A Multi Echelon Inventory Model For A Reparable Item With 1 For 1 Replenishment Distribution Hongmin Li

“A Multi-Echelon Inventory Model for a Reparable Item with one-for-one Replenishment”Steve Graves, 1985Management Science, 31(10)Presented by Hongmin LiThis summary presentation is based on: Graves, Stephen. “A Multi-Echelon Inventory Model for a Repairable Item with One-for-One Replenishment.” Management Science 31 (10), 1985. Problem• Failed item is replaced at the site from the site’s inventory if available; otherwise the shortage lasts until a replacement arrives Repair Depots0from the depot• Failed item is sent to depot for repair. It enters the repair process upon arrival at the depot and goes into the depot Site 1Site 2Site 3inventory upon completionsss123• Depot ships a replacement if available; otherwise, the depot backorders the request and fill it when availableAssumptions• Failure process at each site is a compound Poisson process that depends upon the required no. of working items, not the actual no., thus indep. of the status of the site• One-for-one replenishment• Shipment time from depot to site is deterministic (T )1• Ample repair capacity at the depotNotations“replacement • Qrequests that have i(t)Outstanding orders at site i at time tyet to be filled”• Q(t)Aggregate outstanding orders at the sites at time t• B(t|s0)Backorders at time t at the depot given s0• Di(t1, t2)Failures at site i over the time interval (t1, t2]• D(t1, t2)Aggregate failures at all sites over the time interval (t1, t2]Primary Result• Q(t+T )= B(t|s ) + D(t, t+T )101• No depot backorders at t can arrive at the sites by t+T and no failure occurring after t 1 can be replenished before t+T1• B(t|s ) and D(t, t+T ) are indep. r.v.s since 01depot back orders at t depend only on failures that occur prior to tIn order to find Qi(t)… • Convolve the distribution of B(t|s ) and D(t, 0t+T ) to obtain the distribution of Q(t)1• Disaggregate Q(t) into Q (t)… iB(t|s ) = [Q (t)-s ]+ 000Q (t) the total no. of failed items in the 0system at tAssuming Ample Repair Capacity• Q (t) is the occupancy level in a M|G|∞0queue where the service time includes the in-transit time to the depot T and the 1repair time • Palm’s theoremSteady state distribution of Q0(t) is PoissonDisaggregation of Q(t)• Assuming depot backorders are filled FCFS:The likelihood that any outstanding order is from site i is directly proportional to site i’sfailure rate λi, thus conditional distribution of Qi(t) is binomial.(See equation 3 in the 1985 Graves paper)Comparison to the METRIC approximation• METRIC approximates Q (t) as the ioccupancy level of an independent M|G|∞queue. Thus Q is Poisson. i• METRIC approximation for the case of a deterministic transit time to the sites is equivalent to approximating the depot backorder level by a Poisson r.v.Exact Models• Determine the distribution of Q0– Ample capacityuse Palm’s theorem (assuming Poisson or compound Poisson failure processes): – General shipment time, Poisson failure process, k parallel linesQ0 = In-transit + In-repair + In-repair-queueOccupancy level in an M|M|kan M|G|∞ queuesystemDocument Outline

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