Optimal control in an inventory management problem considering replenishment lead time based upon a non-diffusive stochastic differential equation

An inventory management problem is theoretically discussed for a factory having effects of lead times in replenishing the inventory, where it stocks materials used for its products. It is assumed that the factory can dynamically control the size of ordering materials. By applying the stochastic control theory, the optimal control of the ordering size is derived, in which the expected total cost up to an expiration time is minimized. First, a new stochastic model is constructed for describing an inventory fluctuation of the factory by the use of a non-diffusive stochastic differential equation, where an analytic time is introduced so that the inventory process can be a Markov process even though it is affected by lead times. Next, an optimal control is formulated by introducing an evaluation function quantifying total costs. Based upon them, the Hamilton-Jacobi-Bellman (HJB) equation is derived, whose solution gives the optimal control. Finally, the optimal control is quantitatively examined through numerical solutions of the HJB equation. Numerical results indicate that if time up to an expiration time is short then the optimal control is affected by it, otherwise, the optimal control does not depend on it.