Bounded Memory Peak End Models Can Be Surprisingly Good

Sales promotions provide an important tool in the retail industry and can have a significant impact on a retailer's revenue. In this paper, we study the trade-off between prediction accuracy and optimization complexity. We introduce a compact set of demand features, which includes the last seen price as well as the minimum price seen within a bounded number of past time periods. We refer to this model as the bounded-memory-peak-end model. We compare this model with traditional anchoring based demand models such as the exponential smoothing, peak-end, and bounded memory models. Using ideas from duality theory, we show that even if the true underlying demand exhibits a different demand structure than the bounded memory peak-end model, the suggested bounded-memory-peak-end model can still predict the true demand with provably low prediction error. Moreover, this demand model allows us to solve the promotion planning optimization problem in polynomial time. By analyzing the structure of the optimal promotion policy under the bounded-memory-peak-end demand model, we characterize the optimal separation between consecutive promotions. This allows us to also determine the optimal promotion policy in an intuitive way. In the cases where the demand follows a different model, we still suggest to use the bounded-memory-peak-end and establish what would be the estimation error. We then show how the estimation error translates into a bounded optimality gap on the revenue arising from the optimization aspect of the problem. Finally, inspired by the Gaussian Process, we develop a novel demand prediction procedure that can increase prediction accuracy. In particular, the non-parametric nature of the Gaussian Process allows us to capture demand even when the true underlying true demand model follows a complex functional form. We illustrate the advantages of our approach by testing it on real sales data from a large supermarket retailer and demonstrate 9% relative improvement in WMAPE