=Paper=
{{Paper
|id=Vol-2486/icaiw_istihmr_2
|storemode=property
|title=Influence Networks based Methodology for Consensus Reaching in Group-Decision-Making
Problems
|pdfUrl=https://ceur-ws.org/Vol-2486/icaiw_istihmr_2.pdf
|volume=Vol-2486
|authors=Jorge Ivan Romero-Gelvez,Felix Antonio Cortes-Aldana,Monica Garcia-Melon,Jorge Aurelio Herrera Cuartas,Olmer Garcia Bedoya
}}
==Influence Networks based Methodology for Consensus Reaching in Group-Decision-Making
Problems==
https://ceur-ws.org/Vol-2486/icaiw_istihmr_2.pdf
Influence Networks based Methodology for
Consensus Reaching in Group-Decision-Making
Problems
Jorge Ivan Romero-Gelvez1,3 , Felix Antonio Cortes-Aldana1 , Monica
Garcia-Melon2 , Jorge Aurelio Herrera Cuartas3 , and Olmer Garcia Bedoya3
Universidad Nacional de Colombia
1
Universitat Politecnica de Valencia
2
3
Universidad de Bogotá Jorge Tadeo Lozano, Bogotá
jorgei.romerog@utadeo.edu.co
Abstract. The purpose of this work is to show a way to improve agree-
ment in group decision problems. This work focuses his effort on the
issue refers to assign different importance to every decision-maker. We
propose as a novelty a methodology to assign different levels of impor-
tance to every decision-maker according to their perceived importance
in the group. First, judgments are collected by an html form and use
a proposed method based on SNA and DEMATEL to assign weights
to decision-makers according to their reputation in the decision-group.
Next, we solve the problem using AHP in order to rank the alternatives.
Keywords: Influence · SNA · Consensus · Group-Decision-Making
· MCDA · DEMATEL.
1 Introduction
Group decision-making problems present several issues to reach a consensus be-
tween decision-makers. According to [3,4] there are challenges and open questions
according to represent a different level of importance to decision-makers. We
propose that importance can be represented as weight over every decision-maker
in order to influence a change over decision-makers initial judgments to reach
a consensus agreement level over the main decision-making problem. There are
many methods for weighting criteria, [1,17,29,28,25] Modeling a decision maker’s
preferences as tangible valuations of their value judgments or even quantitative
information represents the core of the multi-criteria decision analysis. Determin-
ing correct valuations for the criteria and alternatives becomes a problem of
vital importance in decision making. Determining this importance has been ap-
proached from multiple perspectives, among some of the best-known approaches,
we can highlight the method of entropy, SWARA, SAW, AHP and ANP, among
others. Even dematel presents a proposal to generate weights from the sets of
importance and influence.
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0)
2019 ICAI Workshops, pp. 280–294, 2019.
Influence Networks based Methodology for Consensus Reaching 281
2 Literature review and basic background
Groups have the potential to produce decisions that are better than individual
ones. However, research and history show that groups can make worse decisions
than individuals. even in the presence of diversity, strong leadership and unlim-
ited time. The influence analysis has been developed from the social sciences,
being psychology one of the branches that has presented more interest due to
an extended use in the dynamics of group decision problems. Its historical evo-
lution is shown below, emphasizing in the end the trend in applications related
to multi-criteria analysis. According to [11] in 1950, many psychologists followed
a research program on communication and social influence that focused on the
foundations of power and influence in groups. Later, in 1970, interest in the field
of group dynamics start to increase. While the increasing use of formal models
tilts the field towards a concern with theoretical issues, the field is still far from
having an adequate balance between theory, method and data. In the 1990s,
[19] the most active lines of research in small groups were no longer in social
psychology but in organizational psychology. However, even in organizational
psychology, there has been a decline in work in human relationships and tradi-
tions of group dynamics exemplified by [16]. During the 1990s, more work was
done in intergroup relationships, based on social cognition approaches, than in
intergroup relationships, in which the structural characteristics of groups are rec-
ognized and treated. At the same time, intergroup relationship research moves
to other fields of application. The social influence network theory presents a for-
malization of the social process of attitude changes that develop in a network
of interpersonal influence [10,9,11]. [21] suggests, that the emergence of inter-
personal influence is among the basic postulates of social psychological theory:
People’s attitudes are usually formed in interpersonal settings in which influen-
tial positions on issues disagree and may change. The social influence network
(SIN) has been continuously developed since the 1950s by [14,13,6,10]. It is one
of the important fields directly related to group decision making and SNA. In
this co-related direction, recent studies on CGDM have been introduced with
the incorporation of the theory of social influence, derived from social networks.
Brunelli et al. He addressed the evaluation of consensus considering the strengths
of the influence of experts in a social network through a measure of centrality
of their own vector with a blurred adjacency relationship approach. In [15], a
leadership-based consensus procedure was developed, where opinion managers
can give advice and influence the formation of a social network’s judgment to
achieve consensus.
3 Methodology
For solve large group decision problems we propose a hybrid methodology based
on [24,12,23,22] using Social network analysis/DEMATEL for assign weights to
DM and analytic hierarchy process (AHP) for solve the decision-making problem.
As novelty we include a weighting rule (eq. 15)
282 J. Romero et al.
Fig. 1: Step by step methodology
The methodology can be seen in the fig. 1. First, we generate the visualization
an analysis of data obtained by all decision-makers (DM). We collect all data
by a html questionnaire shared with several D-M, in order to obtain the value-
judgments of them. For data visualization and analysis of all decision makers we
use JULIA programming language with the packages Taro, DataFrames and
ExcelReaders. Second, we assign weights to all decision-makers as the arithmetic
mean of SNA and proposed DEMATEL. Finally, we assign weights to all decision-
makers and solve the problem with an MCDA technique, in this case, we propose
to use AHP by an own developed package in Julia 4 . The use of SNA, DEMATEL
and AHP is explained as follows.
3.1 Social Network Analysis
According to [2] social network analysis is a collection of techniques under a
methodology that allows us to create social structures by using graph theory
and network analysis. Social structures can represent several kinds of different
relationships between actors and assign weight to those relationships. It can rep-
resent the spread of disease or virus, interpersonal relationships, trust, influence,
among many others. According to [18] SNA practice, it implies following an
analysis structure that suggests the following steps (adapted for our context in
decision making by the authors):
– Identification: In this paragraph we notice the advice from [2] and define
relational states as ”relational cognition” in perceptual aspects. (Influence
among them)
– Analysis: Measure the level of proximity for every decision-maker in the
influence network, to assign them different levels of importance.
– interpretation of information: Assign weights for every decision-maker to
use their influence in a dynamic group decision-making problem and improve
consensus in their judgments.
4
https://github.com/jorgeiv500/AnalyticHierarchyProcess
Influence Networks based Methodology for Consensus Reaching 283
There are many proximity measures, in this work we use the measure [27].
The proximity can be compute as follows:
0 g−1
Cc (ni ) = Pg (1)
j=1 d(ni , nj )
Where: d(ni , nj ) is the distance between the actor ni and the actor nj , g is
the sum of actors present in the network. The proximity measure proposed by
Sabidussi can be seen as the average inverse distance between the actor (inter-
ested party in our problem) and all other actors. Proximity is the inverse measure
of centrality because large values indicate that a node has high peripheral value,
and small values indicate greater centrality in a node. A higher proximity index
indicates less importance for each actor.
Decision-makers weight calculation The importance index is represented as
the weight Wj of each DM when an opposite measure I is normalized as follows:
1
I= (2)
Cc0 (ni )
Being I the importance of each DM ni represented as the influence over all the
DM and Wi is the weight for each DM.
Ii
Wi = P (3)
j Ii
3.2 Decision-Making Trial and Evaluation Laboratory (DEMATEL)
Decision Making Trial and Evaluation Laboratory, was developed by [7,8] based
on structural modeling for solving complicated and intertwined problems. Ac-
cording to [20], this method has his core on graph theory which can divide
multiple criteria into a cause-and-effect group and the causal relationships in
a network. This technique has extended use, so it can calculate the relations
between criteria for Analytic-network-problems and other MCDM methods. In
this method each node represents an evaluation item (like criteria, people, alter-
natives, among others), and arcs represent the strength of their relations. First,
using a four values scale where every influencing factor is denoted by:
– No influence −→ 0
– Low influence −→ 1
– Medium influence −→ 2
– High influence −→ 3
– Very high influence −→ 4
We invite experts denoted as Hi to obtain the direct influence matrices BH .
Every decision-maker give their judgments over the evaluation items with the
scale presented before.
284 J. Romero et al.
0 b12 b1n
BH = b21 0 b2n (4)
xkn1 xkn2 0
Where bi,j represents the direct influenced matrix determined by the H − th
expert.
Next, we compute the average matrix as the arithmetic mean of all BH
matrices.
PH h
h 1
1
PH
0 h=1 b 12HH Ph=1 b1n
1 PH h 1
A = [bij ]n×n = H b
h=1 21 0 H
H h
h=1 b2n
(5)
1 1
PH H
h h
0
P
H h=1 bn1 H h=1 bn2
Where A is the average matrix. Then, using eq 6 we normalize the average
matrix A. The normalized initial direct-relation matrix R can be obtained as
follows:
1
R= A (6)
v
Where the normalizing factor v is given by
Xn n
X
v= max bij , bij , (7)
i,j=1,..,n
j=1 i=1
Later, we compute the total relation matrix T . The powers of R represent the
indirect effects between any factors. A continuous decrease of the indirect effects
of factors along with the powers of matrix R, such as R2 , R3 , R4 , ...., Rn like a
Markov chain matrix, guarantees convergent solutions to the matrix inversion.
Then, the total relation matrix X is given as follows:
T = R + R2 + R3 + . . . = R(I − R)−1 (8)
Where I is n × n unit matrix.
The total effect that directly and indirectly exerted by the ith factor, is
denoted by ri , could be calculated as follows:
n
X
ri = tij (9)
j=1
The total effect including direct and indirect effects received by the jht factor,
id denoted by cj could be calculated as follows:
n
X
cj = tij (10)
i=1
Influence Networks based Methodology for Consensus Reaching 285
Decision-makers weigths calculation According with [1,5] DEMATEL can
be used in order to compute the weights of criteria in MCDM problems as follows.
n n
ri + ci = t+
X X
i = ti,j + tj,i (11)
j=1 j=1
where t+
i represents the importance factor of every evaluated item.
n
X n
X
ri − ci = t−
i = ti,j − tj,i (12)
j=1 j=1
where t−
i represents the influence factor of every evaluated item.
wi = ((t+ )2 + (t− )2 )1/2 (13)
where wi represents euclidean distance of every evaluated factor, and consider
the importance t+ −
i and influence ti of all nodes.
wi
Wi = Pn (14)
i=1 wi
Finally, we normalize matrix wi and obtain the weights for all item under eval-
uation.
Proposed method for weighting based on DEMATEL influence t− i
Given that we want to see the influence of every DM into all other participants,
we propose: from eq 12 obtain influence values of every decision-maker, similar to
centrality values we obtain the capacity of change judgments over other decision-
makers. Later, we show their similarities with a numeric example in the last
section of this paper. We propose only use the influence t−
i , as positive values by
adding the double of the absolute value from a minimum of all negative values
in set t−
i .
−
Pn wi if any ti ∃{R<0}
wi
W (ti ) = i=1
− (15)
Pnti − if t−i ∈ {R ≥ 0}
t
i=1 i
where W (ti ) is the final weigth for every decision-maker, t− i is the influence
vector and ωi = t−i + 2| min n
t−
i=1 i | are positive values from t−
i .
3.3 Analytic hierarchy process
For solving the DM problem, we propose the use of AHP, as an accepted and
used often in problems that include subjective judgments of people. According
to Saaty [26,25] the AHP is a useful tool to structure complex problems that
influence multiple criteria and at the same time classify a set of alternatives in
order of importance. Initially a hierarchical structure is made where the main
286 J. Romero et al.
decision problem is identified, then the criteria and sub-criteria that are taken
into account for the decision are identified.The last level corresponds to the set
of alternatives that will be evaluated concerning each of the criteria and sub-
criteria. This evaluation is carried out through a series of binary comparisons in
a matrix n x n, where n is the number of elements to be compared. In order to
make the comparison, a scale is required. He proposed a scale between 1 and 9
where each intermediate value has an interpretation for the decision-maker (see
Table.1).
Table 1: Saaty scale
Relative Intensity Definition
1 Equal importance
3 Moderate importance of one element over another
5 Strong importance of one element over another
7 Very strong importance of one element over another
9 Extreme importance of one element over another
Values 2, 4, 6 and 8 are intermediate values that can be used in some cases.
The next step is to find the relative priorities of the criteria and / or the alterna-
tives. This step is based on the eigenvector theory. For example if a comparison
matrix is A, then:
Aw = λmax w (16)
Where w corresponds to the column vector of the relative weights obtained
by making the average of each line of the normalized comparison matrix.
The value of λmax is obtained by adding the column vector corresponding to
the multiplication of the original comparison matrix with the column vector of
relative weights.
n
X
λ max = Aw (17)
i
Because comparisons are made subjectively, a consistency index is required to
measure the consistency of the person making the ratings. The consistency index
and the consistency ratio CR are calculated as follows:
λmax − n CI
CI = CR = (18)
n−1 RI
Where the RI inconsistency ratio is a comparison constant that depends on
the size of the paired comparison matrix for sizes of n = 9 (our criteria x criteria
matrix) RI = 1.45
Influence Networks based Methodology for Consensus Reaching 287
4 Solution method
– Decision-Making Problem Formulation: We use a HTML survey 5 form
in order to collect judgments for 6 Decision Makers.
– Data Management and implementation SNA-DEMATEL: The im-
plementation of SNA was made with Gephi and calculating in Julia Language
the Sabidussi proximity measure.
– Decision-Making Problem solution: For solve the DM problem, we first
compare and assign a weight to every DM in order to give them different
levels of importance. Next, we solve the problem using AHP and arithmetic
mean aggregation of their judgments.
– IDE: IPython/IJulia/Jupyter-notebooks.
4.1 Case Study in environmental decisions
Decision-Making Problem Formulation: The primary purpose of this appli-
cation is to compare DEMATEL with SNA to assign weights to decision-makers.
To solve the DM-Importance issue, we analize the problem presented in [24]. This
problem includes new six decision-makers as stakeholders in a group decision-
making problem. The main problem is to prioritize environmental issues in a
natural park. The criteria describe as follows.
Criteria The set of criteria is given in order to prioritize the problems given as
conservation objectives for the Cocuy National Natural Park.
– Keep the eco-systemic connectivity of forest areas and wilderness.
– Preserve habitats and populations of endemic species.
– Keep the water supply that feeds the river basins.
– Protect the Uwa territory that overlaps with the Park.
– Protect outstanding scenic values.
Alternatives After discussing with the panel of decision-makers, it was agreed
to use 6 problems present in the National natural park.
– Socio-Political fragmentation and loss of traditional knowledge of Uwa com-
munity.
– Logging, burning and clearing of vegetation to maintain pastures and crops
between the Andean forests in the eastern sector.
– Infrastructure without environmental impact studies and mitigation mea-
sures (deposits, canals, bridges, roads)
– Extensive grazing in the park.
– Clogging and rapid drying of peatlands, lakes and springs
– Tourism poorly managed in the park.
5
https://www.1ka.si/
288 J. Romero et al.
Data Management and implementation SNA-DEMATEL: In order to
implement SNA and DEMATEL we make a square matrix for every DM col-
lecting the perceived importance of each other DM. Next, according with [24]
we calculate the influence index for every DM, by asking the decision makers
the following question: “Q1.Which stakeholder do you think may agree with your
opinion regarding the ranking of the goals of Cocuy National Park?”. Each DM
gives his/her opinion (their own beliefs) in order to build the square matrix
following the scale:
– No agreement with your opinion = 0
– Low level of agreement with your opinion = 1
– Medium level of agreement with your opinion = 2
– High level of agreement with your opinion = 3
– Very high level of agreement with your opinion = 4
The judgments for every decision maker (also called matrix A See eq.5) can
be observed in Table 2 their aggregation and normalization (also called matrix
D See eq.6) can be observed in Table 3.
0.00 2.33 2.33 2.33 2.50 3.67 0.00 0.18 0.18 0.18 0.19 0.28
1.83 0.00 2.50 3.00 2.67 2.17 0.14 0.00 0.19 0.23 0.20 0.16
1.00 2.00 0.00 1.50 1.83 1.00 0.08 0.15 0.00 0.11 0.14 0.08
A= D = 0.13 0.24 0.14 0.00 0.13 0.14
1.67 3.17 1.83 0.00 1.67 1.83
1.67 1.67 1.83 2.17 0.00 3.00 0.13 0.13 0.14 0.16 0.00 0.23
1.50 1.50 1.83 1.83 1.83 0.00 0.11 0.11 0.14 0.14 0.14 0.00
Table 3: Aggregation A and Normalization D, also called D matrix
DM1 DM2 DM3 DM4 DM5 DM6
DM1 0 0.2 0.2 0.2 0.214 0.314
DM2 0.16 0 0.214 0.257 0.229 0.186
DM3 0.09 0.171 0 0.129 0.157 0.086
DM4 0.14 0.271 0.157 0 0.143 0.157
DM5 0.14 0.143 0.157 0.186 0 0.257
DM6 0.13 0.129 0.157 0.157 0.157 0
From Table 3 we apply DEMATEL and Social Network Analysis in sub sec-
tion Decision-making problem solution as follows.
Influence Networks based Methodology for Consensus Reaching 289
Table 2: Influence values for every decision maker DM
(a) Decision maker 1 (b) Decision maker 2
DM1 DM2 DM3 DM4 DM5 DM6 DM1 DM2 DM3 DM4 DM5 DM6
DM1 0 3 3 3 3 4 DM1 0 2 3 3 2 4
DM2 2 0 3 3 2 2 DM2 2 0 3 3 3 3
DM3 1 1 0 2 1 1 DM3 1 2 0 2 2 0
DM4 1 3 3 0 1 3 DM4 2 4 1 0 2 2
DM5 2 1 2 3 0 3 DM5 1 2 2 1 0 3
DM6 1 2 2 1 2 0 DM6 2 1 2 3 2 0
(c) Decision maker 3 (d) Decision maker 4
DM1 DM2 DM3 DM4 DM5 DM6 DM1 DM2 DM3 DM4 DM5 DM6
DM1 0 3 2 2 3 4 DM1 0 2 2 3 3 3
DM2 2 0 2 3 3 0 DM2 1 0 3 3 3 3
DM3 1 3 0 1 2 1 DM3 1 2 0 0 2 2
DM4 2 3 2 0 2 1 DM4 1 3 2 0 2 2
DM5 1 2 2 2 0 4 DM5 2 2 1 2 0 3
DM6 1 2 1 1 2 0 DM6 2 1 2 2 2 0
(e) Decision maker 5 (f) Decision maker 7
DM1 DM2 DM3 DM4 DM5 DM6 DM1 DM2 DM3 DM4 DM5 DM6
DM1 0 2 2 0 2 3 DM1 0 2 2 3 2 4
DM2 2 0 2 3 2 2 DM2 2 0 2 3 3 3
DM3 1 2 0 2 2 1 DM3 1 2 0 2 2 1
DM4 2 3 1 0 3 2 DM4 2 3 2 0 0 1
DM5 2 1 2 2 0 3 DM5 2 2 2 3 0 2
DM6 2 1 2 2 2 0 DM6 1 2 2 2 1 0
0.38 2.63 0.40
0.75 1.33 0.20
1.50 0.67
Wi = 0.10
Ci = Ii = (19)
1.43 0.70 0.11
1.36 0.73 0.11
2.00 0.50 0.08
Fig. 2: Influence Net-
work in Gephi®
290 J. Romero et al.
Decision-Making Problem solution: For SNA analysis we take the D ma-
trix and represent the strength of all influence-relations in a graph (with gephi
software) as we can see in Fig 2. Their importance is calculated by Sabidussi
centrality/proximity measure [27]. Once we have the Sabidussi centrality Ci we
apply the eq.2 and 3 in order to calculate the weight (importance) for every de-
cision maker (see eq. 19 ). Next, we aplly DEMATEL also starting from matrix
D, next using eq 6 we obtain the direct relations matrix, then we also get the
total relation matrix T from eq 8 (see Table 4)
0.44 0.73 0.72 0.74 0.73 0.85
0.53 0.55 0.69 0.74 0.71 0.72
0.34 0.49 0.35 0.46 0.47 0.45
T = (20)
0.47 0.68 0.59 0.49 0.58 0.63
0.47 0.58 0.58 0.62 0.46 0.69
0.40 0.50 0.51 0.53 0.51 0.42
In order to identify the relevant relations between decision-makers we give a
Table 4: total relation matrix T , r and c values
DM1 DM2 DM3 DM4 DM5 DM6 r
DM1 0.439451 0.731664 0.717209 0.741951 0.733089 0.846074 4.209
DM2 0.533578 0.549456 0.690697 0.742677 0.70559 0.720201 3.942
DM3 0.339858 0.48739 0.345787 0.464839 0.472036 0.449336 2.559
DM4 0.473047 0.676942 0.588607 0.488866 0.584842 0.6284 3.441
DM5 0.466728 0.582645 0.57874 0.617774 0.461313 0.688785 3.396
DM6 0.402952 0.503124 0.510324 0.527189 0.514534 0.424408 2.883
c 2.656 3.531 3.431 3.583 3.471 3.757
threshold value β = 0.6 (0.1 over the arithmetic mean of every bij given in the
matrix T ). In Table 4 we can see all important relations between decision-makers
(in green) and represent them in DEMATEL (see Fig. 3) relations graph. The
final ranking for DEMATEL importance and influence can be observer in Table
5b as follows:
Finally, we solve the problem using AHP though superdecisions-software 6 .
Table 5a shows the weight compatibility between the proposed SNA and DEMA-
TEL approach. Weights obtained using the proposed DEMATEL approach have
been compared with those obtained with Social network analysis by heightening-
sabidussi proximity measure (see Fig. 4 as follows).
Comparing them, there are very similar and present compatibility in their
order of preference, obtaining the same selection results when apply AHP method
as can see in Figs 5 and 6.
6
https://www.superdecisions.com
Influence Networks based Methodology for Consensus Reaching 291
Fig. 3: DEMATEL importance r + c = t+ −
i and influence r − c = ti with their
Relationships from X matrix
Table 5: Influence values for every decision maker DM
(a) DM final weights (b) Influence r − c and Importance
r+c
Sabudussi Proposed
SNA Wi Dematel Wi r c r+c r-c
DM1 0.40 0.31 DM1 4.21 2.66 6.87 1.55
DM2 0.20 0.21 DM2 3.94 3.53 7.47 0.41
DM3 0.10 0.08 DM3 2.56 3.43 5.99 -0.87
DM4 0.11 0.15 DM4 3.44 3.58 7.02 -0.14
DM5 0.11 0.16 DM5 3.40 3.47 6.87 -0.08
DM6 0.08 0.08 DM6 2.88 3.76 6.64 -0.87
Fig. 4: Decision-makers final weigths
292 J. Romero et al.
Fig. 5: AHP Decision-model using Superdecisions software
(a) AHP model results using SNA (b) AHP model results using DEMATEL
Fig. 6: AHP model results with proposed models using Superdecisions
Now, we replace weights with arithmetic mean of two proposed approaches
and obtain as result: Awi = {0.360.200.090.130.140.08} and alternatives solution
Fig. 7: Final Ranking with Superdecisions
Influence Networks based Methodology for Consensus Reaching 293
ranking for the problem: Si = {0.140.180.110.320.140.12} . As can see in Fig. 7
Problem 5 is the most important problem in this numerical example, and there
is an agreement of every decision-makers over the solution, reaching consensus
over the final rank.
5 Conclusions
– Sabidussi proximity measure give ssimilar results to Normalized DEMATEL
proposed weighting method, both proposals are normalized and proportional.
– It is possible to influence change over judgments in a decision-makers group
by an iterative and dynamic way, showing the values of the actor that have
major influence over all.
– People tend to agree with judgments from the most influential actor, given
their importance perception over some participants in the decision-making
group.
– Group decision making presents several issues according to every different
context in decision-making formulation. However, one of the open challenges
can be solved by our proposal in a very effective way, giving to every decision-
maker a perceived-influence by other members of the decision group.
– In addition, to compute weights for every decision-maker, social network
analysis and DEMATEL give the analyst in the group decision problem
the capacity of increasing their knowledge about hidden relationships over
participants and give an easy way to reach agreements faster than other
panel-based-methods.
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