Optimal Supply Diversification Under General Supply Risks

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. The necessity to employ multiple suppliers arises from the fact that when an order is placed with any of the suppliers, only a random fraction of the order size is usable. The model considers a single demand season with a given demand distribution, where all supplies need to be ordered simultaneously before the start of the season. The suppliers differ from one another in terms of their yield distributions, their procurement costs, and capacity levels. The planning model determines which of the potential suppliers are to be retained and what size order is to be placed with each. We consider two versions of the planning model: in the first, the service constraint model (SCM), the orders must be such that the available supply of usable units covers the random demand during the season with (at least) a given probability. In the second version of the model, the total cost model (TCM), the orders are determined so as to minimize the aggregate of procurement costs and end-of-the-season inventory and shortage costs. In the classical inventory model with a single, fully reliable supplier, these two models are known to be equivalent, but the equivalency breaks down under multiple suppliers with unreliable yields. For both the service constraint and total cost models, we develop a highly efficient procedure that generates the optimal set of suppliers as well as the optimal orders to be assigned to each. Most importantly, these procedures generate a variety of important qualitative insights, for example, regarding which sets of suppliers allow for a feasible solution, both when they have ample supply and when they are capacitated, and how various model parameters influence the selected set of suppliers, the aggregate order size, and the optimal cost values.