Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(6)

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

whichthepreferredparttypeforproductionispartiregardlessoftheidling¯i={x:minj=1,2µj jv(x)=µi iv(x)}.Thenahedgingpointdecision:B¯1={x:x1<sB(x2)}foranincreasingcurvesB(x2)policyissuchthatBwhere ∞≤sB(x2)≤∞andI={x:x1<sI(x2)}forsomedecreasingcurve,sI(x2),where0≤sI(x2)≤∞.Thehedgingpointzistheunique¯2includesthestatesonthestatesuchthatsB(z2)=sI(z2)=z1.Thus,BswitchingcurveandregionIincludestheidlingcurveandthehedgingpoint.Becausestateswithx1>z1orx1>z2aretransient,thestationarybehaviorofahedgingpointpolicyisentirelycharacterizedbyitshedgingpointandtheportionofitsswitchingcurvesB(x2)withx2≤z2.

WewillusethefollowingresulttocomparetheexpectationofW=X1/µ1+X2/µ2,theaggregatedworkloadofthesystem,fordi erentpolicies.Property1followsfromtheunderlyingnatureofhedgingpointpolicies;seeLemma2ofdeV´ericourtetal.[10].

Property1Considertwohedgingpointpoliciesπ1andπ2andtheirrespec-tivehedgingpointszπ1,zπ2.Then,

π1π1π2π2z1z2z1z2E[W] E[W]=+ µ1µ2µ1µ2π1π2

whereEπiisthethesteady-stateexpectationgiventhehedgingpointpolicyπi,i∈{1,2}.

Wewillalsousethefactthat,givenµ1b1>µ2b2,theoptimalswitchingcurveintheregionx2<0isaverticallinewhosepositionisexpressedbyasimpleequation(seeFigure1).ApplyingTheorem1ofdeV´ericourtandal.

[10],theoptimalhedgingpointpolicysatis es

C(x)=π 1

2mifx1<z1,x2<0

m,x2<0,ifx1≥z1

m=z1 µh1+b2µ2 1 ln() h1+b1 lnρ1(3)

and x denotesthegreatestintegerlessthanorequaltox.

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