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In group decision making scenarios, where multiple anonymous agents interact, as is the case of social networks, the uncertainty in the provided information as well as the diversity in the experts’ opinions make of them a real challenge from the point of view of information aggregation and consensus achievement. This contribution addresses these two main issues in the following way: On the one hand, in order to deal with highly uncertainty group decision making scenarios, whose main particularity is that some of their experts may not be able to provide any single judgment about an alternative, the proposed approach estimates these missing information using the preferences coming from other trusted similar experts who present high degrees of confidence and consistency. On the other hand, with the objective of increasing the consensus among the agents involved in the decision making process, a feedback based influence network has been proposed. In this network, the influence between the agents is calculated by means of a dynamic combination of the inter agents trust, their self confidence, and their similarity. Thanks to this influence network our approach is able to recognize and isolate malicious users adjusting their influence according to the trust degree between them.
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Trust based group decision making in environments with extreme
Atefeh Taghavia, Esfandiar Eslamib, Enrique Herrera-Viedma*c,1,, Raquel Ure˜na*d,
aDepartment of Mathematics, Graduate University of Advanced Technology, Kerman, Iran.
bFaculty of Mathematics and computer, Shahid Bahonar University of Kerman, Kerman, Iran.
cAndalusian Research Institute on Data Science and Computational Intelligence (DaSCI), University of
Granada, Granada, Spain
dInstitute of Artificial Intelligence (IAI),School of Computer Science and Informatics, De Montfort
University, Leicester, UK
In group decision making scenarios, where multiple anonymous agents interact, as is the
case of social networks, the uncertainty in the provided information as well as the diversity
in the experts’ opinions make of them a real challenge from the point of view of information
aggregation and consensus achievement. This contribution addresses these two main issues in
the following way: On the one hand, in order to deal with highly uncertainty group decision
making scenarios, whose main particularity is that some of their experts may not be able
to provide any single judgement about an alternative, the proposed approach estimates these
missing information using the preferences coming from other trusted similar experts who present
high degrees of confidence and consistency. On the other hand, with the objective of increasing
the consensus among the agents involved in the decision making process, a feedback based
influence network has been proposed. In this network, the influence between the agents is
calculated by means of a dynamic combination of the inter agents trust, their self confidence,
and their similarity. Thanks to this influence network our approach is able to recognize and
isolate malicious users adjusting their influence according to the trust degree between them.
Keywords: Group decision making, Uncertainty, Incomplete information, Ignorance
situations, Intuitionistic fuzzy preference relations., Consensus, Trust
1. Introduction
Nowadays we are living the apogee of the Internet based technologies and consequently
web 2.0 communities, where a large number of users interacts in real time is a generalized
This type of social networks communities constitute a challenging scenario from the point
of view of Group Decision Making (GDM) approaches, because it involves a large number of
agents coming from different backgrounds with different levels of knowledge and influence. In
this type of scenarios, there exist two main key issues that require attention. The consensus
and the uncertainty in the experts’ opinions or preferences.
In many decision making situations, it is desired or even required to reach an agreement
between the experts involved. To do so, in most of the occasions, it is necessary to carry out
an iterative negotiation process between the experts with the objective of bringing closer their
Corresponding author
Email addresses: (Atefeh Taghavi),
(Esfandiar Eslami), (Enrique Herrera-Viedma*), (Raquel
Ure˜na* )
Preprint submitted to Knowledge Based Systems. October 24, 2019
points of view to eventually reach a solution accepted by the majority of them. The higher
the consensus level, the higher the agreement and consent with the final selected answer. In
the literature, there exist various consensus processes that iteratively provide some advice or
recommendations to the experts in order to increase the global consensus level. These types of
recommendations are widely known as feedback mechanisms [1–4], or when there are proposed
to group of experts they are denominated as Group Recommender Systems [5]. This last one
may take into account social aspects like user personality and interpersonal trust.
However, these iterative feedback approaches present the problem of experts’ non cooper-
ative behaviour [3]. That is, the experts may present a reluctance or even refuse to accept
the feedback provided by the system. With this regard, it has been observed in opinion dy-
namics theory that people tend to accept more easily those opinions coming from confident
and similar peers [6]. With this premise in mind, in [6] it has been presented a social network
based consensus approach that estimates each expert’s degree of coherence with the opinion
provided, widely known as consistency and his/her self-confidence to develop an inter experts
similarity network with the goal of providing recommendations based on other highly confident
and consistent similar experts opinions. In [7], the preference relation with self-confidence is
The second important issue considered in this contribution is uncertainty in the information
provided by the experts. Uncertainty may be reflected in various different ways, from the
expert being unsure of the given answer [8] to the extreme case of missing information [9–
14]. For the first case, an interesting way of dealing with the inherent hesitation or vagueness
in the experts’ opinions consists of taking advantage of Atanassov’s intuitionistic fuzzy sets
[15, 16] by allowing the experts to explain their preferences by means of intuitionistic fuzzy
preference relations (IFPRs). The particular case of extreme uncertainty in decision making
scenarios is the one in which the experts are not able to provide any preference rating about
an alternative, for different reasons ranging from lack of knowledge to lack of time or interest,
resulting in preference relations with some of their values missing or unknown [10, 17]. Various
studies remark the negative effects of not taking into consideration the incomplete information
in social networks based decision making. In [18] the impact of missing data in a scientific
collaboration network and in a random bipartite graph has been analyzed concluding that
there are three main missing data mechanisms: ”network boundary specification (non-inclusion
of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice
design)”. Afterward in [19] the ”effect of non-response on the structural properties of social
networks, and the ability of some simple imputation techniques to treat the missing network
data” have been studied pointing out that simple imputation procedures have large negative
effects and demonstrating by numerous simulations the importance of estimating the missing
For the case of GDM with extreme uncertainty as far as the authors know it has not been
proposed any consensus approach. However, not taking into consideration the unknown pref-
erences, in the consensus process could lead to serious biased. For this aim, in the literature,
various estimation approaches have been presented. Most of them use their own experts’ pref-
erences levering the logic transitivity between them [8, 20]. An exhaustive review of these
approaches has been provided in [11]. The main limitation of these transitivity based comple-
tion techniques is that they are applicable only when at least one comparative judgment about
each of the alternatives is provided. Nevertheless in real world decision making [21–24], very
often there are situations in which not any judgment about an alternative is provided and so
the transitivity properties cannot be used for the estimation. This scenario, denominated as
total ignorance situations, has been considered by Alonso et al. in [9], where they proposed
both individual and social strategies to estimate the missing information. As their name in-
dicates, individual strategies estimate the missing information without considering any other
information from other experts using a random initialization of the missing values and applying
transitivity afterward, while social strategies take advantage of the information provided by the
rest of the experts. These approaches present the disadvantage that they may provide solutions
not accepted by the given experts since they might be very far from their given opinions. In
[25], a method based on the concept of social influence network that deals with incomplete
fuzzy preference relation by taking into account the effects of social influence in the network of
decision making has been proposed.
In this contribution, we address these two main issues, consensus and uncertainty in two
main steps: Firstly, in order to deal with uncertainty, we propose a new approach to estimate
the missing preference opinions, expressed as intuitionistic fuzzy preference relations, able to
work even in total ignorance situations. To do so, the experts are firstly clustered depending
on the similarity of their preferences. Then, a new aggregation operator, the TCCI-IOWA
operator that leverage the inter-experts trust, self-confidence, and consistency estimates and
fuse the missing preferences in each of the clusters.
Secondly, a new feedback mechanism based on the trust propagation is proposed with the
goal of increasing the consensus degree. This proposal is based on the premise that people
tend to be more influenced by the opinions or behaviors of similar trusted peers [6, 26]. In
addition, the proposed values to the experts, besides improving the consensus, could improve
the estimated data.
The rest of the paper is set out as follows: The next section reviews some basic necessary
information and backgrounds. In Section 3 the proposed approach that deals with consensus
and missing information is presented. In order to illustrate the way of operation of the proposed
approach, an example is presented in section 4. Finally, in section 5, we outline the conclusion
from this work and we introduce the future research challenges.
2. Preliminaries
In order to make this contribution as self contained as possible, this section is devoted to
providing some necessary definitions and backgrounds used in the rest of the paper.
2.1. Expression of preference in group decision making
In any decision making problem, the people involved in the process are asked to express their
preferences on the set of feasible alternatives (X). Among the various preference elicitations
methods, the pairwise comparison has been proved as the most effective [27]. This comparison
of two alternatives can lead to the preference of one alternative, indifference or incomparability.
A preference relation integrates these three possible states into a single value as follows:
Definition 1 (Preference Relation (PR) [10]).”A preference relation Ris a binary relation
defined on the set Xthat is characterized by a function µp:X×XD, where D is the domain
of representation of preference degrees provided by the decision maker.”
When the cardinality of Xis small, Rmay be conveniently represented by an n×nmatrix
R= (rij), with rij =µp(xi, xj) being interpreted as the degree or intensity of preference of
alternative xiover xj. The elements of Rcan be of a numeric or linguistic nature, i.e., could
represent numeric or linguistic preferences, respectively.
Definition 2 (Fuzzy Preference Relation [1]).”A fuzzy preference relation Ron a finite set of
alternatives Xis a fuzzy set on the product set in X×Xthat is characterized by a membership
function µR:X×X[0,1]. Usually, the preference relation simply represented by the n×n
matrix R= (rij), being rij =µR(xi, xj)(i, k 1, ..., n)interpreted as the preference degree or
intensity of the alternative xiover xj:
rij = 1 is the maximum degree of preference for xiover xj
rij (0.5,1) indicates a definite preference for xiover xj
rij = 0.5indicates indifference between xiand xj
rij +rji = 1 i, j ∈ {1, . . . , n}
is indicated that we have a reciprocal fuzzy preference relation.”
Multiplicative consistency property can be used in estimating the preference value between
a pair of alternatives. For this aim, let (xi, xj) with (i<j) be a pair of alternatives, and xk
(k6=i, j) be an intermediate alternative. Then:
ij =rik ·rkj ·rji
rjk ·rki
in which the denominator should not be zero.
The average of all possible mrk
ij of the pair of alternatives (xi, xj) can be interpreted as their
total estimated value based on multiplicative transitivity:
mrij =
; (2)
where R01
ij ={k6=i, j|(rik , rkj )/R01},R01 ={(1,0),(0,1)}, and #R01
ij is the cardinality of
ij . So, MR = (mrij ), can be constructed.
Definition 3 (Multiplicative Consistency [8]).A fuzzy preference relation R= (rij)is multi-
plicative consistent if and only if R=MR.
By using the similarity between the values rij and mrij the level of consistency of a fuzzy
preference relation can be measured at three different levels: pair of alternatives, alternatives,
and relation [10]:
Level 1. Consistency Index of pair of alternatives.
CLij = 1 d(rij , mrij )i, j (3)
Where d is a suitable distance function.
Level 2. Consistency Level of alternatives.
j=1; i6=j
Level 3. Consistency Level of a fuzzy preference relation.
CL =
The concept of intuitionistic fuzzy set introduced by Atanasov in [16] as an extension of
fuzzy sets, allows covering vagueness/uncertainty. Afterward, intuitionistic fuzzy preference
relations have been defined to deal with uncertainty in preference elicitation [28] as follows:
Definition 4 (Intuitionistic Fuzzy Preference Relation [8]).”An intuitionistic fuzzy preference
relation Bon a finite set of alternatives X={x1, . . . , xn}is characterised by a membership
function µB:X×X[0,1] and a non-membership function νB:X×X[0,1] such that
0µB(xi, xj) + νB(xi, xj)1(xi, xj)X×X.
with µB(xi, xj) = µij interpreted as the certainty degree up to which xiis preferred to xj; and
νB(xi, xj) = νij interpreted as the certainty degree up to which xiis non-preferred to xj.”
If µii =νii = 0.5i∈ {1, . . . , n}and µji =νij i, j ∈ {1, . . . , n}then Bis reciprocal.
As it has been proved in [8], the intuitionistic fuzzy preference relations are isomorphic to
the set of asymmetric fuzzy preference relations and so eqs. (1, 2) defined for the former, work
as well for intuitionistic fuzzy preference relations.
2.2. Missing information in Group decision making scenarios
Definition 5 (Incomplete fuzzy preference relation [1, 10]).An incomplete fuzzy preference
relation is a fuzzy preference relation with a partial membership function.
Definition 6 (Partial function [1]).A function f:XYis a partial function whenever
some elements in Xdo not map onto an element in the set Y. So, if each element in Xmaps
onto an element in Y, then fis a total function.
Definition 7 (Total ignorance fuzzy preference relation [9]).Let E={e1, e2, ..., em}be a
group of experts and X={x1, x2, ..., xn}is a set of alternatives. The experts expressed their
preferences by means of a set of incomplete FPRs, {R1, R2, ..., Rm}. An ignorance situation
occurred when at least one of the experts ehE, does not provide any single preference value
for at least one alternative xiX:
i, h|Rh
where Rh
iis the ith row and ith column of the preference matrix for the expert eh.
So, xiis called the unknown or ignored alternative for eh.
2.3. Expert’s self-confidence degree
By using the hesitancy degree of a reciprocal intuitionistic fuzzy preference relation Ure˜na
et al. in [8] introduce the concept of experts’ confidence degree, which can be measured at
three different levels namely: pair of alternatives, alternatives, and relation levels.
Given a reciprocal intuitionistic fuzzy preference relation, B= (bij)=(hµij, νij i), the levels
of confidence degrees are defined as follows:
Definition 8 (Expert’s self-confidence for pair of alternatives [8]).For a given intuitionistic
preference value bij the confidence level is measured as:
CF Lij = 1 τij (6)
where τij = 1 µij νij is the hesitancy degree associated to bij.
It is clear that CF Lij = 1 τij =µij +νij. This means when C F Lij = 1 then there is no
hesitation, τij = 0. In short, the lower hesitation value the more confidence degree and vice
Definition 9 (Experts’ confidence for an alternative [8]).the confidence level associated to the
alternative xiis defined as:
CF Li=Pn
j=1,j6=i(C F Lij +CF Lj i)
2(n1) (7)
If Bis reciprocal, i.e. C F Lij =CF Lji ,i, j, we have
CF Li=Pn
j=1,j6=iC F Lij
Definition 10 (Experts’ confidence for a relation [8]).For a reciprocal intuitionistic fuzzy
preference relation B, the confidence level is:
i=1 CF Li
2.4. Consensus process
The majority of the existing consensus processes applied to GDM scenarios consists of
an iterative questioning process to control and modify the experts’ opinions to increase the
consensus degree until reaching the desired level.
The consensus degree is usually measured at three different levels of pairs of alternatives,
alternatives, and relation. Thus, when fuzzy preference relations are presented by decision
makers, the consensus degree is calculated as follows [1–3]:
1. For every pair of decision makers ek, el(k= 1, ..., m1, l =k+1, ..., m) a similarity matrix
SM kl = (smkl
ij ) is defined as: smkl
ij = 1 − |rk
ij rl
2. By aggregating all the (m1) ×(m2) similarity matrices, a consensus matrix, CM =
(cmij) is obtained.
cmi,j =φ(smkl
ij ), k = 1, ..., m 1, l =k+ 1, ..., m.
Based on the nature of the GDM problem to solve, different aggregation operators could
be used as φ.
3. Three different levels is defined for consensus degrees:
(a) Consensus degree on pairs of alternatives, (xi, xj)
called cpij is defined as the element of the collective similarity matrix CM ; means:
cpij =cmij (10)
(b) Consensus degree on the alternatives, xiis defined as cai:
j=1;j6=i(cpij +cpji)
2(n1) (11)
(c) Consensus degree on the relation, called cr, is:
cr =Pn
i=1 cai
As mentioned before, cr is used to control the consensus situation. The value cr = 1 shows
full agreement. As much as cr is closer to 1,the consensus among all the decision makers’
opinions is higher.
2.5. Aggregation Step
The collective fuzzy preference relation of all individual preferences of the experts is used to
estimate the desired solution. To do so, it is necessary to merge all the individual preferences
by using a suitable aggregation operator. It is important to use a suitable aggregation operator.
Del Moral et al., in [29], showed that the consensus reached by the experts how influenced by
the use of different aggregation operators with different distance functions. One of the most
widely used operators is Yager’s Ordered Weighted Averaging (OWA) operator [30] or one of
its extended versions such as the Induced OWA (IOWA) [31] defined as follows.
Definition 11 (IOWA operator [8]).”An IOWA operator of dimension mis a function Φw:
(R×R)mRto which a set of weights or weighting vector is associated, W= (w1, ..., wm),
such that wi[0,1] and Piwi= 1 is expressed as follows:
Φw(hu1, p1i, ..., hum, pmi) =
being σ:{1, ..., m} −→ {1, ..., m}a permutation such that uσ(i)>uσ(i+1),i= 1, ..., m 1.
In [8] Ure˜na et al. introduced the consistency and confidence IOWA (CC-IOWA) operator
which employs both criteria to reorder the preferences and to calculate the weights.
2.6. Feedback mechanism
In a consensus reaching process, the feedback mechanism is one of the most important
steps, and consist of generating personalized advice to the experts to reach better consensus
by modifying their preferences. Here, according to the experts’ trust propagation, we create
a feedback mechanism to reach the highest consensus. First, the experts are ranked in three
groups based on their trust score. Then the recommendation process is run to generating the
advice. Each group receives some advice, the higher ranked group will have a lower amount of
2.7. KNN Algorithm
One of the simplest and most frequently used supervised classification methods is the K-
Nearest Neighbors algorithm (or KNN algorithm) [32–34]. KNN works based on the similarity
measure, which is the minimum distance from the classes. By using a suitable distance function
such as Euclidean, Manhattan, Minkowski or Cosine measures, it determines which of the
classes in K, is the nearest neighbor for the new case. Although KNN has been proposed as a
supervised algorithm, there are several articles that use this approach in an unsupervised KNN.
For example when the value of ”K” is not clear, [35], or when there is a large dataset [36, 37].
In this contribution, KNN has been used in an unsupervised way since we do not know a priori
the numbers of clusters. Therefore what KNN does in this case, is to select which instances,
i.e. the experts, in our data are k-nearest to the point, i.e. the expert with ignored data; we
are polling for using the euclidean distance as it is not sensitive to the linear association.
2.8. Trust and distrust
Trust and distrust play a key role in social network based systems, allowing relationships
between users that have an effect on their decisions. There are many researchers working on
this topic, a recent survey has been published in [26]. In 2006 De Cock et al proposed to use
the intuitionistic fuzzy relation to model the network of trust between sources [38]. So, this
method handles both main problems, i.e. ignorance, and vagueness, which means to trust or
distrust as well as trust as a matter of degree. Also, used a trusted third party to deriving
trust information. On the same line, in [39], a method to draw out the trust/ distrust values
for users that are just connected through the network but do not know each other directly has
been proposed. In this sense, they discussed bilattice-based aggregation approaches and it’s
improvement by using ordered weighted averaging techniques, and combination of knowledge
defects. Wu et al. proposed in [40] a consensus model based on trust propagation for a social
network with incomplete linguistic information that includes a visual feedback process to guar-
antee the consensus achievement. The research [41] tackles with the inconsistency existing in
the information provided by the experts proposing a trust induced recommendation mechanism
that generates exclusive advice for the inconsistent experts. Afterward, [42] addressed a visual
interaction consensus model for social network group decision-making which deals with dual
trust propagation. To do so, at first, the trust matrix is completed by constructing the trust
relationships between every two experts based on the transitivity property of trust. By defin-
ing three levels of consensus degree the inconsistent experts are identified. Then based on the
obtained trust network, a feedback mechanism is defined to give advice to the inconsistent ex-
perts to achieve higher levels of consensus. After all, a visual adoption mechanism is presented
providing the experts with some visual information about their individual consensus positions
before and after adopting the recommendation advice [42]. In the following we present the
definition of trust, distrust and trust score that will be used within this contribution:
Definition 12 (Trust Function (TF) [42]).An ordered tuple γ= (t, d)where t, d [0,1] and
t, d are representing the trust and distrust degrees respectively, will be referred to as a trust
function value. The set of trust function values (TFs), or trust function, will be denoted by
Γ = {γ= (t, d)|t, d [0,1]}.
Intuitionistic trust function (ITFs), which is more natural in the real world, is defined by
adding the extra condition 0 t+d1 to the TFs’ definition. Here, in this research, we
will use ITFs. To be able to ordering and comparing the values of TFs, the concepts of trust
score and hesitation degree (knowledge degree) are defined.
Definition 13 (Trust Score (TS) [42]).The trust score associated with an ordered pair of
trust/distrust values γ= (t, d)is:
T S(γ) = td+ 1
Definition 14 (Hesitation Degree (HD) [42]).Hesitation degree or knowledge degree of a trust
function value γ= (t, d)is defined as: H D(γ) = (1 td)2.
By using the concept of TS and hesitation degree of TFs, an order relation in the set of
TFs could be defined.
Definition 15 (Order Relation of TFs [40]).Let γ1= (t1, d1)and γ2= (t2, d2), be TFs values.
γ1=γ2if and only if T S(γ1) = T S (γ2)HD(γ1) = HD(γ2).
γ1precedes γ2;γ1γ2; if and only if one of the following conditions is true:
[1.] T S(γ1)< T S(γ2).
[2.] T S(γ1) = T S (γ2)HD(γ1)> HD(γ2).
γ1succeeds γ2;γ1γ2; if and only if one of the following conditions is tru:
[1.] T S(γ1)> T S(γ2).
[2.] T S(γ1) = T S (γ2)HD(γ1)< HD(γ2).
3. The proposed algorithm
In this section, we propose a trust based decision making approach that allows estimating
the missing values even in the case of total ignorance situations and as well as including a trust
based feedback mechanism to increase the agreement between the experts.
Let X={x1, x2, ..., xn}, n 2 be a set of alternatives evaluated by mexperts, E=
{e1, e2, ..., em}. Each expert used a reciprocal intuitionistic fuzzy preference relation; B=
(bij) = (hµij , νij i),06µij +νij 61 which can be presented by R= (rij = (hµiji)). Let us
suppose that ekprovides the following incomplete intuitionistic fuzzy preference relation, in
which no judgements has been provided at all about an alternative:
h0.5,0.5i hµ12, ν12i · · · x· · · hµ1n, ν1ni
hµ21, ν21i h0.5,0.5i · · · x· · · hµ2n, ν2ni
x x · · · h0.5,0.5i · · · x
hµn1, νn1i hµn2, νn2i · · · x· · · h0.5,0.5i
We assume that the given intuitionistic fuzzy preference relation is reciprocal and so, ac-
cording to [8] Bkis equivalent to a fuzzy preference relation expressed as follows:
0.5µ12 · · · x· · · µ1n
µ21 0.5· · · x· · · µ2n
x x · · · 0.5· · · x
µn1µn2· · · x· · · 0.5
In a nutshell, the proposed approach is composed of the following steps: First of all the
confidence, consistency and trust score of each expert are computed. Then all the experts are
clustered using the KNN algorithm, with the objective of obtaining the different groups of like
minded experts.
Afterward, the completion of the missing information is done at a cluster level, aggregating
the opinions of those experts in the cluster with different weights depending on their trust,
confidence, and consistency. To do so a new IOWA operator that trades off these three criteria,
the T CC I OW A, is defined as follows:
Definition 16 (TCC-IOWA operator).Let a set of experts, E={e1, ..., em}provide pref-
erences about a set of alternatives, X={x1, ..., xm}using the reciprocal intuitionistic fuzzy
preference relations {B1, ..., Bm}.A trusted consistency and confidence IOWA (TCC-IOWA)
operator of dimension m;Φtcc
w, is an IOWA operator whose set of order inducing values is the
set of trust/consistency/confidence index values, {T CC I1, ..., T C CIm}, associated with the set
of experts. Then, the collective reciprocal intuitionistic fuzzy preference relation Btcc = (btcc
ij ) =
ij , νtcc
ij i)is computed as follows:
ij =Φtcc
w(hT CCI1, µ1
iji, ..., hT C CI m, µm
ij i) =
ij (15)
ij =Φtcc
w(hT CCI1, ν1
iji, ..., hT C CI m, νm
ij i) =
ij (16)
T CCIh= (δ1).T Sh+ (δ2).CLh+ (δ3).C F Lh,(17)
T CCIσ(h)>T CCI σ(h+1), wσ(h)>wσ(h+1) >0 (h∈ {1, ..., m 1})
and Pm
h=1 wh= 1.
The parameters T Sh, CLh, C F Lhare the trust score, the consistency level and the confidence
level associated to Bh, respectively. δi[0,1], i = 1,2,3are parameters to control the weights
of trust, consistency and confidence criteria in the inducing variable and P3
i=1 δi= 1.
The weights of the TCC-IOWA operator are obtained as follows:
i=1 T CCIσ(i)
i=1 T CCIσ(i)
T) (18)
in which T=Pm
i=1 T CCIiand Qis the membership function of the linguistic quantifier. A
scheme illustrating how this completion phase is carried out is depicted in Fig. 1.
Computing Trust confidence
consistency Index
Computing the weighted vector
associated to the IOWA operator (𝒘𝒉)
Clustering the experts by using
KNN algorithm
Consensus process
containing feedback
Trust Score
Intuitionistic Fuzzy
Preference Relations
Figure 1: Estimating missing values.
Note that this social based completion approach is designed to increase the agreement of
the like minded experts. Once the completion phase has been completed, the consensus level
will be assessed, and if it is not high enough, the feedback based consensus approach will start.
The proposed consensus approach consists of a feedback mechanism that produces recom-
mendations to experts. In this step, the experts are clustered on three groups based on their
trust degree: high trusted experts, medium trusted experts and low trusted experts [6]. The
high trusted experts receive less recommendation to change their preferences. low trusted ex-
perts have received the most recommendations. The different steps included in this phase are
summarized in Fig. 2.
High trusted experts
Medium trusted experts
Low trusted experts
Consensus Process
Feedback Mechanism
Selection Process
Ranked Trusted Experts
Figure 2: Feedback mechanism.
Based on the given facts, the proposed approach is presented in detail in the following
3.1. Missing information estimation
This first phase is devoted to the estimation of the missing values. To do so, this phase is
composed of the following steps:
Step 1 : Computing trust score (T S ) as in eq. (14) for each expert.
Step 2 : Computing confidence measures for each expert, eq. (9).
Step 3 : Computing consistency measures for each expert, eq. (5).
Step 4 : Clustering the experts by KNN algorithm using the Euclidean distance. To do so,
the similarity between the experts is calculated using their preference relations, this way
experts with similar preferences are included in the same cluster.
Step 5 : Computing Trust/Confidence/Consistency index (T C CI ), by using eq. (17).
Step 6 : Computing the weighting vector (wh) associated to an IOWA operator, like T C CI
I OW A operator, eq. (18).
Step 7 : Missing information estimation,
Let Nkbe the neighbor of expert ek(means the experts who are most similar to ekand are
in the same cluster). Using the weights from the previous step, wh, after normalizing the
weights of the set Nkif necessary, and using the following equations which are obtained
from eqs. (15, 16); the missing values are computed as follows:
ij =X
ij (19)
ij =X
ij (20)
3.2. Feedback proposal to reach consensus
In this second phase, the consensus degree is computed in three different levels, according
to Eqs. (10), (11) and (12).
If the consensus level does not reach a minimum threshold of θ, means cr < θ, then an
iterative feedback process is activated:
3.2.1. Feedback calculation
By using two thresholds λ1, λ2, the experts are clusters to three groups: Elow experts with
low trust degree, Emed experts with medium trust degree and Ehigh experts with high trust
degree. Elow ={ei|T Si< λ1},Emed ={ei|λ1T Si< λ2},Ehigh ={ei|λ2T Si}.
Identify low-trusted experts’ preferences to be changed:
1. Computing the threshold α1;α1=Pn
j=1,j6=icpij )/(n2n).
2. The set of pairs of alternatives, A, with a consensus degree smaller than a threshold,
α1, are identified: A={(i, j )|cpij < α1}.
3. The set of controversial preferences which to be changed by each expert ehELow
is: P CHh
Low =A.
Identify medium-trusted experts’ preferences to be changed:
1. Obtaining the alternatives to be changed, XCH, at first:
XCH ={i|cai< α2}and α2=
2. The pairs of alternatives to be changed are: P={(i, j)|iX CH cpij < α1}.
3. The set of preference values, P CH h
med , that are required to be modified is:
med ={(i, j)P|sah
i< β1}
where, β1=Pm
h=1 sah
Identify high-trusted experts’ preferences to be changed:
1. The set of alternatives to be changed are identified as: XCH ={i|cai< α2}
2. The pairs of alternatives to be changed are supposed as:
P={(i, j)|iX C H cpij < α1}.
3. The set of preference values that are required to be modified is:
high ={(i, j)P|sah
i< β1sph
ij < β2}
where, β2=Pm
h=1 sph
ij/m ,ehEhigh.
It is worth mentioning that our approach provides recommendations based on other similar
and trustworthy experts, and as it has been proved in the literature [40–42] that people are
more open to taking the recommendations that come from like-minded experts.
4. An Illustrative Example
The feasibility and effectiveness of the proposed approach in the real world are demonstrated
by a small simulated real example, which could be extended to a much larger example in the
real world.
The municipality wants to build a new park in the city. For this purpose, Four zones are
designated. Four experts were asked to evaluate the tenders and announce the final result. Each
expert expresses his/her preferences with reciprocal intuitionistic fuzzy preference relations. In
the real world, this can be done through citizen selections through social media or city council
voting. Here we have limited the problem to four experts.
Example 1. let X={x1, x2, x3, x4}be a set of alternatives evaluated by the set of experts,
E={e1, e2, e3, e4}, by using the following reciprocal intuitionistic fuzzy preference relations.
Note that the first expert presents a total ignorance situation since no judgment has been provided
for alternative 3.
(0.5,0.5) (0.4,0.3) (n, n) (0.2,0.7)
(0.3,0.4) (0.5,0.5) (n, n) (0.3,0.4)
(n, n) (n, n) (0.5,0.5) (n, n)
(0.7,0.2) (0.4,0.5) (n, n) (0.5,0.5)
(0.5,0.5) (0.4,0.45) (0.35,0.33) (0.3,0.4)
(0.45,0.4) (0.5,0.5) (0.45,0.4) (0.31,0.38)
(0.33,0.35) (0.4,0.45) (0.5,0.5) (0.4,0.55)
(0.4,0.3) (0.38,0.31) (0.55,0.4) (0.5,0.5)
(0.5,0.5) (0.5,0.4) (0.45,0.2) (0.4,0.3)
(0.4,0.5) (0.5,0.5) (0.6,0.3) (0.5,0.4)
(0.2,0.45) (0.3,0.6) (0.5,0.5) (0.35,0.4)
(0.3,0.4) (0.4,0.5) (0.4,0.35) (0.5,0.5)
(0.5,0.5) (0.4,0.5) (0.45,0.4) (0.43,0.3)
(0.5,0.4) (0.5,0.5) (0.5,0.4) (0.5,0.3)
(0.4,0.45) (0.4,0.5) (0.5,0.5) (0.5,0.4)
(0.3,0.43) (0.3,0.5) (0.4,0.5) (0.5,0.5)
As aforementioned, according to [8], the intuitionistic fuzzy preference relation can be pre-
sented as a fuzzy preference relation:
0.5 0.4n0.2
0.3 0.5n0.3
n n 0.5n
0.7 0.7n0.5
0.5 0.4 0.35 0.3
0.45 0.5 0.45 0.31
0.33 0.4 0.5 0.4
0.4 0.38 0.55 0.5
0.5 0.5 0.45 0.4
0.4 0.5 0.6 0.5
0.2 0.3 0.5 0.35
0.3 0.4 0.4 0.5
0.5 0.4 0.45 0.43
0.5 0.5 0.5 0.5
0.4 0.4 0.5 0.5
0.3 0.3 0.4 0.5
4.1. Phase 1: estimating the missing values
Computing consistency measure By omitting the third row and column of the R1and using mul-
tiplicative transitivity we have:
0.11 0.7
1.05 0.08
0.2 1.4
0.42 0.33 0.3
0.43 0.46 0.31
0.34 0.39 0.4
0.4 0.37 0.56
0.44 0.4 0.53
0.46 0.48 0.56
0.24 0.38 0.24
0.23 0.38 0.57
0.44 0.43 0.41
0.45 0.55 0.5
0.42 0.36 0.52
0.31 0.3 0.38
Consistency Level of the fuzzy preference relations; using formula (5); we obtain:
CL1= 0.46, CL2= 0.99, CL3= 0.92, C L4= 0.98
The confidence level associated with each reciprocal intuitionistic fuzzy preference relation;
formula (9)
CF L1= 0.8, C F L2= 0.79, C F L3= 0.8, C F L4= 0.84
Now, by considering the following trust/distrust matrix, the trust score (TS) of each expert
is computed.
T dT =
(0.5,0.43) (0.6,0.4) (0.8,0.19)
(0.8,0.17) (0.44,0.55) (0.7,0.3)
(0.53,0.4) (0.7,0.3) (0.32,0.65)
(0.62,0.3) (0.44,0.5) (0.29,0.7)
By using formula (14), we have:
T S1= 0.47, T S2= 0.646, T S 3= 0.653, T S 4= 0.53
Now, by considering δ1= 0.4, δ2= 0.3, δ1= 0.3, and using formula (17), the trust/
confidence/ consistency index will be:
T CCI1= 0.57, T CC I2= 0.79, T CC I3= 0.78, T C CI4= 0.76
By ordering the TCC index from the biggest to the lowest, σ1= 2, σ2= 3, σ3= 4, σ4= 1
and using the linguistic quantifier ”most of”, Q(r) = r1/2, in formula (18); the weighting vector
is generated:
w1= 0.1, w2= 0.52, w3= 0.21, w4= 0.16
The KNN algorithm shows that the two nearest neighbors to the first expert are experts 2 and
3. Since w2= 0.52, w3= 0.21, by normalizing these weights and using formulas (15) and (16)
the ignored values are estimated as follows:
(0.5,0.5) (0.4,0.3) (0.38,0.29) (0.2,0.7)
(0.3,0.4) (0.5,0.5) (0.49,0.37) (0.3,0.4)
(.29,0.38) (0.37,0.49) (0.5,0.5) (0.38,51)
(0.7,0.2) (0.4,0.5) (0.51,0.38) (0.5,0.5)
4.2. Phase 2: Consensus Process
For the first expert, the complete estimated fuzzy preference relation is as follows and the
rest of the experts’ preferences remains the same.
0.5 0.4 0.38 0.2
0.3 0.5 0.49 0.3
0.29 0.37 0.5 0.38
0.7 0.4 0.51 0.5
Let θ= 0.94 be the threshold for consensus level and the maximum round of the process be
7. As T S computed before, T S 1= 0.47, T S2= 0.646, T S 3= 0.653, T S 4= 0.53, so the
experts are divided into three groups based on their trusted degree:
Elow ={e1}, Emed ={e4}, Ehigh ={e2, e3}
Therefore, we could assign the following values as the importance weight values:
wI(e1) = 0.15, wI(e2)=0.31, wI(e3) = 0.31, wI(e4)=0.23
By using formula (10,11,12), the consensus degree are computed in three levels: cr = 0.9. Since
cr < θ, so the recommendation mechanism is activated. Here, just the first round is illustrated
by details.
4.2.1. Feedback Mechanism: First Round
Identify low-trusted experts’ preferences to be changed: the threshold α1is computed, α1=
0.903. Identifying the set of pairs of alternatives with a consensus degree smaller than a thresh-
old, α1:
The set of preferences to be changed by each expert ehELow is: P CHh
Low =A. Since
Elow ={e1}, so we have:
0.5 0.45 0.43 0.25
0.35 0.5 0.54 0.35
0.34 0.32 0.5 0.43
0.65 0.35 0.46 0.5
In order to identify the medium-trusted experts’ preferences to be changed, we have: α2=
0.903,XCH ={1,4}and the set of the pairs of alternatives to be changed is: P={(1,4)}.
So, the medium trusted experts fuzzy preference matrix will be:
Table 1: Consensus Degrees for seven Rounds
Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
Consensus Degrees; cr = 0.91153 0.92014 0.92264 0.93139 0.93653 0.93681 0.93931
0.5 0.4 0.45 0.38
0.5 0.5 0.5 0.5
0.4 0.4 0.5 0.5
0.3 0.3 0.4 0.5
To identify the high-importance experts’ controversial preferences, the set of alternatives to
be changed is XCH ={1,4}but the set of alternatives to be changed is empty. Means for the
experts Ehigh ={e2, e3}the fuzzy preference relations not changed. In this round the new cr is
obtained cr = 0.91153.
The results after seven rounds are available in the following table:
The fuzzy preference relations values for each expert after seven rounds will be as following:
0.5 0.45 0.43 0.35
0.45 0.5 0.54 0.45
0.34 0.32 0.5 0.43
0.35 0.35 0.46 0.5
0.5 0.4 0.35 0.3
0.45 0.5 0.45 0.31
0.33 0.4 0.5 0.4
0.4 0.38 0.5 0.5
0.5 0.5 0.45 0.4
0.4 0.5 0.6 0.5
0.2 0.3 0.5 0.4
0.3 0.4 0.45 0.5
0.5 0.4 0.45 0.38
0.5 0.5 0.5 0.5
0.4 0.4 0.5 0.45
0.35 0.3 0.45 0.5
5. Discussion
Nowadays thanks to the worldwide expansion of internet based technologies, many inter-
actions between people are carried out by means of social network based communities. These
cyber scenarios facilitate the communication between millions of users, in real time, no matter
their backgrounds. Therefore from the point of view of group decision making, social networks
constitute not only a great opportunity, but also pose various research challenges. Among
them, we can highlight the anonymity of these systems whose open nature difficults the devel-
opment of trust relationships and the uncertainty in the information that arises from the fact
that very different users with different profiles interact. In order to overcome these issues, in
this contribution we have proposed, for the first time a trust and consensus based approach
for group decision making that deals with uncertainty in the experts opinions by taking advan-
tage of the intuitionistic fuzzy preference relations and by estimating missing information even
when no values have been provided by an expert, what is known in the specialized literature
as a total ignorance situation. From the point of view of the missing information estimation
in [8], Urena et al. proposed an approach to estimate the incomplete reciprocal intuitionistic
preference relations, based on consistency and confidence. In our current contribution, we go
further, introducing a new methodology that deals with total ignorance situations estimating
the unknown information by means of the preferences from the most trusted similar experts
who present high degrees of confidence and consistency. With respect to the trust estimation, in
comparison with [40–42], besides taking into account the trust and distrust degree of agents, as
mentioned in subsection (2.8), our approach uses as well other properties such as the similarity
between experts and also, the self-confidence and consistency property of each one. Therefore,
the proposed approach is more reliable and more successful in evaluating missing values and
increasing the level of consensus.
6. Conclusions
In this contribution, we have introduced a consensus based approach for group decision
making that deals with uncertainty in the experts opinions by taking advantage of the intu-
itionistic fuzzy preference relations and by estimating missing information even when no values
have been provided by an expert, what is known in the specialized literature as a total ignorance
situation. The main novelties introduced in this contribution are the following:
The proposed approach is suitable to deal with highly uncertainty group decision making
situations in which the experts may not be able to provide any single judgment about an
alternative. To overcome this situation, we propose to estimate the missing information
taking into consideration the opinions coming from other trusted similar experts who
present high degrees of confidence and consistency.
In order to increase the consensus among the agents, in this contribution, we develop
a feedback based network where the influence between the agents in the negotiation is
calculated by means of a dynamic combination between the inter agents’ trust, the self
confidence, and the similarity. To do so, we propose a new aggregation operator that
balance in each iteration these three criteria, allocating more influence to those highly
trusted experts that provide the most consistent solution.
The developed approach has been designed to recognize and isolate malicious users since
their influence in the feedback network is adjusted according to the trust degree between
them. So, low trust users will have no influence on the negotiations. Furthermore,
the proposed IOWA based operator, which dynamically calculates both the ordering of
the opinions and the experts’ recommendations weights, has been proved to avoid the
malicious user to ”learn how the system works” and so prevents them from manipulating
the negotiations to reach consensus.
As future work, we plan to explore the application of the proposed approach with real
social network datasets more specifically in the context of e-marketing and e-health scenarios.
Moreover, the use of Hesitant Fuzzy Preference Relations as a way to model uncertainty [43]
will be evaluated.
The authors would like to acknowledge the financial support from the EU project H2020-
MSCA-IF-2016-DeciTrustNET-746398 and the National Spanish project TIN2016-75850-P.
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... In the existing research on incomplete information, Taghavi et al. [42] considered the trust relationship and the complement of incomplete information, but did not build a trust network and the expression of the decision matrix used intuitionistic fuzzy numbers. Wu et al. [30] and Wu et al. [44] considered the completion of the incomplete trust network, but did not consider the completion of the incomplete decision matrix, and the evaluation form of the decision matrix is in the form of deterministic numbers or fuzzy numbers, which does not conform to people's usual habit of using linguistic to express preference information. ...
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It is necessary to consider the trust relationship among experts in the process of group decision-making, however the trust network and preference information among experts may be incomplete. Therefore, this paper proposes an incomplete probabilistic linguistic multi-attribute group decision-making method based on a three-dimensional trust network. Firstly, the two-dimensional trust network is extended to the three-dimensional form, and the probabilistic linguistic term sets are used to express the trust relationship and degree among experts. On this basis, considering the situation of incomplete information, the trust transfer function is designed to complete the establishment of the trust network. Secondly, in order to complete the incomplete probabilistic linguistic decision preference information of experts, the relative trust of experts and the cosine similarity of preference relations are comprehensively considered. Then, the least average method is used to determine the evaluation information that needs to be adjusted, and different opinion adoption factors are set for a personalized recommendation. Finally, an evaluation case of the national wetland park pilot and comparative analysis are used to demonstrate the effectiveness and applicability of the proposed method.
... The first study focused on the representation of trust. There are discrete values [14][15][16], continuous values [17,18], fuzzy logic values (including interval values [19][20][21], intuitionistic fuzzy values (IFVs) [22][23][24][25], Pythagorean fuzzy values (PFVs) [26], interval-valued Pythagorean fuzzy values (IVPFVs) [27]), and other trust representations. The second is the trust propagation method. ...
... Therefore, we will consider the situation in which evaluation information is missing in future work. Second, the SNGDM method is implemented in a multi-attribute environ-ment, and whether it can be extended to other decision-making environments remains to be seen, such as fuzzy preference relationships [22,[48][49][50]. At the same time, it is yet to be determined as to whether the SNGDM method can be extended to solve evaluation decision-making problems in other industries, such as venture capital evaluation, green supply chain management, e-learning course selection, selection of the best substitutes for biopesticides, site assessment, digitalization in logistics and retail, and weapon selection decisions [27,[51][52][53][54][55][56][57]. ...
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As q-rung orthopair fuzzy set (q-ROFS) theory can effectively express complex fuzzy information, this study explores its application to social network environments and proposes a social network group decision-making (SNGDM) method based on the q-ROFS. Firstly, the q-rung orthopair fuzzy value is used to represent the trust relationships between experts in the social network, and a trust q-rung orthopair fuzzy value is defined. Secondly, considering the decreasing and multipath of trust in the process of trust propagation, this study designs a trust propagation mechanism by using its multiplication operation in the q-ROFS environment and proposes a trust q-ROFS aggregation approach. Moreover, based on the trust scores and confidence levels of experts, a new integration operator called q-rung orthopair fuzzy-induced ordered weighted average operator is proposed to fuse experts’ evaluation information. Additionally, considering the impact of consensus interaction on decision-making results, a consensus interaction model based on the q-ROF distance measure and trust relationship is proposed, including consistency measurement, identification of inconsistent expert decision-making opinions and a personalized adjustment mechanism. Finally, the SNGDM method is applied to solve the problem of evaluating online teaching quality.
Residual energy of devices is a key factor to determine whether Internet of Things (IoT) applications can proceed smoothly. Partnership cheating behavior of malicious nodes easily distorts the trust evaluation results only based on the one-to-one relationship between nodes, which greatly reduces the security of data transmission. To balance the excessive energy loss of mobile devices and improve the reliability of data transmission, this article proposes an energy-constrained forwarding scheme based on group trustworthiness between communities (ECF-GT) for mobile IoT, which does everything possible to increase the reliability of data forwarding by improving the accuracy of relay selection. First, it leverages nodal residual energy ratio and two common social attributes between nodes to measure the unidirectional trustworthiness between devices, and constructs the community structure by using our published Dynamic Discovery of Trustworthiness Overlapping Community method. Then, based on the one-to-many, many-to-one, many-to-many trustworthiness between nodes and/or communities, a group dynamic trustworthiness assessment method is proposed to enhance the reliability of evaluating relays. Finally, ECF-GT combines nodal residual energy and group trustworthiness to design forwarding utilities and rules. The experimental results show the effectiveness of ECF-GT. For example, when the number of malicious nodes accounts for 25% of the total number of nodes, ECF-GT can save about 17.3% energy on average comparing to other three well-known forwarding models on Reality dataset.
Consensus reaching is an iterative and dynamic process that supports group decision-making models by guiding decision-makers towards modifying their opinions through a feedback mechanism. Many attempts have been recently devoted to the design of efficient consensus reaching processes, especially when the dynamism is dependent on time, which aims to deal with opinion dynamics models. The emergence of novel methodologies in this field has been accelerated over recent years. In this regard, the present work is concerned with a systematic review of classical dynamic consensus and opinion dynamics models. The most recent trends of both models are identified and the developed methodologies are described in detail. Challenges of each model and open problems are discussed and worthwhile directions for future research are given. Our findings denote that due to technological advancements, a majority of recent literature works are concerned with the large-scale group decision-making models, where the interactions of decision-makers are enabled via social networks. Managing the behavior of decision-makers and consensus reaching with the minimum adjustment cost under social network analysis have been the top priorities for researchers in the design of classical consensus and opinion dynamics models.
With increasing attention paid to the application of agent-based modeling in decision-making issues, a large number of related studies have been published in management science and operational research areas. This study adopts multiple methods, including bibliometric mapping, text mining, and qualitative analysis, to comprehensively review relevant research to explore knowledge frontiers and evolution in decision-making research based on agent-based modeling (DM-ABM). This review is based on 1190 relevant journal articles from the Web of Science Core Collection dataset and 37,167 collectively cited references of the articles. The top ten most-cited studies that constitute the intellectual milestones of DM-ABM were identified. Keywords and research topics develop rapidly; recent research have paid most attention to the keywords “model,” “system,” and “simulation” and topics “learning,” “contracts,” “protocols,” and “self-learning.” The top 24 references with the strongest citation bursts were displayed to show that the area was increasingly active from 2001 to 2010. Transition points were mapped to reveal the top five studies with the highest betweenness centrality, which considerably influences knowledge evolution. Then, the top three clusters are identified as the frontier areas and analyzed by text mining, including intelligent agents, model validation, and collaborative decision making. Finally, the most recent research in this field is investigated, and four future research directions are proposed: the advanced intelligence of agents, approach to reality, group decision making, innovative modeling methodologies and diversified applications.
Purpose The purpose of this study is to build a consensus model of social network group decision-making (SNGDM) based on improved PageRank algorithm. By objectively and fairly measuring the evaluation ability of participants in the decision-making process, the authors can improve the fairness and authenticity of the weight solution of decision-makers (DM) in the decision-making process. This ensures the reliability of the final group consensus results. Design/methodology/approach This study mainly includes six parts: preference expression, calculation of DM's weight, preference aggregation, consensus measurement, opinion adjustment and alternative selection. First, Pythagorean fuzzy expression is introduced to express the preference of DMs, which expands the scope of preference expression of DMs. Second, based on the social network structure among DMs, the process of “mutual judgment” among DMs is increased to measure the evaluation ability of DMs. On this basis, the PageRank algorithm is improved to calculate the weight of DMs. This makes the process of reaching consensus more objective and fair. Third, in order to minimize the evaluation difference between groups and individuals, a preference aggregation model based on plant growth simulation algorithm (PGSA) is proposed to aggregate group preferences. Fourth, the consensus index of DMs is calculated from three levels to judge whether the consensus degree reaches the preset value. Fifth, considering the interaction of DMs in the social network, the evaluation value to achieve the required consensus degree is adjusted according to the DeGroot model to obtain the overall consensus. Finally, taking the group preference as the reference, the ranking of alternatives is determined by using the Pythagorean fuzzy score function. Findings This paper proposes a consensus model of SNGDM based on improved PageRank algorithm to aggregate expert preference information. A numerical case of product evaluation is introduced, and the feasibility and effectiveness of the model are explained through sensitivity analysis and comparative analysis. The results show that this method can solve the problem of reaching consensus in SNGDM. Originality/value Different DMs may have different judgment criteria for the same decision-making problem, and the angle and depth of considering the problem will also be different. By increasing the process of mutual evaluation of DMs, the evaluation ability of each DM is judged only from the decision-making problem itself. In this way, the evaluation opinions recognized by most DMs will form the mainstream of opinions, and the influence of corresponding DMs will increase. Therefore, in order to improve the fairness and reliability of the consensus process, this study measures the real evaluation ability of DMs by increasing the “mutual judgment” process. On this basis, the defect of equal treatment of PageRank algorithm in calculating the weight of DMs is improved. This ensures the authenticity and objectivity of the weight of DMs. That is to improve the effectiveness of the whole evaluation mechanism. This method considers both the influence of DMs in the social network and their own evaluation level. The weight of DMs is calculated from two aspects: sociality and professionalism. It provides a new method and perspective for the calculation of DM’s weight in SNGDM.
In order to improve the reliability of alternative ranking and group consensus adaptively, this paper develops an automatically interactive group decision-making (GDM) framework under the VIKOR environment. The proposed framework contains an interaction network mechanism based on social network analysis (SNA) and an opinion interaction mechanism based on Brain Strom Optimization (BSO) algorithm. Firstly, to connect isolated decision-makers (DMs), interaction relationships among DMs are built based on the SNA and similarity measurement. Secondly, the BSO algorithm is introduced to design the opinion interaction mechanism, including subgroup identification model and preference iteration model. The preference iteration model contains an adaptive preference update rule (UR), iteration stop rule (ISR), and solution selection rule (SSR) to enhance group consensus adaptively rather than force DMs to change the preference. Thirdly, the proposed interactive group VIKOR (G-VIKOR) method is applied in SARS-cov-2 nucleic acid detection sites evaluation. Finally, consistency and comparison analysis are developed to illustrate the validity and robustness of evaluation results.
The trust degrees among individuals play an important role in social network group decision making (SNGDM). They are assumed to be unchanged with the time in most of extant SNGDM studies. However, in some SNGDM scenarios, the trust degrees of individuals will evolve with the time due to the changes of the opinion similarities among individuals. In this paper, we discuss the consensus reaching issue in SNGDM with trust evolution. Specifically, we first formulate a consensus reaching problem with trust evolution, and then propose its resolution framework. In this framework, we establish the trust evolution model, in which the trust degrees among individuals at the next time are determined by the individuals’ historical trust degrees and the opinion similarities at this time. Based on this, we design a novel consensus reaching process, called CRP-OD-TE, which first includes the opinion dynamics-based endogenous feedback mechanism, and the trust evolution-based exogenous feedback mechanism. Furthermore, two applications are provided to illustrate the proposed CRP-OD-TE. Finally, we complete some simulations and comparative analysis. The results show that (1) the introduction of opinion dynamics into CRP as the endogenous feedback mechanism can accelerate the process of consensus reaching; and (2) the evolution of trust degrees has a significant impact on the group consensus reaching, which may be beneficial or destructive.
The energy storage process becomes very important due to the imbalances in energy supply and demand. Therefore some factors need to be considered to increase the efficiency of the energy storage system, such as cost–benefit analysis and technological improvements. This study aims to examine the inventive problem-solving capacities for renewable energy storage investments. A new model is suggested for this objective by considering fuzzy decision-making methodology. It is concluded that prior action and dynamicity are the most essential capacities of renewable energy storage investments. Additionally, dynamicity plays the most critical role when all factors are considered in renewable energy investment projects in a collaborative manner. Hence, it is recommended that the companies should mainly consider the initial developments of the storage facilities. Moreover, location selection for effective energy storage should also be considered to increase the performance of these investments.
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Nowadays, in the social network–based decision‐making processes, like the ones involved in e‐commerce and e‐democracy, multiple users with different backgrounds may take part and diverse alternatives might be involved. This diversity enriches the process, but at the same time, increases the uncertainty of opinions. This uncertainty can be considered from two different perspectives: (i) the uncertainty in the meaning of the words given as preferences, that is, motivated by the heterogeneity of the decision makers; and (ii) the uncertainty inherent to any decision‐making process that may lead to an expert not being able to provide all their judgments. The main objective of this study is to address these two types of uncertainty. To do so, the following approaches are proposed: First, to capture, process, and keep the uncertainty in the meaning of the linguistic assumption, the Interval Type‐2 Fuzzy Sets are introduced as a way to model the experts' linguistic judgments. Second, a measure of the coherence of the information provided by each decision maker is proposed. Finally, a consistency‐based completion approach is introduced to deal with the uncertainty presented in the expert judgments. The proposed approach is tested in an e‐democracy decision‐making scenario.
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On-line platforms foster the communication capabilities of the Internet to develop large-scale influence networks in which the quality of the interactions can be evaluated based on trust and reputation. So far, this technology is well known for building trust and harnessing cooperation in on-line marketplaces, such as Amazon ( and eBay ( However, these mechanisms are poised to have a broader impact on a wide range of scenarios, from large scale decision making procedures, such as the ones implied in e-democracy, to trust based recommendations on e-health context or influence and performance assessment in e-marketing and e-learning systems. This contribution surveys the progress in understanding the new possibilities and challenges that trust and reputation systems pose. To do so, it discusses trust, reputation and influence which are important measures in networked based communication mechanisms to support the worthiness of information, products, services opinions and recommendations. The existent mechanisms to estimate and propagate trust and reputation, in distributed networked scenarios, and how these measures can be integrated in decision making to reach consensus among the agents are analysed. Furthermore, it also provides an overview of the relevant work in opinion dynamics and influence assessment, as part of social networks. Finally, it identifies challenges and research opportunities on how the so called trust based network can be leveraged as an influence measure to foster decision making processes and recommendation mechanisms in complex social networks scenarios with uncertain knowledge, like the mentioned in e-health and e-marketing frameworks.
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This article aims to study the influence of the group attitude on the consensus reaching process in group decision making (GDM). To do that, the attitudinal consensus index (ACI) is defined to aggregate individual consensus levels to form a collective one. This approach allows for the implementation of the group attitude in a continuous state ranging from a pessimistic attitude to an optimistic attitude. Then, ACI is used to build a stop policy to control feedback for consensus, which can be regarded as a generation of the traditional polices: ‘minimum disagreement policy’ and ‘indifferent disagreement policy’. A sensitivity analysis method with visual simulation is proposed to check the adjustment cost and consensus level with different attitudinal parameters. The main conclusion from this analysis is that the bigger the attitudinal parameter implemented is, the bigger the adjustment cost and consensus level are. The visual information facilitates the inconsistent expert keeping a balance between the attitudinal parameter to implement and the adjustment cost and consensus level, which in practice translates into full control of such implementation based on the decision maker's willingness.
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Nowadays we are living the apogee of the Internet based technologies and consequently web 2.0 communities, where a large number of users interact in real time and share opinions and knowledge, is a generalized phenomenon. This type of social networks communities constitute a challenge scenario from the point of view of Group Decision Making approaches, because it involves a large number of agents coming from different backgrounds and/or with different level of knowledge and influence. In these type of scenarios there exists two main key issues that requires attention. Firstly, the large number of agents and their diverse background may lead to uncertainty and or inconsistency and so, it makes difficult to assess the quality of the information provided as well as to merge this information. Secondly, it is desirable, or even indispensable depending on the situation, to obtain a solution accepted by the majority of the members or at least to asses the existing level of agreement. In this contribution we address these two main issues by bringing together both decision Making approaches and opinion dynamics to develop a similarity-confidence-consistency based Social network that enables the agents to provide their opinions with the possibility of allocating uncertainty by means of the Intuitionistic fuzzy preference relations and at the same time interact with like-minded agents in order to achieve an agreement.
Ordered Weighted Averaging (OWA) operators, a family of aggregation functions, are widely used in human decision-making schemes to aggregate data inputs of a decision maker's choosing through a process known as OWA aggregation. The weight allocation mechanism of OWA aggregation employs the principle of linear ordering to order data inputs after the input variables have been rearranged. Thus, OWA operators generally cannot be used to aggregate a collection of n inputs obtained from any given convex partially ordered set (poset). This poses a problem since data inputs are often obtained from various convex posets in the real world. To address this problem, this paper proposes methods that practitioners can use in real-world applications to aggregate a collection of n inputs from any given convex poset. The paper also analyzes properties related to the proposed methods, such as monotonicity and weighted OWA aggregation on convex posets.
Group Recommender Systems are special kinds of Recommender Systems aimed at suggesting items to groups rather than individuals taking into account, at the same time, the preferences of all (or the majority of) members. Most existing models build recommendations for a group by aggregating the preferences for their members without taking into account social aspects like user personality and interpersonal trust, which are capable of affecting the item selection process during interactions. To consider such important factors, we propose in this paper a novel approach to group recommendations based on fuzzy influence-aware models for Group Decision Making. The proposed model calculates the influence strength between group members from the available information on their interpersonal trust and personality traits (possibly estimated from social networks). The estimated influence network is then used to complete and evolve the preferences of group members, initially calculated with standard recommendation algorithms, toward a shared set of group recommendations, simulating in this way the effects of influence on opinion change during social interactions. The proposed model has been experimented and compared with related works.
In large-scale group decision making (GDM), non-cooperative behavior in the consensus reaching process (CRP) is not unusual. For example, some individuals might form a small alliance with the aim to refuse attempts to modify their preferences or even to move them against consensus to foster the alliance's own interests. In this paper, we propose a novel framework based on a self-management mechanism for non-cooperative behaviors in large-scale consensus reaching processes (LCRPs). In the proposed consensus reaching framework, experts are classified into different subgroups using a clustering method, and experts provide their evaluation information, i.e., the multi-criteria mutual evaluation matrices (MCMEMs), regarding the subgroups based on subgroups' performance (e.g., professional skills, cooperation, and fairness). The subgroups' weights are dynamically generated from the MCMEMs, which are in turn employed to update the individual experts' weights. This self-management mechanism in the LCRP allows penalizing the weights of the experts with non-cooperative behaviors. Detailed simulation experiments and comparison analysis are presented to verify the validity of the proposed framework for managing non-cooperative behaviors in the LCRP.
Decision situations in which several individual are involved are known as group decision-making (GDM) problems. In such problems, each member of the group, recognizing the existence of a common problem, tries to come to a collective decision. A high level of consensus among experts is needed before reaching a solution. It is customary to construct consensus measures by using similarity functions to quantify the closeness of experts preferences. The use of a metric that describes the distance between experts preferences allows the definition of similarity functions. Different distance functions have been proposed in order to implement consensus measures. This paper examines how the use of different aggregation operators affects the level of consensus achieved by experts through different distance functions, once the number of experts has been established in the GDM problem. In this situation, the experimental study performed establishes that the speed of the consensus process is significantly affected by the use of diverse aggregation operators and distance functions. Several decision support rules that can be useful in controlling the convergence speed of the consensus process are also derived.
This study is concerned with group decision making contexts in which linguistic preference relations are used to provide the evaluations of results. On the one hand, granulation of linguistic terms, which are used as entries of the preference relations, is carried out for the purpose of dealing with the linguistic information. Formally, the problem is expressed as a multi-objective optimization task in which a performance index composed of the weighted averaging of the criteria of consensus and consistency is maximized via an appropriate association of the linguistic terms with information granules formed as intervals. On the other hand, once the linguistic terms are made operational by mapping them to the corresponding intervals, a selection process, in which the consistency achieved by each agent is also considered, is employed with intent to construct the solution to the decision problem under consideration. An experimental study is reported by demonstrating the main features of the proposed approach. Furthermore, some drawbacks and advantages are also analyzed.
The hesitant fuzzy preference relation (HFPR) is a useful tool for decision makers to elicit their preference information over a set of alternatives. In this paper, it is first proposed an approach to deriving a priority weight vector from an incomplete HFPR using the logarithmic least squares method. Based on the priority weight vector, the consistency index of an incomplete HFPR is defined, which calculates the average deviation between the priority weight vector and all elements of the incomplete HFPR. For an incomplete HFPR which is unacceptably consistent, an automatic algorithm is developed to improve the consistency. These results are then extended to propose a new procedure for group analytic hierarchy process to deal with multi-criteria group decision making problems. The feasibility and effectiveness of the proposed approaches are demonstrated by some numerical examples.