An Adaptive Consensus Model to Manage Non-Cooperative Behaviors in Large Group Decision-Making with Probabilistic Linguistic Information

Abstract

To address challenges in complex group decision-making (GDM)—specifically preference fuzziness, intricate subgroup segmentation, and non-cooperative behavior—this study proposes an adaptive consensus model based on probabilistic linguistic term sets (PLTSs). By integrating fuzzy C-means (FCM) clustering with a Gaussian mixture model (GMM), a fuzzy Gaussian mixture model (FGMM) is constructed to achieve soft segmentation of expert preference distributions. On this basis, an adaptive consensus feedback mechanism is developed, which dynamically integrates interactive and automated adjustment strategies via multi-level consensus thresholds, thereby balancing decision efficiency and quality. To identify and control non-cooperative behaviors, a cooperation index and a three-tier management strategy, which incorporates intra-group negotiation, weight penalties and an exit-delegation mechanism, were introduced. In the case of strategic decision-making of new energy vehicles (NEV), after four rounds of feedback iterations, the group consensus level increased from the initial 0.316 to 0.804, reaching the preset threshold and verifying the effectiveness of the consensus mechanism. Compared with the existing literature methods, the framework in this paper achieves more comprehensive integration and innovation in four aspects: preference expression, clustering mechanism, consensus feedback and behavior management.

1. Introduction

In modern management science, social computing, and emergency management, complex decision-making environments often require the aggregation of expertise and experience from multiple experts. However, due to inherent differences among decision-makers in professional expertise, vested interests, cognitive patterns, and risk preferences, groups frequently exhibit divergent opinions, making it difficult to reach a unified conclusion spontaneously. This challenge has spurred the development of systematic research on consensus-reaching processes (CRP). CRP aim to progressively align decision-makers’ preferences through multiple rounds of discussion, negotiation, and communication. A key take lies in designing effective feedback mechanisms to facilitate consensus [1]. Within the existing literature, most CRPs employ “soft consensus” as the level of agreement within decision-making processes, rather than the unattainable “hard consensus”.

Due to the complexity and ambiguity of GDM problems, experts may struggle provide precise numerical information due to limited knowledge or hesitation. Therefore, scholars have proposed the concept of “fuzzy sets” to express the uncertainty inherent in expert opinions [2]. Subsequently, numerous studies have introduced various extensions of fuzzy sets for decision-making problems, including interval-valued intuitionistic fuzzy sets [3,4], hesitant fuzzy sets [5], and others [6]. Typically, decision-makers express their preferences for a set of alternatives to prioritize them, by using intuitive linguistic terms like “slightly better” or “good” to describe uncertain information. Such evaluation terms constitute a linguistic term set (LTS). However, the LTS framework lacks the capacity to explicitly indicate the weight of different linguistic terms. To address this limitation, Pang et al. [7] proposed the concept of Probabilistic Linguistic Term Sets (PLTSs). As a novel linguistic variable, PLTS serves as an extended expression of hesitant fuzzy linguistic terms, enabling decision-makers to convey multiple linguistic terms alongside with their corresponding probability information [8]. Consequently, conducting in-depth research on PLTSs holds significant theoretical and practical importance.

In recent years, numerous scholars have conducted multifaceted explorations of PLTSs [9,10]. For instance, Tan et al. [1] integrated probabilistic linguistic information with social network analysis to propose a novel dynamic trust mechanism. Similarly, Jin et al. [11] primarily addressed the consistency of probabilistic linguistic preference relations (PLPRs) and solved the trust-driven issue of expert weighting. Building upon the integration of PLTSs and PLPRs, Han et al. [12] introduced innovations concerning trust risk neglect and excessive reliance on group opinions in consensus problems. Considering scenarios where decision-makers may fail to provide complete probabilistic information, scholars have conducted in-depth research on incomplete PLPRs [13,14,15,16]. Furthermore, PLTSs have played a significant role in mitigating non-cooperative behavior issues within group decision-making [14,17,18]. Given the methodological advantages of PLTSs in handling uncertain preference information, this study adopts PLTSs as its methodological foundation.

Early studies often treated groups as homogeneous entities. In recent years, however, scholars have recognized the existence of subgroup structures within larger collectives. Consequently, social network analysis and clustering methods have been introduced to identify these subgroups. Traditional clustering methods, such as K-means clustering [19] and hierarchical clustering, are based on distance metrics to definitively assign each decision-maker to a specific cluster. However, applying these traditional “hard” clustering methods directly to GDM scenarios presents significant limitations. Meanwhile, as decision-making environments grow increasingly complex, scholars have begun exploring more flexible and interpretable clustering techniques. Examples include overlapping community clustering [20,21], spectral clustering [22], fuzzy C-means (FCM) clustering [23], and deep learning-based clustering, among others. Although these methods demonstrate significant advantages over traditional clustering approaches, they often retain inherent limitations. For instance, the FCM still exhibits shortcomings in terms of cluster shape adaptability and noise resistance. To overcome the limitations of individual methods, recent studies have attempted to achieve complementary advantages through method integration. For instance, Wang et al. [24] developed a convex clustering method combining the strengths of K-means and hierarchical clustering to enable adaptive subgroup formation and automatic determination of the optimal number of clusters. Within this research context, this study combines FCM with the Gaussian mixture model (GMM). This approach aims to leverage the theoretical strengths of GMM in modeling probability distributions alongside the flexibility of FCM in characterizing membership degrees. This integration enables more effective handling of fuzzy boundaries within expert preference information and facilitates the identification of complex patterns in group preference structure.

Furthermore, in the design of consensus models, decision-making efficiency and quality are two critical metrics. Existing consensus strategies are typically categorized into automated strategies [25] and interactive approaches [26]. In interactive strategies, the moderator provides decision-makers with recommended preference values; whereas in automated strategies, the moderator uses specific algorithms or rules to automatically adjust decision-makers’ preference values to enhance the level of consensus [27]. However, interactive strategies often lack decision-making efficiency, while automated strategies may compromise decision-making quality. To address this trade-off, this study proposes an adaptive consensus model that synthesizes both interactive and automated strategies. By implementing different strategies based on the specific level of consensus achieved in each round, the model balances both decision-making efficiency and quality.

Additionally, non-cooperative behavior within decision-making processes cannot be overlooked. Existing research in this domain can be broadly categorized into two main approaches: one focuses on managing such behavior through trust relationships and social networks, while the other emphasizes behavioral management using consensus models and optimization methods. The former emphasizes leveraging trust relationships [28,29,30], social network structures [31,32] or coordination mechanisms [33] to mitigate non-cooperative behavior. The latter concentrates on developing sophisticated consensus models and optimization or frameworks, such as hierarchical consensus [27,34], convergent behavior [35], self-organizing mechanisms [36] and robust consensus [37,38].

Consequently, this study employs a Fuzzy Gaussian mixture model (FGMM) based on PLTS to construct an adaptive consensus feedback framework that incorporates non-cooperative behavior management. The primary contributions of this study are summarized as follows: (1) A novel FGMM is proposed by integrating FCM and GMM to overcome the limitations of the standalone approaches, thereby enhancing the clustering performance and robustness. (2) An enhanced adaptive consensus feedback framework is developed that synergistically integrates interactive and automated strategies, achieving a sound balance between decision-making efficiency and quality is ach. (3) By introducing a cooperation index to quantify experts’ cooperation level and implementing a three-tier management strategy, non-cooperative behavior among experts is effectively addressed.

In summary, this study constructs a more practical adaptable group decision-making consensus model through method integration and mechanism innovation. Case studies and multi-angle comparative analysis of new energy vehicle development strategies show that the proposed model is better than traditional methods in terms of consensus stability, behavior management and robustness of decision-making results.

The remainder of this study is organized as follows. Section 2 reviews fundamental concepts of PLTSs and clustering methods. Section 3 formulates the propose an adaptive consensus-reaching framework. Section 4 discusses the identification and management of non-cooperative behavior. Section 5 presents a case study and numerical comparisons. Finally, Section 6 concludes the paper and discusses potential future research directions.

2. Preliminaries

2.1. Probabilistic Linguistic Term Sets

PLTSs mitigate the loss of original linguistic information by integrating probabilistic parameters into the hesitation fuzzy linguistic itemset framework. It overcomes the limitations of traditional linguistic evaluation methods by allowing decision-makers to express judgments and preferences over alternative options using multiple linguistic terms alongside their corresponding probability values. This structure not only avoids the unreasonableness of assigning equal weights to all possible terms but also comprehensively and precisely reflects the actual information states and cognitive disparities among decision-makers.

Definition 1. 

Let $S = \{s_0,{ s}_1, \dots {, s}_{2\mathsf{\tau }}\}$ denote a set of linguistic terms, where τ is a positive integer. Then, PLTS can be defined as

$$L\left(p\right)=\left\{L^k\left(p^k\right)\left|L^k\in S,p^k\ge 0,\right.k=1,2,\cdots ,\#L\left(p\right),{\displaystyle \sum _{k=1}^{\#L\left(p\right)}{p}^k\le 1}\right\}$$

(1)

where  $\left\{s_0,{ s}_1, \dots , s_{2\tau }\right\}$ denotes the set of linguistic terms expressed by the decision-maker. L^k(p^k) represents the k-th linguistic term L^k and its probability p^k, while #L(p) indicates the number of linguistic terms in L(p).

Definition 2. 

Given a PLTS L(p) satisfying ${\sum }_{k=1}^{\#L\left(p\right)}{p}^k\le 1$, the normalized probabilistic language term set (NPLTS) L^N can be defined as

$$L^N(p)=\left\{L^{N(k)}(p^{N(k)})\left|k=1,2,\cdots ,\#L(p)\right.\right\}$$

(2)

$$p^{N(k)}={\displaystyle \frac{p^k}{{\displaystyle \sum _{k=1}^{\#L(p)}{p}^k}}},\hspace{1em}k=1,2,\cdots ,\#L(p)$$

(3)

Standardized operations can effectively redistribute the weights of incomplete probabilistic information relative to the original PLTS. From the PLTS and the NPLTS, the scoring function and deviation degree of the PLTS can be derived.

Definition 3. 

Let $L(p) = \{L^k(p^k)|k = 1, 2, ..., \#L(p)\}$ is a PLTS, the scoring function S(L(p)) for the PLTS is defined as follows:

$$S(L(p))=s_{\alpha }$$

(4)

$$\alpha ={\displaystyle \frac{{\displaystyle \sum _{k=1}^{\#L(p)}{r}^k{p}^k}}{{\displaystyle \sum _{k=1}^{\#L(p)}{p}^k}}}$$

(5)

where r^k is the subscript of linguistic term L^k, and  $p^k$ is the probability of the corresponding term. α is the weighted average of all term subscripts, weighted by their normalized probabilities. For example, in the LTS S = {s₋₂, s₋₁, s₀, s₁, s₂}, if S(L(p)) = s_1.2, the overall evaluation tends toward “better”.

Definition 4. 

From the scoring function, the formula for the deviation ϕ of PLTS is defined as in Equation (6). Deviation degree refers to the weighted standard deviation, measuring the degree of deviation of each language term’s subscript from the mean value α:

$$\varphi (L(p))={\displaystyle \frac{({\displaystyle \sum _{K=1}^{\#L(P)}{(p^k(r^k-\alpha ))}^2{)}^{\frac{1}{2}}}}{{\displaystyle \sum _{k=1}^{\#L(p)}{p}^k}}}$$

(6)

According to Definitions 3 and 4, the comparison rules between any two linguistic terms L₁(p) and L₂(p) are as follows:

(1) 

If $S(L_1(p))>S(L_2(p))$, then L₁(p) > L₂(p)

(2) 

If $S(L_1(p))<S(L_2(p))$, then L₁(p) < L₂(p)

(3) 

If $S(L_1(p))=S(L_2(p))$, then

(a) 

If $\varphi (L_1(p))>\varphi (L_2(p))$, then L₁(p) < L₂(p)

(b) 

If $\varphi (L_1(p))<\varphi (L_2(p))$,then L₁(p) > L₂(p)

(c) 

If $\varphi (L_1(p))=\varphi (L_2(p))$,then L₁(p) ~ L₂(p)

Definition 5. 

Let $L_1(p) = \{L_1^{k_1}(p^{k_1})|k_1 = 1, 2, \dots , \#L_1(p)\}$ and $L_2(p)=\{L_2^{k_2}(p^{k_2})|k_2=1, 2, \dots , \#L_2(p)\}$ be any two PLTS, $L_1^N\left(p\right)$ and $L_2^N\left(p\right)$ be their corresponding NPLTS. The distance between two PLTS can be calculated as follows:

$$d\left(L_1\left(p\right),L_2\left(p\right)\right)=d\left(L_1^N\left(p\right),L_1^N\left(p\right)\right)={\displaystyle \sum _{K=1}^{\#L}\left(p\left(r_1^{N(k_1)},r_2^{N(k_2)}\right)d\left(r_1^{N(k_1)},r_2^{N(k_2)}\right)\right)}$$

(7)

$$d\left(r_1^{N(k_1)},r_2^{N(k_2)}\right)={\displaystyle \frac{\left(r_1^{N(k_1)}-r_2^{N(k_2)}\right)}{T}}$$

(8)

$$p\left(r_1^{N(k_1)},r_2^{N(k_2)}\right)=p\left(r_1^{N(k_1)}\right)p\left(r_2^{N(k_2)}\right)=p_1^{N(k_1)}{p}_2^{N(k_2)}$$

(9)

where T represents the total count of the LTS S, $\#L = \#L_1\left(p\right) \times \#L_2\left(p\right)$.

Definition 6. 

Let $X = \{x_1,{ x}_2, \dots ,{ x}_n\}$ denote a discrete set of alternative options, and $E=\{e_1,{ e}_2, \dots , e_n\}$ denote the set of decision-makers. Each decision-maker provides a probability language preference relation (PLTR) matrix $M={\{L_{\mathit{ij}}(p)\}}_{n\times n}$ for the option set. Each element $L_{\mathit{ij}}\left(p\right)$ represents a PLTS, indicating the decision-maker’s preference for alternative x_i over x_j. Let ${L\left(p\right)}_l^i \in M_l$ and ${L\left(p\right)}_l^j \in M_l$ be two row vectors of PLTR M, the similarity degree between ${L\left(p\right)}_l^i$ and ${L\left(p\right)}_l^j$ can be defined as

$$sim\left(L{\left(p\right)}_l^i,L{\left(p\right)}_l^j\right)=1-d\left(L{\left(p\right)}_l^i,L{\left(p\right)}_l^j\right)$$

(10)

Definition 7. 

Aggregate the comparison information between each alternative and all other alternatives to obtain a value representing the overall preference for that alternative, known as the comprehensive preference value (CPV). This is typically accomplished using the probabilistic language-hybrid weighted geometry (PLHWG) operator. This operator aggregates the preference relations $L_{\mathit{ij}}(p)$ between alternative i and all other alternatives (j = 1, 2, …, m), outputting a new, single PLTS, namely CPV_i, which represents the overall superiority of alternative i:

$$CP{V}_i=PLHW{G}_{\lambda ,\eta }\left(L_{i1}\left(p\right),L_{i2}\left(p\right),\dots ,L_{in}\left(p\right)\right)$$

(11)

where λ represents the weight vector, and η is the weight vector related to integration.

2.2. Clustering Methods

2.2.1. Fuzzy C-Means

FCM is a prominent soft clustering method originally introduced by Dunn [39] in 1974. Unlike hard clustering, FCM uses membership degrees to represent the relationships between data points, thereby determining the cluster to which each data point belongs.

Definition 8. 

FCM uses fuzzy partitioning to allow data points to belong to multiple clusters in the form of membership degrees, thereby minimizing the objective function, which is expressed as in Equation (12):

$$J={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^k{u}_{ij}^v{‖x_i-c_j‖}^2}}$$

(12)

$${\displaystyle \sum _{j=1}^k{u}_{ij}=1,u_{ij}\in \left[0,1\right]}$$

(13)

where u_ij represents the membership degree of point x_i to cluster j, v denotes the fuzzy factor, and c_j is the cluster center. To minimize the objective function J while satisfying the constraints, the Lagrange multiplier method is applied to the objective function, thereby obtaining the membership degree matrix U and the cluster centers c_j.

The key step of the FCM algorithm involves continuous iterative update of cluster centers to minimize intra-cluster distances. The primary advantage lies in generating membership degree information, making it suitable for scenarios with fuzzy boundaries. However, it requires pre-setting the number of clusters, and determining the optimal cluster count is challenging. Furthermore, the initial center points are selected randomly, which introduces significant uncertainty and can adversely affect clustering results.

2.2.2. Gaussian Mixture Model

GMM is a clustering algorithm based on probability density functions. It assumes that the dataset is generated by a mixture of K Gaussian distributions, with each Gaussian distribution corresponding to a latent “cluster”. The objective is to infer cluster memberships by estimating distribution parameters through maximum likelihood (MLE) or maximum a posteriori (MAP) estimation.

Definition 9. 

Suppose the observed data $X = \{x_1,{ x}_2, \dots ,{ x}_n\}$ is generated by a mixture of K Gaussian distributions. Then, the probability of generating a data point p(x_n) is

$$p\left(x_n\right)={\displaystyle \sum _{k=1}^K{\pi }_k\cdot \psi (x_n\left|{\mu }_k,{\Gamma }_k\right.)}$$

(14)

$$\psi (x_n\left|{\mu }_k,{\Gamma }_k\right.)={\displaystyle \frac{1}{{(2\pi )}^{D/2}{\left|{\Gamma }_k\right|}^{1/2}}}\times \exp \left(-{\displaystyle \frac{1}{2}}{\left(x_n-{\mu }_k\right)}^T{\Gamma }_k^{-1}(x_n-{\mu }_k)\right)$$

(15)

where μ_k denotes the mean vector,  ${\Gamma }_k$ represents the covariance matrix, π_k represents the covariance matrix, and  $\psi (x_n|{\mu }_k,{\Gamma }_k)$ is the probability density function of the multivariate Gaussian distribution. Parameter estimation for GMM typically employs the Expectation-Maximization (EM) algorithm, where the E-step calculates the likelihood values and the M-step updates the parameters. The responsibility value  $\gamma (z_{\mathit{nk}})$ reflects the “degree of belonging” or “responsibility” of data point x_n to the k-th Gaussian distribution. The objective of the EM algorithm is to maximize the log-likelihood function. The responsibility value γ is calculated as in Equation (16):

$$\gamma \left(z_{nk}\right)=p\left(z_n=k\left|x_n\right.\right)={\displaystyle \frac{{\pi }_k\cdot \psi \left(x_n\left|{\mu }_k,{\Gamma }_k\right.\right)}{{\displaystyle \sum _{j=1}^K{\pi }_j\cdot \psi \left(x_n\left|{\mu }_k,{\Gamma }_k\right.\right)}}}$$

(16)

Update parameters μ_k, Γ_k, and π_k based on the responsibility value.

$$N_k={\displaystyle \sum _{n=1}^N\gamma \left(z_{nk}\right)}$$

(17)

$${\mu }_k^{new}={\displaystyle \frac{1}{N_k}}{\displaystyle \sum _{n=1}^N\gamma \left(z_{nk}\right)x_n}$$

(18)

$${\Gamma }_k^{new}={\displaystyle \frac{1}{N_k}}{\displaystyle \sum _{n=1}^N\gamma \left(z_{nk}\right)\left(x_n-{\mu }_k^{new}\right){\left(x_n-{\mu }_k^{new}\right)}^T}$$

(19)

$${\pi }_k^{new}={\displaystyle \frac{N_k}{N}}$$

(20)

where N_k denotes the number of valid samples. After completing the parameter estimation update, each data point x_n is assigned to the cluster with the maximum responsibility value.

Essentially, GMM is a probabilistic generative clustering method that enables samples to be associated with multiple clusters with varying probabilities. Through the covariance matrix, it can model clusters with diverse shapes (e.g., elliptical, linear, or rotated ones) and provide posterior probabilities (responsibility value γ) for sample membership in each cluster instead of hard assignments. This soft-assignment approach facilitates a robust assessment of uncertainty. However, this method exhibits high computational complexity. The EM algorithm necessitates extensive iterations, and computing the inverse of the covariance matrix becomes computationally intensive for high-dimensional data. Moreover, the EM algorithm is sensitive to initial values and may converge to local optima. Consequently, a meticulous evaluation of these trade-offs is essential when implementing this approach.

3. Consensus Reached Process

3.1. Fuzzy Gaussian Mixture Model

On the basis of the limitations of the two clustering methods mentioned in Section 2.2, this study introduces a hybrid clustering approach that integrates FCM and GMM. By leveraging FCM’s fuzzy membership principle to enhance GMM’s flexible partitioning capability, this approach aims to overcome the intrinsic limitations of standalone method, thereby improving clustering performance and robustness. Simultaneously, the probabilistic distribution assumption of GMM is incorporated into the objective function of FCM. Specifically, the weighted squared Euclidean distance in FCM is replaced with the Mahalanobis distance, effectively introducing Gaussian distribution while retaining FCM’s membership weighting scheme. The integrated objective function is shown in Equation (21):

$$J_{Hybrid}={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^k{u}_{ij}^v{D}^2\left(x_n,{\mu }_k\right)}}$$

(21)

$$D^2\left(x_n,v_k\right)={\left(x_n-{\mu }_k\right)}^T{\Gamma }_k^{-1}\left(x_n-{\mu }_k\right)$$

(22)

In E-step of the GMM, the membership degree u_ij is used as a weight to adjust the calculation of the posterior probability γ. The posterior probability of GMM contains the complete normalization coefficient of the Gaussian distribution, which has a strict probability interpretation. The membership of FGMM adopts a simplified form, focusing more on the fuzzy division of distance measures, while retaining the concept of subgroup prior weights through π_k. In M-step, parameter updates are performed based on the current membership degrees. The updated FGMM membership degrees and parameters are given by Equations (23)–(26):

$$u_{ik}={\displaystyle \frac{{\pi }_k\times \exp \left(-D^2\left(x_n,{\mu }_k\right)\right)}{{\displaystyle \sum _{j=1}^c{\pi }_j\times \exp \left(-D^2\left(x_n,{\mu }_j\right)\right)}}}$$

(23)

$${\mu }_k={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(u_{ik}\right)}^v{x}_n}}{{\displaystyle \sum _{i=1}^n{\left(u_{ik}\right)}^v}}}$$

(24)

$${\Gamma }_k={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(u_{ik}\right)}^v\left(x_n-{\mu }_k\right){\left(x_n-{\mu }_k\right)}^T}}{{\displaystyle \sum _{i=1}^n{\left(u_{ik}\right)}^v}}}$$

(25)

$${\pi }_k={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{u}_{ik}}$$

(26)

All the above update rules can be derived from the optimization of the target function (21). Equation (23) can be considered as a closed-form solution introduced with entropy regularization on the FCM objective function; Equations (24)–(26) are parameter estimation formulas obtained by directly differentiating the objective function.

3.2. Subgroup Weight Determination

In group decision-making, we tend to trust those who share similar views, values, or judgment criteria more readily. Similarity breeds trust, and trust influences weighting. Therefore, this study utilizes an inferred decision-maker similarity matrix to calculate the trust density of each subgroup, assigning subgroup weights based on this trust density. For a subcluster C_i, its trust density TD(C_i) is the average similarity among all its members:

$$TD\left(C_i\right)={\displaystyle \frac{1}{\left|C_i\right|\left(\left|C_i\right|-1\right)}}{\displaystyle \sum _{e_p\in C_i}{\displaystyle \sum _{e_q\in C_i}si{m}_{pq}},\hspace{0.17em}p\ne q}$$

(27)

where $\left|C_i\right|$ represents the number of experts in subgroup C_i, while the denominator denotes the total number of possible expert pairs within the subgroup, used to calculate the average. $\sum \sum {\mathit{sim}}_{\mathit{pq}}$ represents the sum of similarity degrees among all experts within a subgroup.

Higher trust density indicates greater cohesion within the subgroup, warranting greater influence in the final consensus and thus a higher weight assigned to that subgroup. Normalize the trust density of each subgroup to obtain its final weight:

$${\omega }_{C_i}={\displaystyle \frac{TD\left(C_i\right)}{{\displaystyle \sum _{j=1}^{K}TD\left(C_j\right)}}}$$

(28)

3.3. Consensus Measure

It is difficult to achieve complete consensus in current group decision-making problems. Therefore, a consensus threshold is predefined, such that the optimal solution is deemed acceptable once the consensus level meets this threshold. Typically, we can measure consensus levels on two dimensions: (1) Intra-subgroup consensus level: Assessing the consistency of expert opinions within each subgroup. (2) Overall consensus level: Evaluating the consistency of all expert opinions. The later can be achieved by either weighting the average of each subgroup’s consensus level (using subgroup weights) or directly calculating the consensus level across all experts. This study develops a hierarchical consensus measurement framework based on PLTSs, which progresses from the most granular pairwise alternatives to the overall decision-making level and thus comprehensively gauges the level of group consensus. This framework comprises four consensus levels: consensus on alternative pairs, consensus on alternatives, subgroup level consensus, and overall decision-making consensus.

Consensus on alternative pairs refers to the degree of agreement among all experts when comparing each pair of solutions. The consensus on alternative pairs C_ij can be calculated as

$$C_{ij}={\displaystyle \frac{2}{n\left(n-1\right)}}{\displaystyle \sum _{p=1}^{n-1}{\displaystyle \sum _{q=p+1}^{n}sim\left(L_p^{\left(ij\right)},L_q^{\left(ij\right)}\right)}}$$

(29)

where n denotes the total number of experts, sim() represents the similarity function, and $L_p^{\left(\mathit{ij}\right)}$ indicates expert p’s evaluation of the alternative pair (i, j).

Consensus on alternative measures the overall degree of agreement among experts when comparing each individual alternative against all others. There is a total of m alternatives, and the consensus on alternatives CA_i can be calculated as

$$C{A}_i={\displaystyle \frac{1}{m-1}}{\displaystyle \sum _{j=1,j\ne i}^m{C}_{ij}}$$

(30)

Subgroup level consensus considers the measurement of consensus levels both within and between subgroups. This includes intra-subgroup consensus CC_k and inter-subgroup consensus CB_kl:

$$C{C}_k={\displaystyle \frac{1}{\left|C_k\right|\left(\left|C_k\right|-1\right)/2}}{\displaystyle \sum _{p\in C_k}{\displaystyle \sum _{q\in C_k}sim\left(e_p,e_q\right)}}$$

(31)

$$C{B}_{kl}={\displaystyle \frac{1}{\left|C_k\right|\left|C_l\right|}}{\displaystyle \sum _{p\in C_k}{\displaystyle \sum _{q\in C_l}sim\left(e_p,e_q\right)}}$$

(32)

Overall decision-making consensus measures the comprehensive level of agreement on the entire decision-making issue, serving as the ultimate gauge of group opinion alignment. It can be categorized into simple overall consensus CR and weighted overall consensus CW. Simple overall consensus CR is a single aggregation of CA_i. Weighted overall consensus CW is calculated by comprehensively considering CC_k and CB_kl. CC_k is weighted according to its assigned weight, while CB_kl employs a conservative weighting strategy: it takes half of the smaller weight among the relevant subgroups. This prevents excessive amplification of high consensus among a few high-weight subgroups. Finally, normalization ensures values fall within the range [0, 1].

$$CR={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^{m}C{A}_i}$$

(33)

$$CW={\displaystyle \frac{{\displaystyle \sum _{k=1}^K{\omega }_k\times C{C}_k+}{\displaystyle \sum _{k=1}^K\frac{\min \left({\omega }_k,{\omega }_l\right)}{2}\times C{B}_{kl}}}{{\displaystyle \sum _{k=1}^K{\omega }_k+}{\displaystyle \sum _{k=1}^K\frac{\min \left({\omega }_k,{\omega }_l\right)}{2}}}}$$

(34)

All consensus measures above fall within the interval [0, 1]. In CRP, a predefined threshold δ must be set beforehand. If CW ≥ δ, the entire group achieves consensus, and the alternative selection process proceeds. Otherwise, the iterative process should be initiated. The value of δ depends on the specific problem being addressed. Setting a relatively high δ value (e.g., 0.9 or 0.85) yields superior consensus quality. In time-critical decision-making scenarios, δ can be appropriately lowered to improve consensus efficiency, thus allowing the group to rapidly reach a feasible consensus.

3.4. Adaptive Consensus Feedback Mechanism

The CRP is a dynamic iterative process designed to enhance consensus levels. When the group’s consensus level fails to reach a predefined threshold during consensus iteration, a feedback adjustment mechanism is activated to elevate consensus. This study proposes an adaptive feedback mechanism that integrates interactive and automated strategies to enhance group consensus levels. Specifically, when the consensus level is low, an interaction strategy is initiated, requiring the moderator and experts to negotiate and determine preference adjustments. Conversely, as consensus approaches a predefined threshold, it automatically adjusts expert preference values. This mechanism reduces time consumption while ensuring adequate expert participation.

To distinguish between high and low consensus levels, two consensus thresholds, δ₁ and δ₂, need to be established. Based on these thresholds, the group’s consensus level is categorized into three types: (1) CW < δ₁ indicates a low level of consensus. (2) δ₁ ≤ CW < δ₂ indicates a medium level of consensus. (3) δ₂ ≤ CW < δ indicates a high level of consensus.

When CW ≥ δ, the optimal solution is directly computed. When CW < δ, three distinct feedback adjustment mechanisms are proposed to enhance the level of consensus.

(1) 

Feedback adjustment mechanism for low consensus levels

At the outset of the CRP, decision-makers exhibited significant divergence in preferences, resulting in low consensus. When the overall consensus level falls within the preset range CW < δ₁, extensive adjustments are required to accelerate the convergence process. To maintain the enthusiasm and genuine willingness of decision-makers to participate, an interactive feedback strategy is adopted in this context. Under this interactive strategy framework, the first step involves identifying the alternative with the minimum consensus level in each subgroup; subsequently, decision-makers within the corresponding subgroup are required to adjust their preference values for the identified alternatives. To pinpoint the alternatives and their corresponding subgroups for intervention, an alternative-specific consensus threshold ξ₁ is established, defined as the average of the alternative consensus levels as ξ₁:

$${\xi }_1={\displaystyle \frac{1}{m\left(m-1\right)}}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^m{C}_{ij}}}$$

(35)

Identify alternatives with consensus levels below the threshold ξ₁, and generate directional guidance rules to offer specific recommendations. If $L_{\mathit{ij}}^{C_i}(p)$ < $L_{\mathit{ij}}^c(p)$, all experts in subgroup C_i should increase their evaluations of the alternative pair (x_i, x_j). Conversely, they should decrease their evaluations of this pair. If $L_{\mathit{ij}}^{C_i}(p)$ = $L_{\mathit{ij}}^c(p)$, all experts in subgroup C_i are not required to alter their evaluations of the corresponding alternative pair.

(2) 

Feedback adjustment mechanism for medium consensus level

At this stage, the overall consensus level reaches the preset range δ₁ ≤ CW < δ₂. The primary goal is to reduce both the number of experts required to revise their evaluations and the number of preference values needing adjustment. Where appropriate, an automatic adjustment strategy is activated to minimize discussion duration and coordination costs. Calculate the average consensus level ξ₂ across all current alternative pairs, using the same method as in Equation (35). Identify the pairs whose consensus levels fall below this threshold ξ₂. Subsequently, compute the proportion of pairs that require adjustment. If this proportion is less than 20%, implement an automatic adjustment strategy for the relevant experts, with the adjusted preference values derived from Equation (36). If the proportion exceeds 20%, an interactive adjustment approach shall be adopted, with directional recommendations provided in accordance with the procedure of the preceding stage.

$$P_a=P_i+\lambda \left(P_c-P_i\right)$$

(36)

$$\lambda =\min \left({\displaystyle \frac{{\sigma }_i}{d\left(P_i,P_c\right)}},1\right)$$

(37)

where λ is the adjustment coefficient, P_i and P_c represent the individual probabilistic linguistic preference relationship and the collective probabilistic linguistic preference relationship of the experts, respectively. σ_i denotes the predefined acceptable automatic adjustment range for the i-th expert, σ_i ∈ (0,1), signifies the “maximum authority” or “trust level” granted by the experts to the system for automatic adjustments. d (P_i, P_c) is the actual distance between individual preferences and collective preferences.

(3) 

Feedback adjustment mechanism for high consensus level

When the overall consensus level falls within the preset range δ₂ ≤ CW < δ, the primary goal of this stage is to achieve the final consensus threshold $\delta$ with maximum efficiency while preserving existing consensus outcomes. During this stage, the system operates entirely under an automated adjustment strategy. The adjustment process utilizes the maximum acceptable adjustment range ${\sigma }_i$, which represents the highest level of authority that experts grant to the system for automatic adjustments.

4. Identification and Management of Non-Cooperative Behavior

4.1. Identification of Non-Cooperative Behavior

In the decision-making process, experts who exhibit reluctance to adjust their preference values or who make changes in the opposite direction, are categorized as non-cooperative members. This section proposes a method to measure the degree of cooperation for each expert in every iterative round. By introducing a cooperation index (CI), we derive a metric to identify specific typologies of non-cooperative behavior.

Let TA denote the total adjustment amount of the recommended decision-maker in the t-th round of CRP. Let TP denote the deviation between the adjusted recommended preference value and the decision-maker’s actual preference value in the t-th round of CRP. Define the cooperation index CI as

$$C{I}_i^{\left(t\right)}=\left\{\begin{array}{l}1,\hspace{0.17em}\mathrm{if}\hspace{0.17em}T{A}_i^{\left(t\right)}=0\\ 1-{\displaystyle \frac{T{P}_i^{\left(t\right)}}{T{A}_t^{\left(t\right)}}},\hspace{0.33em}\mathrm{otherwise}\end{array}\right.$$

(38)

When ${\mathit{CI}}_i^{\left(t\right)}$ < 0, it indicates that the decision-maker exhibits non-cooperative behavior in the t-th round of the CRP.

4.2. Management of Non-Cooperative Behavior

In this study, a tiered management strategy was adopted for non-cooperative actors.

First level strategy: Intra-group consultation. The non-cooperative expert’s behavior and its subsequent impact on group consensus are disclosed to other members of the respective subgroup. This is followed by the initiation of a subgroup vote to decide whether automatic preference adjustment should be implemented for the expert in question. If more than two-thirds of the members approve, the automatic adjustment is executed with a magnitude of σ_i.

Second level strategy: Weight penalty. If a subgroup fails to reach a consensus (with the agreement rate falling below two-thirds), a weight penalty mechanism will be triggered. The decision weight for that subgroup is reduced using the weight update function in reference [34].

$$\omega {\left(C_i\right)}^{t+1}=\left\{\begin{array}{l}0.9\times \omega {\left(C_i\right)}^t,C{I}_r^{C_i,t}\in \left[-0.25,0\right]\\ 0.7\times \omega {\left(C_i\right)}^t,C{I}_r^{C_i,t}\in \left[-0.5,-0.25\right)\\ 0.5\times \omega {\left(C_i\right)}^t,C{I}_r^{C_i,t}\in \left[-0.75,-0.5\right)\\ 0.3\times \omega {\left(C_i\right)}^t,C{I}_r^{C_i,t}\in \left[-1,-0.75\right)\end{array}\right.$$

(39)

where ${\omega \left(C_i\right)}^t$ represents the weight of subgroup C_i in round t of discussion, and ${\omega \left(C_i\right)}^{t+1}$ represents the adjusted weight of subgroup C_i in round t + 1. After weight penalty application, all expert weights must be renormalized, calculated as in Equation (40), where $\widehat{\omega }{\left(C_i\right)}^{t+1} \in \left[0,1\right]$, $\sum _{i=1}^k\widehat{\omega }{\left(C_i\right)}^{t+1} = 1$.

$$\widehat{\omega }{\left(C_i\right)}^{t+1}={\displaystyle \frac{\omega {\left(C_i\right)}^{t+1}}{{\displaystyle \sum _{i=1}^k\omega {\left(C_i\right)}^{t+1}}}}$$

(40)

Third level strategy: Exit-delegation. If an expert’s cooperation index remains below 0 for three consecutive rounds and their subgroup consequently fails to reach a consensus, the exit-delegation protocol is activated. The expert will then exclude from remaining consensus-reaching iterations. Their decision-making weight will be redistributed proportionally among other subgroups based on respective consensus levels. Specifically, the subgroup demonstrating the highest consensus level will be allocated two-thirds of the delegated weight, while the subgroup with the second-highest consensus level will receive the remaining one-third.

4.3. Decision-Making Process

The steps of the proposed method are outlined below, and the overall decision-making flowchart is presented in Figure 1.

Step 1: Experts expressed their preferences using PLTS to construct the PLPR.

Step 2: Using the FGMM method, all experts are divided into K subgroups by Equations (21)–(26).

Step 3: The trust density of each subgroup is calculated by Equation (27), and then normalized by Equation (28) to obtain the final weight of each subgroup.

Step 4: Calculate the consensus on alternative pairs, consensus on alternatives, subgroup level consensus, and overall decision-making consensus using Equations (29)–(34).

Step 5: Based on the consensus level, select different strategies: If CW < δ₁, adopt the low consensus level feedback adjustment strategy; If δ₁ ≤ CW < δ₂, adopt the medium consensus level feedback adjustment strategy; If δ₂ ≤ CW < δ, adopt the high consensus level feedback adjustment strategy; If CW ≥ δ, proceed to Step 7.

Step 6: Calculate the cooperation index using Equation (38) to identify non-cooperative behavior, and select the corresponding level of behavioral management methods based on the degree of non-cooperation, where the weight penalty and weight renormalization calculations are as shown in Equations (39) and (40). After updating the weights, return to Step 3.

Step 7: Rank the alternative options. Calculate the CPV for each option using Equation (11), then compute its score and deviation using Equations (4)–(6) for each alternative’s CPV. Finally, rank the alternatives using the comparison rules.

Step 8: End.

5. Case Study

5.1. Case Background

China has made a solemn commitment to the international community to “achieve carbon peak before 2030 and carbon neutrality before 2060”. As a core sector in emission reduction efforts, the green and low-carbon transition of the transportation industry is of particular urgency and critical importance. Among the multitude of pathways toward deep decarbonization in the transportation sector, the large-scale adoption and advancement of new energy vehicles (NEV) is widely recognized as a core strategy for fulfilling the dual carbon goals. Simultaneously, curbing over-reliance on imported oil, strengthening national energy security capabilities, and advancing the shift in energy consumption structures toward diversification and localization (including electricity and hydrogen energy) has emerged as an inevitable choice for China’s energy strategy. Against this backdrop, identifying a forward-looking, feasible, and illustrative development pathway for NEV has emerged as a pivotal challenge. Anchored in global technological trajectories and the prevailing realities of China’s automotive industry, four representative and actionable alternative pathways are proposed:

Alternative x₁: Battery Electric Vehicle (BEV) Dominant Strategy. This strategy concentrates resources on prioritizing the development of battery electric vehicles, which offer inherent merits of full-lifecycle zero emissions and high energy efficiency. Nevertheless, it faces notable challenges, such as potential impacts on power grids and bottlenecks in battery recycling systems.

Alternative x₂: Plug-in Hybrid Electric Vehicle (PHEV) Transition Strategy. This approach utilizes PHEV as a bridge from traditional fuel-powered vehicles to full electrification. It boasts core advantages of mitigating driving range anxiety and capitalizing on existing infrastructure. However, a residual drawback persists, as this pathway inevitably generates a degree of tailpipe emissions during operation.

Alternative x₃: Fuel Cell Electric Vehicles (FCEV) Breakthrough Strategy. This strategy centers on hydrogen fuel cell vehicles, aiming to establish a hydrogen energy industrial chain covering the entire spectrum of production, storage, transportation, and refueling. It offers distinctive advantages of full-cycle zero emissions and rapid refueling performance. Nonetheless, it is confronted with prominent challenges, such as massive infrastructure investment requirements and elevated costs associated with green hydrogen production.

Alternative x₄: Multi-technology Integration Strategy. Adopting a technology-neutral stance, this approach simultaneously supports multiple technical pathways including BEV, PHEV, and FCEV, deferring the ultimate selection to market dynamics. It delivers a core advantage of mitigating technology lock-in risks. However, a significant drawback remains, as the dispersion of policy resources may impede the realization of economies of scale.

This study employed a questionnaire survey, inviting 20 experts {e₁, e₂, …, e₂₀} to evaluate the above four alternative pathways {x₁, x₂, x₃, x₄} using the PLTS method. The questionnaire content is presented in Appendix A.

5.2. Method Process and Numerical Results

Step 1: Using a questionnaire survey format, experts expressed their preferences for PLTS to construct the PLPR. Given the large volume of data, only the PLPR of expert 1 are presented in this paper, as shown in

Table 1. The PLPR of expert 1.

	x1	x2	x3	x4
x1	{s0 (1.0)}	{s−1 (0.4), s0 (0.0), s1 (0.1), s2 (0.0), s3 (0.5)}	{s−2 (0.2), s0 (0.0), s2 (0.8)}	{s1 (0.2), s0 (0.1), s1 (0.2), s2 (0.4), s3 (0.1)}
x2	{s3 (0.5), s2 (0.0), s1 (0.1), s0 (0.0), s1 (0.4)}	{s0 (1.0)}	{s2 (0.2), s0 (0.1), s1 (0.2), s2 (0.1), s3 (0.4)}	{s1 (0.1), s1 (0.3), s2 (0.6)}
x3	{s2 (0.8), s2 (0.2)}	{s−3 (0.4), s2 (0.1), s1 (0.2), s0 (0.1), s2 (0.2)}	{s0 (1.0)}	{s−2 (0.5), s−1 (0.5)}
x4	{s−3 (0.1), s2 (0.4), s1 (0.2), s0 (0.1), s1 (0.2)}	{s−2 (0.6), s1 (0.3), s1 (0.1)}	{s1 (0.5), s2 (0.5)}	{s0 (1.0)}

.

Steps 2: Using the FGMM, the 20 experts were divided into three subgroups via Equations (21)–(26). The clustering results is shown in Figure 2. The “MDS 1” and “MDS 2” representations in the figure are the coordinate axes after dimensionality reduction, which are used to approximate the similarity relationship between experts on a two-dimensional plane.

Steps 3: The trust density of each subgroup was calculated and normalized from Equations (27) and (28), yielding the trust density and initial weights for each subgroup, as shown in

Table 2. Subgroup weights based on trust density.

Subgroup	Number of Experts	Experts	Trust Density	Weight
C1	6	e2, e3, e4, e8, e18, e20	0.12	0.30
C2	7	e1, e6, e10, e13, e14, e15, e16	0.13	0.30
C3	7	e5, e7, e9, e11, e12, e17, e19	0.16	0.40

.

Step 4: Using Equations (29)–(34), sequentially calculate the alternative pairs, subgroups, and overall consensus level. The alternative-level and subgroup-level results are shown in Figure 3. Figure 3a shows the degree of consensus in expert preferences between each pair of alternatives, with warmer colors indicating a higher degree of consensus. Figure 3b shows the consensus level of each alternative. Figure 3c represents the level of consensus of each expert within the subgroup, and the internal consensus can explain the rationality of the subgroup division. Figure 3d shows the level of consensus between different subgroups to identify disagreements between subgroups. The final overall consensus CW = 0.316.

Steps 5 and 6: The preset parameters δ₁, δ₂ and δ are 0.6, 0.7 and 0.8, respectively. Enter the adaptive feedback consensus process with a maximum consensus round count t_m = 5.

First round: Since CW < δ₁, the system enters the feedback adjustment phase at a low consensus level. According to Equations (39) and (40), the alternative solutions requiring adjustment are as follows: C₁: (x₁, x₄), (x₂, x₃), (x₂, x₄); C₂; (x₂, x₃), (x₂, x₄); C₃: (x₁, x₂), (x₁, x₃), (x₁, x₄), (x₂, x₄). Adjust the identified alternatives according to the adjustment rules. Continuing with non-cooperative behavior detection, the result showed that no non-cooperative behavior was detected in this round. After adjustment, the new group consensus CW⁽¹⁾ = 0.616 did not reach the predetermined consensus and needs to enter the second round of feedback adjustment.

Second Round: δ₁ < CW⁽¹⁾ < δ₂, entering the feedback adjustment phase with medium consensus. The alternative options requiring adjustment include: C₁: (x₁, x₂), (x₁, x₃), (x₃, x₄); C₂: (x₁, x₂), (x₁, x₃), (x₁, x₄), (x₃, x₄); C₃: (x₂, x₃), (x₃, x₄). Since the proportion of pairs requiring adjustment exceeds 20%, an interactive adjustment strategy is employed to modify the identified pairs according to the adjustment rules. Subsequently, non-cooperative behavior detection is performed, with no instances detected in this round. The adjusted new group consensus CW⁽²⁾ = 0.743, failing to meet the predetermined consensus threshold, triggering a third round of feedback adjustment.

Third Round: δ₂ < CW⁽²⁾ < δ, entering the feedback adjustment phase with high consensus, adopting an automatic adjustment strategy. This round detected non-cooperative behavior from experts e₁ and e₁₁, ${\mathit{CI}}_{e_1}^{\left(3\right)}$ = −0.01, ${\mathit{CI}}_{e_{11}}^{\left(3\right)}$ = −0.05. Subsequently, intra-group consultations were conducted. Subgroups C₁ and C₃, which included the two non-cooperative experts, both voted through consultation. As more than two-thirds of members in each subgroup agreed to automatically adjust their preferences for these experts, an automatic adjustment management strategy was implemented for experts e₁ and e₁₁. After the adjustment, the group consensus CW⁽³⁾ = 0.782. Since the predetermined consensus was not achieved, the system entered the fourth round of feedback adjustment.

Fourth Round: δ₂ < CW⁽³⁾ < δ, entering the feedback adjustment phase with high consensus, adopting an automatic adjustment strategy. Non-cooperative behavior was detected in three experts: e₁, e₁₁ and e₁₅. Following internal deliberation within Subgroup C₁, only one-fifth of members agreed to automatically adjust preferences for expert e₁. This failed to meet the required two-thirds threshold. Consequently, Subgroup C₁ was subjected to a weight penalty strategy, resulting in a reduced weight of ω(C₁)⁴ = 0.27. Following intra-group consultation within subgroup C₃, only one-sixth of members agreed to automatically adjust preferences for expert e₁₁., which is fewer than two-thirds. Consequently, subgroup C₃ was subjected to a weight penalty strategy, resulting in a reduced weight of ω(C₃)⁴ = 0.35; Following deliberation within subgroup C₂, only one-third of members agreed to automatically adjust preferences for expert e₁₅. This also failed to reach the required two-thirds, thus triggering a weight penalty strategy for subgroup C₂. The adjusted weight is ω(C₂)⁴ = 0.28. After weight penalties, all expert weights are renormalized using Equation (40), and the group consensus is recalculated, yielding CW⁽⁴⁾ = 0.804. Final consensus is achieved, completing the adaptive consensus feedback process. The consensus convergence process is illustrated in Figure 4. It shows the consensus level after each iteration.

Step 7: Ranking of alternatives. The CPV, scores, and deviation results for each alternative, calculated using Equations (4)–(6) and (11), are shown in Figure 5. Figure 5a shows the scores of each alternative, and Figure 5b represents the deviation degree of each alternative. According to the comparison rules mentioned after Definitions 3 and 4, when scores differ, they are directly compared; when scores are equal, deviations are compared. The final ranking of alternatives is x₁ > x₂ > x₃ > x₄. The optimal alternative is alternative x₁.

5.3. Sensitivity Analysis

5.3.1. Number of Clusters

The number of clustering subgroups may exert an impact on both the decision-making process and its outcomes, accordingly. A sensitivity analysis was conducted to examine this relationship. As shown in

Table 3. Decision results under different cluster numbers.

K	Initial Consensus	Final Consensus	Iteration Round	Ranking of Alternatives
2	0.318	0.837	2	x1 > x2 > x3 > x4
3	0.316	0.804	4	x1 > x2 > x3 > x4
4	0.333	0.814	6	x1 > x2 > x3 > x4
5	0.255	0.808	2	x3 > x1 > x2 > x4
6	0.283	0.855	6	x1 > x2 > x3 > x4

, from the initial consensus to the final consensus, it can be seen that the final consensus level under all cluster quantity settings can reach the preset consensus threshold δ, Furthermore, the final consensus levels exhibit relative concentration (0.80–0.85), demonstrating the consensus feedback mechanism’s strong adaptability to the initial grouping structure. Additionally, a distinct nonlinear relationship exists between the number of clusters and the number of iteration rounds. When the number of clusters is two or five, consensus thresholds are achieved after only two rounds of iteration, demonstrating high convergence efficiency. When the number of clusters is three, four, and six, the corresponding number of iterations required is four, six, and six. This phenomenon indicates that specific cluster structures can significantly enhance consensus-building efficiency. The exceptionally high efficiency observed with five clusters may stem from this grouping structure coincidentally forming a balanced preference distribution, thereby reducing coordination costs between subgroups. In this study, the number of clusters K = 3 was selected as the number of clusters to balance convergence efficiency and decision quality. Finally, analyzing the ranking stability, the order of alternatives remained fully consistent (x₁ > x₂ > x₃ > x₄) when the number of clusters was two, three, four, or six, demonstrating strong robustness in the decision results. However, when the number of clusters is set to five, a change in ranking occurs (x₃ > x₁ > x₂ > x₄). This is because the grouping structure alters the dynamic equilibrium of preference aggregation. Comprehensive analysis indicates that the method proposed in this study exhibits strong parameter robustness, generating consistent and logically coherent decision outcomes across the majority of clustering configurations.

5.3.2. Consensus Threshold

To classify group consensus levels, three consensus thresholds (δ₁, δ₂, and δ) are preset, with consensus categorized into low, medium, and high tiers, accordingly. By varying the magnitude of these thresholds, the influence of such parameter adjustments on convergence performance and consensus quality is analyzed. The analysis results are shown in

Table 4. Iterative rounds and final consensus under different threshold combinations.

δ1, δ2, δ	Iteration Round	Final Consensus
0.5, 0.6, 0.7	3	0.747
0.6, 0.7, 0.8	4	0.804
0.7, 0.8, 0.9	5	0.911

. When the threshold settings were upgraded from the more lenient combination (0.5, 0.6, 0.7) to the stricter combination (0.7, 0.8, 0.9), the number of iteration rounds correspondingly increased from three rounds to five rounds. This phenomenon indicates that stringent consensus standards necessitate a more thorough preference adjustment process. From the perspective of the final consensus level, as the threshold strictness increases, the consensus quality shows a continuous improvement trend, rising from 0.747 to 0.911. This validates the decisive impact of threshold settings on consensus quality. Performance variations across different threshold combinations enable specific advantages for diverse application scenarios: In decision-making scenarios with high time sensitivity, relaxed thresholds (0.5, 0.6, 0.7) can achieve an acceptable level of consensus within three rounds; For critical decisions prioritizing quality, strict thresholds (0.7, 0.8, 0.9) ensure high-quality consensus outcomes but may incur additional computational costs. The intermediate thresholds (0.6, 0.7, 0.8) provide balanced solutions, achieving high-quality consensus at a moderate iteration cost. Therefore, this study adopts the intermediate thresholds (0.6, 0.7, 0.8) as the data input for the case analysis.

5.4. Comparative Analysis

5.4.1. Comparison of Clustering Methods

To overcome the limitations of single clustering methods, this study employs a hybrid clustering approach that integrates FCM with a GMM. To demonstrate the advantages of this method, the consensus levels obtained using the traditional K-means method are compared with those from the proposed FGMM. As shown in Figure 6, this further demonstrates the stability advantage of the FGMM method. The traditional K-means method exhibits significant fluctuations in consensus levels across different numbers of clusters, particularly when the number of clusters is set to 4 and 6. In these cases, the final consensus level reaches “1”, indicating a hard consensus level, which is not suitable for practical application scenarios. The final consensus level achieved using the FGMM method consistently remained around 0.8, demonstrating strong stability. This suggests that the traditional K-means method may suffer from instability and over-idealization, though it retains computational efficiency advantages under simple clustering structures. Overall, K-means pursues rapid yet potentially superficial consensus, while FGMM focuses on building deep and stable consensus. By integrating probabilistic membership degrees and Mahalanobis distances, FGMM precisely captures subtle differences in expert preferences. Although it requires more iteration rounds, this approach ensures controllable consensus quality.

5.4.2. Comparison of Existing Literature

Through systematic review and comparative analysis of the representative literature in the field of GDM, the GDM method based on FGMM and adaptive consensus feedback proposed in this paper demonstrates its unique advantages and innovation in multiple key dimensions. As shown in

Table 5. Comparison with existing representative literature.

References	Language Preference Expression	Clustering Method	Consensus Feedback Process	Non-Cooperative Behavior
[5]	Hesitant fuzzy sets	Two-stage clustering	×	×
[19]	Incomplete fuzzy preferences	Improved K-means	Two-stage feedback	×
[27]	Intuitionistic fuzzy numbers	Louvain dynamic clustering	Adaptive feedback	√
[41]	PLTS	×	Two-stage feedback	×
[17]	PLTS	Natural clustering	Two-stage feedback	√
[40]	×	Two-stage clustering	Dynamic feedback	×
This paper	PLTS	FGMM	Adaptive feedback	√

Note: “×” and “√” indicate “no” or “yes” for the inclusion of this method in the reference.

, compared to the hesitant fuzzy sets in reference [5], the incomplete fuzzy preferences in reference [19], and the intuitionistic fuzzy numbers in reference [27], probabilistic linguistic sets demonstrate a superior capacity to accurately characterize the cognitive uncertainty of decision-makers operating within complex decision-making environments. Regarding clustering methods, for the improved K-means in reference [19], two-stage clustering in references [5,40], Louvain dynamic clustering in reference [27], and natural clustering in reference [17], the FGMM not only effectively addresses the boundary rigidity issue of traditional hard clustering methods in group partitioning but also enhances the stability of the overall decision-making system. In addition, the proposed improved adaptive consensus feedback mechanism, when contrasted with the two-stage feedback adopted in references [17,19,41] and the dynamic feedback in reference [40], achieves the matching of feedback strength and consensus state through the organic combination of multi-level consensus measurement and dynamic adjustment strategy. Finally, among the listed references, only references [27,41] address the management of non-cooperative behavior. The three-tiered behavioral management strategy proposed in this paper avoids the limitations of a one-size-fits-all approach while enabling targeted management of expert behavior to enhance decision-making quality.

In summary, the proposed framework highly synthesized four components: a probabilistic linguistic terminology set (PLTSs), the FGMM, adaptive consensus feedback, and non-cooperative behavior management. This integration not only addresses the adaptability limitations of existing methods in complex scenarios but also enables precise modeling of expert preference distributions through the fusion of clustering methods. Furthermore, it establishes intelligent regulation within the consensus feedback process, effectively enhancing the decision-making efficiency and quality of group decisions in complex environments.

6. Conclusions

In this study, an integrated framework is constructed by combining PLTSs, FGMM, adaptive consensus feedback mechanism, and non-cooperative behavior management. Specifically, the proposed fusion method introduces a FGMM to improve the flexibility and robustness of clustering boundaries via a dual mechanism involving membership degrees and probability distributions. Furthermore, an adaptive feedback mechanism integrating interactive and automated strategies is developed, together with a three-tier behavior management strategy. This approach not only ensures in-depth expert participation during the low-consensus phase but also enhances adjustment efficiency in the high-consensus phase, thereby effectively alleviating the negative impact of non-cooperative behaviors on the consensus-reaching process. Finally, a case study on the strategic pathways of new energy vehicle development validates the excellent convergence and stability of the proposed method. Notably, after four rounds of iteration, the overall consensus level rose from the initial value of 0.316 to 0.804, satisfying the preset threshold.

Sensitivity analysis and comparative results indicate (1) The number of clusters affects consensus convergence speed, yet consistent decision outcomes are achievable under most clustering configurations, underscoring the structural stability of the model. While certain specific grouping structures may alter the dynamic balance of preference aggregation, it is recommended in practice to compare multiple group cluster numbers and select the most robust consensus convergence result as the basis for final decision-making. (2) Threshold settings exhibit scenario adaptability. For decision-making scenarios with high timeliness requirements, a relaxed threshold can achieve an acceptable level of consensus in fewer rounds; whereas for critical decisions prioritizing quality, a strict threshold can ensure high-quality consensus results, but may require additional computational costs. This demonstrates the model’s flexibility in practical applications. (3) Compared to traditional K-means methods, the proposed FGMM method demonstrates superior performance in consensus stability and quality control. Compared to existing research, the proposed framework exhibits greater innovation and systematicity.

Based on the findings and limitations of this study, future research may be expanded in the following directions: First, while this paper mentions trust relationships, it has not yet deeply coupled dynamic trust evolution mechanisms with the consensus process. Subsequent work could explore incorporating dynamic trust networks into both the clustering and feedback phases. Second, the current model relies on a unified probabilistic language terminology set. Future research may consider introducing multi-granularity or non-uniform language terminology to accommodate more diverse types of decision-makers. Finally, there are a certain number of preset parameters in the model. In the future, it could be considered to study methods based on historical data or combined with machine learning to adaptively calibrate or optimize key parameters in the model, reducing subjectivity.