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

ofX1obeysthesamebalanceequationsunderπandζin{X1≤0}andtheprobabilitiesarethesameuptonormalization:

Pπ(X1=x1|X1≤0)=Pζ(X1=x1),x1≤0.(12)

Using(8)and(12),andtheexponentialassumptions,

µ2c(X1)] Eζ[ c(X1)]=(h1+b2)Pπ(X1=1)Eπ[ µ1µ2 ](Pπ(X1<1) 1)+(b1 b2)Eζ[X1µ1 2bρ1b1 µµ12Pπ(X1=1).(13)=h1 1 ρ1

Letγ2=Pπ(X1=1).Nextweshowthatγ2isequaltotheprobabilityofhavingnoclass1customerinapriorityqueuewherethehighpriorityisgiventoclass2.ConsiderXπ,atrajectorygeneratedbypolicyπ.Observethate1 Xπisthetrajectoryofaqueuewithclass2havinghighpriorityexceptthatclass1isserved rstinstateswherex1≤0andx2≤ 1.Aqueuethatgivesprioritytoclass2canbeconstructedbychangingtheorderofservicesothatclass2isserved rstinthesestates.Thisresequencingdoesnotchangethetimesatwhichthee1 Xπtrajectoryreaches0;hence,itdoesnotchangethetimesatwhichtheXπtrajectoryreachesandleavesstateswithx1=1.Onceagain,ifpolicyζisoptimalthen (10)ing(11),(13)and(37)intheAppendixforγ2,thelastconditionis

Condition4 2bρ1b1 µµ2 µ12h1 γ2 b2≥01 ρ1µ1

YeeandVeatch[13]suggestthatcondition3willbethesamefornparttypesbuttherewillbeadditionalcomplexconditionsintheplaceofCondition4.

5Su cientConditions

Intheprevioussection,wederivedfournecessaryzero-inventoryconditions.Here,weoutlineanargumentthatconditions3and4arealsosu cient.We rstshowthatconditions1and2areredundant:

10