On the Joint Inventory and Pricing Control for a One-Warehouse Multi-Store Problem with Lost Sales : Spiraling Phenomena and Near-Optimal Heuristics

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. Unmet demand at each store is immediately lost. The retailer incurs the usual variable ordering costs, inventory holding costs and lost sales costs, and his 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 heuristics that directly implement the solution of a deterministic approximation of the original stochastic problem. We show that simple re-optimization 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 re-optimize more frequently). Our findings caution against a naive use of simple re-optimizations in the joint inventory and pricing setting with lost sales without first understanding the dynamics of the model. We then propose two heuristics based on the optimal solution of a deterministic relaxation of the original stochastic problem. Our first heuristic computes static prices and order-up-to levels for warehouse and stores, and then replenish each store at the beginning of each batch of periods. Our second heuristic builds on the first heuristic and dynamically changes prices over time. The dynamic pricing mechanism combines the idea of linear rate adjustment and random error averaging. We show that, under proper choices of control parameters, both heuristics have near-optimal performance when demand is Poisson and the annual market size is large, with the second heuristic outperforming the first one