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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
uncertainty
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
Abstract
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
phenomenon.
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: taghavi.atefe@gmail.com (Atefeh Taghavi), Esfandiar.Eslami@uk.ac.ir
(Esfandiar Eslami), viedma@decsai.ugr.es (Enrique Herrera-Viedma*), raquel.urena@dmu.ac.uk (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
defined.
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
data.
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
2
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:
3
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
when
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:
mrk
ij =rik ·rkj ·rji
rjk ·rki
(1)
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 =
P
kR01
ij
mrk
ij
#R01
ij
; (2)
where R01
ij ={k6=i, j|(rik , rkj )/R01},R01 ={(1,0),(0,1)}, and #R01
ij is the cardinality of
R01
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.
CLi=
n
X
j=1; i6=j
CLij
n1.(4)
Level 3. Consistency Level of a fuzzy preference relation.
CL =
n
X
i=1
CLi
n.(5)
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:
4
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
i=
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
versa.
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)
5
If Bis reciprocal, i.e. C F Lij =CF Lji ,i, j, we have
CF Li=Pn
j=1,j6=iC F Lij
n1(8)
Definition 10 (Experts’ confidence for a relation [8]).For a reciprocal intuitionistic fuzzy
preference relation B, the confidence level is:
CF LB=Pn
i=1 CF Li
n(9)
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
ij|.
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:
cai=Pn
j=1;j6=i(cpij +cpji)
2(n1) (11)
(c) Consensus degree on the relation, called cr, is:
cr =Pn
i=1 cai
n(12)
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.
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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) =
m
X
i=1
wi.pσ(i)(13)
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
advice.
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
7
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
2.(14)
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).
8
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:
Bk=
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:
Rk=
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 ) =
(hµtcc
ij , νtcc
ij i)is computed as follows:
µtcc
ij =Φtcc
w(hT CCI1, µ1
iji, ..., hT C CI m, µm
ij i) =
m
X
h=1
whσ(h)
ij (15)
νtcc
ij =Φtcc
w(hT CCI1, ν1
iji, ..., hT C CI m, νm
ij i) =
m
X
h=1
whσ(h)
ij (16)
9
T CCIh= (δ1).T Sh+ (δ2).CLh+ (δ3).C F Lh,(17)
where
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:
wh=Q(Ph
i=1 T CCIσ(i)
T)Q(Ph1
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
consistency
measures
Computing
confidence
measures
Computing Trust confidence
consistency Index
(TCCI)
Computing the weighted vector
associated to the IOWA operator (𝒘𝒉)
(TCC-IOWA )
Clustering the experts by using
KNN algorithm
Consensus process
containing feedback
mechanism
Computing
Trust Score
Experts
discussion
Preferences
Intuitionistic Fuzzy
Preference Relations
(IFPR)
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.
10
High trusted experts
Medium trusted experts
Low trusted experts
Consensus Process
Feedback Mechanism
Check
Consensus
Level
Low
Consensus
Level
Selection Process
Ranked Trusted Experts
Discussion
Preferences
High
Consensus
Level
Problem
Figure 2: Feedback mechanism.
Based on the given facts, the proposed approach is presented in detail in the following
subsections:
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:
µk
ij =X
hNk
whh
ij (19)
νk
ij =X
hNk
whh
ij (20)
11
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
i=1(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=
n
X
i=1
cai
n
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:
P CHh
med ={(i, j)P|sah
i< β1}
where, β1=Pm
h=1 sah
i/m,ehEmed.
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:
P CHh
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.
12
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.
P1=
(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)
P2=
(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)
P3=
(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)
P4=
(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:
R1=
0.5 0.4n0.2
0.3 0.5n0.3
n n 0.5n
0.7 0.7n0.5
R2=
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
R3=
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
13
R4=
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:
MR1=
0.11 0.7
1.05 0.08
0.2 1.4
MR2=
0.42 0.33 0.3
0.43 0.46 0.31
0.34 0.39 0.4
0.4 0.37 0.56
MR3=
0.44 0.4 0.53
0.46 0.48 0.56
0.24 0.38 0.24
0.23 0.38 0.57
MR4=
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
14
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:
P1=
(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.
R1=
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:
A={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}
The set of preferences to be changed by each expert ehELow is: P CHh
Low =A. Since
Elow ={e1}, so we have:
R1=
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:
15
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
R4=
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:
R1=
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
R2=
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
R3=
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
R4=
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
16
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.
Acknowledgments
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.
17
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... In 1965, Zadeh (Zadeh 1965) presented fuzzy sets (FSs) theory to model the uncertain information. So far the applications of FSs have spread to many fields (Eftekhari et al. 2022;Mokhtia et al. 2021Mokhtia et al. , 2020Saberi-Movahed et al. 2022;Taghavi et al. 2020). FSs are expressed by a degree of membership T A ðxÞ which indicates the degree of belonging of an element x to the set A. To take the non-membership degree into, (Atanassov 1986) proposed an intuitionistic fuzzy set (IFS) by adding a non-membership function to FS. IFS is an extension of FSs, and it has two main functions such as the membership function T A ðxÞ and the non-membership function F A ðxÞ. ...
... Definition 3 (Taghavi et al. 2020;Torra 2009) Let X be a fixed set; a hesitant fuzzy set (HFS) M on X is in terms of a function that when applied to X returns a subset of ½0; 1; which can be represented as the following mathematical symbol: Recently, (Ye 2014) defined a single-valued neutrosophic hesitant fuzzy set (SVNHFS) as combination of the SVNS and HFS as follows: ...
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In order to handle simultaneously the cardinal and ordinal information in decision-making process, QUALIFLEX (QUALItative FLEXible multiple criteria method) is a very well-known decision-making approach. In this work, we extend the classical QUALIFLEX method to neutrosophic environment and develop a neutrosophic QUALIFLEX (N-QUALIFLEX) method that uses the newly defined distance-based comparison approach. It is highly effective in solving multi-criteria decision problems in which both ratings of alternatives on criteria and weights of criteria are single-valued neutrosophic numbers (SVNNs), and their aggregated values are single-valued neutrosophic hesitant fuzzy numbers (SVNHFNs). A neutrosophic hesitancy index (NHI) of a SVNHN is introduced based on degrees of the truth-membership, indeterminacy-membership and falsity-membership, which is used to measure the degree of hesitancy of SVNHN. Considering the NHIS of SVNHFNs, we propose a distance-based comparison approach to determine the magnitude of the SVNHFNs. Then, we apply the comparison approach to define the concordance/discordance index, the weighted concordance/discordance index and the comprehensive concordance/discordance index that are steps of the developed N-QUALIFLEX. By taking all possible permutations of alternatives with respect to the level of concordance/discordance into account, we determine the order of alternatives in final decision. Finally, a practical example on antivirus mask selection over the COVID-19 pandemic is provided to present the effectiveness and applicability of the proposed method, and a comparative study is conducted to show the advantages of the proposed method over other existing methods.
... We compare this study with the existing group decisionmaking methods as shown in Table 6. Since Chu et al. [37] make large-group decision-making, the lack of unknown information in Taghavi et al. [42] is inconsistent with this study, therefore, not given the sorting results. A comprehensive comparison of the ranking results shows that the ranking results of the alternatives obtained in this article and other methods are basically the same, and the optimal solutions are the same. ...
... 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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... 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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