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

4Zero-inventoryConditions

Whenx2≥0,nogeneralequationfortheswitchingcurveisknown.Nev-ertheless,iftheoptimalhedgingpointisat0,theoptimalswitchingcurveisfullycharacterizedbythestraightlinex1=0.Thesystemistheninamake-to-ordermode.Wedenotebypolicyζ,thiszero-inventorypolicywhosehedgingpointisat0.Theassociatedcontrolisde nedby

0ifx=0

Cζ(x)= 1ifx1<0 2ifx1=0,x2<0 (4)

Wepresentfourconditionsthatmustholdforζtobeoptimal.

The rstconditioncomesfromthepartialcharacterizationoftheoptimal

mswitchingcurve.Ifpolicyζisoptimalthenz1mustbeequaltozero,and

weobtainfrom(3),

Condition1

h1µ1+b2µ2>(h1+b1)λ1

Toderivetheotherconditions,weevaluatethee ectofthreepolicyperturbationsdepictedinFigure2.Ifpolicyζisoptimal,theseperturbationsincreasetheaveragecost.

4.1RightShift

Weconsiderthepolicyπde nedbyarightshiftofpolicyζ.Thehedgingpointofpolicyπis(1,0)andtheswitchingcurveisthelinex1=1.Itisclear ]=Eζ[b2X2];theonlydi erenceinaveragecostisduetoX1.thatEπ[b2X2Thecontrolofthe rstparttypecorrespondstoahedgingpointpolicyinasinglepart-typesystem,wherethehedgingpointisoneforpolicyπandzeroforpolicyζ,soPπ(X1=x1)=Pζ(X1=x1+1)forx1≤0.Theincrementalaveragecostis

gπ gζ=h1Pπ(X1=1) b1Pπ(X1≤0)

=h1(1 ρ1) b1ρ1.(5)

Ifpolicyζisoptimal,then(5)mustbenonnegative,whichisthesecondcondition:

7