Content uploaded by Nannan Wu
Author content
All content in this area was uploaded by Nannan Wu on Sep 06, 2020
Content may be subject to copyright.
Applied Soft Computing Journal 93 (2020) 106363
Contents lists available at ScienceDirect
Applied Soft Computing Journal
journal homepage: www.elsevier.com/locate/asoc
Water–Energy–Food nexus evaluation with a social network group
decision making approach based on hesitant fuzzy preference relations
Nannan Wu a, Yejun Xu a,∗, Xia Liu a,b, Huimin Wang a, Enrique Herrera-Viedma c,d
aBusiness School, Hohai University, Nanjing 211100, PR China
bAndalusian Research Institute in Data Science and Computational Intelligence (DaSCI), University of Granada, Granada 18071, Spain
cDepartment of Computer Science and Artificial Intelligence, University of Granada, 18071, Spain
dDepartment of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia
article info
Article history:
Received 18 January 2019
Received in revised form 24 October 2019
Accepted 28 April 2020
Available online 8 May 2020
Keywords:
Water–Energy–Food
Social network group decision making
Trust relationship
Self-confidence
Hesitant fuzzy preference relations
abstract
With the rapid increase and development of global population and economic, the Water–Energy–Food
(WEF) nexus evaluation which is related to the sustainable development of human has become a
hotspot. Whereas, the study of the WEF nexus evaluation from the perspectives of social network group
decision making (SNGDM) is still a challenge. Hence, this paper aims to develop a trust-based SNGDM
approach with hesitant fuzzy preference relations to the WEF nexus evaluation. In the proposed model,
a new fuzzy adjacency matrix and trust score matrix based on expert’s self-confidence are defined to
imply experts’ trust relationship and trust score, respectively. To improve the reliability of the final
decision(s), an iterative algorithm is presented to improve the consistency of experts’ evaluations.
Subsequently, the individual evaluation can be aggregated into a group one by using the trust score
induced ordered weighted averaging operator while the trust scores of experts are the induced factors.
Additionally, an algorithm is utilized to achieve a high level consensus in SNGDM of the WEF nexus
evaluation. Finally, some comparison analyses and discussions show the feasibility and validity of the
proposed method.
©2020 Elsevier B.V. All rights reserved.
1. Introduction
Nowadays, a rapidly rising global population and growing
prosperity are putting unsustainable pressures on resources and
environment. Recently, many researchers have increasingly em-
phasized the importance of the complex relationships among
water, energy and food (called WEF nexus) [1,2]. The WEF nexus,
which was propelled into international discussions on sustainable
development in the Bonn conference of 2011 [3], highlights the
importance of integrative solutions that secure resource supplies
and meet demands sustainably.
As far as we know, one of the most crucial problems con-
cerning the WEF nexus evaluation is how to choose a reasonable
alternative of allocating water resources to energy industry and
food industry. Furthermore, in the management system of coor-
dinating water, energy and food, experts may involve multiple
government departments. Generally, different expert may have
different culture and knowledge backgrounds, the decision mak-
ing and negotiation are important steps to obtain the alternative
that is most consistent with the interests of the people. Thus,
in some cases, the WEF nexus evaluation actually can be seen
∗Corresponding author.
E-mail address: xuyejohn@hhu.edu.cn (Y. Xu).
as a group decision making (GDM) problem [4]. That means, in
real WEF nexus evaluation cases, multiple experts are invited to
participate into the decision making process, provide their evalu-
ations on a finite set of alternatives (possible solutions), and then
to obtain a best alternative by aggregating their evaluations [5–
11].
With the development of information and network technol-
ogy, experts involved in a GDM process may have previous his-
tory of interaction or social network connection with others
and developed opinions rely on the reliability of other experts’
evaluations [12–14]. Hence, the GDM based on social network
analysis (SNA) which can be called social network group decision
making (SNGDM) has been received increasing attention [15–19].
In an SNGDM problem, the relationships among social objects
with social network connections, such as experts of an institu-
tion, employees of a corporation, and members of the United
Nations, can be analyzed by SNA [20–22]. In addition, prefer-
ence relation is an effective tool to express expert’s preference
information over the set of alternatives in SNGDM. Up to now,
many preference relations have been introduced as well as apply
to decision making events [23–27]. However, in real decision
making situations, experts may have different cultures, educa-
tional backgrounds, and personal interest preferences. There are
also fuzzy and hesitant natures in human judgment. Thus, when
https://doi.org/10.1016/j.asoc.2020.106363
1568-4946/©2020 Elsevier B.V. All rights reserved.
2N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
expressing their evaluations, experts may have several possible
values and hesitate before giving decisions. To deal with the
expert’s fuzziness and hesitancy, the hesitant fuzzy preference
relation (HFPR) have been proposed as well as applied to GDM
problems [7,28,29]. In an HFPR, expert’s evaluation consists of
hesitant fuzzy elements (HFEs), which denote all possible pref-
erence values and can be utilized to effectively express expert’s
hesitant and fuzzy information in GDM problems [7,28–30].
To date, many studies related to SNGDM have been presented.
For example, Brunelli, et al. [31] presented the problem of consen-
sus evaluation by endogenously computing the importance of the
experts based on their influence strength in the social network.
Pérez, et al. [32] presented three new social network analysis
based induced ordered weighted averaging (IOWA) operators
that take advantage of the linguistic trustworthiness information.
Chu, et al. [33] addressed GDM problem with fuzzy preference
relations where the experts have directed social network connec-
tions. Wu and Chiclana [34] developed an SNA trust–consensus
model with interval valued fuzzy reciprocal preference relation,
which is one of the first efforts in combining trust degree and con-
sensus level into GDM problem. Furthermore, Wu, et al. [14] put
forward a trust-based theoretical framework to consensus build-
ing within a networked social group using incomplete linguistic
information.
As we all know, the shortages of water, energy and food
could cause social instability and irreparable environmental dam-
age. Therefore, it would be of great importance to find a win–
win solution in the WEF nexus evaluation [35]. To do so, some
researches have been proposed regarding the WEF nexus. For
instance, Bizikova, et al. [36] identified several key areas for
intervention in promoting the WEF nexus. Gain and Wada [37]
discussed the impact factors of the WEF nexus. Daher and Mo-
htar [38] introduced a framework and a set of methodologies
that define the linkages between the interconnected resources
of water, energy and food, and enable explicit corresponding
quantifications.
All the existing WEF nexus studies have made an important
progress in relieving the resource pressure as well as promoting
the sustainable development of society. However, most of them
discuss the WEF nexus from the perspective of engineering and
system rather than management. In other words, the study of
the WEF nexus evaluation from the perspectives of SNGDM is
still a challenging work. Based on the aforementioned analyses,
the research defects in the existing methods for the WEF nexus
evaluation from the perspectives of SNGDM can be listed as
follows:
(1) Many researches assume that experts are completely in-
dependent in the WEF nexus evaluation. In other words,
they are not within any social network connections. Mean-
while, several studies are ignoring the expert’s fuzziness
and hesitancy in WEF nexus evaluation.
(2) The existing studies are only focus on group consensus
while ignore the consistency of individuals’ evaluations.
As we all know, the consistency is recognized as experts
are being neither random nor illogical in their expres-
sion of pairwise comparisons. Moreover, consistency has
direct influence on the ranking results of final decision.
Lack of consistency in preference information can lead to
unreliable results and misleading ranking of alternatives.
(3) Most of the existing researchers mostly assume that the
experts make decisions based on the trust relationship
with others, while the experts’ self-confidence over their
evaluations are often neglected. In real SNGDM situations,
experts’ judgment depends not only on their trust in oth-
ers, but also on their self-confidence in their evaluations.
Self-confidence as one of the human psychological be-
haviors has important impact on decision making. Thus,
to consider the self-confidence of experts in the SNGDM
process still an important aspect.
To fill the gaps mentioned above, in this study, we focus on
the trust-based SNGDM with HFPRs and its application on the
WEF nexus evaluation. The main novelties and contributions of
this paper are highlighted as follows:
(1) Consider the WEF nexus evaluation from the perspectives
of SNGDM, a new fuzzy adjacency matrix based on the
concept of edges proposed to imply the trust relationship
among expert, and a trust score matrix based on the ex-
pert’s degree of self-confidence is used to measure the trust
scores of experts.
(2) To deal with the expert’s fuzziness and hesitancy in SNGDM,
HFPRs are used to express experts’ evaluations. Mean-
while, a novel trust score induced ordered weighted av-
eraging (TS-IOWA) operator is presented to aggregate the
individual’s evaluation into a collective one.
(3) A trust-based decision support model is present to im-
prove the consistency of individual’s evaluation, as well
as to promote an accepted consensus can be achieved. In
addition, integrating the concept of deviation, an objective
thresholds determination method of individual consistency
indexes and group consensus indexes are presented.
The effectiveness of the research in this paper is demonstrated
by a case of study of the WEF nexus evaluation. Moreover, some
comparative analyses and discussions reveal that the degree of
self-confidence and social network connections can impact the
final ranking of alternatives.
The remainder of this paper is organized as follows. Section 2
introduces a basic description of trust-based SNA and HFPRs.
Section 3presents the trust-based SNGDM support model. In Sec-
tion 4, the case study on the WEF nexus evaluation is considered.
Section 5is devoted to the comparisons and discussions. Finally,
the conclusions are summarized in Section 6.
2. Preliminaries
This section reviews some related knowledge regarding the
trust-based social network analysis, and HFPRs.
2.1. Trust-based social network analysis
SNA [20–22] mainly concerns about the relationships between
social entitles, like the members of organizations, corporations,
or nations [39,40]. In SNA, the structural and locational prop-
erties, such as centrality, prestige, and trust relationship, can
be examined. Thus, the SNA has been successfully applied to a
wide range of areas including economics, marketing, and social
sciences [41]. Generally, there are three main elements in SNA: a
set of experts {e1,e2, ..., em}, the relations among them, and the
expert attributes. The network concepts can be referred in unified
manners:
(1) Sociometric, in which relation data are often presented
using a two-ways matrix called socio-matrix;
(2) Graph theoretical, the network is regarded as a graph of
nodes connected by lines;
(3) Algebraic, several distinct relations are distinguished, and
the combinations of relations are represented.
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 3
Table 1
Different representations of the network concept.
The detailed information is shown in Table 1. Specially, the
entries in socio-matrix indicate whether two experts are related
or not. The asymmetric adjacency matrix is employed to repre-
sent the asymmetrical relationships among experts. Furthermore,
the graphs comprising of nodes connected by directed lines, in
which the line goes from expert ekto egis regarded as different
from the line that goes from egto ek,k,g=1,2,...,m. The case
is similar to the algebraic expression.
Obviously, the manners of network concept in Table 1 only
focus on whether the expert ekis related to expert eg, while
ignore the strength or intensity of the relationship. Therefore,
these representations have three main limitations:
(1) Gradualness. In the case of trust, it may not be appropri-
ate to model uncertainties interrelated with relationship
representations. Human beings are not only rational ‘trust’
and ‘distrust’, but trusting someone ‘very much’ or ‘more
or less’, which is explicated as a gradual phenomenon
indeed [34];
(2) Indirectness. Some experts might not have a direct trust
relationship with others because they usually do not know
each other, like experts e1and e6. Due to the connection
of social network, an indirect trust to an unknown expert
can be obtained by a chain of trusted third partner (TTP),
like expert e5. Generally, there may be more than one TTP,
and the relationship strength is negatively related to the
number of the lines of the path of TTPs [14];
(3) Personality. In these three manners, the expert’s strength
or intensity of the trust relationship to herself or him-
self has not been depicted clearly in the interaction of
decision-making. Generally, human decision-making is not
only related to trust in others, but also depends on the
expert’s self-confidence on her or his judgments.
To overcome the limitations mentioned above, the relative
concepts in SNGDM are formally introduced as follows:
Definition 1 ([42,43]).Let E= {e1,e2, ..., em}be a set of experts.
A social network is defined by a directed graph G=(E,L), where
Lis the ordered pair of elements of Eand edge (ek,eg)∈Ldefines
that expert ek∈Edirectly trusts expert eg∈E. In addition,
assume that edge (ek,ek)∈Land (ek,ek) indicates that expert
ektrusts herself or himself directly.
Definition 2 ([42,43]).A sequence of edges (ek,ek1)(ek1,ek2)···
(ekn−1,eg), k,g=1,2, ..., min a social network G=(E,L) is
called a trust path from expert ekto expert eg, and it is denoted
as ek→eg,k,g=1,2, ..., m.
Based on the concepts of edge, this study proposes a novel
fuzzy adjacency matrix to present trust relationships among ex-
perts in SNGDM. The relevant definitions are given below:
Definition 3. A fuzzy adjacency matrix Ron Eis a trust relation-
ship in E×Ewith a membership function µR:E×E→ [0,1],
µR(ek,eg)=rkg , which is used to denote the trust relationship
and is defined as:
µR(ek,eg)=rkg =1
min #(ek,eg),(ek,eg)∈L
0,(ek,eg)/∈L,(1)
where (ek,eg) means the edge from ekto eg; min #(ek,eg) in-
dicates the minimum number of the edges among trust paths
ek→eg; (ek,eg)/∈Lindicates that there exists no edge from
ekto eg. Note that the trust relationship from ekto egdecreases
with the increase of the number of the edges of the trust path.
Moreover, no transitivity and reciprocity condition are required.
Then, the value rkg is interpreted as:
rkg =
0,ekis distrusted to eg
0<rkg <1,ekis a certain degree of indirect trust to eg
1,ekis directly trusted to eg
(2)
With directed social network connections, the experts who
are prestigious tend to receive many nominations or choice [18].
Many approaches have been put forward to obtain the impor-
tance degree of experts from the trust relationships. However,
the expert’s degree of self-confidence has not been considered
in the interaction under the SNGDM. Hence, the degree of self-
confidence is addressed in the following definitions.
Definition 4. Assume that expert ek,k=1,2, ..., mgives the
degree of self-confidence βi∈ [0,1]to own opinion, and she/he
distributes (1−βi) trust across other experts. Then, the trust score
matrix is defined as
π=(πkj)m×m=
β1π12 ... π1m
π21 β2... π2m
... ... ... ...
πm1πm2... βm
,(3)
where
πkg =
βk,k=g
(1−βk)rkg
m
g=1,g=krkg ,k= g,m
g=1,g=krkg = 0
0,k= g,m
g=1,g=krkg =0
,(4)
indicates the trust degree that expert ekgives to eg.
Definition 5. Let π=(πkg )m×mbe the trust score (TS) matrix,
then
TS(eg)=m
k=1πkg
m,g=1,2, ..., m,(5)
represents the normalized trust score of expert eg.
2.2. Hesitant fuzzy preference relation
As is well known, fuzzy sets were first introduced by Zadeh [44]
as effective tools for modeling vagueness and uncertainty. There
are many fuzzy-based methods, including fuzzy logic [45] and
adaptive neuro fuzzy inference system [46,47]. Based on the
4N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
concept of fuzzy logic, Orlovsky [48] proposed a fuzzy preference
relation (FPR) to generalize crisp preference in a decision making
situation. Furthermore, hesitant fuzzy set (HFS) and HFPR are
considered extensions of the fuzzy set and FPR. The definitions
of HFS and HFPR are given below:
Definition 6 ([49]).Let Xbe a fixed set, an HFS on Xcan be
presented as follows:
Γ={⟨x,hΓ(x)⟩|x∈X},(6)
in which hΓ(x) is a set of values in [0,1], implies the possible
membership degrees of the element x∈Xto the set Γ.
HFS first proposed by Torra [49], for convenience, Xia and
Xu [29] called hΓ(x) as hesitant fuzzy element (HFE) and assume
that h= {γσ(s)|s=1,2, ..., #h},h1= {γσ(s)
1|s=1,2, ..., #h1},
and h2= {γσ(2)
2|s=1,2,...,#h2}(#h1=#h2), some basic
operations on the HFEs have been defined in Zhang, et al. [30],
as follows:
h1⊕h2=
γσ(s)
1∈h1,γ σ(s)
2∈h2γσ(s)
1+γσ(s)
2,(7)
h1⊖h2=
γσ(s)
1∈h1,γ σ(s)
2∈h2γσ(s)
1−γσ(s)
2,(8)
λh=
γσ(s)∈hλγ σ(s)(λ > 0),(9)
h⊕a=
γσ(s)∈hγσ(s)+a(a∈(−∞,+∞)),(10)
where γσ(s)
1and γσ(s)
2are the sth elements in h1and h2, respec-
tively, and #his the number of values in h.
Definition 7 ([29]).For an HFE h,
m(h)=1
#h
s=#h
s=1
γσ(s),(11)
is called the score function of h. For h1and h2, if m(h1)>m(h2),
then h1>h2; if m(h1)=m(h2), then h1=h2.
Definition 8 ([50]).An HFPR Hon a set of alternatives Xis
indicated by a matrix H=(hij)n×n⊂X×X, where hij =
{hσ(s)
ij ,s=1,2, ..., #hij}is an HFE implying all the possible
degrees to which xiis preferred to xj. Moreover, hij satisfies
hσ(s)
ij +hσ(s)
ji =1,hσ(s)
ii =0.5,#hij =#hji,i,j=1,2, ..., n.(12)
where hσ(s)
ij is the sth value in hij, #hij is the number of values in
hij.
Remark 1. In [29,30], the values in hij are assumed to be arranged
in the ascending order in the upper triangular matrix, i.e., hσ(s)
ij <
hσ(s+1)
ij , and the values are assumed to be arranged in the descend-
ing order in the lower triangular matrix, i.e., hσ(s+1)
ji <hσ(s)
ji . In this
study, we use the definition in [50], which is slightly different
from the definition of the HFPR in [29,30]. The elements are not
rearranged in a descending or ascending order, because it will
be contradictory in the consistency process, detailed explanation
is in [51]. Furthermore, in most cases, give two HFEs h1and h2,
we have that #h1= #h2. In terms of Eqs. (7)–(10), the same
length of the HFE is required [52]. To solve this problem, some
methods have been proposed and utilized in decision making
problems [53]. For simplify, this paper assumes that #hij =#hji =
#hfor all i,j=1,2, ..., n.
Definition 9 ([50]).Let H=(hij)n×nbe an HFPR. If
˜
hij =1
nn
⊕
k=1hik ⊕hkj⊖1
2,i,j=1,2, ..., n,∀k∈N,(13)
then ˜
H=(˜
hij)n×nis a consistent HFPR.
To obtain important properties in establishing the thresh-
olds of consistency and consensus for expert’s evaluations (see
Theorem 3), the distance between two HFPRs are defined as
follows:
Definition 10. Let h1= {γσ(s)
1|s=1,2, ..., #h},h2= {γσ(s)
2|s=
1,2, ..., #h}be two HFEs. Then, the distance between them can
be computed by
D(h1,h2)=1
#h
#h
s=1γσ(s)
1−γσ(s)
22
, γ σ(s)
1∈h1, γ σ(s)
2∈h2.
(14)
Theorem 1. Let h1, h2and h3are any three HFEs, then
(1) 0 ≤D(h1,h2)≤1. Especially, D(h1,h2)=0⇔h1=h2;
(2) D(h1,h2)=D(h2,h1);
(3) D(h1,h3)≤D(h1,h2)+D(h2,h3).
The proof of Theorem 1 is provided in Appendix.
Definition 11. Given two HFPRs H1=(hij,1)n×nand H2=
(hij,2)n×n. Then, the distance between H1and H2can be calculated
by
D(H1,H2)=2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)2.(15)
Theorem 2. Let H1, H2and H3are any three HFPRs, then
(1) 0 ≤D(H1,H2)≤1. Especially, D(H1,H2)=0⇔H1=H2;
(2) D(H1,H2)=D(H2,H1);
(3) D(H1,H3)≤D(H1,H2)+D(H2,H3).
The proof of Theorem 2 is provided in Appendix.
3. A trust-based decision support model for group decision
making
This section proposes a trust-based decision support model
to improve individual consistency and help experts reach con-
sensus. In Section 3.1, the individual consistency improvement
is introduced. Section 3.2 presents the aggregation of individual
evaluations using TS-IOWA operator. Group consensus reaching
is indicated in Section 3.3. Finally, the framework of trust-based
SNGDM model is presented in Section 3.4
3.1. Individual consistency improving stage
Consistency measure is an important step in GDM problems
since the inconsistent evaluations provided by experts may be
result in unreliable decision result(s) [54–58]. Usually, the con-
sistency improvement contains three steps: (1) Consistency mea-
sure; (2) The consistency threshold setting; (3) The improvement
of consistency. To do so, based on the definitions of deviation and
Dong, et al. [59]’s work on the consistency of linguistic preference
relation, the specific description regarding the above three steps
are given below:
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 5
3.1.1. Individual consistency measure
Definition 12. Consistency index at matrix level. CIM: Given an
HFPR H=(hij)n×n, the corresponding consistent HFPR ˜
H=
(˜
hij)n×ncan be obtained by Eq. (13). Then, CIMcan be computed
by
CIM(H)=D(H,˜
H)=2
#hn(n−1)
n
j>i
n
i=1
#h
s=1
(γσ(s)
ij − ˜γσ(s)
ij )2.
(16)
Eq. (16) has a definite physical implication and reflects the
deviation degree between the HFPR Hand the consistent HFPR.
Obviously, the smaller the value of CIM(H), the more consis-
tent HFPR H. In particular, if CIM(H)=0, then His a completely
consistent HFPR.
Definition 13. Consistency index at element level CIE: Given
an HFPR H=(hij)n×n, the ˜
H=(˜
hij)n×nis the corresponding
consistent HFPR obtained by Eq. (13). Then, the CIE(hij) is utilized
to measure the distance between hij and ˜
hij, can be computed by
CIE(hij)=D(hij ,˜
hij)
=1
#h
#h
s=1γσ(s)
ij − ˜γσ(s)
ij 2
, γ σ(s)
ij ∈hij,˜γσ(s)
ij ∈˜
hij.(17)
Eq. (17) reflects the deviation degree between the HFE hij in
HFPR Hand the HFE ˜
hij in consistent HFPR. Obviously, the smaller
the value of CIE(hij), the closer hij is to ˜
hij. In particular, if CIE(hij )=
0, then hij is a completely equal to ˜
hij.
3.1.2. Estimation of the consistency thresholds
In general, the completely consistent HFPRs are difficult to be
obtained in real GDM problems since the limited rationality of
human decision-making. Thus, this paper presents the acceptable
consistency for GDM problems. By Definition 12, one finds that
CIM(H) virtually reflects the deviation degree between Hand ˜
H,
where ˜
His a completely consistent HFPR. Let εs
ij =γσ(s)
ij − ˜γσ(s)
ij ,
then
CIM(H)=D(H,˜
H)=2
#hn(n−1)
n
j>i
n
i=1
#h
s=1
(εs
ij)2,(18)
CIE(hij)=1
#h
#h
s=1εs
ij2,i<j.(19)
Experts often have certain consistency tendency in pairwise
comparisons, the values of εs
ij relatively centralizes the domain
close to zero [60]. Thus, we assume that εs
ij,i<j,s=1,2, ..., #h
is independent normally distributed with mean 0 and standard
deviation σ, i.e., εs
ij ∼N(0, σ 2), which is discussed by Dong,
et al. [59].
Theorem 3. (#hn(n−1)
2σ(CIM(H)))2is chi-square distributions with
#hn(n−1)
2degrees of freedom, and (#h
σ(CIE(hij)))2is chi-square distri-
butions with #h degrees of freedom. Namely, (#hn(n−1)
2σ(CIM(H)))2∼
χ2(#hn(n−1)
2)and (#h
σ(CIE(hij)))2∼χ2(#h), on the condition that εs
ij,
i<j, s =1,2, ..., #h, is independent normally distributed with
mean 0 and standard deviation σ, namely εs
ij ∼N(0, σ 2).
The proof of Theorem 3 is provided in Appendix.
If we further assume that σ2=σ2
0, namely, εs
ij ∼N(0, σ 2
0),
then, the consistency measure is to test hypothesis Ψ0versus
hypothesis Ψ1:
Ψ0:σ2≤σ2
0;
Ψ1:σ2> σ 2
0.
The freedom degrees of the estimators (#hn(n−1)
2σ(CIM(H)))2and
(#h
σ(CIE(hij)))2are #hn(n−1)
2and #h, respectively. Furthermore, this
is a one-sided right-tailed test, one can have the critical value
λα(n) of χ2(n) distribution at the significance levelα. In this case,
we have
CIM=2σ0
#hn(n−1)λα(#hn(n−1)
2),(20)
CIE=σ0
#hλα(#h).(21)
If CIM(H)≤CIM, we conclude that His of acceptable con-
sistency; otherwise, the His inconsistent. Then, the inconsistent
HFPR will be adjusted based on CIE. The reason for setting up
two consistency indexes is that CIM≤2
n(n−1) CIE≤CIE, con-
sistency threshold at element level CIEcan be set slightly larger
consistency than the threshold at matrix level CIM, to give more
freedom for the experts to modify their preference evaluation.
According to the actual situation, experts can set different
values for αand σ2
0.Table 2 shows the values of CIMand CIEfor
different #hand nwhen setting α=0.05 and σ0=0.1. Note
that the values of CIEis related to #hand does not change with
n. The estimation of consistency threshold provides an important
reference for experts. But in the real decision, the threshold
setting does not need to be strictly in accordance with the values
in Table 2 and can be within a reasonable range.
3.1.3. Consistency improving algorithm
To ensure the reliable of the experts’ evaluations as well
as the reasonable of the final decision(s), it would be of great
importance to improve the inconsistent information of experts.
In this study, when an HFPR His of unacceptable consistency,
i.e., CIM(H)>CIM, the Hwill be returned to the expert. Then,
she/he will be suggested to make some modifications for the eval-
uations. This process is repeated until the predefined consistency
threshold is satisfied within the maximum number of iterations.
The detailed consistency improvement processes are depicted in
Algorithm 1, and the flowchart is presented in Fig. 1.
3.2. Aggregation stage
In the aggregation process, the individual preference informa-
tion can be aggregated into a collective one by using aggregation
operators. The induced ordered weighted averaging (IOWA) op-
erator is an effective tool to be used in the aggregation process in
GDM. The detailed definitions are given below:
Definition 14 ([61,62]).An m-dimensional IOWA operator is a
function Φw:(R×R)m→R, to which a weighting vector W=
(w1, w2, ..., wm) is associated, such that m
k=1wk=1, and
wk∈ [0,1]. The IOWA operator is defined to aggregate the set of
first arguments of a list of 2-tuples {⟨hij,1,u1⟩, . . . ., ⟨hij,m,um⟩}
based on the expression:
Φw⟨hij,1,u1⟩, . . . ., ⟨hij,m,um⟩=
m
k=1
wkhij,σ (k),(22)
where σis a permutation such that hij,σ (k)in ⟨hij,k,uk⟩is the kth
largest value in the set {hij,1,hij,2, ..., hij,m}.
6N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
Table 2
The threshold values CIMand CIEfor different #hand n.
n=3n=4n=5n=6n=7n=8n=9CIE
#h=1 0.0931 0.0591 0.0428 0.0333 0.0272 0.0230 0.0198 0.1646
#h=2 0.0591 0.0382 0.0280 0.0221 0.0182 0.0154 0.0134 0.1075
#h=3 0.0457 0.0299 0.0221 0.0174 0.0144 0.0123 0.0107 0.0833
Algorithm 1
Input: Individual HFPRs H=(hij)n×n, the thresholds CIMand CIE, the iteration parameter δ, and the maximum number of iterations
tmax.
Output: The modified HFPR ˜
H=(˜
hij)n×n, individual consistency indexes CIM(˜
H).
Step 1. Let H(t)=(h(t)
ij )n×n,t=0.
Step 2. Using Eq. (13) to construct the consistent HFPR ˜
H(t)=(˜
h(t)
ij )n×n,
Step 3. Calculating CIM(H(t)) using Eq. (16). If CIM(H(t))>CIMand t≤tmax, go to Step 4. Otherwise, go to the Step 5.
Step 4. Find the position iτand jτof the element θ(t)
iτjτ=CIE(h(t)
ij )>CIE. Let H(t+1) =h(t+1)
ij n×n
, where
h(t+1)
ij =δh(t)
ij +(1 −δ)˜
h(t)
ij ,if i =iτ,j=jτ
h(t)
ij ,otherwise
H(t)= H(t+1) and t=t+1. Then, go to Step 2.
Step 5. Let ˜
H=H(t), output the adjusted HFPR ˜
H, and individual consistency indexes CIM(˜
H).
Fig. 1. Flowchart of the consistency improving for individual HFPRs.
Chiclana, et al. [61] proposed a special IOWA operator called
importance induced ordered weighted averaging (I-IOWA) op-
erator where the expert is assigned an importance degree to
her/him opinions. The I-IOWA operator applies the ordering of
the argument values based on the importance of the information
sources.
Definition 15 ([61]).An m-dimensional I-IOWA operator is a
function ΦI
w:(R×R)m→R, to aggregate the first arguments
of a set of m3-tuples ⟨hij,k,uk, vk⟩:hij,kis the argument value
to be aggregated; ukis an importance degree associated to the
argument value hij,k; and vkis the order inducing values. In this
case, the aggregation is
ΦI
w⟨hij,1,u1, v1⟩, . . . ., ⟨hij,m,um, vm⟩=
m
k=1
wkhij,σ (k),(23)
where
wk=QS(k)
S(m)−QS(k−1)
S(m),(24)
with S(k)=k
l=1uσ(l)and σis a permutation such that vσ(k)in
⟨hij,k,uk, vk⟩is the kth largest value in the set {v1, v2, ..., vm}.
The function Qis denoted as Basic Unit-interval Monotone mem-
bership function, for simplicity, Q(x)=x1/2[63].
If the importance degree of each expert both as a weight
associated to the argument values and as the order inducing
values, then vk=uk. The ordering of the preference values is
first induced by the ordering of importance degree of the experts
from most to least important one, and the weights obtained using
Eq. (24), which reduces to:
wk=QS(k)
S(m)−QS(k−1)
S(m),(25)
with S(k)=k
l=1uσ(l)and σis a permutation such that uσ(k)in
⟨hij,k,uk⟩is the kth largest value in the set {u1,u2, ..., um}.
In this paper, the trust score is considered as the important
degree of expert. Thus, the trust score induced ordered weighted
averaging operator named TS-IOWA, which is defined as follows:
Definition 16. TS-IOWA operator: An m-dimensional TS-IOWA
operator is a function ΦTS
w:(R×R)m→R, to aggregate the
arguments of a set of 2-tuples list {⟨hij,1,u1⟩,...., ⟨hij,m,um⟩}:
hij,kis the argument value to aggregate; ukis both as a weight
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 7
associated to the argument values and as the order inducing
values. In this case, the aggregation is
Φw⟨hij,1,u1⟩, . . . ., ⟨hij,m,um⟩=
m
k=1
wkhij,σ (k),(26)
in which the weights obtained by using Eq. (25),S(k)=k
l=1TSσ(l)
and σis a permutation such that TSσ(k)in ⟨hij,σ (k),TSσ(k)⟩is the
kth largest value in the set {TS1,TS2, ..., TSm}.
Let Hk=(hij,k)n×n,k=1,2,..., mbe a set of individual
HFPRs with acceptable consistency given by a set of experts
E= {e1,e2, ..., em}and W=(w1, w2, ..., wm) be the ex-
perts’ weight vector derived by Eq. (25) associated with TS-IOWA
operator, which satisfied m
k=1wk=1, wk∈ [0,1]. Then, the
collective group HFPR Hc=(hij,c)n×nis computed as follows:
Hc=ΦTS
w⟨w1,Hσ(1)⟩,⟨w2,Hσ(2) ⟩,...,⟨wm,Hσ(m)⟩
=
m
k=1
wkHσ(k),(27)
where hij,c= ⊕m
k=1wkhij,σ (k).
3.3. Group consensus reaching stage
Consensus implies the unanimous and full opinion of all the
experts concerning all the feasible alternatives. However, it is
difficult to reach a unanimous consensus and it is necessary to
accept a ‘soft’ consensus in real GDM [64–67]. Thus, when the
trust-based collective preference matrix is obtained, group con-
sensus indexes at three levels, decision matrix level, alternative
level and element level, are calculated for each expert to give
more freedom for adjusting their preference information [68].
3.3.1. Group consensus index
When the values of consensus indexes at decision matrix
level are all within the thresholds for each expert, the resolution
process of consensus is finished. Otherwise, the experts whose
values of consensus indexes at preference matrix level exceed the
predetermined thresholds are distinguished and invited to modify
some of their opinions based on the alternative level and element
level. The relative definitions are given as follows:
Define 17. Group consensus index (GCI) at matrix level: The
GCI of expert ekconcerns the group at decision matrix level is
GCI(Hk)=2
n(n−1)
n
j>i
n−1
i=1
(D(hij,k,hij,c))2
=2
#hn(n−1)
n
j.>i
n−1
i=1
#h
s
(γσ(s)
ij,k−γσ(s)
ij,c)2.(28)
Define 18. Consensus of alternative level (CA): The CA of
expert ekwith regard to the group on the alternatives xiis
CAi,k=1
n
n
j=1
(D(hij,k,hij,c))2=1
#hn
n
j=1
#h
s
(γσ(s)
ij,k−γσ(s)
ij,c)2.
(29)
Define 19. Consensus of element level (CE): The CE of DM ekwith
regard to the group on the alternatives xito xjis
CEij,k=Dhij,k,hij,c
=1
#h
#h
s=1γσ(s)
ij,k−γσ(s)
ij,c2
, γ σ(s)
ij,k∈hij,k, γ σ(s)
ij,c∈hij,c.(30)
Fig. 2. Flowchart of the group consensus reaching process.
Remark 2. The smaller the value of GCI(Hk)(0 ≤GCI (Hk)≤1),
the greater the consensus level between expert ekand group.
Specially, if GCI(Hk)=0, this means the expert ekhas unanimous
opinion with the group expert. This case in practice is rare.
Therefore, a soft consensus is accepted as a threshold GCI is set
as a minimum consensus level to achieve.
3.3.2. Consensus reaching process
If GCI(Hk)≤GCI , for all k=1,2, ..., m, then the solu-
tion of consensus is derived by using an appropriate selection
process. Otherwise, an interaction procedure to help expert ek
with the bigger values of GCI, i.e., GCI(Hk)>GCI , to ‘see’ their
consensus position in the group. Furthermore, a trust-based rec-
ommendation is proposed to produce personalized advice for ex-
perts on how to modify their opinions to reach group consensus.
Mathematically, these steps are modeled as follows:
Step 1. Experts with group consensus index at matrix level
exceed the threshold GCI are identified: ML =k|GCI(Hk)>GCI ;
Step 2. For experts k∈ML, the consensus indexes at alter-
native xi,i=1,2,..., n, exceed the threshold CA are determined:
AL =(k,i)|(k∈ML)∧(CAi,k>CA);
Step 3. For (k,i)∈AL, the consensus index at element level
exceed the threshold CE are distinguished: .
Step 4. The recommendation produces provide personalized
advice for expert ekto change the evaluation hij,kto a value ˆ
hij,k.
ˆ
hij,k=(1 −ρ)hij,k⊕ρhij,c,(31)
in which ρ∈ [0,1]is a parameter to control the degree of advice
and hij,cis the trust-based collective opinion.
Note that these steps are repeated until the predefined GCI
threshold is satisfied within the maximum number of iterations.
The detailed group consensus reaching process is indicated in
Algorithm 2 To improve visualization, it is presented in Fig. 2.
3.3.3. Selection process
When individual consistency and group consensus are
achieved, a selection process is used to calculate the final score
8N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
Algorithm 2.
Input: The HFPRs Hk=(hij,k)n×n,k=1,2,...,m, the trust score TS(ek), k=1,2,...,m, the thresholds GCI ,CA,CE, the iteration
parameter ρ, and the maximum number of iterations ¯
tmax.
Output: The modified HFPRs ˆ
Hk=(ˆ
hij,k)n×n,k=1,2,...,mand GCI(ˆ
Hk), k=1,2,...,m.
Step 1. Let H(t)
k=(h(t)
ij,k)n×n,t=0.
Step 2. Using Eqs. (25)–(27) to obtained a group HFPR H(t)
c=(h(t)
ij,c)n×n. Then, based on Eqs. (28)-(30), calculating GCI(H(t)
k), CA(t)
i,k,
CE(t)
ij,k, and identifying the sets
ML(t)=kGCI(H(t)
k)>GCI ,(32)
AL(t)=(k,i)(k∈ML(t))∧(CA(t)
i,k>CA),(33)
EL(t)=(k,i,j)((k,i)∈AL(t))∧(CE (t)
ij,k>CE).(34)
Step 3. If ML(t)= ∅ and t≤¯
tmax, go to Step 4. Otherwise, go to the Step 5.
Step 4. Find the element (kτ,iτ,jτ) of EL(t). Let H(t+1)
k=h(t+1)
ij,kn×n
, where
h(t+1)
ij,k=ρh(t)
ij,k+(1 −ρ)h(t)
ij,c,if i =iτ,j=jτ,and k =kτ
h(t)
ij,k,otherwise ,
H(t)= H(t+1) and t=t+1. Then, go to Step 2.
Step 5. Let ˆ
Hk=H(t)
k,k=1,2,...,m. Output the adjusted HFPRs ˆ
Hk, group consensus indexes GCI(ˆ
Hk), k=1,2,...,m.
Fig. 3. The framework of trust-based SNGDM model.
for each alternative. In this paper, we use the hesitant fuzzy
averaging (HFA) operator [30],
hi,c=HFA(hi1,c,hi2,c, ..., hin,c)=1
n
n
⊕
j=1hij,c,(35)
to aggregate all of the preference evaluations in the ith row
of Hc=(hij,c)n×nand then obtain the preference degree hi,c
of the alternative xi,i=1,2, ..., n. Furthermore, calculate
score function m(hi,c) of hi,c,i=1,2,...,n, by Eq. (11). Rank
alternative xiaccording to m(hi,c), i=1,2, ..., nand then select
the optimal alternative(s).
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 9
3.4. Framework of trust-based social network decision-making sup-
port model
The SNGDM problem in this paper is how to achieve consen-
sus in selecting the best alternative(s) from the set of feasible
alternatives based on the preference information provided by a
group of experts socially networked with trust connection. Recall
that the relative sets are
(a) the set of nalternatives denoted as X= {x1,x2, ..., xn};
(b) the set of mexperts, indicated as E= {e1,e2, ..., em}, who
with the degree of self-confidence βk, 0 ≤βk≤1, in a
social network;
(c) the set of HFPRs Hk=(hij,k)n×n,i,j=1,2, ..., n,k=
1,2, ..., m.
Three main processes, individual consistency improving, pref-
erences aggregation, and group consensus reaching, are involved
in the framework of trust-based decision support model, which
is depicted in Fig. 3. The specific steps are as follows:
Step 1. Experts in the social network express their evaluations
(HFPRs) over the set of alternatives and the degrees of self-
confidence on their own opinions, then the trust network can
be constructed. Subsequently, each expert’s trust score can be
computed via fuzzy adjacency relationship and trust score matrix
by Eqs. (1)–(5).
Step 2. Using Eqs. (13)–(16) to compute the consistency in-
dexes for experts. If the CIMis acceptable for each expert, i.e.,
CIM(Hk)<CIM, for all k=1,2, ..., m. The individual HFPRs
will aggregated into a temporal group preference matrix by using
TS-IOWA operator. Otherwise, the inconsistent individual HFPR(s)
will be modified by Algorithm 1.
Step 3. Calculating GCIs using Eq. (28). If GCIs are acceptable
for all experts, i.e., GCI(Hk)<GCI ,k=1,2, ..., m. The temporal
group evaluation is the final group opinion. Otherwise, consensus
identification and Algorithm 2 will be utilized to reach a soft
consensus.
Step 4. After consensus is achieved in the SNGDM problems,
then a selection process is activated to produce the final ranking
of alternatives.
4. Case study of the WEF nexus evaluation
In this section, a practical example concerned about WEF
nexus evaluation is used to illustrate the advantage of the deci-
sion support model proposed in this paper.
4.1. The descriptions of the WEF nexus evaluation
Suppose that there are four potentially alternative {x1,x2,x3,
x4}on the WEF nexus evaluation, seven experts, {e1,e2, ..., e7},
who involve different government departments and have certain
social network connections described in Fig. 4. Three kinds of
social trust relationship are addressed and simulated:
(1) Experts might not have a direct trust relationship with each
other, like e1and e6; but an indirect trust can be achieved via a
chain of trusted third partner, like e4or e5;
(2) The direct trust relationships between experts, such as e4
and e6;
(3) Experts have a direct trust relationship with only one
expert in the social network, and no contact with other experts,
like e7.
Experts make decisions not only depends on the trust-based
social network connections, but also on the degrees of self-
confidence over their own evaluations. Furthermore, due to dif-
ferent knowledge, experiences, abilities, and expectation, experts
are usually not certain about a preference evaluation but has hes-
itancy among several possible evaluation values, when comparing
two alternatives. Thus, the experts may provide HFPRs when they
assess pairwise alternatives.
The degrees of self-confidence, HFPRs and other parameters
are given as follows:
(1) The degrees of self-confidence β=(β1, β2, . . . , β7)=
(0.5,0.5,0.5,0.5,0.5,0.5,0.5);
(2) The experts’ HFPRs:
H1=
{0.5} {0.3,0.3,0.3} {0.5,0.7,0.8} {0.4,0.4,0.4}
{0.7,0.7,0.7} {0.5} {0.7,0.9,0.9} {0.8,0.8,0.8}
{0.5,0.3,0.2} {0.3,0.1,0.1} {0.5} {0.6,0.7,0.7}
{0.6,0.6,0.6} {0.2,0.2,0.2} {0.4,0.3,0.3} {0.5}
,
H2=
{0.5} {0.1,0.3,0.8} {0.1,0.4,0.7} {0.1,0.1,0.1}
{0.9,0.7,0.2} {0.5} {0.7,0.8,0.8} {0.1,0.3,0.5}
{0.9,0.6,0.3} {0.3,0.2,0.2} {0.5} {0.5,0.6,0.9}
{0.9,0.9,0.9} {0.9,0.7,0.5} {0.5,0.4,0.1} {0.5}
,
H3=
{0.5} {0.3,0.5,0.5} {0.7,0.7,0.7} {0.7,0.8,0.8}
{0.7,0.5,0.5} {0.5} {0.2,0.3,0.4} {0.5,0.6,0.6}
{0.3,0.3,0.3} {0.8,0.7,0.6} {0.5} {0.7,0.8,0.9}
{0.3,0.2,0.2} {0.5,0.4,0.4} {0.3,0.2,0.1} {0.5}
,
H4=
{0.5} {0.4,0.5,0.6} {0.3,0.4,0.4} {0.5,0.7,0.7}
{0.6,0.5,0.4} {0.5} {0.3,0.3,0.3} {0.6,0.7,0.8}
{0.7,0.6,0.6} {0.7,0.7,0.7} {0.5} {0.8,0.9,0.9}
{0.5,0.3,0.3} {0.4,0.3,0.2} {0.2,0.1,0.1} {0.5}
,
H5=
{0.5} {0.7,0.8,0.8} {0.4,0.5,0.5} {0.7,0.7,0.7}
{0.3,0.2,0.2} {0.5} {0.3,0.4,0.4} {0.6,0.8,0.9}
{0.6,0.5,0.5} {0.7,0.6,0.6} {0.5} {0.4,0.6,0.6}
{0.3,0.3,0.3} {0.4,0.2,0.1} {0.6,0.4,0.4} {0.5}
,
H6=
{0.5} {0.4,0.7,0.7} {0.7,0.8,0.8} {0.1,0.1,0.1}
{0.6,0.3,0.3} {0.5} {0.7,0.8,0.8} {0.3,0.5,0.5}
{0.3,0.2,0.2} {0.3,0.2,0.2} {0.5} {0.4,0.6,0.7}
{0.9,0.9,0.9} {0.7,0.5,0.5} {0.6,0.4,0.3} {0.5}
,
H7=
{0.5} {0.3,0.5,0.7} {0.4,0.5,0.6} {0.4,0.7,0.7}
{0.7,0.5,0.3} {0.5} {0.3,0.5,0.5} {0.6,0.8,0.8}
{0.6,0.5,0.4} {0.7,0.5,0.5} {0.5} {0.6,0.8,0.9}
{0.6,0.3,0.3} {0.4,0.2,0.2} {0.4,0.2,0.1} {0.5}
;
(3) The thresholds are set at CIM=0.03, CIE=0.05, GCI =
0.03, CA =0.03, and CE =0.05;
(4) The iteration parameters and the maximum number of
iterations are set at δ=ρ=0.9 and tmax =¯
tmax =
50, to control the degree and cost of recommendations,
respectively.
4.2. The application of the proposed method on the WEF nexus
evaluation
The detailed application process of the trust-based SNGDM
support model on the WEF nexus evaluation is as follows:
Step 1. Based on the social network connections described in
Fig. 4, using Eqs. (1)–(5) to calculate the fuzzy adjacency matrix
R, trust scores π, and experts’ trust scores as:
R=
111111/20
0 1 1/2 1/311/20
0 1 1 1/4 1/2 1/30
01/21 1 1/31 0
01/211/21 1 0
01/3 1/211/41 0
11/2 1/2 1/2 1/2 1/31
,
10 N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
Fig. 4. The social network connection.
π=
0.5 0.1111 0.1111 0.1111 0.1111 0.0556 0
0 0.5 0.1072 0.0714 0.2143 0.1072 0
0 0.2400 0.5 0.0600 0.1200 0.0799 0
0 0.0882 0.1765 0.5 0.0588 0.1766 0
0 0.0833 0.1667 0.0833 0.5 0.1667 0
0 0.0799 0.1200 0.2400 0.0600 0.5 0
0.15 0.0750 0.0750 0.0750 0.0750 0.0500 0.5
,
TS =(TS1,TS2, ..., TS7)
=(0.0929,0.1682,0.1795,0.163,0.1627,0.1623,0.0714).
Step 2. Eqs. (13) and (16) are utilized to obtain individual con-
sistency indexes, then CIM(H1)=0.0196, CIM(H2)=0.0406,
CIM(H3)=0.0142, CIM(H4)=0.0068, CIM(H5)=0.0188, ,
CIM(H7)=0.0124. Since CIM(H2)>0.03 and CIM(H6)>0.03, the
consistency of the evaluations for the experts e2and e6need to
be improved. Using Algorithm 1, the modified results are shown
in Table 3.
Step 3. Using Eqs. (25)–(27) to obtain a group HFPR and weights
based on the trust scores and TS-OWA operator, where
Hc
=
{0.5} {0.33,0.5,0.59 } {0.5,0.6,0.65} {0.5,0.58,0.6}
{0.67,0.5,0.41} {0.5} {0.37,0.46,0.5} {0.46,0.6,0.65}
{0.5,0.4,0.35} {0.63,0.54,0.5} {0.5} {0.6,0.71,0.81}
{0.5,0.42,0.4} {0.54,0.4,0.35} {0.4,0.29,0.19} {0.5}
,
and
W=(w1, w2, ..., w7)
=(0.4237,0.166,0.125,0.106,0.0935,0.0495,0.0364).
Furthermore, calculating GCIs using Eq. (28), the results show
in the first column of Table 4. Since GCI(H1)>GCI , and GCI(H6)>
GCI(ba), therefore, Algorithm 2 is activated to help experts e1and
e6to ‘see’ their consensus position in the group and modify their
evaluations.
From the second column in Table 4, we can see that CA(0)
i,1>CA,
CA(0)
i,6>CA for all i=1,2,3,4. Then, the set of CE(0)
ij,k>CE
are identified. As show in the third column in Table 4, only the
elements in the positions with red color are revised using the
recommendation mechanism (Eq. (31)) at the first iteration. After
two iterations, the modified HFPRs, group HFPR are ˆ
H1,ˆ
H6and ˆ
Hc
(see Box I).
The GCIs for the modified HFPRs are shown in the fourth
column in Table 4. Due to GCI(ˆ
Hk)<GCI for all k=1,2, ..., m,
the collective group preference matrix is the final group decision
matrix.
Step 4. After the acceptable group consensus reached, using
the HFA operator and score function, i.e., Eqs. (11) and (35), to
calculate the score of alternatives, we have
ˆ
h1,c
ˆ
h2,c
ˆ
h3,c
ˆ
h4,c
=
{0.46,0.55,0.59}
{0.49,0.51,0.51}
{0.56,0.54,0.54}
{0.49,0.40,0.36}
and
m(ˆ
h1,c)
m(ˆ
h2,c)
m(ˆ
h3,c)
m(ˆ
h4,c)
=
0.53
0.51
0.55
0.41
.
Therefore, the final ranking of alternatives is: x3≻x1≻x2≻
x4.
5. Impact analysis of the degree of self-confidence and social
network connection
In Section 4, each expert’s degree of self-confidence is as-
sumed as βk=0.5, for k=1,2, ..., 7, and the structure of social
networks is fixed. In practice, experts may be unconfident or
overconfident for their evaluations in a dynamic social network.
Thus, this section further investigates whether the experts’ de-
grees of self-confidence and social network connections can im-
pact the ranking results of alternatives. In Section 5.1, the impact
of the degree of confidence in SNGDM is provided. And then, the
impact of social network connections is provided in Section 5.2.
5.1. The impact of the degree of confidence
Due to different position, background, and experience, experts
cannot always be neutral about their own evaluations, i.e., βk=
0.5, k=1,2, ..., 7. Therefore, based on the social network
connections in Fig. 4, we consider extra three situations:
(1) All the experts are unconfident with β=(0.1,0.1,0.1,0.1,
0.1,0.1,0.1);
(2) All the experts are overconfident with β=(0.9,0.9,0.9,
0.9,0.9,0.9,0.9);
(3) Experts with different degrees of self-confidence 0.1,0.1).
The results in second row to fourth row of Table 5 show that
the experts’ degrees of self-confidence can impact the experts’
TSs and weights when the individual HFPRs are aggregated by
utilizing TS-IOWA operator. Obviously, we can see that when
experts with the same degree of self-confidence, the alternative
ranking has not been changed, while the rankings were changed
when experts with different degree of confidence. Therefore,
we conclude that in most cases, the experts’ degrees of self-
confidence can change the final ranking result to some extent,
but not always.
5.2. The impact of social network connections
To further investigate the impact of social network connec-
tions on the rankings of alternatives. We consider the following
two additional circumstances:
(1) The seven experts are completely independent without any
considering social network connections and they are assigned the
same weight;
(2) The social network connections in the sixth row and first
column are changed partially compared with the social network
connections in Fig. 4. First, the direction of social network con-
nection between e1and e3be changed. Second, the directed social
network connection from e5to e3be removed.
The computational procedures are same as in Section 4.2.
Then, combining the corresponding results in Table 5, it is obvious
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 11
ˆ
H1=
{0.5} {0.31,0.34,0.36} {0.50,0.68,0.77} {0.42,0.43,0.44}
{0.69,0.66,0.64} {0.5} {0.64,0.82,0.82} {0.73,0.76,0.77}
{0.50,0.32,0.23} {0.36,0.18,0.18} {0.5} {0.60,0.70,0.72}
{0.58,0.57,0.56} {0.27,0.24,0.23} {0.40,0.30,0.28} {0.5}
,
ˆ
H6=
{0.5} {0.40,0.66,0.66} {0.67,0.77,0.77} {0.20,0.25,0.25}
{0.60,0.34,0.34} {0.5} {0.63,0.71,0.72} {0.30,0.50,0.50}
{0.33,0.23,0.23} {0.37,0.29,0.28} {0.5} {0.41,0.58,0.67}
{0.80,0.75,0.75} {0.70,0.50,0.50} {0.59,0.42,0.33} {0.5}
,
ˆ
Hc=
{0.5} {0.33,0.50,0.59} {0.50,0.59,0.65} {0.50,0.59,0.61}
{0.67,0.50,0.41} {0.5} {0.36,0.45,0.49} {0.45,0.59,0.65}
{0.50,0.41,0.35} {0.64,0.55,0.51} {0.5} {0.60,0.72,0.81}
{0.50,0.41,0.39} {0.55,0.41,0.35} {0.40,0.28,0.19} {0.5}
.
Box I.
Table 3
The modified results of individual consistency improvement.
ekh(0)
ij,k→h(t)
ij,kCI(t)
M(Hk)
e2h(0)
12,2→ {0.12,0.28,0.71},h(0)
21,2→ {0.88,0.72,0.29},
h(0)
14,2→ {0.06,0.13,0.25},h(0)
41,2→ {0.94,0.87,0.75},
h(0)
23,2→ {0.63,0.73,0.71},h(0)
32,2→ {0.37,0.27,0.29},
h(0)
34,2→ {0.45,0.51,0.75},h(0)
43,2→ {0.55,0.49,0.25},
CI(0)
M(H2)=0.0406
.
.
.
CI(16)
M(H2)=0.0273
e6h(0)
12,6→ {0.40,0.66,0.66},h(0)
21,6→ {0.60,0.34,0.34},
h(0)
13,6→ {0.67,0.77,0.77},h(0)
31,6→ {0.33,0.23,0.23},
h(0)
14,6→ {0.13,0.17,0.17},h(0)
41,6→ {0.87,0.83,0.83},
h(0)
23,6→ {0.69,0.77,0.77},h(0)
32,6→ {0.31,0.23,0.23},
h(0)
34,6→ {0.36,0.54,0.63},h(0)
43,6→ {0.64,0.46,0.37},
CI(0)
M(H2)=0.0336
.
.
.
CI(10)
M(H2)=0.0272
Table 4
The identifying results.
GCI(H(0)
k)CA(0)
i,kCE(0)
k=(CE(0)
ij,k)n×nGCI(ˆ
Hk)
GCI(H(0)
1)=0.0325
GCI(H(0)
2)=0.0254
GCI(H(0)
3)=0.0148
GCI(H(0)
4)=0.0210
GCI(H(0)
5)=0.0244
GCI(H(0)
6)=0.0318
GCI(H(0)
7)=0.0130
CA(0)
1,1=0.0412
CA(0)
2,1=0.0732
CA(0)
3,1=0.0593
CA(0)
4,1=0.0441
CA(0)
1,6=0.0648
CA(0)
2,6=0.0500
CA(0)
3,6=0.0567
CA(0)
4,6=0.0678
CE(0)
1=
00.10 0.05 0.07
0.10 00.13 0.05
0.05 0.13 0 0.04
0.07 0.05 0.04 0
CE(0)
6=
0 0.02 0.04 0.14
0.0200.09 0.05
0.04 0.09 00.06
0.14 0.05 0.06 0
GCI(ˆ
H1)=0.0267
GCI(ˆ
H2)=0.0260
GCI(ˆ
H3)=0.0142
GCI(ˆ
H4)=0.0206
GCI(ˆ
H5)=0.0242
GCI(ˆ
H6)=0.0275
GCI(ˆ
H7)=0.0127
that all of the experts’ trust scores, weights and the rankings of
alternatives are changed compared with the results in Section 4.2.
Therefore, we conclude that social network connection is a crucial
factor in trust-based SNGDM.
Recently, large scale group decision making (LSGDM) prob-
lems have become a hotspot [69–71]. Compared with the tradi-
tional group decision making, there are a large number of experts
or decision makers in LSGDM problems. Furthermore, due to
time pressure, lack of knowledge, or the expert’s limited expe-
rience related with the problem domain, experts may provide
the incomplete assessment information in decision making [28,
72]. Thus, in the future, we will try to consider LSGDM and
incomplete assessment information in the proposed trust-based
SNGDM method to enrich its contents.
6. Conclusions
SNGDM has become a research hotspot. This paper mainly
focuses on the trust-based SNGDM with HFPRs and its applica-
tion on the WEF nexus evaluation. The major contributions are
summarized as follows:
(1) The WEF nexus evaluation from the perspectives of SNGDM
is discussed. Meanwhile, we allow experts use HFPRs to express
12 N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
Table 5
Rankings of alternatives in trust-based SNGDM.
Social network connections Self-confidence, Trust scores and Weights Rankings
β=(0.1,0.1,0.1,0.1,0.1,0.1,0.1)
TS =(0.0529,0.1885,0.2088,
0.1791,0.1787,0.1778,0.0143)
w=(0.4569,0.1734,0.1288,
0.1098,0.0969,0.0270,0.0072)
x3≻x1≻x2≻x4
β=(0.9,0.9,0.9,0.9,0.9,0.9,0.9)
TS =(0.1329,0.1479,0.1502,
0.1469,0.1468,0.1467,0.1286)
w=(0.3876,0.1584,0.1211,
0.1022,0.0901,0.0741,0.0665)
x3≻x1≻x2≻x4
β=(0.9,0.9,0.5,0.5,0.5,0.1,0.1)
TS =(0.1671,0.2304,0.1768,
0.1781,0.1410,0.0922,0.0143)
w=(0.4800,0.1591,0.1260,
0.1024,0.0778,0.0476,0.0072)
x2≻x3≻x4≻x1
β=(0.5,0.5,0.5,0.5,0.5,0.5,0.5)
TS =(0.0714,0.0714,0.0714,
0.0714,0.0714,0.0714,0.0714)
w=(0.1429,0.1429,0.1429,
0.1429,0.1429,0.1429,0.1429)
x2≻x3≻x1≻x4
β=(0.5,0.5,0.5,0.5,0.5,0.5,0.5)
TS =(0.1443,0.1531,0.1403,
0.1579,0.1564,0.1595,0.0886)
w=(0.3994,0.1640,0.1249,
0.1034,0.0864,0.0765,0.0453)
x2≻x1≻x4≻x3
their evaluations so as to deal with their fuzziness and hesitancy
in decision process well.
(2) Consider both the trust relationships and self-confidence
among experts, we proposed the new definitions of fuzzy ad-
jacency matrix, trust score matrix and trust score function to
apply to the WEF nexus evaluation. In addition, based on the
definition of trust score, a novel TS-OWA operator is proposed
to aggregate the individuals’ evolutions into a collective
one.
(3) In order to improve the reliability of the final decision
as well as obtain a high level of consensus in the SNGDM for
WEF nexus evaluation, two automatic iterative algorithms are
presented in this study. One is for the consistency improvement
for individual expert. And another to be used to achieve an
acceptable consensus in the WEF nexus evaluation.
Finally, from the results of the impact analysis, the degree
of self-confidence and social network connection could affect
the final ranking of alternatives in the WEF nexus evaluation.
Therefore, we can conclude that DM’s self-confidence and social
network connection are crucial factors in trust-based SNGDM.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgments
This research was partly supported by National Key R&D Pro-
gram of China under Grant Numbers 2017YFC0404600/
2017YFC0404601, National Natural Science Foundation of China
(NSFC) under Grant Numbers 71871085 and 71471056, and also
supported by the grant TIN2016-75850-R from the Spanish Min-
istry of Economy and Competitiveness with FEDER funds.
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 13
(2) D(h1,h2)=1
#h
#h
s=1γσ(s)
1−γσ(s)
22
=1
#h
#h
s=1γσ(s)
2−γσ(s)
12
=D(h2,h1).
(3) D(h1,h3)=1
#h
#h
s=1γσ(s)
1−γσ(s)
32
=1
#h
#h
s=1γσ(s)
1−γσ(s)
2+γσ(s)
2−γσ(s)
32
=1
#h
#h
s=1γσ(s)
1−γσ(s)
22
+
#h
s=1γσ(s)
2−γσ(s)
32
+2
#h
s=1γσ(s)
1−γσ(s)
2γσ(s)
2−γσ(s)
3,
Box II.
D(h1,h3)≤1
#h#h
s=1γσ(s)
1−γσ(s)
22
+#h
s=1γσ(s)
2−γσ(s)
32
+2#h
s=1γσ(s)
1−γσ(s)
22#h
s=1γσ(s)
2−γσ(s)
32
=1
#h(#hD(h1,h2))2+(#hD(h2,h3))2+2(#h)2D(h1,h2)D(h2,h3)
=(D(h1,h2)+D(h2,h3))2=D(h1,h2)+D(h2,h3).
Box III.
Appendix
Proof of Theorem 1.(1) Since 0 ≤(γσ(s)
1−γσ(s)
2)2≤1, for all
s=1,2,...,#hand #his a positive integer greater than or equal
to 1, then
0=0
#h≤1
#h
#h
s=1γσ(s)
1−γσ(s)
22
≤√#h
#h=1
√#h≤1,
i.e., 0 ≤D(h1,h2)≤1.
Especially, D(h1,h2)=0⇔γσ(s)
1−γσ(s)
2=0, for all s=
1,2,...,#h. Then, γσ(s)
1=γσ(s)
2, for all s=1,2,...,#hindicates
h1=h2, i.e., D(h1,h2)=0⇔h1=h2; (See (2) and (3) in Box II.)
Based on Cauchy-inequality, one can obtain that
#h
s=1
(γσ(s)
1−γσ(s)
2)(γσ(s)
2−γσ(s)
3)2
≤#h
s=1
(γσ(s)
1−γσ(s)
2)2#h
s=1
(γσ(s)
2−γσ(s)
3)2,
#h
s=1
(γσ(s)
1−γσ(s)
2)(γσ(s)
2−γσ(s)
3)
≤
#h
s=1
(γσ(s)
1−γσ(s)
2)2
#h
s=1
(γσ(s)
2−γσ(s)
3)2,
Then, see D(h1,h3) given in Box III.
This completes the proof of Theorem 1.
Proof of Theorem 2.(1) For H1,H2, since 0 ≤D(hij,1,hij,2)2≤1,
for all i,j=1,2,...,nand nis a positive integer greater than 1,
then
0≤2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)2
≤2
n(n−1)n(n−1)
2=2
n(n−1) ≤1,
i.e.,0 ≤D(H1,H2)≤1.
Especially, D(H1,H2)=0⇔hij,1−hij,2=0, for all i,j=
1,2,...,n. Then, hij,1=hij,2, for all i,j=1,2,...,nindicates
H1=H2, i.e., D(H1,H2)=0⇔H1=H2; see (2) and (3) in Box IV.
Based on Cauchy-inequality, one can obtain that D(H1,H3) as in
Box V.
This completes the proof of Theorem 2.
Proof of Theorem 3.Because
(#hn(n−1)
2σ(CIM(H)))2=
n
j>i
n
i=1
#hij
s=1
(εs
ij
σ)2,(36)
(#h
σ(CIE(hij)))2=
#h
s=1εs
ij
σ2
,(37)
14 N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363
(2) D(H1,H2)=2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)2
=2
n(n−1)
n
j>i
n
i=1D(hij,2,hij,1)2=D(H2,H1).
(3) D(H1,H3)=2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,3)2≤2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)+D(hij,2,hij,3)2
=2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)2+
n
j>i
n
i=1D(hij,2,hij,3)2+2
n
j>i
n
i=1
D(hij,1,hij,2)D(hij,2,hij,3)
Box IV.
D(H1,H3)≤2
n(n−1)
n
j>i
n
i=1D(hij,1,hij,2)2+
n
j>i
n
i=1D(hij,2,hij,3)2+2
n
j>i
n
i=1D(hij,1,hij,2)2
n
j>i
n
i=1D(hij,2,hij,3)2
=2
n(n−1)n(n−1)
2D(H1,H2)2
+n(n−1)
2D(H1,H2)2
+2n(n−1)
22
D(H1,H2)D(H2,H3)
=(D(H1,H2)+D(H2,H3))2=D(H1,H2)+D(H2,H3).
Box V.
and εs
ij
σ∼N(0,1), then,
(#hn(n−1)
2σ(CIM(H)))2∼χ2(#hn(n−1)
2),(38)
(#h
σ(CIE(hij)))2∼χ2(#h).(39)
The proof of Theorem 3 is completed.
References
[1] M. Gulati, I. Jacobs, A. Jooste, D. Naidoo, S. Fakir, The water–energy–food
security nexus: Challenges and opportunities for food security in South
Africa, Aquatic Procedia 1 (2013) 150–164.
[2] A. Karnib, A quantitative assessment framework for water, energy and food
nexus, Comput. Water Energy Environ. Eng. 6 (2017) 11–13.
[3] H. Hoff, Understanding the Nexus. Background Paper for the Bonn
2011 Conference: The Water, Energy and Food Security Nexus, Sweden,
Stockholm Environment Institute (SEI), Stockholm, 2011.
[4] J. Halbe, C. Pahlwostl, M.A. Lange, C. Velonis, Governance of transitions to-
wards sustainable development – the water–energy–food nexus in Cyprus,
Water Int. 40 (2015) 877–894.
[5] Z.S. Xu, X.Q. Cai, Group consensus algorithms based on preference
relations, Inform. Sci. 181 (2011) 150–162.
[6] Z.S. Xu, Some models for deriving the priority weights from interval fuzzy
preference relations, European J. Oper. Res. 184 (2008) 266–280.
[7] M.M. Xia, Z.S. Xu, Managing hesitant information in GDM problems under
fuzzy and multiplicative preference relations, Int. J. Uncertain. Fuzzy 37
(2013) 865–897.
[8] Y.J. Xu, K.W. Li, H.M. Wang, Distance-based consensus models for fuzzy
and multiplicative preference relations, Inform. Sci. 253 (2013) 56–73.
[9] Y.C. Dong, Z.G. Ding, L. Martínez, F. Herrera, Managing consensus based on
leadership in opinion dynamics, Inform. Sci. 397 (2017) 187–205.
[10] N. Capuano, F. Chiclana, H. Fujita, E. Herrera-Viedma, V. Loia, Fuzzy group
decision making with incomplete information guided by social influence,
IEEE Trans. Fuzzy Syst. 26 (2018) 1704–1718.
[11] Y.J. Xu, Q.Q. Wang, F.J. Cabrerizo, E. Herrera-Viedma, Methods to improve
the ordinal and multiplicative consistency for fuzzy reciprocal preference
relations, Appl. Soft Comput. 67 (2018) 479–493.
[12] L.G. Pérez, F. Mata, F. Chiclana, Social network decision making with
linguistic trustworthiness–based induced OWA operators, Int. J. Intell. Syst.
29 (2015) 1117–1137.
[13] R. Ureña, G. Kou, Y.C. Dong, F. Chiclana, E. Herrera-Viedma, A review on
trust propagation and opinion dynamics in social networks and group
decision making frameworks, Inform. Sci. 478 (2019) 461–475.
[14] J. Wu, F. Chiclana, E. Herrera-Viedma, Trust based consensus model for
social network in an incomplete linguistic information context, Appl. Soft
Comput. 35 (2015) 827–839.
[15] J. Wu, F. Chiclana, H. Fujita, E. Herrera-Viedma, A visual interaction
consensus model for social network group decision making with trust
propagation, Knowl.-Based Syst. 122 (2017) 39–50.
[16] Y.C. Dong, Z.G. Ding, F. Chiclana, E. Herrera-Viedma, Dynamics of public
opinions in an online and offline social network, IEEE Trans. Big Data
(2017) 1–11.
[17] M.R. Ureña, F. Chiclana, G. Melancon, E. Herrera-Viedma, A social net-
work based approach for consensus achievement in multiperson decision
making, Inf. Fusion 47 (2019) 72–87.
[18] Y.C. Dong, Q.B. Zha, H.J. Zhang, G. Kou, H. Fujita, F. Chiclana, E. Herrera-
Viedma, Consensus reaching in social network group decision making:
Research paradigms and challenges, Knowl.-Based Syst. 162 (2018) 3–13.
[19] X. Liu, Y.J. Xu, M. Rosana, F. Herrera, Social network group decision making:
Managing self-confidence-based consensus model with the dynamic im-
portance degree of experts and trust-based feedback mechanism, Inform.
Sci. 505 (2019) 215–232.
[20] R.A. Hanneman, M. Riddle, Introduction to Social Network Methods,
University of California, Riverside, 2005.
[21] H.P. Scott, Social Network Analysis: A Handbook, Sage Publications Ltd.,
London, 2000.
[22] S. Wasserman, K. Faust, Social Network Analysis: Methods and
Applications, Cambridge University Press, 2009.
[23] Y.J. Xu, R. Patnayakuni, H.M. Wang, The ordinal consistency of a fuzzy
preference relation, Inform. Sci. 224 (2013) 152–164.
[24] Y.J. Xu, F. Ma, F.F. Tao, H.M. Wang, Some methods to deal with unac-
ceptable incomplete 2-tuple fuzzy linguistic preference relations in group
decision making, Knowl.-Based Syst. 56 (2014) 179–190.
N. Wu, Y. Xu, X. Liu et al. / Applied Soft Computing Journal 93 (2020) 106363 15
[25] S. Alonso, F.J. Cabrerizo, F. Chiclana, Group decision making with
incomplete fuzzy preference relations, Int. J. Intell. Syst. 24 (2010)
201–222.
[26] N.N. Wu, Y.J. Xu, D.M. Kilgour, Water allocation analysis of the Zhanghe
river basin using the graph model for conflict resolution with incomplete
fuzzy preferences, Sustainability 11 (2019) 1099.
[27] Y.J. Xu, M.Q. Li, F.J. Cabrerizo, F. Chiclana, E. Herrera-Viedma, Algorithms
to detect and rectify multiplicative and ordinal inconsistencies of fuzzy
preference relations, IEEE Trans. Syst. Man Cybern.: Syst. (2019) http:
//dx.doi.org/10.1109/TSMC.2019.2931536.
[28] Y.J. Xu, L. Chen, F. Herrera, H.M. Wang, Deriving the priority weights from
incomplete hesitant fuzzy preference relations in group decision making,
Knowl.-Based Syst. 99 (2016) 71–78.
[29] M.M. Xia, Z.S. Xu, Hesitant fuzzy information aggregation in decision
making, Internat. J. Approx. Reason. 52 (2011) 395–407.
[30] Z.M. Zhang, C. Wang, X.D. Tian, A decision support model for group
decision making with hesitant fuzzy preference relations, Knowl.-Based
Syst. 86 (2015) 77–101.
[31] M. Brunelli, M. Fedrizzi, M. Fedrizzi, Fuzzy m-ary adjacency relations in
social network analysis: Optimization and consensus evaluation, Inf. Fusion
17 (2014) 36–45.
[32] L.G. Pérez, F. Mata, F. Chiclana, G. Kou, E. Herrera-Viedma, Modelling
influence in group decision making, Soft Comput. 20 (2016) 1653–1665.
[33] J.F. Chu, X.W. Liu, Y.M. Wang, Social network analysis based approach to
group decision making problem with fuzzy preference relations, J. Intell.
Fuzzy Syst. 31 (2016) 1258–1271.
[34] J. Wu, F. Chiclana, A social network analysis trust–consensus based
approach to group decision-making problems with interval-valued fuzzy
reciprocal preference relations, Knowl.-Based Syst. 59 (2014) 97–107.
[35] A. Smajgl, J. Ward, L. Pluschke, The water-food-energy nexus-realising a
new paradigm, J. Hydrol. 533 (2016) 533–540.
[36] L. Bizikova, D. Roy, D. Swanson, H.D. Venema, M. McCandless, The
water–energy–food security nexus: Towards a practical planning and
decision-support framework for landscape investment and risk manage-
ment, in: The International Institute for Sustainable Development, IISD,
Canada, 2013.
[37] A.K. Gain, Y. Wada, Assessment of future water scarcity at different spatial
and temporal scales of the Brahmaputra River basin, Water Resour. Manag.
28 (2014) 999–1012.
[38] B.T. Daher, R.H. Mohtar, Water–energy–food (WEF) Nexus Tool 2.0: guiding
integrative resource planning and decision-making, Water Int. 40 (2015)
748–771.
[39] A.W. Wolfe, Social network analysis: Methods and applications, Am. Ethnol.
24 (1995) 136–137.
[40] G.F. Templeton, X. Luo, T.R. Giberson, N. Campbell, Leader personal in-
fluences on membership decisions in moderated online social networking
groups, Decis. Support Syst. 54 (2012) 655–664.
[41] C. Kaiser, S. Schlick, F. Bodendorf, Warning system for online market
research – Identifying critical situations in online opinion formation,
Knowl.-Based Syst. 24 (2011) 824–836.
[42] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan
press LTD, 1976.
[43] K.H. Rosen, Discrete Mathematics and Its Application, sixth ed.,
McGraw-Hill, 2007.
[44] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.
[45] L.A. Zadeh, Fuzzy logic, Computer 21 (2008) 83–93.
[46] W. Caesarendra, T. Wijaya, T. Tjahjowidodo, B.K. Pappachan, A. Wee,
M.I. Roslan, Adaptive neuro-fuzzy inference system for deburring stage
classification and prediction for indirect quality monitoring, Appl. Soft
Comput. 72 (2018) 565–578.
[47] I. Güler, E.D. Ubeyli, Adaptive neuro-fuzzy inference system for classifica-
tion of EEG signals using wavelet coefficients, J. Neurosci. Methods 148
(2005) 113–121.
[48] S.A. Orlovsky, Decision-making with a fuzzy preference relation, Fuzzy Sets
and Systems 1 (1978) 155–167.
[49] V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010) 529–539.
[50] Y.J. Xu, F.J. Cabrerizo, E. Herrera-Viedma, A consensus model for hes-
itant fuzzy preference relations and its application in water allocation
management, Appl. Soft Comput. 58 (2017) 265–284.
[51] Y.J. Xu, C.Y. Li, X.W. Wen, Missing values estimation and consensus build-
ing for incomplete hesitant fuzzy preference relations with multiplicative
consistency, Int. J. Comput. Intel. Syst. 11 (2018) 101–119.
[52] Z.S. Xu, M.M. Xia, Distance and similarity measures for hesitant fuzzy sets,
Inform. Sci. 181 (2011) 2128–2138.
[53] B. Zhu, Z.S. Xu, J.P. Xu, Deriving a ranking from hesitant fuzzy preference
relations under group decision making, IEEE Trans. Cybern. 44 (2017)
1328–1337.
[54] Y.J. Xu, X. Liu, H.M. Wang, The additive consistency measure of fuzzy
reciprocal preference relations, Int. J. Mach. Learn. Cybern. 9 (2018)
1141–1152.
[55] Y.J. Xu, F. Ma, F. Herrera, Revisiting inconsistent judgements for incomplete
fuzzy linguistic preference relations: Algorithms to identify and rectify
ordinal inconsistencies, Knowl.-Based Syst. 163 (2019) 305–319.
[56] Y.J. Xu, N.N. Wu, A two-stage consensus reaching model for group decision
making with reciprocal fuzzy preference relations, Soft Comput. 23 (2019)
8057–8073.
[57] Y.J. Xu, F. Herrera, Visualizing and rectifying different inconsistencies for
fuzzy reciprcoal preference relations, Fuzzy Sets and Systems 362 (2019)
85–109.
[58] C.C. Li, Y.C. Dong, Y.J. Xu, F. Chiclana, E. Herrera-Viedma, F. Herrera,
An overview on managing additive consistency of reciprocal preference
relations for consistency-driven decision making and fusion: Taxonomy
and future directions, Inf. Fusion 52 (2019) 143–156.
[59] Y.C. Dong, Y.F. Xu, H.Y. Li, On consistency measures of linguistic preference
relations, European J. Oper. Res. 189 (2008) 430–444.
[60] P. de Jong, A statistical approach to Saaty’s scaling method for priorities,
J. Math. Psych. 28 (1997) 467–478.
[61] F. Chiclana, E. Herrera-Viedma, F. Herrera, S. Alonso, Some induced ordered
weighted averaging operators and their use for solving group decision-
making problems based on fuzzy preference relations, European J. Oper.
Res. 182 (2007) 383–399.
[62] R.R. Yager, Induced aggregation operators, Fuzzy Sets and Systems 137
(2003) 59–69.
[63] R.R. Yager, Quantifier guided aggregation using OWA operators, Int. J. Intell.
Syst. 11 (1996) 49–73.
[64] G. Pasi, R.A.M. Pereira, A decision making approach to relevance feedback
in information retrieval: A model based on soft consensus dynamics, Int.
J. Intell. Syst. 14 (1999) 105–122.
[65] Y.C. Dong, G.Q. Zhang, W.C. Hong, Y.F. Xu, Consensus models for AHP group
decision making under row geometric mean prioritization method, Decis.
Support Syst. 49 (2010) 281–289.
[66] Y.C. Dong, C.C. Li, Y.F. Xu, X. Gu, Consensus-based group decision making
under multi-granular unbalanced 2-tuple linguistic preference relations,
Group Decis. Negot. 24 (2015) 217–242.
[67] Y.C. Dong, C. Xia, F. Herrera, Minimizing adjusted simple terms in the
consensus reaching process with hesitant linguistic assessments in group
decision making, Inform. Sci. 297 (2015) 95–117.
[68] J. Wu, L.F. Dai, F. Chiclana, H. Fujita, E. Herrera-Viedma, A minimum
adjust cost feedback mechanism based consensus model for group decision
making under social network with distribute linguistic trust, Inf. Fusion 41
(2018) 232–242.
[69] X. Liu, Y.J. Xu, R. Montes, R.X. Ding, F. Herrera, Alternative ranking-
based clustering and reliability index-based consensus reaching process
for hesitant fuzzy large scale group decision making, IEEE Trans. Fuzzy
Syst. 27 (2019) 159–171.
[70] Y.J. Xu, X.W. Wen, W.C. Zhang, A two-stage consensus method for
large-scale multi-attribute group decision making with an application to
earthquake shelter selection, Comput. Ind. Eng. 116 (2018) 113–129.
[71] Z. Zhang, C.H. Guo, L. MartÍnez, Managing multigranular linguistic distribu-
tion assessments in large-scale multiattribute group decision making, IEEE
Trans. Syst. Man Cybern.: Syst. 47 (2017) 3063–3076.
[72] Y.J. Xu, X.W. Wen, Z.Q. Zhang, Missing values estimation for incomplete
uncertain linguistic preference relations and its application in group
decision making, J. Intell. Fuzzy Syst. 36 (2019) 1809–1822.