Multi-Period Stock Allocation Via Robust Optimization

In this paper we re-visit a long-standing multi-echelon inventory al-location problem from a robust optimization perspective. We formulate the problem as a one warehouse, N-retailer, multi-period, 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. Stock may be transferred from the central warehouse to the retailers instantaneously and without cost at the beginning of each period for which the central warehouse still has stock on hand. No other stock transfers are allowed. Under this set-up, the only motive for holding inventory at the central warehouse for allocation in future periods is the so-called risk-pooling motive. The dynamic programming formulation of this problem requires a state space too large for practical computation. Various approximation methods have been proposed for variants of this problem. We apply robust optimization to this problem extending the typical uncertainty set to capture the risk pooling phenomenon and extending the inventory policy to allow for an adaptive, non-anticipatory shipment policy. We show how to represent the uncertainty set compactly so that it grows by no more than the square of the number of retailers. The problem can be solved using Benders decomposition in the general case. In the special case of no initial retailer inventories, two periods, and identical retailers, a relaxed form of the problem admits a closed form solution with surprising insights. Summarizing the experimental results of the paper, we see both confirmation of the value of the robust optimization approach as well as managerial insights into the design and operation of multi-echelon inventory systems