Fluid Models for Production-Inventory Systems

Keqi Yan: Fluid Models for Production-Inventory Systems (Under the direction of Professor Vidyadhar G. Kulkarni) We consider a single stage production-inventory system whose production and demand rates are modulated by a finite state Markov chain called the environment. Supplementary orders can be placed from external suppliers when needed. We model this system by a fluid-flow system and derive the limiting distribution of the bivariate process (fluid level, environment state). We present a stochastic decomposition property for this fluid model and hence prove that the classical deterministic EconomicOrder-Quantity (EOQ) policy is still optimal in this stochastic environment under certain assumptions. We extend the results to more general models: 1. When backlogging is allowed, we investigate the optimal reorder-point / orderquantity (r, q) policy. We prove that for a given order quantity q, the optimal reorder point r∗(q) can be explicitly given by the well-known newsboy solution. We also show that in a special case the optimality of the deterministic EOQ policy with backlogging holds. 2. When the order quantity can be environment-dependent, we derive the limiting distribution and then calculate the optimal order quantity for each ordering state. 3. When there are positive leadtimes, we consider three cases: orders are processed iii sequentially, in parallel, or the leadtimes have different distributions depending on the number of outstanding orders. Assuming there exists an upper limit N for the number of outstanding orders, this model generalizes the emergency-supply model, selective lost-sales model, and can also be an approximation of infinite-supplier model when N is large enough or the probability that there are N outstanding orders is small. We derive the limiting distribution and the optimal (r, q) policy. We prove that for a given q, the optimality of the newsboy solution for r still holds. We also illustrate numerically how to calculate the optimal N∗ which balances the backlogging cost and emergency-supply / lost-sale cost.