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

78 Pages Posted: 29 Sep 2020 Last revised: 15 Jun 2022

See all articles by Yanzhe (Murray) Lei

Yanzhe (Murray) Lei

Queen's University - Smith School of Business

Sheng Liu

Rotman School of Management

Stefanus Jasin

University of Michigan, Stephen M. Ross School of Business

Andrew Vakhutinsky

Oracle Labs

Date Written: September 7, 2020

Abstract

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 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 heuristic policies 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). This finding cautions against a naive use of simple re-optimizations 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 warehouse and stores, and then replenish 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.

Suggested Citation

Lei, Yanzhe (Murray) and Liu, Sheng and Jasin, Stefanus and Vakhutinsky, Andrew, Joint Inventory and Pricing for a One-Warehouse Multi-Store Problem: Spiraling Phenomena, Near Optimal Policies, and the Value of Dynamic Pricing (September 7, 2020). Available at SSRN: https://ssrn.com/abstract=3688561 or http://dx.doi.org/10.2139/ssrn.3688561

Yanzhe (Murray) Lei

Queen's University - Smith School of Business ( email )

Smith School of Business - Queen's University
143 Union Street
Kingston, Ontario K7L 3N6
Canada

Sheng Liu

Rotman School of Management ( email )

105 St. George st
Toronto, ON M5S 3E6
Canada

Stefanus Jasin (Contact Author)

University of Michigan, Stephen M. Ross School of Business ( email )

701 Tappan Street
Ann Arbor, MI MI 48109
United States

Andrew Vakhutinsky

Oracle Labs ( email )

MA
United States

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