A continuous time, deterministic, nonstationary model of economic ordering

Abstract

The paper considers the problem of economic ordering for a deterministic, nonstationary environment in continuous time. Previous work on the topic is reviewed. The specification of the cost criterion common in inventory theory is called in question for nonstationary situations as far as interest cost is concerned. It is proposed to account for interest by discounting rather than in a holding cost expression. The main interest of the paper is in three versions of the problem: First an unconstrained version, for which inventory is allowed to become negative (backlogging model), second a model in which inventory is constrained to be nonnegative (non-backlogging model), and third a nonbacklogging model with a storage space constraint. For the first two problems necessary optimality conditions are derived which are based on control theory for continuous time systems with jumps in the state trajectories, especially on Blaquière's impulsive maximum principle. These conditions reduce the problem of finding an optimal ordering plan, i.e. an unknown number of optimal ordering times and for each of them an optimal order size to a one parameter search problem. Due to the possibility of multiple solutions of the optimality conditions for each ordering time, one cannot in general identify a unique candidate ordering plan for each value of the search parameter, but only a tree-structured set of such plans. The optimality conditions for the first two problem versions and for a fourth one with a storage space constraint but without a non-backlogging constraint are eventually combined to yield a solution of the storage space constrained non-backlogging version.