Stochastic properties of a production-inventory process with planned production using transform methodology

The paper considers a stochastic inventory process in which demand is generated by individual events separated by independent stochastic time intervals whereas production takes place in batches of possibly varying sizes at different points in time. The resulting processes are analysed using Laplace transform methodology. Expressions for inventory and stockouts are developed applying the Theory of Residues. An example includes Poisson-distributed demand intervals. Assuming costs for set-ups, for carrying inventory and for stockouts, conditions are developed for determining the optimal production policy, on the one hand, when applying the traditional average cost measure, on the other, when applying the annuity stream principle.