IPA Derivatives for Make-to-Stock Production-Inventory Systems With Lost Sales

This note applies the stochastic fluid model (SFM) paradigm to a class of single-stage, single-product make-to-stock (MTS) production-inventory systems with stochastic demand and random production capacity, where the finished-goods inventory is controlled by a continuous-time base-stock policy and unsatisfied demand is lost. This note derives formulas for infinitesimal perturbation analysis (IPA) derivatives of the sample-path time averages of the inventory level and lost sales with respect to the base-stock level and a parameter of the production rate process. These formulas are comprehensive in that they are exhibited for any initial inventory state, and include right and left derivatives (when they differ). The formulas are obtained via sample path analysis under very mild regularity assumptions, and are inherently nonparametric in the sense that no specific probability law need be postulated. It is further shown that all IPA derivatives under study are unbiased and fast to compute, thereby providing the theoretical basis for online adaptive control of MTS production-inventory systems.