Zero-Inventory Conditions For a Two-Part-Type Make-to-Stock(20)

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

Assembling(30),(31)and(32),we nallyobtain

γ2=1 ρ[λ λ1σBP1(λ2)].λ2

(33)Foraqueuewhereclass2hasthehighpriority,theprobabilityγ2ofhaving

noclass1customerinthesystemcanthendirectlybeobtainedfrom(33):

γ2= 1 ρ[λ λ2σBP2(λ1)]λ1(34)

whereσBP2istheLaplacetransformoftheclass2busyperiod,whenclass2hasthehighpriority.

BCaseoftheM/M/1

Iftheservicetimeisexponentiallydistributed,wehavefromGrossandHarris

[2],2µ1 σBP1(s)=.(35)2λ1+µ1+s+(λ1+µ1+s) 4λ1µ1

From(33),(34)and(35),weobtainthefollowingexpressionsforγ2andγ2

γ2=

γ2= 1 ρ 2λ1µ1 λ λ2λ+µ1+(λ+µ1)2 4λ1µ11 ρ 2λ2µ2 . λ λ1λ+µ2+(λ+µ2)2 4λ2µ2 (36)(37)

References

[1]T.Bielecki,andP.RKumar,OptimalityofZero-inventoryPoliciesfor

UnreliableManufacturingSystems,OperationsResearch36(1988)532-546.

[2]D.GrossandC.M.Harris,FundamentalsofQueueingTheory,John

WileyandSons(1985).

[3]A.Ha,OptimalDynamicSchedulingPolicyforaMake-to-StockPro-

ductionSystem,OperationsResearch45(1997)42-53.

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