Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(9)

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

lowprioritycustomerinanM/M/1priorityqueue(withourparameters).Expression(36)forγ2isobtainedintheAppendix.Again,ifpolicyζisoptimal,then(6)isnonnegative:

Condition3

1 γ2≤h2

h2+b2

4.3RightExtension

Finally,weassumethatpolicyπisde nedbyarightextensionofpolicyζ.Thehedgingpointofpolicyπisstatee1andtheswitchingcurveisthestraightlinex1=0extendedbythesegment[0,e1].Theassociatedcontrolisthengivenby

0ifx=e1Cπ(x)=1ifx1<0 2if0≤x1≤1

Let

c (x1)=(h1+ orandx=0x2<0(7)µ2µ2b2)x++(b b2)x 111µ1µ1(8)

andrecallthatw=x1/µ1+x2/µ2.Then,forx2≤0,

c(x)=h1x+1+b1x1 b2x2µ2µ2µ2 =(h1+b2)x++(b b)x b2x1 b2x21211µ1µ1µ1

=c (x1) µ2b2w(9)

(10)

(11)andgπ gζ=Eπ[ c(X1)] Eζ[ c(X1)] µ2b2(Eπ[W] Eζ[W]).Eπ[W] Eζ[W]=1.µ1FromProperty1,

Now,weonlyneedthemarginaldistributionofX1underpoliciesπandζ.Observethat,underπ,theX1transitionintensitiesareindependentofX2exceptthosebetweenX1=0andX1=1.Thus,themarginaldistribution

9