Selection of Wind Turbine Blade Material Using Intuitionistic Fuzzy TOPSIS Method ^†

Abstract

Utilizing turbines for wind energy is crucial for sustainable power. The effectiveness of turbines heavily relies on the blades, making material selection vital for their design. This study employs multi-criteria decision making (MCDM), particularly intuitionistic fuzzy (IF) in the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, to facilitate material selection. The Intuitionistic Fuzzy Weighted Averaging (IFWA) operation aggregates the ideas of the decision-makers (D-Ms), prioritizing criteria and alternatives. The combined method, blending the Intuitionistic Fuzzy Set (IFS) and TOPSIS, proves highly efficient in determining the most suitable material among alternatives based on established criteria. The findings highlight carbon fiber as the premier choice, suggesting its adoption for enhanced wind turbine blades.

1. Introduction

Wind energy is becoming a favored renewable source due to its unlimited potential for providing clean energy. This energy is captured by wind turbines located in coastal areas or regions with a constant wind flow exceeding approximately 6 m/s [1]. In wind power generators, a converter harnesses the wind’s kinetic energy, relying on a rotor driven by turbine blades. Therefore, the optimal efficiency of wind turbines hinges on meticulous design, optimization, and the choice of suitable materials for the blades. The structural health of turbine blades is critical to performance, necessitating attention to design aspects that ensure the blade’s endurance, shape, and geometrical dimensions throughout its lifespan [2,3].

The selection of blade material poses a significant challenge for the manufacturing industry, given that wind turbine blades play a crucial role in the turbine’s functionality. Wind blades are vital components, not merely because they convert the wind’s kinetic energy but also because a single blade failure can lead to turbine malfunction and a complete operational halt, incurring substantial costs for the industry [4]. Material choice is pivotal, as it directly impacts component cost and durability, with design engineers considering various criteria. Opting for low-density materials is preferable, as this ensures good strength against multiple types of loadings, including impact, compressive loads, tensile loads, and fatigue. Wind turbine blades must withstand diverse environmental conditions, including drizzle, moisture, wetness, dirt, bugs and flies, and contaminants [5,6]. The chosen material should resist these elements without succumbing to corrosion or wear and tear while also being easily accessible with minimal transportation costs. The material should possess good stiffness and manufacturability. Particularly in developing countries, the emphasis is on low-cost materials and manufacturing processes. Researchers are actively exploring material options for wind turbine blades using various methods [3]. However, current research concentrates on MCDM.

To tackle the complexities inherent in material selection, considerable attention has been directed toward employing MCDM techniques [2,4,6]. The TOPSIS method, which takes into account both the positive ideal solution (PIS) and the negative ideal solution (NIS), operates on the basis that the selected option should be closer to the PIS and farther from the NIS. While TOPSIS excels in ranking solutions, it has limitations, as D-Ms must subjectively judge the weighting of various criteria. Nevertheless, when integrated with other methodologies, TOPSIS delivers a more logical and clear solution. This study suggests an innovative approach to material selection challenges through the IF MCDM approach with the TOPSIS method. The IFS, suggested by Atanassov [7], offers an appropriate framework for handling uncertainty, applied extensively in decision-making problems. In batch decision-making scenarios, aggregating proficient opinions is crucial for effective evaluation. The IFWA operation is employed to consolidate the opinions of individual D-Ms and assess the significance of criteria and alternatives. The integration of TOPSIS with IFS holds significant promise for success in the material selection procedure. This research seeks to address the following inquiries:

Q1. What criteria are typically considered in the choice of the material for wind turbine blades?

Q2. Could the application of the IFS TOPSIS approach impact the process of selecting materials for wind turbine blades?

The organization of the current research is outlined as follows. Section 2 offers a summary of the pertinent literature. The research methodology is detailed in Section 3 and Section 4. Section 5 outlines the framework development for the study. The discussion and findings are presented in Section 6. Conclusions, implications, and prospects for forthcoming research are given in Section 7.

2. Review of Existing Literature

This section provides a concise overview of the materials available for wind turbine blades, the various criteria influencing the selection of the right material, and the impact of MCDM methodologies on the process of choosing suitable materials.

2.1. Material Used for Wind Turbine Blade

In the 19th century, wind turbine blades often used wood and canvas, offering good fatigue traits and wide availability [5,6]. Yet, due to wood’s low stiffness and strength limitations, it was eventually replaced by galvanized steel. Steel, composed of carbon and iron, emerged as a favorable choice owing to its resilience against strain, compression, and fatigue. Nonetheless, its high density became a significant drawback. In response to the weight-related challenges of steel, aluminum emerged as a lighter alternative. Despite its lower density, aluminum proved weaker than steel and incurred higher costs, coupled with low fatigue resistance [6,8]. Consequently, aluminum was eventually replaced by composite materials, which are currently broadly utilized for wind turbine blades. This composite material involves the reinforcement of fibers (glass fiber (E-glass), carbon fiber, and so on) into a matrix comprising polyester vinyl resin, resin, or thermoplastic resin. Carbon fibers, constituted of genuine carbon forming a mesh called graphite, provide an exceptional mixture of great strength, low elasticity, low density, and low weight. Carbon fibers exhibit high tensile and compressive strength and high resistance to impact, corrosion, and fatigue.

2.2. Studies Related to MCDM Methodologies for the Material Selection of Wind Turbine Blades

Numerous methodologies have been employed in previous studies to discern the most suitable material for wind turbine blades, with a focus on MCDM techniques, as summarized in

Table 1. Applied methodologies for wind turbine blade material selection.

No.	Methodology	Reference
1	MCDM, TOPSIS, and Fuzzy Linguistic variables	[5]
2	FANP method	[9]
3	AHP method	[6]
4	PROMETHEE-GAIA method	[2]
5	AHP and TOPSIS techniques	[4]
6	MARCOS method and BWM method	[1]
7	AHP, Simple Additive Weighting (SAW) method, Weighted Product Method (WPM), TOPSIS method, R-method	[10]

.

Scholars have consistently turned to MCDM as an effective means of material selection for the blades of wind turbines. Babu et al. [5] devised the MCDM framework with fuzzy linguistic variables to identify the optimal material for turbine blades. Maity and Chakraborty [9] implemented a fuzzy analytic network process (FANP)-based approach for the selection of materials suitable for impulse turbine blades in wind and wave energy extraction. Maskepatil, Gandigude, and Kale [6] suggested an analytic hierarchy process (AHP) for the selection of material in small wind turbine blades. Zindani, Maity, and Bhowmik [2] deliberated on the choice of suitable materials for the blades of wind turbines using the PROMETHEE-GAIA approach, a widely employed decision-making aid. Okokpujie et al. [4] evaluated four alternatives—stainless steel, aluminum alloy, mild steel, and glass fiber—to determine the optimal material for wind turbine blades, utilizing AHP and TOPSIS techniques for sustainable energy generation. Mondal et al. [1] applied the measurement of alternatives and ranking according to the compromise solution (MARCOS) method to select the optimal composite for blade manufacturing, concurrently incorporating the Best Worst Method (BWM) for criterion weights. Garmode et al. [10] employed four suitable mathematical techniques for material selection in small wind turbine blades, utilizing AHP, the entropy weight method, and their average, in addition to TOPSIS and the R-method for ranking materials.

The IFS, as suggested by Atanassov [7] is designed to effectively handle situations involving vagueness. The application of IFSs has extended to various domains, including logic programming [11], medical diagnosis [12], supplier selection [13], decision-making [14], and evaluation function [15]. This study incorporated the methodology of IFS, a valuable mathematical tool widely used by researchers and academics across various disciplines.

3. Construction of the IFS

To formulate the definition of an IFS, let us consider that A belongs to the IFS within a finite set Z. The definition of IFS A, where ${\mu }_A\left(z\right):Z\to \left[0,1\right]$signifies the membership function and ${\nu }_A\left(z\right):Z\to \left[0,1\right]$ denotes the non-membership function, is as follows:

$$A=\left\{〈z,{\mu }_A\left(z\right),{\nu }_A\left(z\right)〉|z\in Z\right\},\mathrm{and}0\le {\mu }_A\left(z\right)+{\nu }_A\left(z\right)\le 1,$$

(1)

IFS introduces a third piece of data, known as the hesitation degree or IF index. Let ${\pi }_A\left(z\right)$ represent the degree of hesitation regarding whether z belongs to set A or not. This hesitation degree, ${\pi }_A\left(z\right)$, can be expressed for every $z\in Z$ as follows:

$${\pi }_A\left(z\right)=1-{\mu }_A\left(z\right)-{\nu }_A\left(z\right),\mathrm{and}0\le {\pi }_A\left(z\right)\le 1,$$

(2)

where when ${\pi }_A\left(z\right)$ is small, the certainty of knowledge about z is higher, and when ${\pi }_A\left(z\right)$ is large, knowledge about z becomes more doubtful. When ${\mu }_A\left(z\right)=1-{\nu }_A\left(z\right)$ holds for all elements of the universe, the conventional concept of the fuzzy set is reinstated.

If A and B represent the IFSs of set Z, the multiplication operator is specified in the following manner [7]:

$$A⨂B=\left\{{\mu }_A\left(z\right).{\mu }_B\left(z\right),{\nu }_A\left(z\right)+{\nu }_B\left(z\right)-{\nu }_A\left(z\right).{\nu }_B\left(z\right)|z\in Z\right\},$$

(3)

4. Intuitionistic Fuzzy TOPSIS

To initiate the IF TOPSIS process, consider Z as a set of criteria $Z=\left\{Z_1,Z_2,\dots ,Z_n\right\}$ and A as a set of alternatives $A=\left\{A_1,A_2,\dots ,A_m\right\}$. The application of the IF TOPSIS method involves the following steps:

Step 1: Compute the weights assigned by D-Ms: Suppose the decision group comprises l D-M, with their significance expressed as linguistic expressions in intuitionistic fuzzy numbers (IFNs). Let $D_k=\left[{{\mu }_k{,\nu }_k,\pi }_k\right]$ represent an IFN for the evaluation provided by the k-th D-M. The weight of the k-th D-M can then be determined as follows:

$${\lambda }_k={\displaystyle \frac{\left({\mu }_k+{\pi }_k\left(\frac{{\mu }_k}{{\mu }_k+{\nu }_k}\right)\right)}{{\sum }_{k=1}^l\left({\mu }_k+{\pi }_k\left(\frac{{\mu }_k}{{\mu }_k+{\nu }_k}\right)\right)}},\mathrm{where}{\sum }_{k=1}^l{\lambda }_k=1$$

(4)

Step 2: Build the aggregated intuitionistic fuzzy decision matrix (IFDM) according to the judgments of the D-Ms: Let $R^{\left(k\right)}={\left(r_{ij}^{\left(k\right)}\right)}_{m\times n}$ represent an IFDM for each D-M. $\lambda =\{{\lambda }_1,{\lambda }_2,{\lambda }_3,\dots ,{\lambda }_l\}$ denotes the weight of each D-M, and ${\sum }_{k=1}^l{\lambda }_k=1,{\lambda }_k\in [0,1]$. In the context of group decision-making, merging individual decision opinions into a collective opinion requires the construction of an aggregated IFDM. The IFWA operator, as suggested by Xu [16], is employed for this purpose. Let $R={\left(r_{ij}\right)}_{m\times n}$, where

$$\begin{gathered} r_{ij}={IFWA}_{\lambda }\left(r_{ij}^{\left(1\right)},r_{ij}^{\left(2\right)},\dots ,r_{ij}^{\left(l\right)}\right)={\lambda }_1{r}_{ij}^{\left(1\right)}⨁{\lambda }_2{r}_{ij}^{\left(2\right)}⨁{\lambda }_{31}{r}_{ij}^{\left(3\right)}⨁\dots ⨁{\lambda }_l{r}_{ij}^{\left(l\right)} \\ =\left[1-{\prod }_{k=1}^l{\left(1-{\mu }_{ij}^{\left(k\right)}\right)}^{{\lambda }_k},{\prod }_{k=1}^l{\left({\nu }_{ij}^{\left(k\right)}\right)}^{{\lambda }_k},{\prod }_{k=1}^l{\left(1-{\mu }_{ij}^{\left(k\right)}\right)}^{{\lambda }_k}-{\prod }_{k=1}^l{\left({\nu }_{ij}^{\left(k\right)}\right)}^{{\lambda }_k}\right] \end{gathered}$$

(5)

where $r_{ij}=\left({\mu }_{Ai}\left(z_j\right),{\nu }_{Ai}\left(z_j\right),{\pi }_{Ai}\left(z_j\right)\right),\left(i=1,2,\dots ,m;j=1,2,\dots ,n\right)$. In accordance with Equation (5), the aggregated IFDM can be described as follows:

$$R=\left[\begin{array}{ccc}{r}_{11}& \cdots & r_{1n}\\ ⋮& \ddots & ⋮\\ r_{m1}& \cdots & r_{mn}\end{array}\right],$$

(6)

Step 3: Calculate the weights assigned to the criteria: Assigning weights to the criteria entails acknowledging that not all criteria hold equal significance. Let W denote a set of grades representing the significance of each criterion. The process of obtaining W entails consolidating the opinions of individual D-Ms regarding the significance of each criterion.

The weight assigned to the criterion $Z_j$ by the k-th D-M, denoted as $w_j^{\left(k\right)}=\left[{\mu }_j^{\left(k\right)},{\nu }_j^{\left(k\right)},{\pi }_j^{\left(k\right)}\right]$, is represented as an IFN. The calculation of criterion weights involves employing the IFWA operator:

$$\begin{gathered} w_{ij}={IFWA}_{\lambda }\left(w_j^{\left(1\right)},w_j^{\left(2\right)},\dots ,w_j^{\left(l\right)}\right)={\lambda }_1{w}_j^{\left(1\right)}⨁{\lambda }_2{w}_j^{\left(2\right)}⨁{\lambda }_{31}{w}_j^{\left(3\right)}⨁\dots ⨁{\lambda }_l{w}_j^{\left(l\right)} \\ =\left[1-{\prod }_{k=1}^l{\left(1-{\mu }_j^{\left(k\right)}\right)}^{{\lambda }_k},{\prod }_{k=1}^l{\left({\nu }_j^{\left(k\right)}\right)}^{{\lambda }_k},{\prod }_{k=1}^l{\left(1-{\mu }_j^{\left(k\right)}\right)}^{{\lambda }_k}-{\prod }_{k=1}^l{\left({\nu }_j^{\left(k\right)}\right)}^{{\lambda }_k}\right], \end{gathered}$$

(7)

$W=\left[w_1,w_2,w_3,\dots ,w_j\right]$ where $w_j=\left({\mu }_j,{\nu }_j,{\pi }_j\right),\left(j=1,2,\dots ,n\right)$,

Step 4: Formulation of the aggregated weighted IFDM: Once the criterion weights (W) and the combined IFDM are defined, the creation of the aggregated weighted IFDM is carried out on the basis of the following definition by Atanassov [7]:

$$\begin{gathered} R^{\prime }=R⨂W=\left\{〈x,{\mu }_{A_i}\left(z\right).{\mu }_W\left(z\right),{\nu }_{A_i}\left(z\right)+{\nu }_W\left(z\right)-{\nu }_{A_i}\left(z\right).{\nu }_W\left(z\right)〉|z\in Z\right\}, \\ {\pi }_{A_i}w\left(z\right)=1-{\nu }_{A_i}\left(z\right)-{\nu }_W\left(z\right)-{\mu }_{A_i}\left(z\right).{\mu }_W\left(z\right)+{\nu }_{A_i}\left(z\right).{\nu }_W\left(z\right), \end{gathered}$$

(8)

Subsequently, the ensuing matrix establishes the aggregated weighted IFDM:

$$R^{\prime }=\left[\begin{array}{ccc}{r}_{11}^{\prime }& \cdots & r_{1n}^{\prime }\\ ⋮& \ddots & ⋮\\ r_{m1}^{\prime }& \cdots & r_{mn}^{\prime }\end{array}\right]=\left[\begin{array}{ccc}\left({\mu }_{A_{1}W}\left(z_1\right),{\nu }_{A_{1}W}\left(z_1\right),{\pi }_{A_{1}W}\left(z_1\right)\right)& \cdots & \left({\mu }_{A_{1}W}\left(z_n\right),{\nu }_{A_{1}W}\left(z_n\right),{\pi }_{A_{1}W}\left(z_n\right)\right)\\ ⋮& \ddots & ⋮\\ \left({\mu }_{A_{m}W}\left(z_1\right),{\nu }_{A_{m}W}\left(z_1\right),{\pi }_{A_{m}W}\left(z_1\right)\right)& \cdots & \left({\mu }_{A_{m}W}\left(z_n\right),{\nu }_{A_{m}W}\left(z_n\right),{\pi }_{A_{m}W}\left(z_n\right)\right)\end{array}\right],$$

(9)

where $r_{ij}^{\prime }=\left({\mu }_{ij}^{\prime },{\nu }_{ij}^{\prime },{\pi }_{ij}^{\prime }\right)=\left({\mu }_{A_{i}W}\left(z_j\right),{\nu }_{A_{i}W}\left(z_j\right),{\pi }_{A_{i}W}\left(z_j\right)\right)$ represents a component within the aggregated weighted IFDM.

Step 5: Acquire the PIS and NIS in the realm of the IFS: Assume that $J_1$ represents the benefit criteria and $J_2$ denotes the cost criteria. $A^{\star }$ signifies the PIS in IF and $A^{-}$ stands for the NIS in IF. The derivation of $A^{\star }$ and $A^{-}$ is as follows:

$$A^{\star }=\left({\mu }_{A^{\ast }W}\left(z_j\right),{\nu }_{A^{\ast }W}\left(z_j\right)\right)\mathrm{and}{A}^{-}=\left({\mu }_{A^{-}W}\left(z_j\right),{\nu }_{A^{-}W}\left(z_j\right)\right),$$

(10)

where:

$${\mu }_{A^{\ast }W}(z_j)=\left(\left(\underset{i}{\mathit{max}}{\mu }_{A_i.W}\left(z_j\right)|j\in J_1\right),\left(\underset{i}{\mathit{min}}{\mu }_{A_i.W}\left(z_j\right)|j\in J_2\right)\right),$$

(11)

$${\nu }_{A^{\ast }W}(z_j)=\left(\left(\underset{i}{\mathit{min}}{\nu }_{A_i.W}\left(z_j\right)|j\in J_1\right),\left(\underset{i}{\mathit{max}}{\nu }_{A_i.W}\left(z_j\right)|j\in J_2\right)\right),$$

(12)

$${\mu }_{A^{-}W}(z_j)=\left(\left(\underset{i}{\mathit{min}}{\mu }_{A_i.W}\left(z_j\right)|j\in J_1\right),\left(\underset{i}{\mathit{max}}{\mu }_{A_i.W}\left(z_j\right)|j\in J_2\right)\right),$$

(13)

$${\nu }_{A^{-}W}(z_j)=\left(\left(\underset{i}{\mathit{max}}{\nu }_{A_i.W}\left(z_j\right)|j\in J_1\right),\left(\underset{i}{\mathit{min}}{\nu }_{A_i.W}\left(z_j\right)|j\in J_2\right)\right),$$

(14)

Step 6: Determine the separation degrees and the relative closeness coefficient (RCC): IFS alternatives differ in distance measures; the current study uses normalized Euclidean distance [17] to assess separation degrees, $S_{i^{+}}$ and $S_{i^{-}}$, from the PIS and NIS in IF:

$$S^{\ast }=\sqrt{{\displaystyle \frac{1}{2n}}\sum _{j=1}^n\left[{\left({\mu }_{A_{i}W}\left(z_j\right)-{\mu }_{A^{\ast }W}\left(z_j\right)\right)}^2+{\left({\nu }_{A_{i}W}\left(z_j\right)-{\nu }_{A^{\ast }W}\left(z_j\right)\right)}^2+{\left({\pi }_{A_{i}W}\left(z_j\right)-{\pi }_{A^{\ast }W}\left(z_j\right)\right)}^2\right]},$$

(15)

$$S^{-}=\sqrt{{\displaystyle \frac{1}{2n}}{\sum }_{j=1}^n\left[{\left({\mu }_{A_{i}W}\left(z_j\right)-{\mu }_{A^{-}W}\left(z_j\right)\right)}^2+{\left({\nu }_{A_{i}W}\left(z_j\right)-{\nu }_{A^{-}W}\left(z_j\right)\right)}^2+{\left({\pi }_{A_{i}W}\left(z_j\right)-{\pi }_{A^{-}W}\left(z_j\right)\right)}^2\right]},$$

(16)

The RCC for a substitute $A_i$ in relation to the PIS in IF realms A* is formulated as follows:

$$C_{i^{\ast }}={\displaystyle \frac{S_{i^{-}}}{S_{i^{+}}+S_{i^{-}}}}\mathrm{where}0\le C_{i^{\ast }}\le 1,$$

(17)

Step 7: Determine the ranking of the alternatives: After obtaining the RCC for each alternative, the ranking is established on the basis of the descending order of $C_{i^{\ast }}$s.

5. Implementation of Developed Framework

The selection of criteria plays a crucial role in the evaluation procedure, as emphasized in the literature review. Given a clear understanding of the criteria and available alternatives, this study focuses on five specific criteria, including lightweight/density (Z₁), price/cost (Z₂), tensile strength (Z₃), corrosion resistance (Z₄), and durability (Z₅), and four different materials as alternatives, including stainless steel (A₁), aluminum alloy (A₂), carbon fiber (A₃), and glass fiber/E-glass (A₄). The linguistic expressions used to rate both the D-Ms and criteria are depicted in

Table 2. Linguistic expressions to assess the significance of the criteria and D-Ms.

Linguistic Expressions	IFNs
Very significant (VS)	(0.9, 0.1, 0)
Significant (S)	(0.75, 0.2, 0.05)
Medium (M)	(0.50, 0.45, 0.05)
Insignificant (IS)	(0.35, 0.45, 0.05)
Very insignificant (VIS)	(0.1, 0.9, 0)

.

To determine the optimal material among four options, three D-Ms were involved in the assessment. The material selection process involved the following stages:

Phase 1: Calculate the weights assigned by D-Ms.

The evaluation of the D-Ms’ importance begins with linguistic expressions. The linguistic expressions used to rate both D-Ms and criteria are provided in Table 2. The determination of D-Ms’ importance involves considering their field of expertise, educational background, and experience, as outlined in

Table 3. Background information of the decision-makers.

No.	Working Area	Educational Level	Experience	Linguistic Expressions	Weights
D-M1	Mechanical engineering and renewable energy system	PhD	15 years	Very significant	0.406
D-M2	Energy system engineering	PhD	10 years	Significant	0.406
D-M3	Mechanical engineering	Master	5 years	Medium	0.238

. Equation (4) was applied to calculate the weights of the D-Ms. The significance level of the D-Ms in group decisions is displayed in Table 3.

Phase 2: Build the aggregated IFDM according to the judgments of D-Ms.

The linguistic expressions for evaluating the substitutes are outlined in

Table 4. Linguistic expressions for assessing the alternatives.

Linguistic Expressions	IFNs
Extremely high (EH)/Extremely beneficial (EB)	[1.0, 0.0]
Very very high (VVH)/Very very beneficial (VVB)	[0.9, 0.1]
Very high (VH)/Very beneficial (VB)	[0.8, 0.1]
High (H)/Beneficial (B)	[0.7, 0.2]
Medium-high (MH)/Medium beneficial (MB)	[0.6, 0.3]
Fair (F)/Fair (F)	[0.5, 0.4]
Medium-low (ML)/Medium unbeneficial (MU)	[0.4, 0.5]
Low (L)/unbeneficial (U)	[0.25, 0.6]
Very low (VL)/Very unbeneficial (VU)	[0.1, 0.75]
Very very low (VVL)/Very very unbeneficial (VVU)	[0.1, 0.9]

.

The assessments given by the D-Ms for the four alternatives are shown in

Table 5. The evaluations of the alternatives.

Criteria	Material	D-M1	D-M2	D-M3
Z1	A1	H	VVH	VH
	A2	VH	H	H
	A3	MH	MH	MH
	A4	VH	F	MH
Z2	A1	VH	VH	VH
	A2	H	VH	H
	A3	EH	MH	MH
	A4	VH	MH	H
Z3	A1	B	MB	MB
	A2	VB	MB	F
	A3	EB	VB	VVB
	A4	VVB	F	F
Z4	A1	B	MB	VB
	A2	VB	MB	B
	A3	EB	VVB	VB
	A4	EB	VB	B
Z5	A1	VB	VVB	VB
	A2	VVB	VB	B
	A3	VVB	MB	B
	A4	VB	F	F

.

The formation of the aggregated IFDM, R (as per Equation (6)), resulted from combining the opinions of the D-Ms in the following manner:

$$R=\begin{array}{c}\\ \begin{array}{c}{A}_1\\ \begin{array}{c}{A}_2\\ \begin{array}{c}{A}_3\\ A_4\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{\mathrm{Z}}_1\\ \left[\begin{array}{c}(0.816,0.133,0.052)\\ \begin{array}{c}(0.746,0.151,0.104)\\ \begin{array}{c}(0.6,0.3,0.1)\\ (0.673,0.213,0.114)\end{array}\end{array}\end{array}\right.\end{array}\begin{array}{c}{\mathrm{Z}}_2\\ \begin{array}{c}(0.8,0.1,0.1)\\ \begin{array}{c}(0.74,0.156,0.103)\\ \begin{array}{c}(1,0,0)\\ (0.718,0.174,0.108)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{\mathrm{Z}}_3\\ \begin{array}{c}(0.644,0.254,0.101)\\ \begin{array}{c}(0.682,0.206,0.113)\\ \begin{array}{c}(1,0,0)\\ (0.74,0.228,0.032)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{\mathrm{Z}}_4\\ \begin{array}{c}(0.698,0.196,0.106)\\ \begin{array}{c}(0.718,0.174,0.108)\\ \begin{array}{c}(1,0,0)\\ (1,0,0)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{\mathrm{Z}}_5\\ \left.\begin{array}{c}(0.844,0.1,0.056)\\ \begin{array}{c}(0.834,0.118,0.048)\\ \begin{array}{c}(0.787,0.174,0.038)\\ (0.655,0.228,0.117)\end{array}\end{array}\end{array}\right],\end{array}$$

(18)

Phase 3: Calculate the weights assigned to the criteria.

The significance of the criteria, as expressed in linguistic terms by the D-Ms, is shown in

Table 6. The significance of the criteria.

Criteria	D-M1	D-M2	D-M3
Z1	I	VI	VI
Z2	VI	VI	I
Z3	M	I	M
Z4	VI	M	U
Z5	VI	M	M

.

Subsequently, the linguistic expressions were translated into IFNs based on the corresponding linguistic expressions in Table 2. To calculate the importance of each criterion, the views of the D-Ms concerning the criteria were combined using Equation (7).

$$W_{\left\{Z_1,Z_2,Z_3,Z_4,Z_5\right\}}={\left[\begin{array}{c}(0.855,0.133,0.013)\\ \begin{array}{c}(0.876,0.118,0.006)\\ \begin{array}{c}(0.609,0.337,0.054)\\ \begin{array}{c}(0.723,0.262,0.015)\\ (0.740,0.244,0.016)\end{array}\end{array}\end{array}\end{array}\right]}^T,$$

(19)

Phase 4: Formulation of the aggregated weighted IFDM.

After establishing the weights for the criteria and assessing the alternatives, the aggregated weighted IFDM R′ was created using Equation (8) in the following manner:

$$R^{\prime }=\begin{array}{c}\\ \begin{array}{c}{A}_1\\ \begin{array}{c}{A}_2\\ \begin{array}{c}{A}_3\\ A_4\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{Z}_1\\ \left[\begin{array}{c}(0.697,0.247,0.055)\\ \begin{array}{c}(0.637,0.263,0.099)\\ \begin{array}{c}(0.513,0.393,0.094)\\ (0.576,0.317,0.107)\end{array}\end{array}\end{array}\right.\end{array}\begin{array}{c}{Z}_2\\ \begin{array}{c}(0.701,0.206,0.093)\\ \begin{array}{c}(0.648,0.256,0.096)\\ \begin{array}{c}(0.876,0.118,0.006)\\ (0.629,0.272,0.099)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{Z}_3\\ \begin{array}{c}(0.393,0.506,0.102)\\ \begin{array}{c}(0.415,0.473,0.111)\\ \begin{array}{c}(0.609,0.337,0.054)\\ (0.451,0.488,0.061)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{Z}_4\\ \begin{array}{c}(0.505,0.406,0.089)\\ \begin{array}{c}(0.519,0.39,0.09)\\ \begin{array}{c}(0.723,0.262,0.015)\\ (0.723,0.262,0.015)\end{array}\end{array}\end{array}\end{array}\begin{array}{c}{Z}_5\\ \left.\begin{array}{c}(0.624,0.32,0.056)\\ \begin{array}{c}(0.617,0.333,0.05)\\ \begin{array}{c}(0.583,0.376,0.041)\\ (0.485,0.416,0.099)\end{array}\end{array}\end{array}\right],\end{array}$$

(20)

Phase 5: Acquire the PIS and NIS in the realm of the IFS.

Tensile strength, corrosion resistance, and durability are the benefit criteria J1 = {Z₃, Z₄, Z₅}, while lightweight (density) and cost are the cost criteria J2 = {Z₁, Z₂}. Following that, the PIS in IF and the NIS in IF were found by utilizing Equations (10)–(14), as follows:

$$\begin{gathered} A^{\ast }=\left\{\left(0.513,0.393,0.107\right),\left(0.629,0.272,0.099\right),\left(0.609,0.337,0.054\right), \\ \left(0.723,0.262,0.015\right),\left(0.624,0.32,0.041\right)\right\} \end{gathered}$$

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$$\begin{gathered} A^{-}=\left\{\left(0.697,0.247,0.055\right),\left(0.876,0.118,0.006\right),\left(0.393,0.506,0.111\right), \\ \left(0.505,0.406,0.09\right),(0.485,0.416,0.099)\right\} \end{gathered}$$

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Phase 6: Determine the separation measures and RCC.

The separation degrees, both positive and negative, based on the normalized Euclidean distance for every alternative, are quantified in

Table 7. Measures of separation and RCC for each alternative.

Alternative	${\mathit{S}}^{\mathbf{\ast }}$	${\mathit{S}}^{-}$	${\mathbf{C}}_{{\mathbf{i}}^{\mathbf{\ast }}}$	Rank
A1	0.148	0.088	0.372	4
A2	0.125	0.106	0.460	3
A3	0.099	0.150	0.602	1
A4	0.095	0.140	0.597	2

.

To prioritize the substitutes, we initially computed the RCC and then graded the four alternatives in descending order on the basis of C_is. The alternatives were ranked as A₃ > A₄ > A₂ > A₁.

6. Discussion and Findings

In this case study, the selection of materials was determined using the IFS TOPSIS method. Three D-Ms assessed four alternatives—stainless steel, aluminum alloy, carbon fiber, and glass fiber—on the basis of five criteria—density, cost, tensile strength, corrosion resistance, and durability—to establish their ranking, and carbon fiber was chosen as the best material, followed by glass fiber. This finding aligns with the findings of Mondal et al. [1] in a related study. The researchers utilized the MARCOS method to identify the optimal composite material for blade manufacturing. Their study also incorporated the BWM to determine criterion weights, examining eight alternatives and nine criteria, some of which overlapped with the criteria considered in our study. Similarly, this result corresponds with the outcomes reported by Maity and Chakraborty [9] in their related study. The authors employed an FANP-based approach to select materials suitable for impulse turbine blades in wind and wave energy extraction.

7. Conclusions

This research introduces the MCDM approach for material selection using the IF technique for TOPSIS. The IFS provides an effective means to address hesitancy. In the assessment procedure, evaluations for each alternative in relation to each criterion and the weights designated to each criterion were articulated using linguistic expressions characterized by IFNs. An IF averaging operator was employed to amalgamate D-Ms’ ideas. Following the calculation of the PIS in IF and the NIS in IF constructed using Euclidean distance, the RCCs for alternatives were determined, leading to their ranking. The results indicate that carbon fiber emerges as the optimal material for wind turbine blades. This system can be developed to include more D-Ms and criteria to diminish biases. Subsequent research endeavors could involve enhanced D-M engagement in all facets of the TOPSIS technique. This study conducted an analysis utilizing five criteria derived from input provided by three experts. In future research, additional decision makers and criteria could be taken into account. In addition, the ranking of materials can be compared with other methods, like Fuzzy TOPSIS and Rough TOPSIS.