Weighted OWA (Ordered Weighted Averaging Operator) Preference
Aggregation for Group Multicriteria Decisions
GEORGIOS RIGOPOULOS
Division of Mathematics-Informatics and Statistics-Econometrics, Department of Economics
National and Kapodistrian University of Athens
Athens
GREECE
Abstract: - Group decision making is an integral part of operations and management functions in almost every
business domain with substantial applications in finance and economics. In parallel to human decision makers,
software agents operate in business systems and environments, collaborate, compete and perform algorithmic
decision-making tasks as well. In both settings, information aggregation of decision problem parameters and
agent preferences is a necessary step to generate group decision outcome. Although plenty aggregation
information approaches exist, overcomplexity of the underlying aggregating operation, in most of them, is a
drawback, especially for human based group decisions in practice. In this work we introduce an aggregation
method for group decision setting, based on the Weighted Ordered Averaging Operator (WOWA). The
aggregation is applied on decision maker preferences, following the majority concept to generate a unique set
of preferences as input for the decision algorithm. We present the theoretical construction of the model and an
application at a group multicriteria assignment decision problem, along with detailed numerical results. The
proposed method contributes in the field, as it offers a novel approach that is simple and intuitive, and avoids
overcomplexity during group decision process. The method can be also easily deployed into artificial
environments and algorithmic decision-making mechanisms.
Key-Words: - Group Decision Making; Weighted Ordered Averaging Operator (WOWA); Multicriteria
Analysis; Financial Classification.
Received: May 27, 2022. Revised: March 14, 2023. Accepted: April 12, 2023. Published: May 8, 2023.
1 Introduction
Group decision making has been studied in a variety
of settings and environments for long, and although
it might be considered as a mature research domain,
it yet remains active and evolving. Key reasons for
the interest in the domain include among others the
ongoing increasing complexity of real-world
problems, the increasing data volumes used as input,
the usage of artificial agents that take decisions in
collaborative environments instead of humans. All
the above make informed decision-making process a
challenging task. Recent works [1] demonstrate the
extent of the domain and review the direction of
theoretical research towards uncertainty, fuzzy sets
and rough sets, but also touch upon the challenges in
technology adoption [2] of applications and systems
in the domain as well.
A subset of group decision research focuses on
multicriteria decision problems, as they constitute
the majority of real-world settings, where multiple
overlapping criteria need to be considered prior to a
decision. A variety of methodologies and decision
support systems have been introduced with many
practical applications, especially in financial
domain. Zopounidis et. al. [3] present a thorough
review of decision support methodologies focusing
mainly on finance and multicriteria settings. What is
evident from existing research, is that multicriteria
analysis is a valid way to handle inherent
complexity of group decisions and model problems
with large numbers of parameters and participants,
that often leads to overly challenging settings. Salo
et. al. reviewed a large number of academic works
on multicriteria methods utilization for group
decisions and conclude that the potential is very
high as multicriteria analysis provides a structured
way for problem formulation, guides members to
understand requirements effectively and also
express their preferences reflecting their individual
decision model [4]. Other works also indicate the
applicability of multicriteria analysis to assist group
decision making in a variety of problems, resulting
in numerous methodologies and group decision
support systems [5], [6], [7], [8].
In existing research, a group decision problem in
multicriteria setting can be modelled under two
major approaches:
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.2
Georgios Rigopoulos
E-ISSN: 2769-2477
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1) Aggregation of individual decisions. In this
approach, individual multicriteria models are
developed per decision maker and capture
individuals’ preferences. Each group member
formulates a multicriteria problem defining
the parameter values according to her
preferences. The model is solved resulting
into an individual solution set. Next, the
individual solutions are aggregated by
aggregation operators providing thus the
group solution.
2) Aggregation of preferences. In the second
approach, a multicriteria model is developed
for the entire team. Each group member
defines a set of parameter values that are
aggregated by appropriate operators,
providing finally a group parameter value set.
The muticriteria method is then applied on
this group parameter set and the solution
expresses group preference.
Both approaches have some positive and negative
aspects, related to complexity, information loss, and
consensus, to name a few. The aggregation
operation that is applied can lead to partial
information loss or may be overcomplex for
decision makers to understand the impact or
contribution of their preferences to the outcome. So,
a question that arises in such problems is the choice
of the most appropriate aggregation operator or
process to express group preferences and process
them to reach an acceptable outcome. Several works
in the field introduce a variety of approaches, with
Yager’s work [9], [10] in the nineties being seminal
in the field. Yager introduced the Ordered Weighted
Averaging family of operators (OWA) and since
them it has been used extensively either in
multicriteria problems or group decision problems.
Since then, additional families of operators have
been introduced, including fuzzy and linguistic
operators along with various combinations of them.
Interested readers are advised to follow the thorough
bibliographic analysis of Mesa et. al. [11], that
presents a detailed review of the developments in
this field reflecting its evolution and directions for
the future.
Following the above stream of work, we present
here a novel aggregation method for group decision
multicriteria classification problems utilizing the
Weighted OWA (Ordered Weighted Averaging)
operator. The group classification problem refers to
the assignment of a set of alternatives in a number
of categories. So, having a set of alternatives, a set
of categories and a set of evaluation criteria, the aim
is to assign alternatives to categories with respect to
their score on the evaluation criteria according to
group members’ preferences. In our approach we
utilize WOWA operator for the aggregation of
individual preferences calculating an aggregated set
of group parameters, that is used as input for the
classification algorithm. The multicriteria
classification algorithm we use is based on the
concept of inclusion/exclusion of an action with
respect to a category [12], [13].
The process is briefly the following. The group
facilitator proposes a set of parameters to the group
members. Next, each group member evaluates the
proposed parameter set and expresses her
preferences in numeric format. The individual
preferences are aggregated by WOWA operator and
a set of group parameters is generated. The
classification algorithm is applied, using the group
parameter set as input, for the classification of
alternatives and group members evaluate derived
results. In case of low level of group consensus,
parameters are redefined partially or in total and
aggregation phase is repeated. The method is novel
in the field, it is intuitive enough and makes easy for
decision makers to interpret and also estimate the
impact of their preferences on the result.
In this work we focus on the aggregation procedure
of group member preferences, presenting the
approach, as well as a detailed numeric example,
which demonstrates its applicability to real world
problems. Initially, we present necessary theoretical
background information on OWA and WOWA
operators, as well as a brief overview of the
NexClass multicriteria classification algorithm we
utilize. The aggregation approach is presented in the
next section along with a detailed example and
explanations of the steps. Finally, we conclude by
summarizing key findings and considerations for the
future.
2 Background
2.1 OWA operator (Ordered Weighted
Averaging Operator)
OWA operator was initially introduced by Yager [9]
and was developed and discussed further in several
works since then. It remains a very important and
intuitive approach for its simplicity and constitutes
the basis for several families of operators that have
been introduced since then.
An OWA operator of dimension is a mapping
function ℜℜ, which has a weighting vector
associated with it, such as
, and aggregates a set of
values according to the following
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.2
Georgios Rigopoulos
E-ISSN: 2769-2477
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expression
,where
is a permutation of set
, such as ,
, (e.g. is the i-highest value in set
).
A basic property of OWA is the reordering of
arguments according to their values, which
associates a weight to particular positions in the
ordered set of values and not to the values. OWA
operators are commutative, monotonic and
idempotent, following the basic properties of
averaging operators. Weight vector definition is a
central issue for the OWA operator, and it impacts
the outcome. Yager proposes two methods for its
estimation [9]. The first approach uses a kind of
training approach using some training data, while
the second one assigns semantics on the weights.
Following the second approach, weights can express
the concept of fuzzy majority on the aggregation of
the values with OWA. In this approach weights can
be obtained by using a functional form of linguistic
quantifiers. In this case a quantifier is defined as a
function where ,
and for . For a given value
, the is the degree to which
satisfies the fuzzy concept being represented by the
quantifier. Based on function the OWA weight
vector is given by
.
Following this approach, the quantifier determines
the weighting vector according to the semantics
associated with the operator from function . Zadeh
[9] defined membership function of quantifier by
the expression
with
. The most common quantifiers used
are ‘most’, ‘at least half’, ‘as many as possible’ with
parameters equal to
respectively. For example the fuzzy majority
concept can be expressed by using quantifier
‘most’ with values for the
calculation of OWA weights.
The fuzzy majority approach with OWA
aggregation has been utilized as is or with variations
on group decisions, where the objective was the
maximization of group consensus, since this
approach is more appropriate than simple averaging
operators, as it takes into account the majority
concept and can model a variety of group settings.
2.2 WOWA operator (Weighted OWA)
WOWA operator was introduced by Torra [14], [15]
in order to extend OWA based aggregation in a way
to consider weights of sources in addition to weights
of values. It has been used in decision support for
aggregation of preferences and consensus
generation [16], [17], [18].
A WOWA operator of dimension is a mapping
function , which has two weight
vectors associated with it, with
, (which expresses the
values importance in analogy to OWA weights) and
with
,
(which expresses the importance of sources in
analogy to a weighted average operator), and
aggregates a set of values with the
following expression WOWA
, where is a
permutation of set such that
, , (e.g. is the i-highest
value in set ), and and
is the weight vector of
WOWA operator.
Weights are defined as
w* -w* , where is a
monotone increasing function which interpolates
points with the point .
Calculation of can be executed either from
direct definition of function , or from the
definition of the vector initially,
and calculation of the interpolation function
next.
Following the second approach, for the evaluation
of the function from the weight vector
an interpolation method is required.
From available methods the one to be used, has to
define a monotonous and bounded function (e.g.
polynomial) when input data are monotonous and
bounded. WOWA operator can be considered as
generalization of weighted mean and OWA
operators, since for equivalent sources’ weights it
coincides with OWA, while for equivalent values’
weights it coincides with weighted mean.
Although the definition of weights is a process that
is not straightforward and can vary among various
implementations, the WOWA aggregation approach
is intuitive enough WOWA operator is quite
efficient for the aggregation of member preferences
in group setting. It allows aggregation of values
considering members’ importance, and the
definition of zones of different importance which
express variations of majority values.
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.2
Georgios Rigopoulos
E-ISSN: 2769-2477
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3. WOWA aggregation method
3.1 Proposed group decision aggregation
process
The key focus of this work is the introduction of an
aggregation method for group decisions.
Specifically, we are interested in classification
decisions, where a number of alternatives are
assigned to predefined categories, based on the
aggregated preferences of a group of decision
makers. The decision process can be divided in the
aggregation phase, where member preferences are
collected and aggregated, and the classification
phase, where a classification algorithm is applied in
the group preferences. The aggregation phase is
linked to the algorithm used in the classification
phase, as each algorithm usually requires a set of
parameters in a specific format. So, the decision
problem formulation is based on the algorithm first,
and then the aggregation process generates the
appropriate input for the algorithm.
In this work we focus on group classification
problems, where categories are nominal and
predefined and group members provide their
preferences on a number of attributes for the
alternatives to be classified. The specific
classification setting, has been approached by
NexClass classification algorithm [12]. NexClass is
a multicriteria method and decision support system
that was introduced to address nominal
classification problems using the concept of fuzzy
inclusion degree [12], and we shall use it in this
work for group decision setting for the classification
of a set of alternatives into predefined classes
according to their performance at a number of
criteria.
The inclusion/exclusion of an alternative from a
category is determined by evaluating the fuzzy
inclusion degree of the alternative for the specific
category, following concordance/non-discordance
concepts. The categories are defined by an entrance
threshold, which can be considered as the least
typical representative alternative that satisfies the
inclusion requirements. The objective of the
algorithm is to classify actions to categories in a
way to consider inclusion/exclusion concept.
The application of NexClass algorithm in group
classification problems requires the definition of the
parameters for a set of decision makers. To use the
group parameters as input for the algorithm,
appropriate aggregation is required. In this work we
use a WOWA aggregation approach to generate the
input values for the algorithm.
The following parameters are required for the group
decision making process:
1) A set of group decision makers
,m,...m and corresponding importance
weights assigned to each one.
2) A set of evaluation criteria
generated from problem
requirements and their corresponding
assigned weights.
3) A set of categories for
the classification of alternatives. Categories
are defined by their entrance thresholds
and their scores to evaluation criteria .
4) A set of alternatives for
classification, defined by their performance
on the evaluation criteria
.
5) Preference, indifference and veto threshold
values for each criterion.
In group decision environment we consider that a
facilitator drives the process and initiates the
parameters and criteria. Facilitator defines the
alternatives and evaluation criteria and also assigns
member importance weights. In a more generic
setting, problem formulation could be also a group
process, requiring consensus, but for simplicity we
keep it as a separate procedure. However, in the
majority of business decisions the problem is more
or less structured and decision makers are asked to
contribute by providing their preferences. So, after
the initiation of parameters facilitator informs
members to submit their preferences. In this phase,
group members express their preferences on the
proposed parameter set. Specifically, group
members provide
1) Preferred values for each alternative per
criterion for all combinations of criteria and
alternatives. Those values reflect their
preferences on the alternatives.
2) Preferred values on the threshold per
criterion, that define the baseline levels for
assignment to a category.
Member preferences are either expressed or
converted in numeric values. After the preference
collection phase, we apply a WOWA aggregation
process for all the individual member values for
thresholds and alternatives, and the aggregated
values are used as input for NexClass algorithm.
For the calculation of aggregated values with
WOWA, we follow the approach below:
1) We consider the fuzzy majority concept
(although it can be modified following the
problem requirements) and use the values
representing the ‘most’
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.2
Georgios Rigopoulos
E-ISSN: 2769-2477
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Volume 3, 2023
value for the quantifier
and evaluate the
weights of the OWA operator from the
expression
.
2) Following the approach proposed by Torra
[15] we calculate WOWA weights
by interpolating a set of points
defined by the set
and calculate the
function as required
3.2 WOWA aggregation steps
In the following we summarize the proposed
aggregation steps for the aggregation of group
preferences. A classification problem can be defined
by the following initial parameters:
1) A group of members ,j 1,..., as
decision makers and corresponding
importance weights
,
2) A set of evaluation criteria
,
3) A set of categories , i 1,..., for
the classification of actions,
4) A set of alternatives for
classification.
The objective is to classify the alternatives
in appropriate categories with respect to
member preferences. The aggregation process is as
follows:
Step 1: We consider,
1. Values to be aggregated as provided
by members,
2. Members’ weights .
Step 2: We calculate the associated WOWA weights
by means of OWA. For the
calculation we consider the fuzzy majority concept
and use the values representing
the ‘most’ value for the quantifier
and evaluate the weights of
the OWA operator from the expression
.
Step 3: We calculate WOWA weights
following the approach proposed by
Torra [15]. Initially we calculate the set of points
that will be connected. This set is defined as
. Next the set
of points is interpolated and function is
calculated.
Step 4: With respect to the sets of weights
we aggregate the set of values
as WOWA
.
3.3 Illustrating example
To demonstrate the proposed model and aggregation
process, we present a detailed example for a group
decision problem, following the steps presented in
the previous section. We assume a classification
problem with the following initial parameters:
1) A group of seven members ,j
1,...,7
2) A set of group member corresponding
importance weights as
,
3) A set of eight evaluation criteria
,
4) A set of four categories , i 1,...,4
for the classification of actions,
5) A set of six alternatives
for classification.
The objective is to classify the alternatives
in appropriate categories
, i 1,...,4, with respect to member preferences
on alternatives performance on the evaluation
criteria. In general group members provide values
for a series of parameters that we aggregate to reach
to a group value. In the following we present the
aggregation approach as applied only to criteria
acceptance and criteria weights, since the same
procedure is applied to the rest of values.
Step 1: Initially, each member expresses his
opinion indicating acceptance level in a linguistic
scale {Extremely High, High, Medium, Low,
Extremely Low}, on the set of criteria . These
International Journal of Computational and Applied Mathematics & Computer Science
DOI: 10.37394/232028.2023.3.2
Georgios Rigopoulos
E-ISSN: 2769-2477
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values are converted to numeric values ranging from
5 to 1 as below.
ij
.
Step 2: We calculate the associated WOWA weights
by means of OWA. For the
calculation we consider the fuzzy majority concept
and the resulting values are as follows:
I
W
1
0
2
0
3
0.257
4
0.285
5
0.285
6
0.171
7
0
Step 3: The set of points
for the interpolation function is
calculated as
1,
,w
,0
,
,w
,0
,
,w
,0.257
...........
,
,w...
,1
1,1
Based on these points the interpolation function is
is calculated using the algorithm used by Torra
[15]. Next, we calculate the set of WOWA weights
as follows:
1, ωw*w*0.2
...........
, ωw*
-w*
WOWA weights are thus
.
Step 4: Next WOWA values are calculated as
WOWA
. For example
for the first criterion we have
WOWA
Aggregation result for the set of criteria is the
following:
ij
, while results using OWA and
Weighted mean aggregation are ij
and
ij
respectively.
Acceptance result for criteria and are relative
low and thus are excluded from problem We follow
the same procedure for categories.
Members’ preferences on criteria weights are
expressed on numeric values as:
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Georgios Rigopoulos
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ij
Aggregation results are depicted in the table below
(Table 1), compared to results from alternative
aggregation approaches.
Table 1. Aggregation results
OWA
Weighte
d Mean
WOWA
Arithmetic
Mean
16.055
16.900
16.417
17
27.717
27.000
26.030
27
8.370
8.000
7.894
8
13.055
13.890
13.417
14
8.199
9.100
8.732
9
25.343
25.100
24.659
26
The above steps are then repeated for all parameters:
criteria, actions’ scores, categories’ thresholds as
well as indifference, preference and veto thresholds.
Then the aggregated actions’ scores, criteria weights
and categories’ thresholds, is the input parameter set
for the multicriteria classification algorithm, which
is applied next.
4. Conclusion
Group decision making is a very critical part of
today’s automated or semi-automated procedures in
large systems and settings. Apart from humans,
robots and algorithms are also taking part in
decisions, with some involving critical ones. As
such the domain of group decision making needs to
provide appropriate algorithms and procedures for
both agent types especially for automated decision
making. Aggregation of information is of critical
importance, as it can infer bias in the process. Many
works propose complex approaches, that lack
intuition and are not easy to be adopted by group
members. In this work we propose a methodology
for aggregation that is based on the intuitive
majority rule and we utilize an operator from the
OWA family. Group decision problems are
inherently complex, but we believe that this
approach has merits and can be easily
communicated to group members.
We focused on classification decisions, where
aggregation of members’ preferences is executed at
the parameter level and used WOWA operator for
the aggregation of individual values. We presented
details of the aggregation methodology as well as a
detailed example for a classification problem
demonstrating its usage for real life problems. The
methodology can be easily applied to support group
decisions in a variety of environments as it is
intuitive and easy to explain to decision makers.
However, since the methodology requires a relative
substantial number of parameters, it is possible that
group members who are not familiar enough with
the methodology will be confused. Thus, the
number of criteria and parameters should be kept to
a number, which will minimize complexity without
however loosing critical problem parameters.
This approach can be included in automated
decision settings for classification decisions, where
software agents can be utilized. It is an emerging
field and developments in algorithmic decision
making clearly demonstrate this directions. So, this
work will be further developed in the future to reach
a wider domain and become more parametric, so as
to be included in a software system or become part
of some intelligent software agent.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Georgios Rigopoulos is the sole author of the
research covered in this paper.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author haV no conflicts of interest to declare
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(Attribution 4.0 International, CC BY 4.0)
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DOI: 10.37394/232028.2023.3.2
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