Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(4)

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

i.WealsodenotebyX(t)=(X1(t),X2(t))theassociatedrandomvariable.LetCπbethecontrolassociatedwithaMarkovpolicyπ.

Cπ(x)=

0whentheactionistoidle1whentheactionistoproducetype1

2whentheactionistoproducetype2

Weconsideraunitholdingcosthiandanon-zerounitbackordercostbiperunitoftimefortypei.Intherestofthepaperwealsoassumewithoutlossofgeneralitythatµ1b1>µ2b2.Inthestatex,thesystemincursacost + +rateofc(x)=2i=1ci(xi)withci(xi)=hixi+bixi(wherexi=max(xi,0)andx+i= min(xi,0)).Theobjectiveisthento ndthepolicywhichminimizesthelongrunaveragecost

1π tlimsupEx0[c(X(t))dt].0t→∞t(1)

πwhereExdenotestheexpectationgiventhecontrolpolicyπandinitialstate0x0.

Theoptimalaveragecostrateg andtherelativevaluefunctionv(x)satisfythefollowingdynamicprogrammingoptimalityequations(seeVeatchandWein[9]):

1g =c(x)+λ1v(x e1)+λ2v(x e2)+µv(x)v(x)+ΛΛ

+min(0,µ1 1v(x),µ2 2v(x)),[](2)

wheree1istheunitvectoralongx1,e2istheunitvectoralongx2, iv(x)=v(x+ei) v(x),µ=max(µ1,µ2)andΛ=λ1+λ2+µ.

Wedenotebyλ=λ1+λ2,thetotalarrivalrate.ρi=λi/µiistheutilizationratefortypei,andρ=ρ1+ρ2isthetotalutilizationrate.Intherestofthepaper,weassumethatρislessthanone.

Anoptimalpolicysatisfying(2)existsifastablepolicyincurringa nitecostexists(seeforinstanceWeberandStidham[11]).Notethenthataprioritypolicywhichstatestoproduceifandonlyifademandiswaitingisequivalenttoamulti-classpriorityqueue.Sinceρ<1thissystemisstableandanoptimalpolicyexists.

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