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Algorithms2020,13,154;doi:10.3390/a13060154www.mdpi.com/journal/algorithms
Article
AnAlgorithmforFuzzyNegations
Based‐IntuitionisticFuzzyCopulaAggregation
OperatorsinMultipleAttributeDecisionMaking
StylianosGiakoumakisandBasilPapadopoulos*
DepartmentofCivilEngineeringSectionofMathematicsandInformatics,DemocritusUniversityofThrace,
67100Kimeria,Greece;sgiakoum@civil.duth.gr
*Correspondence:papadob@civil.duth.gr;Tel.:+30‐254‐1079‐747
Received:29May2020;Accepted:23June2020;Published:26June2020
Abstract:Inthispaper,wedevelopanovelcomputationmodelofIntuitionisticFuzzyValueswith
theusageoffuzzynegationsandArchimedeancopulas.Thisnovelcomputationmodel’sstructure
isbasedontheextensionoftheexistingoperationsofintuitionisticfuzzyvalueswithsomeclasses
offuzzynegations.Manypropertiesoftheproposedoperationsareinvestigatedandproved.
Additionally,inthispaperweintroducetheconceptsofintuitionisticfuzzyArchimedeancopula
weightedarithmeticandgeometricaggregationoperatorsbasedonfuzzynegations,includinga
furtheranalysisoftheirproperties.Finally,usingacasestudyfromanalreadypublishedpaperwe
foundthatourmethodhasmanyadvantages.
Keywords:intuitionisticfuzzysets;fuzzynegations;copula;intuitionisticfuzzyArchimedean
copulaaggregationoperators;multipleattributedecisionmaking
1.Introduction
Atanassov[1]introducedthenotionIntuitionisticFuzzySet(IFS),whichgeneralizesthenotion
ofFuzzySetproposedbyZadeh[2].BasedonthefactthatIFSisappropriateforcasesdealingwith
uncertaintyandvagueness,manyauthorshaveappliedthemtodecisionmaking.Inthecaseof
multipleattributedecisionmakingtheoperationallawsofIntuitionisticFuzzyValues(IFVs)
proposeanelementaryandremarkabletopic.Atanassov[1,3]introducedthebasicoperationallaws
andpresentedsomeoftheirproperties.Beliakovetal.[4]developedsomeoperationsbyusing
additivegeneratorsoftheproductt‐norm.ZhaoandWei[5]proposedtheintuitionisticfuzzy
Einsteinhybridaggregationoperators.Atanassov,PasiandYager[6]contributedtothecaseof
multi‐criteriagroupdecisionmaking,withtheattributesbeingintuitionisticfuzzynumbersandthe
correspondingweightsbeingcrispnumericalvalues;theirmethodisappliedinthecontextofpublic
relationsandmasscommunication.WangandLi[7]proposedthescorefunctionandtheweighted
scorefunctionmethods.OuyangandPedrycz[8]proposedamodelforintuitionisticfuzzymulti‐
attributesdecisionmakingthatdealswiththedegreesofmembershipandnonmembership
individuallythatisappliedinmultiplecriteriasupplierselectionproblems.Taoetal.[9]gavean
intuitionisticfuzzycopulaarithmeticaggregationoperatorinmulti‐attributedecisionmaking.Liu
andLi[10],extendedpowerBonferronimeantointerval‐valuedintuitionisticfuzzynumbersand
appliedittomultipleattributegroupdecisionmakingintheevaluationofairquality.Seikhand
Mandal[11],introducedintuitionisticfuzzyDombiweightedaveragingoperator,intuitionistic
fuzzyDombihybridaveragingoperator,intuitionisticfuzzyDombiweightedgeometricoperator,
intuitionisticfuzzyDombiorderedweightedgeometricoperatorandintuitionisticfuzzyDombi
hybridgeometricoperator.Xian,GuoandChai[12],introducedanintuitionisticfuzzy
Algorithms2020,13,1542of20
linguistic‐inducedgeneralizedorderedweightedaveragingoperatorandmadeanapplicationabout
takingtargetedmeasuresinpovertyalleviation(TPA).Shi,YangandXiao[13],introducedthe
intuitionisticfuzzypowergeometricHeronianmeanoperatorandtheweightedintuitionisticfuzzy
powergeometricHeronianmeanoperator.Zou,ChenandFan[14],introducedtheimproved
intuitionisticfuzzyweightedgeometricoperatorofintuitionisticfuzzyvalues.Forfurther
contributions,referencesareprovided[15–22].
Näther[23]statesthatcopulasandt‐normsoftencoincide.Nelsen[24]andAlsinaetal.[25]
contributedtotheintroductionandapplicationofcopulas.Beliakovetal.[26],hassetcopulasas
conjunctivefunctions.Inaddition,theselectionofthemostsuitableconjunctivefunctionandhowto
fitadditivegeneratorsforArchimedeant‐normsandArchimedeancopulasisdepictedinNäther
[23].
Thepresentpaperaimstoinvestigatetheintuitionisticfuzzycopulaaggregationoperators,
withthecombinationoffuzzynegations,inordertodevelopmultipleattributedecisionmaking
methodsforIFVs.Toaccomplishthat,theconstructionofnoveloperationsforIFVsisessential.Asa
result,thecombinationofcopulasandfuzzynegations[27]producesnewoperations,whichallow
ustoconstructnovelintuitionisticfuzzyArchimedeancopulaweightedarithmeticandgeometric
aggregationoperationswiththeusageofarithmeticandgeometricmeans.
Asaresult,theremainderofthepaperisorganizedasfollows.Section2includessomebasic
conceptsofcopulatheory,fuzzynegationsandintuitionisticfuzzysetsandalsoourdevelopmenton
noveloperationsforIFVswiththecombinationoftheabove.Thepropertiesoftheproposed
operationsareinvestigated.InSection3weprovidethenewintuitionisticfuzzyArchimedean
copulaweightedarithmeticandgeometricaggregationoperators,includingafurtheranalysisof
theirproperties.InSection4weprovideanalgorithmtoaccomplishthemultipleattributedecision
makingprocedure.InSection5weprovideapracticalexampletomaterializetheapplicationofthe
proposedapproach.Inthediscussionsection,theresultsofourworkareexplainedthoroughlyand
arecomparedwithpreviousstudies.InSection7,weprovidesomeconcludingremarksandfuture
researchdirections.
2.Preliminaries
ThisSectionprovidesnoveloperationsforIFVs,whichareestablishedthroughtheconceptsof
copula,co‐copula,Archimedeancopulaandfuzzynegations.
Sklar[28]presentedtheconceptofcopula,asamathematicaltooltocombineprobability
distributions,framingthedependencestructurewithinrandomvariableswiththeusageoftheir
CDFs.
Definition1.[28]A2‐dimensionalcopulaisafunctionwithdomain
0,1 0,1andrange
0,1 ,grounded
and2‐increasing,i.e.,satisfiesthefollowingboundaryandmonotonicityconditions:
1. ( ,0) (0, ) 0.Cx C y
2. (,1) , (1, ) .Cx xC y y
3. 11 2 2 1 2 2 1
(, ) ( , ) (, ) ( , ) 0.Cx y Cx y Cx y Cx y
where,
121 2
,, , , , 0,1xyx x y y and121 2
,
x
xy y
.
Below,thedefinitionofco‐copulaisgiven,accordingtoCherubinietal.’s[29]work:
Definition2.[29]Theco‐copulaofacopulaisdefinedas:
*(, ) 1 (1 ,1 )Cxy C x y (1)
AccordingtoNelsen[24]theco‐copulaisnotacopula,butmaybeconstructedbytheusageofa
copula.Fréchet[30]andHoeffding[31]introducedtheFréchet–Hoeffding[24]boundsofcopulasfor
any,[0,1],xymax{ 1,0} ( , ) min{ , }.
x
y Cxy xy Atthispoint,themethodfollowsthedefinition
Algorithms2020,13,1543of20
ofArchimedeancopulas,whichmaybeconstructedwithsignificanteaseandensuresome
remarkableproperties.
Definition3.[24]Let
beacontinuous,strictlydecreasingfunctionon[0,1] [0, ]
with(1) 0
and[1]
thepseudo‐inverseof
,whichisdefinedas:
1
[1] (), 0 (0),
() 0, (0) ,
xx
xx
(2)
Then,2
:[0,1] [0,1]C,[1]
(, ) (( ) ())Cxy x y
correspondstotheconditionsofDefinition1,hence
CisacopulaandCiscalledArchimedeancopula.
Inthecase,whereCisstrictlyincreasing,(0)
and[1] 1
,then1
(, ) ( () ())Cxy x y
.In
thatcase
iscalledstrictgeneratorandCiscalledstrictArchimedeancopula.
InTable1,someArchimedeancopulasarelisted.
Table1.Inref.[24]ArchimedeanCopulas.
Name()t
(,)CxyParameter
Gumbel(ln)t
1
exp( [( ln ) ( ln ) ] )xy
1
Clayton1t
1
(1)xy
0
Ali‐Mikhail‐Haq1(1)
ln t
t
1(1)(1)
xy
xy
11
Joeln [1 (1 ) ]t
1
1 [(1 ) (1 ) (1 ) (1 ) ]xyxy
1
Thestudyoftheapplicationsofcopulasinfuzzysetsisofgreatimportance,giventhatspecific
copulasaret‐normsandspecificco‐copulasaret‐conormsandviceversa.Moreover,Näther[23]
mentionedinhisworkthatthecombinationofprobabilisticinformationorfuzzyinformationhardly
matters.Below,theAtanassov’s[1]definitionforIntuitionisticFuzzySetsisprovided:
Definition4.[1]Let
X
beareferenceset,anIntuitionisticFuzzySet(IFS)
A
on
X
isdefinedas:
{, (), ()| },
AA
Ax x xxX
(3)
where,thefunctions()
A
x
and()
A
x
denotethedegreesofmembershipandnon‐membershipofthe
element
x
Xtotheset
A
,respectivelywith:
0()1
Ax
,0()1
Ax
,0()()1
AA
xx
and() 1 () ()
AAA
x
xx
iscalleddegreeof
indeterminacyof
x
to
A
.Additionally,,
iscalledIntuitionisticFuzzyValue(IFV)andthesetof
allIFVsisV.
Definition5.[32]Adecreasingfunction:[0,1] [0,1]niscalledafuzzynegationif(0) 1nand
(1) 0n.Afuzzynegationniscalled:Strict,ifitisstrictlydecreasingandcontinuousandStrong,ifitisan
involution,i.e.,(()) , [0,1]nnx x x .
Table2,proposestwoparametricfamiliesoffuzzynegations:
Table2.Inref.[32]FuzzyNegations.
NameFuzzyNegationParameter
Sugenoclass1
() 1
x
nx
x
(1, )
Yagerclass1/
() (1 )nx x
(0, )
Algorithms2020,13,1544of20
Moreover,theoperationsofIFVsinTaoetal.’s[9]work,arebeingextendedthroughthe
adaptationoffuzzynegations.
Definition6.Let,
and,
betwoIFVsand0kaparameter.
1. Additionoperation
*(, ),(,)
CCC
1
( [ ( ( )) ( ( ))])nn n
1
,[() ()].
2. Multiplicationoperation
*
(, ), (,)
CCC
1[( ) ( )]
1
, ( [ ( ( )) ( ( ))]) .nn n
3. ScalarMultiplicationoperation
11
([(())]),[()].kn kn k
4. Poweroperation
11
[()],( [(())]).
kknkn
Byreplacingthe
parameterwiththeappropriatevalue,theTaoetal.’s[9]operationsofIFVs
areproduced.Moreover,afurtherinvestigationisheldforthepropertiesoftheproposed
operations.
Theorem1.Let,
and,
betwoIFVsand0karealvaluedparameter.Thenovel
operationsofIFVsareclosedortheirvaluesarealsoIFVs,i.e.,,,,
k
CC
kV
.
ProofofTheorem1.Wehave[0,1]
and0()1n
.Since
isstrictlydecreasingthen:
(1)(())(0)n
0(()) ,n
(4)
Similarly,as[0,1]
weget:
0(()) ,n
(5)
WeaddEquations(4)and(5)andweget:
0(())(()) ,nn
(6)
Basedonthefactthat1
isalsostrictlydecreasing,weobtain:
11 1
( ) ( ( ( )) ( ( ))) (0 )nn
1
0 ((( )) (( )))1nn
1
0(((())(())))1nn n
*
0(,)1.C
Morover,byDefinition1,as , [0,1]
,
0(,)1C
holds.
Inaddition:
*
0(,)(,)1.CC
Thus,theresultoftheoperationis:
Algorithms2020,13,1545of20
.
CV
AsEquation(3)isconcerned:
11
( [ ( ( ))]), [ ( )]kn kn k
For1kweget:
11
( [ ( ( ))]), [ ( )]nn
(( )),nn
,,
thatholds.
For2kweget:
11
( [ ( ( )) ( ( ))]), [ ( ) ( )]
Cnn n
11
( [2 ( ( ))]), [2 ( )] 2 ,
Cnn
thatholds.
Weassumethatforknholds.Thenfor1kn
:
11
( 1) [ [ ( ( ))]], ,[()]
C C
kn n n nn n
11 11
( [ ( ( [ [ ( ( ))]]))]) ( ( )), ( ( ( ( ))) ( ))kn nn nn n n
11
( [ ( ( ))]) ( ( )), (( 1) ( ))kn nn n n
11
( [( 1) ( ( ))]), (( 1) ( )) ( 1) .kn n n n n
Hence,bymathematicalinductiontheproofiscomplete.Theproofsthattherestofthe
operationsbelongtothesetVaresimilar.
Nextweinvestigatethepropertiesoftheproposedoperations:□
Theorem2.Let ,V
,,V
andrealvaluedparameters 12
,, 0,kk k then:
1. .
CC
2. .
CC
3. () .
CC
kkk
4. () .
kk k
CC
5. 1212
().
C
kkkk
6. 1212
.
kkkk
C
7. () () ( ).
CC
nnn
8. () () ( ).
CC
nnn
Algorithms2020,13,1546of20
9.
() ( ).
k
kn n
10. (()) ( ).
k
nnk
ProofofTheorem2.Thecases1and2areobvious,becauseofArchimedeancopulasandthe
correspondingco‐copulascommutativity.
1. *11
( , ), ( , ) ( [ ( ( )) ( ( ))]), [ ( ) ( )]
CCC nn n
11*
( [(( )) (( ))]), [( ) ()] ( , ),(, ) .
C
nn n C C
2. *
(, ), (,)
CCC
*
(,),(,) .
C
CC
3. 11 11
( ) ( [ ( ( ( [ ( ( )) ( ( ))])))]), [ ( [ ( ) ( )])]
C
knknnnn k
11
( ) ( [ [ ( ( )) ( ( ))]]), [ [ ( ) ( )]]
C
knknn k
11
( ) ( [ ( ( )) ( ( ))]), [ ( ) ( )]
C
knknknkk
11 1 11 1
( ) ( [ ( ( ( ( )))) ( ( ( ( ))))]), [ ( ( ( ))) ( ( ( )))]
C
k n kn kn k k
11 1 11 1
( ) ( [ ( ( ( ( ( ( )))))) ( ( ( ( ( ( ))))))]), [ ( ( ( ))) ( ( ( )))]
C
k n nn kn nn kn k k
() .
CC
kkk
4. 11 1 1
( ) [ ( [ ( ) ( )])], ( [ ( ( ( [ ( ( )) ( ( ))])))])
k
Cknknnnn
11 1 1
[ [ ( ) ( )]], ( [ [ ( ( )) ( ( ))]]) [ ( ) ( )]], ( [ ( ( ))knknnkknkn
11 1 1 1
( ( ))]) [ ( ( ( ))) ( ( ( )))], ( [ ( ( ( ( ( ( ))))))kn k k n nn kn
1
( ( ( ( ( ( ))))))]) .
kk
C
nn k n
5. 11 1 1
12 1 1 2 2
( [ ( ( ))]), [ ( )] ( [ ( ( ))]), [ ( )]
CC
kknknknknk
11 1 11 1
1212
( [ ( ( ( [ ( ( ))]))) ( ( ( [ ( ( ))])))]), [ ( [ ( )]) ( [ ( )])]nnnkn nnkn k k
111
12 12 12
( [ ( ( )) ( ( ))]), [ ( ) ( )] ( [( ) ( ( ))]),nkn kn k k n kkn
1
12 12
,[( )()]( ).kk kk
6. 12
11 1 1
11 2 2
[ ( )], ( [ ( ( ))]) [ ( )], ( [ ( ( ))])
kk
CC
knkn knkn
11 1 1 1 1
12 1 2
[ ( [ ( )]) ( [ ( )])], ( [ ( ( ( [ ( ( ))]) ( ( ( [ ( ( ))])))])k k n nnkn nnkn
11 1
12 1 2 12
[ ( ) ( )], ( [ ( ( )) ( ( ))]) [( ) ( )],kknknkn kk
12
1
12
,( [( )(())]) .
kk
nkkn
7. 11
( ) ( ) ( [ ( ( ( ))) ( ( ( )))]), [ ( ( )) ( ( ))]
C
nnnnn nn n n
11 1 1
( [ ( ) ( )]), [ ( ( )) ( ( ))] ( [ ( ) ( )], ( [ ( ( ))nnnnnn
( ( ))]) ) ( ).
C
nn
8. 11
( ) ( ) [ ( ( )) ( ( ))], ( [ ( ( ( ))) ( ( ( )))])
C
nn n nnnn nn
1111
[ ( ( )) ( ( ))], ( [ ( ) ( )]) ( ( [ ( ( )) ( ( ))]), [ ( )nnn nnnn
()]) ( ).
C
n
9. 11 11
( ) ( ), ( ) ( [ ( ( ( )))]), [ ( ( ))] ( [ ( )]), [ ( ( ))]kn k n n n k n n k n n k k n
11
( [ ( )], ( [ ( ( ))]) ) ( ).
k
nk nkn n
Algorithms2020,13,1547of20
10. 11 11
( ( )) [ ( ( ))], ( [ ( ( ( )))]) [ ( ( ))], ( [ ( )])
k
n knnknn knnk
11
( ( [ ( ( ))]), [ ( )] ) ( ).nn kn k nk
□
InordertoachievethederivationofdifferentoperationsforIFVs,theusageofgenerator
functionsfromTable1andoffuzzynegationsfromTable2,isaccomplished.Thoseoperationsare
listedinTable3:
Table3.Operationsforintuitionisticfuzzyvalues(IFVs)viaSugenoFuzzyNegation.
NameOperationsFormulas
GumbelAddition
operation
1
1
1
1
1
1 exp( [( ln( )) ( ln( )) ] )
11
,exp( [( ln ) ( ln ) ] )
1
1
1 exp ( [( ln( )) ( ln( )) ] )
11
Multiplicatio
noperation
1
1
1
1
1
1 exp( [( ln( )) ( ln( )) ] )
11
exp( [( ln ) ( ln ) ] ), 1
1
1 exp( [( ln( )) ( ln( )) ] )
11
Scalar
Multiplicatio
noperation
1
1
1
1
1 exp( [ ( ln( )) ] )
1,exp( [ ( ln ) ] )
1
1 exp( [ ( ln( )) ] )
1
k
k
k
Power
operation
1
1
1
1
1 exp( [ ( ln( )) ] )
1
exp( [ ( ln ) ] ), 1
1exp([(ln( ))])
1
k
k
k
ClaytonAddition
operation
1
1
1
1
1
1(( ) ( ) 1)
11 ,( 1)
1
1
1(( ) ( ) 1)
11
Multiplicatio
noperation
1
1
1
1
1
1(( ) ( ) 1)
11
(1), 1
1
1(( ) ( ) 1)
11
Scalar
Multiplicatio
noperation
1
1
1
1
1(( ) 1)
1,( 1)
1
1(( ) 1)
1
k
k
k
Power
operation
1
1
1
1
1(( ) 1)
1
(1), 1
1(( ) 1)
1
k
k
k
Ali‐Mikhail‐Ha
q
Addition
operation
1
1
11
11
1
1(1 )(1 )
11
,
1
11(1 )(1 )
11
1( )
1
1
1(1 )(1 )
11
Algorithms2020,13,1548of20
Multiplicatio
noperation
1
1
11
11
1
1(1 )(1 )
11
,1
1
1(1 )(1 )
11
1( )
1
1
1(1 )(1 )
11
Scalar
Multiplicatio
noperation
1
()
1
11
1(1 )
1,
11(1 )
()
1
1( )
1
1(1 )
1
k
k
k
k
k
k
Power
operation
1
()
1
11
1(1 )
1
,1
1(1 ) ()
1
1( )
1
1(1 )
1
k
k
k
kk
k
JoeAddition
operation
1
1
1
11
11
[(1)(1)(1)(1)]
1111 ,1[(1)(1)(1)(1)]
11
11
1[(1 )(1 )(1 )(1 )]
1111
Multiplicatio
noperation
1
1
1
11
11
[(1)(1)(1)(1)]
1111
1[(1)(1)(1)(1)], 11
11
1[(1)(1)(1)(1)]
1111
Scalar
Multiplicatio
noperation
1
1
1
11
[(1)((1))]
11 ,1[(1)((1))]
11
1 [ (1 ) ( (1 ) ) ]
11
k
k
k
k
k
k
Power
operation
1
1
1
11
[(1)((1))]
11
1[(1 ) ((1 ))], 11
1 [ (1 ) ( (1 ) ) ]
11
k
k
k
k
k
k
InthefollowingSection,weproposethedevelopmentofintuitionisticfuzzyArchimedean
copulaweightedarithmeticandgeometricaggregationoperators,whichisachieved.Furthermore,
thepropertiesofthoseextendedaggregationoperatorsareinvestigated.
3.NovelIntuitionisticFuzzyArchimedeanCopulaWeightedArithmeticandGeometric
AggregationOperators
3.1.IntuitionisticFuzzyArchimedeanCopulaWeightedArithmeticAggregationOperator
Definition7.Let,
ii
i
,1,...,in,beacollectionofIFVsand1
{ ,..., }
n
ww w
betheweighting
vectorofi
,with01
i
w
and11
n
i
iw
.AnintuitionisticfuzzyArchimedeancopulaweighted
Algorithms2020,13,1549of20
arithmeticaggregation(w
IFACWAA )operatorofdimensionnisamapping, :n
w
I
FACWAA V V,according
to:11
( ,..., ) .
wn
n
Ci i i
IFACWAA w
BasedontheproposedadditionandscalarmultiplicationoperationsofIFVs,thefollowing
theoremisexpressed.
Theorem3.Let ,
ii
i
,1,...,in,beacollectionofIFVsand1
{ ,..., }
n
ww w
betheassociated
weightingvector,with01
i
w
and11
n
i
iw
.Then,theaggregatedvalue,basedonthew
IFACWAA
operator,canbeexpressedas:
1
1
1([ (())]( ,..., ),)i
n
iwn
i
IFACW n nAwA
1
1
[()]
i
n
i
i
w
Thevalueofw
IFACWAA operatorisalsoanIFV.
ProofofTheorem3.For 1n,weget:
11
1
1
1
11
( [ ( ( ))]) , ]() [ ( )
wnwnIFAC wWAA
Butwehavethat 11w,soweget:
11
11
1
( ) ( [ ( ( ))]), [ ( )]
wnnIFACWAA
11
1
() ,
w
IFACWAA
11
() ,
w
IFACWAA
as 1,V
1
()
w
IFACWAA V
.
Next,wesupposethatfor nkitholds,sowecalculate:
1
1
1([ (())]( ,..., ),)i
k
iwk
i
IFACW n nAwA
1
1
[()]
i
k
i
i
w
For 1nk
,weget:
1
111111
( ,..., , ( ))()
kk
Ci i i Ci i i C k kwkk wwwIFACWAA
11
11 11
11
11
( [ ( ( ))]), [ ( )] ( [ ( ( ))]), [ ( )]
ii kk
kk
iiCkk
ii
nwn w nwn w
1
11 1
1
1
( [ ( ( ( [ ( ( ))]))) ( ( ( [ ( ( ))])))]),
i k
k
ik
i
nnn wn nnwn
1 1
11 1 1
11
11
, [ ( [ ( )]) ( [ ( )])] ( [ ( ( )) ( ( ))]),
ik ik
kk
ik ik
ii
wwnwnwn
1
1
1
1
,[ ()]) ( )]
ik
k
ik
i
ww
11
11
11
])([ (()), [ ()]
ii
kk
ii
ii
nwn w
Itholds,soasaresult,bymathematicalinductiontheproofiscomplete.□
Algorithms2020,13,15410of20
Proposition4. Let ,
ii
i
,1,...,in,beacollectionofIFVsand1
{ ,..., }
n
ww w
betheassociated
weightingvector,with01
i
w
and11
n
i
iw
.Thenw
IFACWAA operatoris:
1. Idempotent,i.e.,if ,,
ii
,then:for𝑖1, … , 𝑛,1
( ,..., ) .
wn
IFACWAA
2. Bounded,i.e.,for𝑖1, … , 𝑛,1
( ,..., ,)0,1 1 0
wn
IFACWAA
.
3. Commutative,i.e.,foranypermutationof,
ii
i
,() ()
() ,
fi fi
fi
,with 1, ..., in,
where 𝑓:1,2, … , 𝑛→1,2, … , 𝑛,1
( ,..., )
wn
IFACWAA
(1) ( )
( ,..., )
wf fn
IFACWAA
.
ProofofProposition4.1.1
1
1([ (())]( ,..., ),)i
n
iwn
i
IFACW n nAwA
1
1
[()]
i
n
i
i
w
.
As
i
and
i
weget: 1
1
1
( ,..., ) ( [ ( ( ))]),
wn
n
i
i
InwnFACWAA
1
1
[()]
n
i
i
w
11
11
( [ ( ( )) ]), [ ( ) ]
nn
ii
ii
nn w w
11
( [ ( ( ))]), [ ( )]nn
,
.
TheproofsofEquations(2)and(3)areobviousbasedonthepropertiesofArchimedeancopulas
andtheircorrespondingco‐copulas.□
3.2.IntuitionisticFuzzyArchimedeanCopulaWeightedGeometricAggregationOperator
Definition8. Let,
ii
i
,1, ..., in,beacollectionofIFVsand1
{ , ..., }
n
ww w
betheweighting
vectorof i
,with01
i
w
and11
n
i
iw
.AnintuitionisticfuzzyArchimedeancopulaweightedgeometric
aggregation(w
IFACWGA )operatorofdimensionnisamapping, :n
w
I
FACWGA V V,accordingto:
11
( ,..., ) i
w
wn
n
Ci i
IFAC W GA
.
BasedontheproposedmultiplicationandpoweroperationsofIFVs,thefollowingtheoremis
expressed.
Theorem5. Let ,
ii
i
,1,...,in,beacollectionofIFVsand1
{ ,..., }
n
ww w
betheassociated
weightingvector,with01
i
w
and 11
n
i
iw
.Then,theaggregatedvalue,basedonthew
IFACWGA
operator,canbeexpressedas: 1
1
1
( ,..., ) [ ( )],
i
wn
n
i
i
wIFACWGA
1
1
([ (())])
i
n
i
i
nwn
.
Thevalueofw
IFACWGA operatorisalsoanIFV.
ProofofTheorem5.For 1n,weget:
11
11
11 1
( ) [ ( )], ( [ ( ( ))]) ,
w
IFACWGA w n w n
butwehavethat
11wsoweget:
11
11
1
( ) [ ( )], ( [ ( ( ))]) .
w
IFACWGA n n
Asaresult,wehave:
11
11
() , .
w
IFACWGA
As1,V
1
()
w
IFACWGA V
.
Next,supposethatfornkitholds.
For1nk
wecalculate:
1
1111
1
1
(, ( ) ( )..., ) iik
www
kk
Ci i Ci i Ckkw
IFACWGA
11
11 11
11
11
[ ( )], ( [ ( ( ))]) [ ( )], ( [ ( ( ))])
iikk
kk
iiCkk
ii
wn wn w nwn
Algorithms2020,13,15411of20
1
11 1 1 1
1
1 1
[ ( [ ( )]) ( [ ( )])], ( [ ( ( ( [ ( ( ))])))
ik i
k k
ik i
i i
wwnnnwn
1 1 1
11 1
111
11
( ( ( [ ( ( ))])))]) [ ( ) ( )], ( [ ( ( )) ( ( ))])
kikik
kk
kikik
ii
nn w n w w n w n w n
11
11
11
[()],([(())])
ii
kk
ii
ii
wn wn
Itholds,sobymathematicalinductiontheproofiscomplete.□
Remark1. w
IFACWGA operatorpossessesthesamepropertiesas,
w
IFACWAA suchasidempotency,
boundarypropertyandcommutativity.
Inordertodepictsomeoftheproducedw
IFACWAA andw
IFACWGA operatorsweusethe
proposedoperatorsforIFVsasdescribedinTable3,whichareexpressedintheTable4andTable5,
respectively.
Table4.SomeSugeno‐BasedIntuitionisticFuzzyArchimedeanCopulaWeightedArithmetic
AggregationOperators.
Namew
IFAC WAA
Gumbel
1
1
1
11
1
1
1 exp[[ [ ln( )] ] ]
1,exp[[ [ ln ] ] ]
1
1(exp[[ [ln( )]]])
1
i
i
i
i
i
n
in
i
i
n
i
i
i
w
w
w
Clayton
1
1
1
11
1
1
1( ( ))
1,( ( ) )
1
1(( ( )))
1
i
i
i
i
i
n
in
i
i
n
i
i
i
w
w
w
Ali‐Mikhail‐Haq
1
1
1
1
11
1(1 )
1
()
1
11
,
11(1)
1( ) ()
1
1(1 )
1
()
1
1
i
ii
i
i
ii
ii
ii
i
i
n
w
i
n
w
i
n
w
i
Joe
1/
11/
1/ 1
1
1
(1 (1 (1 ) ) )
1,1 (1 (1 (1 ) ) )
1
1(1(1 (1(1 ))))
1
ii
ii
i
ii
i
n
w
n
iw
n
wi
i
Table5.SomeSugeno‐BasedIntuitionisticFuzzyArchimedeanCopulaWeightedGeometric
AggregationOperators.
Namew
IFACWGA
Gumbel
1
11
1
1
1
1
1 exp[[ [ ln( )] ] ]
1
exp[[ [ ln ] ] ], 1
1 (exp[[ [ ln( )] ] ])
1
i
i
i
i
i
n
i
ni
in
i
i
i
w
w
w
Algorithms2020,13,15412of20
Clayton
1
11
1
1
1
1
1( ( ))
1
(()), 1
1(( ( ))
1
i
i
i
i
i
n
i
ni
in
i
i
i
w
w
w
Ali‐Mikhail‐Haq
1
1
1
1
11
1(1 )
1
()
1
1
1,
1(1 ) 1
1( )
() 1
1(1 )
1
()
1
1
i
ii
i
i
ii
i
i
ii
i
i
n
w
i
n
w
i
n
w
i
Joe
1/
1
1/
1/
1
1
1
(1 (1 (1 ) ) )
1
1(1 (1(1 ))), 1
1(1(1 (1(1 ))))
1
ii
i
i
i
ii
i
n
w
ni
w
n
w
i
i
4.w
IFAC WAA andw
IFACWGA OperatorsinMADM
InthissectionanalgorithmforMultiple‐AttributeDecisionMaking(MADM)withIFVsis
introduced.Let1
{ ,..., }
m
Uu ubethesetofalternativesand1
{ ,..., }
n
CC Cbethesetofattributes,
withaweightingvector1
{ ,..., }
n
ww w
,with01
i
w
and
1
1
n
i
i
w
.Theintuitionisticfuzzy
decisionmatrix(,)
ij ij m n
R
where,,
ij ij ij
with1,...,imand1,..., njrepresent
thedegreethati
usatisfies
j
C,suchthat,[0,1]
ij ij
,with1,
ij ij
i.e.,
11 11 1 1
11
,,
,,
nn
mm mnmn
R
(7)
Inordertoobtaintheweightingvectorofattributes,themodifiedmaximizingdeviations
methodforMADMwithintuitionisticfuzzyinformation[9]canbeused.AsMADMwith
intuitionisticfuzzyinformationisaproceduretoendupwiththebestalternative,
i
uU1, ..., ,im
thecomparisonofIFVsisachievedbythefollowingconcept.
ForanyIFV,,
ChenandTan[33]proposedascorefunction
s
andHongandChoi
[34]proposedanaccuracyfunction h,i.e.,()s
and()h
,respectively.Thelarger
thescoreis,thegreatertheIFVwouldbe.Inthecasethatthescorefunction’svaluesfortwoIFVsare
equal;theaccuracyfunctioncanprovidemorespecificresult.XuandYager[35],basedonscoreand
accuracyfunctions,gaveatotalorderforIFVs.
BasedontheaboveandinspiredbyMADMmodelin[9],wederivethefollowingalgorithm.
Algorithms2020,13,15413of20
Algorithm1MultipleAttributeDecisionMakingModel
BEGIN
1:Obtainthenormalizedintuitionisticfuzzydecisionmatrix,i.e.,
(,)
ij ij m n
R
.
2:Determinetheweightingvectoroftheattributes.
3:AggregatetheIFVsusingtheproposed w
IFACWAA or w
IFACWGA
operators.
4:Sorttheoptionsaccordingtoscorefunctionandaccuracyfunction.
5:Selectthebestalternative,byrankingthevaluesof,
i
1, ..., im.
END
Inordertodemonstratetheproposedmethod,weprovidethefollowingnumericalexample,
whichisadaptedfromTaoetal’s[9]work,inordertocomparetheirmethodwiththepresent
extensionofit.
5.APracticalExampleforMADMwithIFVs.
Aresearchfacilityhasarrangedtopurchaseanelectronicdevice.Thepeopleincharge,selected
thefourmostappropriatemodelsforfurtherconsideration,afterthemarketresearch.Thoseare
indicatedasthefouralternatives1234
{, , , }.UuuuuTheevaluationofthealternativesisachieved
bytheconsiderationoffiveattributesbythedecisionmaker(DM).Thoseattributesarelistedas
follows:thelayoutoftheproduct(1
C),thetechnicalassistance(2
C),thebrand(3
C),theprice(4
C)
andthequalityoftheproduct(5
C).
Step1.Thenormalizedintuitionisticfuzzydecisionmatrixcanbeobtainedasfollows:
45
0.4,0.3 0.5,0.2 0.7, 0.2 0.4,0.6 0.6, 0.2
0.6,0.1 0.4, 0.3 0.3, 0.5 0.6,0.2 0.5, 0.3
() 0.5,0.4 0.6, 0.1 0.6, 0.2 0.7, 0.1 0.3,0.6
0.6,0.3 0.4, 0.5 0.5, 0.3 0.8, 0.2 0.5, 0.2
ij
R
.
inwhich,theelement,,
ij ij
with1,..., 4iand 1,...,5j,representstheintuitionistic
membershipandnon‐membershipdegreeofthei‐thalternativethatsatisfiesthej‐th
attribute.
Step2.Inthepresentwork,weuseTaoetal.’s[9],modifiedmaximizingdeviationsmethod,in
ordertoderivethefiveweightingsoftheattributes,andsincethenumericalexampleisthe
sameweget: 10.2162,w20.2703,w30.1892,w40.1081,w50.2162.w
Step3.Theaggregatedresultsaregivenbytheusageof w
IFACWAA and w
IFACWGA operatorsandare
listedin
Algorithms2020,13,15414of20
Table6andTable7.
Step4.Thescorefunction’sresultsforeachalternativeandforeachtypeof w
IFACWAA and
w
IFACWGA operatorsarelistedinTable8andTable9.
Step5.TherankingofthealternativesisachievedandisdepictedinTable10.
Algorithms2020,13,15415of20
Table6.TheaggregationresultsusingintuitionisticfuzzyArchimedeancopulaweightedarithmetic
aggregation(w
IFACWAA ).
Alternatives
Gumbel
1
0.3
Clayton
1
0.3
Ali‐Mikhail‐Haq
1
0.3
Joe
1
0.3
1
u0.5394,0.2459 0.5560,0.2337 0.5356,0.2488 0.5394,0.2459
2
u0.4777,0.2494 0.4908,0.2126 0.4736,0.2544 0.4777,0.2494
3
u0.5388,0.2267 0.5556,0.1776 0.5338,0.2357 0.5388,0.2267
4
u0.5444,0.3020 0.5731,0.2846 0.5395,0.3056 0.5444,0.3020
Table7.TheaggregationresultsusingintuitionisticfuzzyArchimedeancopulaweightedgeometric
aggregation(w
IFACWGA ).
Alternatives
Gumbel
1
0.3
Clayton
1
0.3
Ali‐Mikhail‐Haq
1
0.3
Joe
1
0.3
1
u0.5156,0.2762 0.5045,0.2976 0.5197,0.2693 0.5156,0.2762
2
u0.4534,0.2940 0.4387,0.3097 0.4576, 0.2868 0.4534,0.2940
3
u0.5049,0.3176 0.4823,0.3581 0.5113,0.3012 0.5049,0.3176
4
u0.5152,0.3306 0.5045,0.3443 0.5196,0.3252 0.5152,0.3306
Table8.Thescorefunction’sresultsusing.
w
IFACWAA
Alternatives
Gumbel
1
0.3
Clayton
1
0.3
Ali‐Mikhail‐Haq
1
0.3
Joe
1
0.3
1
u0.29350.32230.28680.2935
2
u0.22830.27820.21920.2283
3
u0.31210.37800.29810.3121
4
u0.24240.28840.23390.2424
Table9.Thescorefunction’sresultsusingw
IFACWGA .
Alternatives
Gumbel
1
0.3
Clayton
1
0.3
Ali‐Mikhail‐Haq
1
0.3
Joe
1
0.3
1
u0.23940.20690.25040.2394
2
u0.15940.12900.17080.1594
3
u0.18730.12430.21010.1873
4
u0.18460.16020.19440.1846
Table10.Rankingofthealternatives.
Typew
IFAC WAA w
IFACWGA
Gumbel314 2
uuuu 134 2
uu u u
Clayton314 2
uuuu 142 3
uu u u
Ali‐Mikhail‐Haq31 4 2
uuuu 134 2
uu u u
Joe314 2
uuuu 134 2
uu u u
Algorithms2020,13,15416of20
Toillustratetheaffectionof
parameterintheprocedureas
parameterisfixedto1(and‐1
inAli‐Mikhail‐Haqcase)weutilizedthefollowinggraphsthataredepictedinFigure1andFigure2
forthe w
IFACWAA andinFigure3andFigure4forthe w
IFACWGA .
(A)(B)
Figure1.(A)Gumbel‐type w
IFACWAA scoreresult.(B)Clayton‐type w
IFACWAA scoreresult.
(A)(B)
Figure2.(A)Ali‐Mikhail‐Haq‐type w
IFACWAA scoreresult.(B)Joe‐type w
IFACWAA scoreresult.
(A)(B)
Figure3.(A)Gumbel‐type w
IFACWGA scoreresult.(B)Clayton‐type w
IFACWGA scoreresult.
Algorithms2020,13,15417of20
(A)(B)
Figure4.(A)Ali‐Mikhail‐Haq‐type w
IFACWGA scoreresult.(B)Joe‐type w
IFACWGA scoreresult.
6.Discussion
ThepresentpaperestablishesspecificnoveloperationallawsofIFVs,asthegeneralizationof
theexistingcopula‐basedoperationallaws,withthecontributionoffuzzynegations.Additionally,
thepaperprovidestheextensionoftheproposedoperationstothearithmeticmeanandtothe
geometricmeanofIFVs.
Firstofall,thereplacementofthe
parameterwiththenumericalvaluezeroinSugenoclass,
orwiththenumericalvalueoneinYagerclass,inDefinitions6–8andthecorrespondingtables
(Table3–Table5),providesuswiththeoperationlawsandtheaggregationoperatorsofTaoetal.’s
[9]work.Asaresult,thegeneralizationoftheexistingoperatorsbytheproposedoperationallaws
andtheextensionoftheexistingaggregationoperatorsareverified.Morespecifically,theextension
oftheexistingaggregationoperatorsisbasedonthereformoftheexistingbasicoperational
environment.
RegardingtheprovidedpracticalexampleforMADMwithIFVs,developedinordertopoint
outtheadvantagesandtheflexibilityoftheproposedalgorithm,wemaycallforththefollowing
comparisonwith[9].
Tocontinuewiththerankingofthealternativesin[9],theusageofGumbel‐typeaggregation
operatorsuggeststhethirdalternative(3
u)asthemostadequateone.Inaddition,theClayton‐,
Ali‐Mikhail‐Haq‐andJoe‐typeofaggregationoperatorsprovidesthesamealternative,respectively,
throughtheotherthreealternativesaredevelopedwithadifferentorder.Ontheotherhand,inour
approach,asTable10shows,thereisnodifferenceintheorderofthefouralternativesforeachtype
ofaggregationoperatorand,asaresult,theorderproblemisdistinguished.
Inaddition,inTable10,theorderofthealternativesisprovided,suggestingtheusageof
w
IFACWGA operatorsforeachcopulatype,whichrepresentsanewbestalternative.Specifically,
Gumbel‐,Ali‐Mikhail‐Haq‐ andJoe‐typeaggregationresultsdemonstratethatthemostadequate
alternativeis1
uand 3
ufollows.Therestrankingremainsthesamewiththew
IFACWAA operators
ranking.However,inthecaseofClayton‐typew
IFACWGA rankinginrespectofthethree
alternativesistotallydifferent,throughthefirstoneremains1
udemonstratingthemostadequate
choice.Thisperspectivemaybeconsideredreasonable,giventhattheparameter
doesnotaffect,
neitherthemembershipdegree,northenonmembershipdegreeintheaggregationprocessandthis
factmaybeobservedintheClayton‐typescorefunction’splot.
Anotherremarkablefactabouttheproposedaggregationoperatorsisthattheyprovidetwo
parametersaffecting(inmostcases)theresults,suggestingmorechoicesandflexibilityforthe
decisionmakers.Forthedescriptionoftheaffectionof
parameter,Figure1‐Figure4havebeen
utilized,as
parameterisfixedto1(and‐1intheAli‐Mikhail‐Haqcase).
InFigure1,apparentlytheGumbel‐andClayton‐typeofw
IFACWAA operatorssuggestdistinct
scorefunctionsforeachalternative,with3
ualwaysrepresentingthemostadequatechoiceand2
u
Algorithms2020,13,15418of20
theleastadequateone.ThesameholdsfortheJoe‐ andAli‐Mikhail‐Haq‐typeofw
IFACWAA
operators,asitisdepictedinFigure2.InFigure3,theGumbel‐typeofw
IFACWGA suggests1
uas
themostadequatechoiceand2
utheleastadequateone,throughalternatives3
uand4
uthat
meetacrossoverpoint.InFigure4,wemayobservethatJoe‐typeofw
IFACWGA proposesthesame
alternatives,respectively,astheGumbel‐typeapproach,butalsoacrossoverpointforthe
alternatives3
uand 4
u.
7.Conclusions
Thepapercitedthegeneralizationofcopula‐basedoperationsofIntuitionisticFuzzyValues
(IFVs)viafuzzynegations.Additionally,novelaggregationoperatorswereproducedfromthenew
operationsofIFVs,withtheirpropertiesbeingfurtherinvestigated.Asaresult,analgorithmis
suggested,whichmaybeutilizedinMultipleAttributeDecisionMaking(MADM)processes.
ThemainadvantageofourworkisthattheaggregationoperatorsofIFVsprovidedinthe
suggestedalgorithmareunivariateparametric,thereforevariousintuitionisticfuzzyArchimedean
copulaweightedarithmeticandgeometricoperatorscouldbeobtained,witheachonepotentially
beingmoreappropriateforthedecisionmakers.Combinedwiththemostappropriatecopula[23]
foreachMADMcase,morespecifiedaggregatoroperatorswouldbeprovidedtoexpressmore
accuratelyadecisionmaker’sattitude.
Inthefutureouraimistocombinetheproposedoperationswithfuzzynegationsviaconic
sections[36],inordertoproducemodifiedfamiliesofaggregatoroperationsofIFVs.Furthermore,
wearewillingtoadapttheproposedoperations,followingtheappropriatetransformation,inother
typesoffuzzysets,suchashesitantfuzzysets[37],andunbalancedlinguistictermsets[38]and
neutrosophicsets[39],inordertoconstructalgorithmsformultipleattributedecisionmakingand
multipleattributegroupdecisionmaking.
AuthorContributions:Investigation,S.G.andB.P.;supervision,B.P.Allauthorshavereadandagreedtothe
publishedversionofmanuscript.
Funding:Thisresearchreceivednoexternalfunding.
Acknowledgments:TheauthorsareverythankfultotheEditorandtheRefereesfortheircorrections,valuable
commentsandsuggestionsinordertoimprovethequalityofthepresentpaper.
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.
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