A Multi-Echelon Inventory Model for a Reparable Item With One for-one Replenishment

“A Multi-Echelon Inventory Model for a Reparable Item with onefor- one Replenishment” Problem Repair Depot Site 1 s1 • 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 available s0 Site 2 Site 3 s2 s3 Assumptions • 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 depot Notations “replacement requests that have yet to be filled” • 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] Primary 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 t In order to find Qi(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 t Assuming 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 T1 and the repair time • Palm’s theorem Steady state distribution of Q0(t) is Poisson Disaggregation 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’s failure rate λi, thus conditional distribution of Qi(t) is binomial. Comparison to the METRIC approximation • METRIC approximates Qi(t) as the occupancy level of an independent M|G|∞ queue. Thus Qi is 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 Models • Determine the distribution of Q0 – Ample capacity use Palm’s theorem (assuming Poisson or compound Poisson failure processes): – General shipment time, Poisson failure process, k parallel lines Q0 = In-transit + In-repair + In-repair-queue Occupancy level in an M|G|∞ queue an M|M|k system Exact 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 orders Heavy Computational burden Approximate 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: Approximate Model • From B(s0) = [Q0-s0]+ (See equations 6 and 7 in the 1985 Graves paper) • Thus we can compute the mean and variance of B(s0) for all values of s0 of interest given the distribution of Q0 Approximate Model • Next approximate the distribution of Qi by a negative binomial distribution (See equations 8, 9, and 10 in the 1985 Graves paper) Justification – Exploratory investigation suggests that the distribution of Qi will be unimodal and will have variance strictly greater than its mean Test 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% Summary • 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 rule