Optimal Sizing and Timing of Modular Capacity Expansions

We build a discrete time, serially correlated stochastic demand, nonstationary, finite horizon, capacity expansion model that includes (1) economies of scale in capacity costs, (2) positive expansion lead times, and (3) a fixed maximum cumulative capacity, called the shell size. When the shell size is finite, we have what is called the modular approach to capacity expansion, in which a shell is built along with the first module, and additional modules (of varying sizes) can be added at less cost than entire new plants of their respective sizes. (When the shell size is infinite, we have the traditional approach in which every expansion is a new plant.) We show that an (s, S) expansion point, expansion level, policy is optimal: Capacity is expanded to S if and only if the initial level is below s. These parameters depend on the period, the last observed demand, and the shell size. We also show that the optimal value of the enterprise is a convex decreasing function of the lump-sum cost incurred at the time of each expansion. We illustrate the model with a numerical example taken from semiconductor manufacturing.