Joint Inventory and Pricing for a One-Warehouse Multistore Problem: Spiraling Phenomena, Near Optimal Policies, and the Value of Dynamic Pricing

We consider a joint inventory and pricing problem with one warehouse and multiple stores in which the retailer makes a one-time decision on the amount of inventory to be placed at the warehouse at the beginning of the selling season, followed by periodic joint replenishment and pricing decisions for each store throughout the season. Demand at each store follows a Poisson distribution, and unmet demand is immediately lost. The retailer incurs the usual variable ordering, inventory holding, and lost sales costs, and the retailer's objective is to maximize the expected total profits. The optimal control (or policy) for this problem is unknown and numerically challenging to compute. We first analyze the performance of two popular and simple heuristic policies that directly implement the solution of a deterministic approximation of the original stochastic problem. We show that simple reoptimization of the deterministic problem may yield a very poor performance by causing a "spiraling down" movement in price trajectory, which, in turn, yields a "spiraling up" movement in expected lost sales quantity (i.e., the expected lost sales quantity continues to increase as we reoptimize more frequently). This finding cautions against a naive use of simple reoptimizations without first understanding the dynamics of the model. We then propose two improved heuristic policies based on the optimal solution of a deterministic relaxation of the original stochastic problem. Our first heuristic policy computes static prices and order-up-to levels for the warehouse and stores and then replenishes each store at the beginning of each batch of periods. Our second heuristic policy builds on the first heuristic and dynamically adjusts prices over time based on realized demands. We show that both policies have near-optimal performance when the annual market size is large with the second policy outperforming the first one. Finally, we also prove a fundamental theoretical lower bound on the performance of any policy that relies on static prices. This lower bound highlights the true value of dynamic pricing, whose effect on performance in our setting cannot be duplicated by simply implementing a more sophisticated replenishment policy.