An Efficient Algorithm for Dynamic Pricing Using a Graphical Representation

Abstract

We study a multi-period, multi-item pricing problem faced by a retailer. The objective is to maximize the total profit by choosing optimal prices. This problem is faced by many retailers (e.g., supermarkets) who need to set the prices for multiple items at the beginning of the selling season. Data is used to estimate demand, and then pricing decisions need to be set in tractable way, while satisfying several important business rules. The strength of our work lies in the graphical reformulation we introduce, which allows us to use ideas from combinatorial optimization. In contrast with some previous work, we do not impose assumptions on the structure of the demand function. The complexity of our method depends linearly on the number of time periods but is exponential in the model memory (number of past prices that affect current demand) and in the number of items. Consequently, for problems with large memory, we show that the profit maximization problem is NP-hard by presenting a reduction from the Traveling Salesman Problem. We introduce the discrete reference price model which is a discretized version of the commonly used reference price model, accounting for an exponential smoothed contribution of all past prices. This discrete model allows us to capture the fact that customers do not form reference prices with infinite precision. For this model, we show that the problem can be solved efficiently with low runtimes. We then approximate several common demand functions by using the discrete reference price model. Next, we extend the reference price model to handle cross-item effects among multiple items using the notion of a virtual reference price. To allow the scalability of our approach, we cluster the different items into blocks and show how to incorporate global business constraints which are important and challenging in practice. Finally, we apply our solution approach using demand models calibrated with supermarket data and show that we can solve realistic size instances in a few minutes.

Keywords: Retail Pricing, Layered Graph, Reference Price Model, Multi-item Pricing