Neutrosophic Association Rule Mining Algorithm for Big Data Analysis

2018·9.2 MB·English

by  Mohamed Abdel-Basset, Mai Mohamed, Florentin Smarandache, Victor Chang;

SS
symmetry
Article
Neutrosophic Association Rule Mining Algorithm for
Big Data Analysis
MohamedAbdel-Basset1,* ID,MaiMohamed1,FlorentinSmarandache2,* ID andVictorChang3
1 DepartmentofOperationsResearch,FacultyofComputersandInformatics,ZagazigUniversity,
Sharqiyah44519,Egypt;[email protected]
2 Math&ScienceDepartment,UniversityofNewMexico,Gallup,NM87301,USA
3 InternationalBusinessSchoolSuzhou,Xi’anJiaotong-LiverpoolUniversity,Wuzhong,
Suzhou215123,China;[email protected]
* Correspondence:[email protected](M.A.-B.);[email protected](F.S.)
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:5March2018;Accepted:9April2018;Published:11April2018
Abstract: BigDataisalarge-sizedandcomplexdataset,whichcannotbemanagedusingtraditional
dataprocessingtools. Miningprocessofbigdataistheabilitytoextractvaluableinformationfrom
theselargedatasets. Associationruleminingisatypeofdataminingprocess,whichisindentedto
determineinterestingassociationsbetweenitemsandtoestablishasetofassociationruleswhose
supportisgreaterthanaspecificthreshold. Theclassicalassociationrulescanonlybeextractedfrom
binarydatawhereanitemexistsinatransaction, butitfailstodealeffectivelywithquantitative
attributes, through decreasing the quality of generated association rules due to sharp boundary
problems. Inordertoovercomethedrawbacksofclassicalassociationrulemining,weproposein
thisresearchanewneutrosophicassociationrulealgorithm. Thealgorithmusesanewapproach
forgeneratingassociationrulesbydealingwithmembership,indeterminacy,andnon-membership
functions of items, conducting to an efficient decision-making system by considering all vague
association rules. To prove the validity of the method, we compare the fuzzy mining and the
neutrosophicmining.Theresultsshowthattheproposedapproachincreasesthenumberofgenerated
associationrules.
Keywords: neutrosophicassociationrule;datamining;neutrosophicsets;bigdata
1. Introduction
Theterm‘BigData’originatedfromthemassiveamountofdataproducedeveryday. Eachday,
Googlereceivescca. 1billionqueries,Facebookregistersmorethan800millionupdates,andYouTube
countsupto4billionviews, andtheproduceddatagrowswith40%everyyear. Othersourcesof
dataaremobiledevicesandbigcompanies. Theproduceddatamaybestructured,semi-structured,
orunstructured. Mostofthebigdatatypesareunstructured;only20%ofdataconsistsinstructured
data. Therearefourdimensionsofbigdata:
(1) Volume: bigdataismeasuredbypetabytesandzettabytes.
(2) Velocity: theacceleratingspeedofdataflow.
(3) Variety: thevarioussourcesandtypesofdatarequiringanalysisandmanagement.
(4) Veracity: noise,abnormality,andbiasesofgeneratedknowledge.
Consequently,Gartner[1]outlinesthatbigdata’slargevolumerequirescost-effective,innovative
formsforprocessinginformation,toenhanceinsightsanddecision-makingprocesses.
Prominentdomainsamongapplicationsofbigdataare[2,3]:
Symmetry2018,10,106;doi:10.3390/sym10040106 www.mdpi.com/journal/symmetry
Symmetry2018,10,106 2of19
(1) Businessdomain.
(2) Technologydomain.
(3) Healthdomain.
(4) Smartcitiesdesigning.
These various applications help people to obtain better services, experiences, or be healthier,
bydetectingillnesssymptomsmuchearlierthanbefore[2]. Somesignificantchallengesofmanaging
andanalyzingbigdataare[4,5]:
(1) AnalyticsArchitecture: Theoptimalarchitecturefordealingwithhistoricandreal-timedataat
thesametimeisnotobviousyet.
(2) Statisticalsignificance: Fulfillstatisticalresults,whichshouldnotberandom.
(3) Distributedmining: Variousdataminingmethodsarenotfiddlingtoparalyze.
(4) Timeevolvingdata: Datashouldbeimprovedovertimeaccordingtothefieldofinterest.
(5) Compression: Todealwithbigdata,theamountofspacethatisneededtostoreishighlyrelevant.
(6) Visualization: Themainmissionofbigdataanalysisisthevisualizationofresults.
(7) Hiddenbigdata:Largeamountsofbeneficialdataarelostsincemoderndataisunstructureddata.
Duetotheincreasingvolumeofdataatamatchlessrateandofvariousforms,weneedtomanage
andanalyzeuncertaintyofvarioustypesofdata. Bigdataanalyticsisasignificantfunctionofbigdata,
whichdiscoversunobservedpatternsandrelationshipsamongvariousitemsandpeopleinterestona
specificitemfromthehugedataset. Variousmethodsareappliedtoobtainvalid,unknown,anduseful
models from large data. Association rule mining stands among big data analytics functionalities.
The concept of association rule (AR) mining already returns to H’ajek et al. [6]. Each association
rule in database is composed from two different sets of items, which are called antecedent and
consequent. A simple example of association rule mining is “if the client buys a fruit, he/she is
80% likely to purchase milk also”. The previous association rule can help in making a marketing
strategyofagrocerystore. Then, wecansaythatassociationrule-miningfindsallofthefrequent
itemsindatabasewiththeleastcomplexities. Fromalloftheavailablerules,inordertodeterminethe
rulesofinterest,asetofconstraintsmustbedetermined. Theseconstraintsaresupport,confidence,
lift, and conviction. Support indicates the number of occurrences of an item in all transactions,
while the confidence constraint indicates the truth of the existing rule in transactions. The factor
“lift”explainsthedependencyrelationshipbetweentheantecedentandconsequent. Ontheother
hand, the conviction of a rule indicates the frequency ratio of an occurring antecedent without a
consequentoccurrence. Associationrulesminingcouldbelimitedtotheproblemoffindinglarge
itemsets,wherealargeitemsetisacollectionofitemsexistinginadatabasetransactionsequaltoor
greaterthanthesupportthreshold[7–20]. In[8],theauthorprovidesasurveyoftheitemsetmethods
fordiscoveringassociationrules. Theassociationrulesarepositiveandnegativerules. Thepositive
association rules take the form X →Y, X ⊆ I, Y ⊆ I and X∩Y = ϕ, where X,Y are antecedent
andconsequentand I isasetofitemsindatabase. Eachpositiveassociationrulemayleadtothree
negativeassociationrules, →Y, X →Y, and X →Y. Generatingassociationrulesin[9]consists
oftwoproblems. Thefirstproblemistofindfrequentitemsetswhosesupportsatisfiesapredefined
minimum value. Then, the concern is to derive all of the rules exceeding a minimum confidence,
basedoneachfrequentitemset.Sincethesolutionofthesecondproblemisstraightforward,mostofthe
proposedworkgoesinforsolvingthefirstproblem. Anapriorialgorithmhasbeenproposedin[19],
which was the basis for many of the forthcoming algorithms. A two-pass algorithm is presented
in [11]. It consumes only two database scan passes, while a priori is a multi-pass algorithm and
needsuptoc+1databasescans,wherecisthenumberofitems(attributes). Associationrulesmining
isapplicableinnumerousdatabasecommunities. Ithaslargeapplicationsintheretailindustryto
improvemarketbasketanalysis[7]. Streaming-Rulesisanalgorithmdevelopedby[9]toreportan
associationbetweenpairsofelementsinstreamsforpredictivecachinganddetectingthepreviously
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undetectablehitinflationattacksinadvertisingnetworks. Runningminingalgorithmsonnumerical
attributesmayresultinalargesetofcandidates. Eachcandidatehassmallsupportandmanyrules
havebeengeneratedwithuselessinformation,e.g.,theageattribute,salaryattribute,andstudents’
grades. Manypartitioningalgorithmshavebeendevelopedtosolvethenumericalattributesproblem.
The proposed algorithms faced two problems. The first problem was the partitioning of attribute
domainintomeaningfulpartitions. Thesecondproblemwasthelossofmanyusefulrulesdueto
thesharpboundaryproblem. Consequently,somerulesmayfailtoachievetheminimumsupport
thresholdbecauseoftheseparatingofitsdomainintotwopartitions.
Fuzzy sets have been introduced to solve these two problems. Using fuzzy sets make the
resultedassociationrulesmoremeaningful. Manyminingalgorithmshavebeenintroducedtosolve
the quantitative attributes problem using fuzzy sets proposed algorithms in [13–27] that can be
separatedintotwotypesrelatedtothekindofminimumsupportthreshold,fuzzyminingbasedon
single-minimumsupportthreshold,andfuzzyminingbasedonmulti-minimumsupportthreshold[21].
Neutrosophictheorywasintroducedin[28]togeneralizefuzzytheory. In[29–32],theneutrosophic
theoryhasbeenproposedtosolveseveralapplicationsandithasbeenusedtogenerateasolution
based on neutrosophic sets. Single-valued neutrosophic set was introduced in [33] to transfer the
neutrosophictheoryfromthephilosophicfieldintothemathematicaltheory,andtobecomeapplicable
inengineeringapplications. In[33],adifferentiationhasbeenproposedbetweenintuitionisticfuzzy
setsandneutrosophicsetsbasedontheindependenceofmembershipfunctions(truth-membership
function,falsity-membershipfunction,andindeterminacy-membershipfunction). Inneutrosophic
sets,indeterminacyisexplicitlyindependent,andtruth-membershipfunctionandfalsity-membership
functionareindependentaswell.Inthispaper,weintroduceanapproachthatisbasedonneutrosophic
sets for mining association rules, instead of fuzzy sets. Also, a comparison resulted association
rulesinbothofthescenarioshasbeenpresented. In[34],anattempttoexpresshowneutrosophic
sets theory could be used in data mining has been proposed. They define SVNSF (single-valued
neutrosophicscorefunction)toaggregateattributevalues. In[35],analgorithmhasbeenintroduced
tominingvagueassociationrules. Itemspropertieshavebeenaddedtoenhancethequalityofmining
associationrules. Inaddition, almostsolditems(itemshasbeenselectedbythecustomer, butnot
checkedout)wereaddedtoenhancethegeneratedassociationrules. AH-pairDatabaseconsisting
of a traditional database and the hesitation information of items was generated. The hesitation
informationwascollected,dependingonlineshoppingstores,whichmakeiteasiertocollectthattype
ofinformation, whichdoesnotexistintraditionalstores. Inthispaper, wearethefirsttoconvert
numericalattributes(items)intoneutrosophicsets. Whilevagueassociationrulesaddnewitemsfrom
thehesitatinginformation,ourframeworkaddsnewitemsbyconvertingthenumericalattributesinto
linguisticterms. Therefore,thevagueassociationruleminingcanberunontheconverteddatabase,
whichcontainsnewlinguisticterms.
ResearchContribution
Detectinghiddenandaffinitypatternsfromvarious,complex,andbigdatarepresentsasignificant
roleinvariousdomainareas,suchasmarketing,business,medicalanalysis,etc. Thesepatternsare
beneficial for strategic decision-making. Association rules mining plays an important role as well
in detecting the relationships between patterns for determining frequent itemsets, since classical
associationrulescannotusealltypesofdatafortheminingprocess. Binarydatacanonlybeusedto
formclassicalrules,whereitemseitherexistindatabaseornot. However,whenclassicalassociation
rules deal with quantitative database, no discovered rules will appear, and this is the reason for
innovatingquantitativeassociationrules. Thequantitativemethodalsoleadstothesharpboundary
problem, where the item is below or above the estimation values. The fuzzy association rules are
introducedtoovercometheclassicalassociationrulesdrawbacks. Theiteminfuzzyassociationrules
hasamembershipfunctionandafuzzyset. Thefuzzyassociationrulescandealwithvaguerules,
butnotinthebestmanner,sinceitcannotconsidertheindeterminacyofrules. Inordertoovercome
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drawbacks of previous association rules, a new neutrosophic association rule algorithm has been
introduced in this research. Our proposed algorithm deals effectively and efficiently with vague
rulesbyconsideringnotonlythemembershipfunctionofitems,butalsotheindeterminacyandthe
falsityfunctions. Therefore,theproposedalgorithmdiscoversallofthepossibleassociationrulesand
minimizesthelosingprocessesofrules,whichleadstobuildingefficientandreliabledecision-making
system. By comparing our proposed algorithm with fuzzy approaches, we note that the number
ofassociationrulesisincreased,andnegativerulesarealsodiscovered. Theseparationofnegative
association rules from positive ones is not a simple process, and it helps in various fields. As an
example, inthemedicaldomain, bothpositiveandnegativeassociationruleshelpnotonlyinthe
diagnosisofdisease,butalsoindetectingpreventionmanners.
Therestofthisresearchisorganizedasfollows. Thebasicconceptsanddefinitionsofassociation
rules mining are presented in Section 2. A quick overview of fuzzy association rules is described
inSection3. TheneutrosophicassociationrulesandtheproposedmodelarepresentedinSection4.
A case study of Telecom Egypt Company is presented in Section 5. The experimental results and
comparisonsbetweenfuzzyandproposedassociationrulesarediscussedinSection6. Theconclusions
aredrawninSection7.
2. AssociationRulesMining
Inthissection,weformulatethe|D|transactionsfromtheminingassociationrulesforadatabase
D.Weusedthefollowingnotations:
(i) I = {i ,i ,...i }representsallthepossibledatasets,calleditems.
1 2 m
(ii) TransactionsetTisthesetofdomaindataresultingfromtransactionalprocessingsuchasT⊆I.
(iii) ForagivenitemsetX⊆IandagiventransactionT,wesaythatTcontainsXifandonlyifX⊆T.
(iv) σ : the support frequency of X, which is defined as the number of transactions out of D that
X
containX.
(v) s: thesupportthreshold.
Xisconsideredalargeitemset,ifσ ≥ |D|×s. Further,anassociationruleisanimplicationof
X
theform X ⇒Y,whereX ⊆ I, Y ⊆ I andX∩Y = ϕ.
Anassociationrule X ⇒Y isaddressedinDwithconfidencecifatleastctransactionsoutofD
containbothXandY.Therule X ⇒Y isconsideredasalargeitemsethavingaminimumsupportsif:
σX∪Y ≥ |D|×s.
For a specific confidence and specific support thresholds, the problem of mining association
rulesistofindoutalloftheassociationruleshavingconfidenceandsupportthatislargerthanthe
correspondingthresholds. Thisproblemcansimplybeexpressedasfindingallofthelargeitemsets,
wherealargeitemsetLis:
L = {X|X ⊆ I∧σ ≥ |D|×s}.
X
3. FuzzyAssociationRules
Miningofassociationrulesisconsideredasthemaintaskindatamining. Anassociationrule
expressesaninterestingrelationshipbetweendifferentattributes.Fuzzyassociationrulescandealwith
bothquantitativeandcategoricaldataandaredescribedinlinguisticterms,whichareunderstandable
terms[26].
LetT = {t ,...,t }beadatabasetransactions. Eachtransactionconsistsofanumberofattributes
1 n
(items). Let I = {i ,...,i } be a set of categorical or quantitative attributes. For each attribute i ,
1 m k
(k =1,...,m),weconsider{n ,...,n }associatedfuzzysets. Typically,adomainexpertdetermines
1 k
themembershipfunctionforeachattribute.
Thetuple< X,A >iscalledthefuzzyitemset,whereX⊆I(setofattributes)and Aisasetof
fuzzysetsthatisassociatedwithattributesfromX.
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Followingisanexampleoffuzzyassociationrule:
IFsalaryishighandageisoldTHENinsuranceishigh
Beforetheminingprocessstarts,weneedtodealwithnumericalattributesandpreparethemfor
theminingprocess. Themainideaistodeterminethelinguistictermsforthenumericalattributeand
definetherangeforeverylinguisticterm. Forexample,thetemperatureattributeisdeterminedbythe
linguisticterms{verycold,cold,cool,warm,hot}. Figure1illustratesthemembershipfunctionofthe
temperatureattribute.
Figure1.Linguistictermsofthetemperatureattribute.
Themembershipfunctionhasbeencalculatedforthefollowingdatabasetransactionsillustrated
inTable1.
Table1.MembershipfunctionforDatabaseTransactions.
Transaction Temp. MembershipDegree
T1 18 1cool
T2 13 0.6cool,0.4cold
T3 12 0.4cool,0.6cold
T4 33 0.6warm,0.4hot
T5 21 0.2warm,0.8cool
T6 25 1warm
Weaddthelinguisticterms{verycold,cold,cool,warm,hot}tothecandidatesetandcalculate
thesupportforthoseitemsets. Afterdeterminingthelinguistictermsforeachnumericalattribute,
thefuzzycandidatesethavebeengenerated.
Table 2 contains the support for each itemset individual one-itemsets. The count for every
linguistictermhasbeencalculatedbysummingitsmembershipdegreeoverthetransactions. Table3
showsthesupportfortwo-itemsets. Thecountforthefuzzysetsisthesummationofdegreesthat
resultedfromthemembershipfunctionofthatitemset. Thecountfortwo-itemsethasbeencalculated
bysummingtheminimummembershipdegreeofthe2items. Forexample,{cold,cool}hascount0.8,
whichresultedfromtransactionsT2andT3. FortransactionT2,membershipdegreeofcoolis0.6and
membershipdegreeforcoldis0.4,sothecountforset{cold,cool}inT2is0.4. Also,T3hasthesame
countfor{cold,cool}. So,thecountofset{cold,cool}overalltransactionsis0.8.
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Table2.1-itemsetsupport.
1-itemset Count Support
Verycold 0 0
Cold 1 0.17
Cool 2.8 0.47
Warm 1.6 0.27
Hot 0.6 0.1
Table3.2-itemsetsupport.
2-itemset Count Support
{Cold,cool} 0.8 0.13
{Warm,hot} 0.4 0.07
{warm,cool} 0.2 0.03
Insubsequentdiscussions,wedenoteanitemsetthatcontainskitemsask-itemset. Thesetofall
k-itemsetsinLisreferredasL .
k
4. NeutrosophicAssociationRules
Inthissection,weoverviewsomebasicconceptsoftheNSsandSVNSsovertheuniversalsetX,
andtheproposedmodelofdiscoveringneutrosophicassociationrules.
4.1. NeutrosophicSetDefinitionsandOperations
Definition 1 ([33]). Let X be a space of points and x∈X. A neutrosophic set (NS) A in X is definite by a
truth-membershipfunction T (x),anindeterminacy-membershipfunction I (x)andafalsity-membership
A A
function F (x). T (x), I (x)and F (x)arerealstandardorrealnonstandardsubsetsof]−0,1+[. Thatis
A A A A
T (x): X→]−0,1+[, I (x): X→]−0,1+[andF (x): X→]−0,1+[. Thereisnorestrictiononthesumof
A A A
T (x), I (x)andF (x),so0− ≤supT (x)+sup I (x)+supF (x)≤3+.
A A A A A A
Neutrosophic is built on a philosophical concept, which makes it difficult to process during
engineeringapplicationsortouseittorealapplications. Toovercomethat,Wangetal.[31],definedthe
SVNS,whichisaparticularcaseofNS.
Definition2. LetXbeauniverseofdiscourse. Asinglevaluedneutrosophicset(SVNS) AoverXisanobject
takingtheform A={(cid:104)x,T (x), I (x),F (x)(cid:105): x∈X},whereT (x): X→[0,1], I (x): X→[0,1]andF (x):
A A A A A A
X→[0,1]with0≤T (x)+ I (x)+F (x)≤3forallx∈X.TheintervalsT (x),I (x)andF (x)represent
A A A A A A
thetruth-membershipdegree,theindeterminacy-membershipdegreeandthefalsitymembershipdegreeofxtoA,
respectively.Forconvenience,aSVNnumberisrepresentedbyA=(a,b,c),wherea,b,c∈[0,1]anda+b+c≤3.
Definition3(Intersection)([31]).FortwoSVNSsA=(cid:104)T (x),I (x),F (x)(cid:105)andB=(cid:104)T (x),I (x),F (x)(cid:105),
A A A B B B
theintersectionoftheseSVNSsisagainanSVNSswhichisdefinedasC= A∩Bwhosetruth,indeterminacyand
falsity membership functions are defined as T (x) = min(T (x),T (x)), I (x) = min(I (x),I (x)) and
C A B A A B
F (x) = max(F (x),F (x)).
C A B
Definition4(Union)([31]). FortwoSVNSs A = (cid:104)T (x),I (x),F (x)(cid:105)and B = (cid:104)T (x),I (x),F (x)(cid:105),
A A A B B B
theunionoftheseSVNSsisagainanSVNSswhichisdefinedasC = A∪Bwhosetruth,indeterminacyand
falsitymembershipfunctionsaredefinedasT (x) = max(T (x),T (x)), I (x) = max(I (x),I (x))and
C A B A A B
F (x) = min(F (x),F (x)).
C A B
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Definition5(Containment)([31]). Asinglevaluedneutrosophicset AcontainedintheotherSVNSB,
denotedby A ⊆ BifandonlyifT (x) ≤ T (x), I (x) ≤ I (x)andF (x) ≥ F (x)forallxinX.
A B A B A B
Next,weproposeamethodforgeneratingtheassociationruleundertheSVNSenvironment.
4.2. ProposedModelforAssociationRule
Inthispaper,weintroduceamodeltogenerateassociationrulesofform:
X →Y whereX∩Y = ϕandX,Yareneutrosophicsets.
Our aim is to find the frequent itemsets and their corresponding support. Generating an
associationrulefromitsfrequentitemsets,whicharedependentontheconfidencethreshold,arealso
discussed here. This has been done by adding the neutrosophic set into I, where I is all of the
possibledatasets,whicharereferredasitems. So I = N∪Mwhere Nisneutrosophicsetand Mis
classicalsetofitems. Thegeneralformofanassociationruleisanimplicationoftheform X →Y,
whereX ⊆ I, Y ⊆ I,X∩Y = ϕ.
Therefore,anassociationrule X →Y isaddressedinDatabaseDwithconfidence‘c’ifatleastc
transactionsoutofDcontainsbothXandY. Ontheotherhand,therule X →Y isconsideredalarge
itemsethavingaminimumsupportsifσX∪Y ≥ |D|×s. Furthermore,theprocessofconvertingthe
quantitativevaluesintotheneutrosophicsetsisproposed,asshowninFigure2.
Figure2.Theproposedmodel.
The proposed model for the construction of the neutrosophic numbers is summarized in the
followingsteps:
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Step1 Setlinguistictermsofthevariable,whichwillbeusedforquantitativeattribute.
Step2 Definethetruth,indeterminacy,andthefalsitymembershipfunctionsforeachconstructed
linguisticterm.
Step3 ForeachtransactiontinT,computethetruth-membership,indeterminacy-membershipand
falsity-membershipdegrees.
Step4 Extend each linguistic term l in set of linguistic terms L into T , I , and F to denote truth-
L L L
membership,indeterminacy-membership,andfalsity-membershipfunctions,respectively.
Step5 Foreachk-itemsetwherek={1,2,...,n},andnnumberofiterations.
• calculate count of each linguistic term by summing degrees of membership for each
i=t
transactionasCount(A) = ∑ µ (x)whereµ isT ,I orF .
A A A A A
i=1
• calculatesupportforeachlinguisticterms = Count(A) .
No.of trnsactions
Step6 Theaboveprocedurehasbeenrepeatedforeveryquantitativeattributeinthedatabase.
In order to show the working procedure of the approach, we consider the temperature as an
attribute and the terms “very cold”, “cold”, “cool”, “warm”, and “hot” as their linguistic terms
to represent the temperature of an object. Then, following the steps of the proposed approach,
constructtheirmembershipfunctionasbelow:
Step1 Theattributetemperature’hassetthelinguisticterms“verycold”,“cold”,“cool”,“warm”,
and“hot”,andtheirrangesaredefinedinTable4.
Table4.Linguistictermsranges.
LinguisticTerm CoreRange LeftBoundaryRange RightBoundaryRange
VeryCold −∞–0 N/A 0–5
Cold 5–10 0–5 10–15
Cool 15–20 10–15 20–25
Warm 25–30 20–25 30–35
Hot 35–∞ 30–35 N/A
Step2 Based on these linguistic term ranges, the truth-membership functions of each linguistic
variablearedefined,asfollows:

 1 ; for x ≤0
Tvery−cold(x) = (5−x)/5 ; for0< x <5
 0 ; for x ≥5

 1(15−x)/5 ;; ffoorr510≤<xx≤<1015
T (x) =
cold  0x/5 ;; ffoorr0x<≥x15<or5x ≤0

 1(25−x)/5 ;; ffoorr2105<≤ xx <≤2250
T (x) =
cool  0(x−10)/5 ;; footrhe1r0w<isex <15

 1(35−x)/5 ;; ffoorr2350≤< xx ≤<3305
T (x) =
warm  0(x−20)/5 ;; footrhe2r0w<isex <25
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
 1 ; for x ≥35
T (x) = (x−30)/5 ; for30< x <35
hot
 0 ; otherwise
Thefalsity-membershipfunctionsofeachlinguisticvariablearedefinedasfollows:

 0 ; for x ≤0
Fvery−cold(x) = x/5 ; for0< x <5 ;
 1 ; for x ≥5

 0(x−10)/5 ;; ffoorr510≤<xx≤<1105
F (x) =
cold  1(5−x)/5 ;; ffoorr0x<≥x15<or5x ≤0

 0(x−20)/5 ;; ffoorr2105<≤ xx <≤2250
F (x) =
cool  1(15−x)/5 ;; footrhe1r0w<isex <15

 0(x−30)/5 ;; ffoorr2350≤< xx ≤<3305
F (x) =
warm  1(25−x)/5 ;; footrhe2r0w<isex <25

 0 ; for x ≥35
F (x) = (35−x)/5 ; for30< x <35
hot
 1 ; otherwise
Theindeterminacymembershipfunctionsofeachlinguisticvariablesaredefinedasfollows:

 0(x+2.5)/5 ;; ffoorr−x ≤2.5−≤2.5x ≤2.5
I (x) =
very−cold  0(7.5−x)/5 ;; ffoorr2x.5≥≤7.x5≤7.5

(x+2.5)/5 ; for2.5≤ x ≤2.5
 (7.5−x)/5 ; for2.5≤ x ≤7.5
I (x) = (x−7.5)/5 ; for7.5≤ x ≤12.5
cold
 0(17.5−x)/5 ;;oftohre1rw2.i5se≤ x ≤17.5

(x−7.5)/5 ; for7.5≤ x ≤12.5
 (17.5−x)/5 ; for12.5≤ x ≤17.5
I (x) = (x−17.5)/5 ; for17.5≤ x ≤22.5
cool
 0(27.5−x)/5 ;;oftohre2rw2.i5se≤ x ≤27.5

(x−17.5)/5 ; for17.5≤ x ≤22.5
 (27.5−x)/5 ; for22.5≤ x ≤27.5
Iwarm(x) = (x−27.5)/5 ; for27.5≤ x ≤32.5
 0(37.5−x)/5 ;;oftohre3rw2.i5se≤ x ≤37.5
Symmetry2018,10,106 10of19

 (x−27.5)/5 ; for27.5≤ x ≤32.5
I (x) = (37.5−x)/5 ; for32.5≤ x ≤37.5
hot
 0 ;otherwise
ThegraphicalmembershipdegreesofthesevariablesaresummarizedinFigure3. Thegraphical
falsity degrees of these variables are summarized in Figure 4. Also, the graphical indeterminacy
degreesofthesevariablesaresummarizedinFigure5. Ontheotherhand,foraparticularlinguistic
term,‘Cool’inthetemperatureattribute,theirneutrosophicmembershipfunctionsarerepresentedin
Figure6.
Figure3.Truth-membershipfunctionoftemperatureattribute.
Figure4.Falsity-membershipfunctionoftemperatureattribute.
Figure5.Indeterminacy-membershipfunctionoftemperatureattribute.