Stability and optimization of a production inventory system under prioritized base-stock control

A make-to-stock system producing two part-types with stochastic processing times and random demands is considered. A prioritized base-stock policy is used to control the production to meet exogenous Poisson demands, where the unmet high-priority demands are partially backordered due to their limited patience and the unmet low-priority demands are fully backordered. Based on the matrix analytical method, a necessary and sufficient condition for the stability of the system is provided. The explicit stationary distribution is derived using the spectral expansion approach. Interesting steady-state performance measures such as stock-out probabilities and lost-sale fraction are then calculated. By investigating the structural properties of the objective functions, simple procedures are presented to find the optimal base-stock levels either to minimize the expected cost or to satisfy the stock-out probability and the lost-sale fraction constraints. In addition, the optimization problem with respect to the maximum backlog level for part-type one and two base-stock levels is addressed and a solution procedure is presented. Numerical examples are given to demonstrate the results.