We were involved in an applied research project where a manufacturer of a bulk industrial chemical that was produced in the southern hemisphere needed to ship its product in huge tanker vessels to ports in the northern hemisphere. Demand is buffered by tanks at the on-shore terminals. It is crucial to produce without interruption, so the tanks cannot be allowed to get too full; on the other hand, the company would like to satisfy all demand, so the tanks cannot get too empty.
We were asked to determine "optimal" inventories in this situation. Classic inventory theory suggests that an (s,S) policy should be optimal. However, this would require shipping orders of arbitrary sizes. In reality, shipments are always in full tankers.
If there is just one size Q of tanker, then F. Chen determined that an intuitive extension of an (s,S) policy is optimal. Here we consider a further extension where there are two tanker sizes, Q and 2Q. In this case we show that a reorder point policy is still optimal, with an added wrinkle: at the margin, there is a choice about whether the last tanker should be of size Q or 2Q. We also consider some more realistic extensions of this model where there is a more general set of possible tanker sizes.