Social influence on group decision-making

6304 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 51, NO. 10, OCTOBER 2021

Consensus Reaching and Strategic Manipulation in Group Decision Making With Trust Relationships

Yucheng Dong , Quanbo Zha , Hengjie Zhang , and Francisco Herrera , Senior Member, IEEE

Abstract—To date, a large number of consensus reaching processes (CRPs) have been reported in group decision making (GDM). Trust relationships should be an essential ele- ment in interactions among a group of individuals, leading to the evolution of individuals’ preferences. Therefore, in this article, we present a trust relationships CRP with a feedback mechanism which consists of two approaches of facilitating consensus reach- ing: 1) the leader-based preference adjustment and 2) the trust relationships improvement. In the trust relationships CRP, we build a bridge between opinion dynamics and GDM to highlight the role of the leaders and trust relationships improvements in the GDM problems. Furthermore, we present a new strategic manipulation issue, called trust relationship manipulation, and discuss some clique-based strategies to manipulate trust rela- tionships to obtain the desired ranking of the alternatives in the GDM problems. Finally, the detailed simulation experiments are proposed to justify our proposal.

Index Terms—Consensus, group decision making (GDM), opinion dynamics, strategic manipulation, trust relationships.

I. INTRODUCTION

A GROUP decision-making (GDM) problem canbe defined as a decision-making situation where a group of individuals try to find a collective solu- tion to a decision-making problem regarding multiple alternatives [15], [16], [39], [47], [71]. Two processes are often needed in GDM [13], [43], [45]: 1) consensus reaching process (CRP) and 2) selection process. The CRP aims at improving the consensus degree among the individuals before the use of the selection process. The selection process is utilized to aggregate the preferences of individuals into

Manuscript received March 26, 2019; accepted December 7, 2019. Date of publication January 13, 2020; date of current version September 16, 2021. This work was supported in part by the NSF of China under Grant 71871149, and in part by Sichuan University under Grant sksyl201705 and Grant 2018hhs- 58. This paper was recommended by Associate Editor J. Lu. (Corresponding author: Quanbo Zha.)

Yucheng Dong is with the Center for Network Big Data and Decision- Making, Business School, Sichuan University, Chengdu 610065, China (e-mail: [email protected]).

Quanbo Zha is with the School of Management Science and Real Estate, Chongqing University, Chongqing 400045, China (e-mail: [email protected]).

Hengjie Zhang is with the Business School, Hohai University, Nanjing 211100, China (e-mail: [email protected]).

Francisco Herrera is with the Andalusian Research Institute on Data Science and Computational Intelligence, University of Granada, 18071 Granada, Spain, and also with the Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia (e-mail: [email protected]).

Color versions of one or more figures in this article are available at https://doi.org/10.1109/TSMC.2019.2961752.

Digital Object Identifier 10.1109/TSMC.2019.2961752

a collective preference to achieve a solution. In practice, it is very important to reach a consensus regarding the group decision result. To date, a lot of CRPs have been developed in different decision context, such as CRPs with different preference representation formats [12], [31], [33], [46], [49]–[52], [81]; CRPs based on minimum adjustments or cost [6], [11], [22], [37], [38], [75], [82], [83]; CRPs driven by consistency and consensus measures [1], [3], [4], [30], [60], [61], [76]; CRPs considering the atti- tudes of individuals [4], [19], [63], [65]; and CRPs under dynamic/Web context [19], [32], [52], [66].

In the GDM problems, there often exist trust relation- ships among individuals, and thus the trust relationships GDM (TRGDM) has been investigated extensively in recent years. The recent results in TRGDM were reviewed by Dong et al. [23] and Ureña et al. [72]. In the existing TRGDM models, the use of trust relationships mainly focuses on three aspects.

1) Analyze the Weights/Influences of Individuals: Gupta [40] adopted the induced ordered weighted averaging operator to find the influences of individuals through the trust relationships. Wu et al. [75] calculated the individual weights in GDM under social network with distributed linguistic trust.

2) Incomplete Preference Values Estimation: Capuano et al. [10] and Liang et al. [55] investigated the TRGDM problems with incomplete information, in which trust relationships were adopted to estimate missing values in their models.

3) Feedback Mechanism: Liu et al. [57] presented a trust- induced recommendation mechanism to ask individuals modify their preference values according to the prefer- ences of individuals they trust. Wu et al. [74] reported a recommendation rule with the consideration of the weights/influence of individuals in the trust relation- ships.

Although the existing models in TRGDM are very useful, there are still some limitations.

1) Limitation 1: The existing TRGDM models ignore inter- actions among individuals with trust relationships, which lead to the evolution of individuals’ preferences. The phenomenon of preference evolution has been well-justified in the discipline of opinion dynamics. In the extant literature, many opinion dynamics models have been proposed to model the evolu- tion of preferences, such as DeGroot model [7], [17], major- ity rule [53], Friedkin and Johnsen model [35], [36], Voter model [27], bounded confidence model [18], [41], and the

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6305

opinion dynamics with a social network [28], [34], [58], [70]. A recent survey on opinion dynamics can be found in [24]. As shown in Section III, the leaders and trust relationships improvements play key roles in the evolution of preferences in TRGDM, and thus it is of necessity to build a robust bridge between opinion dynamics and GDM, and to develop a novel CRP to highlight the roles of leaders and trust relationships improvements in the TRGDM problems.

2) Limitation 2: In some situations, the individuals are often dishonest and want to manipulate the decision result in a GDM problem. The existing strategic manipulation mainly observed that the individuals dishonestly express preferences to achieve the desired ranking. For example, Pelta and Yager [64] and Yager [79], [80] studied the pref- erences strategic manipulation behaviors in the aggregation process of GDM. Dong et al. [21] and Liu et al. [56] stud- ied the incomplete multiple attribute DM problem, in which the minimum, maximum, and average ranking positions of the alternative are computed. Moreover, Palomares et al. [63] and Xu et al. [77] designed the moderator-based approaches to manage the preference strategic manipulation behaviors in the CRP of GDM. Recently, Dong et al. [25], [26] developed a self-management mechanism to manage prefer- ence strategic manipulation behaviors in a CRP. However, in TRGDM, individuals can manipulate trust relationships to achieve the desired ranking of alternatives, which, in this arti- cle, we call trust relationship manipulation (TRM). To our knowledge, the TRM issue has not been analyzed although there are a lot of trust relationship strategic manipulation phenomena in TRGDM.

To overcome the limitations of the TRGDM models ana- lyzed above, we propose a novel consensus reaching model with the use of trust relationships and analyze the TRM in the TRGDM. This proposal is carried out according to the following.

1) As individual preferences will evolve during a CRP in TRGDM, we bridge the gap between opinion dynamics and GDM, showing the key roles of the leaders and trust relationships improvements in the TRGDM problems.

2) Through the analysis of trust relationships, we pro- pose a trust relationships CRP with a new feedback mechanism which consists of two approaches of facil- itating consensus reaching: a) leader-based preference adjustment and b) trust relationships improvement.

3) We propose a new strategic manipulation issue, TRM, and we study some clique-based strategies to manipulate trust relationships to obtain the desired ranking of the alternatives.

Particularly, we present the detailed experimental simula- tions and comparison analysis to justify our proposal.

The remainder of this article is organized as follows. Section II describes the basic knowledge regarding the tra- ditional CRP framework, trust relationships, and opinion dynamics. Next, Section III builds a bridge between opinion dynamics and GDM to motivate this article. Then, Section IV proposes a novel trust relationships CRP to facilitate a con- sensus in TRGDM. Furthermore, the TRM issue is analyzed based on the simulation experiments in Section V. Moreover,

Fig. 1. Traditional CRP framework.

the advantages and limitations of our model are discussed in Section VI. Finally, the conclusions are presented in Section VII.

II. PRELIMINARIES

In this section, we introduce the basic knowledge regarding the CRP framework, trust relationships in GDM, and opinion dynamics with trust relationships.

A. Traditional CRP Framework

In this article, we consider the following GDM problem. Let X = {x1, x2, . . . , xn}(n ≥ 2) be a finite set of alternatives and D = {d1, d2, . . . , dm}(m ≥ 2) be the set of individu- als in a GDM problem. Each individual dk ∈ D offers their preference over the set of alternatives X, and without loss of generality, in this article, the individual dk is assumed to provide their preference over the pairwise (xi, xj) using addi- tive preference relation (also called fuzzy preference relation) Pk = (pkij)n×n, where pkij ∈ [0, 1] denotes the preference degree of alternative xi over xj and p

k ij + pkji = 1 for ∀i, j ∈ {1, . . . , n}.

Most of the existing CRPs follow a general framework which is shown in Fig. 1, and it includes two key phases.

1) Consensus Measure: Consensus measure analyzes the consensus degree in the group [44], [62]. A widely used approach is measuring the distances between individual pref- erences and collective preference [14], [46].

Let {π1, π2, . . . , πm} be the weights of a set of individuals, where πk ≥ 0 is the weight associated to the individ- ual dk and

∑m k=1 πk = 1. Then an aggregation operator f

is employed to obtain a collective additive preference rela- tion Pc = (pcij)n×n. Two highly used operators in GDM are the weighted average (WA) operator and the ordered weighted average operator [78]. In this article, we use the WA operator, i.e.,

pcij = fπ (

p1ij, p 2 ij, . . . , p

m ij

) =

m∑

k=1 πkp

k ij, for i, j = 1, . . . , n. (1)

And the consensus degrees of individual dk and the group can be calculated as

CD(dk) = ∑n−1

i=1 ∑n

j=i+1 (

1 − |pkij − pcij| )

n · (n − 1)/2 (2)

CD = m∑

k=1 πkCD(dk). (3)

Clearly, CD ∈ [0, 1] and a larger CD value indicates a higher consensus degree. Let μ ∈ [0, 1] be a consen- sus threshold, and we argue that individuals have reached an

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6306 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 51, NO. 10, OCTOBER 2021

acceptable consensus if CD ≥ μ. Then, the temporal collective preference relation can be treated as the final preference rela- tion of the group. Otherwise, the individuals will be suggested to adjust their preferences to improve the group consensus degree.

2) Preference Adjustment: Preference adjustment is to pro- vide suggestions that are used by the individuals to adjust their preferences in order to reach a higher consensus degree. In CRPs, identification rule (IR) and direction rule (DR) are two classical rules [9], [44], [62], which are introduced below.

1) IR: Identify the individual(s) whose consensus degree CD(dk) is below the consensus threshold μ, i.e.,

I = {dk|CD(dk) < μ; k ∈ {1, . . . , m}}. (4) 2) DR: To increase the consensus degree, the individuals

identified based on the IR are suggested to adjust their preferences. The adjusted preference p̄kij is suggested to follow the DR and is described as follows:

{ p̄kij ∈

[ min

( pkij, p

c ij

) , max

( pkij, p

c ij

)] , i ≥ j

p̄kij = 1 − p̄kji, i < j (5)

when an acceptable consensus is achieved, the selection pro- cess will be utilized to derive the ranking of alternatives from the final collective preference relation with a consensus. To avoid confusion, the final collective preference relation with a consensus will still be denoted as Pc = (pcij)n×n. The rank- ing of alternative can be generated based on the dominance degree Qi of alternative xi over other alternatives, where the alternatives with higher values are ranked higher [42]. In this article, the dominance degree of alternative xi is calculated by WA as follows:

Qi = fσ ( pci1, p

c i2, . . . , p

c in

) = ∑n

j=1 p c ij

n . (6)

B. Trust Relationships in GDM

In the following, we introduce the basic knowledge regard- ing trust relationships as Definitions 1–4 [8], [20], [48], [69].

Definition 1: The trust relationships can be defined by a directed graph G(D, E), where nodes represent the individ- uals D = {d1, d2, . . . , dm}, and E is a set of edges in which the edge (dk, dl) ∈ E means that individual dk directly trust individual dl.

Definition 2: The adjacency matrix A = (akl)m×m is a 0-1 matrix. If there is an edge from individual dk to individ- ual dl in G(D, E), the (k, l)th entry in the matrix A = (akl)m×m is 1; otherwise, it is 0, i.e.,

akl = {

1, (dk, dl) ∈ E 0, (dk, dl) /∈ E. (7)

For example, the trust relationships among {d1, d2, . . . , d5} can be described a directed graph in Fig. 2, and its adjacent matrix is

A =

⎛

⎜ ⎜ ⎜ ⎜ ⎝

− 1 0 1 1 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1 0 1 0 0 −

⎞

⎟ ⎟ ⎟ ⎟ ⎠

.

Fig. 2. Directed graph.

TABLE I USE OF TRUST RELATIONSHIPS IN TRGDM

Definition 3: A sequence of edges (dk, dk1 ) (dk1 , dk2 ) , . . . , (dkm−1 , dkm )(dkm , dl) in G(D, E) is known as a directed path from individual dk to individual dl, and it is denoted as dk → dl.

Definition 4: The in-degree centrality C(dk) of individual dk is defined as

C(dk) = m∑

l=1,l �=k alk. (8)

The in-degree centrality index of an individual is usually used to reflect their social influence in trust relationships anal- ysis, and the larger in-degree centrality index of an individual indicates higher social influence (Wasserman and Faust, 1994).

In recent years, the TRGDM has been investigated exten- sively. As shown in Table I, trust relationships are mainly used in three aspects in TRGDM.

There exist some approaches to identify trust relationships (e.g., [5] and [68]). But, in the existing research paradigms of the TRGDM, the trust relationships are assumed to be given or known [23], [72]. Therefore, in this article, we follow the existing research line of the TRGDM and assume that the trust relationships are known.

C. Opinion Dynamics With Trust Relationships

In the opinion dynamics, individuals will be influenced by other individuals. In general, the preferences of individ- uals will be reformed to a certain extent by considering the preferences of others.

Let ok,t ∈ R be the preference of individual dk at t round. We use symbol wkl ≥ 0 to denote the weight that individual dk assigns to individual dl, where

∑m l=1 wkl = 1.

Thus, the preference evolution of individual dk can be formulated as follows:

ok,t+1 = wk1o1,t + wk2o2,t + · · · + wkmom,t,t = 0, 1, 2, . . . (9)

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6307

Fig. 3. Trust relationships G(D, E) in Example 1.

Equation (9) can be equivalently presented as follows:

Ot+1 = W · Ot, t = 0, 1, 2, . . . (10)

where W = (wkl)m×m and Ot = (o1,t, o2,t, . . . , om,t)T ∈ Rm. When W is fixed, (9) [also (10)] is called the DeGroot model

which is a classical model in opinion dynamics [17]. In [20], the trust relationships-based DeGroot (TRDG) model has been introduced, which will be used as the basis of this article. In the TRDG model, individual dk will refer to the preference of individual dl when dk trusts dl; otherwise, dk will not refer to it.

Let A = (akl)m×m be the adjacency matrix of trust relation- ships G(D, E). Each individual dk gives the self-confidence degree βk ∈ (0, 1) to their own preference and distributes (1 − βk) across the trusted individuals. Let ωkl be the weight that dk assigns to dl.

Then the evolved preference ok,t+1 can be described as

ok,t+1 = βkok,t + m∑

l=1,l �=k ωkl × ol,t (11)

where ∑m

l=1,l �=k ωkl = 1 − βk; ωkl ∈ (0, 1 − βk] if akl = 1; and ωkl = 0 if akl = 0.

In the TRDG model, βk and ωkl are both fixed across the time or with preferences, and some results of the TRDG model (see Dong et al. [20]) are introduced below.

Definition 5: All individuals reach a consensus if for any O0 ∈ Rm there exists c ∈ R such that limt→∞ ok,t = c (k = 1, 2, . . . , m).

In the TRDG model, c is called the consensus opin- ion/preference.

Definition 6: In G(D, E), let dk → dl be a directed path from individual dk to individual dl. Then, the individual in the set {dk|dl → dk, for all dl ∈ D/{dk}} is defined as a leader. If an individual is not a leader, then the individual is a follower.

In this article, let DleaderG be the set of leaders and D follower G

be the set of followers in G(D, E), respectively. Lemma 1 [20]: When the preferences of individuals evolve

as the TRDG model, all individuals in G(D, E) will form a consensus preference if and only if there exist leaders in G(D, E), i.e., DleaderG �= ∅.

Lemma 2 [20]: The consensus preference is determined by the initial preferences of leaders by c = ∑dk∈DleaderG δko

k,0, where δk ≥ 0.

TABLE II SIX DATA SETS FOR THE INITIAL PREFERENCES pk,012 (k = 1, 2, . . . , 12)

III. MOTIVATION: BRIDGE BETWEEN OPINION DYNAMICS AND GDM

In the section, we use Example 1 to reveal the key roles of the leaders and trust relationships improvements in the TRDG model and to motivate our CRP study in GDM.

Example 1: Consider a GDM problem in which 12 indi- viduals D = {d1, d2, . . . , d12} are involved with the trust relationships G(D, E) as shown in Fig. 3, which consist of two subtrust relationships G(1) and G(2). Let A = (akl)12×12 be the adjacency matrix of the trust relationships G(D, E). Based on Definition 6, the individuals {d1, d2, d3} and {d8, d9} are sub- trust relationship leaders in two subtrust relationships G(1) and G(2), respectively. Moreover, in G(D, E) there is no leader.

The individuals provide their preferences over n alternatives using the additive preference relations Pk,0 = (pk,0ij )n×n at the initial time, where pk,0ij is the preference over the pairwise (xi, xj) at the initial time. To save space, in this example, we just use the preferences in the (1, 2)th entry.

Following the TRDG model, we consider that the preference of individual dk(k = 1, 2, . . . , 12) in the (1, 2)th entry will evolve as

pk,t+112 = βkpk,t12 + m∑

l=1,l �=k ωkl × pl,t12. (12)

First, we use simulations to show the important roles of the leaders in the evolution of preferences pk,t12 . We set six data sets for initial preferences pk,012 (k = 1, 2, . . . , 12), which are listed in Table II. Let the self-confidence degrees of indi- viduals βk = 0.65(k = 1, 2, . . . , 12), the weights ωkl = (1 − βk)/

∑12 l=1 akl(k = 1, 2, . . . , 12; k �= l) (i.e., ω41 = 0.35),

and the weights of individuals πk = 1/12(k = 1, 2, . . . , 12). Notably, when using different data sets and setting different values for ωkl, βk, and πk, we can obtain similar observations.

Fig. 4 shows the evolution of the preferences of individ- uals in the (1, 2)th entry under six groups of data sets. In Fig. 4(a)–(c), the initial preferences of the subtrust rela- tionship leaders {d1, d2, d3, d8, d9} in the (1, 2)th entry

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6308 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 51, NO. 10, OCTOBER 2021

Fig. 4. Evolution of individuals’ preferences pk,012 (k = 1, 2, . . . , 12) under six groups of data sets. (a) Data set 1. (b) Data set 2. (c) Data set 3. (d) Data set 4. (e) Data set 5. (f) Data set 6.

are all {0.26, 0.30, 0.34, 0.80, 0.74} and we observe that no matter how the initial preferences of the followers, {d4, d5, d6, d7, d10, d11, d12} change the individuals’ final preferences are stable at two same preference values. In Fig. 4(d)–(f), the initial preferences of the leaders, {d1, d2, d3, d8, d9}, are all {0.46, 0.54, 0.64, 0.26, 0.34}, we obtain the observation same to Fig. 4(a)–(c).

The observation can be explained based on Lemmas 1 and 2. 1) Because there is no leader in G(D, E), all individual can-

not form a full consensus preference in the framework of the TRDG model. Instead, the individuals’ final pref- erences are stable at two same preference values because there exist subtrust relationship leaders in G(1) and G(2).

2) The two final preference values are determined by the initial preferences of leaders {d1, d2, d3} and {d8, d9} in the two subtrust relationships G(1) and G(2) in Fig. 3, respectively, and the initial preferences of the followers do not influence the final preference values.

These observations show that leaders play a more important role in the evolution of preferences than followers.

Furthermore, we show the important role of the trust rela- tionships improvements in the evolution of preferences pk,t12 . We consider the following three cases to improve the trust relationships among 12 individuals.

Relationships Case 1: Add two edges (d1, d8) and (d8, d1) in trust relationships G(D, E).

Relationships Case 2: Add two edges (d4, d11) and (d11, d4) in trust relationships G(D, E).

Relationships Case 3: Add two edges (d10, d5) and (d5, d10) in trust relationships G(D, E).

Fig. 5(a)–(c), respectively, shows that the evolution of the preferences of pk,012 (k = 1, 2, . . . , 12) under three trust rela- tionships improvement cases, when setting initial preferences pk,012 based on data set 1. Meanwhile, Fig. 5(d)–(f) shows the evolution of the preferences with data set 4 under three cases of trust relationships improvements, respectively.

From Fig. 5, we obtain the following observations. 1) All 12 individuals will form a full consensus when pro-

moting the mutual trust between individuals d1 and d8 [see Fig. 5(a) and (d)].

Fig. 5. Evolution of the preferences of individuals under three trust relation- ships improvements. (a) Data set 1 and relationships case 1. (b) Data set 1 and relationships case 2. (c) Data set 1 and relationships case 3. (d) Data set 4 and relationships case 1. (e) Data set 4 and relationships case 2. (f) Data set 4 and relationships case 3.

2) Individual preferences will still be fragmented, when promoting the mutual trust between individuals d4 and d11 (or d5 and d10) [see Fig. 5(b), (c), (e), and (f)].

The observation can be explained based on Lemma 1: adding the mutual trust between individuals d1 and d8 pro- duces leaders, {d1, d2, d3, d8, d9}, in the trust relationship G(D, E), which indicates all individuals in the trust relation- ship G(D, E) will form a consensus. On the other hand, the other two trust relationships improvements do not produce leaders in the trust relationship G(D, E) to form a consen- sus. This means that the trust relationships improvements play a key role to form a consensus. Particularly, we should choose the proper improvements to promote consensus reaching.

In a GDM problem, the CRP is a consensus-oriented dis- cussion process, and the traditional CRP employs the IR and DR to guide the group to reach a consensus. A fact is that the individuals’ preferences will evolve during a CRP. However, the traditional CRP with the IR and DR only considers the influence of the collective preference on the preference evolu- tion of individuals, and it ignores that the individuals will also interact with each other in the group. In other words, the CRP individuals will refer to the preferences of the individuals they trust to reform their own preferences, which has been justified well in the TRDG model. Therefore, the existing CRPs with the IR and DR in TRGDM ignore the following two issues.

1) The leaders and followers have distinct roles to form a consensus, but the IR identifies the individual(s) con- tributing less to reach a high consensus degree among the group, without the consideration of the difference between leaders and followers in trust relationships.

2) The DR provides some suggestions to help individuals, identified by IR, modify their preferences. But the DR just considers the preference adjustments and ignores the role of trust relationships improvements in consensus reaching.

Therefore, it is of necessity to build a robust bridge between opinion dynamics and GDM, and to develop a novel CRP to highlight the roles of leaders and trust relationships improvements in TRGDM.

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6309

Fig. 6. Framework of TRC.

IV. CRP WITH THE USE OF TRUST RELATIONSHIPS

In this section, we propose a novel CRP with the use of trust relationships in the TRGDM context. In this article, we abbre- viated the proposed CRP with the use of trust relationships as the TRC and the traditional CRP as the TC.

A. Framework of TRC

Let X = {x1, x2, . . . , xn}(n ≥ 2) be a finite set of alter- natives and D = {d1, d2, . . . , dm} be the set of individuals in a GDM problem. Each individual dk expresses their prefer- ence over X with an additive preference relation Pk = (pkij)n×n (k = 1, 2, . . . , m), where pkij is the preference over the pair- wise (xi, xj). And let the trust relationship G(D, E) with the adjacency matrix A = (akl)m×m present the trust relationships among the individuals.

The framework of TRC is described in Fig. 6, and it consists of three key phases: 1) consensus measure; 2) leader-based preference adjustment; and 3) trust relationships improvement. Among these phases, the consensus measure is the same to that in the TC [i.e., (1)–(3)], and the others are described as follows.

1) Leader-Based Preference Adjustment: The leader-based preference adjustment is employed to suggest leaders to adjust their preferences to pursue a higher consensus degree, including two rules: a) leader IR and b) leader DR. The two rules are elaborated in Section IV-C.

2) Trust Relationships Improvement: The trust relationships improvement is used to manage trust relationships to facilitate consensus reaching and includes two rules: a) relationship IR and b) relationship recommenda- tion rule, which will be also introduced in detail in Section IV-C.

When reaching the established consensus, the selection pro- cess, which is same to one in the TC [i.e., (6)], will be utilized to obtain the ranking of alternatives.

In the following, in Section IV-B, we set a theoretical basis of TRC, in Section IV-C, we introduce the implementation of TRC in detail, and we justify the effectiveness of TRC through the simulation experiment in Section IV-D.

B. Theoretical Basis of TRC

We set a theoretical basis of the TRC based on the results of the TRDG model presented in [20].

Let DleaderG be the set of leaders and D follower G be the set of

followers in the trust relationships G(D, E). Theorem 1: Let Pk,t = (pk,tij )n×n(k = 1, 2, . . . , m) and pc,tij =∑m k=1 πk ·pk,tij , where pk,tij is the preference over pairwise (xi, xj)

at round t and πk ≥ 0 is the weight associated with individual dk. Let CD

t = ∑mk=1 πkCDt(dk)/m be the group consensus degree, where CDt(dk) =

∑n−1 i=1

∑n j=i+1(1 − |pk,tij − pc,tij |)/(n ·

(n − 1)/2). When the preference value for pk,tij evolves as (11), i.e., pk,t+1ij = βkpk,tij +

∑m l=1,l �=k ωkl × pl,tij , limt→+∞ CDt = 1

if and only if the set of leaders is nonempty, i.e., DleaderG �= ∅. Proof: The proof of Theorem 1 is provided in Appendix A. Theorem 1 indicates that a full consensus among the indi-

viduals can be reached in the TRGDM with the TRDG model evolving under the condition that there are leaders in the trust relationship G(D, E), and also shows that the individuals can- not reach a full consensus if there is no leader in the trust relationship.

Therefore, we can improve trust relationships to create lead- ers to facilitate a consensus in the TRGDM problems which includes a two-step procedure.

1) Trust Relationships Partition: Based on Theorem 1, a consensus among the individuals in the trust rela- tionship G(D, E) cannot be formed in the TRDG model when DleaderG = ∅. When DleaderG = ∅, a network partition algorithm [20] with the time com- plexity O(n3) is proposed to divide the trust rela- tionship G(D, E) into subtrust relationships, M = {G(1)(D(1), E(1)), . . . , G(r)(D(r), E(r))}, which present three properties as follows.

a) Completeness: D = ⋃ri=1 D(i). This means that all individuals in G(D, E) are allocated to its subtrust relationships.

b) Leadership: The individuals in each subtrust rela- tionship G(τ )(D(τ ), E(τ )) ∈ M can reach a con- sensus, respectively, i.e., let Dleader

G(τ ) be the set of

leaders in G(τ )(D(τ ), E(τ )), and Dleader G(τ )

�= ∅. c) Disjointness: Let G(τ )(D(τ ), E(τ )) and

G(s)(D(s), E(s)) be two subtrust relationships in the set M = {G(1)(D(1), E(2)), . . . , G(r)(D(r), E(r))}, and let G(τ,s)(D(τ,s), E(τ,s)) be the union of the two subtrust relationships, where D(τ,s) = D(τ ) ∪ D(s) and E(τ,s) = {(di, dj)|(di, dj) ∈ E; di, dj ∈ D(τ,s)}. Furthermore, the individuals in G(τ,s) cannot reach a consensus, i.e., let Dleader

G(τ,s) be the set of leaders

in G(τ,s) and Dleader G(τ,s)

= ∅. 2) Adding Edges: According to the properties of the sub-

trust relationships presented above, we can add some edges to create leaders in the group. Adding an edge between two individuals means that trust between them is established. Specifically, in the TRGDM, Theorem 2 is used to add edges in the trust relationship G(D, E) when DleaderG = ∅.

Theorem 2: Let G(τ )(D(τ ), E(τ )) and G(s)(D(s), E(s)) be two subtrust relationships in M = {G(1)(D(1), E(1)) , . . . , G(r)(D(r), E(r))} generated by the network partition algo- rithm [20], and let G(τ,s)(D(τ,s), E(τ,s)) be the union of the

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Fig. 7. Illustration for the process of adding edges.

two subtrust relationships G(τ )(D(τ ), E(τ )) and G(s)(D(s), E(s)). Then, adding an edge from the leader dk ∈ DleaderG(τ ) to another leader dl ∈ DleaderG(s) produces a new graph of trust relationships Ḡ(τ,s)(D(τ,s), Ē(τ,s)), where Ē(τ,s) = (dk, dl)

⋃ E(τ,s). Thus,

Dleader Ḡ(τ,s)

�= ∅ and Dleader Ḡ(τ,s)

= Dleader G(s)

. Proof: The proof of Theorem 2 is provided in Appendix A. We use Example 2 to illustrate the network partition

algorithm [20] and the process of adding edges. Example 2: There is no leader in the trust relationship

G(D, E) described in Fig. 7, and we divide G(D, E) into two subtrust relationships G(1)(D(1), E(2)) and G(2)(D(2), E(2)) using the network partition algorithm [20], where D(1) = {d1, d2, d3} and D(2) = {d4, d5, d6}. Then, these three properties are satisfied: 1) D = D(1) ∪ D(1); 2) d2, d3 are leaders in G(1), and d5, d6 are leaders in G

(1); and 3) DleaderG = ∅. According to Theorem 2, we can add edges to obtain leaders

in the trust relationship G(D, E). For example, individuals d5 and d6 will be the new generated leaders when adding the edge (d3, d6) into G(D, E) in Fig. 7.

C. Implementation of TRC

In this section, we introduce the implementation of the TRC in detail. We propose two approaches (i.e., leader-based preference adjustment and trust relationships improvement) to facilitate consensus reaching, and they are described as follows.

1) Leader-Based Preference Adjustment: We identify the leader(s) in each subtrust relationship in M = {G(1) (D(1), E(1)), . . . , G(r)(D(r), E(r))}, based on the use of the network partition algorithm, and let Dleader

G(τ ) be the set of lead-

ers in G(τ )(D(τ ), E(τ ))(τ ∈ {1, . . . , r}). Then, let LA be the set of leaders with acceptable consensus degrees and let LU be the set of leaders with unacceptable consensus degrees, i.e.,

LA = {dk|dk ∈ DleaderG(τ ) andCD(dk) ≥ μ;k ∈ {1, . . . , m} τ ∈ {1, . . . , r}} (13)

LU = {dk|dk ∈ DleaderG(τ ) andCD(dk) < μ;k ∈ {1, . . . , m} τ ∈ {1, . . . , r}}. (14)

Compared with the preference adjustment rules in the TC, the leader-based preference adjustment centers on the prefer- ence adjustments of leaders to promote a consensus, and it includes the following two rules.

1) Leader IR (LIR): The LIR identifies the leader(s) dk ∈ Dleader

G(τ ) (τ ∈ {1, . . . , r}) whose consensus degree

CD(dk) < μ, i.e., dk ∈ LU [see (14)].

2) Leader DR (LDR): The LDR provides the adjustment suggestions to the identified leader(s) dk ∈ LU , sug- gesting their adjusted preference p̄kij(i, j = 1, 2, . . . , n) to be closer to pcij, i.e., if dk ∈ LU , then p̄kij ∈ [ min(pkij, p

c ij), max(p

k ij, p

c ij)](i ≥ j) and p̄kij = 1 −

p̄kji(i < j). 2) Trust Relationships Improvement: Based on Theorems 1

and 2, we propose the trust relationships improvements to help the group pursue a consensus, and it includes the following two rules.

1) Relationship IR (RIR): The RIR is used to identify the potential trust relationships that can promote the group to reach a consensus. As described in Theorem 2, adding edges between leaders in two different subtrust relation- ships, identified by the network partition algorithm, can produce leaders in the union of the two subtrust relation- ships. Therefore, the leaders dk ∈ LU are recommended to trust other leaders dl ∈ LA in other subtrust relation- ships. Then, the potential trust relationships identified by the RIR can be defined by

H = { (dk, dl)|dk ∈ DleaderG(τ ) , dl ∈ DleaderG(s) ∀G(τ )

× G(s) ∈ M, τ �= s }

∩ { (dk, dl)|dk ∈ LU , dl ∈ LA

} . (15)

2) Relationship Recommendation Rule (RRR): The RRR is used to recommend the edges which are easier to add. Generally, people will be more willing to accept these preferences which are similar to their own preferences [41]. Therefore, the RRR recommends adding those edges which are not only identified by RIR but also connect the individuals with the largest preference similarity between them.

Then, the RRR can be described as follows. Let

sdkl = ∑n−1

i=1 ∑n

j=i+1 1 − |pkij − plij| n(n − 1)/2 (16)

be the similarity degree between the individuals dk and dl. Then, the larger value of sdkl ∈ [0, 1] indicates larger similar- ity between dk and dl. If sd

kl = maxy{sdky|(dk, dy) ∈ H}, we may recommend the leader dk ∈ LU to trust the leader dl ∈ LA. Thus, the trust relationships, recommended by the RRR, can be defined by

R = {

(dk, dl)|dk ∈ LU , dl ∈ {

dz|sdkz = max y

{ sdky|(dk, dy) ∈ H

}}}

.

(17)

Following the framework of TRC, the leader-based prefer- ence adjustment, and the trust relationships improvement, the TRC model is formally presented in Algorithm 1. In general, the probability of enjoying the maximum similarity is very low at the same time. If this happens, any pair of trust relationships can be recommended to the leaders.

D. Simulation Experiment I

In this section, we justify the effectiveness of the TRC in promoting consensus through Simulation Experiment I. In

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6311

Algorithm 1 TRC

Input: The individual preferences {P1, P2, . . . , Pm}, the graph of the trust relationships G(V, E), the weights of individuals {π1, π2, . . . , πm}, the established maximum round T , and the consensus threshold μ.

Output: The ranking of alternatives. Step 1: Let t = 0. Let Pk,t = (pk,tij )n×n = (pkij)n×n(k =

1, 2, . . . , m) and Gt(D, Et) = G(D, E). Step 2: Aggregate the preferences {P1,t, P2,t, . . . , Pm,t} to obtain

Pc,t = (pc,tij )n×n at round t based on Eq. (1), i.e.,

pc,tij = m∑

k=1 πk · pk,tij .

Step 3: Based on Eq. (3), we compute the consensus degree of the group

CDt = m∑

k=1 πk CD

t (dk), where

CDt(dk) = n−1∑

i=1

n∑

j=i+1 (1 − |pk,tij − p

c,t ij |)/(n · (n − 1)/2).

If CDt ≥ μ or t ≥ T , go to Step 7; otherwise go to the next step.

Step 4: Let Mt = {G(1,t)(D(1,t), E(1,t)), . . . , G(r,t)(D(r,t), E(r,t))} be the set of subgraphs identified by Network Partition Algorithm, and let Dleader

G(τ ) be the set of leaders in

G(τ,t)(D(τ,t), E(τ,t)) (τ ∈ {1, . . . , r}). Classify the leaders in G(τ,t)(D(τ,t), E(τ,t)) into two groups:

LA,t = {dk |dk ∈ DleaderG(τ,t) and CDt (dk ) ≥ μ; k ∈ {1, . . . , m}, τ ∈ {1, . . . , r}} LU,t = {dk |dk ∈ DleaderG(τ,t) and CDt (dk ) < μ; k ∈ {1, . . . , m}, τ ∈ {1, . . . , r}}.

Step 5: (a) For the leaders dk ∈ LU,t , we sug- gest the adjusted preferences are as pk,t+1ij ∈ [ min(pk,tij , p

c,t ij ), max(p

k,t ij , p

c,t ij )](i ≥ j) and

pk,t+1ij = 1 − p k,t+1 ji (i < j);

(b) Identify the potential trust relationships,

Ht = {(dk , dl)|dk ∈ DleaderG(τ,t) , dl ∈ DleaderG(s,t) ∀G(τ,t), G(s,t) ∈ M, τ �= s} ∩{(dk , dl)|dk ∈ LU,t , dl ∈ LA,t },

and recommend the leader dk ∈ LU,t to trust another leader dl ∈ LA,t , where sdkl,t = maxy {sd

ky,t|(dk, dy) ∈ Ht} and

sdky,t = n−1∑

i=1

n∑

j=i+1 (1 − |pk,tij − p

y,t ij |)/(n(n − 1)/2).

Step 6: Let Gt+1(D, Et+1) be the adjusted trust relationships and t = t + 1, then go to Step 2.

Step 7: Let Qi = ∑n

j=1 p c,t ij /n. Then, output the ranking of

alternatives derived from the evaluation values Qi.

Simulation Experiment I, we randomly generate the prefer- ences of individuals, their self-confidence degrees, and the trust relationships among them. Then, the leader-based pref- erence adjustments and trust relationships improvements are employed to guide the group to build a consensus.

The settings in Simulation Experiment I are as follows.

1) Generation of Trust Relationships: The trust relation- ships are the randomly generated Erdos-Rényi (ER) random graphs [29], and the parameter b is the possibility of gen- erating an edge between two individuals in the ER random graphs.

2) Implementation of Preference Adjustment: The prefer- ence adjustments are set as the automatic adjustment processes described as follows.

Suppose that the possibility of the individuals accepting the preference adjustment suggestions generated by the leader- based preference adjustment is ρ in each round. Therefore, we consider two cases.

1) During the CRP, the individuals who accepted the sug- gestions will not only be influenced by the suggestions but also by the individuals they trust. Let kl (k, l = 1, 2, . . . , m; k �= l) be the weight dk assigns to dl. If akl = 1, kl ∈ (0, 1 − βk]; otherwise, kl = 0. Let k(m+1) be the weight dk assigns to the suggestion Pc,t = (pc,tij )n×n. In this simulation, these weights sat- isfy that k(m+1) +

∑m l=1,l �=k kl = 1 − βk. Thus, the

automatic adjustment process of preference is set as {

pk,t+1ij = βk pk,tij + ∑m

l=1,l �=k kl × pl,tij + k(m+1)pc,tij , i ≥ j pk,t+1ij = 1−pk,t+1ji , i < j.

(18)

2) The other individuals will only be influenced by the indi- viduals they trust. Let ωkl be the weight dk assigns to dl, where

∑m l=1,l �=k ωkl = 1 − βk(k, l = 1, 2, . . . , m).

Moreover, ωkl ∈ (0, βk] if akl = 1 and ωkl = 0 if akl = 0. Then, the automatic adjustment process of preference is set as

{ pk,t+1ij = βk pk,tij +

∑m l=1,l �=k ωkl × pl,tij , i ≥ j

pk,t+1ij = 1 − pk,t+1ji , i < j. (19)

3) Implementation of Trust Relationships Improvement: Suppose that the possibility of the individuals accepting the recommendations from the trust relationships improvement is η in each round. If the individual dk accepts the recom- mendation to trust another individual dl, we reset akl = 1.

In Simulation Experiment I, two indicators of investigating the effectiveness of the TRC are involved:

1) the consensus degree of the group CDt at round t; 2) the number of suggested (or identified) individuals αt at

round t. In a CRP, it is a desired property that consensus degree of

the group CDt can be improved quickly, and the numbers of suggested (or identified) individuals αt should be small.

Moreover, to compare the TRC with the TC, we present a modified version of Simulation Experiment I, call Simulation Experiment I′. Simulation Experiments I and I′ are both included in Appendix B.

In Simulation Experiments I and I′, we set n = 4, πk = 1/m(k = 1, 2, . . . , m), T = 10, b = 0.025, and μ = 0.85. Then, we set different values for m, ρ, and η, and run 1000 times to obtain the average values for CDt

and αt, respectively. The experimental results are presented in Figs. 8 and 9.

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Fig. 8. Average CDt values under different m, ρ, and η values. (a) m = 20, ρ = 0.1. (b) m = 20, ρ = 0.2. (c) m = 20, ρ = 0.3. (d) m = 30, ρ = 0.1. (e) m = 30, ρ = 0.2. (f) m = 30, ρ = 0.3.

Fig. 9. Average αt values under different m, ρ, and η values. (a) m = 20, ρ = 0.1. (b) m = 20, ρ = 0.2. (c) m = 20, ρ = 0.3. (d) m = 30, ρ = 0.1. (e) m = 30, ρ = 0.2. (f) m = 30, ρ = 0.3.

From Figs. 8 and 9, we obtain the following observations. 1) When setting different values for m, ρ, and η, the CDt

values in the TRC are greater than those of the TC, and the αt values in the TRC are smaller than those of the TC. Therefore, compared with the TC, the TRC can reach a consensus more efficiently.

2) As the η value increases, the average CDt values increase and the average αt values decrease. This means that the more trust relationships are improved, the faster the con- sensus is reached and the fewer individuals are suggested to adjust.

V. TRUST RELATIONSHIP MANIPULATION

There are lots of relationship manipulation phenomena in the real-life GDM problems. However, to our knowledge, the TRM issue has not been investigated in the literature. In this section, we study the clique-based TRM, in which an indi- vidual manipulates trust relationships to assemble a clique to achieve their desired ranking in a TRGDM problem. Specifically, in Section V-A, we analyze some common clique- based manipulation strategies of trust relationships in real life,

and in Section V-B, a simulation experiment is designed to explore the effect of the clique-based TRM.

A. Clique-Based Strategic Manipulation

The existing strategic manipulation mainly studied the situ- ation in which the individuals dishonestly express preferences to pursue their desired ranking (e.g., [21], [25], [26], [56], [63], [64], [77], [79], and [80]). However, because the preference evolution with trust relationships strongly impacts the results in CRPs (see Sections III and IV), in a TRGDM problem, an individual has opportunities to manipulate trust relationships to achieve its desired ranking. In our real-world TRGDM prob- lems, there are some common strategies to manipulate trust relationships, called clique-based strategic manipulation, and they are described as follows.

1) Assemble a Clique: It is common to see these kind of behaviors in the sense that the individuals are easily teamed up with others with the same purpose in order to pursue their interests. Thus, if an individual intends to manipulate trust relationships to achieve their desired ranking, the manipulator can team up with the other individuals who have the same or similar preferences as their desired ranking to form a clique, denoted as D

clique G ⊆ D. Furthermore, the manipulator will try

to make the individuals in D clique G trust each other, i.e., add

edges in the set {(dk, dl)|dk, dl ∈ DcliqueG ; (dk, dl) /∈ E}. 2) Increase the Social Influence of the Clique: In a trust

relationship G(D, E), the social influence of an individual is defined by the in-degree centrality of the individual. In the TRGDM, the preferences of individuals will often be influ- enced by other individuals, and thus the manipulator can increase the social influence of the clique aiming at better influencing the preferences of other individuals toward the pur- pose of the manipulator. Generally, a manipulator will try to make leaders in a trust relationship or its subtrust relationships to trust the individuals, dk ∈ DcliqueG , in the clique.

3) Increase the Self-Confidence of the Clique: The self- confidence of the individual dk is described by the parameter βk in the TRDG model (see Section II-C). Increasing the self- confidence of an individual can reduce the influence of other individuals have on its own preference. Thus, the manipulator will hope to increase the self-confidence of the individuals in the clique, i.e., increase the βk value if dk ∈ DcliqueG .

Example 3: In Fig. 7, we suppose that the individuals want to select the best from four alternatives, the individ- ual d1 wants to manipulate trust relationships to achieve his desired ranking, and the individuals d2 and d3 have the same preference as the individual d1 regarding the best alternative.

1) The manipulator d1 may build a clique D clique G =

{d1, d2, d3}, and then try to make the individuals d1, d2, d3 trust each other, i.e., add edges (d2, d1), (d3, d1), and (d1, d3) into the graph of trust relationships described in Fig. 7.

2) The manipulator will hope to increase the influence of the clique D

clique G = {d1, d2, d3}. Because the leaders

in subtrust relationship G(1) are individuals d5 and d6, the manipulator will try to build trusts from {d5, d6} to

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6313

{d1, d2, d3}. For example, the manipulator may prefer to add the potential edges (d5, d2) and (d6, d3) into the trust relationship.

3) The manipulator will try to improve the self-confidence of the individuals in the clique D

clique G = {d1, d2, d3},

i.e., increase the βk values for k = 1, 2, 3.

B. Simulation Experiment II

In this section, we investigate the effect of the clique-based TRM on CRPs via Simulation Experiment II, which is used to implement the clique-based manipulation strategies presented in Section V-A at the beginning of the CRP. In Simulation Experiment II, we randomly generate the individual prefer- ences, their self-confidence degrees, and the trust relationships among them. Based on these generated data, we investigate whether the proposed clique-based manipulation strategies can achieve desired goal.

Simulation Experiment II is included in Appendix B, and the settings in Simulation Experiment II are as follows.

1) Parameter λ: Let Dleader G(τ )

be the set of leaders in G(τ )(D(τ ), E(τ )) ∈ M(τ ∈ {1, 2, . . . , r}), where M is generated by the network partition algorithm. In Simulation Experiment II, the manipulator will try to advise the leader dl ∈ DleaderG(τ ) to trust dk ∈ D

clique G

to increase the social influence of the clique, where sdlk = maxy{sdly|dl ∈ DleaderG(τ ) anddl /∈ D

clique G , dy ∈

D clique G ; τ ∈ {1, 2, . . . , r}}. And we suppose the prob-

ability that the leader dl will accept the suggestion is λ. Particularly, λ = 0 means that the manipulation strategy to increase the social influence of the clique will not be used.

2) Confidence Threshold θ : In Simulation Experiment II, when dk ∈ DcliqueG and βk < θ , the manipulator will increase the βk value to θ to increase the self-confidence of the clique. In particular, θ = 0 indicates that the manipulation strategy to increase the self-confidence of the clique will be not used.

3) Without loss of generality, the manipulator hopes that the alternative x1 is the best alternative.

In Simulation Experiment II, the success indicator SIt = 1 indicates that the best alternative at t round is alternative x1, which means that the manipulation is successful. On the other hand, SIt = 0 means that the manipulation has failed.

Let m = 20, n = 4, πk = 1/20(k = 1, 2, . . . , 20), T = 20, and ρ = 0.3. When setting μ to be close to 1 and setting different b, λ, and θ values for Simulation Experiment II, we run this simulation 1000 times to obtain the average values SIt. The average SIt value shows the success indicator of manip- ulation in round t in Simulation Experiment II. The average values for SIt are shown in Fig. 10, when setting different values for input parameters in Simulation Experiment II.

Because the preference relations are randomly and uni- formly generated in Simulation Experiment II, the average SIt value that alternative x1 is the best will be 1/4 without TRM. From Fig. 10, the following observations are found.

Fig. 10. Average success indicator SIt in Simulation Experiment II. (a) θ = 0; b = 0.05. (b) θ = 0.3; b = 0.05. (c) θ = 0.6; b = 0.05. (d) θ = 0.9; b = 0.05. (e) θ = 0; b = 0.01. (f) θ = 0.3; b = 0.01. (g) θ = 0.6; b = 0.01. (h) θ = 0.9; b = 0.01. (i) θ = 0; b = 0.02. (j) θ = 0.3; b = 0.02. (k) θ = 0.6; b = 0.02. (l) θ = 0.9; b = 0.02.

1) When only implementing the strategy of forming a clique (i.e., λ = 0 and θ = 0), the success indi- cators SIt of manipulation increase to over 45% as round t increases, which means that forming a clique can significantly increase the possibility of successful manipulation.

2) When increasing the influence of the clique (i.e., λ �= 0 and θ = 0) or increasing the self-confidence of the clique (i.e., λ = 0 and θ �= 0), the success indicator SIt of manipulation increases obviously under different b val- ues, which implies that both increasing the influence and the self-confidence of the clique can further improve the effect of manipulation.

3) When the b value increases, the effect of TRM decreases. The number of edges in the trust relationship increases when the b value increases, which indicates that it will be more difficult to conduct manipulation if there are more high-trust relationships among the individuals.

4) When round t is larger than 10, the success indica- tor tends to be stable. This is because when t is large enough, the essential law of opinion dynamics works to form a stable state [24].

All these observations show that the clique-based TRM which is ignored in the literature is an important aspect to consider when manipulating CRP results.

VI. COMPARISON: ADVANTAGES AND LIMITATIONS

In this section, we discuss the advantages and limitations of the TRC.

There are some obvious differences between the TRC and the TC. Compared with the TC, in the TRC model we improve the use of trust relationships in a TRGDM problem by con- sidering the interactions and preference evolution among the

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individuals with trust relationships, and we develop new feed- back mechanisms: 1) the leader-based preference adjustment and 2) the trust relationships improvement.

1) Advantages: The improvements are described in the following.

a) In the TC, the IR is used to identify the individuals with low consensus degrees, and the DR offers them directions of preference adjustments [9], [44], [62]. Compared with the TC, in the TRC, we develop the leader-based pref- erence adjustment rules to highlight the role of leaders in the consensus reaching.

b) In the existing CRPs in the TRGDM problems, the trust relationships are mainly used to determine the weights of individuals [2], [23], [72]. Compared with these models, the TRC improves the use of the trust relationships in the TRGDM problems. Specifically, some trust relationships improvement rules are proposed to help the group pursue a con- sensus, which increases the efficiency of consensus reaching.

c) The existing strategic manipulation studies focus on that the individuals dishonestly express prefer- ences to achieve the desired ranking in the GDM problems (e.g., [21], [25], [26], [56], [63], [64], [77], [79], and [80]). However, these studies ignore the relationship manipulation issue in the real-life GDM problems. In this article, we present a new strategic manipulation issue, called TRM, which strategically manipulates the trust relationships to assemble a clique to pursue the desired ranking in a TRGDM problem.

d) Detailed simulation experiments have been proposed to justify the TRC, showing that the TRC is more efficient than the TC in consensus reaching in TRGDM.

2) Limitations: Some limitations are found in the following. a) Large groups may appear in the TRGDM prob-

lems, where preferences clustering is often nec- essary when considering the large-scale CRP problems [54], [59], [63], [67], [85]. However, the TRC follows the paradigm of the traditional CRP, and do not study the large-scale CRP problems.

b) In this article, we follow the existing TRGDM models [2], [23], [72] and assume that the trust relationships are given or known. But it should be studied how to identify trust relationships in TRGDM.

c) In natural disasters, emergency decision making is often needed [84]. However, trust relationship rec- ommendation would be not very efficient in this situation.

VII. CONCLUSION

In this article, we developed a new TRGDM consen- sus reaching model considering the preference evolution of individuals with trust relationships, which utilizes the

leader-based preference adjustments and trust relationships improvements to facilitate consensus reaching. Furthermore, we present and investigate the TRM issue and analyze some clique-based strategies of TRM for pursuing desired ranking.

Among the main conclusions, we must point out the importance of the leaders and trust relationships improve- ments in TRGDM as they strongly influence the CRP results in the TRGDM problems. Therefore, it is fundamen- tal to analyze the use of trust relationships and the TRM in TRGDM.

Meanwhile, we argue that two future research avenues are interesting.

1) The TRM is unwelcome in the TRGDM problems. It would be an interesting future study to develop a method to detect and manage such behavior in TRGDM.

2) Emergency decision making is often needed for natural disasters [84]. It would to be interesting to study a data- driven CRP for emergency GDM with trust relation- ships.

APPENDIX A PROOFS

Proof of Theorem 1: Lemma 1 (see [20, Corollary 1]) describes that when the preferences of individuals iterate as (11) (i.e., TRDG model), all individuals in the trust rela- tionship G(D, E) will form the same preference if and only if there are leaders in G(D, E), i.e., DleaderG �= ∅.

Sufficiency: If DleaderG �= ∅, the preference pk,tij over pairwise (xi, xj) at round t associated with the individual dk evolves as (11), based on Lemma 1, for any individual dk, we have

lim t→+∞ p

k,t ij = pc,tij . (20)

Then, according to (2), for any individual dk, we have the consensus degree of dk at round t

lim t→+∞ CD

t (dk) = 1. (21)

According to (3), and thus we have the consensus degree

lim t→+∞ CD

t = 1. (22)

Necessity: If limt→+∞ CDt = 1, then we can induce that limt→+∞ CDt(dk) = 1 for any individual dk based on (3).

According to (2), for the individual dk, we have limt→+∞ pk,tij = pc,tij . And considering the preference pk,tij over pairwise (xi, xj) at round t associated with the individual dk evolves as (11), we have limt→+∞ pk,tij = pc,tij for any original value of pk,tij .

Thus, based on Lemma 1, we have DleaderG �= ∅. This completes the proof of Theorem 1. Proof of Theorem 2: Let G(τ )(D(τ ), E(τ )) and

G(s)(D(s), E(s)) be two subtrust relationships of G(D, E), and let G(τ,s)(D(τ,s), E(τ,s)) be the union of G(τ )

(D(τ ), E(τ )) and G(s)(D(s), E(s)), where E(τ,s) = {(di, dj)| (di, dj) ∈ E; di, dj ∈ D(τ,s)} and D(τ,s) = D(τ )

⋃ D(s).

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DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6315

Add a directed edge (dk, dl) from leader dk ∈ DleaderG(τ ) to leader dl ∈ DleaderG(s) to produce a new trust relationship Ḡ(τ,s)(D(τ,s), Ē(τ,s)), where Ē(τ,s) = (dk, dl)

⋃ E(τ,s).

For any individual di ∈ D(τ,s) and dj ∈ DleaderG(s) , we have the following.

1) If di ∈ D(s), we have di → dj according to Definition 6. 2) If di ∈ D(τ ), there is a path di → dk for all di ∈ D(τ )

naturally. According to the added directed edge (dk, dl), we have di → dl for all di ∈ D(τ ). Because of dl ∈ Dleader

G(s) and dj ∈ DleaderG(s) , and thus we have di → dj for

all di ∈ D(τ ). Based on the above reasons, we have di → dj for

all di ∈ D(τ,s). We use Definition 6 and consider that dj is an arbitrary individual in the set of Dleader

G(s) . Hence, we have

Dleader Ḡ(τ,s)

= Dleader G(s)

�= ∅. This completes the proof of Theorem 2.

APPENDIX B SIMULATION EXPERIMENTS

1) Simulation Experiment: Input: The number of individuals m, the number

of alternatives n, the weights of individuals {π1, π2, . . . , πm}, the established maximum round T , the consensus threshold μ, and the parameters b , ρ, and η.

Output: The consensus degree CDt and the number of suggested individuals αt at round t.

Step 1: Generate initial data: 1) generate a directed ER graph G(D, E) with probability b, and let A = (akl)m×m be the corresponding adjacent matrix; 2) generate βk uniformly and randomly from interval [0, 1]; 3) generate Pk = (pkij)n×n, where pkij is uni- formly and randomly from interval [0, 1] for i ≥ j, pkii = 0.5, and pkij = 1 − pkji for i < j;and 4) let γkl be a trust value from dk to dl. If akl = 0, then set γkl = 0; otherwise, generate γkl uniformly and ran- domly from interval (0, 1]. Let γk(m+1) ∈ (0, 1] be the trust value dk assigns to the suggestions. Then, we can obtain

kl = (γkl · (1 − βk)) / m+1∑

y=1,y �=k γky and

ωkl = (γkl · (1 − βk)) / m∑

y=1,y �=k γky.

Step 2: Let t = 0, αt = 0, Pk,t = Pk, Gt(D, Et) = G(D, E) and At = A.

Step 3: Same to step 2 to compute Pc,t = (pc,tij )n×n in Algorithm 1.

Step 4: Compute CDt same as step 3 in Algorithm 1. If t = T , go to step 7; otherwise, go to the next step.

Step 5: First, same to step 4 in Algorithm 1. Then, let sk,tbe a 0-1 variable. If dk ∈ LU,t, then sk,t = 1; otherwise, sk,t = 0. Then, obtain the number of suggested individuals αt = ∑mk=1 sk,t.

Step 6: Let dk ∈ LU,t accept the preference adjustment and trust relationships improvement with probability ρ and η, respectively. Then:

1) if dk ∈ LU,t and they accept the preference adjustment, compute the evolved preference pk,t+1ij based on (18), i.e., p

k,t+1 ij = βk pk,tij +∑m

l=1,l �=k kl × pl,tij + k(m+1)pc,tij (i ≥ j) and pk,t+1ij = 1 − pk,t+1ji (i < j); otherwise, com- pute pk,t+1ij using (19), i.e., p

k,t+1 ij = βk pk,tij +∑m

l=1,l �=k ωkl × pl,tij (i ≥ j) and pk,t+1ij = 1 − pk,t+1ji (i < j);

2) if dk ∈ LU,t and they accept to trust dl ∈ LA,t, where

sdkl,t = max y

{sdky,t|(dk, dy) ∈ Ht}

sdky,t = n−1∑

i=1

n∑

j=i+1 (1 − |pk,tij − py,tij |)/(n(n − 1)/2)

and

Ht = {(dk, dy)|dk ∈ DleaderG(τ,t) , dy ∈ DleaderG(s,t) ∀G(τ,t), G(s,t), τ �= s}

∩{(dk, dy)|dk ∈ LU,t, dy ∈ LA,t} then we adopt the following two-step pro- cedure: a) reset atkl = 1 and b) randomly regenerate the γkl from interval (0, 1], and reset the weights

ky = (γky · (1 − βk)) / m+1∑

z=1,z�=k γkz and

ωky = (γky · (1 − βk)) / m∑

z=1,z�=k γkz.

Let Gt+1(D, Et+1) be the adjusted trust rela- tionships and At+1 be the corresponding adja- cent matrix. Let t = t + 1, then go to step 3.

Step 7: Output CDt and αt(t = 1, 2, . . . , T). 2) Simulation Experiment I′: We replace steps 5 and 6 in

Simulation Experiment I with steps 5′ and 6′, respectively, to obtain Simulation Experiment I′, which is used to analyze the effectiveness of TC. Steps 5′ and 6′ are presented below.

Step 5′: Based on (4), identify the individuals dk ∈ It, where It = {dk|CD(dk) < μ; k ∈ {1, . . . , m}}. Let sk,tbe a 0-1 variable. If dk ∈ It, then sk,t = 1; otherwise, sk,t = 0. Then, obtain the number of suggested individuals αt = ∑mk=1 sk,t.

Step 6′: Let dk ∈ It accept the preference adjustment with probability ρ. If dk ∈ It and they accept the pref- erence adjustment, compute the evolved preference pk,t+1ij same as step 6(a) in Simulation Experiment I. Let t = t + 1, then go to step 3.

3) Simulation Experiment II: Input: The number of individuals m, the number

of alternatives n, the weights of individuals

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6316 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 51, NO. 10, OCTOBER 2021

{π1, π2, . . . , πm}, the established maximum round T , the consensus threshold μ, and the parameters b, ρ, and λ, and the confidence threshold θ .

Output: The success indicator of manipulation SIt

(t = 0, 1, . . . , T) at round t. Step 1: Same to step 1 in Simulation Experiment I to

generate initial data. Step 2: Let t = 0 and Pk,t = Pk. Step 3: Let the individuals whose best alternative is x1

form a clique, D clique G ⊂ D, and then add the

edges {(di, dj)|di, dj ∈ DcliqueG ; (di, dj) /∈ E} into the trust relationship G(D, E) to produce a new trust relationship Ḡ(D, Ē).

Step 4: Let M = {Ḡ(1)(D(1), Ē(1)), . . . , Ḡ(r)(D(r), Ē(r))} be the subtrust relationships generated by the network partition algorithm, and Dleader

Ḡ(τ ) be the set of leaders

in Ḡ(τ )(D(τ ), Ē(τ )). Let the leader dl ∈ DleaderḠ(τ ) trust the individual dk ∈ DcliqueG with the probability λ, where

sdlk = max y

{sdly|dl ∈ DleaderG(τ ) anddl /∈ D clique G

dy ∈ DcliqueG ; τ ∈ {1, . . . , r}}. Let ¯̄G(D, ¯̄E) be the new trust relationship.

Step 5: If dk ∈ DcliqueG and βk < θ , then increase βk to θ . Step 6: Obtain the collective preference Pc,t = (pc,tij )n×n

same as step 2 in Algorithm 1. Based on (6), we calculate the evaluation values Qi =

∑n j=1 p

c,t ij /n

(i = 1, 2, . . . , n) from Pc,t to obtain the rank- ing of the alternatives. If alternative x1 is the best alternative, let SIt = 1; otherwise, SIt = 0.

Step 7: Based on (4), identify the individuals dk ∈ It, where It = {dk|CDt(dk) < μ; k ∈ {1, . . . , m}}.

Step 8: First, compute the evolved preference pk,t+1ij same as step 6′ in Simulation Experiment I′. Then, if t = T , then go to step 9; otherwise, let t = t + 1 and go to step 6.

Step 9: Output SIt(t = 0, 1, . . . , T).

REFERENCES

[1] J. Aguarón, M. T. Escobar, and J. M. Moreno-Jiménez, “The precise consistency consensus matrix in a local AHP-group decision making context,” Ann. Oper. Res., vol. 245, pp. 245–259, Oct. 2016.

[2] S. Alonso, I. J. Pérez, F. J. Cabrerizo, and E. Herrera-Viedma, “A lin- guistic consensus model for Web 2.0 communities,” Appl. Soft Comput., vol. 13, no. 1, pp. 149–157, 2013.

[3] A. Altuzarra, J. M. Moreno-Jiménez, and M. Salvador, “A Bayesian priorization procedure for AHP-group decision making,” Eur. J. Oper. Res., vol. 182, no. 1, pp. 367–382, 2007.

[4] A. Altuzarra, J. M. Moreno-Jiménez, and M. Salvador, “Consensus building in AHP-group decision making: A Bayesian approach,” Oper. Res., vol. 58, no. 6, pp. 1755–1773, 2010.

[5] D. Artz and Y. Gil, “A survey of trust in computer science and the semantic Web,” Web Semantic, vol. 5, no. 2, pp. 58–71, 2007.

[6] D. Ben-Arieh, T. Easton, and B. Evans, “Minimum cost consensus with quadratic cost functions,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 39, no. 1, pp. 210–217, Jan. 2009.

[7] R. L. Berger, “A necessary and sufficient condition for reaching a con- sensus using Degroot’s model, J. Amer. Stat. Assoc., vol. 76, no. 374, pp. 415–418, 1981.

[8] J. A. Bondy and U. S. R. Murty, Graph Theory With Applications. New York, NY, USA: Macmillan Press, 1976.

[9] F. J. Cabrerizo, J. M. Moreno, I. J. Pérez, and E. Herrera-Viedma, “Analyzing consensus approaches in fuzzy group decision making: Advantages and drawbacks,” Soft Comput., vol. 14, no. 5, pp. 451–463, 2010.

[10] N. Capuano, F. Chiclana, H. Fujita, E. Herrera-Viedma, and V. Loia, “Fuzzy group decision making with incomplete information guided by social influence,” IEEE Trans. Fuzzy Syst., vol. 26, no. 3, pp. 1704–1718, Jun. 2018.

[11] D. Cheng, Z. L. Zhou, F. X. Cheng, Y. F. Zhou, and Y. J. Xie, “Modeling the minimum cost consensus problem in an asymmetric costs context,” Eur. J. Oper. Res, vol. 270, no. 3, pp. 1122–1137, 2018.

[12] X. Chen, H. J. Zhang, and Y. C. Dong, “The fusion process with het- erogeneous preference structures in group decision making: A survey,” Inf. Fusion, vol. 24, pp. 72–83, Jul. 2015.

[13] F. Chiclana, J. M. T. Garcia, M. J. Del Moral, and E. Herrera-Viedma, “A statistical comparative study of different similarity measures of con- sensus in group decision making,” Inf. Sci., vol. 221, pp. 110–123, Feb. 2013.

[14] K. A. Choudhury, R. Shankar, and M. K. Tiwari, “Consensus-based intelligent group decision-making model for the selection of advanced technology,” Decis. Support Syst., vol. 42, no. 3, pp. 1776–1799, 2006.

[15] W. D. Cook, “Distance-based and ad hoc consensus models in ordinal preference ranking,” Eur. J. Oper. Res., vol. 172, no. 2, pp. 369–385, 2006.

[16] W. D. Cook and M. Kress, “Ordinal ranking with intensity of prefer- ence,” Manag. Sci., vol. 31, no. 1, pp. 26–32, 1985.

[17] M. H. Degroot, “Reaching a consensus,” J. Artif. Soc. Soc. Simulat., vol. 69, no. 345, pp. 118–121, 1974.

[18] J. C. Dittmer, “Consensus formation under bounded confidence,” Nonlin. Anal., vol. 47, no. 7, pp. 4615–4621, 2001.

[19] Q. X. Dong and O. Cooper, “A peer-to-peer dynamic adaptive consensus reaching model for the group AHP decision making,” Eur. J. Oper. Res., vol. 250, no. 2, pp. 521–530, 2016.

[20] Y. C. Dong, Z. G. Ding, L. Martínez, and F. Herrera, “Managing consen- sus based on leadership in opinion dynamics,” Inf. Sci., vols. 397–398, pp. 187–205, Aug. 2017.

[21] Y. C. Dong, Y. T. Liu, H. M. Liang, F. Chiclana, and E. Herrera-Viedma, “Strategic weight manipulation in multiple attribute decision making,” Omega, vol. 75, pp. 154–164, Mar. 2018.

[22] Y. C. Dong, Y. F. Xu, H. Y. Li, and B. Feng, “The OWA-based consensus operator under linguistic representation models using position indexes,” Eur. J. Oper. Res., vol. 203, no. 2, pp. 455–463, 2010.

[23] Y. C. Dong et al., “Consensus reaching in social network group deci- sion making: Research paradigms and challenges,” Knowl. Based Syst., vol. 162, pp. 3–13, Dec. 2018.

[24] Y. C. Dong, M. Zhan, G. Kou, Z. G. Ding, and H. M. Liang, “A survey on the fusion process in opinion dynamics,” Inf. Fusion, vol. 43, pp. 57–65, Sep. 2018.

[25] Y. C. Dong, H. J. Zhang, and E. Herrera-Viedma, “Integrating experts’ weights generated dynamically into the consensus reaching process and its applications in managing non-cooperative behaviors,” Decis. Support Syst., vol. 84, pp. 1–15, Apr. 2016.

[26] Y. C. Dong, S. H. Zhao, H. J. Zhang, F. Chiclana, and E. Herrera-Viedma, “A self-management mechanism for non-cooperative behaviors in large-scale group consensus reaching processes,” IEEE Trans. Fuzzy Syst., vol. 26, no. 6, pp. 3276–3288, Dec. 2018.

[27] R. Durrett et al., “Graph fission in an evolving voter model,” Proc. Nat. Acad. Sci. USA, vol. 109, no. 10, pp. 3682–3687, 2012.

[28] Y.-H. Eom and D. L. Shepelyansky, “Opinion formation driven by PageRank node influence on directed networks,” Physica A Stat. Mech. Appl., vol. 436, pp. 707–715, Oct. 2015.

[29] P. Erdõs and A. Rényi, “On the evolution of random graphs,” Publ. Math. Inst. Hungarian Acad. Sci., vol. 5, no. 1, pp. 17–60, 1960.

[30] M. T. Escobar, J. Aguarón, and J. M. Moreno-Jiménez, “Some exten- sions of the precise consistency consensus matrix,” Decis. Support Syst., vol. 74, pp. 67–77, Jun. 2015.

[31] Z. P. Fan, J. Ma, Y. P. Jiang, Y. H. Sun, and L. Ma, “A goal programming approach to group decision making based on multiplicative preference relations and fuzzy preference relations,” Eur. J. Oper. Res., vol. 174, no. 1, pp. 311–321, 2006.

[32] M. Fedrizzi, M. Fedrizzi, and R. A. Pereira, “Soft consensus and network dynamics in group decision making,” Int. J. Intell. Syst., vol. 14, no. 1, pp. 63–77, 1999.

[33] M. Fedrizzi, J. Kacprzyk, and S. Zadroz̀ny, “An interactive multi-user decision support system for consensus reaching processes using fuzzy logic with linguistic quantifiers,” Decis. Support Syst., vol. 4, no. 3, pp. 313–327, 1988.

Authorized licensed use limited to: UNIV OF DERBY. Downloaded on May 16,2022 at 07:34:11 UTC from IEEE Xplore. Restrictions apply.

DONG et al.: CONSENSUS REACHING AND STRATEGIC MANIPULATION IN GDM WITH TRUST RELATIONSHIPS 6317

[34] J. R. P. French, “A formal theory of social power,” Psychol. Rev., vol. 63, no. 3, pp. 181–194, 1956.

[35] N. E. Friedkin and E. C. Johnsen, “Social influence and opinions,” J. Math. Soc., vol. 15, nos. 3–4, pp. 193–206, 1990.

[36] N. E. Friedkin and E. C. Johnsen, “Social influence networks and opinion change,” Adv. Group Process., vol. 16, no. 1, pp. 1–29, 1999.

[37] Z.-W. Gong, J. Forrest, and Y.-J. Yang, “The optimal group consensus models for 2-tuple linguistic preference relations,” Knowl. Based Syst., vol. 37, pp. 427–437, Jan. 2013.

[38] Z. W. Gong, H. H. Zhang, J. Forrest, L. Li, and X. Xu, “Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual,” Eur. J. Oper. Res., vol. 240, no. 1, pp. 183–192, 2015.

[39] S. Greco, M. Kadziñski, V. Mousseau, and R. Sñski, “Robust ordi- nal regression for multiple criteria group decision: UTAGMS -GROUP and UTADISGMS -GROUP,” Decis. Support Syst., vol. 52, no. 3, pp. 549–561, 2012.

[40] M. Gupta, “Consensus building process in group decision making—An adaptive procedure based on group dynamics,” IEEE Trans. Fuzzy Syst., vol. 26, no. 4, pp. 1923–1933, Apr. 2018.

[41] R. Hegselmann and U. Krause, “Opinion dynamics and bounded confi- dence models, analysis, and simulation,” J. Artif. Soc. Soc. Simul., vol. 5, no. 3, pp. 1–33, 2002.

[42] F. Herrera and E. Herrera-Viedma, “Choice functions and mechanisms for linguistic preference relations,” Eur. J. Oper. Res., vol. 120, no. 1, pp. 144–161, 2000.

[43] E. Herrera-Viedma, S. Alonso, F. Chiclana, and F. Herrera, “A consen- sus model for group decision making with incomplete fuzzy preference relations,” IEEE Trans. Fuzzy Syst., vol. 15, no. 5, pp. 863–877, Oct. 2007.

[44] E. Herrera-Viedma, F. J. Cabrerizo, J. Kacprzyk, and W. Pedrycz, “A review of soft consensus models in a fuzzy environment,” Inf. Fusion, vol. 17, pp. 4–13, May 2014.

[45] E. Herrera-Viedma, L. Martinez, F. Mata, and F. Chiclana, “A consen- sus support system model for group decision-making problems with multigranular linguistic preference relations,” IEEE Trans. Fuzzy Syst., vol. 13, no. 5, pp. 644–658, Oct. 2005.

[46] E. Herrera-Viedma, F. Herrera, and F. Chiclana, “A consensus model for multiperson decision making with different preference structures,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 32, no. 3, pp. 394–402, May 2002.

[47] D. S. Hochbaum and A. Levin, “Methodologies and algorithms for group-rankings decision,” Manag. Sci., vol. 52, no. 9, pp. 1394–1408, 2006.

[48] R. A. Horn and C. R. Johnson, Matrix Analysis, 2th ed. Cambridge, U.K.: Cambridge Univ. Press, 1994.

[49] J. Kacprzyk and M. Fedrizzi, “A ‘soft’ measure of consensus in the setting of partial (fuzzy) preferences,” Eur. J. Oper. Res., vol. 34, no. 3, pp. 316–325, 1988.

[50] J. Kacprzyk, M. Fedrizzi, and H. Nurmi, “Group decision making and consensus under fuzzy preferences and fuzzy majority,” Fuzzy Sets Syst., vol. 49, no. 1, pp. 21–31, 1992.

[51] J. Kacprzyk, M. Fedrizzi, and H. Nurmi, “Soft degrees of consensus under additive preferences and fuzzy majorities,” in Consensus Under Fuzziness, J. Kacprzyk, H. Nurmi, and M. Fedrizzi, Eds. Boston, MA, USA: Kluwer, 1997, pp. 55–81.

[52] J. Kacprzyk and S. Zadroz̀ny, “Soft computing and Web intelligence for supporting consensus reaching,” Soft Comput., vol. 14, no. 8, pp. 833–846, 2010.

[53] P. L. Krapivsky and S. Redner, “Dynamics of majority rule in two-state interacting spin systems,” Phys. Rev. Lett., vol. 90, no. 23, 2003, Art. no. 238701.

[54] Á. Labella, Y. Liu, R. M. Rodríguez, and L. Martínez, “Analyzing the performance of classical consensus models in large scale group decision making: A comparative study,” Appl. Soft Comput., vol. 67, pp. 677–690, Jun. 2018.

[55] Q. Liang, X. W. Liao, and J. P. Liu, “A social ties-based approach for group decision-making problems with incomplete additive preference relations,” Knowl. Based Syst., vol. 119, pp. 68–86, Mar. 2017.

[56] Y. T. Liu, H. J. Zhang, Y. Z. Wu, and Y. C. Dong, “Ranking range based approach to MADM under incomplete context and its application in venture investment evaluation,” Technol. Econ. Develop. Econ., vol. 25, no. 5, pp. 877–899, 2019.

[57] Y. Liu, C. Y. Liang, F. Chiclana, and J. Wu, “A trust induced recommen- dation mechanism for reaching consensus in group decision making,” Knowl. Based Syst., vol. 119, pp. 221–231, Mar. 2017.

[58] Z. H. Liu, J. F. Ma, Y. Zeng, L. Yang, Q. P. Huang, and H. L. Wu, “On the control of opinion dynamics in social networks,” Physica A Stat. Mech. Appl., vol. 409, pp. 183–198, Sep. 2014.

[59] C.-C. Li, Y. C. Dong, and F. Herrera, “A consensus model for large-scale linguistic group decision making with a feedback recommendation based on clustered personalized individual semantics and opposing consensus groups,” IEEE Trans. Fuzzy Syst., vol. 27, no. 2, pp. 221–233, Feb. 2019.

[60] J. M. Moreno-Jiménez, J. Aguarón, and M. T. Escobar, “The core of consistency in AHP-group decision making,” Group Decis. Negotiation, vol. 17, no. 3, pp. 249–265, 2008.

[61] J. M. Moreno-Jiménez, J. Aguarón, A. Raluy, and A. Turón, “A spread- sheet module for consistent AHP-consensus building,” Group Decis. Negotiation, vol. 14, no. 2, pp. 89–108, 2005.

[62] I. Palomares, F. J. Estrella, L. Martínez, and F. Herrera, “Consensus under a fuzzy context: Taxonomy, analysis framework AFRYCA and experimental case of study,” Inf. Fusion, vol. 20, pp. 252–271, Nov. 2014.

[63] I. Palomares, L. Martínez, and F. Herrera, “A consensus model to detect and manage noncooperative behaviors in large-scale group deci- sion making,” IEEE Trans. Fuzzy Syst., vol. 22, no. 3, pp. 516–530, Jun. 2014.

[64] D. A. Pelta and R. R. Yager, “Decision strategies in mediated multiagent negotiations: An optimization approach,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 635–640, May 2010.

[65] I. J. Pérez, F. J. Cabrerizo, S. Alonso, and E. Herrera-Viedma, “A new consensus model for group decision making problems with non- homogeneous experts,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 44, no. 4, pp. 494–498, Apr. 2014.

[66] I. J. Pérez, F. J. Cabrerizo, and E. Herrera-Viedma, “A mobile decision support system for dynamic group decision-making problems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 40, no. 6, pp. 1244–1256, Nov. 2010.

[67] F. J. Quesada, I. Palomares, and L. Martínez, “Managing experts? Behavior in large-scale consensus reaching processes with uninorm aggregation operators,” Appl. Soft Comput., vol. 35, pp. 873–887, Oct. 2015.

[68] J. A. Recio-García, L. Quijano, and B. Díaz-Agudo, “Including social factors in an argumentative model for group decision support systems,” Decis. Support Syst., vol. 56, pp. 48–55, Dec. 2013.

[69] K. H. Rosen, Discrete Mathematics and Its Application, 6th ed. New York, NY, USA: McGraw-Hill, 2007.

[70] R.-Y. Tian and Y.-J. Liu, “Isolation, insertion, and reconstruction: Three strategies to intervene in rumor spread based on supernetwork model,” Decis. Support Syst., vol. 67, pp. 121–130, Nov. 2014.

[71] Z. F. Tao, X. Liu, H. Y. Chen, and L. G. Zhou, “Using new version of extended t−norms and s-norms for aggregating interval linguis- tic labels,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 47, no.12, pp. 3284–3298, Dec. 2017.

[72] R. Ureña, G. Kou, Y. C. Dong, F. Chiclana, and E. Herrera-Viedma, “A review on trust propagation and opinion dynamics in social networks and group decision making frameworks,” Inf. Sci., vol. 478, pp. 461–475, Apr. 2019.

[73] S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications. Cambridge, U.K.: Cambridge Univ. Press, 1994.

[74] J. Wu, F. Chiclana, H. Fujita, and E. Herrera-Viedma, “A visual interaction consensus model for social network group decision mak- ing with trust propagation,” Knowl. Based Syst., vol. 122, pp. 39–50, Apr. 2017.

[75] J. Wu, L. F. Dai, F. Chiclana, H. Fujita, and E. Herrera-Viedma, “A min- imum adjustment cost feedback mechanism based consensus model for group decision making under social network with distributed linguistic trust,” Inf. Fusion, vol. 41, pp. 232–242, May 2018.

[76] Z. B. Wu and J. P. Xu, “Managing consistency and consensus in group decision making with hesitant fuzzy linguistic preference relations,” Omega, vol. 65, pp. 28–40, Dec. 2016.

[77] X.-H. Xu, Z.-J. Du, and X.-H. Chen, “Consensus model for multi- criteria large-group emergency decision making considering non- cooperative behaviors and minority opinions,” Decis. Support Syst., vol. 79, pp. 150–160, Nov. 2015.

[78] R. R. Yager, “On ordered weighted averaging aggregation operators in multicriteria decision making,” IEEE Trans. Syst., Man, Cybern., Syst., vol. 18, no. 1, pp. 183–190, Jan./Feb. 1988.

[79] R. R. Yager, “Penalizing strategic preference manipulation in multi-agent decision making,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 393–403, Jun. 2001.

Authorized licensed use limited to: UNIV OF DERBY. Downloaded on May 16,2022 at 07:34:11 UTC from IEEE Xplore. Restrictions apply.

6318 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 51, NO. 10, OCTOBER 2021

[80] R. R. Yager, “Defending against strategic manipulation in uninorm- based multi-agent decision making,” Eur. J. Oper. Res., vol. 141, no. 1, pp. 217–232, 2002.

[81] B. W. Zhang, Y. C. Dong, and E. Herrera-Viedma, “Group decision making with heterogeneous preference structures: An automatic mecha- nism to support consensus reaching,” Group Decis. Negotiation, vol. 28, no. 3, pp. 585–617, 2019.

[82] H. J. Zhang, Y. C. Dong, and X. Chen, “The 2-rank consensus reaching model in the multigranular linguistic multiple-attribute group decision- making,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 48, no. 12, pp. 2080–2094, Dec. 2018.

[83] H. J. Zhang, Y. C. Dong, F. Chiclana, and S. Yu, “Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design,” Eur. J. Oper. Res., vol. 275, no. 2, pp. 580–598, 2019.

[84] L. Zhou, X. H. Wu, Z. S. Xu, and H. Fujita, “Emergency decision mak- ing for natural disasters: An overview,” Int. J. Disaster Risk Reduction, vol. 27, pp. 567–576, Mar. 2018.

[85] Z. Zhang, C. H. Guo, and L. Martínez, “Managing multi-granular linguistic distribution assessments in large-scale multi-attribute group decision making,” IEEE Trans. Syst., Man, Cybern. Syst., vol. 47, no. 11, pp. 3063–3076, Nov. 2017.

Yucheng Dong received the B.S. and M.S. degrees in mathematics from Chongqing University, Chongqing, China, in 2002 and 2004, respec- tively, and the Ph.D. degree in management from Xi’an Jiaotong University, Xi’an, China, in 2008.

He is currently a Professor with the Business School, Sichuan University, Chengdu, China. His current research interests include decision analy- sis, human dynamics, big data analytics, social network, and risk analysis. He has published over

100 international journal papers in Decision Support Systems, the European Journal of Operational Research, IEEE TRANSACTIONS ON BIG DATA, IEEE TRANSACTIONS ON CYBERNETICS, IEEE TRANSACTIONS ON FUZZY SYSTEMS, IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS, IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, IEEE TRANSACTIONS ON RELIABILITY, Omega, and Scientific Data.

Prof. Dong was a recipient of the Highly Cited Researcher by Clarivate Analytics in the field of computer science. He is an Editorial Board Member of Information Fusion, and an Area Editor/Associate Editor of Computers and Industrial Engineering, Group Decision and Negotiation, and IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS.

Quanbo Zha received the B.S. and M.S. degrees in vehicle engineering from Southwest Jiaotong University, Chengdu, China, in 2012 and 2015, respectively, and the Ph.D. degree in manage- ment science from the Business School, Sichuan University, Chengdu, in 2019.

He is currently a Postdoctoral Researcher with the School of Management Science and Real Estate, Chongqing University, Chongqing, China. His research results have been published in refereed journals, including the IEEE TRANSACTIONS ON

SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, Knowledge-Based Systems, and IEEE TRANSACTIONS ON COMPUTATIONAL SOCIAL SYSTEMS, among others. His research interests include opinion dynamics, social network, and decision support systems.

Hengjie Zhang received the B.S. degree from the Chongqing University of Posts and Telecommunications, Chongqing, China, in 2012, and the Ph.D. degree in management science and engineering from Sichuan University, Chengdu, China, in 2017.

He is a Lecturer with the Business School, Hohai University, Nanjing, China. His research results have been published in refereed journals and conference proceedings, including Applied Soft Computing, Decision Support Systems, IEEE TRANSACTIONS

ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, IEEE TRANSACTIONS ON FUZZY SYSTEMS, Information Fusion, Knowledge-Based Systems, Soft Computing, among others. His research interests include decision support systems and consensus reaching process.

Francisco Herrera (Senior Member, IEEE) received the M.Sc. and Ph.D. degrees in mathematics from the University of Granada, Granada, Spain, in 1988 and 1991, respectively.

He is a Professor with the Department of Computer Science and Artificial Intelligence, University of Granada, where he is the Director of the Andalusian Research Institute in Data Science and Computational Intelligence. He has been the Supervisor of 46 Ph.D. students. He has published more than 450 journal papers, receiving

more than 77 000 Google Scholar citations with an H-index of 137. His current research interests include among others, computational intelligence (including fuzzy modeling, computing with words, evolutionary algorithms, and deep learning), information fusion and decision making, and data science (including data preprocessing, prediction, singular problems, and big data).

Prof. Herrera received the several honors and awards, among others: the 2010 Spanish National Award on Computer Science ARITMEL to the “Spanish Engineer on Computer Science,” the International Cajastur “Mamdani” Prize for Soft Computing (Fourth Edition, 2010), the IEEE TRANSACTIONS ON FUZZY SYSTEM Outstanding 2008 and 2012 Paper, the 2011 Lotfi A. Zadeh Prize Best Paper Award (IFSA Association), the 2013 AEPIA Award to a scientific career in Artificial Intelligence, the 2014 XV Andalucía Research Prize Maimónides, the 2017 Andalucía Medal (by the regional government of Andalucía), and the 2018 “Granada: Science and Innovation City” Award. He has been selected as a Highly Cited Researcher by Clarivate Analytics in the fields of Computer Science and Engineering since 2014. He is currently the Editor-in-Chief of Information Fusion (Elsevier). He acts as an Editorial Member of a dozen of journals. He is an Academician in the Spanish Royal Academy of Engineering. He was an ECCAI Fellow in 2009 and an IFSA Fellow in 2013.

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