IPA Derivatives for Make-to-Stock Production-Inventory Systems With Backorders Under the (R,r) Policy

This paper addresses Infinitesimal Perturbation Analysis (IPA) in the class of Make-to Stock (MTS) production-inventory systems with backorders under the continuous-review (R,r) policy, where R is the stock-up-to level and r is the reorder point. A system from this class is traditionally modeled as a discrete system with discrete demand arrivals at the inventory facility and discrete replenishment orders placed at the production facility. Here, however, we map an underlying discrete MTS system to a Stochastic Fluid Model (SFM) counterpart in which stochastic fluid-flow rate processes with piecewise constant sample paths replace the corresponding traditional discrete demand arrival and replenishment stochastic processes, under very mild regularity assumptions. The paper then analyzes the SFM counterpart and derives closed-form IPA derivative formulas of the time-averaged inventory level and time-averaged backorder level with respect to the policy parameters, R and r, and shows them to be unbiased. The obtained formulas are comprehensive in the sense that they are computed for any initial inventory state and any time horizon, and are simple and fast to compute. These properties hold the promise of utilizing IPA derivatives as an ingredient of offline design algorithms and online management and control algorithms of the class of systems under study.