Abstract
We analyze a planning model for a firm or public organization that needs to cover uncertain demand for a given item by procuring supplies from multiple sources. Each source faces a random yield factor with a general probability distribution. The model considers a single demand season. All supplies need to be ordered before the start of the season. The planning problem amounts to selecting which of the given set of suppliers to retain, and how much to order from each, so as to minimize total procurement costs while ensuring that the uncertain demand is met with a given probability. The total procurement costs consist of variable costs that are proportional to the total quantity delivered by the suppliers, and a fixed cost for each participating supplier, incurred irrespective of his supply level. Each potential supplier is characterized by a given fixed cost and a given distribution of his random yield factor. The yield factors at different suppliers are assumed to be independent of the season's demand, which is described by a general probability distribution.
Determining the optimal set of suppliers, the aggregate order and its allocation among the suppliers, on the basis of the exact shortfall probability, is prohibitively difficult. We have therefore developed two approximations for the shortfall probability. Although both approximations are shown to be highly accurate, the first, based on a large-deviations technique (LDT), has the advantage of resulting in a rigorous upper bound for the required total order and associated costs. The second approximation is based on a central limit theorem (CLT) and is shown to be asymptotically accurate, whereas the order quantities determined by this method are asymptotically optimal as the number of suppliers grows. Most importantly, this CLT-based approximation permits many important qualitative insights.