Note: Generalized Notions of Concavity with an Application to Capacity Management

We introduce a generalization of K-concavity termed weak (K₁, K₂)-concavity and show how it can be used to analyze certain dynamic systems arising in capacity management. We show that weak (K₁, K₂)-concavity has two fundamental properties that are relevant for the analysis of such systems: First, it is preserved for linear interpolations; second, it is preserved for certain types of linear extensions. In capacity management problems where both buying and selling capacity involve a fixed cost plus a proportional cost/revenue term, interpolations and extensions are fundamental building blocks of the optimality analysis. In the context of the capacity management problem studied by Ye and Duenyas (2007), we show that weak (K₁, K₂)-concavity is sufficient to prove the general structure of the optimal policy established in that paper.