On the replenishment policy considering less expensive but non-committed supply

We study an inventory system that replenishes its stock through two shipping services, of which the non-committed shipping service can only ship up to a random capacity that is not known until shipment. By transforming the non-convex optimization in the dynamic program to a convex optimization problem, we show that the optimal policy for each period is determined by a quota for the non-committed shipping and a base-stock level for the committed shipping.