Green Supplier Selection Based on Sequential Group Three-Way Decision Making

Abstract

Environmental protection and sustainable development have become the consensus of all countries in the world. Enterprises must pay attention to the impact on the environment in their operations. Therefore, the selection of green suppliers has become a crucial issue for companies. Supplier selection is a dynamic and complex multi-attribute group decision-making process. The decision results have the tripartite characteristics of “accepted”, “rejected” and “pending further investigation”, and experts need to constantly negotiate in the decision-making process to achieve consensus. In view of the above characteristics, this study constructs a sequential group three-way decision making (TWDM) method to support green supplier selection. Firstly, we review the existing literature on the evaluation criteria and selection methods of green suppliers. In this process, we construct an evaluation attribution system including the following four aspects: product formation, service level, development capability, and green level. Secondly, combining with the sequential and group characteristics of decision making, we propose a multiple-attribute sequential group TWDM method based on a multi-level granularity structure. The weight of each decision maker is determined by his influence weight and interval-valued intuitionistic fuzzy entropy weight. The attribute weight is determined by entropy weight and subjective weight. By using the VIsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method improved by grey relation analysis (GRA), we calculate conditional probabilities. Since the cumulative prospect theory can reflect the risk attitude of decision makers, we apply it to calculate the decision thresholds. Finally, we apply a case of Q automobile manufacturer to verify the effectiveness, applicability and feasibility of the method. The innovation of this study is to construct social networks at each granularity level and introduce an expert information exchange model to promote group consensus. The sequential group TWDM method provides a new reference and idea for the selection of green suppliers.

1. Introduction

With the development of the world economy and the acceleration of the industrialization process, issues such as excessive resource waste and severe environmental pollution have become increasingly prominent. Balancing economic and social development with ecological preservation has become a critical concern shared worldwide. In 1987, the World Commission on Environment and Development introduced the concept of sustainable development in a report titled “Our Common Future”. It is necessary to raise environmental awareness and encourage simultaneous economic and environmental development [1].

The concept of green supply chain management (GSCM) [2] can be traced back to Green Procurement introduced by Webb in 1994 [3]. Subsequently, the Manufacturing Research Association at Michigan State University formally proposed the concept of GSCM in their study on “Environmentally Responsible Manufacturing”. GSCM integrates the concepts of environmental protection and resource conservation into the entire supply chain management process. Its purpose is to reduce the harm and adverse impact of products on the environment in the process of production, use and disposal, improve the resource utilization efficiency, and minimize environmental pollution.

GSCM not only guides enterprises to purchase raw materials and products with lower pollution emissions and higher environmental performance, but also encourages suppliers to take active environmental protection measures. According to recent statistics released by the European Union, about 70% of consumers consider a company’s environmental actions and reputation when choosing products. Therefore, whether from the perspective of social benefits or economic benefits, enterprises should strengthen GSCM and select green suppliers.

Green supplier selection is a multi-attribute decision making (MADM) problem. However, existing methods are primarily binary decision making, where suppliers are evaluated and ranked based on multiple attributes, and then a certain number of suppliers are accepted and others are rejected. Although the binary decision making is simple and quick, specifying a fixed number of suppliers may not be a wise choice. On the one hand, it carries the risk of misclassification, as the quality of suppliers within the specified quantity may vary significantly, while the quality difference between a supplier outside the quantity and a supplier within the quantity may be small. On the other hand, supplier selection is a long-term and dynamic process, so a certain number of “pending” suppliers should be allowed, which can incentivize suppliers to continuously improve their performance for acceptance. It can be seen that binary decision making is not suitable for green supplier selection. In contrast, the three-way decision making (TWDM) divides the suppliers into three types, namely “accepted”, “rejected” and “pending further investigation”, and adopts the “three-division and rule” strategy, which is more realistic and can avoid the loss of error cost. Therefore, green supplier selection tends to be a TWDM problem. Due to the seriousness of green supplier selection, it is usually a group decision making (GDM) process implemented by an expert group. In traditional GDM, when experts have different opinions, they often need to carry out multiple rounds of discussion and adjustment to obtain relatively consistent evaluation results, which is often very time-consuming. Therefore, how to quickly and efficiently obtain objective and consistent conclusions based on the evaluation results of each expert has become an important issue in the selection of green suppliers. To solve the above problems, this article presents a sequential group three-way decision-making method for selecting green suppliers. Compared with the traditional methods, this method divides suppliers into three ways, namely, “accepted”, “rejected” and “pending further investigation”, which not only enhances the flexibility and adaptability of supplier selection, but also enhances the initiative of suppliers to improve performance. In addition, this method improves the efficiency of group decision-making by introducing dynamic factors such as multi-level granularity, expert social network, and expert information exchange. Simultaneously, it enriches GSCM theory and provides decision-making ideas for enterprises’ green supplier selection.

The rest of this article is organized as follows. Section 2 conducts a literature review. Section 3 provides the relevant preliminary knowledge and the specific process of the sequential group TWDM method. Section 4 presents a case study to validate the effectiveness of the proposed method. Finally, the conclusions are presented in Section 5.

2. Literature Review

In this section, we conduct a comprehensive literature review on traditional TWDM and sequential TWDM. We delve into their definitions, explore their application areas and examine current research progress.

2.1. Green Supplier Selection

To select green suppliers, scholars have constructed evaluation attribute systems from various perspectives. For instance, Dickson [4] identified 23 criteria, with the most significant being quality, delivery, performance history, warranty and claims policy. Xu et al. [5] conducted a comprehensive assessment of suppliers’ production processes, products, and environmental management, including waste discharge, harmful substance utilization, energy usage, resource utilization, recyclability of products, eco-friendly packaging and other attributes. Freeman and Chen [6] used a semi-structured interview method to identify environmental criteria for green procurement strategy assessment. Akcan and Tas [7] defined 11 environmental criteria for green supplier assessment. In summary, the research results of scholars reveal the key attributes for evaluating green suppliers, as shown in

Table 1. Green supplier evaluation attributes adopted by scholars.

Criteria	Sub-Criteria	References
Product information	Product price	[2,6,8]
	Product quality	[4,6,9]
Service level	Delivery reliability	[6,10,11,12]
	Cooperation experience and compatibility	[13,14,15]
	After-sales service	[9,13,16]
Development capability	Financial status	[4,9,13]
	R&D and innovation capability	[8,10,11,16]
	Staff quality and treatment	[12,17]
	Credibility	[9,14,18]
Green level	Environmental capital investment	[15,16,19]
	Waste disposal	[7,13,17]
	Environmental technology and use of materials	[6,10,19]
	Environmental certification	[11,16,18]
	Green management strategy	[7,16,20]

.

Scholars have proposed various methods for green supplier evaluation. These methods can be categorized into three types: comparative evaluation using the analytic hierarchy process (AHP) [21,22], relative performance evaluation based on data envelopment analysis (DEA) [23,24,25] and MADM methods [26]. Because MADM methods do not rely solely on qualitative comparison like AHP, nor do they need to divide attributes into input and output attributes like DEA, they are more widely used.

Table 2. MADM methods for green supplier evaluation.

MADM Method	Attribute Weighting	Expert Weighting	References
TOPSIS	AHP	No	[27]
	Fuzzy AHP	No	[11]
	Fuzzy entropy	No	[28]
	AHP, entropy	No	[29]
	The maximizing deviation	SWM	[30]
VIKOR	AHP	SWM	[31]
	DEMATEL	No	[32]
GRA	RWGCA	SWM	[33]
	BWM, FGCM	Delphi method	[34]
ELECTRE	Entropy-weighted distance measure	Entropy	[35]
	Maximizing deviation	IVHF-MPSI	[36]
PROMETHEE	Revised Simos’ procedure	SWM	[37]
	ANP	SWM	[38]

provides an overview of the primary MADM methods employed in supplier selection. These methods include the technique for order preference by similarity to an ideal solution (TOPSIS), VIsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR), grey relation analysis (GRA), elimination and choice translating reality (ELECTRE), and preference ranking organization method for enrichment evaluation (PROMETHEE). Additionally, various attribute weighting methods are used, such as AHP, entropy, the maximizing deviation, DEMATEL (decision-making trial and evaluation laboratory), RWGCA (relative weighting grey correlation analysis), BWM (best worst method), FGCM (fuzzy grey correlation method), entropy-weighted distance measure, revised Simos’ procedure and ANP (analytic network process). Expert weighting methods encompass SWM (subjective weighting method), Delphi, entropy and IVHF-MPSI (interval-valued hesitant fuzzy-multiplicative preference set identification).

2.2. Sequential Three-Way Decision Making

To overcome the limitations of binary decision making such as inaccurate classification and high error cost, Yao et al. [39,40] introduced the TWDM theory on the basis of rough set theory. This theory classifies a domain into three distinct categories: positive, negative, and boundary domains. It has been widely developed in recent years. For instance, Wang et al. [41] presented a three-way MADM model based on probabilistic dominance relations and intuitionistic fuzzy sets. Mondal et al. [42] proposed a TWDM model under incomplete information systems. Moreover, they used regret theory and prospect theory to devise three-way decision rules [43]. Qadir et al. [44] introduced a new TWDM technique under an intuitionistic double hierarchy linguistic term set. Furthermore, TWDM theory has been applied in various fields, such as healthcare [45,46], investment [47], text categorization [48] and group decision making [49].

Considering the increasing information in practical scenarios, Yao [50] proposed a multi-granularity computing method, i.e., the sequential TWDM method. When the available information at a particular granularity level is insufficient to make confident decisions about ambiguous alternatives, opting for non-commitment is a suitable choice. At the next granularity level, decision-makers gather more information to enhance their understanding of alternatives and reduce misclassification costs. Sequential TWDM has attracted researchers’ attention due to its potential to dynamically deal with the uncertainty of complex problems and reduce the overall decision cost.

For the case of dynamically changing data under single dimensional granularity, the research results related to TEDM can be divided into the following two categories.

The first category involves adding or removing attributes, and it is worth noting that changes in attributes can significantly affect the granularity of the knowledge space. For example, Zhang et al. [51] introduced an incremental method for updating the relationship matrix. Li et al. [52] presented a dominance-based rough set approach (DRSA) to update approximations. Liu et al. [53] developed an incremental approach for updating approximations in probabilistic rough sets. Zhang et al. [54] devised incremental methods for updating rough approximations within interval-valued information systems when attributes are generalized.

The second category involves updating and modifying condition or decision attribute values, which is common in real-world applications. This includes scenarios such as inserting missing values, correcting erroneous values, and refining or coarsening attribute values. For example, Zhang et al. [55] proposed a TWDM model for handling dynamic decision-making cases involving attribute value updates. Chen et al. [56] focused on strategies for maintaining approximations when attribute values change in incomplete ordered decision systems. They also explored rough set-based methods for updating decision rules when attribute values are coarsened or refined [57]. Luo et al. [58] developed dynamic rough set approaches for set-valued decision systems. Li and Li [59] introduced an incremental approach of preserving approximations in the DRSA when attribute values change. Zeng et al. [60] explored fuzzy rough set approaches for incrementally updating approximations.

This literature review reveals that scholars have conducted extensive research on green supplier evaluation and proposed various MADM methods. However, there is limited research on supplier selection using the TWDM method, especially the sequential group TWDM method. For the sequential group TWDM, scholars mainly focused on the cases in which attributes are added or removed and the values of conditional or decision attributes are updated or modified. In reality, there are more cases where attributes and attribute values remain unchanged, but large differences in expert opinions lead to inefficient decision-making. To address these cases, we propose a sequential group TWDM method for green supplier selection. The key contributions of this article are as follows:

(1)

We present a green supplier selection process based on the sequential group TWDM. This involves constructing a nested sequence of attribute sets, enabling decision-making at different levels of granularity and offering insights for enterprises’ green supplier selection.

(2)

At each granularity level, we construct a social network and use experts’ willingness to exchange information to automatically adjust values with significant deviations. This process facilitates achieving expert consensus and enhances decision-making efficiency.

3. Methodology

In this section, we present the preliminaries and develop the framework for the sequential group TWDM method. In the preliminary part (Section 3.1), we start by defining traditional TWDM, and introduce the concept of sequential TWDM as a specific branch. Next, we describe interval-valued intuitionistic fuzzy sets for data processing, which is the approach employed in this study. In Section 3.2, we propose the sequential group TWDM method, which forms the basis for the subsequent supplier selection process.

3.1. Preliminaries

3.1.1. Traditional Three-Way Decision Making

Assuming ${apr}_{(\alpha ,\beta )}=(U,R)$ is a probabilistic rough approximation space, for X ⊆ U, the upper and lower (α, β) approximation sets of ${apr}_{(\alpha ,\beta )}$ are [61] ${\underset{\_}{apr}}_{\left(\alpha ,\beta \right)}\left(X\right)=\left\{x\in U\left|\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\ge \alpha \right.\right\}$, ${\overline{apr}}_{\left(\alpha ,\beta \right)}\left(X\right)=\left\{x\in U\left|\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)>\beta \right.\right\}$, respectively, where 0 ≤ β ≤ α ≤ 1, and [x] is the equivalence class of X with respect to R. The domain U can be divided into positive domain ${POS}_{\left(\alpha ,\beta \right)}(X)$, boundary domain ${BND}_{\left(\alpha ,\beta \right)}(X)$ and negative domain ${NEG}_{\left(\alpha ,\beta \right)}(X)$ [39]. Yao [62,63] introduced Bayesian theory and loss function into probabilistic rough sets and proposed a TWDM model. The state set $\Omega =\left\{Z,Z^C\right\}$ indicates that the object belongs or does not belong to concept Z. The action set $A=\left\{a_P,a_B,a_N\right\}$ indicates a binary of acceptance, non-commitment, or rejection. Let ${\lambda }_{PZ}$, ${\lambda }_{BZ}$ and ${\lambda }_{NZ}$ represent the losses caused by actions $a_P$, $a_B$ and $a_N$, respectively, when an object belongs to Z, and ${\lambda }_{P{Z}^C}$, ${\lambda }_{B{Z}^C}$ and ${\lambda }_{N{Z}^C}$ represent the losses caused by taking the above three actions when an object does not belong to Z. Generally, assuming the following relationships hold [64], ${\lambda }_{PZ}\le {\lambda }_{BZ}<{\lambda }_{NZ},{\lambda }_{N{Z}^C}\le {\lambda }_{B{Z}^C}<{\lambda }_{P{Z}^C}$, the decision rules are as follows:

(P): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\ge \alpha$ and $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\ge \gamma$, x ∈ POS(X);

(B): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)<\alpha$ and $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)>\beta$, x ∈ BND(X);

(N): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\le \beta$ and $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\le \gamma$, x ∈ NEG(X).

Where:

$$\alpha =\frac{{\lambda }_{P{Z}^C}-{\lambda }_{B{Z}^C}}{\left({\lambda }_{P{Z}^C}-{\lambda }_{B{Z}^C}\right)+\left({\lambda }_{BZ}-{\lambda }_{PZ}\right)},\beta =\frac{{\lambda }_{B{Z}^C}-{\lambda }_{{NZ}^C}}{\left({\lambda }_{B{Z}^C}-{\lambda }_{{NZ}^C}\right)+\left({\lambda }_{NZ}-{\lambda }_{BZ}\right)},\gamma =\frac{{\lambda }_{P{Z}^C}-{\lambda }_{{NZ}^C}}{\left({\lambda }_{P{Z}^C}-{\lambda }_{{NZ}^C}\right)+\left({\lambda }_{NZ}-{\lambda }_{PZ}\right)}.$$

(1)

According to the decision rule (B), $0\le \beta \le \gamma \le \alpha \le 1$ can be derived [65]. Therefore, the above decision rules can be simplified into the following rules:

(P1): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\ge \alpha$, x ∈ POS(X);

(B1): If $\beta <\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)<\alpha$, x ∈ BND(X);

(N1): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\le \beta$, x ∈ NEG(X).

If $0\le \alpha \le \gamma \le \beta \le 1$, the decision rules (P), (B) and (N) degenerate into binary rules:

(P2): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)\ge \gamma$, x ∈ POS(X);

(N2): If $\mathrm{Pr}\left(X\left|\left[x\right]\right.\right)<\gamma$, x ∈ NEG(X).

3.1.2. Sequential Three-Way Decision Making

Yao [50] proposed a sequential TWDM model by extending the TWDM to multi-step decision making. The sequential TWDM is a dynamic method that depends on multi-level granularity to make decisions on objects. It is mainly derived from three aspects. First, in uncertain environments, human cognition is a multi-stage process, and the evaluation information typically progresses from extensive to detailed. Second, when making decisions at multiple levels in sequence, at each level, people need to decide whether to accept, reject, or not commit to alternative solutions. Third, the cost of decision information acquisition and processing, which is not considered in the traditional TWDM, can be reduced if a definite decision can be made with an acceptable error for alternative solutions at a lower granularity level.

$IS=(U,C,P,f)$ is an information system, and $\{C^1,C^2,\cdots ,C^K\}$ is a nested attribute set, i.e., $C^1\subset C^2\subset \cdots \subset C^K=C$. $\forall x\in U$, $R_{C^K}\subset \cdots \subset R_{C^2}\subset R_{C^1},{\left[x\right]}_{R_{C^K}}\subseteq \cdots \subseteq {\left[x\right]}_{R_{C^2}}\subseteq {\left[x\right]}_{R_{C^1}}$. The sequential multi-level granularity structure constructed from the nested attribute set sequences is denoted as $GS=\left({GS}^1,{GS}^2,\cdots ,{GS}^k,\cdots {GS}^K\right)$, where ${GS}^k$ represents the granularity structure of the k-th level. ${GS}^k=(U^k,Z^k,C^k,P^k\left(x\right),{\alpha }^k,{\beta }^k)$, where $U^k$, $Z^k$ and $C^k$ are the sets of decision objects, object classification and attributes, respectively. $P^k\left(x\right)$ is the evaluation function, i.e., the conditional probability, and ${(\alpha }^k,{\beta }^k)$ represents a pair of thresholds satisfying ${\alpha }^k>{\beta }^k$. The positive domain $POS\left(Z^k\right)$, boundary domain $BND\left(Z^k\right)$ and negative domain $NEG\left(Z^k\right)$ can be defined as [66] $POS\left(Z^k\right)=\left\{x\in U^k|P^k\left(x\right)\ge \alpha \right\},$ $BND\left(Z^k\right)=\left\{x\in U^k|\beta <P^k\left(x\right)<\alpha \right\}$, and $NEG\left(Z^k\right)=\left\{x\in U^k|P^k\left(x\right)\le \beta \right\}$, respectively.

All attributes can be classified and a nested attribute set can be created by dividing the sequential decision granularity levels. As the nested sequence level increases, the number of attributes and the evaluation information gradually increases. This means that for options at a lower granularity level that cannot be classified definitively due to insufficient evaluation information, decisions can be made at a higher granularity level with more evaluation information. However, if the appropriate decision has already been made at a lower granularity level, there is no need to invest additional time and resources in making the decision at a higher granularity level.

3.1.3. Interval-Valued Intuitionistic Fuzzy Sets

Interval-valued intuitionistic fuzzy sets can effectively deal with uncertainty and ambiguity by using interval membership and interval non-membership [67]. Expressing attribute values using interval-valued intuitionistic fuzzy numbers (IVIFNs) can minimize information loss and allow for more precise supplier performance evaluation.

Let X be a non-empty set, and an interval-valued intuitionistic fuzzy set $A$ on X is expressed as $A=\left\{<x,\left[{\mu }_A^L\left(x\right),{\mu }_A^R\left(x\right)\right],\left[{\nu }_A^L\left(x\right),{\nu }_A^R\left(x\right)\right]>|x\in X\right\}$. It satisfies the conditions: $0\le {\mu }_A^L\left(x\right)\le {\mu }_A^R\left(x\right)\le 1$, $0\le {\nu }_A^L\left(x\right)\le {\nu }_A^R\left(x\right)\le 1$, and $0\le {\mu }_A^R\left(x\right)+{\nu }_A^R\left(x\right)\le 1$. Where ${\mu }_A^R\left(x\right)$ and ${\mu }_A^L\left(x\right)$ represent the upper and lower boundaries of the membership degree of the element x belonging to $A$, respectively; ${\nu }_A^R\left(x\right)$ and ${\nu }_A^L\left(x\right)$ represent the upper and lower boundaries of the non-membership degree of the element x belonging to $A$, respectfully. The hesitation degree of element $x$ is ${\pi }_A\left(x\right)=\left[{\pi }_A^L\left(x\right),{\pi }_A^R\left(x\right)\right]=[1-{\mu }_A^R\left(x\right)-{\nu }_A^R\left(x\right),1-{\mu }_A^L\left(x\right)-{\nu }_A^L\left(x\right)]$.

Given IVIFNs $\alpha =\left(\left[a,b\right],[c,d]\right)$, ${\alpha }_1=\left(\left[a_1,b_1\right],[c_1,d_1]\right)$ and ${\alpha }_2=\left(\left[a_2,b_2\right],[c_2,d_2]\right)$, their operations laws are summarized as follow:

(1)

${\alpha }_1+{\alpha }_2=\left(\left[a_1+a_2-a_1{a}_2,b_1+b_2-b_1{b}_2\right],[c_1{c}_2,d_1{d}_2]\right)$

(2)

${\alpha }_1\otimes {\alpha }_2=\left(\left[a_1{a}_2,b_1{b}_2\right],[c_1+c_2-c_1{c}_2,d_1+d_2-d_1{d}_2]\right)$

(3)

$\lambda \alpha =\left(\left[1-{\left(1-a\right)}^{\lambda },1-{\left(1-b\right)}^{\lambda }\right],[c^{\lambda },d^{\lambda }]\right),\lambda \ge 0$

(4)

${\alpha }^{\lambda }=\left(\left[a^{\lambda },b^{\lambda }\right],\left[1-{\left(1-c\right)}^{\lambda },1-{\left(1-d\right)}^{\lambda }\right]\right),\lambda \ge 0$

The Euclidean distance between ${\alpha }_1$ and ${\alpha }_2$ is [68]:

$$d\left({\alpha }_1,{\alpha }_2\right)=\sqrt{\frac{{\left(a_1-a_2\right)}^2+{\left(b_1-b_2\right)}^2+{\left(c_1-c_2\right)}^2+{\left(d_1-d_2\right)}^2}{4}}.$$

(2)

The score function is an important concept introduced by Atanassov when proposing intuitionistic fuzzy sets [67]. It combines membership and non-membership degrees in ranking alternatives during decision-making. When decision-makers use the score function, their goal is to seek the optimal alternative with higher approval, lower disapproval, and greater certainty. Given an IVIFN $\alpha =\left(\left[a,b\right],[c,d]\right)$, the score function $S\left(\alpha \right)$ is specified as follows:

$$S\left(\alpha \right)=\frac{a+b-c-d}{{\pi }_{\mathsf{\alpha }}^L\left(x\right)+{\pi }_{\mathsf{\alpha }}^R\left(x\right)+2}.$$

(3)

where $\left[{\pi }_{\alpha }^L\left(x\right),{\pi }_{\alpha }^R\left(x\right)\right]$ is the hesitation degree of element $x$. The larger the scoring function value, the better the corresponding alternative.

3.2. Sequential Group TWDM

Let $E=\left\{e_1,e_2,\cdots e_H\right\}$ be the expert set composed of H experts, $X=\{x_1,x_2,\cdots x_N\}$ be the evaluation object set composed of N alternatives, and $C=\{c_1,c_2,\cdots ,c_M\}$ be the evaluation attribute set. The evaluation matrix given by the h-th expert is $X^h={\left[x_{nm}^h\right]}_{N\times M}$, where $x_{nm}^h=(\left[u_{nm}^{hL},u_{nm}^{hR}\right],[v_{nm}^{hL},v_{nm}^{hR}])$. The sequential group TWDM process is shown in Figure 1.

3.2.1. Construction of Multi-Level Granularity Structure

For attribute $c_m$, the similarity matrix between experts is defined as $SIM=[{s^{ij}\left(c_m\right)]}_{H\times H}$, where:

$$s^{ij}\left(c_m\right)=1-\frac{\sum _{n=1}^{N}d\left(x_{nm}^i,x_{nm}^j\right)}{N},i,j=\mathrm{1,2},\cdots ,H.$$

(4)

Here, $d\left(x_{nm}^i,x_{nm}^j\right)$ represents the Euclidean distance between the two IVIFNs $x_{nm}^i$ and $x_{nm}^j$. The larger the $s^{ij}\left(c_m\right)$, the closer the evaluation of attribute $c_m$ by experts $e_i$ and $e_j$.

The consistency of attribute $c_m$ is defined as the average of the similarity between all experts, and the formula is as follows:

$$CD\left(c_m\right)=\frac{\sum _{i=1}^H\sum _{j=1,j\ne i}^H{s}^{ij}\left(c_m\right)}{H\left(H-1\right)},m=\mathrm{1,2},\cdots ,M.$$

(5)

Based on the consistency of each attribute, a nested sequence of attribute sets can be constructed, as shown in Algorithm 1.

Algorithm 1: Construction of the nested sequences of attribute sets

Input: Evaluation matrices of H experts: $X^h={\left[x_{nm}^h\right]}_{N\times M},h\in \left\{\mathrm{1,2},\cdots ,H\right\}$ Threshold for differences in attribute consistency: $\eta$ Output: The nested attribute set $\{C^1,C^2,\cdots ,C^K\}$ 1: Calculate the consistency of each attribute: $CD\left(c_m\right),m\in \left\{\mathrm{1,2},\cdots ,M\right\}$. 2: k = 0; 3: $C^k=\varnothing$; 4: $C^k=\varnothing ;$ 5: while $C^k\ne C^K$ do 6:  k++ 7:  $C_{new}=\varnothing$ 8:  ${CD}_{max}=\max \left\{CD\left(c_m\right)\right\}$ 9:  if ${CD}_{max}=0$ then 10:   Select any $c_m$ from the attribute set $C^K-C^{k-1}$ to add $C_{new}$; 11: else 12:   for $c_m\in C^K-C^{k-1}$ do 13:    if ${CD}_{max}-CD\left(c_m\right)<=\eta$ then 14:     Add $c_m$ to $C_{new}$ 15:    end if 16:   end for 17:  end if 18:  $C^k=C^{k-1}\cup C_{new}$ 19: end while 20: return $C^1\subset C^2\subset \cdots \subset C^k\subset \cdots \subset C^K$

Based on the nested sequence of attribute sets, a multi-level granularity structure $GS=({GS}^1,{GS}^2,\cdots ,{GS}^k,\cdots {GS}^K)$ can be obtained.

3.2.2. Expert Consensus at a Single Granularity Level

(1)

Constructing expert social network

Social networks use experts as nodes and examine their connections through edges. We construct a social network at each granularity level. In order to correct evaluations with significant deviations, interconnected experts often conduct multiple rounds of discussion and constantly adjust their opinions through information exchange until a consensus is reached. Generally, the higher the similarity between an expert’s evaluation value and other experts, the closer it is to the final consensus, and vice versa. Taking the process of constructing a social network in the t-th round discussion of granularity level ${GS}^k$ as an example, the similarity between the two experts is as follows:

$${{(s}^{ij})}^{kt}=1-\frac{\sum _{n=1}^{N_k}\sum _{m=1}^{M_k}d\left({{(x}_{nm}^i)}^{kt},({x_{nm}^j)}^{kt}\right)}{N_k·M_k},i,j=\mathrm{1,2},\cdots ,H.$$

(6)

where $N_k$ and $M_k$ represent the number of elements in the decision object set $U^k$ and the attribute set $C^k$, respectively.

A directed graph $G^{kt}=(V^{kt},F^{kt})$ can be constructed to represent the social network between experts, where $V^{kt}$ and $F^{kt}$ represent the nodes and edges of the graph, respectively. The adjacency matrix of the digraph is denoted as ${AM}^{kt}={\left[{\left({am}^{ij}\right)}^{kt}\right]}_{H\times H}$, where:

$${\left({am}^{ij}\right)}^{kt}=\left\{\begin{array}{c}1,{\left(s^{ij}\right)}^{kt}\ge {CD}^{kt};\\ 0,{\left(s^{ij}\right)}^{kt}<{CD}^{kt}.\end{array}\right.$$

(7)

Here, ${CD}^{kt}$ is the group consensus degree in the t-th round discussion of ${GS}^k$.

$${CD}^{kt}=\frac{\sum _{i=1}^H\sum _{j=1,j\ne i}^H{\left(s^{ij}\right)}^{kt}}{H\left(H-1\right)}$$

(8)

${\left({am}^{ij}\right)}^{kt}=1$ means that expert $e_i$ has an influence on expert $e_j$, and ${\left({am}^{ij}\right)}^{kt}=0$ means that $e_i$ has no influence on expert $e_j$.

Based on the social network $G^{kt}$, a personal influence matrix ${IID}^{kt}={\left[{\left({IID}^{ij}\right)}^{kt}\right]}_{H\times H}$ reflecting the interaction between two experts is obtained, where:

$${\left({IID}^{ij}\right)}^{kt}=\left\{\begin{array}{c}{{(s}^{ij})}^{kt},{\left({am}^{ij}\right)}^{kt}=1;\\ 0,{\left({am}^{ij}\right)}^{kt}=0.\end{array}\right.$$

(9)

The group influence of experts in the entire group is ${GID}^{kt}={\left[{\left({GID}^i\right)}^{kt}\right]}_{1\times H}$, where:

$${\left({GID}^i\right)}^{kt}=\frac{\sum _{j=1,j\ne i}^H{\left({IID}^{ij}\right)}^{kt}}{H-1}$$

(10)

${\left({GID}^i\right)}^{kt}$ represents the impact of expert $e_i$ on the whole group in the t-th round discussion of granularity level ${GS}^k$.

(2)

Reaching consensus through information exchange among experts.

Taking the process of reaching consensus at the granularity level of ${GS}^k$ as an example, since experts have reached consensus on the previous attribute set $C^{k-1}$, they only need to reach consensus on the attribute set $C^k-C^{k-1}$. Each expert interacts with the experts who influence them in the constructed social network. This can enhance consensus under the attribute set $C^k-C^{k-1}$. The following information exchange rule is proposed:

$${{(x}_{nm}^i)}^{k\left(t+1\right)}={\theta }_i{\sum }_{e_h\in D^i}{\omega }_h{{(x}_{nm}^h)}^{kt}+\left(1-{\theta }_i\right){{(x}_{nm}^i)}^{kt}$$

(11)

where $D^i$ is the set of experts that have an influence on expert $e_i$, and ${\theta }_i$ is the exchange willingness of $e_i$, $0\le {\theta }_i\le 1$. The closer ${\theta }_i$ is to 0, the poorer the willingness of $e_i$ to exchange and the more likely he is not to make adjustments. The closer ${\theta }_i$ approaches 1, the closer $e_i$ tends to approach the comprehensive value of other experts. ${\omega }_h$ is the weight of expert $e_h$, which is calculated as follows:

$${\omega }_h=\frac{{\left({IDD}^{hi}\right)}^{kt}}{{\sum }_{e_h\in D^i}{\left({IDD}^{hi}\right)}^{kt}}.$$

(12)

After the completion of the t-th round information exchange, the consensus degree of the expert group can be calculated. Given a group consensus threshold $\delta$ and a consensus difference threshold $ϵ$, this article takes $\delta =0.9$ and $ϵ=0.001$, if ${CD}^{kt}\ge \delta$ or ${CD}^{kt}-{CD}^{k(t-1)}\le ϵ$, the experts reach consensus; otherwise, they need to further exchange information with other experts in the (t + 1)-th round discussion.

3.2.3. Aggregation of Experts’ Evaluation Matrix and Attribute Weighting

(1)

Expert weighting

The determination of expert weight is crucial in group decision making. We apply the influence of experts in the group and the degree of certainty of the evaluation information to comprehensively determine expert weight. If an expert has a greater influence on other experts, he should have a larger weight. In addition, if there is less information redundancy in an expert’s evaluation matrix, the weight of the expert should also be larger. The group influence of expert $e_h$ after reaching consensus under the granularity level ${GS}^k$ is ${GID}^h$. ${\tilde{\omega }}_{h1}$ is the weight determined based on the influence of expert $e_h$ on other experts, and the formula is as follows:

$${\tilde{\omega }}_{h1}=\frac{{GID}^h}{\sum _{h=1}^H{GID}^h}.$$

(13)

${\tilde{\omega }}_{h2}$ is the expert weight determined by the effectiveness of information. The entropy method is an objective weighting method, which uses information entropy to evaluate the uncertainty and information content of data. The smaller the entropy value, the smaller the uncertainty included in the expert evaluation, and the greater the weight of the expert. The formula for calculating expert weights using entropy method is as follows:

$${\tilde{\omega }}_{h2}=\frac{1-E^h}{{\sum }_{h=1}^H(1-E^h)},E^h=\frac{1}{M_k}{\sum }_{m=1}^{M_k}{E}_m^h.$$

(14)

where $E_m^h$ is the entropy of the m-th attribute, and its formula is [69]:

$$E_m^h=\frac{1}{N_k}{\sum }_{n=1}^{N_k}\frac{2-2{\left({∆}_A\left(x_{nm}^h\right)\right)}^3+{\left({\pi }_A^L\left(x_{nm}^h\right)\right)}^3+{\left({\pi }_A^R\left(x_{nm}^h\right)\right)}^3}{4},m=\mathrm{1,2},\cdots ,M_k.$$

(15)

here ${∆}_A\left(x_{nm}^h\right)$ is the ambiguity of $x_{nm}^h$ belonging to A, and its formula is [63]:

$${∆}_A\left(x_{nm}^h\right)=\sqrt{\frac{{\left({\mu }_A^L\left(x_{nm}^h\right)-{\nu }_A^L\left(x_{nm}^h\right)\right)}^2+{\left({\mu }_A^R\left(x_{nm}^h\right)-{\nu }_A^R\left(x_{nm}^h\right)\right)}^2}{2}}.$$

(16)

The expert weight ${\tilde{\omega }}_h$ is obtained by synthesizing ${\tilde{\omega }}_{h1}$ and ${\tilde{\omega }}_{h2}$ with the weight coefficients $\tau$ and $1-\tau$, respectively.

(2)

Aggregation of expert evaluation matrix

If experts reach a consensus on the evaluation results at the granularity level ${GS}^k$, interval-valued intuitionistic fuzzy power aggregation (IVIFPWA) operator can be applied to aggregate the evaluation matrix of H experts and obtain the group comprehensive evaluation matrix $X^k={\left[{{(x}_{nm}^{\prime })}^k\right]}_{N_k\times M_k}$, where [70]:

$$\begin{array}{c}{{(x}_{nm}^{\prime })}^k=\left(\left[{\mu }_{nm}^{L^{\prime }},{\mu }_{nm}^{R^{\prime }}\right],\left[v_{nm}^{L^{\prime }},v_{nm}^{R^{\prime }}\right]\right)=\sum _{h=1}^H{\tilde{\omega }}_h{\left(x_{nm}^h\right)}^k\\ =\left(\left[1-{\prod }_{h=1}^H{\left(1-u_{nm}^{hL}\right)}^{{\tilde{\omega }}_h},1-{\prod }_{h=1}^H{\left(1-u_{nm}^{hR}\right)}^{{\tilde{\omega }}_h}\right],\left[{\prod }_{h=1}^H{{(\nu }_{nm}^{hL})}^{{\tilde{\omega }}_h},{\prod }_{h=1}^H{{(\nu }_{nm}^{hL})}^{{\tilde{\omega }}_h}\right]\right).\end{array}$$

(17)

where ${\left(x_{nm}^h\right)}^k$ is the evaluation value of expert $e_h$ on the m-th attribute of the n-th evaluation object after reaching a consensus under ${GS}^k$.

(3)

Attribute weighting

Based on the aggregated expert evaluation matrix, the objective weight of each attribute is calculated using the entropy method [71], as follows:

$${\omega }_{m1}=\frac{1-E_m}{{\sum }_{m=1}^{M_k}{(1-E}_m)},m=\mathrm{1,2},\cdots ,M_k.$$

(18)

where $E_m$ is the entropy of the m-th attribute, and the calculation formula is the same as formula (15).

Assuming that the subjective weight of the m-th attribute obtained by AHP, ANP or other subjective weighting method is ${\omega }_{m2}$, the combined weight ${\omega }_m$ is obtained by synthesizing ${\omega }_{m1}$ and ${\omega }_{m2}$ with the weight coefficients $\alpha$ and $1-\alpha$, respectively.

3.2.4. Determination of Conditional Probability

Determining conditional probability is crucial in TWDM. In cases where the decision attribute values are unknown beforehand, scholars use methods like TOPSIS, GRA, and VIKOR to estimate conditional probability. The VIKOR method improved by GRA can not only make full use of the trend information of samples to reflect the internal law of sample data, but also take into account the maximization of group utility and the minimization of individual regret, which is more in line with decision-making characteristics. We use this method to determine conditional probability as follows [72]:

Step 1: Determine the positive ideal point $X^{+}=(x_1^{+},x_2^{+},\cdots ,x_{M_k}^{+})$ and negative ideal point $X^{-}=(x_1^{-},x_2^{-},\cdots ,x_{M_k}^{-})$, where $x_m^{+}={\mathrm{m}\mathrm{a}\mathrm{x}}_n{{(x}_{nm}^{\prime })}^k=\left(\left[{\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mu }_{nm}^{L\prime },{\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mu }_{nm}^{R\prime }\right],\left[{\mathrm{m}\mathrm{i}\mathrm{n}}_n{v}_{nm}^{L\prime },{\mathrm{m}\mathrm{i}\mathrm{n}}_n{v}_{nm}^{R\prime }\right]\right)$, $x_m^{-}={\mathrm{m}\mathrm{i}\mathrm{n}}_n{{(x}_{nm}^{\prime })}^k=\left(\left[{\mathrm{m}\mathrm{i}\mathrm{n}}_n{\mu }_{nm}^{L\prime },{\mathrm{m}\mathrm{i}\mathrm{n}}_n{\mu }_{nm}^{R\prime }\right],\left[{\mathrm{m}\mathrm{a}\mathrm{x}}_n{v}_{nm}^{L\prime },{\mathrm{m}\mathrm{a}\mathrm{x}}_n{v}_{nm}^{R\prime }\right]\right)$.

Step 2: Based on the VIKOR method, the group utility value $S_n$, individual regret value $R_n$, optimal utility value $S^{+}$, worst utility value $S^{-}$, optimal regret value $R^{+}$ and worst regret value $R^{-}$ are determined as follows:

$$S_n={\sum }_{m=1}^{M_k}{\omega }_m\frac{d\left(x_m^{+},x_{nm}^{\prime }\right)}{d\left(x_m^{+},x_{nm}^{\prime }\right)},R_n={\mathrm{m}\mathrm{a}\mathrm{x}}_m{\omega }_m\frac{d\left(x_m^{+},x_{nm}^{\prime }\right)}{d\left(x_m^{+},x_m^{-}\right)},n=\mathrm{1,2},\cdots ,N_k.$$

(19)

$$S^{+}={\mathrm{m}\mathrm{i}\mathrm{n}}_n{S}_n,S^{-}={\mathrm{m}\mathrm{a}\mathrm{x}}_n{S}_n,R^{+}={\mathrm{m}\mathrm{i}\mathrm{n}}_n{R}_n,R^{-}={\mathrm{m}\mathrm{a}\mathrm{x}}_n{R}_n.$$

(20)

Step 3: Calculate the grey correlation degree between the object to be evaluated and the positive ideal point and negative ideal point as follows:

$${\epsilon }_n^{+}=\frac{1}{M_k}{\sum }_{m=1}^{M_k}{\epsilon }_{nm}^{+},{\epsilon }_n^{-}=\frac{1}{M_k}{\sum }_{m=1}^{M_k}{\epsilon }_{nm}^{-},n=\mathrm{1,2},\cdots ,N_k.$$

(21)

where ${\epsilon }_{nm}^{+}$ and ${\epsilon }_{nm}^{-}$ are grey relational coefficients:

$${\epsilon }_{nm}^{+}=\frac{{\mathrm{m}\mathrm{i}\mathrm{n}}_n{\mathrm{m}\mathrm{i}\mathrm{n}}_m\left[{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{+}\right)\right]+\rho {\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mathrm{m}\mathrm{a}\mathrm{x}}_m\left[{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{+}\right)\right]}{{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{+}\right)+\rho {\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mathrm{m}\mathrm{a}\mathrm{x}}_j\left[{\omega }_{j}d\left(x_{nm}^{\prime },x_m^{+}\right)\right]}.$$

(22)

$${\epsilon }_{nm}^{-}=\frac{{\mathrm{m}\mathrm{i}\mathrm{n}}_n{\mathrm{m}\mathrm{i}\mathrm{n}}_m\left[{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{-}\right)\right]+\rho {\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mathrm{m}\mathrm{a}\mathrm{x}}_m\left[{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{-}\right)\right]}{{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{-}\right)+\rho {\mathrm{m}\mathrm{a}\mathrm{x}}_n{\mathrm{m}\mathrm{a}\mathrm{x}}_m\left[{\omega }_{m}d\left(x_{nm}^{\prime },x_m^{-}\right)\right]}.$$

(23)

Here, ρ ∈ [0, 1] is the distinguishing coefficient and usually takes 0.5.

Step 4: Based on GRA, the group utility value $S_n^{\prime }$, individual regret value $R_n^{\prime }$, optimal utility value $S^{\prime +}$, worst utility value $S^{\prime -}$, optimal regret value $R^{\prime +}$ and worst regret value $R^{\prime -}$ are determined as follows:

$$S_n^{\prime }=\frac{{\epsilon }_n^{-}}{{\epsilon }_n^{+}},R_n^{\prime }={\mathrm{m}\mathrm{a}\mathrm{x}}_m\frac{{\epsilon }_{nm}^{-}}{{\epsilon }_{nm}^{+}},n=\mathrm{1,2},\cdots ,N_k.$$

(24)

$$S^{\prime +}={\mathrm{m}\mathrm{i}\mathrm{n}}_n{S}_n^{\prime },S^{\prime -}={\mathrm{m}\mathrm{a}\mathrm{x}}_n{S}_n^{\prime },R^{\prime +}={\mathrm{m}\mathrm{i}\mathrm{n}}_n{R}_n^{\prime },R^{\prime -}={\mathrm{m}\mathrm{a}\mathrm{x}}_n{R}_n^{\prime }.$$

(25)

Step 5: Determine the gain ratio of each object as follows:

$$Q_n=\sigma \frac{{\delta }_n-{\delta }^{+}}{{\delta }^{-}-{\delta }^{+}}+\left(1-\sigma \right)\frac{{\phi }_n-{\phi }^{+}}{{\phi }^{-}-{\phi }^{+}},n=\mathrm{1,2},\cdots ,N_k.$$

(26)

where, σ is the compromise coefficient between group utility and individual regret, 0 ≤ σ ≤ 1; ${\delta }_n={S_n\ast S}_n^{\prime }$, ${\phi }_n=R_n\ast R_n^{\prime }$, ${\delta }^{+}=S^{+}\ast S^{\prime +}$, ${\delta }^{-}=S^{-}\ast S^{\prime -}$, ${\phi }^{+}=R^{+}\ast R^{\prime +}$, ${\phi }^{-}=R^{-}\ast R^{\prime -}$.

Step 6: According to the meaning of gain ratio, the closer the group utility value is to the optimal utility value, the closer the individual regret value is to the optimal regret value, and the smaller the gain ratio value is. That is, the smaller the gain ratio, the better the alternative. Furthermore, the value of the gain ratio is between 0 and 1. Therefore, we can use the value of 1 minus the gain ratio as the conditional probability. On one hand, the value is between 0 and 1, satisfying the requirement of probability. On the other hand, the smaller the gain ratio of an alternative, the greater the conditional probability that it belongs to state Z. The formula for conditional probability is as follows:

$$P\left(Z\left|x_n\right.\right)=1-Q_n,n=\mathrm{1,2},\cdots ,N_k.$$

(27)

3.2.5. Determination of Decision Thresholds and TWDM Rules

In the traditional TWDM, the decision thresholds are determined by the loss function. However, decisions are often accompanied by risks and uncertainties. Many experiments have shown that cumulative prospect theory (CPT) can well predict and explain the behavior of decision makers in the face of risks and uncertainties [73,74,75]. Therefore, we introduce CPT into TWDM to reflect the risk preference of decision makers. CPT is based on the value function rather than the loss function. The larger the value function, the better the alternative [76].

(1)

Determination of value function

Let ${{\lambda }_{PZ}}^h$, ${{\lambda }_{BZ}}^h$ and ${{\lambda }_{NZ}}^h$ represent the benefits caused by actions $a_P$, $a_B$ and $a_N$, respectively, when an object belongs to Z. ${{\lambda }_{P{Z}^C}}^h$, ${{\lambda }_{B{Z}^C}}^h$ and ${{\lambda }_{N{Z}^C}}^h$ represent the benefits caused by taking the above three actions when an object does not belong to Z. Generally, the following relationships hold ${{\lambda }_{NZ}}^h<{{\lambda }_{BZ}}^h\le {{\lambda }_{PZ}}^h,{{\lambda }_{P{Z}^C}}^h<{{\lambda }_{B{Z}^C}}^h\le {{\lambda }_{{NZ}^C}}^h$. Consider another reasonable conditions, that is, the result of partitioning an object belonging to Z into the NEG domain is strictly less than that of partitioning it into the POS domain, and the result of partitioning an object not belonging to Z into the POS domain is strictly less than that of partitioning it into the NEG domain, so the following relationship holds: ${{\lambda }_{P{Z}^C}}^h<{{\lambda }_{PZ}}^h,{{\lambda }_{NZ}}^h<{{\lambda }_{{NZ}^C}}^h$. Given reference point ${\lambda }^{\ast }$ and benefit $\lambda$, the value function is as follows:

$$g\left(\lambda \right)=\left\{\begin{array}{c}{\left(\lambda -{\lambda }^{\ast }\right)}^{\varrho },\lambda \ge {\lambda }^{\ast };\\ -\varsigma {\left({\lambda }^{\ast }-\lambda \right)}^{\varpi },\lambda <{\lambda }^{\ast }.\end{array}\right.$$

(28)

Generally, $\varrho =\varpi =0.88,\varsigma =2.25$. Let ${g_{\circ \ast }}^h=g\left({{\lambda }_{\circ \ast }}^h\right),\circ =\left\{P,B,N\right\},\ast =\{Z,Z^C\}$. Since the value function is a monotonically increasing function, the following relations hold: ${g_{NZ}}^h<{g_{BZ}}^h\le {g_{PZ}}^h,{g_{P{Z}^C}}^h<{g_{B{Z}^C}}^h\le {g_{{NZ}^C}}^h,{g_{P{Z}^C}}^h<{g_{PZ}}^h,{g_{NZ}}^h<{g_{{NZ}^C}}^h$.

The group value can be obtained by combining the individual value of each expert with the corresponding weight, and the formula is as follows:

$$g_{\circ \ast }={\sum }_{h=1}^H{\tilde{\omega }}_h{g_{\circ \ast }}^h.$$

(29)

By comparing the values of $g_{\circ Z}$ and $g_{\circ Z^C}$, all the cases for the weight functions ${\pi }_{\circ }(P\left(Z\left|x_n\right.\right))$ and ${\pi }_{\circ }(P\left(Z^C\left|x_n\right.\right))$ can be obtained as follows:

$${\pi }_{\circ }\left(P\left(Z\left|x_n\right.\right)\right)=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}{w}^{+}\left(P\left(Z\left|x_n\right.\right)\right),0\le g_{\circ Z^C}\le g_{\circ Z};\\ 1-w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),0\le g_{\circ Z}<g_{\circ Z^C};\end{array}\\ 1-w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{\circ Z^C}\le g_{\circ Z}<0;\\ w^{-}\left(P\left(Z\left|x_n\right.\right)\right),g_{\circ Z}<g_{\circ Z^C}<0;\end{array}\\ w^{+}\left(P\left(Z\left|x_n\right.\right)\right),g_{\circ Z^C}<0\le g_{\circ Z};\\ w^{-}\left(P\left(Z\left|x_n\right.\right)\right),g_{\circ Z}<0\le g_{\circ Z^C}.\end{array}\right.$$

(30)

$${\pi }_{\circ }\left(P\left(Z^C\left|x_n\right.\right)\right)=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}1-w^{+}\left(P\left(Z\left|x_n\right.\right)\right),0\le g_{\circ Z^C}\le g_{\circ Z};\\ w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),0\le g_{\circ Z}<g_{\circ Z^C};\end{array}\\ w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{\circ Z^C}\le g_{\circ Z}<0;\\ 1-w^{-}\left(P\left(Z\left|x_n\right.\right)\right),g_{\circ Z}<g_{\circ Z^C}<0;\end{array}\\ w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{\circ Z^C}<0\le g_{\circ Z};\\ w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{\circ Z}<0\le g_{\circ Z^C}.\end{array}\right.$$

(31)

where:

$$w^{+}\left(p_i\right)=\frac{{p_i}^{\varphi }}{{\left({p_i}^{\varphi }+{\left(1-p_i\right)}^{\varphi }\right)}^{1/\varphi }},w^{-}\left(p_i\right)=\frac{{p_i}^{\phi }}{{\left({p_i}^{\phi }+{\left(1-p_i\right)}^{\phi }\right)}^{1/\phi }}.$$

(32)

Generally, $\varphi =0.61,\phi =0.69$.

The cumulative prospect values $\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.$, $\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.$ and $\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.$ are as follows:

$$\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.=\left\{\begin{array}{l}{g}_{PZ}\ast w^{+}\left(P\left(Z\left|x_n\right.\right)\right)+g_{P{Z}^C}\ast \left(1-w^{+}\left(P\left(Z\left|x_n\right.\right)\right)\right),0\le g_{P{Z}^C}\le g_{PZ};\\ g_{PZ}\ast w^{+}\left(P\left(Z\left|x_n\right.\right)\right)+g_{P{Z}^C}\ast w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{P{Z}^C}<0\le g_{PZ};\\ g_{PZ}\ast \left(1-w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right)\right)+g_{P{Z}^C}\ast w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{P{Z}^C}<g_{PZ}<0;\end{array}\right.$$

(33)

$$\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.=\left\{\begin{array}{l}{g}_{BZ}\ast w^{+}\left(P\left(Z\left|x_n\right.\right)\right)+g_{B{Z}^C}\ast \left(1-w^{+}\left(P\left(Z\left|x_n\right.\right)\right)\right),0\le g_{B{Z}^C}\le g_{BZ};\\ g_{BZ}\ast \left(1-w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right)\right)+g_{B{Z}^C}\ast w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),0\le g_{BZ}<g_{B{Z}^C};\\ \begin{array}{c}{g}_{BZ}\ast \left(1-w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right)\right)+g_{B{Z}^C}\ast w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{B{Z}^C}\le g_{BZ}<0;\\ g_{BZ}\ast w^{-}\left(P\left(Z\left|x_n\right.\right)\right)+g_{B{Z}^C}\ast \left(1-w^{-}\left(P\left(Z\left|x_n\right.\right)\right)\right),g_{BZ}<g_{B{Z}^C}<0;\\ \begin{array}{c}{g}_{BZ}\ast w^{+}\left(P\left(Z\left|x_n\right.\right)\right)+g_{B{Z}^C}\ast w^{-}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{B{Z}^C}<0\le g_{BZ};\\ g_{BZ}\ast w^{-}\left(P\left(Z\left|x_n\right.\right)\right)+g_{B{Z}^C}\ast w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{BZ}<0\le g_{B{Z}^C}.\end{array}\end{array}\end{array}\right.$$

(34)

$$\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.=\left\{\begin{array}{l}{g}_{NZ}\ast \left(1-w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right)\right)+g_{N{Z}^C}\ast w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),0\le g_{NZ}<g_{N{Z}^C};\\ g_{NZ}\ast w^{-}\left(P\left(Z\left|x_n\right.\right)\right)+g_{N{Z}^C}\ast \left(1-w^{-}\left(P\left(Z\left|x_n\right.\right)\right)\right),g_{NZ}<g_{N{Z}^C}<0;\\ g_{NZ}\ast w^{-}\left(P\left(Z\left|x_n\right.\right)\right)+g_{N{Z}^C}\ast w^{+}\left(P\left(Z^C\left|x_n\right.\right)\right),g_{NZ}<0\le g_{N{Z}^C}.\end{array}\right.$$

(35)

CP_PB = $\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.-\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.$, CP_BN = $\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.-\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.$ and CP_PN = $\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.-\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.$ are all strictly monotonically increasing [75]. Define ${\alpha }^{\prime }$, ${\beta }^{\prime }$ and ${\gamma }^{\prime }$ as the intersection of $\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.$ and $\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.$, $\mathrm{C}\mathrm{P}(a_B\left|x_n)\right.$ and $\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.$ and $\mathrm{C}\mathrm{P}(a_P\left|x_n)\right.$ and $\mathrm{C}\mathrm{P}(a_N\left|x_n)\right.$, respectively, then the values of ${\alpha }^{\prime }$, ${\beta }^{\prime }$ and ${\gamma }^{\prime }$ exist and are unique. Since the weight functions are the nonlinear transformation of probabilities, ${\alpha }^{\prime }$, ${\beta }^{\prime }$ and ${\gamma }^{\prime }$ do not have analytic solutions. Therefore, we apply the dichotomy to calculate the zeros of CP_PB, CP_BN and CP_PN, and obtain the threshold solutions. By comparing the values of ${\alpha }^{\prime }$ and ${\beta }^{\prime }$, we determine whether it is a TWDM problem and obtain the TWDM rules or binary rules.

If ${\alpha }^{\prime }>{\beta }^{\prime }$, the TWDM rules are as follows:

(P3): If $P\left(Z\left|x_n\right.\right)\ge {\alpha }^{\prime }$, $x_n$ ∈ POS(X);

(B3): If ${\beta }^{\prime }<P\left(Z\left|x_n\right.\right)<{\alpha }^{\prime }$, $x_n$ ∈ BND(X);

(N3): If $P\left(Z\left|x_n\right.\right)\le {\beta }^{\prime }$, $x_n$ ∈ NEG(X).

If ${\alpha }^{\prime }\le {\beta }^{\prime }$, the binary rules are as follows:

(P4): If $P\left(Z\left|x_n\right.\right)\ge {\gamma }^{\prime }$, $x_n$ ∈ POS(X);

(N4): If $P\left(Z\left|x_n\right.\right)<{\gamma }^{\prime }$, $x_n$ ∈ NEG(X).

4. Case Study

In this section, a case study of an automobile manufacturer will be used to verify the feasibility of the proposed method. As the core component of new energy vehicles, power batteries contain a large number of toxic and harmful substances, which will cause serious pollution to the environment if not handled properly. Automobile manufacturer Q needs to evaluate 12 green suppliers $x_1,x_2,\cdots x_{12}$ of new energy power batteries. Six experts $e_1,e_2,\cdots e_6$ are invited to form an expert group. This group includes two procurement supervisors, two technical supervisors, one production department manager, and one professor, all of whom have rich relevant knowledge and experience. An attribute system including 14 attributes $c_1,c_2,\cdots ,c_{14}$ (see Table 1) is constructed.

4.1. The Process and Results of Sequential Group TWDM

Six experts provide preliminary evaluation data containing different types, which are uniformly converted into IVIFNs, as shown in Appendix A. The consistency of the 14 attributes is 0.947, 0.821, 0.653, 0.833, 0.668, 0.832, 0.834, 0.680, 0.683, 0.832, 0.648, 0.678, 0.950 and 0.878, respectively. Set the consistency difference threshold η = 0.1, it can be seen that the consistency differences between $c_1$, $c_{14}$ and $c_{13}$ are less than 0.1. Therefore, the first level granularity attribute set ${GS}^1=\{c_1,c_{13},c_{14}\}$ is obtained. Similarly, different granularity attribute sets with remaining attributes can be obtained. A nested sequence containing three granularity levels is obtained: $GS=({GS}^1,{GS}^2,{GS}^3)$, where, ${GS}^2=\left\{c_1,c_2,c_4,c_6,c_7,c_{10},c_{13},c_{14}\right\}$ and ${GS}^3=\{c_1,c_2,c_3,c_4,{c_5,c}_6,c_7,c_8,c_9{c}_{10},c_{11},c_{12},c_{13},c_{14}\}$.

The group TWDM is made at each granularity level. The calculated group consensus degree of ${GS}^1$ is 0.925, which is greater than the consensus threshold 0.9, so a group consensus was reached. According to the Formulas (13)–(16), the calculation results of expert weights are shown in

Table 3. The calculation results of expert weights in ${GS}^1$.

	${\mathit{G}\mathit{I}\mathit{D}}^{\mathit{h}}$	${\tilde{\mathit{\omega }}}_{\mathit{h}1}$	${\mathit{E}}^{\mathit{h}}$	${\tilde{\mathit{\omega }}}_{\mathit{h}2}$	${\tilde{\mathit{\omega }}}_{\mathit{h}}$
$e_1$	0.927	0.167	0.454	0.171	0.169
$e_2$	0.927	0.167	0.435	0.164	0.166
$e_3$	0.927	0.167	0.442	0.167	0.167
$e_4$	0.926	0.167	0.437	0.165	0.166
$e_5$	0.919	0.166	0.445	0.168	0.167
$e_6$	0.924	0.167	0.437	0.165	0.166

. Here, the weight coefficients of ${\tilde{\omega }}_{h1}$ and ${\tilde{\omega }}_{h2}$ are both 0.5.

The IVIFPWA operator is used to aggregate the evaluation matrix of six experts, and the results are shown in

Table 4. The group comprehensive evaluation results in ${GS}^1$.

Supplier	${\mathit{c}}_1$	${\mathit{c}}_{13}$	${\mathit{c}}_{14}$
$x_1$	([0.298, 0.298], [0.702, 0.702])	([0.270, 0.270], [0.730, 0.730])	([0.621, 0.621], [0.000, 0.000])
$x_2$	([0.310, 0.310], [0.690, 0.690])	([0.323, 0.323], [0.677, 0.677])	([0.679, 0.679], [0.000, 0.000])
$x_3$	([0.248, 0.248], [0.752, 0.752])	([0.261, 0.261], [0.739, 0.739])	([0.633, 0.633], [0.000, 0.000])
$x_4$	([0.265, 0.265], [0.735, 0.735])	([0.282, 0.282], [0.718, 0.718])	([0.626, 0.626], [0.113, 0.113])
$x_5$	([0.286, 0.286], [0.714, 0.714])	([0.277, 0.277], [0.723, 0.723])	([0.681, 0.681], [0.000, 0.000])
$x_6$	([0.316, 0.316], [0.684, 0.684])	([0.288, 0.288], [0.712, 0.712])	([0.653, 0.653], [0.000, 0.000])
$x_7$	([0.311, 0.311], [0.689, 0.689])	([0.292, 0.292], [0.708, 0.708])	([0.746, 0.746], [0.000, 0.000])
$x_8$	([0.283, 0.283], [0.717, 0.717])	([0.316, 0.316], [0.684, 0.684])	([0.622, 0.622], [0.103, 0.103])
$x_9$	([0.311, 0.311], [0.689, 0.689])	([0.282, 0.282], [0.718, 0.718])	([0.713, 0.713], [0.108, 0.108])
$x_{10}$	([0.289, 0.289], [0.711, 0.711])	([0.260, 0.260], [0.740, 0.740])	([0.604, 0.604], [0.112, 0.112])
$x_{11}$	([0.259, 0.259], [0.741, 0.741])	([0.286, 0.286], [0.714, 0.714])	([0.668, 0.668], [0.093, 0.093])
$x_{12}$	([0.258, 0.258], [0.742, 0.742])	([0.301, 0.301], [0.699, 0.699])	([0.626, 0.626], [0.141, 0.141])

. By using entropy method, the objective weights of $c_1$, $c_{13}$ and $c_{14}$ are 0.3072, 0.0278 and 0.0143, respectively.

Considering that there may be interdependence among attributes in the evaluation attribute system, we use the ANP method to determine the subjective weights of attributes [77]. Combined with expert opinions, we construct the network structure of the attributes as shown in Figure 2. Based on the weightless hypermatrix and weighted hypermatrix, we apply the software Super Decisions (Version 3.2) to calculate the subjective weights from $c_1$ to $c_{14}$ as 0.3072, 0.2877, 0.0093, 0.0028, 0.0076, 0.0386, 0.0876, 0.0071, 0.1392, 0.0231, 0.0035, 0.0442, 0.0278 and 0.0143, respectively.

Assuming that the subjective weight of each attribute remains unchanged at all granularity levels, the normalized subjective weights of $c_1$, $c_{13}$ and $c_{14}$ are 0.8797, 0.0795 and 0.0408, respectively. The weight coefficients of subjective and objective are both 0.5, and the combined weights of $c_1$, $c_{13}$ and $c_{14}$ are 0.6007, 0.2003 and 0.1990, respectively.

According to Formulas (19)–(27), the group utility value $S_n$, individual regret value $R_n$, the group utility value $S_n^{\prime }$ of GRA, individual regret value $R_n^{\prime }$ of GRA, the gain ratio $Q_n$ and conditional probability $P\left(Z\left|x_n\right.\right)$ of each alternative in ${GS}^1$ are calculated, as shown in

Table 5. The calculation results of conditional probability in ${GS}^1$.

Supplier	${\mathit{S}}_{\mathit{n}}$	${\mathit{R}}_{\mathit{n}}$	${\mathit{S}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{R}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{Q}}_{\mathit{n}}$	$\mathit{P}\left(\mathit{Z}\left|{\mathit{x}}_{\mathit{n}}\right.\right)$
$x_1$	0.507	0.241	0.995	1.424	0.139	0.861
$x_2$	0.550	0.284	1.016	1.366	0.477	0.523
$x_3$	0.539	0.274	1.015	1.415	0.432	0.568
$x_4$	0.569	0.301	1.021	1.304	0.578	0.422
$x_5$	0.558	0.287	1.031	1.368	0.552	0.448
$x_6$	0.523	0.254	1.007	1.392	0.253	0.747
$x_7$	0.586	0.311	1.057	1.311	0.764	0.236
$x_8$	0.552	0.290	1.006	1.316	0.447	0.553
$x_9$	0.603	0.324	1.060	1.234	0.813	0.187
$x_{10}$	0.536	0.265	1.007	1.326	0.299	0.701
$x_{11}$	0.589	0.320	1.041	1.290	0.756	0.244
$x_{12}$	0.576	0.313	1.024	1.277	0.632	0.368

.

Six experts give the benefits expressed by IVIFNs under different states and actions, as shown in

Table 6. The benefits under different states and actions of six experts.

	${\mathit{e}}_{\mathbf{1}}$	${\mathit{e}}_{\mathbf{2}}$	${\mathit{e}}_{\mathbf{3}}$
${{\lambda }_{PZ}}^h$	([0.951, 0.982], [0.007, 0.018])	([0.800, 0.990], [0.005, 0.010])	([0.890, 0.97], [0.005, 0.010])
${{\lambda }_{BZ}}^h$	([0.300, 0.320], [0.555, 0.650])	([0.250, 0.300], [0.508, 0.600])	([0.200, 0.300], [0.500, 0.600])
${{\lambda }_{NZ}}^h$	([0.015, 0.100], [0.805, 0.836])	([0.020, 0.100], [0.800, 0.850])	([0.010, 0.200], [0.750, 0.800])
${{\lambda }_{P{Z}^C}}^h$	([0.042, 0.085], [0.818, 0.840])	([0.050, 0.100], [0.800, 0.900])	([0.035, 0.090], [0.689, 0.910])
${{\lambda }_{B{Z}^C}}^h$	([0.425, 0.536], [0.250, 0.355])	([0.400, 0.560], [0.200, 0.400])	([0.280, 0.632], [0.280, 0.350])
${{\lambda }_{{NZ}^C}}^h$	([0.865, 0.900], [0.000, 0.066])	([0.800, 0.900], [0.010, 0.100])	([0.820, 0.903], [0.005, 0.008])
	${\mathit{e}}_{\mathbf{4}}$	${\mathit{e}}_{\mathbf{5}}$	${\mathit{e}}_{\mathbf{6}}$
${{\lambda }_{PZ}}^h$	([0.950, 0.980], [0.001, 0.015])	([0.880, 0.980], [0.005, 0.010])	([0.850, 0.950], [0.010, 0.015])
${{\lambda }_{BZ}}^h$	([0.300, 0.400], [0.500, 0.600])	([0.200, 0.300], [0.600, 0.700])	([0.300, 0.350], [0.550, 0.650])
${{\lambda }_{NZ}}^h$	([0.020, 0.080], [0.738, 0.911])	([0.035, 0.200], [0.764, 0.800])	([0.030, 0.120], [0.800, 0.840])
${{\lambda }_{P{Z}^C}}^h$	([0.050, 0.100], [0.750, 0.860])	([0.050, 0.110], [0.870, 0.875])	([0.040, 0.110], [0.774, 0.888])
${{\lambda }_{B{Z}^C}}^h$	([0.335, 0.628], [0.252, 0.372])	([0.430, 0.540], [0.350, 0.400])	([0.350, 0.630], [0.355, 0.360])
${{\lambda }_{{NZ}^C}}^h$	([0.850, 0.900], [0.005, 0.010])	([0.870, 0.981], [0.005, 0.010])	([0.950, 0.950], [0.005, 0.015])

.

The score function is used to defuzzy the above data, and the results are shown in

Table 7. The benefits after defuzzification.

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$
${{\lambda }_{PZ}}^h$	0.934	0.809	0.868	0.932	0.868	0.816
${{\lambda }_{BZ}}^h$	−0.269	−0.238	−0.250	−0.182	−0.364	−0.256
${{\lambda }_{NZ}}^h$	−0.680	−0.686	−0.598	−0.688	−0.604	−0.674
${{\lambda }_{P{Z}^C}}^h$	−0.691	−0.721	−0.648	−0.652	−0.757	−0.691
${{\lambda }_{B{Z}^C}}^h$	0.146	0.148	0.115	0.140	0.096	0.115
${{\lambda }_{{NZ}^C}}^h$	0.783	0.726	0.756	0.776	0.860	0.904

.

Assuming the reference point ${\lambda }^{\ast }=-0.2$, the values in different states are calculated, as shown in

Table 8. The values in different states.

	${\mathit{e}}_1$	${\mathit{e}}_2$	${\mathit{e}}_3$	${\mathit{e}}_4$	${\mathit{e}}_5$	${\mathit{e}}_6$
${g_{PZ}}^h$	1.117	1.008	1.060	1.115	1.060	1.014
${g_{BZ}}^h$	−0.214	−0.127	−0.161	0.029	−0.458	−0.178
${g_{NZ}}^h$	−1.180	−1.193	−1.001	−1.196	−1.013	−1.167
${g_{P{Z}^C}}^h$	−1.204	−1.268	−1.109	−1.118	−1.343	−1.203
${g_{B{Z}^C}}^h$	0.393	0.395	0.362	0.387	0.343	0.362
${g_{{NZ}^C}}^h$	0.985	0.935	0.961	0.979	1.053	1.091

. By weighted synthesis of the values of six experts, the group values can be obtained as: $g_{PZ}=1.063$, $g_{P{Z}^C}=-1.208$, $g_{BZ}=-0.185$, $g_{B{Z}^C}=0.374$, $g_{NZ}=-1.125$ and $g_{{NZ}^C}=1.001$.

According to Formulas (33)–(35) and the definition of decision thresholds, the decision thresholds are obtained as ${\alpha }^{\prime }=0.604$, ${\beta }^{\prime }=0.335$ and ${\gamma }^{\prime }=0.507$. It can be seen that the condition ${\alpha }^{\prime }>{\beta }^{\prime }$ is satisfied, so TWDM rules (P3)–(N3) can be obtained. According to the conditional probability of each green supplier, it can be seen that x₁, x₆ and x₁₀ belong to the POS(X), x₂, x₃, x₄, x₅, x₈ and x₁₂ belong to the BND(X), and x₇, x₉ and x₁₁ belong to the NEG(X). Therefore, suppliers x₇, x₉ and x₁₁ are excluded, and the remaining suppliers enter the calculation of the next granularity level.

In ${GS}^2$, the initial group consensus degree is 0.830, which is less than the consensus threshold 0.9. Therefore, the expert social network is constructed and information exchange is carried out based on Formulas (10) and (11). It is assumed that the exchange willingness of experts is 0.5. After the first round of exchange, the group consensus degree was 0.880, less than 0.9, so the information exchange continued. After the second round of exchange, the group consensus degree is 0.902, greater than 0.9, so a group consensus is reached. Figure 3 shows the information exchange process.

According to the group TWDM steps, the weights of six experts are calculated as 0.166, 0.169, 0.168, 0.162, 0.167 and 0.168, respectively. The combined weights of attributes c₁, c₂, c₄, c₆, c₇, c₁₀, c₁₃ and c₁₄ are 0.253, 0.243, 0.067, 0.091, 0.121, 0.076, 0.076 and 0.073, respectively, and the group values are $g_{PZ}=1.062$, $g_{P{Z}^C}=-1.208$, $g_{BZ}=-0.185$, $g_{B{Z}^C}=0.373$, $g_{NZ}=-1.125$, and $g_{{NZ}^C}=1.001$. The calculation results of conditional probability are shown in

Table 9. The calculation results of conditional probability in ${GS}^2$.

Supplier	${\mathit{S}}_{\mathit{n}}$	${\mathit{R}}_{\mathit{n}}$	${\mathit{S}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{R}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{Q}}_{\mathit{n}}$	$\mathit{P}\left(\mathit{Z}\left|{\mathit{x}}_{\mathit{n}}\right.\right)$
$x_1$	0.693	0.207	1.004	1.346	0.660	0.340
$x_2$	0.708	0.206	1.013	1.424	0.782	0.218
$x_3$	0.644	0.133	1.017	1.236	0.167	0.833
$x_4$	0.700	0.170	1.049	1.310	0.641	0.359
$x_5$	0.682	0.212	0.988	1.377	0.616	0.384
$x_6$	0.702	0.207	1.023	1.417	0.790	0.210
$x_8$	0.709	0.211	1.024	1.570	0.928	0.072
$x_{10}$	0.653	0.171	1.002	1.288	0.331	0.669
$x_{12}$	0.625	0.157	0.978	1.245	0.093	0.907

.

The calculated decision thresholds are ${\alpha }^{\prime }=0.604$, ${\beta }^{\prime }=0.336$ and ${\gamma }^{\prime }=0.507$. According to the conditional probability of each green supplier, x₃, x₁₀ and x₁₂ belong to the POS(X), x₁, x₄ and x₅ belong to the BND(X) and x₂, x₆ and x₈ belong to the NEG(X). Therefore, suppliers x₂, x₆ and x₈ are excluded, and the remaining suppliers enter the calculation of the next granularity level.

In ${GS}^3$, the initial group consensus degree is 0.691, less than the consensus threshold of 0.9, so the expert social network is constructed and information exchange is carried out. After the fourth and fifth exchanges, the group consensus degree is 0.873 and 0.874, respectively, with a difference of less than or equal to the consensus difference threshold of 0.001, thus achieving group consensus. Figure 4 shows the information exchange process.

The weights of the six experts are calculated as 0.170, 0.160, 0.170, 0.171, 0.169 and 0.160, respectively. The weights of the 14 attributes from c₁ to c₁₄ are 0.189, 0.180, 0.039, 0.042, 0.037, 0.059, 0.081, 0.037, 0.105, 0.048, 0.035, 0.055, 0.049 and 0.045, respectively. The group values are $g_{PZ}=1.063$, $g_{P{Z}^C}=-1.207$, $g_{BZ}=-0.185$, $g_{B{Z}^C}=0.373$, $g_{NZ}=-1.124$, and $g_{{NZ}^C}=1.000$. The calculation results of conditional probability are shown in

Table 10. The calculation results of conditional probability in ${GS}^3$.

Supplier	${\mathit{S}}_{\mathit{n}}$	${\mathit{R}}_{\mathit{n}}$	${\mathit{S}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{R}}_{\mathit{n}}^{\mathbf{\prime }}$	${\mathit{Q}}_{\mathit{n}}$	$\mathit{P}\left(\mathit{Z}\left|{\mathit{x}}_{\mathit{n}}\right.\right)$
$x_1$	0.621	0.153	1.015	1.489	0.720	0.280
$x_3$	0.608	0.109	1.054	1.319	0.310	0.690
$x_4$	0.638	0.124	1.059	1.412	0.676	0.324
$x_5$	0.621	0.153	1.019	1.524	0.764	0.236
$x_{10}$	0.591	0.127	1.022	1.428	0.312	0.688
$x_{12}$	0.578	0.114	1.012	1.323	0.040	0.960

.

The calculated decision thresholds are ${\alpha }^{\prime }=0.603$, ${\beta }^{\prime }=0.336$ and ${\gamma }^{\prime }=0.507$. According to the conditional probability of each green supplier, x₃, x₁₀ and x₁₂ belong to the POS(X), and x₁, x₄ and x₅ belong to the NEG(X).

Figure 5 illustrates the decision results of different granularities. First, when making decisions on ${GS}^1$, based on the decision rule and the values of the decision thresholds ${\alpha }^{\prime }$, ${\beta }^{\prime }$ and ${\gamma }^{\prime }$, it can be seen that x₁, x₆ and x₁₀ belong to the POS(X), x₂, x₃, x₄, x₅, x₈ and x₁₂ belong to the BND(X), and x₇, x₉ and x₁₁ belong to the NEG(X). Next, we transfer the suppliers in the acceptance and boundary domains to the next granularity level for decision making. Similarly, after decision making on ${GS}^2$, we obtain the following results: x₃, x₁₀ and x₁₂ in POS(X), x₁, x₄ and x₅ in BND(X), and x₂, x₆ and x₈ in NEG(X). We then continue to make decisions on ${GS}^3$ and end up with the results: x₃, x₁₀ and x₁₂ in POS(X), and x₁, x₄ and x₅ in NEG(X). Therefore, if enterprise Q selects three green suppliers as partners, it should select x₃, x₁₀ and x₁₂, and there is no substantial difference between them. If the number of selections is less than three, it can select x₁₂, x₃ and x₁₀ in order of conditional probability from highest to lowest. If more than three suppliers are selected, x₄, x₁ and x₅ are selected in descending order of conditional probabilities in ${GS}^3$. If more than six suppliers are selected, on the basis of the above 6 suppliers, x₂, x₆ and x₈ are selected according to the order of conditional probabilities in ${GS}^2$ from the highest to the lowest, and so on. From another perspective, if nine suppliers are selected, the remaining suppliers except x₇, x₉ and x₁₁ can be selected at the end of the first granularity level, without spending time and cost to perform subsequent calculations. If six suppliers are selected, the calculation stops at the second granularity level, which can significantly improve decision making efficiency.

Overall, by constructing a multilevel granular structure and social networks, group consensus as well as the three-way classification of alternatives can be quickly achieved, and the risk of misclassification is significantly reduced. In addition, by dividing the decision-making problem into a series of sub-problems, the green supplier selection can be completed quickly at a lower granularity level, which saves the decision time and cost.

4.2. Sensitivity Analysis

4.2.1. Influence of Group Consensus Threshold $\delta$

Figure 6 shows the information exchange process as the group consensus threshold $\delta$ increases from 0.82 to 0.92. The corresponding TWDM results are shown in

Table 11. TWDM results under different group consensus thresholds.

$\mathit{\delta }$		0.82	0.84	0.86	0.88	0.90	0.92
GS1	POS	x1, x6, x10	x1, x6, x10	x1, x6, x10	x1, x6, x10	x1, x6, x10	x1, x6, x10
	BND	x2, x3, x5, x8, x12	x2, x3, x4, x5, x8, x12	x2, x3, x4, x5, x8, x12	x2, x3, x4, x5, x8, x12	x2, x3, x4, x5, x8, x12	x2, x3, x4, x5, x8, x12
	NEG	x4, x7, x9, x11	x7, x9, x11	x7, x9, x11	x7, x9, x11	x7, x9, x11	x7, x9, x11
GS2	POS	x3, x12	x3, x10, x12	x3, x10, x12	x3, x10, x12	x3, x10, x12	x3, x10, x12
	BND	x4, x10	x4, x5	x4, x5	x4, x5	x1, x4, x5	x1, x4, x5
	NEG	x1, x2, x5, x6, x8	x1, x2, x6, x8	x1, x2, x6, x8	x1, x2, x6, x8	x2, x6, x8	x2, x6, x8
GS3	POS	x12	x3, x10, x12	x3, x10, x12	x3, x10, x12	x3, x10, x12	x3, x10, x12
	BND	x3, x4
	NEG	x10	x4, x5	x4, x5	x4, x5	x1, x4, x5	x1, x4, x5

.

It can be seen that (1) with the increase in the group consensus threshold, the number of information interactions increases. (2) The results are the same for consensus thresholds of 0.84, 0.86, and 0.88, while the results are the same for consensus thresholds of 0.90 and 0.92. Unlike the previous case, in the latter case supplier x₁ ultimately belongs to the NEG domain. Finally, (3) when the consensus threshold is 0.82, the decision results are significantly different from those under other consensus thresholds. It can be concluded that the higher the consensus threshold, the higher the accuracy of the decision result, but within a certain range, the decision result is stable. Therefore, the expert group needs to select an appropriate consensus threshold considering the accuracy of the decision results and the workload of the data processing process.

4.2.2. Influence of Reference Point ${\lambda }^{\ast }$

The choice of reference point ${\lambda }^{\ast }$ affects the value function of CPT, which in turn affects the decision threshold and leads to different classification results. Considering the range of data in Table 8 and the test results, we set ${\lambda }^{\ast }$ between −0.6 and 0.2. The decision thresholds and the TWDM results under different reference points are shown in Figure 7 and

Table 12. TWDM results under different reference points.

${\mathit{\lambda }}^{\mathbf{*}}$		−0.6	−0.4	−0.2	0	0.2
GS1	POS	x1, x6, x10	x1, x6, x10	x1, x6, x10	x1, x3, x6, x10	x1, x2, x3, x6, x8, x10
	BND	x2, x3, x5, x8	x2, x3, x4, x5, x8, x12	x2, x3, x4, x5, x8, x12	x2, x4, x5, x8
	NEG	x4, x7, x9, x11, x12	x7, x9, x11	x7, x9, x11	x7, x9, x11, x12	x4, x5, x7, x9, x11, x12
GS2	POS	x3, x10	x3, x10, x12	x3, x10, x12	x3, x10	x3, x10
	BND		x1, x4, x5	x1, x4, x5	x4, x5
	NEG	x1, x2, x5, x6, x8	x2, x6, x8	x2, x6, x8	x1, x2, x6, x8	x1, x2, x6, x8
GS3	POS		x3, x10, x12	x3, x10, x12	x3, x10
	BND	x3, x10			x4, x5
	NEG		x1, x4, x5	x1, x4, x5		x3, x10

, respectively.

It can be seen that (1) as the reference point gradually increases from −0.6 to 0.2, the value of ${\alpha }^{\prime }$ first increases and then decreases, the value of ${\beta }^{\prime }$ changes slightly, and the value of ${\gamma }^{\prime }$ first decreases and then increases. When the reference points are −0.6, −0.4, −0.2 and 0, ${\alpha }^{\prime }$ > ${\beta }^{\prime }$ > ${\gamma }^{\prime }$, and the decision-making problems are TWDM. When the reference point is 0.2, ${\alpha }^{\prime }$ < ${\beta }^{\prime }$, and the problem is degraded to binary decision making. (2) The decision results are the same when the reference points are −0.4 and −0.2, and there are no suppliers in the final BND domain. The results are the same when the reference points are −0.6 and 0, but different from the former, there are suppliers in the BND domain. And (3) when the reference point is 0.2, the decision results are significantly different compared to the first two cases. However, it can also be seen that although x₃ and x₁₀ belong to the NEG domain in this case, they are still the most preferred alternatives.

4.3. The Implications of the Analysis Results

The results show that the sequential group TWDM method proposed in this study can provide a green supplier selection scheme that meets the requirements. Attributes are divided into a nested sequence containing three granularity levels based on the difference threshold of attribute consistency. At each granularity, we determine the number of interactions on the social network based on the consensus threshold, as well as the conditional probability value of each supplier. According to the reference point in the CPT, we determine the expert value function, TWDM thresholds and rules. Finally, combining the results of supplier classification in the POS, NEG and BND domains, we can provide a feasible scheme according to the requirements of supplier selection.

The sensitivity analysis of expert consensus threshold shows that increasing expert threshold is beneficial to improve the accuracy of results. For example, the consensus threshold is increased from 0.88 to 0.90, and supplier x₁ is explicitly marked as not optional. This is consistent with the reality that the more stringent the consensus requirement is, the more able it is to weed out the ambiguous suppliers. However, as can be seen from the number of expert interactions on social networks, the larger the consensus threshold, the more interactions, which means the more time and cost. This is also in line with reality, raising the consensus requirement, and the discussion of experts to reach the threshold will inevitably increase. Therefore, in practice, experts can set a uniform consensus threshold and automatically perform interactive calculations according to the above steps, without the need for successive interactions. Of course, experts can also make appropriate adjustments to the thresholds according to the actual situation and achieve satisfactory decisions that take less time and cost.

The sensitivity analysis of reference point in CPT shows that reference point has a complex effect on the classification results. The reason is that the reference point affects the thresholds of the TWDM and the classification results by influencing the value function of each expert. For example, when the reference point is −0.2, the first batch of suppliers includes x₃, x₁₀ and x₁₂, and the second batch of suppliers includes x₁, x₄ and x₅. But when the reference point is 0, neither x₁₂ nor x₁ is in the first two batches. Therefore, in practice, experts need to discuss the value of the reference point within the range of the benefit values given by all experts. If necessary, they can compare the classification results under different values of reference point to determine the final classification of suppliers.

5. Conclusions

In this article, we proposed a green supplier selection process based on sequential group TWDM. We construct a granularity structure with nested sequences of attributes. We prioritize decision making under attributes with the high expert agreement to improve decision making efficiency and reduce interaction costs. We also introduced CPT into TWDM and derived the corresponding decision rules to better describe the risk attitudes and preferences of decision makers and improve classification accuracy. The case study demonstrates that in decision making at each granularity level, social network-based information exchange can automatically adjust experts’ large deviation and quickly reach group consensus. Compared with the traditional selection methods, the sequential group TWDM method can obtain three categories of “accepted”, “rejected” and “pending further investigation” for green suppliers and improve the classification accuracy. At the same time, it can avoid continuous discussion on the results with significant differences among experts in traditional decision making and improve management efficiency. The sequential group TWDM method provides a new idea and method for green supplier selection.

This study provides some insights into the problem of green supplier selection. However, it can be improved in the following aspects. Firstly, this article summarizes a green supplier evaluation index system, but this system may not be suitable for all types of enterprises. The index system should be improved according to the specific industry to form a targeted evaluation index. Secondly, the article constructs a social network based on the similarity of experts’ opinions. It assumes that the experts’ willingness to interact remains the same to simplify the computations. In reality, relationships and behaviors among experts are diverse, and interaction willingness can change during the information exchange process. Therefore, future work should extend the method by examining various expert relationships and behaviors. Thirdly, the weight determination methods of attributes or experts can continue to be improved. For example, applying the weighting method based on intuitionistic fuzzy cross entropy to reflect the similarity between attributes or expert opinions can improve the objectivity of weights. Finally, how to integrate the case of increasing or decreasing the number of attributes or changing the values of attributes into the case of sequential TWDM is also a major direction in the future.