A stochastic model of multi-level/multi-stage capacity-constrained production–inventory systems

A great deal of research has been done on capacity-constrained production–inventory systems, most of which concerns deterministic demand situations and single-product systems. In this paper we present a model of multi-level capacity-constrained system when external demand is stochastic. Unlike the traditional total cost objective, adopted in the vast majority of capacity-constrained production–inventory models, the (expected) Net Present Value is here used as the objective function. Dynamic programming is chosen as the solution procedure. Numerical examples are given to explain the model and to illustrate features when changing available capacity. The Laplace transform together with input–output analysis are employed as tools to construct the model. This approach has been used in previous research to formulate a theoretical base for Material Requirements Planning systems. The paper provides a further argument for the use of transforms in combination with matrix representations of product structures and capacity requirements, and it extends previous theory in the direction of capacity considerations combined with uncertainty in external demand. Dynamic programming is also shown to be a practical method for the multi-stage optimisation involved. The numerical examples further illustrate, for instance, the natural propensity for subordinate items to be lot sized in a more lumpy way than their parents, and also how the marginal benefit of capacity increments follows the law of diminishing returns. Also comparisons are made with solutions from the deterministic equivalence model, using average demand as a proxy.