Development of the ELECTRE Method Under Pythagorean Fuzzy Sets Based on Existing Correlation Coefficients for Cotton Fabric Selection

ABSTRACT

Cotton fabric selection is a challenging task in the garment product design and development process, and the selection of optimal alternative under the presence of multiple decision criteria becomes complex, and hence it is considered as a multi-criteria decision-making (MCDM) problem. In addition, the selection process involves fuzziness and uncertainty. In this study, Pythagorean fuzzy sets (PFSs) are introduced to handle uncertain information. Elimination and choice translating reality (ELECTRE) is a well-known outranking method for solving MCDM problems. Therefore, we extend the ELECTRE method under the PFS environment, and a correlation-based closeness coefficient is proposed to compare Pythagorean fuzzy numbers (PFNs). This paper applies the proposed PF-ELECTRE approach in solving a practical case involving the ranking cotton fabrics. To exhibit the superiority and robustness of the suggested method, sensitivity analysis is performed to examine the impacts of weights variation, as well as a comparative analysis is carried out between the PF-ELECTRE with several existing MCDM methods. The research contributes to the advancement and development of outranking MCDM methods through a novel PF-ELECTRE approach that utilizes the weighted correlation coefficient. Moreover, the developed method can obtain reliable results and can be used to other textile domains.

摘要

纯棉面料的选择是服装产品设计和开发过程中一项具有挑战性的任务，在多个决策标准存在的情况下，最优方案的选择变得复杂，因此被认为是一个多标准决策问题. 此外，选择过程还涉及到模糊性和不确定性. 在本研究中，引入勾股模糊集（PFSs）来处理不确定信息. 消除和选择翻译现实（ELECTRE）是解决MCDM问题的一种众所周知的高级方法. 因此，我们在PFS环境下扩展了ELECTRE方法，并提出了一种基于相关性的贴近度系数来比较勾股模糊数. 本文将所提出的PF-ELECTRE方法应用于解决一个涉及棉织物分级的实际案例. 为了展示所建议方法的优越性和稳健性，进行了灵敏度分析，以检查权重变化的影响，并在PF-ELECTRE与几种现有的MCDM方法之间进行了比较分析. 这项研究通过一种利用加权相关系数的新型PF-ELECTRE方法，为MCDM方法的进步和发展做出了贡献. 此外，所开发的方法可以获得可靠的结果，并可用于其他纺织领域.

KEYWORDS:

• Cotton fabric selection
• multi-criteriadecision-making
• Pythagorean fuzzy sets
• ELECTRE
• correlation coefficient
• comparative analysis

关键词:

• 纯棉面料选择
• 多准则决策
• 勾股模糊集
• 驻极体
• 相关系数
• 比较分析

Introduction

Cotton fabric is widely used in the textile area, and the process of ranking and selecting an optimal cotton fabric involving a set of properties is complex and considered a multi-criteria decision-making (MCDM) issue. The comfort of cotton fabric is affected by several physical properties including benefit-type properties, where higher values indicate better properties. In contrast, cost-type properties mean that lower values correspond to better characteristics (Moslehi et al. 2023). Therefore, selecting the appropriate fabric to fulfill the desired end-use demands of clothing comfort is a challenging task.

To address the MCDM issue, three types of approaches are commonly used: scoring methods, outranking methods, and compromising methods. Numerous scholars have applied these methods to in the textile area. The scoring method includes the analytical hierarchy process (AHP), weighted product model (WPM), and weighted sum model (WSM). Gupta (2022) used the WSM and WPM to assess the overall durability of handmade carpets, respectively. The compromising method encompasses several techniques, including the Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR), technique for order of preference by similarity to an ideal solution (TOPSIS), and Evaluation based on Distance from Average Solution (EDAS). Akgül et al. (2022) applied the TOPSIS method to choose the optimal dyeing process. Ak, Yucesan, and Gul (2022) applied the VIKOR method to prioritize the hazards in the textile production industry. The outranking method comprises techniques such as the preference ranking organization method for enrichment evaluation (PROMETHEE), elimination and choice translating reality (ELECTRE). Iftikhar et al. (2021) utilized the AHP and PROMETHEE approaches to address the woven fabric selection problem. Uddin et al. (2019) employed the ELECTRE technique to assess the obstacles to implementing a green supply chain. Guarnieri and Trojan (2019) applied the AHP and ELECTRE approaches to aid the supplier selection process. Kuo et al. (2019) used ELECTRE for multi-quality optimization in the fabrication process of hot melt pressure-sensitive adhesives. Out of these methods, the ELECTRE method is widely used in solving MCDM problem.

The ELECTRE approach is a collection of outranking-based methods that is the most applicable when there are numerous alternatives and conflicting criteria (Martyn and Kadziński 2023). It expresses the preference of one option over another utilizing outranking relations and derives a more accurate and appropriate set of alternatives by removing properties that were outranked. All decision-aided methods based on outranking relationships rely on concordance and discordance to determine support for and opposition against the outranked cases. The classical ELECTRE approach, which has numerous variants, including ELECTRE-II, III, IV, IS, and TRI depending on the nature of the issue, has been extensively applied utilized for decision-making in diverse real-life domains. However, only a few studies have applied the ELECTRE method in the textile sector. A summary of the ELECTRE methods is shown in Table 1. This indicates that the calculation procedures of the ELECTRE I and II approaches are simple and suitable for the selection and ranking of alternatives.

Table 1. Comparison of various ELECTRE methods.

Method	Criteria classification	Problematics	Weights	Characteristics	Applications
ELECTRE I	True-criteria	Choosing	Yes	Concordance and discordance Boolean matrices A concordance-discordance Boolean matrix to rank the alternatives	Water management; Business management Network selection
ELECTRE II	True-criteria	Ranking	Yes	Two different outranking relations; An embedded outranking relations sequence	Water management; Supplier selection Software evaluation
ELECTRE III	Pseudocriteria	Ranking	Yes	The indifference and preference thresholds definition.	Water management; Waste management Business management
ELECTRE IV	Pseudocriteria	Ranking	No	The relative importance coefficients of criteria are not needed.	Energy management
ELECTRE TRI	Pseudocriteria	Sorting	Yes	Each action is assigned to one of k prespecified ordered categories	Business management; Energy management
ELECTRE IS	Pseudocriteria	Choosing	Yes	The decision parameters are intervals.	Energy management; Manufacturing systems

In real-life situations, decisions are often made under uncertain circumstances due to the complexity of MCDM problem and the imprecise and vague nature of decision information. Traditional evaluation methods based on crisp values struggle to express uncertainty. To address this problem, Atanassov (1986) launched the concept of intuitionistic fuzzy sets (IFSs), which incorporate membership and nonmembership degrees. IFSs have gained popularity in addressing fuzzy MCDM problems. Pythagorean fuzzy sets (PFSs), pioneered by Yager (2013), are characterized by the property that the sum of squares of membership and nonmembership degrees is less than one. As an extension of IFSs, PFSs are capable of more adequately and accurately handling ambiguous information. Due to the advantages of ELECTRE, various scholars have extended the ELECTRE method in the context of PFS. Related works regarding PF-ELECTRE can be seen in Table 2. Singh and Bisht (2021) applied the hybrid method of ELECTRE I and VIKOR to solve a real-world case related to related to location selection under PFS. Akram, Luqman, and Alcantud (2021) integrated the PFS with ELECTRE I approach to measure risk ratings. Akram, Ilyas, and Garg (2021) proposed a group decision support algorithm based on the ELECTRE II method under using the PFS structure. Akram, Ilyas, and Al-Kenani (2021) implemented the PF-ELECTRE III approach to address a real-world haze management problem. Chen (2020) proposed a PF-ELECTRE approach that utilizes a novel developed PFS Chebyshev distance measure to address the bridge-superstructure construction issue. Akram, Ilyas, and Garg (2020) formulated a PF-ELECTRE I method that employs the strong, midrange, and weak PF concordance and discordance sets. In another study, Chen (2018) developed the ELECTRE method under the interval-valued PFS (IVPF) using a risk attitude assignment model to solve the investment problem. Peng and Yang (2016) developed the IVPFS-ELECTRE to tackle the investment problem. From the above research, the ELECTRE method has been successfully incorporated into the PFS, and it has been used in several areas, indicating that the PF-ELECTRE method can be utilized effectively to address real-world issues. However, there are few studies examining the use of PF-ELECTRE to fabric selection. In addition, these developed methods are good, but it could find that most of these studies are difficult for untrained decision-makers to implement since these methods involve complex computational process. Lastly, most of the previous research applied the score and accuracy function to compare PFNs.

Table 2. Studies related to PF-ELECTRE.

Authors	Methodology	Outranking sets	Comparison methods	Applications
Singh and Bisht (2021)	ELECTRE I	Strong, midrange and weak concordance/discordance sets	score function, accuracy function and hesitancy degree	Location selection
Akram, Luqman, and Alcantud (2021)	ELECTRE I	Concordance and discordance sets	score and accuracy functions	Risk ratings
Akram, Ilyas, and Garg (2021)	ELECTRE II	Concordance, indifferent and discordance sets	score and accuracy functions	Supplier selection
Akram, Ilyas, and Al-Kenani (2021)	ELECTRE III	Concordance and discordance sets	Score function	Haze management
Chen (2020)	ELECTRE II	Chebyshev metric-based net concordance/discordance sets	scalar function	Bridge-superstructure construction
Akram, Ilyas, and Garg (2020)	ELECTRE I	Strong, midrange and weak concordance/discordance sets	score function, accuracy function and hesitancy degree	Waste management
Chen (2018)	ELECTRE II	Concordance and discordance sets	risk attitude-based score function	Investment problem
Peng and Yang 2016	ELECTRE I	Strong, medium, and weak concordance/discordance sets	score function and accurate function	Investment problem

Elaborate research has been conducted on the fundamentals of PFSs, including distance measures (Wang et al. 2023), aggregation operators (Hussain et al. 2023), similarity measures (Zhou et al. 2023), decision-making technologies (Li et al. 2023), and applications of PFSs (Bulut and Özcan 2023; Ye and Chen 2022a). The correlation coefficient is also a crucial factor in decision-making problems. A measure of interdependency can be used to validate the interconnectedness of two variables in correlation analysis. Garg (2016) initiated the concept of correlation coefficient in the PFSs environment to analyze the interdependency between two PFSs, and applied it to address MCDM issues under conditions of uncertainty. Chen (2019) developed a new PF compromise approach that employs Pearson-like correlation coefficients. Thao (2020) proposed a novel variance and covariance-based correlation coefficient, which was then utilized to address medical diagnosis issues. Singh and Ganie (2020) developed several PFS correlation coefficients for clustering analysis, medical diagnosis, and pattern recognition, but these methods did not include all the typical parameters of PFSs. Some statistical methods for estimating the PFS correlation coefficient have been investigated in medical diagnosis (Ejegwa et al. 2022) and classification of building materials (Yan et al. 2023). From the above analysis, it is evident that the correlation coefficient in PFSs has been widely studied. However, few studies have applied the closeness coefficient to make comparisons between two PFNs. Furthermore, there has been little research that incorporated the correlation coefficient within the framework of PF-ELECTRE method.

The main motivations of this research are threefold: (1) The ELECTRE approach is a highly efficient framework for diverse applications to figure out the optimal alternative. ELECTRE II is a more dependable and capable tool for identifying the best choice and provide an overall ranking. (2) PFSs are an important tool to express the uncertainty in broader space and superior to IFSs. They can better handle the uncertainty and ambiguity in the evaluation and easily be integrated with MCDM methods. Therefore, the integration of ELECTRE under the PFSs environment is suitable for addressing MCDM issues. (3) Unlike the previous PF-ELECTRE method, the correlation coefficient is also an essential tool for comparing PFSs. However, few studies have incorporated it into the ELECTRE method.

For this purpose, this research proposes a novel ELECTREE method integrating the correlation coefficient under the PF environment to handle the cotton fabric selection problem. We use the directional correlation coefficient to compute the alternative between the fixed positive-ideal solution (PIS) and negative-ideal solution (NIS), and we perform pairwise comparison using the closeness coefficient to generate the concordance and discordance matrices. Then, the Hamming distance is utilized to determine the discordance index, while the overall precedence index is derived from the net concordance and discordance indices. Our proposed PF-ELECTRE method is specifically applied to the issue of cotton fabric selection, which has become a new research hotspot and is addressed by various MCDM techniques. Finally, to evaluate the robustness and reliability of the PF-ELECTRE method, a sensitivity analysis was conducted by altering the criteria weights to observe their effects on the alternative rankings. In addition, a comparative analysis was conducted with other methods, such as WPM, EDAS (Mitra 2022b), multi-objective optimization by ratio analysis (MOORA) (Mitra 2022a), PF-WPM and Pythagorean fuzzy TOPSIS (PF-TOPSIS) (Ye and Chen 2022b), which have been done to deal with the same problem. The Spearman rank correlation analysis results indicate that the proposed PF-ELECTRE method can obtain reasonable ranking results.

The novelties of this research are as follows: (1) In contrast to the previous studies of PF-ELECTRE methods that apply the score function, accuracy function and indeterminacy to compare PFNs to construct the concordance/discordance sets, this research constructs the closeness coefficient-based concordance/discordance sets. (2) In previous research, the ELECTRE I method was applied; however, it cannot provide a complete ranking of the alternatives. In contrast, the ELECTRE II approach can accurately and conveniently obtain an optimal alternative and an overall ranking order. (3) The classical ELECTRE II approach involves three types of outranking sets, namely concordance, indifference, and discordance sets, but we simplify the outranking sets into concordance and discordance sets, which makes the calculation process more flexibility. (4) The compromising method operates on the principle of identifying an alternative that is closest to the PIS while farthest from the NIS. In contrast, the outranking approach is based on pairwise comparisons of alternative under certain criteria, which can obtain the outranking relation between the alternatives and provide a better understanding of their inner relationships.

Our research has made several significant contributions as follows: (1) Due to the advantages of PFS in describing the fuzziness, and the ELECTRE is a widely used MCDM method, this research develops a novel method, named as PF-ELECTRE technique. (2) This is the first study to incorporate the weighted directional correlation coefficient and correlation-based closeness coefficient of PFSs into the ELECTRE framework to compare PFNs. (3) A case concerning the selection of cotton fabric, adapted from the existing literature to facilitate a comparison of our methodology with other existing methods, is studied by applying the proposed PF-ELECTRE method to confirm its applicability and feasibility. (4) Results from the sensitivity analysis and comparative analysis show that our developed approach outperformed other existing methods, making it a superior approach for addressing MCDM issues in the textile sector.

The remainder of this study is structured as follows: Section 2 presents a briefly introduction to the fundamental concepts of PFS. Section 3 discusses several correlation coefficients and a correlation-based closeness coefficient. Section 4 proposes the PF-ELECTRE approach. In Section 5, the effectiveness of the proposed approach is demonstrated by solving a numerical case study on cotton fabric selection, followed by sensitivity and comparative analyses. Finally, this study is concluded with a discussion of limitations and future recommendations.

Preliminaries

Fundamental concepts of PFSs

Definition 1

(Yager 2013) Let $X=\left\{x_1,x_2,\dots ,x_n\right\}$ be a fixed set; a PFS $P$ is given as:

(1) $P=\left\{x,{\mu }_P\left(x\right),{\nu }_P\left(x\right)|x\in X⟩\right\}$(1)

where ${\mu }_P\left(x\right)$ and ${\nu }_P\left(x\right)$ denote the membership and nonmembership degrees, respectively. Equation 1(1) $P=\left\{x,{\mu }_P\left(x\right),{\nu }_P\left(x\right)|x\in X⟩\right\}$(1) satisfies the following conditions: ${\mu }_P:X\to \left[0,1\right]$, ${\nu }_P:X\to \left[0,1\right]$, and $0\le {\left({\mu }_P\left(x\right)\right)}^2+{\left({\nu }_P\left(x\right)\right)}^2\le 1$. The degree of the hesitancy condition is:

(2) ${\pi }_P\left(x\right)=\sqrt{1-{\left({\mu }_P\left(x\right)\right)}^2-{\left({\nu }_P\left(x\right)\right)}^2}$(2)

Zhang and Xu (2014) simplified PFS by introducing the Pythagorean fuzzy number (PFN), which refers to the pair $\left({\mu }_P\left(x\right),{\nu }_P\left(x\right)\right)$, denoted as $p=\left({\mu }_P,{\nu }_p\right)$.

Yager (2014) proposed a new way to express a PFN using a pair of parameters $\left(r_P,d_P\right)$, where $r_P$ and $d_P$ denote the strength and direction of commitment, which satisfy $0\le r_P,d_P\le 1$, respectively. The relationships among, ${\mu }_P$, ${\nu }_P$, $r_P$ and $d_P$ are expressed as follows:

(3) ${\mu }_P=r_{P}cos\theta$(3)

(4) ${\nu }_P=r_{P}sin\theta$(4)

(5) ${\mu }_P^2+{\nu }_P^2=r_P^2$(5)

where $\theta$ denotes the intersection angle, and $\theta =\left(1-d_P\right)\frac{\pi }{2}\in \left[0,\frac{\pi }{2}\right]$.

Definition 2

(Zhang and Xu 2014) Let $p_1=\left({\mu }_1,{\nu }_1\right)$ and $p_2=\left({\mu }_2,{\nu }_2\right)$ be two PFNs. The operations on these PFNs can be represented as follows:

(6) $p_1\oplus p_2=\left(\sqrt{{\mu }_1^2+{\mu }_2^2-{\mu }_1^2{\mu }_2^2},{\nu }_1{\nu }_2\right)$(6)

(7) $p_1\otimes p_2=\left({\mu }_1{\mu }_2,\sqrt{{\nu }_1^2+{\nu }_2^2-{\nu }_1^2{\nu }_2^2}\right)$(7)

(8) $\lambda p=\left(\sqrt{1-{\left(1-{\mu }_p^2\right)}^{\lambda }},{\nu }_p^{\lambda }\right),\lambda >0$(8)

(9) $p^{\lambda }=\left({\mu }_p^{\lambda },\sqrt{1-{\left(1-{\nu }_p^2\right)}^{\lambda }},\right),\lambda >0$(9)

Definition 3

(Li and Zeng 2018) Let $p_1=\left({\mu }_1,{\nu }_1\right)$ and $p_2=\left({\mu }_2,{\nu }_2\right)$ be two PFNs. In addition, the strength and direction of commitment are added to express the PFNs. The normalized Hamming distance between $p_1$ and $p_2$ is expounded as below:

(10) $d\left(p_1,p_2\right)=\frac{1}{4}\left(\left|{\mu }_1-{\mu }_2\right|+\left|{\nu }_1-{\nu }_2\right|+\left|r_1-r_2\right|+\left|d_1-d_2\right|\right)$(10)

where the intersection angle is calculated by $\theta =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\nu }_P/{\mu }_P\right)$, then the corresponding strength is obtained by $r_P=\sqrt{{\mu }_P^2+{\nu }_P^2}$, and the direction is computed by $d_P=1-\frac{2}{\pi }\theta$.

Correlation coefficients of PFSs

The correlation coefficient plays a crucial role in determining the interdependency, similarity, and interrelationship of two PFNs, and is determined by the information energy and correlation. Numerous scholars have developed correlation coefficients for PFSs.

Definition 4

(Garg 2016) For PFSs $P_1=\{x,{\mu }_{P_1}\left(x\right),{\nu }_{P_1}\left(x\right)|x\in X\}$ and $P_2=\{x,{\mu }_{P_2}\left(x\right),\text{break}{\nu }_{P_2}\left(x\right)|x\in X\}$, the correlation coefficient ($CC$) of $P_1$ and $P_2$ can be described as follows:

(11) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2{({\nu }_{P_2}(x_i))}^2+{({\pi }_{P_1}(x_i))}^2{({\pi }_{P_2}(x_i))}^2}{\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^4+{({\nu }_{P_1}(x_i))}^4+{({\pi }_{P_1}(x_i))}^4}\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^4+{({\nu }_{P_2}(x_i))}^4+{({\pi }_{P_2}(x_i))}^4}}$(11)

Definition 5

(Ejegwa 2021) For PFSs $P_1=\{x,{\mu }_{P_1}\left(x\right),{\nu }_{P_1}\left(x\right)|x\in X\}$ and $P_2=\{x,{\mu }_{P_2}\left(x\right),\text{break}{\nu }_{P_2}\left(x\right)|x\in X\}$, the correlation coefficient of $P_1$ and $P_2$ is defined as:

(12) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n({({\mu }_{P_1}(x_i))}^{\frac{k}{2}}{({\mu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\nu }_{P_1}(x_i))}^{\frac{k}{2}}{({\nu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\pi }_{P_1}(x_i))}^{\frac{k}{2}}{({\pi }_{P_2}(x_i))}^{\frac{k}{2}})}{max({{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^k+{({\nu }_{P_1}(x_i))}^k+{({\pi }_{P_1}(x_i))}^k,{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^k+{({\nu }_{P_2}(x_i))}^k+{({\pi }_{P_2}(x_i))}^k)}$(12)

Based on Eq. (12)(12) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n({({\mu }_{P_1}(x_i))}^{\frac{k}{2}}{({\mu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\nu }_{P_1}(x_i))}^{\frac{k}{2}}{({\nu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\pi }_{P_1}(x_i))}^{\frac{k}{2}}{({\pi }_{P_2}(x_i))}^{\frac{k}{2}})}{max({{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^k+{({\nu }_{P_1}(x_i))}^k+{({\pi }_{P_1}(x_i))}^k,{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^k+{({\nu }_{P_2}(x_i))}^k+{({\pi }_{P_2}(x_i))}^k)}$(12) , Ejegwa and Awolola (2021) proposed a modified correlation coefficient as follows:

(13) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n({({\mu }_{P_1}(x_i))}^{\frac{k}{2}}{({\mu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\nu }_{P_1}(x_i))}^{\frac{k}{2}}{({\nu }_{P_2}(x_i))}^{\frac{k}{2}}+{({\pi }_{P_1}(x_i))}^{\frac{k}{2}}{({\pi }_{P_2}(x_i))}^{\frac{k}{2}})}{Aver({{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^k+{({\nu }_{P_1}(x_i))}^k+{({\pi }_{P_1}(x_i))}^k,{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^k+{({\nu }_{P_2}(x_i))}^k+{({\pi }_{P_2}(x_i))}^k)}$(13)

where $k\le 4$.

Definition 6

(Lin et al. 2021) Let $P=\left\{x,{\mu }_P\left(x\right),{\nu }_P\left(x\right)|x\in X\right\}$ denote a PFS. The information energy of $P$ is defined as:

(14) $T\left(P\right)={\sum }_{i=1}^n\left({\left({\mu }_P\left(x_i\right)\right)}^4+{\left({\nu }_P\left(x_i\right)\right)}^4+{\left(r_P\left(x_i\right)\right)}^4+{\left(d_P\left(x_i\right)\right)}^4\right)$(14)

Definition 7

(Lin et al. 2021) For PFSs $P_1=\{x,{\mu }_{P_1}\left(x\right),{\nu }_{P_1}\left(x\right)|x\in X\}$ and $P_2=\{x,{\mu }_{P_2}\left(x\right),\text{break}{\nu }_{P_2}\left(x\right)|x\in X\}$, the correlation of two PFSs can be defined as:

(15) $CC\left(P_1,P_2\right)={\sum }_{i=1}^n\left({\left({\mu }_{P_1}\left(x_i\right)\right)}^2{\left({\mu }_{P_2}\left(x_i\right)\right)}^2+{\left({\nu }_{P_1}\left(x_i\right)\right)}^2{\left({\nu }_{P_2}\left(x_i\right)\right)}^2+{\left(r_{P_1}\left(x_i\right)\right)}^2{\left(r_{P_2}\left(x_i\right)\right)}^2+{\left(d_{P_1}\left(x_i\right)\right)}^2{\left(d_{P_2}\left(x_i\right)\right)}^2\right)$(15)

Definition 8

(Lin et al. 2021) For PFSs $P_1=\{x,{\mu }_{P_1}\left(x\right),{\nu }_{P_1}\left(x\right)|x\in X\}$ and $P_2=\{x,{\mu }_{P_2}\left(x\right),\text{break}{\nu }_{P_2}\left(x\right)|x\in X\}$, the directional correlation coefficient of $P_1$ and $P_2$ is described as follows:

(16) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2{({\nu }_{P_2}(x_i))}^2+{(r_{P_1}(x_i))}^2{(r_{P_2}(x_i))}^2+{(d_{P_1}(x_i))}^2{(d_{P_2}(x_i))}^2}{\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2+{(r_{P_1}(x_i))}^2+{(d_{P_1}(x_i))}^2}\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_2}(x_i))}^2+{(r_{P_2}(x_i))}^2+{(d_{P_2}(x_i))}^2}}$(16)

Based on the above analysis, it is evident that the correlation coefficient proposed by Lin et al. (2021) considers the strength and direction of commitment and effectively employs the squared degree of ${\mu }_P\left(x\right)$, ${\nu }_P\left(x\right)$, $r_P\left(x\right)$ and $d_P\left(x\right)$. Therefore, we incorporate the correlation coefficient developed by Lin et al. (2021) into the PF-ELECTRE approach.

Considering that decision-makers often use specific reference points to anchor their subjective judgments, the preference of a decision-maker can be expressed as “as close as possible” or “as far as possible” using a positive ideal or a negative ideal as a reference point, respectively. Predictably, the selection of these reference points can affect the strength and even the ranking of preferences. To facilitate anchoring judgments, we identified fixed PF-PIS and PF-NIS.

Correlation-based closeness coefficient

This subsection introduces the correlation-based PFS closeness coefficient $CCC\left(P_i\right)$ to solve uncertain MCDM problems involving PF information. Particularly, the purpose of determining the final ordering of preference among competing alternatives is to establish the PFS correlation-based closeness coefficient. Given that the PF correlation coefficient is specified in terms of positive and negative values, $CC\left(P_i,P^{+}\right)$ indicates a positive relation between $P_i$ and $P^{+}$, whereas $CC\left(P_i,P^{-}\right)$ suggests a negative relation. Generally, the larger $CC\left(P_i,P^{+}\right)$is and the smaller $CC\left(P_i,P^{-}\right)$ is, the better $P_i$ is. $CCC\left(P_i\right)$ measures how close $P_i$ is to $P^{+}$ while also being far from $P^{-}$.

Definition 9

(Chen 2018) Let $P_i$ be the PF characteristic of the alternative. For the ideal PF solutions $P^{+}$ and $P^{-}$, $CCC\left(P_i\right)$ is defined as:

(17) $CCC\left(P_i\right)=\frac{CC\left(P_i,P^{+}\right)}{CC\left(P_i,P^{+}\right)+CC\left(P_i,P^{-}\right)}$(17)

The PF-ELECTRE method

In this section, we present a novel ELECTRE method that is designed to address the MCDM problems within the PFS context. The flowchart illustrating the proposed PF-ELECTRE approach is delineated in Figure 1. For each pair of candidates, according to the core structure of the ELECTRE method, we obtain the directional correlation coefficient between each alternative with PIS and NIS. Based on this correlation coefficient, we calculate the relative closeness coefficient to establish both the concordance sets and discordance sets. In addition, we incorporate the criteria weights into the concordance indices and construct Hamming distance-based comparison indices of discordance. The PF-ELECTRE I method involves constructing a Boolean matrix to obtain the partial outranking relationship, whereas for the PF-ELECTRE II method, we determine the complete outranking relationship by computing the overall precedence index.

Figure 1. Framework of PF-ELECTRE I and II methods.

Problem description

Let $A=\left\{A_1,A_2,\dots ,A_m\right\}\left(m\ge 2\right)$be a set of alternatives, and $C=\left\{C_1,C_2,\dots ,C_n\right\}\left(n\ge 2\right)$be a set of evaluation criteria, where the criteria weight are described as $w={\left(w_1,w_2,\dots ,w_n\right)}^T$ and fulfil the conditions: $0\le w_j\le 1$ and ${\sum }_{j=1}^n{w}_j=1$. In general, the criteria $C$ can be divided into two mutually exclusive subsets, namely, the subsets of benefit criteria $C_I$ and cost criteria $C_{\mathrm{I}\mathrm{I}}$.

Process of the PF-ELECTRE methodology

According to the above analysis, the decision-making process of our proposed PF-ELECTRE I is described as follows:

Step I-1: Construct the PFN decision matrix $P={\left(p_{ij}\right)}_{m\times n}$,

(18) $P={\left(p_{ij}\right)}_{m\times n}=\begin{array}{cc}\hspace{0.17em}& \begin{array}{cccc}{C}_1& \begin{array}{cc}\hspace{0.17em}& \hspace{0.17em}\end{array}& \begin{array}{cccc}{C}_2& \hspace{0.17em}& \cdots & \hspace{0.17em}\end{array}& C_n\end{array}\\ \begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_m\end{array}& \left[\begin{array}{cccc}\left({\mu }_{p_{11}},{\nu }_{p_{11}}\right)& \left({\mu }_{p_{12}},{\nu }_{p_{12}}\right)& \dots & \left({\mu }_{p_{1n}},{\nu }_{p_{1n}}\right)\\ \left({\mu }_{p_{21}},{\nu }_{p_{21}}\right)& \left({\mu }_{p_{22}},{\nu }_{p_{22}}\right)& \dots & \left({\mu }_{p_{2n}},{\nu }_{p_{2n}}\right)\\ ⋮& ⋮& \ddots & ⋮\\ \left({\mu }_{p_{m1}},{\nu }_{p_{m1}}\right)& \left({\mu }_{p_{i2}},{\nu }_{p_{i2}}\right)& \dots & \left({\mu }_{p_{mn}},{\nu }_{p_{mn}}\right)\end{array}\right]\end{array}$(18)

Step I-2: Identify the fixed PIS ($x_j^{+}$) and NIS ($x_j^{-}$) on every criterion:

$x_j^{+}=\left\{P_j\left(1.0,0.0,1.0,1.0\right)|x\in X\right\}\left(j=1,2,\dots ,n\right)$

$x_j^{-}=\left\{P_j\left(0.0,1.0,1.0,0.0\right)|x\in X\right\}\left(j=1,2,\dots ,n\right).$

Step I-3: Obtain the directional correlation coefficient value between each alternative $p_{ij}$ and $x_j^{+}$ and $x_j^{-}$ using Eq. (16)(16) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2{({\nu }_{P_2}(x_i))}^2+{(r_{P_1}(x_i))}^2{(r_{P_2}(x_i))}^2+{(d_{P_1}(x_i))}^2{(d_{P_2}(x_i))}^2}{\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2+{(r_{P_1}(x_i))}^2+{(d_{P_1}(x_i))}^2}\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_2}(x_i))}^2+{(r_{P_2}(x_i))}^2+{(d_{P_2}(x_i))}^2}}$(16) , as shown below:

(19) $C{C}_{ij}\left(P_{ij},x_j^{+}\right)={\sum }_{j=1}^{n}CC\left(C_j\left(p_i\right),C_j\left(x_j^{+}\right)\right)$(19)

(20) $C{C}_{ij}\left(P_{ij},x_j^{-}\right)={\sum }_{j=1}^{n}CC\left(C_j\left(p_i\right),C_j\left(x_j^{-}\right)\right)$(20)

Step I-4: Based on the correlation coefficient $CC$, we apply Eq. (17)(17) $CCC\left(P_i\right)=\frac{CC\left(P_i,P^{+}\right)}{CC\left(P_i,P^{+}\right)+CC\left(P_i,P^{-}\right)}$(17) to calculate the relative correlation-based closeness coefficient $CCC\left(P_i\right)$.

Step I-5: Based on the correlation-based closeness coefficient $CCC\left(P_i\right)$, we make pairwise comparisons between the alternatives to generate the concordance and discordance matrices. The PF concordance sets $CS$ consist of the criteria that $A_i$ is better than $A_j$, which is defined as:

(21) $CS=\left\{\begin{array}{c}CCC\left(p_{i_{1}j}\right)>CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_I\\ CCC\left(p_{i_{1}j}\right)<CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_{\mathrm{I}\mathrm{I}}\end{array}\right.\begin{array}{c}\hspace{0.17em}\\ \hspace{0.17em}\end{array}$(21)

The PF discordance sets $DS$ contain all criteria for which $A_i$ is worse than or equal to $A_j$, which is defined as:

(22) $DS=\left\{\begin{array}{c}CCC\left(p_{i_{1}j}\right)\le CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_I\\ CCC\left(p_{i_{1}j}\right)\ge CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_{\mathrm{I}\mathrm{I}}\end{array}\right.\begin{array}{c}\hspace{0.17em}\\ \hspace{0.17em}\end{array}$(22)

Step I-6: Determine the concordance index $C{I}_{CS}\left(A_i\right)$. $C{I}_{CS}\left(A_i\right)$ is calculated as the summation of the PF weights of the criteria in the concordance set as follows:

(23) $C{I}_{CS}\left(A_i\right)={\sum }_{j\ast }{w}_{j\ast }$(23)

where $j\ast$ are criteria contained in the concordance set $CS$.

Step I-7: Determine the discordance index $D{I}_{DS}\left(A_i\right)$. The Hamming distance-based $D{I}_{DS}\left(A_i\right)$ is defined as follows:

(24) $D{I}_{DS}\left(A_i\right)=\frac{{max}_{j\in DS}\left\{w_j\times d\left(A_{mj},A_{nj}\right)\right\}}{{max}_{j\in }\left\{d\left(A_{mj},A_{nj}\right)\right\}}$(24)

where $d\left(A_{mj},A_{nj}\right)$ is the distance between two PFNs defined in Eq. (10)(10) $d\left(p_1,p_2\right)=\frac{1}{4}\left(\left|{\mu }_1-{\mu }_2\right|+\left|{\nu }_1-{\nu }_2\right|+\left|r_1-r_2\right|+\left|d_1-d_2\right|\right)$(10) .

Step I-8: Calculate the concordance/discordance levels. This paper designates ${\overline{CI}}_{cs}$ as the average of the elements in the concordance matrix:

(25) ${\overline{CI}}_{CS}=\frac{{\sum }_{i=1}^m{\sum }_{j=1}^{m}C{I}_{CS}\left(A_i,A_j\right)}{m\left(m-1\right)}$(25)

Using the concordance outranking relationships, the concordance-based Boolean matrix $B_{CS}$ is constructed as follows:

(26) $B_{CS}=\left[\begin{array}{cccc}-& b_C\left(A_1,A_2\right)& \cdots & b_C\left(A_1,A_m\right)\\ b_C\left(A_2,A_1\right)& -& \cdots & b_C\left(A_2,A_m\right)\\ ⋮& ⋮& \ddots & ⋮\\ b_C\left(A_m,A_1\right)& b_C\left(A_m,A_2\right)& \cdots & -\end{array}\right]$(26)

where each entry $b_C\left(A_{i_1},A_{i_2}\right)$ (for $i_1\ne i_2$) is defined as follows:

(27) $b_C\left(A_{i_1},A_{i_2}\right)=\left\{\begin{array}{c}1\mathrm{i}\mathrm{f}C{I}_{CS}\left(A_{i_1},A_{i_2}\right)\ge {\overline{CI}}_{CS}\\ 0\mathrm{i}\mathrm{f}C{I}_{CS}\left(A_{i_1},A_{i_2}\right)<{\overline{CI}}_{CS}\hspace{0.17em}\end{array}\right.$(27)

The Hamming distance-based discordance index $D{I}_{DS}\left(A_i,A_j\right)$ reflects the degree to which alternative $A_i$ is less favorable than $A_j$. The average discordance level ${\overline{DI}}_{DS}$:

(28) ${\overline{DI}}_{DS}=\frac{{\sum }_{i=1}^m{\sum }_{j=1}^{m}D{I}_{DS}\left(A_i,A_j\right)}{m\left(m-1\right)}$(28)

Then, using the discordance outranking relationships, we generate the discordance-based Boolean matrix $B_{DS}$ as follows:

(29) $B_{DS}=\left[\begin{array}{cccc}-& b_D\left(A_1,A_2\right)& \cdots & b_D\left(A_1,A_m\right)\\ b_D\left(A_2,A_1\right)& -& \cdots & b_D\left(A_2,A_m\right)\\ ⋮& ⋮& \ddots & ⋮\\ b_D\left(A_m,A_1\right)& b_D\left(A_m,A_2\right)& \cdots & -\end{array}\right]$(29)

where each entry $b_D\left(A_i,A_j\right)$ is defined as follows:

(30) $b_D\left(A_i,A_j\right)=\left\{\begin{array}{c}1\mathrm{i}\mathrm{f}D{I}_{DS}\left(A_{i_1},A_{i_2}\right)\le {\overline{DI}}_{DS}\\ 0\mathrm{i}\mathrm{f}D{I}_{DS}\left(A_{i_1},A_{i_2}\right)>{\overline{DI}}_{DS}\hspace{0.17em}\end{array}\right.$(30)

Step I-9: Build the outranking Boolean matrix $B_O$. $B_O$ is obtained by multiplying the elements in $B_{CS}$ and $B_{DS}$ peer-to-peer as follows:

(31) $B_O=\left[\begin{array}{cccc}-& b_O\left(A_1,A_2\right)& \cdots & b_O\left(A_1,A_m\right)\\ b_O\left(A_2,A_1\right)& -& \cdots & b_O\left(A_2,A_m\right)\\ ⋮& ⋮& \ddots & ⋮\\ b_O\left(A_m,A_1\right)& b_O\left(A_m,A_2\right)& \cdots & -\end{array}\right]$(31)

where each entry $b_O\left(A_i,A_j\right)$ is obtained as follows:

(32) $b_O\left(A_i,A_j\right)=b_C\left(A_i,A_j\right)\cdot b_D\left(A_i,A_j\right)$(32)

Based on the obtained results of $B_O$, we complete the ranking process of the PF-ELECTRE I approach.

Meanwhile, this research proposes the PF-ELECTRE II approach, which employs the notions of net concordance indices (NCI), net discordance indices (NDI), as well as overall precedence indices (OPI), to determine a comprehensive ranking order. The first seven steps remain consistent with the PF-ELECTRE I method, and the next steps are as follows.

Step II-8: Determine the $NCI\left(A_i\right)$ and $NDI\left(A_i\right)$ as follows:

(33) $NCI\left(A_i\right)=\frac{{\sum }_{i=1,i_2\ne 1}^{n}C{I}_{CS}\left(A_i,A_{i_2}\right)}{m-1}$(33)

(34) $NDI\left(A_i\right)=\frac{{\sum }_{i=1,i_2\ne 1}^{n}D{I}_{DS}\left(A_i,A_{i_2}\right)}{m-1}$(34)

Step II-9: Construct the overall precedence index$OPI\left(A_i\right)$. In this step, $OPI\left(A_i\right)$ is calculated as follows:

(35) $OPI\left(A_i\right)=NCI\left(A_i\right)\cdot NDI\left(A_i\right)$(35)

Step II-10: Rank the alternative. The overall preference orders are determined in descending order of their $OPI\left(A_i\right)$ values.

Case study and comparative analysis

In this section, we demonstrate the effectiveness and practicability of the proposed PF-ELECTRE approach through its application to a real-world problem of cotton fabric selection. To further demonstrate the effectiveness of our approach, we conducted sensitivity and comparative analyses.

Data description

The case study described by Alam and Ghosh (2013) aims to select the best cotton fabric from a collection of 13 cotton fabrics based on four physical properties: fabric cover (FC), thickness (FT), areal density (GSM), and porosity (FP). According to Alam and Ghosh (2013), the warp and weft counts, threads density, FT and GSM are given. the areal density of all fabric samples has been assessed by using a standard GSM cutter. Warp and weft thread densities were assessed using a pick glass. Meanwhile, the warp and weft counts were measured with the help of a Beesely yarn balance. In addition, the FC was determined by using the following Eqs. (36)(36) $FC=n_1{d}_1+n_2{d}_2-n_1{n}_2{d}_1{d}_2,$(36) and (37(37) $d=\frac{1}{28\sqrt{Ne}}.$(37) ). Moreover, the FP was obtained by using Eq. (38)(38) $FP=1-G/\rho h$(38) .

(36) $FC=n_1{d}_1+n_2{d}_2-n_1{n}_2{d}_1{d}_2,$(36)

(37) $d=\frac{1}{28\sqrt{Ne}}.$(37)

where n=threads/inch, subscripts 1 and 2 denote the warp and weft directions, respectively. Moreover, the variable d refers to the yarn diameter in inch, and Ne refers to the yarn count.

(38) $FP=1-G/\rho h$(38)

where G refers to the fabric GSM, ρ is the fiber density, and h is the thickness. In addition, the fiber density of cotton was assigned to the 1.5 × 10⁶ g/cm³.

As all the criteria directly impact the thermal comfort of clothing in cold climates, they are all benefit-type criteria. The crisp dataset was transformed into a PFS decision matrix by Ye and Chen (2022b). The transformation process makes the values of each property into nine grades, which corresponding to the nine-point Pythagorean fuzzy linguistic scale and simulates the evaluation process of decision makers. By applying Eqs. (3)(3) ${\mu }_P=r_{P}cos\theta$(3) , (4(4) ${\nu }_P=r_{P}sin\theta$(4) ) and (5(5) ${\mu }_P^2+{\nu }_P^2=r_P^2$(5) ), we can easily obtain the corresponding $r_P$ and $d_P$. Table 3 shows the detailed PFN matrix.

Table 3. The PFN decision matrix and corresponding closeness coefficient.

Sample	FC	$CCC\left(p_{ij}\right)$	FT	$CCC\left(p_{ij}\right)$	GSM	$CCC\left(p_{ij}\right)$	FP	$CCC\left(p_{ij}\right)$
	$p_{ij}\left(=\left({\mu }_{ij},{\nu }_{ij},r_{ij},d_{ij}\right)\right)$		$p_{ij}\left(=\left({\mu }_{ij},{\nu }_{ij},r_{ij},d_{ij}\right)\right)$		$p_{ij}\left(=\left({\mu }_{ij},{\nu }_{ij},r_{ij},d_{ij}\right)\right)$		$p_{ij}\left(=\left({\mu }_{ij},{\nu }_{ij},r_{ij},d_{ij}\right)\right)$
A1	(0.5000,0.8000,0.9434,0.3556)	0.4033	(0.2500,0.9200,0.9534,0.1689)	0.3175	(0.1000,0.9900,0.9950,0.0641)	0.2939	(1.0000,0.0000,1.0000,1.0000)	0.7101
A2	(0.7000,0.6000,0.9220,0.5487)	0.5255	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.5000,0.8000,0.9434,0.3556)	0.4033
A3	(1.0000,0.0000,1.0000,1.0000)	0.7101	(1.0000,0.0000,1.0000,1.0000)	0.7101	(1.0000,0.0000,1.0000,1.0000)	0.7101	(0.4000,0.8700,0.9575,0.2744)	0.3598
A4	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.6000,0.7100,0.9296,0.4467)	0.4593
A5	(0.2500,0.9200,0.9534,0.1689)	0.3175	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.4000,0.8700,0.9575,0.2744)	0.3598
A6	(0.4000,0.8700,0.9575,0.2744)	0.3598	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9700,0.9751,0.0654)	0.2941
A7	(0.4000,0.8700,0.9575,0.2744)	0.3598	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.5000,0.8000,0.9434,0.3556)	0.4033
A8	(0.4000,0.8700,0.9575,0.2744)	0.3598	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.1000,0.9900,0.9950,0.0641)	0.2939	(0.4000,0.8700,0.9575,0.2744)	0.3598
A9	(0.5000,0.8000,0.9434,0.3556)	0.4033	(0.2500,0.9200,0.9534,0.1689)	0.3175	(0.4000,0.8700,0.9575,0.2744)	0.3598	(0.1000,0.9900,0.9950,0.0641)	0.2939
A10	(0.6000,0.7100,0.9296,0.4467)	0.4593	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.1000,0.9900,0.9950,0.0641)	0.2939
A11	(1.0000,0.0000,1.0000,1.0000)	0.7101	(0.2500,0.9200,0.9534,0.1689)	0.3175	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.6000,0.7100,0.9296,0.4467)	0.4593
A12	(1.0000,0.0000,1.0000,1.0000)	0.7101	(0.4000,0.8700,0.9575,0.2744)	0.3598	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.7000,0.6000,0.9220,0.5489)	0.5255
A13	(1.0000,0.0000,1.0000,1.0000)	0.7101	(0.2500,0.9200,0.9534,0.1689)	0.3175	(0.1000,0.9700,0.9751,0.0654)	0.2941	(0.7000,0.6000,0.9220,0.5489)	0.5255

Decision procedure via the proposed method

This subsection utilizes the suggested PF-ELECTRE approaches to address the above MCDM problem.

First, we apply the PF-ELECTRE I methodology to address the problem. The detailed procedure is outlined as follows:

Step I-1: Calculate the criteria weight by the AHP technique. The criteria weights, as indicated by Alam and Ghosh (2013), are determined to be $w={\left(0.221,0.322,0.186,0.271\right)}^T$.

Step I-2: Identify the PF-PIS $x^{+}$ and PF-NIS $x^{-}$. We set the corresponding $x^{+}$ and $x^{-}$ as the fixed number 1 or 0 as follows:

$x^{+}=\left\{\left(1.0,0.0,1.0,1.0\right),\left(1.0,0.0,1.0,1.0\right),\left(1.0,0.0,1.0,1.0\right),\left(1.0,0.0,1.0,1.0\right)\right\}$

$x^{-}=\left\{\left(0.0,1.0,1.0,0.0\right),\left(0.0,1.0,1.0,0.0\right),\left(0.0,1.0,1.0,0.0\right),\left(0.0,1.0,1.0,0.0\right)\right\}$

Step I-3: By applying Eq. (16)(16) $CC(P_1,P_2)=\frac{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2{({\nu }_{P_2}(x_i))}^2+{(r_{P_1}(x_i))}^2{(r_{P_2}(x_i))}^2+{(d_{P_1}(x_i))}^2{(d_{P_2}(x_i))}^2}{\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_1}(x_i))}^2+{({\nu }_{P_1}(x_i))}^2+{(r_{P_1}(x_i))}^2+{(d_{P_1}(x_i))}^2}\sqrt{{{\sum }_{i=1}}^n{({\mu }_{P_2}(x_i))}^2+{({\nu }_{P_2}(x_i))}^2+{(r_{P_2}(x_i))}^2+{(d_{P_2}(x_i))}^2}}$(16) , we compute the relative directional correlation coefficient of the alternative between PF-PIS and PF-NIS.

Step I-4. According to Eq. (17)(17) $CCC\left(P_i\right)=\frac{CC\left(P_i,P^{+}\right)}{CC\left(P_i,P^{+}\right)+CC\left(P_i,P^{-}\right)}$(17) , we calculate the correlation-based closeness coefficient. Table 3 presents the detailed results.

Step I-5. Compute the concordance and discordance sets using Eqs. (21)(21) $CS=\left\{\begin{array}{c}CCC\left(p_{i_{1}j}\right)>CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_I\\ CCC\left(p_{i_{1}j}\right)<CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_{\mathrm{I}\mathrm{I}}\end{array}\right.\begin{array}{c}\hspace{0.17em}\\ \hspace{0.17em}\end{array}$(21) and (22(22) $DS=\left\{\begin{array}{c}CCC\left(p_{i_{1}j}\right)\le CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_I\\ CCC\left(p_{i_{1}j}\right)\ge CCC\left(p_{i_{2}j}\right)\mathrm{f}\mathrm{o}\mathrm{r}{c}_j\in C_{\mathrm{I}\mathrm{I}}\end{array}\right.\begin{array}{c}\hspace{0.17em}\\ \hspace{0.17em}\end{array}$(22) ), respectively. The results of $CS$ and $DS$ are presented in Tables 4 and 5.

Table 4. Results of closeness coefficient-based concordance sets.

Sample	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13
A1	—	{2,4}	{4}	{1,2,4}	{1,2,4}	{1,2,4}	{1,2,4}	{1,2,4}	{4}	{2,4}	{4}	{4}	{4}
A2	{1}	—	{4}	{1}	{1,2,4}	{1,2,4}	{1,2}	{1,2,4}	{1,4}	{1,4}	Ø	Ø	Ø
A3	{1,2,3}	{1,2,3}	—	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{2,3}	{2,3}	{2,3}
A4	Ø	{4}	{4}	—	{2,4}	{2,4}	{2,4}	{2,4}	{4}	{4}	Ø	Ø	Ø
A5	Ø	Ø	Ø	{1}	—	{4}	Ø	Ø	{4}	{4}	Ø	Ø	Ø
A6	Ø	Ø	Ø	{1}	{1}	—	Ø	Ø	{4}	{4}	Ø	Ø	Ø
A7	Ø	Ø	{4}	{1}	{1,4}	{4}	—	{4}	{4}	{4}	Ø	Ø	Ø
A8	Ø	Ø	Ø	{1}	{1}	{4}	Ø	—	{4}	{4}	Ø	Ø	Ø
A9	{3}	{2,3}	Ø	{1,2,3}	{1,2,3}	{1,2,3}	{1,2,3}	{1,2,3}	—	{2,3}	{3}	{3}	{3}
A10	{1,3}	{3}	Ø	{1,3}	{1,2,3}	{1,2,3}	{1,2,3}	{1,2,3}	{1}	—	Ø	Ø	Ø
A11	{1,3}	{1,2,3,4}	{4}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,4}	{1,2,4}	—	Ø	Ø
A12	{1,2,3}	{1,2,3,4}	{4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,4}	{1,2,4}	{2,4}	—	{2}
A13	{1,3}	{1,2,3,4}	{4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{1,4}	{1,2,4}	{4}	Ø	—

Table 5. Results of closeness coefficient-based discordance sets.

Sample	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13
A1	—	{1,3}	{1,2,3}	{3}	{3}	{3}	{3}	{3}	{1,2,3}	{1,3}	{1,2,3}	{1,2,3}	{1,2,3}
A2	{2,3,4}	—	{1,2,3}	{2,3,4}	{3}	{3}	{3,4}	{3}	{2,3}	{2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A3	{4}	{4}	—	{4}	{4}	Ø	{4}	{4}	Ø	Ø	{1,4}	{1,4}	{1,4}
A4	{1,2,3,4}	{1,2,3}	{1,2,3}	—	{1,3}	{1,3}	{1,3}	{1,3}	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A5	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{2,3,4}	—	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A6	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{2,3,4}	{2,3,4}	—	{1,2,3,4}	{1,2,3,4}	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A7	{1,2,3,4}	{1,2,3,4}	{1,2,3}	{2,3,4}	{2,3}	{1,2,3}	—	{1,2,3}	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A8	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}	{2,3,4}	{2,3,4}	{1,2,3}	{1,2,3,4}	—	{1,2,3}	{1,2,3}	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A9	{1,2,4}	{1,4}	{1,2,3,4}	{4}	{4}	{4}	{4}	{4}	—	{1,4}	{1,2,4}	{1,2,4}	{1,2,4}
A10	{2,4}	{1,2,4}	{1,2,3,4}	{2,4}	{4}	{4}	{4}	{4}	{2,3,4}	—	{1,2,3,4}	{1,2,3,4}	{1,2,3,4}
A11	{2,4}	Ø	{1,2,3}	{4}	Ø	Ø	Ø	Ø	{2,3}	{3}	—	{1,2,3,4}	{1,2,3,4}
A12	{4}	Ø	{1,2,3}	Ø	Ø	Ø	Ø	Ø	{3}	{3}	{1,3}	—	{1,3,4}
A13	{2,4}	Ø	{1,2,3	Ø	Ø	Ø	Ø	Ø	{2,3}	{3}	{1,2,3}	{1,2,3,4}	—

Step I-6: Calculate the concordance index $C{I}_{CS}$ and construct the concordance matrix. The concordance matrix, consisting of concordance indices calculated by employing Eq. (23)(23) $C{I}_{CS}\left(A_i\right)={\sum }_{j\ast }{w}_{j\ast }$(23) , is shown in Table 6.

Table 6. Results of concordance indices and discordance indices.

Outcome of concordance indices
Sample	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13	Average
A1	—	0.5930	0.2710	0.8140	0.8140	0.8140	0.8140	0.8140	0.2710	0.5930	0.2710	0.2710	0.2710	0.5509
A2	0.2210	—	0.2710	0.2210	0.8140	0.8140	0.5430	0.8140	0.4920	0.4920	0.0000	0.0000	0.0000	0.3902
A3	0.7290	0.7290	—	0.7290	0.7290	1.0000	0.7290	0.7290	1.0000	1.0000	0.5080	0.5080	0.5080	0.7415
A4	0.0000	0.2710	0.2710	—	0.5930	0.5930	0.5930	0.5930	0.2710	0.2710	0.0000	0.0000	0.0000	0.2880
A5	0.0000	0.0000	0.0000	0.2210	—	0.2710	0.0000	0.0000	0.2710	0.2710	0.0000	0.0000	0.0000	0.0862
A6	0.0000	0.0000	0.0000	0.2210	0.2210	—	0.0000	0.0000	0.2710	0.2710	0.0000	0.0000	0.0000	0.0820
A7	0.0000	0.0000	0.2710	0.2210	0.4920	0.2710	—	0.2710	0.2710	0.2710	0.0000	0.0000	0.0000	0.1723
A8	0.0000	0.0000	0.0000	0.2210	0.2210	0.2710	0.0000	—	0.2710	0.2710	0.0000	0.0000	0.0000	0.1046
A9	0.1860	0.5080	0.0000	0.7290	0.7290	0.7290	0.7290	0.7290	—	0.5080	0.1860	0.1860	0.1860	0.4504
A10	0.4070	0.1860	0.0000	0.4070	0.7290	0.7290	0.7290	0.7290	0.2210	—	0.0000	0.0000	0.0000	0.3448
A11	0.4070	1.0000	0.2710	0.7290	1.0000	1.0000	1.0000	1.0000	0.4920	0.8140	—	0.0000	0.0000	0.6428
A12	0.7290	1.0000	0.2710	1.0000	1.0000	1.0000	1.0000	1.0000	0.8140	0.8140	0.5930	—	0.3220	0.7953
A13	0.4070	1.0000	0.2710	1.0000	1.0000	1.0000	1.0000	1.0000	0.4920	0.8140	0.2710	0.0000	—	0.6879
Outcome of discordance indices
Sample	A1	A2	A3	A4	A5	A6	A7	A8	A9	A10	A11	A12	A13	Average
A1	—	0.2716	1.1592	0.0000	0.0000	0.0000	0.0000	0.0000	0.1755	0.0921	0.8840	0.8840	0.8840	1.0014
A2	1.0840	—	1.2874	0.2065	0.0000	0.0000	0.0000	0.0000	0.5324	0.0329	0.8840	0.8840	0.8840	0.5968
A3	0.8570	0.1016	—	0.2145	0.0000	0.0000	0.1016	0.0000	0.0000	0.0000	0.2147	0.3372	0.3372	1.2333
A4	1.0840	0.8840	1.2874	—	0.5782	0.4974	0.8840	0.8840	0.6717	0.8840	0.8840	0.8840	0.8840	0.5121
A5	1.0840	0.8840	1.2880	1.0840	—	0.4369	0.9292	0.8840	0.7502	0.8840	0.8840	0.8840	0.8840	0.2989
A6	1.0840	1.0840	1.2880	1.0840	1.0840	—	1.0840	1.0840	0.7440	0.8840	0.8840	0.8840	0.8840	0.0901
A7	1.0840	0.8840	1.2880	0.4787	0.0000	0.0000	—	0.0000	0.5324	0.5308	0.8840	0.8840	0.8840	0.5209
A8	1.0840	0.8840	1.2880	0.9096	0.0000	0.0000	1.0840	—	0.7440	0.7417	0.8840	0.8840	0.8840	0.4183
A9	1.0840	1.0840	1.2880	1.0840	1.0840	0.0669	1.0840	1.0840	—	0.4161	0.8840	0.8840	0.8840	0.4661
A10	1.0840	1.0840	1.2880	1.0840	0.8402	0.0797	1.0840	1.0840	0.7440	—	0.8840	0.9678	0.9678	0.3721
A11	0.9392	0.0000	1.1598	0.0000	0.0000	0.0000	0.0000	0.0000	0.2330	0.0000	—	1.2469	1.0840	0.6809
A12	0.7742	0.0000	1.0188	0.0000	0.0000	0.0000	0.0000	0.0000	0.2330	0.0000	0.0000	—	0.0000	0.9093
A13	0.7742	0.0000	1.1598	0.0000	0.0000	0.0000	0.0000	0.0000	0.2330	0.0000	0.0000	1.2880	—	0.7884

Step I-7: Calculate the discordance index $D{I}_{DS}$ and construct the discordance matrix. $D{I}_{DS}$ is calculated using Eq. (24)(24) $D{I}_{DS}\left(A_i\right)=\frac{{max}_{j\in DS}\left\{w_j\times d\left(A_{mj},A_{nj}\right)\right\}}{{max}_{j\in }\left\{d\left(A_{mj},A_{nj}\right)\right\}}$(24) ; the distance involved in $D{I}_{DS}$ is calculated by employing Eq. (10)(10) $d\left(p_1,p_2\right)=\frac{1}{4}\left(\left|{\mu }_1-{\mu }_2\right|+\left|{\nu }_1-{\nu }_2\right|+\left|r_1-r_2\right|+\left|d_1-d_2\right|\right)$(10) , which are shown in Table 6.

Step I-8: Computation of concordance level ${\overline{CI}}_{cs}$ and discordance level ${\overline{DI}}_{DS}$. According to Eqs. (25)(25) ${\overline{CI}}_{CS}=\frac{{\sum }_{i=1}^m{\sum }_{j=1}^{m}C{I}_{CS}\left(A_i,A_j\right)}{m\left(m-1\right)}$(25) and (28(28) ${\overline{DI}}_{DS}=\frac{{\sum }_{i=1}^m{\sum }_{j=1}^{m}D{I}_{DS}\left(A_i,A_j\right)}{m\left(m-1\right)}$(28) ), we calculate the concordance and discordance cutoff levels and then construct $B_{CS}$ and $B_{DS}$ by using Eqs. (27)(27) $b_C\left(A_{i_1},A_{i_2}\right)=\left\{\begin{array}{c}1\mathrm{i}\mathrm{f}C{I}_{CS}\left(A_{i_1},A_{i_2}\right)\ge {\overline{CI}}_{CS}\\ 0\mathrm{i}\mathrm{f}C{I}_{CS}\left(A_{i_1},A_{i_2}\right)<{\overline{CI}}_{CS}\hspace{0.17em}\end{array}\right.$(27) and (30(30) $b_D\left(A_i,A_j\right)=\left\{\begin{array}{c}1\mathrm{i}\mathrm{f}D{I}_{DS}\left(A_{i_1},A_{i_2}\right)\le {\overline{DI}}_{DS}\\ 0\mathrm{i}\mathrm{f}D{I}_{DS}\left(A_{i_1},A_{i_2}\right)>{\overline{DI}}_{DS}\hspace{0.17em}\end{array}\right.$(30) ), respectively, as follows:

${\text{rmB}}_{CS}=\left[\begin{array}{c}-101111101000\\ 0-00111111000\\ 11-1111111111\\ 000-111100000\\ 0000-00000000\\ 00000-0000000\\ 000010-000000\\ 0000000-00000\\ 01011111-1000\\ 000011110-000\\ 0101111111-00\\ 11011111111-0\\ 010111111100-\end{array}\right],{\text{rmB}}_{DS}=\left[\begin{array}{c}-101111111000\\ 0-01111111000\\ 01-1111111111\\ 000-110000000\\ 0000-10000000\\ 00000-0000000\\ 000111-111000\\ 0000110-00000\\ 00000100-1000\\ 000001000-000\\ 0101111111-00\\ 01011111111-1\\ 010111111110-\end{array}\right]$

Step I-9: Construct the aggregated outranking matrix. The effective outranking Boolean matrix $B_o$ is derived using Eq. (32)(32) $b_O\left(A_i,A_j\right)=b_C\left(A_i,A_j\right)\cdot b_D\left(A_i,A_j\right)$(32) as follows:

$B_o=\left[\begin{array}{c}-101111101000\\ 0-00111111000\\ 01-1111111111\\ 000-111100000\\ 0000-00000000\\ 00000-0000000\\ 000010-000000\\ 0000000-00000\\ 01011111-1000\\ 000011110-000\\ 0101111111-00\\ 11011111111-0\\ 010111111100-\end{array}\right]$

Based on the outcome in $B_O$, we find that Sample $A_3$ is the best alternative since it is not outranked by any other alternatives, which is consistent with the best alternative obtained by Alam and Ghosh (2013).

Next, we will apply the PF-ELECTRE II approach to solve the same problem. Note that Steps II-1 to II-7 are identical to Steps I-1 to I-7.

Step II-8: Compute the net concordance/discordance indices. By employing Eqs. (33)(33) $NCI\left(A_i\right)=\frac{{\sum }_{i=1,i_2\ne 1}^{n}C{I}_{CS}\left(A_i,A_{i_2}\right)}{m-1}$(33) and (34(34) $NDI\left(A_i\right)=\frac{{\sum }_{i=1,i_2\ne 1}^{n}D{I}_{DS}\left(A_i,A_{i_2}\right)}{m-1}$(34) ), we obtain the $NCI\left(A_i\right)$and $NDI\left(A_i\right)$ presented in the last column and row of Tables 6.

Step II-9: Calculate the overall precedence indices. By employing Eq. (35)(35) $OPI\left(A_i\right)=NCI\left(A_i\right)\cdot NDI\left(A_i\right)$(35) , we calculate the overall precedence indices as: $OPI\left(A_1\right)=0.5517$,$OPI\left(A_2\right)=0.2328$,$OPI\left(A_3\right)=0.9145$, $OPI\left(A_4\right)=0.1475$, $OPI\left(A_5\right)=0.0258$, $OPI\left(A_6\right)=0.0074$, $OPI\left(A_7\right)=0.0898$, $OPI\left(A_8\right)=0.0438$, $OPI\left(A_9\right)=0.2099$, $OPI\left(A_{10}\right)=0.1283$, $OPI\left(A_{11}\right)=0.4376$, $OPI\left(A_{12}\right)=0.7231$, and $OPI\left(A_{13}\right)=0.5424$.

Step II-10: Rank the alternatives. The alternatives are ranked in the descending order of their precedence indices as follows: $A_3>A_{12}>A_1>A_{13}>A_{11}>A_2>A_9>A_4>A_{10}>A_7>A_8>A_5>A_6$.

The overall ranking order obtained from PF-ELECTRE II method indicates that alternative $A_3$ is the best, while alternative $A_6$ is the worst, which is consistent with the findings of Alam and Ghosh (2013). This reinforces the efficacy of the proposed approach. Furthermore, the proposed PF-ELECTRE method provides more detailed information about the ranking order, especially the partial outranking relationship that can be obtained through PF-ELECTRE I. Moreover, the proposed method entails easy calculations, thus making it a practical and efficient approach for MCDM problems.

Sensitivity analysis

The aim of the sensitivity analysis is to assess the effect of the most influential criterion on the ranking performance of the model when it is increased or decreased. It can provide valuable insights for enhancing decision-making processes, and make them more effective and reliable. Therefore, this subsection performs a sensitivity analysis to investigate the impact of changes in criteria weights on the final rankings and assess the robustness and stability of the proposed approach under various models.

To conduct the criteria weight change analysis, we adjust the weight of best criterion deviation ranges from 0.1 to 0.9 with a 10% variation. The weights of other criteria are proportionally corrected. The outcomes of criteria weight change analysis are presented in Table 7 which show that the criterion of FP maintains the first order when the criterion ranks from 0.1 to 0.2, while the original best criterion FT falls to the worst rank. In addition, when $w\ge 0.3$, the original best criterion of FT remains the first rank and the other criteria remains the same ranking order. These findings indicate that the variations in criteria weights did not significantly impact the overall criteria rankings.

Table 7. Sensitivity analysis and ranking results of AHP weights and decision alternatives.

Main criteria	Original	Variation in weight of the best criterion
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
FT	0.322(1)	0.100(4)	0.200(4)	0.300(1)	0.400(1)	0.500(1)	0.600(1)	0.700(1)	0.800(1)	0.900(1)
FC	0.221(3)	0.293(2)	0.261(2)	0.228(3)	0.196(3)	0.163(3)	0.130(3)	0.098(3)	0.065(3)	0.033(3)
GSM	0.186(4)	0.247(3)	0.219(3)	0.192(4)	0.165(4)	0.137(4)	0.110(4)	0.082(4)	0.055(4)	0.027(4)
FP	0.271(2)	0.360(1)	0.320(1)	0.280(2)	0.240(2)	0.200(2)	0.160(2)	0.120(2)	0.080(2)	0.040(2)
Sorting results of decision alternatives
A1	4	4	3	3	3	3	4	5	5	4
A2	6	6	6	7	7	7	7	7	7	6
A3	3	3	1	1	1	1	1	1	1	3
A4	8	8	8	8	8	8	8	8	8	8
A5	12	12	12	12	12	12	12	12	12	12
A6	13	13	13	13	13	13	13	13	13	13
A7	10	10	10	10	10	10	10	10	10	10
A8	11	11	11	11	11	11	11	11	11	11
A9	7	7	7	6	6	4	3	3	3	7
A10	9	9	9	9	9	9	9	9	9	9
A11	5	5	5	5	5	6	6	6	6	5
A12	1	1	2	2	2	2	2	2	2	1
A13	2	2	4	4	4	5	5	4	4	2

After generating the criteria weight changes, we analyzed the corresponding influence on the overall ranking of alternatives. Table 7 summarizes the results by presenting the ranking orders of each alternative under different conditions. When $w\le 0.2$, there exists steady judgment of optimal alternative $A_{12}$, the worst alternative remains $A_6$, and the corresponding ranking order is $A_{12}>A_{13}>A_3>A_1>A_{11}>A_2>A_9>A_4>A_{10}>A_7>A_8>A_5>A_6$. When $w\ge 0.8$, it indicates that $A_3$ is identified as the most optimal cotton fabric, the identification of the worst alternative $A_6$ remains stable, and the overall ranking order remains steady as $A_3>A_{12}>A_9>A_{13}>A_1>A_{11}>A_2>A_4>A_{10}>A_7>A_8>A_5>A_6$. When $0.3\le w\le 0.7$, the alternative $A_3$ has the first rank and alternative $A_6$ has the last. Moreover, alternative $A_9$ has dramatic ranking changes, it moved up to seventh, sixth, fourth and third rankings in the ranges of [0.1, 0.3], [0.4, 0.5], [0.6, 0.7), [0.7, 1.0), respectively. In addition, samples $A_4$, $A_5$, $A_6$, $A_7$, $A_8$ and $A_{10}$ maintain their rankings in all scenarios. According to Table 7, it can be inferred that changes in the criteria weights have a minimal effect on the PF-ELECTRE method. These changes observed are insufficient to impact the ranking order of the alternatives, implying that the alternatives have clearly defined mutual advantages, and the ranking order remains stable and reliable.

Comparative analysis

In this subsection, we conduct a comparative analysis with different methods under different decision-making environments. The first set of methods are traditional MCDM approaches with crisp values, including WPM, TOPSIS, EDAS, and MOORA. The second set of methods are PFS methods, including PF-WPM, PF-TOPSIS, and the proposed PF-ELECTRE. Various scholars have applied the TOPSIS, EDAS, MOORA and PF-TOPSIS methods to solve the same cotton fabric selection issue. WPM is a typical scoring method and is an effective approach to solve single and multi-dimensional MCDM problems that uses multiplication to rank alternatives instead of addition. TOPSIS and EDAS are typical compromising methods. TOPSIS evaluates the ideal solution that is close to the PIS and farthest from the NIS. Similar to the TOPSIS method, EDAS calculates the positive distance and negative distance from the average solution. MOORA is based on simple ratio analysis and reference point. It assists decision-makers in optimizing multiple conflicting criteria or objectives while adhering to specific constraints. All these methods have the notable advantages with their simplicity. Additionally, all the PF-WPM, PF-TOPSIS and PF-ELECTRE methods are extensions of traditional MCDM methods in the PFS context. PF-WPM employs the PF weighted geometric aggregation operator and applies the score function to compare PFNs. PF-TOPSIS is an extension of TOPSIS under the PFS environment, the concepts of closeness coefficient and distance measurements are applied in the calculation process of the PF-TOPSIS approach.

The obtained ranking results are presented in Table 8, and Figure 2 illustrates the ranking order of cotton fabric selection and the results of comparative analyses. Despite some fluctuations in the ranking of middle-order alternatives, the results of this analysis remain consistent enough to identify the same best alternative A₃ and worst alternative A₆ across all methods. Samples A₅ and A₈ have the same ranking in all the methods. The results further demonstrate the advantage of scientific validity of the developed approach while dealing with decision-making scenarios. Although TOPSIS, EDAS and MOORA methods have the importance advantages with their simplicity, they exist distinct limitations that they rely on aggregating function that measures closeness to a reference point. The selection of different reference point will affect the ranking orders, and they cannot deal with the uncertainty. In addition, these methods could not provide the superiority relationships between two alternatives. Compared to the compromising methods that rely on approaching an ideal solution, compromise solutions are reached through mutual concessions. The results obtained by PF-ELECTRE II ensures all relevant information is considered, and the structured decision-making process is consistent and repeatable. In addition, the proposed PF-ELECTRE method enables pairwise comparisons of multiple criteria. This allows decision-makers to accurately understand the outranking relationship between two alternatives under a specific criterion. Finally, the PF-ELECTRE II produce a more accurate overall outranking relations of alternatives. The PF-ELECTRE II calculation process is straightforward and easy to implement. Nevertheless, as the number of criteria and alternatives rises, the calculations involved become more complex and this increases the probability of errors. Additionally, the PF-ELECTRE approach requires a high level of expertise from decision-makers. They should have sufficient knowledge or training to utilize the method effectively.

Figure 2. Sorting results of cotton fabric selection by different approaches.

Table 8. Comparison results of various methods for the problem of cotton fabric selection.

Methods Sample	WPM	TOPSIS	EDAS	MOORA	PF-WPM	PF-TOPSIS	PF-ELECTRE II
		(Chakraborty and Chatterjee 2017)	(Mitra 2022b)	(Mitra 2022a)		(Ye and Chen 2022b)
A1	6	6	6	5	5	5	3
A2	8	9	8	8	6	6	6
A3	1	1	1	1	1	1	1
A4	7	7	7	7	10	9	8
A5	12	12	12	12	11	12	12
A6	13	13	13	13	13	13	13
A7	10	10	10	10	8	8	10
A8	11	11	11	11	9	11	11
A9	5	2	5	6	7	7	7
A10	9	8	9	9	12	10	9
A11	4	5	4	4	4	4	5
A12	2	3	2	2	2	2	2
A13	3	4	3	3	3	3	4
Spearman rank correlation results
WPM	——	0.962	1.000	0.995	0.901	0.951	0.945
TOPSIS		——	0.962	0.940	0.802	0.863	0.874
EDAS			——	0.995	0.901	0.951	0.945
MOORA				——	0.912	0.962	0.967
PF-WPM					——	0.973	0.923
PF-TOPSIS						——	0.967
PF-ELECTRE II							——

To ensure objectivity in our assessment, we conducted a Spearman rank correlation analysis to evaluate the correlation coefficients between the compared methods, as presented in Table 8. The results indicate that, in all cases, the Spearman rank correlation coefficient is greater than 0.8. This suggests that the proposed PF-ELECTRE II method has a strong rank correlation with other methods. The WPM and EDAS displays the highest correlation coefficient of 1, while the TOPSIS and PF-WPM methods have the least correlation coefficient of 0.802. The proposed PF-ELECTRE II method has a high correlation with other methods, indicating the proposed ranking is validated and credible. Both the MOORA and PF-TOPSIS between the PF-ELECTRE methods have the same correlation coefficient of 0.967. Compared to the correlation between PF-TOPSIS and other methods, the proposed PF-ELECTRE has a higher correlation than PF-TOPSIS, indicating that the proposed method can obtain reliable results under the PFS environment. Therefore, the proposed method can serve as a valuable tool for making perceptive and logical decision when selecting cotton fabric in textile industry. In addition, our findings reveal that the correlations between methods with crisp values are higher than those under the PFSs environment. This could be attributed to the information loss that occurs during the transformation of crisp dataset to PFNs. As alternatives may have distinct attribute values for the same criterion, these values could be placed in the same interval when transformed into PFNs, potentially resulting in a loss of information. In light of the above analysis, we could conclude that the proposed PF-ELECTRE method is capable of generating reasonable results and providing valuable information to aid in the selection of cotton fabric.

Conclusion

In this research, we extended the ELECTRE I and II methods into the PFS environment, constituting the novel PF-ELECTRE method. We constructed PF concordance and discordance sets by utilizing the directional correlation coefficient and correlation-based closeness coefficient. Subsequently, we derived PF concordance and discordance matrices, which were utilized to formulate strong and weak outranking relations. Ultimately, we constructed the aggregated outranking matrix to evaluate the alternatives and establish their overall ranking. We illustrated the features and validity of the proposed method through a numerical case involving the selection of cotton fabric. To validate the results, we conducted a sensitivity analysis to verify that the proposed method has stability. Additionally, we conducted a comparative analysis of the proposed method in comparison to existing methods, and the results demonstrated that our proposed approach has excellent efficacy. Moreover, Spearman correlation analysis demonstrates that the proposed method outperforms other methods, and the results for cotton fabric selection are rational. Through different sensitivity and comparative analyses, it demonstrated that the PF-ELECTRE approach has good stability in the assessment of the MCDM problem. Furthermore, this approach is adaptable to a broader range of MCDM issues in the textile industry with ease.

Some limitations of this study are as follows: (1) it only employs the Hamming distance as a distance measurement for PFSs, whereas there are numerous other distance measurements that could be applied. (2) This study applies the PFSs to evaluate alternatives. However, given the growing complexity of decision-making processes, it may be necessary to consider other fuzzy sets as well. (3) The proposed approach was only applied to the selection of cotton fabric in this study, and further investigation is needed to determine its applicability to other textile MCDM problems.

For future work, we intend to expand our study by applying other distance measurements, such as Euclidean distance, Chebyshev distance, and Minkowski distance. Additionally, we aim to extend our approach to other types of fuzzy sets, including hesitant Pythagorean fuzzy sets (HPFSs) and q-rung orthopair fuzzy sets (q-ROFs). To enhance its practicability of our proposed method, we will also explore its potential for addressing other MCDM issues in the textile industry, including textile green supply chain selection, textile waste management.

• Extend the ELECTRE method under Pythagorean fuzzy environment named PF-ELECTRE.

• Apply the correlation-based closeness coefficient to compare the PFNs.

• A novel PF-ELECTRE approach is proposed for the selection of cotton fabrics.

• Conduct a comparative analysis between several different MCDM methods.

• Comparison results show the flexibility and effectiveness of the proposed approach.

Compliance with ethical standards

This article does not contain any studies with human participants or animals that were performed by any of the authors.

Acknowledgements

The authors acknowledge the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The corresponding author is grateful for grant funding support from the National Science and Technology Council, Taiwan (NSTC 111-2410-H-182-012-MY3) and Chang Gung Memorial Hospital, Linkou, Taiwan (BMRP 574) during the completion of this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Science and Technology Council, Taiwan [NSTC 111-2410-H-182-012-MY3]; Chang Gung Memorial Hospital, Linkou, Taiwan [BMRP 574].