from a wind turbine.
	Kilowatt cost ( $C_{7}$ ) [Xu et al. (2022); Li et al. (2024); Ma et al. (2021); Huang et al. (2020); Pang et al. (2021)]	Cost	It directly affects the investment and construction cost of the wind turbine.
	KWH investment ( $C_{8}$ ) [Xu et al. (2022); Li et al. (2024); Zhao (2022); Bosch et al. (2019)]	Cost	The investment required per unit of electricity generated.
Historical performance ( $B_{4}$ )	Sales performance ( $C_{9}$ ) [Wang et al. (2022); Yu et al. (2022); Ma et al. (2021); Zhao (2022); Sagbansua and Balo (2017)]	Benefit	It measures wind turbine sales.
Historical performance ( $B_{4}$ )	Operating performance ( $C_{10}$ ) [Ma et al. (2021); Huang et al. (2020); Pang et al. (2021)]	Benefit	It reflects the wind turbine’s performance and indicates satisfaction.
After-sales service of manufacturer ( $B_{5}$ )	Maintainability ( $C_{11}$ ) [Sagbansua and Balo (2017)]	Benefit	The wind turbine is maintainable if it malfunctions.
After-sales service of manufacturer ( $B_{5}$ )	Warranty ( $C_{12}$ ) [Yu et al. (2022); Ma et al. (2021); El Fadli et al. (2025)]	Benefit	A long warranty period reduces maintenance costs and improves economic efficiency.

These secondary indicators are categorized into cost and benefit attributes based on project optimization objectives. Specifically, the rated wind speed ( $C_{4}$ ), kilowatt cost ( $C_{7}$ ), and KWH investment ( $C_{8}$ ) are defined as cost attributes to minimize capital expenditures and improve operational efficiency. Conversely, the remaining nine indicators are designated as benefit attributes to maximize lifetime asset performance. Furthermore, these indicators exhibit coupled engineering correlations and mutual trade-offs. For instance, expanding the swept area ( $C_{3}$ ) increases annual power generation ( $C_{6}$ ) but simultaneously necessitates more complex structural designs ( $C_{1}$ , $C_{2}$ ) and higher initial investments ( $C_{7}$ , $C_{8}$ ). These intricate multi-criteria conflicts demonstrate the necessity of employing an advanced MCDM model integrated with objective weighting to achieve a balanced and rigorous alternative ranking.

Calculate indicator weights

In the decision-making process, indicators are categorized into quantitative ones derived from objective data and qualitative ones assessed by expert language. Firstly, the qualitative indicators are scored by experts through knowledge and experience on a scale of 1 to 9, with 1 indicating the lowest and 9 the highest. Then, the twelve secondary indicators are standardized using the range method. Finally, to address the structural correlations among the indicators, the objective weight of each criterion is determined by PCA, as shown in Section 2.3.

Integration of expert opinions

The selection process always involves the evaluation of indicators by multiple experts, and effectively integrating the advice from different experts can make the selection results more accurate.

Step 1. Collect the evaluation information of experts. The linguistic evaluations of different alternatives by experts are quantified in the form of CBFNs based on Table 3, and then the resulting data is collected.

Step 2. Construct the CBF decision matrices. The decision matrix for each expert is obtained as follows:

D_{k} = {[d_{i j}^{k}]}_{m \times n} = [\begin{array}{l} d_{11}^{k} & \dots & d_{1 n}^{k} \\ ⋮ & ⋱ & ⋮ \\ d_{m 1}^{k} & \dots & d_{m n}^{k} \end{array}],

(19)

where

d_{i j}^{k} = 〈 {[P_{l i j}^{k}, P_{u i j}^{k}], [N_{l i j}^{k}, N_{u i j}^{k}]}, {λ_{i j}^{k}, μ_{i j}^{k}} 〉

, representing the kth expert evaluation of the alternative

M_{i}

under the attribute

C_{j}

Step 3. Standardize the CBF decision matrices. The indicators are classified into two types: the cost indicator $τ_{c}$ and the benefit indicator $τ_{b}$ , as shown in Table 2. The two types of indicators in the decision matrix $D_{k}$ should be converted into the same type for subsequent calculations. The standardized formula is as follows:

D_{k}^{'} = {[u_{i j}^{k}]}_{m \times n} = {\begin{cases} {(d_{i j}^{k})}^{c}, j \in τ_{c}, \\ d_{i j}^{k}, j \in τ_{b} . \end{cases}

(20)

Table 3.

Language quantitative information evaluation form.

Assessment	Abbreviations	Value
Very good	VG	[0.8001, 1.0000], [‒0.1999, −0.0001], 0.9000, −0.1000
Good	G	[0.6001, 0.8000], [‒0.3999, −0.2000], 0.7000, −0.3000
Average	A	[0.4001, 0.6000], [‒0.5999, −0.4000], 0.5000, −0.5000
Poor	P	[0.2001, 0.4000], [‒0.7999, −0.6000], 0.3000, −0.7000
Very poor	VP	[0.0000, 0.2000], [‒1.0000, −0.8000], 0.1000, −0.9000

See formula (5) for details.

Step 4. Aggregate the CBF decision matrices from different experts. Based on the expert weights and the P-CBFWG operator (defined in Section 2.2), the individual CBF decision matrices $D_{k}^{'}$ are aggregated into a single comprehensive CBF decision matrix $D$ that retains the evaluations of all experts:

D = {[u_{i j}]}_{m \times n} = [\begin{array}{l} u_{11} & \dots & u_{1 n} \\ ⋮ & ⋱ & ⋮ \\ u_{m 1} & \dots & u_{m n} \end{array}] .

The ranking of the alternatives using CBF-ELECTRE-I+

The traditional CBF-ELECTRE-I method [Jamil and Riaz (2022)] exhibits certain limitations, particularly in relation to the discordance matrix and its time complexity. These issues result in less accurate evaluations and inefficiencies in the decision-making process. To overcome these shortcomings, the CBF-ELECTRE-I+ model is proposed to enhance the ranking of the alternatives. Presume that decision-makers are responsible for accessing m-alternatives, WTGi, under each of n-attributes, $C_{j}$ . The footsteps of CBF-ELECTRE-I+ model are provided as follows:

Step 1. Calculate the CBF weighted matrix by taking multiplication of weight vector and decision matrix as below

V = D \otimes W = {[v_{i j}]}_{m \times n} = [\begin{array}{l} v_{11} & v_{12} & \dots & v_{1 n} \\ v_{21} & v_{22} & \dots & v_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ v_{m 1} & v_{m 2} & \dots & v_{m n} \end{array}],

(21)

where

W = {[w_{1}, w_{2}, . . ., w_{n}]}^{T}

is the weight vector of the attributes, satisfying the conditions

0 \leq w_{j} \leq 1

and

\sum_{j = 1}^{n} ‍ w_{j} = 1

v_{i j} = 〈 {[A_{l i j}, A_{u i j}], [B_{l i j}, B_{u i j}]}, {ξ_{i j}, η_{i j}} 〉

1 \leq i \leq m

1 \leq j \leq n

[A_{l i j}, A_{u i j}] = w_{j} * [P_{l i j}, P_{u i j}]

[B_{l i j}, B_{u i j}] = w_{j} * [N_{l i j}, N_{u i j}]

ξ_{i j} = w_{j} * λ_{i j}

η_{i j} = w_{j} * μ_{i j}

Step 2. Determine the bipolar concordance set $C_{α β}$ and the bipolar discordance set $D_{α β}$ are as follows:

C_{α β} = {1 \leq j \leq n | t_{α j} \geq t_{β j}, α \neq β, α, β = 1, 2, . . ., m},

(22)

D_{α β} = {1 \leq j \leq n | t_{α j} < t_{β j}, α \neq β, α, β = 1, 2, . . ., m},

(23)

where

t_{i j} = A_{l i j} + A_{u i j} + B_{l i j} + B_{u i j} + ξ_{i j} + η_{i j}

Step 3. Construct the CBF concordance matrix $X$ and CBF discordance matrix $Y$ .

The CBF concordance indices are calculated using formula (14), then the CBF concordance matrix is

X = [\begin{array}{l} ‒ & x (1, 2) & x (1, 3) & \dots & x (1, m) \\ x (2, 1) & ‒ & x (2, 3) & \dots & x (2, m) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x (m, 1) & x (m, 2) & x (m, 3) & \dots & ‒ \end{array}]

The CBF discordance indices are calculated as:

y (α, β) = \frac{\max_{j \in D_{α β}} \sqrt{\begin{array}{l} \frac{1}{6} ({(A_{l α j} ‒ A_{l β j})}^{2} + {(A_{u α j} ‒ A_{u β j})}^{2} \\ + {(B_{l α j} ‒ B_{l β j})}^{2} + {(B_{u α j} ‒ B_{u β j})}^{2} \\ + {(ξ_{α j} ‒ ξ_{β j})}^{2} + {(η_{α j} ‒ η_{β j})}^{2}) \end{array}}}{\max_{j} \sqrt{\begin{array}{l} \frac{1}{6} ({(A_{l α j} ‒ A_{l β j})}^{2} + {(A_{u α j} ‒ A_{u β j})}^{2} \\ + {(B_{l α j} ‒ B_{l β j})}^{2} + {(B_{u α j} ‒ B_{u β j})}^{2} \\ + {(ξ_{α j} ‒ ξ_{β j})}^{2} + {(η_{α j} ‒ η_{β j})}^{2}) \end{array}}} .

(24)

Thus, the CBF discordance matrix is

Y = [\begin{array}{l} ‒ & y (1, 2) & y (1, 3) & \dots & y (1, m) \\ y (2, 1) & ‒ & y (2, 3) & \dots & y (2, m) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ y (m, 1) & y (m, 2) & y (m, 3) & \dots & ‒ \end{array}] .

Step 4. Determine the corrected CBF discordance matrix $Y^{'}$ using the formula (16).

Y^{'} = [\begin{array}{l} ‒ & y^{'} (1, 2) & y^{'} (1, 3) & \dots & y^{'} (1, m) \\ y^{'} (2, 1) & ‒ & y^{'} (2, 3) & \dots & y^{'} (2, m) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ y^{'} (m, 1) & y^{'} (m, 2) & y^{'} (m, 3) & \dots & ‒ \end{array}]

Step 5. The aggregate dominance matrix $Z = z (α, β)$ is calculated using the formula (17) as follows:

Z = [\begin{array}{l} ‒ & x (1, 2) y^{'} (1, 2) & x (1, 3) y^{'} (1, 3) & \dots & x (1, m) y^{'} (1, m) \\ x (2, 1) y^{'} (2, 1) & ‒ & x (2, 3) y^{'} (2, 3) & \dots & x (2, m) y^{'} (2, m) \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x (m, 1) y^{'} (m, 1) & x (m, 2) y^{'} (m, 2) & x (m, 3) y^{'} (m, 3) & \dots & ‒ \end{array}]

Step 6. Compute the significance $A S_{i}$ of each alternative using the formula (18). Alternative OWTs are then ranked in ascending order of $A S_{i}$ , where the minimum value represents the optimal choice.

Offshore wind turbine selection

This section introduces the application of the proposed hybrid decision-making system in the OWT selection in a certain sea area of China. Table 4 [Li et al. (2024)] provides detailed information about the first phase project of offshore wind power in this sea area. After comprehensively analyzing the main technical parameters of OWTs, the offshore wind resource features, and the current supply market situation, we preliminarily selected nine wind turbine units to meet the bidding requirements of the project, namely, WTG1 to WTG9. The raw data of quantitative indicators [Li et al. (2024)] and technical parameters related to alternative OWTs are shown in Table 5.

Table 4.

Demonstration project of stage one.

Parameter	Introduction
Installed capacity	100 MW
Annual average wind speed	Over 7.4 m/s
Power density	Over 400 W/m²
Comprehensive wind shear index	About 0.09
Turbulence intensity	Small
Annual main wind direction	NE ENE, about 24.8%
Annual main wind direction	S SSW, about 20.2%

Table 5.

The data of quantitative indicators and technical parameters.

Indicator	WTG1	WTG2	WTG3	WTG4	WTG5	WTG6	WTG7	WTG8	WTG9
Rated power	3	3	3	3.6	5	5	5	6	3.6
Swept area	6358	7854	6362	10568	12861	10387	12469	18617	11300
Rated wind speed	13	13	15	12	12.5	12	13	11	12.5
Rated voltage	690	690	690	690	6300	3000	950	6300	750
Annual power generation	22298	22908	21031	25302	24500	21445	23168	29620	27043
Capacity factor	0.25	0.256	0.235	0.287	0.280	0.245	0.264	0.331	0.306
Kilowatt cost	4380	4560	5800	4800	5400	6500	7000	5298	6500
KWh investment	6.38	6.33	7.18	5.86	5.99	7.62	7.27	5.63	6.06

Establishment of the indicator system

Section 3.1.1 has identified the criteria system for OWT selection, including five main criteria and twelve secondary indicators. Initially, the qualitative indicators are rated by three experts, and the comprehensive scoring data is detailed in Table 6. Next, the secondary indicators are standardized by applying range method, and the data is shown in Table 7. Finally, the weight of each indicator is assigned by PCA, and the steps are as follows.

Step 1. The contribution rate of each principal component computed by formula (9) is shown in Table 8. It can be seen that the cumulative contribution rate of the first three principal components is 91.27%, so that they preserve most of the information in the indicators. Therefore, the first three principal components are extracted for subsequent calculations.

Step 2. The component loadings of the first three principal components are given in Table 9. Then, the corresponding coefficients are calculated through formula (10) and presented in Table 10.

Step 3. The coefficients and the final weights of twelve indicators are shown in Table 11.

Table 6.

The comprehensive scoring data of qualitative indicators by experts.

Indicator	WTG1	WTG2	WTG3	WTG4	WTG5	WTG6	WTG7	WTG8	WTG9
Generator and gearbox	9	6	7	9	9	7	9	9	9
Professional safety design of wind turbine	8.2	8	7.9	8.8	8.6	7.9	8	9	8.4
Wind class (IEC)	9	8.5	8.9	8.5	9	8.7	8.8	8.5	9
Sales performance	8.3	8.1	7.7	9	8.9	8.1	8	8.9	8.5
Operating performance	7.9	7.7	7.7	8.9	8.7	7.7	7.9	9	7.9
Maintainability	8.7	8.1	8.2	8.8	8.8	8.1	8.3	9	8.6
Warranty	8.2	8	7.9	8.8	8.6	7.9	8	9	8.4

Table 7.

The data of the standardized indicators using range method.

	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$	$C_{6}$	$C_{7}$	$C_{8}$	$C_{9}$	$C_{10}$	$C_{11}$	$C_{12}$
WTG1	1.0000	0.2727	0.0000	0.5000	1.0000	0.1475	1.0000	0.6231	0.4615	0.1538	0.6667	0.2727
WTG2	0.0000	0.0909	0.1220	0.5000	0.0000	0.2185	0.9313	0.6482	0.3077	0.0000	0.0000	0.0909
WTG3	0.3333	0.0000	0.0003	0.0000	0.8000	0.0000	0.4580	0.2211	0.0000	0.0000	0.1111	0.0000
WTG4	1.0000	0.8182	0.3434	0.7500	0.0000	0.4973	0.8397	0.8844	1.0000	0.9231	0.7778	0.8182
WTG5	1.0000	0.6364	0.5305	0.6250	1.0000	0.4039	0.6107	0.8191	0.9231	0.7692	0.7778	0.6364
WTG6	0.3333	0.0000	0.3287	0.7500	0.4000	0.0482	0.1908	0.0000	0.3077	0.0000	0.0000	0.0000
WTG7	1.0000	0.0909	0.4985	0.5000	0.6000	0.2488	0.0000	0.1759	0.2308	0.1538	0.2222	0.0909
WTG8	1.0000	1.0000	1.0000	1.0000	0.0000	1.0000	0.6496	1.0000	0.9231	1.0000	1.0000	1.0000
WTG9	1.0000	0.4545	0.4031	0.6250	1.0000	0.7000	0.1908	0.7839	0.6154	0.1538	0.5556	0.4545

Integration of expert opinions

In this study, three experienced experts ( $E_{1}$ , $E_{2}$ , and $E_{3}$ ) are invited to assess the indicators related to nine alternatives. To ensure fairness, investors regard the opinions of the three experts as equally significant, meaning the weight vector of the experts is $w = (1 / 3, 1 / 3, 1 / 3)$ . The following is the detailed calculation process.

Step 1. The evaluation information of the three experts is collected based on Table 3. For instance, Table 12 presents the evaluation results of the three experts on WTG3, where ${P, A}$ indicates the frequency of ‘poor’ and ‘average’ appearances is equivalent, and ${G} : {V G} = 4 : 1$ signifies that ‘good’ occurs four times as frequently as ‘very good’.

Table 8.

The contribution rate of each indicator.

Component	Eigenvalue	Contribution Rate	Cumulative Contribution Rate
1	7.891	0.6576	0.6576
2	1.599	0.1333	0.7909
3	1.462	0.1218	0.9127
4	0.386	0.0322	0.9449
5	0.321	0.0268	0.9716
6	0.173	0.0144	0.9860
7	0.125	0.0104	0.9965
8	0.042	0.0035	1.0000
9	0	0.0000	1.0000
10	0	0.0000	1.0000
11	0	0.0000	1.0000
12	0	0.0000	1.0000

Table 9.

Component loadings of the first three principal components.

Indicator	Component
Indicator	1	2	3
Generator and gearbox	0.684	0.533	0.368
Professional safety design of wind turbine	0.990	−0.050	0.047
Swept area	0.752	0.366	−0.481
Rated wind speed	0.761	0.043	−0.449
Wind class (IEC)	−0.281	0.579	0.717
Annual power generation	0.889	0.146	−0.177
Kilowatt cost	0.289	−0.847	0.394
KWh investment	0.858	−0.291	0.262
Sales performance	0.950	−0.062	0.078
Operating performance	0.935	−0.070	−0.008
Maintainability	0.924	0.084	0.338
Warranty	0.990	−0.050	0.047

Table 10.

The coefficients of the principal components.

Indicator	Component
Indicator	1	2	3
Generator and gearbox	0.2435	0.4215	0.3044
Professional safety design of wind turbine	0.3524	−0.0395	0.0389
Swept area	0.2677	0.2894	−0.3978
Rated wind speed	0.2709	0.0337	−0.3713
Wind class (IEC)	−0.0999	0.4579	0.5929
Annual power generation	0.3165	0.1155	−0.1464
Kilowatt cost	0.1029	−0.6698	0.3259
KWh investment	0.3054	−0.2301	0.2167
Sales performance	0.3382	−0.0493	0.0645
Operating performance	0.3328	−0.0554	−0.0066
Maintainability	0.3289	0.0664	0.2795
Warranty	0.3524	−0.0395	0.0389

Table 11.

Indicator weight in the decision-making system.

Criteria	Secondary indicators	Coefficient	Weight
Technology	Generator and gearbox	0.2775	0.1151
	Professional safety design of wind turbine	0.2533	0.1051
	Swept area	0.182	0.0755
Matching with the wind resources	Rated wind speed	0.1507	0.0625
Matching with the wind resources	Wind class (IEC)	0.0739	0.0307
Economy	Annual power generation	0.2253	0.0934
	Kilowatt cost	0.0197	0.0082
	KWh investment	0.2154	0.0893
Historical performance of wind turbine	Sales performance	0.2451	0.1016
Historical performance of wind turbine	Operating performance	0.2308	0.0957
After-sales service of manufacturer	Maintainability	0.2842	0.1179
After-sales service of manufacturer	Warranty	0.2533	0.1051

Example 1

For WTG3, the evaluation of expert $E_{3}$ under factor $C_{2}$ is ${P, A}$ , which needs to be quantified. According to Table 3, the quantified assessment standard is

\begin{array}{l} {P} = [0.2001, 0.4000], [‒ 0.7999, ‒ 0.6000], 0.3000, ‒ 0.7000, \\ {A} = [0.4001, 0.6000], [‒ 0.5999, ‒ 0.4000], 0.5000, ‒ 0.5000, \end{array}

and the probability of assessment standard

{P}

{A}

is 0.5. We use the weight to replace the probability of

{P}

and

{A}

in evaluation level

{P, A}

and use the P-CBFWG operator to aggregate assessment standards

{P}

and

{A}

. The quantification result of the

{P, A}

obtained by fusion is calculated as follows:

\begin{array}{l} CBFN 1 = [0.200 1^{0.5} * 0.400 1^{0.5}, 0 . 4^{0.5} * 0 . 6^{0.5}], \\ [‒ (1 ‒ (1 ‒ 0.799 9^{0.5}) * (1 ‒ 0.599 9^{0.5})), ‒ (1 ‒ (1 ‒ 0 . 6^{0.5}) * (1 ‒ 0 . 4^{0.5}))], \\ 0 . 3^{0.5} * 0 . 5^{0.5}, ‒ (1 ‒ (1 ‒ 0 . 7^{0.5}) * (1 ‒ 0 . 5^{0.5})) . \\ = [0.2829, 0.4899], [‒ 0.7171, ‒ 0.5101], 0.3873, ‒ 0.6127 . \end{array}

Table 12.

Assessment results of WTG3 from experts.

Criteria	Experts	Assessment grade	Criteria	Experts	Assessment grade
$C_{1}$	$E_{1}$	{A}	$C_{7}$	$E_{1}$	{A}
	$E_{2}$	{P,A}		$E_{2}$	{A}
	$E_{3}$	{A}		$E_{3}$	{A}
$C_{2}$	$E_{1}$	{A}	$C_{8}$	$E_{1}$	{A}
	$E_{2}$	{A}		$E_{2}$	{P,A}
	$E_{3}$	{P,A}		$E_{3}$	{P,A}
$C_{3}$	$E_{1}$	{P}	$C_{9}$	$E_{1}$	{P}
	$E_{2}$	{P,A}		$E_{2}$	{P,A}
	$E_{3}$	{A}		$E_{3}$	{P,A}
$C_{4}$	$E_{1}$	{A}	$C_{10}$	$E_{1}$	{P}
	$E_{2}$	{A}		$E_{2}$	{P}
	$E_{3}$	{P,A}		$E_{3}$	{P}
$C_{5}$	$E_{1}$	{G}	$C_{11}$	$E_{1}$	{P}
	$E_{2}$	{G}:{VG}=4:1		$E_{2}$	{P,A}
	$E_{3}$	{G,VG}		$E_{3}$	{P,A}
$C_{6}$	$E_{1}$	{P}	$C_{12}$	$E_{1}$	{P}
	$E_{2}$	{P}		$E_{2}$	{P,A}
	$E_{3}$	{VP,P}		$E_{3}$	{P,A}

Example 2

For WTG3, the evaluation of expert $E_{2}$ under factor $C_{5}$ is ${G} : {V G} = 4 : 1$ and needs to be quantified. According to Table 3, the quantified assessment standard is

{G} = [0.6001, 0.8000], [‒ 0.3999, ‒ 0.2000], 0.7000, ‒ 0.3000,

{V G} = [0.8001, 1.0000], [‒ 0.1999, ‒ 0.0001], 0.9000, ‒ 0.1000 .

After replacing probabilities with weights, the weight of ${G}$ is 0.8, and the weight of ${V G}$ is 0.2. Finally, the P-CBFWG operator is used to fuse the assessment standards ${G}$ and ${V G}$ to obtain a quantified result of ${G} : {V G} = 4 : 1$ . The calculation process is as follows:

\begin{array}{l} CBFN 2 = [0.600 1^{0.8} * 0.800 1^{0.2}, 0 . 8^{0.8} * 1^{0.2}], \\ [‒ (1 ‒ (1 ‒ 0.399 9^{0.8}) * (1 ‒ 0.199 9^{0.2})), ‒ (1 ‒ (1 ‒ 0 . 2^{0.8}) * (1 ‒ 0.000 1^{0.2}))], \\ 0 . 7^{0.8} * 0 . 9^{0.2}, ‒ (1 ‒ (1 ‒ 0 . 3^{0.8}) * (1 ‒ 0 . 1^{0.2})) \\ = [0.6356, 0.8365], [‒ 0.3644, ‒ 0.1635], 0.7361, ‒ 0.2639 . \end{array}

Step 2. The CBF decision matrices $D_{1}$ , $D_{2}$ , and $D_{3}$ are constructed corresponding to each expert. For example, the decision matrix of expert $E_{1}$ is shown in Table 13. Then, the standardized CBF decision matrices $D_{1}^{'}$ , $D_{2}^{'}$ , and $D_{3}^{'}$ are obtained with the formula (20).

Step 3. The matrices $D_{1}^{'}$ , $D_{2}^{'}$ , and $D_{3}^{'}$ are aggregated into the comprehensive CBF decision matrix $D$ with equal weight ( $\frac{1}{3}$ each) using the P-CBEWG operator to obtain the assessment of the WTGi by all experts, as shown in Table 14.

The ranking of alternatives using CBF-ELECTRE-I+

Step 1. Substitute the attribute weights and the data obtained from Table 14 into formula (21) to obtain the CBF weighted decision matrix as shown in Table 15.

Step 2. Determine the CBF bipolar concordance set $C_{α β}$ and the CBF bipolar discordance set $D_{α β}$ according to Section 3.3, as shown in Tables 16 and 17, respectively.

Step 3. The bipolar fuzzy concordance matrix $X$ and the bipolar fuzzy discordance matrix $Y$ are constructed as

X = [\begin{array}{l} ‒ & 0.7671 & 0.8094 & 0.2957 & 0.2876 & 0.5762 & 0.5762 & 0.2876 & 0.4177 \\ 0.7626 & ‒ & 0.8094 & 0.1600 & 0.1518 & 0.5762 & 0.4711 & 0.1825 & 0.2820 \\ 0.4863 & 0.3712 & ‒ & 0.1906 & 0.1906 & 0.3874 & 0.1825 & 0.1906 & 0.1825 \\ 0.8175 & 0.9107 & 0.8094 & ‒ & 0.6188 & 0.7964 & 0.6188 & 0.4177 & 0.8094 \\ 0.8482 & 0.8482 & 0.8094 & 0.5879 & ‒ & 0.7668 & 0.8094 & 0.3080 & 0.8091 \\ 0.4238 & 0.4238 & 0.8018 & 0.3812 & 0.3087 & ‒ & 0.3877 & 0.1906 & 0.5417 \\ 0.6848 & 0.6848 & 0.9375 & 0.4863 & 0.4863 & 0.6877 & ‒ & 0.2957 & 0.6930 \\ 0.8175 & 0.8175 & 0.8094 & 0.9693 & 0.8987 & 0.8718 & 0.8094 & ‒ & 0.8094 \\ 0.6130 & 0.7180 & 0.8175 & 0.3992 & 0.2841 & 0.5816 & 0.4121 & 0.1906 & ‒ \end{array}],

Y = [\begin{array}{l} ‒ & 0.1604 & 0.3929 & 1 & 1 & 0.6057 & 1 & 1 & 1 \\ 1 & ‒ & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & ‒ & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0.6329 & 0.9916 & ‒ & 1 & 0.9842 & 1 & 1 & 0.2669 \\ 0.6228 & 0.5459 & 0.9069 & 0.1575 & ‒ & 0.9002 & 1 & 1 & 0.1432 \\ 1 & 0.6438 & 0.5966 & 1 & 1 & ‒ & 1 & 1 & 1 \\ 0 & 0 & 0.1370 & 1 & 1 & 0.3899 & ‒ & 1 & 1 \\ 0.8199 & 0.9357 & 1 & 0.0711 & 1 & 1 & 1 & ‒ & 0.3824 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & ‒ \end{array}] .

Step 4. The corrected CBF discordance matrix $Y^{'}$ is reckoned as

Y^{'} = [\begin{array}{l} ‒ & 0.8396 & 0.6071 & 0 & 0 & 0.3943 & 0 & 0 & 0 \\ 0 & ‒ & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ‒ & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.3671 & 0.0084 & ‒ & 0 & 0.0158 & 0 & 0 & 0.7331 \\ 0.3772 & 0.4541 & 0.0931 & 0.8425 & ‒ & 0.0998 & 0 & 0 & 0.8568 \\ 0 & 0.3562 & 0.4034 & 0 & 0 & ‒ & 0 & 0 & 0 \\ 1 & 1 & 0.8630 & 0 & 0 & 0.6101 & ‒ & 0 & 0 \\ 0.1801 & 0.0643 & 0 & 0.9289 & 0 & 0 & 0 & ‒ & 0.6176 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ‒ \end{array}] .

Step 5. The aggregate dominance matrix $Z = z (α, β)$ is

Z = [\begin{array}{l} ‒ & 0.6440 & 0.4914 & 0 & 0 & 0.2272 & 0 & 0 & 0 \\ 0 & ‒ & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ‒ & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.3113 & 0.0068 & ‒ & 0 & 0.0125 & 0 & 0 & 0.5933 \\ 0.3199 & 0.3852 & 0.0754 & 0.4953 & ‒ & 0.0766 & 0 & 0 & 0.6932 \\ 0 & 0.1510 & 0.3234 & 0 & 0 & ‒ & 0 & 0 & 0 \\ 0.5797 & 0.6848 & 0.8091 & 0 & 0 & 0.4196 & ‒ & 0 & 0 \\ 0.1283 & 0.0545 & 0 & 0.8028 & 0 & 0 & 0 & ‒ & 0.4999 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ‒ \end{array}] .

Step 6. The significance of each alternative and the corresponding ranking order based on formula (18) are outlined in Table 18. Ranking the significance values $A S_{i}$ in ascending order identifies WTG8 as the optimal choice.

Comparative analysis

As shown in Figure 4, based on the evaluation result of the CBF-ELECTRE-I+ model in Section 4.3 and the rule that a smaller significance value indicates greater importance for the corresponding OWTs, WTG8 is the optimal choice. Meanwhile, the traditional CBF-ELECTRE-I model is applied to OWT selection as well and compared with the novel CBF-ELECTRE-I+ model. It is evident from Table 19 that the ranking results of the two models are quite similar, with WTG8 yielding the optimal result. To facilitate a better comparison between the two models, we calculated the additive inverse of the significance of the CBF-ELECTRE-I+ model and created a dual line chart comparing it with the traditional CBF-ELECTRE-I, as shown in Figure 5. The graph shows that CBF-ELECTE-I+ exhibits a larger difference between the highest and second-highest points in the final result compared to CBF-ELECTE-I. This distinction makes it easier to select the optimal solution, thereby achieving higher selection accuracy.

Figure 4.

Ranking result using CBF-ELECTRE-I+.

Table 13.

The CBF decision matrix by $E_{1}$ .

Table 14.

Assessment results of all experts on schemes.

	$C_{1}$	$C_{2}$	$C_{3}$
WTG1	[0.3560,0.5604], [‒0.6440,–0.4396],0.4588,‒0.5412	[0.6422,0.8432], [‒0.3578,–0.1568],0.7428,‒0.2572	[0.2805,0.4874], [‒0.7195,–0.5126],0.3848,‒0.6152
WTG2	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.3582,0.5624], [‒0.6418,–0.4376],0.4609,‒0.5391	[0.2805,0.4874], [‒0.7195,–0.5126],0.3848,‒0.6152
WTG3	[0.3560,0.5604], [‒0.6440,–0.4396],0.4588,‒0.5412	[0.3582,0.5624], [‒0.6418,–0.4376],0.4609,‒0.5391	[0.2805,0.4874], [‒0.7195,–0.5126],0.3848,‒0.6152
WTG4	[0.4409,0.6428], [‒0.5591,–0.3572],0.5420,‒0.4580	[0.6422,0.8432], [‒0.3578,–0.1568],0.7428,‒0.2572	[0.3551,0.5596], [‒0.6449,–0.4404],0.4579,‒0.5421
WTG5	[0.5839,0.7847], [‒0.4161,–0.2153],0.6843,‒0.3157	[0.6422,0.8432], [‒0.3578,–0.1568],0.7428,‒0.2572	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG6	[0.4409,0.6428], [‒0.5591,–0.3572],0.5420,‒0.4580	[0.3560,0.5604], [‒0.6440,–0.4396],0.4588,‒0.5412	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG7	[0.5839,0.7847], [‒0.4161,–0.2153],0.6843,‒0.3157	[0.6422,0.8432], [‒0.3578,–0.1568],0.7428,‒0.2572	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG8	[0.6911,0.8926], [‒0.3089,–0.1074],0.7919,‒0.2081	[0.6422,0.8432], [‒0.3578,–0.1568],0.7428,‒0.2572	[0.7247,0.9261], [‒0.2753,–0.0740],0.8254,‒0.1746
WTG9	[0.4409,0.6428], [‒0.5591,–0.3572],0.5420,‒0.4580	[0.4925,0.6953], [‒0.5075,–0.3047],0.5941,‒0.4059	[0.2812,0.4881], [‒0.7188,–0.5119],0.3855,‒0.6145
	$C_{4}$	$C_{5}$	$C_{6}$
WTG1	[0.3677,0.5696], [‒0.6323,–0.4304],0.4687,‒0.5313	[0.5625,0.7641], [‒0.4375,–0.2359],0.6634,‒0.3366	[0.2249,0.4283], [‒0.7751,–0.5717],0.3270,‒0.6730
WTG2	[0.3677,0.5696], [‒0.6323,–0.4304],0.4687,‒0.5313	[0.3582,0.5624], [‒0.6418,–0.4376],0.4609,‒0.5391	[0.2249,0.4283], [‒0.7751,–0.5717],0.3270,‒0.6730
WTG3	[0.4323,0.6350], [‒0.5677,–0.3650],0.5335,‒0.4665	[0.6405,0.8415], [‒0.3595,–0.1585],0.7411,‒0.2589	[0.0000,0.3581], [‒1.0000,–0.6419],0.2518,‒0.7482
WTG4	[0.2590,0.4627], [‒0.7410,–0.5373],0.3610,‒0.6390	[0.4268,0.6282], [‒0.5732,–0.3718],0.5276,‒0.4724	[0.4268,0.6282], [‒0.5732,–0.3718],0.5276,‒0.4724
WTG5	[0.2897,0.4954], [‒0.7103,–0.5046],0.3932,‒0.6068	[0.5625,0.7641], [‒0.4375,–0.2359],0.6634,‒0.3366	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG6	[0.2590,0.4627], [‒0.7410,–0.5373],0.3610,‒0.6390	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.0000,0.3581], [‒1.0000,–0.6419],0.2518,‒0.7482
WTG7	[0.3677,0.5696], [‒0.6323,–0.4304],0.4687,‒0.5313	[0.6405,0.8415], [‒0.3595,–0.1585],0.7411,‒0.2589	[0.2249,0.4283], [‒0.7751,–0.5717],0.3270,‒0.6730
WTG8	[0.2590,0.4627], [‒0.7410,–0.5373],0.3610,‒0.6390	[0.3582,0.5624], [‒0.6418,–0.4376],0.4609,‒0.5391	[0.5625,0.7641], [‒0.4375,–0.2359],0.6634,‒0.3366
WTG9	[0.2897,0.4954], [‒0.7103,–0.5046],0.3932,‒0.6068	[0.5625,0.7641], [‒0.4375,–0.2359],0.6634,‒0.3366	[0.4268,0.6282], [‒0.5732,–0.3718],0.5276,‒0.4724
	$C_{7}$	$C_{8}$	$C_{9}$
WTG1	[0.2143,0.2535], [‒0.8476,–0.6437],0.2550,‒0.7450	[0.2650,0.4690], [‒0.7350,–0.5310],0.3673,‒0.6327	[0.4553,0.6577], [‒0.5447,–0.3423],0.5566,‒0.4434
WTG2	[0.2143,0.2535], [‒0.8476,–0.6437],0.2550,‒0.7450	[0.2650,0.4690], [‒0.7350,–0.5310],0.3673,‒0.6327	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG3	[0.4000,0.5999], [‒0.6000,–0.4001],0.5000,‒0.5000	[0.4691,0.6744], [‒0.5309,–0.3256],0.5713,‒0.4287	[0.2511,0.4569], [‒0.7489,–0.5431],0.3547,‒0.6453
WTG4	[0.2143,0.2535], [‒0.8476,–0.6437],0.2550,‒0.7450	[0.3071,0.1056], [‒0.8944,‒0.6929],0.2063,‒0.7937	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG5	[0.3364,0.5393], [‒0.6636,–0.4607],0.4379,‒0.5621	[0.2678,0.1615], [‒0.8705,–0.6659],0.2325,‒0.7675	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG6	[0.4986,0.7028], [‒0.5014,–0.2972],0.6006,‒0.3994	[0.6745,0.9260], [‒0.3255,–0.0740],0.7808,‒0.2192	[0.2805,0.4874], [‒0.7195,–0.5126],0.3848,‒0.6152
WTG7	[0.6539,0.7102], [‒0.4119,–0.2105],0.6887,‒0.3113	[0.4691,0.6744], [‒0.5309,–0.3256],0.5713,‒0.4287	[0.3152,0.5218], [‒0.6848,–0.4782],0.4194,‒0.5806
WTG8	[0.2659,0.4699], [‒0.7341,–0.5301],0.3681,‒0.6319	[0.3352,0.3200], [‒0.7679,–0.5379],0.3511,‒0.6489	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG9	[0.4986,0.7028], [‒0.5014,–0.2972],0.6006,‒0.3994	[0.3677,0.5696], [‒0.6323–0.4304],0.4687,‒0.5313	[0.2812,0.4881], [‒0.7188,–0.5119],0.3855,‒0.6145
	$C_{10}$	$C_{11}$	$C_{12}$
WTG1	[0.3188,0.5253], [‒0.6812,–0.4747],0.4229,‒0.5771	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2511,0.4569], [‒0.7489,–0.5431],0.3547,‒0.6453
WTG2	[0.3188,0.5253], [‒0.6812,–0.4747],0.4229,‒0.5771	[0.4553,0.6577], [‒0.5447,–0.3423],0.5566,‒0.4434	[0.2511,0.4569], [‒0.7489,–0.5431],0.3547,‒0.6453
WTG3	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2511,0.4569], [‒0.7489,–0.5431],0.3547,‒0.6453	[0.2511,0.4569], [‒0.7489,–0.5431],0.3547,‒0.6453
WTG4	[0.6118,0.8121], [‒0.3882,–0.1879],0.7120,‒0.2880	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758
WTG5	[0.5625,0.7641], [‒0.4375,–0.2359],0.6634,‒0.3366	[0.4882,0.6910], [‒0.5118,–0.3090],0.5898,‒0.4102	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG6	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.3152,0.5218], [‒0.6848,–0.4782],0.4194,‒0.5806	[0.4570,0.6593], [‒0.5430,–0.3407],0.5583,‒0.4417
WTG7	[0.2249,0.4283], [‒0.7751,–0.5717],0.3270,‒0.6730	[0.2805,0.4874], [‒0.7195,–0.5126],0.3848,‒0.6152	[0.3551,0.5596], [‒0.6449,–0.4404],0.4579,‒0.5421
WTG8	[0.6405,0.8415], [‒0.3595,–0.1585],0.7411,‒0.2589	[0.5227,0.7253], [‒0.4773,–0.2747],0.6242,‒0.3758	[0.5246,0.7272], [‒0.4754,–0.2728],0.6261,‒0.3739
WTG9	[0.4290,0.6304], [‒0.5710,–0.3696],0.5298,‒0.4702	[0.2812,0.4881], [‒0.7188,–0.5119],0.3855,‒0.6145	[0.3551,0.5596], [‒0.6449,–0.4404],0.4579,‒0.5421

Table 15.

The weighted CBF decision matrix.

	$C_{1}$	$C_{2}$	$C_{3}$
WTG1	[0.0410,0.0645], [‒0.0741,–0.0506],0.0528,‒0.0623	[0.0675,0.0886], [‒0.0376,–0.0165],0.0780,‒0.0270	[0.0212,0.0368], [‒0.0543,–0.0387],0.0290,‒0.0464
WTG2	[0.0460,0.0691], [‒0.0690,–0.0460],0.0575,‒0.0575	[0.0376,0.0591], [‒0.0674,–0.0460],0.0484,‒0.0566	[0.0212,0.0368], [‒0.0543,–0.0387],0.0290,‒0.0464
WTG3	[0.0410,0.0645], [‒0.0741,–0.0506],0.0528,‒0.0623	[0.0376,0.0591], [‒0.0674,–0.0460],0.0484,‒0.0566	[0.0212,0.0368], [‒0.0543,–0.0387],0.0290,‒0.0464
WTG4	[0.0507,0.0740], [‒0.0643,–0.0411],0.0624,‒0.0527	[0.0675,0.0886], [‒0.0376,–0.0165],0.0780,‒0.0270	[0.0268,0.0422], [‒0.0487,–0.0332],0.0346,‒0.0409
WTG5	[0.0672,0.0903], [‒0.0479,–0.0248],0.0788,‒0.0363	[0.0675,0.0886], [‒0.0376,–0.0165],0.0780,‒0.0270	[0.0395,0.0547], [‒0.0360,–0.0207],0.0471,‒0.0284
WTG6	[0.0507,0.0740], [‒0.0643,–0.0411],0.0624,‒0.0527	[0.0374,0.0589], [‒0.0677,–0.0462],0.0482,‒0.0569	[0.0395,0.0547], [‒0.0360,–0.0207],0.0471,‒0.0284
WTG7	[0.0672,0.0903], [‒0.0479,–0.0248],0.0788,‒0.0363	[0.0675,0.0886], [‒0.0376,–0.0165],0.0780,‒0.0270	[0.0395,0.0547], [‒0.0360,–0.0207],0.0471,‒0.0284
WTG8	[0.0795,0.1027], [‒0.0355,–0.0124],0.0911,‒0.0239	[0.0675,0.0886], [‒0.0376,–0.0165],0.0780,‒0.0270	[0.0547,0.0699], [‒0.0208,–0.0056],0.0623,‒0.0132
WTG9	[0.0507,0.0740], [‒0.0643,–0.0411],0.0624,‒0.0527	[0.0517,0.0731], [‒0.0533,–0.0320],0.0624,‒0.0426	[0.0212,0.0368], [‒0.0543,–0.0386],0.0291,‒0.0464
	$C_{4}$	$C_{5}$	$C_{6}$
WTG1	[0.0230,0.0356], [‒0.0395,–0.0269],0.0293,‒0.0332	[0.0173,0.0234], [‒0.0134,–0.0072],0.0203,‒0.0103	[0.0210,0.0400], [‒0.0724,–0.0534],0.0306,‒0.0629
WTG2	[0.0230,0.0356], [‒0.0395,–0.0269],0.0293,‒0.0332	[0.0110,0.0172], [‒0.0197,–0.0134],0.0141,‒0.0165	[0.0210,0.0400], [‒0.0724,–0.0534],0.0306,‒0.0629
WTG3	[0.0270,0.0397], [‒0.0355,–0.0228],0.0333,‒0.0291	[0.0196,0.0258], [‒0.0110,–0.0049],0.0227,‒0.0079	[0.0000,0.0335], [‒0.0934,–0.0600],0.0235,‒0.0699
WTG4	[0.0162,0.0289], [‒0.0463,–0.0336],0.0226,‒0.0399	[0.0131,0.0193], [‒0.0176,–0.0114],0.0162,‒0.0145	[0.0399,0.0587], [‒0.0536,–0.0347],0.0493,‒0.0441
WTG5	[0.0181,0.0310], [‒0.0444,–0.0315],0.0246,‒0.0379	[0.0173,0.0234], [‒0.0134,–0.0072],0.0203,‒0.0103	[0.0374,0.0561], [‒0.0561,–0.0374],0.0467,‒0.0467
WTG6	[0.0162,0.0289], [‒0.0463,–0.0336],0.0226,‒0.0399	[0.0184,0.0245], [‒0.0123,–0.0061],0.0215,‒0.0092	[0.0000,0.0335], [‒0.0934,–0.0600],0.0235,‒0.0699
WTG7	[0.0230,0.0356], [‒0.0395,–0.0269],0.0293,‒0.0332	[0.0196,0.0258], [‒0.0110,–0.0049],0.0227,‒0.0079	[0.0210,0.0400], [‒0.0724,–0.0534],0.0306,‒0.0629
WTG8	[0.0162,0.0289], [‒0.0463,–0.0336],0.0226,‒0.0399	[0.0110,0.0172], [‒0.0197,–0.0134],0.0141,‒0.0165	[0.0526,0.0714], [‒0.0409,–0.0220],0.0620,‒0.0315
WTG9	[0.0181,0.0310], [‒0.0444,–0.0315],0.0246,‒0.0379	[0.0173,0.0234], [‒0.0134,–0.0072],0.0203,‒0.0103	[0.0399,0.0587], [‒0.0536,–0.0347],0.0493,‒0.0441
	$C_{7}$	$C_{8}$	$C_{9}$
WTG1	[0.0017,0.0021], [‒0.0069,–0.0053],0.0021,‒0.0061	[0.0237,0.0419], [‒0.0657,–0.0474],0.0328,‒0.0565	[0.0463,0.0668], [‒0.0554,–0.0348],0.0566,‒0.0451
WTG2	[0.0017,0.0021], [‒0.0069,–0.0053],0.0021,‒0.0061	[0.0237,0.0419], [‒0.0657,–0.0474],0.0328,‒0.0565	[0.0407,0.0610], [‒0.0610,–0.0407],0.0508,‒0.0508
WTG3	[0.0033,0.0049], [‒0.0049,–0.0033],0.0041,‒0.0041	[0.0419,0.0602], [‒0.0474,–0.0291],0.0510,‒0.0383	[0.0255,0.0464], [‒0.0761,–0.0552],0.0361,‒0.0656
WTG4	[0.0017,0.0021], [‒0.0069,–0.0053],0.0021,‒0.0061	[0.0274,0.0094], [‒0.0799,–0.0619],0.0184,‒0.0709	[0.0531,0.0737], [‒0.0485,–0.0279],0.0634,‒0.0382
WTG5	[0.0027,0.0044], [‒0.0054,–0.0038],0.0036,‒0.0046	[0.0239,0.0144], [‒0.0778,–0.0595],0.0208,‒0.0686	[0.0531,0.0737], [‒0.0485,–0.0279],0.0634,‒0.0382
WTG6	[0.0041,0.0057], [‒0.0041,–0.0024],0.0049,‒0.0033	[0.0603,0.0827], [‒0.0291,–0.0066],0.0697,‒0.0196	[0.0285,0.0495], [‒0.0731,–0.0521],0.0391,‒0.0625
WTG7	[0.0053,0.0058], [‒0.0034,–0.0017],0.0056,‒0.0025	[0.0419,0.0602], [‒0.0474,–0.0291],0.0510,‒0.0383	[0.0320,0.0530], [‒0.0696,–0.0486],0.0426,‒0.0590
WTG8	[0.0022,0.0038], [‒0.0060,–0.0043],0.0030,‒0.0052	[0.0299,0.0286], [‒0.0686,–0.0481],0.0314,‒0.0580	[0.0531,0.0737], [‒0.0485,–0.0279],0.0634,‒0.0382
WTG9	[0.0041,0.0057], [‒0.0041,–0.0024],0.0049,‒0.0033	[0.0328,0.0509], [‒0.0565,–0.0384],0.0419,‒0.0475	[0.0286,0.0496], [‒0.0731,–0.0520],0.0392,‒0.0625
	$C_{10}$	$C_{11}$	$C_{12}$
WTG1	[0.0305,0.0503], [‒0.0652,–0.0454],0.0405,‒0.0552	[0.0472,0.0707], [‒0.0707,–0.0471],0.0589,‒0.0589	[0.0264,0.0480], [‒0.0787,–0.0571],0.0373,‒0.0678
WTG2	[0.0305,0.0503], [‒0.0652,–0.0454],0.0405,‒0.0552	[0.0537,0.0775], [‒0.0642,–0.0403],0.0656,‒0.0523	[0.0264,0.0480], [‒0.0787,–0.0571],0.0373,‒0.0678
WTG3	[0.0192,0.0383], [‒0.0766,–0.0574],0.0287,‒0.0670	[0.0296,0.0538], [‒0.0883,–0.0640],0.0418,‒0.0760	[0.0264,0.0480], [‒0.0787,–0.0571],0.0373,‒0.0678
WTG4	[0.0586,0.0777], [‒0.0372,–0.0180],0.0681,‒0.0276	[0.0616,0.0855], [‒0.0562,–0.0324],0.0736,‒0.0443	[0.0549,0.0762], [‒0.0501,–0.0289],0.0656,‒0.0395
WTG5	[0.0538,0.0731], [‒0.0419,–0.0226],0.0635,‒0.0322	[0.0575,0.0814], [‒0.0603,–0.0364],0.0695,‒0.0483	[0.0420,0.0630], [‒0.0630,–0.0420],0.0525,‒0.0525
WTG6	[0.0192,0.0383], [‒0.0766,–0.0574],0.0287,‒0.0670	[0.0371,0.0615], [‒0.0807,–0.0564],0.0494,‒0.0684	[0.0480,0.0693], [‒0.0571,–0.0358],0.0587,‒0.0464
WTG7	[0.0215,0.0410], [‒0.0742,–0.0547],0.0313,‒0.0644	[0.0331,0.0574], [‒0.0848,–0.0604],0.0454,‒0.0725	[0.0373,0.0588], [‒0.0678,–0.0463],0.0481,‒0.0570
WTG8	[0.0613,0.0805], [‒0.0344,–0.0152],0.0709,‒0.0248	[0.0616,0.0855], [‒0.0562,–0.0324],0.0736,‒0.0443	[0.0551,0.0764], [‒0.0499,–0.0287],0.0658,‒0.0393
WTG9	[0.0411,0.0603], [‒0.0546,–0.0354],0.0507,‒0.0450	[0.0331,0.0575], [‒0.0847,–0.0603],0.0454,‒0.0724	[0.0373,0.0588], [‒0.0678,–0.0463],0.0481,‒0.0570

Table 16.

The CBF bipolar concordance set $C_{α β}$ .

$β$	1	2	3	4	5	6	7	8	9
$C_{1 β}$	—	{2,3,4,5,6,7,8,9,10,12}	{1,2,3,6,9,10,11,12}	{2,4,5,7,8}	{2,4,5,8}	{2,4,6,9,10,11}	{2,4,6,9,10,11}	{2,4,5,8}	{2,4,5,9,11}
$C_{2 β}$	{1,3,4,6,7,8,10,11,12}	—	{1,2,3,6,9,10,11,12}	{4,7,8}	{4,8}	{2,4,6,9,10,11}	{4,6,9,10,11}	{4,5,8}	{4,9,11}
$C_{3 β}$	{1,3,4,5,7,8,12}	{3,4,5,7,8,12}	—	{4,5,7,8}	{4,5,7,8}	{2,4,5,6,10}	{4,5,8}	{4,5,7,8}	{4,5,8}
$C_{4 β}$	{1,2,3,6,7,9,10,11,12}	{1,2,3,4,5,6,7,9,10,11,12}	{1,2,3,6,9,10,11,12}	—	{2,6,9,10,11,12}	{1,2,4,6,9,10,11,12}	{2,6,9,10,11,12}	{2,4,5,9,11}	{1,2,3,6,9,10,11,12}
$C_{5 β}$	{1,2,3,5,6,7,9,10,11,12}	{1,2,3,5,6,7,9,10,11,12}	{1,2,3,6,9,10,11,12}	{1,2,3,4,5,7,8,9}	—	{1,2,3,4,6,9,10,11}	{1,2,3,6,9,10,11,12}	{2,4,5,7,9}	{1,2,3,4,5,9,10,11,12}
$C_{6 β}$	{1,3,5,7,8,12}	{1,3,5,7,8,12}	{1,3,6,7,8,9,10,11,12}	{1,3,4,5,7,8}	{3,5,7,8,12}	—	{3,8,11,12}	{4,5,7,8}	{1,3,5,7,8,11,12}
$C_{7 β}$	{1,2,3,4,5,6,7,8,12}	{1,2,3,4,5,6,7,8,12}	{1,2,3,5,6,7,8,9,10,11,12}	{1,2,3,4,5,7,8}	{1,2,3,4,5,7,8}	{1,2,3,4,5,6,7,9,10}	—	{2,4,5,7,8}	{1,2,3,4,5,7,8,9,12}
$C_{8 β}$	{1,2,3,6,7,9,10,11,12}	{1,2,3,5,6,7,9,10,11,12}	{1,2,3,6,9,10,11,12}	{1,2,3,4,6,7,8,9,10,11,12}	{1,2,3,6,8,9,10,11,12}	{1,2,3,4,6,9,10,11,12}	{1,2,3,6,9,10,11,12}	—	{1,2,3,6,9,10,11,12}
$C_{9 β}$	{1,3,5,6,7,8,10,12}	{1,2,3,5,6,7,8,10,12}	{1,2,3,6,7,9,10,11,12}	{1,4,5,6,7,8}	{4,5,6,7,8}	{1,2,4,6,7,9,10}	{6,10,11,12}	{4,5,7,8}	—

Table 17.

The CBF bipolar discordance set $D_{α β}$ .

$β$	1	2	3	4	5	6	7	8	9
$D_{1 β}$	—	{1,11}	{4,5,7,8}	{1,3,6,9,10,11,12}	{1,3,6,7,9,10,11,12}	{1,3,5,7,8,12}	{1,3,5,6,7,8,12}	{1,3,6,7,9,10,11,12}	{1,6,7,8,10,12}
$D_{2 β}$	{2,5,9}	—	{4,5,7,8}	{1,2,3,5,6,9,10,11,12}	{1,2,3,5,6,7,9,10,11,12}	{1,3,5,7,8,12}	{1,2,3,5,6,7,8,12}	{1,2,3,6,7,9,10,11,12}	{1,2,5,6,7,8,10,12}
$D_{3 β}$	{2,6,9,10,11}	{1,6,9,10,11}	—	{1,2,3,6,9,10,11,12}	{1,2,3,6,9,10,11,12}	{1,3,7,8,9,11,12}	{1,2,3,5,6,7,9,10,11,12}	{1,2,3,6,9,10,11,12}	{1,2,6,7,9,10,11,12}
$D_{4 β}$	{4,5,8}	{4,8}	{4,5,7,8}	—	{1,3,4,5,7,8}	{3,5,7,8}	{1,3,4,5,7,8}	{1,3,6,7,8,10,12}	{4,5,7,8}
$D_{5 β}$	{4,8}	{4,8}	{4,5,7,8}	{6,10,11,12}	—	{5,7,8,12}	{4,5,7,8}	{1,3,6,8,10,11,12}	{6,7,8}
$D_{6 β}$	{2,4,6,9,10,11}	{2,4,6,9,10,11}	{2,4,5}	{2,6,9,10,11,12}	{1,2,4,6,9,10,11}	—	{1,2,4,5,6,7,9,10}	{1,2,3,6,9,10,11,12}	{2,4,6,9,10}
$D_{7 β}$	{9,10,11}	{9,10,11}	{4}	{6,9,10,11,12}	{6,9,10,11,12}	{8,11,12}	—	{1,3,6,9,10,11,12}	{6,10,11}
$D_{8 β}$	{4,5,8}	{4,8}	{4,5,7,8}	{5}	{4,5,7}	{5,7,8}	{4,5,7,8}	—	{4,5,7,8}
$D_{9 β}$	{2,4,9,11}	{4,9,11}	{4,5,8}	{2,3,9,10,11,12}	{1,2,3,9,10,11,12}	{3,5,8,11,12}	{1,2,3,4,5,7,8,9}	{1,2,3,6,9,10,11,12}	—

Table 18.

Significance of alternative OWTs and ranking order.

Significance	Ranking order
8.3096	WTG3
7.8453	WTG2
6.3311	WTG9
6.1564	WTG1
5.8649	WTG6
4.3280	WTG4
4.2671	WTG7
2.9013	WTG5
2.0634	WTG8

Table 19.

Ranking results of CBF-ELECTRE-I+ and CBF-ELECTRE-I.

Methods	Ranking results	Best decision
CBF-ELECTRE-I	8,5,7,4,1,6,9,2,3	WTG8
CBF-ELECTRE-I+	8,5,7,4,6,1,9,2,3	WTG8

Figure 5.

Comparison between CBF-ELECTRE-I and CBF-ELECTRE-I+ in the offshore wind turbine selection.

Besides, to evaluate the efficiency of the CBF-ELECTRE-I+ model, we compared its time complexity with that of the CBF-ELECTRE-I. Although both models have an overall time complexity of $O (m^{2})$ , the traditional algorithm requires five calculations, each with a time complexity of $O (m^{2})$ , to determine the threshold and construct the concordance and discordance dominance matrices. In contrast, the improved model only requires a single calculation with a time complexity of $O (m^{2})$ to determine the modified discordance matrix, making it more efficient in OWT selection.

In practice, the first bidding of the project saw the selection of seventeen WTG8 units, each with a hub height of 110 m and a total installed capacity of 102 MW. This outcome confirms the effectiveness of our proposed selection method.

Conclusion

The efficiency of offshore wind development and the long-term safety and financial returns of offshore wind farms are closely related to OWTs. In practice, the capital expenditures of OWTs constitute the largest proportion of the entire offshore wind development cycle. Therefore, OWT selection is crucial in the early stages of wind farm development. However, in the decision-making process of OWT selection, insufficient consideration of uncertain environments, expert linguistic ambiguity, and lack of more nuanced and intuitive ranking can impact the accuracy of selecting OWTs. To address these issues, this study proposes a novel decision-making system based on CBFS and the improved ELECTRE-I.

The decision-making system in this study consists of four main stages: determining the indicator system for the decision-making model, obtaining the comprehensive CBF decision matrix by quantifying expert assessments with the CBFS scheme while integrating expert opinions, and ranking the alternatives using CBF-ELECTRE-I+. In the first stage, the evaluation criteria system is constructed through literature research and expert advice, containing five main criteria and twelve secondary indicators. Besides, factors such as rated power and safety level are mainly considered as the screening conditions, and nine alternatives are identified for further selection. Then the corresponding weight of each indicator is determined through PCA. In the second stage, owing to the fact that most experts tend to express their opinions in a qualitative manner, we introduce CBFS to construct the CBF decision matrices, which can represent the opinion of each expert and effectively address the ambiguity of expert scoring. Specifically, all entries in the CBF decision matrices are CBFNs determined by the viewpoints of experts. The P-CBFWG operator is employed to aggregate the CBF decision matrices into a single comprehensive CBF decision matrix that contains the evaluations of all experts. In the sorting process, the optimized ELECTRE-I method is combined with CBFS to form a novel CBF-ELECTRE-I+ model. This model calculates the importance of each alternative OWT, resulting in a nuanced ranking process and accurate ranking outcomes.

Finally, the proposed method is applied to actual bidding projects for selecting OWTs and compared with the traditional CBF-ELECTRE-I model. Both CBF-ELECTRE-I and CBF-ELECTRE-I+ have selected the optimal OWT from nine alternatives, which is consistent with the actual situation. However, CBF-ELECTRE-I+ exhibits higher selection accuracy and computational efficiency. Therefore, the decision-making system presented in this study can effectively assist decision-makers in selecting the optimal OWT and also extends the application scenarios of CBFS in the field of management decision-making.

Footnotes

ORCID iD

Li Xu

Credit statement

Fei Chen: Data curation, Methodology, Software, Writing – original draft; Li Xu: Methodology, Supervision, Writing – review & editing; Yanqi Wu: Validation, Formal analysis; Kuisong Jiang: Investigation, Visualization.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data will be made available upon reasonable request.*

Appendix

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$D_{1}$	$C_{1}$	$C_{2}$	$C_{3}$
WTG1	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG2	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG3	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG4	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127
WTG5	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG6	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.4001,0.6000], [‒0.5999,‒0.4000],0.5000,‒0.5000	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG7	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG8	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000
WTG9	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127
	$C_{4}$	$C_{5}$	$C_{6}$
WTG1	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG2	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG3	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG4	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG5	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG6	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG7	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG8	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000
WTG9	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
	$C_{7}$	$C_{8}$	$C_{9}$
WTG1	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG2	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG3	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG4	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG5	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG6	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG7	[0.2298,0.4338], [‒0.7702,–0.5662],0.3323,‒0.6677	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG8	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.6929,0.8944], [‒0.3071,–0.1056],0.7937,‒0.2063	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG9	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127
	$C_{10}$	$C_{11}$	$C_{12}$
WTG1	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG2	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG3	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000
WTG4	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG5	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG6	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.4001,0.6000], [‒0.5999,–0.4000],0.5000,‒0.5000
WTG7	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2001,0.4000], [‒0.7999,–0.6000],0.3000,‒0.7000	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127
WTG8	[0.6001,0.8000], [‒0.3999,–0.2000],0.7000,‒0.3000	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084
WTG9	[0.4900,0.6928], [‒0.5100,–0.3072],0.5916,‒0.4084	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127	[0.2829,0.4899], [‒0.7171,–0.5101],0.3873,‒0.6127