Research Article Material Selection in Engineering Design...

Research ArticleMaterial Selection in Engineering DesignUsing Choquet Integral-Based Linguistic Operators underHybrid Environment

Anhua Peng Xiqin Wen and Kaibo Wu

Engineering Training Center Huaihai Institute of Technology Lianyungang Jiangsu 222005 China

Correspondence should be addressed to Anhua Peng pah7301126com

Received 27 March 2015 Revised 1 July 2015 Accepted 14 July 2015

Academic Editor Kalyana C Veluvolu

Copyright copy 2015 Anhua Peng 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

The performance of phase changematerials directly influences the performance and cost of thermal energy storage and it is the firstimportant task to select the suitable phase change materials for use in a particular kind of applications Due to the decision makerrsquosknowledge field and the nature of evaluated attributes assessments are alwayswith different formats whichwere first unified into thelinguistic terms in the basic linguistic term set Two-additive fuzzy measures were used to model criteria interactions by pairs andthe special expressions of Marichal entropy and Choquet integral were derived more convenient to use in practice Fuzzy measureswere identified based on the maximum of Marichal entropy and based on the Choquet integral the linguistic hybrid weightedgeometric averaging with interaction was developed for integrating the individual attributesrsquo ratings The detailed decision makingprocedure was illustrated with the material 332Cu as the optimal solution which by comparison is reasonable and trustworthy

1 Introduction

Engineering design draws on tens of thousands of materialsand on many hundreds of processes to shape join and finishthem One aspect of optimized design of a product or systemis that of selecting from this vast menu the materials andprocesses that best meet the needs of the design maximizingits performance and minimizing its cost The selection ofthe most appropriate materials not only affects the capabilityof manufacturing systems and satisfaction of customers butalso impacts environmental issues Furthermore materialselection is the prerequisite for a chain of different engineer-ing selection problems for instance process selection andmachine selection As pointed out by Tawancy et al in [1]the variation inmaterial and design resulted in significant dif-ference in service performanceThe pipe using heat-resistantcasting steel failed after only 22000 h of service while thatusing wrought INCOLOY alloy 800H remained in operationfor 83000 hTherefore material selection plays an importantrole in product cost and performance throughout its life cycleAn ever increasing variety of materials is available today witheach having its own characteristics applications advantages

and limitations There is no material which satisfies all therelevant properties For example some materials are goodenough to satisfy cost-related criteria but not so good interms of some mechanical criteria while some are good tosatisfy a set of thermal criteria but not suitable in terms ofcost and so on The large number of materials available todesigners coupled with complex interrelationship betweenthe different selection parameters often makes the materialselection process a difficult task The traditional materialselection methods such as those based only on designersrsquoexperiences try-and-error methods or analogymethods areoften made in the following way one chooses between a fewmaterials which have been used for similar situations beforeThis often leads to a conservative choice and one also missesnewly developedmaterials whichmay be suitable for the newmodified situation

To ease out thematerial selection procedure andmake theright decision a systematic and efficient approach is requiredAccording to literature retrieval these methods can roughlybe classified as material selection charts knowledge-basedmethods and multiattribute decision making (MADM)Ashby has suggested material selection chart also known as

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2015 Article ID 943795 12 pageshttpdxdoiorg1011552015943795

2 Journal of Control Science and Engineering

Ashby chart for selectingmaterials in a given application andit is widely used in the literature as [2] However drawing theAshby chart requires a broad engineering knowledge whichsometimes makes it difficult for a practitioner to employ themethod and the material selection procedure is performedbased on two performance indices per chart Consequentlyif more than two performance indices are required to beconsidered then it should be done using a sequential processIn addition Ashbyrsquos charts normally offer a range or a list ofmaterials to the designers to choose from so they can onlybe used in material screening not in material ranking Fuzzyinference systems (FIS) and genetic algorithm (GA) are twotypical knowledge-based methods and can be found widelyused in material selection as in [3 4] However a limitationof the FIS is that the inclusion of a new criterion increasesexponentially the number of decision rules of an inferencesystem The main drawback of GA is that it requires users tohave a level of specialized knowledge that is likely to be wellbeyond that possessed by most managers and organizationaldecision makers Also a severe drawback of GA is that somefeasible solutions cannot be generated by crossover operation[5] As stated in [6] this research provides evidence that theMADM approaches have potential to greatly improve thematerial selection methodology which motivates this paperto useMADM to address the phase change material selectionproblem

Much literature using MADM deals with the materialselection However in most of the literature on the materialselection only one kind of ratings for attributes was consid-ered In the literature [7] three kinds of ratings for attributesare considered exact values intervals and linguistic termsbut employing themethod of computing the interval distanceto normalize although simple in calculation loses a lot ofuseful information As stated in [8] some of these attributescan be expressed as numbers like density or thermal conduc-tivity some are Boolean such as the ability to be recycledsome like resistance to corrosion can be expressed onlyas a ranking (eg poor adequate and good) and somecan only be captured in text and images Moreover in thematerial selection process especially in the initial screeningstage the growing complexity and uncertainty of decisionsituations make it less and less possible for a decision makerto consider all relevant aspects of a problem and necessitatethe participation of multiple experts in decision making toconsider every aspect completely draw on collective wisdomabsorb all useful ideas and finally improve decision makingresultsDue to the decisionmakerrsquos knowledge field attitudesmotivations and personality and the nature of evaluatedattributes the decision makers may provide the assessmentswith different formats Such a type of MADM problemsis called the fuzzy heterogeneous MADM problems withwhich seldom literature deals [9] Consequently it is verynecessary and important to develop a normalization methodwhich dealswith the fuzzy heterogeneous information In thispaper a method is proposed to transform the heterogeneousinformation to linguistic terms in the basic linguistic term set(BLTS)

Many rankingmethods have been developed to aggregateeach attributersquos rating for all alternatives which can be

classified as two different approaches compensatory andnoncompensatory models Whether compensatory methodsor noncompensatory methods most of the ranking methodsregard attributersquos relationships as independent To all intentsand purposes the relationships among many attributesexhibit interdependences with various degrees such as therelationship between hardness and elasticmodulus increasedhardness usually leading to decreased elastic modulus andthat between strength and elongation at break increasedstrength usually leading to decreased elongation at breakThis has also given rise to the attention of many experts Asargued by Jahan et al in [10] it can be highlighted that thecorrelation between criteria is realistic in material selectionthus ranking of materials without attention to the depen-dency of material properties causes doubtable final solutionAs proposed by Karande et al in [11] future researchmay aimat improving these methods so that the possible correlationbetween the considered criteria can be taken into account forarriving at the best material selection decision Liu et al in[12] proposed that considering the interrelationship of thematerial indices is one of the subjects that should receivesome more attention in the process of material selectionIt is indeed true for a decision making model consideringinterdependences among attributes is more scientific accu-rate than that not considering interdependences which isonly a special case in decision making problems Jahan et alin [10] proposed the correlation effects weighting to mitigatethe effect of interdependences where the attribute with thegreater intercriteria correlation with the other attributes wasassigned a smaller correlation effects weighting Peng andXiao in [7] proposed the analytic network process (ANP) arelatively new MADM method based on analytic hierarchyprocess (AHP) to consider the feedback and interactionswithin and between sets of design criteria and alternativesHowever the ANP can only identify whether or not acriterion is affected by the corresponding control criterionbut cannot identify whether the interactions between any twocriteria are positive (superadditive) or negative (subadditive)and with ANP decision makers must construct so manycomparison matrices which incurs great burdens on thedecision makers

In 1974 Sugeno introduced the concept of fuzzy mea-sures substituting the additive rigid constraints in classicaltheory of probability withmonotonywithweaker constraintsand in the process of MADM employing the integrationoperators based on fuzzymeasures and integral not only takesinto account the relative weights but also flexibly representsand treats any interactions among attributes To the bestknowledge of the authors to date no paper on materialselection has used them to deal with the interdependencesamong attributes Some literature as in [13] applied Choquetintegrals to supplier selection but under the presuppositionthat the fuzzy measures are already known or are onlysubjectively identified by experts yet actually whether ornot the fuzzy measures are accurate directly determines theaccuracy of fuzzy integrals and therefore how to determinethe fuzzy measures is the key step Literature [14] and soforth employed 120582 fuzzy measures to identify fuzzy measuresfor each attribute or attribute coalition but although it can

Journal of Control Science and Engineering 3

greatly reduce the difficulty in identifying fuzzy measuresit can only express one kind of interactions either all withpositive interactions or with negative interactions abatingthe power of interaction expressions and violating the actualsituations In this paper to better capture the interactionsamong attributes two-additive fuzzy measures were used tomodel criteria interactions by pairs and to derive the specialexpressions of Marichal entropy and Choquet integral moreconvenient to use in practice Fuzzy measures were identifiedbased on the maximum of Marichal entropy Two Choquetintegral-based operators were proposed to obtain the overallratings of each alternative which were then used to sort allalternatives

2 Transforming Hybrid Informationinto Linguistic Terms in BLTS

21 Linguistic Terms When an attribute is related to qualita-tive aspects it may be difficult to qualify it using some valuesand it is very convenient to express with linguistic terms (egwhen evaluating chemical stability of a material terms likeldquovery goodrdquo ldquo goodrdquo ldquoaveragerdquo ldquo badrdquo or ldquovery badrdquo can beused) Suppose 119878 = 1199040 1199041 119904119867 is a finite and total discreteterm set where the middle term represents ldquoaveragerdquo that isa probability of approximately 05 and the remaining termsare ordered symmetrically around it As for the properties ofa linguistic term refer to [15]

With literature retrieval four ways can be found to treatthe linguistic variables (i) based on the extension principle(ii) based on the symbolic model (iii) based on virtuallinguistic terms and (iv) based on 2-tuple fuzzy linguisticrepresentation (119904

119896 119886119896) (where 119904

119896is a linguistic label from a

predefined linguistic term set 119878 119886119896 119886119896isin [minus05 05) denotes

the value of symbolic translation particularly 119886119896= 0 in a

predefined linguistic term set) Since the first two methodstake an approximation process this inevitably produces theconsequent information loss and hence the lack of preci-sion In comparison 2-tuple fuzzy linguistic representationinvolves no approximation process does not give rise toinformation loss is explicit enough in physicalmeanings andtherefore is used in this paper Let 120573 120573 isin [0 119867] be the resultof an aggregation of the indices of a set of labels assessedin a linguistic term set 119878 that is the result of a symbolicaggregation operation with 119867 + 1 as the cardinality of set119878 Then 120573 can be represented as 2-tuple (119904

119896 119886119896) using the

function Δ [16]

Δ [0 119867] 997888rarr 119878times [minus05 05)

Δ (120573) =

119904119896

119896 = round (120573)

119886119896= 120573 minus 119896 119886

119896isin [minus05 05)

(1)

where round(120573) is the usual round operation 119904119896has the

closest index label to120573 and 119886119896denotes the difference between

120573 and 119896 in 0 1 119867 Conversely let (119904119896 119886119896) be a 2-tuple

linguistic term then (119904119896 119886119896) can be represented as equivalent

numerical value 120573 isin [0 119867] using the inverse function Δminus1

[16]

Δminus1 119878 times [minus05 05) 997888rarr [0 119867]

Δminus1(119904119896 119886119896) = 119896 + 119886

119896= 120573

(2)

22 Making the Linguistic Terms Uniformed For groupdecision making problems experts may express linguisticpreferences over attributes or alternatives with differentcardinalities so in the process of information integrationwe should first uniform the linguistic terms with differentcardinalities into the ones in the BLTS Let 119878119867

[0119867minus1] be BLTSwith cardinality of 119867 and source linguistic term (119904

119866

119896 119886119896) in

set 119878119866[0119866minus1] can be equivalently transformed into (1199041015840

1198961015840 1198861198961015840) in

the BLTS using the following function [7]

(1199041015840

1198961015840 1198861198961015840) = Δ (120573

1015840) = Δ(

Δminus1(119904119866

119896 119886119896) (119867 minus 1)

119866 minus 1) (3)

The transformation function enjoys good properties ofthe one-to-one characteristic and simple calculation processand can do the inverse operation

23 Transforming the Other Information

231 Normalization Suppose 119891119894119895is the rating of alternative

119900119894(119894 = 1 2 119898) in respect of criterion 119888

119895(119895 = 1 2 119899)

Generally criteria can be classified into two types benefit(Ω1) and cost (Ω2) criteria The larger the value of analternative on the benefit criterion the better the alternativewhile the smaller the value of an alternative on the costcriterion the better the alternative If 119891

119894119895is a triangular fuzzy

number then it is denoted by 119906119894119895

(119909) = (119887119897

119894119895 119887119898

119894119895 119887119906

119894119895) or

119894119895=

(119887119897

119894119895 119887119898

119894119895 119887119906

119894119895) for short whose membership function is given as

follows

119906119894119895

(119909) =

(119909 minus 119887119897

119894119895)

(119887119898119894119895minus 119887119897

119894119895)

if 119887119897119894119895le 119909 lt 119887

119898

119894119895

1 if 119909 = 119887119898

119894119895

(119887119906

119894119895minus 119909)

(119887119906119894119895minus 119887119898

119894119895)

if 119887119898119894119895lt 119909 le 119887

119906

119894119895

(4)

If 119891119894119895is an interval number then it is denoted by 119906

ℎ119894119895

(119909) =

(ℎ119897

119894119895 ℎ119906

119894119895) or ℎ

119894119895= [ℎ

119897

119894119895 ℎ119906

119894119895] for short whose membership

function is given as follows

119906ℎ119894119895

(119909) =

1 if ℎ119897119894119895le 119909 le ℎ

119906

119894119895

0 if 119909 isin others(5)

Since the physical dimensions and measurements of the119899 attributes are different the raw data need to be normalized


For a triangular fuzzy number 119894119895= (119887

119897

119894119895 119887119898

119894119895 119887119906

119894119895) it can be

normalized as follows [9]

lowast

119894119895

=

(119887119897

119894119895

119887119906119894max

119887119898

119894119895

119887119906119895max

119887119906

119894119895

119887119906119895max

) if 119895 isin Ω1

(1 minus119887119906

119894119895

119887119906119895max

1 minus119887119898

119894119895

119887119906119895max

1 minus119887119897

119894119895

119887119906119895max

) if 119895 isin Ω2

(6)

where 119887119906119895max = max

forall119894119887119906

119894119895| 119894 = 1 2 119898

For an interval fuzzy number ℎ119894119895= [ℎ

119897

119894119895 ℎ119906

119894119895] it can be


ℎlowast

119894119895=

[

[

ℎ119897

119894119895

ℎ119906119895max

ℎ119906

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω1

[

[

1 minusℎ119906

119894119895

ℎ119906119895max

1 minusℎ119897

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω2

(7)

where ℎ119906119895max = max

forall119894ℎ119906

119894119895| 119894 = 1 2 119898

For a real number 119892119894119895 it can be normalized as follows [9]

119892lowast

119894119895=

119892119894119895

119892119894max

if 119895 isin Ω1

1 minus119892119894119895

119892119894max

if 119895 isin Ω2

(8)

where 119892119895max = max

forall119894119892119894119895| 119894 = 1 2 119898

232 Transformation of Normalization Data Size into BLTSFor convenience 119906

119873(119909) is hereafter employed to express the

membership function of the triangular interval and realnumbers Letting 119865(S

119867) be the set of fuzzy sets defined in S

119867

the BLTS 119906119873(119909) is transformed into119865(S

119867) using the function

120591119873119878119867

[17]

120591119873119878119867

119891lowast

119894119895997888rarr 119865 (S

119867)

120591119873119878119867

(119891lowast

119894119895) = (119904

1015840

119896 120574119873

119896) | 119896 isin 0 1 119867

120574119873

119896= max

forall119909

min 119906119873 (119909) 119906119904

119896

(119909)

(9)

Note that 120574119873119896is not related to 1198861015840

119896at all but represents the

extent to which 119906119873(119909) belongs to fuzzy linguistic term 119904

1015840

119896

Supposing the BLTS is 1198787 = (1199041015840

0 1199041015840

1 1199041015840

119896 119904

1015840

6) with mem-bership function of triangular fuzzy numbers 119906

1199041015840

119896

(119909) definedas 119906

1199041015840

119896

(119909) = (119887119897

119896 119887119898

119896 119887119906

119896) = (max(119896 minus 1)7 0 1198967min(119896 +

1)7 1) (119896 = 0 1 6) the transformations of a triangularfuzzy number an interval number and a real number into thelinguistic terms are illustrated respectively in Figures 1ndash3

s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N6 = 0

120574N2

120574N3 120574N4

120574N5

ublowast119894119895

Figure 1 Illustration of transforming a triangular number to alinguistic term in BLTS

s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N5 = 120574N6 = 0

120574N2120574N3 = 10000

120574N4uhlowast119894119895

(x)

Figure 2 Illustration of transforming an interval to a linguistic termin the BLTS

233 Transformation of 119865(S119867) into a 2-Tuple Linguistic

Representation 119865(S119867) is transformed into a 2-tuple linguistic

representation using the following equation [17]

120594 119865 (S119867) 997888rarr [0 119867]

120594 (119865 (S119867)) = 120594 ((119904

1015840

119896 120574119873

119896) 119896 = 0 1 119867)

=sum119867

119896=0 119896120574119873

119896

sum119867

119896=0 120574119873

119896

= 120573

119903119894119895= (119904

1015840

119897 1198861015840

119897) = Δ (120594 (119865 (S

119867))) = Δ (120573)

(10)

3 Fuzzy Measures

31 Basic Concepts

Definition 1 (see [18]) Let 119875(119862) be the power set of 119862 =

1198881 1198882 119888

119895 119888

119899 a discrete fuzzy measure on 119875(119862) is a

set function 120583 119875(119862) rarr [0 1] satisfying the followingconditions

(i) boundedness 120583(120601) = 0 120583(119862) = 1(ii) monotonicity if 119861

1 1198612isin 119875(119862) and 119861

1sube 119861

2then

120583(1198611) le 120583(119861

2)

From the perspective of MADM 120583(1198611) represents the

strength of coalition of 1198611 Intuitively we could get the


s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N2 = 120574N5 = 120574N6 = 0

120574N3

120574N4

uglowast119894119895(x)

blowastij

Figure 3 Illustration of transforming a real number to a linguisticterm in the BLTS

following results about any coalition 1198611 1198612isin 119875(119862) 119861

1cap119861

2=

120601 If 120583(1198611) + 120583(119861

2) lt 120583(119861

1cup 119861

2) 119861

1and 119861

2exhibit a negative

synergetic interaction between them if 120583(1198611) + 120583(119861

2) gt

120583(1198611cup119861

2) 119861

1and 119861

2exhibit a positive synergetic interaction

between them and if 120583(1198611) + 120583(119861

2) = 120583(119861

1cup 119861

2) 119861

1and

1198612are considered to be independent which is also called an

additivemeasure In order to avoid heavy notations hereafterwe denote 120583(119888

119894 119888

119896) 120583(119879 cup 119888

119894 119888119895) 119868(119888

119894 119888

119896) and

119898120592(119888119894 119888

119896) respectively with 120583(119894 sdot sdot sdot 119896) 120583(119879119894119895) 119868(119894 sdot sdot sdot 119896)

and119898120592(119894 sdot sdot sdot 119896)

Although fuzzy measures constitute a flexible tool formodeling the importance of coalitions they are not easy tohandle in a practical problem since we generally need to find2119899 minus 2 values for 119899 criteria In most of the practical problemsan expert can guess the importance of singletons or of pairs ofelements but not that of subsets of more elements So in thispaper 2-additive fuzzy measure was used to identify fuzzymeasures which coincides with habits of thought and is abetter trade-off between modeling accuracy and algorithmcomplexity for only 119899(119899 + 1)2 real coefficients are requiredto define a 2-additive fuzzy measure

Definition 2 (see [19]) Let 120583 be a fuzzy measure on 119875(119862)TheShapley value for every 119888

119895is defined as

119868 (119895) = sum

119879sube119862119888119895

(119899 minus |119879| minus 1) |119879|

119899[120583 (119879119895) minus 120583 (119879)] (11)

where | sdot | denotes the cardinality of a set 119868(119895) can beinterpreted as the importance of element 119888

119895with regard to

interactions A basic property of 119868(119895) issum119899119895=1

119868(119895) = 120583(119862) = 1and if the relationship of all attributes exhibits independencethen 119868(119895) = 120583(119895)

32 Two-Additive Fuzzy Measures

Definition 3 (see [19]) Set function 119898120592(119861) (119861 isin 119875(119862)) is

called Mobius representation and the relationship betweenthe Mobius representation and fuzzy measure is

119872120592 (119861) = sum

119879sube119861

(minus1)|119861|minus|119879|

120583 (119879) forall119861 isin 119875 (119862) (12)

Inversely for a given Mobius representation the corre-sponding fuzzy measure can be calculated as follows

120583 (119879) = sum

119861sube119879

119898120592 (119861) (13)

For any coalition 119879 isin 119875(119862) and |119879| gt 119896 119898120592(119879) = 0 and

there exists at least one 119879 (|119879| = 119896) while 119898120592(119879) = 0 which

we call 119896-order additive fuzzy measure Obviously if 119896 = 119899the fuzzy measure is a general fuzzy measure if 119896 = 2 thenwe call it 2-order fuzzy measure and if 119896 = 1 then it reducesto an additive measure According to (12) and (13) we have120583(119895) = 119898

120592(119895)119898

120592(120601) = 120583(120601) = 0

33 From Mobius to Interaction Index

Definition 4 (see [19]) For a givenMobius representation thecorresponding interaction index can be calculated as

119868 (119879) =

119899minus|119879|

sum

119896=0

1119896 + 1

sum

119861sube119862119879

|119861|=119896

119898120592 (119879 cup119861) (14)

where 119862 119879 is the set difference between 119862 and 119879 For the2-order fuzzy measure we can obtain the following results

119898120592 (119894) = 119868 (119894) minus

12sum

119895isin119862119894

119868 (119894119895) 119868 (119894119895) = 119898120592(119894119895) (15)

The relationship of 119888119894 119888119895exhibits a positive synergetic

interaction then 119868(119894119895) gt 0 and the stronger the positivesynergetic interaction the greater the value of 119868(119894119895) If itexhibits a negative synergetic interaction then 119868(119894119895) lt 0 Ifit is considered to be independence then 119868(119894119895) = 0

34 Entropy of Fuzzy Measure

Definition 5 (see [20]) For coalition 119862 = 1198881 1198882 119888

119895

119888119899 the entropy of fuzzy measure is defined as

119867119872(120583) =

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595 [120583 (119861119895) minus 120583 (119861)] (16)

where 120595(119909) = minus119909 ln119909 120574119861(|119862|) = (|119862|minus |119861|minus1)|119861||119862| forall119861 isin

119875(119862) and sum119861sube119862119888

119895

120574119861(|119862|) = 1

According to (16) 120583(119861119895) minus 120583(119861) = sum119879sube119861cup119895

119898120592(119879) minus

sum119879sube119861

119898120592(119879) = sum

119879sube119861119898120592(119879 cup 119895) and for 2-additive fuzzy

measure sum119879sube119861

119898120592(119879 cup 119895) can be simplified as sum

119879sube119861119898120592(119879 cup

119895) = 119898120592(119895) + sum

119894isin119861119898120592(119894119895) and substituting (15) into 119898

120592(119895) +

sum119894isin119861

119898120592(119894119895) then 120583(119861119895) minus 120583(119861) can be further simplified as

120583 (119861119895) minus 120583 (119861) = 119868 (119895) minus12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895) (17)

Compared with (16) (17) is the special case under theconditions of 2-order fuzzy measure more convenient andeasier to use


35 Procedures to Identify 2-Additive Fuzzy Measure Based onthe Maximum Entropy

Step 1 (identifying Shapley values) Since the sum of allattributesrsquo Shapley values satisfies sum

119899

119895=1119868(119895) = 1 it is

reasonable to consider the Shapley values as the weightswith no regard to dependence AHP the most widely usedapproach to identify weights is employed here and for itscalculation procedure one can refer to [21]

Step 2 (determining the range of 119868(119894119895)) According to the liter-ature [22]forall119894 119895 isin 119862 if |119868(119894119895)| = 119905

119894119895le 2min119868(119894) 119868(119895)(119899minus1)

then the nonnegativity of fuzzy measures can be ensured Assuch the interaction index 119868(119894119895) should be limited to [minus119905

119894119895 119905119894119895]

In this paper [minus119905119894119895 119905119894119895] was evenly divided into five parts

[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905

119894119895] which respectively represent the five types

of interactions significantly positive synergetic interactionpositive synergetic interaction independence negative syn-ergetic interaction and significantly negative synergeticinteraction According to decision makersrsquo cognition on theinteraction of any two attributes they determine one type ofinteraction and the corresponding range of 119868(119894119895)

Step 3 Determine 119868(119894119895) by solving the following optimizationmodel

max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)

Step 4 Determine 120583(119895) in accordance with (15)

4 Linguistic Aggregation Operatorswith Interaction

Up to now many aggregation operators have been devel-oped to aggregate linguistic ratings as linguistic orderedweighted geometric averaging operator [23] extended 2-tuple weighted geometric operator [24] and so forth How-ever most of the existent linguistic aggregation operatorsonly consider the addition of the importance of individ-ual attributes that is presupposing the relationships of allattributes are independent Based on the basic Choquetintegrals [14] this paper develops the operators consideringthe interactions between attributes

Definition 6 Linguistic weighted geometric averaging withinteraction (LWGAI) operator of dimension 119899 is a mapping

LWGAI S119899 rarr S which is defined as

(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)

where 119891(119888(119895)) is the reordering of 119891(1198881) 119891(1198882) 119891(119888119895)

119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888

(2)) ge sdot sdot sdot ge 119891(119888(119899)) 119862

(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0

(1) If all the elements in119862 are independent then 120583(119862(119895))minus

120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899

119895=1 120583(119888119895) = 1 120583(119888119895) can be considered

equal to 120596119895 the weight of the attribute 119888

119895 In this

situation the LWGAI operator is reduced to linguisticweighted geometric averaging (LWGA) operator andcan conversely be considered the extension of LWGAoperator with regard to interactions

(2) According to (13) for 2-additive fuzzy measure120583(119862

(119895)) minus 120583(119862

(119895minus1)) can be transformed into119898120592(119888(119895)) +

sum119895minus11198951=1119898120592(119888(119895) 119888(1198951)) and for the convenience of com-

putation (19) can be transformed into LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further

according to (15) can be transformed into

LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)

) has been calculated by the optimizationmodel (18)

In (20) 120583(119862(119895)) minus 120583(119862

(119895minus1)) can be considered actuallyweighting the rating of attribute 119888

(119895)itself In group decision

making different decision makers maybe provide differentratings to the identical attributes especially to the qualitativeones with some ratings unduly high but some ones undulylow Therefore in the process of integrating the individualexpertsrsquo ratings into the collective one we should not onlyassign weights to the ratings themselves to embody expertsrsquoauthority but assign weights to the ordered positions ofratings to mitigate the influence of unfair ratings on thedecision results by weighting these ratings with small valuesWith this consideration we develop the following operator

Definition 7 Linguistic hybrid weighted geometric averagingwith interaction (LHWGAI) operator of dimension 119899 is amapping LHWGAI S119899 rarr S which is defined as

(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888

(2)) ge sdot sdot sdot ge 120601(119888(119899)) In 120601(119888

119895) 120588

119895is used

to weight the ordered position of rating of attribute 119888119895with

120588119895isin [0 1] andsum119899

119895=1 120588119895 = 1 and 119899 is the balancing coefficient

(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888

119895) Therefore the LWGAI

operator is a special case of the LHWGAI whichreflects not only the importance degree of the givenratings of an attribute but also their ordered positions

(2) According to normal distribution the further a valueis apart from the mean value the smaller the value ofits probability density function is while the closer avalue is to the mean value the greater the value of itsprobability density function is which coincides withnotion mentioned above Therefore the followingformula is employed to determine the weights toweight the ordered position of the rating of attribute119888119895[25]

120588119895=

exp (minus (Δminus1 (119904119895 119886119895) minus 119902

119895)221205902

119899)

sum119899

119895=1 exp (minus (Δminus1 (119904119895 119886119895) minus 119902119895)221205902

119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean

value of Δminus1(119904119895 119886119895) (119895 = 1 2 119899) and 120590

119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902

119895)2 the standard deviation

5 Applications to Selection ofPhase Change Materials

Taking full advantage of solar power is one of the mostimportant means to mitigate energy shortages resourcedepletion environmental pollution and so forth broughtabout by traditionally thermal power generation but due today alternating with night climate change and solar energyradiation intensity fluctuating with time within a day solarenergy is an intermittent not a stable energy source Conse-quently integration of the solar power with thermal energystorage (TES) is necessary for its effective utilization as it canstore solar power and release it whenever necessary resultingin capacity buffer stable power output and increased annualutilization rate There are three types of TES sensible heatstorage phase change heat storage (latent heat storage) andthermochemical energy storage [2] Of the three types phasechange heat storage can store and release heat with almostno change in temperature and enjoys the following goodproperties stable output temperature and energy and greaterdensity in heat storage There are a large number of phasechange materials available to designers who when selectingmaterials are required to take into account a large numberof material selection criteria depending on the applicationsand the performance of phase change materials directly

influences the performance and cost of TES As such it is acomplex time consuming yet urgent problem to select thesuitable phase change materials for use in a particular kind ofTES applications Figure 4 illustrates the procedure for phasechange material selection

51 Identifying the Evaluating Criteria and Raw Data Gen-erally analyzing and translating the design requirements(expressed as constraints and objectives) into required mate-rialrsquos properties (attributes) is the first step and then wedivide the requiredmaterial properties into ldquorigidrdquo and ldquosoftrdquorequirements Any material with one property that cannotsatisfy the ldquorigidrdquo requirements can first be eliminated High-temperature molten salt and aluminum-base alloy are twokinds of the most potential phase change materials but thehigh-temperaturemolten salt suffers from lower thermal con-ductivity and solid-liquid delamination and it is eliminatedfrom the candidates not satisfying the design requirementsand constraints The five kinds of aluminum-base alloy arelisted as the candidate materials that is 35Mg6Zn (119900

1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which

the number in front of an elemental symbol expresses the per-centage of the corresponding elemental symbol The ratingsof the primary selection materials against any attribute arechecked and the attributes withmuch less distinction degreethough they may be important can be removed since theyhave little effect on the final ranking results As a result thesix attributes (119888

1 economies 119888

2 chemical stability 119888

3 phase

change latent heat 1198884 density 119888

5 thermal conductivity and 119888

6

corrosivity) employed tomaterial ranking are listed inTable 1in which the criteriarsquos types expressions and requirementsare also described Listed in Table 2 are the raw ratings 119891119905

119894119895

the rating of alternative 119900119894in respect of attribute 119888

119895provided

by expert 119890119905

52 Transforming Heterogeneous Information into LinguisticTerms in BLTS In this paper suppose the BLTS is 1198787 =

(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840

6) For ratings of attribute price (119888

4) with

real values according to (8) compute 119906119892lowast

1198944

(119909) for ratings ofattribute 119888

3with interval values according to (7) compute

119906ℎlowast

1198943

(119909) and for ratings of attribute 1198885with triangular fuzzy

numbers according to (6) compute 119906lowast

1198945

(119909) Subsequentlyaccording to (9) compute 120574119873

119896(119896 = 0 1 6) and finally

according to (10) compute (1199041015840119897 1198861015840

119897) For ratings of attributes

1198881 119888

2 and 119888

6 which are linguistic terms with different

cardinalities according to (3) make the linguistic termsuniformed Table 3 shows the normalized calculation results

53 Integrating the Individual Ratings of Each Expert

531 Identifying the Expert Shapley Values Sincesum4119905=1

119868119890(119905) =

1 119868119890(119905) is equivalent to the corresponding weight 120596

119890(119905) with

no regard to interactions Suppose I119890 I119890

= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP


Design requirementsand constraints

Identifying the initial evaluation

criteriaMaterial screening

According to the ldquorigidrdquo criteria

Identifying the feasible solutions

Ratings of the initial criteria

Distinction degrees

Identifying the final evaluation criteria

Identifying the Shapley values

According to the expertrsquos cognitionAHP

Determining the range of the interaction indexes

Solving the optimization model (18) The exact interaction indexes between any two experts

Fuzzy measures for experts

Ratings of the final criteria

Individual linguistic terms in BLTS

Collective linguistic terms

LHWGAI operator Weights of ordered positions of ratings

LWGAI operator

Overall ratings of each alternativeRanking alternatives

The exact interaction indexes between any two attributes Fuzzy measures for attributes

According to (15)

Figure 4 Procedure for phase change material selection

Table 1 Criteriarsquos types expressions and requirements

Criteria Types Expressions Requirements

1198881 economies Cost Linguistic terms

The less the ratings the better the attributes The ratings of material costsubject to various factors are hardly exactly identified and so expressedin linguistic terms according to the knowledge of experts

1198882 chemical stability Beneficial Linguistic terms

Materials with good chemical stability though subject to repeated heatabsorption and heat release do not experience the problems ofsegregation side reaction and chemolysis and hence smaller attenuationin the heat storage capacity

1198883 phase change latent heat Beneficial Intervals Under the same phase change temperature the greater the phase change

latent heat is the more the energy can be stored

1198884 density Beneficial Exact values

For the materials with approximately equal phase change latent heat thegreater the density is the greater the heat per volume can be storedwhich lowers the cost of the heat storage equipment

1198885 thermal conductivity Beneficial Triangular values

Greater thermal conductivity implies quicker speed in the process of heatstorage and extraction and better performance in conductivity and thatabsorbing or releasing the same heat requires less temperature gradient

1198886 corrosivity Cost Linguistic terms

Materials with smaller high-temperature corrosivity are compatible withmany other materials which implies a wide range of material selectionfor the heat storage pieces of equipment lowering their cost

532 Identifying the Interaction Indexes between Expertsand Their Fuzzy Measures Since the expert preferences arerelated to expertrsquos social status prestige knowledge struc-tures expectations and so forth consequently in groupdecision making the preferences among experts maybeexhibit interactions If these respects of experts are similarthe relationship of experts exhibits a negative synergeticinteraction resulting in overestimation if neglected whileif they are greatly different it exhibits a positive synergetic

interaction resulting in underestimation if neglected Deter-mine the range of the interaction indexes [minus119905

119894119895 119905119894119895] between

experts as shown in Table 4 By solving the optimizationmodel (18) the interaction indexes can be obtained 119868lowast

119890(12) =

minus00272 119868lowast119890(13) = 00296 119868lowast

119890(14) = minus00080 119868lowast

119890(23) =

minus00117 119868lowast119890(24) = 00272 and 119868lowast

119890(34) = 00381 According to

(15) determine the single expertrsquos Mobius representation forexample119898

120592119890(1) = 119868

119890(1) minus 05 times (119868lowast

119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =

02220 minus 05 times (minus00272 + 00296 minus 00080) = 02228 and


Table 2 Raw ratings on each attribute with respect to each alternative and each expert

PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593

533 Calculating the Collective Ratings For the attribute 1198884

since the 4 experts provide the same ratings on each alter-native the individual ratings are also the collective ratingsFor the attributes 119888

1 1198882 1198883 1198885 and 119888

6 respectively according

to (22) calculate respectively for the five alternatives theweight vectors (120588) of ordered position of individual expertrsquosratingsThe number of these ordered position weight vectorsamounts to 25 According to Definition 7 calculate thecollective ratings 119903

119894119895shown in the bottom of Table 3

54 Calculating the Overall Ratings of Each AlternativeWith the same method used in identifying each expertrsquosShapley values we can obtain the attributersquos Shapleyvalues as I

119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =

(02076 03170 01225 01549 01011 00968) and the fuzzymeasure of attributes as 120583

119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and

120583119888(6) = 00917 According to Definition 6 the overall

ratings of each alternative 119911119894are calculated as 119911

1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654

In the light of the results the best material is 1199002(332Cu)

followed successively by 1199005 1199004 1199003 and 119900

1

6 Conclusions

(1) This study has contributed to the material selec-tion literature by (i) considering hybrid informationincluding real values interval values triangular fuzzynumbers and linguistic variables with different car-dinalities and proposing a method to transform theheterogeneous information to linguistic terms in theBLTS (ii) providing a feasible and effective methodto determine 2-additive fuzzy measures based onthe principle of maximum entropy and further sim-plifying the expressions of Marichal entropy andChoquet integral which after simplification is moreconvenient to use in practice and (iii) proposing theLHWGAI operator considering not only the interac-tions between attributes but the ordered positions ofthe attribute ratings

(2) Compared with [26] in which the interaction indexis elicited according directly to expertrsquos cognitionlacking any objective basis however in this paperfirst merely identify the range of it according toexpertrsquos cognition and then further determine itsexact value according to the principle of maximumentropy embodying the expertrsquos cognition and enjoy-ing the good objectiveness


Table 3 Individual and corrective normalized ratings

PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840

4 05000)Collective normalized ratings

1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)

Table 4 Interaction coefficient ranges between any two experts

119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901

[00000 00000] [minus00816 minus00272] [00296 00888] [minus00296 00296]1198902

[minus00816 minus00272] [00000 00000] [minus00272 00272] [00272 00816]1198903

[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]

(3) The expressions of attribute ratings in this paper onlycover linguistic terms real values interval valuesand triangular fuzzy numbers There exist the otherexpressions such as interval-valued fuzzy numbers[27] and uncertain linguistic terms [28] In additionthis paper only involves benefit and cost criteria butsometimes the other attribute types such as target-based criteria also need to be considered As to

the target-based criteria how to normalize the fuzzyhybrid ratings is another important future researchdirection

(4) The proposed decision model for multiple attributematerial selection considering attribute interactionsunder hybrid environment is a general method andcan be easily extended to deal with othermanagement


decision making problems such as strategic manage-ment human resource management supply chainmanagement and investment management

Nomenclature

(119904119866

119896 119886119896) The 2-tuple linguistic representation in

a linguistic term set with cardinalitybeing 119866

(1199041015840


the basic linguistic term set119900119894 Alternative 119894

119888119895 Criterion 119895

119890119905 Expert 119905

119891119905

119894119895 The raw rating of alternative 119900

119894in

respect of 119888119895provided by 119890

119905

119903119905

119894119895 The normalized rating of alternative 119900

119894

in respect of 119888119895provided by 119890

119905

119903119894119895 The collective normalized rating of

alternative 119900119894in respect of 119888

119895

119911119894 The overall rating of alternative 119900

119894

119906119894119895

(sdot) Membership function120583119890(sdot) Fuzzy measure for an expert

120583119888(sdot) Fuzzy measure for an attribute

119868119890(sdot) Shapley value for an expert

119868119888(sdot) Shapley value for an attribute

119898120592119890(sdot) Mobius representation for an expert

119868119890(119894119895) Interaction index between any two

experts119868119888(119894119895) Interaction index between any two

attributesMADM Multiattribute decision makingBLTS The basic linguistic term setLWGAI Linguistic weighted geometric

averaging with interactionLHWGAI Linguistic hybrid weighted geometric

averaging with interactionANP Analytic network processAHP Analytic hierarchy process

Conflict of Interests

On behalf of the coauthors the authors declare that there isno conflict of interests regarding the publication of this paper

References

[1] H M Tawancy A Ul-Hamid A I Mohammed and NM Abbas ldquoEffect of materials selection and design on theperformance of an engineering productmdashan example frompetrochemical industryrdquo Materials amp Design vol 28 no 2 pp686ndash703 2007

[2] S Khare M DellrsquoAmico C Knight and S McGarry ldquoSelectionof materials for high temperature sensible energy storagerdquo SolarEnergy Materials amp Solar Cells vol 115 pp 114ndash122 2013

[3] K-P Lin H-P Ho K-CHung and P-F Pai ldquoCombining fuzzyweight average with fuzzy inference system for material sub-stitution selection in electric industryrdquo Computers amp IndustrialEngineering vol 62 no 4 pp 1034ndash1045 2012

[4] L Anojkumar M Ilangkumaran and V Sasirekha ldquoCompara-tive analysis of MCDM methods for pipe material selection insugar industryrdquo Expert Systems with Applications vol 41 no 6pp 2964ndash2980 2014

[5] C Wu and D Barnes ldquoFormulating partner selection criteriafor agile supply chains a Dempster-Shafer belief acceptabilityoptimisation approachrdquo International Journal of ProductionEconomics vol 125 no 2 pp 284ndash293 2010

[6] A Jahan M Y Ismail S M Sapuan and F Mustapha ldquoMate-rial screening and choosing methodsmdasha reviewrdquo Materials ampDesign vol 31 no 2 pp 696ndash705 2010

[7] A-H Peng and X-M Xiao ldquoMaterial selection usingPROMETHEE combined with analytic network processunder hybrid environmentrdquo Materials amp Design vol 47 pp643ndash652 2013

[8] M F Ashby Y J M Brechet D Cebon and L Salvo ldquoSelectionstrategies for materials and processesrdquoMaterials amp Design vol25 no 1 pp 51ndash67 2004

[9] D-F Li and S-P Wan ldquoFuzzy heterogeneous multiattributedecision making method for outsourcing provider selectionrdquoExpert Systems with Applications vol 41 no 6 pp 3047ndash30592014

[10] A Jahan F Mustapha S M Sapuan M Y Ismail and MBahraminasab ldquoA framework for weighting of criteria in rank-ing stage of material selection processrdquo International Journal ofAdvanced Manufacturing Technology vol 58 no 1ndash4 pp 411ndash420 2012

[11] P Karande S K Gauri and S Chakraborty ldquoApplications ofutility concept and desirability function formaterials selectionrdquoMaterials amp Design vol 45 pp 349ndash358 2013

[12] H-C Liu L-X Mao Z-Y Zhang and P Li ldquoInduced aggre-gation operators in the VIKOR method and its application inmaterial selectionrdquo Applied Mathematical Modelling vol 37 no9 pp 6325ndash6338 2013

[13] GUWei X F Zhao R Lin andH JWang ldquoGeneralized trian-gular fuzzy correlated averaging operator and their applicationto multiple attribute decision makingrdquo Applied MathematicalModelling vol 36 no 7 pp 2975ndash2982 2012

[14] M-L Tseng J H Chiang and L W Lan ldquoSelection ofoptimal supplier in supply chain management strategy withanalytic network process and choquet integralrdquo Computers andIndustrial Engineering vol 57 no 1 pp 330ndash340 2009

[15] Z B Wu and Y H Chen ldquoThe maximizing deviation methodfor group multiple attribute decision making under linguisticenvironmentrdquo Fuzzy Sets and Systems vol 158 no 14 pp 1608ndash1617 2007

[16] F Herrera and L Martinez ldquoA 2-tuple fuzzy linguistic repre-sentation model for computing with wordsrdquo IEE Transactionson Fuzzy Systems vol 8 no 66 pp 746ndash752 2000

[17] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[18] C Q Tan ldquoA multi-criteria interval-valued intuitionistic fuzzygroup decision making with Choquet integral-based TOPSISrdquoExpert Systems with Applications vol 38 no 4 pp 3023ndash30332011

[19] M Grabisch ldquo119896-order additive discrete fuzzy measures andtheir representationrdquo Fuzzy Sets and Systems vol 92 no 2 pp167ndash189 1997


[20] J-L Marichal ldquoEntropy of discrete Choquet capacitiesrdquo Euro-pean Journal of Operational Research vol 137 no 3 pp 612ndash6242002

[21] R V Rao and J P Davim ldquoA decision-making frameworkmodel for material selection using a combined multipleattribute decision-makingmethodrdquoThe International Journal ofAdvanced Manufacturing Technology vol 35 no 7-8 pp 751ndash760 2008

[22] J-Z Wu Q Zhang and S-J Sang ldquoSupplier evaluation modelbased on fuzzy measures and choquet integralrdquo Journal ofBeijing Institute of Technology vol 19 no 1 pp 109ndash114 2010

[23] 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

[24] 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

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

[26] G Buyukozkan O Feyzioglu and M S Ersoy ldquoEvaluation of4PL operating models a decision making approach based on 2-additive Choquet integralrdquo International Journal of ProductionEconomics vol 121 no 1 pp 112ndash120 2009

[27] S-J Chen and S-M Chen ldquoFuzzy risk analysis based onmeasures of similarity between interval-valued fuzzy numbersrdquoComputers amp Mathematics with Applications vol 55 no 8 pp1670ndash1685 2008

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

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Ashby chart for selectingmaterials in a given application andit is widely used in the literature as [2] However drawing theAshby chart requires a broad engineering knowledge whichsometimes makes it difficult for a practitioner to employ themethod and the material selection procedure is performedbased on two performance indices per chart Consequentlyif more than two performance indices are required to beconsidered then it should be done using a sequential processIn addition Ashbyrsquos charts normally offer a range or a list ofmaterials to the designers to choose from so they can onlybe used in material screening not in material ranking Fuzzyinference systems (FIS) and genetic algorithm (GA) are twotypical knowledge-based methods and can be found widelyused in material selection as in [3 4] However a limitationof the FIS is that the inclusion of a new criterion increasesexponentially the number of decision rules of an inferencesystem The main drawback of GA is that it requires users tohave a level of specialized knowledge that is likely to be wellbeyond that possessed by most managers and organizationaldecision makers Also a severe drawback of GA is that somefeasible solutions cannot be generated by crossover operation[5] As stated in [6] this research provides evidence that theMADM approaches have potential to greatly improve thematerial selection methodology which motivates this paperto useMADM to address the phase change material selectionproblem

Much literature using MADM deals with the materialselection However in most of the literature on the materialselection only one kind of ratings for attributes was consid-ered In the literature [7] three kinds of ratings for attributesare considered exact values intervals and linguistic termsbut employing themethod of computing the interval distanceto normalize although simple in calculation loses a lot ofuseful information As stated in [8] some of these attributescan be expressed as numbers like density or thermal conduc-tivity some are Boolean such as the ability to be recycledsome like resistance to corrosion can be expressed onlyas a ranking (eg poor adequate and good) and somecan only be captured in text and images Moreover in thematerial selection process especially in the initial screeningstage the growing complexity and uncertainty of decisionsituations make it less and less possible for a decision makerto consider all relevant aspects of a problem and necessitatethe participation of multiple experts in decision making toconsider every aspect completely draw on collective wisdomabsorb all useful ideas and finally improve decision makingresultsDue to the decisionmakerrsquos knowledge field attitudesmotivations and personality and the nature of evaluatedattributes the decision makers may provide the assessmentswith different formats Such a type of MADM problemsis called the fuzzy heterogeneous MADM problems withwhich seldom literature deals [9] Consequently it is verynecessary and important to develop a normalization methodwhich dealswith the fuzzy heterogeneous information In thispaper a method is proposed to transform the heterogeneousinformation to linguistic terms in the basic linguistic term set(BLTS)

Many rankingmethods have been developed to aggregateeach attributersquos rating for all alternatives which can be

classified as two different approaches compensatory andnoncompensatory models Whether compensatory methodsor noncompensatory methods most of the ranking methodsregard attributersquos relationships as independent To all intentsand purposes the relationships among many attributesexhibit interdependences with various degrees such as therelationship between hardness and elasticmodulus increasedhardness usually leading to decreased elastic modulus andthat between strength and elongation at break increasedstrength usually leading to decreased elongation at breakThis has also given rise to the attention of many experts Asargued by Jahan et al in [10] it can be highlighted that thecorrelation between criteria is realistic in material selectionthus ranking of materials without attention to the depen-dency of material properties causes doubtable final solutionAs proposed by Karande et al in [11] future researchmay aimat improving these methods so that the possible correlationbetween the considered criteria can be taken into account forarriving at the best material selection decision Liu et al in[12] proposed that considering the interrelationship of thematerial indices is one of the subjects that should receivesome more attention in the process of material selectionIt is indeed true for a decision making model consideringinterdependences among attributes is more scientific accu-rate than that not considering interdependences which isonly a special case in decision making problems Jahan et alin [10] proposed the correlation effects weighting to mitigatethe effect of interdependences where the attribute with thegreater intercriteria correlation with the other attributes wasassigned a smaller correlation effects weighting Peng andXiao in [7] proposed the analytic network process (ANP) arelatively new MADM method based on analytic hierarchyprocess (AHP) to consider the feedback and interactionswithin and between sets of design criteria and alternativesHowever the ANP can only identify whether or not acriterion is affected by the corresponding control criterionbut cannot identify whether the interactions between any twocriteria are positive (superadditive) or negative (subadditive)and with ANP decision makers must construct so manycomparison matrices which incurs great burdens on thedecision makers

In 1974 Sugeno introduced the concept of fuzzy mea-sures substituting the additive rigid constraints in classicaltheory of probability withmonotonywithweaker constraintsand in the process of MADM employing the integrationoperators based on fuzzymeasures and integral not only takesinto account the relative weights but also flexibly representsand treats any interactions among attributes To the bestknowledge of the authors to date no paper on materialselection has used them to deal with the interdependencesamong attributes Some literature as in [13] applied Choquetintegrals to supplier selection but under the presuppositionthat the fuzzy measures are already known or are onlysubjectively identified by experts yet actually whether ornot the fuzzy measures are accurate directly determines theaccuracy of fuzzy integrals and therefore how to determinethe fuzzy measures is the key step Literature [14] and soforth employed 120582 fuzzy measures to identify fuzzy measuresfor each attribute or attribute coalition but although it can






119896 119886119896) (where 119904





119896 119886119896) using the

function Δ [16]


Δ (120573) =

119904119896

119896 = round (120573)

119886119896= 120573 minus 119896 119886

119896isin [minus05 05)

(1)






[16]


Δminus1(119904119896 119886119896) = 119896 + 119886

119896= 120573

(2)



119866

119896 119886119896) in


1198961015840 1198861198961015840) in


(1199041015840

1198961015840 1198861198961015840) = Δ (120573

1015840) = Δ(

Δminus1(119904119866

119896 119886119896) (119867 minus 1)

119866 minus 1) (3)





119895(119895 = 1 2 119899)




(119909) = (119887119897

119894119895 119887119898

119894119895 119887119906

119894119895) or

119894119895=

(119887119897

119894119895 119887119898

119894119895 119887119906


follows

119906119894119895

(119909) =

(119909 minus 119887119897

119894119895)

(119887119898119894119895minus 119887119897

119894119895)

if 119887119897119894119895le 119909 lt 119887

119898

119894119895

1 if 119909 = 119887119898

119894119895

(119887119906

119894119895minus 119909)

(119887119906119894119895minus 119887119898

119894119895)

if 119887119898119894119895lt 119909 le 119887

119906

119894119895

(4)


ℎ119894119895

(119909) =

(ℎ119897

119894119895 ℎ119906

119894119895) or ℎ

119894119895= [ℎ

119897

119894119895 ℎ119906



119906ℎ119894119895

(119909) =

1 if ℎ119897119894119895le 119909 le ℎ

119906

119894119895





119897

119894119895 119887119898

119894119895 119887119906

119894119895) it can be


lowast

119894119895

=

(119887119897

119894119895

119887119906119894max

119887119898

119894119895

119887119906119895max

119887119906

119894119895

119887119906119895max

) if 119895 isin Ω1

(1 minus119887119906

119894119895

119887119906119895max

1 minus119887119898

119894119895

119887119906119895max

1 minus119887119897

119894119895

119887119906119895max

) if 119895 isin Ω2

(6)

where 119887119906119895max = max

forall119894119887119906

119894119895| 119894 = 1 2 119898


119897

119894119895 ℎ119906

119894119895] it can be


ℎlowast

119894119895=

[

[

ℎ119897

119894119895

ℎ119906119895max

ℎ119906

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω1

[

[

1 minusℎ119906

119894119895

ℎ119906119895max

1 minusℎ119897

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω2

(7)


forall119894ℎ119906

119894119895| 119894 = 1 2 119898


119892lowast

119894119895=

119892119894119895

119892119894max

if 119895 isin Ω1

1 minus119892119894119895

119892119894max

if 119895 isin Ω2

(8)


forall119894119892119894119895| 119894 = 1 2 119898





119867



120591119873119878119867

[17]

120591119873119878119867

119891lowast

119894119895997888rarr 119865 (S

119867)

120591119873119878119867

(119891lowast

119894119895) = (119904

1015840

119896 120574119873

119896) | 119896 isin 0 1 119867

120574119873

119896= max

forall119909

min 119906119873 (119909) 119906119904

119896

(119909)

(9)




1015840

119896


0 1199041015840

1 1199041015840

119896 119904

1015840


1199041015840

119896

(119909) definedas 119906

1199041015840

119896

(119909) = (119887119897

119896 119887119898

119896 119887119906

119896) = (max(119896 minus 1)7 0 1198967min(119896 +


s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N6 = 0

120574N2

120574N3 120574N4

120574N5



s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N5 = 120574N6 = 0

120574N2120574N3 = 10000

120574N4uhlowast119894119895

(x)





120594 119865 (S119867) 997888rarr [0 119867]

120594 (119865 (S119867)) = 120594 ((119904

1015840

119896 120574119873

119896) 119896 = 0 1 119867)

=sum119867

119896=0 119896120574119873

119896

sum119867

119896=0 120574119873

119896

= 120573

119903119894119895= (119904

1015840

119897 1198861015840

119897) = Δ (120594 (119865 (S

119867))) = Δ (120573)

(10)

3 Fuzzy Measures

31 Basic Concepts


1198881 1198882 119888

119895 119888




1 1198612isin 119875(119862) and 119861

1sube 119861

2then

120583(1198611) le 120583(119861

2)




s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N2 = 120574N5 = 120574N6 = 0

120574N3

120574N4

uglowast119894119895(x)

blowastij



1cap119861

2=

120601 If 120583(1198611) + 120583(119861

2) lt 120583(119861

1cup 119861

2) 119861

1and 119861

2exhibit a negative


2) gt

120583(1198611cup119861

2) 119861

1and 119861



2) = 120583(119861

1cup 119861

2) 119861

1and



119894 119888

119896) 120583(119879 cup 119888

119894 119888119895) 119868(119888

119894 119888

119896) and

119898120592(119888119894 119888





119895is defined as

119868 (119895) = sum

119879sube119862119888119895

(119899 minus |119879| minus 1) |119879|

119899[120583 (119879119895) minus 120583 (119879)] (11)








119872120592 (119861) = sum

119879sube119861

(minus1)|119861|minus|119879|

120583 (119879) forall119861 isin 119875 (119862) (12)


120583 (119879) = sum

119861sube119879

119898120592 (119861) (13)




120592(119895)119898

120592(120601) = 120583(120601) = 0



119868 (119879) =

119899minus|119879|

sum

119896=0

1119896 + 1

sum

119861sube119862119879

|119861|=119896

119898120592 (119879 cup119861) (14)


119898120592 (119894) = 119868 (119894) minus

12sum

119895isin119862119894

119868 (119894119895) 119868 (119894119895) = 119898120592(119894119895) (15)





119895


119867119872(120583) =

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595 [120583 (119861119895) minus 120583 (119861)] (16)


119875(119862) and sum119861sube119862119888

119895

120574119861(|119862|) = 1


119898120592(119879) minus

sum119879sube119861

119898120592(119879) = sum




119879sube119861119898120592(119879 cup

119895) = 119898120592(119895) + sum


120592(119895) +

sum119894isin119861


120583 (119861119895) minus 120583 (119861) = 119868 (119895) minus12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895) (17)





119899

119895=1119868(119895) = 1 it is



119894119895le 2min119868(119894) 119868(119895)(119899minus1)


119894119895 119905119894119895]


[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905




max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)






(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)


119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888


(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0


120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899



119895 In this



(119895)) minus 120583(119862




Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further


LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)


In (20) 120583(119862(119895)) minus 120583(119862





(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119896 119886119896) (where 119904





119896 119886119896) using the

function Δ [16]


Δ (120573) =

119904119896

119896 = round (120573)

119886119896= 120573 minus 119896 119886

119896isin [minus05 05)

(1)






[16]


Δminus1(119904119896 119886119896) = 119896 + 119886

119896= 120573

(2)



119866

119896 119886119896) in


1198961015840 1198861198961015840) in


(1199041015840

1198961015840 1198861198961015840) = Δ (120573

1015840) = Δ(

Δminus1(119904119866

119896 119886119896) (119867 minus 1)

119866 minus 1) (3)





119895(119895 = 1 2 119899)




(119909) = (119887119897

119894119895 119887119898

119894119895 119887119906

119894119895) or

119894119895=

(119887119897

119894119895 119887119898

119894119895 119887119906


follows

119906119894119895

(119909) =

(119909 minus 119887119897

119894119895)

(119887119898119894119895minus 119887119897

119894119895)

if 119887119897119894119895le 119909 lt 119887

119898

119894119895

1 if 119909 = 119887119898

119894119895

(119887119906

119894119895minus 119909)

(119887119906119894119895minus 119887119898

119894119895)

if 119887119898119894119895lt 119909 le 119887

119906

119894119895

(4)


ℎ119894119895

(119909) =

(ℎ119897

119894119895 ℎ119906

119894119895) or ℎ

119894119895= [ℎ

119897

119894119895 ℎ119906



119906ℎ119894119895

(119909) =

1 if ℎ119897119894119895le 119909 le ℎ

119906

119894119895





119897

119894119895 119887119898

119894119895 119887119906

119894119895) it can be


lowast

119894119895

=

(119887119897

119894119895

119887119906119894max

119887119898

119894119895

119887119906119895max

119887119906

119894119895

119887119906119895max

) if 119895 isin Ω1

(1 minus119887119906

119894119895

119887119906119895max

1 minus119887119898

119894119895

119887119906119895max

1 minus119887119897

119894119895

119887119906119895max

) if 119895 isin Ω2

(6)

where 119887119906119895max = max

forall119894119887119906

119894119895| 119894 = 1 2 119898


119897

119894119895 ℎ119906

119894119895] it can be


ℎlowast

119894119895=

[

[

ℎ119897

119894119895

ℎ119906119895max

ℎ119906

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω1

[

[

1 minusℎ119906

119894119895

ℎ119906119895max

1 minusℎ119897

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω2

(7)


forall119894ℎ119906

119894119895| 119894 = 1 2 119898


119892lowast

119894119895=

119892119894119895

119892119894max

if 119895 isin Ω1

1 minus119892119894119895

119892119894max

if 119895 isin Ω2

(8)


forall119894119892119894119895| 119894 = 1 2 119898





119867



120591119873119878119867

[17]

120591119873119878119867

119891lowast

119894119895997888rarr 119865 (S

119867)

120591119873119878119867

(119891lowast

119894119895) = (119904

1015840

119896 120574119873

119896) | 119896 isin 0 1 119867

120574119873

119896= max

forall119909

min 119906119873 (119909) 119906119904

119896

(119909)

(9)




1015840

119896


0 1199041015840

1 1199041015840

119896 119904

1015840


1199041015840

119896

(119909) definedas 119906

1199041015840

119896

(119909) = (119887119897

119896 119887119898

119896 119887119906

119896) = (max(119896 minus 1)7 0 1198967min(119896 +


s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N6 = 0

120574N2

120574N3 120574N4

120574N5



s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N5 = 120574N6 = 0

120574N2120574N3 = 10000

120574N4uhlowast119894119895

(x)





120594 119865 (S119867) 997888rarr [0 119867]

120594 (119865 (S119867)) = 120594 ((119904

1015840

119896 120574119873

119896) 119896 = 0 1 119867)

=sum119867

119896=0 119896120574119873

119896

sum119867

119896=0 120574119873

119896

= 120573

119903119894119895= (119904

1015840

119897 1198861015840

119897) = Δ (120594 (119865 (S

119867))) = Δ (120573)

(10)

3 Fuzzy Measures

31 Basic Concepts


1198881 1198882 119888

119895 119888




1 1198612isin 119875(119862) and 119861

1sube 119861

2then

120583(1198611) le 120583(119861

2)




s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N2 = 120574N5 = 120574N6 = 0

120574N3

120574N4

uglowast119894119895(x)

blowastij



1cap119861

2=

120601 If 120583(1198611) + 120583(119861

2) lt 120583(119861

1cup 119861

2) 119861

1and 119861

2exhibit a negative


2) gt

120583(1198611cup119861

2) 119861

1and 119861



2) = 120583(119861

1cup 119861

2) 119861

1and



119894 119888

119896) 120583(119879 cup 119888

119894 119888119895) 119868(119888

119894 119888

119896) and

119898120592(119888119894 119888





119895is defined as

119868 (119895) = sum

119879sube119862119888119895

(119899 minus |119879| minus 1) |119879|

119899[120583 (119879119895) minus 120583 (119879)] (11)








119872120592 (119861) = sum

119879sube119861

(minus1)|119861|minus|119879|

120583 (119879) forall119861 isin 119875 (119862) (12)


120583 (119879) = sum

119861sube119879

119898120592 (119861) (13)




120592(119895)119898

120592(120601) = 120583(120601) = 0



119868 (119879) =

119899minus|119879|

sum

119896=0

1119896 + 1

sum

119861sube119862119879

|119861|=119896

119898120592 (119879 cup119861) (14)


119898120592 (119894) = 119868 (119894) minus

12sum

119895isin119862119894

119868 (119894119895) 119868 (119894119895) = 119898120592(119894119895) (15)





119895


119867119872(120583) =

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595 [120583 (119861119895) minus 120583 (119861)] (16)


119875(119862) and sum119861sube119862119888

119895

120574119861(|119862|) = 1


119898120592(119879) minus

sum119879sube119861

119898120592(119879) = sum




119879sube119861119898120592(119879 cup

119895) = 119898120592(119895) + sum


120592(119895) +

sum119894isin119861


120583 (119861119895) minus 120583 (119861) = 119868 (119895) minus12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895) (17)





119899

119895=1119868(119895) = 1 it is



119894119895le 2min119868(119894) 119868(119895)(119899minus1)


119894119895 119905119894119895]


[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905




max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)






(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)


119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888


(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0


120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899



119895 In this



(119895)) minus 120583(119862




Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further


LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)


In (20) 120583(119862(119895)) minus 120583(119862





(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119897

119894119895 119887119898

119894119895 119887119906

119894119895) it can be


lowast

119894119895

=

(119887119897

119894119895

119887119906119894max

119887119898

119894119895

119887119906119895max

119887119906

119894119895

119887119906119895max

) if 119895 isin Ω1

(1 minus119887119906

119894119895

119887119906119895max

1 minus119887119898

119894119895

119887119906119895max

1 minus119887119897

119894119895

119887119906119895max

) if 119895 isin Ω2

(6)

where 119887119906119895max = max

forall119894119887119906

119894119895| 119894 = 1 2 119898


119897

119894119895 ℎ119906

119894119895] it can be


ℎlowast

119894119895=

[

[

ℎ119897

119894119895

ℎ119906119895max

ℎ119906

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω1

[

[

1 minusℎ119906

119894119895

ℎ119906119895max

1 minusℎ119897

119894119895

ℎ119906119895max

]

]

if 119895 isin Ω2

(7)


forall119894ℎ119906

119894119895| 119894 = 1 2 119898


119892lowast

119894119895=

119892119894119895

119892119894max

if 119895 isin Ω1

1 minus119892119894119895

119892119894max

if 119895 isin Ω2

(8)


forall119894119892119894119895| 119894 = 1 2 119898





119867



120591119873119878119867

[17]

120591119873119878119867

119891lowast

119894119895997888rarr 119865 (S

119867)

120591119873119878119867

(119891lowast

119894119895) = (119904

1015840

119896 120574119873

119896) | 119896 isin 0 1 119867

120574119873

119896= max

forall119909

min 119906119873 (119909) 119906119904

119896

(119909)

(9)




1015840

119896


0 1199041015840

1 1199041015840

119896 119904

1015840


1199041015840

119896

(119909) definedas 119906

1199041015840

119896

(119909) = (119887119897

119896 119887119898

119896 119887119906

119896) = (max(119896 minus 1)7 0 1198967min(119896 +


s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N6 = 0

120574N2

120574N3 120574N4

120574N5



s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N5 = 120574N6 = 0

120574N2120574N3 = 10000

120574N4uhlowast119894119895

(x)





120594 119865 (S119867) 997888rarr [0 119867]

120594 (119865 (S119867)) = 120594 ((119904

1015840

119896 120574119873

119896) 119896 = 0 1 119867)

=sum119867

119896=0 119896120574119873

119896

sum119867

119896=0 120574119873

119896

= 120573

119903119894119895= (119904

1015840

119897 1198861015840

119897) = Δ (120594 (119865 (S

119867))) = Δ (120573)

(10)

3 Fuzzy Measures

31 Basic Concepts


1198881 1198882 119888

119895 119888




1 1198612isin 119875(119862) and 119861

1sube 119861

2then

120583(1198611) le 120583(119861

2)




s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N2 = 120574N5 = 120574N6 = 0

120574N3

120574N4

uglowast119894119895(x)

blowastij



1cap119861

2=

120601 If 120583(1198611) + 120583(119861

2) lt 120583(119861

1cup 119861

2) 119861

1and 119861

2exhibit a negative


2) gt

120583(1198611cup119861

2) 119861

1and 119861



2) = 120583(119861

1cup 119861

2) 119861

1and



119894 119888

119896) 120583(119879 cup 119888

119894 119888119895) 119868(119888

119894 119888

119896) and

119898120592(119888119894 119888





119895is defined as

119868 (119895) = sum

119879sube119862119888119895

(119899 minus |119879| minus 1) |119879|

119899[120583 (119879119895) minus 120583 (119879)] (11)








119872120592 (119861) = sum

119879sube119861

(minus1)|119861|minus|119879|

120583 (119879) forall119861 isin 119875 (119862) (12)


120583 (119879) = sum

119861sube119879

119898120592 (119861) (13)




120592(119895)119898

120592(120601) = 120583(120601) = 0



119868 (119879) =

119899minus|119879|

sum

119896=0

1119896 + 1

sum

119861sube119862119879

|119861|=119896

119898120592 (119879 cup119861) (14)


119898120592 (119894) = 119868 (119894) minus

12sum

119895isin119862119894

119868 (119894119895) 119868 (119894119895) = 119898120592(119894119895) (15)





119895


119867119872(120583) =

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595 [120583 (119861119895) minus 120583 (119861)] (16)


119875(119862) and sum119861sube119862119888

119895

120574119861(|119862|) = 1


119898120592(119879) minus

sum119879sube119861

119898120592(119879) = sum




119879sube119861119898120592(119879 cup

119895) = 119898120592(119895) + sum


120592(119895) +

sum119894isin119861


120583 (119861119895) minus 120583 (119861) = 119868 (119895) minus12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895) (17)





119899

119895=1119868(119895) = 1 it is



119894119895le 2min119868(119894) 119868(119895)(119899minus1)


119894119895 119905119894119895]


[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905




max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)






(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)


119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888


(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0


120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899



119895 In this



(119895)) minus 120583(119862




Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further


LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)


In (20) 120583(119862(119895)) minus 120583(119862





(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










s9984000 s9984001 s9984002 s9984003 s9984004 s9984005 s9984006

0 016 034 050 066 084 1

x

120574N0 = 120574N1 = 120574N2 = 120574N5 = 120574N6 = 0

120574N3

120574N4

uglowast119894119895(x)

blowastij



1cap119861

2=

120601 If 120583(1198611) + 120583(119861

2) lt 120583(119861

1cup 119861

2) 119861

1and 119861

2exhibit a negative


2) gt

120583(1198611cup119861

2) 119861

1and 119861



2) = 120583(119861

1cup 119861

2) 119861

1and



119894 119888

119896) 120583(119879 cup 119888

119894 119888119895) 119868(119888

119894 119888

119896) and

119898120592(119888119894 119888





119895is defined as

119868 (119895) = sum

119879sube119862119888119895

(119899 minus |119879| minus 1) |119879|

119899[120583 (119879119895) minus 120583 (119879)] (11)








119872120592 (119861) = sum

119879sube119861

(minus1)|119861|minus|119879|

120583 (119879) forall119861 isin 119875 (119862) (12)


120583 (119879) = sum

119861sube119879

119898120592 (119861) (13)




120592(119895)119898

120592(120601) = 120583(120601) = 0



119868 (119879) =

119899minus|119879|

sum

119896=0

1119896 + 1

sum

119861sube119862119879

|119861|=119896

119898120592 (119879 cup119861) (14)


119898120592 (119894) = 119868 (119894) minus

12sum

119895isin119862119894

119868 (119894119895) 119868 (119894119895) = 119898120592(119894119895) (15)





119895


119867119872(120583) =

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595 [120583 (119861119895) minus 120583 (119861)] (16)


119875(119862) and sum119861sube119862119888

119895

120574119861(|119862|) = 1


119898120592(119879) minus

sum119879sube119861

119898120592(119879) = sum




119879sube119861119898120592(119879 cup

119895) = 119898120592(119895) + sum


120592(119895) +

sum119894isin119861


120583 (119861119895) minus 120583 (119861) = 119868 (119895) minus12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895) (17)





119899

119895=1119868(119895) = 1 it is



119894119895le 2min119868(119894) 119868(119895)(119899minus1)


119894119895 119905119894119895]


[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905




max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)






(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)


119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888


(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0


120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899



119895 In this



(119895)) minus 120583(119862




Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further


LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)


In (20) 120583(119862(119895)) minus 120583(119862





(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119899

119895=1119868(119895) = 1 it is



119894119895le 2min119868(119894) 119868(119895)(119899minus1)


119894119895 119905119894119895]


[06119905119894119895 119905119894119895] [02119905

119894119895 06119905

119894119895] [minus02119905

119894119895 02119905

119894119895] [minus06119905

119894119895 minus02119905

119894119895] and

[minus119905119894119895 minus06119905




max 119867119872(120583)

=

119899

sum

119895=1sum

119861sube119862119888119895

120574119861 (|119862|) 120595[119868 (119895) minus

12

sum

119894isin119862119861

119868 (119894119895) +12sum

119894isin119861

119868 (119894119895)]

st I (119894119895) isin [minus119905119894119895 119905119894119895]

119868 (119895) = 120596119895

119894 119895 = 1 2 119899 119894 = 119895

(18)






(119904119896 119886119896) = LWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(19)


119891(119888119899) with 119891(119888

119895) = Δ

minus1(119904119895 119886119895) (119895 = 1 2 119899) such that

119891(119888(1)) ge 119891(119888


(119895)= (119888

(1) 119888(2) 119888(119895))119891(119888

(119899+1)) = 0 and 120583(119862(0)) = 0


120583(119862(119895minus1)) = 120583(119888

(119895)) (119904

119896 119886119896) = LWGAI(sdot) =

Δ(prod119899

119895=1119891(119888

(119895))120583(119888(119895))) = Δ(prod

119899

119895=1119891(119888

119895)120583(119888119895)) and since

120583(119862) = sum119899



119895 In this



(119895)) minus 120583(119862




Δ(prod119899

119895=1119891(119888(119895))[119898120592(119888(119895))+sum119895minus11198951=1

119898120592(119888(119895)119888(1198951))]) which further


LWGAI (sdot) = Δ(

119899

prod

119895=1119891 (119888

(119895))[120583(119888(119895))+sum119895minus11198951=1

119868(119888(119895)119888(1198951))]

) (20)

where 119868(119888(119895) 119888(1198951)


In (20) 120583(119862(119895)) minus 120583(119862





(119904119896 119886119896) = LHWGAI ((1199041 1198861) (1199042 1198862) (119904119899 119886119899))

= Δ(

119899

prod

119895=1120601 (119888

(119895))[120583(119862(119895))minus120583(119862

(119895minus1))])

(21)



120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120601(119888119899) with 120601(119888

119895) = Δ

minus1(119904119895 119886119895)119899times120588119895 (119895 = 1 2 119899) such

that 120601(119888(1)) ge 120601(119888


119895) 120588

119895is used


120588119895isin [0 1] andsum119899


(1) If 120588 = (1119899 1119899 1119899) then 120601(119888119895) = Δ

minus1(119904119895

119886119895)119899times1119899

= Δminus1(119904119895 119886119895) = 119891(119888




120588119895=


119895)221205902

119899)

sum119899


119899)

119895 = 1 2 119899

(22)

where 119902119895

= (1119899)sum119899119895=1 Δ

minus1(119904119895 119886119895) is the mean


119899=

radic(1(119899 minus 1)) sum119899119895=1(Δ

minus1(119904119895 119886119895) minus 119902






1)

332Cu (1199002) 12Si (119900

3) 5Si30Cu (119900

4) and 34Mg (119900

5) in which


1 economies 119888


3 phase



6


119894119895


119895provided

by expert 119890119905


(1199041015840

0 1199041015840

1 119904

1015840

119896 119904

1015840


4) with


1198944



119906ℎlowast

1198943



1198945


119896(119896 = 0 1 6) and finally



1198881 119888

2 and 119888





119868119890(119905) =


119890(119905) with


= (119868119890(1) 119868

119890(2)

119868119890(3) 119868

119890(4)) = (02220 02040 02860 02880) was calculated

using AHP








Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Distinction degrees











LWGAI operator



According to (15)


















119894119895 119905119894119895] between


119890(12) =


119890(14) = minus00080 119868lowast

119890(23) =


119890(34) = 00381 According to


120592119890(1) = 119868


119890(12) + 119868lowast

119890(13) + 119868lowast

119890(14)) =




PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











PCMs 1198881

1198882

1198883kJkg 119888

4kgm3

1198885W(msdotK) 119888

6

119891119905

1198941 119891119905

1198942 119891119905

1198943 119891119905

1198944 119891119905

1198945 119891119905

1198946Expert 1

1199001

11990452 119904

51 [300 320] 2380 [170 190 220] 119904

32

1199002

11990454 119904

53 [340 355] 3424 [105 130 155] 119904

31

1199003

11990451 119904

52 [540 570] 2700 [120 155 180] 119904

31

1199004

11990454 119904

54 [400 430] 2730 [160 175 205] 119904

32

1199005

11990453 119904

51 [310 350] 2300 [175 205 225] 119904

31

Expert 21199001

11990472 119904

72 [290 310] 2380 [165 190 225] 119904

73

1199002

11990475 119904

74 [320 340] 3424 [95 130 145] 119904

75

1199003

11990472 119904

73 [480 530] 2700 [110 145 185] 119904

74

1199004

11990474 119904

72 [380 420] 2730 [155 185 210] 119904

72

1199005

11990476 119904

76 [300 340] 2300 [185 210 230] 119904

76

Expert 31199001

11990454 119904

72 [310 330] 2380 [145 175 210] 119904

52

1199002

11990454 119904

76 [350 370] 3424 [105 130 160] 119904

51

1199003

11990452 119904

73 [470 490] 2700 [115 125 150] 119904

53

1199004

11990451 119904

72 [420 450] 2730 [165 190 220] 119904

54

1199005

11990453 119904

74 [305 355] 2300 [155 195 210] 119904

53

Expert 41199001

11990475 119904

32 [305 350] 2380 [157 176 205] 119904

51

1199002

11990476 119904

31 [330 370] 3424 [105 125 150] 119904

54

1199003

11990473 119904

31 [445 480] 2700 [118 135 165] 119904

52

1199004

11990472 119904

32 [430 460] 2730 [158 205 238] 119904

51

1199005

11990474 119904

31 [335 350] 2300 [165 208 235] 119904

53

according to (13) 120583119890(1) = 119898

120592119890(1) = 02228 Similarly 120583

119890(2) =

02098 120583119890(3) = 02580 and 120583

119890(4) = 02593



1 1198882 1198883 1198885 and 119888





119888= (119868

119888(1) 119868

119888(2) 119868

119888(3) 119868

119888(4) 119868

119888(5) 119868

119888(6)) =


119888(1) = 01773 120583

119888(2) = 03212

120583119888(3) = 01122 120583

119888(4) = 01402 120583

119888(5) = 00878 and



1= 31116

1199112= 42852 119911

3= 33714 119911

4= 35565 and 119911

5= 40654



1

6 Conclusions





PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











PCMs 1198881

1198882

1198883

1198884

1198885

1198886

119903119905

1198941 119903119905

1198942 119903119905

1198943 119903119905

1198944 119903119905

1198945 119903119905

1198946Expert 1

1199001

1199041015840

3 (1199041015840

2 minus05000) (1199041015840

3 03143) (1199041015840

4 01771) (1199041015840

5 00729) 1199041015840

61199002

1199041015840

6 (1199041015840

4 05000) (1199041015840

4 minus03409) (1199041015840

6 00000) (1199041015840

3 04732) 1199041015840

31199003

(1199041015840

2 minus05000) 1199041015840

3 (1199041015840

6 minus02475) (1199041015840

5 minus03105) (1199041015840

4 00429) 1199041015840

31199004

1199041015840

6 1199041015840

6 (1199041015840

3 03777) (1199041015840

5 minus02591) (1199041015840

5 minus02023) 1199041015840

61199005

(1199041015840

4 05000) (1199041015840

2 minus05000) (1199041015840

3 04954) (1199041015840

4 00583) (1199041015840

5 02671) 1199041015840

3Expert 2

1199001

1199041015840

2 1199041015840

2 (1199041015840

3 04294) (1199041015840

4 01771) (1199041015840

5 minus00072) 1199041015840

31199002

1199041015840

5 1199041015840

4 (1199041015840

4 minus02844) (1199041015840

6 00000) (1199041015840

3 01823) 1199041015840

51199003

1199041015840

2 1199041015840

3 (1199041015840

6 minus03709) (1199041015840

5 minus03105) (1199041015840

4 minus01591) 1199041015840

41199004

1199041015840

4 1199041015840

2 (1199041015840

4 04890) (1199041015840

5 minus02591) (1199041015840

5 minus01744) 1199041015840

21199005

1199041015840

6 1199041015840

6 (1199041015840

4 minus03990) (1199041015840

4 00583) (1199041015840

5 03415) 11990476

Expert 31199001

1199041015840

6 1199041015840

2 (1199041015840

4 minus00771) (1199041015840

4 01771) (1199041015840

5 minus01399) 1199041015840

31199002

1199041015840

6 1199041015840

6 (1199041015840

4 minus04021) (1199041015840

6 00000) (1199041015840

4 minus03590) (1199041015840

2 minus05000)1199003

1199041015840

3 1199041015840

3 (1199041015840

6 minus02033) (1199041015840

5 minus03105) (1199041015840

4 minus04221) (1199041015840

4 05000)1199004

(1199041015840

2 minus05000) 1199041015840

2 (1199041015840

5 03542) (1199041015840

5 minus02591) (1199041015840

5 01186) 1199041015840

61199005

(1199041015840

4 05000) 1199041015840

4 (1199041015840

4 02085) (1199041015840

4 00583) (1199041015840

5 00600) (1199041015840

4 05000)Expert 4

1199001

1199041015840

5 1199041015840

6 (1199041015840

4 02311) (1199041015840

4 01771) (1199041015840

5 minus04538) (1199041015840

2 minus05000)1199002

1199041015840

6 1199041015840

3 (1199041015840

4 03925) (1199041015840

6 00000) (1199041015840

3 01537) 1199041015840

61199003

1199041015840

3 1199041015840

3 (1199041015840

6 03131) (1199041015840

5 minus03105) (1199041015840

3 04685) 1199041015840

31199004

1199041015840

2 1199041015840

6 (1199041015840

6 minus04682) (1199041015840

5 minus02591) (1199041015840

5 00219) (1199041015840

2 minus05000)1199005

1199041015840

4 1199041015840

3 (1199041015840

4 minus03020) (1199041015840

4 00583) (1199041015840

5 00579) (1199041015840


1199031198941 119903

1198942 1199031198943 119903

1198944 1199031198945 119903

1198946

1199001

(1199041015840

4 minus01124) (1199041015840

2 01048) (1199041015840

4 minus03736) (1199041015840

4 01771) (1199041015840

5 minus04084) (1199041015840

3 minus03356)1199002

(1199041015840

6 minus01942) (1199041015840

4 minus01945) (1199041015840

4 minus00574) (1199041015840

6 00000) (1199041015840

3 02254) (1199041015840

3 02416)1199003

(1199041015840

2 04012) (1199041015840

3 00000) (1199041015840

6 minus03105) (1199041015840

5 minus03105) (1199041015840

4 minus03918) (1199041015840

3 04325)1199004

(1199041015840

3 minus04403) (1199041015840

3 03624) (1199041015840

5 minus01034) (1199041015840

5 minus02591) (1199041015840

5 minus01505) (1199041015840

3 01930)1199005

(1199041015840

5 minus3456) (1199041015840

3 03961) (1199041015840

4 minus01105) (1199041015840

4 00583) (1199041015840

5 01908) (1199041015840

5 minus03026)


119868119890(119894119895) 1198901 1198902 1198903 1198904

1198901



[00296 00888] [minus00272 00272] [00000 00000] [00381 01143]1198904

[minus00296 00296] [00272 00816] [00381 01143] [00000 00000]






Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nomenclature

(119904119866



(1199041015840



119888119895 Criterion 119895

119890119905 Expert 119905

119891119905


119894in


119905

119903119905


119894


119905



119895


119894

119906119894119895













References
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Material Selection in Engineering Design...

Documents

Transcript of Research Article Material Selection in Engineering Design...