Multiperiod Stock Allocation via Robust Optimization

We consider a one-warehouse, N-retailer, multiperiod, stock allocation problem in which holding costs are identical at each location and no stock is received from outside suppliers for the duration of the planning horizon. No shipments are allowed between retailers. The only motive for holding inventory at the central warehouse for allocation in future periods is the so-called risk pooling motive. We apply robust optimization to this problem extending the inventory policy to allow for an adaptive, nonanticipatory shipment policy. We consider two alternatives for the uncertainty set, one in which risk pooling is implicit and another for which risk pooling is explicit. The explicit risk pooling uncertainty set grows by no more than the square of the number of retailers. The general problem can be solved using Benders' decomposition. A special case gives rise to closed-form solutions for both uncertainty set alternatives. The explicit risk pooling uncertainty set leads to a square root law in which the optimal stock to reserve at the central warehouse grows with the square root of the number of retailers. The experimental results confirm the value of the robust optimization approach and provide managerial insights into the operation of such systems.