Consensus stability among composite decision makers in the framework of hesitant fuzzy graph model with application to doctor–patient disputes

Abstract

Doctor–patient disputes are inevitable in human beings’ social lives. The rapid development of social media makes doctor–patient disputes easier to spiral out of control. One of the ensuing problems is that the size of the parties in conflict has increased, and conflicts between individuals are more easily transformed into conflicts between groups and organizations. Thus, a new concept called composite decision-makers (CDMs) that represent a set of stakeholders or organizations with common interest goals is proposed in the graph model for conflict resolution (GMCR) with hesitant fuzzy preference relations (HFPRs). To be specific, first, this research uses HFPRs to represent the preferences of individuals within a group, and thereby defines a consensus measurement method based on grey relational analysis. Then, the K-means clustering method is improved based on social trust network and grey relational analysis. Next, based on the clustering results and consensus measures, a DeGroot opinion evolution model is proposed to simulate the interaction of opinions between subgroups. The above described method can be used to reach consensus among CDMs and thereby obtain a preference ranking over all states for CDMs for carrying out conflict analysis. CDMs’ HFPRs has been transformed to be group preferences that are represented by fuzzy preferences. Crucially, a set of novel consensus stability definitions among CDMs is proposed in the hesitant fuzzy graph model structure. Finally, the new proposed stability definitions are applied to doctor–patients disputes in China to assist the government in promoting reconciliation between doctors and patients to enhance social harmony.

Introduction

Conflicts or disputes are one of the inevitable events in real life human social life, ranging from arguments among family members to military wars between nations. Particularly, the harm of doctor–patient disputes is widespread and far-reaching, affecting the normal operation of the doctor–patient relationship and even the entire harmonious society¹. As the world’s largest developing country, various reasons such as inadequate legal systems for medical security, uneven distribution of medical resources, blocked channels for patients to protect their rights, and one-sided media coverage have led to doctor–patient disputes becoming a major obstacle for China to build a harmonious society². Obviously, the reasonable resolution of conflicts not only helps to ease tensions among different stakeholders, but also contributes to building a harmonious society and promoting economic development^3,4,5. In order to assist decision-makers (DMs) in real-life conflicts to achieve the above described ultimate solution goal, various formal methods have been designed, including game theory and its extension model^6,7,8,9,10 as well as the graph model for conflict resolution (GMCR)¹¹. Compared to other existing conflict analysis methods, GMCR is considered a flexible and simple approach for resolving conflicts for many benefits¹². The advantages of GMCR can be summarized as three aspects: First, conflicts can involve multiple DMs, and each DM is allowed to have access to different strategic options; Second, the modeling and analysis process in GMCR only requires DMs’ relative preferences; Third, GMCR not only can deal with both transitive and intransitive preference information but also character both DM’s reversible and irreversible movement between feasible states¹³. This section describes the harmfulness of doctor–patient disputes and the superiority of GMCR as the latest conflict analysis method in resolving doctor–patient disputes.

The basic research structure of GMCR has been reviewed in the following paper. Up until now, the superiority of GMCR makes it rich in both theoretical research and practical applications. To be specifically, the theoretical research of GMCR has been carried out from three research insights: forward perspectives, behavioral perspective, and inverse perspective¹⁴. Within the forward perspectives of GMCR, original GMCR has been extended to different forms with uncertain preferences to resolve conflicts in complex uncertain environments^15,16. In other words, various extended GMCR models and their stability definitions effectively help DMs obtain practical and feasible conflict policy recommendations in uncertain decision making environments^17,18,19,20. Forward GMCR has always been the most fundamental and important research direction in GMCR as it ensures the basic purpose of conflict analysis methods. Behavioral perspective of GMCR focuses on the impact of the DMs’ personal characteristics such as attitude, power, and emotion on the development and resolution of conflicts. The difficulty of behavior GMCR lies in how to characterize the interaction process of personal characteristics among DMs. As for the inverse perspective of GMCR, many different approaches have been integrated into inverse GMCR to obtain required preference relations, which can be used to ensure a given state to be stable under a specific stability definition^21,22,23. The research on the inverse problem of GMCR is meaningful and practical. However, existing research methods still have significant shortcomings and are worthy of further in-depth study²⁴. Recently, within the forward GMCR, hesitant fuzzy preference relations (HFPRs) have been integrated into GMCR to characterize the hesitation and uncertainty of DM’s preferences at the same time. The biggest advantage of HFPRs is that it can describe the uncertain preferences of composite decision makers (CDMs), which constituted of several single DMs. In other words, CDMs is proposed in hesitant fuzzy GMCR to describe a group of individuals with differing decision characteristics–conservative or aggressive by Wu et al.²⁵ and Wu et al.²⁶, and any one of CDMs can make a decision on behalf of the group. However, due to differences in professional knowledge and life backgrounds, the preferences among individual DM in CDMs may be slightly different, making it difficult to reach consensus. By consulting existing reference literature on GMCR, we can clearly find that existing research rarely considers situation where individual opinions within CDMs cannot reach a consensus. It is obvious that the traditional GMCR cannot resolve conflicts with CDMs in uncertain environments. At the same time, GMCR with HFPRs provides the theoretical basis for researching consensus stability among CDMs in conflicts.

The consensus problem (i.e. common understanding or agreements) is a critical issue worth attention in GMCR, which can greatly influence DM’s judgment and strategies when facing a conflict^27,28,29. Thus, in this section, we will review the traditional consensus research in decision making fields. As is known to all, consensus issues have always been a research hotspot³⁰, aiming to assist the achievement of consensus among a set of DMs. Until today, consensus problems continue to deepen. For example, Guo, et al.³¹ proposed multi-dimensional multi-round minimum-cost consensus models and designed different distance measures to assess the distance between DMs in the consensus reaching process. Liao, et al.³² proposed a double alpha-cut-based consensus measure for group decision making and apply the method to a case study of selecting reverse logistics suppliers. The study of consensus issues among DMs is not only beneficial for promoting consensus on interests and reducing confrontation among all stakeholders involved in decision-making process, but also for the effective implementation of policy recommendations and gaining more support from people³³. It is obvious that the research methods and solving steps of consensus issues can be used to guide and promote consensus research among CDMs within GMCR³⁴. Note that the consensus problem studied in this research means the equilibrium of conflicts that are accepted by all DMs. The traditional consensus problem means decision-making not only seeks the majority agreement of participants, but also aims to resolve and alleviate the opposition of a minority in order to achieve the decision with the most agreement. In this paper, we aim to achieve consensus among CDMs before conducting conflict analysis in GMCR with HFPRs and apply it to resolve real-life conflicts. It is evident that the consensus research in traditional decision making field is a research hotspot. Thus, studying consensus within the framework of GMCR is a meaningful research direction and has significant implications for the development of conflict analysis theory.

Next, various methods have been introduced in GMCR to aggregate the opinions of individuals within the CDMs to be one preference vector. Within complex conflicts involving multiple DMs, each stakeholder may not represent an individual DM but represents an organization. It is evident that each individual in organization often rarely represent true preference tendency of CDMs. For example, stakeholders in real life doctor–patient disputes include doctor, patient, and local government. Among them, doctor and the local government are supposed to be an independent entity, while the patients are composed of multiple DMs and belong to a CDM. When the opinions of individuals within the CDMs cannot be unified, it is difficult to determine the DM’s true preferences and conduct conflict analysis. Before conducting conflict analysis, it is necessary to measure consensus and obtain a total preference ranking within the CDMs. Thus, in this article, a new internal consensus level measurement method has been proposed based on the grey relational analysis to measure consensus in CDMs within hesitant fuzzy GMCR. Then, K-means clustering method has been improved based on grey relational analysis and social trust network, with the aim of clustering individuals from the perspectives of opinion grey relational degree and social relationships, reducing the complexity of the consensus process. Next, a DeGroot model has been proposed that can simulate the evolution of opinions between subgroups, which can balance the influence of each subgroup on the final opinion and eliminate the dominant role of some subgroups in the opinion evolution process. It is evident that the introduction of multiple different methods is aimed at better characterizing the preference aggregation of CDMs, and thus enabling the generation of stability definitions within the framework of GMCR. Note that CDMs’ HFPRs has been transformed to be group preference that represents by fuzzy preferences. Finally, a new set consensus stability definition is proposed in hesitant fuzzy GMCR to describe preference interaction among CDMs where individual opinions cannot be unified. It is not only promotes the development of GMCR in complex uncertain environments but also first research internal consensus within GMCR. The preference vector of CDMs obtained in this paper becomes the basic condition for generating consensus stability definitions in GMCR.

In the end, the new proposed consensus stability definitions are applied to doctor–patient disputes to demonstrate its scientific validity. Nowadays, the resolution of doctor–patient relationships has always been a difficult problem in social management, mainly due to the increasing trend of doctor–patient relationships³⁵. The increasing number of doctor–patient disputes has had a very negative impact on the harmonious development of society in China and even around the world. The doctor–patient relationship, as a special interpersonal relationship established between doctors and patients in medical activities, has always been a social focus issue. In the context of such a tense doctor–patient relationship, relevant government agencies need reasonable guidance to reform the existing management and operation mechanisms and effectively improve the increasingly tense doctor–patient relationship³⁶. The formulation and implementation of policies require reasonable and effective theoretical support. It is evident that certain DM in doctor–patient disputes may consist of different individual stakeholders. The complexity of internal consensus among CMs exacerbates the difficulty of doctor–patient disputes. Faced this problems, the novel proposed method in this paper is used to apply in doctor–patient disputes to obtain reasonable policy recommendations. Finally, the contributions of this paper are summarized as follows:

• (1) Within the GMCR, a novel method is proposed to achieve internal consensus among CDMs to carry out conflict analysis. In other words, this paper first uses clustering methods and opinion evolution models to obtain CDMs’ preferences.

• (2) This article first introduces social network to realize CDM’s consensus reaching, and then we can more reasonably judge the preference characteristics and types of CDMs based on social relationship.

• (3) This paper proposes a set of novel consensus stability definitions among CDMs in GMCR with HFPRs, which greatly enriches the consensus research on GMCR in uncertain environments.

• (4) The new proposed GMCR is applied to real-life doctor–patient disputes to illustrate its scientificity and effectiveness. Conflict analysis result indicates that the government needs to actively investigate patient needs and regulate medical institutions and personnel to ensure the fairness and objectivity of conflict resolution.

The remaining articles are organized as follows: “Preliminaries” introduces basic concepts of GMCR with HFPRs, social trust network and DeGroot model. In “Internal consensus analysis among CDMs in GMCR withHFPRs”, internal consensus analysis among DMs in GMCR has been carried out. Then, in “Consensus stability among CDMs in GMCR with HFPRs”, a set of consensus stability definitions among CDMs is first constructed in traditional GMCR. Then, in “Application to the doctor–patient disputes in China”, a real-life doctor–patient dispute in China is applied to prove the validity of proposed method. Finally, in “Conclusions and future work”, some conclusions and future works of this research have been described.

Preliminaries

GMCR

Traditional GMCR includes four elements: DMs, feasible states, directed graph, and preferences relations, and it is represented as GMCR = < N, S, {(Γ_k, Θ_k):k∈N$\}$ > ³⁷. The specific contents are described as follows:

• (1) DMs: Assuming a total of n stakeholders participate in conflict, a set of DMs can be represented by a bounded nonempty set N = {1, 2, …, k, …, q, …., n$\}$.

• (2) Feasible states: DM k’s courses of action are called options. An option may or may not be selected in a particular scenario. All feasible combinations of options constitute the set of feasible states S={s₁, s₂, …, s_m$\}$. For example, suppose that there exist two DMs N={1, 2$\}$ and three options o₁, o₂, o₃ in conflicts. Option o₁ is determined by DM 1, and options o₂ and o₃ are controlled by DM 2. Mathematically, there are 2³=8 states. If all 8 states are feasible states, then the set of feasible state is S={s₁, s₂, …, s₈$\}$. However, if DM2 cannot simultaneously select options 1 and 2, the number of feasible states needs to be reduced.

• (3) Directed graph: Each vertex in directed graph Γ_k represents feasible state. The movement between two feasible states is represented as arc. In particular, the movement controlled by each DM is represented by arcs with different characteristics.

• (4) Preference relation: “~_k” represents that DM k neither prefers state s_i nor s_j. “≻_k” represents that DM k prefers one state another state. Θ_k satisfies completeness, ≻_k satisfies asymmetry, ~_k satisfies reflexivity and symmetry.

Definition 1 ³⁸

According to a given DM k and state s, the reachable list is defined as follows.

$${R_k}(s)=\{ {s_i} \in S:(s,{s_i}) \in {A_k}\}$$

(1)

In the process of forming a state in GMCR, if only one option selection for focus DM among two states is different, and then there exists a directed arc between these two states in the directed graph. Therefore, for a state s, all the associated states connected by using a directed arc are called a reachable set. For example, there exists four states S={s₁, s₂, s₃, s₄$\}$in conflicts. For a state s₁, if there exist directed arc among states s₁, s₃, s₄, and thereby the reachable states from state s₁ for DM k is equal to R_k(s₁)={s₃, s₄$\}$.

Definition 2 ³⁸

Within the GMCR, according to Eq. (1), improvement list is defined as follows.

$$R_{k}^{+}(s)=\{ {s_i} \in {R_k}(s):{s_i}{ \succ _k}s\}$$

(2)

The common point between Definitions 1 and 2 is that the distance between the reachable states and the initial state is one-step movement, and the difference is that the reachable state in Definition 2 belongs to the improvement state. Definitions 1 and 2 are constructed for a single DM.

In general, conflicts in real-life involve multiple DMs, and its modeling and analyzing process is more complex. Thus, within the GMCR with multiple DMs, suppose that there exist a coalition H ⊆ N, |H| ≥ 2. Similar to Definitions 1 and 2, the reachable list and unilateral improvement list for H are represented as R_H(s) and $\:{\text{R}}_{\text{H}}^{+}\text{(}\text{s}\text{)}$, respectively. Note that the legal sequence in R_H(s) and $\:{\text{R}}_{\text{H}}^{+}\text{(}\text{s}\text{)}$ represents that a DM cannot move twice in a row. All DMs that are determined in the legal sequence of R_H(s) are represented as Ω_H(s, s_i). Similarly, there exists similar DMs in improvement reachable list, represented as $\:{\Omega}_{\text{H}}^{+}\text{(}\text{s},{\text{s}}_{\text{i}}\text{)}$.

Definition 3 ³⁹

Let a coalition H⊆N and a state s. Thus, the steps to obtain a reachable list R_H(s) are shown as follows.

• (1) If $k \in H$ and ${s_1} \in {R_k}(s)$, then ${s_1} \in {R_H}(s)$ and $k \in {\Omega _H}(s,{s_1})$.

• (2) If $k \in H$, ${s_1} \in {R_H}(s)$, and ${s_2} \in {R_k}({s_1})$, then we obtain ${\Omega _H}(s,{s_1}) \ne \{ k\}$,${s_2} \in {R_H}(s)$ and $k \in {\Omega _H}(s,{s_2})$.

Definition 4 ³⁹

Let a coalition H ⊆ N and a state s. Thus, the steps of reachable improvement list $\:{\text{R}}_{\text{H}}^{+}\text{(}\text{s}\text{)}$ are shown as follows.

• (1) If $k \in H$ and ${s_1} \in R_{k}^{+}(s)$, then ${s_1} \in R_{H}^{+}(s)$ and $k \in \Omega _{H}^{+}(s,{s_1})$.

• (2) If $k \in H$, ${s_1} \in R_{H}^{+}(s)$, and ${s_2} \in R_{k}^{+}({s_1})$, then we obtain $\Omega _{H}^{+}(s,{s_1}) \ne \{ k\}$, ${s_2} \in R_{H}^{+}(s)$ and $k \in \Omega _{H}^{+}(s,{s_2})$.

Within traditional GMCR, four basic stability definitions have been proposed to show DM’s interaction and resolve conflicts^40,41.

Definition 5

A state s is Nash stability (Nash, R) stable for the focus DM k if and only if (iff) $R_{k}^{+}(s)=\emptyset$.

Definition 6

A state s is general metarationality (GMR) stable for DM k iff for each ${s_1} \in R_{k}^{+}(s)$ there is at least one ${s_2} \in {R_{N - \{ k\} }}({s_1})$ satisfying ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$, where a coalition $N - \{ k\}$ represents that all the other DMs except DM k.

Definition 7

A state s is symmetric metarationality (SMR) stable for DM k iff for each ${s_1} \in R_{k}^{+}(s)$ there is at least one ${s_2} \in {R_{N - \{ k\} }}({s_1})$ satisfying ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$, and ${s_3}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$ for all ${s_3} \in {R_k}({s_2})$.

Definition 8

A state s is sequential stability (SEQ) stable for DM k iff for each state ${s_1} \in R_{k}^{+}(s)$ there is at least one ${s_2} \in R_{{N - \{ k\} }}^{+}({s_1})$ satisfying ${s_2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \prec } s$.

Remark 1

If N in Definitions 5−8 just includes two DMs, Definitions 5 –8 are considered to be four basic stability definitions in GMCR with two DMs.

HFPRs

Hesitant fuzzy set (HFS) is developed from fuzzy set and is considered to be a practical tool to determine whether an element belongs to a set influenced by a hesitation among different values^42,43,44. A set of alternatives in group decision making is equal to S in GMCR. We first review the original definitions of fuzzy preferences.

Definition 9

Let S be a set of states in GMCR or be a set of alternations in decision-making process, the fuzzy preferences over S is characterized by fuzzy binary relations with membership function ${\mu _R}:S \times S \to$[0, 1]. It is represented by a matrix $\Re ={({r_{ij}})_{m \times m}}$, where r_ij is used to represent the preference degree between two states s_i and s_j. r_ij satisfies the following condition:

$${r_{ij}} + {r_{ji}} = 1,{r_{ii}} = 0.{{5}},i,j = {{1}},{{ 2}}, \ldots ,m$$

(3)

It is obvious that the larger the value of r_ij, the more likely that s_i is preferred to s_j. Particularly, r_ij = 1 represents that one strictly prefers s_i than s_j, at the same time, r_ij = 0 represents that one strictly prefers s_j than s_i.

Definition 10

Let S be a set of states, and thereby the HFS over S is defined as follows:

$$A=\left\{ {\left\langle {s,{h_A}(s)} \right\rangle {\text{|}}s \in S} \right\}$$

(4)

where h_A(s) is used to represent a set of numerical values in [0, 1]. In other words, it represents the possible membership degree of a given state s to the set A. h_A(s) is named hesitant fuzzy element.

In order to describe CDMs’ preference between two states in GMCR by using HFS, Wu, et al.²⁵ first use HFPRs to replace crisp preferences in GMCR to describe DM’s complex preference.

Definition 11

Let S be a set of states. Suppose that k∈N is a CDM. Then, the HFPRs over S can be defined as ${P^k}={(p_{{ij}}^{k})_{m \times m}}$, where $p_{{ij}}^{k}=\{ p_{{ij,1}}^{k},p_{{ij,2}}^{k},...,p_{{ij,f}}^{k},...,p_{{ij,l}}^{k}\}$. If CDM k is composed of a single individual, then l = 1; if DM k is a CDM, then l > 1, which is equal to the number of individuals within the CDM. $p_{{ij,f}}^{k}$ is the fth individual’s preference values between states s_i and s_j, and it possesses the following conditions:

$$p_{ij,f}^k + p_{ji,f}^k = 1,\,{\text{and}}\,{p_{ii,f}} = 0.5\,{\text{for}}\,{\text{all}}\,i,j = {{1}},{{ 2}},{{ }} \ldots ,m,f = {{1}},{{ 2}},{{ }} \ldots ,l,k \in N$$

(5)

Obviously, $p_{{ij,f}}^{k}>0.5$ means that individual f prefers state s_i than state s_j.

Social trust network

Basic concept of social trust network

With the development of social media software, different individuals or groups can more easily and conveniently establish connections, which also affect the evolution of various conflicts. DMs in conflicts can establish connections through social media to form social relations. Furthermore, the internal individuals in CDMs also can establish complex social relations. To reasonably analyze the social relations of different subjects’ opinions and apply them to the process of conflict analysis, the crucial work in the social trust network is to construct trust transmission mechanism among CDMs⁴⁵. Recall that social trust network is a quantitative analysis method based on mathematical methods and graph theory⁴⁶. Three forms of social trust network are shown in Table 1.

Table 1 Different expression of social trust network.

In Table 1, the social relations expressed in matrix form contain only 0 and 1 elements. Among them, 0 represents complete distrust and 1 represents complete trust. However, matrix expression method in social trust network has strong limitations. First, it cannot reflect the uncertainty in the DM’s trust relationship. Second, not all DMs will express a clear trust relationship. For example, suppose that DM 2 and DM 4 in Table 1 do not have a trust relationship, and the trust relationship needs to be established through the trust propagation mechanism. In other words, the use of 0 and 1 to express the trust relationship cannot establish the trust propagation between DMs. To solve the above problems, the representation of trust relationship has been widely studied. Based on this, the representation based on trust and distrust among DMs proposed by Wu, et al.⁴⁷ is introduced as follows:

Definition 12

The µ=(t, d), t, d ∈ [0,1] is used to represent the trust relationship between DMs, where t is the trust degree and d is the distrust degree. Thus, the social relations among n DMs can be represented by the following matrix.

$$A=\left[ {\begin{array}{*{20}{c}} - &{{\mu _{12}}}& \cdots &{{\mu _{1n}}} \\ {{\mu _{21}}}& - & \ldots &{{\mu _{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{\mu _{n1}}}&{{\mu _{n2}}}& \cdots & - \end{array}} \right]$$

Furthermore, trust degree t and distrust degree d also can form matrix, represented as trust degree matrix T and distrust degree matrix D, respectively.

Definition 13

The trust score is a mapping on the binary array µ=(t, d), and it is calculated as follows:

$$ts(\mu )=(t - d+1)/2$$

(6)

Then, trust scores among DMs can be obtained by using Eq. (5), and it constitute the trust score matrix:

$$TSM=\left[ {\begin{array}{*{20}{c}} - &{t{s_{12}}}& \cdots &{t{s_{1n}}} \\ {t{s_{21}}}& - & \ldots &{t{s_{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {t{s_{n1}}}&{t{s_{n2}}}& \cdots & - \end{array}} \right]$$

(7)

where ts_kq is used to represent the trust score determined between DM q and DM k.

Next, the trust score obtained among DM q and all other DMs is calculated as follows:

$$T{S_q} = \frac{{\sum\limits_{k = 1,k \ne q}^n {t{s_{kq}}} }}{{n - 1}},q = {{1}},{{ 2}}, \ldots ,n$$

(8)

Furthermore, the value of TS_q reflects DM q’s social status in the group to some extent.

In order to comprehensively consider the impact of trust and relation strength on DMs’ social relationships, two situations are combined as a comprehensive evaluation value between DMs. Suppose that the trust relationship between DM k and DM q is called φ_kq = (t_kq, s_kq), and the calculation method for the comprehensive evaluation value is shown as follows:

$$c{e_{kq}} = {t_{kq}} \times {s_{kq}},q = {\rm{1}},{\rm{ 2}},{\rm{ }} \ldots ,n$$

(9)

The comprehensive evaluation values between all DMs form a comprehensive evaluation matrix, which is represented as follows:

$$CE=\left[ {\begin{array}{*{20}{c}} - &{c{e_{12}}}& \cdots &{c{e_{1n}}} \\ {c{e_{21}}}& - & \ldots &{c{e_{2n}}} \\ \vdots & \vdots & \ddots & \vdots \\ {c{e_{n1}}}&{c{e_{n2}}}& \cdots & - \end{array}} \right]$$

Evaluation method of missing trust relationship

When two DMs need other DMs to act as intermediaries to establish a trust relationship, as the number of intermediaries’ increases, the level of trust between DMs will decrease, while the level of distrust will gradually increase. Based on this characteristic, Archimedean t-norm and t-conorm operators can be used to simulate the process of trust propagation⁴⁸. Therefore, the exponential operator is used to determine the weights of the trust path and calculate the final trust relationship⁴⁹. At the same time, the trust level and distrust level between DMs are also introduced in the paper.

Definition 14

Suppose that there are v trust paths between DMs k and q, let ${\phi _{ijk}}=({t_{ijk}},{d_{ijk}})$ be the trust relationship between states i and j through given path ν, then all trust relationships on the trust paths form a set ${\Phi _{ij}}=\{ {\phi _{ij1}},{\phi _{ij2}},...,{\phi _{ijv}}\}$. If the number of intermediaries on each path forms another set {z₁, z₂, …, z_v$\}$, then the weight calculation method on each path is as follows:

$${w_f} = EO({z_f}) = \frac{{\exp (\eta {z_f})}}{{\sum\nolimits_{g = 1}^v {\exp (\eta {z_g})} }},f = {{1}}, \ldots ,v$$

(10)

Then, the trust relationship can be determined as follows:

$${\phi _{ij}} = ({t_{ij}},{d_{ij}}) = \sum\limits_{f = 1}^v {\frac{{\exp (\eta {z_f})}}{{\sum\nolimits_{g = 1}^v {\exp (\eta {z_g})} }}} {\phi _{ijf}},i,j = {{1}},{{ 2}}, \ldots ,m$$

(11)

Obviously, η is variable coefficient. When η > 0, w_f is positively correlated with z_f; when η < 0, w_f is negatively correlated with z_f. Since the value of z_f is inversely proportional to the trust propagation efficiency, η < 0 is crucial condition when calculating the weight. The lack of trust between DMs can be filled by using the above-described way.

DeGroot model of social trust network

In the process of decision making, DMs constantly exchange opinions. The opinions of DMs constantly evolve and integrate, and the opinion evolution model can be used to simulate this process. In the opinion evolution model, DMs exchange and evolve opinions according to established mechanisms, and finally reach consensus or split. Among them, DeGroot model is a more common and basic opinion evolution model⁵⁰. To be specifically, the opinions of each DM in a group are constantly influenced by other individuals, thus making their own opinions constantly evolve and forming new opinions. DMs are influenced to different degrees by other individuals in the process of opinion evolution, which are determined by the relationship between them, communication methods and other factors.

Definition 15

Suppose that a total of n DMs participate in the process of opinion evolution. The preference relation of DM k at time point T is denoted as $p_{k}^{{(T)}}$, and the preference relation of DM k at time point T + 1 can be expressed as follows:

$$p_k^{(T + 1)} = \sum\limits_{q = 1}^n {{w_{kq}}p_q^{(T)}} ,T = 0,{{ 1}},{{ 2}}, \ldots$$

(12)

Furthermore, Eq. (12) can be represented as follows:

$$P^{{(T + 1)}} = W \times P^{{(T)}} ,T = 0,{\text{ 1}},{\text{ 2}},{\text{ }} \ldots$$

(13)

where W=(w_ij)_n×n, ${P^{(T)}}=(p_{1}^{{(T)}},p_{2}^{{(T)}},...,p_{n}^{{(T)}})$. It is evident that Eqs. (12) and (13) are the basic DeGroot model.

If the social trust network is taken as a factor that determines the degree of influence among DMs, the weights in DeGroot model can be determined as follows:

$${w_{kq}}=\frac{{s{d_k} \cdot t{s_{kq}}}}{{\sum\limits_{{q=1,q \ne k}}^{n} {t{s_{kq}}} }}$$

where k, q represent different DMs.

Then, DeGroot model based on social trust network is expressed as follows:

$$P_{k}^{{(T + 1)}} = st_{k} p_{k}^{{(T)}} + \sum\limits_{{q = 1,q \ne k}}^{n} {w_{{kq}} p_{q}^{{(T)}} } ,T = 0,{\text{ 1}},{\text{ 2}},{\text{ }} \ldots$$

(14)

In the DeGroot model based on social trust networks, the trust relationship between DMs determines the degree to which each individual is influenced by other individuals. Therefore, Eq. (14) calculates weights based on the trust and confidence values between DMs. In addition, the confidence level of DMs determines their degree of adherence to their own views. In the process of opinion evolution, DMs not only retain their own views, but also are inevitably influenced by external factors, thus constantly changing their views. DeGroot model of social networks is a quantitative description and representation of the iterative process of viewpoint evolution.

Internal consensus analysis among CDMs in GMCR with HFPRs

The increase in the number of the DMs in conflicts will increase the complexity of conflict analysis, which increases the difficulty of obtaining conflict resolutions. As the number of DMs in conflicts increases, the differences of opinions among individuals become common situation. Thus, it is particularly important to resolve differences in individual opinions before conducting conflict analysis. Otherwise, on the one hand, it may lead to DMs being unable to clarify their own position and viewpoint in conflict, thereby reducing their advantage in the conflict; On the other hand, even if the process of resolving individual differences and consensus reaching is skipped to analyze conflicts, the resulting conflict equilibrium cannot satisfy all individuals. Furthermore, individual dissatisfaction with the equilibrium of conflicts can further lead to the intensification and intensification of disputes, deviating from the original intention of conflict analysis.

In this paper, in order to reduce the complexity of conflict analysis caused by significant differences in individual preferences among DMs, this paper constructs a consensus measurement model for CDMs based on grey relational analysis, clustering methods, social trust networks, and opinion evolution models. In addition, to avoid the dominant influence of highly influential individuals on the group during the consensus process, this paper uses clustering methods to identify highly influential subgroups and improves the traditional K-means method. The main process of the internal consensus reaching model among CDMs is shown in Fig. 1.

Fig. 1

Flow chart of internal consensus model for CDMs.

Consensus level analysis based on grey relational analysis

Methods for determining group preferences

The preferences of individuals over all feasible state in conflict can be converted into DM’s ranking vector. After the preference of an individual DM and the group preference of the CDMs are transformed, the curve similarity between individual preference and group preference can be measured by using grey relational analysis method. Before measuring the consensus level of each individual, it is necessary to determine the group preference. If there are l individuals within the CDM k, then the HFPRs of focus CDM is ${P^k}={(p_{{ij}}^{k})_{m \times m}}$, where $p_{{ij}}^{k}$ = {$p_{{ij,1}}^{k}$, $p_{{ij,2}}^{k}$, …, $p_{{ij,f}}^{k}$, …, $p_{{ij,l}}^{k}$$\}$. It is represented as internal individual preferences of CDM k. Based on the social trust network and the index operator, the determination of group preference is as follows:

Step 1

Social trust matrix based on trust and distrust is used to represent the social relationships among individuals within CDMs. Then, Eq. (5) is used to obtain the trust score matrix within CDM, and Eq. (7) is used to obtain the comprehensive trust score of each individual, represented as a vector TS_k={TS₁, TS₂, …, TS_f, …, TS_l$\}$.

Step 2

Individuals in a group with higher comprehensive trust scores often have greater influence, so they should have higher weights. The weight result is proportional to the value of the comprehensive trust score. Equation (11) is used to calculate the weight of individuals, thereby there exists.

$$w_{f} = EO(TS_{f} ) = \frac{{\exp (\eta TS_{f} )}}{{\sum\limits_{{d = 1}}^{l} {\exp (\eta TS_{d} )} }},f = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

Let $\eta$> 0, so that the weight result is proportional to the comprehensive trust score value, and the weight vector can be obtained as w_k = {w₁, w₂, …, w_f, …, w_l$\}$^T.

Step 3

Based on HFPRs, Eq. (10) is used to calculate group opinions, represented as $P_{c}^{k}={(p_{{ij,c}}^{k})_{m \times m}}$, where

$$p_{{ij,c}}^{k} = \sum\limits_{{f = 1}}^{l} {\frac{{\exp (\eta TS_{f} )}}{{\sum\nolimits_{{d = 1}}^{l} {\exp (\eta TS_{d} )} }}p_{{ij,f}}^{k} } ,i,j = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,m$$

Step 4

The group opinions is transformed as ranking vectors, represented as $P_{c}^{k}={(p_{{ij,c}}^{k})_{m \times m}}$, and then the ranking score of feasible state s_i is shown as follows.

$$s_{i} = \frac{{\left( {\prod\limits_{{i = 1}}^{m} {p_{{ij}} } } \right)^{{1/m}} }}{{\sum\nolimits_{{j = 1}}^{m} {\left( {\prod\limits_{{i = 1}}^{m} {p_{{ij}} } } \right)^{{1/m}} } }},i,j = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,m$$

(15)

Based on the ranking scores, all the feasible states are ranked in descending order to obtain the ranking vector of the group.

In addition, the hesitation fuzzy threshold of each individual DM within the CDMs is not the same in conflicts. To unify the threshold of the group, based on the weight vector obtained in Step 2, Eq. (10) is used to calculate the group threshold, and the result is as follows:

$${\lambda ^k}=\sum\limits_{{f=1}}^{l} {\frac{{\exp (\eta T{S_f})}}{{\sum\nolimits_{{d=1}}^{l} {\exp (\eta T{S_d})} }}\lambda _{f}^{k}}$$

(16)

Consensus measurement method based on grey relational analysis

After converting DM’s HFPRs into individual ranking vectors using Eq. (15), grey relational analysis is used to calculate the curve similarity between various preferences. The calculation method of curve similarity is described as follows.

Definition 16

Suppose that there are l individuals within the CDM k, the individual’s ranking vector over all states is shown as follows:

$$\left\{ \begin{gathered} {O_1}=({o_1}(1),{o_1}(2),...,{o_1}(m)) \hfill \\ {O_2}=({o_2}(1),{o_2}(2),...,{o_2}(m)) \hfill \\ \vdots \hfill \\ {O_l}=({o_l}(1),{o_l}(2),...,{o_l}(m)) \hfill \\ \end{gathered} \right.$$

Furthermore, the grey relational coefficient (GRC) between individual f and individual d with respect to feasible state s_i can be calculated as follows.

$$GRC_{{fd}}^{k}=\frac{{\gamma +\varsigma M}}{{{\Delta _{fd}}(i)+\varsigma M}},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \varsigma \in (0,1)$$

(17)

where ∆_fd(i)=|o_f(i) − o_d(i)|, $\gamma =\mathop {\hbox{min} }\limits_{{fd}} \mathop {\hbox{min} }\limits_{i} {\Delta _{fd}}(i)$, $M=\mathop {\hbox{max} }\limits_{{fd}} \mathop {\hbox{max} }\limits_{i} {\Delta _{fd}}(i)$, $\varsigma$ is a variable coefficient.

Next, GRC between individual f and individual d over all feasible states is shown as follows:

$$GRC_{{fd}} = \frac{{\sum\nolimits_{{i = 1}}^{m} {r_{{fd}}^{i} } }}{m},f,d = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(18)

The curve similarity between the preferences of individual f and d is positively correlated with the value of the GRC, and it satisfies GRC_fd ∈ [0, 1].

In the process of reaching group consensus, individual opinions will only be recognized as group opinions and reach consensus with the group when the similarity between individual opinions and group opinions reaches a certain level. If the group is denoted as c, the curve similarity between individual f’s opinion and the group c’s opinion is GRC_fc. When GRC_fd ≥ η, it shows that individual f will accept the group c’s opinion and reach consensus with the group, where η is called consensus threshold.

K-means clustering algorithm based on social trust network and grey relational analysis

The composition of CDMs in conflicts is often complex and uncertain. In smaller conflicts, CDMs may be small organizations or groups, but in larger conflicts, large organizations often participate. The number of individuals within CDMs is relatively large, which greatly increases the complexity of internal consensus building. Thus, the K-means clustering method is used to divide internal individuals into different interest groups, which can reduce the complexity of consensus reached; On the other hand, it is convenient to find subgroups with greater influence in the group and appropriately weaken their influence, so that the opinions of each subgroup can be fully considered in the consensus process, avoiding some individuals from being dissatisfied with the final conflict equilibrium solution due to their opinions being ignored, leading to further escalation of the conflict.

However, traditional K-means clustering methods face the following shortcomings: First, there is no basis for selecting clustering centers; second, in group decision-making in a social trust network environment, Euclidean distance is often based on social trust relationships and used as the basis for dividing subgroups, which cannot comprehensively consider the similarity of opinions between individuals. Based on this, the improvement of the K-means algorithm is described as follows:

Step 1

Equation (7) is used to calculate the comprehensive trust score of individual f, denoted as TS_f.

Step 2

Equations (17) and (18) is used to calculate the similarity between individual f’s opinion and group opinion, and record as GRC_fc;

Step 3

Based on the individual’s GRC and comprehensive trust score, the authority score (AS) of an individual is defined in⁴⁹ it is shown as follows.

$$AS_{f} = \alpha \cdot TS_{f} + (1 - \alpha )GRC_{{fc}} ,f = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(19)

where α is a variable parameter that determines the impact of the GRC and the comprehensive trust score on the AS.

Suppose that the variance of {GRC_1c, GRC_2c, …, GRC_lc$\}$ and {TS₁, TS₂,…, TS_l$\}$ are V_GRC and V_TS, respectively. Then, we have

$$\alpha =\frac{{{V_{TS}}}}{{{V_{TS}}+{V_{GRC}}}} \in [0,1]$$

(20)

.

Step 4

Determining the number of subgroups as;

Step 5

Equation (20) is used to obtain the set of AS {AS₁, AS₂,…, AS_l$\}$, and then select the individual with the highest AS to be one cluster center, the individual with the lowest AS to be another cluster center, and the other −2 cluster centers can be randomly selected.

After determining the cluster center, it is necessary to divide the remaining non central individuals into subgroups through clustering rules. Euclidean distance method based on social trust networks is often used in the K-means algorithm.

Definition 17

Suppose that the individuals serving as cluster centers form a set E_c, and the remaining individuals form a set E_r based on the social trust network, the Euclidean distance between the cluster center δ ∈E_c and the non-cluster center individuals ∈E_r is as follows:

$$d_{\xi }^{\delta } = \sqrt {\sum\nolimits_{{q = 1,q \ne \xi }}^{l} {(ts_{{\xi q}} - ts_{{\delta q}} )^{2} } } ,\delta ,\xi = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(21)

Obviously, Euclidean distance only divides subgroups based on social relationships between individuals, without considering the similarity of individual opinions.

Before considering the GRC and Euclidean distance comprehensively, it is necessary to first perform forward processing on the data. The GRC is positive data, while the Euclidean distance is negative data, let $\bar {d}_{q}^{l}=M - d_{\xi }^{\delta }$, where M = max$\{ d_{\xi }^{\delta }\}$, and $d_{\xi }^{\delta }$ is also a positive type data. Then, they can be combined to define the GRC-Euclidean index (GE), which is calculated as follows.

$$GE_{\xi }^{\delta } = \beta \cdot \bar{\xi }f + (1 - \beta ) \cdot GRC_{{\xi \delta }} ,\delta ,\xi = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(22)

where β is the weight coefficient, β ∈ [0, 1].

The detailed process of improved K-means clustering algorithm can be found in Algorithm 1.

Algorithm 1. Improved K-means clustering algorithm.

It is necessary to determine the social trust relationships between individuals before clustering. However, if there are unrelated types of social relationships between individuals, their trust relationships are often difficult to complete. At this point, the only available information is the similarity of opinions between the two individuals. Therefore, opinion similarity is used to evaluate the social trust relationship between such individuals, with the value of trust being the GRC between the two, and the value of distrust being (1 − GRC).

Consensus decision-making method based on opinion evolution model

The clustering method divides individuals within CDMs into different subgroups, where individuals within these subgroups have similar opinions and social relationships. Different subgroups have different influences on the group. When there is a subgroup with greater influence in the group, the entire opinion evolution process will be dominated by that subgroup, and the influence of other subgroups will gradually weaken during the evolution process. To avoid this situation, this research first proposes a subgroup identification method based on grey relational analysis, which can identify subgroups with significant influence in the group. Then, the DeGroot model based on inter subgroup opinion interaction was proposed, which can simulate the opinion interaction process between high influence subgroups and low influence subgroups, enabling both parties to coordinate their opinions and appropriately weaken the dominant role of high influence subgroups in opinion interaction.

Subgroup identification method based on grey relational analysis

To balance the influence of each subgroup on group opinions, a subgroup identification method has been designed based on grey relational analysis before conducting opinion evolution. It identifies subgroups based on the grey relational degree between subgroup opinions and group opinions. If the grey relational between subgroups is high, it indicates that in the process of opinion integration, the subgroup plays a significant decisive role in determining group opinions, and its influence is high. Suppose that a group is divided into subgroups, the subgroup set is denoted as G = {G₁, G₂, …, G_h, …, G$\}$.

Step 1

Equations (5) and (7) are used to calculate the trust scores of individuals within subgroup G_h;

Step 2

Group preference determination method in “Methods for determining group preferences” is used to calculate the preference information of subgroups;

Step 3

Equations (17) and (18) is used to calculate the GRC between each subgroup preference and group preference. The GRC between all subgroup preferences and group preferences form a set {GRC₁, GRC₂, …, GRC$\}$.

Step 4

GRC_h is ranked in descending order, and it is used to obtain a new set $\{ GR{C_{{\sigma _{(1)}}}},GR{C_{{\sigma _{(2)}}}},...,GR{C_{{\sigma _{(\varphi )}}}}\}$, for ∀h (h = 1, 2, …, ), satisfying $GR{C_{{\sigma _{(\varphi )}}}}>GR{C_{{\sigma _{(\varphi +1)}}}}$;

Step 5

As for $GR{C_{{\sigma _{(1)}}}}$, its corresponding subgroup is identified as a high impact subgroup, denoted as G_max. As for $GR{C_{{\sigma _{(\varphi )}}}}$, its corresponding subgroup is identified as a low impact subgroup, denoted as G_min.

DeGroot model based on opinion interaction between subgroups

The subgroup identification method distinguishes between high influence subgroups and low influence subgroups. To balance the influence of each subgroup and avoid the domination of group opinions by high influence subgroups, DeGroot model is selected to describe opinion interaction between subgroups. Through negotiation and interaction between high influence subgroups and low influence subgroups, the two types of subgroups are balanced, so that the final group opinion is accepted by each subgroup and consensus is reached.

Let G={G₁, G₂, …, G_h, …, G$\}$ be the clustered set of subgroups, G_max is a high impact subgroup, G_min is a low impact subgroup, and the model is established as follows. Suppose that there exist DM f∈G_max, d∈G_min, where f, d = 1, 2, …, l. Then, we have.

$$p_{{ij}}^{{f,(T)}} = st_{f} \times (p_{{ij}}^{{f,(T - 1)}} ) + \sum\limits_{{d \in G_{{\min }} }} {w_{{fd}} \times (p_{{ij}}^{{d,(T - 1)}} )} ,i,j \in \left[ {{\text{1}},m} \right],T = 0,{\text{ 1}},{\text{ 2}}, \ldots$$

(23)

where ${w_{fd}}=\frac{{s{d_f} \cdot t{s_{fd}}}}{{\sum\limits_{{d \in {G_{\hbox{min} }}}} {t{s_{fd}}} }}$.

$$p_{{ij}}^{{d,(T)}} = st_{d} \times (p_{{ij}}^{{d,(T - 1)}} ) + \sum\limits_{{d \in G_{{\max }} }} {w_{{df}} \times (p_{{ij}}^{{f,(T - 1)}} )} ,i,j \in \left[ {{\text{1}},m} \right],T = 0,{\text{ 1}},{\text{ 2}},{\text{ }} \ldots$$

(24)

where ${w_{df}}=\frac{{s{d_d} \cdot t{s_{df}}}}{{\sum\limits_{{f \in {G_{\hbox{max} }}}} {t{s_{df}}} }}$.

In the process of opinion evolution, the degree to which DMs retain their own opinions is determined by their personal confidence values. In many studies on the DeGroot opinion evolution model, confidence values are subjectively given by researchers and lack objective evaluation criteria. Therefore, this article supplements the objective determination method of confidence value to ensure the rationality of the opinion evolution process.

Definition 18

Let $r_{f}^{{in}}$ be the degree of trust in individual d, and $r_{f}^{{out}}$ be the degree of trust out of individual d, which can be determined as follows:

$$st_{d} = \frac{{\gamma _{d}^{{in}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {t_{{fd}} } }}{{\gamma _{d}^{{in}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {t_{{fd}} } + \gamma _{d}^{{out}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {d_{{fd}} } }},d = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(25)

At the same time, the non-confidence of individual can be expressed as follows:

$$sd_{d} = 1 - st_{d} = \frac{{\gamma _{d}^{{out}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {d_{{fd}} } }}{{\gamma _{d}^{{in}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {t_{{fd}} } + \gamma _{d}^{{out}} \cdot \sum\nolimits_{{f \ne d,f = 1}}^{n} {d_{{fd}} } }},d = {\text{1}},{\text{ 2}},{\text{ }} \ldots ,l$$

(26)

The degree of trust represents the number of individuals d trusts in the group, which can reflect the degree to which individuals are influenced by others. Individuals with higher degrees of trust are often more inclined to change their opinions. Therefore, the degree of trust is positively correlated with the level of distrust; the degree of trust represents the number of other individuals in a group who trust individual d. The opinions of individuals who are trusted by more people are often more likely to play a role in the process of opinion evolution. The degree of trust is positively correlated with confidence levels.

After each evolution of opinions, new subgroup opinions and group opinions can be obtained. At the same time, due to changes in opinions, the grey relational between each subgroup preference and group preference will also change after iteration, which will lead to changes in subgroup recognition results. The consensus level analysis method based on grey relational analysis has been introduced. When using this method to measure consensus, the threshold is a variable parameter. When the threshold is low, the consensus achievement speed is faster. However, the role of the model in balancing the influence of each subgroup may not be fully utilized. In order to balance the influence of subgroups under the premise of reaching consensus, the balance index is defined as follows.

Definition 19

The high impact subgroup after the Tth iteration is denoted as $G_{{\hbox{max} }}^{T}$, and its grey relational with the group is $GRC_{{\hbox{max} }}^{T}$; the low impact subgroup is denoted as $G_{{\hbox{min} }}^{T}$, and its grey relational with the group is $GRC_{{\hbox{min} }}^{T}$, and the equilibrium index after the Tth iteration is B^T=$GRC_{{\hbox{max} }}^{T}$−$GRC_{{\hbox{min} }}^{T}$. When B^T ≤ ε, it is considered that the influence of the subgroup on the group has reached an equilibrium state, and ε is the threshold for measuring the influence of the subgroup.

After the Tth iteration, as for ∀h(h = 1, 2, …, ), $GRC_{f}^{T} \geqslant \eta$, and B^T ≤ ε, and then all subgroups reach a consensus, and the influence of each subgroup is balanced, the opinion evolution ends; Otherwise, the research needs to continue the process of opinion evolution between subgroups. The overall process of the internal consensus model for CDMs can be summarized as Algorithm 2.

Algorithm 2. Internal consensus reaching method for CDMs

Consensus stability among CDMs in GMCR with HFPRs

Basic concepts for constructing consensus stability definitions

It is obvious the previous section mainly studies the consensus reaching model within CDMs based on HFPRs. Based on this, it is possible to eliminate internal differences among individuals and obtain a unified opinion among CDMs, thereby enabling the process of conflict analysis to proceed smoothly. At the same time, the difference in the influence of subgroups on the group opinion also decreases below the threshold, and consensus is reached among all subgroups. Finally, the group opinion is represented by using fuzzy preference. Below, various new underlying definitions are proposed to character different stakeholders’ interest interaction and provide theoretical basis to design consensus stability in GMCR with HFPRs.

Definition 20

The improved reachable set in consensus stability from s for DM k is represented as follows:

$$R_{k}^{C}(s)=\{ {s_1} \in S,(s,{s_1}) \in {A_k}\}$$

(27)

It is obvious that the $R_{k}^{C}(s)$ is equal to ${R_k}(s)$ in traditional GMCR.

In conflict, when DMs shift the current state towards a more favorable one through strategic changes, they usually face countermeasures from their opponents. When its opponent adopts a strategic shift to counteract, it will cause further changes in the conflict state. Usually, the strategic shift adopted by the opponent of the DM often leads to a shift in the conflict state towards a direction that is more unfavorable to the DM. Thus, a new concept called hesitation fuzzy threshold is proposed to resolve above described situation.

Definition 21

Let k∈N be a CDM in conflict and fuzzy preferences $\Re ={(r_{{ij}}^{k})_{m \times m}}$ be the aggregated CDMs’ group preference. Thus, the consensus relative strength of preferences is defined to show the intensity of preference for a state (relative to another), called as ${r^k}({s_i},{s_j})$, and it is abbreviated as $r_{{ij}}^{k}$, satisfying ${r^k}$∈[− 1, 1]. Furthermore, the hesitation fuzzy threshold of CDMs is defined as ${\lambda ^{CDMs}}$. Thus, CDM k chooses to movement from state s_i to s_j iff $r_{{ij}}^{k}$>${\lambda ^k}$.

Definition 22

The improved reachable set in constructing consensus stability from s for DM k is defined as follows.

$$R_{k}^{{+,C}}(s)=\{ {s_1} \in R_{k}^{C}(s),{s_1}{ \succ _k}s\}$$

(28)

Next, conflict problems typically involve two or more DMs with different interest goals and wishes. Thus, the improved reachable set for a coalition is difficult to obtain and is defined as follows.

Definition 23

Let s be the initial state, $\Omega _{H}^{+}(s,{s_1})$ is the set of DMs who move from state s to state s_i by taking compliant actions. Thus, the rules for determining $R_{H}^{{+,c}}(s)$ are shown as follows:

• (1) If $k \in H$ and ${s_1} \in R_{k}^{{+,C}}(s)$, then ${s_1} \in R_{H}^{{+,C}}(s)$ and $k \in \Omega _{H}^{+}(s,{s_1})$.

• (2) If $k \in H$, ${s_1} \in R_{H}^{{+,C}}(s)$, and ${s_2} \in R_{k}^{{+,C}}(s)$, then we obtain $\Omega _{H}^{+}(s,{s_1}) \ne \{ k\}$,${s_2} \in R_{H}^{{+,C}}(s)$ and $k \in \Omega _{H}^{+}(s,{s_2})$.

The above describe definitions are used to provide underlying concepts for constructing consensus stability definitions. To clearly show the difference between new proposed GMCR and traditional GMCR, a flow chart is described as follows.

Four consensus stability definitions among CDMs in GMCR with HFPRs

A set of new consensus stability definitions among CDMs has been defined in the structure of GMCR with HFPRs.

Definition 24

A state s is considered to be consensus Nash stability (CNash) for k in GMCR with HFPRs iff there exists $R_{k}^{{+,c}}(s)=\emptyset$, represented as $s \in S_{k}^{{CNash}}$.

Definition 25

A state s is considered to be consensus general metarationality (CGMR) for k in GMCR with HFPRs iff for any state ${s_1} \in R_{k}^{{+,c}}(s)$, there is at least one state ${s_2} \in {R_{N - \{ k\} }}({s_1})$ that satisfies ${r^k}({s_2},s)$<${\lambda ^k}$, represented as $s \in S_{k}^{{CGMR}}$.

Definition 26

A state s is considered to be consensus symmetric metarationality (CSMR) for k in GMCR with HFPRs iff for any state ${s_1} \in R_{k}^{{+,c}}(s)$, there is at least one state ${s_2} \in {R_{N - \{ k\} }}({s_1})$ that satisfies ${r^k}({s_2},s)$<${\lambda ^k}$ and for any state ${s_3} \in {R_k}({s_2})$ that satisfies ${r^k}({s_3},s)$<${\lambda ^k}$, represented as $s \in S_{k}^{{CSMR}}$.

Definition 27

A state s is considered to be consensus sequential (CSEQ) for k in GMCR with HFPRs iff for any state ${s_1} \in R_{k}^{{+,c}}(s)$, there is at least one state ${s_2} \in R_{{N - \{ k\} }}^{{+,c}}({s_1})$ that satisfies ${r^k}({s_2},s)$<${\lambda ^k}$, represented as $s \in S_{k}^{{CSEQ}}$.

It is obvious that CNash in GMCR with HFPRs is considered to be strong stability definitions, which means that does not need to consider the reactions and attitudes of the opponent DMs. At the same time, the remaining three consensus stability definitions are considered as weak stability definitions.

Remark 2

If N only includes two DMs, represented as N = {k, q$\}$, Definitions 24−27 also can be used to describe DM’s interaction in conflicts. In other words, Definitions 24−27 is universal and widely applicable.

Remark 3

If DM’s preferences in Definitions 24−27 are replaced by fuzzy preferences, Definitions 24−27 can be considered to be fuzzy stability definitions in GMCR. Furthermore, Definitions 24−27 are a special situation of fuzzy stability definitions.

Application to the doctor–patient disputes in China

Background

China has frequent medical disputes and a complex social environment. The tense doctor–patient relationship stems from uneven allocation of medical resources, leading to long waiting times for patients and difficulty in ensuring service quality. The influence of public opinion has intensified the complexity of medical disputes, making it difficult for doctors to cope with external pressure. The problems with the medical system and weak legal awareness have also contributed to the spread of doctor–patient disputes to a certain extent. The underlying reason lies in the lack of trust between doctors and patients, and the future development trend may further deteriorate. To address this issue, it is necessary to establish a more equitable mechanism for allocating medical resources, strengthen communication and trust building between doctors and patients, improve the quality of medical services, and enhance the promotion and enforcement of relevant laws and regulations. Only by comprehensively promoting the harmonious development of doctor–patient relationships can we effectively resolve medical disputes and ensure social stability and medical safety.

Basic elements of GMCR

In doctor–patient disputes, the options faced by patients depend on the nature of the conflict and the communication situation between both parties. If patients maintain rationality and calmness in doctor–patient disputes, they can often adopt some strategies to solve the problem and ensure their own rights and interests. The choice of doctors in doctor–patient disputes requires comprehensive consideration of medical ethics, laws, patient emotions, and hospital management requirements. Effective communication, rational response, and seeking support and assistance when necessary are key factors for doctors to alleviate conflicts and protect their own rights. The government’s goal is to promote reconciliation among all parties involved in doctor–patient disputes by issuing policies. Thus, the feasible states of doctor–patient disputes are shown as follows:

Table 2 Feasible States of doctor–patient disputes.

Furthermore, according to Table 2, we can obtain the directed graph of doctor–patient disputes. Each state in Table 2 becomes corresponding vertexes in Fig. 2. Thus, the movement between vertexes in Fig. 2 is equal to movement between corresponding states. The movements between two states are used to represent arcs in directed graph. Red arcs in Fig. 2 represent DM 1 Doctor, black arcs in Fig. 2 represent DM 2 Patient, and blue arcs in Fig. 2 represent DM 3 Government.

Fig. 2

The directed graph of the doctor–patient disputes.

DMs 1 and 3, doctor and government, as official institutions, often have a clear decision making attitude, while as DM 2 patients, their strategic choices are influenced by the interests of internal members, and there are significant differences in the preference states of internal individuals for conflict states. Thus, the preference relations over all states for DMs 1 and 3 are supposed to be fuzzy preferences. The preference relation over all states for DM 2 is supposed to be HFPRs, as shown in Table 3.

Table 3 DM’s preferences in doctor–patient disputes.

Consensus reaching among DMs and stability analysis

Based on the HMPRs matrix of CDMs, we use the solving model proposed in this paper to process the preferences of its internal individuals. The trust degree matrix and distrust degree matrix between individuals within the CDM are represented as T₂ and D₂, as shown in the following.

$${T_2}=\left[ {\begin{array}{*{20}{c}} - & - &{0.3}& - &{0.3}& - &{0.7}& - &{0.8}&{0.3} \\ {0.8}& - & - &{0.6}& - &{0.4}&{0.3}&{0.9}& - & - \\ - &{0.6}& - & - &{0.3}& - & - & - &{0.8}&{0.9} \\ - & - &{0.6}& - & - &{0.5}&{0.2}& - & - & - \\ {0.6}&{0.8}& - &{0.7}& - & - & - &{0.8}& - &{0.4} \\ - & - & - &{0.7}&{0.8}& - &{0.6}& - &{0.3}& - \\ {0.6}& - & - & - & - &{0.9}& - &{0.3}& - & - \\ - &{0.5}&{0.3}& - & - & - & - & - &{0.4}& - \\ {0.7}&{0.1}& - &{0.9}& - & - &{0.5}&{0.4}& - & - \\ - & - & - & - &{0.3}&{0.5}& - & - &{0.5}& - \end{array}} \right]$$

$${D_2}=\left[ {\begin{array}{*{20}{c}} - & - &{0.3}& - &{0.3}& - &{0.7}& - &{0.8}&{0.3} \\ {0.8}& - & - &{0.6}& - &{0.4}&{0.3}&{0.9}& - & - \\ - &{0.6}& - & - &{0.3}& - & - & - &{0.8}&{0.9} \\ - & - &{0.6}& - & - &{0.5}&{0.2}& - & - & - \\ {0.6}&{0.8}& - &{0.7}& - & - & - &{0.8}& - &{0.4} \\ - & - & - &{0.7}&{0.8}& - &{0.6}& - &{0.3}& - \\ {0.6}& - & - & - & - &{0.9}& - &{0.3}& - & - \\ - &{0.5}&{0.3}& - & - & - & - & - &{0.4}& - \\ {0.7}&{0.1}& - &{0.9}& - & - &{0.5}&{0.4}& - & - \\ - & - & - & - &{0.3}&{0.5}& - & - &{0.5}& - \end{array}} \right]$$

Based on the social trust relationships between individuals represented by the above matrix, the steps for achieving internal consensus among CDMs are determined based on Algorithm 2 and Fig. 3, including Step 1: Internal individual social trust network, Step 2: Cluster, Step 3: Internal social trust network in subgroup, Step 4: Determining of preference information of each subgroup, and Step 5: Opinion evolution between subgroups. After iteration, the curve similarity between the opinions of each subgroup and the group opinion reaches the threshold, and at the same time, the difference in the influence of subgroups on the group opinion also decreases below the threshold. The subgroups reach consensus, and the opinion evolution process ends. To save paper space, the specific calculation process has been shown in Appendix. The final group preference information is shown as fuzzy preferences, as shown in the following.

Fig. 3

The modeling and analysis stage of consensus stability definitions in GMCR with HFPRs.

$${\Re _2}=\left[ {\begin{array}{*{20}{c}} {0.500}&{0.595}&{1.000}&{0.736}&{0.978}&{0.886}&{1.000}&{0.983}&{1.000} \\ {0.405}&{0.500}&{0.965}&{0.319}&{0.616}&{0.855}&{0.988}&{0.487}&{0.525} \\ {0.000}&{0.035}&{0.500}&{0.000}&{0.007}&{0.103}&{0.705}&{0.024}&{0.036} \\ {0.264}&{0.681}&{1.000}&{0.500}&{0.974}&{0.793}&{1.000}&{0.976}&{0.851} \\ {0.022}&{0.384}&{0.993}&{0.026}&{0.500}&{0.509}&{1.000}&{0.424}&{0.560} \\ {0.114}&{0.145}&{0.968}&{0.207}&{0.491}&{0.500}&{0.990}&{0.482}&{0.511} \\ {0.000}&{0.012}&{0.463}&{0.000}&{0.000}&{0.009}&{0.500}&{0.014}&{0.076} \\ {0.017}&{0.513}&{0.976}&{0.024}&{0.576}&{0.518}&{0.986}&{0.500}&{0.846} \\ {0.000}&{0.475}&{0.964}&{0.149}&{0.440}&{0.489}&{0.924}&{0.1549}&{0.500} \end{array}} \right]$$

Finally, according to the consensus stability Definitions 24−27, the consensus stability analysis result of doctor–patient disputes is shown in Table 4.

Table 4 The consensus stability analysis result of doctor–patient disputes.

As shown in Table 4, four states such as s₄, s₅, s₈, and s₉ are considered as consensus equilibrium state. To be specific, the two states of s₄ and s₈ indicate that DM 1 doctor adjusts the command to alleviate the burden faced by DM 2 patient. State s₅ indicates that DM 1 doctor insists on the original command, while DM 2 patient delays and avoids it. In addition, state s₉ indicates that the DM 2 patient has given up communication with the DM 1 doctor and is directly appealing. The research results indicate that the patient’s preference for concessions has a greater impact on the outcome of medical disputes and the development of the situation. Scientific guidance of patients’ rational rights protection is the key to successfully mediating medical disputes. In addition, due to the long legal litigation cycle, private negotiation for compensation and third-party mediation intervention are still effective means for both doctors and patients to resolve conflicts.

Comparative analysis among GMCR with HFPRs and traditional GMCR and GMCR with fuzzy preferences

GMCR with HFPRs, original GMCR^11,39and fuzzy GMCR¹⁷ are selected to carry out comparative analysis from there aspects, as shown in Table 5. Both traditional GMCRs and fuzzy GMCR are the most classic graph modeling methods. Therefore, the comparative analysis of the three types of GMCR is persuasive.

Table 5 The comparison among GMCR with hfprs, traditional graph model, and GMCR with fuzzy preferences.

As described in Table 6, it is obvious that three different GMCR models hold the completeness of stability analysis result of conflicts. Particularly, GMCR with HFPRs not only allows characterizing DM’s uncertainty preferences but also designing a new preference ranking methods. Compared with existing conflict analysis methods, GMCR with HFPRs proposed in this paper not only proposes new ideas for obtaining DMs’ preference ranking, but also provides new set of definitions for stability analysis in uncertain environments. The above research conclusions are of great significance for expanding the research ideas of GMCR and can also promote the research scope of conflict analysis approaches.

The significance of research

In doctor–patient disputes, the attitudes and viewpoints conveyed by CDMs can have a significant impact on the cognition and judgment of stakeholders, thereby affecting the development of doctor–patient conflicts. In real-life, almost all stakeholders believe that the preference consensus of CDMs has a significant impact on the resolution of doctor–patient disputes. The one-sided understanding of medical disputes by CDMs can exacerbate doctor–patient disputes and lead to a deterioration of doctor–patient relationships. So when mediating medical disputes, it is not only important to pay attention to the preferences and attitudes of individual DM, but also to the guiding role of CDMs. CDMs should firmly uphold the correct moral stance, build bridges for doctor–patient communication, and guide doctor–patient disputes to develop in a favorable direction.

This article finds that the concessions made by patients have a greater impact on the effectiveness of doctor–patient dispute mediation. Therefore, how to reduce the cost of rights protection and guide patients to protect their rights rationally is the key to resolving medical disputes and promoting the healthy development of doctor–patient relationships. Therefore, when mediating medical disputes, it is necessary to open up channels for rights protection, reduce the cost of rights protection, and actively guide patients to protect their rights rationally and think rationally through publicity and education, patient participation, and other means, in order to prevent the phenomenon of “compensation for every disturbance”.

Conclusions and future work

According to the real life development situations of doctor–patient disputes in China and reality survey data, this article first constructs a set of new consensus stability definitions among CDMs in the framework of hesitant fuzzy graph model based on the conflict analysis theory, and effectively predicts the development and provides a reasonable solution of doctor–patient conflicts using stability analysis methods. On this basis, many conflict analysis works were conducted on the equilibrium state of doctor–patient conflicts, exploring the impact of different concessions between the medical and patient sides on the development of doctor–patient dispute situations and game results. Based on the consensus stability analysis results, a countermeasure plan for dispute mediation was formulated. The research results indicate that the patient’s preference for concessions has a greater impact on the outcome of medical disputes and the development of the situation. Scientific guidance of patients’ rational rights protection is the key to successfully mediating medical disputes. In addition, due to the long legal litigation cycle, private negotiation for compensation and third-party mediation intervention are still effective means for both doctors and patients to resolve conflicts.

It is obvious that the proposed stability definitions in GMCR with HFPRs are defined from the view of logical representation. However, the definition of logical stability faces the drawback of high computational complexity and difficulty in application. To resolve this shortcoming, matrix representation of stability definitions is proposed in GMCR paradigm. Thus, matrix representation of consensus stability definitions among CDMs in the framework of hesitant fuzzy graph model is a crucial work in the future research. Third party mediation provides an equal and fair communication platform for both doctors and patients, which help to eliminate misunderstandings and barriers between the two parties, promote open communication, and jointly seek solutions. Particularly, inverse perspective can provide decision making support for third party medication in doctor–patient disputes. Therefore, another future work is to develop inverse analysis work in consensus stability definitions among CDMs in the framework of hesitant fuzzy graph model. Finally, CDMs may not only exist in doctor–patient conflicts, but also in other fields such as environmental and military conflicts. Therefore, the research conclusions of this article have generalizability and replicability, and can provide theoretical basis for DMs in different fields.