Dynamic Pricing Through Sampling Based Optimization

Abstract

In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm offers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among different products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the firm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for different robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a confidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty.Finally, we compare the revenue performance of the different models using numerical simulations, exploring the behavior of each model under different sample sizes and sampling distributions.

Dynamic Pricing and Learning: Historical Origins, Current Research, and New Directions

“…A potential drawback of these robust approaches is that no learning takes place, despite the accumulation of sales data. Lobel and Perakis (2011) attempt to bridge the gap between robust and data-driven approaches to dynamic pricing, by considering a setting where the uncertainty set is deduced from data samples. A robust extension of Caldentey (2009) andvan Roy (2010), where finite inventory is sold during an infinite time horizon, is studied by Li et al (2009).…”

A constraint sampling approach for multi-stage robust optimization

“…Then, constraint sampling techniques prove attractive as they circumvent the added complexity, thereby yielding problems that can potentially be solved with less computational effort. While the benefits of combining decision rule and sampling techniques have been explored by Calafiore and Nilim [2004], Bertsimas and Caramanis [2007], Skaf and Boyd [2009] and Lobel and Perakis [2010], several issues remain to be addressed. First, tractability results have only been provided for the case of polynomial decision rules.…”