Consider a central depot (or plant) which supplies several locations experiencing random demands. Orders are placed (or production is initiated) periodically by the depot. The order arrives after a fixed lead time, and is then allocated among the several locations. (The depot itself does not hold inventory.) The allocations are finally received at the demand points after another lag. Unfilled demand at each location is backordered. Linear costs are incurred at each location for holding inventory and for backorders. Also, costs are assessed for orders placed by the depot. The object is to minimize the expected total cost of the system over a finite number of time periods.
This system gives rise to a dynamic program with a state space of very large dimension. We show that this model can be systematically approximated by a single-location inventory problem. All the qualitative and quantitative results for such problems can then be applied.
Federgruen, Awi, and P. Zipkin. "Approximations of dynamic, multilocation production and inventory problems." Management Science 30, no. 1 (January 1984): 69-84.
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