Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(2)

We consider the dynamic scheduling of a two-part-type make-tostock production system using the model of Wein [12]. Exogenous demand for each part type is met from finished goods inventory; unmet demand is backordered. The control policy determines which pa

withshortcycletimes,amake-to-stockenvironmentmayberequiredtorapidlyandreliably llorders.Assumingthatbuildingstockinadvanceisfeasible,operationalandeconomicparametersdeterminewhetheritshouldbebuiltinadvanceorwhetheramake-to-orderpolicyispreferable.

Forasimplemodelwithoneproductionstage,asingleparttype,andbackorderingcoststhetradeo isknown:Zeroinventoryisoptimaliftheprobabilityofbackorderingademandislessthanacertaincostratio.Inmulti-part-typesystems,thedecisionismademorecomplexbythevarietyofsequencingcontrolsthatarepossible.Wegivenecessaryandsu cientconditionsontheparametersofatwopart-typesystemforazero-inventorypolicytobeoptimal.ThemodelweconsideristhatofWein[12].Asingleproductionfacilityproducesstockinordertosatisfyarrivingdemands.Fortractability,theproductionsystemismodeledbyanexponentialserverandthedemandsarriveaccordingtoindependentPoissonprocesses.However,partialresultsforgeneralproductiontimesarepresentedinthelastsection.Thismodelisthemake-to-stockversionofamulti-classqueue.Weconsideranymake-to-stockpolicy;i.e.,productioncapacitycanbedynam-icallyallocatedbetweenparttypes.Theobjectiveistominimizeaverageholdingandbackordercosts.

Thestructureoftheoptimalpolicyisonlypartiallyknownforthisprob-lem.Wein[12]providesinsightsintothestructureoftheoptimalpolicyusingaBrownianapproximationforthismulti-classmake-to-stockqueuingcontrolproblem.Ha[3]showsthattheoptimalpolicyisahedgingpointpolicyinthecaseofin nitehorizondiscountedcost.Ahedgingpointpolicyischaracterizedbyswitchingandidlingcurveswhichsplitthestatespaceinthreeregions:onewherethemachineisidleandthetwoothersthatdeter-minetheparttypetoproduce.Theintersectionofthesetwocurvesiscalledthehedgingpoint.Thehedgingpointandaswitchingcurvearesu cienttodescribethestationarybehaviorofthepolicy.deV´ericourtetal.[10]showthattheoptimalswitchingcurveisastraightlineinaparticularsubsetofthestatespace.Numericalstudiesindicatethattheremainderofthecurvehasnosimplestructure.HeuristicshavebeenproposedbyWein[12].VeatchandWein[9]andPe na-PerezandZipkin[4]proposee ectiveheuristicstoapproximatethiscurvealongwiththeoptimalhedgingpoint.

However,ifzeroinventory(produceonlywhendemandsarewaitinginthesystem)isoptimal,Ha[3]impliesthattheproductioncapacityisallocatedamongtheparttypesaccordingtoa“cµ”rule.Thiscompletelyspeci esthecontrolofthesystem.Inthispaper,wederivenecessaryandsu cientcon-

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