The nature of optimal policies for deterministic finite-horizon inventory models

Abstract

This paper concerns a general formulation for a longstanding problem. This is the problem of determining a replenishment schedule that minimises the total cost of stocking and holding an inventory in a deterministic finite-horizon inventory model. The formulation permits the treatment of seemingly unrelated models in a single framework. These include classical lot-size models, batching models, repair models, recovery models and others. Admissible control policies are restricted to a partition of some closed interval on the real line. The solution of a mixed integer nonlinear programming problem (MINLP) delivers the optimal partition. It is shown that the MINLP possesses an optimal solution under very mild conditions. The theory of submodular functions on a lattice is the key to handling the integer variable. This theory permits the recovery and generalisations of earlier results on the interleaving property of optimal partitions and a convexity property of the value of the objective function. Past intractable inventory models with demand driven by a general differential equation and in the absence and in the presence of shortages and inflation are solved. This generalises a number of existing results in the literature.