A maximizing dominance degree model based on trust relationships and TODIM method to reach group consensus

Abstract

To obtain the suitable alternative(s) for the organization, this paper proposes a more practical method to solve the decision-making problems in society. That is combined with the TODIM (TOmada de decisão interativa multicrit $\overset{´}{e}$ rio). The maximizing dominance degree model to reach consensus is proposed with two following components: (1) constructing the complete trust relationships network; (2) the maximizing dominance degree feedback mechanism to reach group consensus. Therefore, firstly owing to the complexity of the trust relationships network, judging the direct and indirect trust propagation paths among the decision makers (DMs) to construct the complete trust relationships network and identifying the highest value of Trust Score (TS) as the leader is possible. Then identify the inconsistent DM based on the established consensus index. During the feedback process, inconsistent DMs adopt the feedback mechanism based on the dominance degree of the leader until the group consensus is reached. Later, the corresponding ranking result is calculated by the TODIM method. Finally, a numerical example is applied to illustrate the effectiveness and feasibility of the optimal model.

Keywords

Trust relationship leader TODIM dominance degree deviation degree

1 Introduction

Nowadays, it is meaningful to regard the relationship between individuals and group as a visual social network. In the visual social network, the social connections among the individuals are clear, so a group of DMs shows their preferences on a set of alternatives to select the best solution. This process is called social network group decision-making (SN-GDM) [1 –4]. The core of GDM lies in how to make use of the decision information obtained, and according to the decision information type, the integrity of decision information, and the method of information fusion. It is finally converted to the comprehensive sorting of alternatives. Based on decision-making information types, decision-making mainly includes fuzzy decision-making [5], intuitive fuzzy multi-attribute decision-making [6 –9], hesitating multi-attribute decision-making [10, 11], interval value information multi-attribute decision-making [12], and so on.

The traditional GDM model is based on the expected utility theory. However, in real life, some problems may exist due to the complexity and uncertainty of the social network and the different knowledge backgrounds of DMs. These may limit the ability to process information and lead to inconsistency problems. The decision made at this time may affect the final result and cause losses [13 –21]. Considering a new trend of SN-GDM today, DMs rely on the opinions of their close friends or people with similar interests from social media, and social media plays an important role in human communication [22]. This is called trust relationship, while the traditional GDM models neglect it [4 , 23–25]. Furthermore, trust relationships among networked DMs are considered as a new reliable source [26, 27]. By discovering and excavating trust relationships in social networks, trust social network among experts will be generated. Next, we analyze and study the established trust network according to its nature.

As one of the multi-attribute decision-making methods, the classical TODIM method considers the DM’s subjective psychological behavior based on prospect theory, then calculates the dominance degree of alternatives over other alternatives to rank and prioritize [28 –30]. Compared with the pure use of prospect theory, TODIM method has fewer parameters and simple calculation [31 –33]. It can effectively capture the psychological behavior of DMs and directly select other comparison alternatives as a reference point. The decision making process is more convenient and objective [34, 35]. So far, many scholars have paid attention to the classical TODIM method, and some achievements have been made one after another. Consider the risk and uncertainty in multi-criteria decision making(MCDM) problems, Fan et al. [36] proposed a hybrid approach combining TODIM and fuzzy numbers to handle these issues. To address the problems of weights of DMs and criteria are completely unknown, Qin et al. [37] extended the classical TODIM method in triangular linguistic to solve MCDM problem. Yu et al. [38] constructed nonlinear programming with intuitionistic linguistic numbers to solve MCDM problems with the DM’s psychological behavior considered. Although the classical TODIM method resolves the problems of DMs in the above issues, it ignored trust relation among DMs in the social network. Most of the above researches on TODIM decision making regard experts as independent individuals with no relationship with each other and ignore the social network relationship among DMs. However, in the real social network, DMs are not completely independent individuals, and there is no guarantee that decisions will be made completely rational. Hence, it is necessary to propose an extended TODIM model based the trust relationship in social network.

To obtain the suitable alternative(s) for the group, combined with the TODIM method, this paper’s proposed method is more realistic than the previous papers. With the rise of social development, DMs are more closely connected, creating the social network environment of GDM problems, especially in large-scale social networks. However, as the complexity of the large-scale social networks, the method proposed has limitations. Then, in the consensus reaching process (CRP), aggregating individual preferences into a collective one for deriving a final ranking solution is important [23]. However, different from theoretical research, each individual could not keep entirely rational in the real society. To solve this problem, some scholars put forward methods to improve it. Considering the consequences of experts’ incomplete rationality, Wu et al. [39] proposed decision-making method based on experts’ maximum self-esteem to solve this problem. There are three possible states, tolerance behavior, rationalist behavior and conflict behavior. Take three possible states into consideration, Cao et al. [40] proposed a bidirectional feedback mechanism for consensus driven by the behavior of DMs. To reduce the impact of irrationality, in the process of consensus reaching, Wu et al. [41] used DMs to evaluate with interval-valued fuzzy reciprocal preference relations.

In addition, the key issue in CRP is all DMs should reach group consensus; otherwise, turn to the feedback process. Most existing research on feedback mechanisms aims to minimize the adjustments between original and adjusted preferences or consensus costs [42 –45]. However, the feedback mechanism we adopted is based on the dominance degree in this paper.

This paper proposes a maximizing dominance degree model based on trust relationships and the TODIM method to reach group consensus. According to the trust relationships network of DMs, the inconsistent DMs modify their preference matrix with their own trust degree, and then use the TODIM decision-making method to obtain the ranking results of the alternatives. This optimal model could fully combine the trust relationship among social network groups and ensure the original preference values as much as possible. The key issues in SN-GDM would be: (1) the constructing of trust relationships between DMs, and (2) how to reach group consensus during the feedback mechanism for each DM.

The rest of this paper is set out as follows: Section 2 introduces the steps related to TODIM (Section 2.1.) and builds a complete social network (Section 2.2. & Section 2.3. & Section 2.4.). Section 3 describes how to reach consensus in judgment and judge inconsistent experts (Section 3.1.), and make feedback modifications (Section 3.2.), then make the selection process (Section 3.3.). Section 4 presents the steps of the maximizing dominance degree model based on trust relationships and the TODIM method to reach group consensus. Section 5 explains the model through a numerical example (Section 5.1.), carries out comparative analysis (Section 5.2.1.) and sensitivity analysis (Section 5.2.2.), then the advantages and conclusions are as following.

2 Preliminaries

This section introduces several basic concepts and definitions related to TODIM and trust social network analysis.

2.1 TODIM method

In this section, we introduce the classical TODIM method. The classical TODIM method is used to solve the crisp value of the multi-attribute decision problems. The related description would follow the steps below.

In a decision-making problem, the set of alternatives is A = { A₁, A₂, . . . , A_m } where A_i represents the i - th alternative; C = { C₁, C₂, . . . , C_n } is the set of the attributes where C_j represents the j - th attribute; the corresponding weight vector is w = { w₁, w₂, . . . , w_n } , w_j denotes the importance degree of the attribute C_j, which satisfies the following conditions $\sum_{j = 1}^{n} w_{j} = 1$ and w_j ∈ [0, 1]. Suppose the decision-making matrix is $X_{u} = {[x_{ij}^{u}]}_{m \times n}$ where $x_{ij}^{u}$ denotes the evaluation value of the attribute w_j by the DM e_u. The classical TODIM method mainly involves the following steps:

Step 1: Normalize the decision matrix $X_{u} = {[x_{ij}^{u}]}_{m \times n}$ into $X_{u}^{'} = {[x_{ij}^{u'}]}_{m \times n}$ , i = 1, 2, . . . , m, j = 1, 2, . . . , n.

Step 2: Calculate the relative weight w_jr of attribute C_j to the reference attribute C_r expressed as follows:

$w_{jr} = \frac{w_{j}}{w_{r}}$ (1) where w_r = max { w_j|j ∈ N } .

Step 3: Calculate the dominance degree $Φ_{j}^{u} (A_{i}, A_{k})$ of each alternative A_i over A_k with respect to criterion C_j as follows:

$\begin{matrix} Φ_{j}^{u} (A_{i}, A_{k}) \\ = {\begin{matrix} \sqrt{(x_{ij}^{u'} - x_{kj}^{u'}) w_{jr} / - (\sum_{j = 1}^{n} w_{jr})} \begin{matrix} \begin{matrix} \end{matrix} & x_{ij}^{u'} - x_{kj}^{u'} > 0 \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} & \begin{matrix} \end{matrix} & x_{ij}^{u'} - x_{kj}^{u'} = 0 \end{matrix} \end{matrix} \\ - \frac{1}{θ} \sqrt{(x_{kj}^{u'} - x_{ij}^{u'}) (\sum_{j = 1}^{n} w_{jr}) / - w_{jr}} \begin{matrix} \end{matrix} \begin{matrix} x_{kj}^{u'} - x_{ij}^{u'} > 0 \end{matrix} \end{matrix} \end{matrix}$ (2)

where θ (θ > 0) denotes the attenuation factor of the losses to present the sensitivity of decision makers to losses. The value of this parameter θ varies according to the situation. The smaller θ value is, the higher the DM’s aversion to loss is. Vice versa, the larger θ value is, the lower the DM’s aversion to loss is. Therefore, if the value of θ is big enough, all the losses of DM are negligible [46].

Step 4: Calculate the comprehensive dominance degree about each alternative A_i over A_k by using the following equation:

$δ (A_{i}, A_{k}) = \sum_{j = 1}^{n} Φ_{j}^{u} (A_{i}, A_{k}), i, k \in M$ (3)

Step 5: Calculate the DM’s overall dominance degree of alternative A_i using the following expression:

$ξ (A_{i}) = \frac{\sum_{k = 1}^{m} δ (A_{i}, A_{k}) - \min_{i \in M} {\sum_{k = 1}^{m} δ (A_{i}, A_{k})}}{max_{i \in M} {\sum_{k = 1}^{m} δ (A_{i}, A_{k})} - \min_{i \in M} {\sum_{k = 1}^{m} δ (A_{i}, A_{k})}}$ (4)

Step 6: Rank alternatives according to the overall dominance degree of alternatives. The larger the value of ξ (A_i) is, the better alternative A_i is.

2.2 Trust social network analysis

DMs in social network build trust relationships and form networks which are called trust networks. Since there are direct and indirect paths in the trust social network, the analysis of trust propagation process is meaningful. Wu et al. [13] proposed a dual trust propagation to analyze the paths.

Firstly, identify all the direct and indirect trust relationships between the DMs to build a complete social network.

Definition 1. (Trust Function (TF)). The trust score (t, d) is an element in the set [0, 1] × [0, 1] where t, d ∈ [0, 1]. And the former component t denotes the trust value, and the latter component d denotes the distrust value. TF will be referred to as a trust function value.

The trust relationship matrix can be represented by TD = [(t_uv, d_uv)] _r*r, where (t_uu, d_uu) = (1, 0), uv = 1, 2, . . . , r, φ present the component is not existed, $TD = [\begin{matrix} (1, 0) & (t_{12}, d_{12}) & \dots & (t_{1 u}, d_{1 u}) \\ (t_{21}, d_{21}) & (1, 0) & \dots & (t_{2 u}, d_{2 u}) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (t_{u 1}, d_{u 1}) & (t_{u 2}, d_{u 2}) & \dots & (1, 0) \end{matrix}]$ .

Definition 2. (Einstein product E_⊗ and Einstein sum E_⊕). Einstein product and Einstein sum are used to illustrate the general method of trust propagation.

$E_{\otimes} = t \otimes d = \frac{t \cdot d}{1 + (1 - t) \cdot (1 - d)}$ (5)

$E_{\oplus} = t \oplus d = \frac{t + d}{1 + t \cdot d}$ (6)

In a trust social network, if e₁ trust e₂ directly and e₂ trust e₃ directly, then e₁ trust e₃ indirectly, which is called transitivity of the trust relationship, as shown in Fig. 1.

Fig. 1

Transitivity of trust relationship.

Due to different backgrounds and other causes in a social network, some DMs may not directly trust others, which means they cannot give the orthopairs of trust/distrust values directly. Hence a trust propagation chain to find out trust/distrust relationships via dual trust propagation operator is proposed, which can help to identify indirect trust DMs. However, one key issue is to construct the right trust propagation operator. Therefore, it is possible to devise the trust propagation chain to determine whether an unknown DM can be trusted.

2.3 Trust social network analysis

Definition 3 [13]. (dual trust propagation operator). This dual trust propagation operator Z based on Einstein product E_⊗ and Einstein sum E_⊕ illustrates the general approach to trust propagation. For (t₁, d₁) and (t₂, d₂) in the social network, the definition of Z is as follows

$\begin{matrix} Z ((t_{1}, d_{1}), (t_{2}, d_{2})) = (E_{\otimes} (t_{1}, t_{2}), E_{\oplus} (d_{1}, d_{2})) \\ \begin{matrix} \begin{matrix} = & (\frac{t_{1} \cdot t_{2}}{1 + (1 - t_{1}) (1 - t_{2})}, \frac{d_{1} + d_{2}}{1 + d_{1} \cdot d_{2}}) \end{matrix} \end{matrix} \end{matrix}$ (7)

However, in real social networks, there are more than three experts involved in decision-making. In this case, the corresponding Z is calculated as follows.

$\begin{matrix} Z ((t_{1}, d_{1}), (t_{2}, d_{2}), . . ., (t_{r}, d_{r})) \\ = (E_{\otimes} (t_{1}, t_{2}, . . ., t_{r}), E_{\oplus} (d_{1}, d_{2}, . . ., d_{r})) \\ = (\frac{2 \prod_{i = 1}^{r} t_{i}}{\prod_{i = 1}^{r} (2 - t_{i}) + \prod_{i = 1}^{r} t_{i}}, \\ \frac{\prod_{i = 1}^{r} (1 + d_{i}) - \prod_{i = 1}^{r} (1 - d_{i})}{\prod_{i = 1}^{r} (1 + d_{i}) + \prod_{i = 1}^{r} (1 - d_{i})}) \end{matrix}$ (8)

As illustrated in Fig. 2, for the other possible indirect path: L₂ : e_u → e₂ → e₃ → e_v, the corresponding value:

Fig. 2

Trust propagation from e_u to e_v.

$\begin{matrix} Z ((t_{3}, d_{3}), (t_{4}, d_{4}), (t_{5}, d_{5})) \\ = (E_{\otimes} (t_{3}, t_{4}, t_{5}), E_{\oplus} (d_{3}, d_{4}, d_{5})) \\ = (\frac{2 t_{3} t_{4} t_{5}}{(2 - t_{3}) (2 - t_{4}) (2 - t_{5}) + t_{3} t_{4} t_{5}}, \\ \frac{(1 + d_{3}) (1 + d_{4}) (1 + d_{5}) - (1 - d_{3}) (1 - d_{4}) (1 - d_{5})}{(1 + d_{3}) (1 + d_{4}) (1 + d_{5}) + (1 - d_{3}) (1 - d_{4}) (1 - d_{5})}) \end{matrix}$

In the realistic trust social network, experts’ direct and indirect trust relationships are unknown [13, 47]. To analyze the indirect relationships, [13] investigates a dual trust propagation operator based on the t-norm Einstein product and the t-norm Einstein sum to illustrate the general practice until the completed trust relationships network is obtained. The approach to constructing complete trust relationship network is an important issue in SN-GDM.

As shown in Fig. 2, there exist two paths from e_u to e_v: L₁ : e_u → e₁ → e_v; L₂ : e_u → e₂ → e₃ → e_v. Actually, in a completely social network, there might exist multiple paths for an indirect relationship. To reduce the attenuation of information, the average of the multiple trust propagation paths is computed. As shown in Fig. 2, the average value for Z from e_u to e_v is

$Z = \frac{Z ((t_{1}, d_{1}), (t_{2}, d_{2})) + Z ((t_{3}, d_{3}), (t_{4}, d_{4}), (t_{5}, d_{5}))}{2}$

2.4 Trust social network analysis

When the complete trust social network is constructed, then distinguish the most trusted DM in this social network. Besides, according to the constructed social network, the importance degree of each DM is calculated within the group. The trust score (TS) and the importance degree of each DM are calculated as follows.

Definition 4 [13]. (Trust Score (TS)) TS represents the general advantage of t over corresponding d for each DM, that is, the strict trust value of each DM. The definition is as follows.

$TS = \frac{t - d + 1}{2}$ (9)

Obviously, the TS index reflects the corresponding importance degree of DMs, and it could be a reliable resource in deriving importance degree for each DM.

Definition 5. The weight values of each DM are assigned by using TS values, the importance degree of each DM in the social network is computed as follows [48].

$λ_{T}^{σ (h)} = Q (\frac{T (σ (h))}{T (σ (k))}) - Q (\frac{T (σ (h - 1))}{T (σ (k))})$ (10)

Where $T (σ (h)) = \sum_{l = 1}^{h} {TS}_{σ (l)}$ , σ is a permutation to rank the ordering of the TS values, TS_l is the l - th largest element in {TS₁, . . . , TS_k}. Q is an increasing monotone function Q (r) = r^a (a ∈ [01]), such that Q (0) =0, Q (1) =1 [48].

The higher the trust score a DM is, the more important degree is assigned to the DM. The collective decision matrices could be aggregated using the above social network based $w_{T}^{u}$ as follows.

Definition 6. Let ${X^{(u)} = [x_{ij}^{(u)}]_{m * n}, u = 1, 2, . . ., r}$ be collective trust decision matrices given by a set of DMs E = {e₁, e₂, . . . , e_r} within this social network. Combined with Equation (10) The collective decision matrix guided by trust relationship is $X^{c} = (x_{ij}^{c})_{m \times n}$ with the element as follows [49],

$x_{ij}^{c} = \sum_{u = 1}^{r} λ_{T}^{σ (h)} \cdot x_{ij}^{(u)}$ (11)

The TODIM method considers the perception of DMs by the dominance degree matrix, so it is necessary to calculate the collective dominance degree matrix.

Definition 7. Let $λ_{T}^{σ (h)}$ be the weight of each individual DM. The dominance degree $Φ_{j}^{u} (A_{i}, A_{k})$ denote each alternative A_i over A_k with respect to criterion C_j as shown in Equation (2), the corresponding collective dominance degree is defined as follows.

$δ^{c} (A_{i}, A_{k}) = \sqrt{\sum_{u = 1}^{r} λ^{(u)}} • Φ_{j}^{(u)} (A_{i}, A_{k})$ (12) where $λ^{(u)}$ given by Equation (10).

Proof. Combined with Equations (2) and (11), according to the quantity relationship between $x_{ij}^{'}$ and $x_{kj}^{'}$ , there are 3 situations existed.

When $x_{ij}^{u'} - x_{kj}^{u'} > 0$ ,

$\begin{matrix} δ^{c} (A_{i}, A_{k}) = \sqrt{(x_{ij}^{c} - x_{kj}^{c}) w_{jr} / - (\sum_{j = 1}^{n} w_{jr})} \\ \begin{matrix} \begin{matrix} \end{matrix} & = \end{matrix} \sqrt{(\sum_{u = 1}^{r} λ^{(u)} \cdot x_{ij}^{(u)} - \sum_{u = 1}^{r} λ^{(u)} \cdot x_{ij}^{(u)}) w_{jr} / - (\sum_{j = 1}^{n} w_{jr})} \end{matrix}$ $\begin{matrix} \begin{matrix} = \end{matrix} \sqrt{\sum_{u = 1}^{r} λ^{(u)} (x_{ij}^{(u)} - x_{ij}^{(u)}) w_{jr} / - (\sum_{j = 1}^{n} w_{jr})} \\ \begin{matrix} = \end{matrix} \sqrt{\sum_{u = 1}^{r} λ^{(u)}} \cdot \sqrt{(x_{ij}^{(u)} - x_{ij}^{(u)}) w_{jr} / - (\sum_{j = 1}^{n} w_{jr})} \\ \begin{matrix} = \end{matrix} \sqrt{\sum_{u = 1}^{r} λ^{(u)}} \cdot Φ_{j}^{u} (A_{i}, A_{k}) \end{matrix}$

□

In addition, there are still two cases, $x_{ij}^{u'} - x_{kj}^{u'} = 0$ and $x_{kj}^{u'} - x_{ij}^{u'} > 0$ , the steps about both two situations are similar to those in the first case, so the proof is omitted.

3 Consensus reaching process and decision procedure

Once the collective decision matrix based trust relationship is obtained, the consensus index could be calculated. Before proceeding to select the optimal solution, DMs in the trust social network would pre-set a threshold. When the consensus index is below the pre-set threshold, the resolution process is carried out. Otherwise, make adjustments to the inconsistent DMs according to the feedback mechanism, ensuring all DMs can reach group consensus. While all the DMs in the social network reach consensus, the next step could be taken. As explained above, the DMs in the group may have inconsistent opinions among them. Therefore, before the aggregation step, the DMs in the group should reach consensus. The identified inconsistent DMs mean contribute less to the group, but these DMs may not know how much should be revised to maximize the advantage. Therefore, this paper proposes a feedback mechanism based on the maximization of dominance to help DMs make visual adjustments.

3.1 Consensus index

Most of the current studies about consensus measurement are based on decision matrix. However, because the TODIM method considers the reference dependence and loss avoidance behavior of each individual, consensus degree measurement based on dominance degree matrix can better consider the feelings of DMs and fit the reality.

Let $δ^{(u)} = [Φ_{j}^{u} (A_{i}, A_{k})]_{m * n}$ and $δ^{(c)} = [Φ_{j}^{c} (A_{i}, A_{k})]_{m * n}$ be the dominance degree matrices of e_u and the group respectively. The corresponding expression is defined by calculating the deviation degree (DD) between the DMs to measure the consensus index.

$\begin{matrix} {DD}^{u} = d (δ^{(u)}, δ^{(c)}) \\ \begin{matrix} = {(\prod_{i = 1}^{m} \prod_{j = 1}^{n} \frac{max (Φ_{j}^{u} (A_{i}, A_{k}), Φ_{j}^{c} (A_{i}, A_{k}))}{min (Φ_{j}^{u} (A_{i}, A_{k}), Φ_{j}^{c} (A_{i}, A_{k}))})}^{\frac{1}{m \times n}} \end{matrix} \end{matrix}$ (13)

The smaller the DD^u is, the smaller the deviation degree between e_u and the group, the greater the consensus index among individual DM. Generally speaking, 0 ⩽ DD^u ⩽ 1. While DD^u = 0, there is no deviation between the individual and the collective. However, this is not possible in a real society. When the value of DD^u lower than the established threshold γ, that means the DM has reached a consensus. Otherwise, identify the DMs whose DD^u value is higher than the threshold, and these DMs need to modify their decision matrices in the next step. The relative modified method is introduced as follows.

Identify the DMs whose DD^u value is higher than the threshold γ:

$EXPCH = {u | {DD}^{u} > γ}$ (14)

3.2 Recommendation mechanism for generating advice

Given e^t ∈ EXPCH, the feedback mechanism is used to help the inconsistent DMs to revise. Due to the DMs are not completely rational in the real social network, the DMs would refer the recommendations of the DMs with the highest trust scores in the group to make changes. The relevant calculations are as follows:

Definition 7. (trust leader (e_l)) In the trust social network, the DM with the highest TS is the trust leader e_l. As shown in Equation (9), the definition is as follows.

$e_{l} = max_{r} {{TS}_{r}}$ (15) with e_l ∈ E, l = 1, 2, . . . , r. When the trust leader e_l is identified, feedback adjustment can be carried out.

$Φ^{t *} (A_{i}) = (1 - α) \times Φ^{t} (A_{i}) + α \times Φ^{l} (A_{i})$ (16)

α (α ∈ [0, 1]) is a parameter to control the degree between the group consensus and individual independence. When α = 0, inconsistent DMs keep the original matrix unchanged when making modifications. When α = 1, this means inconsistent DMs fully adopted the e_l’s advice.

However, the DMs prefer to maximize their interests while ensuring group consensus. Therefore, how to maximize the dominance degree is a key issue to the group consensus process. This article proposes the maximum dominance degree feedback mechanism to determine the value α.

For the DMs in the group, the optimization model is defined as follows.

$\begin{matrix} max \sum_{i = 1}^{m} \sum_{j = 1}^{n} ξ (A_{i}) \\ s . t . {\begin{matrix} Φ_{j}^{t *} (A_{i}, A_{k}) = (1 - α) \times Φ_{j}^{t} (A_{i}, A_{k}) + α \times Φ_{j}^{l} (A_{i}, A_{k}) \\ δ (A_{i}, A_{k}) = \sum_{j = 1}^{n} Φ_{j} (A_{i}, A_{k}) \\ {DD}^{M} ⩽ γ \\ {DD}^{NM} ⩽ γ \\ 0 ⩽ α ⩽ 1 \end{matrix} \end{matrix}$ (17)

By resolve the model (17), the value α can be determined. Then, the DMs could keep the balance between the group consensus and individual independence.

4 The consensus model for maximizing dominance degree based on trust relationship

Combined with the TODIM, the consensus model for maximizing dominance degree based on trust relationship has the following steps. The complete flow chart is shown as Fig. 3 below. The notations used in this paper are listed in Table 1.

Fig. 3

Maximizing dominance degree consensus model for SN-GDM with trust propagation.

Table 1

The notations listed in this paper

Notations	Implications
A _i	The i - th alternative
C _j	The j - th attribute
w _j	The importance degree of C_j
$X_{u} = {[x_{ij}^{u}]}_{m \times n}$	The decision-making matrix
$X_{u}^{'} = {[x_{ij}^{u'}]}_{m \times n}$	The normalized decision-making matrix
e _u	The DM
$Φ_{j}^{u} (A_{i}, A_{k})$	The dominance degree of A_i over A_k
δ (A_i, A_k)	The comprehensive dominance degree of A_i over A_k
ξ (A_I)	The overall dominance degree of A_i
TF	Trust function
TS	Trust score
E _⊗	Einstein product
E _⊕	Einstein sum
$λ_{T}^{σ (h)}$	The importance degree of each DM
$X^{c} = (x_{ij}^{c})_{m \times n}$	The collective decision matrix
δ^c (A_i, A_k)	The collective dominance degree
DD ^u	The deviation degree
α	The feedback parameter

Before proceeding to the CRP, the first thing is to build a complete social network, as described in Equations (5), (6), (7), (8) above. Then,

Step 1: The decision matrix $X_{u} = [x_{ij}^{u}]_{m * n}$ of each DM is normalized and $X_{u}^{'} = [x_{ij}^{u'}]_{m * n}$ is obtained, with i = 1, . . . , m ; j = 1, . . . , n ; u = 1, 2, . . . , r.

Step 2: Calculate the relative weight of c_j to c_r by Equation (1).

Step 3: Calculate the TS value of each DM by Equation (9), and select the leader e_l by Equation (15) in the group according to the largest TS value.

Step 4: Calculate dominance degree $Φ_{j}^{u} (A_{i}, A_{k})$ of each alternative A_i over A_k with respect to the criterion C_j by Equation (2). In existing research on TODIM, θ = 1 and θ = 2.5 are used at most. In this article, the value of the selected attenuation factor is θ = 1.

Step 5: Calculate the comprehensive dominance degree regarding each alternative A_i over A_k by Equation (3). And the ranking of the corresponding alternatives of the DM is calculated.

Step 6: Calculate the corresponding weight of each individual by Equation (10), then aggregate the collective dominance degree matrix $δ^{(c)} = [Φ_{j}^{c} (A_{i}, A_{k})]_{m * n}$ by Equation (11), and calculate the corresponding ranking of the group.

Step 7: Calculate DD^u to determine whether it is lower than 0.63, and make corresponding modifications to the DMs greater than 0.63. Otherwise, turn to Step 9.

Step 8: By Equations (16) and (17), the DMs who modified could calculate the value of α, then turn to Step 4.

Step 9: Based on the value of α from the previous step, calculate the collective dominance degree matrix and the corresponding ranking after the adjustment.

Step 10: End.

5 Example

5.1 Numerical example

Due to the wide variety and huge quantity of agricultural products and the high requirements for the function of sales channels, group decision making is required when selecting agricultural product suppliers. This article considers an agricultural product supplier selection problem. A company needs to select two of the six suppliers A₁, A₂, A₃, A₄ to supply, to maximize the benefits of the company. The company invited five DMs e₁, e₂, e₃, e₄, e₅ to evaluate the six attributes provided by the four alternatives A₁, A₂, A₃, A₄, including service level c₁, brand value c₂, cost c₃, quality c₄, supply capability c₅, and market prospects c₆. This article assumes the weight vector of the property as w = (0.1, 0.15, 0.2, 0.25, 0.2, 0.1) ^T, and the judgment information of each DM on the alternatives is given in the form of a matrix. The trust relationship among the DMs e₁, e₂, e₃, e₄, e₅ as shown in Fig. 4.

Fig. 4

The trust relationship among the DMs in the group.

$X_{1} = [\begin{matrix} 0.8 & 0.3 & 0.6 & 0.65 & 0.65 & 0.7 \\ 0.65 & 0.6 & 0.45 & 0.8 & 0.75 & 0.7 \\ 0.75 & 0.6 & 0.3 & 0.2 & 0.7 & 0.2 \\ 0.65 & 0.85 & 0.65 & 0.4 & 0.2 & 0.75 \end{matrix}]$

$X_{2} = [\begin{matrix} 0.6 & 0.5 & 0.4 & 0.5 & 0.15 & 0.55 \\ 0.4 & 0.4 & 0.2 & 0.4 & 0.25 & 0.5 \\ 0.4 & 0.2 & 0.4 & 0.5 & 0.5 & 0.4 \\ 0.4 & 0.15 & 0.15 & 0.5 & 0.4 & 0.15 \end{matrix}]$ $X_{3} = [\begin{matrix} 0.4 & 0.5 & 0.5 & 0.5 & 0.4 & 0.6 \\ 0.5 & 0.4 & 0.4 & 0.2 & 0.3 & 0.5 \\ 0.25 & 0.5 & 0.4 & 0.6 & 0.25 & 0.2 \\ 0.6 & 0.3 & 0.3 & 0.2 & 0.4 & 0.6 \end{matrix}]$ $X_{4} = [\begin{matrix} 0.5 & 0.6 & 0.1 & 0.4 & 0.3 & 0.2 \\ 0.6 & 0.3 & 0.2 & 0.5 & 0.8 & 0.6 \\ 0.5 & 0.3 & 0.2 & 0.5 & 0.2 & 0.4 \\ 0.2 & 0.6 & 0.3 & 0.4 & 0.3 & 0.6 \end{matrix}]$ $X_{5} = [\begin{matrix} 0.3 & 0.2 & 0.7 & 0.2 & 0.5 & 0.2 \\ 0.4 & 0.5 & 0.2 & 0.7 & 0.2 & 0.3 \\ 0.3 & 0.5 & 0.4 & 0.3 & 0.5 & 0.55 \\ 0.4 & 0.3 & 0.3 & 0.7 & 0.3 & 0.3 \end{matrix}]$

As shown by expression (5), (6), (7), (8), the corresponding TD matrix is $TD = [\begin{matrix} (1, 0) & (0.9, 0.2) & (0.22, 0.78) & (0.52, 0.56) & (0.32, 0.73) \\ (0.9, 0.2) & (1, 0) & (0.22, 0.78) & (0.6, 0.4) & (0.38, 0.62) \\ (0.6, 0.3) & (0.8, 0.1) & (1, 0) & (0.44, 0.48) & (0.27, 0.68) \\ (0.8, 0.2) & (0.7, 0.2) & (0.45, 0.55) & (1, 0) & (0.7, 0.3) \\ (0.6, 0.2) & (0.7, 0.1) & (0.7, 0.3) & (0.27, 0.68) & (1, 0) \end{matrix}]$

Step 1: Normalized the decision matrices X₁X₂X₃X₄X₅ into $X_{1}^{'} X_{2}^{'} X_{3}^{'} X_{4}^{'} X_{5}^{'}$ .

Step 2: The weight vector of the property is assumed as w = (0.1, 0.15, 0.2, 0.25, 0.2, 0.1) ^T, the corresponding value of w_r is 0.25, hence the relative weight is (0.4, 0.6, 0.8, 1, 0.8, 0.4) ^T.

Step 3: By Equation (9), the TS value of each DM is calculated.

TS₁ = 0.875; TS₂ = 0.937; TS₃ = 0.522;

TS₄ = 0.875; TS₅ = 0.541.

By contrast, $e_{2} = max_{r} {{TS}_{r}}$ , hence the leader in the social network is e₂.

Step 4: Calculate the dominance degree for each DM, this article takes e₁ as an example, as shown in Table 2.

Table 2

The dominance degree of A_i over A_k when e₁ are targeting attributes c₁

e ₁	c ₁
	A ₁	A ₂	A ₃	A ₄
A ₁	0	–3.16	–1.83	–3.16
A ₂	–2.23	0	–1.58	0.22
A ₃	–1.58	0.16	0	0.27
A ₄	–3.16	–2.23	–2.74	0

Step 5: Combine the data from Table 2, calculate the comprehensive dominance degree of A_i over A_k, as illustrated in Table 3. At this point, the ranking result of e₁ is A₁ ≻ A₂ ≻ A₄ ≻ A₃ as shown in Table 4.

Table 3

The comprehensive dominance degree of A_i over A_k of e₁

e ₁	A ₁	A ₂	A ₃	A ₄
A ₁	0	–3.90	–8.22	–5.82
A ₂	–3.25	0	–6.89	–3.17
A ₃	–1.25	–1.35	0	–3.40
A ₄	–3.34	–3.53	–7.61	0

Table 4

The initial ranking result of e₁

A ₁	A ₂	A ₃	A ₄
1.00	0.94	0	0.69

Step 6: The group comprehensive dominance is derived by Tables 2 and 3, as given by Table 5. The weight of DM is assigned by the value of TS as shown in Equation (10).The corresponding weight of the DMs are: λ₁ = 0.23; λ₂ = 0.42; λ₃ = 0.1; λ₄ = 0.13; λ₅ = 0.12.

Table 5

The initial ranking of the collective dominance degree matrix

A ₁	A ₂	A ₃	A ₄
1.00	0.61	0.14	0

And the collective dominance degree matrix δ^(c) is calculated as shown in Table 5.

The ordering corresponding to δ^(c) is A₁ ≻ A₂ ≻ A₃ ≻ A₄.

Step 7: Calculate DD^u to determine whether it is lower than 0.63, and make corresponding modifications to the identified DMs. $\begin{matrix} {DD}^{1} = 0.65; {DD}^{2} = 0.56; {DD}^{3} = 0.55; \\ {DD}^{4} = 0.60; {DD}^{5} = 0.62 . \end{matrix}$

Comparative with 0.63, DD¹ is greater than 0.63, so the DM who needs to be modified is e₁, i.e. EXPCH = e₁.

Step 8: In the previous step, the identified inconsistent DM who get the feedback mechanism is e₁. Therefore, by Equation (17), calculate the value of α is 0.58.

When α = 0.58, the collective comprehensive dominance degree of e₁ is changed, the new comprehensive dominance degree is shown in Table 6.

Table 6

The comprehensive dominance degree of A_i over A_k of e₁

e ₁	A ₁	A ₂	A ₃	A ₄
A ₁	0	–7.09	–7.50	–8.64
A ₂	–1.21	0	–4.23	–4.47
A ₃	–1.36	–3.78	0	–5.43
A ₄	–1.62	–2.98	–2.55	0

Step 9: Since the value of α is calculated, the TODIM method is used to evaluate the sorting results of the final solution.

Calculate the collective dominance degree matrix and the corresponding ranking as shown in Tables 7 and 8. The new collective comprehensive dominance degree is shown in Table 7.

Table 7

The collective comprehensive dominance degree of A_i over A_k after the adjustment

A ₁	A ₂	A ₃	A ₄
A ₁	0	–13.55	–12.11	–14.73
A ₂	–6.96	0	–8.92	–9.38
A ₃	–5.52	–7.92	0	–12.36
A ₄	–5.88	–6.31	–6.52	0

Table 8

The ranking of group after the adjustment

A ₁	A ₂	A ₃	A ₄
1.00	0.48	0.49	0

Then, after the adjustment, the collective ranking of the group is obtained as shown in Table 8.

The ranking result is A₁ ≻ A₃ ≻ A₂ ≻ A₄, which is different from before.

Step 10: End.

5.2 Discussion

In order to illustrate the rationality and effectiveness of this method proposed, the classical TODIM method is used to solve the numerical example below for comparative analysis, and sensitivity analysis for the effect of individual independence parameters on group and individual.

5.2.1 Comparative analysis

(1) Comparative analysis of group

As could be seen in Fig. 5, from the perspective of group, the ranking result after the adjustment is A₁ ≻ A₃ ≻ A₂ ≻ A₄, whereas the initial ranking result is A₁ ≻ A₂ ≻ A₃ ≻ A₄. When α = 0, the alternatives should be chose are A₁A₂; when α = 0.58, the alternatives should be selected are A₁A₃.

Fig. 5

The group ranking before and after the adjustment.

When α = 0, the dominance degree of A₂ is much higher than that of A₃. The DMs in the group don’t reach consensus. As the value α grows, the dominance degree of A₂ decreased while A₃ increased. When α reaches 0.58, the dominance degree of A₃ exceeds that of A₂. And at this point, the DMs in the group has reached consensus, which is more realistic than before.

(2) Comparative analysis of e₁

In Fig. 6, from the perspective of e₁, the ranking of e₁ after the adjustment is A₁ ≻ A₂ ≻ A₄ ≻ A₃, while the origin ranking is A₁ ≻ A₂ ≻ A₃ ≻ A₄.

Fig. 6

The ranking of e₁ before and after the adjustment.

Whatever α takes, the alternatives e₁ would prefer are A₁A₂. Even so, the dominance degree of each alternative has greatly changed. As could be seen in Fig. 6, as the value of α grows, the dominance degree of A₃ increases while A₂ and A₄ decreased sharply.

(3) Comparative analysis of the other method

To verify the rationality and effectiveness of this proposed method, we calculated the data from this paper using the method of [50]. The collective ranking of the group is shown in Table 9.

Table 9

The ranking of group in other method

A ₁	A ₂	A ₃	A ₄
1.00	0.35	0.17	0

The ranking result is A₁ ≻ A₂ ≻ A₃ ≻ A₄, at this point the alternatives would be selected are A₁A₂. However, the alternatives before the adjustment i.e. the TODIM method, are A₁A₂. As we have stated above, using TODIM directly not only ignores the feelings of the DMs, but also does not accord with the reality. That is the advantage of the method we proposed.

5.2.2 Sensitivity analysis

(1) Sensitivity analysis of group

We calculated the final group sort result under seven values respectively for sensitivity analysis. The trend shown in Fig. 7 shows different values of α have an impact on the final sort result, and this effect changes as the values of α change.

Fig. 7

The effect of different α parameter values on the final group sort result.

When α = 0, the final group sort result is: A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.1, the final group sort result is: A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.3, the final group sort result is: A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.5, the final group sort result is: A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.58, the final group sort result is: A₁ ≻ A₃ ≻ A₂ ≻ A₄;

When α = 0.7, the final group sort result is: A₁ ≻ A₃ ≻ A₂ ≻ A₄;

When α = 0.9, the final group sort result is: A₁ ≻ A₃ ≻ A₂ ≻ A₄.

When α = 0, i.e. the initial group sort result, and α = 0.1, α = 0.3, α = 0.5, the alternatives could be chose are A₁A₂. However, this situation changed when the α value increased to 0.58. When α = 0.58/0.7/0.9, the alternatives selected are A₁A₃. Besides, α = 0.58 means the identified DM, i.e. e₁, keeps balance between the group consensus and individual independence. With the increase of α, the dominance degree of A₃ increases while that of A₂ decreases, and A₁ and A₄ remain unchanged.

(2) Sensitivity analysis of e₁

As shown in Fig. 8, when α takes different values, the final sort will change.

Fig. 8

The effect of different α values on the final sort result for e₁.

When α = 0, the sort at this point isA₁ ≻ A₂ ≻ A₄ ≻ A₃;

When α = 0.1, the sort at this point is A₁ ≻ A₂ ≻ A₄ ≻ A₃;

When α = 0.3, the sort at this point is A₁ ≻ A₂ ≻ A₄ ≻ A₃;

When α = 0.5, the sort at this point is A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.58, the sort at this point is A₁ ≻ A₂ ≻ A₃ ≻ A₄;

When α = 0.7, the sort at this point is A₁ ≻ A₃ ≻ A₂ ≻ A₄;

When α = 0.9, the sort at this point is A₁ ≻ A₃ ≻ A₂ ≻ A₄.

When α = 0, 0 .1, 0.3, 0 .5, 0.58, the alternatives e₁ would be willing to choose are A₁A₂, and when α = 0.7/0 .9, the alternatives e₁ would prefer A₁A₃. Although the choice is the same when α = 0/0 . 1/0 .3 and α = 0.5/0.58, the overall ranking has changed. Compared with when α = 0/0 . 1/0 .3, the dominance degree of A₃ has more advantages than A₄ when α = 0.5 and α = 0.58. While the value of α increase to 0.7, the dominance degree of A₃ is lower than that of A₂, and it lasts until α = 0.9.

6 Advantages of the proposed approach

From the analysis above, the main advantages of the proposed optimization model could be summarized as follows.

(1) Considering the direct and indirect trust relationships among the DMs are realistic, because DM would refer others’ matrix to make adjustments due to the complexity and rapidity of real social networks, which also improves the efficiency of evaluation.

(2) In the process of GDM, we also consider the feelings of the supervisors. When selecting the final plan, the supervisors hope to choose the alternatives with the largest overall dominance degree, so DMs take the maximum dominance degree as the objective function when making decisions.

(3) The TODIM decision-making method takes into account the reference dependence and loss avoidance behaviors of DMs. Therefore the feedback adjustment model based on dominance degree is proposed in this paper. DMs make modifications based on the degree of dominance, which can more directly show the behavior of DMs when dealing with risks. The feedback parameter also reflects the trust relationship between DMs and others through the social network.

7 Conclusions

To address the group consensus problems in social networks, this paper proposes a model for maximizing dominance degree based on trust relationships and TODIM method in the consensus reaching process.

(1) The trust relationships play an important role in the whole decision-making process. Identify the potential direct and indirect trust relationships among the DMs and construct the completed trust network. In this trust relationships network, the identified inconsistent DMs would change based on the trust relationships with the leader e_l.

(2) The optimal model to achieve group consensus with the maximizing dominance degree is established. The maximizing dominance degree model is proposed to determine the approximate feedback parameter α. Once this α is found, the degree to which inconsistent experts refer to e_l in the feedback process is obtained, then calculated by the TODIM method. The proposed optimal model raises the consistency of the final ranking result and makes up for the shortcomings brought about by the limited rationality of the DMs and the inadequate understanding of the alternatives. Besides, this improves DMs’ decision-making efficiency and effectively maintains the original judgments to sort and select the alternatives for the group. This provides a new way to solve the group consensus problem of TODIM multi-attribute decision-making which considers the trust relationships among DMs in social network. In future research, considering the group consensus using TODIM group decision-making methods based on trust relationship in large-scale groups would be meaningful.

However, this paper also has the following limitations, which can be considered in future. As the complexity of the large-scale social networks, the method proposed has limitations. And as there are many types of languages, future research should consider a detailed classification of languages. Adding the Pythagorean theorem and sines and cosines is also worth studying in further research. Besides, this paper only discusses the situation with known attribute weights, but it is usually difficult to obtain the attribute weight vector in some real-life decision making, that is what we should study in further research. Consider the loss avoidance mentality of DMs, the decision-making risk study is not extensive enough which should be make further research. In addition, the length of trust chain would also have an impact on trust spread, so how to improve the efficiency of communication is also an issue we should pay attention to.

Footnotes

Acknowledgments

This paper is strongly supported by the Natural Science Foundation of Fujian Province, China (2020J01463), the Humanity and Social Science Youth foundation of Ministry of Education (19YJC630022), the Starting Research Fund from the Fuzhou University (GXRC201905).

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