Research Article Linguistic Weighted Aggregation …downloads.hindawi.com/journals/mpe/2015/485923.pdfResearch Article Linguistic Weighted Aggregation under Confidence Levels ChonghuiZhang,

Research ArticleLinguistic Weighted Aggregation under Confidence Levels

Chonghui Zhang1 Weihua Su23 Shouzhen Zeng24 and Linyun Zhang5

1College of Statistics and Mathematics Zhejiang Gongshang University Hangzhou 310018 China2College of Mathematics and Statistics Zhejiang University of Finance and Economics Hangzhou 310018 China3Center for Applied Statistics Renmin University of China Beijing 100872 China4College of Computer and Information Zhejiang Wanli University Ningbo 310015 China5Research Institute of Economic and Social Development Zhejiang University of Finance and Economics Hangzhou 310018 China

Correspondence should be addressed to Shouzhen Zeng zszzxl163com

Received 8 July 2014 Revised 17 December 2014 Accepted 21 December 2014

Academic Editor Kalyana C Veluvolu

Copyright copy 2015 Chonghui Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We develop some new linguistic aggregation operators based on confidence levels Firstly we introduce the confidence linguisticweighted averaging (CLWA) operator and the confidence linguistic ordered weighted averaging (CLOWA) operatorThese two newlinguistic aggregation operators are able to consider the confidence level of the aggregated arguments provided by the informationproviders We also study some of their properties Then based on the generalized means we introduce the confidence generalizedlinguistic ordered weighted averaging (CGLOWA) operatorThemain advantage of the CGLOWAoperator is that it includes a widerange of special cases such as the CLOWA operator the confidence linguistic ordered weighted quadratic averaging (CLOWQA)operator and the confidence linguistic ordered weighted geometric (CLOWG) operator Finally we develop an application of thenew approach in a multicriteria decision-making under linguistic environment and illustrate it with a numerical example

1 Introduction

Information aggregation is a technique that analyzes theinformation in order to provide a final result A variety ofaggregation operators have been developed in the past fewdecades One of the most common aggregation operators isthe weighted average (WA) operator [1] Another interestingone is the ordered weighted averaging (OWA) operatororiginally introduced by Yager [2] Its main advantage isthat it provides a parameterized family of aggregation oper-ators between the minimum and the maximum [3] Sincetheir introduction many new extensions of the WA andOWA operator have been proposed such as the weightedOWA (WOWA) operator [4] the OWA weighted averaging(OWAWA) operator [5] the induced ordered weighted aver-aging (IOWA) operator [6] the generalized OWA (GOWA)operator [7] the induced generalized OWA (IGOWA) opera-tor [8] the power ordered weighted average (POWA) oper-ator [9] and the continuous generalized ordered weightedaveraging (CGOWA) operator [10]

Due to time pressure and decisionsmakerrsquos limited exper-tise related with the problem domain decision informationabout alternatives is often uncertain or fuzzy As a result itis more suitable to provide preferences by using linguisticvariables rather than numerical onesThe use of the linguisticvariable provides a direct way to assess vague environmentswhere the information is provided by using expressions suchas low high good or bad [11 12] In order to aggregate thelinguistic information many linguistic aggregation operatorshave been developed such as the linguistic weighted orderedweighted averaging (LOWA) operator [13 14] linguisticordered weighted geometric averaging (LOWGA) operator[15] the induced linguistic generalized ordered weightedaveraging (ILGOWA) operator [16] the linguistic generalizedordered weighted averaging (LGOWA) operator [17] andthe linguistic generalized power ordered weighted average(LGPOWA) operator [18] Recently some new aggregationoperators are developed to aggregate linguistic informationFor example Zhou and Chen [19] developed the induced lin-guistic continuous ordered weighted geometric (ILCOWG)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 485923 7 pageshttpdxdoiorg1011552015485923

2 Mathematical Problems in Engineering

operator which is very suitable for group decision-making(GDM) problems taking the form of uncertain multiplicativelinguistic preference relations Wei et al [20] developed anew aggregation operator called the belief structure gen-eralized linguistic hybrid averaging (BS-GLHA) operatorWang et al [21] developed some linguistic cloud aggregationoperators including the cloud weighted arithmetic averag-ing (CWAA) operator cloud ordered weighted arithmeticaveraging (COWA) operator and cloud hybrid arithmetic(CHA) operator Based on probabilistic information andinduced aggregation operators Merigo et al [22] developedthe induced linguistic probabilistic ordered weighted average(ILPOWA) which uses probabilities and OWA operators inthe same formulation considering the degree of importancethat each concept has in the formulation

Most of the existing linguistic aggregation operators donot consider the confidence level of the aggregated argumentsprovided by the decision makers However in many realdecision-making problems such as the blind peer reviewof doctoral dissertation in China the evaluation experts arerequested to provide two types of information such as theperformance of the evaluation objects and the familiaritywiththe evaluation areas (called confidence levels) [23 24] Toovercome this issue Yu [24] and Xia et al [23] developedsome induced aggregation operators under confidence levelswhich can take into account the confidence levels of theaggregated arguments while after reviewing the existingliterature it seems that there is no investigation on linguisticinformation aggregation under belief levels which is aninteresting and important issue In this paper we focus onthe linguistic information aggregation issue in the situationwhere the confidences levels of the aggregated arguments areasked to be considered and we develop a series of linguisticaggregation operators considering the confidence levels ofthe aggregated arguments such as the confidence linguisticweighted averaging (CLWA) operator the confidence linguis-tic ordered weighted averaging (CLOWA) operator and theconfidence generalized linguistic ordered weighted averaging(CGLOWA) operator We also apply the developed operatorsto decision-making with linguistic information Finally anillustrative example has been given to show the developedmethod

This paper is organized as follows First we briefly reviewsomebasic concepts such as the linguistic approach to be usedthroughout the paper the LWA and the LOWA operatorSecond we present the CLWA CLOWA and CGLOWAoperator Third we discuss the applicability of the CGLOWAoperator with a multicriteria decision-making example andwe end the paper summarizing the main conclusions

2 Preliminaries

This section briefly reviews the linguistic approach thelinguistic weighted average and the linguistic OWAoperator

21 Linguistic Approach The linguistic approach is an approx-imate technique which represents qualitative aspects aslinguistic values by means of linguistic variables For compu-tational convenience let 119878 = 119904

120572| 120572 = 1 2 119905 be a finite

and totally ordered discrete term set where 119904120572represents a

possible value for a linguistic variable For example a set ofnine terms 119878 could be given as follows

119878 = 1199041= EL 119904

2= VL 119904

3= L 1199044= LM 119904

5= M

1199046= MH 119904

7= H 119904

8= VH 119904

9= EH

(1)

Note that EL means extremely low VL very low L low LMlow-medium M medium MH medium-high H high VHvery high and EH extremely high Usually in these casesit is required that in the linguistic term set there exist thefollowing

(i) a negation operatorNeg(119904119894) = 119904119895such that 119895 = 119892+1minus119894

(ii) the set ordered 119904119894le 119904119895if and only if 119894 le 119895

(iii) Max operator Max(119904119894 119904119895) = 119904119894if 119904119894ge 119904119895

(iv) Min operator Min(119904119894 119904119895) = 119904119894if 119904119894le 119904119895

In order to preserve all the given information Xu [25]extended the discrete term set 119878 to a continuous term set119878 = 119904

120572| 120572 isin [1 119905] where if 119904

120572isin 119878 then we call 119904

120572the

original term and otherwise we call 119904120572the virtual term In

general the decision maker uses the original linguistic termsto evaluate alternatives and the virtual linguistic terms canonly appear in the actual calculation [26]

Consider any two linguistic terms 119904120572 119904120573isin 119878 and 120583 gt 0

we define some operational laws as follows

(i) 120583119904120572= 119904120583120572

(ii) 119904120572+ 119904120573= 119904120573+ 119904120572= 119904120572+120573

(iii) (119904120572)120583

= 119904120572120583

(iv) 119904120572times 119904120573= 119904120573times 119904120572= 119904120572120573

Note that this model is very useful for computing withwords because it is very easy to use and it follows a similarmethodology as the numerical information

22 The Linguistic Weighted Average The linguistic weightedaverage (LWA) [14] is a linguistic aggregation operator thatuses the weighted average under uncertain environmentsassessed with linguistic information It is defined as follows

Definition 1 A LWA operator of dimension 119899 is a mappingLWA 119878119899 rarr 119878 which has an associated weighting vector 119882with 119908

119894isin [0 1] and sum

119899

119894=1119908119894= 1 such that

LWA (1199041205721 1199041205722 119904

120572119899) =

119899

sum

119894=1

119908119894119904120572119894 (2)

23 The Linguistic OWA Operator The OWA operator is anonlinear operator and an important feature is the reorder-ing step It has been studied in many documents and it has awide range of applications [27ndash38] By extending the OWAoperator assessed with linguistic information the LOWAoperator can be defined as follows

Mathematical Problems in Engineering 3

Definition 2 A LOWA operator of dimension 119899 is a mappingLOWA 119878119899 rarr 119878 which has an associated weighting vector119882 with 119908


119899

119895=1119908119895= 1 such that

LOWA (1199041205721 1199041205722 119904

120572119899) =

119899

sum

119895=1

119908119895119904120573119895 (3)

where 119904120573119895is the 119895th largest of the 119904

120572119894

The LOWA operator has been also extended underdifferent frameworks including the use of Dempster-Shafertheory [20 39] induced aggregation operators [16 26 40]generalized aggregation operators [17 41] distance measures[42] and power aggregation operators [18] However most ofthe existing linguistic aggregation operators do not considerthe confidence level of the aggregated arguments providedby the information providers Therefore in the next sectionwe will develop some new linguistic aggregation operatorswhich can take into account the confidence levels of theaggregated arguments

3 Linguistic Information AggregationOperators under Confidence Levels

In some real decision-making problems the evaluationexperts are requested to provide two types of informationsuch as the performance of the evaluation objects and thefamiliarity with the evaluation areas (called confidence lev-els) In this Section we investigate the linguistic informationaggregationmethods under confidence levels and proposed aseries of new linguistic aggregation operators

31 Confidence Linguistic Weighted Averaging (CLWA) Oper-ator In the following we propose the confidence linguisticweighted averaging (CLWA) operator and study the desirableproperties of the proposed operator The definition of theCLWA operator is given as follows

Definition 3 A CLWA operator of dimension 119899 is a mappingCLWA 119877119899 times 119878

119899

rarr 119878 which has an associated weightingvector119882 with 119908


119899

119894=1119908119894= 1 such that

CLWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119894(119897119894119904120572119894)

(4)

where 119897119894is the belief levels of linguistic variable 119904

120572119894 0 le 119897

119894le 1

In particular if 1198971= 1198972= sdot sdot sdot = 119897

119899= 1 then the CLWAoperator

reduces to the LWA operator

In the following example we present a simple numericalexample showing how to use the CLWA operator in anaggregation process

Example 4 Assume the following arguments in an aggre-gation process 119860 = (119904

1 1199044 1199043 1199047) with the following confi-

dence levels vector being (06 05 09 06) and the followingweighting vector119882 = (03 02 01 04) thenCLWA (⟨06 119904

1⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

253

(5)

Let 119904120572119894

and 119904120573119894

(119894 = 1 2 119899) be two collections oflinguistic variables and 119897

119894(0 le 119897

119894le 1) and 119908

119894(0 le 119908

119894le 1)

the confidence levels and weight of them respectively withsum119899

119894=1119908119894= 1Then we can easily prove the CLWA operator has

the properties of monotonicity boundness and idempotencyas follows

(1) (Monotonicity) If 119904120572119894le 119904120573119894for all 119894 then

CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le CLWA (⟨1198971 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(6)

(2) (Boundness) Consider

min119894

(119897119894119904120572119894) le CLWA (⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le max

119894

(119897119894119904120572119894)

(7)

(3) (Idempotency) If 119904120572119894= 119904120572for all 119894 then

CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (8)

32 Confidence Linguistic Ordered Weighted Aggregation(CLOWA) Operator In this section we introduce the ideaof OWA [2] into confidence linguistic information aggrega-tion problem and propose the confidence linguistic orderedweighted averaging (CLOWA) operator

Definition 5 Let (1199041205721 1199041205722 119904

120572119899) be a collection of linguistic

sets and 119897119894(0 le 119897

119894le 1) the confidence levels linguistic

variable 119904120572119894 A CLOWA operator of dimension 119899 is a mapping

CLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119895(119897119895119904120572119895)

(9)


119894119904120572119894 In particular if 119897

1=

1198972= sdot sdot sdot = 119897

119899= 1 then the CLOWA operator reduces to the

LOWA operator



dence levels vector being (06 05 09 06) and the followingweighting vector119882 = (03 02 01 04) then

CLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

224

(10)

The CLOWA operator is a mean or averaging operator Thisis a reflection of the fact that the operator is monotonicidempotent bounded and commutativeThese properties areproven in the following theorems

Theorem 7 (monotonicity) Let 119891 be the CLOWA operator if119904120572119894le 119904120573119894for all 119894 then

119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le 119891 (⟨119897

1 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(11)


Theorem 8 (idempotency) Let 119891 be the CLOWA operator119904120572119894= 119904120572for all 119894 then

119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (12)

Theorem 9 (bounded) Let 119891 be the CLOWA operator then

min119894

(119897119894119904120572119894) le 119862119871119874119882119860(⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le max119894

(119897119894119904120572119894)

(13)

Theorem 10 (commutativity) Let 119891 be the CLOWA operatorThen

119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119891 (⟨119897

1015840

1 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899

⟩)

(14)

where (⟨11989710158401 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899

⟩) is any permutation of the argu-ments (⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) Note that the CLWA operator

does not have this property

33 Confidence Generalized Linguistic Ordered WeightedAveraging (CGLOWA) Operator The confidence generalizedlinguistic ordered weighted averaging (CGLOWA) operatoris an extension of the OWA operator that uses the maincharacteristics of both the CLOWA and the GOWA oper-ator Then we can obtain a generalization that includesthe CLOWA operator and many other situations such asconfidence linguistic ordered weighted geometric (CLOWG)operator and confidence linguistic ordered weighted har-monic averaging (CLOWHA) operator It can be defined asfollows

Definition 11 Let (1199041205721 1199041205722 119904


sets and 119897119894(0 le 119897119894le 1) the confidence levels linguistic variable

119904120572119894 A CGLOWA operator of dimension 119899 is a mapping

CGLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CGLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩)

= (

119899

sum

119894=1

119908119895(119897119895119904120572119895)120582

)

1120582

(15)


119894119904120572119894and 120582 is a parameter

such that 120582 isin (minusinfin 0) cup (0 +infin) In particular if 1198971= 1198972=

sdot sdot sdot = 119897119899

= 1 then the CGLOWA operator reduces to theGLOWA operator [17]

Example 12 Assume the following arguments in an aggrega-tion process 119860 = (119904

minus2 1199040 1199043 119904minus1) with the following confi-

dence levels vector being (06 05 09 06) and the followingweighting vector 119882 = (03 02 01 04) And without loss ofgenerality let 120582 = 2 then

CGLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

270

(16)

Note that the different aggregated results can be obtainedif the parameter 120582 takes different values The selection of theparticular parameter 120582 depends on the particular interest ofthe decision maker in the specific problem considered

Similar to the CLOWA operator the CGLOWA operatoris also monotonic idempotent bounded and commutativeIf we analyze different values of the parameter 120582 we obtain agroup of particular cases For example we have the following

(i) If120582 = 1 theCGLOWAoperator becomes theCLOWAoperator

(ii) If120582 = 2 weget theconfidence linguisticorderedweightedquadratic averaging (CLOWQA) operator

(iii) If 120582 rarr 0 we form the confidence linguistic orderedweighted geometric (CLOWG) operator

(iv) If120582 = minus1 theconfidence linguisticorderedweightedhar-monic averaging (CLOWHA) operator is obtained

4 An Application of the Proposed Operator toMulticriteria Decision-Making

The CGLOWA (or CLOWA) operator is applicable in awide range of situations such as decision-making statisticsengineering and economics In summary all of the studiesthat use the OWA operator can be revised and extended byusing this new approach

In the following we are going to develop a decision-making method about the use of the CGLOWA in a multicri-teria decision-making problem For a multicriteria decision-making problem let 119883 = 119883

1 1198832 119883

119898 be a discrete set

of alternatives and let 119866 = 1198661 1198662 119866

119899 be the set of

criteria The main steps of the decision-making methods areas follows

Step 1 The decision makers provide their evaluations andbelief levels about the alternative 119883

119894under the attribute 119866

119895

forming the decision matrix 119863 = (119886119894119895)119898times119899

where 119886119894119895isin 119878 is a

preference value which takes the form of linguistic variable

Step 2 The decision makers provide their belief levels of theevaluations expressed by 119897

119894119895(119894 = 1 2 119898 119895 = 1 2 119899)


119894119895 119894 =

1 2 119898 119895 = 1 2 119899

Step 3 Utilize the CGLOWA (or some of its special cases)operator to aggregate the evaluations 119886

119894119895for each alternative

119883119894 Consider

119886119894= CGLOWA (⟨119897

1198941 1198861198941⟩ ⟨119897

119894119899 119886119894119899⟩) 119894 = 1 2 119898

(17)

Step 4 Rank all the alternatives119883119894(119894 = 1 2 119898) and select

the best one(s) in accordance with 119886119894(119894 = 1 2 119898)

Example 13 Let us consider a blind peer review of doc-toral dissertation evaluation problem in Chinarsquos university(adapted from [24]) In many Chinese universities thedoctoral dissertation will be reviewed by several expertsanonymously And they will review dissertation according to


Table 1 Linguistic fuzzy decision making metric

1198661

1198662

1198663

1198664

1198665

1198831

1199043

1199045

1199044

1199047

1199049

1198832

1199045

1199046

1199046

1199047

1199048

1198833

1199042

1199044

1199047

1199048

1199047

1198834

1199044

1199047

1199045

1199046

1199046

1198835

1199043

1199048

1199047

1199046

1199049

five criteria including topic selection and literature review(1198661) innovation (119866

2) theory basis and special knowledge

(1198663) capacity of scientific research (119866

4) and theses writing

(1198665) Unlike many existing evaluation methods the experts

not only required to provide the evaluation results of the doc-toral dissertation but also asked to give the degrees to whichthey are familiar with the research topics (called belief levels)Suppose there are five declarations that need to be reviewedby expert the degree of familiarity of the five declarationsprovided by expert is 119897

119895= (08 07 09 08 06) Suppose the

experts give the decisionmatrix under a linguistic frameworkof nine linguistic terms in the set as explained in Section 21shown in Table 1

Suppose that the weighting vector associating with theCGLOWA operator is 119882 = (011 024 030 024 011)which is derived by using the normal distribution basedmethod [1] With this information it is possible to aggregatethe available information in order to take a decision Byusing some key particular cases of the CGLOWA operatorsto aggregate the linguistic variables for five declarations theaggregated results and the ranking of the declarations areshown in Table 2

As we can see depending on the aggregation operatorused the results and decisions may be different Thereforethe decision about which declarations to select may be alsodifferent Note that in this example the optimal choice is thesame for all aggregation operators

If we do not consider the confidence levels factor inother words all the criteria of the evaluated objects aretreated with sure familiar by the decision maker then ourproposed CGLOWA operator is reduced to the existingGLOWAoperator [17]The aggregated results of this exampleby the types of theGLOWAoperators such as linguisticOWA(LOWA) operator the linguistic OWG (LOWG) operatorthe linguistic OWHA (LOWHA) operator and the linguisticordered weighted quadratic averaging (LOWQA) operatorare listed in Table 3

It can be easily seen that the ranking of the five doctoraldissertations obtained by the proposed CLGOWA operatoris similar to the result by the LGOWA operator Howeverin real-life decision process the decision maker(s) may notbe familiar with the doctoral dissertation absolutely To dealwith such situations the proposed CLGOWA operator isuseful tool From the above analysis we can see that the

main advantage of using the CGLOWA operator is thatit can consider the belief levels of the decision maker(s)Another main advantage is that it includes a wide range ofparticular cases such as the CLOWA the CLOWQA and theCLOWG operator Due to the fact that each particular familyof CGLOWA operator may give different results the decisionmaker(s) will select for his decision the one that is closest tohis interests

5 Concluding Remarks

In this paper we have developed some new linguistic infor-mation aggregation operators by introducing the belief levelsThe confidence linguistic weighted averaging (CLWA) oper-ator the confidence linguistic ordered weighted averaging(CLOWA) operator and the confidence generalized linguis-tic ordered weighted averaging (CGLOWA) operator havebeen introduced We have studied various properties of thedeveloped operators We have also presented an applicationof the CGLOWA operator to a multicriteria decision-makingproblem concerning the dissertation evaluation problemWe have seen that the CGLOWA is very useful becauseit represents very well the uncertain information by usinglinguistic labels as well as considering the belief levels ofthe decision makers Moreover it includes a wide range ofparticular cases such as the CLOWA the CLOWQA and theCLOWG operator Due to the fact that each particular familyof CGLOWA operator may give different results the decisionmaker gets amore complete view of the decision problem andis able to select the alternative that is closest to his interests

It is worth pointing out that we can extend the confi-dence aggregation using a similar method to deal with theother fuzzy situations where the arguments are expressed ininterval values or triangular fuzzy values In future researchwe expect to develop further extensions by adding newcharacteristics in the problem such as the use of inducingvariables or probabilistic aggregations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the editor and anony-mous reviewers who gave useful suggestions and consider-ably helped in improving the paper This paper is supportedby the MOE Project of Humanities and Social Sciences(nos 14YJC910006 13JJD910002) Zhejiang Province NaturalScience Foundation (no LQ14G010002) Statistical ScientificKey Research Project of China (no 2013LZ48) Key ResearchCenter of Philosophy and Social Science of Zhejiang Provinceand Modern Port Service Industry and Creative CultureResearch Center Zhejiang Provincial Key Research Basefor Humanities and Social Science Research (Statistics)


Table 2 Aggregated results by the CGLOWA and the rankings of alternatives

1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833

Table 3 Aggregated results by the GLOWA operator and the rankings of alternatives

1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833

Projects in Science and Technique of Ningbo Municipal (no2012B82003) and Ningbo Natural Science Foundation (no2013A610286)

References

[1] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005

[2] R R Yager ldquoOn ordered weighted averaging aggregation oper-ators in multicriteria decisionmakingrdquo IEEE Transactions onSystems Man and Cybernetics B Cybernetics vol 18 no 1 pp183ndash190 1988

[3] J M Merigo and A M Gil-Lafuente ldquoNew decision-makingmethods and their application in the selection of financialproductsrdquo Information Sciences vol 180 pp 2085ndash2094 2010

[4] V Torra ldquoTheweightedOWAoperatorrdquo International Journal ofIntelligent Systems vol 12 no 2 pp 153ndash166 1997

[5] J M Merigo ldquoA unified model between the weighted averageand the induced OWA operatorrdquo Expert Systems with Applica-tions vol 38 no 9 pp 11560ndash11572 2011

[6] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEE Transactions on Systems Man and CyberneticsB Cybernetics vol 29 no 2 pp 141ndash150 1999

[7] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[8] J M Merigo and A M Gil-Lafuente ldquoThe induced generalizedOWA operatorrdquo Information Sciences vol 179 no 6 pp 729ndash741 2009

[9] R R Yager ldquoThe power average operatorrdquo IEEE Transactions onSystems Man and Cybernetics A Systems and Humans vol 31no 6 pp 724ndash731 2001

[10] L-G Zhou and H-Y Chen ldquoContinuous generalized OWAoperator and its application to decisionmakingrdquo Fuzzy Sets andSystems vol 168 pp 18ndash34 2011

[11] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000

[12] L A Zadeh ldquoThe concept of a linguistic variable and its applica-tion to approximate reasoningmdashIrdquo Information Sciences vol 8no 3 pp 199ndash249 1975

[13] F Herrera E Herrera-Viedma and J L Verdegay ldquoA sequentialselection process in group decision making with a linguistic

assessment approachrdquo Information Sciences vol 85 no 4 pp223ndash239 1995

[14] Z S Xu ldquoEOWA and EOWG operators for aggregating linguis-tic labels based on linguistic preference relationsrdquo InternationalJournal of Uncertainty Fuzziness and Knowledge-Based Systemsvol 12 no 6 pp 791ndash810 2004

[15] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004

[16] J M Merigo A M Gil-Lafuente L-G Zhou and H-Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 no 4 pp 531ndash549 2012

[17] J M Merigo A M Gil-Lafuente L G Zhou and H Y ChenldquoGeneralization of the linguistic aggregation operator and itsapplication in decision makingrdquo Journal of Systems Engineeringand Electronics vol 22 no 4 pp 593ndash603 2011

[18] LG Zhou andHYChen ldquoA generalization of the power aggre-gation operators for linguistic environment and its applicationin group decision makingrdquo Knowledge-Based Systems vol 26pp 216ndash224 2012

[19] L G Zhou and H Y Chen ldquoThe induced linguistic continuousordered weighted geometric operator and its application togroup decision makingrdquo Computers amp Industrial Engineeringvol 66 pp 222ndash232 2013

[20] G-WWei X Zhao and R Lin ldquoSome hybrid aggregating oper-ators in linguistic decision making with Dempster-Shafer beliefstructurerdquo Computers amp Industrial Engineering vol 65 no 4pp 646ndash651 2013

[21] J Q Wang P Lu H Y Zhang and X H Chen ldquoMethod ofmulti-criteria group decision-making based on cloud aggre-gation operators with linguistic informationrdquo Information Sci-ences vol 274 pp 177ndash191 2014

[22] J M Merigo M Casanovas and P Daniel ldquoLinguistic groupdecision making with induced aggregation operators and prob-abilistic informationrdquo Applied Soft Computing vol 24 pp 669ndash678 2014

[23] M M Xia Z S Xu and N Chen ldquoInduced aggregation underconfidence levelsrdquo International Journal of Uncertainty Fuzzi-ness and Knowledge-Based Systems vol 19 no 2 pp 201ndash2272011

[24] D J Yu ldquoIntuitionistic fuzzy information aggregation underconfidence levelsrdquo Applied Soft Computing Journal vol 19 pp147ndash160 2014


[25] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[26] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006

[27] G Beliakov A Pradera and T Calvo Aggregation Functions AGuide for Practitioners Springer Berlin Germany 2007

[28] S P Wan ldquo2-Tuple linguistic hybrid arithmetic aggregationoperators and application to multi-attribute group decisionmakingrdquo Knowledge-Based Systems vol 15 pp 31ndash40 2013

[29] G W Wei ldquoUncertain linguistic hybrid geometric mean oper-ator and its application to group decision making under uncer-tain linguistic environmentrdquo International Journal of Uncer-tainty Fuzziness Knowledge-Based Systems vol 17 pp 251ndash2672009

[30] G W Wei and X F Zhao ldquoSome induced correlated aggre-gating operators with intuitionistic fuzzy information and theirapplication tomultiple attribute group decisionmakingrdquoExpertSystems with Applications vol 39 no 2 pp 2026ndash2034 2012

[31] Y J Xu and H MWang ldquoThe induced generalized aggregationoperators for intuitionistic fuzzy sets and their application ingroup decision makingrdquo Applied Soft Computing Journal vol12 no 3 pp 1168ndash1179 2012

[32] X H Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggrega-tion operatorsrdquo Information Fusion vol 14 pp 108ndash116 2013

[33] D J Yu ldquoIntuitionistic fuzzy geometric Heronian mean aggre-gation operatorsrdquo Applied Soft Computing Journal vol 13 no 2pp 1235ndash1246 2013

[34] R R Yager J Kacprzyk and G Beliakov Recent Developmentson the Ordered Weighted Averaging Operators Theory andPractice Springer Berlin Germany 2011

[35] S Z Zeng ldquoSome intuitionistic fuzzy weighted distance mea-sures and their application to group decision makingrdquo GroupDecision and Negotiation vol 22 no 2 pp 281ndash298 2013

[36] S Z Zeng J M Merigo and W H Su ldquoThe uncertainprobabilistic OWA distance operator and its application ingroup decision makingrdquo Applied Mathematical Modelling vol37 no 9 pp 6266ndash6275 2013

[37] H M Zhang ldquoThe multiattribute group decision makingmethod based on aggregation operators with interval-valued2-tuple linguistic informationrdquo Mathematical and ComputerModelling vol 56 no 1-2 pp 27ndash35 2012

[38] H M Zhang ldquoSome interval-valued 2-tuple linguistic aggrega-tion operators and application in multiattribute group decisionmakingrdquo Applied Mathematical Modelling vol 37 no 6 pp4269ndash4282 2013

[39] J MMerigo M Casanovas and LMartınez ldquoLinguistic aggre-gation operators for linguistic decision making based on theDempster-Shafer theory of evidencerdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 18 no3 pp 287ndash304 2010

[40] J Lan Q Sun Q Chen and Z Wang ldquoGroup decision makingbased on induced uncertain linguistic OWA operatorsrdquo Deci-sion Support Systems vol 55 no 1 pp 296ndash303 2013

[41] B Peng C Ye and S Z Zeng ldquoUncertain pure linguistic hybridharmonic averaging operator and generalized interval aggre-gation operator based approach to group decision makingrdquoKnowledge-Based Systems vol 36 pp 175ndash181 2012

[42] S Z Zeng and W H Su ldquoLinguistic induced generalizedaggregation distance operators and their application to decisionmakingrdquo Economic Computer and Economic Cybernetics Studiesand Research vol 46 pp 155ndash172 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


operator which is very suitable for group decision-making(GDM) problems taking the form of uncertain multiplicativelinguistic preference relations Wei et al [20] developed anew aggregation operator called the belief structure gen-eralized linguistic hybrid averaging (BS-GLHA) operatorWang et al [21] developed some linguistic cloud aggregationoperators including the cloud weighted arithmetic averag-ing (CWAA) operator cloud ordered weighted arithmeticaveraging (COWA) operator and cloud hybrid arithmetic(CHA) operator Based on probabilistic information andinduced aggregation operators Merigo et al [22] developedthe induced linguistic probabilistic ordered weighted average(ILPOWA) which uses probabilities and OWA operators inthe same formulation considering the degree of importancethat each concept has in the formulation

Most of the existing linguistic aggregation operators donot consider the confidence level of the aggregated argumentsprovided by the decision makers However in many realdecision-making problems such as the blind peer reviewof doctoral dissertation in China the evaluation experts arerequested to provide two types of information such as theperformance of the evaluation objects and the familiaritywiththe evaluation areas (called confidence levels) [23 24] Toovercome this issue Yu [24] and Xia et al [23] developedsome induced aggregation operators under confidence levelswhich can take into account the confidence levels of theaggregated arguments while after reviewing the existingliterature it seems that there is no investigation on linguisticinformation aggregation under belief levels which is aninteresting and important issue In this paper we focus onthe linguistic information aggregation issue in the situationwhere the confidences levels of the aggregated arguments areasked to be considered and we develop a series of linguisticaggregation operators considering the confidence levels ofthe aggregated arguments such as the confidence linguisticweighted averaging (CLWA) operator the confidence linguis-tic ordered weighted averaging (CLOWA) operator and theconfidence generalized linguistic ordered weighted averaging(CGLOWA) operator We also apply the developed operatorsto decision-making with linguistic information Finally anillustrative example has been given to show the developedmethod

This paper is organized as follows First we briefly reviewsomebasic concepts such as the linguistic approach to be usedthroughout the paper the LWA and the LOWA operatorSecond we present the CLWA CLOWA and CGLOWAoperator Third we discuss the applicability of the CGLOWAoperator with a multicriteria decision-making example andwe end the paper summarizing the main conclusions

2 Preliminaries

This section briefly reviews the linguistic approach thelinguistic weighted average and the linguistic OWAoperator

21 Linguistic Approach The linguistic approach is an approx-imate technique which represents qualitative aspects aslinguistic values by means of linguistic variables For compu-tational convenience let 119878 = 119904

120572| 120572 = 1 2 119905 be a finite

and totally ordered discrete term set where 119904120572represents a

possible value for a linguistic variable For example a set ofnine terms 119878 could be given as follows

119878 = 1199041= EL 119904

2= VL 119904

3= L 1199044= LM 119904

5= M

1199046= MH 119904

7= H 119904

8= VH 119904

9= EH

(1)

Note that EL means extremely low VL very low L low LMlow-medium M medium MH medium-high H high VHvery high and EH extremely high Usually in these casesit is required that in the linguistic term set there exist thefollowing

(i) a negation operatorNeg(119904119894) = 119904119895such that 119895 = 119892+1minus119894

(ii) the set ordered 119904119894le 119904119895if and only if 119894 le 119895

(iii) Max operator Max(119904119894 119904119895) = 119904119894if 119904119894ge 119904119895

(iv) Min operator Min(119904119894 119904119895) = 119904119894if 119904119894le 119904119895

In order to preserve all the given information Xu [25]extended the discrete term set 119878 to a continuous term set119878 = 119904

120572| 120572 isin [1 119905] where if 119904

120572isin 119878 then we call 119904

120572the

original term and otherwise we call 119904120572the virtual term In

general the decision maker uses the original linguistic termsto evaluate alternatives and the virtual linguistic terms canonly appear in the actual calculation [26]

Consider any two linguistic terms 119904120572 119904120573isin 119878 and 120583 gt 0

we define some operational laws as follows

(i) 120583119904120572= 119904120583120572

(ii) 119904120572+ 119904120573= 119904120573+ 119904120572= 119904120572+120573

(iii) (119904120572)120583

= 119904120572120583

(iv) 119904120572times 119904120573= 119904120573times 119904120572= 119904120572120573

Note that this model is very useful for computing withwords because it is very easy to use and it follows a similarmethodology as the numerical information

22 The Linguistic Weighted Average The linguistic weightedaverage (LWA) [14] is a linguistic aggregation operator thatuses the weighted average under uncertain environmentsassessed with linguistic information It is defined as follows

Definition 1 A LWA operator of dimension 119899 is a mappingLWA 119878119899 rarr 119878 which has an associated weighting vector 119882with 119908


119899

119894=1119908119894= 1 such that

LWA (1199041205721 1199041205722 119904

120572119899) =

119899

sum

119894=1

119908119894119904120572119894 (2)

23 The Linguistic OWA Operator The OWA operator is anonlinear operator and an important feature is the reorder-ing step It has been studied in many documents and it has awide range of applications [27ndash38] By extending the OWAoperator assessed with linguistic information the LOWAoperator can be defined as follows




119899

119895=1119908119895= 1 such that

LOWA (1199041205721 1199041205722 119904

120572119899) =

119899

sum

119895=1

119908119895119904120573119895 (3)


120572119894






119899



119899

119894=1119908119894= 1 such that

CLWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119894(119897119894119904120572119894)

(4)


120572119894 0 le 119897

119894le 1








1⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

253

(5)

Let 119904120572119894

and 119904120573119894


119894(0 le 119897

119894le 1) and 119908

119894(0 le 119908

119894le 1)





CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le CLWA (⟨1198971 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(6)


min119894

(119897119894119904120572119894) le CLWA (⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le max

119894

(119897119894119904120572119894)

(7)


CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (8)


Definition 5 Let (1199041205721 1199041205722 119904


sets and 119897119894(0 le 119897



CLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119895(119897119895119904120572119895)

(9)


119894119904120572119894 In particular if 119897

1=

1198972= sdot sdot sdot = 119897


LOWA operator




CLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

224

(10)



119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le 119891 (⟨119897

1 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(11)



119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (12)


min119894

(119897119894119904120572119894) le 119862119871119874119882119860(⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le max119894

(119897119894119904120572119894)

(13)


119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119891 (⟨119897

1015840

1 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899

⟩)

(14)

where (⟨11989710158401 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899


1 1199041205721⟩ ⟨119897




Definition 11 Let (1199041205721 1199041205722 119904




CGLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CGLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩)

= (

119899

sum

119894=1

119908119895(119897119895119904120572119895)120582

)

1120582

(15)









CGLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

270

(16)










1 1198832 119883







119895


where 119886119894119895isin 119878 is a



119894119895(119894 = 1 2 119898 119895 = 1 2 119899)


119894119895 119894 =

1 2 119898 119895 = 1 2 119899



119883119894 Consider

119886119894= CGLOWA (⟨119897

1198941 1198861198941⟩ ⟨119897

119894119899 119886119894119899⟩) 119894 = 1 2 119898

(17)






1198661

1198662

1198663

1198664

1198665

1198831

1199043

1199045

1199044

1199047

1199049

1198832

1199045

1199046

1199046

1199047

1199048

1198833

1199042

1199044

1199047

1199048

1199047

1198834

1199044

1199047

1199045

1199046

1199046

1198835

1199043

1199048

1199047

1199046

1199049







119895= (08 07 09 08 06) Suppose the












Acknowledgments




1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833


1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899

119895=1119908119895= 1 such that

LOWA (1199041205721 1199041205722 119904

120572119899) =

119899

sum

119895=1

119908119895119904120573119895 (3)


120572119894






119899



119899

119894=1119908119894= 1 such that

CLWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119894(119897119894119904120572119894)

(4)


120572119894 0 le 119897

119894le 1








1⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

253

(5)

Let 119904120572119894

and 119904120573119894


119894(0 le 119897

119894le 1) and 119908

119894(0 le 119908

119894le 1)





CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le CLWA (⟨1198971 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(6)


min119894

(119897119894119904120572119894) le CLWA (⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le max

119894

(119897119894119904120572119894)

(7)


CLWA (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (8)


Definition 5 Let (1199041205721 1199041205722 119904


sets and 119897119894(0 le 119897



CLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩) =

119899

sum

119894=1

119908119895(119897119895119904120572119895)

(9)


119894119904120572119894 In particular if 119897

1=

1198972= sdot sdot sdot = 119897


LOWA operator




CLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

224

(10)



119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) le 119891 (⟨119897

1 1199041205731⟩ ⟨119897

119899 119904120573119899⟩)

(11)



119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (12)


min119894

(119897119894119904120572119894) le 119862119871119874119882119860(⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le max119894

(119897119894119904120572119894)

(13)


119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119891 (⟨119897

1015840

1 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899

⟩)

(14)

where (⟨11989710158401 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899


1 1199041205721⟩ ⟨119897




Definition 11 Let (1199041205721 1199041205722 119904




CGLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CGLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩)

= (

119899

sum

119894=1

119908119895(119897119895119904120572119895)120582

)

1120582

(15)









CGLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

270

(16)










1 1198832 119883







119895


where 119886119894119895isin 119878 is a



119894119895(119894 = 1 2 119898 119895 = 1 2 119899)


119894119895 119894 =

1 2 119898 119895 = 1 2 119899



119883119894 Consider

119886119894= CGLOWA (⟨119897

1198941 1198861198941⟩ ⟨119897

119894119899 119886119894119899⟩) 119894 = 1 2 119898

(17)






1198661

1198662

1198663

1198664

1198665

1198831

1199043

1199045

1199044

1199047

1199049

1198832

1199045

1199046

1199046

1199047

1199048

1198833

1199042

1199044

1199047

1199048

1199047

1198834

1199044

1199047

1199045

1199046

1199046

1198835

1199043

1199048

1199047

1199046

1199049







119895= (08 07 09 08 06) Suppose the












Acknowledgments




1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833


1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119904

120572 (12)


min119894

(119897119894119904120572119894) le 119862119871119874119882119860(⟨119897

1 1199041205721⟩ ⟨119897

119899 119904120572119899⟩)

le max119894

(119897119894119904120572119894)

(13)


119891 (⟨1198971 1199041205721⟩ ⟨119897

119899 119904120572119899⟩) = 119891 (⟨119897

1015840

1 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899

⟩)

(14)

where (⟨11989710158401 1199041015840

1205721

⟩ ⟨1198971015840

119899 1199041015840

120572119899


1 1199041205721⟩ ⟨119897




Definition 11 Let (1199041205721 1199041205722 119904




CGLOWA 119877119899 times 119878119899



119899

119895=1119908119895= 1 such that

CGLOWA (⟨1198971 1199041205721⟩ ⟨1198972 1199041205722⟩ ⟨119897

119899 119904120572119899⟩)

= (

119899

sum

119894=1

119908119895(119897119895119904120572119895)120582

)

1120582

(15)









CGLOWA (⟨06 1199041⟩ ⟨05 119904

4⟩ ⟨09 119904

3⟩ ⟨06 119904

7⟩) = 119904

270

(16)










1 1198832 119883







119895


where 119886119894119895isin 119878 is a



119894119895(119894 = 1 2 119898 119895 = 1 2 119899)


119894119895 119894 =

1 2 119898 119895 = 1 2 119899



119883119894 Consider

119886119894= CGLOWA (⟨119897

1198941 1198861198941⟩ ⟨119897

119894119899 119886119894119899⟩) 119894 = 1 2 119898

(17)






1198661

1198662

1198663

1198664

1198665

1198831

1199043

1199045

1199044

1199047

1199049

1198832

1199045

1199046

1199046

1199047

1199048

1198833

1199042

1199044

1199047

1199048

1199047

1198834

1199044

1199047

1199045

1199046

1199046

1198835

1199043

1199048

1199047

1199046

1199049







119895= (08 07 09 08 06) Suppose the












Acknowledgments




1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833


1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198661

1198662

1198663

1198664

1198665

1198831

1199043

1199045

1199044

1199047

1199049

1198832

1199045

1199046

1199046

1199047

1199048

1198833

1199042

1199044

1199047

1199048

1199047

1198834

1199044

1199047

1199045

1199046

1199046

1198835

1199043

1199048

1199047

1199046

1199049







119895= (08 07 09 08 06) Suppose the












Acknowledgments




1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833


1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198831

1198832

1198833

1198834

1198835

RankingCLOWA (120582 = 1) 119904

410119904480

119904432

119904423

119904524

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWQA (120582 = 2) 119904423

119904483

119904464

119904446

119904537

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

CLOWG (120582 rarr 0) 119904396

119904477

119904396

119904439

119904507

1198835≻ 1198832≻ 1198834≻ 1198831= 1198833

CLOWHA (120582 = minus1) 119904382

119904473

119904356

119904434

119904484

1198835≻ 1198832≻ 1198834≻ 1198831≻ 1198833


1198831

1198832

1198833

1198834

1198835

RankingLOWA (120582 = 1) 119904

546119904635

119904584

119904556

119904678

1198835≻ 1198832≻ 1198833≻ 1198834≻ 1198831

LOWQA (120582 = 2) 119904574

119904640

119904615

119904571

119904697

1198835≻ 1198832≻ 1198833≻ 1198831≻ 1198834

LOWG (120582 rarr 0) 119904518

119904630

119904541

119904559

119904652

1198835≻ 1198832≻ 1198834≻ 1198833≻ 1198831

LOWHA (120582 = minus1) 119904492

119904625

119904486

119904552

119904618

1198832≻ 1198835≻ 1198834≻ 1198831≻ 1198833


References




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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