We study a single period multiproduct inventory problem with substitution and proportional costs and revenues. We considerNproducts andNdemand classes with full downward substitution, i.e., excess demand for classican be satisfied using productjfori≥j. We first discuss a two-stage profit maximization formulation for the multiproduct substitution problem. We show that a greedy allocation policy is optimal. We use this to write the expected profits and its first partials explicitly. This in turn enables us to prove additional properties of the profit function and several interesting properties of the optimal solution. In a limited computational study using two products, we illustrate the benefits of solving for the optimal quantities when substitution is considered at the ordering stage over similar computations without considering substitution while ordering. Specifically, we show that the benefits are higher with high demand variability, low substitution cost, low profit margins (or low price to cost ratio), high salvage values, and similarity of products in terms of prices and costs.

Supply chain management is becoming an increasingly important issue, especially when in most industries the cost of materials purchased comprises 40–60% of the total sales revenue. Despite the benefits cited for single sourcing in the popular literature, there is enough evidence of industries having two/three sources for most parts. In this paper we address the operational issue of quantity allocation between two uncertain suppliers and its effects on the inventory policies of the buyer. Based on the type of delivery contract a buyer has with the suppliers, we suggest three models for the supply process. Model I is a one-delivery contract with all of the order quantity delivered either in the current period with probability β, or in the next period with probability 1 – β. Model II is also a one-delivery contract with a random fraction of the order quantity delivered in the current period; the portion of the order quantity not delivered is cancelled. Model III is similar to Model II with the remaining quantity delivered in the next period. We derive the optimal ordering policies that minimize the total ordering, holding and penalty costs with backlogging. We show that the optimal ordering policy in period n for each of these models is as follows: for x ≥ ū n , order nothing; for v̄ n ≤ x < ū n , use only one supplier; and for x < v̄ n , order from both suppliers. For the limiting case in the single period version of Model I, we derive conditions under which one would continue ordering from one or the other or both suppliers. For Model II, we give sufficient conditions for not using the second (more expensive) supplier when the demand and yield distributions have some special form. For the single period version of Models II and III with equal marginal ordering costs we show that the optimal order quantities follow a ratio rule when demand is exponential and yields are either normal or gamma distributed.