Production planning over time and the nature of the expectation and planning horizon

Given the amounts required of a certain commodity in each of T future periods of time, we examine the problem of how production should be scheduled over time in order to satisfy these requirements at the lowest possible cost. A general solution to this problem is exhibited under certain "reasonable" assumptions concerning the nature of production and storage costs, and a convenient graphical method of securing the solution is outlined. One significant aspect of the solution is that under certain conditions (and especially if the cost of carrying inventories is "large" relative to the marginal cost of production), the optimal plan consists of a sequence of plans covering successive intervals of the entire horizon and having the following property: the production schedule for each interval is identical with the optimum plan for the corresponding interval considered separately; it depends, in other words, on requirements within the interval itself, but is essentially independent of requirements for earlier or later periods. It is shown that these results have significant implications concerning the length of the expectation and planning horizon relevant to current production decisions and that they throw some light on the rationale of certain observed business practices.