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

We consider a joint inventory and pricing problem with one warehouse and multiple stores, in which the retailer needs to make 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. To deal with this, we propose a heuristic control based on the optimal solution of a deterministic relaxation of the original stochastic problem. The construction of our heuristic combines four ideas: (1) order-up-to control, (2) dynamic pricing with linear rate adjustment, (3) replenishment batching, and (4) random errors averaging. We show for a particular choice of control parameters that the heuristic is close to optimal when demand is Poisson and the annual market size is large. In addition to analyzing our proposed heuristic, we also analyze the performance of some popular and simple heuristics that directly implement the solution of the deterministic approximation. We show that simple re-optimization of 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 keeps increasing as we re-optimize more frequently). This cautions against the use of simple re-optimizations in the joint inventory and pricing setting with lost sales.