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

 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

= N, s

= VL, s

= L, s

= M,

= H, s

= VH, s

= 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

) = s

such that j =

g+1−i.

 The set is ordered: s

≤ s

if and only if i ≤ j.

 Max operator: Max(s

, s

) = s

if s

≥ s

 Min operator: Min(s

, s

) = s

if s

≤ s

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

| s

< s

≤ s

∈ [1, t]},

where, if s

∈ S, we call s

the original linguistic

term, otherwise, we call s

the virtual one.

Consider any two linguistic terms s

, s

∈ Ŝ, and

∈ [0, 1], we define some operational laws

as follows (Xu, 2004a; 2004b):



= s

μα

 s

⊕ s

= s

⊕ s

= s

 (s

)

= s

 s

⊗ s

= s

⊗ s

= 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

→ S, which has an associated

weighting vector W such that w

∈ [0, 1] and

∑

, then:

LOWA(s

, s

, …, s

) =

∑

(1)

where s

is the jth largest of the s

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

→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

∈ [0,1], then:

GOWA(a

, a

,…, a

) =

⎟

⎠

⎞

⎜

⎝

⎛

∑

(2)

where b

is the jth largest of the a

, and

is a

parameter such that

∈ (−∞, ∞).

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

→S that has an associated

weighting vector W of dimension n such that the

sum of the weights is 1 and w

∈ [0,1], then:

LGOWA(s

, …, s

) =

⎟

⎠

⎞

⎜

⎝

⎛

∑

(3)

where s

is the jth largest of the s

, 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

= w*

−

j+1

, where w

is the jth weight of the DLGOWA and w*

−

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

≥ s

, for all

, then, LGOWA(s

, …, s

) ≥ LGOWA(s

, …,

). It is bounded because the LGOWA aggregation

is delimitated by the minimum and the maximum:

Min{s

} ≤ LGOWA(s

, …, s

) ≤ Max{s

}. It is

idempotent because if s

= s

, for all s

, then,

LGOWA(s

, …, s

) = s

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:

(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

and w

= 0, for all j ≠ 1. The linguistic minimum is

obtained if w

= 1 and w

= 0, for all j ≠ n. More

generally, if w

= 1 and w

= 0, for all j ≠ k, we get

for any

, LGOWA(s

, …, s

) = b

, where b

is the

kth largest argument a

. The LGM is found when w

= 1/n, for all a

. 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

= w

= 0, and for all others w

= 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

= 1/m for k ≤ j* ≤ k + m − 1 and w

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/n)(1 −

) +

, and w

= (1/n)(1 −

) for j = 2 to n with

∈ [0,

1]. The “andlike” S-LGOWA operator is found

when w

= (1/n)(1 −

) +

and w

= (1/n)(1 −

) for

j = 1 to n − 1 with

∈ [0, 1]. Finally, the

generalized S-LGOWA operator is obtained when

= (1/n)(1 − (

)) +

, w

= (1/n)(1 − (

)) +

, and w

= (1/n)(1 − (

)) for j = 2 to n − 1 where

∈ [0, 1] and

≤ 1. Note that if

= 0, the

generalized S-LGOWA operator becomes the

“andlike” S-LGOWA operator and if

= 0, it

becomes the “orlike” S-LGOWA operator.

4.2 Analysing the Parameter λ

If we analyze different values of the parameter

, 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

= 1, we get the LOWA operator.

LGOWA(s

, …, s

) =

∑

(5)

Note that if w

= 1/n, for all a

, we get the LA and

if the ordered position of i = j, the LWA.

When

= 0, we get the LOWG operator.

LGOWA(s

, …, s

) =

∏

(6)

If w

= 1/n, for all a

, we get the linguistic

geometric average (LGA) and if i = j, for all a

, the

linguistic weighted geometric average (LWGA).

When

= −1, we get the LOWHA operator.

LGOWA(s

, …, s

) =

∑

(7)

Note that if w

= 1/n, for all a

, we get the

linguistic harmonic mean (LHM) and if i = j, for all

, the linguistic weighted harmonic mean (LWHM).

When

= 2, we get the LOWQA operator.

LGOWA(s

, …, s

) =

2/1

⎟

⎠

⎞

⎜

⎝

⎛

∑

(8)

If w

= 1/n, for all a

, we get the linguistic LQA

and if i = j, for all a

, 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

→ S that has an

associated weighting vector W of dimension n such

that the sum of the weights is 1 and w

∈ [0,1], then:

QLOWA(s

, …, s

) =

(

)

⎟

⎠

⎞

⎜

⎝

⎛

∑

−

sgwg

(9)

where s

is the jth largest of the s

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222

As we can see, we replace s

with a general

continuous strictly monotone function g(s

). In this

case, the weights of the ascending and descending

versions are also related by w

= w*

−

j+1

, where w

the jth weight of the Quasi-DLOWA and w*

−

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

= Strategy 1.

 A

= Strategy 2.

 A

= Strategy 3.

 A

= Strategy 4.

 A

= 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:

= Very bad, N

= Bad, N

= Regular, N

= Good,

= Very good. The linguistic expected results

depending on the situation N

and the alternative A

are shown in Table 1.

Table 1: Linguistic payoff matrix.

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

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

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

Min

⎬A

LGA

⎬A

LQA

⎬A

LWA

⎬A

LOWA

⎬A

ALOWA

⎬A

LOWG

⎬A

LOWQA

⎬A

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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