Optimal and near-optimal (s,S) inventory policies for levy demand processes

A Levy jump process is a continuous-time, real-valued stochastic process which has independent and stationary increments, with no Brownian component. We study some of the fundamental properties of Levy jump processes and develop (s,S) inventory models for them. Of particular interest to us is the gamma-distributed Levy process, in which the demand that occurs in a fixed period of time has a gamma distribution. We study the relevant properties of these processes, and we develop a quadratically convergent algorithm for finding optimal (s,S) policies. We develop a simpler heuristic policy and derive a bound on its relative cost. For the gamma-distributed Levy process this bound is 7.9% if backordering unfilled demand is at least twice as expensive as holding inventory. Most easily-computed (s,S) inventory policies assume the inventory position to be uniform and assume that there is no overshoot. Our tests indicate that these assumptions are dangerous when the coefficient of variation of the demand that occurs in the reorder interval is less than one. This is often the case for low-demand parts that experience sporadic or spiky demand. As long as the coefficient of variation of the demand that occurs in one reorder interval is at least one, and the service level is reasonably high, all of the policies we tested work very well. However even in this region it is often the case that the standard Hadley–Whitin cost function fails to have a local minimum.