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Ordinal and joint feedback mechanisms to support group consensus based on
interval-valued number with self-confidence
Feixia Jia, Francisco Chiclanab,d, Mi Zhouc, Enrique Herrera-Viedmad,e, Jian Wua
aSchool of Economics and Management, Shanghai Maritime University, Shanghai 201306, China
bInstitute of Artificial Intelligence, Faculty of Computing, Engineering and Media, De Montfort University, Leicester,
UK
cSchool of Management, Hefei University of Technology, Hefei, Anhui, China
dAndalusian Research Institute on Data Science and Computational Intelligence (DaSCI), University of Granada,
Granada, Spain
eDepartment of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah
21589, Saudi Arabia.
Abstract
This article proposes a framework of consensus reaching process based on interval-valued number with
self-confidence involving two feedback mechanisms: (1) the ordinal feedback mechanism (OFM), and
(2) the joint feedback mechanism(JFM). Firstly, a new concept called the interval-valued number with
self-confidence (IVN-SC) is defined, which allows experts to express self-confidence (SC) when provid-
ing their evaluations by interval-valued number (IVN). In OFM, three types of discordant behaviours
are explored from the perspective of IVN consensus and SC consensus, and then the corresponding
personalized mechanisms for discordant behaviours are implemented to guarantee they both reach
consensus threshold. While in JFM, the comprehensive consensus degree by combing the IVN and
SC is presented to measure the agreement level among experts, and then a joint optimisation model
with IVN-SC is activated to support the discordant experts to reach the threshold value of group
consensus. It explores that the JFM is less strict than the OFM because the former contains the
compensatory aggregation operator while the latter does not. Finally, an illustrative example and
comparative analysis are devised to testify the effectiveness of the proposed models.
Keywords: Group Decision Making, Consensus, Adjustment Cost, Self-confidence, Ordinal
Feedback Mechanism, Joint Feedback Mechanism.
1. Introduction
Group decision making (GDM) can be defined as a scenario where a group of experts (decision
makers) participate in decision-making providing their opinions regarding a finite set of alternatives
under criteria, which then aims to derive a common group solution after the individual opinions
are fused or aggregated [1, 2]. Achieving a high-level consensus among experts through consensus
reaching process (CRP) is the key to GDM problems [3, 4]. Until now, there have been various CRPs
Email addresses: Fxji6198@163.com (Feixia Ji), chiclana@dmu.ac.uk (Francisco Chiclana), zhoumi@hfut.edu.cn
(Mi Zhou), viedma@decsai.ugr.es (Enrique Herrera-Viedma), jyajian@163.com (Jian Wu)
Preprint submitted to Information Fusion July 30, 2022
implemented in different decision-making environments [5, 6, 7]. In CRP, the feedback mechanism
is a critical stage for improving consensus and reducing conflicts, as the optimal modification setting
is sought for assessments of the detected discordant experts [8]. In realistic GDM problems, coping
with uncertainty is always a challenging issue, and therefore various forms of representation have
been proposed and widely used in the literature, which include, but are not limited to: fuzzy sets [9],
intuitionistic fuzzy sets [10, 11, 12], hesitant fuzzy sets [13, 14], personalised individual semantics [15]
and trust functions [16, 17].
Self-confidence, as one of the psychological implication of human self-statement, has gradually
attracted increasing attention in the field of decision-making [18, 19, 20, 21, 22, 23]. Firstly, Liu et al.
[24] proposed a new kind of preference relation, which is named by self-confident preference relation.
Dong et al. [25] validated that the self-confident preference relation could reach higher quality of
final decisions than incomplete preference relations in most situations. To explore the consensus issue
in the GDM with self-confident preference relations, Liu et al. [26] designed a novel iteration-based
consensus building framework to manage both the preference values and the self-confidence. Liu et
al. [27] developed a consensus model by considering overconfidence behaviours and discussed the
overconfidence behaviours detection and management. The above studies have strongly demonstrated
that it would be of great importance to consider experts’ self-confidence levels in decision-making
analysis. Inspired by this interesting expression with self-confidence, this article proposes the interval-
valued number with self-confidence (IVN-SC) expression tool. Elements in an IVN-SC are composed
of two parts, the first one is the assessment evaluated by interval-valued number regarding a finite
set of alternatives under criteria, while the second part represents the self-confidence level associated
with the first part, which is defined on a linguistic scale.
In certain GDM issues, such as the review of journal papers, one first goes about rating the
paper and then gives a self-confident score for familiarity with the field. In this case, there are two
dimensions of consensus: (1) rating value consensus and (2) self-confidence consensus. How do the
two dimensions of consensus affect the achievement of group consensus is a matter for further study.
Thus, the feedback mechanisms considering self-confidence behaviours are applied to two scenarios of
facilitating different consensus requirements: (1) the ordinal feedback mechanism (OFM) respectively
identifies and manages IVN and SC consensus to guarantee they both reach consensus threshold, and
(2) the joint feedback mechanism (JFM) includes the compensation aggregation operator to combine
IVN and SC consensus to provide joint feedback for IVN-SC.
(1) In OFM, an identification standard is provided to detect three types of discordant behaviours
involving IVN consensus and SC consensus. Subsequently, corresponding mechanisms are imple-
mented to help discordant experts with different discordant behaviours in achieving consensus.
The comparative analysis is devised to show that the adjustment cost generated by different im-
plementation sequences in the OFM is relatively stable, while OFM will cause over-adjustment
2
due to the iteration-based feedback mechanism to guarantee both the IVN and SC consensus reach
acceptable thresholds.
(2) In JFM, comprehensive consensus degree (CCD) combining the IVN and SC is presented to
measure the agreement level among experts. Subsequently, a joint optimisation model with IVN-
SC in GDM is built, which is able to help the discordant experts reach the threshold value of
consensus. The comparative analysis is proposed to verify that the comprehensive parameter of
control has impact on the identification of discordant experts and JFM has a lower consensus
adjustment cost than OFM.
To achieve a mutual agreement in GDM problems and improve the validity of the feedback mech-
anism, a more reasonable policy is to generate personalized feedback advice in the interactive process
[28, 29, 30, 31, 32]. Recently, Wu et al. [17] investigated a two-fold personalized feedback mechanism
to support discordant experts to achieve threshold of consensus in social network GDM. The advantage
of this method is that it achieves a balance between group consensus and individual personality for
generating personalized recommendation advice based on social network. However, under the assess-
ment information of IVN-SC, how does the corresponding personalized feedback mechanism work is a
challenge to address in GDM problems. Hence, this study continues the research line of personalized
feedback mechanism and proposes a framework of CRP with two personalized feedback mechanisms
that enable different consensus requirements to manage self-confidence behaviours in GDM with IVN-
SC. In this CRP, experts are allowed to express self-confidence levels when providing their assessment
evaluated by interval-valued number. Then, OFM and JFM personalized feedback mechanisms with
self-confidence behaviours are proposed to pursuit higher consensus. It is found that JFM is less
strict than OFM because the former contains the compensatory aggregation operator while the latter
does not. Following this, an illustrative example and comparative analysis are devised to validate the
consensus effectiveness of the proposed models.
The remainder of the paper is organised as follows: Section 2 offers some basic knowledge regarding
2-tuple linguistic model and the minimum adjustments (or cost) consensus model. Section 3 intro-
duces the detailed proposals for consensus reaching with self-confidence behaviours, which includes
the concept of interval-valued number with self-confidence (IVN-SC), description framework of OFM
and JFM with IVN-SC, consensus degree measurement at three different decision matrix levels, the
proposed OFM and JFM consensus models and selection process. Following this, a numerical example
and comparative analysis corresponding to OFM and JFM models are conducted in Section 4. Finally,
Section 5 concludes the study and discusses the research directions for the future.
3
2. Preliminaries
This section includes the basic knowledge regarding the 2-tuple linguistic model and the minimum
adjustments (or cost) consensus model.
2.1. 2-tuple linguistic scale model
Fuzzy (cardinal) linguistic methods deal with qualitative problems represented by language vari-
ables [33, 34], although they are associated with important limitations regarding the information
aggregation result not matching in general any of the original linguistic terms, which means that an
approximate matching value is required [35, 36]. To address this, Herrera et al. [37] proposed an
ordinal linguistic model, known as the 2-tuple linguistic model (sλ, α), which is based on the concept
of symbolic translation α∈[−0.5,0.5), as captured in the below definition.
Definition 1 (2-Tuple Linguistic Scale Model). [37]. Let S={so, s1, ..., sg}be a linguistic term
set and β∈[0, g] be a number in the granularity interval of the linguistic term set S, then the 2-
tuple that expresses the equivalent information to β∈[0, g ] is obtained with the following function
∆ : [0, g]→S×[−0.5,0.5), where
∆ (β) = (sλ, α),with
λ=round (β)
α=β−λ;α∈[−0.5,0.5) .
The inverse function of ∆ is ∆−1:S→[0, g] with ∆−1((sλ, α)) = λ+α. Obviously, the symbolic
translation αreflects the deviation of the linguistic term sλand the equivalent numerical value β. An
ordering and a negation operator on the set of 2-tuples linguistic values were presented in [26, 37] as
follows:
•2-tuple ordering: (su, σ)<(sl,ℵ)⇐⇒ u<l∨[u=l∧σ < ℵ].
•2-tuple negation operator: N eg ((sλ, α)) = ∆g−∆−1(sλ, α).
2.2. Minimum adjustments (or cost) consensus model
The basic model and origins of the minimum adjustment (or cost) consensus model are presented
in this section.
In GDM issues, the optimal modification setting is sought for assessments during the consen-
sus reaching process, with the purpose of reaching a higher consensus state. Ben-Arieh and Eas-
ton [38] developed the concept of consensus cost and designed a consensus reaching algorithm to
assist decision makers in reaching higher consensus. Dong et al. [39] proposed a minimum adjust-
ments consensus model within a linguistic framework, hereby represented in numerical form: let
{t1, t2, ..., tm}and {t1, t2, ..., tm}be the original and adjusted assessments provided by decision makers
E={e1, e2, ..., em}, respectively,
4
Min
m
X
k=1
dtk, tk
s.t.
tc=Ft1, t2, ..., tm
CL tk, tc≥β, k = 1, ..., m.
(1)
Where {t1, t2, ..., tm}are decision variables, and the objective function is to minimize the distance
between the original and adjusted assessments. The first constraint is to derive the collective evaluation
tcfrom the modified evaluation of decision makers {t1, t2, ..., tm}utilizing the aggregation function F.
The second constraint ensures that the consensus level of all decision makers is not lower than the
predefined value of β.
Estimating the adjustment distance or the consensus cost is an essential indicator for the efficiency
of GDM. To this end, Zhang et al. [40] designed a specific optimization model regarding minimum cost
consensus model. Wu et al. [41] proposed a minimum adjustment cost feedback mechanism based on
distributed linguistic trust in social network GDM. The advantage of this method is that it can lessen
the possibility that discordant experts are forced to modify their assessment information by selecting
the parameters of control that minimize the adjustment cost. Also, a comprehensive work on feedback
mechanism paradigms is investigated in [42]. Numerous consensus models with minimum adjustments
(or cost) have been devised and the impact of self-confidence behaviours on the adjustment cost of
the GDM consensus reaching process, as aimed in this study, is not being accounted for.
3. OFM and JFM models with IVN-SC
This section describes in detail the proposals for interval-valued number with self-confidence (IVN-
SC) consensus reaching with self-confidence behaviours to facilitate resolution.
3.1. Interval-valued number with self-confidence
In realistic GDM problems, it is hard to ensure that each expert can express her/his opinions in a
precise or unambiguous way [30, 43, 44]. This difficulty is due to the unpredictability of decision events
and the limitation of cost and knowledge [8, 28, 29]. Interval-valued number (IVN) is an effective tool
to address uncertain problems in GDM [45]. A brief description of an IVN is introduced below:
Definition 2 (Interval-Valued Number (IVN)). [45]. ϕ= [oL, oR] = o:oL6oR, o ∈Ris
called an IVN, where oLand oRare the left and right limit of the interval ϕon the real line R,
respectively. If oL=oR, then ϕ= [o, o] is a real number.
Zhang [46] provided the score and accuracy functions to compare the two interval-valued two-tuple
variables, which is based on [47, 48]. This article adjusts the score and accuracy functions initiated
from Zhang [46], and the modified score and accuracy functions are given below.
5
Definition 3 (Score Function). [46]. For an IVN ϕ= [oL, oR], the corresponding score function
is:
S(ϕ) = oL+oR
2.(2)
Definition 4 (Accuracy Function). [48]. For an IVN ϕ= [oL, oR], the corresponding accuracy
function is:
H(ϕ) = oL+oR−1.(3)
Definition 5 (Order Relation of IVNs). [46]. Let ϕ1= [o1L, o1R] and ϕ2= [o2L, o2R] be two
IVNs, S(ϕ) be the score and H(ϕ) be the accuracy of IVNs. The ordering relation that ϕ1< ϕ2is
true if and only of one of the following conditions:
(1) S(ϕ1)< S(ϕ2) ;
(2) S(ϕ1) = S(ϕ2)∨H(ϕ1)< H(ϕ2) .
A set of nine linguistic terms of Table 1 was proposed by Liu et al. [26], SSL ={lλ|λ= 0,1, ..., 8},
to characterize experts’ self-confidence levels.
Table 1: Description of nine linguistic terms of set SSL .
Linguistic label Semantic meaning
l0None confident
l1Very low confident
l2Low confident
l3Slightly low confident
l4Medium confident
l5Slightly high confident
l6High confident
l7Very high confident
l8Perfect confident
This article assumes linguistic model to characterize the experts’ self-confidence levels in GDM
involving a finite set of alternatives A={a1, ..., ap}and a finite set of criteria C={c1, ..., cq}as per
the definition below.
Definition 6 (Interval-Valued Number with Self-Confidence (IVN-SC)). An IVN-SC P=
(ϕij , sij )p×qis a matrix with elements including two components: the first one ϕij being assessment
evaluated by interval-valued number (IVN) on alternative aiunder the criterion cj, while the second
one sij ∈SSL being the self-confidence (SC) associated with the first component ϕij .
6
Example 1. Let A={a1, a2, a3}be the set of alternatives, and C={c1, c2, c3}be the set of criteria,
with SSL as described above. Then, an expert may provide Table 2 as her/his IVN-SC matrix, with
Table 2: The example IVN-SC decision matrix.
c1c2c3
a1([0.3,0.5] , l3) ([0.3,0.6] , l5) ([0.6,0.7] , l4)
a2([0.5,0.6] , l3) ([0.4,0.6] , l7) ([0.4,0.7] , l6)
a3([0.4,0.6] , l6) ([0.4,0.8] , l7) ([0.2,0.3] , l6)
the element ϕ12 = [0.3,0.6] interpreted as follows: the expert’s assessment information regarding the
alternative a1under the criterion c2is evaluated by an IVN [0.3,0.6], while the expert’s self-confidence
level s12 =l5on this IVN is “slightly high confident”as per the provided linguistic value l5.
For the fusion of information, the aggregating operators are proposed, one of which is defined as
the interval-valued two-tuple weighted average (IVTWA) operator [46]. This idea also can be used to
extend to interval-valued number with self-confidence (IVN-SC) cases as follows:
Definition 7 (IVN-SC Weighted Average Operator). Let ˜
R={([o1L, o1R], lt1),([o2L, o2R], lt2),
..., ([omL, omR ], ltm)}be with ω= (ω1, ω2, ..., ωm)Tas the corresponding weighting vector, where
[oiL, oiR ]∈IV N (R), ωi∈[0,1] and Pm
i=1 ωi= 1, the IVN-SC weighted average operator is defined as:
W A(˜
R) = ([
m
X
i=1
ωioiL,
m
X
i=1
ωioiR],∆[
m
X
i=1
ωi∆−1(lti)]).(4)
3.2. Framework of OFM and JFM with IVN-SC
As the efficiency of consensus and the reliability of final decisions are of great importance in realistic
GDM problems, discordant behaviours must be reliably managed [44, 49, 50]. In a CRP, the consensus
level of experts is predefined to judge whether or not to implement the feedback mechanism, and the
identification standard to discordant experts is also in place to determine when the CRP needs to
come to an end [51, 52].
Since group decision makers may have different knowledge backgrounds, cultures and goals, it is
not appropriate to set the same identification standard to discordant experts whose consensus degree is
lower than the accepted level. Thus in this article, the feedback mechanisms considering self-confidence
behaviours are applied to two scenarios of facilitating different consensus requirements: (1) ordinal
feedback mechanism (OFM) with three discordant behaviours, and (2) joint feedback mechanism
(JFM) with IVN-SC. Moreover, the OFM for consensus can be seen as stricter than the JFM because
the latter contains the compensatory aggregation operator while the former does not.
To study the influence of OFM and JFM consensus models with self-confidence behaviours in
GDM problems, this article allows experts to use IVN-SC for expressing their assessment information.
7
Subsequently, the OFM and JFM consensus models are presented by using the minimum adjustment
cost to detect and manage experts’ discordant assessment evaluated by IVN-SC in CRPs. As illustrated
in Fig.1, the proposed framework of OFM and JFM with IVN-SC involves the following key procedures:
Expe rts
Dete ct the Disc ordant
Be haviours
The O rdinal Fee dback
Me chanism with
Discordant Be haviours
JFM CPRs
OFM CPRs
Opi nion s
Dec ision Making Ma trix
with IVN-SC
Co nsensus Degre e
Both Interval-Value d Numbe r
and Self-Co nfidence
Co nsensus
Ide ntific atio n
The Jo int Fee dback
Me chanism with
IVN-S C
Final Group IVN -SC
Alternatives Ranking
Reac hing C ons ens us
Se le ction
Proce s s
No
No
Yes Yes
Co nsensus
Ide ntific atio n
Figure 1: The framework of OFM and JFM models with IVN-SC.
Step 1: IVN-SC consensus degree at three decision matrix levels
At this step, detailed in Section 3.3, a method for measuring IVN-SC consensus degree among
experts is provided, which is also used as an effective tool to identify the discordant experts with the
lowest contributions to the consensus.
Step 2: OFM and JFM consensus models driven on minimum adjustment cost
At this step, elaborated in Section 3.4 and Section 3.5, two consensus models (OFM and JFM)
with self-confidence behaviours to generate personalized modifications are presented.
Step 3: Selection Process
At this step, illustrated in Section 3.6, the ordering of alternatives is generated from the obtained
IVN-SC score.
In this article, it is assumed that a group of mexperts {ek:k= 1, ..., m}provide opinions on p
alternatives {ai:i= 1, ..., p}with respect to the qcriteria {cj:j= 1, ..., q}. For the sake of simplicity,
let M={1,2, .., m},P={1,2, ..., p}and Q={1,2, ...q}. Several key notations are listed in Table 3.
3.3. Measuring consensus degree at the three different decision matrix levels
Consensus degree aims to measure the agreement level among the group experts, which is also used
as an effective tool to identify the discordant experts in the group. This article proposes consensus
degree based on IVN-SC at the three different levels of a decision matrix: (1) alternative element level;
(2) alternative level; and (3) decision matrix level.
8
Table 3: Parameters and decision variables of the proposed OFM and JFM models.
Parameters Description
βConsensus boundary, i.e. group consensus degree threshold value
ηComprehensive coefficient in JFM
DB ={DBI, DBII , DBIII}Discordant behaviours in OFM. i.e. DBI={eh|AT Dh< β ∧ASDh≥β}
DET/DES Discordant experts subgroup regarding IVN/SC in OFM
DE T /DES Experts subgroup not included in the DET/DES in OFM
EI CH Discordant experts subgroup in JFM
AT Dh
(t)/ASDh
(t)(t= 1,2,3) Consensus degree of discordant experts after the tth iteration in OFM
CC Dh/CC DhConsensus degree of discordant experts before/after JFM
DE T Eh/DESEhElement set of IVN/SC to adjust in OFM
IJShElement set of IVN-SC to adjust in JFM
FOF M ={FIV N , FSC , FI V N−SC }Adjustment cost of discordant behaviours in OFM
FJF M Adjustment cost of discordant behaviours in JFM
Decision Variables Description
(θh, γh) Ordinal feedback parameters for discordant expert ehin OFM, h∈M
αhJoint feedback parameters for discordant expert ehin JFM, h∈M
Ph= (ϕh
ij , sh
ij )p×qOriginal IVN-SC of expert eh, h ∈M
RP h= (Rϕh
ij , Rsh
ij )p×qAdjusted IVN-SC of expert eh, h ∈M
ωhThe weight of expert eh, h ∈M
B=ϕij , sij p×qThe decision matrix IVN-SC of collective
Let Pk=ϕk
ij , sk
ij p×qand Ps=ϕs
ij , ss
ij p×qbe two IVN-SCs provided by experts ekand
es(k, s ∈M), respectively.
•Level 1. Element of alternative. The IVN consensus degree between experts ekand esat
the alternative element (ai, cj) is defined as:
T Eij ϕk
ij , ϕs
ij = 1 −
okL
ij −osL
ij |+|okR
ij −osR
ij
2.(5)
The IVN consensus degree of expert ekwith respect to the group of experts at the alternative
element (ai, cj) is:
AT Ek
ij =1
m−1
m
X
s=1,s6=k
T Eij ϕk
ij , ϕs
ij .(6)
The SC consensus degree between experts ekand esat the alternative element (ai, cj) is defined
as:
SEij sk
ij , ss
ij = 1 −
∆−1sk
ij −∆−1ss
ij
g.(7)
The SC consensus degree of expert ekwith respect to the group of experts at the alternative
9
element (ai, cj) is:
ASEk
ij =1
m−1
m
X
s=1,s6=k
SEij sk
ij , ss
ij .(8)
•Level 2. Alternative. The IVN consensus degree of expert ekto the group of experts on
alternative aiis:
AT Ak
i=1
q
q
X
j=1
AT Ek
ij .(9)
The SC consensus degree of expert ekwith respect to the group of experts on alternative aiis:
ASAk
i=1
q
q
X
j=1
ASEk
ij .(10)
•Level 3. Decision matrix. The IVN consensus degree of expert ekwith respect to the group
of experts is:
AT Dk=1
p
p
X
i=1
AT Ak
i.(11)
The SC consensus degree of expert ekwith respect to the group of experts is:
ASDk=1
p
p
X
i=1
ASAk
i.(12)
Generally, a consensus threshold β∈[0.5,1) is often predefined to achieve a qualified majority and
assure that the decision-making output is acceptable for the group of experts [53, 54]. Without loss
of generality, this article assumes the threshold value of β= 0.8.
3.4. Ordinal feedback mechanism for consensus (OFM)
In some decision cases, the group decision is made under the conditions of sufficient budget, enough
time and high-required consensus. Therefore, OFM consensus model is developed, which guarantees
that both the assessment evaluated by interval-valued number (IVN) and self-confidence (SC) reach
acceptable thresholds. To do that, the proposed OFM includes two steps: (1) Identification of three
discordant behaviours (Section 3.4.1), and (2) Generation of recommended advice by discordant be-
haviours (Section 3.4.2), which are described in detail below.
3.4.1. OFM discordant behaviours
This article assumes that both IVN and SC consensus degree are equally important in GDM with
IVN-SC, and thus experts’ discordant behaviours can be discussed from the perspective of their IVN
consensus and of their SC consensus. For an expert eh, if AT Dh≥βand ASDh≥β, then the expert
ehis called a concordant expert because she/he has reached acceptable consensus in both IVN and
SC. Otherwise, the expert ehfeatures one of the following discordant behaviours:
10
Discordant behaviour I (DBI).If an expert ehonly achieves an acceptable consensus in SC, i.e.
ASDh≥β, but not in IVN, i.e. AT Dh< β , then the expert ehhas dissimilar interval-valued
numbers with respect to the group of experts while expressing similar self-confidence levels. We
say that such an expert ehfeatures discordant behaviour I.
Discordant behaviour II (DBII ).If an expert ehonly achieves an acceptable consensus in IVN,
i.e. AT Dh≥β, but not in SC, i.e. ASDh< β , then the expert ehhas dissimilar self-confidence
levels with respect to the group of experts while expressing similar interval-valued numbers. We
say that such an expert ehfeatures discordant behaviour II.
Discordant behaviour III (DBIII).If an expert ehhas not reached acceptable consensus in IVN
and SC, i.e. AT Dh< β and ASDh< β , then the expert ehhas dissimilar interval-valued
numbers and self-confidence levels with respect to the group of experts. We say that such an
expert ehfeatures discordant behaviour III.
Algorithm 1 identifies the discordant experts and their alternatives and corresponding elements
where IVN and SC consensus levels discordance occurs, respectively.
Algorithm 1: Discordant Identification mechanism of IVN-SC decision matrix in OFM.
begin
Input: The experts’ consensus degree at three levels: AT Eh
ij , AT Ah
iand AT Dhin IVN and
ASEh
ij , ASAh
iand ASDhin SC;
The consensus threshold β;
Output: Discordant experts, alternatives and elements in IVN and SC, respectively;
1The discordant experts are identified:
DET =h|AT Dh< β;
DES =h|ASDh< β;
2The discordant alternatives are identified:
DE T Ah=i|h∈DET ∧AT Ah
i< β;
DESAh=i|h∈DES ∧ASAh
i< β;
3The discordant alternatives elements are identified:
DE T Eh=n(i, j)|i∈D ET Ah∧AT Eh
ij < βo;
DESEh=n(i, j )|i∈DESAh∧ASEh
ij < βo;
end
3.4.2. OFM generation of recommended advice by discordant behaviours
Once the discordant experts are identified, the feedback mechanism is activated to generate rec-
ommended advice for their identified discordant elements in D ET E and DESE to increase the cor-
11
responding consensus degree. Therefore, each discordant behaviour feedback mechanism is developed
next.
(1) DBIFeedback: The set of discordant experts in this case is DBI=DET ∩DES, and they
receive the personalized adjustment, Rϕh
ij = [RohL
ij , RohR
ij ], only for the IVN, ϕh
ij = [ohL
ij , ohR
ij ], of their
elements in DET E h:
Rϕh
ij =1−θh·ϕh
ij +θh·ϕij
=h1−θh·ohL
ij +θh·oL
ij ,1−θh·ohR
ij +θh·oR
ij i(13)
being ϕij = [oL
ij , oR
ij ] = h1
mPm
h=1 ohL
ij ,1
mPm
h=1 ohR
ij ithe group of experts’ IVNs average at the time,
while θh∈[0,1] is a feedback parameter that is used to control the degree of acceptance of the
recommended advice by the discordant experts eh.
The consensus of discordant expert is proved to reach higher level after adopting the feedback
advices in [55]. In the following, it also can prove that the consensus level of concordant expert also
becomes higher.
Lemma 1. [55]. For discordant expert eh, i.e., AT Dh< β, it is AT Dh> AT D h, where AT Dhis
the consensus degree for ehusing the new evaluation Rϕh.
Lemma 1 shows that the consensus degree for discordant expert will improve after she/he receives
the feedback advice, and thus it can be used to prove the following Theorem.
Theorem 1. Assuming that there is one discordant expert ehfeaturing DBI, who changes her/his
IVN for element (ai, cj)from ϕh
ij to Rϕh
ij , then the new overall consensus degree of concordant experts
on IVN is greater than original consensus degree.
Proof. Without loss of generality, it can be assumed that ehis the discordant expert who features
DBIand others are concordant experts, we have:
AT Dh=1
(m−1) pq
m
X
s=1,s6=h
p
X
i=1
q
X
j=1
T Eij ϕh
ij , ϕs
ij ,
where
T Eij ϕh
ij , ϕs
ij = 1 −dϕh
ij , ϕs
ij = 1 −
ohL
ij −osL
ij |+|ohR
ij −osR
ij
2.
For simplicity, we denote dϕh
ij , ϕs
ij =
ϕh
ij −ϕs
ij
. After ehchanges her/his IVN for element (ai, cj)
from ϕh
ij to Rϕh
ij , the new overall consensus degree of concordant experts become:
m
X
s=1,s6=h
AT Ds
(1) =
m
X
s=1,s6=h
1−1
(m−1) pq
p
X
i=1
q
X
j=1
Rϕh
ij −ϕs
ij
+
m
X
l=1,l6=h
ϕl
ij −ϕs
ij
= (m−1) −1
(m−1) pq
p
X
i=1
q
X
j=1
m
X
s=1,s6=h
Rϕh
ij −ϕs
ij
+
m
X
s=1,s6=h
m
X
l=1,l6=h
ϕl
ij −ϕs
ij
.
12
And the new consensus degree of discordant expert ehis:
AT Dh
(1) = 1 −1
(m−1) pq
p
X
i=1
q
X
j=1
m
X
s=1,s6=h
Rϕh
ij −ϕs
ij
.
It is proved that AT Dh
(1) > AT Dhin Lemma 1, thus
p
X
i=1
q
X
j=1
m
X
s=1,s6=h
Rϕh
ij −ϕs
ij
<
p
X
i=1
q
X
j=1
m
X
s=1,s6=h
ϕh
ij −ϕs
ij
.
Therefore m
X
s=1,s6=h
AT Ds
(1) >
m
X
s=1,s6=h
AT Ds.
The scenario with multiple discordant experts can be regarded as a sequence of successive steps
in which one discordant expert changes her/his evaluation. At each step in this sequence, the new
consensus of the discordant experts will be greater than the previous step, as will be the new overall
consensus degree of concordant experts.
Definition 8 (OFM DBIAdjustment Cost). The OFM DBIadjustment cost of the discordant
experts’ acceptance of the recommended advice is
FIV N =X
h∈DBI
(i,j)∈DE T Eh
Rϕh
ij −ϕh
ij
=X
h∈DBI
(i,j)∈DE T Eh
RohL
ij −ohL
ij |+|RohR
ij −ohR
ij
.
(14)
A linear function of variables (x1, x2, . . . , xn) is a function of the form m1x1+m2x2+· · · +mnxn
where m1,m2, . . . , mnare constants. The following result shows that the OFM DBIadjustment cost
is a linear function of the set of parameters of control {θ1, θ2, . . . , θ#DBI}.
Proposition 1. The OFM DBIadjustment cost is a linear function of the set of parameters of control
{θ1, θ2, . . . , θ#DBI}.
Proof.
FIV N =X
h∈DBI
(i,j)∈DE T Eh
Rϕh
ij −ϕh
ij
=X
h∈DBI
(i,j)∈DE T Eh
1−θh·ϕh
ij +θh·ϕij −ϕh
ij
=X
h∈DBI
θh·X
(i,j)∈DE T Eh
ϕh
ij −ϕij
.
Denoting mh=P(i,j)∈DE T Eh
ϕh
ij −ϕij
(≥0), therefore, it is obvious that FIV N is a linear function
of {θ1, θ2, . . . , θ#DBI}.
13
The above result means that an increase on one of the parameters of control, while the others are
not decreased, increases the value of the OFM DBIadjustment cost since none of its constants mi
are negatives.
Thus, the personalized feedback mechanism can lessen the possibility that discordant experts are
forced to modify their assessment information by selecting the parameters of control that minimise the
OFM DBIadjustment cost, which translates into preserving the discordant experts’ original opinions
to the greatest extent while reaching the consensus threshold value of the group.
The OFM for DBIis therefore expressed as the following optimization model with decision vari-
ables {θ1, θ2, . . . , θ#DBI}:
M(1)
Min FIV N :X
h∈DBI
θh·X
(i,j)∈DE T Eh
ϕh
ij −ϕij
s.t.
AT Dh
(1) ≥β(h∈DBI)
AT Dk
(1) ≥βk∈DE T
Rϕh
ij =1−θh·ϕh
ij +θh·ϕij (i, j)∈D ET Eh
0≤θh≤1.
(15)
(2) DBI I Feedback: The set of discordant experts in this case is DBI I =DE T ∩DES, and they
receive the personalized adjustment, Rsh
ij , only for the self-confidence values, sh
ij , of their elements in
DESEh:
Rsh
ij = ∆ 1−γh·∆−1sh
ij +γh·∆−1(sij ),(16)
being sij = ∆ 1
mPm
h=1 ∆−1sh
ij the group of experts’ self-confidence average value at the time,
while γh∈[0,1] is a feedback parameter that is used to control the degree of acceptance of the
recommended advice by the discordant experts eh.
Theorem 2. Assuming that there is only one discordant expert ehfeaturing DBI I , who changes
her/his SC for element (ai, cj)from sh
ij to Rsh
ij , then the new overall consensus degree of concordant
experts on SC is greater than original consensus degree.
Proof. The proof is similar to that of Theorem 1. We omit the proof here.
Like in OFM DBI, the OFM DBI I adjustment cost is a linear function of the set of parameters
of control {γ1, γ2, . . . , γ#DBI I }with non-negative parameters.
Definition 9 (OFM DBI I Adjustment Cost). After the discordant experts accept the recom-
mended advice, the adjustment cost of the feedback for DBII can be calculated as:
FSC =X
h∈DBII
(i,j)∈DE SE h
∆−1Rsh
ij −∆−1sh
ij
g
=X
h∈DBII
γh·X
(i,j)∈DE SE h
∆−1sh
ij −∆−1(sij )
g
.
(17)
14
The OFM for DBI I is expressed as the following model with decision variables {γ1, γ2, . . . , γ#D BII }:
M(2)
Min FSC :X
h∈DBII
γh·X
(i,j)∈DE SE h
∆−1sh
ij −∆−1(sij )
g
s.t.
ASDh
(2) ≥β(h∈DBI I )
ASDk
(2) ≥βk∈DES
Rsh
ij = ∆ 1−γh·∆−1sh
ij +γh·∆−1(sij )(i, j)∈DESEh
0≤γh≤1.
(18)
(3) DBIII Feedback: The set of discordant experts in this case is DBIII =DET ∩DES, and
they receive the personalized adjustment for their interval-valued numbers and the self-confidence
values of their elements in D ET E hand in DESEh, respectively:
Rϕh
ij =1−θh·ϕh
ij +θh·ϕij ;Rsh
ij = ∆ 1−γh·∆−1sh
ij +γh·∆−1(sij ),(19)
being ϕij ,sij ,θh, γhas previously defined.
Theorem 3. Assuming that there is only one discordant expert ehfeaturing DBIII, who changes
her/his IVN from ϕh
ij to Rϕh
ij , and SC from sh
ij to Rsh
ij , then the new overall consensus degrees of
concordant experts on IVN and SC are greater than original consensus degree.
Proof. The proof is similar to that of Theorem 1. We omit the proof here.
Definition 10 (OFM DBIII Adjustment Cost). The OFM DBIII adjustment cost of the discor-
dant experts’ acceptance of the recommended advice is
FIV N −SC =X
h∈DBII I
(i,j)∈DE T Eh∪DE SE h
Rϕh
ij −ϕh
ij
+
∆−1Rsh
ij −∆−1sh
ij
g
.(20)
Proposition 2. The OFM DBIII adjustment cost is the sum of two functions, each one a linear
functions of the set of parameters of control {θ1, θ2, . . . , θ#DBII I }and {γ1, γ2, . . . , γ#DBI II }with non-
negative parameters, respectively.
Proof.
FIV N −SC =X
h∈DBII I
(i,j)∈DE T Eh∪DE SE h
Rϕh
ij −ϕh
ij
+
∆−1Rsh
ij −∆−1sh
ij
g
=X
h∈DBII I
θh·X
(i,j)∈DE T Eh
ϕh
ij −ϕij
+X
h∈DBII I
γh·X
(i,j)∈DE SE h
∆−1sh
ij −∆−1(sij )
g
15
Denoting
mh=X
(i,j)∈DE T Eh
ϕh
ij −ϕij
(≥0),
nh=X
(i,j)∈DE SE h
∆−1sh
ij −∆−1(sij )
g(≥0),
it is obvious that FIV N −SC is a sum of two linear functions with non-negative parameters.
The OFM for DBIII is therefore expressed as the following optimization model with decision
variables {θ1, θ2, . . . , θ#DBII I }and {γ1, γ2, . . . , γ#DBII I }:
M(3)
Min FIV N −SC :X
h∈DBII I
θh·X
(i,j)∈DE T Eh
ϕh
ij −ϕij
+Ph∈DBII I γh·P(i,j)∈DES Eh
|∆−1(sh
ij )−∆−1(sij )|
g
s.t.
AT Dh
(3) ≥β, AS Dh
(3) ≥β(h∈DBIII)
AT Dk
(3) ≥β, AS Dk
(3) ≥β(k /∈DBIII)
Rϕh
ij =1−θh·ϕh
ij +θh·ϕij (i, j)∈D ET Eh
Rsh
ij = ∆ 1−γh·∆−1sh
ij +γh·∆−1(sij )(i, j)∈DESEh
0≤θh≤1,0≤γh≤1.
(21)
The OFM with three types of discordant behaviours is summarized in Algorithm 2.
3.5. Joint feedback mechanism (JFM)
In order to eliminate over-adjustment as well as to reduce unnecessary adjustment cost in OFM
with IVN-SC, a joint feedback mechanism (JFM) for consensus is proposed. To do that, the concept
of comprehensive consensus degree (CCD) is introduced:
Definition 11 (Comprehensive Consensus Degree at Element level (CE)). The comprehen-
sive consensus degree between experts ekand esat the element (ai, cj) is defined as a linear combination
of the IVN and SC consensus degree at the element level given in Eq. (5) and Eq. (7):
CEks
ij =η·T Eks
ij + (1 −η)·SEks
ij .(22)
As usual, η∈[0,1] is a parameter that can be used to control how much IVN consensus degree is in
the comprehensive consensus degree. Recall that we already mentioned the equal importance of both
IVN and SC, and therefore we are using the comprehensive parameter of control value η= 0.5 in this
article.
The comprehensive consensus degree of expert ekat the three levels of a decision matrix are:
Level 1. Comprehensive consensus degree at element.
CCEk
ij =1
m−1
m
X
s=1,s6=k
CEks
ij .(23)
16
Algorithm 2: The OFM with discordant behaviours in the order of DBI−DBII −DBIII.
begin
Input: The original IVN-SC Ph=ϕh
ij , sh
ij p×q;
The criteria weighting vector of individual expert: V= (v1, v2, ..., vq);
The consensus threshold β;
Output: The boundary feedback parameters: θh, γhand ranking of alternatives.;
1Compute consensus degree at three levels for the individual experts: AT Eh
ij , AT Ah
iand
AT Dhin IVN and ASEh
ij , ASAh
iand ASDhin SC;
2if ∃AT Dh< β ∨ASDh< β then
3Feedback for discordant behaviour I:
if ∃AT Dh< β ∧ASDh≥βthen
4.1. Apply Algorithm 1 to determine the discordant experts set DBI;
4.2. Build the optimisation Model (15) and solve the IVN feedback parameters;
end if
4Compute the consensus degree AT Dh
(1) and ASDh
(1), respectively;
5Feedback for discordant behaviour II:
if ∃ASDh
(1) < β ∧AT Dh
(1) ≥βthen
6.1. Apply Algorithm 1 to determine the discordant experts set DBI I ;
6.2. Build the optimisation Model (18) and solve the SC feedback parameters;
end if
6Compute the consensus degree AT Dh
(2) and ASDh
(2), respectively;
7Feedback for discordant behaviour III:
if ∃AT Dh
(2) < β ∧ASDh
(2) < β then
8.1. Apply Algorithm 1 to determine the discordant experts set DBIII;
8.2. Build the optimisation Model (21) and solve the IVN-SC feedback parameters;
end if
8Compute the consensus degree AT Dh
(3) and ASDh
(3), respectively;
end if
9Switch on the selection process;
10 Rank the alternatives;
end
17
Level 2. Comprehensive consensus degree at alternative.
CCAk
i=1
q
q
X
j=1
CCEk
ij .(24)
Level 3. Comprehensive consensus degree at matrix.
CCDk=1
p
p
X
i=1
CCAk
i.(25)
Following our previous methodology, the JFM uses the comprehensive consensus degree concept
to identify discordant experts, as well as their respective discordant alternative and the elements of
alternatives.
3.5.1. JFM identification of discordant IVN-SC decision matrix elements
Algorithm 3 describes the JFM identification of discordant decision elements.
Algorithm 3: JFM identification of discordant IVN-SC decision matrix elements.
begin
Input: The experts’ comprehensive consensus degree: CCEh
ij , CCAh
iand CCDh;
The consensus threshold β;
Output: Discordant experts, alternatives and elements in the IVN-SC decision matrix:
EI CH, AI C and IJS;
1The discordant experts are identified:
EI C H =h|C CDh< β;
2The discordant alternatives are identified:
AICh=i|h∈EI C H ∧CCAh
i< β;
3The discordant elements of alternatives are identified:
IJSh=n(i, j)|i∈AI C h∧C CEh
ij < βo;
end
3.5.2. JFM Generation of recommended advice
For the identified elements in IJSh, the corresponding discordant experts receive the following
advice:“Evaluation ϕh
ij , sh
ij should be closer to Rϕh
ij , Rsh
ij .”
Rϕh
ij , Rsh
ij =1−αh·ϕh
ij +αh·ϕij ,∆1−αh·∆−1sh
ij +αh·∆−1(sij ),(26)
where ϕij =1
mPm
h=1 ϕh
ij and sij = ∆ 1
mPm
h=1 ∆−1sh
ij are the group average of the original
IVN and SC assessment values; and αhis the JFM comprehensive parameter for discordant expert eh.
Theorem 4. Assuming that there is only one discordant expert ehwho changes her/his IVN-SC from
ϕh
ij , sh
ij to Rϕh
ij , Rsh
ij , then the new overall consensus degree of concordant experts on IVN-SC
is greater than original consensus degree.
18
Proof. The proof is similar to that of Theorem 1. We omit the proof here.
Like in Section 3.4.2, the adjustment cost for JFM is defined and it is shown that is a linear
transformation of the set of parameters of control {α1, α2, . . . , α#EI CH }with non-negative parameters.
Definition 12 (JFM Adjustment Cost). The JFM adjustment cost of the discordant experts’
acceptance of the recommended advice is
FJF M =X
h∈EI CH
(i,j)∈I J Sh
Rϕh
ij ,
∆−1Rsh
ij
g
−
ϕh
ij ,
∆−1sh
ij
g
=X
h∈EI CH
αh·
X
(i,j)∈I J Sh
ϕh
ij −ϕij
+
∆−1sh
ij −∆−1(sij )
g
.
(27)
Usually, the higher the individual consensus levels are, the less the adjustment opinions will be
required, and therefore the less adjustment cost the experts will bear. Therefore, the JFM with IVN-
SC as the following Model with decision variables {α1, α2, . . . , α#EIC H }will generate personalized
feedback advice in both IVNs and SCs:
M(4)
Min FJF M :X
h∈EI CH
αh·
X
(i,j)∈I J Sh
ϕh
ij −ϕij
+
∆−1sh
ij −∆−1(sij )
g
s.t.
CCDh≥β(h∈EICH)
CCDk≥β(k= 1,2...m, k 6=h)
0≤αh≤1
Rϕh
ij =1−αhϕh
ij +αhϕij ,(i, j)∈I J Sh
Rsh
ij = ∆ 1−αh·∆−1sh
ij +αh·∆−1(sij )
.
(28)
Hence, in less strict GDM problems, a more reasonable policy would be to generate joint person-
alized feedback advice with IVN-SC in the interactive process according to individual comprehensive
consensus degree in CCD. As such, the JFM for consensus can be seen as less strict than OFM
because the former contains the compensatory aggregation operator while the latter does not.
The JFM driven by minimum adjustment cost with IVN-SC in GDM problems is therefore sum-
marized in Algorithm 4.
3.6. Selection process
When experts reach the group consensus threshold value, the selection process is activated to
derive the final ordering of the considered alternatives [56, 57, 58]. This includes the computation of
the experts’ weights based on their final CCD values, which using Yager’s OWA operator proposal
[59, 60] translates into:
Wσ(k)
T=QT(σ(k))
T(σ(m))−QT(σ(k−1))
T(σ(m)) ,(29)
19
Algorithm 4: The JFM for GDM with IVN-SC.
begin
Input: The original IVN-SC Ph=ϕh
ij , sh
ij p×q;
The criteria weighting vector of individual expert: V= (v1, v2, ..., vq);
The consensus threshold β;
Output: The boundary feedback parameters αhand ranking of alternatives.;
1Apply Eqs (23-25) to compute experts comprehensive consensus degree at three levels:
CCEh
ij ,CCAh
i, CCDh;
2The JFM for eh;
3if ∃CCDh< β then
3.1. Apply Algorithm 3 to determine the set the discordant experts EI C H and its
corresponding set of discordant elements IJ Sh;
3.2. Build the optimisation Model (28);
3.3. Obtain the joint feedback parameter for discordant expert eh:αh;
3.4. Apply Eq. (26) to generate personalized feedback advice for discordant expert eh;
else
4Switch on the selection process;
5Rank the alternatives;
end if
end
being
T(σ(k)) =
k
X
l=1
CCDσ(l), T (σ(s)) =
s
X
l=1
CCDσ(l).
σthe permutation such that
CCDσ(1) ≥CCDσ(2) ≥ · · · ≥ CCDσ(k)≥ · · · ≥ C CDσ(m)
and Qthe Yager’s linguistic quantifier Q(x) = xa(a≥0) representing the fuzzy majority concept of
‘most’ in [61]. In particular, the parameter a=1
2is implemented herein to obtain the experts’ weight-
ing vector since this weight determination method guarantees that experts with higher comprehensive
consensus degree will be allocated higher weight degree, which is in agreement with the overall aim of
achieving a highly consensus based final decision [62].
Once the experts weights are derived, the collective IVN-SC decision matrix B=ϕij , sij p×qis
computed by fusing the individual IVN-SC matrices through IVN-SC weighted average operator as
per Eq.(4), whose rows are subsequently fused using the criterion weighting vector to derive a weighted
collective overall experts’ IVN-SCs for each alternative B=(ϕ1, s1),(ϕ2, s2),...,(ϕp, sp), which are
20
fused by using the following IVN-SC score:
IV SC (ai) = ϕi·∆−1(si)
g= [oL
i·∆−1(si)
g, oR
i·∆−1(si)
g].(30)
The performance of the alternatives would be obtained in the form of interval-valued number by
combining corresponding self-confidence as IV SC (ai). In order to rank the alternatives to identify
which one is the best, this article then compares the score and accuracy of the aggregated evaluations
by using the ranking method through Definition(3-5).
4. Numerical example and comparative analysis
This section provides an illustrative example to show the applicability of the proposed OFM and
JFM consensus models in consensus reaching process and analyzes the performance of the proposed
OFM and JFM through comparative analysis.
4.1. An illustrative example
As far as we know, one of the most crucial issues regarding the sustainable supplier selection for
enterprises is to choose a reasonable supplier. Thus, in some cases, the supplier selection actually can
be seen as a GDM problem. When selecting strategic suppliers, which refer to the few suppliers that
are necessary for the enterprises’ strategic development, a strict OFM method is appropriate since it
guarantees both IVN and SC levels reach consensus. However, when choosing general suppliers, the
requirements may not be that strict, and then JFM may be implemented. Therefore, the consensus
requirements change constantly with different levels of suppliers in selection.
Herein, it assumes that an enterprise decides to choose sustainable suppliers, and that after pre-
evaluation, four possible alternative suppliers remain for further evaluation A={a1, a2, a3, a4}, which
is conducted by ten experts from the enterprise and supply chain field E={e1, ..., e10}. Considering
the quality and efficiency of the final decision, the organizer provides the experts with the following
four important criteria: c1– technological compatibility, c2– social acceptability, c3– economy, and
c4– environment.
Step 1– Experts’ IVN-SC matrices. Below, the ten experts’ IVN-SC matrices from which the
best supplier is to be selected by consensus are listed:
P1=
c1c2c3c4
a1([0.4,0.6] , l1) ([0.6,0.7] , l5) ([0.4,0.5] , l3) ([0.6,0.8] , l7)
a2([0.3,0.4] , l2) ([0.4,0.5] , l8) ([0.3,0.7] , l6) ([0.4,0.6] , l4)
a3([0.4,0.8] , l5) ([0.5,0.7] , l6) ([0.3,0.4] , l5) ([0.1,0.4] , l5)
a4([0.1,0.4] , l7) ([0.2,0.4] , l3) ([0.1,0.6] , l2) ([0.4,0.6] , l8)
;
21
P2=
c1c2c3c4
a1([0.6,0.9] , l3) ([0.6,0.7] , l4) ([0.5,0.7] , l3) ([0.4,0.6] , l7)
a2([0.3,0.6] , l3) ([0.2,0.5] , l7) ([0.5,0.6] , l7) ([0.4,0.7] , l3)
a3([0.3,0.5] , l5) ([0.2,0.4] , l7) ([0.2,0.3] , l6) ([0.2,0.5] , l3)
a4([0,3,0.7] , l8) ([0.3,0.4] , l2) ([0.2,0.6] , l2) ([0.3,0.7] , l6)
;
P3=
c1c2c3c4
a1([0.4,0.5] , l3) ([0.6,0.8] , l5) ([0.6,0.9] , l4) ([0.4,0.5] , l7)
a2([0.4,0.6] , l2) ([0.3,0.6] , l6) ([0.3,0.6] , l5) ([0.5,0.9] , l3)
a3([0.3,0.6] , l6) ([0.4,0.8] , l7) ([0.1,0.2] , l6) ([0.2,0.5] , l4)
a4([0.3,0.5] , l6) ([0.1,0.5] , l3) ([0.3,0.4] , l0) ([0.2,0.5] , l7)
;
P4=
c1c2c3c4
a1([0.3,0.6] , l5) ([0.5,0.7] , l5) ([0.7,1.0] , l5) ([0.3,0.5] , l7)
a2([0.6,0.8] , l2) ([0.6,0.7] , l5) ([0.1,0.5] , l5) ([0.4,0.7] , l4)
a3([0.6,0.8] , l5) ([0.6,0.9] , l8) ([0.3,0.4] , l7) ([0.3,0.4] , l4)
a4([0.5,0.9] , l7) ([0.6,0.8] , l8) ([0.5,0.7] , l8) ([0.5,0.6] , l6)
;
P5=
c1c2c3c4
a1([0.1,0.2] , l5) ([0.2,0.4] , l8) ([0.1,0.3] , l7) ([0.3,0.5] , l7)
a2([0.7,0.9] , l3) ([0.1,0.4] , l8) ([0.4,0.5] , l7) ([0.3,0.5] , l3)
a3([0.2,0.4] , l7) ([0.2,0.4] , l8) ([0.3,0.5] , l7) ([0.1,0.6] , l5)
a4([0.2,0.6] , l8) ([0.1,0.2] , l3) ([0.3, .0.7] , l1) ([0.1,0.4] , l7)
;
P6=
c1c2c3c4
a1([0.6,0.9] , l2) ([0.5,0.6] , l5) ([0.4,0.7] , l3) ([0.5,0.6] , l6)
a2([0.4,0.5] , l3) ([0.4,0.7] , l6) ([0.4,0.5] , l7) ([0.3,0.6] , l3)
a3([0.5,0.7] , l6) ([0.4,0.7] , l7) ([0.2,0.3] , l6) ([0.2,0.6] , l4)
a4([0.1,0.7] , l7) ([0.2,0.7] , l2) ([0.6,0.7] , l1) ([0.4,0.8] , l7)
;
P7=
c1c2c3c4
a1([0.8,1.0] , l4) ([0.4,0.8] , l4) ([0.5,0.6] , l3) ([0.6,0.8] , l7)
a2([0.5,0.7] , l3) ([0.3,0.8] , l6) ([0.2,0.3] , l6) ([0.5,0.6] , l2)
a3([0.4,0.5] , l6) ([0.3,0.6] , l7) ([0.2,0.4] , l7) ([0.2,0.5] , l4)
a4([0.1,0.4] , l6) ([0.1,0.3] , l3) ([0.3,0.5] , l1) ([0.3,0.6] , l7)
;
P8=
c1c2c3c4
a1([0.9,1.0] , l3) ([0.6,0.9] , l7) ([0.6,0.7] , l3) ([0.2,0.8] , l7)
a2([0.3,0.7] , l8) ([0.3,0.4] , l7) ([0.5,0.8] , l1) ([0.6,0.7] , l7)
a3([0.4,0.5] , l7) ([0.5,0.6] , l8) ([0.1,0.3] , l5) ([0.4,0.7] , l4)
a4([0.2,0.3] , l8) ([0.1,0.3] , l3) ([0.1,0.3] , l1) ([0.3,0.7] , l7)
;
P9=
c1c2c3c4
a1([0.6,0.7] , l5) ([0.4,0.6] , l5) ([0.6,0.8] , l3) ([0.4,0.6] , l6)
a2([0.4,0.5] , l4) ([0.4,0.6] , l7) ([0.4,0.7] , l7) ([0.4,0.6] , l4)
a3([0.3,0.6] , l5) ([0.3,0.5] , l7) ([0.2,0.4] , l5) ([0.2,0.5] , l5)
a4([0.2,0.4] , l8) ([0.2,0.5] , l4) ([0.4,0.9] , l2) ([0.5,0.6] , l7)
;
22
P10 =
c1c2c3c4
a1([0.4,0.8] , l4) ([0.5,0.9] , l5) ([0.7,0.8] , l6) ([0.3,0.7] , l6)
a2([0.5,0.8] , l2) ([0.2,0.7] , l7) ([0.6,0.8] , l5) ([0.1,0.5] , l2)
a3([0.4,0.8] , l7) ([0.4,0.6] , l2) ([0.7,0.9] , l1) ([0.7,1.0] , l5)
a4([0.4,0.6] , l6) ([0.2,0.3] , l2) ([0.2,0.8] , l1) ([0.2,0.5] , l6)
.
Step 2– Measuring consensus. This is done at the three different decision matrix levels.
Level 1.Consensus degree of experts at the element of alternative.
AT E1=
0.772 0.878 0.772 0.800
0.783 0.861 0.850 0.906
0.856 0.844 0.872 0.817
0.828 0.872 0.811 0.889
;ASE1=
0.653 0.903 0.861 0.958
0.833 0.819 0.833 0.847
0.875 0.792 0.819 0.889
0.903 0.875 0.819 0.833
;
AT E2=
0.772 0.878 0.850 0.889
0.850 0.850 0.850 0.894
0.867 0.778 0.872 0.883
0.828 0.839 0.844 0.878
;ASE2=
0.847 0.819 0.861 0.958
0.861 0.903 0.806 0.847
0.875 0.875 0.847 0.889
0.875 0.819 0.819 0.833
;
AT E3=
0.761 0.867 0.806 0.867
0.872 0.883 0.861 0.783
0.878 0.833 0.794 0.883
0.850 0.850 0.800 0.856
;ASE3=
0.847 0.903 0.833 0.958
0.833 0.875 0.806 0.875
0.903 0.875 0.847 0.833
0.847 0.875 0.736 0.889
;
AT E4=
0.739 0.889 0.728 0.856
0.806 0.772 0.772 0.894
0.778 0.722 0.872 0.828
0.661 0.583 0.800 0.856
;ASE4=
0.792 0.903 0.778 0.958
0.833 0.764 0.806 0.847
0.875 0.819 0.792 0.833
0.903 0.347 0.153 0.889
;
AT E5=
0.483 0.667 0.550 0.856
0.717 0.772 0.861 0.850
0.778 0.778 0.839 0.839
0.861 0.806 0.856 0.767
;ASE5=
0.792 0.625 0.583 0.764
0.861 0.819 0.806 0.847
0.847 0.819 0.792 0.778
0.875 0.875 0.847 0.944
;
AT E6=
0.772 0.867 0.828 0.867
0.850 0.861 0.861 0.883
0.844 0.867 0.872 0.872
0.817 0.794 0.756 0.822
;ASE6=
0.764 0.903 0.861 0.903
0.861 0.875 0.806 0.875
0.903 0.875 0.847 0.917
0.903 0.819 0.847 0.944
;
AT E7=
0.672 0.856 0.828 0.800
0.861 0.828 0.728 0.883
0.878 0.867 0.883 0.883
0.828 0.850 0.833 0.900
;ASE7=
0.847 0.819 0.861 0.958
0.861 0.875 0.833 0.792
0.903 0.875 0.792 0.917
0.847 0.875 0.847 0.944
;
23
AT E8=
0.628 0.833 0.850 0.800
0.850 0.839 0.806 0.828
0.878 0.856 0.839 0.783
0.806 0.850 0.711 0.878
;ASE8=
0.847 0.736 0.861 0.958
0.333 0.903 0.361 0.514
0.847 0.819 0.819 0.917
0.875 0.875 0.847 0.944
;
AT E9=
0.783 0.844 0.839 0.889
0.850 0.872 0.861 0.906
0.878 0.844 0.883 0.883
0.850 0.861 0.756 0.856
;ASE9=
0.792 0.903 0.861 0.903
0.778 0.903 0.806 0.847
0.875 0.875 0.819 0.889
0.875 0.792 0.819 0.944
;
AT E10 =
0.783 0.844 0.806 0.867
0.839 0.850 0.761 0.761
0.856 0.878 0.483 0.517
0.817 0.861 0.811 0.856
;ASE10 =
0.847 0.903 0.694 0.903
0.833 0.903 0.806 0.792
0.847 0.347 0.375 0.889
0.847 0.819 0.847 0.889
.
Level 2. Consensus degree of experts at the alternatives.
AT A1=0.806 0.850 0.847 0.850 ;ASA1=0.844 0.833 0.844 0.858 ;
AT A2=0.847 0.861 0.850 0.847 ;ASA2=0.872 0.861 0.858 0.851 ;
AT A3=0.825 0.850 0.847 0.839 ;ASA3=0.885 0.847 0.885 0.851 ;
AT A4=0.803 0.811 0.800 0.725 ;ASA4=0.858 0.813 0.830 0.573 ;
AT A5=0.639 0.800 0.808 0.822 ;ASA5=0.691 0.833 0.809 0.885 ;
AT A6=0.833 0.864 0.864 0.797 ;ASA6=0.858 0.854 0.885 0.878 ;
AT A7=0.789 0.825 0.878 0.853 ;ASA7=0.872 0.840 0.872 0.878 ;
AT A8=0.778 0.831 0.839 0.811 ;ASA8=0.851 0.528 0.851 0.885 ;
AT A9=0.839 0.872 0.872 0.831 ;ASA9=0.865 0.833 0.865 0.858 ;
AT A10 =0.825 0.803 0.683 0.836 ;ASA10 =0.837 0.833 0.615 0.851 .
Level 3. Consensus degree at the decision matrix.
1) IVN consensus degree of experts:
AT D1= 0.838; AT D2= 0.851; AT D 3= 0.840; AT D4= 0.785; AT D5= 0.767;
AT D6= 0.840; AT D7= 0.836; AT D 8= 0.815; AT D9= 0.853; AT D10 = 0.787.
2) SC consensus degree of experts:
ASD1= 0.845; ASD2= 0.860; ASD3= 0.867; ASD4= 0.768; ASD5= 0.805;
ASD6= 0.869; ASD7= 0.865; ASD8= 0.779; ASD9= 0.855; ASD10 = 0.784.
As not all experts reach the consensus threshold β= 0.8, the OFM and JFM are activated.
24
4.1.1. OFM consensus model
From Algorithm 1,we have: DET ={4,5,10};DES ={4,8,10}.
DE T E4={(4,1),(4,2)};DE T E5={(1,1),(1,2),(1,3)};DE T E10 ={(3,3),(3,4)};
DESE4={(4,2),(4,3)};DESE8={(2,1),(2,3),(2,4)};DESE10 ={(3,2),(3,3)}.
Applying Algorithm 2, the proposed models can be effectively and rapidly solved by GRG method
in several software tools (e.g., Matlab and Excel) to obtain a local optimal solution that satisfies all
constraints and optimal conditions.
(a) DBIFeedback (b) DBII Feedback (c) DBII I Feedback
Figure 2: OFM visual consensus simulation of discordant experts.
•DBIFeedback: DBI={5}.The optimal feedback parameter obtained applying Model (15) is
θ5= 0.520 with adjustment cost FIV N = 1.044, and corresponding personalized advice for e5:
- IVN [0.1,0.2] about a1under c1should be closer to [0.3,0.4], with SC l5unchanged.
- IVN [0.2,0.4] about a1under c2should be closer to [0.3,0.5], with SC l8unchanged.
- IVN [0.1,0.3] about a1under c3should be closer to [0.3,0.5], with SC l7unchanged.
Implementing the above advice, new consensus degree of experts in DET and DES would be:
AT D4
(1) = 0.788; AT D5
(1) = 0.800; AT D8
(1) = 0.818; AT D10
(1) = 0.790;
ASD4
(1) = 0.768; ASD5
(1) = 0.805; ASD8
(1) = 0.779; ASD10
(1) = 0.784.
The threshold of consensus is still not reached by all experts, DBII feedback is applied next.
•DBI I Feedback: DBII ={8}.The optimal feedback parameter obtained applying Model (18)
is γ8= 0.212 with adjustment cost FSC = 0.342, and personalized advice for e8:
- SC l8about a2under c1should be closer to l7, with IVN [0.3,0.7] unchanged.
- SC l1about a2under c3should be closer to l2, with IVN [0.5,0.8] unchanged.
- SC l7about a2under c4should be closer to l6, with IVN [0.6,0.7] unchanged.
25
Implementing the above advice, new consensus degree of experts in DET and DES would be:
AT D4
(2) = 0.788; AT D5
(2) = 0.800; AT D8
(2) = 0.818; AT D10
(2) = 0.790;
ASD4
(2) = 0.771; ASD5
(2) = 0.807; ASD8
(2) = 0.800; ASD10
(2) = 0.786.
The threshold of consensus is still not reached by all experts, DBIII feedback is applied next.
•DBIII Feedback: DBIII ={4,10}.The optimal feedback parameters obtained applying Model
(21) are (θ4, γ4) = (0.239,0.334) and (θ10 , γ10) = (0.138,0.148) with adjustment cost FIV N−SC =
1.233, the corresponding personalized advice for e4:
- IVN-SC on alternative a4under criterion c1should be closer to ([0.4,0.8] , l7).
- IVN-SC on alternative a4under criterion c2should be closer to ([0.5,0.7] , l6).
- IVN-SC on alternative a4under criterion c3should be closer to ([0.5,0.7] , l6).
The corresponding personalized advice for e10 are:
- IVN-SC on alternative a3under criterion c2should be closer to ([0.4,0.6] , l3).
- IVN-SC on alternative a3under criterion c3should be closer to ([0.6,0.8] , l2).
- IVN-SC on alternative a3under criterion c4should be closer to ([0.6,0.9] , l5).
Implementing the above advice, new consensus degree of experts in DET and DES would be:
AT D4
(3) = 0.800; AT D5
(3) = 0.802; AT D8
(3) = 0.820; AT D10
(3) = 0.800;
ASD4
(3) = 0.800; ASD5
(3) = 0.812; ASD8
(3) = 0.804; ASD10
(3) = 0.800.
A visual consensus simulation of discordant experts before and after the OFM is depicted in Fig.2
by using some software tools, such as Visio. It can be seen that all discordant experts reach the group
consensus threshold after OFM, with following personalized feedback adjustment cost value being
FOF M = 2.619.
4.1.2. JFM consensus model
Firstly, the individual comprehensive consensus degrees are computed:
CCD1= 0.841; CCD2= 0.856; CCD3= 0.854; CCD4= 0.776; CC D5= 0.786;
CCD6= 0.854; CCD7= 0.851; CCD8= 0.797; CCD9= 0.854; CC D10 = 0.785.
From Algorithm 3, we have: EIH ={4,5,8,10};
IJS4={(4,1),(4,2),(4,3)};IJS5={(1,1),(1,2),(1,3)};
IJS8={(2,1),(2,3),(2,4)};IJS10 ={(3,2),(3,3),(3,4)}.
26
Table 4: The collective IVN-SC decision matrix.
c1c2c3c4
a1([0.6,0.8] ,(l3,+0.2)) ([0.5,0.7] ,(l5,+0.1)) ([0.5,0.7] ,(l4,−0.5)) ([0.4,0.6] ,(l7,−0.5))
a2([0.4,0.6] ,(l3,+0.1)) ([0.3,0.6] ,(l7,−0.5)) ([0.4,0.6] ,(l6,+0.1)) ([0.4,0.6] ,(l3,+0.3))
a3([0.4,0.6] ,(l6,−0.2)) ([0.4,0.6] ,(l7,−0.1)) ([0.2,0.4] ,(l6,−0.3)) ([0.2,0.6] ,(l4,+0.2))
a4([0.2,0.6] ,(l7,+0.1)) ([0.2,0.5] ,(l3,−0.1)) ([0.4,0.6] ,(l2,−0.5)) ([0.4,0.7] ,(l7,−0.1))
Applying Algorithm 4, we have the following optimal feedback parameter from Model (28) α4=
0.314, α5= 0.158, α8= 0.000, α10 = 0.169, with adjustment cost :FJF M = 2.024. Implementing the
corresponding personalized IVN-SC advice as per (26), the new consensus degree of experts would
reach the group consensus threshold:
CCD1= 0.846; CCD2= 0.861; CCD3= 0.859; CCD4= 0.800; CC D5= 0.800;
CCD6= 0.859; CCD7= 0.856; CCD8= 0.800; CCD9= 0.859; CC D10 = 0.800.
4.1.3. Selection process
Following the feedback IVN-SC advice from JFM consensus model, the experts’ weighting vector
using the linguistic quantifier Q(x) = x1
2is ω= (0.068,0.321,0.086,0.055,0.054,0.102,0.075,0.054,0.133,0.054)T.
The collective IVN-SC decision matrix is listed in Table 4. Using the criterion weighting vector
ν= (0.2,0.3,0.1,0.4)T, the weighted collective overall experts’ IVN-SCs on the set of alternatives
become:
Bc=
IV N −SC
a1([0.485,0.685] ,(l5,+0.1))
a2([0.375,0.617] ,(l5,−0.5))
a3([0.306,0.576] ,(l5,+0.5))
a4([0.274,0.589] ,(l5,+0.2))
.
The IVSC score values of the alternatives:
IV SC (a1) = [0.309,0.436]; IV SC (a2) = [0.212,0.349];
IV SC (a3) = [0.209,0.394]; IV SC (a4) = [0.177,0.382].
Comparing the score and accuracy of IV SC (ai) by using the ranking method through Definition(3-
5). Lead to the ranking a1a3a2a4, making alternative a1the best sustainable supplier for
the considered enterprise.
4.2. Comparative analysis
4.2.1. Impact of the comprehensive parameter of control in JFM
A sensitive analysis is conducted to discuss the influence on the adjustment cost in JFM of the
comprehensive parameter of control ηused in computing the comprehensive consensus degree as per
27
(22). Different ηvalues from 0 to 1 are used with the data of the illustrative example in Section 4.1.1.
To directly demonstrate the impact of the comprehensive parameter, the results obtained are depicted
in Fig.3, where the blue line represents the final adjustment cost and four dotted lines indicate the
trend of JFM adjustment parameters.
Figure 3: The adjustment cost of different value of η.
As illustrated in Fig.3, it is clear that JFM adjustment cost reaches the highest value of 2.376 when
η= 0.8, which is still lower than the obtained OFM adjustment cost 2.619 with the data in Section
4.1.1. Moreover, the comprehensive parameter of control ηhas an effect on the JFM adjustment
parameter and may even decrease it to zero (taking expert e8as an example), which means that an
expert that was initially identified as discordant is now concordant. Therefore, the comprehensive
parameter of control ηhas impact on the identification of discordant experts since the JFM uses the
comprehensive consensus degree concept to identify discordant experts, as well as their respective
discordant alternative and the elements of alternatives.
4.2.2. The adjustment cost generated by different implementation sequences in the OFM
This article refers to Method 1 in this section to the application of the JFM consensus model.
Using number 1 to represent DBI, number 2 for DBI I , and number 3 for DBIII, there are six
possible sequential applications of the OFM consensus model: Method 2(1-2-3); Method 3(2-1-3);
Method 4(3-1-2); Method 5(3-2-1); Method 6(1-3-2) and Method 7(2-3-1). The adjustment cost with
the data of Section 4.1.1 for the above seven methods is depicted in Fig.4 and numerically in Table
5. Method 6(1-3-2) is the OFM model with lowest adjustment cost, however, is still higher than JFM
consensus model (Method 1), which indicated that JFM is less strict than the OFM models due to
the implementation of a compensatory aggregation which the latter lacks.
The over-adjustment cost of discordant behaviours management by OFM can be understood better
by analysing the trend chart of AT D and ASD of discordant experts in Method 2. From Fig.5(a), it
28
Figure 4: The adjustment cost of different methods.
Table 5: The detailed consensus results using different sequence of OFM.
Method θ4, γ4 θ5,0 0, γ4 θ10 , γ10FO F M Cost
2 (0.230,0.334) (0.520,0) (0,0.212) (0.138,0.148) 2.619
3 (0.230,0.334) (0.520,0) (0,0.212) (0.138,0.148) 2.619
4 (0.303,0.360) (0.474,0) (0,0.164) (0.192,0.178) 2.732
5 (0.303,0.360) (0.474,0) (0,0.164) (0.192,0.178) 2.732
6 (0.230,0.360) (0.520,0) (0,0.164) (0.138,0.178) 2.611
7 (0.303,0.360) (0.474,0) (0,0.164) (0.192,0.178) 2.732
Method α4α5α8α10 FJ F M Cost
1 (0.314,0.314) (0.158,0.158) (0.000,0.000) (0.169,0.169) 2.024
is clear that the values of the AT D of all discordant experts increase when the number of iteration
increase. It means that experts who feature DBIhave already reached the acceptable threshold
boundary after the first iteration, but their consensus degrees will constantly increase in the second
and the third iterations for DBI I and DBIII, which eventually causes over-adjustment and unnecessary
adjustment cost. A similar conclusion is clearly noticed from Fig.5(b).
4.2.3. Impact of experts’ self-confidence levels on alternative rankings
In realistic GDM problems, a complete interval-valued number matrix denotes that the expert has
full self-confidence in her/his evaluations, and thus the corresponding self-confidence levels for experts
are the same. Therefore, an IVN is actually a particular case of IVN-SC. Here, this article compares
the feedback model without considering SC values with the proposed JFM consensus model proposed
29
(a) ATD of discordant expert. (b) ASD of discordant expert.
Figure 5: The trend chart of ATD and ASD in OFM of Method 2.
in this article. To do this, Step 1 and Step 2 in Algorithm 4 are replaced by the below Step 1A and
Step 2A, respectively.
Step 1A. Compute consensus degree at three levels AT E h
ij , AT Ah
i, AT Dhfor individual experts in
IVN without considering the self-confidence levels of experts.
Step 2A. Identify the discordant experts E T H and discordant elements AP Shin IVN and solve the
personalized optimisation model:
M(5)
MinF1:Ph∈ET H εh·Pi,j∈AP Sh|ϕh
ij −ϕij |
s.t.
AT Dh≥β(h∈E T H)
AT Dk≥β(k= 1,2...t, k 6=h)
Rϕh
ij =1−εhϕh
ij +εhϕij i, j ∈AP S h
0≤εh≤1.
(31)
With the data of Section 4, using Step 1A and Step 2A leads to: ET H ={4,5,10}and
AP S4={(4,1) ,(4,2)};AP S 5={(1,1) ,(1,2) ,(1,3)};AP S10 ={(3,3) ,(3,4)}.
From Fig.6, it is obvious that the numbers of discordant experts are different, e8was identified as a
discordant expert in JFM, while is no longer discordant when SC is not considered, which demonstrates
that the self-confidence level has an influence on the identification of discordant experts.
Using Eq.(15) to adjust the discordant IVN, the collective IVN decision matrix and final alterna-
tive ranking for the feedback model without considering SC are given in Table 6. The rankings of
alternatives are different, which means that self-confidence levels of experts may influence the final
decision in GDM problem outcome.
5. Conclusion
This article presents personalized feedback mechanisms OFM and JFM based on IVN-SC with
self-confidence behaviours to reach consensus in GDM. The main advantages and differences with
respect to existing approaches are summarized as follows:
30
(a) Feedback without SC (b) Feedback with SC
Figure 6: Consensus simulation of discordant experts by different mechanisms.
Table 6: Alternative ranking of the special case of the IVN-SC (i= 1,2,3,4).
Collective PcSaiRanking of Alternatives
Pc=
a1[0.6,0.8] [0.5,0.7] [0.5,0.7] [0.4,0.6]
a2[0.4,0.6] [0.3,0.6] [0.4,0.6] [0.4,0.6]
a3[0.4,0.6] [0.4,0.6] [0.2,0.4] [0.2,0.6]
a4[0.2,0.6] [0.2,0.5] [0.4,0.7] [0.4,0.7]
0.586
a1a2a3a4
0.496
0.442
0.433
(1) A new concept of interval-valued number with self-confidence (IVN-SC) is defined, which allows
experts to express self-confidence levels with linguistic term when providing their interval-valued
numbers as evaluation. The ordinal feedback mechanism (OFM) is proposed, in which an iden-
tification standard is given to detect the experts’ discordant behaviours involving interval-valued
number (IVN) consensus and self-confidence (SC) consensus. Subsequently, the corresponding
personalized mechanisms are implemented to help discordant experts with different discordan-
t behaviours in achieving consensus. Finally, the analysis of sequencing is devised for OFM to
demonstrate that the adjustment cost of OFMs in different implementation sequences is relatively
stable. In conclusion, the OFM guarantees both the IVN and SC reach acceptable thresholds
while iteration-based feedback mechanism will cause over-adjustment.
(2) The joint feedback mechanism (JFM) with IVN-SC is proposed. In JFM, a comprehensive con-
sensus degree (CCD), which combines the IVN and SC consensus degree in a compensative way,
is presented to measure the consensus level among experts. It is regarded as a reliable source
of importance associated to experts in determining their aggregation weights. Subsequently, a
joint optimisation model with IVN-SC in GDM is built, which allows discordant experts to select
different feedback parameters according to individual CCD and aim at reaching the consensus
31
boundary. Finally, a sensitivity analysis with the comprehensive parameter ηis proposed to verify
the rationality of the proposed mechanism. As such, the JFM can be seen as less strict than the
OFM because it contains a compensatory aggregation operator which the latter lacks.
In traditional GDM problems, the number of participants is assumed to be small [16, 43, 41]. Large-
scale group decision making (LSGDM), as a special from of GDM, has attracted increasing attention
driven by the advancement and development of technology and the demand of society [63, 64, 65, 66].
Therefore, it is worth discussing whether the proposed feedback mechanisms can be used to discuss
LSGDM problems to solve real-life problems. As clustering analysis process (CAP) has a crucial role
in dealing with LSGDM since it enables the breakdown of large group into several different subgroups,
how to perform CAP in LSGDM requires further research. Furthermore, the effect of social network
structural relationship among the subgroups in LSGDM remains to be investigated.
Acknowledgements
The authors are very grateful to the editors and anonymous referees for their valuable comments
and suggestions to improve considerably the quality of this paper. This work was sponsored by
National Natural Science Foundation of China (NSFC) (No. 71971135, 72071056 and 71910107002),
and was supported in part by the Spanish State Research Agency under Project PID2019-103880RB-
I00/AEI/10.13039/501100011033.
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