Weighted Aggregated Sum Product Assessment Method Based on Aczel–Alsina T-Norm and T-Conorm Under Bipolar T-Spherical Fuzzy Information: Design Scheme Selection Application

Abstract

Selecting optimal design solutions is inherently complex due to multiple criteria encompassing users’ uncertain needs, experiences, and costs. This process must manage uncertainty and ambiguity, making developing a scientific, rational, and efficient guidance method imperative. Bipolar T-spherical fuzzy sets (BTSFS), a hybrid of bipolar fuzzy sets and T-spherical fuzzy sets, effectively handle the bipolarity inherent in all elements. In this work, we propose a Weighted Aggregated Sum Product Assessment (WASPAS) method based on BTSFS and the Aczel–Alsina T-norm (AATN) and T-conorm (AATCN) to address the problem of selecting conceptual design solutions. We first establish operational rules for BTSFS using AATN and AATCN and introduce weighted aggregation operators (BTSFAAWA) and geometric aggregation operators (BTSFAAWG) while examining fundamental properties, such as idempotency, boundedness, and monotonicity. Subsequently, we propose a two-stage BTSFS-based WASPAS method; criterion weights are calculated using the BTSFAAWA operator, and final rankings are obtained through comprehensive calculations using both the weighted sum method (WSM) based on BTSFAAWA and the weighted product method (WPM) based on BTSFAAWG. Finally, we validate the effectiveness of our method through a case study of the selection of cultural and creative products. Sensitivity and comparative analyses are conducted to demonstrate the advantages of our approach.

1. Introduction

In the modern product development process, product solution selection is regarded as a critical link to ensure that the final design can meet both market demand and user expectations [1]. In recent years, the widespread adoption of digital transformation and AI-assisted design methods has significantly enhanced the product solution selection process [2,3]. These technologies boost design efficiency and optimize resource allocation through data analysis and pattern recognition, freeing designers to focus more on creativity and innovation. Although technological advancements have made the selection of design solutions more efficient, in practical applications, how to quickly and effectively choose among numerous possible solutions has become a significant challenge in design practice [4]. For example, the creativity and complexity of user needs in the design process are often difficult to fully capture through algorithms [3,5]. In addition, data quality and model bias may also hurt decision outcomes. If the data used are not comprehensive enough or have biases, the final decision may result in product design not genuinely meeting the actual needs of users and may even lead to market failure [6]. Therefore, product solution selection is a classic multi-criteria group decision making (MCGDM) problem in such a challenging context [7,8]. MCGDM can comprehensively consider the interests of all parties and multiple standards, enhance the scientificity and rationality of decision making by assigning weights to different criteria, and integrate decision makers’ (DMs) opinions [9,10], helping design teams make wiser choices in complex decision making environments [11,12].

Uncertainty and ambiguity in decision making are important issues in selecting product design solutions [8,13,14]. Fuzzy set (FS) [15] or intuitionistic fuzzy set (IFS) [16] methods provide a solid theoretical basis for evaluating the uncertainty of product design solutions. However, they still have some shortcomings in the evaluation of the design scheme. IFSs mainly focus on membership and non-membership but ignore the uncertainty of DM, which limits their applicability when facing complex and diverse design decision scenarios. In addition, models with uncertainty in DM, such as spherical fuzzy set (SFS) [17,18] or T-spherical fuzzy set (TSFS) [19], have limitations in handling bipolar information (such as extreme views of support and opposition) and cannot fully capture the multidimensional features of complex user needs, thereby affecting the accuracy of decision making. In contrast, the BTSFS [20] provides a more comprehensive framework that simultaneously considers membership, non-membership, and uncertainty while considering both supporting and opposing bipolar information. By flexibly adjusting the parameter t, DMs can control the range of these elements, thereby enhancing the model’s adaptability. This advantage enables BTSFS to have higher flexibility and accuracy in handling complex decisions, especially in product design scheme selection, which can effectively integrate different types of evaluation information and optimize subsequent product development and optimization. Therefore, BTSFS is expected to compensate for the shortcomings of traditional fuzzy set methods and provide stronger decision support for complex product design and market demand.

The WASPAS method [21] is a multi-criteria decision making method that primarily relies on the WSM and the WPM to calculate the score of the final solution. The WASPAS method aims to provide a comprehensive evaluation scheme by simultaneously considering the weights and ratings of multiple criteria. Therefore, its fuzzy extension method has been widely used in engineering fields, such as industrial robot selection [22] and renewable energy scheme evaluation [23]. However, the existing WASPAS methods for fuzzy set extension have an insufficient ability to handle complex bipolar fuzzy information, making it difficult to integrate multiple types of information for effective decision making. In addition, the WASPAS method requires introducing weighted aggregation operators and geometric operators to adapt to fuzzy environments, thereby effectively integrating decision information. Existing fuzzy extended WASPAS methods are also unable to handle BTSF information effectively.

Applying AATN and AATCN to fuzzy aggregation operators can effectively handle uncertainty and fuzziness and improve the reliability of aggregation results, as the AATN and AATCN have good mathematical properties that help ensure consistency in the aggregation process [24]. In addition, AATN and AATCN can more flexibly express decision makers’ preferences through parameters, making the aggregation operator more in line with practical needs. The aggregation operators developed based on AATN and AATCN have been successfully applied in fuzzy environments, such as IFS [25,26], Pythagorean fuzzy set (PFS) [27], q-rung orthopair fuzzy set (q-ROFS) [28], and TSFS [29].

Although the BTSFS theory has demonstrated significant advantages, research on its aggregation operators and extended WASPAS methods is still insufficient, indicating that the potential of BTFS has not been fully explored and requires further exploration. Faced with the complexity of the market environment, DM needs to rely on more effective tools to make design decisions. Developing aggregation operators for BTSFS and WASPAS-based methods can improve decision quality under uncertainty and in the face of complex criteria, thus enhancing decision flexibility and adaptability. Therefore, our research aims to develop a WASPAS method based on BTSFS to provide decision makers with practical and effective support for optimizing product solution selection. Overall, our work contributions are as follows:

• We developed the BTSFAAWA and BTSFAAWG operators based on AATN and AATCN to flexibly aggregate BTFS information. We also verified the idempotency, boundedness, and monotonicity properties of these operators.
• We extended the WASPAS method to the BTSFS environment by proposing the BTSF-WASPAS method. Utilizing the developed BTSFAAWA operator to calculate criteria weights, we ultimately ranked the design schemes through comprehensive calculations using both the BTSFAAWA and BTSFAAWG operators.
• We validated the effectiveness and flexibility of the BTSF-WASPAS method in practical applications through a case study of cultural and creative products, demonstrating the advantages of BTSFS in complex MCGDM.
• We conducted a sensitivity analysis to explore the effects of parameter changes in the BTSFAAWA operator, the BTSFAAWG operator, BTSFS, and WASPAS methods on the final ranking results. Through comparison with the TSFS-based WASPAS method, we validated the effectiveness of the BTSF-WASPAS method and highlighted its advantages.

The remaining part of our work is organized as follows. Section 2 introduces the background of our research, including the study of the MCGDM method and the WASPAS method in the product design scheme selection, and it summarizes the research gap and the motivation. The third section describes the basic theory, the proposed aggregation operator, and the BTSF-WASPAS method. The fourth section showcases the implementation process of our method through real cases. The fifth section discusses the impact of the parameters of our method on the decision results through a sensitivity analysis and a comparative analysis. Finally, the sixth section summarizes our work.

2. Background

2.1. Research on the Fuzzy MCGDM Method for Product Scheme Selection

The fuzzy MCGDM method has been widely used in product solution selection because it can effectively address the fuzziness and uncertainty in the decision making process [7,8,11,12]. When facing complex market environments and diverse user demands, traditional decision making methods often fail to fully consider various standards and stakeholder perspectives. Therefore, researchers continue to explore fuzzy multi-criteria group decision making methods to enhance DM’s scientific and flexible nature. These methods can handle the opinions and weights of different decision makers and accurately describe each solution’s advantages and disadvantages through fuzzy logic. For example, Lin et al. [15] proposed a fuzzy analytic hierarchy process (AHP) and TOPSIS to calculate and evaluate fashion design schemes. Yan et al. [6] introduced the Evidence-Based Reasoning (ER) theory of uncertain reasoning into fuzzy AHP to calculate the performance score of alternative solutions, focusing on solving the problem of ranking reversal in the product solution decision making process. The VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje) method, based on a rough set improvement proposed by Tiwari et al. [13], evaluates product design concepts and selects appropriate concept combination schemes. The focus of this method is not a consideration of the cost and benefit characteristics of design standards but rather determining the optimal concept that meets the constraints imposed by the design team on the design standards and that satisfies the maximum customer preference. Chen et al. [30] proposed a new rough fuzzy data envelopment analysis (DEA) method to select appropriate intelligent service system design schemes, integrating the advantages of fuzzy numbers in capturing interpersonal uncertainty and the feasibility of rough numbers in perceiving interpersonal uncertainty. Zhu et al. [31] proposed a novel AHP based on rough numbers to determine the weights of each evaluation criterion. Then, they proposed an improved VIKOR based on coarse numbers to evaluate design concept alternatives. Cui et al. [7] proposed a conceptual design decision method considering multi-granularity heterogeneous evaluation semantics and uncertain beliefs. It generates random numbers based on the distribution of cloud models to construct an advantage matrix and uses the VIKOR method to select the optimal solution. Huang et al. [32] proposed a mixed-preference-based MCGDM method that utilizes spherical Z fuzzy numbers to solve the problem of fuzzy information in green product design. Liu et al. [17] proposed an information axiom method that integrates spherical fuzzy AHP and axiomatic design to evaluate the human–machine interface design scheme of fitness equipment. Zhang et al. [33] proposed a hybrid MCGDM method that combines the Best–Worst method (BWM) and evaluation based on the distance from the average solution (EDAS) to obtain the optimal design solution. Geng et al. [16] proposed a new IFS-based BWM-DEA model that considers the preferences of DM to calculate the efficiency scores of alternative solutions and rank them based on their efficiency scores. Regarding the confidence level of DM, Aydoğan et al. [8] proposed an axiomatic design method based on Z-number to evaluate conceptual design schemes. Similarly, Liu et al. [34] proposed a design scheme selection method that considered DM confidence level through Z-number. They obtained the priority of the scheme through AHP and the technique for order preference by similarity to ideal solution (TOPSIS) method. Jing et al. [14] proposed a conceptual scheme decision model based on interval-valued IFS considering diversified customer preference distributions. The evaluation data of the same preference in the scheme are integrated, and the comprehensive satisfaction of the scheme is obtained through the interval-valued IFS-weighted aggregation operator, thereby determining the optimal scheme. Yang et al. [35] proposed the q-ROF power Muirhead mean operator and the q-ROF weighted power Muirhead mean operator method for evaluating the traceability system of agricultural products in blockchain technology. We have summarized the research on using the MCGDM method for product design scheme evaluation, including the criterion weight calculation method, the scheme ranking method, and application cases, as shown in

Table 1. MCGDM method for product design scheme evaluation.

Study	Uncertainty Information Representation	Weight Calculation	Ranking Method	Application
[15]	FS	AHP	TOPSIS	Fashion design
[6]	FS	AHP	AHP	Rotor and bearing system in turbo generator design
[13]	Rough set	Aggregation method	VIKOR	Testing rigmachine design
[30]	Fuzzy rough set	DEA	DEA	Product service system design
[31]	Fuzzy rough set	AHP	VIKOR	Lithography tool design
[7]	Cloud model	maximum deviation model	VIKOR	Design of tree climbing and pruning machine
[32]	SFS	comprehensive analysis	Gray relation analysis	Green product design
[17]	SFS	AHP	Axiomatic design	Fitness product design
[33]	q-ROFS	BWM	EDAS	Refrigerator product design
[16]	IFS	BWM	DEA	Laptop design
[8]	Z-number	-	Axiomatic design	Pen design
[34]	Z-number	AHP	TOPSIS	Kitchen garbage can design
[14]	IFS	Aggregation method	Aggregation method	Mobile phone design
[35]	q-ROFS	Aggregation method	Aggregation method	Agricultural product system design

.

2.2. Research on WASPAS and Its Fuzzy Extension Method

The WASPAS method, which extends through fuzzy sets, has made significant progress in decision science research. This method combines fuzzy set theory with MCGDM methods, providing an effective tool for dealing with uncertainty and complexity. In recent years, researchers have explored the applicability of fuzzy WASPAS methods in different application scenarios, particularly in areas like product solution selection [36], supply chain management [37], and risk assessment [38]. With the development of relevant theories and the increase in practical needs, the fuzzy WASPAS method has gradually evolved into various variants aiming to improve the flexibility and accuracy of decision making. Specifically, Stanujkić et al. [39] proposed an extension of the WASPAS method using IFSs and combined it with the Hamming distance method to consider the efficiency and usability of the proposed method in evaluating example websites. Keshavarz-Ghorabaee [40] proposed an extension of the WASPAS method using Fermatan fuzzy sets (FFSs) and a combined simple multi-attribute rating technique (SMART) to deal with information uncertainty. Senapati et al. [36] established the WASPAS method for MCGDM in a picture fuzzy set environment. They provided scenario studies involving appropriate air conditioning system selection. Kutlu Gundogdu et al. [22] extended the traditional WASPAS method to the SFS environment. They demonstrated its application through an industrial robot selection problem. Thilagavathy et al. [41] developed Maclaurin symmetric mean aggregation operators based on the Hamacher operations of SFS. They defined this operator to develop WASPAS for solving MCGDM problems in a spherical fuzzy environment. Arunodaya et al. [37] proposed an integrated method based on the WASPAS approach to solving the MCGDM problems with hesitant fuzzy information. They applied the proposed method to the problem of selecting green suppliers. Meng et al. [38] used the fuzzy AHP method to evaluate and rank environmental, social, and governance standards and sub-standards and the fuzzy WASPAS method to evaluate and rank key investment strategies for green finance development.

2.3. Research Gaps and Motivations

We have summarized the research gap through a literature review, as follows:

• Although the fuzzy MCGDM method has been widely used in product design scheme selection, existing models still have shortcomings in dealing with uncertainty. Some existing models only focus on membership and non-membership [14,16,35], failing to fully consider the uncertainty of DM in complex selection, which limits its effectiveness in practical applications.
• Some existing models can reflect DM’s uncertainty degree [17,32]. However, they have shortcomings in expressing bipolar information of support and opposition. They cannot fully capture key decision factors in complex user needs, thus affecting the final decision’s accuracy.
• The existing fuzzy set extended WASPAS methods have an insufficient ability to handle bipolar information, making it difficult to effectively integrate multiple types of information for complex decision making. It is urgent to improve their flexibility and reliability through a more comprehensive information aggregation theoretical framework.

Based on these research gaps, we have summarized our research motivation as follows:

• Given the shortcomings of existing research, it is essential to develop a new decision making tool that can simultaneously consider membership, non-membership, and uncertainty degrees. BTSFS provides a more comprehensive framework and more comprehensive fuzzy information processing capabilities. Compared to IFS, PFS, and SFS methods, BTFS can simultaneously handle membership, non-membership, and uncertainty degrees and express bipolar information of support and opposition. The parameter t can adjust the range of uncertainty, which makes BTFS more flexible than traditional fuzzy sets, and it more accurately reflects the true intention of DM, enabling it to cope with more complex and diverse design options.
• In addition, developing effective and flexible aggregation operators is the key to solving the information fusion process. Most existing research has not extensively expanded the use of aggregation operators. By introducing the advantages of AATN and AATCN, combined with BTSFS theory, we aim to develop new aggregation operators to enhance the application effect of the WASPAS method in practical decision making and improve the scientificity and accuracy of decision making. This innovation fills the gap in the existing research and provides new possibilities for the future development of fuzzy MCGDM methods.

3. Method

3.1. Preliminaries

We have summarized the research gap through a literature review, as follows. Since Zadeh put forward the fuzzy set theory, many researchers have used different fuzzy extensions to deal with uncertainty, and they have discussed them in relation to different applications. SFS [42] and TSFS [19] consider membership, non-membership, and hesitancy degrees simultaneously, so they have more space to describe uncertainty than an ordinary fuzzy set. BTSFS [20], as an extended model of SFS, incorporates the DM’s opinions on fuzzy set parameters into the model in an interval rather than a single value, which can control the uncertain range through parameter t and provide more flexibility. In this section, we recall some basic definitions of BTSFSs, AATNs, and AATCNs for further development of this paper.

Definition 1 ([19]). 

In a TSFS $\mathcal{A}$ of the universe of discourse, X is given by

$$\mathfrak{I}=\left\{〈x,\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right)〉\left|x\in X\right.\right\},$$

(1)

where $0\le s_{\mathcal{A}}\left(x\right)\le 1$, $0\le d_{\mathcal{A}}\left(x\right)\le 1$, $0\le i_{\mathcal{A}}\left(x\right)\le 1$, $0\le {\left(s_{\mathcal{A}}\left(x\right)\right)}^t+{\left(d_{\mathcal{A}}\left(x\right)\right)}^t+{\left(i_{\mathcal{A}}\left(x\right)\right)}^t\le 1$, and $t$ is a positive integer. For each $x\in X$, $s_{\mathcal{A}}\left(x\right)$, $t_{\mathcal{A}}\left(x\right)$, and $i_{\mathcal{A}}\left(x\right)$ are the membership degree, the non-membership degree, and the uncertainty degree of $x$ to $\mathcal{A}$.

Definition 2 ([43]).

Let X be a nonempty set. A bipolar fuzzy set $\mathcal{B}$ in X is an object having the form

$$\mathcal{B}=\left\{〈x,\left({\mu }_{\mathcal{B}}^{\mathcal{P}}\left(x\right),{\mu }_{\mathcal{B}}^{\mathcal{N}}\left(x\right)\right)〉\left|x\in X\right.\right\},$$

(2)

where $0\le {\mu }_{\mathcal{B}}^{\mathcal{P}}\left(x\right)\le 1$ and $-1\le {\mu }_{\mathcal{B}}^{\mathcal{N}}\left(x\right)\le 0$. For each $x\in X$, ${\mu }_{\mathcal{B}}^{\mathcal{P}}\left(x\right)$ and ${\mu }_{\mathcal{B}}^{\mathcal{N}}\left(x\right)$ are the positive membership degree and the negative membership degree of $x$ to $\mathcal{B}$.

Definition 3 ([43]).

In a BTSFS $\mathfrak{I}$ of the universe of discourse, X is given by

$$\mathfrak{I}=\left\{〈x,\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right)〉\left|x\in X\right.\right\},$$

(3)

where $0\le s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\le 1$, $0\le d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\le 1$, $0\le i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\le 1$, $-1\le s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\le 0$, $-1\le d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\le 0$, $-1\le i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\le 0$, with restrictions, $0\le {\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t+{\left(d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t+{\left(i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t\le 1$, $-1\le \left(-{\left|s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t\right)+\left(-{\left|d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t\right)+\left(-{\left|i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t\right)\le 0$, $0\le {\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t+{\left(d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t+{\left(i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)\right)}^t+{\left|s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t+{\left|d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t+{\left|i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right|}^t\le 2$, and $t$ is a positive integer. For each $x\in X$, $s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)$, $d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)$, and $i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right)$ are the positive membership degree, the non-membership degree, and the uncertainty degree of $x$ to $\mathfrak{I}$, respectively. Similarly, for each $x\in X$, $s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)$, $d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)$, and $i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)$ are the negative membership degree, the non-membership degree, and the uncertainty degree of $x$ to $\mathfrak{I}$, respectively. For a BTSFS $\mathfrak{I}$, the pair $\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right)$ is called a bipolar T-spherical fuzzy number (BTSFN). For convenience, the pair $\left(s_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{P}}\left(x\right),s_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),d_{\mathfrak{I}}^{\mathcal{N}}\left(x\right),i_{\mathfrak{I}}^{\mathcal{N}}\left(x\right)\right)$ is denoted by $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$, where $s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}}\in \left[0, 1\right]$, $s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\in \left[-1, 0\right]$, $0\le {\left(s^{\mathcal{P}}\right)}^t+{\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\le 1$, $-1\le \left(-{\left|s^{\mathcal{N}}\right|}^t\right)+\left(-{\left|d^{\mathcal{N}}\right|}^t\right)+\left(-{\left|i^{\mathcal{N}}\right|}^t\right)\le 0$, and $0\le {\left(s^{\mathcal{P}}\right)}^t+{\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t+{\left|s^{\mathcal{N}}\right|}^t+{\left|d^{\mathcal{N}}\right|}^t+{\left|i^{\mathcal{N}}\right|}^t\le 2$.

Definition 4.

The score function of BTSFN $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$ is defined as

$$\mathbb{S}\left(\mathfrak{s}\right)={\displaystyle \frac{1+{\left(s^{\mathcal{P}}\right)}^{\mathrm{t}}\left(1-{\left(d^{\mathcal{P}}\right)}^{\mathrm{t}}-{\left(i^{\mathcal{P}}\right)}^{\mathrm{t}}\right)-{\left|s^{\mathcal{N}}\right|}^{\mathrm{t}}\left(1-{\left|d^{\mathcal{N}}\right|}^{\mathrm{t}}-{\left|i^{\mathcal{N}}\right|}^{\mathrm{t}}\right)}{2},}$$

(4)

where $\mathbb{S}\left(\mathfrak{s}\right) \in [0,1].$

Definition 5.

The accuracy function of BTSFN $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$ is defined as

$$\mathbb{A}\left(\mathfrak{s}\right)={\displaystyle \frac{{\left(s^{\mathcal{P}}\right)}^{\mathrm{t}}+{\left(d^{\mathcal{P}}\right)}^{\mathrm{t}}+{\left(i^{\mathcal{P}}\right)}^{\mathrm{t}}{+\left|s^{\mathcal{N}}\right|}^{\mathrm{t}}+{\left|d^{\mathcal{N}}\right|}^{\mathrm{t}}+{\left|i^{\mathcal{N}}\right|}^{\mathrm{t}}}{2},}$$

(5)

where $\mathbb{A}\left(\mathfrak{s}\right)\in \left[\mathrm{0,1}\right]$.

Definition 6.

Let ${\mathfrak{s}}_1=\left(s_1^{\mathcal{P}},d_1^{\mathcal{P}},i_1^{\mathcal{P}},s_1^{\mathcal{N}},d_1^{\mathcal{N}},i_1^{\mathcal{N}}\right)$ and ${\mathfrak{s}}_2=\left(s_2^{\mathcal{P}},d_2^{\mathcal{P}},i_2^{\mathcal{P}},s_2^{\mathcal{N}},d_2^{\mathcal{N}},i_2^{\mathcal{N}}\right)$ be any two BTSFNs, and then an ordered relation between these two BTSFNs is established as follows:

(1) 

If $\mathbb{S}\left({\mathfrak{s}}_1\right)<\mathbb{S}\left({\mathfrak{s}}_2\right)$, then ${\mathfrak{s}}_1\prec {\mathfrak{s}}_{2 }$ (${\mathfrak{s}}_1$ is inferior to ${\mathfrak{s}}_2$);

(2) 

If $\mathbb{S}\left({\mathfrak{s}}_1\right)<\mathbb{S}\left({\mathfrak{s}}_2\right)$, then ${\mathfrak{s}}_1\prec {\mathfrak{s}}_2$ (${\mathfrak{s}}_1$ is superior to ${\mathfrak{s}}_2$);

(3) 

If $\mathbb{S}\left({\mathfrak{s}}_1\right)<\mathbb{S}\left({\mathfrak{s}}_2\right)$, then

• If $\mathbb{A}\left({\mathfrak{s}}_1\right)<\mathbb{A}\left({\mathfrak{s}}_2\right)$, then ${\mathfrak{s}}_1\prec {\mathfrak{s}}_2$ (${\mathfrak{s}}_1$ is inferior to ${\mathfrak{s}}_2$);
• If $\mathbb{A}\left({\mathfrak{s}}_1\right)>\mathbb{A}\left({\mathfrak{s}}_2\right)$, then ${\mathfrak{s}}_1\succ {\mathfrak{s}}_2$ (${\mathfrak{s}}_1$ is superior to ${\mathfrak{s}}_2$);
• If $\mathbb{A}\left({\mathfrak{s}}_1\right)=\mathbb{A}\left({\mathfrak{s}}_2\right)$, then ${\mathfrak{s}}_1\sim {\mathfrak{s}}_2$ (${\mathfrak{s}}_1$ is equivalent to ${\mathfrak{s}}_2$).

Definition 7 ([44]).

For any pair of real numbers u and v, the AATN ${\left(T_A^{\eta }\right)}_{\eta \in \left(0,+\infty \right)}$ and AATCN ${\left(S_A^{\eta }\right)}_{\eta \in \left(0,+\infty \right)}$ are ascertained by

$$T_A^{\eta }\left(u,v\right)=e^{-{\left({\left(-ln\left(u\right)\right)}^{\eta }+{\left(-ln\left(v\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}},$$

(6)

$$S_A^{\eta }\left(u,v\right)=1-e^{-{\left({\left(-ln\left(1-u\right)\right)}^{\eta }+{\left(-ln\left(1-v\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}},$$

(7)

where $\eta \ge 1$ and $\left(u,v\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,1}\right]$. Here, $\eta$ is the operational parameter of Aczel–Alsina operations.

In this paper, the AATN and AATCN will be utilized to develop aggregation operators in a BTSF environment.

3.2. Operation Rules of BTSFN Based on AATN and AATCN

In this section, we first develop some operational rules for BTSFNs with the assistance of AATN and AATCN. Next, the proposed operational rules are utilized to develop weighted averaging and a weighted geometric aggregation operator.

Definition 8.

Assume that $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$, ${\mathfrak{s}}_1=\left(s_1^{\mathcal{P}},d_1^{\mathcal{P}},i_1^{\mathcal{P}},s_1^{\mathcal{N}},d_1^{\mathcal{N}},i_1^{\mathcal{N}}\right)$, and ${\mathfrak{s}}_2=\left(s_2^{\mathcal{P}},d_2^{\mathcal{P}},i_2^{\mathcal{P}},s_2^{\mathcal{N}},d_2^{\mathcal{N}},i_2^{\mathcal{N}}\right)$ are any three BTSFNs, and the AATN and AATCN operations are defined as follows.

(1) 

Addition of two BTSFNs:

$${\mathfrak{s}}_1⨁{\mathfrak{s}}_2=\left(\begin{array}{c}\begin{array}{c}\sqrt[t]{1-e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\end{array}\\ \sqrt[t]{e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \begin{array}{c}-\sqrt[t]{1-e^{-{\left({\left(-ln\left(1-{\left|s_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left|s_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{e^{-{\left({\left(-ln\left({\left|d_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left({\left|d_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\left(-ln\left({\left|d_1^{\mathcal{N}}\right|}^t+{\left|i_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left({\left|d_2^{\mathcal{N}}\right|}^t+{\left|i_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left({\left|d_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left({\left|d_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\end{array}\right)$$

(8)

(2) 

Multiplication of two BTSFNs:

$$\begin{array}{c}{\mathfrak{s}}_1⨂{\mathfrak{s}}_2=\left(\begin{array}{c}\begin{array}{c}\sqrt[t]{e^{-{\left({\left(-ln\left({\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{1-e^{-{\left({\left(-ln\left(1-{\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\end{array}\\ \sqrt[t]{e^{-{\left({\left(-ln\left(1-{\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left(1-{\left(d_1^{\mathcal{P}}\right)}^t-{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(d_2^{\mathcal{P}}\right)}^t-{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\left(-ln\left({\left|s_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left({\left|s_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{1-e^{-{\left({\left(-ln\left(1-{\left|d_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left|d_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\left(-ln\left(1-{\left|d_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left|d_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left(1-{\left|d_1^{\mathcal{N}}\right|}^t-{\left|i_1^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left|d_2^{\mathcal{N}}\right|}^t-{\left|i_2^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\right)\end{array}$$

(9)

(3) 

Multiplication of a crisp value $\lambda$ ($\lambda$ > 0):

$$\begin{array}{c}\begin{array}{c}\lambda \mathfrak{s}=\left(\begin{array}{c}\sqrt[t]{1-e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\\ \begin{array}{c}-\sqrt[t]{1-e^{-{\left(\lambda {\left(-ln\left(1-{\left|s^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left|d^{\mathcal{N}}\right|}^t+{\left|i^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\end{array}\right)\end{array}\end{array}$$

(10)

(4) 

Exponent ($\lambda >0$) of a BTSFN:

$$\begin{array}{c}\begin{array}{c}\begin{array}{c}{\mathfrak{s}}^{\lambda }=\left(\begin{array}{c}\sqrt[t]{e^{-{\left(\lambda {\left(-ln\left({\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{1-e^{-{\left({\lambda \left(-ln\left(1-{\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \sqrt[t]{e^{-{\left(\lambda {\left(-ln\left(1-{\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left(1-{\left(d^{\mathcal{P}}\right)}^t-{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\\ \begin{array}{c}-\sqrt[t]{e^{-{\left(\lambda {\left(-ln\left({\left|s^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{1-e^{-{\left({\lambda \left(-ln\left(1-{\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left(\lambda {\left(-ln\left(1-{\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left(1-{\left|d^{\mathcal{N}}\right|}^t-{\left|i^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\end{array}\right)\end{array}\end{array}\end{array}$$

(11)

Theorem 1.

If $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$, ${\mathfrak{s}}_1=\left(s_1^{\mathcal{P}},d_1^{\mathcal{P}},i_1^{\mathcal{P}},s_1^{\mathcal{N}},d_1^{\mathcal{N}},i_1^{\mathcal{N}}\right)$, and ${\mathfrak{s}}_2=\left(s_2^{\mathcal{P}},d_2^{\mathcal{P}},i_2^{\mathcal{P}},s_2^{\mathcal{N}},d_2^{\mathcal{N}},i_2^{\mathcal{N}}\right)$ are any three BTSFNs and $\lambda >0$ is a real number, then $\lambda \mathfrak{s}$, ${\mathfrak{s}}^{\lambda }$, ${\mathfrak{s}}_1⨁{\mathfrak{s}}_2$, and ${\mathfrak{s}}_1⨂{\mathfrak{s}}_2$ are also BTSFNs.

Proof of Theorem 1.

(1)

Let $\lambda \mathfrak{s}=\left(a^{\mathcal{P}},b^{\mathcal{P}},c^{\mathcal{P}},o^{\mathcal{N}},p^{\mathcal{N}},q^{\mathcal{N}}\right)$. Then, from Definition 8, we have

$$\begin{gathered} a^{\mathcal{P}}=\sqrt[t]{1-e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},b^{\mathcal{P}}=\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\mathrm{and}{c}^{\mathcal{P}}= \\ \sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}. \end{gathered}$$

Because $\mathfrak{s}$ is a BTSFN, $0\le {\left(s^{\mathcal{P}}\right)}^t+{\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\le 1$, we can obtain $1-{\left(s^{\mathcal{P}}\right)}^t\ge {\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\ge 0$. Then,

$$\begin{array}{cc}\hfill -ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)& \ge -ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\Rightarrow e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & \ge e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & \Rightarrow e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\ge 0\Rightarrow 0\hfill \\ & \le {\left(a^{\mathcal{P}}\right)}^t+{\left(b^{\mathcal{P}}\right)}^t+{\left(c^{\mathcal{P}}\right)}^t\hfill \\ & =1-e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}+e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & +e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left(d^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & =1-e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}+e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & =1-\left(e^{-{\left(\lambda {\left(-ln\left(1-{\left(s^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\lambda \left(-ln\left({\left(d^{\mathcal{P}}\right)}^t+{\left(i^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\right)\le 1\hfill \end{array}$$

Similarly,

$$\begin{gathered} o^{\mathcal{N}}=-\sqrt[t]{1-e^{-{\left(\lambda {\left(-ln\left(1-{\left|s^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},p^{\mathcal{N}}=-\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}, \mathrm{a}\mathrm{n}\mathrm{d} q^{\mathcal{N}} \\ =-\sqrt[t]{e^{-{\left({\lambda \left(-ln\left({\left|d^{\mathcal{N}}\right|}^t+{\left|i^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left(\lambda {\left(-ln\left({\left|d^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}} \end{gathered}$$

Because $\mathfrak{s}$ is a BTSFN, $-1\le \left(-{\left|s^{\mathcal{N}}\right|}^t\right)+\left(-{\left|d^{\mathcal{N}}\right|}^t\right)+\left(-{\left|i^{\mathcal{N}}\right|}^t\right)\le 0\Rightarrow {0\le \left|s^{\mathcal{N}}\right|}^t+{\left|d^{\mathcal{N}}\right|}^t+{\left|i^{\mathcal{N}}\right|}^t\le 1$, we can obtain $-1\le \left(-{\left|o^{\mathcal{N}}\right|}^t\right)+\left(-{\left|p^{\mathcal{N}}\right|}^t\right)+\left(-{\left|q^{\mathcal{N}}\right|}^t\right)\le 0$. Therefore, $\mathsf{\lambda }\mathfrak{s}$ is a BTSFN.

(2)

Let ${\mathfrak{s}}_1⨁{\mathfrak{s}}_2=\left({\mathrm{a}}^{\mathcal{P}},b^{\mathcal{P}},{\mathrm{c}}^{\mathcal{P}},o^{\mathcal{N}},p^{\mathcal{N}},q^{\mathcal{N}}\right)$. Then, from Definition 8, we have

$$\begin{gathered} {\mathrm{a}}^{\mathcal{P}}=\sqrt[t]{1-e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},b^{\mathcal{P}}=\sqrt[t]{e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}, \mathrm{a}\mathrm{n}\mathrm{d} \\ {\mathrm{c}}^{\mathcal{P}}=\sqrt[t]{e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}} \end{gathered}$$

Because ${\mathfrak{s}}_1$ and ${\mathfrak{s}}_2$ are BTSFNs, $0\le {\left(s_1^{\mathcal{P}}\right)}^t+{\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\le 1$ and $0\le {\left(s_2^{\mathcal{P}}\right)}^t+{\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\le 1$. We can obtain $1-{\left(s_1^{\mathcal{P}}\right)}^t\ge {\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t$ and $1-{\left(s_2^{\mathcal{P}}\right)}^t\ge {\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t$. Then,

$$\begin{gathered} -ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\ge -ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right) \mathrm{and}-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\ge -ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right) \\ \Rightarrow {\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\ge {\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\mathrm{and} {\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\ge \\ {\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta } \\ \Rightarrow {\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta } \\ \ge {\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta } \\ \Rightarrow e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\ge e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}} \\ \Rightarrow e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}} \\ \ge 0 \end{gathered}$$

Now,

$$\begin{array}{cc}\hfill 0& \le {\left({\mathrm{a}}^{\mathcal{P}}\right)}^t+{\left({\mathrm{b}}^{\mathcal{P}}\right)}^t+{\left({\mathrm{c}}^{\mathcal{P}}\right)}^t\hfill \\ & =1-e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & +e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & +e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & -e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & =1-e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & +e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\hfill \\ & =1-\left(e^{-{\left({\left(-ln\left(1-{\left(s_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left(1-{\left(s_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\left(-ln\left({\left(d_1^{\mathcal{P}}\right)}^t+{\left(i_1^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }+{\left(-ln\left({\left(d_2^{\mathcal{P}}\right)}^t+{\left(i_2^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}\right)\hfill \\ & \le 1\hfill \end{array}$$

Similarly, we can obtain $-1\le \left(-{\left|o^{\mathcal{N}}\right|}^t\right)+\left(-{\left|p^{\mathcal{N}}\right|}^t\right)+\left(-{\left|q^{\mathcal{N}}\right|}^t\right)\le 0$. Therefore, ${\mathfrak{s}}_1⨁{\mathfrak{s}}_2$ is a BTSFN.

(3)

We can prove in the same way that ${\mathfrak{s}}^{\lambda }$ and ${\mathfrak{s}}_1⨂{\mathfrak{s}}_2$ are BTSFNs. □

Theorem 2.

Let $\lambda$, ${\lambda }_1$, and ${\lambda }_2>0$. Then,

$${\mathfrak{s}}_1⨁{\mathfrak{s}}_2={\mathfrak{s}}_2⨁{\mathfrak{s}}_1$$

(12)

$${\mathfrak{s}}_1⨂{\mathfrak{s}}_2={\mathfrak{s}}_2⨂{\mathfrak{s}}_1$$

(13)

$$\mathsf{\lambda }\left({\mathfrak{s}}_1⨁{\mathfrak{s}}_2\right)=\mathsf{\lambda }{\mathfrak{s}}_1⨁\mathsf{\lambda }{\mathfrak{s}}_2$$

(14)

$${\mathsf{\lambda }}_1\mathfrak{s}⨁{\mathsf{\lambda }}_2\mathfrak{s}=\left({\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2\right)\mathfrak{s}$$

(15)

$${\left({\mathfrak{s}}_1⨂{\mathfrak{s}}_2\right)}^{\mathsf{\lambda }}={{\mathfrak{s}}_1}^{\mathsf{\lambda }}⨂{{\mathfrak{s}}_2}^{\mathsf{\lambda }}$$

(16)

$${\mathfrak{s}}^{{\mathsf{\lambda }}_1}⨂{\mathfrak{s}}^{{\mathsf{\lambda }}_2}={\mathfrak{s}}^{\left({\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2\right)}$$

(17)

3.3. BTSFAAWA and BTSFAAWG Operators

In this section, we develop BTSFAAWA and BTSFAAWG operators using Definition 8. The BTSFAAWA and BTSFAAWG operators are utilized to aggregate the decision BTSF information from several DMs.

Definition 9.

Suppose ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (h = 1, 2, …, n) is a group of BTSFNs. Then, the BTSFAAWA operator is a function $BTSFAAWA:{BTSFN}^n\to BTSFN$ with regard to $w=\left(w_1,w_2,...,w_n\right);w\in \left[\mathrm{0,1}\right]; {\sum }_{h=1}^n{w}_h=1$ is defined by

$$\begin{array}{cc}\hfill BTSFAAWA\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)& =w_1{\mathfrak{s}}_1⨁w_2{\mathfrak{s}}_2⨁\dots ⨁w_n{\mathfrak{s}}_n\hfill \\ & =\left(\begin{array}{c}\sqrt[t]{1-e^{-{\left({\sum }_{h=1}^n{w}_h{\left(-ln\left(1-{\left(s_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{e^{-{\left({\sum }_{h=1}^n{w_h\left(-ln\left({\left(d_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \sqrt[t]{e^{-{\left({\sum }_{h=1}^n{w_h\left(-ln\left({\left(d_h^{\mathcal{P}}\right)}^t+{\left(i_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\sum }_{h=1}^n{w}_h{\left(-ln\left({\left(d_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{1-e^{-{\left({\sum }_{h=1}^n{w}_h{\left(-ln\left(1-{\left|s_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{e^{-{\left({\sum }_{h=1}^n{w_h\left(-ln\left({\left|d_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\sum }_{h=1}^n{w_h\left(-ln\left({\left|d_h^{\mathcal{N}}\right|}^t+{\left|i_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\sum }_{h=1}^n{w}_h{\left(-ln\left({\left|d_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\right)\hfill \end{array}$$

(18)

We might demonstrate the accompanying properties effectively by utilizing the BTSFAAWA operator.

Theorem 3.

(Idempotency). If all of the BTSFNs in the collection ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (h = 1, 2, …, n) are identical to $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$, then $BTSFAAWA\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)=\mathfrak{s}$.

Theorem 4.

(Boundedness). Let ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (h = 1, 2, …, n) be the collection of BTSFNs. Let ${\mathfrak{s}}^{-}=min〈\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)〉=\left(s^{\mathcal{P}-},d^{\mathcal{P}-},i^{\mathcal{P}-},s^{\mathcal{N}-},d^{\mathcal{N}-},i^{\mathcal{N}-}\right)$, and ${\mathfrak{s}}^{+}=min〈\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)〉=\left(s^{\mathcal{P}+},d^{\mathcal{P}+},i^{\mathcal{P}+},s^{\mathcal{N}+},d^{\mathcal{N}+},i^{\mathcal{N}+}\right)$, Then, ${\mathfrak{s}}^{-}\le BTSFAAWA\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)\le {\mathfrak{s}}^{+}$.

Theorem 5.

(Monotonicity). Let ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (i = 1, 2, …, n) and ${\mathfrak{s}\prime }_h=\left({s\prime }_h^{\mathcal{P}},{d\prime }_h^{\mathcal{P}},{i\prime }_h^{\mathcal{P}},{s\prime }_h^{\mathcal{N}},{d\prime }_h^{\mathcal{N}},{i\prime }_h^{\mathcal{N}}\right)$ (I = 1, 2, …, n) be two sets of BTSFNs with ${\mathfrak{s}}_h\le {\mathfrak{s}\prime }_h,\forall h=\mathrm{1,2},\dots ,n.$ Then, $BTSFAAWA\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)\le BTSFAAWA\left({\mathfrak{s}\prime }_1,{\mathfrak{s}\prime }_2,\dots ,{\mathfrak{s}\prime }_n\right)$.

Definition 10.

Suppose ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$(h = 1, 2, …, n) is a group of BTSFNs. Then, the BTSFAAWG operator is a function $BTSFAAWG:{BTSFN}^n\to BTSFN$ with regard to $w=\left(w_1,w_2,\dots ,w_n\right);w\in \left[\mathrm{0,1}\right]; {\sum }_{i=1}^n{w}_i=1$ is defined as follows.

We can demonstrate the accompanying properties effectively by utilizing the BTSFAAWG operator.

$$\begin{array}{cc}\hfill BTSFAAWG\left({\mathfrak{s}}_1,{\mathfrak{s}}_2\dots .,{\mathfrak{s}}_n\right)& ={{\mathfrak{s}}_1}^{w_1}⨂{{\mathfrak{s}}_2}^{w_2}⨂\dots ⨂{{\mathfrak{s}}_n}^{w_n}\hfill \\ & =\left(\begin{array}{c}\sqrt[t]{e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln\left({\left(s_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\sqrt[t]{{1-e}^{-{\left({\sum }_{i=1}^n{w_i\left(-ln\left({1-\left(d_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \sqrt[t]{e^{-{\left({\sum }_{i=1}^n{w_i\left(-ln\left(1-{\left(d_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln\left(1-{\left(d_h^{\mathcal{P}}\right)}^t-{\left(i_h^{\mathcal{P}}\right)}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ \begin{array}{c}-\sqrt[t]{e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln\left({\left|s_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},-\sqrt[t]{{1-e}^{-{\left({\sum }_{i=1}^n{w_i\left(-ln\left({1-\left|d_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}},\\ -\sqrt[t]{e^{-{\left({\sum }_{i=1}^n{w_i\left(-ln\left(1-{\left|d_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}-e^{-{\left({\sum }_{i=1}^n{w}_i{\left(-ln\left(1-{\left|d_h^{\mathcal{N}}\right|}^t-{\left|i_h^{\mathcal{N}}\right|}^t\right)\right)}^{\eta }\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\eta $}\right.}}}}\end{array}\end{array}\right)\hfill \end{array}$$

(19)

Theorem 6.

(Idempotency). If all of the BTSFNs in the collection ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (h = 1, 2, …, n) are identical to $\mathfrak{s}=\left(s^{\mathcal{P}},d^{\mathcal{P}},i^{\mathcal{P}},s^{\mathcal{N}},d^{\mathcal{N}},i^{\mathcal{N}}\right)$, then $BTSFAAWG\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)=\mathfrak{s}$.

Theorem 7.

(Boundedness). Let ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (h = 1, 2, …, n) be the collection of BTSFNs. Let ${\mathfrak{s}}^{-}=min〈\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)〉=\left(s^{\mathcal{P}-},d^{\mathcal{P}-},i^{\mathcal{P}-},s^{\mathcal{N}-},d^{\mathcal{N}-},i^{\mathcal{N}-}\right)$ and ${\mathfrak{s}}^{+}=min〈\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)〉=\left(s^{\mathcal{P}+},d^{\mathcal{P}+},i^{\mathcal{P}+},s^{\mathcal{N}+},d^{\mathcal{N}+},i^{\mathcal{N}+}\right)$. Then, ${\mathfrak{s}}^{-}\le BTSFAAWG\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)\le {\mathfrak{s}}^{+}$.

Theorem 8.

(Monotonicity). Let ${\mathfrak{s}}_h=\left(s_h^{\mathcal{P}},d_h^{\mathcal{P}},i_h^{\mathcal{P}},s_h^{\mathcal{N}},d_h^{\mathcal{N}},i_h^{\mathcal{N}}\right)$ (I = 1, 2, …, n) and ${\mathfrak{s}\prime }_h=\left({s\prime }_h^{\mathcal{P}},{d\prime }_h^{\mathcal{P}},{i\prime }_h^{\mathcal{P}},{s\prime }_h^{\mathcal{N}},{d\prime }_h^{\mathcal{N}},{i\prime }_h^{\mathcal{N}}\right)$ (I = 1, 2, …, n) be two sets of BTSFNs with ${\mathfrak{s}}_h\le {\mathfrak{s}\prime }_h,\forall h=\mathrm{1,2},\dots ,n.$ Then, $BTSFAAWG\left({\mathfrak{s}}_1,{\mathfrak{s}}_2,\dots ,{\mathfrak{s}}_n\right)\le BTSFAAWG\left({\mathfrak{s}\prime }_1,{\mathfrak{s}\prime }_2,\dots ,{\mathfrak{s}\prime }_n\right)$.

3.4. BTSF-WASPAS Method

In this section, we introduce our proposed BTSF-WASPAS method. The entire process of our proposed BTSF-WASPAS method is shown in Figure 1. Suppose n design alternatives (DAs) ${DA}_i=\left\{{DA}_1,{DA}_2,\dots ,{DA}_n\right\}$ need to be evaluated by k DMs ${DM}_s=\left\{{DM}_1,{DM}_2,\dots ,{DM}_k\right\}$ with weights ${\omega }_s=\left\{{\omega }_1,{\omega }_2,\dots ,{\omega }_k\right\}$.

The BTSF-WASPAS method is thoroughly explained as follows.

• Step 1: Formation of individual BTSF decision matrices.

DMs evaluate the ability of the design schemes based on the selected criteria and express their ideas using linguistic information in the form of BTSFNs. The perception of DMs is represented by the BTSF decision matrix ${\tilde{\mathfrak{D}}}^{\left(s\right)}$ shown in Equation (20) combined with the linguistic terms shown in

Table 2. Linguistic terms and related BTSFNs.

Linguistic Terms	BTSFNs
Extreme Importance (EXI), Extremely Good (EG)	(0.95, 0.05, 0, −0.05, −0.95, 0)
High Importance (HI), Very Good (VG)	(0.8, 0.2, 0.1, −0.2, −0.8, −0.1)
Importance (I), Good (G)	(0.65, 0.35, 0.3, −0.35, −0.65, −0.3)
Fair (F), Medium (M)	(0.5, 0.5, 0.5, −0.5, −0.5, −0.5)
Slightly Low Importance (SLI), Poor (P)	(0.35, 0.65, 0.3, −0.65, −0.35, −0.3)
Low Importance (LI), Very Poor (VP)	(0.2, 0.8, 0.1, −0.8, −0.2, −0.1)
Very Low Importance (VLI), Extremely Poor (EP)	(0.05, 0.95, 0, −0.95, −0.05, 0)

.

For m criteria and n DAs, decision matrix ${\tilde{\mathfrak{D}}}^{\left(s\right)}$ of each DM is as follows,

$${\tilde{\mathfrak{D}}}^{\left(s\right)}={\left[{{\mathfrak{s}}_{ij}}^{\left(s\right)}\right]}_{m\times n}=\left[\begin{array}{ccc}\left({\left(s_{11}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(d_{11}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(i_{11}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(s_{11}^{\mathcal{N}}\right)}^{\left(s\right)},{\left(d_{11}^{\mathcal{N}}\right)}^{\left(s\right)},{\left(i_{11}^{\mathcal{N}}\right)}^{\left(s\right)}\right)& \cdots & {{\mathfrak{s}}_{1n}}^{\left(s\right)}\\ ⋮& \ddots & ⋮\\ \left({\left(s_{m1}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(d_{m1}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(i_{m1}^{\mathcal{P}}\right)}^{\left(s\right)},{\left(s_{m1}^{\mathcal{N}}\right)}^{\left(s\right)},{\left(d_{m1}^{\mathcal{N}}\right)}^{\left(s\right)},{\left(i_{m1}^{\mathcal{N}}\right)}^{\left(s\right)}\right)& \cdots & {{\mathfrak{s}}_{mn}}^{\left(s\right)}\end{array}\right],$$

(20)

where $i=1, 2, \dots , m$ and $j=1, 2, \dots ,n$.

• Step 2: Aggregation of DMs’ opinions.

We aggregate the independent perspectives of DMs through the BTSFAAWA operator to obtain solutions acceptable to all DMs in the decision group, thus constructing an aggregated BTSF decision matrix $\tilde{\mathfrak{D}}$.

$$\tilde{\mathfrak{D}}={\left[{\mathfrak{s}}_{ij}\right]}_{m\times n}=\left[\begin{array}{ccc}\left(s_{11}^{\mathcal{P}},d_{11}^{\mathcal{P}},i_{11}^{\mathcal{P}},s_{11}^{\mathcal{N}},d_{11}^{\mathcal{N}},i_{11}^{\mathcal{N}}\right)& \cdots & \left(s_{1n}^{\mathcal{P}},d_{1n}^{\mathcal{P}},i_{1n}^{\mathcal{P}},s_{1n}^{\mathcal{N}},d_{1n}^{\mathcal{N}},i_{1n}^{\mathcal{N}}\right)\\ ⋮& \ddots & ⋮\\ \left(s_{m1}^{\mathcal{P}},d_{m1}^{\mathcal{P}},i_{m1}^{\mathcal{P}},s_{m1}^{\mathcal{N}},d_{m1}^{\mathcal{N}},i_{m1}^{\mathcal{N}}\right)& \cdots & \left(s_{mn}^{\mathcal{P}},d_{mn}^{\mathcal{P}},i_{mn}^{\mathcal{P}},s_{mn}^{\mathcal{N}},d_{mn}^{\mathcal{N}},i_{mn}^{\mathcal{N}}\right)\end{array}\right]$$

(21)

In $\tilde{\mathfrak{D}}$, any entry ${\mathfrak{s}}_{ij}=\left(s_{ij}^{\mathcal{P}},d_{ij}^{\mathcal{P}},i_{ij}^{\mathcal{P}},s_{ij}^{\mathcal{N}},d_{ij}^{\mathcal{N}},i_{ij}^{\mathcal{N}}\right)$ describes a collective reflection of all DMs about the DAs using BTSFAAWA, described in Equation (18), which is represented by the following equation:

$${\mathfrak{s}}_{ij}=BTSFAAWA\left({{\mathfrak{s}}_{ij}}^{\left(1\right)},{{\mathfrak{s}}_{ij}}^{\left(2\right)},\dots ,{{\mathfrak{s}}_{ij}}^{\left(k\right)}\right)$$

(22)

• Step 3: Evaluate the weight of the criteria.

To determine the weight of the criteria, each DM evaluates the importance of each criterion using the linguistic terms in Table 2. DMs assess the decision criteria as

$$\tilde{\mathcal{C}}={\left[{\tilde{\mathfrak{c}}}_{jk}\right]}_{n\times k}=\left[\begin{array}{ccc}\left(s_{11}^{\mathcal{P}},d_{11}^{\mathcal{P}},i_{11}^{\mathcal{P}},s_{11}^{\mathcal{N}},d_{11}^{\mathcal{N}},i_{11}^{\mathcal{N}}\right)& \cdots & \left(s_{1k}^{\mathcal{P}},d_{1k}^{\mathcal{P}},i_{1k}^{\mathcal{P}},s_{1k}^{\mathcal{N}},d_{1k}^{\mathcal{N}},i_{1k}^{\mathcal{N}}\right)\\ ⋮& \ddots & ⋮\\ \left(s_{j1}^{\mathcal{P}},d_{j1}^{\mathcal{P}},i_{j1}^{\mathcal{P}},s_{j1}^{\mathcal{N}},d_{j1}^{\mathcal{N}},i_{j1}^{\mathcal{N}}\right)& \cdots & \left(s_{jk}^{\mathcal{P}},d_{jk}^{\mathcal{P}},i_{jk}^{\mathcal{P}},s_{jk}^{\mathcal{N}},d_{jk}^{\mathcal{N}},i_{jk}^{\mathcal{N}}\right)\end{array}\right]$$

(23)

In Equation (23), each row represents a criterion, and each column evaluates the importance of ${DM}_k$ for the criterion.

• Step 4: Determine the aggregated criteria weights.

We aggregate DMs’ opinions regarding the criteria through Equation (18) and obtain the aggregated fuzzy criteria importance vector, as shown below.

$$\mathcal{C}=\left[{\mathfrak{c}}_j\right]=\left[\begin{array}{c}\left(s_1^{\mathcal{P}},d_1^{\mathcal{P}},i_1^{\mathcal{P}},s_1^{\mathcal{N}},d_1^{\mathcal{N}},i_1^{\mathcal{N}}\right)\\ ⋮\\ \left(s_j^{\mathcal{P}},d_j^{\mathcal{P}},i_j^{\mathcal{P}},s_j^{\mathcal{N}},d_j^{\mathcal{N}},i_j^{\mathcal{N}}\right)\end{array}\right]$$

(24)

In $\mathcal{C}$, any entry ${\mathfrak{c}}_j$ describes a collective reflection of all DMs regarding criteria importance, and it can be expressed as

$${\mathfrak{c}}_j=BTSFAAWA\left({\tilde{\mathfrak{c}}}_{j1},{\tilde{\mathfrak{c}}}_{j2},\dots ,{\tilde{\mathfrak{c}}}_{jk}\right)$$

(25)

• Step 5: Calculation of the importance weight of criteria.

We perform defuzzing on the aggregated fuzzy criteria importance vector based on the score function given in Equation (4).

$${\tilde{w}}_j=\mathbb{S}\left({\mathfrak{c}}_j\right)$$

(26)

Afterwards, we can obtain the normalized criteria weights by using Equation (27).

$$w_{\mathrm{j}}={\displaystyle \frac{{\tilde{w}}_j}{\sum {\tilde{w}}_j}}$$

(27)

• Step 6: Compute the results of the WSM part.

We calculated the result ${\tilde{Q}}_i^{\left(1\right)}$ of the WSM part of BTSF-WASPAS using the proposed BTSFAAWA operator, as shown below.

$${\tilde{Q}}_i^{\left(1\right)}=BTSFAAWA\left({\mathfrak{s}}_{i1},{\mathfrak{s}}_{i2},\dots ,{\mathfrak{s}}_{ij}\right)=w_1{\mathfrak{s}}_{i1}⨁w_2{\mathfrak{s}}_{i1}⨁\dots ⨁w_{\mathrm{j}}{\mathfrak{s}}_{ij}$$

(28)

• Step 7: Compute the results of the WPM part.

We calculated the result ${\tilde{Q}}_i^{\left(2\right)}$ of the WPM part of BTSF-WASPAS using the proposed BTSFAAWG operator, as shown below.

$${\tilde{Q}}_i^{\left(2\right)}=BTSFAAWG\left({\mathfrak{s}}_{i1},{\mathfrak{s}}_{i2},\dots ,{\mathfrak{s}}_{ij}\right)=w_1{\mathfrak{s}}_{i1}⨁w_2{\mathfrak{s}}_{i1}⨁\dots ⨁w_{\mathrm{j}}{\mathfrak{s}}_{ij}$$

(29)

• Step 8: Obtain the final score of BTSF-WASPAS.

Based on the results of the WSM and WPM sections, we add the two parts together to obtain the final WASPAS score.

$${\tilde{Q}}_i=\mathbb{S}\left(\lambda {\tilde{Q}}_i^{\left(1\right)}⨁\left(1-\lambda \right){\tilde{Q}}_i^{\left(2\right)}\right)$$

(30)

4. Results

Cultural and creative products (CCPs) are product forms that combine cultural elements with modern design and achieve cultural and commercial value through creative expression [45]. CCPs are typically based on specific historical, regional, or artistic cultures. Moreover, CCP design incorporates innovative design concepts to meet consumers’ needs for personalized, emotional, and cultural experiences. In our case, we illustrate the process of our proposed method by evaluating the teapot design in CCP. The criteria considered are C1, the product modeling structure, C2, the color combination, C3, the cultural connotation, C4, the style’s consistency, and C5, the aesthetics of the product design. There are four DAs to be evaluated, as shown in Figure 2. We invited four experts $\left\{{DM}_1,{DM}_2,{DM}_3,{DM}_4\right\}$ in the field of engineering design to evaluate these four proposals based on five criteria. Based on their work experience, we assign weight vectors as $\left\{\mathrm{0.2,0.25,0.4,0.15}\right\}$.

First, based on the linguistic terms in Table 2, each DM provides evaluations of four DAs under five criteria, as shown in the

Table 3. Evaluations of all DMs.

DMs	DAs	C1	C2	C3	C4	C5
DM1	A1	VG	VG	G	G	VG
	A2	VG	VG	VG	M	VG
	A3	G	G	M	P	G
	A4	G	G	M	P	G
DM2	A1	G	G	VG	G	G
	A2	G	VG	VG	P	G
	A3	VG	VG	P	M	VG
	A4	G	G	M	M	G
DM3	A1	M	G	G	P	M
	A2	G	G	M	P	G
	A3	G	VG	P	M	G
	A4	M	M	P	P	M
DM4	A1	G	G	G	G	G
	A2	G	G	M	M	G
	A3	VG	VG	P	G	VG
	A4	G	G	M	P	G

. We aggregate these evaluations using the BTSFAAWA operator by considering the importance levels of decision makers. In this case, we take $t=4$ and $\eta =3$. An aggregated decision matrix is achieved, as in

Table 4. Aggregated decision matrix.

DAs	C1	C2	C3	C4	C5
A1	(0.7228, 0.3241, 0.2379, −0.4647, −0.5763, −0.4918)	(0.7273, 0.299, 0.2235, −0.3436, −0.6673, −0.3004)	(0.757, 0.264, 0.1777, −0.3355, −0.6878, −0.2993)	(0.6264, 0.3911, 0.3426, −0.4649, −0.5679, −0.4768)	(0.65, 0.35, 0.3, −0.35, −0.65, −0.3)
A2	(0.7273, 0.299, 0.2235, −0.3436, −0.6673, −0.3004)	(0.7624, 0.2568, 0.1687, −0.3331, −0.6936, −0.2986)	(0.7893, 0.217, 0.1207, −0.3064, −0.7468, −0.2789)	(0.6264, 0.3911, 0.3426, −0.4649, −0.5679, −0.4768)	(0.7862, 0.2218, 0.1264, −0.3121, −0.7373, −0.285)
A3	(0.7365, 0.2892, 0.2101, −0.3418, −0.6721, −0.3003)	(0.8914, 0.1532, 0.0735, −0.4764, −0.5651, −0.5257)	(0.4405, 0.6021, 0.4421, −0.6395, −0.3688, −0.3188)	(0.4801, 0.541, 0.4912, −0.6084, −0.4182, −0.3741)	(0.5801, 0.4527, 0.4201, −0.4912, −0.5189, −0.4976)
A4	(0.6253, 0.407, 0.3292, −0.606, −0.4496, −0.347)	(0.4601, 0.5756, 0.4709, −0.63, −0.3851, −0.3361)	(0.5676, 0.4769, 0.4296, −0.579, −0.4621, −0.4157)	(0.4482, 0.5926, 0.4542, −0.6366, −0.374, −0.3242)	(0.5889, 0.4432, 0.4066, −0.4887, −0.5241, −0.4964)

. The linguistic importance weights of the criteria assigned by DMs are shown in

Table 5. Importance weights of the criteria.

Criteria	DM1	DM2	DM3	DM4
C1	F	I	LI	F
C2	F	F	SLI	I
C3	EXI	HI	EXI	HI
C4	I	HI	F	SLI
C5	I	F	I	I

. The weight of each criterion obtained by using the BTSFAAWA operator and the score function is presented in

Table 6. Aggregation of criteria weights.

Criteria	Aggregated BTSF Values	Crisp Score	Normalized Weights
C1	(0.5864, 0.4757, 0.4016, −0.7532, −0.296, −0.1713)	0.395	0.1355
C2	(0.5658, 0.4925, 0.4202, −0.6079, −0.4254, −0.3696)	0.4819	0.1653
C3	(0.9339, 0.0735, 0.0227, −0.1853, −0.8475, −0.0979)	0.88	0.3019
C4	(0.7313, 0.3251, 0.2195, −0.5653, −0.5026, −0.4259)	0.595	0.2041
C5	(0.6366, 0.374, 0.3242, −0.4482, −0.5926, −0.4542)	0.5628	0.1931

.

By using Equation (28), we calculated the results of the WSM part, as shown in

Table 7. Results of the WSM part.

DAs	${\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(1\right)}$	$\mathit{\lambda }{\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(1\right)}$
A1	(0.7217, 0.3112, 0.234, −0.4272, −0.6269, −0.4342)	(0.6865, 0.396, 0.279, −0.4036, −0.6903, −0.4488)
A2	(0.7657, 0.2547, 0.1629, −0.4098, −0.6662, −0.4328)	(0.7301, 0.3377, 0.203, −0.3871, −0.7244, −0.4423)
A3	(0.7169, 0.3516, 0.2366, −0.5974, −0.4559, −0.3779)	(0.6818, 0.4362, 0.2757, −0.5658, −0.5361, −0.4151)
A4	(0.572, 0.484, 0.4135, −0.605, −0.4306, −0.3734)	(0.5415, 0.5622, 0.448, −0.5731, −0.5124, −0.4143)

. Similarly, by using Equation (29), we calculated the results of the WPM part, as shown in

Table 8. Results of the WPM part.

DAs	${\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(2\right)}$	$\left(1-\mathit{\lambda }\right){\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(2\right)}$
A1	(0.6872, 0.3518, 0.3046, −0.3715, −0.6563, −0.3472)	(0.6527, 0.4364, 0.3523, −0.3508, −0.7159, −0.3567)
A2	(0.7204, 0.3439, 0.3008, −0.3375, −0.7166, −0.2978)	(0.6853, 0.4287, 0.3494, −0.3187, −0.7676, −0.3009)
A3	(0.5289, 0.555, 0.433, −0.4432, −0.6292, −0.3458)	(0.5002, 0.6267, 0.4576, −0.4188, −0.6923, −0.3583)
A4	(0.5197, 0.5449, 0.4373, −0.5732, −0.4735, −0.4416)	(0.4915, 0.6176, 0.4635, −0.5426, −0.5525, −0.4787)

. Finally, we combined the results of the WSM and WPM parts to calculate the final WASPAS score for each DA, as shown in

Table 9. Final WASPAS score.

DAs	$\mathit{\lambda }{\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(1\right)}⨁\left(1-\mathit{\lambda }\right){\tilde{\mathit{Q}}}_{\mathit{i}}^{\left(2\right)}$	Score Values
A1	(0.7073, 0.3298, 0.2651, −0.4092, −0.6407, −0.4031)	0.6118
A2	(0.7479, 0.2906, 0.2144, −0.39, −0.6885, −0.3986)	0.6463
A3	(0.6827, 0.4163, 0.2856, −0.5669, −0.5151, −0.3923)	0.5578
A4	(0.5529, 0.5109, 0.4249, −0.5916, −0.4503, −0.4027)	0.4849

($\lambda$ = 0.5). According to the results of BTSF-WASPAS, A2 is the optimal design, and the ranking of the four DAs is A2 > A1 > A3 > A4. A1 adopts a streamlined body and a curved spout, making its shape beautiful and practical. At the same time, the smooth lines of the body and the ergonomic handle design enhance the user experience, indicating a balance between shape and function. The teapot’s color is mainly soft blue and brown, highlighting the beauty of nature and tradition through the combination of colors. The landscape texture design of the teapot conveys the natural aesthetic concept of traditional Chinese culture. It highlights the profound cultural heritage of the product. Overall, the experts unanimously believed that this design’s elements, including the spout, the body, and the lid, maintain consistency in style and pleasing aesthetics, meeting consumers’ expectations for product design coherence.

5. Sensitivity Analysis and Comparative Analysis

We conducted a sensitivity analysis in this section to investigate the impact of the aggregation operator parameter $\eta$ given in Equation (18), the membership, non-membership, and uncertain degree parameter $t$ controlling BTSFS given in Equation (3), and the parameter $\lambda$ given in Equation (30) controlling the WPM and WSM parts in WASPAS in relation to the decision results. Specifically, we calculated the final score values for each DA by changing parameter $\eta$ from 1 to 10, parameter $t$ from 1 to 10, and parameter $\lambda$ from 0.1 to 0.9, as shown in Figure 3, Figure 4, and Figure 5, respectively. Overall, in this case, with different parameters changing, A2 is the optimal DA. Specifically, in Figure 3, as parameter $\eta$ increases, the score values of A1 and A2 gradually approach each other, while the scores of A3 and A4 gradually move away. A4 is relatively stable and has always been the worst DA. In Figure 4, as parameter $t$ increases, the score values of the four DAs become closer, indicating that a considerable parameter $t$ may decrease ranking resolution in our method. Nevertheless, A2 remains the optimal DA. A4 is still the worst DA. In Figure 5, as parameter $\lambda$ increases, the score values of the four DAs become closer and farther apart, which means that the results calculated through the WPM part will increase the difference in the score values of DAs and have better discriminative performance. Therefore, our method can achieve more flexible and effective results by adjusting parameters according to actual needs.

We also conducted a comparative analysis. We followed the method described in Section 3 and compared the final score values of the WASPAS method based on TSFS (TSF-WASPAS) [46]. The linguistic terms used based on TSFS change the negative membership, non-membership, and uncertainty shown in Table 2 to (0,0,0). Regarding the parameter settings for TSFS, we maintain consistency with the BTSF-WASPAS method, where $t$ = 4 and $\lambda$= 0.5. Afterward, the weighted aggregation operator of TSFS is used to calculate the parts of WSM and WPM. We compared the existing TSFS aggregation operators based on AATN and AATCN, which are the TSF-weighted average (TSFAAWA) and the weighted geometry (TSFAAWG) operators [29]. We set the parameters for TSFAAWA and TSFAAWG to $\eta$= 3 and obtained the final score results using TSFAAWA and TSFAAWG operators. In addition, we also compared the final score value of BTSF-WASPAS with the final score value obtained through the BTSFAAWA and BTSFAAWG operators. All final score results are shown in Figure 6.

The comparative analysis results show that A2 is the optimal DA among all methods, and A4 is the worst. It is worth noting that the difference between A3 and A4 in the TSFAAWG operator’s results is minimal. Overall, the BTSF-WASPAS method is robust. Due to the combination of BTSFAAWA and BTSFAAWG operators, it has higher flexibility in parameters, which increases the possibility of meeting practical needs. In addition, BTSF can consider the negative part of the information, that is, the bipolar information that combines the positive and negative parts, which can enhance the robustness of decision making even in situations of high uncertainty or significant environmental changes, thus ensuring the robustness of decision making. The BTSF method comprehensively analyzes different scenarios, meaning the selection of solutions is based on current advantages and in consideration of potential future challenges. Therefore, our BTSF-WASPAS method not only helps evaluate the overall performance of DAs by balancing benefits and risks but also enhances the robustness of decision making and the ability to cope with complex environments.

The MCGDM method based on BTSFS has significant advantages under the product design scheme selection problem. Compared to traditional fuzzy sets, rough sets, IFSs, q-ROFs, and SFSs, this method can simultaneously handle positive and negative information (bipolarity) and comprehensively capture the complex emotions of DMs. It introduces hesitation and parameter t, making the description of uncertainty and hesitation more refined and enhancing the ability to distinguish between solutions. We summarize the advantages of BTSFS as shown in

Table 10. Comparison between BTSFS and other methods.

Comparison Dimension	FS [15]	Rough Set [13]	IFS [16]	q-ROFS [33]	SFS [17]	BTSFS
Handling Positive and Negative Information (Bipolarity)	No	No	No	No	No	Yes
Representation of Membership and Non-Membership Degrees	Membership degree only	Upper and lower approximations	Yes	Yes	Yes	Yes
Consideration of Hesitancy Degree	No	No	Implicit	Implicit	Yes	Yes (with parameter t)
Flexibility of Aggregation Operators	Basic	Limited	Moderate	High	High	Very high
Suitability for Complex Decision Environments	Low	Moderate	Moderate	High	High	Very high

.

This method also supports advanced aggregation operators, which can effectively integrate the opinions of multiple evaluators and are suitable for complex multi-criteria decision making environments. The BTSFS method reduces information loss, improves decision making accuracy, and enhances group decision making efficiency in information processing. Therefore, it exhibits greater effectiveness in dealing with conflicts of opinion or incomplete information from DMs.

6. Conclusions

In this work, we developed aggregation operators based on BTSF and WASPAS methods from the perspective of MCGDM to address the problem of selecting modern product design solutions. We developed BTSFAAWA and BTSFAAWG aggregation operators based on AATN and AATCN and verified their critical mathematical properties. Afterward, we extended the traditional WASPAS method to the BTSF environment and proposed the BTSF-WASPAS method. We validated the effectiveness of our method through the CCP design scheme selection problem. In our results, design scheme A2 has the best overall performance among the five CCP design criteria we defined. Afterward, we conducted a comparative and sensitivity analysis, and the results showed that our method is robust. Compared with existing models, our method considers bipolar information and provides a more comprehensive representation of uncertainty and hesitancy, leading to more robust and effective results in MCGDM.

Our work also has some limitations. Ranking schemes solely through the BTSFAAWA operator may reduce the discriminative power of the ranking. Additionally, our method may be somewhat complex, and it requires guidance from relevant personnel during use. In the future, we will develop an integrated decision support system to improve the usability of our methods. The BTSF aggregation operator we propose can also be used in other MCGDM methods, such as AHP, to achieve more effective and accurate decision making.