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LargescalenestingofirregularpatternsusingcompactneighborhoodalgorithmS.K.Cheng,K.P.RaoThetypicalnestingtechniquethatiswidelyusedisthegeometricaltiltingofasinglepatternorselectedclusterstepbystepfromtheoriginalpositiontoanorientationof1808,i.e.orthogonalpacking.However,thisisablindsearchofbeststocklayoutand,geometrically,itbecomesinef®cientwhenseveralpatternentitiesareinvolved.Also,itisnothighlysuitableforhandlingpatternswitharangeoforientationconstraints.Inthispaper,analgorithmisproposedwhichcombinesthecompactneighborhoodalgorithmCNAwiththegeneticalgorithmGAtooptimizelargescalenestingprocesseswiththeconsiderationofmultipleorientationconstraints.2000ElsevierScienceS.A.Allrightsreserved.KeywordsCuttingstockproblemNestingCompactneighborhoodalgorithmGeneticalgorithmOrientationconstraints1.IntroductionThecuttingstockproblemisofinteresttomanyindustrieslikegarment,paper,shipbuilding,andsheetmetalindustries.GilmoreandGomory7haveinitiatedtheresearchworktosolvetherectangularcuttingstockproblembyusinglinearprogramming.Fortheirregularcase,Adamowicz1attemptedtouseaheuristicapproachwhichdividestheproblemintotwosubproblems,calledclusteringandnesting.Clusteringistospecifyacollectionofpatternsthat®twelltogetherbeforenestingontoagivenstock.Nestingofpatternsorclusterscanbebroadlydividedintotwobroadcategories,namely,smallscaleandlargescale.Thedifferencebetweenthemisthelevelofduplicationoftheclusteronthegivenstock.Forsmallscalenesting,weonlyneedto®ndtheinterorientationrelationshipbetweentheselectedclusterandthegivenstock4.However,theproblembecomesmorecomplicatedforlargescalenestingsincetheinterspacerelationshipbetweentheduplicatedclustersshouldalsobeconsidered.Traditionally,twobasictechniquesarepopularlyusedforgeneratingthistypeofnestinghexagonalapproximationandorthogonalnesting.Atypicalpattern,showninFig.1a,withbothconcaveandconvexfeatures,isselectedtoexplainthesetechniques.TheCorrespondingauthor.Tel.85227888409fax85227888423.Emailaddressmekpraocityu.edu.hkK.P.Raopatterncontourisplottedwiththehelpofadigitizer,asshowninFig.1b,andhasanareaApof74.44sq.units.InthehexagonalapproximationsuggestedbyDoriandBenBassat5,thepatternis®rstapproximatedusingaconvexpolygonwhichisfurtherapproximatedbyanotherconvexpolygonwithfewernumberofentitiesuntilanhexagonalenclosureisobtained,asshowninFig.1c.Thehexagonisthenpavedonagivenstockwithnooverlappingoftheformer6.TheresultantlayoutgeneratedbyuseofthistechniqueisgiveninFig.1e.Itisreadilyevidentthatthetechniqueisnothighlyef®cientduetothepoorapproximationperformance,especiallyinthecaseofhighlyirregularpatterns.Anotherproblemisthatthepatternorclustercanassumetwopositionsonly0or1808,withnoexploitationorconsiderationofotherpermissiblerangeoforientations.Inthesecondtechnique,usedbyNee9,thenestingprocessisachievedbyapproximatingasinglepattern/clusterbyarectangleasshowninFig.1d.Thisrectangleisthenduplicatedinanorthogonalway,resultinginthelayoutshowninFig.1f.Thistechniquecanbeeasilyappliedwhentherearenoorpartialorientationconstraints,i.e.thesinglepatternorclustercanrotatewithinacertainrangewhile®ttingitonthestock.Likethehexagonalapproximation,themaindisadvantageofthisapproachisthatthealgorithmsperformanceishighlydependentontheshapeofpatterns.Moreover,inthecaseofmultipleorientationconstraints,the09240136/00/±seefrontmatter2000ElsevierScienceS.A.Allrightsreserved.PIIS0924013600004027136S.K.Cheng,K.P.Rao/JournalofMaterialsProcessingTechnology1032000135±140Fig.1.aThechosen¯atpatternfordemonstratingtheworkingprincipleofCNAalgorithmbpatterncontourobtainedbydigitizerchexagonalapproximationdorthogonalapproximationelayoutgeneratedbyusinghexagonalapproximationyieldingastockutilizationof60.05flayoutgeneratedbyusingorthogonalapproximationyieldingastockutilizationof67.14andglayoutgeneratedbyusingCNAyieldingastockutilizationof74.10.timetakentoestimateasuitablerotationangleforthepatternsisalwaysmuchlonger.Inordertoincreasetheaccuracyandspeedofnesting,ChengandRao4proposedacompactneighborhoodalgorithmCNAthatconsiderstherelationshipbetweenthenumberofneighborsandthesharingspacebetweenthem.Fig.1gshowsatypicallayoutgeneratedusingCNAwhichnormallyyieldshigherpackingdensitywhencomparedwiththeorthogonalandhexagonalapproximations.However,CNA,initspresentform,hasbeenmainlydesignatedfornestingofpatternswiththeconsiderationoffullorientationconstraints,andisnotidealforsituationswheremorefreedomisavailableintheorientationofpatterns.Thisstudyisaimedatimprovingthe¯exibilityofCNAbyincorporatingtheavailablefreedomintheorientationofpatternsandageneticalgorithmGAthatfollowsnaturalrulestooptimizethegeneratedlayouts.ThenewtechniqueistranslatedintoacomputerprogramwritteninCobjectorientedlanguage.Thenewalgorithmcanhandletheproblemofnestingtwodimensional¯atpatternsofanyshapecontaininglinesegmentsandarcs.Withthehelpofatypicalexample,theenhancedcapabilitiesofCNAandtheassociatedcomputerprogramwillbedemonstratedinthispaper.2.DescriptionofcompactneighborhoodalgorithmCNAACNA4tracksthecharacteristicsoftheevolvingneighborhoodswhenthepatternsaremovedtoformdifferentarrangements,assummarizedschematicallyinFig.2a±c.Asthesheardisplacementincreases,theupperandlowerneighborstendtocollapseduetothechangeincrystallizationdirections.Finally,amostcompactstructureandanumericalvalueformaterialyield,calleduniversalcompactutilizationUCU,canbeobtained.NomatterFig.2.TypicalneighborhoodstructuresforcircularpatternsÐaformationoforthogonalpackingunitcellwithNn8andApu16r2bshearingoflayersleadingtoshearedorthogonalpackingandcbestcompactstructurewithhexagonalpackingunitcellwithNn6andAu63r2,whereAuistheareaofaunitcell,rtheradiusofcircularpatternandNnisthenumberofneighborstoconstructtheunitcell.S.K.Cheng,K.P.Rao/JournalofMaterialsProcessingTechnology1032000135±140137Fig.3.aStepsinvolvedinthegenerationofselfslidingpathtocreateaneighborhoodandboptimalneighborhoodstructurewithhexagonalpackingunitcellwithaUCUof83.07.whetherthepatterncanberotatedornot,UCUindicatestheupperlimitofyieldthatmaybepossiblewithanychosenstockandhencecanberegardedasanindexforstoppingcriteriainthenestingprocess.Themainstepsinvolvedin®ndingthecompactneighborhoodare1generatingaselfslidingpathorano®tpolygonNFP1,asshowninFig.3a,whichguidestherelativemovementbetweentwopatternswiththeconsiderationofnooverlappingand2de®ningthecrystallizationdirections,asshowninFig.3b,thatprovideessentialdataforbuildingthewholeneighborhoodby®llingthegivenstockduringlargescalenesting.3.ProposedalgorithmforlargescalenestingTheproposedtechniquesofenhancingthecapabilitiesofCNAbytakingadvantageofageneticalgorithmaredealtinthissection.A¯atpatterncanbedividedintoentitiesoflinesegmentsandarcs.Polygonalrepresentationmethods2expandthisstructureto®lltheentirestock.Fornestingofpatternswithfullorientationconstraint,itisonlynecessarytodecideanestingvectorCDnthatde®neswheretheneighborhoodshouldbetranslatedaroundthegivenstock.However,inthecaseofnestingofpatternswithlimitedornoorientationlimitations,theproblembecomesmorecomplicatedduetoanincreaseinthepossiblecombinationsthatweneedtoconsider.Inthiscase,the®rststepÐwhichisglobalwithorwithoutorientationlimitationsÐistotranslatetheneighborhoodtoanarbitrarypositioninsidethegivenstock,i.e.de®ningavectorCDn.Afterward,anestingangleynistobedeterminedsothatagoodorientationisselectedfortheneighborhoodtogrow.AlltherequiredgeometricaloperationsaresummarizedinFig.4.ItiscriticaltooptimizeCDnandynwhichcan®nallyleadtoamostcompactneighborhoodstructure.Itisbelievedthattherearenouniquemathematicalstepstocalculatetheseparametersforanytypeofstock.Inaddition,wecannotacceptanexhaustivesearchbecauseoftheconstraintsposedoncomputationtime,especiallyinthecaseofnestingofpatternswithtoolongacomputationtime,especiallywhilenestingpatternswithmanyentitiesandconcavefeatures.Hence,inthisstudy,arecentpopularoptimizationtechnique,calledGA,isapplied.Themainprincipleisprovidedinthefollowingsection.3.2.GAforoptimizinglayoutsGA8maintainsapopulationofcandidateproblemsolutions.Basedontheirperformance,the®ttestofthesesolutionsnotonlysurvive,and,analogoustosexualreproduction,exchangeinformationwithothercandidatestoformanewgeneration.Beforestartinganygeneticoperation,oneneedstode®nethe®tnessfunctionandthecodingmethod.Asmentionedearlier,thegoalinnestingofpatternsistoreducethescrapby®ttingtheclusterstogethersothattheyoccupyaminimumarea.Torepresentthecompactnessofaparticularlayout,onecanbecon®dentthatthemostdirectwayistorelateitwiththestockyieldfxYypy1canbeusedtorepresentbothconcaveandconvexarcsassetsofstraightlines.Theactualnumberoflinesisdependentontherequiredaccuracylevel.Also,clearanceoroffsetgenerationisanessentialstepthatcontributestowardsthesuccessofCAD/CAMtechnology.Analgorithmtogeneratetherequiredoffset,calledthreepointislandtracingTPITtechnique2,isincorporatedinthepresentnestingsystem.3.1.CNAforlargescalenestingIntheprevioussection,wehavealreadymentionedthebasicstepsinvolvedinobtainingthebestcompactneighborhood,asshowninFig.3b.Ournextconcernisthedeterminationofthebestpositiontoplacethe®rstpatternandxwherexistheareaofthegivenstockandythetotalareaofthepatternsthatcouldbecutoutfromthegivenstock.Codingcandirectlyandindirectlyin¯uencetheoptimizationprocess.Thisisbecauseourmainconcernishowto®xthetranslationpositioni.e.nestingvectorCDnandthedegreeofrotationi.e.nestingangleyn.Theyarethusselectedasthecodingparametersthatguidethepropertiesi.e.correspondingtonaturalchromosomesforexchangeinthegeneticoperatorsofcrossoverandmutation.3.3.ThegeneticoperatorsAsproposedbyHollandetal.8,theGAaimsatoptimizingthesolutionbymimickingnaturesevolutionary138S.K.Cheng,K.P.Rao/JournalofMaterialsProcessingTechnology1032000135±140Fig.4.Translationoftheneighborhoodtoaprede®nedpositionwithnestingvectorCDnandsubsequentrotationinvolvinganestingangleyn.process.LikehumanbeingsatypicalGAcontainsthefollowinggeneticoperators.3.3.1.InitializationAtthebeginning,apopulationofigeneticsolutionsi.e.layoutsaregeneratedbyrandomlyselectingvaluesforalltheparametersi.e.nestingangleynandnestingvectorCDn.Inrealindustrialapplications,thereareusuallysituationsthatlimittherotationofpatternsfreely.Forexample,inthedesignofprogressivedies,therearemanyrestrictedorientationregionsasaresultofthecostofsettingcorrespondingpilots,limitationsofbendingangleforsubsequentsheetmetaloperations,andthelike.Fig.5showstwotypicalpatternswithdifferentrestrictedandunrestrictedorientationregions.Asaresult,thepositionsthatthesepatternscouldassumearelimitedtotwolimitedrangesinthepresentcases,asshowninthesame®gure.Hence,aftertherandomgenerationofanestingangleyn,thefollowinginequalitycheckhastobesatis®edypjYklynypjYkuwithjYkPIY1jNand1kCpj2whereNandCpjarethenumberofpatternstobenestedi.e.p1,p2,FFFFFF,pNandthenumberofrestrictedorientationregionsofpatternpj,respectively.Fig.5.Rotationconstraintswithrespecttothelowerboundofthe®rstunrestrictedregionÐastationarypatternp1yp1,1l08,yp1,1u308,yp1,2l1708,yp1,2u1908andbmovablepatternp2yp2,1l08,yp2,1u608,yp2,2l2008,yp2,2u2308.
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