1 Introduction

Group decision making (GDM) refers to the process by which multiple decision makers (DMs) select the best option from a set of feasible alternatives (Xu et al. 2021; Dong et al. 2022; Zhang and Li 2021). Usually, the individuals involved in decision making are not independent of each other, they are influenced by the opinions of other DMs connected to them in the social network. Therefore, the social network group decision making (SN-GDM) is proposed by Wu et al. in Wu and Chiclana (2014), Wu et al. (2015), Wu et al. (2017) to investigates the social relationships among DMs. In addition, the advancement of social media and information technology facilitates that large number of people in social network participate in decision making processes, which has contributed to social network large-group decision making (SN-LGDM) to become a hot research issue (Lu et al. 2022; Chao et al. 2021; Du et al. 2021). The main characteristics of SN-LGDM include: (1) A group of no fewer than 20 DMs participate in the decision making process; (2) There exist social relationships between the DMs (Yu et al. 2021; Chen et al. 2022). An approach to reduce the dimensional complexity of the large number of DMs involved in SN-LGDM is the application of clustering techniques to classify the DMs into clusters or subgroups sharing similar characteristics (Pan et al. 2022; Wu et al. 2021; Labella et al. 2018).

When there is no group consensus, subgroups obtained through clustering need to interact with each other, in what is known as consensus reaching process (CRP), to approximate their opinions, (Wan et al. 2020; Wu et al. 2022; Wu Wang et al 2021). As a group composed of several DMs, the subgroups exhibits more of a ‘group behavior’ than an ‘individual behavior’ in the process of opinion interaction. Normally, when individuals conflict with group opinions in a group, they are often pressured by the group to show conformity behavior, that is, they show behavior consistent with other members of the group (Bergstrom and Bak-Coleman 2019). However, after the large-scale DMs are divided into several subgroups, each subgroup often has its own goals (Zhang et al. 2021), which are usually different, and the individuals in the subgroup will subordinate to their own small group. Therefore, polarization or conflict of opinions is more likely to occur in SN-LGDM, and then it is difficult to reach consensus. This article aims to explore several behavior related to the CRP of subgroups in SN-LGDM.

The first aspect to consider is the interaction (feedback) mechanism in CRP. The feedback mechanism is effective in promoting the interaction of subgroup opinions to improve the consensus level (Cao et al. 2021; Sun et al. 2022; Cao et al. 2021). The structure of the traditional feedback mechanism is ‘central’ with the collective opinion as the interaction object (Pérez et al. 2014). Recently, a ‘bidirectional’ interaction mode has been developed, which let the paired DMs/subgroups use each other’s judgements as the feedback reference of the interaction (Dong and Cooper 2016; Dong et al. 2022; Tang et al. 2021). Compared with centralized interaction, the bidirectional feedback mechanism does not need a moderator to help DMs/subgroups interact and feedback, and it is applicable to the situation where the initial consensus level is relatively low, in which it is difficult to find a ‘central opinion’ to represent all individual opinions. However, research on the bidirectional feedback mechanism in SN-LGDM is still insufficient at this stage; in addition, few studies have explored the impact of trust and consensus on subgroup interactions. In fact, whether two DMs/subgroups are able to interact is highly related to the opinion similarity and trust level between them; thus, the opinions of the other party will be considered by a DM/subgroup only when their opinions are similar and the DM/subgroup has a high trust in the other party (Li et al. 2022). Hence, consensus and trust among subgroups are actually useful resources that can drive and facilitate the CRP, and they can complement each other according to specific needs. For instance, Yu et al. (2021) proposed a minimum-cost consensus model considering voluntary trust loss in SN-LGDM, in which DM with high trust and low consensus can reduce the consensus cost by voluntarily losing some trust.

Since subgroups will exhibit specific interaction (feedback) behaviors in the CRP of SN-LGDM, it is obvious that this is the second aspect to address (Cheng et al. 2022; Gong et al. 2021; Gao and Zhang 2022). On the one hand, subgroup’s cohesion may be considered as an impact characteristic on the subgroup compromise behavior with regards to their opinion interaction. Indeed, cohesion has attracted the attention of many scholars in recent years, especially on the large-group decision making research context (Labella et al. 2019; Rodríguez et al. 2021), and it has been used mainly on the assignation of weights to subgroups but not on the role it plays on the feedback process. Actually, group cohesion can be seen as a mechanism to maintain the connection and togetherness based on the cooperative behavior, expectations conformity and emotional ties of group members (Rodríguez et al. 2021), which will affect the behavior and decision making of subgroups to a certain extent. On the other hand, subgroups will try to ensure that their opinions are adjusted to the minimum in the CRP (Yu et al. 2021; Yuan et al. 2021; Wu and Tu 2021). When the subgroup opinions is considers as the result of the aggregation of all its members’ opinions, the adjustment of subgroup opinions will invariably imply that the original opinions of each member will also be modified. Therefore, subgroups will minimize their own concessions in the process of opinion interaction in order to protect their independence.

Based on the above discussion, this paper aims to investigate the bidirectional interaction-based hybrid consensus strategies of subgroups, and design a minimum adjustment feedback model to help the interacting subgroups pursue higher consensus. In addition, the personalized individual semantic (PIS) model is used to deal with the ambiguity of different semantics of individuals’ expressed opinions. The main contributions of this article are summarized as follows:

  • The hybrid feedback strategies in bidirectional interaction under different scenarios are investigated. By distinguishing four types of interacting subgroups based on the interaction consensus threshold and the interaction trust threshold, four corresponding feedback strategies are proposed in which the consensus level and trust level of subgroups are regarded as reliable resources to facilitate the achievement of group consensus. Among them, for the subgroup pairs that cannot directly interact, the third-party subgroup is used to assist the completion of the interaction process.

  • A minimum adjustment bidirectional feedback model considering the influence of cohesion on the interaction behavior of subgroups in SN-LGDM is developed. First, the group cohesion is divided into internal cohesion and external cohesion, which are measured considering the internal and external consensus and trust level of subgroups; then a cohesion-based minimum adjustment feedback model is proposed, which can achieve sufficient subgroup interaction and help the subgroups reach consensus effectively with less opinion modification.

The rest of the article is organized as follows: Sect. 2 includes some preliminaries on on SN-LGDM problems, the 2-tuple linguistic model, the PIS model used to represent opinions, and social network analysis (SNA). Section 3 investigates the consensus and trust combined driven bidirectional interaction consensus framework with hybrid strategies, while a minimum adjustment bidirectional feedback model considering cohesion is developed in Sect. 4. An illustrative example of blockchain platform selection in supply chain is presented in Sect. 5. Some simulation and comparison analysis are provided in Sect. 6. Conclusions are drawn in Sect. 7.

2 Preliminaries

Let \( Z=\left\{ z_{1}, z_{2},...,z_{m} \right\} (m\ge 2) \) be a set of alternatives or feasible solutions to the decision making problem of interest; and let \( DM=\left\{ dm_{1},dm_{2},...,dm_{n}\right\} (n\ge 20) \) be a set of DMs who use terms of the linguistic term set \( S=\left\{ S_{0},S_{1},\ldots ,S_{g}\right\} \) to express their pairwise comparative evaluation of alternatives: \( L^k={ (l_{ij}^k)}_{m\times m}, \) where \( l_{ij}^k \in S \) represents the linguistic preference degree of alternative \( z_i \) over \( z_j \) (Li et al. 2021). In order to make the paper self-contained, some preliminaries on the 2-tuple linguistic model, the numerical scale model, the PIS model and social network analysis are provided.

2.1 The 2-tuple Linguistic Model, the Numerical Scale Model, and the PIS Model

The 2-tuple fuzzy linguistic model was developed based on the following specific ordinal representation methodology to allow computing with words (CW) (Herrera and Martínez 2000; Martínez and Herrera 2012): Let \( \beta \in [0,g] \) be a value representing the result of a symbolic aggregation operation of terms in S; the linguistic 2-tuple \( (S_i,\alpha ) \) that represents the equivalent information to \( \beta \) verifies: \( i=round\left( \beta \right) \in \{0,1,\ldots ,g\},\) and \( \alpha = \beta -i \in [-0.5,0.5)\), where round is the usual round operation.

The concept of the numerical scale proposed by Dong et al. (2009) to transform linguistic terms and linguistic 2-tuples into real numbers: A function \( NS:S\rightarrow {\mathbb {R}}\) is called a numerical scale of S,  and \( NS\left( S_i\right) \) is called the numerical index of \( S_i .\) The numerical scale NS for linguistic 2-tuples \(S \times [-0.5,0.5)\) is defined as

$$\begin{aligned} NS (S_{i},\alpha )={\left\{ \begin{array}{ll} &{} NS (S_{i}) +\alpha \cdot (NS (S_{i+1})-NS (S_{i}))\; \alpha \ge 0\\ &{} NS (S_{i}) +\alpha \cdot (NS (S_{i})-NS (S_{i-1}))\; \alpha < 0\\ \end{array}\right. } \end{aligned}$$
(1)

In practice, the same linguistic term may mean different to different DMs. The personalized individual semantics (PIS) model was developed to model the specific semantics of individuals by means of an interval numerical scale and the 2-tuple linguistic model (Li et al. 2018, 2019; Cao et al. 2022), and are obtained by solving the following consistency-driven optimization-based linear programming models (for \(k \in \{1,2,\ldots ,n\}\)):

$$ \begin{gathered} {\text{max}}\quad CI(L^{k} ) = 1 - \frac{{4\sum\nolimits_{\begin{subarray}{l} i,j,h = 1 \\ i < j < h \end{subarray} }^{m} {\left| {NS^{k} (l_{{ij}}^{k} ) + NS^{k} (l_{{jh}}^{k} ) - NS^{k} (l_{{ih}}^{k} ) - 0.5} \right|} }}{{m(m - 1)(m - 2)}} \hfill \\ \;\left\{ {\begin{array}{*{20}c} {NS^{k} (S_{0} ) = 0} & {} \\ {NS^{k} (S_{{\frac{g}{2}}} ) = 0.5} & {} \\ {NS^{k} (S_{g} ) = 1} & {} \\ {NS^{k} (S_{i} ) \in \left[ {\frac{{i - 1}}{g},\frac{{i + 1}}{g}} \right],\quad i \ne 0,\frac{g}{2},g} & {} \\ {NS^{k} (S_{{i + 1}} ) - NS^{k} (S_{i} ) \ge \varepsilon ,\quad i = 0,1,...,g - 1} & {} \\ \end{array} } \right. \hfill \\ \end{gathered} $$
(2)

2.2 Social Network Analysis

SNA is a useful methodology to study the relationships between social entities (Wasserman and Faust 1994). As a reliable social relationship, trust has been extensively investigated in recent years (Liu et al. 2019; Wang et al. 2022). Obviously, DMs in SN-LGDM do not exist independently but exhibit links between them through existing trust relationships. Thus, DMs’ trust relationships is modelled mathematically via a so-called ‘trust network’ in this article (Du et al. 2021). Generally, people can subjectively express their trust or distrust of each other directly, so as to form a directed trust network. In addition, trust relationship can also be established based on the interaction and cooperation between individuals, which translates into an undirected trust network. The sociometric matrix of a directed trust network is not necessarily symmetric, while the sociometric matrix of an undirected trust network is. In this last case, the existence or not of trust relationship between DMs \(dm_h\) and \(dm_k\) is modelled with a simple delta function:

$$\begin{aligned} d_{hk}= {\left\{ \begin{array}{ll} 1 &{} \text {if there exists trust relationship between } dm_{h} \text { and } dm_{k}\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$
(3)

Thus, in our framework, the DMs of a SN-LGDM can be represented by an undirected network graph \(G=\left( N,D\right) \), which consists of a set of nodes (individual DMs), \(N=DM=\left\{ dm_1,dm_2,\ldots ,dm_n\right\} \), connected by a set of edges (trust relationships), \(D=\left\{ d_{hk},h\ne k,h,k=1,...,n\right\} \). In such undirected trust network, the number of edges among DMs/communities is used as a measure of the strength of trust relationship.

3 Consensus and Trust Combined Driven Bidirectional Interaction Framework in SN-LGDM

The following steps are usually carried out when solving an SN-LGDM problem:

Step 1: Opinions gathering. DMs adopt preference expression structures to provide evaluation information on alternatives.

Step 2: DMs clustering. Large-scale DMs are classified into several subgroups based on certain metrics.

Step 3: Consensus reaching process (CRP). The agreement level of DMs is calculated. If the predefined threshold is reached, then the selection process to obtain the optimal solution is activated. If not, subgroups need to implement multiple rounds of opinion interaction to improve their consensus level.

When subgroups adopt the bidirectional interaction mechanism in the CRP, the consensus and trust level between subgroups will have an impact on the interaction process, which will be investigated in this section.

3.1 Community Detection in the Trust Network

DMs clustering is usually applied to reduce the dimension of the SN-LGDM problem, and the Louvain algorithm is often implemented to identifying communities in large networks. The Louvain algorithm is a community detection method based on DMs’ social network relationships that can reveal the hierarchical structure of the community with high efficiency (Xu et al. 2020; Qin et al. 2022).

The Louvain algorithm optimizes the modularity of the network. The modularity measures the structural tightness within the network and and its mathematical formulation for the assumed trust network is

$$\begin{aligned} Q=\frac{1}{2s}\sum _{h,k}\left[ d _{hk}-\frac{t_{h}\cdot t_{k}}{2s} \right] \cdot \delta (c_{h},c_{k}) \end{aligned}$$
(4)

where s is the sum of all the edge weights in the network (when the network is not a weighted graph, all edges have associated the same weight value of 1 and s is the total numbers of edges in the network); \(t_h\) and \(t_k\) are the sum of the weights of the edges attached to node \(dm_h\) and node \(dm_k\), respectively; \(c_h\) and \(c_k\) are the communities of nodes \(dm_h\) and \(dm_k\), respectively; and

$$\begin{aligned} \delta \left( c_h, c_k\right) = {\left\{ \begin{array}{ll} 1 &{} \text {if } c_h = c_k;\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$
(5)

The objective is to maximise Q. To do this, for each node \(dm_h\), the change of modularity is calculated by removing it from its community \(c_h\) and moving it into the community \(c_k\) of each neighbour \(dm_k\) of \(dm_h\):

$$\begin{aligned} \Delta Q=\left[ \frac{\sum _{in}+t_{h,in}}{2s}-\left( \frac{\sum _{tot}+t_{h}}{2s} \right) ^2\right] -\left[ \frac{\sum _{in}}{2s}-\left( \frac{\sum _{tot}}{2s} \right) ^2-\left( \frac{t_{h}}{2s} \right) ^2\right] \end{aligned}$$
(6)

where \(\sum _{in}\) is the sum of weights of the links inside \(c_k\), \(\sum _{tot}\) is the sum of weights of links to nodes in community \(c_k\), \(t_{h,in}\) is the sum of weights of edges linking node \(dm_h\) to the nodes in community \(c_h\). The Louvain algorithm for our SN-LGDM network \(G=\left( N,D\right) \) is provided as Algorithm 1.

figure a

3.2 Consensus and Trust Measure

Suppose that Algorithm 1 clusters the n DMs, \(DM=\left\{ dm_1,dm_2,\ldots ,dm_n\right\} ,\) into l communities, \(C=\left\{ c_1,c_2,\ldots ,c_l\right\} .\) Let \(U^{c_h}={(u_{ij}^{c_h})}_{m\times m}\) be the fuzzy preference relation on the set of m alternatives, \( Z=\left\{ z_{1}, z_{2},...,z_{m} \right\} ,\) of community \(c_h\), which is obtained by aggregating the individual preferences of its nodes: \(U^{c_h}=f(U^{1},\ldots ,U^{\nu _{c_h}})\), where f is the aggregation operator, \(\nu _{c_h} =\#c_h\)(\(\nu _{c_h}\) is the number of DMs in community \(c_h\), it is equal to the cardinality \(\#c_h\)), and \(\{U^{1},\ldots ,U^{\nu _{c_h}}\}\) are the individual fuzzy preference matrices obtained transforming the corresponding linguistic preference relations \(\{L^{1},\ldots ,L^{\nu _{c_h}}\}\) through the PIS model described in Sect. 2.1. The definition of ‘consensus level’ and ‘trust level’ at the community level is defined as follows.

  1. (a)

    Consensus measure is defined as the similarity degree of individual/community preferences (Liu et al. 2022; Xing et al. 2022).

Definition 1

(Consensus Measure) The mutual consensus level (MCL) between communities \(c_h\) and \(c_k\) is

$$ MCL(c_{h} ,c_{k} ) = 1 - \frac{2}{{m(m - 1)}}\sum\limits_{\begin{subarray}{l} i = 1 \\ j > i \end{subarray} }^{{m - 1}} {\left| {u_{{ij}}^{{c_{h} }} - u_{{ij}}^{{c_{k} }} } \right|} $$
(7)

The general consensus level (GCL) of all communities \(C=\left\{ c_1,c_2,\ldots ,c_l\right\} \) is

$$\begin{aligned} GCL=1-\frac{4}{l(l-1)m(m-1)}\sum_{\stackrel{k=1}{h>k}}^{l-1}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| u_{ij}^{c_{h}}- u_{ij}^{c_{k}}\right| \end{aligned}$$
(8)

The mutual consensus level measures the similarity degree between the fuzzy preference matrices between the two communities, and the general consensus level measures the average similarity degree of the fuzzy preference matrices of all communities in the group. Generally, a predefined consensus threshold value \({\overline{MCL}}\) or \({\overline{GCL}}\) is required to reach a state of acceptable consensus (Sun et al. 2022); otherwise, the feedback mechanism is carried out to generate modification suggestions to reach such state of acceptable consensus.

  1. (b)

    Trust measure is defined as the strength of trust relationship between individuals or communities (Yu et al. 2021).

Definition 2

(Trust Measure) Let \(dm_r\in c_h\), \(dm_s\in c_k\), the mutual trust level between communities \(c_h\) and \(c_k\) is

$$\begin{aligned} MTL(c_{h},c_{k})=\frac{\displaystyle \sum _{r=1}^{\nu _{c_{h}}} \sum _{s=1}^{\nu _{c_{k}}} d_{rs}}{\nu _{c_{h}}\nu _{c_{k}}} \end{aligned}$$
(9)

The mutual trust level between communities measures the density of trust relationships between members of the two communities. The more trust relationships exist between the two communities’ members, the higher the mutual trust level.

3.3 Consensus-trust Driven Bidirectional Interaction Mechanism

3.3.1 Bidirectional Interaction Mechanism

The bidirectional interaction mechanism implements a distributed preference adjustment strategy by which the identified pair of individuals modify their opinions towards each other when interacting. Formally, the original preference relations \(U^{c_p}={(u_{ij}^{c_p})}_{m\times m}\) and \(U^{c_q}={(u_{ij}^{c_q})}_{m\times m} \) of \(c_p\) and \(c_q\) are updated through their interaction to \({\overline{U}}^{c_p}={({\overline{u}}_{ij}^{c_p})}_{m\times m}\) and \({\overline{U}}^{c_q}={({\overline{u}}_{ij}^{c_q})}_{m\times m},\) respectively, based on the following bidirectional feedback rules:

$$\begin{aligned} {\overline{u}}_{ij}^{c_{p}}= & {} \eta ^{c_{p}}\cdot u_{ij}^{c_{q}}+(1-\eta ^{c_{p}})\cdot u_{ij}^{c_{p}} \end{aligned}$$
(10)
$$\begin{aligned} {\overline{u}}_{ij}^{c_{q}}= & {} \eta ^{c_{q}}\cdot u_{ij}^{c_{p}}+(1-\eta ^{c_{q}})\cdot u_{ij}^{c_{q}} \end{aligned}$$
(11)

where \( \eta ^{c_p},\eta ^{c_p}\in [0,1]\) are feedback parameters used to control the amount of preference modification. Notice that constraint \(\eta ^{c_p}+\eta ^{c_p}\le 1\) is imposed to avoid unnecessary excessive adjustments.

3.3.2 Mutual Consensus-trust Plot Construction

In realistic decision-making scenarios, not all pairs of communities can interact with each other. Feasible interaction behavior depends on the mutual consensus and mutual trust level between the communities. Generally, communities interaction should happen when the rule assumed for individuals applies: people tend to accept advices from those they trust, and their preferences should be relatively similar to ensure that their opinions will not be modified excessively.

This combined rule of trust and similarity can be formally model with the introduction of an interaction consensus threshold, herein denoted as \({\overline{ICT}},\) and an interaction trust threshold, herein denoted as \({\overline{ITT}}\). These threshold values will represent the lowest acceptable mutual consensus level and mutual trust level for any two communities to interact with each other. Four quadrants (categories) can be generated with the interaction consensus and interaction trust thresholds (see Fig. 1).

Fig. 1
figure 1

Mutual consensus-trust plot

Full size image

Community pairs can be mapped to different positions within the four quadrant. Thus, if the pair of communities \(c_h\) and \(c_k\) \((h,k\in \{1,...,l\})\) are identified to interact with each other, the following four situations based on their mutual consensus-trust values may occur:

  1. (a)

    Quadrant 1–\(MCL\left( c_h,c_k\right) \ge {\overline{ICT}} \text { and } MTL\left( c_h,c_k\right) \ge {\overline{ITT}}.\)In this case, the mutual consensus level and mutual trust level are greater than the corresponding interaction threshold values, which means that the members of the two communities are relatively familiar and trust each other, and the preferences or opinions of the two communities are also relatively similar, therefore, the two communities can accept each other’s opinions as feedback references to adjust their preferences as per (10) and (11).

  2. (b)

    Quadrant 2 – \(MCL\left( c_h,c_k\right) <{\overline{ICT}} \text { and } MTL\left( c_h,c_k\right) \ge {\overline{ITT}}.\) In such a case, there exist interaction and trust between the members of the two communities, but there is a large gap in the opinions of the two communities on alternatives. Usually, their mutual trust level ensures the acceptability of advices from each other and therefore their interaction is possible; however, their low mutual consensus level implies that an excessive adjustment of their original preferences is required if \({\overline{ICT}}\) were to be reached after one round of negotiation, which they may be reluctant to.

  3. (c)

    Quadrant 3 – \(MCL\left( c_h,c_k\right)<{\overline{ICT}} \text { and } MTL\left( c_h,c_k\right) <{\overline{ITT}}.\) In this case, members of the two communities seldom interact with each other and are not familiar with each other, and the opinions and judgments of the two communities are very different. Thus, the two communities cannot accept each other’s opinions as feedback references for mutual adjustment, and a third community is required to be found as a ‘intermediary’ to help the two communities interact.

  4. (d)

    Quadrant 4 – \(MCL\left( c_h,c_k\right) \ge {\overline{ICT}} \text { and } MTL\left( c_h,c_k\right) <{\overline{ITT}}.\) When the opinions or preferences of the two communities are relatively similar but have not reached an acceptable mutual consensus level, if the trust and interaction relationships between the members of the two communities are weak, then the lack of mutual trust will also reduce the acceptability of the advices from each other.

Remark 1

The interaction consensus threshold and interaction trust threshold are set to judge whether the communities can interact. The specific values of the two interaction thresholds can be determined according to the specific decision-making problems and the characteristics of DMs. For example, if the decision-making problem is urgent, then low interaction thresholds can be set to enhance the chances of interaction between communities and speed up the reaching of the predefined consensus threshold value.

4 Minimum Adjustment Bidirectional Feedback Strategies Considering Cohesion

In this section, four different feedback strategies corresponding to the four quadrants of the mutual consensus-trust plot are proposed. A minimum adjustment bidirectional feedback model that considers cohesion is applied in these strategies.

4.1 Cohesion Measure

Given a set of communities, two different community cohesion measures are possible: internal cohesion and external cohesion (Kamis et al. 2019). Internal cohesion reflects the cohesiveness of members within the same community while external cohesion represents the compatibility of a community with the other communities. In both cases, cohesion is computed by combining the corresponding consensus level and trust level of communities:

  • The internal consensus level of community \(c_{h}\) is:

    $$\begin{aligned} CL_{int}({c_{h}})=1-\frac{4}{\nu _{c_h}(\nu _{c_h}-1)m(m-1)}\sum_{\stackrel{k=1}{h>k}}^{\nu _{c_h}-1}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| u_{ij}^{h}- u_{ij}^{k}\right| \end{aligned}$$
    (12)

  • The external consensus level of \(c_{h}\):

    $$\begin{aligned} CL_{ext}({c_{h}})=1-\frac{2}{m(m-1)(l-1)}\sum\limits_{{\mathop {k \ne h}\limits^{{k = 1}} }}^{{l}} {\mathop \sum \limits_{{\mathop {j > i}\limits^{{i = 1}} }}^{{m - 1}} } \left| u_{ij}^{c_{h}}- u_{ij}^{c_{k}}\right| \end{aligned}$$
    (13)

  • The internal trust level of \(c_{h}\):

    $$\begin{aligned} TL_{int}({c_{h}})=\frac{\sum _{h\in c_h}\sum _{k\in c_h}d_{hk}}{\nu _{c_{h}}(\nu _{c_{h}}-1)} \end{aligned}$$
    (14)

  • The external trust level of \(c_{h}\):

    $$\begin{aligned} TL_{ext}({c_{h}})=\frac{\sum _{h\in c_h}\sum _{k\notin c_h}d_{hk}}{\nu _{c_{h}}(l-\nu _{c_{h}})} \end{aligned}$$
    (15)

Cohesion increases when the consensus level increases. In general, common opinions, goals and attitudes of group members will enhance interpersonal attraction and increase the group cohesion. In addition, the trust relationships among group members will also enhance group cohesion. The more trust relationships exist among group members, the higher the possibility of communication and interaction between them, and the higher the group cohesion. Thus, the below expression are proposed to measure the internal and external cohesion of community \(c_h\):

$$\begin{aligned} CC_{int}(c_{h})= & {} CL_{int}({c_{h}})^{\gamma }\cdot TL_{int}({c_{h}})^{\varphi }; \end{aligned}$$
(16)
$$\begin{aligned} CC_{ext}(c_{h})= & {} CL_{ext}({c_{h}})^{\gamma }\cdot TL_{ext}({c_{h}})^{\varphi }. \end{aligned}$$
(17)

As usual, \(\gamma \) and \(\varphi \), which verify \(0\le \gamma ,\varphi \le 1\) and \(\gamma +\varphi =1,\) are weight parameters used to increase/decrease the effect of the consensus level and trust level in community cohesion, respectively. The following definition of community cohesion is proposed (Kamis et al. 2019):

Definition 3

(Community Cohesion) The community cohesion (CC) of \(c_h\) is

$$\begin{aligned} CC_{c_{h}}=\frac{\nu _{c_{h}}\cdot CC_{int}(c_{h})}{n}+\frac{(n-\nu _{c_{h}})\cdot CC_{ext}(c_{h})}{n} \end{aligned}$$
(18)

in which n is the number of DMs, and \(\nu _{c_h} =\#c_h\)(\(\nu _{c_h}\) is the number of DMs in community \(c_h\), it is equal to the cardinality \(\#c_h\)). The community cohesion is actually the integration of internal cohesion and external cohesion according to the number of the community members. The more community members of \(c_h\), the greater the proportion of internal cohesion, and vice versa.

4.2 Feedback Strategies for the Four Quadrants

If the general consensus level of the communities does not reach the preset consensus threshold, it is necessary to adopt the feedback mechanism to generate opinion modification suggestions to increase the general consensus level. In the bidirectional interaction mechanism, the pairwise subgroups with minimum mutual consensus (PMC) (Hou and Triantaphyllou 2019) are identified to implement mutual adjustments of their opinions towards each other:

$$\begin{aligned} PMC=\left\{ (c_{p},c_{q})|MCL(c_{p},c_{q})=\min _{\begin{array}{c} h,k=1,...,l\\ h<k \end{array}}\left\{ MCL(c_{h},c_{k} )\right\} \right\} . \end{aligned}$$
(19)

A pair \((c_{p},c_{q})\in PMC\) may belong to any quadrant in the mutual consensus-trust plot, and therefore corresponding feedback strategies are designed.

  1. (a)

    Strategy for Quadrant 1- Minimum adjustment bidirectional interaction. In this case, the two communities can implement mutual adjustment to reach \({\overline{MCL}}\). In the process of opinion interaction, on the one hand, the compromise degree of communities is related to their own cohesion. It is a recognized phenomenon that a cohesive group with unified goals is less likely to be persuaded than a loose group with conflicting goals, consequently, groups with higher cohesion are less likely to be persuaded to adjust their original judgements. This is modelled by imposing that the amount of adjustment of opinions by communities is inversely related to their cohesion values. On the other hand, the communities often minimize their concessions in order to protect the independence of their opinions.

    Based on the above analysis, a multi-objective programming model of minimum adjustment bidirectional feedback mechanism for \(c_{p}\) and \(c_{q}\) is constructed as follows.

    $$\begin{aligned} {\textbf {min}}&\quad \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{p}}- u_{ij}^{c_{p}}\right| \nonumber \\ {\textbf {min}}&\quad \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{q}}- u_{ij}^{c_{q}}\right| \nonumber \\ s.t.&\left\{ \begin{array}{lr} \displaystyle 1-\frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{p}}- {\overline{u}}_{ij}^{c_q}\right| \ge {\overline{MCL}} &{} (20.1) \\ \displaystyle \sum_{\stackrel{i=1}{j>i}}^{m-1}\left| {\overline{u}}_{ij}^{c_{p}}- u_{ij}^{c_{p}}\right| \cdot CC_{c_{p}}=\sum _{\begin{array}{c} i=1\\ j>i \end{array}}^{m-1} \left| {\overline{u}}_{ij}^{c_{q}}- u_{ij}^{c_{q}}\right| \cdot CC_{c_{q}} &{} (20.2) \\ {\overline{u}}_{ij}^{c_{p}}=\eta ^{c_{p}}\cdot u_{ij}^{c_{q}}+(1-\eta ^{c_{p}})\cdot u_{ij}^{c_{p}} &{} (20.3) \\ {\overline{u}}_{ij}^{c_{q}}=\eta ^{c_{q}}\cdot u_{ij}^{c_{p}}+(1-\eta ^{c_{q}})\cdot u_{ij}^{c_{q}} &{} (20.4) \\ {\overline{u}}_{ij}^{c_{k}}={u}_{ij}^{c_{k}}, k\ne p, q &{} (20.5) \\ 0\le \eta ^{c_{p}}, \eta ^{c_{q}}\le 1; \eta ^{c_{p}}+\eta ^{c_{q}} \le 1 &{} (20.6) \\ \end{array} \right. \end{aligned}$$
    (20)

    The objective function of Model (20) is to minimize the opinion adjustment of the two interacting communities; Expression (20.1) is used to ensure that the two communities reach the mutual consensus threshold; Expression (20.2) is designed to set the opinion adjustment amount of communities inversely related to their cohesion values. In addition, expression (20.3)-(20.4) contribute to obtain the updated opinions of interacting communities after feedback process, and expression (20.5) provides the updated opinions of other communities. Furthermore, expression (20.6) describes the range of the feedback parameters. After the feedback from the Model (20) is implemented by communities \(c_{p}\) and \(c_{q}\), the consensus level needs to be recalculated to judge whether \({\overline{GCL}}\) is reached, so as to determine whether it is necessary to continue with additional iterative adjustments. The analytical expressions of the global optimal solution of Model (20) is provided in Theorem 1.

Theorem 1

The global optimal solution of Model (20) is

$$\begin{aligned} \left( \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{q}}}{CC_{c_{p}}+CC_{c_{q}}}, \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{p}}}{CC_{c_{p}}+CC_{c_{q}}}\right) , \end{aligned}$$

which is achieved at the optimum feedback parameters

$$\begin{aligned} (\eta ^{c_{p}},\eta ^{c_{q}})=\left( \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{q}}}{(CC_{c_{p}}+CC_{c_{q}})\cdot (1-MCL(c_p,c_q))}, \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{p}}}{(CC_{c_{p}}+CC_{c_{q}})\cdot (1-MCL(c_p,c_q))} \right) . \end{aligned}$$

Proof

Using (7)–(10), the objective functions of Model (20) become:

$$\begin{aligned} \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{p}}- u_{ij}^{c_{p}}\right|&= \frac{2 \eta ^{c_{p}}}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| u_{ij}^{c_{q}}- u_{ij}^{c_{p}}\right| \\&=\eta ^{c_{p}} \cdot (1-MCL(c_p,c_q))\\ \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{q}}- u_{ij}^{c_{q}}\right|&= \frac{2 \eta ^{c_{q}}}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| u_{ij}^{c_{p}}- u_{ij}^{c_{q}}\right| \\&=\eta ^{c_{q}} \cdot (1-MCL(c_p,c_q)) \end{aligned}$$

Since \(1-MCL(c_p,c_q)\ge 0\), then minimum of the objective functions are achieved when \(\eta ^{c_{p}}\) and \(\eta ^{c_{q}}\) are minimum, respectively. From constraint (20.1), we have

$$\begin{aligned}&1-\frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| {\overline{u}}_{ij}^{c_{p}}- {\overline{u}}_{ij}^{c_q}\right| \\&\quad = 1-\frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1} \left| (1-\eta ^{c_{p}}-\eta ^{c_{q}}) \cdot ({u}_{ij}^{c_{p}}- {u}_{ij}^{c_q})\right| \ge {\overline{MCL}} \end{aligned}$$

Constraint \(\eta ^{c_{p}}+\eta ^{c_{q}} \le 1\) implies that \(1-\eta ^{c_{p}}-\eta ^{c_{q}} \ge 0\), which implies that (20.1) becomes

$$\begin{aligned}&1-\frac{2 }{m(m-1)}\sum _{i=1,j>i}^{m-1}\left| {u}_{ij}^{c_{p}}- {u}_{ij}^{c_q}\right| \cdot (1-\eta ^{c_{p}}-\eta ^{c_{q}})\\&\quad =1-\frac{2}{m(m-1)}\sum _{i=1,j>i}^{m-1}\left| {u}_{ij}^{c_{p}}- {u}_{ij}^{c_q}\right| \\&\qquad +\left( \eta ^{c_{p}}+\eta ^{c_{q}})\cdot (\frac{2}{m(m-1)}\sum _{i=1,j>i}^{m-1}\left| {u}_{ij}^{c_{p}}- {u}_{ij}^{c_q}\right| \right) \\&\quad =MCL(c_p,c_q)+(\eta ^{c_{p}}+\eta ^{c_{q}})\cdot (1-MCL(c_p,c_q))\ge {\overline{MCL}} \end{aligned}$$

In addition, (20.2) means that \(\eta ^{c_{p}}\cdot CC_{c_{p}}=\eta ^{c_{q}}\cdot CC_{c_{q}},\) and therefore

$$\begin{aligned}&MCL(c_p,c_q)+(\eta ^{c_{p}}+\eta ^{c_{q}})\cdot (1-MCL(c_p,c_q))\\&\quad \ge {\overline{MCL}} \implies \eta ^{c_{p}} \ge \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{q}}}{(CC_{c_{p}}+CC_{c_{q}})\cdot (1-MCL(c_p,c_q))} \end{aligned}$$

It is clear that:

$$\begin{aligned} \min \eta ^{c_{p}}= \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{q}}}{(CC_{c_{p}}+CC_{c_{q}})\cdot (1-MCL(c_p,c_q))}. \end{aligned}$$

Similarly, it is:

$$\begin{aligned} \min \eta ^{c_{q}}= \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{p}}}{(CC_{c_{p}}+CC_{c_{q}})\cdot (1-MCL(c_p,c_q))} \end{aligned}$$

Thus, the global optimal solution of Model (20) is

$$\begin{aligned} \left( \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{q}}}{CC_{c_{p}}+CC_{c_{q}}}, \frac{({\overline{MCL}}-MCL(c_p,c_q))\cdot CC_{c_{p}}}{CC_{c_{p}}+CC_{c_{q}}}\right) . \end{aligned}$$

This completes the proof of Theorem 1. \(\square \)

  1. (b)

    Strategy for Quadrant 2 - Exchange trust for consensus. Since the mutual consensus level of the pair of communities is below the interaction consensus threshold, \({\overline{ICT}},\) reaching the target threshold \({\overline{MCL}}\) through their interaction would require excessive adjustment of their original preferences (the further \(MCL(c_{p},c_{q})\) is from \({\overline{MCL}}\), the more the adjustment). However, since \(MTL\left( c_h,c_k\right) \ge {\overline{ITT}}\), the communities are opened to mutual advices from each other, and their interaction adjustments could be attenuated by deducting the excess trust \(MTL\left( c_p,c_q\right) -{\overline{ITT}}\) from the target consensus threshold \({\overline{MCL}}\) to set a lower one (Yu et al. 2021):

    $$\begin{aligned} {\overline{MCL}}'={\overline{MCL}}-(MTL\left( c_p,c_q\right) -{\overline{ITT}}) \end{aligned}$$
    (21)

    After setting a lower mutual consensus threshold in this way, Model (20) is therefore implemented with \({\overline{MCL}}\) replaced by \({\overline{MCL}}'\) to generate feedback advices.

  2. (c)

    Strategy for Quadrant 3 - Interaction with third-party communities. In this case, the communities cannot interact directly, so third-party communities are identified to serve as the ‘coordinators’ to assist the interaction. Considering trust is the premise of interaction, to ensure acceptability of interaction advices (Dong et al. 2022), the sets of communities with mutual trust with \(c_{p}\) and \(c_{q}\) above the minimum interaction trust threshold are obtained:

    $$\begin{aligned} CF_{c_{p}}=\left\{ c_{h} | MTL(c_{p},c_{h}) \ge {\overline{ITT}}\right\} ;\quad \quad CF_{c_{q}}=\left\{ c_{k} | MTL(c_{q},c_{k}) \ge {\overline{ITT}}\right\} . \end{aligned}$$

    Let us assume that both \(CF_{c_{p}}\) and \(CF_{c_{q}}\) are non-empty. This means that each community have at least one third community they trust enough. It is obvious that \(MCL(c_{p},c_{h}) \ge MCL(c_{p},c_{q}) \ \forall h \in CF_{c_{p}}\) and that \(MCL(c_{q},c_{k}) \ge MCL(c_{p},c_{q}) \forall k \in CF_{c_{q}}.\) Let \(h^{*}\) and \(k^{*}\) be the communities in \(CF_{c_{p}}\) and \(CF_{c_{q}}\) with maximum mutual consensus with communities \(c_{p}\) and \(c_{q}\), respectively, i.e.

    $$\begin{aligned} MCL(c_{p},c_{h^{*}})= & {} \max _{h \in CF_{c_{p}}} MCL(c_{p},c_{h}); \quad \quad \\ MCL(c_{q},c_{k^{*}})= & {} \max _{k \in CF_{c_{q}}} MCL(c_{q},c_{k}). \end{aligned}$$

    Then, the below average opinion of communities \(c_{h^{*}}\) and \(c_{k^{*}}\) is used in the minimum adjustment bidirectional feedback mechanism Model (20) for communities \(c_{p}\) and \(c_{q}:\)

    $$\begin{aligned} u_{ij}^{hk^{*}}=\dfrac{CC_{ext}(c_{h^{*}})\cdot u_{ij}^{c_{h^{*}}}+CC_{ext}(c_{k^{*}}) \cdot u_{ij}^{c_{k^{*}}}}{CC_{ext}(c_{h^{*}})+CC_{ext}(c_{k^{*})}}. \end{aligned}$$
    (22)

    The amount of adjustments for each of the communities \(c_{p}\) and \(c_{q}\) will be adjusted by taking into account the the mutual consensus between their respective opinions and the above average opinions:

    $$\begin{aligned} \overline{MCL_p}'= & {} {\left\{ \begin{array}{ll} {\overline{MCL}} &{} \text {if } MCL\left( c_p,c_{hk^{*}}\right) \ge {\overline{ICT}}\\ {\overline{MCL}}-(MTL\left( c_p,c_{h^{*}}\right) -{\overline{ITT}}) &{} \text {otherwise.} \end{array}\right. }; \\ \overline{MCL_q}'= & {} {\left\{ \begin{array}{ll} {\overline{MCL}} &{} \text {if } MCL\left( c_q,c_{hk^{*}}\right) \ge {\overline{ICT}}\\ {\overline{MCL}}-(MTL\left( c_q,c_{k^{*}}\right) -{\overline{ITT}}) &{} \text {otherwise.} \end{array}\right. }. \end{aligned}$$

    Thus, the following minimum adjustment model is constructed for \(c_{p}\) and \(c_{q}\):

    $$\begin{aligned} {\textbf {min}}&\quad \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| {\overline{u}}_{ij}^{c_{p}}- u_{ij}^{c_{p}}\right| \nonumber \\ {\textbf {min}}&\quad \frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| {\overline{u}}_{ij}^{c_{q}}- u_{ij}^{c_{q}}\right| \nonumber \\ s.t.&\left\{ \begin{array}{lr} \displaystyle 1-\frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| {\overline{u}}_{ij}^{c_{p}}- u_{ij}^{hk^{*}}\right| \ge \overline{MCL_p}' &{} (23.1) \\ \displaystyle 1-\frac{2}{m(m-1)}\sum_{\stackrel{i=1}{j>i}}^{m-1}\left| {\overline{u}}_{ij}^{c_{q}}- u_{ij}^{hk^{*}} \right| \ge \overline{MCL_q}' &{} (23.2) \\ {\overline{u}}_{ij}^{c_{p}}=\eta ^{c_{p}}\cdot u_{ij}^{hk^{*}}+(1-\eta ^{c_{p}})\cdot u_{ij}^{c_{p}} &{} (23.3) \\ {\overline{u}}_{ij}^{c_{q}}=\eta ^{c_{q}}\cdot u_{ij}^{hk^{*}}+(1-\eta ^{c_{q}})\cdot u_{ij}^{c_{q}} &{} (23.4) \\ {\overline{u}}_{ij}^{c_{k}}={u}_{ij}^{c_{k}}, k\ne p, q &{} (23.5) \\ 0\le \eta ^{c_{p}}, \eta ^{c_{q}}\le 1 &{} (23.6) \\ \end{array} \right. \end{aligned}$$
    (23)

    If any of the above sets is empty, then there will be no trusted community to serve as coordinator in the interaction of communities \(c_{p}\) and \(c_{q}\), since at least one of the two communities will not trust any of the other communities. In this case, the opinions of the community with highest external cohesion, \(c_{\chi }(CC_{ext}(c_{\chi })=\max _{h} \left\{ CC_{ext}(c_{h}) \right\} ),\) are proposed to be used as the \(c_{hk^{*}}\) in the above minimum adjustment bidirectional feedback mechanism Model (23) for communities \(c_{p}\) and \(c_{q}.\)

  3. (d)

    Strategy for Quadrant 4- Exchange consensus for trust. In this scenario, the mutual trust level between the two communities is below the minimum interaction trust threshold, which makes the interaction not feasible. However, since the mutual consensus level of the communities is above the minimum interaction consensus threshold, a trust on opinions maybe considered to exist between the communities (Liu et al. 2022) to increase mutual trust level:

    $$\begin{aligned} MTL'\left( c_p,c_q\right) =MTL\left( c_p,c_q\right) +( MCL\left( c_p,c_q\right) -{\overline{ICT}}) \end{aligned}$$
    (24)

    If \(MTL'\left( c_p,c_q\right) \ge {\overline{ITT}}\), then the pair of community would move to Quadrant 1 and Mode l(20) applies; otherwise, Quadrant 3 strategy is applied.

After the application of the proposed bidirectional interaction mechanism, the general consensus level of communities will be acceptable, and the selection process is to be applied to determine the optimal solution of the SN-LGDM problem. Herein, the cohesion and size of communities are combined to assign community weights (Rodríguez et al. 2021) to aggregate the communities’ opinions into a collective one, from which the final ranking of the alternatives is obtained. The whole procedure is summarized in Algorithm 2.

figure b

5 An Illustrative Example of Blockchain Platform Selection in Supply Chain

5.1 Example Description

As a distributed ledge technology, blockchain technology offers a reliable decentralized infrastructure that is not under the control of a central authority (Farshidi et al. 2020). Blockchain technology has the characteristics of decentralization, anti-tampering and distributed storage, which can effectively settle the issues such as privacy security and single point failure (Lai and Liao 2021). Therefore, it has received extensive attention in different business applications, which includes supply chain management (Bai et al. 2021). For example, major technology providers such as IBM and Microsoft provide the infrastructure to use blockchain technology to track products and supply chains for enterprises.

Blockchain technology can potentially improve the management structure and operation of the entire supply chain (Bai et al. 2021), which has a profound impact on the organizations and tiers in the supply chain. However, with the emergence of various blockchain vendors and platforms, the blockchain platform selection of supply chain network has become a highly complex decision-making problem. On the one hand, the supply chain network involves large-scale enterprises and organizations, and users at different positions in the supply chain have different expectations and demands for the introduction of blockchain technology, so it is prone to have opinion conflict. On the other hand, there exists cooperation and trust relationships between supply chain members, and members will also exchange information, which will have an impact on the outcome of decision making.

Suppose a supply chain network producing electronic products needs to introduce blockchain technology to optimize its management. Four blockchain platforms, \(Z=\left\{ z_1,z_2,z_3,z_4\right\} \), have been determined through consultation among supply chain members: Ethereum, JM Morgan Quorum, Hyperledger, and BigchainDB. In order to select the most suitable scheme, twenty DMs \(\left\{ dm_1,dm_2,\ldots ,dm_{20}\right\} \) in the supply chain network are selected to evaluate the four blockchain platform alternatives, whose opinions are represented as linguistic preference relations based on the linguistic term set S with elements: \(S_0\)= extremely poor, \(S_1\) = very poor, \(S_2\) = poor, \(S_3\) = fair, \(S_4\) = good, \(S_5\) =very good, \(S_6\)= extremely good .

$$ \begin{gathered} L^{1} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{1} } & {S_{4} } & {S_{2} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{1} } \\ {} & - & - & {S_{3} } & {S_{4} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{2} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{1} } & {S_{6} } & {S_{0} } \\ {} & - & {S_{3} } & {S_{0} } & {S_{5} } \\ {} & - & - & {S_{3} } & {S_{2} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{3} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{2} } & {S_{1} } & {S_{3} } \\ {} & - & {S_{3} } & {S_{4} } & {S_{2} } \\ {} & - & - & {S_{3} } & {S_{6} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ L^{4} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{5} } & {S_{1} } & {S_{3} } \\ {} & - & {S_{3} } & {S_{6} } & {S_{0} } \\ {} & - & - & {S_{3} } & {S_{4} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{5} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{3} } & {S_{4} } & {S_{6} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{5} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{6} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{2} } & {S_{5} } & {S_{1} } \\ {} & - & {S_{3} } & {S_{0} } & {S_{6} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ L^{7} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{4} } & {S_{5} } & {S_{6} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{1} } \\ {} & - & - & {S_{3} } & {S_{5} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{8} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{2} } & {S_{6} } & {S_{0} } \\ {} & - & {S_{3} } & {S_{1} } & {S_{4} } \\ {} & - & - & {S_{3} } & {S_{3} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{9} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{6} } & {S_{5} } & {S_{4} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{3} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ L^{{10}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{6} } & {S_{1} } & {S_{3} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{2} } \\ {} & - & - & {S_{3} } & {S_{4} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{{11}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{4} } & {S_{6} } & {S_{6} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{5} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{{12}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{5} } & {S_{0} } & {S_{4} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{1} } \\ {} & - & - & {S_{3} } & {S_{6} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ \;L^{{13}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{2} } & {S_{3} } & {S_{6} } \\ {} & - & {S_{3} } & {S_{4} } & {S_{2} } \\ {} & - & - & {S_{3} } & {S_{5} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{{14}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{4} } & {S_{2} } & {S_{1} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{5} } \\ {} & - & - & {S_{3} } & {S_{2} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{{15}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{4} } & {S_{4} } & {S_{5} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{1} } \\ {} & - & - & {S_{3} } & {S_{3} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ \;L^{{16}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{6} } & {S_{1} } & {S_{4} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{0} } \\ {} & - & - & {S_{3} } & {S_{6} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);L^{{17}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{6} } & {S_{1} } & {S_{4} } \\ {} & - & {S_{3} } & {S_{5} } & {S_{1} } \\ {} & - & - & {S_{3} } & {S_{4} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right)L^{{18}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{6} } & {S_{4} } & {S_{6} } \\ {} & - & {_{3} } & {S_{2} } & {S_{5} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & {} & - & - & {S_{3} } \\ \end{array} } \right); \hfill \\ L^{{19}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {_{4} } & {S_{4} } & {S_{5} } \\ {} & - & {S_{3} } & {S_{2} } & {S_{3} } \\ {} & - & - & {S_{3} } & {S_{1} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right);\;L^{{20}} = \left( {\begin{array}{*{20}c} {} & {S_{3} } & {S_{2} } & {S_{3} } & {S_{5} } \\ {} & - & {S_{3} } & {S_{4} } & {S_{6} } \\ {} & - & - & {S_{3} } & {S_{2} } \\ {} & - & - & - & {S_{3} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

5.2 Solution Process

The following steps illustrate the process to solve this SN-LGDM problem.

Step 1: PIS transformation. Following the description of Section. 2.1, let \(\varepsilon =0.01\); the personalized numerical scales of linguistic terms for twenty DMs are derived by solving Model (2), which are listed in Table 1; then the fuzzy preference matrices \(U^k={(u_{ij}^k)}_{4\times 4}, (k=1,...,20)\) are obtained.

Step 2: Community detection. Algorithm 1 is applied to detect four communities as depicted in Fig. 2.

$$\begin{aligned} C_1= & {} \left\{ dm_1, dm_3, dm_4, dm_{10}, dm_{12}, dm_{16}, dm_{17} \right\} ;\\ C_2= & {} \left\{ dm_2, dm_6, dm_8, dm_{14} \right\} ;\\ C_3= & {} \left\{ dm_5, dm_{13}, dm_{20}\right\} ;\\ C_4= & {} \left\{ dm_7, dm_9, dm_{11}, dm_{15}, dm_{18}, dm_{19}\right\} . \end{aligned}$$

Step 3: Consensus and trust measure. As per (Wu et al. 2021), the consensus threshold \({\overline{MCL}}\) and \({\overline{GCL}}\) are set as 0.80; the initial general consensus level of communities as per Definition 1 is 0.70 using, which is lower than \({\overline{GCL}}\), so the feedback mechanism is implemented to improve the consensus level.

The cohesion of communities (\(\gamma =\varphi =0.5\) is adopted) are listed in Table 2, and the initial mutual consensus level matrix(MCLM) and mutual trust level matrix(MTLM) of communities are as follows.

$$ MCLM = \left( {\begin{array}{*{20}c} {} & - & {0.60} & {0.69} & {0.68} \\ {} & {0.60} & - & {0.73} & {0.72} \\ {} & {0.69} & {0.73} & - & {0.79} \\ {} & {0.68} & {0.72} & {0.79} & - \\ \end{array} } \right);\;MTLM = \left( {\begin{array}{*{20}c} {} & - & {0.25} & {0.14} & {0.17} \\ {} & {0.25} & - & 0 & {0.25} \\ {} & {0.14} & 0 & - & {0.22} \\ {} & {0.17} & {0.25} & {0.22} & - \\ \end{array} } \right){\text{ }} $$

Step 4: Interaction mechanism. The interaction mechanism of communities includes two phases, i.e., the dynamic judgment of the feedback strategy and the minimum adjustment bidirectional feedback model solution. The detailed feedback process is described below. First round of interaction.From the initial MCLM, communities \(c_1\) and \(c_2\) are identified as the initial pair of minimum consensus: \(PMC=(1,2)\). The thresholds values \({\overline{ICT}}=0.70\) and \({\overline{ITT}}=0.20\) are set in this article. Since \(MCL\left( c_1,c_2\right) < {\overline{ICT}}\),\(\ MTL\left( c_1,c_2\right) >{\overline{ITT}}\), then the corresponding Quadrant 2 of the mutual consensus-trust plot strategy applies. Since \({\overline{MCL}}'=0.75,\) Model (20) is applied replacing \({\overline{MCL}}\) with \({\overline{MCL}}'=0.75\) to generate feedback suggestions to improve \(MCL\left( c_1,c_2\right) \). The solution of the model is achieved for \(\eta ^{c_{1}}=0.18\) and \(\eta ^{c_{2}}=0.19\), and the updated MCLM is provided below:

$$\begin{aligned} MCLM_{1^{st} Round}=\begin{pmatrix} &{}-&{}0.75&{}0.74&{}0.71 \\ &{}0.75&{}-&{}0.78&{}0.74 \\ &{}0.74&{}0.78&{}-&{}0.79 \\ &{}0.71&{}0.74&{}0.79&{}- \\ \end{pmatrix}; \end{aligned}$$

The general consensus degree is recalculated to become \(GCL=0.75\); thus, the interaction will continue.

Second round of interaction. At this round, the pair of communities with minimum consensus is identified to be \(c_1\) and \(c_4\). Since \(MCL\left( c_1,c_4\right) > {\overline{ICT}}\), \(MTL\left( c_1,c_4\right) < {\overline{ITT}}\), then the corresponding Quadrant 4 of the mutual consensus-trust plot strategy applies, which implies that \(c_1\) and \(c_4\) aretransferred to Quadrant 3 to implement interaction with the third-party communities. Since \(CF_{c_{1}}=\left\{ c_2 \right\} \), \(CF_{c_{4}}=\left\{ c_2, c_3 \right\} \). The community in \(CF_{c_{4}}\) with maximum mutual consensus with \(c_{4}\) is community \(c_3\); thus, the average opinion of \(c_2\) and \(c_3\) obtained by (22) is used as the feedback reference for \(c_1\) and \(c_2\), i.e., \(u_{ij}^{hk^{*}}=u_{ij}^{23^{*}}\).

Given that \(MCL\left( c_1,c_{23^{*}}\right) =0.78\ge {\overline{ICT}}\) and \(MCL\left( c_4,c_{23^{*}}\right) =0.77\ge {\overline{ICT}}\), it is \( \overline{MCL_1}'=\overline{MCL_4}'={\overline{MCL}}\) and Model (23) is solved. The solution is obtained when \(\eta ^{c_{1}}=0.10\) and \(\eta ^{c_{4}}=0.16\); the updated MCLM is

$$\begin{aligned} MCLM_{2^{nd} Round}=\begin{pmatrix} &{}-&{}0.77&{}0.76&{}0.74 \\ &{}0.77&{}-&{}0.78&{}0.78 \\ &{}0.76&{}0.78&{}-&{}0.81 \\ &{}0.74&{}0.78&{}0.81&{}- \\ \end{pmatrix}; \end{aligned}$$

The general consensus degree is recalculated to become \(GCL=0.77\); thus, the interaction still continues.

Third round of interaction. At this round, the pair of communities with minimum consensus is identified to be \(c_1\) and \(c_4\). Their mutual consensus level at this stage has improved. After judgment based on the strategy for Quadrant 4, \(c_1\) and \(c_4\) can be moved to Quadrant 1 to implement mutual interaction, whose solution is obtained when \(\eta ^{c_{1}}=0.12\) and \(\eta ^{c_{4}}=0.11\); leading to the general consensus degree to be \(GCL=0.80\); Thus, the feedback mechanism is terminated.

Table 1 Personalized numerical scales of linguistic terms for twenty DMs
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Fig. 2
figure 2

The result of community detection using Algorithm 1

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Table 2 Cohesion measure
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Table 3 provides the important indicators of each round, and the final ranking of alternatives of the problem is: \(z_1>z_2>z_4>z_3\).

Table 3 Related indicators of each round of feedback
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6 Simulation and Comparison Analysis

6.1 Simulation Analysis

In order to verify the validity of the proposed model, randomized numerical simulation experiments are conducted in this subsection. Specifically, the simulation experiment includes the following steps:

Step 1: The elements of the decision matrix of each community are randomly generated from the interval [0,1] and, without losing generality, the cohesion of each community is also randomly generated from the interval [0.4,0.6].

Step 2: Model (20) is applied to implement interaction between communities until the consensus threshold is reached.

Step 3: Obtain the number of interactions required to reach consensus (denoted as t) and the evolution process of GCL.

Let \(m=4\), the number of communities \(l\in \left\{ 3,4,5 \right\} \), for each number of communities l, the simulation experiment is run 100 times. Since the initial opinion matrices of communities are randomly generated, the initial consensus level obtained varies, and the number of interactions required to reach the consensus threshold will also be different. The percentage of different number of interactions t under different number of communities is shown in Fig. 3. In addition, fixing the value of l, the evolution process of the average GCL under different number of interactions is shown in Fig. 4.

Fig. 3
figure 3

Percentage of different number of interactions under different number of communities

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Fig. 4
figure 4

Average GCL evolution process under different number of interactions

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From Figs. 3 and  4, it can be noticed that the number of interactions required for the group to reach consensus varies under different number of communities. Meanwhile, it can be found that the application of our proposed model to randomly generated decision data can effectively help communities reach group consensus within a limited number of feedback rounds. In other words, the model proposed in this paper is effective in group consensus promotion.

6.2 Comparison and Discussion

In order to illustrate the necessity and rationality of introducing the interaction consensus threshold and interaction trust threshold, the proposed consensus framework (denoted as M1) is compared with the bidirectional feedback mechanism that doesn’t consider the interaction threshold (denoted as M2) based on the previous illustrative example. Specifically, the identified community pairs interact with each other directly adopting Model (20) in M2 without setting the interaction threshold. The external consensus level evolution of communities in two methods are depicted in Fig. 5, and the adjustment degree (\( AD_{c_{h}}=\frac{2}{m(m-1)}\sum _{i=1,j>i}^{m-1}\left| {\overline{u}}_{ij}^{c_{h}}- u_{ij}^{c_{h}}\right| \)) of the pairwise communities implementing interaction in two methods are represented in Fig. 6.

Fig. 5
figure 5

External consensus level evolution of communities in two methods

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Fig. 6
figure 6

Adjustment degree of PMC in M1(blue bar) and M2(green bar)

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As can be seen from Fig. 5, the consensus is reached through three rounds of feedback in both methods, indicating that setting the interaction thresholds will not affect the consensus efficiency. In addition, it can be seen from Fig. 6 that the adjustment degree of interacting communities in each round of M2 varies greatly, the adjustment degree decreases in large gradient from the first round of feedback to the third round, which indicates that the community that interacts first needs to make large-scale opinion modification compared with the community that interacts later, which is unfair. In the actual decision-making, it will reduce the willingness of DMs to accept feedback suggestions, and even produce non-cooperative behavior. However, after the interaction thresholds are set, since communities with low mutual consensus or trust cannot interact, it will not lead to excessive opinion modification, which avoids the problem of M2 to a certain extent.

Furthermore, in order to highlight the distinctive advantages of this article, some discussions are provided regarding the comparison of the proposed consensus framework with existing consensus models.

  1. (a)

    Different from the existing research on the bidirectional interaction mechanism, this paper assumes that not all paired subgroups can implement mutual opinion interaction, since this is related to the mutual consensus level and mutual trust level between them. Therefore, the interaction consensus threshold and the interaction trust threshold are set to classify the specific situation to address, and the corresponding feedback strategies are designed to realize the interaction of subgroups under different scenarios. Specifically, for the subgroups that can interact, a minimum adjustment bidirectional feedback model is proposed to ensure that subgroups reach mutual consensus with minimum opinion modification. For subgroups that cannot interact, the opinions of third-party communities is appropriately identified to assist them in their required interaction.

  2. (b)

    This paper discusses the relationship between trust and consensus, and investigates their impact on the CRP. In fact, trust and consensus are reliable resources that can promote and drive the achievement of group consensus, and they can complement each other under certain conditions. For example, subgroups can use high level of trust in exchange for less adjustment of their opinions, or they can use opinion similarity to build their trust relationships.

  3. (c)

    Existing literature rarely considers the role of cohesion in the feedback process. Cohesion can be divided into internal cohesion and external cohesion, which can reflect the characteristics of groups at different levels, and will affect the group’s interaction and compromise behavior. This paper calculates the community cohesion considering the subgroup’s internal and external consensus and trust level, and it further investigates the effect of cohesion in the feedback mechanism.

7 Conclusion

This article proposes a consensus-trust driven bidirectional minimum adjustment consensus framework with hybrid strategies in SN-LGDM. The main contributions of this article are summarized as follows.

  • This paper investigates the hybrid feedback strategies in the bidirectional interaction mechanism. By introducing the interaction consensus threshold and interaction trust threshold, the interacting subgroups are classified into one of four categories. Corresponding feedback strategies for each category are designed, in which the consensus and trust between subgroups are utilized to facilitate the realization of group consensus.

  • A minimum adjustment bidirectional feedback model considering cohesion is developed in the context of SN-LGDM. The proposed model takes the influence of cohesion on the compromise degree of subgroup into account, and it aims to minimize the amount of opinion modification of interacting subgroups. In addition, with regard to the subgroups that cannot interact directly, the third-party communities in the group is utilized to assist the interaction process.

There are obviously some limitations that require future research efforts. This article only discusses the impact of cohesion on subgroup interaction process, and does not explore the non-cooperative behavior and/or manipulative behavior that subgroups may utilize in the bidirectional interaction of SN-LGDM. In addition, the design and exploration of the theoretical mechanism in bidirectional interaction also worth further research.