“A Multi-Echelon Inventory Model for a Reparable Item with one-for-one Replenishment”ProblemAssumptionsNotationsPrimary ResultIn order to find Qi(t)…Assuming Ample Repair CapacityDisaggregation of Q(t)Comparison to the METRIC approximationExact ModelsExact ModelsApproximate ModelApproximate ModelApproximate ModelTest ProblemSummaryQuestions““A MultiA Multi--Echelon Inventory Model Echelon Inventory Model for a Reparable Item with onefor a Reparable Item with one--forfor--one Replenishment”one Replenishment”Steve Graves, 1985Steve Graves, 1985Management Science, 31(10)Management Science, 31(10)Presented by Hongmin LiPresented 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.ProblemProblemSite 3Site 1 Site 2Repair Depots0s1s2s3• Failed item is replaced at the site from the site’s inventory if available; otherwise the shortage lasts until a replacement arrives from 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 inventory upon completion• Depot ships a replacement if available; otherwise, the depot backorders the request and fill it when availableAssumptionsAssumptions• 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 (T1)• Ample repair capacity at the depotNotationsNotations• Qi(t)Outstanding orders at site i at time t• 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]“replacement requests that have yet to be filled”Primary ResultPrimary Result• Q(t+T1)= B(t|s0) + D(t, t+T1)• No depot backorders at t can arrive at the sites by t+T1 and no failure occurring after t can be replenished before t+T1• B(t|s0) and D(t, t+T1) are indep. r.v.s since depot back orders at t depend only on failures that occur prior to tIn order to find In order to find QQii(t(t))… … • Convolve the distribution of B(t|s0) and D(t, t+T1) to obtain the distribution of Q(t)• Disaggregate Q(t) into Qi(t)… B(t|s0) = [Q0(t)-s0]+ Q0(t) the total no. of failed items in the system at tAssuming Ample Repair CapacityAssuming Ample Repair Capacity• Q0(t) is the occupancy level in a M|G|∞queue where the service time includes the in-transit time to the depot T1and the repair time • Palm’s theoremSteady state distribution of Q0(t) is PoissonDisaggregationDisaggregationof of Q(tQ(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 Comparison to the METRIC approximationapproximation• METRIC approximates Qi(t) as the occupancy level of an independent M|G|∞queue. Thus Qiis Poisson. • 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 ModelsExact 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|G|∞ queuean M|M|ksystemExact ModelsExact Models• For every value of s0– Find distribution of backorders at the depot – Disaggregate using conditional distribution Pr[Qi=j|Q=k] , which is determined by the priority scheme for filling the outstanding ordersHeavyComputational burdenApproximate ModelApproximate Model• From Q(t+T1)= B(t|s0) + D(t, t+T1) and binomial conditional distribution of the outstanding orders at each site, the mean and variance of each site’s outstanding orders can be expressed in terms of the mean and variance of the depot backorder levels: Graves (1983)Approximate ModelApproximate Model• From B(s0) = [Q0-s0]+• Thus we can compute the mean and variance of B(s0) for all values of s0 of interest given the distribution of Q0(See equations 6 and 7in the 1985 Graves paper)Approximate ModelApproximate Model• Next approximate the distribution of Qiby a negative binomial distributionJustification – Exploratory investigation suggests that the distribution of Qiwill be unimodal and will have variance strictly greater than its mean(See equations 8, 9, and 10 in the 1985 Graves paper)Test ProblemTest Problem• Specify the following– the expected repair cycle time– the aggregate demand rate– the depot stock level– the site fill rate• Test shows that METRIC errs 11.5%, the approximate model errs 0.9%SummarySummary• Provides a general framework for determining the distribution of net inventory levels in a multi-echelon system• Presents an approximate model assuming ample repair capacity, Poisson failure process and FCFS