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

percentage quantity to the concept expressed by the

linguistic quantifiers.

( )⎩⎨⎧

≤<<−

≥=

0.4 x 0 9.04.0 8.02

0.9 x 1xxxmostQ

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.

[ ]TnwwwW ,,, 21 K=

Such as [ ]1,0∈iw and ∑=

=n

iiw

1

1 .

( ) ∏=

=n

i

win

ibaaaOWG1

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:

⎟⎠⎞

⎜⎝⎛ −

−⎟⎠⎞

⎜⎝⎛=

niQ

niQwi

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.

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

bbbfi

n

i

win

MM nii bbaaaF1

,,,

121

21)()(,,, KK

where [ ]1,0∈iw with ∑=

=n

iiw

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

θθθθγ i

nii bbfw K

max

max

1min1maxmax

1min

...... δ

δ

δδδ

δ

θγ

θθθγ ii ++

⋅⋅⋅+−

+

where

⎩⎨⎧ ≥

=otherwise

kiifki 0

1 δγ

and

⎩⎨⎧

≥≠+≥

=otherwiseiycardinalitwithitemofnumber

iifiycardinalitwithitemofnumberi

1) ( minδθ

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:

⎪⎩

⎪⎨⎧ ≤−=

otherwisexaaifaadist ji

ji 01),(

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

jjii 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

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

wi

ib!

= 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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