Large-Scale Satisfaction Rating-Driven Selection of New Energy Vehicles: A Basic Uncertain Linguistic Information Bonferroni Mean-Based MCGDM Approach Considering Criteria Interaction

Abstract

The continuous revolution of new energy technologies and the introduction of subsidy policies have promoted green consumers’ willingness to purchase new energy vehicles and automotive online service platforms have disclosed vehicle reputation and consumer satisfaction ratings information. However, due to issues such as uncertain data quality, large data volumes, and the emergence of positive reviews, the cost for potential car buyers to acquire useful decision-making knowledge has increased. Therefore, it is necessary to develop a scientific decision-making method that leverages the advantages of large-scale consumer satisfaction ratings to support potential car buyers in efficiently acquiring credible decision-making knowledge. In this context, the Bonferroni mean (BM) is a prominent operator for aggregating associated attribute information, while basic uncertain linguistic information (BULI) represents both information and its credibility in an integrated manner. This study proposes an embedded-criteria association learning BM operator tailored to large-scale consumer satisfaction ratings-driven scenarios and extends it to the BULI environment to address online ratings aggregation problems. Firstly, to overcome the limitations of BM with weighted interaction (WIBM) when dealing with independent criteria, we introduce an adjusted WIBM operator and extend it to the BULI environment as the BULIWIBM operator. We discuss fundamental properties such as idempotence, monotonicity, boundedness, and degeneracy. Secondly, addressing the constraints on interaction coefficients in BM due to subjective settings, we leverage expert knowledge to explore potential temporal characteristics hidden within large-scale consumer satisfaction ratings and develop a method for learning criteria and interaction coefficients. Finally, we propose a conversion method between user credibility-based ratings and BULI. By combining this method with the proposed adjusted BM operator, we construct a multi-criteria group decision-making (MCGDM) approach for product ranking driven by large-scale consumer satisfaction ratings. The effectiveness and scientific rigor of our proposed methods are demonstrated through solving a new energy vehicle selection problem on an online service platform and conducting comparative analysis. The case analysis and comparative analysis results demonstrate that the interaction coefficients, derived from expert knowledge and 42,520 user ratings, respectively, fell within the ranges of [0.2391, 0.7857] and [0.6546, 1.0]. The comprehensive interaction coefficient lay within the range of [0.4674, 0.7965], effectively mitigating any potential biases caused by subjective or objective factors. In comparison to online service platforms, our approach excels in distinguishing between alternative vehicles and significantly impacts their ranking based on credibility considerations.

1. Introduction

With the continuous innovation of new energy technologies and the government’s ongoing subsidy policies for the new energy vehicle industry, consumers’ awareness of environmental protection has significantly increased and the concept of green consumption is gradually gaining popularity. New energy vehicles, due to their low carbon emissions and environmental friendliness, are favored by more and more consumers. This trend not only reflects the public’s emphasis on environmental protection but also demonstrates the important role of new energy technologies in promoting sustainable development. At the same time, the era of big data has arrived, with exponential data growth becoming a fundamental driver for the development of various industries. According to statistics released by the China International Big Data Industry Expo in 2023, the scale of China’s big data industry expanded to an astonishing CNY 1.57 trillion in 2022 [1]. The emergence of big data has led to a paradigm shift from traditional model-driven decision research to data-driven decision research, particularly in the field of multi-criteria decision-making (MCDM) methods driven by large-scale data. MCDM methods [2] are widely used in complex scenarios to make decisions based on individual preferences across multiple criteria [3,4,5]. Use of big data provides robust empirical support for these MCDM methods [6]. Therefore, effectively utilizing data-driven approaches in constructing decision-making methods holds immense value and significance. Although various online automotive service platforms provide the public with convenient and open channels for automotive reputation information, the uncertainty of information quality and the large volume of data have become major obstacles for green consumers to obtain useful decision-making knowledge at low cost. This research developed a multi-criteria group decision-making (MCGDM) method for new energy vehicle selection, driven by large-scale online consumer satisfaction ratings, from the perspective of information quality.

In the domain of MCDM problems, aggregation functions [7] have been extensively employed as a fundamental theory for implementing information fusion [8]. In the context of product ranking, aggregation functions play a crucial role in consolidating large-scale multi-criteria satisfaction ratings to derive comprehensive ratings. It is worth noting that the product ranking problem is a classic LGMCDM problem, wherein certain correlations between criteria may exist. For instance, during car evaluations, there might be a correlation between interior design and space criteria. The selection of appropriate aggregation functions that can accurately represent the correlation becomes crucial when fusing correlated multiple-criteria ratings. The Bonferroni mean (BM) operator, which serves as a crucial class of aggregation functions [9,10,11], effectively represents the interaction between criteria. Yager initially elucidated the characterization of criteria interaction within the BM, laying a solid groundwork for subsequent research [12]. Moreover, to augment the concurrent representation of criterion interaction and independence, an extended Bonferroni mean (EBM) [13] was proposed. However, this EBM tends to overlook the importance of capturing the extent of interaction between criteria. To address this concern, the BM with weighted interaction (WIBM) [14] has been proposed. The common characteristic of the aforementioned BM operators lies in their primary application to conventional MCDM methods, wherein the assessment and level of interaction among criteria predominantly rely on subjectively established criteria. Previous studies have shown that the BM operator can effectively consider the interaction between criteria in the process of information aggregation, making it an effective method for solving decision problems with criterion associations. However, the current judgment of criterion correlations and setting of criterion interaction coefficients in the BM operator rely entirely on subjective settings, making them susceptible to subjective biases. In dealing with decision problems driven by big data, which provide rich sources of information for setting criterion interaction coefficients, this study aims to optimize data utilization by designing a data-driven approach based on expert knowledge to generate interactive correlation sets and determine standard interaction coefficients. This approach aims to address limitations associated with subjective setting methods.

Undoubtedly, big data offers abundant resources for MCDM research. However, the integration of data quality into the decision method construction process remains a valuable avenue for exploration. It is noteworthy that a novel information theory, known as basic uncertain information (BUI), has been proposed, which enables the simultaneous representation of both the information and its credibility [15,16]. The ability of BUI to represent data reliability in the decision-making process has garnered significant attention from scholars. A prime example of BUI, often employing a linguistic scale rather than an interval scale, is observed when reviewers assess papers submitted to specific journals or conference proceedings. In such cases, the BUI pair $〈week \mathrm{accept};\mathrm{medium}〉$ indicates that the reviewers possess a relatively low acceptance threshold for the paper and have self-assessed their knowledge level as moderate. In order to facilitate the aggregation of BUI pairs, a comprehensive framework for BUI aggregation functions [16] was proposed, laying a solid foundation for further research on aggregation functions. Owing to its inherent advantages in representing information quality, BUI has garnered increasing attention from scholars [17,18,19,20]. Considering the inherent advantages of the OWA aggregation function [21] in effectively capturing both the intrinsic value of information and its relative importance within a given sorting position, Jin et al. [18] extended OWA to a BUI environment and introduced the BUIOWA operator. Considering the inherent volatility of information credibility within a specific range, the technique expanded the notional BUI to interval-type BUI (ItBUI), along with its corresponding weighted average operators [20]. In certain decision-making scenarios, the representation of information credibility may adopt a linguistic approach. To address this issue, unsymmetrical BUI (UBUI) [19] was proposed, along with the corresponding definition of aggregation operators. In complex and uncertain decision-making environments, decision makers tend to rely on qualitative linguistic rather than quantitative information for representing preferences. This preference is driven by the fact that qualitative linguistic information aligns more closely with human cognitive processes and offer decision makers greater flexibility in expressing evaluations of specific objects. In light of the aforementioned factors, scholars have expanded BUI to a 2-tuple linguistic environment and introduced basic uncertain linguistic information (BULI) and its related weighted average operator [22]. They integrated QFD and ELECTRE III methodologies to devise a hybrid LGMCDM approach [23] for addressing material selection problems. In certain decision-making scenarios where decision makers are unable to provide precise linguistic information for representing preferences, researchers have expanded the concept of BULI to interval environment. Consequently, they have introduced interval BULI (IBULI) and proposed a hybrid LGMCDM approach based on the IBULI weighted average operator [24]. In summary, previous studies have predominantly focused on the BULI aggregation operator and MCDM methods, establishing a robust foundation for the development of the BULI decision theory and methodology system. However, these approaches have primarily addressed MCDM problems with independent criteria, while limited research attention has been given to decision problems involving interrelated criteria. Therefore, this study aims to propose the utilization of Bonferroni mean within a BULI framework and demonstrate its fundamental properties.

The aforementioned BULI research primarily focuses on the investigation of decision methods and weighted-averaging operators, with less emphasis placed on decision problems pertaining to attributes or criteria. Hence, this study aims to expand the application of the BM operator within the BULI environment and proposes a corresponding MCGDM approach. The primary objectives and methods of this paper are as follows:

(i)

This study highlights the limitations of the existing BM with a weighted interaction (WIBM) operator in terms of closure, through rigorous theorem proofs and comprehensive case studies. By modifying the conditions for the interaction coefficient in the operator, we propose an adjusted WIBM operator and demonstrate its closure properties conclusively. Furthermore, we introduce an extended WIBM (EWIBM) operator specifically designed to handle linguistic term set information, and rigorously prove its boundedness, monotonicity, and idempotence.

(ii)

To address MCDM problems in the BULI environment more effectively, we propose a novel BULI WIBM (BULIWIBM) operator based on both EWIBM and adjusted WIBM operators. Our analysis reveals that the existing BULIWA operator can be considered as a special form of this proposed operator.

(iii)

In order to overcome the limitations associated with relying solely on expert knowledge for setting criterion interaction coefficients in BM operators and harnessing the advantages offered by large-scale rating data, we design a method for generating criterion interaction sets and coefficients based on collaborative efforts between experts’ knowledge and extensive consumer satisfaction ratings.

(iv)

Comprehensively leveraging publicly disclosed user information from online service platforms, we propose a conversion method between consumer satisfaction ratings and BULIs. Moreover, we introduce an MCGDM method aimed at supporting consumers’ car purchasing decisions.

The paper is structured as follows: Section 2 provides a concise overview of the fundamental concepts of BM and BULI, followed by the introduction of adjusted BM and BULIWIBM in Section 3. A large-scale online consumer satisfaction ratings aggregation decision-making approach for generating product ranking is proposed in Section 4. The case study and comparative analysis are presented in Section 5, while Section 6 offers a summary along with an exposition on future research directions.

2. Preliminaries

2.1. Bonferroni Mean

Bonferroni mean (BM) is widely utilized in multi-criteria decision-making (MCDM) due to its unique ability to express the degree of mutual influence between criteria. Yager [12] was the first to provide a description of attribute interactions within BM. To further enhance its representation of both criterion interaction and independence, an extended version called the extended Bonferroni mean (EBM) was proposed [13]. Building upon this, the Bonferroni mean weighted interaction (WIBM) was developed as a solution to address the issue of subjective setting methods heavily influencing criterion interaction degrees [14]. Some concepts related to BM are stated.

Definition 1. 

Let $X=\{x_i|i=1,2,\dots ,n\}\in {[0,1]}^n$ be a collection of crisp data. Assume $p\ge 0,q\ge 0$ and $p+q>0$. The normalized Bonferroni mean with weighted interaction (WIBM) operator is given by [14]:

$$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}},$$

(1)

where $w_i\in \left[0,1\right]\left(i=1,2,\dots ,n\right)$ is a collection of weights and ${\displaystyle \sum _{i=1}^n{w}_i=1}$. Further, $v_{i,j}$ is a weight that $v_{i,j}={\displaystyle \frac{w_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}$, where $w_{i,i}=0\left(i=1,2,\dots ,n\right)$, $w_{i,j}\in \left[0,1\right]\left(i\ne j,i,j=1,2,\dots ,n\right)$ is a collection of weights and at least one is bigger than 0. In the special case when $w_{i,j}=w_i\cdot w_j\left(i,j=1,2,\dots ,n;i\ne j\right)$, the WIBM reduces to the normalized weighted Bonferroni mean [25]

$$WIB{M}^{p,q}\left(X\right)=WB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{\begin{array}{l}j=1\\ i\ne j\end{array}}^n\frac{w_j}{1-w_i}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}$$

(2)

Furthermore, when $w_j={\displaystyle \frac{1}{n}}\left(j=1,2,\dots ,n\right)$, the WBM reduces to the Bonferroni mean [12]:

$$WB{M}^{p,q}\left(X\right)=B{M}^{p,q}\left(X\right)={\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i^p\left(\frac{1}{n-1}{\displaystyle \sum _{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}$$

(3)

The advantage of the WIBM operator lies in its ability to quantitatively measure the degree of interaction between criterion $C_i$ and criterion $C_j$ through the interaction coefficient $w_{i,j}\in \left[0,1\right]$. $w_{i,j}$ can represent both independent and non-independent scenarios between criterion $C_i$ and criterion $C_j$, providing flexibility for decision makers to set its value. Additionally, this enhances the interpretability of the operator, making it more intuitive in revealing the underlying principles governing interactions between criteria. Furthermore, the parameters $p>0,q>0$ of the WIBM operator exhibit strong degeneracy as they encompass various special forms such as weighted power average, weighted average, and weighted geometric operators.

Theorem 1. 

Let $p>0,q>0$ and $X=\{x_i|i=1,2,\dots ,n\}\in {[0,1]}^n$ be a collection of crisp data; then, the WIBM operator satisfies the fleeing properties:

(1) (Boundedness) if $x_{\max }={\max }_i\left(x_i\right),x_{\min }={\min }_i\left(x_i\right),i=1,2,\cdots ,n$ we have

$$x_{\min }\le WIB{M}^{p,q}\left(X\right)\le x_{\max }$$

(2) (Idempotency) if $x_i=x_0\left(i=1,2,\cdots ,n\right)$, we have

$$WIB{M}^{p,q}\left(X\right)=x_0$$

(3) (Monotonicity) Let $Y=\left\{y_i\left|i=1,2,\cdots ,n\right.\right\}\in {\left[0,1\right]}^n$ be a collection of crisp data, and $x_i\le y_i$, then

$$WIB{M}^{p,q}\left(X\right)\le WIB{M}^{p,q}\left(Y\right).$$

2.2. Basic Uncertain Linguistic Information

Basic uncertain information (BUI) [15] has garnered significant attention for its capacity to simultaneously characterize information and assess its credibility. Furthermore, BUI was extended to a 2-tuple linguistic environment and the concept of BULI was thus proposed [19].

Definition 2. 

The vector ${\tilde{\mathit{X}}}^{\left(n\right)}={\left(\widetilde{x_i}\right)}_{i=1}^n={\left(〈x_i;c_i〉\right)}_{i=1}^n,x_i,c_i\in \left[0,1\right],i=1,2,\dots ,n$ is called a basic uncertain information vector. We denote ${\tilde{\mathit{X}}}^{\left(n\right)}$ by $〈{\mathit{X}}^{\left(n\right)},{\mathit{c}}^{\left(n\right)}〉$, where ${\mathit{X}}^{\left(n\right)}$ is called the value vector and ${\mathit{c}}^{\left(n\right)}$ is called the certainty vector. $〈x_i;c_i〉\in {\left[0,1\right]}^2$ is called the basic uncertain information (BUI) [18].

Definition 3. 

Let $A:{\left[0,1\right]}^n\to \left[0,1\right]$ be a fixed aggregation function, and $P_x:J\to \left[0,1\right]$, $P_c:J\to \left[0,1\right]$ on $J$ such that $P_x〈x;c〉=x$ and $P_c〈x;c〉=c$, mapping $\tilde{A}:J^n\to J$ such that [16]:

BUI (1) for any $〈x_1;c_1〉,\dots ,〈x_n;c_n〉\in J$,

$$\tilde{A}\left(〈x_1;c_1〉,\dots ,〈x_n;c_n〉\right)=〈A\left(x_1,\dots ,x_n\right);c〉,$$

 for some $c\in \left[0,1\right]$; that is,

$$A\left(x_1,\dots ,x_n\right)=P_x\left(\tilde{A}\left(〈x_1;c_1〉,\dots ,〈x_n;c_n〉\right)\right),$$

independent of $c_i$;

BUI (2) for any fixed $X=\left(x_1,\dots ,x_n\right)\in {\left[0,1\right]}^n$, the mapping $A_X:{\left[0,1\right]}^n\to \left[0,1\right]$ given by

$$A_X\left(c_1,\dots ,c_n\right)=P_c\left(\tilde{A}\left(〈x_1;c_1〉,\dots ,〈x_n;c_n〉\right)\right)$$

is an aggregation function;

BUI (3) $A_X\left(c\right)=A_Y\left(c\right)$ for all vectors $x,y,c\in {\left[0,1\right]}^n$ such that $c_i{x}_i=c_i{y}_i$ for all $i=1,\dots ,n$ is called a BUI–aggregation function (related to A).

Definition 4. 

Let $S=\left\{s_{\alpha }|\alpha =0,1,\dots \tau \right\}$ be an linguistic term set (LTS) and $\stackrel{-}{S}$ be the 2-tuple set related with $S$ defined as $\stackrel{-}{S}=S\times [-0.5,0.5)$, and $\psi \in \left[0,\tau \right]$ be a value representing the result of a symbolic aggregation operation. The 2-tuple that expresses the equivalent information to $\psi$ is then obtained as [22]:

$$\begin{gathered} {\Delta }_S:\left[0,\tau \right]\to S\times [0.5,0.5), \\ \mathrm{where} {\Delta }_S\left(\psi \right)=\left(s_i,\alpha \right), \mathrm{with} \left\{\begin{array}{cc}{s}_i,& i=round\left(\psi \right)\\ \alpha =\psi -i,& \alpha \in [-0.5,0.5)\end{array}\right.. \end{gathered}$$

(4)

Definition 5. 

Let $\left(s_k,\alpha \right)$ and $\left(s_l,\gamma \right)$ be two linguistic 2-tuples. Then [22]:

(1) 

if k < l, then (s_k, α) is smaller than (s_l, γ);

(2) 

if k < l, then

if α = γ, then (s_k, α) and (s_l, γ) represent the same information;

if α < γ, then (s_k, α) is smaller than (s_l, γ),

Definition 6. 

Let $S=\left\{s_{\alpha }|\alpha =0,1,\dots \tau \right\}$ be an LTS. The pair $b_i=〈\Delta \left({\Psi }_i\right);c_i〉$, which binarily includes a 2-tuple linguistic term ${\Delta }_S\left({\Psi }_i\right)$ defined by $\stackrel{-}{S}$ and its source reliability $c_i\in \left[0,1\right]$, is called a representation of basic uncertain linguistic information (BULI) [20].

Definition 7. 

Let $b_i=〈{\Delta }_S\left({\Psi }_i\right),c_i〉$ and $b_j=〈{\Delta }_S\left({\Psi }_j\right),c_j〉$ be two arbitrary BULI pairs; the comparison laws between them can be defined by [17]:

$$\begin{gathered} b_i>b_j\iff \left({\Psi }_i{c}_i>{\Psi }_j{c}_j\right)\vee \left(\left({\Psi }_i{c}_i={\Psi }_j{c}_j\right)\wedge \left({\Psi }_i{c}_i>{\Psi }_j{c}_j\right)\right) \\ {b}_i<b_j\iff \left({\Psi }_i{c}_i<{\Psi }_j{c}_j\right)\vee \left(\left({\Psi }_i{c}_i={\Psi }_j{c}_j\right)\wedge \left({\Psi }_i{c}_i<{\Psi }_j{c}_j\right)\right) \\ {b}_i=b_j\iff \left({\Psi }_i={\Psi }_j\right)\wedge \left(c_i=c_j\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{0.17em} \end{gathered}$$

(5)

Definition 8. 

Let $\mathit{B}=\left\{b_i=〈{\Delta }_S\left({\psi }_i\right);c_i〉|{\Delta }_S\left({\psi }_i\right)\in \stackrel{-}{S,c_i\in \left[0,1\right]},i=1,2,\dots ,n\right\}$ be a collection of BULI pairs, in which S refers to a given LTS. The BULI weighted averaging (BULIWA) operator is defined as [17]:

$$BULIWA\left(b_1,\dots ,b_n\right)=BULIWA\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)=〈\Delta \left({\displaystyle \sum _{i=1}^n{w}_i{\psi }_i}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉$$

(6)

where $\mathit{W}={\left(w_1,w_2,\dots ,w_n\right)}^T$ is the weighting vector with $w_i\in \left[0,1\right]$ and ${\displaystyle {\sum }_{i=1}^n{w}_i=1}.$

3. BULIWIBM

In this section, an adjusted WIBM operator is proposed, and this adjusted operator is extended to develop the BULIWIBM operator with BULI pairs.

The Limitations of WIBM Operator

First, some limitations of the WIBM operator are analyzed below.

Assuming that there are no restrictions in Definition 1, such as ‘$w_{i,j}\in \left[0,1\right]\left(i\ne j,i,j=1,2,\dots ,n\right)$ is a collection of weights and at least one of these is bigger than 0’, the WIBM operator is not suitable for handling cases where all variables are independent. The specific reasons are as follows:

Property 1. 

Let $X=\{x_i|i=1,2,\dots ,n\}\in {[0,1]}^n$ be a collection of crisp data. When $w_{i,j}=0\left(i,j=1,2,\dots ,n\right)$, i.e., all inputs are independent, the WIBM operator reduces to the following operator:

$$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}.$$

(7)

and the boundedness of this operator is not satisfied.

Proof of Property 1. 

Since $w_{i,j}=0\left(i,j=1,2,\dots ,n\right)$ and $w_{i,i}=0$, then:

$$v_{i,j}={\displaystyle \frac{w{ }_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}={\displaystyle \frac{0}{0}}=1\left(i,j=1,2,\dots ,n\right),$$

then:

$$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}$$

Next, a simple example is provided to demonstrate this operator does not satisfy the boundedness. □

Example 1. 

Let $X=\left\{0.4,0.5,0.7\right\}$ be a collection of inputs and the related weighted vector is $w={\left(0,3,0.3,0.4\right)}^T$ and, $w_{i,j}=0\left(i,j=1,2,\dots ,n\right)$ and $p=1,q=1$, then:

$$WIB{M}^{1,1}\left(X\right)={\left(\begin{array}{l}0.3\cdot 0.4\cdot \left(0.4+0.5+0.7\right)+0.3\cdot 0.5\cdot \left(0.4+0.5+0.7\right)\\ +0.4\cdot 0.7\cdot \left(0.4+0.5+0.7\right)\end{array}\right)}^{{\displaystyle \frac{1}{2}}}=0.9381>{\max }_i\left\{x_i\right\}=0.7.$$

Therefore, the boundedness of this operator is not satisfied.

Obviously, the aggregated result does not satisfy the boundedness. Therefore, we propose the following adjusted WIBM operator.

Definition 9. 

Let $p>0,q>0$, and $X=\{x_i|i=1,2,\dots ,n\}\in {[0,1]}^n$ be a collection of crisp data. The adjusted WIBM operator is given by

$$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}},$$

(8)

where $w_i\in \left[0,1\right]\left(i=1,2,\dots ,n\right)$ is a collection of weights and ${\displaystyle \sum _{i=1}^n{w}_i=1}$. Further, $v_{i,j}$ is a weight that $v_{i,j}={\displaystyle \frac{w_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}$, and $w_{i,j}\in \left[0,1\right]\left(i\ne j,i,j=1,2,\dots ,n\right)$ is a collection of weight, and

$$w_{i,i}=\left\{\begin{array}{cc}0& \mathrm{if}{\displaystyle {\sum }_{j=1;i\ne j}^n{w}_{i,j}\ne 0}\\ 1& \mathrm{if}{\displaystyle {\sum }_{j=1;i\ne j}^n{w}_{i,j}=0}\end{array}\right..$$

The comparison of $w_{i,i}$ between WIBM and adjusted WIBM is shown in 

Table 1. The comparison of w_i,i between two operators.

WIBM Operator [13]	Adjusted WIBM Operator in Definition 9
$w_{i,i}=0$	$w_{i,i}=\left\{\begin{array}{cc}0& \mathrm{if}{\displaystyle {\sum }_{j=1;i\ne j}^n{w}_{i,j}\ne 0}\\ 1& \mathrm{if}{\displaystyle {\sum }_{j=1;i\ne j}^n{w}_{i,j}=0}\end{array}\right.$

.

Remark 1. 

The adjusted operator will continue to be referred to as WIBM operator for the sake of convenience.

The following property for the adjusted WIBM operator is discussed.

Property 2. 

Let $X=\{x_i|i=1,2,\dots ,n\}\in {[0,1]}^n$ be a collection of crisp data. When $w_{i,j}=0\left(i,j=1,2,\dots ,n;j\ne i\right)$, i.e., all inputs are independent, the adjusted WIBM operator reduces to the weighted power mean operator:

$$WIB{M}^{p,q}\left(X\right)=WP{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^{p+q}}\right)}^{{\displaystyle \frac{1}{p+q}}}.$$

(9)

Proof of Property 2. 

Since $w_{i,j}=0\left(i,j=1,2,\dots ,n\right)$ and $w_{i,i}=\left\{\begin{array}{cc}0& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}\ne 0}\\ 1& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}=0}\end{array}\right.$, then

$$v_{i,j}={\displaystyle \frac{w{ }_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}=\left\{\begin{array}{cc}{\displaystyle \frac{0}{1}}& i\ne j\\ {\displaystyle \frac{1}{1}}& i=j\end{array}\right.\left(i,j=1,2,\dots ,n\right),$$

then

$$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left(x_i^q\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^{p+q}}\right)}^{{\displaystyle \frac{1}{p+q}}}=WP{M}^{p,q}\left(X\right)$$

□

A comparison between the special case of WIBM and the adjusted WIBM is provided in

Table 2. Comparison between WIBM and adjusted WIBM.

WIBM Operator [13]	Adjusted WIBM Operator in Definition 9
When all inputs are independent
$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left(x_i^q+{\displaystyle \sum _{j\ne i}^n{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}$	$WIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left(x_i^q\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}$

.

Example 2 (Continuation of Example 1). 

Let $X=\left\{0.4,0.5,0.7\right\}$ be a collection of inputs,$w_i=\left(0,3,0.3,0.4\right)$ be a collection of weights, and $w_{i,j}=0\left(i,j=1,2,\dots ,n\right)$. The WIBM operator is ($p=1,q=1$):

$$WIB{M}^{1,1}\left(X\right)={\left(\begin{array}{l}0.3\cdot 0.4\cdot 0.4+0.3\cdot 0.5\cdot 0.5\\ +0.4\cdot 0.7\cdot 0.7\end{array}\right)}^{{\displaystyle \frac{1}{2}}}=0.5648<{\max }_i\left\{x_i\right\}=0.7.$$

Property 2 and Example 2 demonstrate that the adjusted operator overcomes the shortcomings of the WIBM operator.

Next, the WIBM operator is extended from the unit interval to the $[0,\tau ]\left(\tau >1\right)$, resulting in the extended WIBM operator.

Definition 10. 

Let $p>0,q>0$ and $X=\{x_i|i=1,2,\dots ,n\}\in {[0,\tau ]}^n\left(\tau >1\right)$ be a collection of crisp data. The extended WIBM (EMIBM) operator is given by

$$EWIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}},$$

(10)

where $w_i\in \left[0,1\right]\left(i=1,2,\dots ,n\right)$ is a collection of weights and ${\displaystyle \sum _{i=1}^n{w}_i=1}$, $v_{i,j}={\displaystyle \frac{w_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}$ and $w_{i,j}\in \left[0,1\right]\left(i\ne j,i,j=1,2,\dots ,n\right)$ is a collection of weights, and

$$w_{i,i}=\left\{\begin{array}{cc}0& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}\ne 0}\\ 1& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}=0}\end{array}\right..$$

Some special cases of the EWIBM operator are discussed.

Property 3. 

When $q\to 0$, the EWIBM operator reduces to the extended weighted power mean:

$$EWIB{M}^{p,q\to 0}\left(X\right)=EWPM\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_i^p}\right)}^{{\displaystyle \frac{1}{p}}}.$$

(11)

When $q\to 0,p\to 0$, the EWIBM operator reduces to extended weighted geometric mean:

$$EWIB{M}^{p\to 0,q\to 0}\left(X\right)=EWGM\left(X\right)={\displaystyle \prod _{i=1}^n{x}_i^{w_i}}.$$

(12)

Theorem 2. 

Let $p>0,q>0$ and $X=\{x_i|i=1,2,\dots ,n\}\in {[0,\tau ]}^n$ be a collection of crisp data. The EWIBM operator satisfies the following properties:

(1) (Idempotency) If $x_i=x_0\left(i=1,\dots ,n\right)$, then

$$EWIB{M}^{p,q}\left(X\right)=x_0.$$

(2) (Monotonicity) If $Y=\left\{y_i|i=1,\dots ,n\right\}\in {\left[0,\tau \right]}^n$ be a collection of crisp data with $x_i\le y_i\left(i=1,2,\dots ,n\right)$, then

$$EWIB{M}^{p,q}\left(X\right)\le EWIB{M}^{p,q}\left(Y\right).$$

(3) (Boundedness) If $x_{\max }={\max }_i\left\{x_i\right\},x_{\min }={\min }_i\left\{x_i\right\}$, then

$$x_{\min }\le EWIB{M}^{p,q}\left(X\right)\le x_{\max }.$$

Proof of Theorem 2. 

(1) Since $x_i=x_0\left(i=1,2,\dots ,n\right)$, then

$$EWIB{M}^{p,q}\left(X\right)={\left({\displaystyle \sum _{i=1}^n{w}_i\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{x}_0^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}={\left({\displaystyle \sum _{i=1}^n{w}_i{x}_0^{p+q}}\right)}^{{\displaystyle \frac{1}{p+q}}}=x_0$$

(2) According to the monotonicity of $WIB{M}^{p,q}\left(X\right)$, then

$$EWIB{M}^{p,q}\left(X\right)\le EWIB{M}^{p,q}\left(Y\right)$$

(3) According to (1) and (2), we have:

$$EWIB{M}^{p,q}\left(x_{\max },x_{\max },\dots ,x_{\max }\right)=x_{\max },EWIB{M}^{p,q}\left(x_{\min },x_{\min },\dots ,x_{\min }\right)=x_{\min }$$

$$EWIB{M}^{p,q}\left(x_{\min },x_{\min },\dots ,x_{\min }\right)\le EWIB{M}^{p,q}\left(X\right)\le EWIB{M}^{p,q}\left(x_{\max },x_{\max },\dots ,x_{\max }\right)$$

□

Based on the proposed EWIBM operator, the BULIWIBM operator for aggregating BULI pairs is defined as follows.

Definition 11. 

Let $\mathit{B}=\left\{b_i=〈{\Delta }_S\left({\psi }_i\right);c_i〉|{\Delta }_s\left({\psi }_i\right)\in \stackrel{-}{S},c_i\in \left[0.1\right],i=1,2,\dots ,n\right\}$ be a collection of BULI pairs to be aggregated, in which $S$ refers to a given LTS. The BULI weighted interaction Bonferroni mean (BULIWIBM) operator based on the EWIBM operator is given by

$$\begin{array}{c}BULIWIBM\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\\ =〈{\Delta }_S\left(EWIB{M}^{p,q}\left({\psi }_1,\dots ,{\psi }_n\right)\right);WA\left(c_1,\dots ,c_n\right)〉\\ =〈\Delta \left({\left({\displaystyle \sum _{i=1}^n{w}_i{\psi }_i^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{\psi }_j^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉\end{array},$$

(13)

where ${\displaystyle {\sum }_{i=1}^n{w}_i=1}$ and $w_i\in \left[0,1\right]\left(i=1,2,\cdots ,n\right)$.

Theorem 3. 

Let $\mathit{B}=\left\{b_i=〈{\Delta }_S\left({\psi }_i\right);c_i〉|{\Delta }_s\left({\psi }_i\right)\in \stackrel{-}{S},c_i\in \left[0.1\right],i=1,2,\dots ,n\right\}$

be a collection of BULI pairs to be aggregated, the BULIWIBM operator satisfied the following properties:

(1) (Idempotency) If $b_i=〈{\Delta }_S\left({\psi }_i\right);c_i〉=b_0=〈{\Delta }_S\left({\psi }_0\right);c_0〉,i=1,2,\dots ,n$, then

$$BULIWIB{M}^{p,q}\left(\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\right)=b_0=〈{\Delta }_S\left({\psi }_0\right);c_0〉.$$

(2) (Monotonicity) Let $\mathit{H}=\left\{〈{\Delta }_S\left({\eta }_i\right);d_i〉,i=1,2,\dots ,n\right\}$ be a collection of BULI pairs with ${\psi }_i\le {\eta }_i,c_i\le d_i,i=1,2,\dots ,n$, then

$$BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\le BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\eta }_1\right);d_1〉,\dots ,〈{\Delta }_S\left({\eta }_n\right);d_n〉\right)$$

(3) (Boundedness) Let $b_{\max }=〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉=〈{\Delta }_S\left({\max }_i\left\{{\psi }_i\right\}\right);{\max }_i\left\{c_i\right\}〉$ and

$$b_{\min }=〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉=〈{\Delta }_S\left({\min }_i\left\{{\psi }_i\right\}\right);{\min }_i\left\{c_i\right\}〉, \mathit{then}$$

$$b_{\min }\le BULIWIB{M}^{p,q}\left(\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\right)\le b_{\max }.$$

Proof of Theorem 3. 

(1) According to the idempotency of EWIBM operator and WA operator, and

$$b_i=〈{\Delta }_S\left({\psi }_i\right);c_i〉=〈{\Delta }_S\left({\psi }_0\right);c_0〉,i=1,2,\dots ,n$$

then

$$BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)=BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_0\right);c_0〉,\dots ,〈{\Delta }_S\left({\psi }_0\right);c_0〉\right)=〈{\Delta }_S\left({\psi }_0\right);c_0〉$$

(2) According to the monotonicity of EWIBM operator and WA operator, and

$${\psi }_i\le {\eta }_i,c_i\le d_i,i=1,2,\dots ,n$$

then

$$EWIB{M}^{p,q}\left({\psi }_1,\cdots ,{\psi }_n\right)\le EWIB{M}^{p,q}\left({\eta }_1,\cdots ,{\eta }_n\right) \mathrm{and} {\displaystyle \sum _{i=1}^n{w}_i{c}_i}\le {\displaystyle \sum _{i=1}^n{w}_i{d}_i}$$

Therefore,

$$BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\le BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\eta }_1\right);d_1〉,\dots ,〈{\Delta }_S\left({\eta }_n\right);d_n〉\right)$$

(3) According to the idempotency of BULIWIBM, we have

$$BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉,\dots ,〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉\right)=〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉$$

$$BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉,\dots ,〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉\right)=〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉$$

According to the monotonicity of BULIWIBM, we have

$$\begin{array}{l}\hspace{1em}BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉,\dots ,〈{\Delta }_S\left({\psi }_{\min }\right);c_{\min }〉\right)\\ \le BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)\\ \le BULIWIB{M}^{p,q}\left(〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉,\dots ,〈{\Delta }_S\left({\psi }_{\max }\right);c_{\max }〉\right)\end{array}$$

The proof is competed. □

Some special cases of the BULIWIBM operator are discussed.

Property 4. 

(1) When $q\to 0$, the BULIWIBM operator reduces to BULIWPM

$$BULIWIB{M}^{p,q\to 0}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)=〈\Delta \left({\left({\displaystyle \sum _{i=1}^n{w}_i{\psi }_i^p}\right)}^{{\displaystyle \frac{1}{p}}}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉.$$

(2) When $q\to 0,p\to 1$, the BULIWIBM operator reduces to BULIWA [17]

$$BULIWIB{M}^{p,q\to 0}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)=〈\Delta \left({\displaystyle \sum _{i=1}^n{w}_i{\psi }_i}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉.$$

(3) When $q\to 0,p\to 0$, the BULIWIBM operator reduces to BULIWGM

$$BULIWIB{M}^{p\to 0,q\to 0}\left(〈{\Delta }_S\left({\psi }_1\right);c_1〉,\dots ,〈{\Delta }_S\left({\psi }_n\right);c_n〉\right)=〈{\Delta }_S\left({\displaystyle \prod _{i=1}^n{\psi }_i^{w_i}}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉.$$

4. Large-Scale Consumer Multi-Criteria Satisfaction Ratings Aggregation Approach

The proposed operator was employed to develop a large-scale online group users multi-criteria product ratings aggregation approach. This section describes the establishment of a user credibility rating system, acquisition of criteria interaction coefficients, and development of a multi-criteria ratings fusion model.

4.1. Large-Scale Consumer Multi-Criteria Satisfaction Ratings Aggregation Problem

Online platforms provide a substantial amount of user feedback information, which serves as a valuable decision-making tool for subsequent consumer purchases. However, the inevitable decline in information credibility due to the big data environment necessitates the importance of comprehensive reliability in consumer satisfaction ratings to assist users in making informed decisions. In this section, we employ basic uncertain linguistic information (BULI) to characterize both consumer satisfaction ratings and the related credibility while utilizing a data-driven weight generation method to effectively aggregate ratings and offer consumers more reliable ranking results.

For the large-scale satisfaction ratings aggregation problem, let $A=\left\{a_1,a_2,\dots ,a_{\mu }\right\}$ be a set of the products, $C=\left\{c_1,c_2,\dots ,c_n\right\}$ be a set of criteria, $G{S}^{\phi }={\left(s_k^{\phi ,j}\right)}_{m_{\phi }\times n}$ be a matrix of group consumer satisfaction ratings for alternative $a_{\phi }\left(\phi =1,2,\cdots ,\mu \right)$, and $s_k^{\phi ,j}$ denote the rating of the $k-th\left(k\in \{1,2,\dots ,m_{\phi }\}\right)$ user for the alternative $a_{\phi }$ under criterion $c_j\left(j\in \{1,2,\dots n\}\right)$.

4.2. Criteria Interaction Coefficient Learning Method

The interaction coefficient of traditional criteria is typically determined directly by experts, and the decision-making trial and evaluation laboratory (DEMATEL) method is widely used for its exceptional information fusion ability [26]. However, big data provides a more reliable database for determining the interaction coefficient. Additionally, we propose an interaction coefficient generation method based on the combination of consumer satisfaction ratings and expert knowledge, taking inspiration from the existing weight calculation method [27] to obtain the interaction relationship between the two criteria.

• Phase I: The interaction coefficient of criterion based on expert knowledge

During this stage, the degree of interaction between criteria is characterized by experts using a set of linguistic terms, which are then quantitatively converted into information output. Subsequently, a criteria interaction matrix is derived from expert knowledge based on the DEMATEL method.

Step 1: Obtain the criteria interaction matrix and criterion weights based on expert knowledge.

Step 1.1: First, collect the expert linguistic matrix $L^u={\left(l_{j_{0}j}^u\right)}_{n\times n}$ evaluating the degree of interaction between $c_j$ and $c_{j_0}$ as assessed by experts $e_u\left(u=1,2,\dots ,r\right)$, where $l_{j_{0}j}^u\in \left\{NC,LC,MC,RHC,HC\right\}$, as shown in

Table 3. The transform between linguistic term set and ratings.

Linguistic Terms of Criteria Interaction	Related Ratings
Negligible correlation (NC)	1
Low correlation (LC)	2
Moderate correlation (MC)	3
Relatively high correlation (RHC)	4
Highly correlated (HC)	5

. Second, based on the conversion method in Table 3, transform the linguistic matrix $L^u={\left(l_{j_{0}j}^u\right)}_{n\times n}$ into rating matrix $X^u={\left(x_{j_{0}j}^u\right)}_{n\times n}$. Finally, obtain the comprehensive criteria interaction matrix $X={\left(x_{j_{0}j}\right)}_{n\times n}$;

Step 1.2: Obtain the normalized matrix $N$ for matrix $X$:

$$N=\lambda X, \lambda ={\displaystyle \frac{1}{{\max }_{1\le j_0\le n}{\displaystyle \sum _{j=1}^n{x}_{j_{0}j}}}}\left(j_0,j\in \left\{1,2,\cdots ,n\right\}\right).$$

(14)

Step 1.3: Obtain the comprehensive matrix $M$:

$$M=\kappa \left(N{\left(E-N\right)}^{-1}\right)={\left(m_{j_{0}j}\right)}_{n\times n},$$

(15)

where $E$ is the $n$-th order unit matrix, and $\kappa ={\displaystyle \frac{1}{{\max }_{1<j,j_0<n}(N)}}$

$$x_{j_{0}j}={\displaystyle \frac{{\displaystyle \sum _{u=1}^r{x}_{j_{0}j}^u}}{r}}.$$

(16)

Step 2: Obtain the set of criteria interaction $I_j$ for criterion $j$:

$$I_j=\left\{c_{j_0}|m_{j_{0}j}\ge \theta \right\},$$

(17)

$$\theta =Mean\left(M\right)+SD\left(M\right)={\displaystyle \frac{1}{n^2}}{\displaystyle \sum _{j_0=1}^n{\displaystyle \sum _{j=1}^n{m}_{j_{0}j}+\sqrt{\frac{1}{n^2}{\displaystyle \sum _{j_0=1}^n{\displaystyle \sum _{j=1}^n{\left(m_{j_{0}j}-Mean\left(M\right)\right)}^2}}}}}.$$

(18)

Step 3: Obtain the criteria interaction matrix $r\_expert$ based on expert knowledge:

$$r\_expert={\left(e_{j_{0}j}\right)}_{n\times n},e_{j_{0}j}=\left\{\begin{array}{cc}0& \mathrm{i}\mathrm{f}\hspace{0.17em}{j}_0=j or m_{j_{0}j}<\theta \\ m_{j_{0}j}& \mathrm{i}\mathrm{f}\hspace{0.17em}{m}_{j_{0}j}\ge \theta \end{array}\right..$$

(19)

Step 4: Calculate criterion weights $w_{j_0}^c$ [28]:

$$w_{j_0}^c={\displaystyle \frac{\sqrt{{\left(R_{j_0}+D_{j_0}\right)}^2+{\left(R_{j_0}-D_{j_0}\right)}^2}}{{\displaystyle \sum _{j_0=1}^n\sqrt{{\left(R_{j_0}+D_{j_0}\right)}^2+{\left(R_{j_0}-D_{j_0}\right)}^2}}}}.$$

(20)

where $R_{j_0}+D_{j_0}$ represents central factors and $R_{j_0}-D_{j_o}$ the causal factors

$$R_{j_0}={\displaystyle \sum _{j=1}^n{m}_{j_{0}j}},D_{j_0}={\displaystyle \sum _{j=1}^n{m}_{j{j}_0}}.$$

(21)

• Phase II: The interaction coefficient of criteria based on consumer satisfaction ratings

To obtain an interaction coefficient that better reflects the interdependence between criteria, the temporal variation of ratings and the successive influence of relationships among criteria ratings based on user feedback are investigated. Subsequently, this interaction coefficient is combined with $r\_expert$ to determine the interaction coefficient for criteria. Here are the detailed steps:

Step 1: Obtain the variation trend vector ${\alpha }^j=\left(t_1^j,t_1^j,\dots ,t_g^j\right)$ for consumer satisfaction ratings $S_j$ under criteria $c_j$:

Step 1.1: For convenience, we provide the following symbol interpretation:

$S_j$ represents a set of consumer satisfaction ratings under the criterion $c_j$, where

$$S_j=\left\{s_{k_{p,q}}^j|k=1,2,\dots ,m,p\in \left\{1,2,\dots ,12\right\},q\in \left\{1,2,\dots ,31\right\}\right\}.$$

(22)

and $s_{k_{p,q}}^j$ is the rating of the k-th user in the $p$-th month and the $q$-th day under criterion $c_j$.

$S_{\sigma }^j=\left\{s_{k_{p,q}}^j|p=\sigma \right\}$ represents the set of consumer satisfaction ratings in the $\sigma$-th month, $S_{{\sigma }_b}^j=\left\{s_{k_{p,q}}^j|p=\sigma ,q\le 15\right\}$ represents a set of consumer satisfaction ratings from the first half of the month, and $S_{{\sigma }_e}^j=\left\{s_{k_{p,q}}^j|p=\sigma ,q>15\right\}$ represents the second half of the month;

Step 1.2: Obtain the ratings of variation trend vector $t_{\sigma }^j$ under criteria $c_j$ for the $\sigma$-th month:

$$t_{\sigma }^j=\left\{\begin{array}{cc}1& b_{\sigma }>e_{\sigma }\\ -1& b_{\sigma }<e_{\sigma }\\ 0& b_{\sigma }=e_{\sigma }\end{array}\right..$$

(23)

where $b_{\sigma }$ is the mean of the rating of $S_{{\sigma }_b}^j$, and $e_{\sigma }$ is the mean of $S_{{\sigma }_e}^j$;

Step 1.3: Obtain the variation trend vector ${\alpha }^j=\left(t_1^j,t_1^j,\dots ,t_h^j\right)$ for $S_j$;

Step 2: Obtain the interaction coefficient of the criteria.

Before providing specific steps, we first present some fundamental definitions.

Definition 12. 

Let ${\alpha }^{j_0}=\left(t_1^{j_0},t_2^{j_0},\dots ,t_h^{j_0}\right)$ and ${\alpha }^j=\left(t_1^j,t_2^j,\dots ,t_h^j\right)$ be the variation trend vector under the criteria $c_{j_o}$ and $c_j$, respectively. The related common sub-vectors are defined as follows:

(1) The common sub-vector and set for ${\alpha }^j$ and ${\alpha }^{j_0}$ are given by:

$${\gamma }_k^{j{j}_0}\subseteq {\alpha }^j\cap {\alpha }^{j_0},\left({\gamma }_{k+1}^{j{j}_0}\not\subset {\gamma }_k^{j{j}_0},k<h\right),$$

(24)

and

$${\Gamma }^{j{j}_0}=\left\{{\gamma }_k^{j{j}_0}|k=1,2,\dots ,k_{j{j}_0}\right\},$$

(25)

respectively, where $k_{j{j}_0}$ is the number of the common sub-vectors.

(2) The longest common sub-vector set is given by:

$$\left\{\begin{array}{l}{\Gamma }_{\max }^{j{j}_0}=\left\{{\gamma }_{l_{\max }}^{j{j}_0}|l_{\max }=1_{\max },2_{\max },\cdots ,L_{\max }^{j{j}_0}\right\}\\ l_{\max }\in \left\{k\left|l_k^{j{j}_0}={\max }_{1\le k\le k_{j{j}_0}}\left\{l_k^{j{j}_0}\right\}\right.\right\}\end{array}\right.,$$

(26)

where $l_k^{j{j}_0}\left(l_k^{j{j}_0}>2\right)$ is the length of the common sub-vector ${\gamma }_{l_{\max }}^{j{j}_0}$.

(3) The position function $\chi :{[1,h]}^2\to \left[1,h\right]$ of the first element ${\gamma }_{l_{\max }}$ of the sub-vector $\alpha$ is given by:

$$\chi \left(\alpha ,{\gamma }_{l_{\max }}^{j{j}_0}\right)=x_{l_{\max }}.$$

Definition 12 is elucidated through the utilization of Example 3.

Example 3. 

Let ${\alpha }^1$ and ${\alpha }^2$ be two variation trend vectors.

$$\begin{gathered} {\alpha }^1=\left(1,0,0,-1,-1,0,0,1,0,-1,1,0,1,-1,1\right), \\ {\alpha }^2=\left(-1,-1,1,0,0,-1,-1,1,0,-1,-1,1,0,1,-1\right).\hspace{1em} \hspace{0.17em} \end{gathered}$$

The related common sub-vector set is calculated:

$${\Gamma }^{12}=\left\{{\gamma }_1^{12},{\gamma }_2^{12},{\gamma }_3^{12}\right\}=\left\{\left(1,0,0,-1,-1\right),\left(1,0,-1\right),\left(-1,1,0,1,-1,1\right)\right\}.$$

The longest common sub-vector set is calculated:

$${\Gamma }_{\max }^{12}=\left\{{\gamma }_{1_{\max }}^{12},{\gamma }_{2_{\max }}^{12}\right\}=\left\{\left(1,0,0,-1,-1,0,0\right),\left(-1,1,0,1,-1,1\right)\right\}.$$

The position function values are calculated:

$$\begin{array}{l}\chi \left({\alpha }^1,{\gamma }_{1_{\max }}^{12}\right)=1,\chi \left({\alpha }^2,{\gamma }_{1_{\max }}^{12}\right)=3\\ \chi \left({\alpha }^1,{\gamma }_{2_{\max }}^{12}\right)=10,\chi \left({\alpha }^2,{\gamma }_{2_{\max }}^{12}\right)=11\end{array}.$$

Example 3 can be illustrated by Figure 1 and Figure 2:

Based on the given definitions, specific steps are provided as follows:

Step 2.1: Obtain the position function values of the longest common sub-vector ${\Gamma }_{\max }^{j{j}_0}$ for ${\alpha }^j$ and ${\alpha }^{j_0}$

$$\chi \left({\alpha }^j,{\gamma }_{l_{\max }}^{j{j}_0}\right)=x_{l_{\max }}^j,\chi \left({\alpha }^{j_0},{\gamma }_{l_{\max }}^{j{j}_0}\right)=x_{l_{\max }}^{j_0}.$$

(27)

Step 2.2: The representation of the longest common sub-vector ${\Gamma }_{\max }^{j{j}_0}$ for ${\alpha }^j$ and ${\alpha }^{j_0}$ can be categorized into the following three scenarios $T^{j\succ j_0}$, $T^{j\prec j_0}$, and $T^{j\sim j_0}$:

$$T^{j\succ j_0}=\left\{{\gamma }_{l_{\max }}^{j{j}_0}|x_{l_{\max }}^{j_0}<x_{l_{\max }}^j,l_{\max }=1_{\max },2_{\max },\cdots ,L_{\max }^{j{j}_0}\right\},$$

(28)

$$\begin{gathered} T^{j\prec j_0}=\left\{{\gamma }_{l_{\max }}^{j{j}_0}|x_{l_{\max }}^{j_0}>x_{l_{\max }}^j,l_{\max }=1_{\max },2_{\max },\cdots ,L_{\max }^{j{j}_0}\right\},\hspace{1em} \\ {T}^{j\sim j_0}=\left\{{\gamma }_{l_{\max }}^{j{j}_0}|x_{l_{\max }}^{j_0}=x_{l_{\max }}^j,l_{\max }=1_{\max },2_{\max },\cdots ,L_{\max }^{j{j}_0}\right\}. \end{gathered}$$

where $T^{j\succ j_0}$ represents the longest common sub-vectors in ${\alpha }^j$ that precede those in ${\alpha }^{j_0}$, $T^{j\prec j_0}$ represents the longest common sub-vectors in ${\alpha }^j$ that follow those in ${\alpha }^{j_0}$, and $T^{j\sim j_0}$ represents the longest common sub-vectors that simultaneously appear in both ${\alpha }^j$ and ${\alpha }^j$.

Step 2.3: Obtain the interaction coefficient of criteria $c_j$ and $c_{j_0}$;

Step 2.3.1: Obtain the distance $d_{l_{\max }}^{j_{0}j}$ between the position function values of the longest common sub-vector for ${\alpha }_{j_0}$ and ${\alpha }_j$:

$$d_{l_{\max }}^{j_{0}j}=\left|x_{l_{\max }}^{j_0}-x_{l_{\max }}^j\right|$$

(29)

The above rule is illustrated by Figure 2 with the vector in Example 3;

Step 2.3.2: Obtain the interaction weight coefficient ${\overline{\varpi }}_{j{j}_0}$ with the following rules:

Case 1: ${\Gamma }_{\max }^{j{j}_0}=T^{j\succ j_0}\cup T^{j\sim j_0}$

$${\overline{\varpi }}_{j_{0}j}={\displaystyle \frac{1}{\left|T^{j\succ j_0}\right|+\left|T^{j\sim j_0}\right|}}\left({\displaystyle \sum _{{\gamma }_{l_{\max }}^{j{j}_0}\in T^{j\succ j_0}}^{ }\frac{l_{\max }^{j{j}_0}}{l_{\max }^{j{j}_0}+d_{l_{\max }}^{j{j}_0}}}+\left|T^{j\sim j_0}\right|\right).$$

(30)

Case 2: ${\Gamma }_{\max }^{j{j}_0}=T^{j\prec j_0}\cup T^{j\sim j_0}$

$${\overline{\varpi }}_{j_{0}j}={\displaystyle \frac{1}{\left|T^{j\prec j_0}\right|+\left|T^{j\sim j_0}\right|}}\left({\displaystyle \sum _{{\gamma }_k^{j{j}_0}\in T^{j\prec j_0}}^{ }\frac{l_{\max }^{j{j}_0}}{l_{\max }^{j{j}_0}+d_{l_{\max }}^{j{j}_0}}}+\left|T^{j\sim j_0}\right|\right).$$

(31)

If $d_{l_{\max }}^{j{j}_0}=0$, let ${\gamma }_{l_{\max }{ }^{\prime }}^{j{j}_0}$ be equal to the first $l_{\max }^{j{j}_0}-1$ elements of ${\gamma }_{l_{\max }}^{j{j}_0}$, then get the set ${\Gamma }_{{\max }^{\left(1\right)}}^{j{j}_0}=\left\{{\gamma }_{l_{\max }{ }^{\prime }}^{j{j}_0}|l_{{\max }^{\prime }}^{j{j}_0}=l_{\max }^{j{j}_0}-1\right\}$. Let the set ${\Gamma }_{{\max }^{\left(1\right)}}^{j{j}_0}$ replace ${\Gamma }_{\max }^{jj0}$ in ${\Gamma }^{\max }$, and then recalculate the coefficients according to the above steps;

Step 3: Obtain the user interaction coefficient matrix $r\_user$.

$$r\_user={\left(u_{j_{0}j}\right)}_{n\times n}, \mathrm{where} u_{j_{0}j}=\left\{\begin{array}{cc}0& \mathrm{i}\mathrm{f} j_0=j orj\notin I_{j_0}\\ {\overline{\varpi }}_{j_{0}j}& \mathrm{i}\mathrm{f} j\in {\mathrm{I}}_{j_0}\end{array}\right..$$

(32)

• Phases III: Obtain the comprehensive interaction coefficients

Step 1: The comprehensive interaction coefficients $w_{j_{0}j}\left(j_0\ne j;j,j_0=1,2,\dots ,n\right)$ demanded by the proposed operator to aggregate the associated criterion information is obtained by the interaction coefficient generation method:

$$w_{j_{0}j}=\epsilon u_{j_{0}j}+\left(1-\epsilon \right)e_{j_{0}j},\epsilon \in \left[0,1\right].$$

(33)

4.3. Establishment of the Transform Method between Consumer Satisfaction Ratings and BULI Pairs

In this section, we consider both the rating and the credibility of users, subsequently transforming them into a BULI matrix.

The process diagram illustrating the conversion method between ratings and BULI pairs is depicted in Figure 3.

• Phases I: Establish the consumer satisfaction rating and user credibility matrix.

Step 1: Establish the consumer satisfaction rating matrix.

Let $G{S}^{\phi }={\left(s_k^{\phi ,j}\right)}_{m_{\phi }\times n}$ be the group consumer satisfaction rating matrix, where $s_k^{\phi ,j}$ indicates the rating of the $\mathrm{k-th} \left(k\in \{1,2,\dots ,m_{{ }^{\phi }}\}\right)$ user $u_k^{\phi }$ for alternative $a_{\phi }\left(\phi \in \{1,2,\cdots ,\mu \}\right)$ under criteria $c_j\left(j\in \{1,2,\dots n\}\right)$;

Step 2: Establish the user credibility matrix:

Step 2.1: Establish the user reputation index system.

According to the user information collected by Autohome.com, the reputation index system was obtained as shown in

Table 4. The reputation index system for users.

Indexes	Indexes Analysis
$z_1^{ }$ Certification index	Based on whether the user has submitted their driving license on Autohome.com.
$z_2^{ }$ Interaction index	Based on the number of consumer satisfaction ratings supported, viewed, and commented on have been posted on the platform.
$z_3^{ }$ Daily utilization index	According to the total mileage and daily mileage of the car displayed for the user on the platform.
$z_4^{ }$ Platform content rating index	Based on ratings of user posts provided by the Autohome.com.

, where $z_r\left(u_k^{\phi }\right)\left(r\in \left\{1,2,3,4\right\}\right)$ is the $r$-th index for user $u_k^{\phi }$;

Step 2.2: Establish the calculation method of reputation index:

Step 2.2.1: Calculate the certification index $z_1\left(u_k^{\phi }\right)$

$$z_1\left(u_k^{\phi }\right)=\left\{\begin{array}{cc}1& u_k^{\phi } \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{a}\mathrm{r} \mathrm{o}\mathrm{w}\mathrm{n}\mathrm{e}\mathrm{r}\\ 0.5& u_k^{\phi } \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{c}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{c}\mathrm{a}\mathrm{r} \mathrm{o}\mathrm{w}\mathrm{n}\mathrm{e}\mathrm{r}\end{array}\right.$$

(34)

Step 2.2.2: Calculate the interaction index $z_2^k\left(u_k^{\phi }\right)$. Depending on the number of views, supports, and comments, the interaction index level is divided. The index rating is obtained according to the classification level as shown in

Table 5. The index $z_2^k\left(u_k^{\phi }\right)$ rating criteria.

Ratings	0.2	0.4	0.6	0.8	1.0
${\lambda }_1$ Number of views	$[1,100)$	$[101,1000)$	$[1001,\mathrm{10,000})$	$[\mathrm{10,001},\mathrm{100,000})$	$>\mathrm{100,000}$
${\lambda }_2$ Number of supports	$0$	$[1,5)$	$[6,10)$	$[11,100)$	$>100$
${\lambda }_3$ Number of comments	$0$	$[1,5)$	$[6,10)$	$[11,100)$	$>100$

.

Calculate the interaction index $z_2^k\left(u_k^{\phi }\right)$:

$$z_2\left(u_k^{\phi }\right)={\displaystyle \frac{1}{3}}\left(s{\lambda }_1^k+s{\lambda }_2^k+s{\lambda }_3^k\right)$$

(35)

where $s{\lambda }_i^k\left(i=1,2,3\right)$ is the rating of ${\lambda }_i^k\left(i=1,2,3\right)$ according to Table 5.

Step 2.2.3: Calculate the daily utilization index $z_3^k\left(u_k^{\phi }\right)$:

$$D{U}_k={\displaystyle \frac{O_k}{c{d}_k-p{d}_k}}.$$

(36)

where daily mileage $D{U}_k$ is calculated from total mileage $O_k$, purchase date $p{d}_k$ and comment date $c{d}_k$ for user $u_k^{\phi }$. The rating of the daily utilization index $z_3^k\left(u_k^{\phi }\right)$ is obtained as follows:

$$z_3\left(u_k^{\phi }\right)=\left\{\begin{array}{cc}0.2& \mathrm{i}\mathrm{f} D{U}_k\in \left[0,25\right]\\ 0.4& \mathrm{i}\mathrm{f} D{U}_k\in (25,50]\\ 0.6& \mathrm{i}\mathrm{f} D{U}_k\in (50,100]\\ 0.8& \mathrm{i}\mathrm{f} D{U}_k>100\end{array}\right..$$

(37)

Step 2.2.4: Calculate the platform content rating index $z_4^k\left(u_k^{\phi }\right)$. The levels of comments are as given by Autohome.com. The rating of $z_4^k\left(u_k^{\phi }\right)$ is shown in

Table 6. The index $z_4^k\left(u_k^{\phi }\right)$ rating rule.

Ratings	0.2	0.4	0.6	0.8
Comment level	Common	Recommendation	Quintessence	Full quintessence

;

Step 2.3: Obtain the reputation rating $\xi \left(u_k^{\phi }\right)$ for user $u_k^{\phi }$:

$$\xi \left(u_k^{\phi }\right)={\displaystyle \frac{1}{4}}{\displaystyle \sum _{r=1}^4{z}_r\left(u_k^{\phi }\right)}$$

(38)

Step 2.4: Calculate the concordance index.

Calculate the concordance index ${\rho }_k^{\phi ,j}$ as follows, for the concordance index rating of the user $u_k^{\phi }$ under the criterion $c_j$ $c_j\left(j\in \{1,2,\dots n\}\right)$ for the alternative $a_{\phi }\left(\phi \in \{1,2,\cdots ,\mu \}\right)$:

$${\rho }_k^{\phi ,j}=1-|{\displaystyle \frac{s_k^{\phi ,j}}{\tau }}-{\displaystyle \frac{1}{m_{\phi }}}{\displaystyle \sum _{k=1}^{m_{\phi }}\frac{s_k^{\phi ,j}}{\tau }}|.$$

(39)

where $s_k^{\phi ,j}\in \left[0,\tau \right]$ means the rating of the user $u_k^{\phi }$ under the criterion $c_j$ for the alternative $a_{\phi }$;

Step 2.5: Obtain the user credibility matrix $C^{\phi }={\left(c_k^{\phi ,j}\right)}_{m_{\phi }\times n}$:

$$c_k^{\phi ,j}=\epsilon \xi \left(u_k^{\phi }\right)+\left(1-\epsilon \right){\rho }_k^{\phi ,j},\epsilon \in \left[0,1\right].$$

(40)

• Phase II: Establish the BULI matrix.

Step 1: Obtain the BULI pair ${\delta }_k^{\phi ,j}$ with respect to rating $s_k^{\phi ,j}\in \left[0,\tau \right]$ and credibility $c_k^{\phi ,j}$ for the user $u_k^{\phi }$:

$${\delta }_k^{\phi ,j}=〈{\Delta }_S\left({\psi }_k^{\phi ,j}\right),c_k^{\phi ,j}〉.$$

(41)

where ${\psi }_k^{\phi ,j}=s_k^{\phi ,j}-1$;

Step 2: Establish the group BULI matrix $G{Z}^{\phi }$ for the alternative $a_{\phi }\left(\phi \in \{1,2,\cdots ,\mu \}\right)$:

$$G{Z}^{\phi }={\left({\delta }_k^{\phi ,j}\right)}_{m_{\phi }\times n}={\left(〈{\Delta }_S\left({\psi }_k^{\phi ,j}\right),c_k^{\phi ,j}〉\right)}_{m_{\phi }\times n}.$$

(42)

4.4. Multi-Criteria BULI Pairs Fusion Model

In this subsection, a multi-criteria BULI pairs fusion model is constructed.

Step 1: Aggregating group BULI matrices to obtain the multi-criteria BULI matrix for alternatives.

Step 1.1: Obtain group BULI matrices $G{Z}^{\phi }\left(\phi =1,2,\cdots ,\mu \right)$ by using the method in Section 4.3:

$$G{Z}^{\phi }={\left({\delta }_k^{\phi ,j}\right)}_{m_{\phi }\times n}=\left(〈{\Delta }_S\left({\psi }_k^{\phi ,j}\right),c_k^{\phi ,j}〉\right)$$

(43)

Step 1.2: Obtain the multi-criteria BULI matrix $Z^{\phi }={\left({\delta }^{\phi ,j}\right)}_{1\times n}={\left(〈{\Delta }_S\left({\Phi }^{\phi ,j}\right),c^{\phi ,j}〉\right)}_{1\times n}$:

$${\Phi }^{\phi ,j}={\displaystyle \frac{1}{m_{\phi }}}{\displaystyle \sum _{k=1}^{m_{\phi }}{\psi }_k^{\phi ,j}},c^{\phi ,j}={\displaystyle \frac{1}{m_{\phi }}}{\displaystyle \sum _{k=1}^{m_{\phi }}{c}_k^{\phi ,j}}$$

(44)

Step 2: Obtain the criterion weights $w_i=w_{j_0}^c$ and the interaction coefficient $w_{i,j}$ according to Section 4.1 and Section 4.2,

Step 3: Obtain comprehensive BULI ${\delta }^{\phi }$ for alternative $a_{\phi }\left(\phi \in \{1,2,\cdots ,\mu \}\right)$ by using the BULIWIBM operator:

$${\delta }^{\phi }=\left(〈{\Delta }_S\left({\Phi }^{\phi }\right),c^{\phi }〉\right)=BULIWIB{M}^{p,q}\left({\delta }^{\phi ,1},{\delta }^{\phi ,2},\dots ,{\delta }^{\phi ,n}\right)=〈\Delta \left({\left({\displaystyle \sum _{i=1}^n{w}_i{\left({\psi }_i^{\phi }\right)}^p\left({\displaystyle \sum _{j=1}^n{v}_{i,j}{\left({\psi }_j^{\phi }\right)}^q}\right)}\right)}^{{\displaystyle \frac{1}{p+q}}}\right);{\displaystyle \sum _{i=1}^n{w}_i{c}_i}〉,$$

(45)

where $v_{i,j}={\displaystyle \frac{w_{i,j}}{{\displaystyle \sum _{j=1}^n{w}_{i,j}}}}$ and $w_{i,i}=\left\{\begin{array}{cc}0& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}\ne 0}\\ 1& \mathrm{if}{\displaystyle {\sum }_{\begin{array}{l}j=1\\ i\ne j\end{array}}^n{w}_{i,j}=0}\end{array}\right.$

Step 4: Based on the obtained BULIs ${\delta }^{\phi }$, the ranking of alternative $a_{\phi }\left(\phi =1,\cdots ,\mu \right)$ is derived:

$$a_{\sigma \left(\phi -1\right)}\succ a_{\sigma \left(\phi \right)}\left(\phi =2,\cdots ,\mu \right),$$

(46)

where ${\delta }^{\sigma \left(\phi -1\right)}\succ {\delta }^{\sigma \left(\phi \right)}\left(\phi =2,\cdots ,\mu \right)$.

4.5. MCGDM Approach for New Energy Vehicle Selection

In this subsection, we propose an approach to large-scale online group users multi-criteria ratings aggregation to address the problem discussed in Section 4.1.

Step 1: Determinate the products’ alternative set $A=\left\{a_1,a_2,\dots ,a_{\mu }\right\}$ and the product evaluation criteria set $C=\left\{c_1,c_2,\dots ,c_n\right\}$, and collect group consumer satisfaction rating matrices $G{S}^{\phi }\left(\phi =1,\cdots ,\mu \right)$ for all alternatives $a_{\phi }\left(\phi =1,\cdots ,\mu \right)$;

Step 2: Employ the criterion interaction coefficient generation method, which combines expert knowledge and large-scale consumer satisfaction ratings as described in Section 4.2, to derive the interaction coefficient $w_{j_{0}j}$ between criterion $c_j$ and criterion $c_{j_0}$;

Step 3: Using the conversion method from Section 4.3, which converts ratings to BULI pairs, transform the group rating matrices $G{S}^{\phi }\left(\phi =1,\cdots ,\mu \right)$ into a group BULI matrices $G{Z}^{\phi }\left(\phi =1,\cdots ,\mu \right)$;

Step 4: Apply the multi-criteria BULI pairs fusion model from Section 4.4 to obtain the comprehensive BULIs ${\delta }^{\phi }$ of the alternatives $a_{\phi }\left(\phi =1,\cdots ,\mu \right)$, as well as the ranking results of the alternatives $a_{\sigma \left(\phi -1\right)}\succ a_{\sigma \left(\phi \right)}\left(\phi =2,\cdots ,\mu \right)$.

The process of the produce ranking approach is depicted in Figure 4.

5. Case Study and Comparative Analysis

To demonstrate the application of the proposed methodology, this section presents a case study on automobile ranking and a comparative analysis.

5.1. Case Study

In this subsection, the proposed model was utilized to rank new energy vehicles based on consumer satisfaction ratings obtained from Automotive.com. The new energy vehicle selection problem has been extensively addressed in existing research using MCDM methods. For instance, the analytic hierarchy process (AHP) has been employed to rank products by extracting features from online reviews [29]. However, most current ranking methods often overlook the reliability of extracted user information. Therefore, BULI representation of information combined with user credibility can provide a more accurate reflection of users’ feedback on products than other ranking methodologies.

The present study is based on a dataset comprising six cars sourced from Automotive.com, namely BMW iX3, Volvo XC40, LEADINGIDEAL L7, AITO M7, Tesla Model 3, and Audi Q4 e-tron, as presented in

Table 7. Products’ data volume.

Alternative	${\mathit{a}}_{\mathrm{1}}$	${\mathit{a}}_{\mathrm{2}}$	${\mathit{a}}_{\mathrm{3}}$	${\mathit{a}}_{\mathrm{4}}$	${\mathit{a}}_{\mathrm{5}}$	${\mathit{a}}_{\mathrm{6}}$
Products	BMW iX3	Volvo XC40	LEADINGIDEAL L7	AITO M7	Tesla Model 3	Audi Q4 e-tron
data	6709	7894	6899	7790	7389	5839

. These specific car models were selected for analysis based on the consumer satisfaction ratings information available.

Step 1: Determinate the products alternative set $A=\left\{a_1,a_2,\dots ,a_6\right\}$ as shown in Table 7, and determinate the product evaluation criteria set $C=\left\{c_1,c_2,\dots ,c_8\right\}$ as shown in

Table 8. The evaluation criteria.

Criteria	Description
$\mathrm{Space} (c_1$)	The interior ridable space owned by the vehicle
$\mathrm{Driving} \mathrm{experience} (c_2$)	The user’s driving experience of the vehicle after a test drive
$\mathrm{Range} (c_3$)	The distance a vehicle can travel on a full charge
$\mathrm{Appearance} (c_4$)	The exterior of the vehicle
$\mathrm{Upholstery} (c_5$)	Interior decoration of the vehicle
$\mathrm{Cos}\mathrm{t} \mathrm{performance} (c_6$)	Vehicle price to configuration ratio
$\mathrm{Intelligence} (c_7$)	Intelligent technologies and connectivity features installed in the vehicle
$\mathrm{Comfortability} (c_8$)	Driving and riding comfort of the vehicle

. Collect group consumer satisfaction rating matrices $G{S}^{\phi }\left(\phi =1,\cdots ,6\right)$ for all alternatives $a_{\phi }\left(\phi =1,\cdots ,6\right)$ from Autohome.com. The detailed presentation of rating data is no longer feasible due to the vast volume of data;

Step 2: Employ the criterion interaction coefficient generation method, which combines expert knowledge and large-scale consumer satisfaction ratings as described in Section 4.2, to derive the interaction coefficient $w_{j_{0}j}$ between criterion $c_j$ and criterion $c_{j_0}$:

Step 2.1: Acquire the criteria weights $w_{j_0}^c$ and the interaction coefficient $r\_expert$ based on expert knowledge.

According to the experts’ rating of the criteria, the comprehensive impact matrix $M$ is obtained from Equations (14)–(16), as shown in

Table 9. The comprehensive impact matrix $M$.

Criteria	${\mathit{c}}_{\mathrm{1}}$	${\mathit{c}}_{\mathrm{2}}$	${\mathit{c}}_{\mathrm{3}}$	${\mathit{c}}_{\mathrm{4}}$	${\mathit{c}}_{\mathrm{5}}$	${\mathit{c}}_{\mathrm{6}}$	${\mathit{c}}_{\mathrm{7}}$	${\mathit{c}}_{\mathrm{8}}$
$c_1$	0.3645	0.5674	0.4725	0.4096	0.3667	0.5225	0.5413	0.5571
$c_2$	0.5624	0.5436	0.5510	0.4584	0.5031	0.7086	0.8074	0.7168
$c_3$	0.4648	0.4703	0.3857	0.3453	0.3775	0.5937	0.6755	0.5291
$c_4$	0.5085	0.5129	0.4175	0.3073	0.3744	0.5885	0.5705	0.5627
$c_5$	0.3995	0.5342	0.3699	0.3105	0.3011	0.4751	0.5265	0.5184
$c_6$	0.5606	0.7102	0.6733	0.5082	0.5531	0.6036	0.8160	0.7137
$c_7$	0.4560	0.5680	0.5698	0.4173	0.4966	0.6566	0.5788	0.6586
$c_8$	0.5343	0.6571	0.4799	0.4320	0.5430	0.6068	0.7626	0.5411

. Based on $M$, the criteria interaction set is obtained, as shown in

Table 10. The set of criterion interaction $I_j$ with respect to $C_j$.

Set	${\mathit{I}}_{\mathrm{1}}$	${\mathit{I}}_{\mathrm{2}}$	${\mathit{I}}_{\mathrm{3}}$	${\mathit{I}}_{\mathrm{4}}$	${\mathit{I}}_{\mathrm{5}}$	${\mathit{I}}_{\mathrm{6}}$	${\mathit{I}}_{\mathrm{7}}$	${\mathit{I}}_{\mathrm{8}}$
Interaction criteria	$c_2,c_7,c_8$	$\begin{array}{l}{c}_1,c_3,c_6,\\ c_7,c_8\end{array}$	$c_6,c_7$	$c_6,c_7,c_8$	$c_2$	$\begin{array}{l}{c}_1,c_2,c_3,\\ c_5,c_7,c_8\end{array}$	$\begin{array}{l}{c}_2,c_3,c_6,\\ c_8\end{array}$	$\begin{array}{l}{c}_1,c_2,c_5,\\ c_6,c_7\end{array}$

, where $\theta =0.6504$,and the set of criteria interaction $I_j$ for criterion $j$ is obtained from Equation (17) as shown in Table 10. Finally, the expert criteria interaction matrix $r\_expert$ is obtained from Equation (19), as shown in

Table 11. The expert criterion interaction matrix $r\_expert$.

Criteria	${\mathit{c}}_{\mathrm{1}}$	${\mathit{c}}_{\mathrm{2}}$	${\mathit{c}}_{\mathrm{3}}$	${\mathit{c}}_{\mathrm{4}}$	${\mathit{c}}_{\mathrm{5}}$	${\mathit{c}}_{\mathrm{6}}$	${\mathit{c}}_{\mathrm{7}}$	${\mathit{c}}_{\mathrm{8}}$
$c_1$	0	0.6953	0	0	0	0	0.6634	0.6827
$c_2$	0.6893	0	0.6753	0	0	0.8684	0.9894	0.8784
$c_3$	0	0	0	0	0	0.7276	0.8278	0
$c_4$	0	0	0	0	0	0.7212	0.6991	0.6895
$c_5$	0	0.6546	0	0	0	0	0	0
$c_6$	0.6870	0.8704	0.8251	0	0.6779	0	1	0.8746
$c_7$	0	0.6961	0.6982	0	0	0.8046	0	0.8071
$c_8$	0.6548	0.8052	0	0	0.6655	0.7436	0.9345	0

. The criteria weights are obtained from Equations (20) and (21):

$$w_{j_0}^c=\left(0.11,0.13,0.11,0.10,0.10,0.14,0.14,0.13\right)$$

Step 2.2: Acquire the interaction coefficient $r\_user$ based on large-scale consumer satisfaction ratings, and calculate the interaction coefficient $w_{j_{0}j}$.

Based on large-scale consumer satisfaction ratings, the user interaction matrix $r\_user$ is obtained from the Equations (22)–(32) as shown in

Table 12. The user criterion interaction matrix $r\_user$.

Criteria	${\mathit{c}}_{\mathrm{1}}$	${\mathit{c}}_{\mathrm{2}}$	${\mathit{c}}_{\mathrm{3}}$	${\mathit{c}}_{\mathrm{4}}$	${\mathit{c}}_{\mathrm{5}}$	${\mathit{c}}_{\mathrm{6}}$	${\mathit{c}}_{\mathrm{7}}$	${\mathit{c}}_{\mathrm{8}}$
$c_1$	0	0.3181	0	0	0	0	0.4837	0.7525
$c_2$	0.6154	0	0.2595	0	0	0.5737	0.5039	0.3706
$c_3$	0	0	0	0	0	0.5943	0.7181	0
$c_4$	0	0	0	0	0	0.3642	0.2813	0.6473
$c_5$	0	0.2813	0	0	0	0	0	0
$c_6$	0.3877	0.3000	0.6667	0	0.6077	0	0.5930	0.2391
$c_7$	0	0.7857	0.2857	0	0	0.7547	0	0.2538
$c_8$	0.6154	0.6471	0	0	0.7500	0.4642	0.5870	0

. Finally, the comprehensive interaction coefficients $w_{j_{0}j}$ are obtained, as shown in

Table 13. The comprehensive criteria interaction matrix.

Criteria	${\mathit{c}}_{\mathrm{1}}$	${\mathit{c}}_{\mathrm{2}}$	${\mathit{c}}_{\mathrm{3}}$	${\mathit{c}}_{\mathrm{4}}$	${\mathit{c}}_{\mathrm{5}}$	${\mathit{c}}_{\mathrm{6}}$	${\mathit{c}}_{\mathrm{7}}$	${\mathit{c}}_{\mathrm{8}}$
$c_1$	0	0.5068	0	0	0	0	0.5735	0.7176
$c_2$	0.6523	0	0.4674	0	0	0.7210	0.7467	0.6245
$c_3$	0	0	0	0	0	0.6610	0.7730	0
$c_4$	0	0	0	0	0	0.5427	0.4902	0.6684
$c_5$	0	0.4679	0	0	0	0	0	0
$c_6$	0.5373	0.5852	0.7459	0	0.6428	0	0.7965	0.5569
$c_7$	0	0.7409	0.4920	0	0	0.7797	0	0.5305
$c_8$	0.6351	0.7262	0	0	0.7078	0.6039	0.7608	0

($\epsilon =0.5$);

Step 3: Translate the ratings into BULI and aggregate the group BULIs to derive the BULI ratings for each individual product:

Step 3.1: Using the conversion method from Section 4.3, which converts ratings to BULI, transform the group rating matrices $G{S}^{\phi }\left(\phi =1,\cdots ,6\right)$ into group BULI matrices $G{Z}^{\phi }\left(\phi =1,\cdots ,6\right)$, then aggregate a group BULI into a multi-criteria BULI matrix $Z^{\phi }$, as shown in

Table 14. Multi-criteria BULI matrix $Z^{\phi }$.

	${\mathit{c}}_{\mathrm{1}}$	${\mathit{c}}_{\mathrm{2}}$	${\mathit{c}}_{\mathrm{3}}$	${\mathit{c}}_{\mathrm{4}}$
$a_1$	<3.824, 0.6841>	<3.527, 0.6631>	<3.656, 0.6692>	<3.768, 0.6783>
$a_2$	<3.869, 0.6700>	<3.634, 0.6501>	<3.403, 0.6395>	<3.846, 0.6675>
$a_3$	<3.693, 0.7790>	<3.465, 0.7472>	<3.717, 0.7817>	<3.803, 0.8006>
$a_4$	<3.5, 0.7649>	<3.518, 0.7693>	<3.359, 0.7509>	<3.835, 0.8199>
$a_5$	<3.733, 0.8005>	<3.34, 0.7624>	<3.218, 0.7513>	<3.725, 0.7995>
$a_6$	<3.511, 0.7697>	<3.693, 0.7904>	<3.418, 0.7557>	<3.777, 0.8064>
	${\mathit{c}}_5$	${\mathit{c}}_6$	${\mathit{c}}_7$	${\mathit{c}}_8$
$a_1$	<3.165, 0.6562>	<3.491, 0.6595>	<2.584, 0.6079>	<2.79, 0.6084>
$a_2$	<3.646, 0.6507>	<3.695, 0.6526>	<3.06, 0.6253>	<3.293, 0.6242>
$a_3$	<3.468, 0.7530>	<3.651, 0.7694>	<1.777, 0.4844>	<1.834, 0.4767>
$a_4$	<3.486, 0.7632>	<3.031, 0.7434>	<2.97, 0.6931>	<3.128, 0.6882>
$a_5$	<3.803, 0.8150>	<2.925, 0.7297>	<3.043, 0.7026>	<2.96, 0.7133>
$a_6$	<3.195, 0.7514>	<3.561, 0.7678>	<2.59, 0.6579>	<2.884, 0.6715>

;

Step 3.2: By employing the BULIWIBM operator in Section 4.4, the multi-criteria BULI matrices $Z^{\phi }$ $\left(\phi =1,\cdots ,6\right)$ are aggregated into comprehensive BULI ${\delta }^{\phi }$ for alternative $a_{\phi }\left(\phi \in \{1,2,\cdots ,6\}\right)$, as illustrated in

Table 15. The alternative comprehensive ratings.

Alternatives	${\mathit{a}}_{\mathrm{1}}$	${\mathit{a}}_{\mathrm{2}}$	${\mathit{a}}_{\mathrm{3}}$	${\mathit{a}}_{\mathrm{4}}$	${\mathit{a}}_{\mathrm{5}}$	${\mathit{a}}_{\mathrm{6}}$
${\delta }^{\phi }$	$〈\begin{array}{l}3.2631,\\ 0.7445\end{array}〉$	$〈\begin{array}{l}3.5034,\hfill \\ 0.7702\hfill \end{array}〉$	$〈\begin{array}{l}2.9864,\hfill \\ 0.5976\hfill \end{array}〉$	$〈\begin{array}{l}3.2815,\hfill \\ 0.6418\hfill \end{array}〉$	$〈\begin{array}{l}3.2448,\hfill \\ 0.6531\hfill \end{array}〉$	$〈\begin{array}{l}3.2720,\hfill \\ 0.6408\hfill \end{array}〉$

;

Step 4: Apply the multi-criteria BULI pairs fusion model from Section 4.4 to obtain the comprehensive BULIs ${\delta }^{\phi }$ of the alternatives $a_{\phi }\left(\phi =1,\cdots ,6\right)$. The ranking results can be obtained as follows, with the best being $a_2$:

$$a_2\succ a_1\succ a_5\succ a_4\succ a_6\succ a_3.$$

5.2. Comparative Analysis

In this subsection, a comparative and analytical examination of the proposed operators and their associated parameters is conducted.

5.2.1. Comparison of Without Regard to Credibility

To assess the impact of credibility on the ranking outcomes, a comparison was made between the product ratings provided by Automotive.com and the rankings proposed in this study, as presented in

Table 16. Comparison of different ranking methods.

Ranking Method	Ranking Results
Automotive.com (without credibility)	$a_4=a_5\succ a_2\succ a_3=a_6\succ a_1$
EWIBM method (without credibility)	$a_2\succ a_6\succ a_1\succ a_5\succ a_4\succ a_3$
BULIWIBM method (with credibility)	$a_2\succ a_1\succ a_5\succ a_4\succ a_6\succ a_3$

.

According to Table 16, the following results were obtained:

(1)

The best alternatives obtained by Automotive.com were $a_4$ and $a_5$, and the best alternative obtained with the WIBM operator was $a_2$.

(2)

The ranking results from Automotive.com failed to differentiate between $a_4$ and $a_5$, as well as between $a_3$ and $a_6$. Nevertheless, the ranking methodology employed in this study based on the WIBM operator successfully discerned such distinctions.

(3)

When applying the WIBM-based ranking method with credibility, the ranking of $a_6$ only marginally surpassed that of $a_3$, whereas without taking credibility into account, it was only slightly inferior to $a_2$. This underscores the paramount importance of incorporating credibility in our evaluation.

5.2.2. Comparison of BM Operator Coefficient Generation Methods

In order to elucidate the impact of different methods for generating the interaction coefficient among various BM operators for the aggregation outcomes, we compared the generation methods presented in

Table 17. Comparison of BM operator coefficient generation methods.

BM Operators	$\mathbf{Whether} \mathbf{There} \mathbf{Is} {\mathit{w}}_{\mathit{i}\mathit{j}}$	${\mathit{w}}_{\mathit{i}\mathit{j}}$ Generation Method
BM [29]	No	/
NWIBM [11]	Yes	Subjective
PFWIBM [25]	Yes	Subjective Gray relational analysis
BULIWIBM	Yes	Subjective Data-drive approach

with those proposed in [12,30,31].

The results presented in Table 17 demonstrate that, in comparison to other BM operators, the operator proposed in this paper possesses a more extensive data foundation and is well-suited for intricate scenarios. Moreover, it enables a more rational aggregation of inputs during the decision-making process.

6. Conclusions

The utilization of a data-driven interaction coefficient in WIBM enables the mitigation of drawbacks associated with subjective coefficient setting, thereby facilitating a more effective characterization of attribute correlation degrees. In this study, we constructed the BULIWIBM operator within the BULI environment by combining a big data-driven interaction coefficient generation method and incorporating information credibility, leading to the development of a large-scale online group consumer satisfaction ratings aggregation approach and ranking method. The advantages of our research are as follows:

(1)

This study uncovers and addresses the limitations of the existing WIBM operator, expanding its applicability beyond the unit interval $\left[0,1\right]$ to a wider range $\left[0,\tau \right]\left(\tau \ge 1\right)$ by proposing the extended WIBM (EWIBM) operator. Additionally, we introduce the BULIWIBM operator, enriching the theoretical framework of MCDM information fusion based on the BM operator;

(2)

The robust data foundation of the big data-based interaction coefficient generation method enables its application potential to extend across various scenarios;

(3)

Incorporating credibility into the consumer satisfaction ratings aggregation process enhances the quality of decision-making outcomes;

(4)

Driven by large-scale consumer satisfaction ratings, this study integrates information credibility and proposes a MCGDM method for selecting new energy vehicles. This method not only provides decision-making references for potential buyers but also offers a research paradigm for product selection studies on online service platforms, such as for hotels and restaurants.

The use of data-driven MCDM methods is increasingly dominant in diverse fields. However, challenges still exist for the BULIWIBM operator, such as high computational complexity and a lack of universal standards for parameter settings when dealing with large-scale data-driven MCDM problems. Furthermore, there is potential to apply more generalized operational rules to enhance the BULIWIBM operator’s effectiveness. Future research should focus on advancing mining and deepening understanding of criteria interactions driven by data while addressing uncertain information. Integrating generalized BULI operational laws [32] and the BM operator [33] into the BULIWIBM operator could improve its generalization capability further. Moreover, it is worth considering decision-making methods that more effectively capture real-time and dynamic characteristics.