A Linear Approach to the Dynamic Inventory Problem

The decision aspects of inventory management revolve around striking some balance between the costs of change and the costs of failing to change. However, in any but the simplest cases of economic order quantities it is difficult to specify precisely how one cost interacts with the other: by how much exactly do additional set-ups reduce the total cost of inventories or wasted production, or the total penalties for delayed deliveries or lost sales--supposing that units are taken out of inventory by some demand function and added to inventories by some production function? This paper analyzes the interaction of the various costs and provides relatively simple optimizing formulae assuming that, mainly, a) demand is reasonably well approximated by a linear function of time; b) production proceeds at a constant rate between one set-up and the next; c) demand and production overlap in time such that the total quantity produced in any one lot interval equals the total quantity consumed in that interval (a unit being consumed as soon as it is ready, though the run from which it comes is not yet completed); and if d) inventories are permitted; that is, surplus production is storable and backorders are acceptable, though both are more or less costly.