In this paper we consider an assembly problem where two critical components are required for assembly of the final product, the demand for which is stochastic. The components can be ordered separately from individual suppliers or in a set (a set refers to the components in the required ratio) from a joint supplier. We consider the case where the assembly stage is free, i.e., the firm procures and stores the components and sells complete sets. The supplier delivery process may be random owing to uncertainty in the production process (e.g., semiconductor industries). We assume that a supplier, with probability β (say), supplies 100% of the order quantity in the current period, and with probability (1 - β) supplies nothing. If there is no delivery during this period, the order is delivered in the next period. The added complexity of coordinating shipments of different components requires careful planning in placing the orders. In the single-period problem, if no order is placed with the joint supplier, the order quantities from the individual suppliers follows an order-up-to policy structure with identical order levels. However, it is optimal to diversify (i.e., order from the joint supplier as well) when the inventory level is below a certain threshold (determined in this paper). With lower initial inventory levels, the firm cannot risk the cost of stockouts if the individual supplier(s) fail to deliver in the current period. With certain conditions on the cost and delivery parameters of the suppliers, we show that the policy structure for the multi-period problem is similar to that of the single-period problem, except that the order up-to-levels are not the same. Intuitively, it might be optimal to order extra components for use in the future. This is a direct consequence of the uncertainty in the delivery timing of the suppliers. Finally we conduct a computational study of the two-period problem and determine the effect of supplier costs and the probability of delivery on the optimal order policy. The policies are intuitive and offer a better understanding of the effect of supply and demand uncertainty on the assembly problem.
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