Research Article Multistage Multiattribute Group Decision ...downloads.hindawi.com/journals/mpe/2016/1687068.pdf · Bale zentis et al. [ ] presented a fuzzy MULTIMOORA ... cients

Research ArticleMultistage Multiattribute Group Decision-Making MethodBased on Triangular Fuzzy MULTIMOORA

Wen-feng Dai12 Qiu-yan Zhong2 and Chun-ze Qi1

1School of Information Engineering Lanzhou University of Finance and Economics Lanzhou 730020 China2Faculty of Management and Economics Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Wen-feng Dai hope2503sinacom

Received 6 April 2016 Revised 26 June 2016 Accepted 3 July 2016

Academic Editor Anna M Gil-Lafuente

Copyright copy 2016 Wen-feng Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposed a new multiattribute group decision-making method in which the period significance coefficients and theattribute significance coefficients are completely unknown and the attribute values are triangular fuzzy numbers At first toobtain the period significance coefficients the period significance coefficients optimization model is constructed according tothe time degree and the differences of the decision information in different periods Then attribute significance coefficients aredetermined by the maximum deviationmethod Based on this alternatives are ranked by the triangular fuzzy ratio systemmethodthe triangular fuzzy reference point method and the triangular fuzzy full multiplicative form respectively The dominance theoryis used for aggregating the subordinate rankings into the final ranking Finally a numerical example is presented to illustrate thefeasibility and effectiveness of the proposed method

1 Introduction

Multiattribute decision-making (MADM) has been widelyused in many practical problems such as emergency planselection supplier selection consumer purchasing selectionand human resource management However when dealingwith many complex problems single decision-maker cannotaccurately consider all aspects of the decision problem whichwill lead to unreasonable decision results In this case mul-tiple decision-makers are required to participate in the deci-sion process and the multiattribute group decision-making(MAGDM) appearsNowadaysMAGDMhas attractedmanyresearchersrsquo attention and has become a hot research topic [1ndash9]

In classical MAGDMmethods the evaluation values andthe attribute significance coefficients are known completelyHowever in the practical decision they are partly knowneven completely unknown To deal with these problemsmany fuzzy decision methods and stochastic decision meth-ods are proposed Parameters in fuzzy decision methods areusually assumed to have given membership functions Simi-larly parameters in the stochastic decision-making methodsare presumed to have appointed probability distributions

But decision-makers may not be capable of determiningmembership functions or probability distributions properlySo many decision methods based on the fuzzy logic theoryare proposed Cao et al [10ndash12] presented decision meth-ods based on interval numbers However when intervalnumbers are used to represent the decision informationthe interval ranges of them are usually too large so as tocover the whole value range The intervals are likely to befurther enlarged after many operations [13ndash15] In additioneach value between the upper and the lower bound of theinterval number is generally considered to have the samevalue opportunity [16] All these disadvantages will leadto the information distortion or deviation Compared withinterval numbers triangular fuzzy numbers not only canrepresent the interval range of the decision information butcan highlight the gravity center having the largest probabilityHence the triangular number can make up the deficiencyof the interval number in the precision and facilitate thedecision-making at least to some extent

The multiobjective optimization on the basis of ratioanalysis (MOORA) a newly developed MADM methodwas firstly proposed by Brauers and Zavadskas for solvingMADM problems [17 18] Then MOORA was extended to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1687068 8 pageshttpdxdoiorg10115520161687068

2 Mathematical Problems in Engineering

MULTIMOORA (MOORA plus the full multiplicative form)[19ndash21] Compared with other MADM methods MULTI-MOORA is easier to understand and implement BesidesMULTIMOORA can facilitate the decision-making processand provide effective rankings [22ndash25] At present MUL-TIMOORA has been expanded into different forms accord-ing to the fuzzy logic theory Brauers et al [19] extendedMULTIMOORA into the fuzzy number environment Liuet al [26] expanded MULTIMOORA based on the fuzzynumber and used it to evaluate the risk of the failure modeBalezentis et al [27] presented a fuzzy MULTIMOORAmethod for linguistic reasoning and used it to deal with acandidate selection problem Datta et al [28] proposed grey-MULTIMOORA and used it to solve the robot selectionproblem

However few studies extended MULTIMOORA intothe triangular fuzzy number environment and obtainedthe final ranking by the dominance theory Balezentis etal [27] extended MULTIMOORA based on the triangu-lar fuzzy number But they only considered the currentdecision information and ignored the importance of thehistorical information In addition the attribute signifi-cance coefficients are given in advance which will leadto unreasonable results Based on this we developed anovel extension of MULTIMOORA method based on con-sidering the current and historical information to solveMAGDM problemsThemajor contribution of our proposedmethod is that it considered the current and the historicalinformation of the decision object So it can describe thedecision object comprehensively and objectively Besides toobtain the stable period significance coefficients the periodsignificance coefficients optimization model is constructedaccording to the time degree and the differences of thedecision information in different periods Hence our pro-posed method can help to obtain more objective decisionresults

The remainder of this paper is organized as followsSection 2 briefly introduces somebasic concepts related to thetriangular fuzzy number Section 3 mainly presents the stepsof the original MULTIMOORA method A novel extensionof MULTIMOORA is proposed in Section 4 In order toillustrate the feasibility and effectiveness of the proposedmethod a numerical example is presented in Section 5Finally conclusions are given in Section 6

2 Triangular Fuzzy Numbers

Compared with the crisp number the triangular fuzzy num-ber is more in accord with the uncertainty of the decisionenvironment and the fuzziness of human thinking Besidescompared with the interval number the triangular fuzzynumber not only can represent the interval range of thedecision information but also can highlight the gravitycenter having the maximum value probability Hence thetriangular number can reduce the information distortion andinformation deviation in the process of decision-makingBased on this this paper represents the decision informationwith triangular fuzzy numbers

Definition 1 A triangular fuzzy number is defined as(119886

119871 119886

119872 119886

119880) and its membership function 120583

(119909) can be

defined as [28]

120583

(119909) =

(119909 minus 119886

119871)

(119886

119872minus 119886

119871)

119886

119871le 119909 le 119886

119872

(119886

119880minus 119909)

(119886

119880minus 119886

119872)

119886

119872le 119909 le 119886

119880

0 otherwise

(1)

in which 119909 isin 119877 and 119886119871 and 119886119880 represent the lower boundand the upper bound of the triangular fuzzy number respectively 0 le 119886119871 le 119886119872 le 119886119880 If 119886119871 119886119872 and 119886119880 are equal degenerates a crisp value

Definition 2 Let = (119886119871 119886119872 119886119880) and 119887 = (119887119871 119887119872 119887119880) betwo triangular fuzzy numbers and let 120582 be a real number theyfollow the following rules [29 30]

(1) + 119887 = (119886119871 + 119887119871 119886119872 + 119887119872 119886119880 + 119887119880)(2) times 119887 = (119886119871 times 119887119871 119886119872 times 119887119872 119886119880 times 119887119880)(3) 120582 = (120582119886119871 120582119886119872 120582119886119880)(4) 120582 = ((119886119871)120582 (119886119872)120582 (119886119880)120582)(5) 119887 = (119887119871119886119880 119887119872119886119872 119887119880119886119871)

Definition 3 Let = (119886119871 119886119872 119886119880) and 119887 = (119887119871 119887119872 119887119880) be twotriangular fuzzy numbers the distance between them couldbe obtained as the following

119889 (

119887)

=

radic

[(119886

119871minus 119887

119871)

2

+ (119886

119872minus 119887

119872)

2

+ (119886

119880minus 119887

119880)

2

]

3

(2)

Definition 4 Let = (119886

119871 119886

119872 119886

119880) be a triangular fuzzy

number its defuzzy formula can be defined as the following

119886 =

119886

119871+ 2119886

119872+ 119886

119880

4

(3)

3 The MULTIMOORA Method

Brauers and Zavadskas extended the MOORA method toMULTIMOORAwhich includes three parts [17ndash19] the ratiosystem method the reference point method and the fullmultiplicative form method

Let 119860 = 119886

1 119886

2 119886

119898 be the alternative set let 119862 =

119888

1 119888

2 119888

119899 be the attribute set let 119881 = (V

119894119895)

119898times119899be the

decision matrix let V119894119895be the evaluation value of alternative

119886

119894under the attribute 119888

119895 119894 = 1 2 119898 119895 = 1 2 119899 let

119881

lowast= (Vlowast119894119895)

119898times119899be the standardized decision matrix and let Vlowast

119894119895

be the standardized form of V119894119895 Hence

Vlowast119894119895=

V119894119895

radicsum

119898

119894=1V2119894119895

(4)

Mathematical Problems in Engineering 3

31 The Ratio System Method The evaluation value of thealternative 119886

119894under the ratio system method can be defined

as the following

119910

119894=

119892

sum

119895=1

Vlowast119894119895minus

119899

sum

119895=119892+1

Vlowast119894119895 (5)

in which 119892 and 119899 minus 119892 represent the numbers of benefitattributes and cost attributes respectively 119908

119895represents the

attribute significance coefficient and 119910119894represents the evalu-

ation value of alternative 119886119894under the ratio system method

Alternatives are sorted according to evaluation values indescending order The optimal alternative is the one havingthe biggest value [13]

119886

lowast

RS = 119886119894 | max119894

119910

119894 (6)

32 The Reference Point Method At first the maximalattribute reference point should be specified by (7) Hence

119903

119895=

max119894

Vlowast119894119895 119895 le 119892

min119894

Vlowast119894119895 119895 gt 119892

(7)

The deviation between the standardized evaluation valueVlowast119894119895and the reference point 119903

119895can be defined as |119903

119895minusVlowast119894119895| So the

evaluation value of 119894th alternative under the reference pointmethod can be represented as the following

119911

119894= max119895

1003816

1003816

1003816

1003816

1003816

119903

119895minus Vlowast119894119895

1003816

1003816

1003816

1003816

1003816

(8)

Obviously the smaller 119911119894is the better alternative 119886

119894will

be The optimal alternative is the one having the smallestevaluation value Hence

119886

lowast

RP = 119886119894 | min119894

119911

119894 (9)

33 The Full Multiplicative Form Method The evaluationvalue of the 119894th alternative under the full multiplicative formcan be defined as the following

119880

119894=

prod

119892

119895=1Vlowast119894119895

prod

119899

119895=119892+1Vlowast119894119895

(10)

in which prod

119892

119895=1Vlowast119894119895

represents the product of all benefitattributesrsquo normalized evaluation values andprod119899

119895=119892+1Vlowast119894119895repre-

sents the product of all cost attributesrsquo normalized evaluationvalues The bigger 119880

119894is the better 119886

119894will be The optimal

alternative can be specified by

119886

lowast

MF = 119886119894 | max119894

119880

119894 (11)

34 The Final Ranking of MULTIMOORA The dominancetheory can integrate several rankings into a compressiveranking according to propositions like equability transitivityand domination and so forthHence the alternative rankingsobtained from Sections 31 32 and 33 can be aggregated intothe final ranking by the dominance theory The dominancetheoryrsquos application in MULTIMOORA is introduced inliteratures [7 8 20]

4 The Proposed Method

41 The Decision-Making Problem Description Let 119860 =

119886

1 119886

2 119886

119898 represent the alternative set let 119862 =

119888

1 119888

2 119888

119899 represent the attribute set let 119904 = (119904

1 119904

2 119904

119899)

represent the attribute significance coefficient with 0 le 119904119895le

1 and sum119899119895=1119904

119895= 1 let 119872 = 119898

1 119898

2 119898

ℎ represent

the decision-maker set let 119879 = 119905

1 119905

2 119905

ℎ represent

the period set let 119896 = (1198961 119896

2 119896

ℎ) represent the period

significance coefficients with 0 le 119896119897le 1 and sumℎ

119897=1119896

119897= 1 let

119881

(119897)

= (V(119897)119894119895)

119898times119899represent the initial decision matrix in period

119905

119897 and let V(119897)

119894119895represent the alternative 119886

119894evaluation value

under the attribute 119888119895in period 119905

119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)

42 The Decision-Making Method

Step 1 (determine the period significance coefficients) Usu-ally the nearer the period approaches the decision pointthe more important the period will be and its significancecoefficients should be bigger In addition there are somefluctuations in the decision-making environment caused bythe uncertainty of the information and the vagueness ofthe human thinking In order to obtain the stable periodsignificance coefficients the two factors mentioned aboveshould be comprehensively considered in the process of theperiod significance coefficients optimal model constructionSuppose 120575 = sum

ℎ

119897=1((ℎ minus 119897)(ℎ minus 1))119896

119897 then 120575 is called

the time degree The smaller 120575 is the more importantinformation closer to the decision point will be The stableperiod significance coefficients can be calculated based on thevalue of 120575 and the distances of alternativesrsquo evaluation valuein different periods

The sum of distances among alternativesrsquo evaluationvalues in period 119905

119897can be computed

119889

119897=

119898

sum

119894=1

119898

sum

119896=1

119899

sum

119895=1

119889 (V(119897)119894119895 V(119897)119896119895) (12)

The volatility of decision environment can be measuredby variance

119863

2(119889

119897119896

119897) =

ℎ

sum

119897=1

[(119889

119897119896

119897minus 119864 (119889

119897119896

119897))]

2

=

1

ℎ

ℎ

sum

119897=1

(119889

119897119896

119897)

2minus

1

ℎ

2(

ℎ

sum

119897=1

119889

119897119896

119897)

2

(13)


In order to obtain the stable period significance coeffi-cients the period significance coefficients optimal model isconstructed as follows

min 119885 = 119863

2(119889

119897119896

119897)

st 120575 =

ℎ

sum

119897=1

ℎ minus 119897

ℎ minus 1

119896

119897

ℎ

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 ℎ

(14)

The decision information in different periods is inte-grated according to the period significance coefficients andthe comprehensive evaluation matrix 119881 is obtained 119881 =

(V119894119895)

119898times119899 V119894119895= (V119871119894119895 V119872119894119895 V119880119894119895)

Step 2 (normalize the decision matrix) The decision matrix

119881 = (V119894119895)

119898times119899should be normalized according to the following

equations [30]

Vlowast119871119894119895=

V119871119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(15)

Vlowast119872119894119895

=

V119872119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(16)

Vlowast119880119894119895=

V119880119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(17)

where Vlowast119894119895= (Vlowast119871119894119895 Vlowast119872119894119895 Vlowast119880119894119895) is the normalized form of V

119894119895=

(V119871119894119895 V119872119894119895 V119880119894119895) 0 le Vlowast119871

119894119895le Vlowast119872119894119895

le Vlowast119880119894119895le 1

Step 3 (calculate the attribute significance coefficients) Theattribute significance coefficients are used to represent theimportance of the attribute which can be calculated by themaximum deviation method The bigger the deviation ofthe attribute value is the bigger the attribute significancecoefficient will be The attribute significance coefficients canbe computed by the following equation

119904

119895=

sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

sum

119899

119895=1sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

(18)

where 119889(Vlowast119894119895 Vlowast119896119895) represents the distance between alternatives

119886

119894and 119886119896under attribute 119888

119895

Step 4 (the triangular fuzzy MULTIMOORA method)

Step 41 (the triangular fuzzy ratio system method)The eval-uation values of alternatives under the triangular fuzzy

ratio system method can be obtained according to (19)Hence

119884

119894=

119892

sum

119895=1

119904

119895Vlowast119894119895minus

119899

sum

119895=119892+1

119904

119895Vlowast119894119895 (19)

in which 119884119894represents the evaluation value of alternative 119886

119894

119884

119894= (119884

119871

119894 119884

119872

119894 119884

119880

119894) 119892 and 119899 minus 119892 represent the numbers of

benefit attributes and cost attributes respectively The bigger

119884

119894is the better alternative 119886

119894will be

The optimal alternative in the triangular fuzzy ratiosystem method can be obtained by (20) Hence

119886

lowast

TRS = 119886119894 | max119894

119884

119894 (20)

Step 42 (the triangular fuzzy reference point method) Thereference point of the 119895th attribute can be determinedaccording to (21) Hence

119895=

+

119895= max119894

Vlowast119894119895 119895 le 119892

minus

119895= min119894

Vlowast119894119895 119895 gt 119892

(21)

Thedeviation between Vlowast119894119895and 119895can be defined as follows

119889

119894119895= 119889 (119904

119895

119895 119904

119895Vlowast119894119895) (22)

Based on this the evaluation values of alternatives underthe triangular fuzzy reference pointmethod can be calculatedas the following equation

119885

119894= max119895

119889

119894119895= max119895

119889 (119904

119895

119895 119904

119895Vlowast119894119895) (23)

According to (9) the smaller 119885119894is the better alternative

119886

119894will be The optimal alternative under the triangular fuzzy

reference point method can be determined by the followingequation

119886

lowast

TRP = 119886119894 | min119894

119885

119894 (24)

Step 43 (the triangular fuzzy full multiplicative form meth-od)The evaluation values of alternatives under the triangularfuzzy full multiplicative form method can be computedaccording to (25) Hence

119880

119894=

prod

119892

119895=1(Vlowast119894119895)

119904119895

prod

119899

119895=119892+1(Vlowast119894119895)

119904119895 (25)

where 119880119894represents the evaluation value of the alternative

119886

119894 The bigger 119880


119894will be Hence

the optimal alternative under the triangular fuzzy full mul-tiplicative form method can be determined by the followingequation

119886

lowast

TMF = 119886119894 | max119894

119880

119894 (26)


Table 1 The decision matrixes in different periods

119888

1119888

2119888

3

119881

(1)

119886

1[54 62 78] [75 86 92] [43 46 52]

119886

2[62 71 82] [69 75 88] [38 45 51]

119886

3[69 76 88] [73 88 95] [43 47 57]

119886

4[66 75 89] [66 75 84] [42 48 56]

119881

(2)

119886

1[64 73 85] [73 84 91] [37 44 49]

119886

2[65 70 84] [65 73 85] [35 41 47]

119886

3[58 65 78] [73 85 93] [43 46 49]

119886

4[76 83 94] [67 75 83] [45 47 52]

119881

(3)

119886

1[54 62 78] [65 76 82] [43 51 58]

119886

2[62 71 82] [59 73 81] [41 48 55]

119886

3[69 76 88] [80 88 95] [46 54 59]

119886

4[66 75 89] [63 75 89] [45 30 62]

Table 2 The comprehensive decision matrix 119881

119888

1119888

2119888

3

119881

119886

1[58 66 80] [69 80 86] [41 48 54]

119886

2[63 71 83] [62 73 83] [38 45 52]

119886

3[65 72 84] [77 87 94] [45 50 55]

119886

4[70 78 91] [65 75 86] [45 50 58]

Step 5 (the final ranking of alternatives) The alternativerankings obtained from Steps 41 42 and 43 can be inte-grated into the final ranking of alternatives according to thedominance theory

5 Numerical Example

An investment company intends to choose and invest the bestone from four alternatives In order to obtain the reasonableresult 5 experts are invited to evaluate alternatives from theeconomic benefits the social benefits and the environmentpollution Let 119860 = 119886

1 119886

2 119886

3 119886

4 represent the alternative

set let 119862 = 119888

1 119888

2 119888

3 represent the attribute set (119888

1and 1198882

are the benefit attributes 1198883is the cost attribute) let 119879 =

119905

1 119905

2 119905

ℎ represent the period set let 119904 = (119904

1 119904

2 119904

119899)

represent the attribute significance coefficients with 0 le 119904119895le

1 andsum119899119895=1119904

119895= 1 let 119896 = (119896

1 119896

2 119896


significance coefficients with 0 le 119896119895le 1 and sum119899

119895=1119896

119895= 1 let

119881

(119897)

= (V(119897)119894119895)

119898times119899represent the decisionmatrix in period 119905

119897 and

let V(119897)119894119895

represent the evaluation value of alternative 119886119894under

attribute 119888119895in period 119905

119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)The decision

matrixes in different periods are constructed according to theevaluation values given by each expert the results are shownin Table 1

Step 1 (determine the period significance coefficients) Thedistances among alternatives in different periods can becalculated by (12) 119889

1= 1522 119889

2= 1786 and 119889

3= 1747

Let 120575 be 03 according to decision-makersrsquo experience Basedon this the period significance coefficients optimal model isconstructed by (14) Hence

min 119885

=

((1522119896

1)

2+ (1786119896

2)

2+ (1747119896

3)

2)

3

minus

(1522119896

1+ 1786119896

2+ 1747119896

3)

2

9

st 119896

1+

1

2

119896

2= 03

3

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 3

(27)

The period significance coefficients can be computedusing lingo 110 119896

1= 0122 119896

2= 0356 and 119896

3= 0522 Based

on this the comprehensive decision matrix 119881 is obtained byaggregating the evaluation values of alternatives in differentperiods the results are shown in Table 2

Step 2 (normalize the comprehensive decision matrix) Thecomprehensive decision matrix 119881 can be normalized to 119881lowast

according to (15)sim(17) the results are shown in Table 3

Step 3 (determine the attribute significance coefficients)According to (18) the attribute significance coefficients are119904 = (0335 0374 0291)


Table 3 The normalized comprehensive decision matrix 119881lowast

119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]

Table 4 The results of the triangular fuzzy ratio system method

119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2

Table 5 The results of the triangular fuzzy reference point method

Distances betweenthe attribute values

and the reference points 119885

119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2

Table 6 The results of the triangular fuzzy full multiplicative formmethod

119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2


Step 41 (the triangular fuzzy ratio system method)The eval-uation values and the ranking of alternatives under thetriangular fuzzy ratio system method can be obtained by (19)and (20) The results are listed in Table 4

Step 42 (the triangular fuzzy reference point method) Theevaluation values and the ranking of alternatives under thetriangular fuzzy reference point method can be obtained by(21)sim(24) and the results are listed in Table 5

Step 43 (the triangular fuzzy full multiplicative form meth-od) The evaluation values and the ranking of alternativesunder the triangular fuzzy full multiplicative form methodcan be obtained according to (25) and (26) and the results areshown in Table 6

Step 5 (the final ranking of alternatives) The rankings ofalternatives obtained by Step 4 can be integrated into the final

ranking by the dominance theory The results are shown inTable 7

Recently many MADM methods are proposed to dealwith the problem under the fuzzy environment Liu [15]proposed an extension of the technique for order prefer-ence by similarity to ideal solution (TOPSIS) based on thetriangular fuzzy number Jiang [16] extended the VIsekri-terijumska optimizacija i KOmpromisno Resenje (VIKOR)into the triangular fuzzy number environment Chatterjeeand Chakraborty [25] used the operational competitivenessrating analysis (OCRA) the preference ranking organizationmethods for enrichment evaluations (PROMETHEE) andF-TOPSIS to solve the material selection problem In orderto illustrate the effectiveness and feasibility of our proposedmethod methods proposed by literatures [15 16 25] are usedto deal with the decision problem mentioned above Theresults are listed in Table 8

All methods listed in Table 8 obtained 1198864as the optimal

alternative which proves our proposed method is effectiveand feasible

6 Conclusion

In the practical decision-making if decision-makers ignorethe historical information of the decision object they willnot comprehensively describe the decision object and obtainreasonable results However most of the existing MAGDMmethods only consider the current information of the deci-sion object Besides decision-makers are difficult to givethe crisp evaluation value because of the complexity of thedecision environment and the fuzziness of human cognitionAiming at these problems mentioned above we put forwarda novel extended method of MULTIMOORA to solve theMAGDM problem

Compared with other existing MAGDM methods theadvantages of our proposed method are described as follows(1) our proposed method considers the information of thedecision object in different periods so it can comprehensivelydescribe the decision object and obtains more reasonableresults (2) in the process of constructing period significancecoefficients optimization model the information influencedegree in different periods on the decision results and thevolatility of the decision environment are considered whichwill help to obtain more stable period significance coeffi-cients (3) triangular fuzzy numbers are used to represent thedecision information which not only can express the intervalranges of the decision information but also can highlightthe gravity center having the largest probability In additiontriangular fuzzy numbers can make up the deficiencies of


Table 7 The final ranking of alternatives

RankingThe triangular fuzzy ratio system

methodThe triangular fuzzy reference point

methodThe triangular fuzzy full multiplicative

form methodThe finalranking

119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1

Table 8 The comparisons between the proposed method and other methods

RankingThe proposed method TF-TOPSIS [15] TF-VIKOR [16] F-VIKOR [25] PROMETHEE [25] OCRA [25]

119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1

the interval number in precision and facilitate the decision-making at least to some extent In this paper all significancecoefficients are exact numbers The future work of this paperis to discuss the significance coefficient being triangular fuzzynumber form

Competing Interests

There are no competing interests related to this paper

References

[1] G-W Wei ldquoA method for multiple attribute group decisionmaking based on the ET-WG and ET-OWG operators with 2-tuple linguistic informationrdquo Expert Systems with Applicationsvol 37 no 12 pp 7895ndash7900 2010

[2] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010

[3] Z P Chen and W Yang ldquoAn MAGDM based on constrainedFAHP and FTOPSIS and its application to supplier selectionrdquoMathematical and Computer Modelling vol 54 no 11-12 pp2802ndash2815 2011

[4] G-WWei ldquoGrey relational analysis method for 2-tuple linguis-tic multiple attribute group decision making with incompleteweight informationrdquo Expert Systems with Applications vol 38no 5 pp 4824ndash4828 2011

[5] Z S Xu ldquoOn method for uncertain multiple attribute deci-sion making problems with uncertain multiplicative preferenceinformation on alternativesrdquo Fuzzy Optimization and DecisionMaking vol 4 no 2 pp 131ndash139 2005

[6] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[7] Z S Xu ldquoApproaches to multiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[8] S-J Chuu ldquoInteractive group decision-making using a fuzzylinguistic approach for evaluating the flexibility in a supplychainrdquo European Journal of Operational Research vol 213 no1 pp 279ndash289 2011

[9] S-M Chen and S-J Niou ldquoFuzzy multiple attributes groupdecision-making based on fuzzy preference relationsrdquo ExpertSystems with Applications vol 38 no 4 pp 3865ndash3872 2011

[10] Q-WCao and JWu ldquoThe extendedCOWGoperators and theirapplication tomultiple attributive group decisionmaking prob-lems with interval numbersrdquo Applied Mathematical Modellingvol 35 no 5 pp 2075ndash2086 2011

[11] T Balezentis and S Zeng ldquoGroup multi-criteria decision mak-ing based upon interval-valued fuzzy numbers an extensionof the MULTIMOORA methodrdquo Expert Systems with Applica-tions vol 40 no 2 pp 543ndash550 2013

[12] D Stanujkic N Magdalinovic R Jovanovic and S StojanovicldquoAn objective multi-criteria approach to optimization usingMOORAmethod and interval grey numbersrdquoTechnological andEconomic Development of Economy vol 18 no 2 pp 331ndash3632012

[13] M Dong S Li and H Zhang ldquoApproaches to group decisionmaking with incomplete information based on power geomet-ric operators and triangular fuzzy AHPrdquo Expert Systems withApplications vol 42 no 21 pp 7846ndash7857 2015

[14] O Aydin ldquoBulanik AHP ile Ankara icin hastane yer secimirdquoJournal of Dokuz Eylul University Faculty of Economics andAdministrative Sciences vol 24 no 2 pp 87ndash104 2009

[15] W-S Liu ldquoResearch on preferredmethod of connectionminingvillage based on AHP-TOPSIS of triangular fuzzy numbersrdquoApplication Research of Computers vol 33 no 2 pp 458ndash4622016

[16] W-Q Jiang ldquoExtension of VIKOR method for multi-criteriagroup decision making problems with triangular fuzzy num-bersrdquo Control and Decision vol 30 no 6 pp 1059ndash1064 2015

[17] W K M Brauers and E K Zavadskas ldquoThe MOORA methodand its application to privatization in a transition economyrdquoControl and Cybernetics vol 35 no 2 pp 445ndash469 2006

[18] P Karande and S Chakraborty ldquoApplication of multi-objectiveoptimization on the basis of ratio analysis (MOORA) method


for materials selectionrdquoMaterials amp Design vol 37 pp 317ndash3242012

[19] W KM Brauers A Balezentis and T Balezentis ldquoMultimoorafor the EU member States updated with fuzzy number theoryrdquoTechnological and Economic Development of Economy vol 17no 2 pp 259ndash290 2011

[20] W K M Brauers and E K Zavadskas ldquoProject managementbyMULTIMOORA as an instrument for transition economiesrdquoTechnological and Economic Development of Economy vol 16no 1 pp 5ndash24 2010

[21] W K M Brauers and E K Zavadskas ldquoMultimoora opti-mization used to decide on a bank loan to buy propertyrdquoTechnological and Economic Development of Economy vol 17no 1 pp 174ndash188 2011

[22] A Hafezalkotob and A Hafezalkotob ldquoComprehensive MUL-TIMOORAmethod with target-based attributes and integratedsignificant coefficients for materials selection in biomedicalapplicationsrdquoMaterials and Design vol 87 pp 949ndash959 2015

[23] W K M Brauers and E K Zavadskas ldquoRobustness of MUL-TIMOORA a method for multi-objective optimizationrdquo Infor-matica vol 23 no 1 pp 1ndash25 2012

[24] R R Yager ldquoOn ordered weighted averaging aggregation oper-ators in multicriteria decisionmakingrdquo IEEE Transactions onSystems Man and Cybernetics vol 18 no 1 pp 183ndash190 1988

[25] P Chatterjee and S Chakraborty ldquoMaterial selection usingpreferential rankingmethodsrdquoMaterials andDesign vol 35 pp384ndash393 2012

[26] H-C Liu X-J Fan P Li and Y-Z Chen ldquoEvaluating therisk of failure modes with extended MULTIMOORA methodunder fuzzy environmentrdquo Engineering Applications of ArtificialIntelligence vol 34 pp 168ndash177 2014

[27] A Balezentis T Balezentis and W K M Brauers ldquoPersonnelselection based on computing with words and fuzzy MULTI-MOORArdquo Expert Systems with Applications vol 39 no 9 pp7961ndash7967 2012

[28] S Datta N Sahu and S Mahapatra ldquoRobot selection basedon greyminusMULTIMOORA approachrdquo Grey Systems Theory andApplication vol 3 no 2 pp 201ndash232 2013

[29] G S Liang and J F Ding ldquoFuzzy MCDM based on the conceptof 120572-cutrdquo Journal of Multi-Criteria Decision Analysis vol 12 no6 pp 299ndash310 2003

[30] J M Merigo and M Casanovas ldquoInduced and uncertain heavyOWA operatorsrdquoComputers and Industrial Engineering vol 60no 1 pp 106ndash116 2011

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


MULTIMOORA (MOORA plus the full multiplicative form)[19ndash21] Compared with other MADM methods MULTI-MOORA is easier to understand and implement BesidesMULTIMOORA can facilitate the decision-making processand provide effective rankings [22ndash25] At present MUL-TIMOORA has been expanded into different forms accord-ing to the fuzzy logic theory Brauers et al [19] extendedMULTIMOORA into the fuzzy number environment Liuet al [26] expanded MULTIMOORA based on the fuzzynumber and used it to evaluate the risk of the failure modeBalezentis et al [27] presented a fuzzy MULTIMOORAmethod for linguistic reasoning and used it to deal with acandidate selection problem Datta et al [28] proposed grey-MULTIMOORA and used it to solve the robot selectionproblem

However few studies extended MULTIMOORA intothe triangular fuzzy number environment and obtainedthe final ranking by the dominance theory Balezentis etal [27] extended MULTIMOORA based on the triangu-lar fuzzy number But they only considered the currentdecision information and ignored the importance of thehistorical information In addition the attribute signifi-cance coefficients are given in advance which will leadto unreasonable results Based on this we developed anovel extension of MULTIMOORA method based on con-sidering the current and historical information to solveMAGDM problemsThemajor contribution of our proposedmethod is that it considered the current and the historicalinformation of the decision object So it can describe thedecision object comprehensively and objectively Besides toobtain the stable period significance coefficients the periodsignificance coefficients optimization model is constructedaccording to the time degree and the differences of thedecision information in different periods Hence our pro-posed method can help to obtain more objective decisionresults

The remainder of this paper is organized as followsSection 2 briefly introduces somebasic concepts related to thetriangular fuzzy number Section 3 mainly presents the stepsof the original MULTIMOORA method A novel extensionof MULTIMOORA is proposed in Section 4 In order toillustrate the feasibility and effectiveness of the proposedmethod a numerical example is presented in Section 5Finally conclusions are given in Section 6

2 Triangular Fuzzy Numbers

Compared with the crisp number the triangular fuzzy num-ber is more in accord with the uncertainty of the decisionenvironment and the fuzziness of human thinking Besidescompared with the interval number the triangular fuzzynumber not only can represent the interval range of thedecision information but also can highlight the gravitycenter having the maximum value probability Hence thetriangular number can reduce the information distortion andinformation deviation in the process of decision-makingBased on this this paper represents the decision informationwith triangular fuzzy numbers

Definition 1 A triangular fuzzy number is defined as(119886

119871 119886

119872 119886

119880) and its membership function 120583

(119909) can be

defined as [28]

120583

(119909) =

(119909 minus 119886

119871)

(119886

119872minus 119886

119871)

119886

119871le 119909 le 119886

119872

(119886

119880minus 119909)

(119886

119880minus 119886

119872)

119886

119872le 119909 le 119886

119880

0 otherwise

(1)

in which 119909 isin 119877 and 119886119871 and 119886119880 represent the lower boundand the upper bound of the triangular fuzzy number respectively 0 le 119886119871 le 119886119872 le 119886119880 If 119886119871 119886119872 and 119886119880 are equal degenerates a crisp value

Definition 2 Let = (119886119871 119886119872 119886119880) and 119887 = (119887119871 119887119872 119887119880) betwo triangular fuzzy numbers and let 120582 be a real number theyfollow the following rules [29 30]

(1) + 119887 = (119886119871 + 119887119871 119886119872 + 119887119872 119886119880 + 119887119880)(2) times 119887 = (119886119871 times 119887119871 119886119872 times 119887119872 119886119880 times 119887119880)(3) 120582 = (120582119886119871 120582119886119872 120582119886119880)(4) 120582 = ((119886119871)120582 (119886119872)120582 (119886119880)120582)(5) 119887 = (119887119871119886119880 119887119872119886119872 119887119880119886119871)

Definition 3 Let = (119886119871 119886119872 119886119880) and 119887 = (119887119871 119887119872 119887119880) be twotriangular fuzzy numbers the distance between them couldbe obtained as the following

119889 (

119887)

=

radic

[(119886

119871minus 119887

119871)

2

+ (119886

119872minus 119887

119872)

2

+ (119886

119880minus 119887

119880)

2

]

3

(2)

Definition 4 Let = (119886

119871 119886

119872 119886

119880) be a triangular fuzzy

number its defuzzy formula can be defined as the following

119886 =

119886

119871+ 2119886

119872+ 119886

119880

4

(3)

3 The MULTIMOORA Method

Brauers and Zavadskas extended the MOORA method toMULTIMOORAwhich includes three parts [17ndash19] the ratiosystem method the reference point method and the fullmultiplicative form method

Let 119860 = 119886

1 119886

2 119886

119898 be the alternative set let 119862 =

119888

1 119888

2 119888

119899 be the attribute set let 119881 = (V

119894119895)

119898times119899be the

decision matrix let V119894119895be the evaluation value of alternative

119886

119894under the attribute 119888

119895 119894 = 1 2 119898 119895 = 1 2 119899 let

119881

lowast= (Vlowast119894119895)

119898times119899be the standardized decision matrix and let Vlowast

119894119895

be the standardized form of V119894119895 Hence

Vlowast119894119895=

V119894119895

radicsum

119898

119894=1V2119894119895

(4)




as the following

119910

119894=

119892

sum

119895=1


119899

sum

119895=119892+1

Vlowast119894119895 (5)






119886

lowast

RS = 119886119894 | max119894

119910

119894 (6)


119903

119895=

max119894

Vlowast119894119895 119895 le 119892

min119894

Vlowast119894119895 119895 gt 119892

(7)





119911

119894= max119895

1003816

1003816

1003816

1003816

1003816

119903

119895minus Vlowast119894119895

1003816

1003816

1003816

1003816

1003816

(8)


119894will


119886

lowast

RP = 119886119894 | min119894

119911

119894 (9)


119880

119894=

prod

119892

119895=1Vlowast119894119895

prod

119899

119895=119892+1Vlowast119894119895

(10)

in which prod

119892

119895=1Vlowast119894119895


119895=119892+1Vlowast119894119895repre-





119886

lowast

MF = 119886119894 | max119894

119880

119894 (11)




119886

1 119886

2 119886


119888

1 119888

2 119888


1 119904

2 119904

119899)


1 and sum119899119895=1119904

119895= 1 let 119872 = 119898

1 119898

2 119898

ℎ represent


1 119905

2 119905

ℎ represent


2 119896



119897=1119896

119897= 1 let

119881

(119897)

= (V(119897)119894119895)


119905

119897 and let V(119897)




119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)



ℎ

119897=1((ℎ minus 119897)(ℎ minus 1))119896





119889

119897=

119898

sum

119894=1

119898

sum

119896=1

119899

sum

119895=1

119889 (V(119897)119894119895 V(119897)119896119895) (12)


119863

2(119889

119897119896

119897) =

ℎ

sum

119897=1

[(119889

119897119896

119897minus 119864 (119889

119897119896

119897))]

2

=

1

ℎ

ℎ

sum

119897=1

(119889

119897119896

119897)

2minus

1

ℎ

2(

ℎ

sum

119897=1

119889

119897119896

119897)

2

(13)



min 119885 = 119863

2(119889

119897119896

119897)

st 120575 =

ℎ

sum

119897=1

ℎ minus 119897

ℎ minus 1

119896

119897

ℎ

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 ℎ

(14)


(V119894119895)

119898times119899 V119894119895= (V119871119894119895 V119872119894119895 V119880119894119895)


119881 = (V119894119895)


equations [30]

Vlowast119871119894119895=

V119871119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(15)

Vlowast119872119894119895

=

V119872119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(16)

Vlowast119880119894119895=

V119880119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(17)


119894119895=

(V119871119894119895 V119872119894119895 V119880119894119895) 0 le Vlowast119871

119894119895le Vlowast119872119894119895

le Vlowast119880119894119895le 1


119904

119895=

sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

sum

119899

119895=1sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

(18)


119886


119895




119884

119894=

119892

sum

119895=1

119904


119899

sum

119895=119892+1

119904

119895Vlowast119894119895 (19)


119894

119884

119894= (119884

119871

119894 119884

119872

119894 119884

119880



119884


119894will be


119886

lowast

TRS = 119886119894 | max119894

119884

119894 (20)


119895=

+

119895= max119894

Vlowast119894119895 119895 le 119892

minus

119895= min119894

Vlowast119894119895 119895 gt 119892

(21)


119889

119894119895= 119889 (119904

119895

119895 119904

119895Vlowast119894119895) (22)


119885

119894= max119895

119889

119894119895= max119895

119889 (119904

119895

119895 119904

119895Vlowast119894119895) (23)


119886



119886

lowast

TRP = 119886119894 | min119894

119885

119894 (24)


119880

119894=

prod

119892

119895=1(Vlowast119894119895)

119904119895

prod

119899

119895=119892+1(Vlowast119894119895)

119904119895 (25)


119886

119894 The bigger 119880


119894will be Hence


119886

lowast

TMF = 119886119894 | max119894

119880

119894 (26)



119888

1119888

2119888

3

119881

(1)

119886

1[54 62 78] [75 86 92] [43 46 52]

119886

2[62 71 82] [69 75 88] [38 45 51]

119886

3[69 76 88] [73 88 95] [43 47 57]

119886

4[66 75 89] [66 75 84] [42 48 56]

119881

(2)

119886

1[64 73 85] [73 84 91] [37 44 49]

119886

2[65 70 84] [65 73 85] [35 41 47]

119886

3[58 65 78] [73 85 93] [43 46 49]

119886

4[76 83 94] [67 75 83] [45 47 52]

119881

(3)

119886

1[54 62 78] [65 76 82] [43 51 58]

119886

2[62 71 82] [59 73 81] [41 48 55]

119886

3[69 76 88] [80 88 95] [46 54 59]

119886

4[66 75 89] [63 75 89] [45 30 62]


119888

1119888

2119888

3

119881

119886

1[58 66 80] [69 80 86] [41 48 54]

119886

2[63 71 83] [62 73 83] [38 45 52]

119886

3[65 72 84] [77 87 94] [45 50 55]

119886

4[70 78 91] [65 75 86] [45 50 58]


5 Numerical Example


1 119886

2 119886

3 119886


set let 119862 = 119888

1 119888

2 119888


1and 1198882


119905

1 119905

2 119905


1 119904

2 119904

119899)


1 andsum119899119895=1119904

119895= 1 let 119896 = (119896

1 119896

2 119896



119895=1119896

119895= 1 let

119881

(119897)

= (V(119897)119894119895)


119897 and

let V(119897)119894119895



119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)The decision



1= 1522 119889

2= 1786 and 119889

3= 1747


min 119885

=

((1522119896

1)

2+ (1786119896

2)

2+ (1747119896

3)

2)

3

minus

(1522119896

1+ 1786119896

2+ 1747119896

3)

2

9

st 119896

1+

1

2

119896

2= 03

3

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 3

(27)


1= 0122 119896

2= 0356 and 119896

3= 0522 Based







119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]


119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2




119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2


119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2










6 Conclusion









119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












as the following

119910

119894=

119892

sum

119895=1


119899

sum

119895=119892+1

Vlowast119894119895 (5)






119886

lowast

RS = 119886119894 | max119894

119910

119894 (6)


119903

119895=

max119894

Vlowast119894119895 119895 le 119892

min119894

Vlowast119894119895 119895 gt 119892

(7)





119911

119894= max119895

1003816

1003816

1003816

1003816

1003816

119903

119895minus Vlowast119894119895

1003816

1003816

1003816

1003816

1003816

(8)


119894will


119886

lowast

RP = 119886119894 | min119894

119911

119894 (9)


119880

119894=

prod

119892

119895=1Vlowast119894119895

prod

119899

119895=119892+1Vlowast119894119895

(10)

in which prod

119892

119895=1Vlowast119894119895


119895=119892+1Vlowast119894119895repre-





119886

lowast

MF = 119886119894 | max119894

119880

119894 (11)




119886

1 119886

2 119886


119888

1 119888

2 119888


1 119904

2 119904

119899)


1 and sum119899119895=1119904

119895= 1 let 119872 = 119898

1 119898

2 119898

ℎ represent


1 119905

2 119905

ℎ represent


2 119896



119897=1119896

119897= 1 let

119881

(119897)

= (V(119897)119894119895)


119905

119897 and let V(119897)




119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)



ℎ

119897=1((ℎ minus 119897)(ℎ minus 1))119896





119889

119897=

119898

sum

119894=1

119898

sum

119896=1

119899

sum

119895=1

119889 (V(119897)119894119895 V(119897)119896119895) (12)


119863

2(119889

119897119896

119897) =

ℎ

sum

119897=1

[(119889

119897119896

119897minus 119864 (119889

119897119896

119897))]

2

=

1

ℎ

ℎ

sum

119897=1

(119889

119897119896

119897)

2minus

1

ℎ

2(

ℎ

sum

119897=1

119889

119897119896

119897)

2

(13)



min 119885 = 119863

2(119889

119897119896

119897)

st 120575 =

ℎ

sum

119897=1

ℎ minus 119897

ℎ minus 1

119896

119897

ℎ

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 ℎ

(14)


(V119894119895)

119898times119899 V119894119895= (V119871119894119895 V119872119894119895 V119880119894119895)


119881 = (V119894119895)


equations [30]

Vlowast119871119894119895=

V119871119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(15)

Vlowast119872119894119895

=

V119872119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(16)

Vlowast119880119894119895=

V119880119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(17)


119894119895=

(V119871119894119895 V119872119894119895 V119880119894119895) 0 le Vlowast119871

119894119895le Vlowast119872119894119895

le Vlowast119880119894119895le 1


119904

119895=

sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

sum

119899

119895=1sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

(18)


119886


119895




119884

119894=

119892

sum

119895=1

119904


119899

sum

119895=119892+1

119904

119895Vlowast119894119895 (19)


119894

119884

119894= (119884

119871

119894 119884

119872

119894 119884

119880



119884


119894will be


119886

lowast

TRS = 119886119894 | max119894

119884

119894 (20)


119895=

+

119895= max119894

Vlowast119894119895 119895 le 119892

minus

119895= min119894

Vlowast119894119895 119895 gt 119892

(21)


119889

119894119895= 119889 (119904

119895

119895 119904

119895Vlowast119894119895) (22)


119885

119894= max119895

119889

119894119895= max119895

119889 (119904

119895

119895 119904

119895Vlowast119894119895) (23)


119886



119886

lowast

TRP = 119886119894 | min119894

119885

119894 (24)


119880

119894=

prod

119892

119895=1(Vlowast119894119895)

119904119895

prod

119899

119895=119892+1(Vlowast119894119895)

119904119895 (25)


119886

119894 The bigger 119880


119894will be Hence


119886

lowast

TMF = 119886119894 | max119894

119880

119894 (26)



119888

1119888

2119888

3

119881

(1)

119886

1[54 62 78] [75 86 92] [43 46 52]

119886

2[62 71 82] [69 75 88] [38 45 51]

119886

3[69 76 88] [73 88 95] [43 47 57]

119886

4[66 75 89] [66 75 84] [42 48 56]

119881

(2)

119886

1[64 73 85] [73 84 91] [37 44 49]

119886

2[65 70 84] [65 73 85] [35 41 47]

119886

3[58 65 78] [73 85 93] [43 46 49]

119886

4[76 83 94] [67 75 83] [45 47 52]

119881

(3)

119886

1[54 62 78] [65 76 82] [43 51 58]

119886

2[62 71 82] [59 73 81] [41 48 55]

119886

3[69 76 88] [80 88 95] [46 54 59]

119886

4[66 75 89] [63 75 89] [45 30 62]


119888

1119888

2119888

3

119881

119886

1[58 66 80] [69 80 86] [41 48 54]

119886

2[63 71 83] [62 73 83] [38 45 52]

119886

3[65 72 84] [77 87 94] [45 50 55]

119886

4[70 78 91] [65 75 86] [45 50 58]


5 Numerical Example


1 119886

2 119886

3 119886


set let 119862 = 119888

1 119888

2 119888


1and 1198882


119905

1 119905

2 119905


1 119904

2 119904

119899)


1 andsum119899119895=1119904

119895= 1 let 119896 = (119896

1 119896

2 119896



119895=1119896

119895= 1 let

119881

(119897)

= (V(119897)119894119895)


119897 and

let V(119897)119894119895



119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)The decision



1= 1522 119889

2= 1786 and 119889

3= 1747


min 119885

=

((1522119896

1)

2+ (1786119896

2)

2+ (1747119896

3)

2)

3

minus

(1522119896

1+ 1786119896

2+ 1747119896

3)

2

9

st 119896

1+

1

2

119896

2= 03

3

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 3

(27)


1= 0122 119896

2= 0356 and 119896

3= 0522 Based







119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]


119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2




119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2


119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2










6 Conclusion









119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min 119885 = 119863

2(119889

119897119896

119897)

st 120575 =

ℎ

sum

119897=1

ℎ minus 119897

ℎ minus 1

119896

119897

ℎ

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 ℎ

(14)


(V119894119895)

119898times119899 V119894119895= (V119871119894119895 V119872119894119895 V119880119894119895)


119881 = (V119894119895)


equations [30]

Vlowast119871119894119895=

V119871119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(15)

Vlowast119872119894119895

=

V119872119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(16)

Vlowast119880119894119895=

V119880119894119895

radic(13)sum

119898

119894=1[(V119871119894119895)

2

+ (V119872119894119895)

2

+ (V119880119894119895)

2

]

(17)


119894119895=

(V119871119894119895 V119872119894119895 V119880119894119895) 0 le Vlowast119871

119894119895le Vlowast119872119894119895

le Vlowast119880119894119895le 1


119904

119895=

sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

sum

119899

119895=1sum

119898

119894=1sum

119898

119896=1119889 (Vlowast119894119895 Vlowast119896119895)

(18)


119886


119895




119884

119894=

119892

sum

119895=1

119904


119899

sum

119895=119892+1

119904

119895Vlowast119894119895 (19)


119894

119884

119894= (119884

119871

119894 119884

119872

119894 119884

119880



119884


119894will be


119886

lowast

TRS = 119886119894 | max119894

119884

119894 (20)


119895=

+

119895= max119894

Vlowast119894119895 119895 le 119892

minus

119895= min119894

Vlowast119894119895 119895 gt 119892

(21)


119889

119894119895= 119889 (119904

119895

119895 119904

119895Vlowast119894119895) (22)


119885

119894= max119895

119889

119894119895= max119895

119889 (119904

119895

119895 119904

119895Vlowast119894119895) (23)


119886



119886

lowast

TRP = 119886119894 | min119894

119885

119894 (24)


119880

119894=

prod

119892

119895=1(Vlowast119894119895)

119904119895

prod

119899

119895=119892+1(Vlowast119894119895)

119904119895 (25)


119886

119894 The bigger 119880


119894will be Hence


119886

lowast

TMF = 119886119894 | max119894

119880

119894 (26)



119888

1119888

2119888

3

119881

(1)

119886

1[54 62 78] [75 86 92] [43 46 52]

119886

2[62 71 82] [69 75 88] [38 45 51]

119886

3[69 76 88] [73 88 95] [43 47 57]

119886

4[66 75 89] [66 75 84] [42 48 56]

119881

(2)

119886

1[64 73 85] [73 84 91] [37 44 49]

119886

2[65 70 84] [65 73 85] [35 41 47]

119886

3[58 65 78] [73 85 93] [43 46 49]

119886

4[76 83 94] [67 75 83] [45 47 52]

119881

(3)

119886

1[54 62 78] [65 76 82] [43 51 58]

119886

2[62 71 82] [59 73 81] [41 48 55]

119886

3[69 76 88] [80 88 95] [46 54 59]

119886

4[66 75 89] [63 75 89] [45 30 62]


119888

1119888

2119888

3

119881

119886

1[58 66 80] [69 80 86] [41 48 54]

119886

2[63 71 83] [62 73 83] [38 45 52]

119886

3[65 72 84] [77 87 94] [45 50 55]

119886

4[70 78 91] [65 75 86] [45 50 58]


5 Numerical Example


1 119886

2 119886

3 119886


set let 119862 = 119888

1 119888

2 119888


1and 1198882


119905

1 119905

2 119905


1 119904

2 119904

119899)


1 andsum119899119895=1119904

119895= 1 let 119896 = (119896

1 119896

2 119896



119895=1119896

119895= 1 let

119881

(119897)

= (V(119897)119894119895)


119897 and

let V(119897)119894119895



119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)The decision



1= 1522 119889

2= 1786 and 119889

3= 1747


min 119885

=

((1522119896

1)

2+ (1786119896

2)

2+ (1747119896

3)

2)

3

minus

(1522119896

1+ 1786119896

2+ 1747119896

3)

2

9

st 119896

1+

1

2

119896

2= 03

3

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 3

(27)


1= 0122 119896

2= 0356 and 119896

3= 0522 Based







119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]


119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2




119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2


119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2










6 Conclusion









119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888

1119888

2119888

3

119881

(1)

119886

1[54 62 78] [75 86 92] [43 46 52]

119886

2[62 71 82] [69 75 88] [38 45 51]

119886

3[69 76 88] [73 88 95] [43 47 57]

119886

4[66 75 89] [66 75 84] [42 48 56]

119881

(2)

119886

1[64 73 85] [73 84 91] [37 44 49]

119886

2[65 70 84] [65 73 85] [35 41 47]

119886

3[58 65 78] [73 85 93] [43 46 49]

119886

4[76 83 94] [67 75 83] [45 47 52]

119881

(3)

119886

1[54 62 78] [65 76 82] [43 51 58]

119886

2[62 71 82] [59 73 81] [41 48 55]

119886

3[69 76 88] [80 88 95] [46 54 59]

119886

4[66 75 89] [63 75 89] [45 30 62]


119888

1119888

2119888

3

119881

119886

1[58 66 80] [69 80 86] [41 48 54]

119886

2[63 71 83] [62 73 83] [38 45 52]

119886

3[65 72 84] [77 87 94] [45 50 55]

119886

4[70 78 91] [65 75 86] [45 50 58]


5 Numerical Example


1 119886

2 119886

3 119886


set let 119862 = 119888

1 119888

2 119888


1and 1198882


119905

1 119905

2 119905


1 119904

2 119904

119899)


1 andsum119899119895=1119904

119895= 1 let 119896 = (119896

1 119896

2 119896



119895=1119896

119895= 1 let

119881

(119897)

= (V(119897)119894119895)


119897 and

let V(119897)119894119895



119897 V(119897)119894119895= (V(119897)119871119894119895 V(119897)119872119894119895

V(119897)119880119894119895)The decision



1= 1522 119889

2= 1786 and 119889

3= 1747


min 119885

=

((1522119896

1)

2+ (1786119896

2)

2+ (1747119896

3)

2)

3

minus

(1522119896

1+ 1786119896

2+ 1747119896

3)

2

9

st 119896

1+

1

2

119896

2= 03

3

sum

119897=1

119896

119897= 1 119896

119897isin [0 1] 119897 = 1 2 3

(27)


1= 0122 119896

2= 0356 and 119896

3= 0522 Based







119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]


119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2




119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2


119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2










6 Conclusion









119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119888

1119888

2119888

3

119881

lowast

119886

1[0388 0445 0543] [0438 0508 0548] [0419 0491 0554]

119886

2[0426 0477 0558] [0395 0465 0528] [0395 0463 0530]

119886

3[0440 0487 0571] [0487 0552 0598] [0458 0516 0566]

119886

4[0469 0525 0613] [0411 0476 0548] [0460 0516 0591]


119884

119894119884

119894Ranking

119886

1[0172 0196 0226] 0197 4

119886

2[0176 0199 0231] 0201 3

119886

3[0196 0220 0250] 0221 1

119886

4[0177 0204 0238] 0206 2




119894Ranking

119888

1119888

2119888

3

119886

1 0026 0018 0008 0026 3119886

2 0016 0031 0000 0031 4119886

3 0012 0000 0015 0015 1119886

4 0000 0026 0017 0026 2


119880

119894119880

119894Ranking

119886

1[0635 0728 0838] 0732 4

119886

2[0639 0733 0849] 0739 3

119886

3[0685 0763 0858] 0767 1

119886

4[0648 0740 0849] 0745 2










6 Conclusion









119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119886

1 4 3 4 4119886

2 3 4 3 3119886

3 2 2 1 2119886

4 1 1 2 1



119886

14 4 4 4 3 4

119886

23 3 3 3 4 2

119886

32 2 2 2 2 3

119886

41 1 1 1 1 1


Competing Interests


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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