A two-stage consensus model based on personalized individual semantics under linguistic preference relations with self-confidence

Abstract

Linguistic preference relations with self-confidence (LPRs-SC) are the preference relation that can reflect the decision maker’s (DM) confidence psychology and has received widespread attention for their simple form and multiple information. Currently, arithmetic studies of LPRs-SC are conducted separately for preference relations and self-confidence. In addition, personalized individual semantics (PIS) is an important tool in large-scale decision-making to reflect the differences in the semantic understanding of DMs. However, the confidence level in LPRs-SC limits the preference relation to a certain extent and the linguistic representations of these two components are usually different. This means that it is not only necessary to propose an arithmetic rule that can express the restrictive relationship between the two but also to construct a model that can extract the PIS of preference relation and confidence respectively. Besides, we constructed a two-stage consensus reaching process (CRP) based on the specificity of the LPRs-SC structure when enhancing group harmony. The process takes self-confidence as an independent source of information, delineates the adjusted categories in detail, and builds an adjustment model accordingly. Finally, the example and comparative analyses verify the merits of the proposed PIS in terms of consistency enhancement and CRP in terms of speed and accuracy harmonization.

Keywords

Personalized individual semantics linguistic preference relations with self-confidence consensus reaching process large scale decision making

1 Introduction

Large-scale decision making (LSDM) is based on the traditional group decision making (GDM) in which more DMs with professional views on the problem to be solved are organized to participate in the process [5]. When the context is more complex or there are more alternatives, it requires more background knowledge and stronger personal ability for the experts to give a direct evaluation, which is obviously time-consuming and demanding. In this case, preference relations have been proposed to cope with such issues by narrowing down the DMs’ focus to pairs of alternatives, and allowing them to make two-by-two comparisons to give their own more favourable evaluations [31, 37]. So far, based on fuzzy preference relations [2 , 30], and multiplicative preference relations [27, 32], scholars developed different types of preference relations for different application scenarios, and applied them to the decision-making process. For example, natural language is an important way for human beings to exchange ideas and communicate opinions. Therefore, researchers have used linguistic terms to represent pairwise comparisons to construct linguistic preference relations (LPRs) [6 , 36]. With the increasingly refined linguistic descriptions of decision-makers, Guo et al. proposed double hierarchy linguistic preference relation (DHLPR) [11], probabilistic double hierarchy linguistic preference relation (PDHLPR) [10] and widely used them in the fields of educational model selection and intelligent healthcare. The double hierarchy linguistic form extends the descriptive dimension of LPR by introducing auxiliary terms and improves the fit of LPR with natural language. This makes it more adaptable to the complex expression requirements of decision makers. However, although the above preference relations continuously improve the expression system from the perspective of DMs’ descriptive habits, they lack the consideration of DMs’ psychological factors.

It is well known that most of the information processed in the decision-making process comes from the evaluations provided by the DM, and the DM’s evaluation of alternatives in a particular form not only conveys the perception of the problem to be solved but includes certain psychological factors. In addition to people’s level of risk aversion [4], researchers often discuss the impact of DM’s self-confidence on the decision-making process [9, 34]. This is because self-confidence can show the DM’s mastery of the decision problem, personal assessment habits, and other hidden information [1, 14]. To better combine LPRs and the psychology of self-confidence, Liu et al. [22] proposed a new form of preference information: LPRs-SC. It consists of two parts: the LPR that expresses the DM’s preference information about alternatives and a measure of self-confidence level for this evaluation of the LPR. For example, the evaluation (good, slightly high) of the alternative pair can be interpreted as the DM’s belief that is better than, and the confidence in this evaluation is slightly higher. LPRs-SC has attracted many researchers to a series of discussions on various aspects of its properties and operations due to its simple format and diversity of information [23, 35]. First and foremost is the consistency that all preference relations must be examined. Xu et al.’s discussion of additive consistency includes not only the definition of consistency indicators but also the arithmetic rules of LPRs-SC and the adjustment program for inconsistent situations [35]. In addition, the CRP can promote team cohesion and increase participant satisfaction. The CRP in the LPRs-SC environment has also been discussed to some extent, Zhu et al. proposed a consensus measure and a new weight allocation method that considers both the experts’ confidence level and preference value [41], but the proposed arithmetic rule does not take into account the effect of confidence level on preference relation and the DM’s bias in language understanding. In fact, to address the second issue, many scholars have discussed it from different perspectives. Chen et al. propose customized individual semantics with the help of flexible conversion of CLEs (comparative linguistic expressions) to HFLTS (hesitant fuzzy linguistic term set) [3]; the personalized semantic extraction method is a numerical scaling function obtained from an optimization-driven model with the aim of maximizing the degree of consistency of preference relations proposed by Li et al. [18]. Jing et al. applied this idea to a heterogeneous LPRs-SC environment and extracted the different understandings of linguistic items by decision-makers [15]. However, the PISs treatment of preference relation and confidence level are the same in this approach. In general, the language used by DMs to describe preferences is different from the language used to describe confidence. Hence, it is not reasonable to consider both parts of the PIS as the same when extracting the PIS for LPRs-SC.

All of the above studies have significantly contributed and advanced the development of large-scale decision-making and preference relations, but the following three research gaps remain:

The current research on the arithmetic rules of LPRs-SC calculates the two parts of preference relations and self-confidence level separately. However, the self-confidence level of LPRs-SC is a restriction on the preference relation, and calculating it separately will destroy the integrity of LPRs-SC. A rule that embodies the limits of confidence levels on preference relations in the arithmetic process is necessary to be proposed.

PIS can be extracted from the information provided by the DM, reflecting the different understanding of the same linguistic variable by different individuals [8 , 40]. Simply put, not everyone thinks that “good” corresponds to a real value of 0.7. As mentioned above, the language set used by DMs to provide ratings of pairwise comparisons is different from the language set used to characterize confidence. Therefore, a model should be proposed to extract the PIS for preference relations and confidence levels separately.

The CRP is an important step in the decision-making process to eliminate the gap of opinion and enhance group consensus [13 , 33]. The specificity of the structure of LPRs-SC will make CRP more complicated. In addition to the fact that DMs with similar preference relations are more likely to exchange preference opinions, DMs with similar levels of self-confidence are also more harmonious in exchanging self-confidence information. Then, when constructing the CRP of LPRs-SC, taking self-confidence level as an accessory information of preference relation and adjusting the whole matrix in the feedback mechanism will result in a redundant adjustment of self-confidence level. Accordingly, it is necessary to construct a more accurate CRP that takes into account the structural characteristics of LPRs-SC.

To fill the above research gaps, the objective of this paper is to propose improved arithmetic rules, PIS extraction models, and two-stage CRPs. Therefore, the main contributions and innovations are emphasized as follows:

We believe that when DMs deal with each other, those who are more confident in their evaluations will hold more power in discourse. Following this idea, we improve the arithmetic rules of LPRs-SC to incorporate the effect of confidence level on preference relations into the operation of preference relations.

We propose a PIS extraction model that can adapt to the two parts of LPRs-SC from different language sets. By defining the consistency of confidence level and using this definition to drive the optimization model, PIS of confidence level can be obtained. Based on this, the consistency index of preference relation is defined, and using this definition to drive the optimization model to obtain the PIS of preference relation.

A two-stage CRP that satisfies the structural characteristics of LPRs-SC is proposed. The measurement stage constructs a two-dimensional coordinate-based consensus level by treating the confident consensus level as independent information. The recognition stage first roughly divides the groups based on the relationship between the consensus level and the group consensus level. Then, the subgroups were classified twice using the K-means method. The feedback phase constructs an adjustment model based on the group classification obtained in the recognition phase separately to improve the accuracy of the feedback process. The results show that the proposed CRP can achieve a higher harmony between the speed of consensus and the amount of adjustment.

The rest of the paper is organized as follows. In Section 2, we introduce the basic definitions, consistency, and consensus metrics of existing LPRs-SC. Based on the lack of existing research, we propose an improved consistency metric and PIS extraction model for LPRs-SC in Section 3. A two-stage hybrid CRP of LPRs-SC based on PIS is constructed in Section 4, and a numerical example and comparatively analyzed is provided in Section 5 to demonstrate the practicability and superiority of the proposed model. Finally, Section 6 analyzes the limitations and future outlook of this paper.

2 Preliminaries

In this section, we systematically present the conversion rules between languages and values, the preference relations used in this paper, and their associated metrics.

2.1 2-tuple linguistic model and numerical scale model

To build a bridge between linguistic terms and real numbers, Herrera and Martínez [12] proposed a 2-tuple linguistic representation model $(s_{l}, α) \in \bar{S} = S \times [- 0.5, 0.5)$ , which solves the problem that numerical values cannot be transformed into linguistic terms in a one-to-one correspondence. S ={ s₀, s₁, ⋯ , s_g } in the model is the linguistic set, while the correspondence between (s_l, α) and the value β ∈ [0, g] is determined by the function Δ and its inverse function Δ^-1: $Δ : [0, g] \to \bar{S}$ $Δ (β) = (s_{l}, α), with {\begin{matrix} s_{l}, & l = round (β) \\ α = β - l & α \in [- 0.5, 0.5) \end{matrix}$ $Δ^{- 1} : \bar{S} \to [0, g]$ $Δ^{- 1} (s_{l}, α) = l + α = β$

Without loss of generality, negation operator Neg ((s_l, α)) = (g - (Δ^-1 (s_l, α))) and round (·) is the usual rounding operation.

The numerical scale model $\bar{NS} : \bar{S} \to R$ is an improved model proposed by Dong et al. to provide a more convenient representation for decision makers with different linguistic understanding and to make the binary language model more generalizable [6, 7]. The numerical scale $\bar{NS}$ on $\bar{S}$ for $(s_{l}, α) \in \bar{S}$ is defined by $\begin{matrix} \bar{NS} (s_{l}, α) \\ = {\begin{matrix} NS (s_{l}) + α \times (NS (s_{l + 1}) - NS (s_{l})), & α ⩾ 0 \\ NS (s_{l}) + α \times (NS (s_{l}) - NS (s_{l - 1})), & α < 0 \end{matrix} \end{matrix}$ where NS (s_l) is called the numerical index of s_l. The inverse operator of an ordered $\bar{NS}$ is denoted by ${\bar{NS}}^{- 1} : R \to \bar{S}$ .

The model takes into account the diversity of conversions between subscripts and real values of linguistic variables, which lays the foundation for subsequent work to reflect personalized information.

In fact, it is unrealistic to know the numerical perception of each DM on linguistic variables in advance, otherwise the use of linguistic variables would be meaningless. Based on this, Li et al. [18] proposed PIS and investigated the application of PIS in GDM, which is dedicated to mining personalized semantic information of DMs from existing decision data.

2.2 Linguistic preference relation with self-confidence

The LPRs-SC proposed by Liu et al. [22] is commonly known as a linguistic preference relation to which a level of self-confidence is added as supplementary information. For convenience and without loss of generality, it is stipulated that DMs use S = {l_α|α = - γ, ⋯ , 0, ⋯ , γ} and S^SL = {s_i|i = 0, 1, ⋯ , g} (g = 2τ) to denote their preference relations and self-confidence levels, respectively. The definition of LPRs-SC is given below:

Definition 1. [22] A matrix T = (t_ij, y_ij) _n×n is called an LPRs-SC if its elements have two components: the first component t_ij ∈ S represents the linguistic preference of the alternative x_i over x_j, and the second component y_ij ∈ S^SL represents the self-confidence level associated with the first component t_ij. The following conditions are assumed: $\begin{matrix} t_{ij} \oplus t_{ji} = l_{0}, t_{ii} = l_{0}, y_{ij} = y_{ji} and y_{ii} = s_{g} for \forall i, j \in N . \end{matrix}$

To measure the consistency of LPRs-SC, the operational laws of 2-tuples of LPRs-SC should be introduced first.

Definition 2. [35] Let (l_i, s_α), (l_j, s_β) be two 2-tuples, l_i and l_j are the linguistic preference values, and s_α, s_β are the corresponding self-confidence levels, where s_α, s_β ∈ S^SL, μ, μ₁, μ₂ ∈ [0, 1]. Then, here are the following rules of operation:

(l_i, s_α) ⊕ (l_j, s_β) = (l_i ⊕ l_j, min { s_α, s_β }) = (l_i+j, min { s_α, s_β });

(l_i, s_α) ⊕ l_k = (l_i, s_α) ⊕ (l_k, s_g) = (l_i+k, s_α);

μ ((l_i, s_α)) = (μl_i, s_α) = (l_μi, s_α);

μ₁ (l_i, s_α) ⊕ μ₂ (l_j, s_β) = (μ₁l_i ⊕ μ₂l_j, min { s_α, s_β }) = (l_{μ₁*i+μ₂*j}, min { s_α, s_β });

(l_i, s_α) = (l_-i, s_g-α).

Remark 1. Obviously, the result of the operational laws of 2-tuples is still the 2-tuples. (1)–(4) are well understood according to the operation rules of linguistic variables. As for rule (5), an example can help to understand it: the expert gives (l₂, s₆) for the comparison of options A₁ and A₂ (S^SL = { s₀, ⋯ , s₆ }), so that means the expert considers A₁ to be more preferable than A₂ and the level of confidence in this evaluation is s₆ (totally confident). Accordingly, the expert’s level of confidence in the evaluation of A₂ as preferable to A₁, i.e., l_-2 suld be s₀ (diffident).

2.3 Consistency and consensus measure of LPRs-SC

The level of consistency is a key indicator of the presence of mutually exclusive information in a preference relation. Based on Definition 2, the definition of the LPRs-SC consistency metric that takes into account the experts’ multiple self-confidence levels is given below:

Definition 3. [35] Let T = (t_ij, y_ij) _N×N and $\hat{z} = {({\hat{z}}_{ij}, {\hat{s}}_{ij})}_{N \times N}$ be two LPRs-SC, where $({\hat{z}}_{ij}, {\hat{s}}_{ij}) = (\frac{1}{N} \oplus_{k = 1}^{N} (t_{ik} \oplus t_{kj}), (\min {y_{ik}, y_{kj} |$ $k = 1, \dots, N} (i \neq j), y_{ij}, {\tilde{s}}_{ij} \in S^{SL}$ . Then the ACI (T) can be defined as follows: $\begin{matrix} \begin{matrix} ACI (T) = \frac{2}{N (N - 1)} \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N} \frac{Δ^{- 1} (y_{ij})}{g} \end{matrix} \\ (\frac{| index (t_{ij}) - index ({\hat{z}}_{ij}) |}{2 γ}) . \end{matrix}$ where Δ^-1 (·) represents language scale functions and index (·) indicates the subscript of the language variable.

A suitable consensus level not only needs to reflect the average level of similarity but needs to consider the effect of variance fluctuations. The consensus measure proposed by Zhong et al. [39] is used in the paper. To make the method more consistent with the LPRs-SC structural characteristics, we use the vectorial consensus level.

Let $T^{k} = {(t_{ij}^{k}, y_{ij}^{k})}_{N \times N}$ be a LPRs-SC provided by the DM e^k (k = 1, ⋯ , M), then the similarity of preference relations sim_p (·) and the similarity of confidence level sim_c (·) between any two DMs can be defined as:

$\begin{matrix} \begin{matrix} \begin{matrix} {sim}_{p} (T^{α}, T^{β}) \end{matrix} \end{matrix} \\ \begin{matrix} = \frac{1}{N (N - 1)} \sum_{i, j = 1}^{N} (1 - | f^{α} (T_{ij}^{α}) - f^{β} (T_{ij}^{β}) |) \end{matrix} \end{matrix}$ (1)

$\begin{matrix} \begin{matrix} \begin{matrix} {sim}_{c} (T^{α}, T^{β}) \end{matrix} \end{matrix} \\ \begin{matrix} = \frac{1}{N (N - 1)} \sum_{i, j = 1}^{N} (1 - | g^{α} (y_{ij}^{α}) - g^{β} (y_{ij}^{β}) |) \end{matrix} \end{matrix}$ (2)

Definition 4. [39] Let $T^{k} = {(t_{ij}^{k}, y_{ij}^{k})}_{N \times N}$ be a LPRs-SC provided by the DM e^k (k = 1, ⋯ , M), then the consensus level (CL_p (T^k) , CL_c (T^k)) of e^k is defined as:

$\begin{matrix} \begin{matrix} {CL}_{p} (T^{k}) = \frac{{(mean (\vec{{SD}_{p}^{k}}))}^{2}}{mean (\vec{{SD}_{p}^{k}}) + var (\vec{{SD}_{p}^{k}})} \end{matrix} \end{matrix}$ (3)

$\begin{matrix} \begin{matrix} {CL}_{c} (T^{k}) = \frac{{(mean (\vec{{SD}_{c}^{k}}))}^{2}}{mean (\vec{{SD}_{c}^{k}}) + var (\vec{{SD}_{c}^{k}})} \end{matrix} \end{matrix}$ (4) where $\vec{{SD}_{p}^{k}} = ({sim}_{p} (T^{k}, T^{1}), \dots, {sim}_{p} (T^{k}, T^{M}))$ $\vec{{SD}_{c}^{k}} = ({sim}_{c} (T^{k}, T^{1}),, \dots, {sim}_{c} (T^{k}, T^{M}))$ $\begin{matrix} mean (\vec{{SD}_{p}^{k}}) = \frac{1}{M - 1} \sum_{h = 1, h \neq k}^{M} {sim}_{p} (T^{k}, T^{h}) \end{matrix}$ $\begin{matrix} mean (\vec{{SD}_{c}^{k}}) = \frac{1}{M - 1} \sum_{h = 1, h \neq k}^{M} {sim}_{c} (T^{k}, T^{h}) \end{matrix}$ $\begin{matrix} var (\vec{{SD}_{p}^{k}}) = \frac{1}{M - 1} \sum_{h = 1, h \neq k}^{M} ({sim}_{p} (T^{k}, T^{h}) \\ - mean (\vec{{SD}_{p}^{k}}))^{2} \end{matrix}$ $\begin{matrix} var (\vec{{SD}_{c}^{k}}) = \frac{1}{M - 1} \sum_{h = 1, h \neq k}^{M} ({sim}_{c} (T^{k}, T^{h}) \\ - mean (\vec{{SD}_{c}^{k}}))^{2} \end{matrix}$

CL_p (T^k) ∈ [0, 1] , CL_c (T^k) ∈ [0, 1], larger CL value indicates a higher level of consensus between the DM e^k and others.

3 A novel consistency index and PIS model of LPRs-SC

In this section, we improve the existing operation rules of LPRs-SC to mine out and apply the effect of self-confidence level on the arithmetic of preference relation. Based on the improvement of the arithmetic rules, a consistency index that conforms to the structural characteristics of LPRs-SC is proposed. The PIS obtained with such consistency metrics can solve the contradiction of two parts of LPRs-SC coming from different language sets but using the same PIS.

3.1 Improved operation rules of LPRs-SC

Although the operational law given in Definition 2 is highly applicable and easy to compute, for the two LPRs-SCs, the effect of the self-confidence level of both parties is not taken into account during the computation of the preference relation. Therefore, the improved law is given below:

Definition 5. Let (l_i, s_α), (l_j, s_β) be two 2-tuples, l_i and l_j are the linguistic preference values, and s_α, s_β are the corresponding self-confidence levels, where s_α, s_β ∈ S^SL, μ, μ₁, μ₂ ∈ [0, 1]. Then, here are the following rules of operation:

$(l_{i}, s_{α}) \oplus (l_{j}, s_{β}) = (l_{i} \oplus l_{j}, min {s_{α}, s_{β}}) = (l_{round (\min (i, j) + \frac{I_{s} (max (i, j))}{α + β} | j - i |)}, min {s_{α}, s_{β}})$ ;

$(l_{i}, s_{α}) \oplus l_{k} = (l_{i}, s_{α}) \oplus (l_{k}, s_{g}) = (l_{round (\min (i, k) + \frac{g}{α + g} | k - i |)}, s_{α})$

μ ((l_i, s_α)) = (μ l_i, s_α) = (l_μi, s_α);

$\begin{matrix} μ_{1} (l_{i}, s_{α}) \oplus μ_{2} (l_{j}, s_{β}) \\ = (μ_{1} l_{i} \oplus μ_{2} l_{j}, min {s_{α}, s_{β}}) \\ = (l_{round (\min (μ_{1} i, μ_{2} j) + \frac{I_{s} (max (μ_{1} i, μ_{2} j))}{α + β} | μ_{2} j - μ_{1} i |)} \\ min {s_{α}, s_{β}}) \end{matrix}$ ;

(l_i, s_α) = (l_-i, s_g-α).

Remark 2. I_s (·) represents the subscript of the language variable corresponding to the value in S^SL. In contrast to Definition 2, this law considers the effect of the confidence levels of both parties in the calculation of the preference relation. The proportion of the confidence level is used as the proportion of the corresponding evaluation in the fusion preference, which can make the evaluation information of the more confident party be considered and adopted more.

To facilitate the understanding of the improvement of arithmetic rules, an example is given below:

Example 1. Let (l_-1, s₃), (l_-1, s₆) , (l₄, s₂) be three 2-tuples, l_-1, l₂ and l₄ are the linguistic preference values, and s₄, s₆, s₂ are the corresponding self-confidence levels, where s₄, s₆, s₂ ∈ S^SL = { s₀, s₁, s₂, s₃, s₄, s₅, s₆ } . If we use the calculation rules in Definition 2, $\begin{matrix} (l_{- 1}, s_{3}) \oplus (l_{4}, s_{2}) = (l_{- 1 + 4}, s_{2}) = (l_{3}, s_{2}) \end{matrix}$ $\begin{matrix} (l_{- 1}, s_{6}) \oplus (l_{4}, s_{2}) = (l_{- 1 + 4}, s_{2}) = (l_{3}, s_{2}) \end{matrix}$

If we use the calculation rules in Definition 5, $\begin{matrix} (l_{- 1}, s_{3}) \oplus (l_{4}, s_{2}) \\ = (l_{round (- 1 + (2 / 5) * 5)}, s_{2}) = (l_{1}, s_{2}) \end{matrix}$ $\begin{matrix} (l_{- 1}, s_{6}) \oplus (l_{4}, s_{2}) \\ = (l_{round (- 1 + (2 / 8) * 5)}, s_{2}) = (l_{0}, s_{2}) \end{matrix}$

It can be seen that the new calculation rules will give different results according to the different self-confidence levels in the 2-tuples. In other words, this operator fully takes into account the influence of the confidence level on the preference relation.

3.2 A novel consistency index and PIS model of LPRs-SC

Since the preference relation and self-confidence level of LPRs-SC are based on different sets of languages, the PIS should be divided into two parts: the PIS of preference information and the PIS of confidence level. Accordingly, there will be two consistency indicators. In addition, the computation rule of preference relation is affected by self-confidence, so the consistency indicator of self-confidence would be defined before giving the consistency indicator of preference relation.

Observing that the difference between the average confidence level of each alternative in the LPRs-SC and the confidence level of each alternative needs to be as small as possible, the following confidence consistency index is constructed in this paper.

Definition 6. Let T = (t_ij, y_ij) _N×N be an LPRs-SC, where t_ij ∈ S, y_ij ∈ S^SL. Then the consistency index of self-confidence SC - CI (T) can be defined as: $\begin{matrix} \begin{matrix} SC - CI (T) = \sum_{\begin{matrix} i, j = 1 \\ i > j \end{matrix}}^{N} | g (y_{ij}) - \frac{1}{2} \cdot \frac{1}{N - 1} (\sum_{\begin{matrix} m = 1 \\ m \neq i \end{matrix}}^{N} g (y_{mi}) \end{matrix} + \sum_{\begin{matrix} m = 1 \\ m \neq j \end{matrix}}^{N} g (y_{mj})) | \frac{1}{C_{N}^{2}} \end{matrix}$ where g (·) denotes the numerical scale function.

The PIS of self-confidence level is solved by minimizing the SC - CI (T) to drive the optimization model (M-1).

$\begin{matrix} min (SC - CI (T^{k})) = \sum_{\begin{matrix} i, j = 1 \\ i > j \end{matrix}}^{N} \frac{1}{C_{N}^{2}} | g^{k} (y_{ij}) \end{matrix}$ $\begin{matrix} - \frac{1}{2} \cdot \frac{1}{N - 1} (\sum_{\begin{matrix} m = 1 \\ m \neq i \end{matrix}}^{N} g^{k} (y_{mi}) + \sum_{\begin{matrix} m = 1 \\ m \neq j \end{matrix}}^{N} g^{k} (y_{mj})) | \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} g^{k} (s_{r}) + g^{k} (s_{2 τ - r}) = 1 r = 0, 1, \dots, 2 τ \\ g^{k} (s_{0}) = 0 \\ g^{k} (s_{τ}) = 0.5 \\ g^{k} (s_{2 τ}) = 1 \\ g^{k} (s_{r}) \in [\frac{r - 1}{2 τ}, \frac{r + 1}{2 τ}], r = 1, 2, \dots, 2 τ - 1; r \neq 0 \\ g^{k} (s_{r + 1}) - g^{k} (s_{r}) ⩾ λ, r = 0, 1, \dots, 2 τ \end{matrix} \end{matrix}$

where g^k (·) is the self-confidence PIS of DM e^k (k = 1, 2, ⋯ , M), the constraints are referenced from Li et al. [20] to ensure generalization and ordering of g^k (·), λ is a small constraint value.

After quantifying the self-confidence level, a consistency indicator of the preference relation under the influence of self-confidence is constructed as follows:

Definition 7. Let T = (t_ij, y_ij) _N×N be a LPRs-SC and $C = {(c_{ij}, s_{ij})}_{N \times N} = (\oplus_{k = 1}^{N} (t_{ik} \oplus t_{kj}),$ min {y_ik, y_kj|k = 1, 2, ⋯ , N}) _N×N be the consistent matrix corresponding to T, where y_ij, s_ij ∈ S^SL. Then, the CI (T) can be defined as follows:

$\begin{matrix} \begin{matrix} \begin{matrix} CI (T) = \end{matrix} 1 - \\ \sqrt[q]{\frac{2}{N (N - 1)} \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N} g (y_{ij}) \cdot {| f (t_{ij}) - f (c_{ij}) |}^{q}} \end{matrix} \end{matrix}$

where f (·) is the numerical scale function and q ∈ [1, ∞).

When the preference value t_ij is more distant from the consistent c_ij, it means that T is more inconsistent, in other words, the consistency degree of T is lower. This is in line with the fact that the higher the value of |f (t_ij) - f (c_ij) |, the lower the value of CI (T). This requires the CI (T) to be as big as possible to drive the optimization model to get f (·). Then, the optimization model (M-2) is constructed as:

$\begin{matrix} max CI (T^{k}) = 1 - \\ \sqrt[q]{\frac{2}{N (N - 1)} \sum_{i = 1}^{N - 1} \sum_{j = i + 1}^{N} g^{k} (y_{ij}) \cdot {| f^{k} (t_{ij}) - f^{k} (c_{ij}) |}^{q}} \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} f^{k} (s_{- t}) + f^{k} (s_{t}) = 1 t = - γ, \dots, γ \\ f^{k} (s_{- t}) = 0 \\ f^{k} (s_{0}) = 0.5 \\ f^{k} (s_{t}) = 1 \\ f^{k} (s_{r}) \in [\frac{t + γ - 1}{2 γ}, \frac{t + γ + 1}{2 γ}], t = - γ + 1, \dots, γ - 1 \\ f^{k} (s_{t + 1}) - f^{k} (s_{t}) ⩾ λ, t = - γ, \dots, γ \end{matrix} \end{matrix}$

where f^k (·) is the preference relation PIS of DM e^k (k = 1, 2, ⋯ , M).

The proposed model not only reflects the different understanding of semantics by DMs but notes the different semantic perceptions of DMs when it comes to preference evaluation versus confidence evaluation.

4 T-stage consensus model with PIS

After the above definition and modelling, different DMs have different f and g. The linguistic variables in LPRs-SC are accordingly transformed into FPRs-SC containing individual semantic information. This section proposes a two-stage CRP to improve the consensus level of the experts. The main approach is to generate a hybrid consensus feedback strategy that considers confidence and preference relations by performing consensus metrics on each of the two parts of the LPRs-SC and categorizing the experts.

4.1 Identification mechanisms

Obviously, the closer the consensus vector (CL_p (T^k) , CL_c (T^k)) is to (1, 1) indicates that the DM’s opinion is more harmonized with others. Following this idea, the group consensus level GCL is constructed.

$\begin{matrix} GCL = 1 - \\ \begin{matrix} \frac{1}{M} \sum_{k = 1}^{M} \sqrt{\frac{1}{2} ({(1 - {CL}_{p} (T^{k}))}^{2} + {(1 - {CL}_{c} (T^{k}))}^{2})} \end{matrix} \end{matrix}$ (5)

where GCL ∈ [0, 1].

The identification mechanism and the feedback mechanism need to be activated when the degree of group consensus level does not reach the desired value α. The identification mechanism is a process of selecting the objects to be adjusted according to certain rules among many DMs. The identification mechanism has an important impact on the subsequent stage.

The clusters of experts are mapped to a two-dimensional coordinate system by two aspects of consensus levels. Therefore, a preference-confidence consensus graph can be constructed to analyse the consensus levels of experts, where X-axis and Y-axis represent the preference relation consensus level CL_p (T^k) and the self-confidence consensus level CL_c (T^k), respectively. In addition, each point in Fig. 1 (a simple example) represents an expert involved in GDM.

Fig. 1

An example of the preference-confidence consensus graph.

The idea behind the construction of the mechanism is to quickly divide the group into two subgroups: the acceptable set G_A and the unacceptable set G_U. Individuals in G_A do not need to be adjusted during the consensus process and are the target direction of the adjustment of G_U. G_U is the group that contributes less to the GCL, in other words, their opinions need to be adjusted. Subsequently, the DMs in G_U are identified in detail to improve the accuracy of the identification mechanism. By analyzing the two consensus levels separately, the DMs in G_U are classified as: G₁ (both preference relation and self-confidence level need to be adjusted), G₂ (only preference relation needs to be adjusted), and G₃ (only self-confidence level needs to be adjusted). The detailed process and model are as follows:

(1) Preliminary Classification

A preliminary classification result {G_A, G_U} is obtained by comparing the normalized consensus coordinates of each DM with the GCL by Equation (5).

If e^k ∈ G_U, there is $\begin{matrix} 1 - \sqrt{\frac{1}{2} ({(1 - {CL}_{p} (T^{k}))}^{2} + {(1 - {CL}_{c} (T^{k}))}^{2})} \\ ⩽ GCL \end{matrix}$

If e^k ∈ G_A, there is $\begin{matrix} 1 - \sqrt{\frac{1}{2} ({(1 - {CL}_{p} (T^{k}))}^{2} + {(1 - {CL}_{c} (T^{k}))}^{2})} \\ > GCL \end{matrix}$

This is a preliminary categorization method designed to roughly and quickly delineate subgroups that need to be adjusted from the population. The dots in the Fig. 1(2) also indicate the DMs involved in the consensus process, and the different colours represent the different categories to which they belong. Orange for G_A, green for G_U. The expression of the circular arc is: $\begin{matrix} 1 - \sqrt{\frac{1}{2} ({(1 - {CL}_{p} (T^{k}))}^{2} + {(1 - {CL}_{c} (T^{k}))}^{2})} \\ = GCL \end{matrix}$

Table 1

Discriminatory conditions and corresponding processing methods of non-cooperative behaviour

Conditions	Judgment	Treatment
$\frac{(\sum_{i, j = 1}^{N} g^{k} (T_{ij}^{k}) - N)}{N (N - 1)} > g^{k} (s_{τ + 2}), τ > 2$	Non-cooperative behaviour	Weight penalty
$\frac{(\sum_{i, j = 1}^{N} g^{k} (T_{ij}^{k}) - N)}{N (N - 1)} > g^{k} (s_{τ + 1}), τ ⩽ 2$
$\frac{(\sum_{i, j = 1}^{N} g^{k} (T_{ij}^{k}) - N)}{N (N - 1)} ⩽ g^{k} (s_{τ + 2}), τ > 2$	Preference relation needs to be adjusted	Conventional optimization model
$\frac{(\sum_{i, j = 1}^{N} g^{k} (T_{ij}^{k}) - N)}{N (N - 1)} ⩽ g^{k} (s_{τ + 1}), τ ⩽ 2$

(2) Precise identification

The K-means classification (classification number K = 2) was performed for preference relationship and self-confidence level respectively, and the G_U was divided into three subsets {G₁, G₂, G₃} (as shown in Fig. 1(3)). The consensus level of preference relation and the self-confidence of DMs in G₁ are both low in the classification results. DMs in G₂, their preference relation is in the lower category, but the confidence consensus is in the higher category. That is, this kind of DMs are confident in their evaluations, but their preference information is not highly consistent with others. DMs in G₃ agree with other experts in terms of preference relations, but in terms of self-confidence, DMs contribute less to the consensus. This indicates that they lack sufficient confidence in their assessment, resulting in a lower overall consensus level than GCL. After classification, the green dots in Fig. 1(2) are divided into three colors. The green in Fig. 1(3) represents G₁, the blue for G₂, and the red dot for G₃.

4.2 Feedback mechanism

After determining the DM e^θ to be adjusted, the feedback model is built separately according to the adjustment needs of different categories. The feedback process for each category can be roughly divided into the following steps: (1) Finding the adjustment direction. Search for the DM e^d with the highest similarity to the adjusted from G_A. (2) Determine the adjustment interval. To fully respect the DM’s willingness to adjust, the adjustment interval is shortened from [0,1] to between e^d and e^θ. (3) Establish the optimization model. Minimize the amount of adjustment as the objective function. The adjustment interval and the corresponding consensus level greater than the threshold are the constraints. The specific process is as follows:

(1) Adjustment model for G₃

For DM $e_{G_{3}}^{θ}$ in G₃, their preference relations have been agreed upon and only require some adjustment to the level of self-confidence. Find e^d in G_A with the highest similarity to the self-confidence matrix of $e_{G_{3}}^{θ}$ ( $\underset{e^{d} \in G_{A}}{argmax} {sim}_{c} (T^{d}, T^{θ})$ ), and let e^d as the adjustment direction of $e_{G_{3}}^{θ}$ . The optimization model (M-3) is constructed as:

$\begin{matrix} min {d (g^{θ} (\tilde{y_{ij}^{θ}}), g^{θ} (y_{ij}^{θ}))} \end{matrix}$ $\begin{matrix} = \min {\sum_{i, j = 1}^{N} | g^{θ} (\tilde{y_{ij}^{θ}}) - g^{θ} (y_{ij}^{θ}) |} \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} \begin{matrix} \tilde{y_{ij}^{θ}} \in [\min (g^{d} (y_{ij}^{d}), g^{θ} (y_{ij}^{θ})) \\ , \max (g^{d} (y_{ij}^{d}), g^{θ} (y_{ij}^{θ}))] \end{matrix} \\ {CL}_{c} (\tilde{T_{G_{3}}^{θ}}) = \frac{{(\underset{e^{l} \in Ω - {G_{1}, G_{3}}}{mean} (\vec{{SD}_{c}^{l}}))}^{2}}{\underset{e^{l} \in Ω - {G_{1}, G_{3}}}{mean} (\vec{{SD}_{c}^{l}}) + \underset{e^{l} \in Ω - {G_{1}, G_{3}}}{var} (\vec{{SD}_{c}^{l}})} > δ \end{matrix} \end{matrix}$

where δ is the expected threshold.

(2) Adjustment model forG₂

For DM $e_{G_{2}}^{θ}$ in G₂, they have a high level of confidence in their preference relation that deviates from GCL. First, it is necessary to judge the non-cooperative behaviour (as shown in the Table 1).

The conventional optimization model is to find the e^d with the highest similarity to $e_{G_{2}}^{θ}$ ‘s preference relation matrix in G_A, and take e^d as the direction of $e_{G_{2}}^{θ}$ . The optimization model (M-4) is constructed as follows:

$\begin{matrix} \begin{matrix} min {d (f^{θ} (\tilde{t_{ij}^{θ}}), f^{θ} (t_{ij}^{θ}))} \\ = \min {\sum_{i, j = 1}^{N} | f^{θ} (\tilde{t_{ij}^{θ}}) - f^{θ} (t_{ij}^{θ}) |} \end{matrix} \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} \tilde{z_{ij}^{θ}} \in [\min (f^{d} (t_{ij}^{d}), f^{θ} (t_{ij}^{θ})), \max (f^{d} (t_{ij}^{d}), f^{θ} (t_{ij}^{θ}))] \\ {CL}_{p} (\tilde{T_{G_{2}}^{θ}}) = \frac{{(\underset{e^{l} \in Ω - {G_{1}, G_{2}}}{mean} (\vec{{SD}_{p}^{l}}))}^{2}}{\underset{e^{l} \in Ω - {G_{1}, G_{2}}}{mean} (\vec{{SD}_{p}^{l}}) + \underset{e^{l} \in Ω - {G_{1}, G_{2}}}{var} (\vec{{SD}_{p}^{l}})} > δ \end{matrix} \end{matrix}$

If there is no non-cooperative behaviour in the group, the contribution of each DM to the GCL is equal. However, if there are DMs with non-cooperative behaviour, they need to be penalized in terms of weight. The idea of this penalty is to minimize the impact of non-cooperative behaviour on the GCL. The specific penalty is as follows: $\begin{matrix} \begin{matrix} λ_{θ^{'}} = \frac{1}{M} ({CL}_{p} (e_{G_{2}}^{θ^{'}}) - GCL); \end{matrix} \end{matrix}$ $\begin{matrix} λ_{Ω - θ^{'}} = \frac{1}{# Ω - θ^{'}} (1 - \sum_{k \in {θ^{'}}} λ_{k}) \end{matrix}$

where λ_θ′ is the weight of DMs with non-cooperative behaviour in $e_{G_{2}}^{θ^{'}}$ , and λ_Ω-θ′ denotes the weight of DMs other than those with non-cooperative behavior. # Ω - θ′ indicates the number of individuals in the set Ω - θ′.

(3) Adjustment model for G₁

Neither the preference relation nor the self-confidence of the DMs in G₁ reaches the GCL. If this class of DMs goes to find the adjustment directions for preference relations and confidence separately, it will destroy the integrity and coordination of the LPRs-SC structure. Therefore, this paper constructs a search method considering sequential ordering to find the adjustment direction e^d for $e_{G_{1}}^{θ}$ : $\begin{matrix} \underset{e^{l} \in G_{A}}{argmax} \\ (rank ({sim}_{p} (T^{θ}, T^{l})) + rank ({sim}_{c} (T^{θ}, T^{l}))) \end{matrix}$ where rank (·) denotes that the elements are sorted from smallest to largest and given ordinal numbers. This search method is for $e_{G_{1}}^{θ}$ to find the DM with the highest combined ranking of preference relation similarity and confidence similarity as e^d.

After finding e^d, the adjustment model (M-5) for the preference relation can be constructed as:

$\begin{matrix} min {d (f^{θ} (\tilde{t_{ij}^{θ}}), f^{θ} (t_{ij}^{θ}))} \end{matrix}$ $\begin{matrix} = \min {\sum_{i, j = 1}^{N} | f^{θ} (\tilde{t_{ij}^{θ}}) - f^{θ} (t_{ij}^{θ}) |} \end{matrix}$ $\begin{matrix} s . t . {\begin{matrix} \tilde{z_{ij}^{θ}} \in [\min (f^{d} (t_{ij}^{d}), f^{θ} (t_{ij}^{θ})), \max (f^{d} (t_{ij}^{d}), f^{θ} (t_{ij}^{θ}))] \\ {CL}_{p} (\tilde{T_{G_{2}}^{θ}}) = \frac{{(\underset{e^{l} \in Ω - {G_{1}, G_{2}}}{mean} (\vec{{SD}_{p}^{l}}))}^{2}}{\underset{e^{l} \in Ω - {G_{1}, G_{2}}}{mean} (\vec{{SD}_{p}^{l}}) + \underset{e^{l} \in Ω - {G_{1}, G_{2}}}{var} (\vec{{SD}_{p}^{l}})} > δ \end{matrix} \end{matrix}$

The adjusted model (M-6) for self-confidenceis:

$\begin{matrix} min {d (g^{θ} (\tilde{y_{ij}^{θ}}), g^{θ} (y_{ij}^{θ}))} = \min {\sum_{i, j = 1}^{N} | g^{θ} (\tilde{y_{ij}^{θ}}) - g^{θ} (y_{ij}^{θ}) |} \\ s . t . {\begin{matrix} \tilde{y_{ij}^{θ}} \in [\min (g^{d} (y_{ij}^{d}), g^{θ} (y_{ij}^{θ})), \max (g^{d} (y_{ij}^{d}), g^{θ} (y_{ij}^{θ}))] \\ {CL}_{c} (\tilde{T_{G_{3}}^{θ}}) = \frac{{(\underset{e^{l} \in Ω - {G_{1}, G_{3}}}{mean} (\vec{{SD}_{c}^{l}}))}^{2}}{\underset{e^{l} \in Ω - {G_{1}, G_{3}}}{mean} (\vec{{SD}_{c}^{l}}) + \underset{e^{l} \in Ω - {G_{1}, G_{3}}}{var} (\vec{{SD}_{c}^{l}})} > δ \end{matrix} \end{matrix}$

5 Two-stage hybrid consensus reaching algorithm based on PIS

Based on the above theory and model, a two-stage CRP algorithm for LPRs-SC based on PIS is proposed (Fig. 2 for the flowchart).

Fig. 2

Flowchart of PIS-based two-stage hybrid CRP.

Algorithm 1
Input: LPRs-SCs $T^{k} = {(T_{ij}^{k}, y_{ij}^{k})}_{N \times N}$ provided by M DM e ^k ( i = 1, 2, ⋯ , N ; j = 1, 2, ⋯ , N ; k = 1, 2, ⋯ , M ) ; α ; δ .
Output: PIS of self-confidence g ^k and preference relation f ^k; Adjusted LPRs-SCs; Number of iterations T ;.
Step 1. PIS extraction process for T ^k.
Step 1.1. The PIS of confidence level g ^k was obtained by driving optimization model (M-1).
Step 1.2. The PIS of preference relation f ^k was obtained by driving optimization model (M-2).
Step 2. Consensus measure stage
Step 2.1. Calculate similarity matrices ${({SD}_{p}^{k})}_{N \times N}$ ${({SD}_{c}^{k})}_{N \times N}$ according to Equations (1), (2).
Step 2.2. Get CL _p ( T ^k) and CL _c ( T ^k) according to Equations (3), (4).
Step 2.3. The GCL is obtained according to Equation (5).
If GCL < α, continue the following steps, otherwise skip to Step 5.
Step 3. Identification mechanism
Step 3.1. Preliminary identification was performed to obtain G _A and G _U.
Step 3.2. By K-means classification (K = 2), the G _U is accurately recognized to obtain G ₁, G ₂ and G ₃.
Step 4. Feedback mechanism
Step 4.1. Search for moderators e ^d according to the different adjustment needs of DM in different categories.
Step 4.2. Solve the optimization model (M-3) –(M-6) to get a new $\tilde{T^{k}}$ and return to Step 2.
Step 5. Output $\tilde{T^{k}}$ , g ^k and f ^k.

6 Illustrative example and comparative analysis

In this section, we will apply the proposed method in a case study to illustrate the usage of the proposed method and compare it with the results without PIS and without precise identification.

6.1 Illustrative example

Online shopping refers to the purchase and transaction of goods or services through the Internet platform. The rapid development of the Internet has made online shopping one of the most common and popular shopping methods in modern life. The considerable profitability has led to more and more merchants flocking to e-commerce platforms, and competition has become more intense. In the age structure of online shoers,eople are the main consumers. College students, as the main force of young people, have huge consumption potential. Therefore, it is important to study the characteristics of merchants that college students value most when they shop online, which will help merchants stand firm in the fierce competition.

We organized a group of 20 college students to form the DMs group {E^k|k = 1, 2, ⋯ , 20 } and conducted a series of surveys during which four alternative characteristics were considered: Celebrity endorsement (A₁), Platform composite score (A₂), Public perception (A₃), Quality of customer service (A₄). And 20 LPRs-SCs T^kwere constructed based on the language set L of preference relations and the language set S of confidence levels (see Appendix A for details). In this example, the parameters required in the process are set as follows: q = 1 ; α = 0.8 ; δ = 0.8. $\begin{matrix} \begin{matrix} L = \\ {\begin{matrix} \begin{matrix} l_{- 4} = extremely poor & l_{- 3} = very poor \\ l_{- 2} = poor & l_{- 1} = slightly poor \\ l_{0} = fair & l_{1} = slightly good \\ l_{2} = good & l_{3} = very good \end{matrix} \\ l_{4} = extremely good \end{matrix}} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} S = \\ {\begin{matrix} \begin{matrix} \begin{matrix} s_{0} = none & s_{1} = very low & s_{2} = low \\ s_{3} = medium & s_{4} = high & s_{5} = very high \end{matrix} \end{matrix} \\ s_{6} = perfect \end{matrix}} \end{matrix} \end{matrix}$

6.2 Two-stage hybrid consensus reaching process based on PIS

The consensus algorithm proposed in Section 5 is utilized to solve this problem.

Step 1. PIS extraction process

First, the PIS g^k for the confidence level was obtained by solving model (M-1), as shown in Table 2. After confirming g^k, the PIS f^k of preference relation is solved by model (M-2), as shown in Table 3.

Table 2
PIS of self-confidence

k g^k (s₀) g^k (s₁) g^k (s₂) g^k (s₃) g^k (s₄) g^k (s₅) g^k (s₆)

1 0 0.1433 0.49 0.5 1

2 0 0.3333 0.49 0.5 0.51 0.6667 1

3 0 0.332 0.416 0.5 584 0.668 1

4 0 0.3333 0.49 0.5 0.51 0.6667 1

5 0 0.3333 0.49 0.5 1

6 0 0.3333 0.49 0.5 0.51 0.6667 1

7 0 0.3333 0.3989 0.5 0.6011 0.6667 1

8 0 0.3333 0.49 0.5 0.51 0.6667 1

9 0 0.3333 0.49 0.5 0.51 0.6667 1

10 0 0.333 0.4666 0.5 0.5334 0.667 1

11 0 0.3321 0.4161 0.5 0.5839 0.6679 1

12 0 0.3331 0.4334 0.5 0.5666 0.6669 1

13 0 0.3333 0.49 0.5 0.51 0.6667 1

14 0 0.1725 0.1902 0.5 0.8098 0.8275 1

15 0 0.3333 0.49 0.5 0.51 0.6667 1

16 0 0.3333 0.49 0.5 0.51 0.6667 1

17 0 0.3333 0.3433 0.5 0.6567 0.6667 1

18 0 0.3333 0.49 0.5 0.51 0.6667 1

19 0 0.3325 0.4163 0.5 0.5837 0.6675 1

20 0 0.3333 0.49 0.5 0.51 0.6667 1

k	g^k (s₁)	g^k (s₂)	g^k (s₃)	g^k (s₄)	g^k (s₅)	g^k (s₆)
1	0.1433	0.49	0.5	1
2	0.3333	0.49	0.5	0.51	0.6667	1
3	0.332	0.416	0.5	584	0.668	1
4	0.3333	0.49	0.5	0.51	0.6667	1
5	0.3333	0.49	0.5			1
6	0.3333	0.49	0.5	0.51	0.6667	1
7	0.3333	0.3989	0.5	0.6011	0.6667	1
8	0.3333	0.49	0.5	0.51	0.6667	1
9	0.3333	0.49	0.5	0.51	0.6667	1
10	0.333	0.4666	0.5	0.5334	0.667	1
11	0.3321	0.4161	0.5	0.5839	0.6679	1
12	0.3331	0.4334	0.5	0.5666	0.6669	1
13	0.3333	0.49	0.5	0.51	0.6667	1
14	0.1725	0.1902	0.5	0.8098	0.8275	1
15	0.3333	0.49	0.5	0.51	0.6667	1
16	0.3333	0.49	0.5	0.51	0.6667	1
17	0.3333	0.3433	0.5	0.6567	0.6667	1
18	0.3333	0.49	0.5	0.51	0.6667	1
19	0.3325	0.4163	0.5	0.5837	0.6675	1
20	0.3333	0.49	0.5	0.51	0.6667	1

Table 3

PIS of preference relation f^k (s_h)

k\h	–4	–3	–2	–1	0	1	2	3	4
1	0	0.115	0.125	0.49	0.5	0.51	0.875	0.885	1
2	0	0.25	0.375	0.385	0.5	0.615	0.625	0.75	1
3	0	0.115	0.125	0.49	0.5	0.51	0.875	0.885	1
4	0	0.25	0.3143	0.49	0.5	0.51	0.6857	0.75	1
5	0	0.25	0.375	0.49	0.5	0.51	0.625	0.75	1
6	0	0.25	0.26	0.49	0.5	0.51	0.74	0.75	1
7	0	0.25	0.375	0.385	0.5	0.615	0.625	0.75	1
8	0	0.25	0.375	0.385	0.5	0.615	0.625	0.75	1
9	0	0.25	0.3151	0.49	0.5	0.51	0.6849	0.75	1
10	0	0.25	0.26	0.49	0.5	0.51	0.74	0.75	1
11	0	0.1424	0.375	0.49	0.5	0.51	0.625	0.8576	1
12	0	0.25	0.375	0.49	0.5	0.51	0.625	0.75	1
13	0	0.1428	0.375	0.385	0.5	0.615	0.625	0.8572	1
14	0	0.25	0.26	0.49	0.5	0.51	0.74	0.75	1
15	0	0.115	0.125	0.49	0.5	0.	0.875	0.885	1
16	0	0.25	0.375	0.49	0.5	0.51	0.625	0.75	1
17	0	0.25	0.26	0.27	0.5	0.73	0.74	0.75	1
18	0	0.1494	0.375	0.385	0.5	0.615	0.625	0.8506	1
19	0	0.0732	0.125	0.49	0.5	0.51	0.875	0.9268	1
20	0	0.25	0.26	0.27	0.5	0.73	0.74	0.75	1

Step 2. Consensus measure stage

Firstly, the preference relation similarity ${({SD}_{p}^{k})}_{N \times N}$ and the confidence level similarity ${({SD}_{c}^{k})}_{N \times N}$ are calculated by Equations (1), (2). Secondly, the consensus level of preference relation CL_p (T^k) and the self-confidence CL_c (T^k) are calculated by Equations (3), (4), respectively, as shown in Table 4. The GCL is obtained from Equation (5): GCL = 0.7573 < α = 0.8. Thus, the identification mechanism and feedback mechanism need to be activated to adjust the DM group so that the GCL reaches the threshold.

Table 4

Initial consensus level of 20 DMs

k	1	2	3	4	5
CL ( T ^k)	(0.62,0.78)	(0.74,0.85)	(0.68,0.84)	(0.68,0.85)	(0.67,0.87)
k	6	7	8	9	10
CL ( T ^k)	(0.73,0.81)	(0.76,0.79)	(0.73,0.84)	(0.76,0.75)	(0.71,0.82)
k	11	12	13	14	15
CL ( T ^k)	(0.73,0.83)	(0.77,0.79)	(0.77,0.84)	(0.76,0.68)	(0.67,0.86)
k	16	17	18	19	20
CL ( T ^k)	(0.67,0.87)	(0.64,0.75)	(0.73,0.85)	(0.68,0.78)	(0.72,0.86)

Step 3. The first iteration in the CRP

The preliminary classification led to the DM group being classified as: $\begin{matrix} G_{A} = {E^{2}, E^{6}, E^{7}, E^{8}, E^{10}, E^{11}, E^{12}, E^{13}, \\ E^{16}, E^{18}, E^{20}}; \end{matrix}$ $\begin{matrix} G_{U} = {E^{1}, E^{3}, E^{4}, E^{5}, E^{9}, E^{14}, E^{15}, E^{17}, E^{19}} . \end{matrix}$

The precise identification allows G_U to be divided into G₁ ={ E¹, E¹⁷, E¹⁹ }, G₂ = { E³, E⁴, E⁵, E¹⁵ } , G₃ = {E⁹, E¹⁴} (As shown in Fig. 3(a)). For the DMs in G₁, {E¹⁸, E⁸, E²} is selected in G_A according to the ordinal ranking as the adjustment direction. The preference relation and self-confidence are respectively adjusted by launching optimization model (M-5) and (M-6). For DMs in G₂, the mean self-confidence level of {E³, E⁴, E⁵, E¹⁵} are {0.584, 0.6095, 0.5261, 0.4706}, which is less than {0.6680, 0.6667, 0.6667, 0.6667}, respectively. That is, neither DM in G₂ is non-cooperative behaviour and a routine identification adjustment for them is sufficient.

Fig. 3

Variation of consensus graph in numerical example.

The { E², E¹², E¹², E⁸ } in G_A with the highest similarity to the preference relation of G₂ will be used as the adjustment direction. Subsequently, the preference relations are adjusted by launching optimization model (M-4). As for DMs in G₃, only self-confidence needs to be adjusted. DM in {E⁷, E⁶}, which has the highest similarity to G₃ ‘s self-confidence level, is selected as the adjustment direction in G_A, and the optimization model (M-3) is performed for G₃ to adjust the self-confidence.

The consensus level of $\tilde{T^{k}}$ was recalculated after the adjustment (CLs are shown in Table 5). The group consensus level GCL = 0.7867 is below the threshold 0.8, initiating the second round of identification mechanism and feedback mechanism.

Table 5

Consensus level of 20 DMs after the first round of adjustments

k	1	2	3	4	5
CL ( T ^k)	(0.75,0.81)	(0.76,0.87)	(0.70,0.85)	(0.71,0.86)	(0.74,0.89)
k	6	7	8	9	10
CL ( T ^k)	(0.75,0.82)	(0.78,0.80)	(0.76,0.86)	(0.77,0.79)	(0.73,0.89)
k	11	12	13	15
CL ( T ^k)	(0.75,0.85)	(0.79,0.81)			(0.77,0.88)
k	16	17	18	19	20
CL ( T ^k)	(0.72,0.89)	(0.68,0.81)	(0.74,0.8	(0.70,0.79)	(0.74,0.87)

Step 4. The second iteration in the CRP

The preliminary classification led to the DM group being classified as: $\begin{matrix} G_{A} = {E^{2}, E^{5}, E^{7}, E^{8}, E^{10}, E^{11}, E^{12}, E^{13} \\ , E^{15}, E^{18}, E^{20}} \end{matrix}$ $\begin{matrix} G_{U} = {E^{1}, E^{3}, E^{4}, E^{6}, E^{9}, E^{14}, E^{16}, E^{17}, E^{19}} \end{matrix}$

The precise identification allows G_U to be divided into: G₁ ={ E¹⁷, E¹⁹ }, G₂ = { E³, E⁴, E¹⁶ } , G₃ = {E¹, E⁶, E⁹, E¹⁴} (shown in Fig. 3(b)).

For the DMs in G₁, {E²⁰, E²} is selected as the adjustment direction in G_A. At the same time, the preference relation and self-confidence are respectively adjusted by launching optimization model (M-5) and (M-6). For DMs in G₂, the mean self-confidence level of G₂ are {0.584, 0.6095, 0.5245}, which is less than {8, 0.667, 0.667}, respectively. That is, neither DM in G₂ is non-cooperative behaviour and a routine identification adjustment for them is sufficient. The { E², E¹², E⁸ } in G_A with the highest similarity to the preference relation of G₂ will be used as the adjustment direction. Subsequently, the preference relations are adjusted by launching optimization model (M-4). As for DMs in G₃, only self-confidence needs to be adjusted. DM in {E¹⁰, E⁸, E⁷, E¹⁰}, which has the highest similarity to G₃’s self-confidence level, is selected as the adjustment direction in G_A, and the optimization model (M-3) is performed for G₃ to adjust the self-confidence.

The consensus level of $\tilde{T^{k}}$ was recalculated after the adjustment (CLs are shown in Table 6). The group consensus level GCL = 0.8128 is higher than the threshold 0.8, ending the process and returning the austed $\tilde{F^{k}} = (\tilde{f^{k} (t_{ij}^{k})}, \tilde{g^{k} (y_{ij}^{k})})$ (shown in the Appendix B) and austed $\tilde{T^{k}}$ (shown in the Appendix C).

Table 6

Consensus level of 20 DMs after the second round of adjustments

k	1	2	3	4	5
CL ( T ^k)	(0.78,0.81)	(0.78,0.87)	(0.80,0.85)	(0.80,0.86)	(0.77,0.89)
k	6	7	8	9	10
CL ( T ^k)	(0.78,0.82)	(0.81,0.80)	(0.78,0.8	(0.80,0.79)	(0.73,0.89)
k	11	12	13	14	15
CL ( T ^k)	(0.78,0.85)	(0.82,0.81)			(0.80,0.88)
k	16	17	18	19	20
CL ( T ^k)	(0.81,0.89)	(0.80,0.81)	(0.77,0.87)	(0.81,0.79)	(0.76,0.87)

6.3 Discussion of results

The CRP is an essential process for overcoming the gap between the opinions of DM groups, and its purpose is to make the whole group’s evaluation of the alternatives more uniform. The blue, green, and red dots in Fig. 4 indicate the original data, the data after the first round of adjustment, and the final data, respectively. Some of the analysis is summarized below:

It can be seen that after two rounds of adjustment, the whole group is more aggregated and all of them are closer to the consensus optimal goal (1, 1). This suggests that the proposed CRP worked as it should and fully aggregated the opinions of the group of DMs.

The result of the decision making is A₃ ≻ A₂ ≻ A₄ ≻ A₁, which means that public perception is the first factor that college students consider when shopping online, followed by platform composite score, quality of customer service, and finally celebrity endorsement. Without access to the real thing, consumers will be more inclined to decide whether to buy a product based on the reviews of other consumers. Many stores have captured this message and used review brushing and cash rebates to increase favorable ratings on platforms. However, the development of the era of big data has led to the rapid expansion of social media, and college students who are willing to share their lives will have purchased high-quality goods or shoddy products on social media platforms. Therefore, despite the platform’s inflated favorable rate, college students can learn about the public perception of goods from a wider range of channels.

Fig. 4

Comparison of the consistency index values with PIS and without PIS.

6.4 Comparative analysis

To verify the f^k and g^k computational models proposed in this paper can make the consistency level of LPRs-SC as high as possible. The following is a comparison of the consistency level of T^k obtained using f^k and g^k with the $\hat{T^{k}}$ obtained using:

$p^{k} = {\frac{i + 4}{8} | i = - 4, \dots, - 1, 0, 1, \dots, 4}$ ;

$q^{k} = {\frac{i}{6} | i = 0, 1, \dots, 5, 6} (k = 1, 2, \dots, 20)$

In addition, the consensus process in Section 5.1 (CRP⁰) is compared with the CRP without precise identification (CRP¹) to explore whether precise identification contributes to the consensus process. It is well known that to evaluate the appropriateness of a CRP, both the speed of consensus increase and the amount of adjustment are indispensable. Sacrificing a large number of adjustments for a rapid increase in consensus speed is not considered, and similarly, conducting multiple rounds of consensus adjustments in order to ensure that the number of adjustments is low enough is undoubtedly a waste of resources. The method proposed in this paper is to sacrifice a small portion of the consensus increase speed in exchange for a substantial reduction in the amount of adjustment to achieve harmony between the two.

Figures 5 and 6 show the comparison of the LPRs-SC consistency metrics with or without PIS extraction and the comparison of the iterative process of CRP with or without precise identification, respectively. Comparative conclusions are analyzed below:

It is obvious that the consistency level of T^k after PIS extraction process is higher than the consistency level of $\hat{T^{k}}$ without PIS extraction (Fig. 5). This indicates that the method proposed in this paper effectively improves the consistency level of LPRs-SC.

To measure this harmony, a metric Rate is given, which is calculated as follows: $\begin{matrix} Rate = \frac{{GCL}_{final} - {GCL}_{initial}}{\frac{1}{2} (\frac{1}{20} \sum_{k = 1}^{20} | \tilde{f^{k} (T_{ij}^{k})} - f^{k} (T_{ij}^{k}) | + \frac{1}{20} \sum_{k = 1}^{20} | \tilde{g^{k} (y_{ij}^{k})} - g^{k} (y_{ij}^{k}) |)} . \end{matrix}$

Fig. 5

Comparison of the consensus iteration process, amount of adjustment, and Rate of CRP with and without accurate identification.

Fig. 6

Consensus iterative process for different values of α.

As can be seen from Fig. 6, CRP with precise identification not only has better consensus iteration speed but also has lower adjustment. This suggests that the inclusion of precise identification allows CRP to reach a higher consensus at a lower adjustment cost which is demonstrated by a higher Rate.

6.5 Sensitivity analysis

The subjectively set threshold value α plays a crucial role in the whole consensus process. To verify the stability of the proposed method, we analyzed the sensitivity with regard to the ranking of alternatives and the speed of consensus iteration by setting α = 0.8, α = 0.82, α = 0.84, α = 0.86, and α = 0.88.

The conclusions of the sensitivity analysis are as follows:

From Fig. 7, it can be seen that all CRPs reach the threshold after the fourth round of adjustment. This not only confirms the stability of the proposed method but also shows that the pre-set threshold has little effect on the speed of consensus reaching. Therefore, the decision planner can set the ideal group consensus level threshold according to the needs of the actual situation.

As can be seen in Fig. 8, the fluctuation of α. does not affect the fact that alternative A₃ resides in the preferred position. The results are also consistent for alternative A₁. residing in the bottom position except for α = 0.88. That is, the pre-set group consensus threshold did not affect college students’ perception of public perceptions as an important factor in their own online shopping, nor did it have much effect on the fact that college students do not care much about celebrity endorsements when shopping online.

Fig. 7

Radar plots of alternative orderings obtained for different values of α.

7 Conclusion

The inclusion of more stakeholders in the decision-making process makes traditional group decision-making more complex. In addition to considering the aggregation of opinions of DMs, the influence of some personalized characteristics and psychological factors on the decision-making process has become more obvious. In this general context, we propose a PIS extraction model and a two-stage hybrid CRP for PIS-based preference relation-confidence level consensus analysis in the context of LPRs-SC. First, since DMs use different sets of languages for their descriptions of preference evaluation and self-confidence psychology, we propose a PIS model for self-confidence level and a PIS model for preference relations influenced by self-confidence level, respectively. Second, considering the specificity of the structure of LPRs-SCs, two-dimensional vectors are used to represent the consensus level of LPRs-SCs in the consensus management stage. This approach treats the self-confidence level as independent information rather than subsidiary information of preference relations. Meanwhile, this paper proposes an accurate identification and feedback mechanism based on preference-confidence consensus. Different feedback models are customized according to the problems of different types of DMs as reflected by the degree of preference-confidence consensus, which improves the accuracy and speed of CRP. The above method is applied to the actual problem of investigating college students’ online shopping tendency factors to verify its feasibility and effectiveness. The experimental results show that the most important factor for college students to choose goods online is public perception, and the least important is celebrity endorsement. This is because college students are more concerned about whether they can buy better quality goods at a lower price when they make online purchases without financial resources. This requires a wide range of information and frequent trials, and college students, as the main component of the young group, are undoubtedly the strongest in the ability to obtain the required information on the Internet. Therefore, college students who are at the front end of information dissemination will highly value the posts enthusiastically shared by those who have already purchased, from which they can obtain a real public attitude. In addition, with the widespread popularization of the Internet in recent years, the spreading speed and breadth of some entertainment scandals have reached unprecedented heights, which makes the credibility of celebrities continue to fall. Although there are still a lot of enthusiastic fans following, as college students who already have basic logical judgment and mature values, they are more likely to trust the useful products they have found that buyers have shared on social platforms than the products recommended by celebrities who are not in the same level of consumption as they are. The results of the case analysis and comparative analysis show that the proposed method can not only extract personalized semantics that make the overall consistency of LPRs-SC higher, but also achieve a higher level of consensus by adjusting as little information as possible.

From the perspective of personalized semantics and CRP, this paper still has several limitations, which are mainly summarized as follows:

DMs are communicating with each other during the CRP, does any decision maker change his/her understanding of the language during the communication process? In other words, whether PIS changes due to communication among DMs is an issue not considered in this paper.

As a matter of fact, DMs with similar PIS will be more time-efficient in communication. However, although the personalized semantics and assertiveness characteristics of DMs are included in the CRP of this paper, the social relationship among DMs is not considered from the perspective of PIS.

Based on these limitations, several potential directions for future research are noted:

It is proposed that when constructing the dynamic PIS, the input FPRs-SC of the current round is converted back to the LPRs-SC by the PIS obtained in the previous round, and then the PIS extraction is performed. This not only keeps the consistency level of LPRs-SC at the highest level but also realizes the dynamic PIS extraction.

Combining the similarity matrix of PIS and social networks can be considered. The social network reflects the social relationships among the participants, while the PIS similarity matrix reflects the hidden linguistic preferences of the participants. By combining linguistic preferences with social networks, the relationships between participants can be considered from both explicit and implicit perspectives.

8 Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 72171002 and No. 61806001).

The Appendix is available in the electronic version of this article:.

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