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CORRESPONDINGAUTHORTEL3031498143FAX3031498180EMAILADDRESSGEORGIADCPERICERTHGRMCGEORGIADISCOMPUTERSRECEIVEDINREVISEDFORM1SEPTEMBER2000ACCEPTED1OCTOBER2000ABSTRACTTHISPAPERPRESENTSANEWMATHEMATICALPROGRAMMINGFORMULATIONFORTHEPROBLEMOFDETERMININGTHEOPTIMALMANNERINWHICHSEVERALPRODUCTROLLSOFGIVENSIZESARETOBECUTOUTOFRAWROLLSOFONEORMORESTANDARDTYPESTHEOBJECTIVEISTOPERFORMTHISTASKSOASTOMAXIMIZETHEPRO“TTAKINGACCOUNTOFTHEREVENUEFROMTHESALES,THECOSTSOFTHEORIGINALROLLS,THECOSTSOFCHANGINGTHECUTTINGPATTERNANDTHECOSTSOFDISPOSALOFTHETRIMAMIXEDINTEGERLINEARPROGRAMMINGMILPMODELISPROPOSEDWHICHISSOLVEDTOGLOBALOPTIMALITYUSINGSTANDARDTECHNIQUESANUMBEROFEXAMPLEPROBLEMS,INCLUDINGANINDUSTRIALCASESTUDY,AREPRESENTEDTOILLUSTRATETHEECIENCYANDAPPLICABILITYOFTHEPROPOSEDMODELSCOPEANDPURPOSEONEDIMENSIONALCUTTINGSTOCKTRIMLOSSPROBLEMSARISEWHENPRODUCTIONITEMSMUSTBEPHYSICALLYDIVIDEDINTOPIECESWITHADIVERSITYOFSIZESINONEDIMENSIONEGWHENSLITTINGMASTERROLLSOFPAPERINTONARROWERWIDTHROLLSSUCHPROBLEMSOCCURWHENTHEREARENOECONOMIESOFSCALEASSOCIATEDWITHTHEPRODUCTIONOFTHELARGERRAWMASTERROLLSINGENERAL,THEOBJECTIVESINSOLVINGSUCHPROBLEMSARETO5P69MINIMIZETRIMLOSSP69AVOIDPRODUCTIONOVERRUNSAND/ORP69AVOIDUNNECESSARYSLITTERSETUPSTHEABOVEPROBLEMISPARTICULARLYIMPORTANTINTHEPAPERCONVERTINGINDUSTRYWHENASETOFPAPERROLLSNEEDTOBECUTFROMRAWPAPERROLLSSINCETHEWIDTHOFAPRODUCTISFULLYINDEPENDENTOFTHEWIDTHOFTHERAWPAPERAHIGHLYCOMBINATORIALPROBLEMARISESINGENERAL,THECUTTINGPROCESSALWAYSPRODUCESINEVITABLETRIMLOSSWHICHHASTOBEBURNEDORPROCESSEDINSOMEWASTETREATMENTPLANTTRIMLOSSPROBLEMSINTHEPAPERINDUSTRYHAVE,INRECENTYEARS,MAINLYBEENSOLVEDUSINGHEURISTICRULESTHEPRACTICALPROBLEMFORMULATIONHAS,THEREFORE,INMOSTCASESBEENRESTRICTEDBYTHEFACTTHATTHESOLUTIONMETHODSOUGHTTOBEABLETOHANDLETHEENTIREPROBLEMCONSEQUENTLY,ONLYASUBOPTIMALSOLUTIONTOTHEORIGINALPROBLEMHASBEENOBTAINEDAND03050548/02/SEEFRONTMATTERP72002ELSEVIERSCIENCELTDALLRIGHTSRESERVEDPIIS0305054800001027VERYOFTENTHISRATHERSIGNI“CANTECONOMICPROBLEMHASBEENLEFTTOAMANUALSTAGETHISWORKPRESENTSANOVELALGORITHMFORECIENTLYDETERMININGOPTIMALCUTTINGPATTERNSINTHEPAPERCONVERTINGPROCESSAMIXEDINTEGERLINEARPROGRAMMINGMODELISPROPOSEDWHICHISSOLVEDTOGLOBALOPTIMALITYUSINGAVAILABLECOMPUTERTOOLSANUMBEROFEXAMPLEPROBLEMSINCLUDINGANINDUSTRIALCASESTUDYAREPRESENTEDTOILLUSTRATETHEAPPLICABILITYOFTHEPROPOSEDALGORITHMP72002ELSEVIERSCIENCELTDALLRIGHTSRESERVEDKEYWORDSINTEGERPROGRAMMINGOPTIMIZATIONTRIMLOSSPROBLEMSPAPERCONVERTINGINDUSTRY1INTRODUCTIONANIMPORTANTPROBLEMWHICHISFREQUENTLYENCOUNTEREDININDUSTRIESSUCHASPAPERISRELATEDWITHTHEMOSTECONOMICMANNERINWHICHSEVERALPRODUCTROLLOFGIVENSIZESARETOBEPRODUCEDBYCUTTINGONEORMOREWIDERRAWROLLSAVAILABLEINONEORMORESTANDARDWIDTHSTHESOLUTIONOFTHISPROBLEMINVOLVESSEVERALINTERACTINGDECISIONSP69THENUMBEROFPRODUCTROLLSOFEACHSIZETOBEPRODUCEDTHISMAYBEALLOWEDTOVARYBETWEENGIVENLOWERANDUPPERBOUNDSTHEFORMERNORMALLYREECTTHE“RMORDERSTHATARECURRENTLYOUTSTANDING,WHILETHELATTERCORRESPONDTOTHEMAXIMUMCAPACITYOFTHEMARKETHOWEVER,CERTAINDISCOUNTSMAYHAVETOBEOEREDTOSELLSHEETSOVERANDABOVETHEQUANTITIESFORWHICH“RMORDERSAREAVAILABLEP69THENUMBEROFRAWROLLSOFEACHSTANDARDWIDTHTOBECUTROLLSMAYBEAVAILABLEINONEORMORESTANDARDWIDTHS,EACHOFADIERENTUNITPRICEP69THECUTTINGPATTERNFOREACHRAWROLLCUTTINGTAKESPLACEONAMACHINEEMPLOYINGANUMBEROFKNIVESOPERATINGINPARALLELONAROLLOFSTANDARDWIDTHWHILETHEPOSITIONOFTHEKNIVESMAYBECHANGEDFROMONEROLLTOTHENEXT,SUCHCHANGESMAYINCURCERTAINCOSTSFURTHERMORE,THEREMAYBECERTAINTECHNOLOGICALLIMITATIONSONTHEKNIFEPOSITIONSTHATMAYBEREALIZEDBYANYGIVENCUTTINGMACHINETHEOPTIMALSOLUTIONOFTHEABOVEPROBLEMISOFTENASSOCIATEDWITHTHEMINIMIZATIONOFTHETRIMAWASTETHATISGENERALLYUNAVOIDABLESINCEROLLSOFSTANDARDWIDTHSAREUSEDHOWEVER,TRIMLOSSMINIMIZATIONDOESNOTNECESSARILYIMPLYMINIMIZATIONOFTHECOSTOFTHERAWMATERIALSROLLSBEINGUSEDESPECIALLYIFSEVERALSTANDARDROLLSIZESAREAVAILABLEAMOREDIRECTECONOMICCRITERIONISTHEMAXIMIZATIONOFTHEPRO“TOFTHEOPERATIONTAKINGACCOUNTOFP69THEREVENUEFROMPRODUCTROLLSSALES,INCLUDINGTHEEECTSOFANYBULKDISCOUNTSP69THECOSTOFTHEROLLSTHATAREACTUALLYUSEDP69THECOSTS,IFANY,OFCHANGINGTHEKNIFEPOSITIONSONTHECUTTINGMACHINEP69THECOSTOFDISPOSINGOFTRIMWASTETHEABOVECONSTITUTESAHIGHLYCOMBINATORIALPROBLEMANDITISNOTSURPRISINGTHATTRADITIONALLYITSSOLUTIONHASOFTENBEENCARRIEDOUTMANUALLYBASEDONHUMANEXPERTISETHESIMPLI“EDVERSIONOFTHISPROBLEMISSIMILARTOTHECUTTINGSTOCKPROBLEMKNOWNINTHEOPERATIONRESEARCHLITERATURE,WHEREANUMBEROFORDEREDPIECESNEEDTOBECUTOBIGGERSTOREDPIECESINTHEMOSTECONOMICFASHIONINTHE1960SANDTHE1970S,SEVERALSCIENTI“CARTICLESWEREPUBLISHEDONTHEPROBLEMOF1042GSCHILLING,MCGEORGIADIS/COMPUTERSTHATRAWROLLSOFTHETYPETTHATPERMITSTHESMALLESTMINIMUM1044GSCHILLING,MCGEORGIADIS/COMPUTERSANDTHATEACHRAWROLLWILLBEUSEDTOPRODUCEPRODUCTROLLSOFASINGLETYPEONLYOVERALL,THISLEADSTOTHEFOLLOWINGUPPERBOUNDONTHENUMBEROFRAWROLLSTHATMAYBEREQUIREDJP13P0P24“P39P9P71P14P16NP13P0P24P71P87MINP82BP13P9P14P82/BP71P881WECANALSOCALCULATEALOWERBOUNDJP13P9P14ONTHEMINIMUMNUMBEROFRAWROLLSTHATARENECESSARYTOSATISFYTHEMINIMUMDEMANDFORTHEEXISTINGORDERSWEDOTHISBYASSUMINGTHATROLLSOFTHETYPETALLOWINGTHEMAXIMUMPOSSIBLEENGAGEMENTBP13P0P24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