Optimal capacity investment decisions with two-sided fixed-capacity adjustment costs

In this paper, we consider the optimal management of capacity when a firm faces fixed costs and variable costs to purchase capacity. The firm can also salvage capacity and receive a variable value per unit capacity salvaged, but faces a (different) fixed cost in this case. Each period, the firm faces a stochastic demand, and maintenance costs for capacity that it decides to keep. The firm would thus like to decide how much capacity it should purchase or salvage each period. We introduce a new concept, which we call (K-1, K-2)-concavity, and show that the profit-to-go function satisfies this property. This enables us to characterize the structure of an optimal policy, which is rather complex, consisting of multiple regions in which different decisions are made. We show how special cases of this problem (e.g., no fixed costs, expansion or contraction not allowed) reduce to well-known results, and how (K-1, K-2)-concavity is a generalization of concavity, K-concavity, and sym-K-concavity. We also show how different lead times for purchasing or salvaging capacity can be integrated into the model. Finally, we extend the model to the case where demand is Markov modulated, and a portion of capacity can deteriorate in each period.