Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(13)

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

coupledtrajectoriesremaininthissegmentwhentype1demandsarriveorproductiontimesarecompleted.However,whenatype2demandarrivestothesystem,thecoupledtrajectoriesmoveintoSegment2. Segment2:

ζπInthissegment,X1<0,andXπ Xζ=e1.IfX1<0,thenboth

policiesproducetype1andthetrajectoriesstayinSegment2.IfππX1=0andX2=0bothpoliciesstillproducetype1,butwhenaproductiontimeiscomplete,thetrajectoriesmoveintoSegment1.IfζπππX1=0andX2<0,thenC1=1andC1=2,andwhenaproductiontimeiscompleted,thetrajectoriesmoveintoSegment3.

ζ Segment3:Inthissegment,X1=0,Xπ Xζ=e2(sothatwealso

πππhaveX1=0).Bothpoliciesproducetype2ifX2<0.IfX2=0,ζπ=1,andwhenaproductiontimeiscompleted,theC1=2andC1trajectoriesmoveintoSegment1,attheirhedgingpoints.Whena

type1demandarrivestothesystem,thecoupledtrajectoriesmoveintoSegment4.Otherwise,theystayinSegment3.

Segment4:

ζInthissegment,X1<0,Xπ Xζ=e2.Bothpoliciesstatetopro-

ducetype1,sothatbothtrajectoriesstayinSegment4untilaservicecompletionmakesthementerSegment3.

Fromthepreviousde nitionofthefoursegments,wehave

Xπ Xζ=

e1inSegment1and2e2inSegment3and4(14)(15)h1inSegment1

b1inSegment2c(Xπ) c(Xζ)= b2inSegment3and4

Ifwedenotebyp1,p2,p3andp4thestationaryprobabilitiesofbeinginSegment1,2,3and4respectively,weobtainfrom(15)

gπ gζ=h1p1 b1p2 b2(p3+p4).

Itremainsthentoderiveexpressionsforp1,p2,p3andp4.

13(16)