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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×X→D, 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

k∈R01

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

n−1.(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

0≤µB(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 = 0.5∀i∈ {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:X−→ Yis 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 eh∈E, does not provide any single preference value

for at least one alternative xi∈X:

∃i, h|Rh

i=∅

where Rh

iis the i−th row and i−th 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(n−1) (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

n−1(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, ..., m−1, 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 (m−1) ×(m−2) 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(n−1) (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.

6

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)m−→ Rto 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+d≤1 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) = t−d+ 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 −t−d)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;γ1≺γ2; 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

wh.µσ(h)

ij (15)

νtcc

ij =Φtcc

w(hT CCI1, ν1

iji, ..., hT C CI m, νm

ij i) =

m

X

h=1

wh.νσ(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(Ph−1

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

h∈Nk

wh.µh

ij (19)

νk

ij =X

h∈Nk

wh.νh

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|λ1≤T Si< λ2},Ehigh ={ei|λ2≤T Si}.

•Identify low-trusted experts’ preferences to be changed:

1. Computing the threshold α1;α1=Pn

i=1(Pn

j=1,j6=icpij )/(n2−n).

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 eh∈ELow

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)|i∈X 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,eh∈Emed.

•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)|i∈X 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< β1∧sph

ij < β2}

where, β2=Pm

h=1 sph

ij/m ,eh∈Ehigh.

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 eh∈ELow 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

References

[1] E. Herrera-Viedma, S. Alonso, F. Chiclana, and F.Herrera, “A consensus model for group

decision making with incomplete fuzzy preference relations,” IEEE Transactions on Fuzzy

Systems, vol. 15, no. 5, pp. 863–877, 2007.

[2] F. Mata, L. Martinez, and E. Herrera-Viedma, “An adaptive consensus support model

for group decision-making problems in a multigranular fuzzy linguistic context,” IEEE

Transactions on Fzzy Systems, vol. 17, no. 2, 2009.

[3] I. J. Perez, F. J. Cabrerizo, S. Alonso, and E. Herrera-Viedma, “A new consensus model

for group decision making problems with non-homogeneous experts,” IEEE Transactions

on Systems, Man, and Cybernetics: Systems, vol. 44, no. 4, 2014.

[4] J. Wu, Q. Sun, H. Fujita, and F. Chiclana, “An attitudinal consensus degree to control

feedback mechanism in group decision making with diﬀerent adjustment cost,” Knowledge-

Based Systems, vol. 164, pp. 265–273, 2019.

[5] N. Capuano, F. Chiclana, E. Herrera-Viedma, H. Fujita, and V. Loia, “Fuzzy group

decision making for inﬂuence-aware recommendations,” Computers in Human Behavior,

vol. 101, pp. 371–379, 2019.

[6] R. Ure˜na, F. Chiclana, G. Melan¸con, and E. Herrera-Viedma, “A social network based

approach for consensus achievement in multiperson decision making,” Information Fusion,

vol. 47, pp. 72–87, 2019.

[7] W. Liu, Y. Dong, F. Chiclana, F. Cabrerizo, and E. Herrera-Viedma, “Group decision-

making based on heterogeneous preference relations with self-conﬁdence,” Fuzzy Optimiza-

tion and Decision Making, vol. 16:4, pp. 429–447, 2017.

[8] R. Ure˜na, F. Chiclana, H. Fujita, and E. Herrera-Viedma, “Conﬁdence-consistency driven

group decision making approach with incomplete reciprocal intuitionistic preference rela-

tions,” Knowledge-Based Systems, vol. 89, pp. 86 – 96, 2015.

[9] S. Alonso, E. Herrera-Viedma, F. Chiclana, and f. Herrera, “Individual and social strategies

to deal with ignorance situations in multi-person decision making,” International Journal

of Information Technology and decision Making, vol. 8, pp. 313–333, 2009.

[10] E. Herrera-Viedma, F. Chiclana, F.Herrera, and S. Alonso, “Group decision-making model

with incomplete fuzzy preference relations based on additive consistency,” IEEE Transac-

tions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 37, no. 1, pp. 176–189,

2007.

[11] R. Ure˜na, F. Chiclana, J. Morente-Molinera, and E. Herrera-Viedma, “Managing incom-

plete preference relations in decision making: A review and future trends,” Information

Sciences, vol. 302(0), pp. 14 – 32, 2015.

[12] J. Wu and F. Chiclana, “Multiplicative consistency of intuitionistic reciprocal prefer-

ence relations and its application to missing values estimation and consensus building,”

Knowledge-Based Systems, vol. 71, no. 0, pp. 187 – 200, 2014.

[13] Z. S. Xu, “Incomplete linguistic preference relations and their fusion,” Information Fusion,

vol. 7, no. 3, pp. 331–337, 2006.

18

[14] Z. Xu, X. Cai, and E. Szmidt, “Algorithms for estimating missing elements of incomplete

intuitionistic preference relations,” International Journal of Intelligent Systems, vol. 26,

no. 9, pp. 787–813, 2011.

[15] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, pp. 87–96,

1986.

[16] K. T. Atanassov, On intuitionistic fuzzy sets theory. Springer-Verlag Berlin Heidelberg,

2012.

[17] S. Alonso, F. Cabrerizo, F. Chiclana, F. Herrera, and E. Herrera-Viedma, “Group decision

making with incomplete fuzzy linguistic preference relations,” International Journal of

Intelligent Systems, vol. 24, pp. 201–222, 2009.

[18] G. Kossinets, “Eﬀects of missing data in social networks,” Social Networks, vol. 28, pp. 247–

268, 2006.

[19] M. Huisman, “Imputation of missing network data: Some simple procedures,” Journal of

Social Structure, vol. 10, no. 1, 2009.

[20] R.Ure˜na, G. Kou, J. Wu, F. Chiclana, and E. Herrera-Viedma, “Dealing with incomplete

information in linguistic group decision-making by means of interval type-2 fuzzy sets,”

International Journal of Intelligent Systems, pp. 1–20, 2019.

[21] R. Burt, “A note on missing network data in the general social survey,” Social networks,

vol. 9, pp. 63–73, 1987.

[22] R. M. Hogarth and H. Kunreuther, “Decision making under ignorance: arguing with your-

self,” Journal of Risk and Uncertainty, vol. 10, pp. 15–36, 1995.

[23] J. L. Schafer and J. W. Graham, “Missing data: our view of the state of the art,” Psycho-

logical Methods, vol. 7, no. 2, pp. 147–177, 2002.

[24] A. Stomakhin, M. B. Short, and A. L. Bertozzi, “Reconstruction of missing data in social

networks based on temporal patterns of interactions,” Inverse Problems, vol. 27, no. 11,

2011.

[25] N. Capuano, F. Chiclana, H. Fujita, E. Herrera-Viedma, and V. Loia, “Fuzzy group de-

cision making with incomplete information guided by social inﬂuence,” EEE Trans. On

Fuzzy Systems, vol. 26, no. 3, pp. 1704–1718, 2018.

[26] R. Ure˜na, G. Kou, Y. Dong, F. Chiclana, and E. Herrera-Viedma, “A review on trust prop-

agation and opinion dynamics in social networks and group decision making frameworks,”

Information Sciences, vol. 478, pp. 461 – 475, 2019.

[27] I. Millet, “The eﬀectiveness of alternative preference elicitation methods in the analytic

hierarchy process,” Journal of Multi-Criteria Decision Analysis, vol. 6, no. 1, pp. 41–51,

1997.

[28] E. Szmidt and J. Kacprzyk, “Using intuitionistic fuzzy sets in group decision making,”

Control Cybernet, vol. 4, no. 31, pp. 1037–1053, 2002.

[29] M. del Moral, F. Chiclana, J. Tapia, and E. Herrera-Viedma, “A comparative study on

consensus measures in group decision making,” Int. J. of Intelligent Systems, vol. 33(8),

pp. 1624–1638, 2018.

19

[30] R. R. Yager, “On ordered weighted averaging aggregation operators in multicriteria deci-

sion making,” IEEE Trans. Syst. Man Cybern., vol. 18, pp. 183–190, Jan. 1988.

[31] R. R. Yager, “Induced aggregation operators,” Fuzzy Sets and Systems, vol. 137, pp. 59–69,

2003.

[32] O. Sutton, Introduction to k Nearest Neighbour Classiﬁcation and Condensed Nearest

Neighbour Data Reduction. University lectures, University of Leicester, 2012.

[33] F. Cabrerizo, J. Morente-Molinera, W. Pedrycz, A. Taghavi, and E. Herrera-Viedma,

“Granulating linguistic information in decision making under consensus and consistency,”

Expert Systems With Applications, vol. 99, pp. 83–92, 2018.

[34] Y. Dong, S. Zhao, H. Zhang, F. Chiclana, and E. Herrera-Viedma, “A self-management

mechanism for non-cooperative behaviors in large-scale group consensus reaching pro-

cesses,” IEEE Trans. On Fuzzy Systems, vol. 26, pp. 3276–3288.

[35] C. Cariou and K. Chehdi, “Unsupervised nearest neighbors clustering with application to

hyperspectral images,” IEEE Journal of Selected Topics in Signal Processing, vol. 9, no. 6,

pp. 1105–1116, 2015.

[36] A. X. Lu and A. M. Moses, “An unsupervised knn method to systematically detect changes

in protein localization in high-throughput microscopy images,” PLoS ONE, vol. 11, no. 7,

2016.

[37] S. Vajda and K. Santosh, “A fast k-nearest neighbor classiﬁer using unsupervised clus-

tering,” Santosh K., Hangarge M., Bevilacqua V., Negi A. (eds) Recent Trends in Image

Processing and Pattern Recognition, RTIP2R 2016, Springer, Singapore, vol. 709, 2017.

[38] M. D. Cock and P. P. D. Silva, “A many valued representation and propagation of trust

and distrust,” Lect. Notes Comput. Sci., vol. 3849, pp. 108–113, 2006.

[39] P. Victor, C. Cornelis, M. D. Cock, and E. Herrera-Viedma, “Practical aggregation oper-

ators for gradual trust and distrust,” Fuzzy Sets Syst., vol. 184, pp. 126–147, 2011.

[40] J. Wu, F. Chiclana, and E. Herrera-Viedma, “Trust based consensus model for social

network in an incomplete linguistic information context,” Applied Soft Computing, vol. 35,

pp. 827–839, 2015.

[41] Y. Liu, C. Liang, F. Chiclana, and J. Wu, “A trust induced recommendation mechanism

for reaching consensus in group decision making,” Knowledge-Based Systems, vol. 119,

pp. 221–231, 2017.

[42] J. Wu, F. Chiclana, H. Fujita, and E. Herrera-Viedma, “A visual interaction consensus

model for social network group decision making with trust propagation,” Knowledge–Based

Systems, vol. 000, pp. 1–12, 2017.

[43] Z. Zhang, X. Kou, W. Yu, and C. Guo, “On priority weights and consistency for incomplete

hesitant fuzzy preference relations,” Knowledge-Based Systems, vol. 143, pp. 115–126, 2018.

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