European Journal of Operational Research 203 (2010) 455–463

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

The OWA-based consensus operator under linguistic representation modelsusing position indexes

Yucheng Dong a,b,*, Yinfeng Xu a,b, Hongyi Li c, Bo Feng d

a Management School, Xi’an Jiaotong University, Xi’an 710049, PR Chinab State Key Lab. for Manufacturing Systems Engineering, Xi’an 710049, PR Chinac Faculty of Business Administration, The Chinese, University of Hong Kong, Shatin, N.T., Hong Kongd School of Business Administration, South China University of Technology, Guangzhou 510640, PR China

a r t i c l e i n f o

Article history:Received 25 March 2008Accepted 19 August 2009Available online 22 August 2009

Keywords:Decision analysisLinguistic representation modelOWA operatorDeviation measureConsensus

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.08.013

* Corresponding author. Address: Management SchXi’an 710049, PR China. Tel.: +86 2982665034.

E-mail address: ycdong@mail.xjtu.edu.cn (Y. Dong

a b s t r a c t

When using linguistic approaches to solve decision problems, we need linguistic representation models.The symbolic model, the 2-tuple fuzzy linguistic representation model and the continuous linguisticmodel are three existing linguistic representation models based on position indexes. Together with thesethree linguistic models, the corresponding ordered weighted averaging operators, such as the linguisticordered weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extendedordered weighted averaging operator, have been developed, respectively. In this paper, we analyze theinternal relationship among these operators, and propose a consensus operator under the continuous lin-guistic model (or the 2-tuple fuzzy linguistic representation model). The proposed consensus operator isbased on the use of the ordered weighted averaging operator and the deviation measures. Some desiredproperties of the consensus operator are also presented. In particular, the consensus operator provides analternative consensus model for group decision making. This consensus model preserves the originalpreference information given by the decision makers as much as possible, and supports consensus pro-cess automatically, without moderator.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The ordered weighted averaging (OWA) operator was intro-duced by Yager (1988). Chiclana et al. (1998, 2001, 2002, 2003,2007), Herrera et al. (2001), Herrera-Viedma et al. (2002) and Donget al. (2008a) studied a group decision making (GDM) model, usingthe OWA operator guided by a relative linguistic quantifier. How-ever, there are some situations in which the information may notbe quantified because of its nature. A number of studies recently fo-cused on GDM under linguistic environment (Ben-Arieh and Chen,2006; Chen and Ben-Arieh, 2006; Delgado et al., 1993, 1998; Donget al., 2009; García-Lapresta et al., 2009; Herrera et al., 1995, 1996b,2005; Herrera and Herrera-Viedma, 2000; Herrera and Martínez,2000; Kacprzyk, 1986; Wang and Hao, 2006; Xu, 2004a,b,c,d).When using linguistic approaches to solve decision problems, weneed linguistic representation models. There are two types of lin-guistic representation models, i.e., the semantic model (Deganiand Bortolan, 1988) and the model using position indexes (Delgadoet al., 1993; Herrera and Martínez, 2000, 2001; Wang and Hao,

ll rights reserved.

ool, Xi’an Jiaotong University,

).

2006; Xu, 2004b,d, 2005). The semantic model is based on theextension principle, and makes operations on the fuzzy numbersthat support the semantics of the linguistic terms. Together withthe semantic model, Zhou et al. (2008) recently presented thetype-1 OWA operators for aggregating uncertain information withuncertain weights induced by type-2 linguistic quantifiers. Themodels using position indexes make computations on the positionindexes of the linguistic terms. At present, there are three linguisticrepresentation models using position indexes:

(1) the symbolic model (Delgado et al., 1993),(2) the 2-tuple fuzzy linguistic representation model (Herrera

and Martínez, 2000, 2001), and(3) the continuous linguistic model (Xu, 2004b,d, 2005).

Under the symbolic model, Herrera et al. (1996b) presentedthe linguistic ordered weighted averaging (LOWA) operator.Ben-Arieh and Chen (2006) developed fuzzy LOWA operator (FLO-WA), based on the LOWA operator. To avoid the information lossof the symbolic model, Herrera and Martínez (2000) furtherproposed a 2-tuple fuzzy linguistic representation model, anddeveloped a 2-tuple ordered weighted averaging (TOWA) opera-tor. Xu (2004d, 2005) presented the concept of the continuous

456 Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463

linguistic sets and developed the extended ordered weightedaveraging (EOWA) operator. In this paper, we called Xu’s methodthe continuous linguistic model. Xu (2004a,c, 2006) also studied theuncertain linguistic aggregation operators, based on the continu-ous linguistic model.

The consensus problem is a very important problem in GDM (Ben-Arieh and Chen, 2006; Bordogna et al., 1997; Chiclana et al., 2008;Dong et al., 2008b; Fan and Chen, 2005; Herrera et al., 1996a; Herre-ra-Viedma et al., 2002, 2005, 2007; Kacprzyk et al., 1992, 1997; Xu,2005). Generally, at the beginning of every GDM problem, decisionmakers’ opinions may differ substantially. Therefore, it is necessaryto develop a consensus process in an attempt to obtain a solution ofconsensus. For GDM using linguistic preference relations, Herreraet al. (1996a) introduced a consensus model, which is based on theuse of fuzzy majority of consensus, represented by means of a linguis-tic quantifier. Herrera-Viedma et al. (2005) further developed a con-sensus support system model for GDM with multi-granular linguisticpreference relations. Xu (2005) defined the concepts of deviationmeasure to characterize the difference between linguistic preferencerelations. Dong et al. (2008b) further developed a consistency mea-sure method for individual linguistic preference relations, definingthe deviation measure of a linguistic preference relation to the setof consistent linguistic preference relations.

The purpose of this paper is to analyze the internal relation-ship among the LOWA operator, the TOWA operator and theEOWA operator, and to develop a new linguistic operator toaggregate linguistic variables, under the continuous linguisticmodel (or the 2-tuple fuzzy linguistic representation model). Thisoperator is based on the use of the OWA operator and the devia-tion measure, and provides an alternative consensus model forGDM using linguistic information. This consensus model pre-serves the original preference information given by the decisionmakers as much as possible, and supports consensus processautomatically, without moderator. The rest of this paper is orga-nized as follows: in Section 2, we summarize some existing OWA-based linguistic aggregation operators using position indexes (i.e.,LOWA, TOWA and EOWA), analyze the internal relationshipbetween the TOWA operator and the EOWA operator, and discussthe information loss of the LOWA operator. In Section 3, theOWA-based consensus operator under the continuous linguisticmodel (or the 2-tuple fuzzy linguistic representation model) isformally proposed. In Section 4, some desired properties of theconsensus operator are shown. In Section 5, three illustrativeexamples are provided. Concluding remarks and future researchagenda are included in Section 6.

2. Several OWA-based operators under linguistic computationalmodels using position indexes

The basic notations and operational laws of linguistic variablesare introduced (see Herrera and Martínez, 2000, 2001; Xu, 2004b,d,2005). Let S ¼ fsaja ¼ �t; . . . ;�1;0;1; . . . ; tg be a linguistic label setwith odd cardinality. The label sa represents a possible value for alinguistic variable. It is required that the linguistic label set shouldsatisfy the following characteristics:

(1) The set is ordered: sa > sb if and only if a > b;(2) There is a negation operator: neg ðsaÞ ¼ s�a.

We call this linguistic label set S the linguistic scale. For exam-ple, S can be defined as:

S ¼ fs�4 ¼ extremely poor; s�3 ¼ very poor; s�2 ¼ poor

s�1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good

s2 ¼ good; s3 ¼ very good; s4 ¼ extremely goodg:

Let s 2 S. We denote I(s) the position index (or lower index) of sin S. For example, if s ¼ sa, then IðsÞ ¼ a. As mentioned above, thesymbolic model, the 2-tuple fuzzy linguistic representation modeland the continuous linguistic model make computations on the po-sition indexes of the linguistic labels. Together with these three lin-guistic representation models, some OWA-based linguisticaggregation operators have been proposed to solve GDM problem.Now, we introduce several interesting OWA-based linguisticaggregation operators, i.e., the LOWA, TOWA and EOWA operators,analyze the internal relationship among them.

2.1. The LOWA operator, the TOWA operator and the EOWA operator

2.1.1. The LOWA operatorThe LOWA operator, presented in Herrera et al. (1996b), is

based on the OWA operator and the convex combination of linguis-tic labels defined by Delgado et al. (1993). Let fa1; . . . ; amg be a setof linguistic variables to aggregate, and then the LOWA operator isdefined as follows:

LOWAwða1; . . . ;amÞ ¼ Cmfwk;sbk;k¼ 1;2; . . . ;mg

¼w1� sb1�ð1�w1Þ�Cm�1fkh;sbh;h¼ 2; . . . ;mg;

where w ¼ fw1;w2; . . . ;wmg is a weighting vector, such that wi P 0and

Pmi¼1wi ¼ 1; kh ¼ wh=

Pmk¼2wk;h ¼ 2; . . . ;m and sbk

is the kthlargest variable in fa1; . . . ; amg. Cm is the convex combination oper-ator of m variables and if m ¼ 2, then it is defined as

C2fwi; bi; i ¼ 1;2g ¼ w1 � sj�ð1�w1Þ � si ¼ sk; sj; si 2 S ðj P iÞ

such that k ¼ minft; iþ roundðw1ðj� iÞÞg, where round is the usualround operation, and b1 ¼ sj; b2 ¼ si.

2.1.2. The TOWA operatorIn order to avoid information loss in aggregating the linguistic

variables, Herrera and Martínez (2000) presented TOWA operator,which is based on their interesting 2-tuple fuzzy linguistic repre-sentation model. Let S ¼ fsaja ¼ �t; . . . ;�1;0;1; . . . ; tg be a linguis-tic label set and y 2 ½�t; t� a value representing the result of asymbolic aggregation operation, then the 2-tuple that expressesthe equivalent information to y is obtained by the followingfunction:

D : ½�t; t� ! S� ½�0:5;0:5Þ

DðyÞ ¼ ðsi; xÞ;withsi; i ¼ roundðyÞx ¼ y� i; x 2 ½�0:5; 0:5Þ

�:

Let r ¼ fðr1; x1Þ; . . . ; ðrm; xmÞg be a set of 2-tuples andfw1;w2; . . . ;wmg be an associated weighting vector that satisfieswi P 0 and

Pmi¼1wi ¼ 1. The TOWA operator for linguistic 2-tuples

is computed as

TOWAwððr1; x1Þ; . . . ; ðrm; xmÞÞ ¼ DXm

i¼1

wiy�i

!;

where y�i is the ith largest of the yi values, and yi ¼ D�1ððri; xiÞÞ.

2.1.3. The EOWA operatorsXu (2004b, 2005) extended the discrete linguistic label set S to a

continuous linguistic label set S ¼ fsaja 2 ½�q; q�g, where qðq P tÞis a sufficiently large positive integer. If sa 2 S, then we call sa theoriginal linguistic label; otherwise, we call it the virtual linguisticlabel. In general, the decision maker uses the original linguistic la-bels to evaluate alternatives, and the virtual linguistic labels canonly appear in operations.

Considering any two linguistic terms sa; sb 2 S, and l;l1;

l2 2 ½0;1�, Xu (2004b, 2005) introduced some new operationallaws as follows:

Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463 457

(1) sa � sb ¼ saþb;

(2) sa � sb ¼ sb � sa;

(3) lsa ¼ sla;

(4) ðl1 þ l2Þsa ¼ l1sa � l2sa;

(5) lðsa � sbÞ ¼ lsa � lsb

Let fa1; . . . ; amg be a set of linguistic variables to aggregate andfw1;w2; . . . ;wmg be a weighting vector, such that wi P 0 andPm

i¼1wi ¼ 1, then the EOWA operator is defined as follows:

EOWAwða1; . . . ; amÞ ¼ w1sb1 �w2sb2 � . . .�wnsbm¼ s�b;

where b ¼Pm

k¼1wkbk, and sbkis the kth largest variable in

fa1; . . . ; amg.Let {a1,. . ., ai,. . ., an} be a set of linguistic variables to aggre-

gate, where ai 2 S. Let sc be the result of aggregating fa1; . . . ; amgusing EOWAw, and ðrc; xcÞ be the result of aggregating fða1;0Þ; . . . ;

ðam;0Þgusing TOWAw. Dong et al. (2009) showed that

D�1ððrc; xcÞÞ ¼ IðscÞ;

Furthermore, let ðsi; xÞ, where si 2 S, be a 2-tuple linguistic label,then the corresponding continuous linguistic label is siþx 2 S. In thisway,

D�1ðsi; xÞ ¼ IðsiþxÞ:

Theorem 1. Let fa1; . . . ; amg be a set of linguistic variables toaggregate, where ai 2 S, and Let ðri; xiÞ be the 2-tuples correspondingto ai. Let fw1;w2; . . . ;wmg be an associated weighting vector thatsatisfies wi P 0 and

Pmi¼1wi ¼ 1. Then

D�1ðTOWAwððr1; x1Þ; . . . ; ðrm; xmÞÞÞ ¼ IðEOWAwða1; . . . ; amÞÞ:

Proof. Since ðri; xiÞ is the 2-tuples corresponding to ai, we havethat D�1ðri; xiÞ ¼ IðaiÞ. Let y�i be the ith largest of the yi values,where yi ¼ D�1ðri; xiÞ ¼ IðaiÞ, then we have that

D�1ðTOWAwððr1; x1Þ; . . . ; ðrm; xmÞÞÞ ¼Xm

i¼1

wiy�i ð1Þ

and

IðEOWAwða1; . . . ; amÞÞ ¼Xm

i¼1

wiy�i : ð2Þ

Consequently, D�1ðTOWAwððr1; x1Þ; . . . ; ðrm; xmÞÞÞ ¼ IðEOWAw

ða1; . . . ; amÞÞ. This completes the proof of Theorem 1. h

Theorem 1 is a generalization of the result presented in Donget al. (2009). It further shows that the EOWA operator has similar-ity to the TOWA operator. The main difference between these twooperators is using different representation format. In the rest ofthis paper, we adopt the EOWA operator and the correspondingrepresentation format (i.e., the continuous linguistic label set).

Yager (1996) suggested a notable way to compute the weights(i.e., wk; k ¼ 1;2; . . . ;m) of the OWA-based operators using linguis-tic quantifiers (Zadeh, 1983), which, in the case of a non-decreasingproportional quantifier Q, is given by

wk ¼ Qðk=mÞ � Qððk� 1Þ=mÞ; 8k

where the membership function of a non-decreasing proportionalquantifier Q is given as follows:

QðxÞ ¼0; 0 6 x < a;x�ab�a ; a 6 x 6 b;1; b < x 6 1;

8><>:

with a; b 2 ½0;1�.

In general, we denote [a, b] a fuzzy linguistic quantifier. When afuzzy linguistic quantifier Q is used to compute the weights of theEOWA operator, it is symbolized by EOWAQ .

2.2. The concepts of deviation measures

Let sa; sb 2 S be two linguistic variables. Xu (2005) defined thedeviation measure between sa and sb as follows:

dðsa; sbÞ ¼ja� bj

T; ð3Þ

where T is the number of linguistic terms in the set S.The deviation measure between sa and sb has a definite physical

implication and reflects the deviation degree between sa and sb.Obviously, the smaller the value of dðsa; sbÞ, the more similar thesetwo linguistic variables. If dðsa; sbÞ ¼ 0, then sa and sb have no dif-ference. Here, the deviation measure between two sets of linguisticvariables are defined below.

Definition 1. Let a ¼ fa1; . . . ; amg and b ¼ fb1; . . . ; bmg be two setsof linguistic variables. We define the deviation measure between aand b as follows:

dða; bÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1m

Xm

i¼1

ðdðai; biÞÞ2vuut : ð4Þ

Dong et al. (2008b) also defined the deviation measure betweentwo linguistic preference relations:

Definition 2. Let A ¼ ðaijÞn�n and B ¼ ðbijÞn�n be two linguisticpreference relations. We define the deviation measure between Aand B as follows:

dðA; BÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

nðn� 1ÞXn

j¼iþ1

Xn

i¼1

ðdðaij; bijÞÞ2vuut : ð5Þ

2.3. The information loss of the LOWA operator

Herrera and Martínez (2000) and Ben-Arieh and Chen (2006)pointed out that the LOWA operator loses some useful informationin the aggregation process. When using the deviation measure be-tween LOWAwða1; . . . ; amÞ and EOWAwða1; . . . ; amÞ, that is dðLOWAw

ða1; . . . ; amÞ;EOWAwða1; . . . ; amÞÞ, to measure the information loss ofthe LOWA operator, we can provide an upper bound for this infor-mation loss.

Theorem 2. Let fa1; . . . ; amg be a set of linguistic variables toaggregate and fw1;w2; . . . ;wmg be a weighting vector, such thatwi P 0 and

Pmi¼1wi ¼ 1, then we have

dðLOWAwða1; . . . ; amÞ; EOWAwða1; . . . ; amÞÞ 6m� 1

2T:

Proof. By induction over the number of arguments to aggregate,the proof can be completed. Without loss of generality, supposethat ai P aj for i > j. For m = 2, let sp; sq be the labels correspondingto LOWAwða1; a2Þ and EOWAwða1; a2Þ, then we have

p ¼ minft; Iða1Þ þ roundðw1ðIða2Þ � Iða1ÞÞÞg; ð6Þ

and

q ¼ w1Iða2Þ þ ð1�w1ÞIða1Þ: ð7Þ

Consequently,

dðLOWAwða1; a2Þ; EOWAwða1; a2ÞÞ ¼1Tjp� qj 6 1

2T: ð8Þ

458 Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463

Suppose that it is true for m-1, i.e.,

dðLOWAwða1; . . . ; am�1Þ; EOWAwða1; . . . ; am�1ÞÞ 6m� 2

2T: ð9Þ

For m, LOWAwða1; . . . ; amÞ ¼ w1 � a1�ð1�w1Þ � Cm�1fkh; ah;h ¼2; . . . ;mg. Let sx ¼ Cm�1fkh; ah;h ¼ 2; . . . ;mg and sy ¼ k2a2 � . . .�khah � . . . kmam. Using induction hypothesis, we obtain

dðsx; syÞ ¼1Tjy� xj 6 m� 2

2T: ð10Þ

Since

LOWAwða1; . . . ; amÞ ¼ w1 � a1�ð1�w1Þ � sx ð11Þ

and

EOWAwða1; . . . ; amÞ ¼ w1a1�; . . . ;�wiai � . . .�wmam

¼ w1a1 � ð1�w1Þsy; ð12Þ

we have that

dðLOWAwða1; . . . ; amÞ;EOWAwða1; . . . ; amÞÞ¼ dðw1 � a1�ð1�w1Þ � sx;w1a1 � ð1�w1Þsx � ð1�w1Þsy�xÞ

¼ 1TjIðw1 � a1�ð1�w1Þ � sxÞ � Iðw1a1 � ð1�w1ÞsxÞ

� ð1�w1ÞIðsy�xÞj 61TðjIðw1 � a1�ð1�w1Þ � sxÞ

� Iðw1a1 � ð1�w1ÞsxÞj þ jy� xjÞ: ð13Þ

By (8) and (10), we have that

dðLOWAw ða1; . . . ; amÞ; EOWAwða1; . . . ; amÞÞ 6m� 1

2T: ð14Þ

This completes the proof of Theorem 2.

Remark 1. In this section, we analyze the internal relationshipbetween the TOWA operator and the EOWA operator, and providean upper bound for the information loss of the LOWA operator.These operators are on the basis of position indexes of linguisticlabels, and only suitable for dealing with linguistic term sets thatare uniformly and symmetrically distributed. Recently, Wang andHao (2006), Herrera et al. (2008) focused on methodologies to dealwith label sets that are not uniformly and symmetrically distrib-uted (i.e., unbalanced linguistic label sets).

3. The OWA-based consensus operator under position indexes

In GDM under linguistic environment, individual decision mak-ers provide a set of linguistic variables fa1; . . . ; amg to representtheir own opinions on an alternative. Using the OWA-based oper-ator to aggregate the linguistic variables fa1; . . . ; amg, we obtain thecollective linguistic variable ac. By calculating dðai; acÞ; i ¼1;2; . . . ;m, we can measure the deviation degree between the indi-vidual opinions and the collective opinion. When max

idðai; ;acÞ > b,

where b is the established threshold, we argue that the decisionmakers’ opinions have unacceptable consensus degree, and suggestto adjust the original linguistic variables to reach consensus. In thissection, we propose an OWA-based linguistic operator to reachconsensus for GDM under the continuous linguistic model (or the2-tuple fuzzy linguistic representation model).

In fact, the main work of reaching consensus is to find a set ofadjusted linguistic variables with an acceptable consensus level,�a ¼ fa1; . . . ; amg. In order to preserve the information ina ¼ fa1; a2; . . . ; amg as much as possible, we hope that the deviationdegree between a and �a is minimal, namely,

min�a

dða; �aÞ; ð15Þ

i.e.,

min�a

1mT2

Xm

i¼1

ðIðaiÞ � IðaiÞÞ2 !

: ð16Þ

At the same time, �a ¼ fa1; . . . ; amg has the acceptable consensuslevel, that is

maxi

dðai; acÞ 6 b; ð17Þ

i.e.,

dðai;EOWAwða1; . . . ; amÞÞ 6 b; i ¼ 1;2; . . . m: ð18Þ

In this way, by decreasing the deviation degree betweenai; ði ¼ 1;2; . . . ;mÞ and EOWAwða1; . . . ; amÞ, an optimization modelto reach consensus can be constructed as follows:

min�a

1mT2

Pmi¼1ðIðaiÞ � IðaiÞÞ2

� �s:t: dðai;EOWAwða1; . . . ; amÞÞ 6 b; i ¼ 1;2; . . . m

8<: : ð19Þ

Note 1. We use maxi

dðai; acÞ to define the consensus level ina ¼ fa1; a2; . . . ; amg. If max

idðai; acÞ 6 b, then we consider

a ¼ fa1; a2; . . . ; amg acceptable consensus; otherwise, we considera unacceptable consensus. Similarly, we may consider using1m

Pmi¼1dðai; acÞ 6 h (in general, h < b) to define the acceptable con-

sensus level. In our future research, we will discuss this consensusdefinition.

Let xi ¼ IðaiÞ. We denote frað1Þ; . . . ;raðiÞ; . . . ;raðmÞg is a per-mutation of f1;2; . . . ;mg such that araði�1Þ P araðiÞ for all i ¼ 2;. . . ;m. (i.e., araðiÞ is the ith largest variable in a. Letfrxð1Þ;...;rxðiÞ;...;rxðmÞg is another permutation of f1;2; . . . ;mgsuch that xrxði�1Þ P xrxðiÞ for all i ¼ 2; . . . ;m. Model (19) can be rede-scribed as follows:

minx

1mT2

Pmi¼1ðIðaiÞ � xiÞ2

� �

s:t: 1T jxi �

Pmi¼1

wixrxðiÞj 6 b; i ¼ 1;2; . . . m

8>><>>: : ð20Þ

We denote model (20) as P1. Solving P1, we obtainai ¼ I�1ðxiÞ; i ¼ 0;1; . . . ;m. Before presenting the approach to solveP1, we introduce a new model:

minx

1mT2

Pmi¼1ðIðaiÞ � xiÞ2

� �

s:t: 1T jxi �

Pmi¼1

wixrxðiÞj 6 b; i ¼ 1;2; . . . m

xraðiÞ � xraði�1Þ 6 0; i ¼ 2; . . . ;m

8>>>>><>>>>>:

: ð21Þ

We denote model (21) as P2.

Lemma 1. P2 is a strictly convex quadratic programming.

Proof. The constraint condition xraðiÞ � xraði�1Þ 6 0ði ¼ 2; . . . ;mÞguarantees that rxðiÞ ¼ raðiÞ. Consequently,

1Tjxi �

Xm

i¼1

wixrxðiÞj ¼1Tjxi �

Xm

i¼1

wixraðiÞj: ð22Þ

Based on (22), P2 can be redescribed as follows:

minx

1mT2

Pmi¼1ðIðaiÞ � xiÞ2

� �

s:t: xi �Pmi¼1

wixraðiÞ 6 Tb; i ¼ 1;2; . . . m

Pmi¼1

wixraðiÞ � xi 6 Tb; i ¼ 1;2; . . . m

xraðiÞ � xraði�1Þ 6 0; i ¼ 2; . . . ;m

8>>>>>>>>>><>>>>>>>>>>:

ð23Þ

Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463 459

Let f ðxÞ ¼ 1mT2

Pmi¼1ðIðaiÞ � xiÞ2. We prove that f ðxÞ is a strictly

convex function. Moreover, raðiÞ completely depends on a, so theconstraint conditions of P2 are linear. Hence, P2 is a strictly convexquadratic programming. This completes the proof of Lemma 1. h

It is well known that the optimal solution to the strictly convexquadratic programming exists and is unique, and can be effectivelysolved by some classical methods such as Lemke algorithm, etc. Inthe following, we present a good linkage of P2 to P1. Let X1 and X2

be the feasible sets corresponding to P1 and P2. Naturally,

X1 � X2: ð24Þ

Lemma 2. Let ap; aq 2 a, and a ¼ fa1; a2; . . . ; amg be the optimal

solution to P1. Then, ap P aq for ap > aq.

Proof. Using reduction to absurdity, we assume that there exist pand q, such that ap > aq and ap < aq. Let a ¼ fa1; . . . ; amg, where

ai ¼aq; for i ¼ p

ap; for i ¼ q

ai; for i – p; q

8><>: :

Since dðai;EOWAwða1; . . . ; amÞÞ 6 b, we find that dðai;EOWAw

ða1; . . . ; amÞÞ 6 b: So, a 2 X1. Moreover, we have that

ðdða; aÞÞ2 � ðdða; aÞÞ2 ¼ 1mT2 ððIðapÞ � IðapÞÞ2 þ ðIðaqÞ

� IðaqÞÞ2 � ðIðapÞ � IðapÞÞ2 � ðIðaqÞ

� IðaqÞÞ2Þ: ð25Þ

Since ap ¼ aq and aq ¼ ap, From (25), we obtain that

ðdða; �aÞÞ2 � ðdða; ��aÞÞ2 ¼ 1mT2 ðIðaqÞ � IðapÞÞðIðapÞ � IðaqÞÞ > 0: ð26Þ

Consequently,

dða; �aÞ > dða; ��aÞ; ð27Þ

which contradicts the fact that �a is the optimal solution to P1. Thiscompletes the proof of Lemma 2. h

Lemma 3. Let ap; aq 2 a and ap ¼ aq. If a ¼ fa1; a2; . . . ; amg is theoptimal solution to P1, then ��a ¼ fa1; a2; . . . ; amg is the optimal solutionto P1, where

ai ¼aq; for i ¼ p

ap; for i ¼ q

ai; for i – p; q

8><>: :

Proof. Since EOWAwða1; a2; . . . ; amÞ ¼ EOWAwða1; a2; . . . ; amÞ, wehave that

maxi

dðai;EOWAwða1; a2; . . . ; amÞÞ

¼ maxi

dðai;EOWAwða1; a2; . . . ; amÞÞ 6 b: ð28Þ

Therefore, a 2 X1. Moreover, when ap ¼ aq, we have thatdða; aÞ ¼ dða; aÞ ¼ min

x2X1

dða; xÞ. Consequently, a ¼ fa1; a2; . . . ; amg is

the optimal solution to P1. This completes the proof of Lemma 3. h

Theorem 3. If a ¼ fa1; a2; . . . ; amg is the optimal solution to P2, thena ¼ fa1; a2; . . . ; amg is the optimal solution to P1.

Proof. Let a� ¼ fa�1; a�2; . . . ; a�mg be an optimal solution to P1. From(24), we have that

dða; a�Þ ¼ minx2X1

dða; xÞ 6 minx2X2

dða; xÞ ¼ dða; �aÞ: ð29Þ

Here, we consider two cases:

Case A: ap–aq for any ap; aq 2 a. In this case, Lemma 2 guaranteesthat a�raðiÞ � a�raði�1Þ 6 0, so a� 2 X2. Consequently,

dða; a�ÞP minx2X2

dða; xÞ ¼ dða; �aÞ: ð30Þ

From (29) and (30), we obtain that dða; a�Þ ¼ dða; �aÞ: The optimalsolution of P2 exists and is unique, so �a ¼ a�.Case B: 9 ap; aq 2 a and ap ¼ aq. Without loss of generality, wesuppose that ai–aj for any ai; aj 2 a=fapg. In this case, Lemma 3guarantees a� ¼ fa�1; a�2; . . . ; a�mg also the optimal solution to P1,where

a�i ¼a�q; for i ¼ p

a�p; for i ¼ q

a�i ; for i – p; q

8><>: :

From Lemma 2, we also know that if ai 2 a=fap; aqg is the kth largestvariable in a, then a�i =a�i is the kth largest variable in a�=a�. Thus,a� 2 X2 or a� 2 X2. Without loss of generality, we suppose thata� 2 X2. Similar to Case A, we have that a ¼ a�. This completes theproof of Theorem 3. h

Theorem 4. Let að1Þ ¼ fað1Þ1 ; . . . ; að1Þm g, að2Þ ¼ fað2Þ1 ; . . . ; að2Þm g be twooptimal solutions to P1, and let a ¼ fa1; . . . ; amg be the optimal solu-tion to P2. Then,

EOWAwðað1Þ1 ; . . . ; að1Þm Þ ¼ EOWAwðað2Þ1 ; . . . ; að2Þm Þ¼ EOWAwða1; . . . ; amÞ:

Proof. We consider two cases:

Case A: ap–aq for any ap; aq 2 a. In this case, from the proof ofTheorem 3, we can find that �a ¼ að1Þ and �a ¼ að2Þ. This means thatEOWAwðað1Þ1 ; . . . ; að1Þm Þ ¼ EOWAwðað2Þ1 ; . . . ; að2Þm Þ ¼EOWAwða1; . . . ; amÞ.

Case B: 9 ap; aq 2 a and ap ¼ aq. In this case, according toLemma 3, we can construct two optimal solutions að1;�Þ; að2;�Þ to P1,satisfying the following conditions:

EOWAwðaðk;�Þ1 ; . . . ; aðk;�Þm Þ ¼ EOWAwðaðkÞ1 ; . . . ; aðkÞm Þ; k ¼ 1;2: ð31Þ

aðk;�ÞraðiÞ � aðk;�Þraði�1Þ 6 0; k ¼ 1;2: ð32Þ

The Eq. (32) guarantees that að1;�Þ; að2;�Þ 2 X2. Similar to the proofof Theorem 3, we have that að1;�Þ ¼ að2;�Þ ¼ �a. Consequently,

EOWAwðað1;�Þ1 ; . . . ; að1;�Þm Þ ¼ EOWAwðað2;�Þ1 ; . . . ; að2;�Þm Þ¼ EOWAwða1; . . . ; amÞ: ð33Þ

From (31), we obtain that

EOWAwðað1Þ1 ; . . . ; að1Þm Þ ¼ EOWAwðað2Þ1 ; . . . ; að2Þm Þ¼ EOWAwða1; . . . ; amÞ: ð34Þ

This completes the proof of Theorem 4. h

The implementation of the proposed consensus reaching modelrequires the following two-step procedure.

Step 1: Solve the optimization model P1 to obtain�a ¼ fa1; a2; . . . ; amg.

Step 2: Obtain the collective linguistic variable ac , by aggregat-ing �a using the EOWA operator.

Note 2. Let S be a linguistic label set. Theorem 4 shows that theproposed consensus reaching model is a mapping: Sn ! S, in es-sence. Thus, we call the proposed consensus reaching model asthe OWA-based consensus operator. When the weighting vector

460 Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463

is w and the established deviation threshold is b, we denote theOWA-based consensus operator as Cb

w.Note 3. Theorem 3 guarantees that the optimal solution to P2 is

also one to P1. Therefore, in the rest of this paper, we replace P1

with P2 in the implementation of the consensus operator.

Theorem 5. Cbwða1; a2; . . . ; amÞ ¼ EOWAwða1; a2; . . . ; amÞ under the

condition that dðai;EOWAða1; a2; . . . ; amÞÞ 6 bði ¼ 1;2; . . . ;mÞ.

Proof. When dðai;EOWAða1; a2; . . . ; amÞÞ 6 bði ¼ 1;2; . . . ;mÞ, theoptimal solution to P1 (or P2) is a. Thus, Cb

wða1; a2; . . . ; amÞ ¼EOWAwða1; a2; . . . ; amÞ. This completes the proof of Theorem 5. h

Remark 2. The consensus operator is a generalization of the OWAoperator. It provides an alternative consensus model for GDM. Itcan preserve the original preference information given by the deci-sion makers as much as possible, and support consensus processautomatically, without moderator. The operations of the EOWAoperator are based on position indexes. Therefore we can similarlydevelop a consensus operator under numerical environment.

4. Properties of the consensus operator

When using Cbw to respectively aggregate a ¼ fa1; a2; . . . ; amg and

b ¼ fb1; b2; . . . ; bmg, let �a ¼ fa1; a2; . . . ; amg and b ¼ fb1; b2; . . . ; bmgbe two sets of the corresponding linguistic variables to be adjusted.Before proposing some desired properties of the consensus opera-tor Cb

w, we first introduce Lemma 4 and Lemma 5.

Lemma 4. minfaigP minfaig and maxfaig 6 maxfaig.

Proof. We first prove that minfaigP minfaig. Using reduction toabsurdity, we assume that minfaig < minfaig. Without loss of gen-erality, we also assumed that P f1;2; . . . ;mg and ap < minfaig forp 2 P. Let �a ¼ fa1; . . . ; amg, where

ai ¼minfaig; for i 2 P

ai; for i2P

�; ð35Þ

we find that

dðEOWAwða1; . . . ; amÞ;maxðaiÞÞ 6 b; ð36Þ

and

dðEOWAwða1; . . . ; amÞ;minðaiÞÞ 6 b: ð37Þ

Thus, �a is a feasible solution to P1. Since

ðdða;�aÞÞ2�ðdða;�aÞÞ2¼ 1T2m

Xi2P

ðIðaiÞ� IðaiÞÞðIðaiÞþ IðaiÞ�2IðaiÞÞ<0;

ð38Þ

we have that dða; �aÞ � dða; �aÞ < 0, which contradicts the fact that �a isthe optimal solution of P1. Similarly, we prove thatmaxfaig 6 maxfaig. This completes the proof of Lemma 4. h

From Lemma 4, we can obtain Corollary 1.

Corollary 1. dðminfaig;maxfaigÞ 6 dðminfaig;maxfaigÞ.

Lemma 5. If ai � bi ¼ s0 and wi ¼ wmþ1�i for all i ¼ 1; . . . ;m, thenai � bi ¼ s0.

Proof. When using Cbw to aggregate fa1; a2; . . . ; amg=fb1; b2; . . . ; bmg,

let Xa=Xb be the feasible sets corresponding to P2. When

�a 2 Xa; ð39Þwe find that

��a 2 Xb ð40Þ

under condition that wi ¼ wiþm�1. Since

dða; �aÞ ¼ minx2Xa

dða; xÞ: ð41Þ

we have that

dðb;��aÞ ¼ minx2Xb

dðb; xÞ: ð42Þ

under the condition wi ¼ wiþm�1.Based on (40) and (42), we have that bi ¼ �ai, that is

ai � bi ¼ s0. This completes the proof of Lemma 5. h

In the following, we introduce some desired properties of theconsensus operator Cb

w.

Property 1. For any consensus operator Cbw

minfaig 6 Cbwða1; a2; . . . ; amÞ 6 maxfaig:

Proof. From the implementation of the consensus operator, wehave that

minfaig 6 Cbwða1; a2; . . . ; amÞ ¼ EOWAwða1; a2; . . . ; amÞ

6 maxfaig: ð43Þ

By Lemma 4, we obtain that minfaig 6 Cbwða1; a2; . . . ; amÞ

6 maxfaig. This completes the proof of Property 1. h

Property 2. Commutativity. Let fa1; a2; . . . ; amg be a set of linguisticvariables and fd1; d2; . . . ; dmg be a permutation of the fa1; a2;

. . . ; amg. Then, for any consensus operator Cbw,

Cbwða1; a2; . . . ; amÞ ¼ Cb

wðd1; d2; . . . ;dmÞ

Property 3. Monotonicity. Let fa1; a2; . . . ; amg be a set of linguisticvariables, let fc1; c2; . . . ; cmg be a second set of linguistic variables,such that 8i; ai P ci. Then, when m 6 2,

Cbwðc1; c2; . . . ; cmÞ 6 Cb

wða1; a2; . . . ; amÞ;

for any consensus operator Cbw.

Proof. Without loss of generality, we first suppose that ai > aj andci > cj for i > j. Property 3 can be obtained by showing ai P ci fori ¼ 1;2; . . . ;m. When m ¼ 1, it is obvious that a1 P c1. Whenm ¼ 2, using reduction to absurdity, we assume that a1 < c1 ora2 < c2. We consider three cases:

Case A: a1 < c1 and a2 < c2. In this case,

ðdða; �aÞÞ2 � ðdða; �cÞÞ2

¼ 1mT2 ða1

2 þ a22 � c1

2 � c22 þ 2a1ðc1 � a1Þ þ 2a2ðc2 � a2ÞÞ

>1

mT2 ða12 þ a2

2 � c12 � c2

2 þ 2c1ðc1 � a1Þ þ 2c2ðc2 � a2ÞÞ

¼ ðdðc; �aÞÞ2 � ðdðc; �cÞÞ2 P 0 ð44Þ

Thus, we have that dða; �aÞ > dða; cÞ, which contradicts the factthat �a is the adjusted preference values in using Cb

w to aggregatefa1; a2; . . . ; amg.

Case B: a1 < c1 and a2 P c2. In this case, let c ¼ fc1; c2g, wherec1 ¼ a1 and c2 ¼ minfa2; c2g, we have that

dðci;EOWAwðc1; c2ÞÞ 6 b for i ¼ 1;2 ð45Þ

and

dðc; cÞ < dðc; cÞ: ð46Þ

This contradicts the fact that c is the adjusted preference valuesin using Cb

w to aggregate fc1; c2; . . . ; cmg.

Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463 461

Case C: a1 P c1 and a2 < c2. In this case, let �a ¼ fa1; a2g, wherea1 ¼ maxfc1; a1g and a2 ¼ c2, similarly, we have that

dðai;EOWAwða1; a2ÞÞ 6 b for i ¼ 1;2 ð47Þ

and

dða; �aÞ < dða; �aÞ: ð48Þ

This contradicts the fact that �a is the adjusted preference valuesin using Cb

w to aggregate fa1; a2; . . . ; amg.Based on the three cases, we have a1 P c1 and a2 P c2.

Consequently, Cbwðc1; c2Þ 6 Cb

wða1; a2Þ; which completes this proofof Property 3. h

Note 4. We guess that the consensus operator Cbw also satisfies

monotonicity when m > 2. However, it is an open problem to com-pletely validate the property.

Property 4. Idempotency. If ai ¼ a for all i ¼ 1; . . . ;m, then for anyconsensus operator Cb

wða1; a2; . . . ; amÞ ¼ a.

Property 5. Let fa1; a2; . . . ; amg be a bag aggregate. If wi ¼ wmþ1�i forall i ¼ 1; . . . ;m, then,

Cbwða1; a2; . . . ; amÞ � Cb

wð�a1;�a2; . . . ;�amÞ ¼ s0:

for any consensus operator Cbw.

Proof. Let Cbwða1; a2; . . . ; amÞ ¼ EOWAwða1; a2; . . . ; amÞ. By Lemma 5,

we have that Cbwð�a1;�a2; . . . ;�amÞ ¼ EOWAwð�a1;�a2; . . . ;�amÞ.

Since wi ¼ wmþ1�i, we have that

EOWAwða1; a2; . . . ; amÞ ¼ �EOWAwð�a1;�a2; . . . ;�amÞ: ð49Þ

Consequently, Cbwða1; a2; . . . ; amÞ � Cb

wð�a1;�a2; . . . ;�amÞ ¼ s0:

This completes the proof of Property 5. h

There are a set of alternatives X ¼ fx1; . . . ; xng, and a set of lin-

guistic preference relations fPð1Þ; . . . ; PðmÞg, where PðkÞ ¼ ðpðkÞij Þn�n,

and pðkÞij represents the preference degree or intensity of alternative

xi over alternative xj for expert ek. In general, we assume that PðkÞ is

reciprocal in the sense that pðkÞij � pðkÞji ¼ s0 (s0 is the median label in

S). Using the consensus operator Cbw, we derive a collective prefer-

ence relation, PðcÞ ¼ ðpcijÞn�n indicating the global preference be-

tween every pair of alternatives. In this case,

pcij ¼ Cb

wðpð1Þij ; . . . ; pðkÞij ; . . . ;pðmÞij Þ

¼ EOWAwðpð1Þij ; . . . ;pðkÞij ; . . . ;pðmÞij Þ: ð50Þ

Theorem 6. When wi ¼ wmþ1�i for all i ¼ 1; . . . ;m, we have that (1)PðkÞðk ¼ 1;2; . . . ;mÞ and PðcÞ are reciprocal linguistic preferencerelations, and (2) dðPðcÞ; PðkÞÞ 6 b.

Proof. From Lemma 5, we have that PðkÞðk ¼ 1;2; . . . ;mÞ arereciprocal linguistic preference relations under the condition thatwi ¼ wmþ1�i for all i ¼ 1; . . . ;m. By Property 5, we have that PðcÞ isalso a reciprocal linguistic preference relation. According to

Definition 2, we obtain that dðPðcÞ; PðkÞÞ 6 b. This completes theproof of Theorem 6. h

5. Illustrative example

In order to show how these theoretical results work in practice,let us consider the following three examples.

5.1. Example 1

Let T ¼ 9;w ¼ ð1=2;1=4;1=4ÞT and a ¼ ðs4; s2; s3ÞT , then we have

LOWAwða1; a2; a3Þ ¼12� s4�

12� C2fkh; ah;h ¼ 2;3g;

where kh ¼ 1=2; h ¼ 2;3. Since C2fkh; ah; h ¼ 2;3g ¼ s3, we havethat

LOWAwða1; a2; a3Þ ¼ s4:

Moreover, we have that

EOWAwða1; a2; a3Þ ¼ s134;

and

TOWAwððs4; 0Þ; ðs2;0Þ; ðs3;0ÞÞ ¼ ðs3;0:25Þ:

Consequently,

IðEOWAwða1; a2; a3ÞÞ ¼ D�1ððs4;0Þ; ðs2; 0Þ; ðs3;0ÞÞ ¼134;

and

dðLOWAwða1; a2; a3Þ; EOWAwða1; a2; a3ÞÞ ¼1

12<

m� 12T

¼ 19:

The above results are consistent with Theorems 1 and 2. Letb ¼ 1

12, we consider using Cbw to aggregate linguistic labels in a.

Step 1: Computing the adjusted preference values�a ¼ fa1; a2; a3g. According to P2, we have that

minx

1243

P3i¼1ðIðaiÞ � xiÞ2

� �

s:t: 19 jxi �

P3i¼1

wixrxðiÞj 6 112 ; i ¼ 1;2; . . . 3

xraðiÞ � xraði�1Þ 6 0; i ¼ 2;3

8>>>>><>>>>>:

ð51Þ

The constraint condition xraðiÞ � xraði�1Þ 6 0; ði ¼ 2;3Þ guaranteesthat raðiÞ ¼ rxðiÞ i ¼ 1;2;3. Since rað1Þ ¼ 1;rað2Þ ¼ 3 andrað3Þ ¼ 2, we can transform model (51) into model (52):

minx

1243 ðð4� x1Þ2 þ ð2� x2Þ2 þ ð3� x3Þ2Þ

s:t:2x1 � x2 � x3 � 3 6 0�2x1 þ x2 þ x3 � 3 6 0�2x1 þ 3x2 � x3 � 3 6 02x1 � 3x2 þ x3 � 3 6 0�2x1 � x2 þ 3x3 � 3 6 02x1 þ x2 � 3x3 � 3 6 0x3 � x1 6 0x2 � x3 6 0

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð52Þ

Solving model (52), we can obtain x1 ¼ 3:72; x2 ¼ 2:43;x3 ¼ 2:85. Since ai ¼ I�1ðxiÞ; i ¼ 0;1; . . . ;m, we have that �a ¼fs3:72; s2:43; s2:85g.

Step 2: Obtain the collective linguistic variable ac. By aggregat-ing �a using EOWAw, we have that ac ¼ s3:18.

5.2. Example 2

Let us consider the example used in Herrera and Martínez(2000). In the example, there is a linguistic pre-establish label set:

SExample1 ¼ fs�3 ¼ N; s�2 ¼ VL; s�1 ¼ L; s0 ¼M; s1 ¼ H; s2 ¼ VH; s3 ¼ Pg:

A distribution company needs to renew its computing system,so it contracts a consulting company to carry out a survey of thedifferent possibilities existing on the market, to decide which isthe best option for its needs. There are four alternatives:

462 Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463

x1 ¼ UNIX; x2 ¼WINDOWS�NT; x3 ¼ AS=400 and x4 ¼ VMS. Theconsulting company has a group of four consultancy departmentsfe1; e2; e3; e4g. They supplied a decision matrix L:

L ¼ ðlijÞ4�4 ¼

s�2 s0 s0 s�1

s0 s�1 s�2 s1

s1 s�2 s0 s0

s1 s1 s�1 s�1

0BBB@

1CCCA;

where li;j denotes the preference degree of the decision maker ei

over xj.Without loss of generality, we set w ¼ ð0:3;0:3;0:2;0:2ÞT .

Applying the LOWAw operator and the EOWAw operator, respec-tively, we obtain the corresponding collective preference vectorsp1 and p2. Let b ¼ 0:1, we also consider using the consensus oper-ator Cb

w to aggregate individual preferences. Then the adjustedindividual preferences and the collective preference vector are Land p3, respectively. L is listed as follows:

L ¼

s�0:5 s�0:2 s�0:25 s�0:8

s�0:35 s�1 s�1:25 s0:4

s0:45 s�1 s�0:25 s0

s0:45 s0:2 s�1:25 s�0:8

0BBB@

1CCCA

The values of p1; p2 and p3 are listed in Table 1. p1 shows that{UNIX, WIN-NT, VMS} are the best options for the distributioncompany. p2 and p3 show that UNIX is the best option for the dis-tribution company.

5.3. Example 3

Let us also consider the example used in Xu (2005). In theexample, there are four decision makers dkðk ¼ 1;2; . . . ;4Þ. Thedecision makers compare five alternatives with respect to certaincriterion by using SExample2

SExample2 ¼ fs�4 ¼ extremely poor; s�3 ¼ very poor; s�2 ¼ poor

s�1 ¼ slightly poor; s0 ¼ fair; s1 ¼ slightly good

s2 ¼ good; s3 ¼ very good; s4 ¼ extremely goodg;

and construct, respectively, the linguistic preference relationsPð1Þ; Pð2Þ; Pð3Þ and Pð4Þ. They are listed as follows:

Pð1Þ ¼

s0 s�1 s3 s�1 s3

s1 s0 s1 s0 s2

s�3 s�1 s0 s�1 s2

s1 s0 s1 s0 s0

s�3 s�2 s�2 s0 s0

0BBBBBB@

1CCCCCCA

Pð2Þ ¼

s0 s1 s2 s0 s4

s�1 s0 s�1 s0 s0

s�2 s1 s0 s�1 s3

s0 s0 s1 s0 s1

s�4 s0 s�3 s�1 s0

0BBBBBB@

1CCCCCCA

Pð3Þ ¼

s0 s0 s3 s1 s3

s0 s0 s�2 s2 s2

s�3 s2 s0 s1 s1

s�1 s�2 s�1 s0 s�1

s�3 s�2 s�1 s1 s0

0BBBBBB@

1CCCCCCA

Pð4Þ ¼

s0 s2 s0 s�1 s2

s�2 s0 s�1 s1 s0

s0 s1 s0 s�1 s2

s1 s�1 s1 s0 s1

s�2 s0 s�2 s�1 s0

0BBBBBB@

1CCCCCCA

When applying the fuzzy quantifier Q ¼ ½0:2;0:8� to computethe weights of Cb

w, we have that w ¼ ð0:0833;0:4167;0:4167;0:0833ÞT . Let b ¼ 1, we consider using the consensus operator Cb

w

Table 1Results of the illustrative example.

UNIX WIN-NT AS/400 VMS

p1 s0 s0 s�1 s0

p2 s0:2 s�0:3 s�0:6 s�0:1

p3 s0:1 s�0:4 s�0:65 s�0:2

to aggregate Pð1Þ; Pð2Þ; Pð3Þ and Pð4Þ. In this case, the consensus oper-ator Cb

w reduce to the EOWA operator. The obtained collective lin-guistic preference relation is PðcÞ:

PðcÞ ¼

s0 s1:1668 s2:6668 s0:2501 s3:3334

s�1:1668 s0 s�0:2499 s1:2501 s1:6668

s�2:6668 s0:2499 s0 s�0:1666 s2:3334

s�0:2501 s�1:2501 s0:1666 s0 s0:7501

s�3:3334 s�1:6668 s�2:3334 s�0:7501 s0

0BBBBBB@

1CCCCCCA

Without loss of generality, we also set b ¼ 0:08. Using the con-sensus operator Cb

w to aggregate Pð1Þ; Pð2Þ; Pð3Þ; Pð4Þ, we obtain the ad-

justed preference relations Pð1Þ; Pð2Þ; Pð3Þ; Pð4Þ, and the collective

linguistic preference relation Pðc0 Þ. They are listed as follows:

Pð1Þ ¼

s0 s�0:2 s2:5 s�0:8 s3

s0:2 s0 s�0:1 s0:2 s1:7

s�2:5 s0:1 s0 s�0:7 s2

s0:8 s�0:2 s0:7 s0 s�0:3

s�3 s�1:7 s�2 s0:3 s0

0BBBBBB@

1CCCCCCA

Pð2Þ ¼

s0 s1 s1:5 s0:2 s3:7

s�1 s0 s�0:8 s0:2 s0:3

s�1:5 s0:8 s0 s�0:7 s2:7

s�0:2 s�0:2 s0:7 s0 s0:7

s�3:7 s�0:3 s�2:7 s�0:7 s0

0BBBBBB@

1CCCCCCA

Pð3Þ ¼

s0 s0 s2:5 s0:4 s3

s0 s0 s�1:5 s1:4 s1:7

s�2:5 s1:5 s0 s0:1 s1:3

s�0:4 s�1:4 s�0:1 s0 s�0:5

s�3 s�1:7 s�1:3 s0:5 s0

0BBBBBB@

1CCCCCCA

Pð4Þ ¼

s0 s1:2 s1:3 s�0:8 s2:3

s�1:2 s0 s�0:8 s1:2 s0:3

s�1:3 s0:8 s0 s�0:7 s2

s0:8 s�1:2 s0:7 s0 s0:7

s�2:3 s�0:3 s�2 s�0:7 s0

0BBBBBB@

1CCCCCCA

Pðc0 Þ ¼

s0 s0:5 s1:9833 s�0:2833 s3

s�0:5 s0 s�0:8 s0:7167 s1

s�1:9833 s0:8 s0 s�0:6333 s2

s0:2833 s�0:7167 s0:6333 s0 s0:1833

s�3 s�1 s�2 s�0:1833 s0

0BBBBBB@

1CCCCCCA

ð1Þ ð2Þ ð3Þ ð4Þ ðc0 Þ

We find that P ; P ; P ; P and P are reciprocal linguisticpreference relations, and max

kdðPðkÞ; Pðc0 ÞÞ ¼ 0:0696 < b ¼ 0:08.

6. Conclusions and future research

In the paper, we analyze the internal relationship of severalinteresting OWA-based linguistic operators based on position in-dexes (i.e., LOWA, TOWA and EOWA), and propose an OWA-basedconsensus operator under the continuous linguistic model (or the2-tuple fuzzy linguistic representation model). The major contribu-tions and findings are as follows:

(1) Based on the result presented in Dong et al. (2009), we furtheranalyze the internal relationship between the EOWA opera-tor and the TOWA operator. Moreover, we also provide anupper bound for the information loss of the LOWA operator.

(2) We develop a consensus operator. This consensus operator isa generalization of the OWA operator, and provides an alter-native consensus model for GDM. It has two characters: (i) it

Y. Dong et al. / European Journal of Operational Research 203 (2010) 455–463 463

preserves the original preference information given by thedecision makers as much as possible, and supports consen-sus process automatically, without moderator; (ii) it satisfiessome desirable properties.

In our future research, we will further investigate the aggrega-tion of linguistic preference relations using the consensus operator,and discuss the related properties. A detailed comparative study ofdifferent consensus models may also be an interesting topic.

Acknowledgements

We are very grateful to the editor and the anonymous refereesfor their valuable comments and suggestions, which have beenvery helpful in improving the paper. Moreover, Yucheng Dongand Yinfeng Xu would like to acknowledge the financial supportof Grants (No. 70801048, 70525004 and 60736027) from NSF ofChina. Hongyi Li would like to acknowledge the financial supportof a Grant (No. CUHK4726/05H) from the Research Grants Councilof the Hong Kong Special Administrative Region, China. Bo Fengwould like to acknowledge the financial support of Grant (No.70901027) from NSF of China.

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