Infinite Horizon Production Planning in Time-Varying Systems with Convex Production and Inventory Costs

We consider the planning of production over the infinite horizon in a system with time-varying convex production and inventory holding costs. This production lot size problem is frequently faced in industry where a forecast of future demand must be made and production is to be scheduled based on the forecast. Because forecasts of the future are costly and difficult to validate, a firm would like to minimize the number of periods into the future it needs to forecast in order to make an optimal production decision today. In this paper, we first prove that under very general conditions finite horizon versions of the problem exist that lead to an optimal production level at any decision epoch. In particular, we show it suffices for the first period infinite horizon production decision to solve for a horizon that exceeds the longest time interval over which it can prove profitable to carry inventory. We then develop a closed-form expression for computing such a horizon and provide a simple finite algorithm to recursively compute an infinite horizon optimal production schedule.