An Extension to Modigliani and Hohn's Planning Horizons Results

This paper deals with the planning horizon issues of the problem of finding the optimal production schedule over a [0, T] period, for a product having: (1) deterministic demand, (2) strictly convex production costs, and (3) linear holding and back-logging costs. A set of solutions defined as extrapolations is defined, and a property of nonintersection of extrapolations is proved. Using this property uniqueness and planning horizon theorems are proved. It is shown that the reason for the planning horizons is the jump (discontinuity) in the derivative of the inventory cost function when inventory is zero. In addition, the paper deals with the following question: What kinds of changes (variations) in demand may occur such that the optimal solution (production schedule) on some time interval will remain unchanged? This question is associated with the following one: What information is needed about demand in order that the optimal solution on some time interval may be found?