Using Linear Programming to Derive Planning Horizons for a Production Smoothing Problem

In a previous paper Kunreuther and Morton examined a single product production planning problem in which demand in each of a finite number of periods is known with certainty, and there are linear costs for holding inventory and for adjusting production levels between periods. This paper reexamines the problem, taking advantage of it being a linear program. In particular, the dual problem has considerable structure, and the patterns of variables in dual feasible solutions have interesting geometric properties that afford a great deal of intuition. The requirement of dual feasibility is a more formal approach to the "marginal cost balancing" notion of Kunreuther and Morton and provides some extension and strengthening of the earlier results. A proof with simplified logical structure for the Kunreuther-Morton planning horizon is indicated. Then a different kind of planning horizon result is given. These, in contrast to the K-M planning horizons, occur when future demands increase, such as at the bottom of a seasonal cycle.