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附录附录1AnImprovedRoughSetApproachtoDesignofGatingSchemeforInjectionMouldingF.Shi,1Z.L.Lou,1J.G.Lu2andY.Q.Zhang11DepartmentofPlasticityEngineering,ShanghaiJiaotongUniversity,P.R.Chinaand2CenterofCAD,NanjingUniversityofChemicalTechnology,P.R.ChinaThegateisoneofthemostimportantfunctionalstructuresinaninjectionmould,asithasadirectinfluenceonthequalityoftheinjectionproducts.Thedesignofagatingschemeincludestheselectionofthetypesofgateandcalculationofthesizesanddeterminationofthelocation,whichdependsheavilyonpriorexperienceandknowledgeandinvolvesatrialanderrorprocess.Duetothevaguenessanduncertaintyinthedesignofagatingscheme,classicalroughsettheoryisnoteffective.Inthispaper,afuzzyroughsetmodelisproposed,whichisnotbasedonequivalentrelationshipsbutonfuzzysimilarityrelationships.AninductivelearningalgorithmbasedonthefuzzyroughsetmodelFRILAisthenpresented.Comparedtodecisiontreealgorithms,theproposedalgorithmcangeneratefewerclassificationrulesmoreover,thegeneratedrulesaremoreconcise.Finally,anintelligentprototypesystemforthedesignofagatingschemebasedonaninducedfuzzyknowledgebaseisdeveloped.Anillustrativeexampleprovestheeffectivenessoftheproposedmethod.KeywordsFuzzyroughsetGatingschemeInjectionmoldIntelligentdesignKnowledgeacquisition1.IntroductionThemanufacturingindustryforplasticproductshasbeengrowingrapidlyinrecentyears,andplasticsareusedwidelytosubstituteformetals.Theinjectionmouldingprocessisthemostpopularmouldingprocessformakingthermoplasticparts.Thefeedingsystem,whichisoneoftheimportantfunctionalstructures,comprisesasprue,aprimaryrunner,asecondaryrunnerandagate.Themoltenplasticflowsfromthemachinenozzlethroughthesprueandrunnersystemandintothecavitiesthroughthegate.Actingastheconnectionbetweentherunnerandthecavity,thegatecaninfluencedirectlythemouldventing,theoccurrenceofjetting,thelocationofweldlines,andwarpage,shrinkageandresidualstresses.Hence,thegatedesignisimportantforassuringthequalityofthemould.Thedesignofagateincludestheselectionofthetypeofgate,calculationofthesizeanddeterminationofthelocation.Andthedesignofagateisbasedontheexperienceandknowledgeofthedesigners.Thedeterminationsofthelocationandsizesaremadebasedonatrialanderrorprocess.Inrecentyears,afeaturemodellingenvironmentandintelligenttechnologyhavebeenintroducedforgatedesign.LeeandKiminvestigatedgatelocationsusingtheevaluationcriteriaofwarpage,weldlinesandizodimpactstrength.Alocalsearchwasusedtodeterminethenodesofthelocationofthegate1.SaxenaandIraniproposedaframeforanonmanifoldtopologybasedenvironment.Aprototypesystemforgatelocationdesignwasdeveloped.Thecriteriaforevaluationwerebasedongeometryrelatedparameters2.Linselectedtheinjectionlocationandsizeofthegateasthemajorcontrolparameters,andchosetheproductperformancedeformationastheoptimisingparameter.Combiningthetechnologiesofabductivenetworksandsimulationannealingoptimisationalgorithms,theoptimalmodelforthelocationandsizeofthegatewasconstructed3,4.Zhouetal.establishedarulesetfordeterminingthelocationofthegatebasedonanalysisoftheplasticparts.Thelocationofthegatewasdeterminedthroughreasoningwithrules5.Pandelidisetal.developedasystemwhichcanoptimisegatelocationbasedontheinitialgatingplans.ThesystemusedMOLDFLOWsoftwareforflowanalysis,andcontrolledthetemperaturedifferentialandthenumberofelementsoverpackedwithanoptimisationstrategy6.DengusedID3anditsmodifiedalgorithmstogeneratetherulesetfortheselectionofthegatetypes7.However,therearemanyfuzzyorvagueattributesintheselectionofthetypes,suchastheattributeoflossofpressurethathastwofuzzylinguisticvariablesi.e.canbehighandmustbelow.TheID3algorithmscannotdealwithfuzzyornoiseinformationefficiently.Itisalsodifficulttocontrolthesizeofthedecisiontreeextractedbythealgorithmsandsometimesverylargetreesaregenerated,makingcomprehensibilitydifficult7,8.Roughsettheoryprovidesanewmathematicalapproachtovagueanduncertaindataanalysis9,10.Thispaperintroducesthetheoryofroughsetsforthedesignofagatingscheme.Theselectionofthetypeofgateisbasedonthetheoryofroughsets.Consideringthelimitationsofroughsets,thispaperproposesanimprovedapproachbasedonroughsettheoryforthedesignofthegatingscheme.Theimprovedroughsetapproachtotheschemedesignwillbegivenfirst.AfuzzyroughsetbasedinductivelearningalgorithmFRILA,whichisappliedintheimprovedapproach,willthenbepresented.Anexampleofthedesignofagatewillfinallybegiven.Table1.Classificationcriteria.ConditionattributesFuzzylinguisticvariablesStyleofplasticpartspDeep,Middle,ShallowShell,Deep,Middle,ShallowTube,Deep,Middle,ShallowRingNumberofcavitiesnSinglecavity,MulticavityLossofpressurelCanbehigh,MustbelowConditionofseparatinggatefrompartsqMustbeeasy,NotrequestspeciallyMachiningperformancemMustbeeasy,Notrequestspecially2.ARoughSetApproachtoGatingSchemeDesign2.1DesignoftheGatingSchemeThemodelofthegatingschemedesigncanbedescribedasfollows.Adecisiontablewith4tuplescanberepresentedasT(U,C,D,T).whereUistheuniverse.C{C1,C2,,Ck}isthesetofconditionattributes,eachofwhichmeasuressomeimportantfeatureofanobjectintheuniverseU.TCk{Tk1,T2k,...,TkSk}isthesetofdiscretelinguisticterms.Inotherwords,TCkisthevaluesetoftheconditionattributes.D{D1,D2,,Dl}isthesetofdecisionattributes,thatis,eachobjectintheuniverseisclassifiedbythesetD.Generally,theconditionattributescanbeclassifiedasfivesets,includingstyleofplasticparts,numberofcavities,lossofpressure,conditionofseparatinggatefrompartsandmachineperformance.ThedetailsofthefiveconditionattributesandcorrespondingvariablesofthefuzzylinguisticareshowninTable1.Fromthetable,itcanbeseenthatmostoftheattributesarevaguesincetheyrepresentahumanperceptionanddesire.Forinstance,shell,tubeandringareselectedfortheclassificationofplasticpartsandtheirfuzzylinguisticvaluesaredeep,middleandshallow,respectively.Fortheattributelossofpressure,canbehighandmustbelowareselectedtoapproximatethefuzzyattribute.AfuzzyruleforgatingschemedesigncanbewritteninthefollowingformIFC1isT1i1ANDCkisTikTHENDisDj1whereTkikisthelinguistictermofconditionattributeCk,andDjisaclasstermofthedecisionattributeD.FuzzyruleswiththeformofEq.1areusedtoperformminmaxfuzzyinference.LetckbethemembershipvalueofanobjectinTkanddbetheforecastvalueofDj,wheredikminckandministheminimumoperator.Iftwoormoreruleshavethesameconclusion,theconclusionwiththelargestvalueofd,whichisalsonamedthecertaintyfactorischosen.Fortheproblemofthegatingschemedesign,afuzzydesignrulecanbedescribedasfollows.IFTypeofplasticpartmiddleshellANDNumberofcavitiessingleANDConditionofseparatinggatefrompartnotrequestespecially2THENGatingschemestraightgateCF0.825Fromtheaboverule,thegatingschemeofthestraightgatewillbeselectedisswithacertaintyfactorof0.825,ifthetypeofpartismiddleshellandthenumberofcavitiesissingleandtheconditionofseparatinggatefrompartisnotrequired.Theaboveisjustlikehumanlanguageandiseasytounderstand.2.2BasicConceptsofRoughSetsInrecentyears,theroughsetRStheory,proposedbyPawlak,hasbeenattractingtheattentionoftheresearchers.ThebasicideaofRSistoclassifytheobjectsofinterestintosimilarityclassesequivalentclassescontainingindiscernibleobjectsviatheanalysisofattributedependencyandattributereduction.Theruleinductionfromtheoriginaldatamodelisdatadrivenwithoutanyadditionalassumptions.Roughsetshavebeenappliedinmedicaldiagnosis,patternrecognition,machinelearning,andexpertsystems10,11.Adecisiontablewitha4tuplecanberepresentedasT,whereUistheuniverse,,CandDarethesetsofconditionanddecisionattributes,respectively,VisthevaluesetoftheattributeainA,andfisaninformationfunction.Assumingasubsetofthesetofattributes,twoobjectsxandyinUareindiscerniblewithrespecttoPifandonlyif,.TheindiscernibilityrelationiswrittenasINDP.U/INDPisusedtodenotethepartitionofUgiventheindiscernibilityrelationINDP.Aroughsetapproximatestraditionalsetsbyapairofsets,whicharethelowerandtheupperapproximationsofthesets.ThelowerandupperapproximationsofasetY.UgivenanequivalencerelationINDParedefinedasfollowsThedefinitionofthelowerapproximationofasetinvolvesaninclusionrelationwherebytheobjectsinanequivalenceclassoftheattributesareentirelycontainedintheequivalenceclassforthedecisioncategory.Thisisthecaseofaperfectorunambiguousclassification.Fortheupperapproximation,theobjectsarepossiblyclassifiedusingtheinformationinattributesetP.Attributereductionisimportantforroughsettheory.Basedontheabovedefinitions,theconceptofreduction,denotedbyREDP,isdefinedasfollowsQ.PisareductionofPifandonlyifINDPINDQ.2.3AnImprovedRoughSetApproachInthedesignofthegatingscheme,itiscrucialtoacquirethefuzzyrulesefficiently.Knowledgeacquisitionisthebottleneck.Aroughsetisappliedtosolvetheproblemforthe
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