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

ThisleadstoadirectextensionofCondition4tothesymmetricalnon-exponentialcase.Hence,using(33)and(34)oftheAppendixtocomputeγ2 andγ2,Conditions2-4forthezero-inventorypolicytobeoptimal,become:

h1

h1+b1h21[ρλ λ1+(1 ρ)λ1σBP1(λ2)]≤λ2h2+b2ρ1b1 b21 ρ(h1 )[λ λ2σBP2(λ1)]≥b2λ11 ρ1ρ1≤(24)(25)(26)

withσBPi,theLaplacetransformofthehighprioritybusyperiodinaM/G/1queue,whereclassihasthehighpriority.FollowingtheproofofProperty2,onecanalsoshowthat(26)implies(24).

Asmentionedearlier,theoptimalpolicyisnotMarkovian,andsince(25)and(26)considerhedgingpointpolicyperturbations,theseconditionsarenotsu cient.Itmaybepossibletoprovethesu ciencyifwerestrictthestudytotheclassofhedgingpointpolicies.

7Numericalresultsandinsights

Inthissectionweexplorethezero-inventoryconditionsnumericallyanddrawsomeinsights.Ourresultssuggestadditionalconditionsunderwhichmake-to-order(MTO)isattractive.Forasinglepart-type,itiswellknownthatwhenutilizationisloworholdingcostsarelargerelativetobackordercosts,noinventoryshouldbeheld(Condition2).Condition3issimilar,butusescostsforthelowpriorityparttypeandreplacesthecombinedutilization(whichthelowprioritypartsees)withthesmaller1 γ2.Condition4capturestheinteractionbetweenparttypes.ToexaminewhenCondition4applies,we rstnotethatitholdsasymptoticallyash1/b2→∞.Evenforh1 ofcomparablesizetoh2,itmayholdif1 γ2issmall(suggestingthattheutilizationislow),andeitherthediscrepancybetweenparttypepriorities(b1µ1 b2µ2)issmallorthehighpriorityparttypehaslowutilization(ρ1

1).

Furtherinsightisgainedbyconsideringthecasewhereparttypesdi eronlyincost.Settingµ1=µ2,Condition4canbewritten

γ2≥ b2

h1+b1

15b1 b21 ρ1(27)