Further Results on Planning Horizons in the Production Smoothing Problem

A generalized version of the Modigliani-Hohn production smoothing model is analyzed, and two types or classes of theorem are proved. (Theorems 1 and 2): The planning horizon (defined as a minimal interval of sufficient length to yield optimal current decisions) depends critically on the existence of "bottleneck" conditions; in the case of the original M-H model, a bottleneck arises out of the inventory nonnegativity constraint, and it is here shown that an additional interesting bottleneck condition arises out of an inventory storage capacity constraint. Whether a particular type of bottleneck defines a horizon will depend, interestingly, on the terminal inventory condition that is imposed for the original optimization process. (Theorems 3 and 4): The sensitivity of horizon length to two economic parameters, the discount factor and the marginal storage cost, will depend on whether the horizon-determining bottleneck is due to a binding inventory capacity constraint, or a binding nonnegativity constraint. The capacity constraint will yield "perverse" horizon sensitivities first noted by Charnes, Dreze and Miller [Charnes, Abraham, Drèze, Jaques, Miller, Merton. 1966. Decision and horizon rules for stochastic planning problems: a linear example. Econometrica 34 307-330.] in their work on the warehousing problem. In light of these results, the importance of horizons as a programming concept can be questioned.