Cost relationships in stochastic inventory systems: A simulation study of the (S, S-1, t=1) model

The analysis of the full stochastic model in which both the demand per unit time and the lead time are stochastic is complex. Analysis of the reduced stochastic inventory models in which only one of the parameters (either the demand per unit time or the lead time) is stochastic and the other is constant is relatively less complex. In this paper we exploit insights from vector analysis and postulate an approximation that expresses the optimal cost of the full stochastic model in terms of the optimal costs of the two reduced models. We demonstrate the adequacy of the cost relationship in the context of one specific type of inventory model--a periodic review (S, S-1, t=1) model--by performing an extensive set of simulations, using the Poisson, the exponential, and the gamma distributions to characterize demand and lead time. We also use the simulation data to develop regression relationships between the cost and an appropriate measure of variability, such as the standard deviation, the variance, or the coefficient of variation. For the cost of the full model, we find in our computations that our approximations have 98.4% accuracy for the Poisson, 96% for the exponential, and 97% for the gamma.