MM-OWG operator

1. Majority Multiplicative Ordered Weighted Geometric Operators
Peláez J.I. Doña J.M. Mesas A.
Department of Languages and Computer Sciences,
University of Malaga, Malaga 29071, Spain.
E-mail: jignacio@lcc.uma.es
Gil A.M.
Department of Financial Economy and Accounting,
University of Granada. Granada 18071, Spain
E-mail: amgil@ugr.es
Abstract
The aggregation of experts’ preferences consists
in combining the individual preferences into a
collective one, where the properties contained in
every individual preference are summarized or
reflected. This is a necessary and very important
task to perform when we want to obtain a final
solution of multicriteria decision-making or
group decision making problems. In these
problems the majority concept plays a main role
in the aggregation process. In this paper we
present a geometric operator to obtain a feasible
majority aggregation value for the decision
making problem.
Keywords: OWG, MM-OWG Operators,
linguistic quantifiers, group decision making,
majority opinion, majority concept .
1 INTRODUCTION
Decision making is a usual task in human activities where
a set of experts work in a decision process to obtain a
final value which is representative of the group. The first
step of this decision process is constituted by the
individual evaluations of the experts; each decision maker
rates each alternative on the basis of an adopted
evaluation scheme [5, 6, 13]. We assume that at the end of
this stage each alternative has associated a performance
judgment on the linguistic scale or numeric scale [4, 13].
The second step consists in determining for each
alternative a consensual value which synthesizes the
individual evaluation. This value must be representative
of a collective estimation and is obtained by the
aggregation of the opinions of the experts [4, 10, 13, 14].
Finally, the process concludes with the selection of the
best alternative/s as the most representative value of
solution of the problem.
One of the main problems in decision making is how to
define operators which considers the majority opinions
from the individual opinions. To obtain a value of
synthesis of the alternatives which is representative of the
opinions of the experts exist diverse approaches in which
are realized an aggregation guided by the concept of
majority, where majority is defined as a collective
evaluation in which the opinions of the most of the
experts involved in the decision problem are considered.
In these approaches the result is not necessarily of
unanimity, but it must be obtained a solution with
agreement among a fuzzy majority of the decision makers
[7, 12, 14].
In the fuzzy approaches to decision making, the concept
of majority is usually modelled by using linguistic
quantifiers such as at least 80% and most. A linguistic
quantifier is formally defined as a fuzzy subset of a
numeric domain [1, 6, 8, 9]. The semantics of such a
fuzzy subset is described by a membership function which
describes the compatibility of a given absolute or

2. percentage quantity to the concept expressed by the
linguistic quantifiers.
( )
⎩
⎨
⎧
≤
<<−
≥
=
0.4x0
9.04.08.02
0.9x1
xxxmostQ
Figure 1: Definition of the linguistic quantifier most.
In group decision making, linguistic quantifiers are used
to indicate a fusion strategy to guide the process of
aggregating the experts’ opinions. The results of this
aggregation process must represent the semantic of the
linguistic quantifier. An example of a linguistic
expression which employs a linguistic quantifier
representing the majority concept is the following: Q
experts are satisfied by solution a, where Q denotes a
linguistic quantifier (for example most) which expresses a
majority concept. If we want to produce a solution which
satisfies this proposition, the experts’ opinions must be
aggregate using an operator which captures the semantics
of the concept expressed by the quantifier Q.
In this paper the problem of constructing a majority
opinion using OWG operators is considered. The paper is
structured as follows: in section, 2 the aggregation with
OWG operators are introduced. In section 3, the MM-
OWG operators for modelling the majority concept are
defined; and finally, the conclusions are exposed.
2 OWG OPERATORS
The OWG operator [2, 3] is defined to aggregate ratio-
scale judgements. It is based on the OWA operator [15]
and on the geometric mean, and therefore, incorporate the
advantage of geometric mean to deal with ratio-scale
judgements. It is defined as a mapping function
F R Rn
: → that has associated a weighting vector W
with length n.
[ ]T
nwwwW ,,, 21 K=
Such as [ ]1,0∈iw and ∑=
=
n
i
iw
1
1.
( ) ∏=
=
n
i
w
in
i
baaaOWG
1
21 ,,, K
with bi being the ith
largest element of the aj.
A fundamental aspect of these operators is the reorder
step of the arguments. As a result of this, the element to
aggregate ai is not associated with a weight wi, but a
weight wi will be associated with an ordered position in
the aggregation.
The weighting vector W is obtained using the same
method that in the OWA operator case. In [14] the use of
the fuzzy quantifier is proposed for representing the
concept of fuzzy majority. That is to obtain the weights
from a functional form of the linguistic quantifiers. In this
case the quantifiers is defined as a function
[ ] [ ]1,01,0: →Q where Q(0)=0, Q(1)=1 and )()( yQxQ ≥
for x>y. For a given value [ ]1,0∈x , the Q(x) is the degree
to which x satisfies the fuzzy concept being represented
by the quantifier. Based on function Q, the OWG vector is
determined from Q in the following way:
⎟
⎠
⎞
⎜
⎝
⎛ −
−⎟
⎠
⎞
⎜
⎝
⎛
=
n
i
Q
n
i
Qwi
1
These weights have the function to increase or decrease
the importance of the different components of the
aggregation according to the semantics associated with
the operator from Q, that is, the quantifier determines the
strategy of construction of the weighting vector.

3. A number of important properties can be associated with
the OWG operator [2, 3].
The OWG operators can be used in different areas.
Usually is can be used in multi-criteria decision making
[2, 3] where we use a ratio scale and we need only to
satisfy some portion of the criteria. Following the
semantic in the aggregation process of the OWG
operators for decision making environment is studied, and
it is shows how this type of representation is not valid to
represent the majority concept.
3 MAJORITY OPERATOR MM-OWG
Majority operators [10, 11, 12] arise because it is
necessary to obtain representative values of the majority
elements to aggregate in some aggregation processes
without the omission of minority values. The most
common aggregation operators [3, 4, 15] over-emphasize
the opinion of the minority as the expense of those of the
majority creating an aggregation that can be considered
imprecise for group decision making problems. The
majority multiplicative operators are defined as OWG
without the use of linguistic quantifiers.
The MM-OWG operator is defined in [12] as:
( ) ( )
∏∏ ==
==
n
i
bbbf
i
n
i
w
in
MM nii
bbaaaF
1
,,,
1
21
21
)()(,,, K
K
where [ ]1,0∈iw with ∑=
=
n
i
iw
1
1
bi is the ith
element of a1,…, an that is ordered in ascender
order by cardinalities.
The weights of an MM-OWG operator are calculated:
Let δi the importance for the element i with δi > 0, then
( ) +
⋅⋅⋅⋅
==
+− min1min1maxmax
min
...
,,1
δδδδ
δ
θθθθ
γi
nii bbfw K
max
max
1min1maxmax
1min
...
... δ
δ
δδδ
δ
θ
γ
θθθ
γ ii
++
⋅⋅⋅ +−
+
where
⎩
⎨
⎧ ≥
=
otherwise
k
i
ifk
i 0
1 δ
γ
and
⎩
⎨
⎧
≥
≠+≥
=
otherwiseiycardinalitwithitemofnumber
iifiycardinalitwithitemofnumber
i
1)( minδ
θ
The majority operators aggregate in function of δi, which
represents generally the importance of the element i using
its cardinality. In the majority processes are considered
the formation of discussion or majority groups depending
on similarities or distances among the experts’ opinions.
All values with a minimum of separation are considered
inside the same group. The calculation method for the
value δi is independent from the definition of the majority
operators. In this work the importance value δi is
calculated by using the distance function:
⎪⎩
⎪
⎨
⎧ ≤−
=
otherwise
xaaif
aadist ji
ji
0
1
),(
The cardinality of ai is the sum of all values dist(ai, aj) for
j = 1…n being n the number of elements to aggregate.
∑=
=
n
j
jii aadist
1
),(δ
The value x model the final size of each group. Socially
this grade is measured by the flexibility of the decision
maker for grouping and reinforcing his/her opinions.
An example of application is the following: Let us
suppose to have the following values to aggregate using
values of the scale of AHP: [1/9 1/9 2 3 9 9 9]. We want
to obtain a fusion opinion which must be representative of
the majority concept. In this example we use the value of

4. x = 1 in the distance function for the calculation of the
cardinalities δi.
Using this operator we only consider 4 elements to
aggregate B = [1/9 2 3 9] with cardinalities [1 1 2 3].
The weighting vector is:
W = [0.042 0.042 0.208 0.708]
Then
MM-OWG = ∏=
n
i
w
i
i
b
!
= 5.589
The solution obtained is a representative value of the
majority concept defined in [10], which is intuitively a
value between 5 and 7. This definition uses a majority
semantic which considers all the elements of the
aggregation.
CONCLUSIONS
In this paper we present the OWG operator MM-OWG.
This operator has included the concept of majority in the
definition of the weighting vector. We observe how the
results produced this operator is appropriate for
aggregation which must represent in the fusion result the
majority concept.
This approach is not able to model majority concepts like
most, at least 80%, etc. For this reason, in the future work
we will use of linguistic quantifiers in the aggregation
process to model these types of semantics.
Acknowledgements
Research Supported in part under project TIC 2002-
119942-E
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