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
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 polices 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.