Prospect theory-based large-scale group decision-making method with heterogeneous preference relations

Abstract

Large-scale group decision-making (LSGDM) issues are increasingly prevalent in modern society across various domains. The preference information has emerged as a widely adopted approach to tackle LSGDM problems. However, a significant challenge lies in facilitating consensus among decision-makers (DMs) with diverse backgrounds while considering their hesitation and psychological behavior. Consequently, there is a pressing need to establish a novel model that enables DMs to evaluate alternatives with heterogeneous preference relations (HPRs). To this end, this research presents a new consensus-building method to address LSGDM problems with HPRs. First, a novel approach for solving collective priority weight is introduced based on cosine similarity and prospect theory. In particular, a new cosine similarity measure is defined for HPRs. Subsequently, a consensus index is provided to gauge the consensus level among DMs by considering their psychological behavior and risk attitudes. Further, a consensus-reaching model is developed to address LSGDM with HPRs. Finally, an instance of supplier selection is presented to demonstrate the practicality and efficacy of the proposed method.

Keywords

Large-scale group decision-making prospect theory heterogeneous preference relations consensus reaching risk attitudes

1 Introduction

With the increasingly complex decision-making environment, large-scale group decision-making (LSGDM), which involves a mass of decision-makers (DMs), has attracted increasing attention from scholars [9 , 47]. As an effective way to express DMs’ opinions, preference relation has been widely used in LSGDM [10 , 63]. DMs express their evaluation opinions utilizing a preference relation by comparing the alternatives in pairs, which is a more precise approach than direct assessments. Given the involvement of abundant individuals with diverse knowledge, experience, and professional backgrounds, different forms of preference relations may be utilized. As a result, the application of HPRs in LSGDM has emerged as a significant research topic in recent years [6 , 62].

However, there has been limited research in the field of heterogeneous LSGDM that considers the hesitation of DMs. Furthermore, most research in this area has been focused on a single preference structure, such as preference order, utility function, fuzzy preference relation (FPR), linguistic preference relation (LPR), or multiplicative preference relation (MPR). Unfortunately, these structures fail to adequately capture the hesitation and uncertainty that DMs experience in complex decision-making environments. In reality, due to the constraints imposed by their knowledge and experience, DMs often struggle to provide accurate evaluation information, especially when multiple factors need to be considered in complex decision-making problems. FPR and MPR only enable DMs to convey a single degree of preference, which fails to capture the nuanced qualitative evaluation information and accompanying hesitation. Especially for LSGDM problems involving a mass of DMs, if the hesitation of DMs is ignored, the decision results may deviate from the original intention of DMs.

As is known to all, in traditional group decision-making (GDM), various preference relations reflecting DMs’ hesitancy have been deeply studied. The most commonly used ones include hesitant fuzzy linguistic preference relations (HFLPRs) [65], intuitionistic fuzzy preference relations (IFPRs) [35], and interval-valued fuzzy preference relations (IVFPRs) [46]. Among this, HFLPRs can express DMs’ qualitative assessment information along with their hesitancy. IFPRs can gauge DMs’ preference degree from both positive and negative perspectives. Lastly, IVFPRs allow for the expression of DMs’ preference degree within a certain range. These three preference relations comprehensively cover the hesitancy and uncertainty that DMs may exhibit in most scenarios. Thus, this research employs HFLPRs, IFPRs, and IVFPRs to measure DMs’ hesitancy and uncertainty within the LSGDM environment. This is the first point of interest in this research.

Since evaluation opinions are given by DMs subjectively, the psychological behavior and risk attitudes of DMs have a significant influence on decision results. As an effective tool to measure the psychological behavior of DMs, prospect theory has been extensively applied in GDM problems [17 , 51]. Nevertheless, the DMs’ psychological behavior is seldom considered in existing heterogeneous LSGDM. Actually, even in complex LSGDM problems involving multiple DMs, the psychological behavior of the DMs still holds significant influence over the decision outcomes. Varied psychological behaviors and attitudes towards risk will result in divergent final evaluation opinions, particularly when DMs demonstrate hesitance and uncertainty during the provision of initial evaluation opinions or in the consensus-reaching process. Additionally, the psychological behavior of DMs during the feedback adjustment phase also affects whether they can accept feedback suggestions. Therefore, this study incorporates prospect theory into heterogeneous LSGDM problems with three distinct preference evaluation forms to align with the psychological behavior of DMs and enhance the accuracy of decision results.

As a crucial component of LSGDM, the consensus-reaching process significantly enhances decision result reliability, making it more widely accepted and recognized by DMs. Currently, the consensus-reaching model in LSGDM has garnered extensive attention [6 , 60]. Tian et al. [43] presented an adaptive consensus-building process to detect the non-cooperative behavior of DMs in heterogeneous LSGDM. Tang et al. [41] gave a consensus-reaching method with hybrid strategies by considering the cohesion of subgroups. Liu et al. [20] built a consensus model by detecting and managing the overconfidence behavior of DMs in an LSGDM environment with self-confidence fuzzy preference relations. Chao et al. [6] proposed a consensus-building model by considering non-cooperative behavior and heterogeneous preference structure in LSGDM and applied it to the financial inclusion project. The management method of personalized individual semantics and consensus in linguistic distribution LSGDM was provided by Xiao et al. [55]. Based on the interactive information feedback mechanism, Nie et al. [27] constructed a consensus-reaching process that considered the non-support degree for minority opinions.

In the consensus-reaching process, a consensus measure is a crucial tool for gauging the level of agreement among DMs [30]. Similarity measures are commonly used to assess the degree of consensus since they efficiently obtain the collective priority vector [5 , 43]. Accordingly, this paper utilizes the cosine similarity measure to evaluate HPRs’ similarity and effectively assess the level of consensus.

The consensus-building models mentioned above effectively address LSGDM problems in various environments, consequently contributing to the advancement of consensus theory. Nonetheless, the study of heterogeneous LSGDM is still at a nascent stage, and the current methods are subject to certain limitations:

(1) Due to the intricate nature of decision-making problems and the constraints posed by DMs’ knowledge and experience, it is natural for DMs to encounter hesitancy and uncertainty when evaluating alternatives. Various fuzzy sets have been proposed to capture and represent such hesitancy and uncertainty [2 , 66]. However, current research on heterogeneous LSGDM inadequately considers DMs’ hesitance. Furthermore, most methods only focus on preference structures with a single membership (such as FPRs and MPRs).

(2) In the LSGDM process, since DMs typically provide subjective decision opinions, the decision results are significantly affected by the psychological behavior and risk attitudes of DMs. Regrettably, current heterogeneous LSGDM methods seldom consider DMs’ psychological behavior and risk attitudes.

Clearly, the existing methods need to be optimized and improved. To this end, this research is dedicated to solving the problem of heterogeneous LSGDM by considering the hesitation, psychological behavior, and risk attitudes of DMs.

With the above discussion, the primary contributions of this article are summarized as follows:

(1) Upon prospect theory, a heterogeneous LSGDM method is proposed that considers the psychological behavior of DMs.

(2) Cosine similarity is employed to measure the consensus level between heterogeneous preference information, which can quickly identify DMs who need adjustment, thereby enhancing consensus efficiency.

(3) A flexible feedback adjustment mechanism is designed, which provides multiple adjustment options for DMs while reflecting their psychological behavior and risk attitude.

The rest of this paper is organized as follows: Section 2 recalls some basic concepts and introduces prospect theory, similarity measure, and three kinds of preference relations (namely, HFLPR, IFPR, and IVFPR). The prospect value function based on three preference relations is defined, and the method for determining the collective priority vector is presented in Section 3. Section 4 establishes the consensus-building model under the heterogeneous preference environment and outlines the general framework. Section 5 provides the application case of the proposed method in supplier selection and offers a comparative analysis. Finally, Section 6 is the concluding remarks.

2 Preliminaries

To accommodate the uncertain preferences of DMs in various fields, three classic preference relations, namely HFLPR, IFPR, and IVFPR, have been employed to measure the hesitancy of DMs. These three preference relations are highly comprehensive as they encompass three well-established expressions of uncertainty: linguistic, interval value, and intuitionistic. It is noteworthy that this research focuses on decision-making problems in uncertain environments, and the three presented preference relations can readily be expanded to incorporate other forms of preference relations that can effectively represent uncertainty, such as hesitant fuzzy preference relations, probabilistic linguistic preference relations, uncertain linguistic preference relations, etc.

Let X = {x₁, x₂, ⋯ , x_n} indicates the collection of comparison objects throughout the text.

2.1 Heterogeneous preference relations

Definition 2.1 [65]. Let S = {s₀, s₁, ⋯ , s_2τ} be a linguistic term set, and H_s be its ordered finite subset, that is, the hesitant fuzzy linguistic term set (HFLTS) on S, the preference relation B = (b_ij) _n×n defined by the HFLTS is called the HFLPR, if it satisfies:

1) $b_{ij}^{σ (l)} + b_{ji}^{σ (l)} = s_{2 τ}, l = 1, 2, \dots, | b_{ij} |$ ;

2) |b_ij| = |b_ji|; 3) b_ii = s_τ;

4) $b_{ij}^{σ (l)} < b_{ij}^{σ (l + 1)}$ ; 5) $b_{ji}^{σ (l + 1)} < b_{ji}^{σ (l)}$ .

where b_ij ∈ H_s, b_ij is the hesitancy degree when x_i is preferred over x_j, |b_ij| is the cardinality of b_ij, and $b_{ij}^{σ (l)}$ is the lth linguistic term in b_ij.

Definition 2.2 [35]. An IFPR on the object set X = {x₁, x₂, ⋯ , x_n} is represented by a matrix C = (c_ij) _n×n, where c_ij = (u_ij, v_ij) is an intuitionistic fuzzy number with the conditions: u_ij, v_ij ∈ [0, 1] , u_ij + v_ij ≤ 1, u_ij = v_ji, u_ii = v_ii = 0.5, π_ij = 1 - u_ij - v_ij, for all i, j = 1, 2, ⋯ , n. u_ij represents the preference degree of the object x_i over x_j; v_ij denotes the preference degree of the object x_j over x_i; and π_ij indicates the indeterminacy degree between x_i and x_j, which can be interpreted as the information that the decision maker is unable or unwilling to give due to the lack of relevant experience and knowledge.

Definition 2.3 [46]. An IVFPR on the set X = {x₁, x₂, ⋯ , x_n} is denoted by a matrix R = (r_ij) _n×n, where $r_{ij} = [{\underline{r}}_{ij}, {\bar{r}}_{ij}]$ is an interval number, indicating that the preference degree of alternative x_i over x_j is between ${\underline{r}}_{ij}$ and ${\bar{r}}_{ij}$ , and it meets the conditions: $0 < {\underline{r}}_{ij} \leq {\bar{r}}_{ij} < 1$ , ${\underline{r}}_{ij} + {\bar{r}}_{ji} = 1, {\underline{r}}_{ii} = {\bar{r}}_{ii} = 0.5$ for all i, j = 1, 2, ⋯ , n.

The above three preference relations can better reflect DMs’ hesitation and uncertainty and have been widely employed in various decision problems. Kahneman and Tversky [13] proposed the prospect theory to better align with DMs’ original opinions and account for their hesitation and psychological behavior.

2.2 Prospect theory

Definition 2.4 [13]. Prospect theory, established by Kahneman and Tversky, is currently an important theory to describe the decision maker’s psychological behavior under risk and uncertainty. The value function can be mathematically formulated as the following expression:

$v (z) = {\begin{matrix} z^{α}, If z \geq 0 \\ - ρ (- z)^{β}, If z < 0 \end{matrix}$ (1) where z indicates the difference between the decision value and the reference point, v (z) represents the prospect value, α is the risk-seeking coefficient and β is the risk-averse coefficient, ρ > 1 denotes the loss averse coefficient and means that decision maker is more sensitive to the losses than to equal gains. In this paper, the parameter values determined by Kahneman and Tversky[13], namely α = β = 0.88, ρ = 2.25, are adopted.

2.3 Cosine similarity measure

In the research of LSGDM based on preference relation, the cosine similarity measure is often used to solve the priority vector of preference relation due to its good properties. In this study, the cosine similarity measure is adopted to measure the similarity of HPRs and compute the collective priority vector.

Definition 2.5 [34]. Let a_j = (a_1j, a_2j, …, a_nj) be two non-negative n-vectors(j = 1, 2), then the cosine similarity measure between a₁ and a₂ can be denoted as follows: $s (a_{1}, a_{2}) = \frac{\sum_{i = 1}^{n} a_{i 1} a_{i 2}}{\sqrt{\sum_{k = 1}^{n} a_{k 1}^{2}} \sqrt{\sum_{k = 1}^{n} a_{k 2}^{2}}} 2$ (2) Obviously, the value of s (a₁, a₂) is between 0 and 1, and the closer the value of s (a₁, a₂) is to 1, the higher the similarity between a₁ and a₂ is.

Due to the inherent structural differences of HPRs, it is challenging to evaluate the similarity of different preference relations directly. Thus, this research adopts a uniform transformation of HFLPRs and IFPRs into IVFPRs to measure the similarity of HPRs. The conversion methods presented in [49] and [52] are employed as follows:

(1) Let S = {s₀, s₁, ⋯ , s_2τ} be a linguistic term set, then hesitant fuzzy linguistic element h = {s_a, s_b, ⋯ , s_c} can be transformed into interval-valued fuzzy number r = [a/2τ, c/2τ], where 0 ≤ a < b < c ≤ 2τ [49]. Thus, the corresponding HFLPR H = (h_ij) _n×n is converted to IVFPR $R^{h} = (r_{ij}^{h})_{n \times n}$ .

(2) Let c_ij = (u_ij, v_ij) is an intuitionistic fuzzy number, then the corresponding isomorphic interval-valued fuzzy number is $r_{ij}^{c} = [u_{ij}, 1 - v_{ij}]$ [52]. Therefore, IVFPR $R^{c} = (r_{ij}^{c})_{n \times n}$ can be transformed from IFPR C = (c_ij) _n×n.

Remark 1. As for the intuitionistic fuzzy number c_ij = (u_ij, v_ij), considering that it has the property of u_ij = v_ji, v_ij = u_ji, the transformation of its non-membership [v_ij, 1 - u_ij] is equivalent to [u_ji, 1 - v_ji]. Thus, the upper and lower triangles of IVFPR R^c correspond to the membership and non-membership degrees of IFPR C, respectively.

After getting the uniform IVFPR, we present the definition of cosine similarity between IVFPRs.

Definition 2.6 Let $R_{1} = (r_{ij}^{1})_{n \times n}$ and $R_{2} = (r_{ij}^{2})_{n \times n}$ be two IVFPRs, where $r_{ij}^{1} = [r_{ij}^{1 -}, r_{ij}^{1 +}]$ , $r_{ij}^{2}$

$= [r_{ij}^{2 -}, r_{ij}^{2 +}]$ , then the cosine similarity between R₁ and R₂ is as follows: $s (R_{1}, R_{2}) = λ s (R_{1}^{-}, R_{2}^{-}) + (1 - λ) s (R_{1}^{+}, R_{2}^{+}) 3$ (3) where $s (R_{1}^{-}, R_{2}^{-}) = \frac{1}{n} \cdot \sum_{j = 1}^{n} \frac{\sum_{i = 1}^{n} r_{ij}^{1 -} r_{ij}^{2 -}}{\sqrt{\sum_{k = 1}^{n} (r_{kj}^{1 -})^{2}} \sqrt{\sum_{k = 1}^{n} (r_{kj}^{2 -})^{2}}}$ (4) $s (R_{1}^{+}, R_{2}^{+}) = \frac{1}{n} \cdot \sum_{j = 1}^{n} \frac{\sum_{i = 1}^{n} r_{ij}^{1 +} r_{ij}^{2 +}}{\sqrt{\sum_{k = 1}^{n} (r_{kj}^{1 +})^{2}} \sqrt{\sum_{k = 1}^{n} (r_{kj}^{2 +})^{2}}}$ (5) λ ∈ [0, 1] is the similarity parameter, set to 0.5 in this article without loss of generality.

Fig. 1

The value functions $v_{ij}^{-}$ and $v_{ij}^{+}$ .

3 Solving the collective priority vector

The ultimate purpose of LSGDM is to select the best alternative, so a method to address the collective priority vector is presented by adopting the prospect theory and HPRs.

3.1 HPRs-based prospect value function

Since both HFLPR and IFPR can be transformed into IVFPR in essence (HFLPR is transformed by the envelope of its hesitant fuzzy linguistic element [32], while IFPR and IVFPR are mathematically and multiplicatively transitivity isomorphisms [52], the specific conversion method is given in Section 2.3), the HPRs are uniformly transformed into IVFPRs and their prospect value (PV) functions are presented in this paper.

Definition 3.1 Let R = (r_ij) _n×n be an IVFPR, where $r_{ij} = [{\underline{r}}_{ij}, {\bar{r}}_{ij}]$ . Considering that the decision-maker is sensitive to the endpoint value of the given interval information, we take the left and right endpoint values as the reference points, respectively, and the prospect value functions are shown as follows: $v_{ij}^{-} = {\begin{matrix} (x_{ij} - {\underline{r}}_{ij})^{α}, x_{ij} \geq {\underline{r}}_{ij} \\ - ρ ({\underline{r}}_{ij} - x_{ij})^{β}, x_{ij} < {\underline{r}}_{ij} \end{matrix}$ $v_{ij}^{+} = {\begin{matrix} (x_{ij} - {\bar{r}}_{ij})^{α}, x_{ij} \geq {\bar{r}}_{ij} \\ - ρ ({\bar{r}}_{ij} - x_{ij})^{β}, x_{ij} < {\bar{r}}_{ij} \end{matrix}$ where $x_{ij} \in [{\underline{r}}_{ij}, {\bar{r}}_{ij}]$ represents the expected preference value of the decision-maker for the uncertainty interval $[{\underline{r}}_{ij}, {\bar{r}}_{ij}]$ . Actually, since $x_{ij} \geq {\underline{r}}_{ij}$ and $x_{ij} < {\bar{r}}_{ij}$ are always true, the Equation (6) can be simplified to $v_{ij}^{-} = (x_{ij} - {\underline{r}}_{ij})^{α}, v_{ij}^{+} = - ρ ({\bar{r}}_{ij} - x_{ij})^{β}$ , In this paper, we set $x_{ij} = (1 - η) {\underline{r}}_{ij} + η {\bar{r}}_{ij}$ , where η ∈ [0, 1], then the corresponding value of x_ji is $η {\underline{r}}_{ji} + (1 - η) {\bar{r}}_{ji}$ , which satisfies the condition that x_ij + x_ji = 1.

To intuitively present the psychological behavior of DMs within the given interval $[{\underline{r}}_{ij}, {\bar{r}}_{ij}]$ , the schematic diagram of the prospect value functions are presented as follows:

The figure on the left represents the value function $v_{ij}^{-}$ , where the reference point is the left endpoint ${\underline{r}}_{ij}$ of the interval. As a result, the right half-axis portion is considered. On the contrary, the figure on the right illustrates the value function $v_{ij}^{+}$ , with the reference point being the right end of the interval, thus focusing on the left part. It is observed that when the value of x_ij approaches the midpoint of the interval, the sensitivity of a decision-maker progressively diminishes. This aligns with people’s intuitive perception.

Note that $v_{ij}^{-}$ indicates the expected gain relative to reference point ${\underline{r}}_{ij}$ , which is invariably positive, while $v_{ij}^{+}$ represents the expected loss concerning reference point ${\bar{r}}_{ij}$ , which is always negative. Thus, preference values corresponding to reference points ${\underline{r}}_{ij}$ and ${\bar{r}}_{ij}$ can be expressed as $o_{ij}^{-} = {\underline{r}}_{ij} + v_{ij}^{-}$ and $o_{ij}^{+} = {\bar{r}}_{ij} + v_{ij}^{+}$ , respectively. By combining the risk preferences of DMs for different reference points, the comprehensive preference value for x_i over x_j can be set as: $o_{ij} = θ o_{ij}^{-} + (1 - θ) o_{ij}^{+}$ where θ ∈ [0, 1] is the risk attitude parameter of DMs. Since the value of o_ij may be negative, in order to facilitate the solution, this paper normalized it to: $o_{ij}^{N} = \frac{e^{o_{ij}}}{e^{o_{ij}} + e^{o_{ji}}}$ Clearly, the conditions $0 < o_{ij}^{N} < 1, o_{ij}^{N} + o_{ji}^{N} = 1, o_{ii}^{N} = 0.5$ are always true, so matrix $O = (o_{ij}^{N})_{n \times n}$ completely satisfies the property of the fuzzy preference relation [28]. Therefore, the comprehensive preference value can be utilized to define the consistency of the IVFPR.

Definition 3.2 Let R = (r_ij) _n×n be an IVFPR and $O = (o_{ij}^{N})_{n \times n}$ be its corresponding normalized comprehensive preference matrix. If we have $o_{ij}^{N} = \frac{w_{i}}{w_{i} + w_{j}}$ then R is called multiplicatively consistent, where w = (w₁, w₂, ⋯ , w_n) is the priority vector of O, satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

An approach for computing the collective priority vector can be acquired by combining the cosine similarity measure with the normalized comprehensive preference matrix of HPRs.

3.2 The solution of priority vector

Assume that the preference relations given by each decision-maker are multiplicatively consistent, and there exists a collective priority vector $w^{'} = (w_{1}^{'}, w_{2}^{'},$ $\dots, w_{n}^{'})$ equal to all the individual priority vectors. Then, the following equation can be established based on the cosine similarity measure. $s (p_{j}, w^{'}) = \frac{\sum_{i = 1}^{n} p_{ij} w_{i}^{'}}{\sqrt{\sum_{i = 1}^{n} p_{ij}^{2}} \sqrt{\sum_{i = 1}^{n} {w^{'}}_{i}^{2}}} = 1$ where $p_{ij} = \frac{o_{ij}^{N}}{1 - o_{ij}^{N}} = \frac{w_{i}}{w_{j}}$ , and p_j = (p_1j, p_2j, ⋯ , p_nj) is the column vector of matrix P = (p_ij) _n×n.

However, finding a priority vector that fully embodies the preferences of all DMs can be challenging in practical GDM problems. Hence, our goal is to ensure that the value of s (p_j, w′) approaches 1, ensuring that the individual priority vector closely aligns with the collective priority vector. To this end, an objective programming model is constructed to derive a priority vector that best reflects the individual preferences as follows: max $Z = \sum_{k = 1}^{Q} σ^{k} \sum_{j = 1}^{n} \sum_{i = 1}^{n} {\bar{p}}_{ij}^{k} {\bar{w}}_{i}$

$s . t . {\sum i = 1 n ({\bar{w}}_{i})^{2} = 10 \leq {\bar{w}}_{i} \leq 1$ (6) where ${\bar{p}}_{ij}^{k} = \frac{p_{ij}^{k}}{\sqrt{\sum_{l = 1}^{n} (p_{lj}^{k})^{2}}}$ , ${\bar{w}}_{i} = \frac{w_{i}^{'}}{\sqrt{\sum_{l = 1}^{n} {w^{'}}_{i}^{2}}}$ , σ^k (k = 1, 2, ⋯ , Q) is the kth decision-maker weight with the condition 0 ≤ σ^k ≤ 1, and Q is the total number of DMs.

The value of ${\bar{w}}_{i}$ can be obtained by solving the Model (11). Further, the collective priority vector $w_{i}^{'} = \frac{{\bar{w}}_{i}}{\sum_{i = 1}^{n} {\bar{w}}_{i}}$ can be derived by combining with constraint $\sum_{i = 1}^{n} w_{i}^{'} = 1$ (namely, $\sum_{i = 1}^{n} {\bar{w}}_{i} \sqrt{\sum_{l = 1}^{n} {w^{'}}_{i}^{2}} = 1$ ).

Remark 2. Model (11) presents the solution method of collective priority vector in a heterogeneous preference environment, which comprehensively considers the hesitance, psychological behavior, and risk attitudes of DMs, thus improving the reliability of decision results. It is easy to prove that Model (11) has a unique analytical solution, which avoids the aggregation operation of individual preference relations and simplifies the LSGDM process to a certain extent.

4 Consensus-building process

This section can be divided into two parts. The first part utilizes the fuzzy C-means algorithm [7] to cluster DMs with similar opinions. The second part delineates the methodology for measuring the consensus level in a heterogeneous preference environment and establishes the consensus-reaching model.

4.1 Fuzzy C-means clustering algorithm

Compared with GDM, a significant feature of LSGDM is the large number of DMs involved. Therefore, in LSGDM, DMs should be clustered first to simplify the process of consensus-reaching and improve decision efficiency. At present, many clustering algorithms have been applied to LSGDM [11 , 54], among which the K-means algorithm and fuzzy C-means algorithm are the most commonly adopted. Considering that there is a certain similarity among heterogeneous preference opinions of DMs, a preference opinion is allowed to belong to different clusters with different membership degrees. Therefore, based on the similarity measure, the fuzzy C-means algorithm with certain fault tolerance is adopted in this paper. The specific steps are as follows:

(1) Determine the number of clusters: in general, the clustering number N is random and can be determined according to the actual decision-making needs.

(2) Select the initial center point: in order to avoid the sensitivity of the random selection method to the initial center point. This paper uses the average of all expert opinions as the initial center, namely, $\bar{R} = ({\bar{r}}_{ij})_{n \times n}$ is regarded as the initial center c¹. Then, the farthest point R^l from c¹ was taken as the second cluster center c², that is, the similarity $s (c^{1}, R^{l}) = min_{k} s (c^{1}, R^{k})$ , where k = 1, 2, . . . , Q, and Q is the total number of DMs. Repeat the process until all sub-group centers c^h (h = 1, 2, . . . , N) are identified.

(3) Calculate membership degree: FCM algorithm needs to compute membership degree to classify DMs, and its calculation formula is as follows: $μ_{C^{h}} (R^{k}) = \frac{(1 / d (c^{h}, R^{k}))^{1 / (b - 1)}}{\sum_{u = 1}^{N} (1 / d (c^{u}, R^{k}))^{1 / (b - 1)}}$ (7) which means the membership degree of decision-maker e_k to subgroup C^h, and where d (c^h, R^k) =1 - s (c^h, R^k) is the distance measure. b is the fuzziness parameter, which can measure the degree of overlap between subgroups [29]. The larger b is, the greater the fuzziness of the cluster will be. Many studies [3 , 64] show that the reasonable value of b should be set as 2.

(4) Clustering: according to Equation (12), the expert e_k is assigned as cluster C^h if R^k has the maximum membership degree to C^h, namely $C^{h} (R^{k}) = max_{h} μ_{C^{h}} (R^{k})$ where h = 1, 2, . . . , N, and N is the number of cluster.

(5) Update centers: take the average of all expert opinions in the corresponding sub-groups $c_{ij}^{h, t + 1} = \frac{1}{| C^{h, t} |} \sum_{R^{k} \in C^{h, t}} r_{ij}^{k}$ where t is the number of iterations, and |C^h,t| is the number of experts in cluster C^h,t.

(6) Termination condition: in this paper, with the parameter ξ approaching 0, the termination condition of the algorithm is set as follows $\frac{\sum_{k = 1}^{Q} \sum_{h = 1}^{N} | μ_{C^{h, t + 1}} (R^{k}) - μ_{C^{h, t}} (R^{k}) |}{Q \cdot N} \leq ξ$ that is, the membership degree is basically unchanged in two consecutive iterations.

4.2 Consensus-reaching

Based on the clustering results of the FCM algorithm, this section presents the consensus-reaching process under the given heterogeneous preference environment, which includes determining the weight of clusters, calculating the consensus degree, and setting up the feedback mechanism.

4.2.1 Determining the weights of clusters and individuals

Through the FCM algorithm introduced above, the population can be clustered into several subgroups, and the opinions of DMs within the subgroup obtained are relatively similar. Therefore, most current researches on the consensus-reaching process regard the subgroup as a whole to discuss the overall consensus level. However, while experts within the subgroup share certain similarities, significant differences still arise when many experts are involved, indicating a comparatively low level of consensus among DMs in the cluster. Thus, Tang et al. [41] proposed a hybrid consensus-reaching strategy by considering intra-subgroup and inter-subgroup consensus levels. Building on this idea, this paper examines the consensus-building process in a heterogeneous preference environment. To achieve this, based on the cosine similarity in Definition 2.6, we present a method for determining expert and subgroup weights.

1) Individual weight: the weight $σ_{u}^{k}$ of expert e_k can be identified as: $σ_{u}^{k} = \frac{c_{k}}{\sum_{e_{k} \in C^{u}} c_{k}}$ (8) where $c_{k} = \sum_{h \neq k, e_{h} \in C^{u}} s (R^{k}, R^{h})$ (9) which can be interpreted as the confidence degree of expert e_k in cluster C^u.

Remark 3. At present, the mainstream thought thinks that the weight of the individual should be 1/|C^u|(|C^u| is the number of experts in cluster C^u) due to the similarity of the opinions of various experts within the subgroup. However, different from the weight determination methods in existing literature, this paper takes into account the cohesion of subgroups, so there will be some differences in the weight of experts even in the same cluster.

2) Subgroup weight: Let $c^{u} = \sum_{R^{k} \in C^{u}} σ_{u}^{k} R^{k}$ be the center of subgroup C^u, then the cohesion of C^u can be expressed as: $s (C^{u}) = \frac{\sum_{R^{k} \in C^{u}} s (R^{k}, c^{u})}{| C^{u} |}$ (10) Combined with the size and cohesion of the subgroup, its weight can be acquired as: $w^{u} = \frac{φ (u)}{\sum_{u = 1}^{N} φ (u)}$ (11) where $φ (u) = \frac{| C^{u} |}{Q} \cdot \frac{s (C^{u})}{\sum_{u = 1}^{N} s (C^{u})}$ (12)Q is the total number of DMs, and N is the number of subgroups.

4.2.2 Calculating consensus degree

Definition 4.1 Let CL (R^k) be the consensus level of expert e_k, then it can be indicated as: $CL (R^{k}) = \frac{1}{n} \sum_{j = 1}^{n} \sum_{i = 1}^{n} {\bar{p}}_{ij}^{k} {\bar{w}}_{i}$ (13) where ${\bar{p}}_{ij}^{k}$ and ${\bar{w}}_{i}$ can be obtained from Model (11). To accelerate the process of consensus-reaching, let $CL (R_{j}^{k}) = \sum_{i = 1}^{n} {\bar{p}}_{ij}^{k} {\bar{w}}_{i}$ represent the consensus level of the jth column of the preference matrix given by the kth decision-maker.

Based on individual consensus level, the calculation method of subgroup consensus degree CL^u can be presented as follows: ${CL}^{u} = \sum_{R^{k} \in C^{u}} σ_{u}^{k} CL (R^{k})$ (14) where $σ_{u}^{k}$ can be calculated by Eqs.(13) and (14).

Further, the overall consensus degree can be derived as: $GCL = \sum_{u = 1}^{N} w^{u} {CL}^{u}$ (15) where w^u can be found as Equation (16).

Clearly, the values of CL (R^k) , CL^u and GCL are all between 0 and 1. If CL (R^k) =1, the individual opinion is consistent with the overall opinion; if CL^u = 1, then the subgroup opinion is consistent with the overall opinion, and the subgroup consensus is reached; if GCL = 1, then all the experts’ opinions are completely consistent, and the overall consensus is reached.

Remark 4. In Equation (18), we take the collective priority vector $w^{'} = (w_{1}^{'}, w_{2}^{'}, \dots, w_{n}^{'})$ as the collective opinion, the matrix P = (p_ij) _n×n transformed from the normalized comprehensive preference matrix as the individual opinion, and then take their cosine similarity as the individual consensus level. As a result, the consensus levels of individuals and subgroups are determined, and the collective priority weight is solved by Model (11), thus avoiding the aggregation of preference opinions of subgroups. Since the comprehensive preference value considers the psychological behavior of DMs, the consensus level obtained by this method is easier to be accepted by DMs.

4.2.3 Feedback regulation

Assume that the thresholds corresponding to CL^u and GCL are $\bar{CL}$ and $\bar{GCL}$ , respectively. If ${CL}^{u} > \bar{CL}$ , the degree of subgroup consensus is acceptable; otherwise, feedback adjustment is carried out to improve the degree of subgroup consensus. If $GCL > \bar{GCL}$ , the overall consensus degree is acceptable; otherwise, feedback adjustment is conducted to increase the overall consensus degree. Based on these two consensus levels, the consensus-reaching process is divided into the following two situations:

1) If $GCL \geq \bar{GCL}$ , then the overall consensus level is reached. Even if there is ${CL}^{u} < \bar{CL}$ , it does not affect the overall consensus level within the acceptable range, so there is no need for experts to adjust their opinions.

2) If $GCL < \bar{GCL}$ , then we need to find out the subgroup that needs to be improved according to ${CL}^{u} < \bar{CL}$ , and then identify the experts within the subgroup who need to be modified through $CL (R^{k}) < \bar{CL}$ . After then, based on $CL (R_{j}^{k})$ , determine the column of preference opinions that expert e_k needs to modify, and finally propose adjustment suggestions. The specific adjustment method is as follows:

Let $R^{k} = (r_{ij}^{k})_{n \times n}$ be the IVFPR given by expert e_k, and $O^{k} = (o_{ij}^{N_{k}})_{n \times n}$ be its corresponding normalized comprehensive preference matrix, where $o_{ij}^{N_{k}} = \frac{e^{o_{ij}^{k}}}{e^{o_{ij}^{k}} + e^{o_{ji}^{k}}} .$

According to Definition 3.2, if IVFPR R^k is multiplicatively consistent, then Equation $o_{ij}^{N_{k}} = \frac{w_{i}}{w_{i} + w_{j}}$ holds. Accordingly, the collective priority vector $w^{'} = (w_{1}^{'}, w_{2}^{'}, \dots, w_{n}^{'})$ derived from Model (11) can be reduced to the normalized comprehensive preference matrix by the formula: $o_{ij}^{N} = \frac{w_{i}^{'}}{w_{i}^{'} + w_{j}^{'}}$ namely, the comprehensive preference matrix $O = (o_{ij}^{N})_{n \times n}$ is multiplicative consistent. Therefore, to improve the consensus level of DMs, we only need to adjust the size of $o_{ij}^{N_{k}}$ to make it close to the value of $o_{ij}^{N}$ . Notice that the magnitude of $o_{ij}^{N_{k}}$ depends on the values of $o_{ij}^{k}$ and $o_{ji}^{k} (i < j)$ , so the monotonicity of $o_{ij}^{k}$ and $o_{ji}^{k}$ with respect to the interval endpoints ${\underline{r}}_{ij}^{k}, {\bar{r}}_{ij}^{k}$ and ${\underline{r}}_{ji}^{k}, {\bar{r}}_{ji}^{k}$ respectively need to be explored (in this paper, $x_{ij} = (1 - η) {\underline{r}}_{ij} + η {\bar{r}}_{ij}$ and $x_{ji} = η {\underline{r}}_{ji} + (1 - η) {\bar{r}}_{ji}$ , satisfy x_ij + x_ji = 1, so the monotonicity of $o_{ij}^{k}$ and $o_{ji}^{k}$ may be different).

1) Monotonicity of $o_{ij}^{k}$ with respect to ${\underline{r}}_{ij}^{k}$ . $\begin{matrix} o_{ij}^{k} = & θ o_{ij}^{k -} + (1 - θ) o_{ij}^{k +} \\ = & θ ({\underline{r}}_{ij} + v_{ij}^{-}) + (1 - θ) ({\bar{r}}_{ij} + v_{ij}^{+}) \\ = & θ ({\underline{r}}_{ij} + (x_{ij} - {\underline{r}}_{ij})^{α}) \\ + (1 - θ) [{\bar{r}}_{ij} - ρ ({\bar{r}}_{ij} - x_{ij})^{β}] \\ = & θ ({\underline{r}}_{ij} + η^{α} ({\bar{r}}_{ij} - {\underline{r}}_{ij})^{α}) \\ + (1 - θ) [{\bar{r}}_{ij} - ρ (1 - η)^{β} ({\bar{r}}_{ij} - {\underline{r}}_{ij})^{β}] \end{matrix}$ When α = β, the partial derivative of $o_{ij}^{k}$ with respect to ${\underline{r}}_{ij}$ is as follows: $\frac{\partial o_{ij}^{k}}{\partial {\underline{r}}_{ij}} = θ + α ({\bar{r}}_{ij} - {\underline{r}}_{ij})^{α - 1} [ρ (1 - θ) (1 - η)^{α} - θ η^{α}]$ Let $\frac{\partial o_{ij}^{k}}{\partial {\underline{r}}_{ij}} > 0$ , it is easy to obtain that $o_{ij}^{k}$ increases monotonically with respect to ${\underline{r}}_{ij}$ when $η < \frac{g}{1 + g}$ , where $g = (\frac{ρ (1 - θ)}{θ})^{\frac{1}{α}}$ . If $η > \frac{g}{1 + g}$ , then $o_{ij}^{k}$ is the increasing function of ${\underline{r}}_{ij}$ when $d > [α (η^{α} - ρ \frac{(1 - θ)}{θ} (1 - η)^{α})]^{\frac{1}{1 - α}}$ , where $d = {\bar{r}}_{ij} - {\underline{r}}_{ij}$ . Otherwise, $o_{ij}^{k}$ is the decreasing function of ${\underline{r}}_{ij}$ . Similarly, the partial derivatives of $o_{ij}^{k}$ with respect to ${\bar{r}}_{ij}$ , $o_{ji}^{k}$ with respect to ${\underline{r}}_{ji}$ and ${\bar{r}}_{ji}$ can be obtained as follows: $\begin{matrix} \frac{\partial o_{ij}^{k}}{\partial {\bar{r}}_{ij}} & = 1 - θ + α ({\bar{r}}_{ij} - {\underline{r}}_{ij})^{α - 1} [θ η^{α} - ρ (1 - θ) (1 - η)^{α}] \\ \frac{\partial o_{ji}^{k}}{\partial {\underline{r}}_{ji}} & = θ + α ({\bar{r}}_{ji} - {\underline{r}}_{ji})^{α - 1} [ρ (1 - θ) η^{α} - θ (1 - η)^{α}] \\ \frac{\partial o_{ji}^{k}}{\partial {\bar{r}}_{ji}} & = 1 - θ + α ({\bar{r}}_{ji} - {\underline{r}}_{ji})^{α - 1} [θ (1 - η)^{α} - ρ (1 - θ) η^{α}] \end{matrix}$ The corresponding monotonicity is also easy to acquire, which is omitted here.

For the sake of discussion, this paper only takes parameter values η = 0.4 and θ = 0.5 as an example to give feedback strategies, so g = 2.5131 and the result

\frac{1}{1 + g} = 0.2846 < η = 0.4 < \frac{g}{1 + g} = 0.7153

can be obtained. In this case, the monotonicity of each function is shown in Table 1.

Table 1
The monotonicity of each function

	${\underline{r}}_{ij}$	${\bar{r}}_{ij}$	${\underline{r}}_{ji}$	${\bar{r}}_{ji}$
$o_{ij}^{k}$	↗	d> 0.3139 : ↗
		d< 0.3139 : ↘
$o_{ji}^{k}$			↗	d> 0.00008 : ↗
				d< 0.00008 : ↘

With these discussions, the following feedback strategies are offered:

1) If $o_{ij}^{N} k <_{o}^{ij} N$ , then when d < 0.3139, the decision-maker is advised to increase $r_{_}^{ij} k$ and decrease ${\bar{r}}_{ji}^{k}$ (This article defaults to d > 0.00008), and when d > 0.3139, it should increase the values of ${\underline{r}}_{ij}^{k}$ , ${\bar{r}}_{ij}^{k}$ and decrease ${\underline{r}}_{ji}^{k}$ , ${\bar{r}}_{ji}^{k}$ simultaneously.

2) If $o_{ij}^{N_{k}} > o_{ij}^{N}$ , then when d < 0.3139, it should decrease ${\underline{r}}_{ij}^{k}$ and increase ${\bar{r}}_{ji}^{k}$ , and when d > 0.3139, it should decrease the values of ${\underline{r}}_{ij}^{k}$ , ${\bar{r}}_{ij}^{k}$ and increase ${\underline{r}}_{ji}^{k}$ , ${\bar{r}}_{ji}^{k}$ simultaneously.

3) In the above cases, the decision-maker can flexibly choose to adjust only the value of the left endpoint ${\underline{r}}_{ij}^{k}$ or only the value of the right endpoint ${\bar{r}}_{ij}^{k}$ , or change the value of both ${\underline{r}}_{ij}^{k}$ and ${\bar{r}}_{ij}^{k}$ at the same time according to his own will.

Remark 5. This research introduces a feedback adjustment mechanism that considers the hesitation and psychological behavior of DMs, which improves upon the general LSGDM method. Notably, this approach incorporates the influence of individual weight and subgroup weight on the consensus level, making it more reasonable and likely to be accepted by DMs. However, existing methods often overlook the cohesion of subgroups when determining weights, treating the weights of experts in the same subgroup as equal, which lacks rationality. Additionally, this paper utilizes three consensus levels to identify experts and alternatives that require modification, leading to a more efficient consensus-reaching process with reduced adjustment cost and improved consensus-reaching efficiency.

In summary, the process of LSGDM with HPRs is given in Table 2.

Table 2

Prospect theory-based LSGDM method with HFPs

Input: HFLPR H = (h_ij) _n×n, IFPR C = (c_ij) _n×n and IVFPR R = (r_ij) _n×n. The thresholds

\bar{CL}

and

\bar{GCL}

Output: The ranking of alternatives.

Step 1: Convert HFLPRs B and IFPRs C to IVFPRs R^h and R^c, respectively.

Step 2: The decision experts are divided into N clusters by FCM algorithm.

Step 3: The weights of experts and subgroups are determined by Eqs.(13) and (16), respectively.

Step 4: Calculate the normalized comprehensive preference value

o_{ij}^{N_{k}}

by Eqs.(6-8).

Step 5: Solve the collective priority vector through Model(11).

Step 6: Compute consensus levels CL (R^k) , CL^u and GCL, if

GCL > \bar{GCL}

, then move to the next step, otherwise give adjustment opinions and return to Step 4.

Step 7: Sort the alternatives according to the collective priority vector w′.

Step 8: End

5 Numerical example and comparative analysis

In this section, a numerical example and comparative analysis are given to demonstrate the feasibility and effectiveness of the proposed method.

5.1 Numerical example

Prefabricated building is gaining worldwide attention due to their importance in sustainable urbanization [12]. It involves the building products constructed by prefabricating building components, parts, and materials in the factory, transporting them to the construction site, and finally assembling them [37]. Compared with the traditional cast-in-place construction method, the prefabricated construction method has the advantages of shortening the construction period, reducing on-site labor, and improving the quality of the building and life cycle environmental performance. In today’s world, where environmental protection is increasingly importance, the traditional construction methods used in the construction industry are no longer adequate to meet the demands of the industry’s transformation and upgrading. At the same time, prefabricated buildings have become widely used in the construction industry because of the advantages mentioned above [36].

The complete prefabricated building industry chain is based on research and development-design-production-construction-operation and maintenance. As the core node of the supply chain, construction companies have a significant impact on integrating the advantageous resources of each enterprise so that various tasks can be effectively and rationally operated. Therefore, how to select suppliers (e.g., component manufacturing enterprise) reasonably will directly affect whether the prefabricated building project can achieve the expected results.

This section provides an illustrative scenario of supplier selection for a construction company to validate the proposed model. Construction Company B, a general contractor undertaking a prefabricated building project, requires a dependable floor supplier. To ensure an informed decision, Company B invites 20 experts (e₁, e₂, …, e₂₀) to compare and evaluate five shortlisted companies (x₁, x₂, x₃, x₄, and x₅) to identify the most suitable floor supplier. The thresholds $\bar{CL}$ and $\bar{GCL}$ are set to 0.95, while the parameter values η and θ are 0.4 and 0.5, respectively. Due to the intricate nature of the factors that need to be taken into account, such as project management competence, enterprise reputation, professional technical capability, business management ability, and supply chain management competence, the experts have provided the following preference information in the form of IVFPRs, IFPRs, and HFLPRs.

$R_{1} = (\begin{matrix} 0.5 & [0.2, 0.4] & [0.5, 0.7] & [0.5, 0.6] & [0.6, 0.8] \\ 0.5 & [0.25, 0.4] & [0.4, 0.65] & [0.7, 0.8] \\ 0.5 & [0.3, 0.4] & [0.35, 0.5] \\ 0.5 & [0.8, 0.9] \\ 0.5 \end{matrix}), R_{2} = (\begin{matrix} 0.5 & [0.2, 0.3] & [0.6, 0.8] & [0.5, 0.7] & [0.6, 0.85] \\ 0.5 & [0.2, 0.4] & [0.4, 0.65] & [0.75, 0.8] \\ 0.5 & [0.3, 0.5] & [0.35, 0.4] \\ 0.5 & [0.75, 0.9] \\ 0.5 \end{matrix})$

$R_{3} = (\begin{matrix} 0.5 & [0.3, 0.5] & [0.6, 0.7] & [0.3, 0.4] & [0.65, 0.8] \\ 0.5 & [0.25, 0.4] & [0.4, 0.65] & [0.3, 0.4] \\ 0.5 & [0.3, 0.4] & [0.35, 0.5] \\ 0.5 & [0.8, 0.9] \\ 0.5 \end{matrix}), R_{4} = (\begin{matrix} 0.5 & [0.5, 0.6] & [0.6, 0.75] & [0.55, 0.6] & [0.3, 0.4] \\ 0.5 & [0.25, 0.46] & [0.12, 0.3] & [0.35, 0.5] \\ 0.5 & [0.27, 0.42] & [0.26, 0.4] \\ 0.5 & [0.8, 0.9] \\ 0.5 \end{matrix})$ (16)

$R_{5} = (\begin{matrix} 0.5 & [0.13, 0.32] & [0.52, 0.71] & [0.51, 0.66] & [0.35, 0.56] \\ 0.5 & [0.18, 0.39] & [0.3, 0.52] & [0.75, 0.95] \\ 0.5 & [0.22, 0.39] & [0.6, 0.75] \\ 0.5 & [0.82, 0.9] \\ 0.5 \end{matrix}), R_{6} = (\begin{matrix} 0.5 & [0.2, 0.4] & [0.23, 0.35] & [0.52, 0.66] & [0.61, 0.87] \\ 0.5 & [0.25, 0.4] & [0.4, 0.65] & [0.57, 0.68] \\ 0.5 & [0.3, 0.4] & [0.32, 0.49] \\ 0.5 & [0.56, 0.72] \\ 0.5 \end{matrix})$

$C_{1} = (\begin{matrix} 0.5 & 〈 0.3, 0.6 〉 & 〈 0.2, 0.5 〉 & 〈 0.35, 0.46 〉 & 〈 0.26, 0.68 〉 \\ 0.5 & 〈 0.25, 0.4 〉 & 〈 0.34, 0.6 〉 & 〈 0.27, 0.58 〉 \\ 0.5 & 〈 0.73, 0.14 〉 & 〈 0.65, 0.25 〉 \\ 0.5 & 〈 0.8, 0.14 〉 \\ 0.5 \end{matrix}), C_{2} = (\begin{matrix} 0.5 & 〈 0.23, 0.6 〉 & 〈 0.6, 0.25 〉 & 〈 0.57, 0.25 〉 & 〈 0.66, 0.32 〉 \\ 0.5 & 〈 0.5, 0.4 〉 & 〈 0.74, 0.16 〉 & 〈 0.77, 0.18 〉 \\ 0.5 & 〈 0.83, 0.12 〉 & 〈 0.73, 0.25 〉 \\ 0.5 & 〈 0.8, 0.14 〉 \\ 0.5 \end{matrix})$ (17)

$C_{3} = (\begin{matrix} 0.5 & 〈 0.3, 0.6 〉 & 〈 0.2, 0.5 〉 & 〈 0.35, 0.46 〉 & 〈 0.26, 0.68 〉 \\ 0.5 & 〈 0.25, 0.4 〉 & 〈 0.34, 0.6 〉 & 〈 0.27, 0.58 〉 \\ 0.5 & 〈 0.73, 0.14 〉 & 〈 0.65, 0.25 〉 \\ 0.5 & 〈 0.8, 0.14 〉 \\ 0.5 \end{matrix}), C_{4} = (\begin{matrix} 0.5 & 〈 0.3, 0.6 〉 & 〈 0.2, 0.5 〉 & 〈 0.35, 0.46 〉 & 〈 0.26, 0.68 〉 \\ 0.5 & 〈 0.25, 0.4 〉 & 〈 0.34, 0.6 〉 & 〈 0.27, 0.58 〉 \\ 0.5 & 〈 0.73, 0.14 〉 & 〈 0.65, 0.25 〉 \\ 0.5 & 〈 0.78, 0.12 〉 \\ 0.5 \end{matrix})$ (18)

$C_{5} = (\begin{matrix} 0.5 & 〈 0.76, 0.16 〉 & 〈 0.82, 0.13 〉 & 〈 0.65, 0.26 〉 & 〈 0.64, 0.18 〉 \\ 0.5 & 〈 0.39, 0.6 〉 & 〈 0.24, 0.6 〉 & 〈 0.28, 0.67 〉 \\ 0.5 & 〈 0.85, 0.1 〉 & 〈 0.9, 0.05 〉 \\ 0.5 & 〈 0.82, 0.13 〉 \\ 0.5 \end{matrix}), C_{6} = (\begin{matrix} 0.5 & 〈 0.23, 0.56 〉 & 〈 0.32, 0.55 〉 & 〈 0.25, 0.58 〉 & 〈 0.3, 0.68 〉 \\ 0.5 & 〈 0.22, 0.5 〉 & 〈 0.34, 0.55 〉 & 〈 0.27, 0.68 〉 \\ 0.5 & 〈 0.83, 0.14 〉 & 〈 0.65, 0.2 〉 \\ 0.5 & 〈 0.81, 0.1 〉 \\ 0.5 \end{matrix})$ (19)

$C_{7} = (\begin{matrix} 0.5 & 〈 0.33, 0.56 〉 & 〈 0.12, 0.7 〉 & 〈 0.25, 0.56 〉 & 〈 0.16, 0.78 〉 \\ 0.5 & 〈 0.75, 0.2 〉 & 〈 0.64, 0.16 〉 & 〈 0.77, 0.18 〉 \\ 0.5 & 〈 0.83, 0.14 〉 & 〈 0.75, 0.2 〉 \\ 0.5 & 〈 0.8, 0.1 〉 \\ 0.5 \end{matrix}), H_{1} = (\begin{matrix} s_{4} & {s_{1}, s_{2}} & {s_{2}, s_{3}, s_{4}} & {s_{5}, s_{6}} & {s_{3}, s_{4}, s_{5}} \\ s_{4} & {s_{3}, s_{4}} & {s_{6}, s_{7}} & {s_{6}, s_{7}, s_{8}} \\ s_{4} & {s_{2}, s_{3}} & {s_{1}, s_{2}} \\ s_{4} & {s_{4}, s_{5}} \\ s_{4} \end{matrix})$ (20)

$H_{2} = (\begin{matrix} s_{4} & {s_{5}, s_{6}, s_{7}} & {s_{5}, s_{6}} & {s_{6}, s_{7}, s_{8}} & {s_{2}, s_{3}, s_{4}} \\ s_{4} & {s_{1}, s_{2}} & {s_{5}, s_{6}} & {s_{6}, s_{7}} \\ s_{4} & {s_{1}, s_{2}, s_{3}} & {s_{3}, s_{4}} \\ s_{4} & {s_{4}, s_{5}, s_{6}} \\ s_{4} \end{matrix}), H_{3} = (\begin{matrix} s_{4} & {s_{5}, s_{6}, s_{7}} & {s_{1}, s_{2}} & {s_{5}, s_{6}, s_{7}} & {s_{4}, s_{5}} \\ s_{4} & {s_{3}, s_{4}} & {s_{6}, s_{7}, s_{8}} & {s_{7}, s_{8}} \\ s_{4} & {s_{1}, s_{2}, s_{3}} & {s_{2}, s_{3}, s_{4}} \\ s_{4} & {s_{5}, s_{6}} \\ s_{4} \end{matrix})$ (21)

$H_{4} = (\begin{matrix} s_{4} & {s_{1}, s_{2}, s_{3}} & {s_{1}, s_{2}} & {s_{5}, s_{5}} & {s_{4}, s_{5}} \\ s_{4} & {s_{5}, s_{6}} & {s_{6}, s_{7}} & {s_{3}, s_{4}} \\ s_{4} & {s_{2}, s_{3}} & {s_{1}, s_{2}, s_{3}} \\ s_{4} & {s_{5}, s_{6}, s_{7}} \\ s_{4} \end{matrix}), H_{5} = (\begin{matrix} s_{4} & {s_{1}, s_{2}, s_{3}} & {s_{4}, s_{5}} & {s_{4}, s_{5}, s_{6}} & {s_{5}, s_{6}} \\ s_{4} & {s_{3}, s_{4}} & {s_{6}, s_{7}} & {s_{6}, s_{6}} \\ s_{4} & {s_{1}, s_{2}} & {s_{2}, s_{3}} \\ s_{4} & {s_{6}, s_{7}} \\ s_{4} \end{matrix})$ (22)

$H_{6} = (\begin{matrix} s_{4} & {s_{1}, s_{2}, s_{3}} & {s_{3}, s_{4}} & {s_{4}, s_{5}} & {s_{4}, s_{5}, s_{6}} \\ s_{4} & {s_{3}, s_{4}} & {s_{6}, s_{7}} & {s_{4}, s_{5}} \\ s_{4} & {s_{2}, s_{3}} & {s_{5}, s_{6}, s_{7}} \\ s_{4} & {s_{5}, s_{5}} \\ s_{4} \end{matrix}), H_{7} = (\begin{matrix} s_{4} & {s_{1}, s_{2}} & {s_{2}, s_{3}, s_{4}} & {s_{5}, s_{6}} & {s_{3}, s_{4}} \\ s_{4} & {s_{2}, s_{2}} & {s_{5}, s_{6}} & {s_{5}, s_{6}, s_{7}} \\ s_{4} & {s_{2}, s_{3}} & {s_{1}, s_{2}} \\ s_{4} & {s_{3}, s_{3}} \\ s_{4} \end{matrix})$ (23)

Among them, R_i (i = 1, 2, …, 6) are the IVFPRs, C_i (i = 1, 2, …, 7) denote the IFPRs, and H_i (i = 1, 2, …, 7) represent the HFLPRs. The linguistic term set employed here is S = {s₀ : extre - mely poor, s₁ : very poor, s₂ : poor, s₃ : slightlypoor, s₄ : fair, s₅ : slightly good, s₆ : good, s₇ : very good, s₈ : extremely good}.

According to Table 2, the selection process of the supplier is as follows:

Step 1: The FCM algorithm is adopted to cluster the transformed IVFPRs. Without loss of generality, the number of clusters is set as 3. The clustering results are shown in Table 3.

Step 2: The weight information is calculated using Eqs. (13) and (16), as shown in Table 3.

Table 3

Clustering results and weight information

Subgroup	The DMs in each subgroup	The weight of experts	The weight of each subgroup
C ¹	{e₄, e₇, e₈, e₉,	{0.1456, 0.1411, 0.1373, 0.1465,	0.3508
	e₁₀, e₁₂, e₁₃}	0.1462, 0.1401, 0.1432}
C ²	{e₁₁}	{1}	0.0514
C ³	{e₁, e₂, e₃, e₅,	{0.0808, 0.0838, 0.0829, 0.0836,	0.5978
	e₆, e₁₄, e₁₅, e₁₆,	0.0838, 0.0853, 0.0844, 0.0846
	e₁₇, e₁₈, e₁₉, e₂₀}	0.0825, 0.0836, 0.0851, 0.0796, }

Step 3: The normalized matrix $O^{k} = (o_{ij}^{N_{k}})_{n \times n}, k = 1, 2, \dots, 20$ can be obtained from the Eqs.(6), (7), and (8).

Step 4: Substituting matrix element value $o_{ij}^{N_{k}}$ into the model (11), ${\bar{w}}_{1} = 0.4669, {\bar{w}}_{2} = 0.4188, {\bar{w}}_{3} = 0.5021, {\bar{w}}_{4} = 0.4618$ and ${\bar{w}}_{5} = 0.3758$ can be obtained, and thus the collective priority vector can be solved as: $w_{1}^{'} = 0.2098, w_{2}^{'} = 0.1882, w_{3}^{'} = 0.2256, w_{4}^{'} = 0.2075, w_{5}^{'} = 0.1689$ .

Step 5: The consensus degree of each subgroup can be computed by Eqs. (18), (19), and (20), as shown in Table 4.

Table 4

Consensus levels CL (R^k) , CL^u and GCL

	C ¹	C ²	C ³
CL ^u	0.9552	0.9426	0.943
CL (R^k)	e₄ : 0.951, e₇ : 0.9622, e₈ : 0.9608,	e₁₁ : 0.9423	e₁ : 0.9635, e₂ : 0.9546, e₃ : 0.9625,
	e₉ : 0.9622, e₁₀ : 0.9627, e₁₂ : 0.9526,		e₅ : 0.9513, e₆ : 0.9654, e₁₄ : 0.9234,
	e₁₃ : 0.9347		e₁₅ : 0.9289, e₁₆ : 0.9185, e₁₇ : 0.9297,
			e₁₈ : 0.9342, e₁₉ : 0.959, e₂₀ : 0.9251
GCL	0.9473

As listed in Table 4, the overall consensus level is 0.9473, which is below 0.95. Thus, the feedback adjustment process is entered. With CL² = 0.9426 and CL³ = 0.943, both values are slightly below 0.95. As a result, expert e₁₁ in Subgroup C² should consider revising their opinions, and experts e₁₄, e₁₅, e₁₆, e₁₇, e₁₈ and e₂₀ in Subgroup C³ should make necessary adjustments to their viewpoints. First, the normalized comprehensive preference matrix $O = (O_{ij}^{N})_{5 \times 5}$ is calculated utilizing Equation (21). Subsequently, $O_{ij}^{N_{11}}$ and $O_{ij}^{N}$ are compared to provide suggestions through the feedback adjustment mechanism detailed in Section 4.2.3. Suppose expert e₁₁ modifies the positions with a significant deviation, yielding the modified IVFPR as follows: $R_{11} = (\begin{matrix} 0.5 & [0.76, 0.84] & [0.6, 0.88] & [0.65, 0.74] & [0.64, 0.82] \\ 0.5 & [0.39, 0.4] & [0.3, 0.35] & [0.26, 0.33] \\ 0.5 & [0.85, 0.88] & [0.9, 0.92] \\ 0.5 & [0.85, 0.87] \\ 0.5 \end{matrix})$ (24)

After modification, the consensus level of each subgroup is 0.9565, 0.9545, and 0.9479, respectively, and the overall consensus level GCL is 0.9513. Since 0.9513 > 0.95 meets the acceptable level, even if the subgroup C₃ does not reach the acceptable consensus level, there is no need to make any changes. The final collective priority vector is w′ = (0.2053, 0.1891, 0.2293, 0.2073, 0.169), that is, x₃ is the best cooperative supplier. As a result, company B should be recommended to choose the third floor supplier under the given evaluation opinion.

5.2 Comparison analyses

Several researchers have proposed various effective solutions for LSGDM problems. This section presents a comparative analysis to illustrate the features and merits of the proposed method.

To ensure a fair comparison, we compare our method (referred to as M1) with that proposed by Wu et al. [53] (referred to as M2), which covers the heterogeneous preference structure utilized in our research. The primary procedures involved in solving the example depicted in Section 5.1 employing M2 are as follows:

Step 1: Compute the priority vectors for each subgroup.

According to the classification method in M2, which is based on the type of preference information, DMs can be categorized into three subgroups: C¹ : {e₁, e₂, e₃, e₄, e₅, e₆}, C² : {e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃}, and C³ : {e₁₄, e₁₅, e₁₆, e₁₇, e₁₈, e₁₉, e₂₀}. The priority vectors for each subgroup, which have been determined via the utilization of the subgroup score matrix, are listed in Table 5.

Table 5
Priority vectors derived from M2

Subgroup weight Priority vector Preference ordering

C ¹ 0.3 (0.1822, 0.188, 0.2227, 0.1196, 0.2874) (5, 3, 2, 1, 4)

C ² 0.35 (0.2294, 0.199, 0.1012, 0.1743, 0.2961) (5, 1, 2, 4, 3)

C ³ 0.35 (0.1979, 0.2256, 0.1914, 0.1958, 0.1893) (2, 1, 4, 3, 5)

C ^o (0.2042, 0.2050, 0.1692, 0.1654, 0.2561) (5, 2, 1, 3, 4)

	Subgroup weight	Priority vector	Preference ordering
C ¹	0.3	(0.1822, 0.188, 0.2227, 0.1196, 0.2874)	(5, 3, 2, 1, 4)
C ²	0.35	(0.2294, 0.199, 0.1012, 0.1743, 0.2961)	(5, 1, 2, 4, 3)
C ³	0.35	(0.1979, 0.2256, 0.1914, 0.1958, 0.1893)	(2, 1, 4, 3, 5)
C ^o		(0.2042, 0.2050, 0.1692, 0.1654, 0.2561)	(5, 2, 1, 3, 4)

Table 5 shows that the initial collective priority vector, weighted by each subgroup’s priority vector, yields (0.2042, 0.2050, 0.1692, 0.1654, 0.2561), indicating that the initial preference order is x₅ ≻ x₂ ≻ x₁ ≻ x₃ ≻ x₄. Notably, this finding diverges significantly from the initial preference order x₃ ≻ x₁ ≻ x₄ ≻ x₂ ≻ x₅ obtained in Step 4 of Section 5.1. This can be attributed to the following aspects:

1) Various clustering methods yield distinct clustering outcomes. Unlike the implementation of the fuzzy C-means algorithm in this study, the utilization of classification according to preference information type in M2 results in significant disparities of opinion within the identical subgroup. This discrepancy undermines the representativeness of the aggregated subgroup preference information.

2) Compared with M2, which only utilizes subgroup size as the subgroup weight, this research integrates both subgroup cohesion and size. As a result, the subgroup with a higher internal consensus level is granted a higher weight.

3) In M2, individual preference information is aggregated to obtain subgroup preference information, and subsequently, the priority vector calculated by each subgroup preference information is weighted as the collective priority vector. However, this aggregate process may result in information loss and overlooks the psychological behavior of DMs in uncertain scenarios. Conversely, model (11) is employed in this study to minimize the difference between the obtained collective priority vector and that obtained by each individual, fully considering the psychological behavior of DMs under uncertain circumstances.

Step 2: Measure consensus level.

In M2, the mean of all individual ordinal consensus levels is treated as the overall consensus level. The corresponding results are presented in Table 6.

Table 6

Ordinal consensus levels acquired by M2

Expert	CL (R^k)	GCL
{e₁, e₂, e₃, e₄, e₅, e₆}	{0.5528, 0.4084, 0.6127, 0.5, 0.6127, 0.6127}	0.4416
{e₇, e₈, e₉, e₁₀, e₁₁, e₁₂, e₁₃}	{0.5, 0.3292, 0.5, 0.5, 0.5, 0.5, 0.3292}
{e₁₄, e₁₅, e₁₆, e₁₇, e₁₈, e₁₉, e₂₀}	{0.5528, 0.2584, 0.2929, 0.3292, 0.1938, 0.1938, 0.5528}

Upon comparing Tables 4 and 6, it becomes clear that the ordinal consensus levels attained by M2 are substantially lower than that obtained by cosine similarity and prospect theory in this study. Moreover, the overall consensus level B falls far below the consensus threshold of 0.95 prescribed in the above example. The maximum non-full ordinal consensus level with five alternatives that can be achieved is 0.7764 [42]. Thus, relying solely on the ordinal consensus measure is likely to mask the true similarity of distinct preference opinions, resulting in the designated consensus threshold being hard to reach. In contrast, the consensus measurement approach with prospect theory and cosine similarity improves the overall consensus level and reflects the preference intensity for distinct alternatives.

Step 3: Feedback regulation.

M2 adhered to the principle of minimum deviation and devised a range of goal programming models to provide DMs with modification recommendations in the event of low consensus levels. Nonetheless, due to the stringent constraints of M2, the availability of viable solutions cannot be ensured. If the score matrix obtained in Step 1 is inputted into the M2, a feasible solution cannot be founded. Thus, attaining consensus by utilizing M2 in the above example is unfeasible.

Compared with M2, this paper’s methodology swiftly identifies DMs requiring modifications and corresponding modification locations by utilizing the consensus levels of the three tiers CL^u, CL (R^k), and $CL (R_{j}^{k})$ . Furthermore, the proposed approach offers detailed modification recommendations for each preference opinion $[{\underline{r}}_{ji}^{k}$ , ${\bar{r}}_{ji}^{k}]$ , providing flexible options for DMs to adjust while enhancing consensus levels. Additionally, since this article’s methodology comprehensively accounts for DMs’ psychological behaviors and risk preferences, the suggestions provided are objectively more likely to be accepted by DMs, thereby facilitating consensus attainment.

Also, the disparities with the current methods can be outlined in the following facets:

1) Comparisons of heterogeneous preference environments.

To deal with HPRs provided by DMs, some approaches have been developed [6 , 43]. Compared with the preference structures (fuzzy preference relations, multiplicative preference relations, linguistic preference relations, utility functions, and preference orders) studied by Tian et al. [43] and Chao et al. [6], the heterogeneous preference environment used in this paper can extract the decision hesitation information more adequately. For one thing, IVFPRs and HFLPRs can better reflect the uncertainty and hesitation of DMs in complex situations. For another, IFPRs provide DMs with both positive and negative two directions to assess alternatives. In contrast to the heterogeneous information (crisp numbers, interval numbers, triangular fuzzy numbers) examined by Li et al. [14], the heterogeneous information analyzed in this paper offers DMs more flexible assessment options (including both quantitative and qualitative assessments).

2) Comparisons between the proposed method and the existing methods.

Compared to the LSGDM methods in references [6 , 62], the proposed method considers the psychological behavior of each decision-maker in the decision-making process based on the prospect theory. Currently, most of the research on LSGDM ignore the impact of DMs’ psychological behavior on decision outcome. Actually, When DMs exhibit hesitation and uncertainty in a complex situation, their psychological behavior significantly impacts the outcome of the decisions [48 , 51]. In this study, the interval endpoint values of the assessment information provided by DMs are utilized as two reference points to gauge the potential loss or gain perceived by DMs about their minimum and maximum preferences. The priority weights of alternatives are solved from the comprehensive preference value to obtain the optimal ranking result. Therefore, in complex decision-making scenarios, the results obtained by this method can be readily accepted by DMs.

3) Comparisons of weight determination methods.

Most studies [11 , 61] utilized the size of a subgroup as a criterion to determine its weight. In other words, the weight assigned to a subgroup increases proportionally with the number of DMs it encompasses. Moreover, all DMs within the same subgroup are given equal weight. Actually, there may be some differences in expert opinions even within the same subgroup, and the degree of cohesion of different subgroups is not the same [40, 41]. For this reason, Tang et al. [41] considered the two criteria of subgroup size and subgroup cohesion, Ma et al. [23] combined the cluster reliability, and Tang et al. [40] integrated the sum of squared errors in the process of hierarchical consensus-reaching. Compared to methods in references [40, 41], this paper not only considers the size and cohesion of subgroups but also presents an approach for determining individual weight. Taking the weighted average of experts’ preferences as the subgroup center to compute the cohesion of a subgroup is more convincing than directly calculating in references [40, 41].

4) Comparisons of consensus-reaching processes.

Unlike methods in references [11, 41], our approach for calculating consensus degree avoids converging from individual preference to collective one. Therefore, there is no need to discuss the subjective and objective adjustment coefficient [11 , 58], and the DMs that need to be modified can be found directly through the feedback regulation mechanism in this paper to give modification suggestions. Compared to feedback adjustment methods in references [6] and [43], this paper presents an adjustment strategy based on the monotony of the function to improve overall consensus levels. This approach reflects DMs’ comprehensive psychological expected value relative to their minimum and maximum preferences in cases of hesitation. Additionally, it offers DMs a more flexible way to modify their opinions by allowing them to adjust either the left or right endpoint according to their preferences, thereby increasing the likelihood of DMs accepting the suggestions. Overall, vis-a-vis the current consensus-reaching models, the features of the proposed model are outlined in Table 7.

Table 7

Comparison with recent consensus-reaching models

References	Consider	Consider	Consider	Consider	Consider	Offer
	hesitation	the	the risk	subgroup	three	multiple
	and	psychological	attitude	cohesion	levels of	adjustment
	heterogeneity	behavior	of DMs		consensus	options
Our stud	✓	✓	✓	✓	✓	✓
[24]	—	—	—	—	✓	—
[22]	—	—	—	✓	—	—
[8]	—	—	—	—	—	—
[45]	—	—	—	—	—	—
[16]	—	—	—	—	✓	—
[1]	—	—	—	—	✓	—
[38]	—	—	—	✓	—	—
[4]	—	—	—	—	✓	—
[15]	—	—	—	—	✓	—

Note. “—” denotes not considered or not involved.

As evident from Table 7, few consensus models simultaneously consider decision makers’ hesitation, heterogeneity, psychological behavior, and risk attitude in both traditional group decision-making [4 , 16] and LSGDM [1 , 45]. Although Trillo et al. [45] utilized a sentiment analysis approach to detect the degree of positivity and aggressiveness of DMs, there exist fundamental distinctions between their study and the psychological behaviors of DMs in uncertain situations analyzed in this paper. The preference attitude discussed by Tan et al. [38] based on PLPRs differs significantly from the risk attitude in this article. The former is characterized by classifying preferences using linguistic terminology, whereas the latter refers to the subjective inclination of DMs in uncertain situations.

Each method possesses its applicable scenarios. The method proposed in this paper is well-suited for complex decision-making problems where DMs cannot provide precise evaluation information. Additionally, it permits different DMs to adopt varying forms of evaluation information. As different approaches possess their respective advantages in distinct scenarios, this article omits quantitative comparisons with existing methods.

6 Conclusions

This research presents a novel method to solve heterogeneous LSGDM problems. Firstly, a novel approach is presented for solving the collective priority vector by incorporating prospect theory and cosine similarity between HPRs. For one thing, the hesitancy and heterogeneity of DMs are considered, and the psychological behavior and risk attitude of DMs are reflected. For another, the method aims to ensure the collective priority vector closely aligns with the priority vector obtained by each individual, facilitating consensus building. Then, DMs with HPRs are segregated into multiple subgroups utilizing the fuzzy C-means algorithm, simplifying the calculation process and accelerating the consensus-reaching procedure. Next, the weights of experts and subgroups are determined by utilizing expert confidence degree and subgroup cohesion. This approach effectively distinguishes the importance of varying experts and subgroups. Moreover, three tiers of consensus levels are introduced through cosine similarity, facilitating the rapid identification of DMs who need to adjust. Further, by analyzing the monotonicity of the comprehensive preference value, a flexible feedback adjustment mechanism is designed, which provides multiple options for DMs to adjust. Finally, the effectiveness of the proposed method is verified by numerical examples, and the strengths of the proposed approach are emphasized through comparative analysis.

Admittedly, there are still some limitations that need to be addressed. For instance, the non-cooperative behavior of DMs remains undiscussed, and there may be a certain amount of information loss in the data conversion process. Furthermore, the consistency of preference information following feedback adjustment is not explored. Hence, it is crucial for future research to focus on managing non-cooperative behavior, minimizing information loss, ensuring preference information consistency [59], and extending the application of the proposed method to a broader range of heterogeneous preference environments.

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Prospect theory-based large-scale group decision-making method with heterogeneous preference relations

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Heterogeneous preference relations

2.2 Prospect theory

3.1 HPRs-based prospect value function

3.2 The solution of priority vector

4.1 Fuzzy C-means clustering algorithm

4.2.1 Determining the weights of clusters and individuals

Table 1 The monotonicity of each function

5.1 Numerical example

References

Table 1
The monotonicity of each function