Efficient optimization of the dual-index policy using Markov chains

This article considers the inventory control of a single product in one location with two supply sources facing stochastic demand. A premium is paid for each product ordered from the faster "emergency" supply source. Unsatisfied demand is backordered and ordering decisions are made periodically. The optimal control policy for this system is known to be complex. For this reason a type of base-stock policy known as the Dual-Index Policy (DIP) is used as the as control mechanism for this inventory system. Under this policy ordering decisions are based on a regular and an emergency inventory position and their corresponding order-up-to levels. Previous work on this policy assumes deterministic lead times and uses simulation to find the optimal order-up-to levels. This article provides an alternate proof for the result that separates the optimization of the DIP in two one-dimensional problems. An insight from this proof allows the model to be generalized to accommodate stochastic regular lead times and provide an approximate evaluation method based on limiting results so that optimization can be done without simulation. An extensive numerical study shows that this approach yields excellent results for deterministic lead times and good results for stochastic lead times.