Optimal inventory policy with two suppliers

We analyze a periodic-review inventory model where the decision maker can buy from either of two suppliers. With the first supplier, the buyer incurs a high variable cost but negligible fixed cost; with the second supplier, the buyer incurs a lower variable cost but a substantial fixed cost. Consequently, ordering costs are piecewise linear and concave. We show that a reduced form of generalized (s, S) policy is optimal for both finite and (discounted) infinite-horizon problems, provided that the demand density is log-concave. This condition on the distribution is much less restrictive than in previous models. In particular, it applies to the normal, truncated normal, gamma, and beta distributions, which were previously excluded. We concentrate on the situation in which sales are lost, but explain how the policy, cost assumptions, and proofs can be altered for the case where excess demand is backordered. In the lost sales case, the optimal policy will have one of three possible forms: a base stock policy for purchasing exclusively at the high variable cost (HVC) supplier; an (sLVC, SLVC) policy for buying exclusively from the low variable cost supplier; or a hybrid (s, SHVC, SLVC) policy for buying from both suppliers.