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

Property2Condition4 Condition2 Condition1

Proof:Thestationaryprobabilityofnolowprioritycustomerinapriorityqueueislessthantheprobabilityofthesameeventwhenthehighpriority classhasbeenwithdrawn.Inparticular,γ2≤1 ρbiningthisboundwithCondition4

h1 1ρ1b1µ21≥b2 1 ρ1µ1γ21 ρ1 ≥0

andCondition2isveri ed.Furthermore,Condition2canbewrittenh1µ1≥(h1+b1)λ1,whichimpliesCondition1.2

Supposeconditions3and4aresatis ed.FromProperty2,Condition1isalsosatis edandtheoptimalswitchingcurveisgivenbyx1=0,intheregionx2<0.Furthermore,Condition3and4implythatanuporrightextensionoftheswitchingcurveincreasestheaveragecost.FollowingVeatchandCaramanis[8],weconsidertheclassofpoliciesπusingtheoptimalswitchingcurvebutvarioushedgingpointsalongthiscurve.Itcanbeshownthatgπdecreasesmonotonicallyasthehedgingpointmovestowardtheoptimalhedgingpoint.Hence,Condition3and4implythatthezero-inventorypolicyisoptimalinthisclassofpolicies,andinfactoptimalamongallpolicies.6GeneralSymmetricProductionTimes

Untilnow,wehaveconsideredanexponentialproductiontime.However,someofthepreviousconditionscanbeextendedwhenanarbitrarydistri-butionisassumed.Weconsiderherethesamesystemasde nedinSection2exceptthattheproductiontimedoesnotdependontheparttypeandweonlyspecifyitsmean1/µ.Wealsoassumethatthetypeofapartischosenattheendofitsproduction.Inthiscase,theoptimalcontrolisnotMarkoviananymore,andcandependonthetime.Werestrictourstudytotheclassofhedgingpointpolicies(whichmaynotbeoptimal).

Property1stillholdsforthissystem.Toseethis,notethat,fori∈{1,2},πiπiz1+z2 X1 X2correspondstothenumberofcustomersinanM/G/1queue.

Ifwecomputeγ2forapriorityM/G/1queue,conditions2and3stillhold.However,Equation(12)ofSection4.3isnottrueingeneral,andthederivationofCondition4cannotbeextendeddirectly.Inthefollowingweproposetoderivethislastconditionusingcoupledtrajectories.

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