THE LINGUISTIC GENERALIZED OWA OPERATOR AND ITS
APPLICATION IN STRATEGIC DECISION MAKING
José M. Merigó and Anna M. Gil-Lafuente
Department of Business Administration, University of Barcelona, Av. Diagonal 690, 08034 Barcelona, Spain
Keywords: Linguistic aggregation operators, Linguistic decision making, Strategic decision making.
Abstract: We introduce the linguistic generalized ordered weighted averaging (LGOWA) operator. It is a new
aggregation operator that uses linguistic information and generalized means in the OWA operator. It is very
useful for uncertain situations where the available information can not be assessed with numerical values
but it is possible to use linguistic assessments. This aggregation operator generalizes a wide range of
aggregation operators that use linguistic information such as the linguistic generalized mean (LGM), the
linguistic weighted generalized mean (LWGM), the linguistic OWA (LOWA) operator, the linguistic
ordered weighted geometric (LOWG) operator and the linguistic ordered weighted quadratic averaging
(LOWQA) operator. We also introduce a new type of Quasi-LOWA operator by using quasi-arithmetic
means in the LOWA operator. Finally, we develop an application of the new approach. We analyze a
decision making problem about selection of strategies.
1 INTRODUCTION
In the literature, we find a wide range of aggregation
operators for fusing the information. A very well
known aggregation operator is the ordered weighted
averaging (OWA) operator (Yager, 1988). The
OWA operator has been studied by a lot of authors
such as (Merigó, 2007; Yager and Kacprzyk, 1997).
Often, when using the OWA operator, it is
considered that the available information is
numerical. However, this may not be the real
situation found in the decision making problem.
Sometimes, the available information is vague or
imprecise and it is not possible to analyze it with
numerical values. Therefore, it is necessary to use
another approach such as a qualitative one that uses
linguistic assessments. In (Herrera et al., 1995), they
introduced the first linguistic version of the OWA
operator. They called it the linguistic OWA
(LOWA) operator. Since then, a lot of new
developments have been suggested about it such as
(Herrera and Herrera-Viedma, 1997; Herrera and
Martínez, 2000; Xu, 2004a; 2004b).
Another interesting extension of the OWA
operator is the generalization that uses generalized
means. This type of aggregation is known as the
generalized OWA (GOWA) operator (Karayiannis,
2000; Yager, 2004). It generalizes a wide range of
aggregation operators such as the OWA, the ordered
weighted geometric (OWG) operator, etc. The
GOWA operator has been further generalized
(Beliakov, 2005) by using quasi-arithmetic means.
The result is the Quasi-OWA operator (Fodor,
1995). For further information on the GOWA
operator, see (Merigó, 2007).
The aim of this paper is to develop a generalized
OWA operator for situations where the available
information can not be assessed with numerical
values but it is possible to use linguistic assessments.
We will call it the linguistic generalized OWA
(LGOWA) operator. This type of linguistic
aggregation operator uses the LOWA operator and
the generalized mean in the same formulation. Then,
it is able to include a wide range of particular cases
such as the LOWA itself, the linguistic OWG
(LOWG) operator, the linguistic average (LA), the
linguistic weighted average (LWA), etc. We further
generalize the LGOWA operator by using quasi-
arithmetic means. The result is the Quasi-LOWA
operator. We should note that recently, a different
linguistic Quasi-OWA operator has been studied in
(Wang and Hao, 2006). We also develop an
application of the new approach in a strategic
decision making problem in order to see its
implementation in the real life.
219
M. Merigó J. and M. Gil-Lafuente A. (2008).
THE LINGUISTIC GENERALIZED OWA OPERATOR AND ITS APPLICATION IN STRATEGIC DECISION MAKING.
In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 219-224
DOI: 10.5220/0001692102190224
Copyright
c
SciTePress
This paper is organized as follows. In Section 2,
we briefly comment some preliminary concepts. In
Section 3, we present the LGOWA operator. Section
4 analyzes different families of LGOWA operators.
In Section 5, we discuss the Quasi-LOWA operator.
Section 6 develops a decision making application of
the new approach. Finally, in Section 7, we
summarize the main conclusions of the paper.
2 PRELIMINARIES
In this Section, we discuss the linguistic approach to
be used throughout the paper, the LOWA operator
and the GOWA operator.
2.1 Linguistic Approach
Usually, people are used to work in a quantitative
setting, where the information is expressed by means
of numerical values. However, many aspects of the
real world cannot be assessed in a quantitative form.
Instead, it is possible to use a qualitative one, i.e.,
with vague or imprecise knowledge. In this case, a
better approach may be the use of linguistic
assessments instead of numerical values. The
linguistic approach represents qualitative aspects as
linguistic values by means of linguistic variables
(Zadeh, 1975).
We have to select the appropriate linguistic
descriptors for the term set and their semantics. One
possibility for generating the linguistic term set
consists in directly supplying the term set by
considering all terms distributed on a scale on which
a total order is defined (Herrera and Herrera-
Viedma, 1997). For example, a set of seven terms S
could be given as follows:
S = {s
1
= N, s
2
= VL, s
3
= L, s
4
= M,
s
5
= H, s
6
= VH, s
7
= P}
Note that N = None, VL = Very low, L = Low, M
= Medium, H = High, VH = Very high, P = Perfect.
Usually, in these cases, it is required that in the
linguistic term set there exists:
A negation operator: Neg(s
i
) = s
j
such that j =
g+1i.
The set is ordered: s
i
s
j
if and only if i j.
Max operator: Max(s
i
, s
j
) = s
i
if s
i
s
j
.
Min operator: Min(s
i
, s
j
) = s
i
if s
i
s
j
.
Different approaches have been developed for
dealing with linguistic information such as (Herrera
and Herrera-Viedma, 1997; Herrera and Martínez,
2000; Xu, 2004a; 2004b). In this paper, we will
follow the ideas of (Xu, 2004a; 2004b). Then, in
order to preserve all the given information, we
extend the discrete linguistic term set S to a
continuous set Ŝ = {s
α
| s
1
< s
α
s
t
,
α
[1, t]},
where, if s
α
S, we call s
α
the original linguistic
term, otherwise, we call s
α
the virtual one.
Consider any two linguistic terms s
α
, s
β
Ŝ, and
μ
,
μ
1
,
μ
2
[0, 1], we define some operational laws
as follows (Xu, 2004a; 2004b):
μ
s
α
= s
μα
.
s
α
s
β
= s
β
s
α
= s
α
+
β
.
(s
α
)
μ
= s
α
μ
.
s
α
s
β
= s
β
s
α
= s
αβ
.
2.2 LOWA Operator
In the literature, we find a wide range of linguistic
aggregation operators (Herrera and Herrera-Viedma,
1997; Herrera et al., 1995; Herrera and Martínez,
2000; Xu, 2004a; 2004b). In this study, we will
consider the LOWA operator developed by Xu
(2004a; 2004b) with its particular cases that include
the linguistic average (LA), among others. Then, we
should point out that the LOWA operator we are
going to use is also known as the extended OWA
(EOWA) operator (Xu, 2004a).
Definition 1. A LOWA operator of dimension n is a
mapping LOWA: S
n
S, which has an associated
weighting vector W such that w
j
[0, 1] and
=
=
n
j
j
w
1
1
, then:
LOWA(s
α
1
, s
α
2
, …, s
α
n
) =
=
n
j
j
j
sw
1
β
(1)
where s
β
j
is the jth largest of the s
α
i
.
2.3 GOWA Operator
The GOWA operator (Karayiannis, 2000; Yager
2004) is a generalization of the OWA operator by
using generalized means. It includes a wide range of
means such as the OWG operator, the ordered
weighted quadratic averaging operator (OWQA),
etc. It can be defined as follows.
Definition 2. A GOWA operator of dimension n is a
mapping GOWA:R
n
R that has an associated
ICEIS 2008 - International Conference on Enterprise Information Systems
220
weighting vector W of dimension n such that the
sum of the weights is 1 and w
j
[0,1], then:
GOWA(a
1
, a
2
,…, a
n
) =
λ
λ
/1
1
=
n
j
jj
bw
(2)
where b
j
is the jth largest of the a
i
, and
λ
is a
parameter such that
λ
(−∞, ).
3 LINGUISTIC GENERALIZED
OWA OPERATOR
The LGOWA operator is an extension of the OWA
operator that uses linguistic information and genera-
lized means. It provides a parameterized family of
linguistic aggregation operators that includes the
LOWA operator, the linguistic maximum, the
linguistic minimum and the linguistic average (LA),
among others. It can be defined as follows.
Definition 3. A LGOWA operator of dimension n is
a mapping LGOWA:S
n
S that has an associated
weighting vector W of dimension n such that the
sum of the weights is 1 and w
j
[0,1], then:
LGOWA(s
α
1
, …, s
α
n
) =
λ
λ
β
/1
1
=
n
j
j
j
sw
(3)
where s
β
j
is the jth largest of the s
α
i
, and λ is a
parameter such that λ (, ).
From a generalized perspective of the reordering
step, we can distinguish between the descending
LGOWA (DLGOWA) operator and the ascending
LGOWA (ALGOWA) operator. The weights of
these operators are related by w
j
= w*
n
j+1
, where w
j
is the jth weight of the DLGOWA and w*
n
j+1
the jth
weight of the ALGOWA operator.
The LGOWA operator is a mean or averaging
operator. This is a reflection of the fact that the
operator is commutative, monotonic, bounded and
idempotent. It is commutative because any
permutation of the arguments has the same
evaluation. It is monotonic because if s
α
i
s
δ
i
, for all
α
i
, then, LGOWA(s
α
1
, …, s
α
n
) LGOWA(s
δ
1
, …,
s
δ
n
). It is bounded because the LGOWA aggregation
is delimitated by the minimum and the maximum:
Min{s
α
i
} LGOWA(s
α
1
, …, s
α
n
) Max{s
α
i
}. It is
idempotent because if s
α
i
= s
α
, for all s
α
i
, then,
LGOWA(s
α
1
, …, s
α
n
) = s
α
.
Another interesting issue to consider is the
attitudinal character of the LGOWA operator. Using
a similar methodology as it was used by (Yager,
2004) for the GOWA operator we can define the
following measure:
α
(W) =
λ
λ
/1
1
1
=
n
j
j
n
jn
w
(4)
Note that other measures could be discussed
such as the entropy of dispersion, the divergence of
W and the balance operator (Merigó, 2007).
4 FAMILIES OF LGOWA
OPERATORS
Different families of linguistic aggregation operators
are found in the LGOWA operator. Basically, we
can classify them in two big groups.
4.1 Analysing the Weighting Vector W
By choosing a different manifestation of the
weighting vector in the LGOWA operator, we are
able to obtain different types of aggregation
operators. For example, we can obtain the linguistic
maximum, the linguistic minimum, the linguistic
generalized mean (LGM) and the linguistic weighted
generalized mean (LWGM).
The linguistic maximum is obtained if w
1
= 1
and w
j
= 0, for all j 1. The linguistic minimum is
obtained if w
n
= 1 and w
j
= 0, for all j n. More
generally, if w
k
= 1 and w
j
= 0, for all j k, we get
for any
λ
, LGOWA(s
α
1
, …, s
α
n
) = b
k
, where b
k
is the
kth largest argument a
i
. The LGM is found when w
j
= 1/n, for all a
i
. The LWGM is obtained when the
ordered position of i is the same than j.
Following a similar methodology as it has been
developed in (Merigó, 2007; Yager, 1993), we could
study other particular cases of the LGOWA operator
such as the step-LGOWA, the window-LGOWA,
the olympic-LGOWA, the centered-LGOWA
operator, the S-LGOWA operator, the median-
LGOWA, the E-Z LGOWA, the maximal entropy
LGOWA weights, the Gaussian LOWA weights, the
minimal variability OWA weights, the nonmono-
tonic LGOWA operator, etc.
THE LINGUISTIC GENERALIZED OWA OPERATOR AND ITS APPLICATION IN STRATEGIC DECISION
MAKING
221
For example, if w
1
= w
n
= 0, and for all others w
j*
= 1/(n 2), we are using the olympic-LGOWA that
it is based on the olympic average (Yager, 1996).
Note that if n = 3 or n = 4, the olympic-LGOWA is
transformed in the median-LGOWA and if m = n 2
and k = 2, the window-LGOWA is transformed in
the olympic-LGOWA.
When w
j*
= 1/m for k j* k + m 1 and w
j*
=
0 for j* > k + m and j* < k, we are using the
window-LGOWA operator. Note that k and m must
be positive integers such that k + m 1 n.
Another interesting family is the S-LGOWA
operator based on the S-OWA operator (Yager,
1993; Yager and Filev, 1994). It can be subdivided
in three classes, the “orlike”, the “andlike” and the
generalized S-LGOWA operator. The “orlike” S-
LGOWA operator is found when w
1
= (1/n)(1
α
) +
α
, and w
j
= (1/n)(1
α
) for j = 2 to n with
α
[0,
1]. The “andlike” S-LGOWA operator is found
when w
n
= (1/n)(1
β
) +
β
and w
j
= (1/n)(1
β
) for
j = 1 to n 1 with
β
[0, 1]. Finally, the
generalized S-LGOWA operator is obtained when
w
1
= (1/n)(1 (
α
+
β
)) +
α
, w
n
= (1/n)(1 (
α
+
β
)) +
β
, and w
j
= (1/n)(1 (
α
+
β
)) for j = 2 to n 1 where
α
,
β
[0, 1] and
α
+
β
1. Note that if
α
= 0, the
generalized S-LGOWA operator becomes the
“andlike” S-LGOWA operator and if
β
= 0, it
becomes the “orlike” S-LGOWA operator.
4.2 Analysing the Parameter λ
If we analyze different values of the parameter
λ
, we
obtain another group of particular cases such as the
usual LOWA operator, the LOWG operator, the
LOWHA operator and the LOWQA operator. Note
that it is possible to distinguish between descending
and ascending orders in all the cases.
When
λ
= 1, we get the LOWA operator.
LGOWA(s
α
1
, …, s
α
n
) =
=
n
j
j
j
sw
1
β
(5)
Note that if w
j
= 1/n, for all a
i
, we get the LA and
if the ordered position of i = j, the LWA.
When
λ
= 0, we get the LOWG operator.
LGOWA(s
α
1
, …, s
α
n
) =
=
n
j
w
j
j
s
1
β
(6)
If w
j
= 1/n, for all a
i
, we get the linguistic
geometric average (LGA) and if i = j, for all a
i
, the
linguistic weighted geometric average (LWGA).
When
λ
= 1, we get the LOWHA operator.
LGOWA(s
α
1
, …, s
α
n
) =
=
n
j
j
j
s
w
1
1
β
(7)
Note that if w
j
= 1/n, for all a
i
, we get the
linguistic harmonic mean (LHM) and if i = j, for all
a
i
, the linguistic weighted harmonic mean (LWHM).
When
λ
= 2, we get the LOWQA operator.
LGOWA(s
α
1
, …, s
α
n
) =
2/1
1
2
=
n
j
j
j
sw
β
(8)
If w
j
= 1/n, for all a
i
, we get the linguistic LQA
and if i = j, for all a
i
, the linguistic weighted
quadratic mean (LWQM).
Note that we could analyze other families by
using different values in the parameter λ and study
these families individually.
5 QUASI-ARITHMETIC MEANS
IN THE LOWA OPERATOR
As it is explained in (Beliakov, 2005), a further
generalization of the GOWA operator is possible by
using quasi-arithmetic means. Following the same
methodology than (Fodor et al., 1995), we can
suggest a similar generalization of the LGOWA
operator by using quasi-arithmetic means. We will
call this generalization the Quasi-LOWA operator.
Note that this generalization is different than (Wang
and Hao, 2006) because it uses a different linguistic
approach. The Quasi-LOWA operator can be
defined as follows.
Definition 4. A Quasi-LOWA operator of dimension
n is a mapping QLOWA: S
n
S that has an
associated weighting vector W of dimension n such
that the sum of the weights is 1 and w
j
[0,1], then:
QLOWA(s
α
1
, …, s
α
n
) =
(
)
=
n
j
j
j
sgwg
1
1
β
(9)
where s
β
j
is the jth largest of the s
α
i
.
ICEIS 2008 - International Conference on Enterprise Information Systems
222
As we can see, we replace s
β
λ
with a general
continuous strictly monotone function g(s
β
). In this
case, the weights of the ascending and descending
versions are also related by w
j
= w*
n
j+1
, where w
j
is
the jth weight of the Quasi-DLOWA and w*
n
j+1
the
jth weight of the Quasi-ALOWA operator.
Note that all the properties and particular cases
commented in the LGOWA operator are also
included in this generalization. For example, we
could study different families of Quasi-LOWA
operators such as the Quasi-LA, the Quasi-LWA, the
Quasi-step-LOWA, the Quasi-window-LOWA, the
Quasi-olympic-LOWA, etc.
6 APPLICATION IN STRATEGIC
DECISION MAKING
In the following, we are going to develop a
numerical example about the use of the LGOWA
operator in a business decision making problem. We
will analyze a strategic decision making problem
where an enterprise is analysing which is the most
appropriate global strategy for them. We will
assume that they consider five alternatives for the
next period. As the environment is very uncertain,
the group of experts of the enterprise is not able to
use numerical information in the analysis. Instead,
they will use linguistic information. Note that other
decision making applications could be developed
with the LGOWA operator such as financial
decision making (Merigó, 2007), human resource
selection (Merigó, 2007), etc.
Assume an enterprise is analyzing its general
policy for the next year and they consider five
possible strategies to follow.
A
1
= Strategy 1.
A
2
= Strategy 2.
A
3
= Strategy 3.
A
4
= Strategy 4.
A
5
= Strategy 5.
In order to evaluate these strategies, the group of
experts considers that the key factor is the economic
situation of the company for the next year. After
careful analysis, the experts have considered five
possible situations that could happen in the future:
N
1
= Very bad, N
2
= Bad, N
3
= Regular, N
4
= Good,
N
5
= Very good. The linguistic expected results
depending on the situation N
i
and the alternative A
k
are shown in Table 1.
Table 1: Linguistic payoff matrix.
N
1
N
2
N
3
N
4
N
5
A
1
S
3
S
6
S
2
S
4
S
5
A
2
S
7
S
3
S
1
S
2
S
6
A
3
S
5
S
4
S
4
S
3
S
4
A
4
S
2
S
3
S
6
S
5
S
4
A
5
S
4
S
2
S
7
S
5
S
2
In this example, we assume that the group of
experts assumes the following weighting vector for
all the cases: W = (0.1, 0.2, 0.2, 0.2, 0.3). Note that
this weighting vector will be used as a weighted
average in the LWA, but for the LOWA, ALOWA,
LOWG and LOWQA, it will be used as the
attitudinal character of the enterprise.
With this information, we can aggregate it in
order to take a decision. First, we consider some
basic linguistic aggregation operators. The results
are shown in Table 2.
Table 2: Linguistic aggregated results 1.
Max Min LA LGA LQA
A
1
S
3
S
6
S
2
S
4
S
5
A
2
S
7
S
3
S
1
S
2
S
6
A
3
S
5
S
4
S
4
S
3
S
4
A
4
S
2
S
3
S
6
S
5
S
4
A
5
S
4
S
2
S
7
S
5
S
2
As we can see, the decision is different
depending on the aggregation operator used.
Now, we are going to consider the results
obtained by using other particular cases of LGOWA
operators such as the LWA, the LOWA, the
ALOWA, the LOWG and the LOWQA operator.
The results are shown in Table 3.
Table 3: Linguistic aggregated results 2.
LWA LOWA ALOWA LOWG LOWQ
A
1
S
3
S
6
S
2
S
4
S
5
A
2
S
7
S
3
S
1
S
2
S
6
A
3
S
5
S
4
S
4
S
3
S
4
A
4
S
2
S
3
S
6
S
5
S
4
A
5
S
4
S
2
S
7
S
5
S
2
As we can see, in this case we also get different
results depending on the aggregation operator used.
Note that more particular cases of the LGOWA
operator could be considered in the analysis such the
ones explained in the previous sections.
Another interesting issue is to establish an
ordering of the strategies. Note that this is useful
when we want to consider more than one strategy in
the analysis. The results are shown in Table 4.
THE LINGUISTIC GENERALIZED OWA OPERATOR AND ITS APPLICATION IN STRATEGIC DECISION
MAKING
223
Table 4: Ordering of the strategies.
Ordering
Max
A
2
=A
5
A
1
=A
4
A
3
Min
A
3
A
1
=A
4
=A
5
A
2
LA
A
1
=A
3
=A
4
=A
5
A
2
LGA
A
3
A
1
=A
4
A
5
A
1
LQA
A
2
A
5
A
1
A
4
A
3
LWA
A
1
A
4
A
3
A
5
A
2
LOWA
A
3
A
1
=A
4
A
5
A
2
ALOWA
A
5
A
1
=A
4
A
3
A
2
LOWG
A
3
A
1
=A
4
A
5
A
2
LOWQA
A
5
A
2
A
1
=A
3
=A
4
As we can see, depending on the linguistic
aggregation used, the ordering is different.
7 CONCLUSIONS
We have presented the LGOWA operator. It is an
aggregation operator that uses linguistic information
and generalized means in the OWA operator. We
have seen that this operator is very useful for
situations where the available information can not be
assessed with numerical values but it is possible to
use linguistic ones. We have studied some of its
main properties and we have found a wide range of
particular cases. We have seen that it is possible to
further generalize it by using quasi-arithmetic means
obtaining the Quasi-LOWA operator.
We have applied the new approach in a business
decision making problem. We have analyzed the
selection of strategies. We have seen that the results
and decisions are different depending on the
particular LGOWA operator used.
In future research, we expect to develop more
extensions of the LGOWA operator by introducing
more characteristics in the problem and applying it
in different business problems. For example, we
could mention the possibility of using different
linguistic approaches and the use of different
extensions of the OWA operator such as the induced
LGOWA operator or the hybrid LGOWA operator.
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