Infinite horizon production scheduling in time-varying systems under stochastic demand

We consider infinite horizon production scheduling under stochastic demand. All problem data are allowed to vary across periods, including demand distributions, costs, and revenues. A forecast horizon, when it exists, is a finite problem horizon with the property that the corresponding first-period optimal production decision remains optimal regardless of demand and cost projections beyond this horizon. Thus, a forecast horizon allows us to reduce the amount of future data we need to forecast to solve for an optimal first decision for the infinite horizon problem. In this paper, we establish the existence of a forecast horizon under the assumptions that (1) costs and revenues are time-varying linear, and (2) demand is never eventually zero. A key result for establishing the existence and computation of forecast horizons is the monotonicity, and hence convergence, of optimal first-period policies as the horizon increases of finite horizon versions of the infinite horizon problem. A closed-form formula is provided for computing a forecast horizon that depends only on the discount factor and uniform upper and lower bounds on demand and unit production and inventory holding costs. In particular, its value is independent of, and determined in advance of, forecasting the demand distribution. We show that the effect of uncertainty in demand is to increase the forecast horizon associated with a deterministic problem by a constant plus a factor equal to one plus the ratio of these upper to lower bounds on per-period demand. The associated forecast horizon can be surprisingly short, even a few days, when the inventory costs are high.