Approximation Algorithms for Dynamic Inventory Management on Networks

Abstract

We provide the first approximation algorithm for dynamic inventory management on a network with stochastic demand and backlogging. Specifically, under a mild cost condition, we prove the cost of a specially designed base-stock policy is less than 1.618 times the cost of an optimal policy. We develop a novel stochastic programming analysis to prove this result: we carefully calibrate two stochastic programs (providing upper and lower bounds on the optimal policy), and compare their objectives. The upper bound arises from a new class of base-stock policies we define to address the currently unresolved issue of how to assign and fulfill backlogs in a system with fulfillment flexibility. We show the optimal policy in this class takes a simple and intuitive form: backlogs are assigned to their lowest cost activities for replenishment, and the ordered resources for those activities are committed to the backlogs for fulfillment. Next, the lower bound stochastic program is derived through a novel cost accounting scheme that captures the trade-off between current inventory decisions and future backlog decisions. We then exploit this trade-off to bound the ratio of the two stochastic program objectives and prove our main result. We also demonstrate our policy's practicality with numerical simulations that show it performs within 1% of optimal on average across a wide range of problem instances. Finally, we show that our techniques and results extend to more general settings, demonstrating their potential for broader applicability. Importantly, our approach extends to problems with lead times, where we prove an approximation guarantee for base-stock policies that is independent of both the lead time and network structure. Thus, our work provides the new managerial insight that properly designed base-stock policies can be effective in network settings.

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