The emergence of carriers that deliver items to geographically dispersed destinations quickly and at a reasonable cost, combined with the low cost of sharing information through networked databases, has opened up new opportunities to better manage inventory. We investigate these benefits in the context of a supply chain in which a manufacturer supplies expensive, low-demand items to vertically integrated or autonomous retailers via one central depot. The manufacturer's lead time is assumed to be due to the geographical distance from the market or a combination of low volumes, high variety, and inflexible production processes. We formulate and solve an appropriate mathematical model based on one-for-one inventory policies in which a replenishment order is placed as soon as the customer withdraws an item. We find that sharing and transshipment of items often, but not always, reduces the overall costs of holding, shipping, and waiting for inventory. Unexpectedly, these cost reductions are sometimes achieved through increasing overall inventory levels in the supply chain. Finally, while sharing of inventory typically benefits all the participants in decentralized supply chains, this is not necessarily the case---sharing can hurt the distributor or individual retailers, regardless of their relative power in the supply chain.Multi-Echelon Systems, Transshipment, Approximation in Inventory Models

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We consider several subclasses of the problem of grouping n items (indexed 1, 2, …, n) into m subsets so as to minimize the function g(S1, …, Sm). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(·) that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S1, …, Sm satisfies a more general condition called “semiconsecutiveness”). Moreover, by restricting attention to “consecutive” (or “semiconsecutive”) partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman.

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