On the optimality of stepwise policies for managing capacity, inventory and backorders

Abstract

We consider the problem of simultaneously managing capacity, inventory and backorders in a multi-mode production environment modelled via Brownian motion. The presence of more than two production modes adds an additional level of complexity: not just when to change modes, but also which mode to change to. We show that, under the two assumptions that demand is the overriding source of variability in the process and that the cost to change from one production mode to another is proportional to the difference in the production capacities, a policy that moves stepwise among the modes minimizes the long-run average cost. Examples demonstrate that if either assumption is violated no policy that moves stepwise among the modes may be optimal. To focus on the complexity of identifying when to change modes and which mode to change to, we restrict our model to simple convex holding and backorder costs and linear processing costs and costs for rejecting demand and idling capacity. We adopt the economic average cost model that allows the manager to reject demand or idle capacity at any time. We demonstrate that under our two assumptions we may impose an ordering assumption on the relative value functions that dramatically simplifies the classic Hamilton-Jacobi-Bellman equations. We show that under the economic average cost model a production mode that is initially unattractive may later become attractive as new modes are added. Our arguments are essentially constructive and lead to a practical algorithm for finding an optimal policy.