Are base-stock policies optimal in inventory problems with multiple delivery modes?

We present a periodic review inventory model with multiple delivery modes. While base-stock policies are optimal for one or two consecutive delivery modes, they are not so otherwise. For multiple consecutive delivery modes, we show that only the fastest two modes have optimal base stocks, and provide a simple counterexample to show that the remaining ones do not. We investigate why the base-stock policy is or is not optimal in different situations.