Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(14)

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

First,itcanbeshownthatp1istheprobabilityofhavingnoclass1customerinapriorityqueuewherethehighpriorityisgiventoclass2(seeSection4.3).Itfollowsthat p1=γ2,(17)whereγ2isgivenby(37)oftheAppendix.

Furthermore,Segments1and3arethesegmentsoftrajectoryXζwhereζζ=0.Hencewehavep1+p3=P(X1=0)=1 ρ1andweobtainX1

p3=1 ρ1 γ2.

Wethenevaluatep2andp4usingthefollowingtwoequations

p1+p2+p3+p4=1p4p3=.p1p2(19)(20) (18)

The rstequationisstraightforward.Toshowthesecondone,wefollowVeatchandCaramanis[8].LetNi(t)bethenumberoftype1demandarrivalsin(0,t]occuringwhiletrajectoriesareinSegment1,2,3and4respectively.LetTi(t)bethetimein(0,t]ingaweaklawoflargenumbers,onecanshowthattheproportionN3(t)/N1(t)approachesT3(t)/T1(t)whent→∞withprobability1.Furthermore,eacharrivaloftype1demandmakesthetrajectoriesmovefromSegment1intoSegment2,orfromSegment3intoSegment4.ItfollowsthatT4(t)/T2(t)andN3(t)/N1(t)approachthesamelimitwhent→∞.SinceT3(t)/T1(t)andT4(t)/T2(t)alsoapproachp3/p1andp4/p2respectively,weobtainEquation(20).Itfollowsfrom(17),(18),(19)and(20)that

p2=

p4ρ1 γ21 ρ1ρ1 =ρ1 γ21 ρ1(21)(22)

Combining(17),(21),(18)and(22)with(16)wecanderivethedi erenceofaveragecosts

ρ1γ2 γ2 b2(1 ).gπ gζ=h1γ2 b11 ρ11 ρ1 (23)

14