Selection of Knitted Fabrics Using a Hybrid BBWM-PFTOPSIS Method

ABSTRACT

Selecting the best knitted fabric with various comfort properties is considered a complicated multi-criteria decision-making (MCDM) issue that involves ambiguity and vagueness. In such scenarios, Pythagorean fuzzy sets (PFSs) provide an effective tool for addressing uncertainty and ambiguity in MCDM problems that contain human subjective evaluations and judgments. First, this research identifies the factors affecting the comfort of knitted fabrics as the evaluation criteria. Second, the Bayesian best-worst method (BBWM) is preferred for less pairwise comparisons and obtains highly reliable results with a probabilistic perspective for determining the criteria weights. Furthermore, due to its logical computation approach and ease of operation, the technique for order preference by similarity to ideal solution (TOPSIS) is commonly utilized for addressing MCDM problems. Therefore, this research proposes an innovative MCDM framework that combines the BBWM technique with Pythagorean fuzzy TOPSIS (PFTOPSIS). The BBWM determines the criteria weights, and the weighted sine similarity-based PFTOPSIS is utilized to rank alternatives. The proposed BBWM-PFTOPSIS approach was employed to solve a real-world case. Moreover, this article conducts a sensitivity analysis and three comparative analyses to reveal the efficiency and reliability of the BBWM-PFTOPSIS approach. The ranking results establish the viability and effectiveness of BBWM-PFTOPSIS.

摘要

选择具有各种舒适性能的最佳针织物被认为是一个复杂的多准则决策问题，涉及模糊性和模糊性. 在这种情况下，勾股模糊集（PFSs）为解决包含人类主观评价和判断的MCDM问题中的不确定性和模糊性提供了一个有效的工具. 首先，本研究确定了影响针织物舒适性的因素作为评价标准. 其次，贝叶斯最佳-最差方法（BBWM）对于较少的成对比较是优选的，并且从概率的角度获得了用于确定标准权重的高度可靠的结果. 此外，由于其逻辑计算方法和易操作性，通过与理想解的相似性进行排序偏好的技术（TOPSIS）通常用于解决MCDM问题. 因此，本研究提出了一种创新的MCDM框架，将BBWM技术与勾股模糊TOPSIS（PFTOPSIS）相结合. BBWM确定标准权重，并利用基于加权正弦相似性的PFTOPSIS对备选方案进行排序. 所提出的BBWM-PFTOPSIS方法被用于解决真实世界的案例. 此外，本文还进行了敏感性分析和三次比较分析，以揭示BBWM-PFTOPSIS方法的有效性和可靠性. 排名结果确定了BBWM-PFTOPSIS的可行性和有效性.

KEYWORDS:

• Knitted fabrics
• Multi-criteria decision-making (MCDM)
• Pythagorean fuzzy set (PFS)
• Bayesian best-worst method (BBWM)
• Technique for order preference by similarity to ideal solution (TOPSIS)
• BBWM-PFTOPSIS

关键词:

• 针织面料
• 多准则决策
• 勾股模糊集 (PFS)
• 贝叶斯最佳-最差方法 (BBWM)
• 基于理想解相似性的排序偏好技术 （TOPSIS）

Introduction

Knitwear has become increasingly important in the garment industry in recent years, and is now an necessary piece of clothing for everyday wear (Lu 2022). Knit garments have always been associated with relaxed clothing and fashion (Salopek Čubrić et al. 2022), such as sportswear, casual wear, and underwear. As textile technology has advanced and people’s living standards have continuously risen, the need for knitted fabrics has extended to style, durability, and clothing comfort. The comfort performance of knitted fabrics directly affects the wearing experience of the human body, and the influencing factors are divided into external factors, which are mainly related to the use environment of the fabric, and internal factors, which are related to the fabric’s material and structure. The thermal and moisture comfort of fabric is mainly manifested in air permeability, moisture permeability, and other abilities. Selecting and ranking candidate knitted fabrics from various alternatives against certain conflicting attributes is a multicriteria decision-making (MCDM) issue.

Some scholars have applied various MCDM-based approaches to address these issues in the textile area. The process includes three steps. First, the evaluation criteria are chosen according to the corresponding problem. Second, the criteria weights are usually obtained via the Analytic Hierarchy Process (AHP) (Okur and Ercan 2022). Last, various approaches, such as the Weighted Sum Method (WSM) (Lahdhiri, Babay, and Jmali 2022), Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) (Danişan, Özcan, and Eren 2022), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) (Atthirawong, Panprung, and Wanitjirattikal 2023), Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) (Simsek et al. 2022), ELimination Et Choice Translating REality (ELECTRE) (Singh et al. 2020), and Measurement of Alternatives and Ranking according to COmpromise Solution (MARCOS) (Mitra 2022), are performed to select the alternatives.

Due to its transparent logic and fewer mathematical operations, TOPSIS, a well-known MCDM tool, more objectively reflects the results, is easier to implement, and has been widely utilized in the textile and apparel area. Various studies have integrated the AHP and TOPSIS to address MCDM issues in the textile area (Bait, Marino Lauria, and Schiraldi 2022; Lahdhiri, Babay, and Jmali 2022), and most related research selects the optimal alternative with the minimum Euclidean distances from the positive ideal solution (PIS) and maximum distance from the negative ideal solution (NIS). Moreover, by integrating various measurements, the TOPSIS approach has been developed in various structures. Some researchers have incorporated similarity measures into the TOPSIS technique. Özlü and Karaaslan (2022) proposed a modified TOPSIS based on a vector similarity measure. Deli, Uluçay, and Polat (2022) introduced normalized similarity, Dice similarity, and Jaccard similarity to TOPSIS. Olgun et al. (2022) developed the cosine similarity measure for the TOPSIS tool.

Although the AHP is extensively utilized to derive criteria weights via standard pairwise comparisons, it can also be employed for large-scale, complicated MCDM problems and integrated with different MCDM techniques. However, the main issue with the AHP is that it lacks consistency in pairwise comparisons. In contrast, the Best-Worst Method (BWM), developed by Rezaei (2015), improves the consistency ratio by requiring fewer comparative data. Despite its many advantages, the BWM has limitations in aggregating the preferences of multiple experts, and it only facilitates individual decision-making (Liu et al. 2023). To eliminate the negative consequences of traditional preferences aggregating approaches in group decision-making scenarios (Yalcin Kavus et al. 2022), Mohammadi and Rezaei (2020) developed the Bayesian BWM (BBWM). BBWM is an extension of the BWM that aggregates evaluations from multiple experts in a probabilistic environment. To minimize information loss and facilitate group decision-making, BBWM utilizes a probabilistic perspective and creedal ranking (Gul, Yucesan, and Ak 2022). Following the initial proposal of the BBWM, it has been widely used in various areas, while few studies have applied the BBWM to the textile field. The above research shows that the BBWM is an efficient method to compute the criteria weights, and it is novel and original to introduce the BBWM method into the textile area.

In real life, the evaluation of certain criteria involves uncertainty. Taking moisture permeability as an example, the areal density and thickness of a fabric affect the moisture permeability, and different combinations of areal density and thickness have different degrees of influence on the moisture permeability, so the evaluation of the moisture permeability involves uncertainty. In this situation, traditional MCDM techniques may have limitations when dealing with uncertain information. Fuzzy sets (FSs) were proposed to express the fuzziness and address the limitations of judgments via crisp numbers. Later, various FS theories were developed to depict qualitative assessment. Yager (2013) introduced Pythagorean fuzzy sets (PFSs), which are characterized by membership and nonmembership grades. Since they are capable of handling the ambiguity and uncertainty in real-world problems, PFSs have gained much interest from various researchers to solve actual MCDM issues, which have been classified into three categories: information measures (inclusion measures, similarity measures, and distance measures) (Ashraf et al. 2023), aggregation operators (AOs) (Khalil and Sharqi 2023), and MCDM techniques (Menekse et al. 2023).

Similarity measure is a prominent tool for estimating the comparability between the pairs of data (Kumar and Kumar 2023). As one of the research hotspots in PFSs, various PFS similarity measures have been studied in previous studies. Some researchers have proposed vector-based similarity measures, such as the Dice similarity measure (Wang, Gao, and Wei 2019), Jaccard similarity measure (Huang, Lin, and Xu 2020), and cosine similarity measure (Wei and Wei 2018), Farhadinia (2022) developed a novel PFS similarity measure by integrating the t-norm and s-norm. Arora and Naithani (2022b, 2022c) proposed several basic logarithmic and weighted logarithmic PF-similarity measures. Moreover, the similarity measures have been extensively utilized to address MCDM problems (Saikia, Dutta, and Talukdar 2023; Verma and Mittal 2023).

Various traditional MCDM approaches have been extended in PFS environment. (Hajiaghaei-Keshteli et al. (2023) utilized the Pythagorean fuzzy TOPSIS (PFTOPSIS) to select the most suitable green supplier. Similarly, Saeidi et al. (2022) employed PFTOPSIS to evaluate sustainable human resource management. Yang et al. (2022) applied PF-VIKOR to evaluate water ecological security. Madhavi et al. (2023) introduced a MCDM model utilizing PF-based VIKOR and TOPSIS methods to address resource deletion attacks in wireless sensor networks. Ye and Chen (2022b) applied PF-PROMETHEE to select cotton woven fabric. Additionally, Akram, Luqman, and Alcantud (2022) applied the hesitant PF-ELECTRE I technique for risk assessment of failure modes and effect analysis. Of all these methods, PFTOPSIS has gained popularity due to its transparent logic and fewer mathematical operations involved in the decision-making process. However, few studies have applied PFTOPSIS in fabric selection (Ye and Chen 2022c). Furthermore, the majority of earlier investigations have concentrated on distance-based PF-TOPSIS, and only a few scholars have utilized similarity measures to the structure of PFTOPSIS (Hussain 2021; Li et al. 2019; Rani et al. 2020).

The motivations behind the study are: (1) Since several factors affect the comfort of knitted fabrics, this research introduces an evaluation framework for guiding general decision-making in evaluating the comfort of knitted fabrics. (2) Compared to traditional TOPSIS, PFTOPSIS considers both PIS and NIS, enabling decision-makers to weigh trade-offs between different criteria and identify solutions that are both optimal and feasible. PFSs’ ability to tolerate imprecision and uncertainty also reduces the impact of outliers on the final ranking, making PFTOPSIS more robust to outliers. Criteria weights are a crucial component of PFTOPSIS, as they determine how each alternative is evaluated based on its performance on each criterion, and the criteria weights are commonly obtained by AHP, BWM and BBWM. (3) Compared to AHP, BBWM is an efficient MCDM technique to evaluate criteria weights, as it provides less pairwise comparisons, simplifies calculation steps, and provides more consistent results. Compared to BWM, BBWM can better handle impression and uncertainty. Moreover, it can be easily amalgamated with other MCDM methods and is readily accessible. These advantages make BBWM a useful method for easily combining with other MCDM methods under fuzzy sets. Despite these findings, the BBWM, which was introduced in 2020, is seldom applied in the textile area, especially in fabric selection. (4) Since PFTOPSIS can be formed by different measurements in various structures, we utilize a similarity measure instead of a distance measure in the PFTOPSIS structure, since similarity measures are commonly used to describe impression and uncertainty under the PFS context. (5) PFSs are better suited to handle imprecision decision-making problems, and this flexibility aids in resolving problems by providing more information on variables and making the process more intelligent. Hence, we apply a framework of BBWM-PFTOPSIS to address a real-world case involving the selection of the appropriate knitted fabric.

This research is aimed at proposing a hybrid method that integrates two well-known MCDM methods named the BBWM and PFTOPSIS to address a real-life MCDM issue of knitted fabric selection, which includes five alternatives. First, the important criteria of the comfort of knitted fabrics are identified. Second, according to an expert’s opinion, we apply the BBWM to calculate the criteria weights. Then, we transform the values of knitted fabrics into Pythagorean fuzzy numbers (PFNs), and weighted sine similarity-based PFTOPSIS is employed to rank the alternatives. Third, we employ sensitivity analysis to illustrate the effectiveness and robustness of BBWM-PFTOPSIS. Fourth, we employ a comparative analysis to verify its efficacy in comparison to other MCDM techniques in the PFS environment, including PF-WSM, PF-ELECTRE, PF-VIKOR, PF weighted arithmetic mean (PFWAM) AO, and PF Aczel-Alsina weighted averaging (PFAAWA) AO. Finally, we perform a Spearman’s rank correlation coefficient analysis to investigate the correlations between the ranks obtained from various MCDM approaches.

According to the above discussions, the novelties of this work are as follows: (1) We construct an extension of weighted sine similarity-based TOPSIS with PFSs, which provides flexibility to decision-makers in defining their uncertainties. (2) A hybrid BBWM-PFTOPSIS model have been developed to obtain the criteria weights and rank alternatives. Different MCDM approaches are employed at different stages of this model. (3) This is the initial application of decision-making in the knitted fabric selection problem under the PFSs environment.

The paper’s contributions are fourfold: (1) Since the evaluation of alternatives includes uncertainty, we use the PFS linguistic term transforming system to make pairwise comparisons and express uncertainty. (2) We apply the BBWM results as a foundation for understanding important criteria or less important criteria for decision-makers with fewer pairwise comparisons. (3) This research proposes a new hybrid method, including the BBWM and weighted sine similarity-based PFTOPSIS, for the first time in knitted fabric selection, which extends prior research and helps prioritize the best alternatives. (4) A practical case of the selection of knitted fabrics is given to illustrate the efficiency and applicability of BBWM-PFTOPSIS, which is pioneering for addressing the fuzzy MCDM issue in fabric selection. Moreover, this method is adaptable to other textile areas.

The remaining sections of this paper are arranged as follows. Section 2 conducts a review of current research on BBWM and PFTOPSIS. Section 3 provides the definition of PFSs. The following section 4 elucidates the research methodology of BBWM-PFTOPSIS. In section 5, a real-world case study on knitted fabrics selection is presented, demonstrating the practical application of the BBWM-PFTOPSIS technique. Lastly, Section 6 presents the conclusions, managerial implications, limitations and future research.

Literature review

In this section, the first subsection presents an overview of previous works on the BBWM, and the second subsection briefly reviews previous studies on PFTOPSIS.

BBWM studies

BWM is an efficient approach for pairwise comparison, as it requires fewer pairwise comparisons and minimizes inconsistency in the comparison data. BBWM, as a modified and upgraded version of BWM, is an MCDM method in which the perception of experts uses a probabilistic method and facilitates more accurate decision-making on the integrated criteria rankings (Mohammadi and Rezaei 2020). Compared to BWM, BBWM offers numerous notable advantages, including the ability to generate a more precise matrix of decision-maker opinions and to require fewer pairwise comparisons (Ahmed et al. 2023). Furthermore, BBWM’s credal ranking provides a straightforward and nuanced approach to determine the dominance of a criterion over others (Debnath et al. 2023). In addition, the Monte-Carlo simulation framework in BBWM, based on a probabilistic perspective, minimizes the loss of information (Gul, Yucesan, and Ak 2022).

The BBWM is a novel MCDM technique that has been used in various sectors, including supply chain management (Afghah et al. 2023; Ahmed et al. 2023), location selection (Hashemkhani Zolfani et al. 2022; Yalcin Kavus et al. 2022), risk assessment (Gupta et al. 2023; Khan et al. 2022), and sustainability assessment (Debnath et al. 2023; Liu et al. 2023). In practical scenarios, BBWM is often used alone or in conjunction with other MCDM methods. Munim et al. (2023) presented a hybrid approach combining BBWM and PROMETHEE to rank the choices of fuel and energy sources. Gul, Yucesan, and Ak (2022) prioritized textile industry risks using BBWM and VIKOR methods. Similarly, Hsu, Kuo, and Liou (2023) developed a hybrid model of BBWM and modified VIKOR to assess urban bikeability. Despite the extensive research on the integration of BWM and MCDM methods, few studies have combined BBWM with TOPSIS.

Moreover, it can be inferred that most of the previous research has been evaluated using crisp values, whereas the decision-making environment is full of uncertainty. Therefore, some scholars have integrated the BBWM with fuzzy MCDM approaches. Yalcin Kavus et al. (2022) combined BBWM and PF Weighted Aggregated Sum-Product Assessment (WASPAS) to address parcel locker location selection issue. Yang et al. (2022) proposed a hybrid model of BBWM and Grey PROMETHEE-AL to assess the effectiveness of medical tourism. Chen et al. (2020) utilized BBWM to obtain aggregated criteria weights, and the distillation algorithm, in conjunction with ELECTRE III, was used to address the projection of probabilistic linguistic term sets for ranking alternatives. The assessment of control measures in risk management was investigated by Gul, Yucesan, and Ak (2022) through the integration of BBWM and fuzzy VIKOR. Modares, Farimani, and Emroozi (2023) and Modares et al. (2023) applied the BBWM and fuzzy TOPSIS to prioritize the suppliers.

Table 1 provides a summary of previous research on BBWM, outlining the combined methods and application areas. The above analysis infers that utilizing BBWM and TOPSIS in resolving real-life problems is becoming increasingly crucial. However, there is limited research on the combination of BBWM with other fuzzy MCDM methods, and few academics have utilized BBWM for fabric selection. Therefore, this paper aims to comprehensively evaluate the criteria weights through a holistic approach.

Table 1. A brief summary about BBWM and PFTOPSIS studies.

BBWM
Authors	MCDM problem	Combined MCDM method	Type of fuzzy sets
			FS	PFSs
Yang et al. (2022)	Medical tourism performance assessment	Grey PROMETHEE	√
Gul, Yucesan, and Ak (2022)	Risk assessment in oil station	Fuzzy VIKOR	√
Chen et al. (2020)	Healthcare waste management	ELECTRE III	√
Modares, Farimani, and Emroozi et al. (2023)	Supply chain management	TOPSIS	√
Modareset al. (2023)	Supplier selection	TOPSIS	√
Yalcin Kavus et al. (2022)	Parcel locker location selection	PF WASPAS		√
Saner, Yucesan, and Gul (2022)	Evaluation of hospital disaster preparedness	VIKOR, TOPSIS	N/A	N/A
Hashemkhani Zolfani et al. (2022)	Location selection	MARCOS	N/A	N/A
PF-TOPSIS
			Type of information measures
Authors	MCDM problem	Combined MCDM method	Distance	Similarity	Correlation
Sarkar and Biswas (2021)	Transportation company selection	Entropy	√
Mahanta and Panda (2021)	Face mask selection	N/A	√
Li et al. (2021)	Advertising platforms selection	Closeness coefficient	√
Sarkar and Biswas (2020)	Selection of the best rail project	Entropy	√
Hussian and Yang (2019)	Ranking of private schools	Hausdorff metric	√
Zhang and Xu (2014)	Service quality evaluation	N/A	√
Garg et al. (2023)	Supplier selection	Hamy mean operators		√
Mishra et al. (2022)	Sustainable biomass crop selection	Score function		√
Rani et al. (2020)	Sustainable recycling partner selection	Trigonometric function		√
Agheli, Adabitabar Firozja, and Garg (2022)	Service quality evaluation	T-norm and S-norm		√
Zulqarnain et al. (2021)	Green supplier selection	N/A			√
Lin, Huang, and Xu (2019)	Firewall productions selection	Entropy			√

Previous studies regarding PF-TOPSIS

The TOPSIS is a popular and widely utilized MCDM method due to its ability to balance PIS and NIS (Hooshangi, Mahdizadeh Gharakhanlou, and Ghaffari Razin 2023), as well as its simple computational process (Ye and Chen 2022c). Since the ambiguity and multiple uncertainties of practical MCDM issues, extending TOPSIS to PFS circumstances would prove beneficial. Additionally, previous studies have highlighted that the TOPSIS approach can be constructed using various measurement units (Kirişci 2023), such as distance measures, correlation measures, and similarity measures.

Various studies have applied PFTOPSIS to address a variety of MCDM problems. After Zhang and Xu (2014) initiated the Hamming distance-based PFTOPSIS, most of the existing studies have integrated Hamming and Euclidean distances into the structure of PFTOPSIS. Moreover, several scholars have developed novel distance measures for PFTOPSIS (Hussian and Yang 2019; Sarkar and Biswas 2020, 2021). Since Garg (2016) introduced the correlation coefficient to the PFS environment, numerous researchers have extended and proposed various correlation coefficients for the PFTOPSIS method (Lin, Huang, and Xu 2019; Zulqarnain et al. 2021). Since Zhang (2016) proposed the similarity measures for PFSs, many academics have developed novel similarity measures and some studies have extended them to PFTOPSIS. Rani et al. (2020) introduced two novel PFS similarity measures and employed them to create a novel PFTOPSIS method. Agheli, Adabitabar Firozja, and Garg (2022) utilized the T-norm and S-norm-based similarity measure into the structure of PFTOPSIS. Garg et al. (2023) proposed a novel PFTOPSIS that utilizes the generalized Dice similarity measure and Hamy mean AO. Table 1 also lists the measures used and examples of PFTOPSIS applications.

According to the above analysis, it is evident that the distance measure, similarity measure, and correlation coefficient measure have been extensively researched. However, few studies have integrated the similarity measure into PFTOPSIS. Moreover, due to the advantages of PFTOPSIS, it is selected in this study to broaden its application to fabric selection problems.

Preliminaries

This section outlines the fundamental principles of PFSs, relevant information measures, and AOs for this research.

Definition

1 (Yager and Abbasov 2013) A PFS $P$ in $X=\left\{x_1,x_2,\dots ,x_n\right\}$ is presented below:

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

where ${\mu }_P$ and ${\nu }_P$: $X\to \left[0,1\right]$ are membership and nonmembership degrees, fulfilling the condition that $0\le {\mu }_P^2\left(x\right)+{\nu }_P^2\left(x\right)\le 1$, and hesitation ${\pi }_P\left(x\right)=\sqrt{1-{\mu }_P^2\left(x\right)-{\nu }_P^2\left(x\right)}$. To simplify matters (Zhang and Xu 2014), called $P$ as a PFN and signified as $p=\left(\mu ,\nu \right)$.

Definition

2 (Zhang and Xu 2014) For a PFN $p=\left(\mu ,v\right)$, then the score function is shown below:

(2) $SF\left(p\right)={\mu }^2-v^2$(2)

For any two PFNs $p_1$ and $p_2$, the laws of comparison can be summarized as: (1) If $SF\left(p_1\right)>SF\left(p_2\right)$, then $p_1\succ p_2$; (2) If $SF\left(p_1\right)=SF\left(p_2\right)$, then $p_1\sim p_2$.

Definition 3

(Zhang and Xu 2014) For two PFNs $p_1$ and $p_2$, the Hamming distance is calculated using Eq.(3)(3) $D\left(p_1,p_2\right)=\frac{1}{2}\left(\left|{\mu }_{p_1}^2-{\mu }_{p_2}^2\right|+\left|{\nu }_{p_1}^2-{\nu }_{p_2}^2\right|+\left|{\pi }_{p_1}^2-{\pi }_{p_2}^2\right|\right)$(3) :

(3) $D\left(p_1,p_2\right)=\frac{1}{2}\left(\left|{\mu }_{p_1}^2-{\mu }_{p_2}^2\right|+\left|{\nu }_{p_1}^2-{\nu }_{p_2}^2\right|+\left|{\pi }_{p_1}^2-{\pi }_{p_2}^2\right|\right)$(3)

Definition

4 (Yager and Abbasov 2013) For a collective of PFNs $p_i=\left({\mu }_i,{\nu }_i\right)$, $w_i$ is the weight vector, the PF weighted averaging (PFWA) AO can be defined as Eq.(4)(4) $PFWA\left(p_1,p_2,\dots ,p_i\right)=\left(\sqrt{1-{\prod }_{i=1}^n{\left(1-{\mu }_i^2\right)}^{w_i}},{\prod }_{i=1}^n{\nu }_i^{w_i}\right)$(4) :

(4) $PFWA\left(p_1,p_2,\dots ,p_i\right)=\left(\sqrt{1-{\prod }_{i=1}^n{\left(1-{\mu }_i^2\right)}^{w_i}},{\prod }_{i=1}^n{\nu }_i^{w_i}\right)$(4)

Definition

5 (Arora and Naithani 2022a) Let $P$ and $Q$ be PFSs of $X=\left\{x_1,x_2,\dots ,x_n\right\}$, the weighted sine similarity measures for PFSs are computed as follows:

(5) $S_{WA}\left(P,Q\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\mu }_P^2\left(x_i\right)-{\mu }_Q^2\left(x_i\right)\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\nu }_P^2\left(x_i\right)-v_Q^2\left(x_i\right)\right|\right\}\right]$(5)

Methodology

This study employed a three-step research methodology. First, evaluation criteria were extracted from the relevant literature. Second, the BBWM was employed for computing the weights of criteria. Last, the weighted sine similarity-based PFTOPSIS approach was utilized for ranking alternatives. Figure 1 illustrated the details of the suggested methodology of this research.

Figure 1. Flowchart of the proposed methodology.

Bayesian BWM method

BBWM is an extension of BMW that aids in assimilation of experts evaluations of with minimum loss. The steps of BBWM are briefly presented in the following:

Step 1. Definition of a certain evaluation criteria set as $C=\left\{C_j\left|j=1,2,\dots n\right.\right\}$.

Step 2. Determination of the best criterion ($C_B$) and worst criterion ($C_W$), respectively.

Step 3. Developing pairwise comparisons between $C_B$ and all other criteria. The vector $P_B$ for the resulting best-to-others is:

(6) $P_B=\left\{p_{B1},p_{B2},\cdots ,p_{Bn}\right\},$(6)

where $p_B$ expresses the significance of $C_B$ over $C_j$ using numbers between 1 and 9 and $p_{BB}=1$.

Step 4. The priorities of other criteria are compared over $C_W$. Similarly, a 9-point numeric scale was employed. The vector $P_W$ for the resulting others-to-worst is:

(7) $P_W=\left\{p_{1W},p_{2W},\cdots ,p_{nW}\right\},$(7)

where $p_W$ presents the preference of $C_j$ over $C_W$ and $p_{WW}=1$.

Step 5. Obtain the criteria weights. $P_B$ and $P_W$ serve as input vectors for calculating, and the criteria weights $w^{\ast }=\left(w_1^{\ast },w_2^{\ast },\dots ,w_n^{\ast }\right)$ are aggregated in a probability perspective:

(8) $\begin{array}{rl}& P_{Bj}\sim multinomial\left(1/w_j\right);P_{jW}\sim multinomial\left(w_j\right);\mathrm{\forall }j=1,2,\dots ,n\\ & w^{\ast }\sim Dir\left(\gamma \times w^{\ast }\right);\gamma \sim gamma\left(0.1,0.1\right);\\ & w^{\ast }\sim Dir\left(1\right).\end{array}$(8)

where multinomial refers to a multinomial distribution. $Dir\left(1\right)$ and Gamma (0.1, 0.1) are the Dirichlet and gamma distributions, respectively. JAGS (Just Another Gibbs Sampler), which is one of the most preferred Monte Carlo techniques, is employed to solve the BBWM model.

PFTOPSIS method

The fundamental principle of PFTOPSIS is that the ideal option is away from Pythagorean fuzzy NIS (PFNIS) and as close as possible to Pythagorean fuzzy PIS (PFPIS). In similarity-based PFTOPSIS, the selected alternative should be highly similar to PFPIS and minimally similar to PFNIS. The followings are the steps to implement weighted sine similarity-based PFTOPSIS:

Step 1. Create a PFN-based decision matrix $A_{ij}={\left(C_j\left(a_i\right)\right)}_{m\times n}$ as follows:

$A_{ij}={\left(C_j\left(a_i\right)\right)}_{m\times n}=\left(\begin{array}{cccc}\left({\mu }_{11},{\nu }_{11}\right)& \left({\mu }_{12},{\nu }_{12}\right)& \cdots & \left({\mu }_{1n},{\nu }_{1n}\right)\\ \\ \left({\mu }_{21},{\nu }_{21}\right)& \left({\mu }_{22},{\nu }_{22}\right)& \cdots & \left({\mu }_{2n},{\nu }_{2n}\right)\\ ⋮& ⋮& ⋮& ⋮\\ \left({\mu }_{m1},{\nu }_{m1}\right)& \left({\mu }_{m2},{\nu }_{m2}\right)& \cdots & \left({\mu }_{mn},{\nu }_{mn}\right)\end{array}\right).$

where $a_i\left(i=1,2,\dots ,m\right)$ represents alternatives and $C_j\left(j=1,2,\dots ,n\right)$ represents criteria.

Step 2. Determine PFPIS ($a_j^{+}$) and PFNIS ($a_j^{-}$) using Eqs.(9)(9) $a_j^{+}=\left\{\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right),\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right)\right\},\hspace{0.17em}$(9) and (10(10) $a_j^{-}=\left\{\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right),\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right)\right\}.$(10) ), respectively, as follows:

(9) $a_j^{+}=\left\{\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right),\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right)\right\},\hspace{0.17em}$(9)

(10) $a_j^{-}=\left\{\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right),\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right)\right\}.$(10)

where $J^{+}$ represents the benefit-type criteria set, and $J^{-}$ represents the cost-type criteria set.

Step 3. Calculate the weighted sine similarity measures $S_i^{+}\left(a_{ij},a_j^{+}\right)$ and $S_i^{-}\left(a_{ij},a_j^{-}\right)$ using Eqs.(11)(11) $S_i^{+}\left(a_{ij},a_j^{+}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{+}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{+}\right)\right)}^2\right|\right\}\right]$(11) and (12(12) $S_i^{-}\left(a_{ij},a_j^{-}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{-}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{-}\right)\right)}^2\right|\right\}\right]$(12) ), respectively:

(11) $S_i^{+}\left(a_{ij},a_j^{+}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{+}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{+}\right)\right)}^2\right|\right\}\right]$(11)

(12) $S_i^{-}\left(a_{ij},a_j^{-}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{-}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{-}\right)\right)}^2\right|\right\}\right]$(12)

Step 4. Calculate the relative closeness (RC). It is computed by using Eq.(13)(13) $R{C}_i=\frac{S_i^{+}\left(a_{ij},a_j^{+}\right)}{S_i^{-}\left(a_{ij},a_j^{-}\right)+S_i^{+}\left(a_{ij},a_j^{+}\right)}.$(13) as follows:

(13) $R{C}_i=\frac{S_i^{+}\left(a_{ij},a_j^{+}\right)}{S_i^{-}\left(a_{ij},a_j^{-}\right)+S_i^{+}\left(a_{ij},a_j^{+}\right)}.$(13)

Step 5. Sort the options in descending order of their RC. The solution having the highest $R{C}_i$ is the best.

Model application and discussion

This research applies the proposed BBWM-PFTOPSIS methods to solve an actual case. A textile company in Jiaxing, China, is mainly engaged in the production and sales of knitted garments. Based on a preliminary market survey, the company aimed to design and produce a piece of early autumn knitted clothing to meet consumers’ needs. Five potential knitted fabrics, made of different kinds of yarns with the same 28/2 metric count and woven in a 1 + 1 purl stitch, were considered for production under four criteria. The BBWM was utilized to assign weights to the selection criteria, while PFTOPSIS approach was utilized to rank the knitted fabrics. The effectiveness and reliability of BBWM-PFTOPSIS were evaluated through sensitivity and three comparative analyses.

Data collection and description

Clothing comfort refers to pleasant psychological, physiological, and physical coordination between individuals and their surroundings (Ye and Chen 2022c). The features of fiber type, yarn properties, and fabric structure impact the fabric’s heat and moisture transmission, which is a critical factor in determining thermos-physiological wear comfort (Mitra 2023), including thermal properties, air permeability, and moisture permeability (Krithika et al. 2020). An overview of the concept of thermos-physiological wear comfort is presented in Table 2. In this research, we identified the thickness, weight, air permeability, and moisture permeability as the evaluation criteria in this research, aligning with the conclusions that the fabrics should have less weight and thickness (Iftikhar et al. 2021), and adequate air permeability and moisture vapor permeability (Mitra et al. 2015).

Table 2. Applied dimensions and criteria for knitted fabrics comfort assessment.

Dimensions	Criteria	Definitions	Research purpose and remarks	References
Structural characteristics	Weight ($C_1$)	Grams per square meter of fabric	Impact on thermal conductivity and air permeability	Oğlakcioğlu and Marmarali (2007); Raja Balasaraswathi and Rathinamoorthy (2022); Çeven and Günaydın (2021)
	Thickness ($C_2$)	Perpendicular distance through the fabric	Impacts thermal conductivity and absorptivity	Noman et al. (2020); Garg, Sikka, and Midha (2022)
Wearability	Air permeability ($C_3$)	Air flow through a fabric under a specific air pressure.	The relationships between air permeability and woven structure	Adamu and Gao (2022); Limeneh et al. (2022); Eryuruk (2021)
	Moisture permeability ($C_4$)	Fabric through water vapor.	The relationships between moisture permeability and fabric structure	Chen et al. (2018); Zhao, Lu, and Li (2021); Eryuruk (2021)

Five samples with dimensions of $8.5\times 5.5$cm were prepared and tested according to a standard method, and the average of the values was calculated. The fabrics were measured for a period of 24 hours under standard atmospheric conditions of 65 ± 2% relative humidity and a temperature maintained at 20 ± 2°C. As per ISO 5084 guidelines, fabric weight should be measured with an electronic balance (HHB822S), and the fabric thickness could be measured with a YG141N tester. The air permeability was measured with a differential pressure of 100 Pa, and the sample test size was 20 $\mathrm{c}{\mathrm{m}}^2$. The team’s self-developed experimental instrument, the “wireless temperature and humidity test system”, measured the fabrics’ moisture permeability.

The collective decision matrix of the alternatives is presented in Table 3. There are four criteria for knitted fabrics: weight ($C_1$), thickness ($C_2$), air permeability ($C_3$), and moisture permeability ($C_4$). Among them, $C_1$ and $C_2$ are considered cost-related properties (the lesser, the better), and $C_3$ and $C_4$ are considered beneficial properties (the higher, the better). The initial case data in Table 2 shows that alternative $A_3$ is the heaviest and that alternative $A_2$ is the lightest under the $C_1$ criterion. Alternative $A_1$ has the maximum fabric thickness, while alternative $A_4$ has the minimum fabric thickness under the $C_2$ criterion. Moreover, alternative $A_2$ has the maximum air permeability, which has a large gap with other alternatives. In addition, alternative $A_1$ has the minimum fabric moisture permeability.

Table 3. The dataset of knitted fabrics and PFN decision matrix.

	Alternatives	$C_1$	$C_2$	$C_3$	$C_4$	Alternatives	$C_1$	$C_2$	$C_3$	$C_4$
The original dataset of knitted fabrics	$A_1$	1.67	1.31	916.34	35.32	$A_4$	1.30	0.70	895.77	53.89
	$A_2$	1.17	0.74	2063.98	52.16	$A_5$	1.25	0.73	945.99	98.21
	$A_3$	1.76	1.11	932.06	40.00
PFSs linguistic terms		Extremely low (EL)	Very low (VL)	Low (L)	Medium low (ML)	Medium (M)	Medium high (MH)	High (H)	Very high (VH)	Extremely high (EH)
Corresponding PFN		(0.10,0.95)	(0.15,0.85)	(0.25,0.75)	(0.35,0.65)	(0.50,0.45)	(0.65,0.35)	(0.75,0.25)	(0.85,0.15)	(0.95,0.10)
Scale for grading the fabric properties	$C_1$	(-∞,1.24]	(1.24,1.30]	(1.30,1.37]	(1.37,1.43]	(1.43,1.50]	(1.50,1.56]	(1.56,1.63]	(1.63,1.69]	(1.69, +∞)
	$C_2$	(-∞,0.768]	(0.768,0.836]	(0.836,0.903]	(0.903,0.971]	(0.971,1.04]	(1.04,1.11]	(1.11,1.17]	(1.17,1.24]	(1.24, +∞)
	$C_3$	(-∞,1030]	(1030,1160]	(1160,1290]	(1290,1410]	(1410,1540]	(1540,1670]	(1670,1800]	(1800,1930]	(1930, +∞)
	$C_4$	(-∞,42.3]	(42.3,49.3]	(49.3,56.3]	(56.3,63.3]	(63.3,70.3]	(70.3,77.2]	(77.2,84.2]	(84.2,91.2]	(91.2, +∞)
	Alternatives	$C_1$	$C_2$	$C_3$	$C_4$	Alternatives	$C_1$	$C_2$	$C_3$	$C_4$
PFN decision matrix	$A_1$	(0.85,0.15)	(0.95,0.10)	(0.10,0.95)	(0.10,0.95)	$A_4$	(0.15,0.85)	(0.10,0.95)	(0.10,0.95)	(0.25,0.75)
	$A_2$	(0.10,0.95)	(0.10,0.95)	(0.95,0.10)	(0.25,0.75)	$A_5$	(0.15,0.85)	(0.10,0.95)	(0.10,0.95)	(0.95,0.10)
	$A_3$	(0.95,0.10)	(0.65,0.35)	(0.10,0.95)	(0.10,0.95)

Application of the BBWM-PFTOPSIS method

Following the flowchart of the proposed BBWM-PFTOPSIS, the calculation process of the knitting fabrics selection problem is as follows:

Steps 1–2. Define the evaluation set, $C_B$ and $C_W$.

Based on the above analysis, the four evaluation criteria are defined. In addition, $C_B$ and $C_W$ are assigned to $C_3$ and $C_2$, respectively.

Steps 3–5. Make comparisons of $C_B$ with other criteria and $C_W$ with other criteria. $P_B$ and $P_W$ are formed and presented in Table 4, and weights are determined. Next, the BBWM model was solved through MATLAB software, which resulted in criteria weights of $w=\left(0.1401,0.0936,0.4553,0.3110\right)$, as shown in Table 4. Weight $C_3$ is 0.4553 at its maximum, followed by $C_4$, $C_1$, and $C_2$.

Table 4. The vectors of BO, OW, and criteria weights.

Best criterion	Worst criterion	Criteria	Best to others	Others-to-worst	Weight	Rank
$C_3$	$C_2$	$C_1$	6	3	0.1401	3
		$C_2$	9	1	0.0936	4
		$C_3$	1	9	0.4553	1
		$C_4$	2	6	0.3110	2

Steps 6–7. Transform the crisp numbers of the knitted fabric into PFNs and create a PFN decision matrix. Following the procedure proposed by Ye and Chen (2022c), we categorized the crisp values into nine grades for each criterion, and then alternatives were graded using the PFN linguistic terms introduced by Büyüközkan and Göçer (2021). Table 3 shows the obtained PFN decision matrix.

Step 8. Determine the $a_j^{+}$ and $a_j^{-}$ of alternatives. According to Eqs.(9)(9) $a_j^{+}=\left\{\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right),\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right)\right\},\hspace{0.17em}$(9) and (10(10) $a_j^{-}=\left\{\left(min\hspace{0.17em}{a}_{ij}\left|j\in J^{-}\right.\right),\left(max\hspace{0.17em}{a}_{ij}\left|j\in J^{+}\right.\right)\right\}.$(10) ), the following results are obtained:

$a_j^{+}=\left\{\left(0.10,0.95\right),\left(0.10,0.95\right),\left(0.95,0.10\right),\left(0.95,0.10\right)\right\},$

$a_j^{-}=\left\{\left(0.95,0.10\right),\left(0.95,0.10\right),\left(0.10,0.95\right),\left(0.10,0.95\right)\right\}.$

Step 9. $S_i^{+}\left(a_{ij},a_j^{+}\right)$ and $S_i^{-}\left(a_{ij},a_j^{-}\right)$ are computed in accordance with Eqs. (11)(11) $S_i^{+}\left(a_{ij},a_j^{+}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{+}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{+}\right)\right)}^2\right|\right\}\right]$(11) and (12(12) $S_i^{-}\left(a_{ij},a_j^{-}\right)=1-\frac{1}{2n}\sum _{i=1}^n{w}_i\left[sin\left\{\frac{\pi }{2}\left|{\left(\mu \left(a_{ij}\right)\right)}^2-{\left(\mu \left(a_j^{-}\right)\right)}^2\right|\right\}+sin\left\{\frac{\pi }{2}\left|{\left(\nu \left(a_{ij}\right)\right)}^2-{\left(\nu \left(a_j^{-}\right)\right)}^2\right|\right\}\right]$(12) ), as shown in Table 5, respectively.

Table 5. The closeness coefficients and rank order of similarity-based PFTOPSIS.

Alternatives	$S_i^{+}$	$S_i^{-}$	$R{C}_i$	Rank
$A_1$	0.7551	0.9948	0.4315	5
$A_2$	0.9327	0.8072	0.5361	1
$A_3$	0.7586	0.9899	0.4338	4
$A_4$	0.8153	0.9210	0.4696	3
$A_5$	0.8826	0.8673	0.5044	2

Step 10. $R{C}_i$ is computed utilizing Eq. (13)(13) $R{C}_i=\frac{S_i^{+}\left(a_{ij},a_j^{+}\right)}{S_i^{-}\left(a_{ij},a_j^{-}\right)+S_i^{+}\left(a_{ij},a_j^{+}\right)}.$(13) , as presented in Table 5.

Step 11. Sort the alternatives. Based on $R{C}_i$, we find that alternative $A_2$ has the maximum value, while $A_1$ has the worst value. Thus, the most appropriate alternative is$A_2$ and the least appropriate alternative is $A_1$. Furthermore, the result indicates that the similarity-based PFTOPSIS method is effective in addressing the practical case.

Sensitivity analysis

Within this research, a sensitivity analysis was conducted to confirm the reliability and effectiveness of the proposed methodology’s outcomes. The sensitivity analysis was performed to investigate various priorities on criteria weights, which could potentially impact the proposed method’s results. To achieve this objective, we modified the criteria weight of the best criterion weight within a feasible range while proportionally weighting the other criteria. The weight of $C_3$ was divided into 19 scenarios, which varied from 0.05 to 0.95 with a 0.05 increment and defined as S1-S19. Table 6 shows the impacts of varying criteria weights on the rankings of both criteria and alternatives.

Table 6. Sensitivity analysis and ranking of the main criterion and alternatives for BBWM-PFTOPSIS.

Main criteria	Original	Variation in weight of the best criterion
		S1 (0.05)	S2 (0.1)	S3 (0.15)	S4 (0.2)	S5 (0.25)	S6 (0.3)	S7 (0.35)	S8 (0.4)	S9 (0.45)	S10 (0.5)	S11 (0.55)	S12 (0.6)	S13 (0.65)	S14 (0.7)	S15 (0.75)	S16 (0.8)	S17 (0.85)	S18 (0.9)	S19 (0.95)
$C_3$	0.4553 (1)	0.0500 (4)	0.1000 (4)	0.1500 (3)	0.2000 (3)	0.2500 (2)	0.3000 (2)	0.35 (2)	0.4000 (1)	0.45 (1)	0.5000 (1)	0.5500 (1)	0.6000 (1)	0.6500 (1)	0.7000 (1)	0.7500 (1)	0.8000 (1)	0.8500 (1)	0.9000 (1)	0.9500 (1)
$C_1$	0.1401 (3)	0.2443 (2)	0.2315 (2)	0.2186 (2)	0.2058 (2)	0.1929 (3)	0.1800 (3)	0.1672 (3)	0.1543 (3)	0.1415 (3)	0.1286 (3)	0.1157 (3)	0.1029 (3)	0.0900 (3)	0.0772 (3)	0.0643 (3)	0.0514 (3)	0.0386 (3)	0.0257 (3)	0.0129 (3)
$C_2$	0.0936 (4)	0.1632 (3)	0.1547 (3)	0.1461 (4)	0.1375 (4)	0.1289 (4)	0.1203 (4)	0.1117 (4)	0.1031 (4)	0.0945 (4)	0.0859 (4)	0.0773 (4)	0.0687 (4)	0.0601 (4)	0.0516 (4)	0.0430 (4)	0.0344 (4)	0.0258 (4)	0.0172 (4)	0.0086 (4)
$C_4$	0.3110 (2)	0.5424 (1)	0.5139 (1)	0.4853 (1)	0.4568 (1)	0.4282 (1)	0.3997 (1)	0.3711 (1)	0.3426 (2)	0.3140 (2)	0.2855 (2)	0.2569 (2)	0.2284 (2)	0.1998 (2)	0.1713 (2)	0.1427 (2)	0.1142 (2)	0.0856 (2)	0.0571 (2)	0.0285 (2)
Total	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
Ranking of alternatives
Alternatives	Original	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10	S11	S12	S13	S14	S15	S16	S17	S18	S19
$A_1$	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5	5
$A_2$	1	2	2	2	2	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1
$A_3$	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4
$A_4$	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3	3
$A_5$	2	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2

The variation in criteria weights and corresponding ranking order are shown in Table 6. At a weight of 0.05 to 0.35, $C_4$ still ranks first. When the $w\in [0.05,0.10]$, $C_3$ is found as the worst criterion. Moreover, when the criterion weight of $C_3$ varied from 0.4 to 0.95, it maintained the highest ranking, and the rankings of the other criteria remained constant as $C_3\succ C_4\succ C_1\succ C_2$. In addition, in most scenarios S3-S19, $C_2$ is the least important criterion. The sensitivity analysis conducted on the criteria weights did not result in any significantly changes to the rankings of other criteria. It can be concluded that the criteria weights variation makes this method highly flexible, allowing decision makers to choose appropriate parameters based on the demands and practical conditions of the problem.

The results of ranks of alternatives are shown in Table 6. In all scenarios, alternatives $A_4$ and $A_3$ keep the third and fourth ranking orders, and $A_1$ is the worst alternative which remains the last ranking order. In addition, the ranking of$A_2$ remained stable at the second or first level even when the weights were changed, showing the least fluctuation among all alternatives. In the scenarios at $w\le 0.25$, it only changes the rankings of alternatives 5 and 2. $A_5$ is the optimal alternative, $A_2$ is second to $A_5$. When the $w\ge 0.3$, $A_2$ is the optimal alternative, and the ranking of alternatives remains constant as $A_2\succ A_5\succ A_4\succ A_3\succ A_1$. Despite the influence of criteria weighting on outcomes, $A_2$ consistently ranks among the top two alternatives. It can be concluded that the variations of criteria weights have little effect on the ranking order that it provides decision makers confidence in decision making. Therefore, we concluded that our introduced BBWM-PFTOPSIS was consistent and robust in decision-makings, and our analysis indicates that $A_2$ is the optimal alternative.

Comparative analysis

This section includes three comparative analyses demonstrating the advancement of BBWM-PFTOPSIS approach. Firstly, we conducted a comparative analysis of existing similarity measures. Secondly, we compared BWM and BBWM. Finally, we compared our approach with other existing PFS MCDM methods.

In the first comparison, the sine similarity measure is compared with other existing similarity measures.

(1) Zhang (2016) proposed the similarity measures for PFSs as follows:

(14) $S_Z\left(P,Q\right)=\frac{1}{n}\sum _{i=1}^n\frac{\left(\left|{\mu }_P^2\left(x_i\right)-v_Q^2\left(x_i\right)\right|+\left|v_P^2\left(x_i\right)-{\mu }_H^2\left(x_i\right)\right|+\left|{\pi }_P^2\left(x_i\right)-{\pi }_Q^2\left(x_i\right)\right|\right)}{\left(\begin{array}{c}\left|{\mu }_P^2\left(x_i\right)-{\mu }_Q^2\left(x_i\right)\right|+\left|{\nu }_P^2\left(x_i\right)-v_Q^2\left(x_i\right)\right|+\left|{\pi }_P^2\left(x_i\right)-{\pi }_Q^2\left(x_i\right)\right|\\ +\left|{\mu }_P^2\left(x_i\right)-v_Q^2\left(x_i\right)\right|+\left|v_P^2\left(x_i\right)-{\mu }_H^2\left(x_i\right)\right|+\left|{\pi }_P^2\left(x_i\right)-{\pi }_Q^2\left(x_i\right)\right|\end{array}\right)}$(14)

(2) Peng, Yuan, and Yang (2017) developed a new similarity measure for PFSs as follows:

(15) $S_P\left(P,Q\right)=1-\sum _{i=1}^n\frac{\left(\left|{\mu }_P^2\left(x_i\right)-{\nu }_P^2\left(x_i\right)-{\mu }_Q^2\left(x_i\right)+v_Q^2\left(x_i\right)\right|\right)}{2n}$(15)

(3) Wei and Wei (2018) introduced a novel PFSs similarity measure as follows:

(16) $S_W\left(P,Q\right)=\frac{1}{n}\sum _{i=1}^n\frac{{\mu }_P^2\left(x_i\right){\mu }_Q^2\left(x_i\right)+{\nu }_P^2\left(x_i\right)v_Q^2\left(x_i\right)}{\sqrt{{\mu }_P^4\left(x_i\right)+{\nu }_P^4\left(x_i\right)}\sqrt{{\mu }_Q^4\left(x_i\right)+v_Q^4\left(x_i\right)}}$(16)

(4) Rani et al. (2020) presented a modified similarity measure for PFSs as follows:

(17) $S_R\left(P,Q\right)=1-\frac{1}{n}\sum _{i=1}^{n}sin\left[\frac{\pi }{2}\left\{max\left|{\mu }_P^2\left(x_i\right)-{\mu }_Q^2\left(x_i\right)\right|,\left|{\nu }_P^2\left(x_i\right)-v_Q^2\left(x_i\right)\right|\right\}\right]$(17)

The data used to make comparisons were adopted from Jun (2011), and the results are presented in Table 7. In cases 1 and 2, analogous counterintuitive issues occur for $S_R\left(P,Q\right)$. In cases 3 and 4, $S_W\left(P,Q\right)$ is unable to deal with the division by zero issue. Furthermore, in cases 1 and 4, $S_Z\left(P,Q\right)$ and $S_P\left(P,Q\right)$ obtained identical results, which is not unreasonable. $S_A\left(P,Q\right)$ is a more reasonable similarity measure without any counterintuitive examples.

Table 7. Comparison results of the existing PF-similarity measures.

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
$P$	$\left(0.3,0.3\right)$	$\left(0.3,0.4\right)$	$\left(1.0,0.0\right)$	$\left(0.5,0.5\right)$	$\left(0.4,0.2\right)$	$\left(0.4,0.2\right)$
$Q$	$\left(0.4,0.4\right)$	$\left(0.4,0.3\right)$	$\left(0.0,0.0\right)$	$\left(0.0,0.0\right)$	$\left(0.5,0.3\right)$	$\left(0.5,0.2\right)$
$S_Z\left(P,Q\right)$	0.5000	0.0000	0.5000	0.5000	0.6000	0.7000
$S_P\left(P,Q\right)$	1.0000	0.9300	0.5000	1.0000	0.9800	0.9550
$S_W\left(P,Q\right)$	1.0000	0.8550	N/A	N/A	0.9950	0.9960
$S_R\left(P,Q\right)$	0.1097	0.1097	1.0000	0.3827	0.1409	0.1409
$S_A\left(P,Q\right)$	0.9451	0.8903	0.5000	0.8087	0.8903	0.9295

In the second comparison, we compared the BWM and BBWM and obtained the criteria weights of BWM ($w_B$) and BBWM ($w_{BB}$) as $w_B=\left(0.1007,0.0537,0.5436,0.3020\right)$ and $w_{BB}=\left(0.1404,0.0936,0.4553,0.3110\right)$, respectively. As depicted in Figure 2, although the criteria weights have changed, their relative significance ranking remains the same. This demonstrates that the BBWM does not altered the importance of each criterion during methodological improvement. In addition, the introduction of Bayesian theory in BWM creates a certain decision-making environment that avoids further expansion of differences among the criteria.

Figure 2. The criteria weights obtained by BWM and BBWM.

In the final comparison, a comparison was conducted between BBWM-PFTOPSIS and other MCDM approaches under the PFS environment, including WSM, VIKOR, ELECTRE, WAM and AAWA AOs. This was done to thoroughly validate the feasibility and practicality of BBWM-PFTOPSIS. The WSM is a commonly employed scoring model for addressing one-dimensional problems, and Yager (2013) expanded this approach to address the PF scenario. Meanwhile, TOPSIS and VIKOR are two widely utilized compromising approaches with similar calculating principles. In addition, ELECTRE is a classical outranking method. The AO is a critical tool in ranking alternatives by assigning comprehensive values to them (Ye and Chen 2022a), and it has also become a research hotspot in the PFS environment. The arithmetic mean AO has been widely examined in various studies, we opted a novel PFWAM AO developed by Kumar and Chen (2023) to compare with other MCDM methods (as shown in Definition 6). In addition, we modified an interval-valued PFS Aczel Alsina AO proposed by Hussain et al. (2023) into the PFS context (as shown in Definition 7). Consequently, we compared PF-WSM, PF-ELECTRE II, PF-VIKOR, PFWAM and PFAAWA with the proposed PFTOPSIS.

Definition

6 (Kumar and Chen 2023) For a collective of PFNs $p_i=\left({\mu }_i,{\nu }_i\right)$, $w_i$ is the weight vector, then the PFWAM AO is defined as:

(18) $PFWAM\left(p_1,p_2,\dots ,p_i\right)=\left(\sqrt{\sum _{i=1}^n{w}_i{\mu }_i^2},\sqrt{1-\sum _{i=1}^n{w}_i\left(1-{\nu }_i^2\right)}\right)$(18)

Definition

7 (Hussain et al. 2023) For a collective of PFNs $p_i=\left({\mu }_i,{\nu }_i\right)$, $w_i$ is the weight vector, then the modified PFAAWA AO is defined as:

(19) $PFAAWA\left(p_1,p_2,\dots ,p_i\right)=\left(\sqrt{1-e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln\left(1-{\mu }_i^2\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}},e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln{\nu }_i\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\right)$(19)

In the PF-VIKOR method, we set the parameters of maximum group utility to 0.5. For the PFAAWA AO, we set the parameter of $\lambda =1$. Table 8 shows that $A_2$ is identified as the best alternative across all approaches, while the worst alternative is not same. In the PFTOPSIS and PF-VIKOR compromising approaches, $A_1$ is the worst alternative of the prioritization. In other methods, $A_4$ is the worst alternative. In addition, $A_5$ and $A_3$ remain the second- and fourth-ranked across all of the scenarios. Despite PF-ELECTRE has the ability to generate an overall ranking through pairwise comparisons, its complex calculation process may be challenging for some decision-makers. Additionally, the WSM method struggles to handle multi-dimensional MCDM problems. Moreover, score function is used to compare PFNs and identify the defuzzified values obtained by PF-WSM and PFS AOs, but it fails to account for the influence of the hesitancy degree. In contrast, the TOPSIS method has a straightforward computation process, adheres to rational logic, and is easily understandable for decision-makers. Therefore, the proposed BBWM-PFTOPSIS approach is undoubtedly feasible and reliable.

Table 8. Comparison with different methods and ranking of alternatives.

Method	PFTOPSIS	PF-WSM	PF-ELECTRE II	PF-VIKOR	PFWAM	PFAAWA	Evaluation Index	Values	Prioritization of alternatives
PFTOPSIS	1.000						Closeness coefficients ($RC$)	$RC\left(A_1\right)=0.4315$, $RC\left(A_2\right)=0.5361$, $RC\left(A_3\right)=0.4338$, $RC\left(A_4\right)=0.4696$, $RC\left(A_5\right)=0.5044$	A2 $\succ$ A5 $\succ$ A4 $\succ$ A3 $\succ$ A1
PF-WSM	.600	1.000					Score function ($SF$)	$SF\left(A_1\right)=-0.0199$, $SF\left(A_2\right)=0.5609$, $SF\left(A_3\right)=-0.0787$, $SF\left(A_4\right)=-0.7268$, $SF\left(A_5\right)=0.3037$	A2 $\succ$ A5 $\succ$ A1 $\succ$ A3 $\succ$ A4
PF-ELECTRE II	.600	1.000	1.000				Precedence indices ($H$)	$H\left(A_1\right)=0.2814$, $H\left(A_2\right)=0.7354$, $H\left(A_3\right)=0.1537$, $H\left(A_4\right)=0.1007$, $H\left(A_5\right)=0.5100$	A2 $\succ$ A5 $\succ$ A1 $\succ$ A3 $\succ$ A4
PF-VIKOR	1.000	.600	.600	1.000			Compromise indices ($Q$)	$Q\left(A_1\right)=1.0000$, $Q\left(A_2\right)=0.0000$, $Q\left(A_3\right)=0.9930$,$Q\left(A_4\right)=0.8428$, $Q\left(A_5\right)=0.6353$	A2 $\succ$ A5 $\succ$ A4 $\succ$ A3 $\succ$ A1
PFWAM	.600	1.000	1.000	.600	1.000		Score function ($SF$)	$SF\left(A_1\right)=-0.5023$,$SF\left(A_2\right)=0.0423$, $SF\left(A_3\right)=-0.5308$, $SF\left(A_4\right)=-0.7435$, $SF\left(A_5\right)=0.3104$	A2 $\succ$ A5 $\succ$ A1 $\succ$ A3 $\succ$ A4
PFAAWA	.600	1.000	1.000	.600	1.000	1.000	Score function ($SF$)	$SF\left(A_1\right)=-0.0199$, $SF\left(A_2\right)=0.5609$, $SF\left(A_3\right)=-0.0787$, $SF\left(A_4\right)=-0.7268$, $SF\left(A_5\right)=0.3037$	A2 $\succ$ A5 $\succ$ A1 $\succ$ A3 $\succ$ A4

Moreover, we apply Spearman’s Rho (rank-order) correlation analysis to compare the ranking order obtained by BBWM-PFTOPSIS and other techniques. According to the results in Table 8, it indicated that the ranking of BBWM-PFTOPSIS technique is reasonably congruent with other approaches (1.0 for PF-VIKOR; 0.6 for PF-ELECTRE II, PF-WSM, PFWAM and PFAAWA) and that the developed BBWM-PFTOPSIS is efficient.

Conclusions and future studies

Conclusions

Selection of the most suitable knitted fabric among the conflicting properties is a challenge of MCDM. PFSs are an effective method for modeling complicated uncertainties in practical MCDM problems. In addition, BBWM is used to minimize the information loss from a probabilistic perspective, which has not been utilized for textile fabric selection. Moreover, TOPSIS is widely utilized in the textile field. Therefore, motivated by these findings, this research solved the knitted fabric selection problem using BBWM-PFTOPSIS. According to the BBWM, air permeability is the most important criterion. PFTOPSIS indicates that $A_2$ has the greatest RC and that it is the best alternative. To test the robustness of all weights and priority ranking, we conducted a sensitivity analysis. In addition, three comparative analyses were performed: (1) Comparisons between the applied weighted sine similarity measure and other existing similarity measures; (2) Comparisons between the BWM and BBWM; and (3) Comparisons between PF-TOPSIS and other typical PF MCDM approaches. All comparative results indicate the feasibility and reliability of BBWM-PFTOPSIS.

Managerial implications

The evaluation of knitted fabrics in the current study encompassed four parameters, with air permeability deemed the most important criterion. This finding enables company practitioners and managers to prioritize air permeability during garments design and production. This study also provides a decision-making framework that assists practitioners and managers in selecting fabrics suitable for a wider range of textile management domain. The BBWM method employs a probabilistic framework to integrate the preferences of multiple experts and establish the ultimate aggregated criteria priorities. Similarly, the PFTOPSIS approach can be used to assess alternatives for each criterion.

Limitations and future studies

Despite the various advantages of the introduced method, this research still has some limitations: (1) In this research, we developed a novel BBWM-PFTOPSIS approach, which showed that the BBWM can minimize information loss and the weighted sine similarity measure can obtain reliable results. The proposed technique was utilized to solve a specific case about knitted fabrics selection, its generalization to other textile areas remains a topic for discussion. (2) This study considered four criteria to evaluate the comfort of knitted fabrics, but there are other properties that could affect wear comfort that were not included in the analysis. (3) The criteria weights and ranking of knitted fabric were identified using BBWM and PFTOPSIS, respectively. While there exist many other MCDM techniques, their integration with BBWM to effectively address the fabric selection issue is worth exploring. Additionally, there are other MCDM tools available to calculate criteria weights. (4) In certain situations, the sum of squares of membership and nonmembership degrees can exceed 1, which cannot be effectively described by PFSs. To represent such instances of uncertainty, other high-order fuzzy sets are available.

In the future, there are several interesting areas of study worth exploring: (1) Researchers can apply BBWM-PFTOPSIS to tackle other MCDM challenges in the textile field, such as product development, waste management and supplier selection, to demonstrate its effectiveness of BBWM-PFTOPSIS, and make its application more general. (2) As the thermos-physiological comfort of knitted fabric is affected by both heat and moisture transmission mechanisms, other decisive factors, such as the properties of constituent yarns, could be considered to evaluate the knitted fabrics. (3) Moreover, researchers can combine the BBWM with other MCDM techniques under different fuzzy environments, such as the scoring method (i.e., PF-WASPAS (Lahane et al. 2023)), compromising method (i.e., Fuzzy rough numbers Aczel-Alsina MARCOS (Pamučar et al. 2023)), and outranking method (i.e., 2-tuple linguistic m-polar fuzzy PROMETHEE (Akram, Noreen, and Pamucar 2023)). The BBWM could be used to determine the criteria weights, and the scoring, compromising, and outranking methods could be applied to select optimal knitted fabrics. In addition, other methods, such as Analytic Network Process (ANP) and Decision Making Trial and Evaluation Laboratory (DEMATEL), can be employed to derive criteria weights. (4) We can expand the TOPSIS method to other fuzzy contexts, such as Fermatean fuzzy sets (Deveci et al. 2023), q-rung orthopair fuzzy sets (Qahtan et al. 2023), Type-2 neutrosophic fuzzy sets (Deveci et al. 2023), and T-spherical fuzzy sets. (5) Rather than using only similarity measures, PFTOPSIS can be enhanced by incorporating other information measures, such as correlation coefficients and dissimilarity measures, into its structure. In addition, the PFS aggregation operators could be used to rank alternatives (Qahtan et al. 2023).

Highlights

• Utilize the Bayesian best-worst method (BBWM) for criteria weight calculation.

• Introduce PF-TOPSIS, an extension of TOPSIS method, tailored for PF environment.

• Develop the weighted sine similarity-based PFTOPSIS to rank alternative options.

• Propose a hybrid BBWM-PFTOPSIS approach to select knitted fabrics.

• Conduct sensitivity and comparative analyses to assess the method’s feasibility.

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 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 authors.

Additional information

Funding

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