Total Ordering Defined on the set of all Intuitionistic Fuzzy Numbers

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by  V. Lakshmana Gomathi Nayagam

Undefined1(2015)1–13 1
IOSPress
Total Ordering Defined on the set of all
Intuitionistic Fuzzy Numbers
6
1
0 V.LakshmanaGomathiNayagam a,Jeevaraj.S a,∗ andGeethaSivaraman b
2
aDepartmentofMathematics,NationalInstituteofTechnologyTiruchirappalli,India,

n E-mail:[email protected]
a
J
bDepartmentofMathematics,St.Joseph’sCollege(Autonomous),Tiruchirappalli,India,
6
Email:[email protected]

]
M
G
.
h Abstract.L.A.Zadehintroducedtheconceptoffuzzysettheoryasthegeneralisationofclassicalsettheoryin1965andfurther
t ithasbeengeneralisedtointuitionisticfuzzysets(IFSs)byAtanassovin1983tomodelinformationbythemembership,non
a
m membershipandhesitancydegreemoreaccuratelythanthetheoryoffuzzylogic.Thenotionsofintuitionisticfuzzynumbersin
differentcontextswerestudiedinliteratureandappliedinreallifeapplications.Problemsindifferentfieldsinvolvingqualitative,
[
quantitative and uncertain information can be modelled better using intutionistic fuzzy numbers introduced in [17] which

1 generalisesintuitionisticfuzzyvalues[1,7,17],intervalvaluedintuitionisticfuzzynumber(IVIFN)[10]thanwithusualIFNs
v [5,11,21].Rankingoffuzzynumbershavestartedinearlyseventiesinthelastcenturyandacompleterankingontheclassof
1 fuzzynumbershaveachievedbyW.WangandZ.Wangonlyon2014.AcompleterankingontheclassofIVIFNs,usingaxiomatic
2 setofmembership,nonmembership,vagueandprecisescorefunctionshasbeenintroducedandstudiedbyGeethaetal.[10].In
6 thispaper,atotalorderingontheclassofIFNs[17]usingdoubleupperdensesequenceintheinterval[0,1]whichgeneralises
1 thetotalorderingonfuzzynumbers(FNs)isproposedandillustratedwithexamples.Examplesaregiventoshowtheproposed
0
methodonthistypeofIFNisbetterthanexistingmethodsandthispaperwillgivethebetterunderstandingoverthisnewtype
.
1 ofIFNs.
0
6 Keywords:Doubleupperdensesequence,totalorderrelation,intuitionisticfuzzynumber,intervalValuedintuitionisticfuzzy
1 number,trapezoidalintuitionisticfuzzynumber.
:
v
i
X
r 1. Introduction methodsforintuitionisticfuzzyvalues,intervalvalued
a
intuitionistic fuzzy numbers have been studied in
Informationsystem(IS)isadecisionmodelused [4,9,12,13,15,18,19,20,26,29,31,33,34]. But till date,
toselectthebestalternativefromallthealternativesin there exists no single method or combination of
handundervariousattributes.Thedatacollectedfrom methodsavailabletorankanytwoarbitraryIFNs.The
theexpertsmaybeincompleteorimprecisenumerical difficulty of defining total ordering on the class of
quantities. To deal with such data, the theory of IFS intuitionisticfuzzynumbersisthatthereisnoeffective
provided by Atanassov [1] aids better. In information tooltoidentifyanarbitrarilygivenintuitionisticfuzzy
system, dominance relation rely on ranking of data, number by finitely many real-valued parameters. In
rankingofintuitionisticfuzzynumbersisinevitable. thiswork,byestablisinganewdecompositiontheorem
Manyresearchershavebeenworkinginthearea for IFSs, any IFN can be identified by infinitely
ofrankingofIFNssincelastcentury.Differentranking many but countable number of parameters. A new
decomposition theorem for intuitionistic fuzzy sets
*Correspondingauthor.Tel.:[phone]; is established by the use of an double upper dense
E-mail:[email protected] sequence defined in [0,1]. Actually there are many
0000-0000/15/$00.00(cid:13)c 2015–IOSPressandtheauthors.Allrightsreserved
2 LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers
doubleupperdensesequencesavailableintheinterval tilltodaytherealgeneralizationofintuitionisticfuzzy
[0,1]. Since the choice of a double upper dense valuesandintervalvaluedintuitionisticfuzzynumbers
sequence is considered as the necessary reference havenotbeenstudiedindetail.Thisproblemmotivate
systems for defining a complete ranking, infinitely us to study this type of intuitionistic fuzzy numbers
many total orderings on the set of all IFNs can be [17]anditsorderingprinciplesforranking.
well defined based on each choice of double upper
dense sequence. After introduction, some necessary
2. Preliminaries
fundamentalknowledgeonorderingandintuitionistic
fuzzy numbers, interval valued intuitionistic fuzzy
Herewegiveabriefreviewofsomepreliminaries.
numbersisintroducedinSection2.InSection3,anew
decomposition theorem for intuitionistic fuzzy sets is Definition2.0.1. (Atanassov[1]).LetXbeanonempty
establisedusingdoubleupperdensesequencedefined set.Anintuitionisticfuzzyset(IFS)AinX isdefined
in the interval [0,1]. Section 4 is used to define total by A = (µA,νA), where µA : X → [0,1] and νA :
ordering on the set of all intuitionistic fuzzy numbers X →[0,1]withtheconditions0≤µA(x)+νA(x)≤
by using double upper dense sequence in the interval 1,∀x∈X.ThenumbersµA(x),νA(x)∈[0,1]denote
[0,1].SeveralexamplesaregiveninSection5toshow the degree of membership and non-membership of x
howthetotalorderingonIFNscanbeusedforranking to lie in A respectively. For each intuitionistic fuzzy
better than some other existing methods. Application subsetAinX,πA(x)=1−µA(x)−νA(x)iscalled
ofourproposedmethodinsolvingintuitionisticfuzzy hesitancydegreeofxtolieinA.
information system problem is shown in section 5 by Definition 2.0.2. (Atanassov & Gargov, [2]). Let
developing a new algorithm. Finally conclusions are D[0,1] be the set of all closed subintervals of the
giveninsection6. interval [0,1] . An interval valued intuitionistic fuzzy
set on a set X (cid:54)= φ is an expression given by A =
1.1. Motivation {(cid:104)x,µ (x),ν (x)(cid:105):x∈X}where
A A
µ : X → D[0,1],ν : X → D[0,1] with the
A A
The capacity to handle dubious and uncertain data condition0<sup µ (x)+sup ν (x)≤1.
x A x A
ismoreeffectivelydonebystretchingoutintuitionistic
Theintervalsµ (x)andν (x)denote,respectively,
A A
fuzzy values to TrIFNs because the membership and
thedegreeofbelongingnessandnon-belongingnessof
non-membershipdegreesarebetterexpressedas
the element x to the set A. Thus for each x ∈ X,
trapezoidal values rather than exact values. TrIFNs
µ (x) and ν (x) are closed intervals whose lower
A A
are generalisation of intuitionistic fuzzy values and
and upper end points are, respectively, denoted by
IVIFNs.Asageneralisation,thesetofTrIFNsshould
µ (x), µ (x) and ν (x), ν (x). We denote
contain the set of all intuitionistic fuzzy values and AL AU AL AU
A={(cid:104)x,[µ (x),µ (x)],[ν (x),ν (x)](cid:105):
IVIFNs. But the existing defintion for TrIFNs [5,11, AL AU AL AU
x∈X}where0<µ (x)+ν (x)≤1.
A A
21] does not contain the set of intuitionistic fuzzy
For each element x ∈ X , we can compute the
values which means the existing definition for TrIFN
unknown degree (hesitance degree) of belongingness
is not the real generalisation of intuitionistic fuzzy
π (x)toAasπ (x) = 1−µ (x)−ν (x) = [1−
A A A A
values. Hence the study about new structure for
µ (x)−ν (x),1−µ (x)−ν (x)].Wedenote
intuitionistic fuzzy number [17] is essential. In the AU AU AL AL
thesetofallIVIFSsinXbyIVIFS(X).AnIVIFvalue
application point of view our proposed method is
isdenotedbyA=([a,b],[c,d])forconvenience.
moreapplicableandmorenaturalwhenitiscompared
Definition 2.0.3. (Atanassov & Gargov, [2]). The
with the existing methodology. More precisely, the
complementAc ofA = (cid:104)x,µ (x),ν (x):x∈X(cid:105)is
existing definition for Trapezoidal intuitionistic fuzzy A A
givenbyAc =(cid:104)x,ν (x),µ (x):x∈X(cid:105).
numbers (TrIFN) present in the literature [5,11,21] A A
is defined as A = (cid:104)(a,b,c,d)(e,f,g,h)(cid:105) with e ≤ Definition2.0.4. (Lakshmanaet.al[17]).Anintuitionis-
a ≤ f ≤ b ≤ c ≤ g ≤ d ≤ h does not tic fuzzy set (IFS) A = (µ ,ν ) of R is said to
A A
generalise even the intuitionistic fuzzy value of the be an intuitionistic fuzzy number if µ and ν are
A A
kind A = (a,c) with a + c ≤ 1 and a < c. That fuzzy numbers. Hence A = (µ ,ν ) denotes an
A A
is, if we write A = (a,c) in trapezoidal intuitionistic intuitionistic fuzzy number if µ and ν are fuzzy
A A
form,wegetA = (cid:104)(a,a,a,a)(c,c,c,c)(cid:105)witha < c numbers with ν ≤ µ c, where µ c denotes the
A A A
whichcontradictstheabovedefinitionforTrIFNs.So complementofµ .
A
LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers 3
Anintuitionisticfuzzynumber Definition2.0.8. (Geethaetal.,[10])ForanyIVIFN
A={(a,b ,b ,c),(e,f ,f ,g)}with(e,f ,f ,g)≤ A = ([a,b],[c,d]),thevaguescorefunctionisdefined
1 2 1 2 1 2
(a,b ,b ,c)cisshowninfig(1). asP(A)= a−b−c+d+ac+bd.
1 2 2
Definition2.0.9. (Geethaetal.,[10])ForanyIVIFN
A = ([a,b],[c,d]), the imprecise score function is
definedasIP(A)= −a+b−c+d−ac+bd.
2
Definition 2.0.10. Let X be a non-empty set. Any
subset of the cartesian product X × X is called a
relation,denotedbyR,onX.WewriteaRbiff(a,b)∈
R. A relation R is called a partial ordering on X if
it is reflexive, antisymmetric, and transitive. A partial
ordering R on X is called total ordering if either
aRb or bRa for any a,b ∈ X. Two total orderings
Fig.1.IntuitionisticFuzzyNumber
R and R are different iff there exist a,b ∈ X with
1 2
a(cid:54)=bsuchthataR bbutbR aoraR bbutbR a.For
1 2 2 1
Definition2.0.5. (Lakshmanaet.al[17])Atrapezoidal any given total ordered infinite set, there are infintely
intuitionistic fuzzy number A is defined by A = many ways to redefine a new total ordering on it. A
{(µA,νA)|x∈R},whereµAandνAaretrapezoidal relation R is called an equivalence relation if it is
fuzzynumberswithνA(x)≤µcA(x). reflexive,symmetricandtransitive.
Wenotethatthecondition(e,f ,f ,g)≤(a,b ,b ,c)c
1 2 1 2
of the trapezoidal intuitionistic fuzzy number A =
{(a,b1,b2,c),(e,f1,f2,g)} where (a,b1,b2,c) and 3. TotalOrderingdefinedonIntuitionisticFuzzy
(e,f1,f2,g) are membership and nonmembership Number
fuzzy numbers of A with either e ≥ b and f ≥ c
2 1
or f2 ≤ a and g ≤ b1 on the legs of trapezoidal Geetha et.al [10] have acheived the total
intuitionisticfuzzynumber. ordering on the set of IVIFN using membership,
nonmembership, vague and precise score functions.
A trapezoidal intuitionistic fuzzy number A =
LetusrecallthetotalorderingdefinedbyGeethaet.al
{(a,b ,b ,c),(e,f ,f ,g)}withe≥b andf ≥cis
1 2 1 2 2 1 [10]onIVIFN.
showninfig(2).
Let A = ([a ,b ],[c ,d ]),B = ([a ,b ],[c ,d ]) ∈
1 1 1 1 2 2 2 2
IVIFN. The total ordering ≤ on IVIFN may be
definedtooneofthefollowingcriterion:
(1).L(A)<L(B),or
(2).L(A)=L(B)but −LG(A)<−LG(B),or
(3).L(A)=L(B)andLG(A)=LG(B)butP(A)<
P(B),or
(4). L(A) = L(B),LG(A) = LG(B) and P(A) =
P(B)but −IP(A)≤−IP(B).
The above way of defining a total ordering is often
referredtoaslexicographicinliterature[8].
Fig. 2. Trapezoidal Intuitionistic Fuzzy Number In this section a new total ordering is defined in
A={(a,b1,b2,c),(e,f1,f2,g)} the set of all intuitionistic fuzzy numbers using the
above ordering and a new decomposition theorem on
Definition 2.0.6. (Lakshmana et al., [16]) For any intuitionisticfuzzysetswhichisgiveninthefollowing
IVIFN A = ([a,b],[c,d]), the membership score subsection.
functionisdefinedasL(A)= a+b−c−d+ac+bd.
2
3.1. UpperDensesequencein(0,1]
Definition2.0.7. (Geethaetal.,[10])ForanyIVIFN
A=([a,b],[c,d]),thenon-membershipscorefunction
The major difficulty in defining total ordering
isdefinedasLG(A)= −a−b+c+d+ac+bd.
2 on the set of all fuzzy numbers is that there is no
4 LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers
effective tool to identify an arbitrarily given fuzzy ={(1,1),(1/2,1/2),(1/3,1/3),(2/3,2/3),
numberbyonlyfinitelymanyreal-valuedparameters. (1/4,1/4),(3/4,3/4),(1/5,1/5),...}. Clearly S is a
Toovercomethisdifficulty,W.Wang,Z.Wang[27]has doubleupperdensesequencein(0,1].
introducedtheconceptofupperdensesequenceinthe
Intheforthcomingsectionsthisdoubleupperdense
interval(0,1],anddefinedthetotalorderingontheset
sequencewillbeveryusefulfordefiningtotalorderings
of all fuzzy numbers using this sequence. This upper
onthesetofIFNs.
densesequencegivesvaluesforαintheα-cutofFNs.
ButforIFNs,twosequencesareneededtogivevalues 3.2. Decompositiontheoremforintuitionisticfuzzy
for α & β in the (α,β)-cut where α ∈ (0,1] & β ∈ numberusingupperdensesequence
[0,1). For convenience upper dense sequence of α is
denoted by S = {α |i=1,2,...} and upper dense In this section, a new decomposition theorem
1 i
sequence of β is denoted by S = {β |i=1,2,...}. for IFSs is established using the double upper dense
2 i
In this section, the upper dense sequence in (0,1] is sequence defined in (0,1]. Before establising a new
brieflyreviewed. decompositiontheoremforintuitionisticfuzzysets,it
is needed for us to define decompostion theorems for
Definition3.1.1. (W.Wang,Z.Wang,[27])Asequence
intuitionisticfuzzysetsusingspecialα−β cuts.
S = {α |i=1,2,...} ⊂ (0,1] is said to be upper
i
Definition 3.2.1. Let X be a nonempty universal
densein(0,1]if,foreverypointx ∈ (0,1]and(cid:15) > 0,
set and A = (µ ,ν ) be an intuitionistic fuzzy
thereexistssomeα ∈S suchthatα ∈[x,x+(cid:15))for A A
i i
set of X with membership function µ and with a
somei.AsequenceS ⊂[0,1)issaidtobelowerdense A
nonmembershipfunctionν .Letα,β ∈(0,1]
in [0,1) if, for every point x ∈ [0,1) and any (cid:15) > 0, A
thereexistsαi ∈S suchthatαi ∈(x−(cid:15),x]forsome 1. The α−β cut, denoted by α−βA is defined by
i. (α−β)A = αµ × βν where αµ = {x ∈
A A A
X|µ (x) ≥ α} and βν = {x ∈ X|ν (x) ≥
From definition 3.1.1 we note that, any A A A
β}. Equivalently (α−β)A = µ−1([α,1]) ×
upper dense sequence in (0,1] is nothing but a dense A
ν−1([β,1])isasubsetof℘(X)×℘(X).
sequence with real number 1 and it is also a lower A
2. The strong alpha beta cut, denoted by (α−β)+A
densesequence.
is defined by (α−β)+A =α+ µ ×β+ ν where
A A
Definition3.1.2. LetS ={α |i=1,2,...}andS = α+µ = {x ∈ X|µ (x) > α} and β+ν =
1 i 2 A A A
{β |i=1,2,...} be two upper dense sequences in {x ∈ X|ν (x) > β}.Equivalently(α−β)+A =
i A
(0,1]thenthedoubleupperdensesequenceisdefined µ−1((α,1])×ν−1((β,1])isasubsetof℘(X)×
A A
asS =(S ,S )={(α ,β )|i=1,2,...}. ℘(X).
1 2 i i
3. ThelevelsetofA,denotedbyL(A)isdefinedby
Example for an upper dense sequence and double
L(A)=L ×L where
upperdensesequencein(0,1]isgivenasfollows. µA νA
L ={α|µ (x)=α, x∈X}and
µA A
Example 3.1.1. Let S1 = {αi|i=1,2...} be the set LνA = {β|νA(x)=β, x∈X} which is a
ofallrationalnumbersin(0,1],whereα = 1,α = subsetof℘([0,1])×℘([0,1]).
1 2
1,α = 1,α = 2,α = 1,α = 3,α = 1,α =
2 3 3 4 3 5 4 6 4 7 5 8 Example3.2.1. LetA=(cid:104)(0.17,0.3,0.47,0.56),
2,α = 3,α = 4,....
5 9 5 10 5 (0.05,0.13,0.16,0.23)(cid:105)beatrapezezoidalintuitionistic
ThensequenceS1isanupperdensesequencein(0,1]. fuzzy number. Then (α−β)A = ([0.17 + (0.3 −
If we allow a number to have multiple occurences in
0.17)α,0.56 − (0.56 − 0.47)α],[0.05 + (0.13 −
the sequence , the general members in upper dense
0.05)β,0.23−(0.23−0.16)β])=([0.17+0.13α,0.56−
sequence S = {s |i=1,2,...} can be expressed
β βi 0.09α],[0.05+0.08β,0.23−0.07β]),∀α,β ∈[0,1].
(cid:108)by(cid:113)sβi = (ki(cid:109)− k−21),i = 1,2,... where k = (α−β)+A = ((0.17+(0.3−0.17)α,0.56−(0.56−
2i+ 1 − 1 . That is, s = 1, s = 1, s = 0.47)α),(0.05 + (0.13 − 0.05)β,0.23 − (0.23 −
4 2 β1 β2 2 β3
0.16)β)) = ((0.17+0.13α,0.56−0.09α),(0.05+
1, s = 1, s = 2, s = 1, s = 1, s =
β4 3 β5 3 β6 β7 4 β8 0.08β,0.23−0.07β)),∀α,β ∈[0,1].
2, s = 3,.... In sequence S , for instance, s is
4 β2 4 β β3 L =({0.17+(0.3−0.17)α,0.56−(0.56−0.47)α},
thesamerealnumberassβ1. {0µ.A05+(0.13−0.05)β,0.23−(0.23−0.16)β)}
Example 3.1.2. Consider S as in example 3.1.1 Let =({0.17+0.13α,0.56−0.09α},
1
S =(S ,S )={(α ,β )|α =β ,α ∈S } {0.05+0.08β,0.23−0.07β}),∀α,β ∈[0,1].
1 2 i i i i i 1
LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers 5
One way of representing a fuzzy set is by special ([a+(b −a)α,c−(c−b )α],[e+(f −e)β,g−
1 2 1
fuzzy sets on α-cuts and another way of representing (g − f )β]) and the special intuitionistic fuzzy set
2
a fuzzy set is by special fuzzy sets on Strong α-cuts. A employed in definition 3.2.2 is defined by
(α,β)
As a generalisaton of fuzzy sets, any intuitionistic its membership ( µ ) and non-membership function
α A
fuzzysetcanalsoberepresentedbytheuseofspecial ( ν )asfollows
β A

intuitionistic fuzzy sets on α − β cuts and special α, x∈[a+(b −a)α,c−(c−b )α]
 1 2
intuitionisticfuzzysetsonstrongα−β cuts.
µ (x)=
α A
Definition 3.2.2. The special intuitionistic fuzzy set 0, otherwise
A = ( µ , ν ) is defined by its membership 
((α,µβ))andnoαn-AmeβmbAershipfunction( ν )asfollows, β, x∈[e+(f1−e)β,g−(g−f2)β]
α A β A ν (x)=
 β A
 α , x∈α µA 0, otherwise
µ (x)=
α A
 0 , otherwise Examples of sets αµA, βνA, αµA and βνA for
three values of α (namely α ,α ,α ) and β (namely
 1 2 3
β , x∈β ν
 A β1,β2,β3)areshowninfigure3.
ν (x)= .
β A
 0 , otherwise
The following decomposition theorems will show
the representation of an arbitrary IFS in terms of the
specialIFSs A.
(α,β)
Theorem 3.1. First Decomposition theorem of an
IFS: Let X be a non-empty set. For an intuitionistic
fuzzysubsetA=(µ ,ν )inX,
A A
(cid:16) (cid:17)
(cid:83) (cid:83) (cid:83)
A = µ , ν , where is
α∈[0,1] α A β∈[0,1]β A
standardunion.
Proof. Let x be an arbitrary element in X and let
µ (x)=a&ν (x)=b.
A A
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17)
(cid:83) (cid:83)
Then µ (x), ν (x) =
α∈[0,1] α A β∈[0,1] β A
(cid:0) (cid:1)
Sup µ (x),Sup ν (x) =
α∈[0,1] α A β∈[0,1] β A Fig.3.IllustrationofFirstDecompositionTheorem
(cid:2) (cid:3)
(max Sup µ (x),Sup µ (x) ,
α∈[0,a] α A α∈(a,1] α A
(cid:2) (cid:3)
max Supβ∈[0,b] βνA(x),Supβ∈(b,1] βνA(x) ). According to theorem 3.1, A is obtained by taking
For each α ∈ [0,a], we have µA(x) = a ≥ α, the standard fuzzy union of sets (αµA, βνA) for all
therefore αµA(x) = α. On the other hand, for each α,β ∈[0,1].
α∈(a,1],wehaveµ (x)=a<αand µ (x)=0.
A α A
Theorem3.2. SecondDecompositiontheoremofan
Similarly,foreachβ ∈[0,b],wehaveν (x)=b≥β,
A IFS: Let X be a non-empty set. For an intuitionistic
therefore ν (x) = β. On the other hand, for each
β A fuzzysubsetAinX,
Tβh∈ere(bfo,r1e],(cid:16)w(cid:16)e(cid:83)haveνA(xµ) =(cid:17)(bx<),(cid:16)β(cid:83)andβνA(xν)(cid:17)=(0x.)(cid:17) A = (cid:16)(cid:83)α∈[0,1] α+µA,(cid:83)β∈[0,1] β+νA(cid:17), where (cid:83) is
α∈[0,1] α A β∈[0,1] β A standardunion.
(cid:0) (cid:1)
= Sup α,Sup β =(a,b)=
α∈[0,a] β∈[0,b]
(µA(x),νA(x)).Hencethetheorem. Proof. Let x be an arbitrary element in X and let
µ (x)=a&ν (x)=b.
A A
To illustrate the above theorem, let us consider a (cid:16)(cid:16)(cid:83) (cid:17) (cid:16)(cid:83) (cid:17) (cid:17)
Then µ (x), ν (x) =
trapezoidal intuitionistic fuzzy number A as in figure α∈[0,1]α+ A β∈[0,1]β+ A
(cid:0) (cid:1)
3. Supα∈[0,1] α+µA(x),Supβ∈[0,1] β+νA(x)
(cid:2) (cid:3)
For each α,β ∈ [0,1], the α − β cut of A = =(max Supα∈[0,a)α+µA(x),Supα∈[a,1]α+µA(x) ,
{(a,b ,b ,c),(e,f ,f ,g)} is given by (α−β)A = max(cid:2)Sup ν (x),Sup ν (x)(cid:3)).
1 2 1 2 β∈[0,b) β+ A β∈[b,1] β+ A
6 LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers
For each α ∈ [0,a), we have µ (x) = a > α, WemayfindarealnumberK ∈S suchthatK ≥α
A 1 1 1
therefore µ (x) = α. On the other hand, for , which implies that K ∈ [α,a) and µ (x) <
α+ A 1 α+ A
each α ∈ [a,1], we have µ (x) = a ≤ α and µ (x) ≤ Sup µ (x). Thus taking the
A K1 A K1∈S1 K1 A
µ (x)=0. supremumwithrespecttoα ∈ (0,µ (x)),weobtain
α+ A A
Similarly,foreachβ ∈[0,b),wehaveν (x)=b>β, µ (x)=Sup µ (x)≤Sup µ (x)
A A α∈(0,a)α+ A (cid:83) K1∈S1K1 A
therefore ν (x) = β. On the other hand, for each =Sup µ (x).i.e.,µ (x)≤ µ (x).
β+ A α∈S1 α A A α∈S1 α A
β ∈[b,1],wehaveν (x)=b≤β and ν (x)=0. Similarlyforeachβ ∈ [0,b),sinceS isupperdense
A β+ A 2
Therefore in [0,1]. We may find a real number K ∈ S such
(cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17) 2 2
(cid:83) µ (x), (cid:83) ν (x) = that K ≥ β , which implies that K ∈ [α,b) and
α∈[0,1] α+ A β∈[0,1] β+ A 2 2
(cid:0) (cid:1) ν (x) < ν (x) ≤ Sup ν (x). Thus
Supα∈[0,a)α,Supβ∈[0,b)β =(a,b)=(µA(x),νA(x)). β+ A K2 A K2∈S2 K2 A
taking the supremum with respect to β ∈ (0,ν (x)),
Hencethetheorem. A
weobtainν (x)=Sup ν (x)≤
A β∈(0,b) β+ A
Theorem 3.3. Third Decomposition theorem of an SupK2∈S2 K2(cid:83)νA(x)=Supβ∈S2 βνA(x).
IFS: Let X be a non-empty set. For an intuitionistic i.e.,νA(x)≤ β∈S2 βνA(x).
fuzzysubsetAinX, HenceA(x)=(µA(x),νA(x))⊆
Astan=dar(cid:16)d(cid:83)unαi∈oLnµ.A αµA,(cid:83)β∈LνA βνA(cid:17), where (cid:83) is (cid:16)(1(cid:83))aαn∈dS1(2α)µcAon(xcl)u,d(cid:83)eβs∈thSe2pβrνoAo(fx. )(cid:17). ......(2)
Proof. The proof of this theorem is similar to the
Theorem3.2. 4. TotalOrderingonthesetofallIntuitionistic
FuzzyNumbers
Regarding IFN as special intuitionistic fuzzy
subset (IFS) of (cid:60), these decomposition theorems The new decomposition theorem established
are also available for IFNs. Since they identify an in section 3 identifies an arbitrary intuitionistic fuzzy
intuitionistic fuzzy number by uncountably infinite number by a countably many real-valued parameters.
realvaluedparametersgenerallyandtherefore It provides us with a powerful tool for defining total
lexicography can not be used anymore, unfortunately order in the class of IFN, by extending lexicographic
noneofthemcanbeusedtodefineatotalorderingon rankingrelationdefinedinGeethaet.al[10].
thesetofIFNs.Thus,establisinganewdecomposition
theorem, which identifies any IFN by only countably
Note 4.0.1. Let A be an intuitionistic fuzzy number.
manyrealvaluedparametersisessential.
LetS = {(α ,β )|i=1,2,...} ∈ [0,1]beandouble
i i
Theorem3.4. FourthDecompositiontheoremofan upper dense sequence. The (αi − βi)-cut of an IFN
IFS: Let A = (µA,νA) be an intuitionistic fuzzy A at each αi,βi , i = 1,2,..., is a combination
subsetofX,andS =(S ,S )beagivendoubleupper of two closed intervals. Denote this intervals by
1 2
densesequencein[0,1]. ([ai,bi],[ci,di]), where [ai,bi] is the α-cut of the
(cid:16)(cid:83) (cid:83) (cid:17) membership function of A and [c ,d ] is the β-cut of
ThenA= µ , ν . i i
α∈S1 α A β∈S2 β A thenon-membershipfunctionofAand
Proof. Let x be an arbitrary element in X and let letC4i−3 = ai+bi−ci−d2i+aici+bidi,
µA(x) = a&νA(x) = b.SinceS1 ⊆ [0,1]&S2 ⊆ C4i−2 = ai+bi−ci−d2i−aici−bidi ,
[µ0A,1(cid:16)a]n,dTh(cid:83)uβs∈wSe2hβaνvAe(cid:83)⊆α(cid:83)∈Sβ1∈[0α,1µ(cid:17)]Aβ⊆νA(cid:83)=α∈ν[A0,1.] αµA = CC44ii−=1 =ai−abii−+bcii−−cdi2+i+d2ai+icai−icbii+dbii,dii=, 1,2,....
So (cid:83) µ ,(cid:83) ν ⊆A ........(1) By fourth decomposition theorem these countably
α∈S1 α A β∈S2 β A (cid:83) manyparameters{C |j =1,2,...}identifythe
Nowwehavetoshowthatµ (x) ≤ µ (x) j
(cid:83) A α∈S1 α A intuitionistic fuzzy number. Using these parameters,
andν (x)≤ ν (x)toprovethetheorem.
A β∈S2 β A wedefinearelationonthesetofallintuitionisticfuzzy
From second decomposition theorem we know that ,
numberasfollows.
foreachx∈X,(µ (x),ν (x))=
A A
(cid:0) (cid:1)
Supα∈[0,1] α+µA(x),Supβ∈[0,1] β+νA(x) = Definition 4.0.3. (Ranking Principle) Let A and B
(cid:0) (cid:1)
Sup µ (x),Sup ν (x) . be any two IFNs. Consider any double upper dense
α∈[0,a) α+ A β∈[0,b) β+ A
Foreachα ∈ [0,a),sinceS isupperdensein(0,1]. sequence S = {(α ,β )|i=1,2,...} in [0,1], using
1 i i
LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers 7
note4.0.1,foreachi=1,2...wehaveC ,C , B = (cid:104)(0.35,0.35,0.4,0.55),(0,0.2,0.25,0.35)(cid:105) be
4i−3 4i−2
C ,C ,whichdescribeasequencesofC (A)and twoTrIFNs.
4i−1 4i j
C (B). If A (cid:54)= B, then there exists j such that The ordering < defined by using double upper dense
j
C (A)(cid:54)=C (B)andC (A)=C (B)forallpositive sequenceS giveninexample3.1.2andthewayshown
j j k k
integers k < j. The ’<’ relation on the set of of all indefinition4.0.3arenowadapted.Wehave(α−β)A=
intuitionisticfuzzynumber(IFN)isdefinedas, ([0.3+0.05α,0.5−0.1α],[0.1+0.1β,0.3−0.05β]),
A<Bifthereexistsj suchthatC (A)<C (B)and (α−β)B =([0.35,0.55−0.15α],[0.2β,0.35−0.1β]).
j j
C (A)=C (B)forallpositiveintegersk <j. For i = 1,(α ,β ) = (1,1), we have C (A) =
k k 1 1 1
0.235 = C (B),C (A) = 0.065 = C (B),C (A) =
TheaboverankingprincipleonIFNsisillustratedin 1 2 2 3
0.085 = C (B), C (A) = −0.065 = C (B). Now
thefollowingexamples. 3 4 4
we have to find for i = 2,(α ,β ) = (1/2,1/2),
2 2
Example4.0.2. Let then C (A) = 0.26125 < C (B) = 0.30125. Since
5 5
A=(cid:104)(0.20,0.30,0.50),(0.35,0.55,0.65)(cid:105) C (A)<C (B).HenceA<B.
5 5
B =(cid:104)(0.17,0.32,0.58),(0.37,0.63,0.73)(cid:105)and
Theorem4.1. Relation<isatotalorderingontheset
C = (cid:104)(0.25,0.40,0.70),(0.45,0.75,0.85)(cid:105) be three
ofallintuitionisticfuzzynumber.
triangularintuitionisticfuzzynumbers(TIFN).
The ordering < defined by using double upper dense
Proof. Claim:<istotalorderingonthesetofIFN.
sequence S given in example 3.1.2 and the way
To prove < is total ordering we need to show the
shown in definition 4.0.3 are now adapted. Thus we
following(1).<isPartialorderingonthesetofIFN
have (α−β)A = ([0.2 + 0.1α,0.5 − 0.2α],[0.35 +
(2).AnytwoelementsofthesetofIFNsarecomparable.
0.2β,0.65−0.1β]),(α−β)B =([0.17+0.15α,0.58−
(1).<ispartialorderingonthesetofIFN:
0.26α],[0.37+.26β,0.73−0.1β]),(α−β)C =([0.25+
(i)<isrefelxive:
0.15α,0.70−0.3α],[0.45+0.3β,0.85−0.1β]).For
It is very clear that the relation < is reflexive for any
i = 1,(α ,β ) = (1,1), C (A) = −0.85,C (B) =
1 1 1 1 A.
−0.1084,C (C) = −0.05. i.e., C (A) < C (B) <
1 1 1 (ii)<isantisymmetric:
C (C).HenceA<B <C.
1 claim:IfA<BandB <AthenA=B.
Example4.0.3. Let SupposeA(cid:54)=B,thenfromthehypothesisA≺Band
A=(cid:104)(0.35,0.35,0.4,0.6),(0.1,0.2,0.3,0.35)(cid:105) B ≺A.Fromthedefinition4.0.3,wecanfindj such
1
B = (cid:104)(0.35,0.35,0.45,0.55),(0,0.3,0.3,0.35)(cid:105) be that C (A) < C (B) and C (A) = C (B) for all
j1 j1 j j
twoTrIFNs. positiveintegersj < j .Similarlyweareabletofind
1
Thelordering<definedbyusingdoubleupperdense j such that C (A) < C (B) and C (A) = C (B)
2 j2 j2 j j
sequence S given in example 3.1.2 and the way for all positive integers j < j . Then j &j must be
2 1 2
shown in definition 4.0.3 are now adapted. We have the same, let it to be j . But C (A) < C (B), and
0 j0 j0
(α−β)A = ([0.35,0.6 − 0.2α],[0.1 + 0.1β,0.35 − C (B) < C (A) this contradicts our hypothesis.
j0 j0
0.05β]),(α−β)B =([0.35,0.55−0.1α],[0.3β,0.35− Therefore our assumption A (cid:54)= B is wrong. Hence
0.05β]). For i = 1,(α ,β ) = (1,1), C (A) = A=B.
1 1 1
0.22 = C (B), C (A) = −0.03,C (B) = 0.02. (iii)<istransitive:
1 2 2
Since C (A) < C (B). Hence A > B. From this To prove (iii), we have to show that if A < B and
2 2
example we come to know that C alone can not B <C thenA<C
1
rank any two given IFNs. With the help of C we LetA,B,C bethreeIFNs.LetusassumeA < B and
2
discriminateAandB. B <C.....(1)
ThereforefromA < B,wecanfindapositiveinteger
The following example shows the importance of
k suchthatC (A) < C (B)andC (A) = C (B)
doubleupperdensesequence.Inthepreviousexamples, 1 k1 k1 k k
forallpositiveintegerk <k .SimilarlyfromB <C,
differentIFNsarerankedbymeansofC andC .But 1
1 2 wecanfindapositiveintegerk suchthatC (B) <
ingeneralC andC aloneneednotbeenoughtorank 2 k2
1 2 C (C) and C (B) = C (C) for all positive integer
theentireclassofintuitionisticfuzzynumberwhichis k2 k k
k < k . Now taking j = min(k ,k ), we have
showninexample4.0.4. 2 0 1 2
C (A) < C (C) and C (A) = C (C) for all
k0 k0 k k
Example4.0.4. Let positive integer k < k . i.e., A < C. Hence < is
0
A=(cid:104)(0.3,0.35,0.4,0.5),(0.1,0.2,0.25,0.3)(cid:105) Transitive.
8 LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers
Thereforefrom(i),(ii),and(iii),weprovedtherelation 5.1. Comparisionbetweenourproposedmethodwith
<isPartialOrderingonthesetofallIFNs. thescorefunctiondefinedinLakshmanaet
(2).AnytwoelementsofthesetofIFNsarecomparable. al.[17]:
For any two IFNs A and B, they are either A =
B, or A (cid:54)= B. In the latter case, there are some In this subsection, our proposed method is
integers j such that C (A) (cid:54)= C (B). Let J = compared with the total score function defined in
j j
{j|C (A)(cid:54)=C (B)}.ThenJ islowerbounded0and Lakshmanaetal.[17],whichisexplainedherewithan
j j
therefore, according to the well ordering property , J illustrativeexample.
has unique smallest element, denoted by j . Thus we
0 Definition5.1.1. (Note1.2[3])
haveC (A) = C (B)forallpositiveintegersj < j ,
j j 0 The membership score of the triangular intuitionistic
and either C (A) < C (B) or C (A) > C (B),
j0 j0 j0 j0 fuzzy number (TIFN) M = {(a,b,c)(e,f,g)} is
that is, either A ≺ B or B ≺ A in this case. So, for
definedbyT(M) = [1+R(M)−L(M)],whereL(M) =
thesetwoIFNs,eitherA < B orB < A.Thismeans 2
1−a andR(M)= c .
thatpartialordering<isatotalorderingonthesetof 1+b−a 1+c−b
allintuitionisticfuzzynumbers.Hencetheproof. Definition5.1.2. (Lakshmanaetal.[17])
Thenewmembershipscoreofthetriangular
Similar to the case of total orderings on the real
intuitionistic fuzzy number M = {(a,b,c)(e,f,g)}
line (−∞,+∞) an the total ordering on the sets of
is defined by NT(M) = [1+NL(M)−NR(M)], where
special types of IFNs shown in section 3, infinitely 2
many different total orderings on the set of IFNs can NL(M)= e andNR(M)= 1−g .
1+e−f 1+f−g
be defined. Even using a given upper dense sequence
Definition5.1.3. (Lakshmanaetal.[17])
in[0,1],therearestillinfinitelymanydifferentwaysto
Thenonmembershipscoreofthetriangular
determineatotalorderingonthesetofIFNs.Anotable
intuitionisticfuzzynumberM = {(a,b,c)(e,f,g)}is
factisthateachofthemisconsistentwiththenatural
definedbyNT (M)=1−NT(M).
c
ordering on the set of all real numbers. This can be
regardedasafundamentalrequirementforanypractice Definition5.1.4. (Lakshmanaetal.[17])
orderingmethodonthesetofallIFNs. LetM =(cid:104)(a,b,c),(e,f,g)(cid:105)beantriangular
intuitionistic fuzzy number. If e ≥ b and f ≥ c,
then the score of the intuitionistic fuzzy number M is
5. Significanceoftheproposedmethod definedby(T,NT ),whereT isthemembershipscore
c
of M which is obtained from (a,b,c) and NT is the
c
Many researchers have proposed different
nonmembershipscoreofM obtainedfrom(e,f,g).
ranking methods on IFNs, but none of them has
covered the entire class of IFNs, and also almost all Definition5.1.5. (Lakshmanaetal.[17])
the methods have disadvantage that at some point of RankingofIntuitionisticFuzzyNumbers:
time they ranked two different numbers as the same. Let M1 = (cid:104)(a1,b1,c1),(e1,f1,g1)(cid:105) and M2 =
In this paper a special type of IFNs which is shown (cid:104)(a2,b2,c2),(e2,f2,g2)(cid:105)betwotriangularintuitionisti
in figure 1 which generalizes IFN more natural in cfuzzynumberswithei ≥ bi &fi ≥ ci.ThenM1 ≤
real scenario. Problems in different fields involving M2 ifthemembershipscoreofM1 ≤themembership
qualitative,quantitativeanduncertaininformationcan score of M2 and the nonmembership score of M1 ≥
be modelled better using this type of IFNs when thenonmembershipscoreofM2.
compared with usual IFNs. Our proposed ranking
Inthisexampledefinition5.1.1to5.1.5aredemonstr
methodonthistypeofIFNwillgivethebetterresults
atedandalsotheillogicalityofLakshmanaetal’s[17]
over other existing methods, and this paper will give
methodisshown.
the better understanding over this new type of IFNs.
This type of IFNs are very much important in real Example5.1.1. LetM =(cid:104)(0,0.2,0.4),(0.4,0.45,.5)(cid:105)
life problems and this paper will give the significant with (0,0.2,0.4)c ≤ (0.4,0.45,0.5) and 0.4 ≥
changeintheliterature.Modelingproblemsusingthis 0.2&0.45≥0.4andN =(cid:104)(0.25,0.25,0.25),(0.4,
type of IFN will give better result. In this subsection 0.45,0.5)(cid:105) with (0.25,0.25,0.25)c ≥ (0.4,0.45,0.5)
ourproposedmethodiscomparedwiththetotalscore and0.4 ≥ 0.25&0.45 ≥ 0.25bethetwotriangular
function defined in Lakshmana et al.[17], which is intuitionisticfuzzynumbers.Usingdefinition5.1.1we
explainedherewithanillustrativeexample. get L(M) = 0.8333 and R(M) = 0.3333 which
LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers 9
impliesT(M)=0.25.Applyingdefinition5.1.2,5.1.3 alternativesinhandundervariousattributes.Thedata
5.1.4 to M we get NL(M) = 0.421053,NR(M) = collected from the experts may be incomplete or
0.526316, NT(M) = 0.447368 and NT (M) = imprecisenumericalquantities.Todealwithsuchdata
c
0.553. Therefore the total score of membership and the thoery of IFS provided by Atanassov [1] aids
nonmembership functions of M are T = 0.25 and better.Ininformationsystem,dominancerelationrely
NTc =0.553respectively.Thentheintuitionisticfuzzy on ranking of data, ranking of intuitionistic fuzzy
score is represented by (T,NTc) = (0.25,0.553). numbersisinevitable.
Similarly using definitions 5.1.1 to 5.1.3, we get
Definition5.3.1. AnInformationSystem
the total score of membership and nonmembership
S = (U,AT,V,f) with V = ∪ V where
functions of N are T = 0.25, and NT = 0.553 a∈AT a
c
V is a domain of attribute a is called trapezoidal
respectively. a
intuitionistic fuzzy information system if V is a set
Therefore from definition 5.1.5, this method ranks
of TrIFN. We denote f(x,a) ∈ V by f(x,a) =
M and N are equal but M & N are different a
(cid:104)(a ,a ,a ,a ),(a (cid:48),a (cid:48),a (cid:48),a (cid:48))(cid:105) where a ,a (cid:48) ∈
triangularintuitionisticfuzzynumbers. 1 2 3 4 1 2 3 4 i i
[0,1].
The total ordering < defined by using double upper
densesequenceD giveninexample3.1.2andthe
(α,β) Thenumericalillustrationisgiveninexample5.5.1.
way shown in definition 4.0.3 are now adapted. For
i = 1 and (α ,β ) = (1,1), we have C (M) = Definition 5.3.2. An TrIFIS, S = (U,AT,V,f)
1 1 1
−0.16andC1(N)=−0.0875. together with weights W = {wa/a ∈ AT} is called
i.e., C (M) < C (N). Hence our proposed method WeightedTrapezoidalIntuitionisticFuzzyInformation
1 1
ranksN asbetterone. System(WTrIFIS)andisdenotedby
S =(U,AT,V,f,W).
5.2. Comparisionoftheproposedmethodwithsome
Definition5.3.3. Leta∈AT beacriterion.Letx,y ∈
existingmethods
U. If f(x,a) > f(y,a) (as per definition 4.0.3) then
Inthissubsectionsignificanceofourproposed x>a ywhichindicatesthatxisbetterthan(outranks)
methodoversomeexistingmethodsareexplainedwith y with respect to the criterion a. Also x =a y means
examples.Thetable1showsthatourproposedmethod thatxisequallygoodasywithrespecttothecriterion
issignificantoverthemethodspresentedin[4,9,12,13, a,iff(x,a)=f(y,a).
15,18,19,20,26,29,31,33,34] , and it is supported by
Definition 5.3.4. Let S = (U,AT,V,f,W) be an
Geethaet.al[10].
WTrIFISandA⊆AT.
For example, let A = ([0.2,0.25],[0.4,0.45]) and
LetB (x,y) = {a∈A|x> y}andletC (x,y) =
A a A
B = ([0.15,0.3],[0.35,0.5])betwoIVIFNs.Thenby
{a∈A|x= y}.Theweightedfuzzydominancerelati-
a
applyingLakshmanaandGeetha[15]approachweget
on WR (x,y) : U × U → [0,1] is defined by
LAGa(nAd)B=arLeGe(qBua)l =whi0c.h45+isδ2(i0l.l7o0g)ictahli.sBimypalpiepslytihnagt WRA(xA,y)=(cid:80)a∈BA(x,y)wa+ (cid:80)a∈CA2(x,y)wa.
the total ordering < defined by using double upper
Definition 5.3.5. Let S = (U,AT,V,f,W) be an
densesequenceD giveninexample3.1.2andthe
(α,β) WTrIFIS and A ⊆ AT. The entire dominance degree
way shown in definition 4.0.3 are now adapted. For
ofeachobjectisdefinedas
i = 1 and (α1,β1) = (1,1), we have C1(A) = WR (x )= 1 (cid:80)|U| WR (x ,y )
−0.10375 and C (B) = −0.09875, i.e., C (A) < A i |U| j=1 A i j
1 1
C (B). Hence A < B which is supported by Geetha
1 5.4. AlgorithmforRankingofobjectsinWTrIFIS
et.al’s approach. The double upper dense sequence
S in example 3.1.2 is used as a necessary reference
Let S = (U,AT,V,f,W) be an WTrIFIS. The
system for our proposed method in table 1 (i.e., for
objectsinU arerankedusingfollowingalgorithm.
i=1,(α ,β )=(1,1)).
1 1
5.3. TrapezoidalIntuitionisticFuzzyInformation Algorithm:5.3
System(TrIFIS) 1. Using Definition 4.0.3 find Cj(cid:48)s accordingly, to
decidewhetherx > x orx > x orx = x for
i a i j a i i a j
Information system (IS) is a decision model alla∈A(A⊆AT)andforallx ,x ∈U.
i j
used to select the best alternative from the all the 2.EnumerateB (x ,x )using
A i j
10 LakshmanaGomathiNayagametal./TotalOrderingDefinedonthesetofallIntuitionisticFuzzyNumbers
Table1
Significanceofproposedmethod
OtherexisitingMethods shortcomingsofexisitingmethods Numericalexample Geethaet.al[10] ProposedMethod
Xu,Z.S.[29]
s(A)=a+b−2c−d A=([a1,b1],[c1,d1]), A=([0,0.3],[0.35,0.65]) L(A)=−0.2525 C1(A)=−0.2525,
h(A)=a+b+2c+d B=([a1−(cid:15),b1+(cid:15)],[c1−(cid:15),d1+(cid:15)]) B=([0.1,0.2],[0.45,0.55]) L(B)=-0.2725 C1(B)=−0.2725
s(A)=s(B)=a1+b1−2c1−d1, s(A)=s(B)=−0.35, L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
h(A)=h(B)=a1+b1+2c1+d1⇒A=B h(A)=h(B)=0.65⇒A=B
DejianYu,et.al.,[9]
S(A)=2+a+2b−c−d A=([a1,b1],[c1,d1]), A=([0,0.3],[0.35,0.65]) L(A)=−0.2525 C1(A)=−0.2525,
B=([a1−(cid:15),b1+(cid:15)],[c1−(cid:15),d1+(cid:15)]) B=([0.1,0.2],[0.45,0.55]) L(B)=-0.2725 C1(B)=−0.2725
S(A)=S(B)=2+a1+b12−c1−d1, S(A)=S(B)=0.65, L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
⇒A=B ⇒A=B
JunYe,[13]
M(A)=a+b−1+c+2d A=([a1,b1],[c1,d1]), A=([0.1,0.15],[0.25,0.35]) L(A)=−0.13625 C1(A)=−0.13625,
B=([a1−(cid:15),b1+(cid:15)],[c1−(cid:15),d1+(cid:15)]) B=([0.05,0.2],[0.20,0.40]) L(B)=-0.13 C1(B)=−0.13
M(A)=M(B)=a1+b1−1+c1+2d1, M(A)=M(B)=−0.45, L(B)>L(A)⇒B>A C1(B)>C1(A)⇒B>A
⇒A=B ⇒A=B
LakshmanaandGeetha[15]
LG(A)=a+b+δ(2−2a−b−c−d),∀δ∈[0,1] A=([a1,b1],[c1,d1]), A=([0.2,0.25],[0.40,0.45]) L(A)=−0.10375 C1(A)=−0.10375,
B=([a1−(cid:15),b1+(cid:15)],[c1−(cid:15),d1+(cid:15)]) B=([0.15,0.30],[0.35,0.50]) L(B)=-0.09875 C1(B)=−0.09875
LG(A)=LG(B)=a1+b1+δ(2−a21−b1−c1−d1), LG(A)=LG(B)=0.45+δ2(0.70), L(B)>L(A)⇒B>A C1(B)>C1(A)⇒B>A
⇒A=B ⇒A=B
ChenandTan[4]
S(A)=µ−ν, A=(a1,b1),B=(a1−(cid:15),b1−(cid:15)),∀(cid:15)∈(0,1] A=(0.6,0.2),B=(0.47,0.07) L(A)=0.52,L(B)=0.4329 C1(A)=0.52,C1(B)=0.4329
whereA=(µ,ν)withµ+ν≤1 S(A)=S(B)=a1−b1⇒A=B S(A)=S(B)=0.4⇒A=B L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
HongandChoi[12]
H(A)=µ+ν, A=(a1,b1),B=(a1−(cid:15),b1+(cid:15)),∀(cid:15)∈(0,1] A=(0.4,0.3),B=(0.6,0.1) L(A)=0.22,L(B)=0.56 C1(A)=0.22,C1(B)=0.56
whereA=(µ,ν)withµ+ν≤1 H(A)=H(B)=a1+b1⇒A=B H(A)=H(B)=0.7⇒A=B L(A)<L(B)⇒A<B C1(A)<C1(B)⇒A<B
Liu[20]
SLi(A)=µ+µ(1−µ−ν), A=(0,b1),B=(0,b2),whereb1<b2 A=(0,0.1),B=(0,0.7) L(A)=−0.1,L(B)=−0.7 C1(A)=−0.1,C1(B)=−0.7
whereA=(µ,ν)withµ+ν≤1 SLi(A)=SLi(B)=0⇒A=B SLi(A)=SLi(B)=0⇒A=B L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
ZhouandWu[33]
SZh(A)=µ−ν+(α−β)(1−µ−ν), A=(a1,b1),B=(a1−(cid:15),b1−(cid:15)),∀(cid:15)∈(0,1]&α=β A=(0.6,0.1),B=(0.5,0) L(A)=0.56,L(B)=0.5 C1(A)=0.56,C1(B)=0.5
whereA=(µ,ν),α,β∈[0,1]withα+β≤1 SZh(A)=SZh(B)=a1−b1⇒A=B SZh(A)=SZh(B)=0.5⇒A=B L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
Linetal[18]
SLin(A)=2µ+ν−1, A=(a1,b1),B=(a1+2(cid:15),b1−(cid:15)),∀(cid:15)∈(0,1] A=(0.25,0.1),B=(0.3,0) L(A)=0.175,L(B)=0.3 C1(A)=0.175,C1(B)=0.3
whereA=(µ,ν)withµ+ν≤1 SLin(A)=SLin(B)=2a1+b1−1⇒A=B SLin(A)=SLin(B)=−0.4⇒A=B L(A)<L(B)⇒A<B C1(A)<C1(B)⇒A<B
Wangetal[26]
SW(A)=µ−ν−(1−µ2−ν), A=(a1,b1),B=(a1+3(cid:15),b1+(cid:15)),∀(cid:15)∈(0,1] A=(0.34,0.2),B=(0.44,0.5) L(A)=0.208,L(B)=0.16 C1(A)=0.208,C1(B)=0.16
whereA=(µ,ν)withµ+ν≤1 SW(A)=SW(B)=3a1−2b1−1⇒A=B SW(A)=SW(B)=−0.09⇒A=B L(A)>L(B)⇒A>B C1(A)>C1(B)⇒A>B
L.Linetal[19]
SL(A)=µ2+32(µ+ν)−1, A=(a1,b1),B=(a1+2(cid:15),b1−23(cid:15)),∀(cid:15)∈(0,1] A=(0.52,0.31),B=(0.62,0.377) L(A)=0.3712,L(B)=0.47674 C1(A)=0.3712,C1(B)=0.47674
whereA=(µ,ν)withµ+ν≤1 SL(A)=SL(B)=4a1+23b1−2⇒A=B SL(A)=SL(B)=0.505⇒A=B L(A)<L(B)⇒A<B C1(A)<C1(B)⇒A<B
Ye[31]
SY(A)=µ(2−µ−ν)+(1−µ−ν)2, - A=(0.3,0.6),B=(0.3,0.5) L(A)=−0.12,L(B)=−0.05 C1(A)=−0.12,C1(B)=−0.05
whereA=(µ,ν)withµ+ν≤1 SY(A)=SY(B)=0.32⇒A=B L(A)<L(B)⇒A<B C1(A)<C1(B)⇒A<B
ZhangandYu[34]
KK((AAii,,AA+−))==(cid:114)(cid:114)aaii22++1212((bbaciii22i++++22dcbciiii2)2)++ddii22 AB==(([[ab11,,ba11]],,[[ba11,,ba11])])w,itha1>b1 AB==(([[00..23,,00..23]],,[[00..23,,00..23]])) LL((BA))==00..0049 CC11((BA))==00..0049,
WhereA−=([0,0],[1,1]), K(A,A−)=K(B,A−)=√12, K(A,A−)=K(B,A−)=√12, L(A)<L(B)⇒A<B C1(A)<C1(B)⇒A<B
A+=([1,1],[0,0]) K(A,A+)=K(B,A+)=√12⇒A=B K(A,A+)=K(B,A+)=√12⇒A=B
B (x ,x ) = {a∈A|x > x } and C (x ,x ) Example5.5.1. Inthisexampleweconsideraselection
A i j i a j A i j
usingC (x ,x )={a∈A|x = x }. problemofthebestsupplierforanautomobilecompany
A i j i a j
3.Calculate the weighted fuzzy dominance relation from the available alternatives {xi|i=1to10} of
using WR (x,y) : U × U → [0,1] defined by pre evaluated 10 suppliers, based on WTrIFIS with
A
WR (x ,x )=(cid:80) w +(cid:80)a∈CA(xi,xj)wa. attributes{aj|j =1to5}asproductquality,relation-
A i j a∈BA(xi,xj) a 2 shipcloseness,deliveryperformance,socialresponsib-
4.Calculatetheentiredominancedegreeofeachobject
ilityandlegalissue.
usingWR (x )= 1 (cid:80)|U| WR (x ,x ).
A i |U| j=1 A i j AnTrIFISwithU ={x1,x2,...,x10},AT ={a1,a2,...
5. The objects are ranked using entire dominance
,a } is given in Table 2, and weights for the each
5
degree.ThelargerthevalueofWRA(xi),thebetteris attribute Wa is given by W = {wa|a∈AT} =
theobject. {0.3,0.2,0.15,0.17,0.18}.
Inthetable2,f(x ,a )=(a ,a ,a ,a ),(c ,c ,c
i i 1 2 3 4 1 2 3
5.5. NumericalIllustration
,c )denotesthetrapezoidalintuitionisticfuzzynumbers
4
which include intuitionistic fuzzy values and interval
Inthissubsection,Algorithm5.4isillustratedbyan valued intuitionstic fuzzy numbers and they stand for
example5.5.1. the evaluation of alternative x under the criteria a
i i