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Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory Costs

Industrial Engineering, University of Washington, Seattle, Washington 98195
Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109
archis{at}u.washington.edu
rlsmith{at}umich.edu

We consider an infinite horizon production planning problem with nonstationary, convex production and inventory costs. Backlogging is allowed, unlike as in related previous work, and inventory cost is interpreted as backlogging cost when inventory is negative. We create finite horizon truncations of the infinite horizon problem and employ classic results on convex production planning to derive a closed-form formula for the minimum forecast horizon. We show that optimal production levels are monotonically increasing in the length of horizon, leading to solution convergence and a rolling horizon procedure for delivering an infinite horizon optimal production plan. The minimum forecast horizon formula is employed to illustrate how cost parameters affect how far one must look into the future to make an infinite horizon optimal decision today.

Key Words: minimum forecast horizons; convex network flow; production planning
History: Received: June 14, 2005; accepted: January 31, 2008.