Dynamic Pricing with Infrequent Inventory Replenishments

We consider a joint pricing and inventory control problem in which pricing can be adjusted more frequently than inventory ordering decisions. More specifically, the pricing decision is adjusted every period, whereas new inventory is ordered every epoch, with each epoch consisting of multiple periods. This setting is motivated by many examples, especially among online retailers, in which prices are much easier to change than inventory levels, because changing the latter is subject to prior arrangements or logistical constraints. In this setting, the retailer determines the inventory level at the beginning of each epoch and solves a dynamic pricing problem within that epoch, with no further replenishment opportunities. The optimal pricing and inventory control policy is obtained from an intricate dynamic program (DP), in which the inventory level is the state variable and the pricing policy is characterized as a function of the inventory level. We consider the situation in which the demand-price function and the distribution of random demand noise are both unknown to the retailer, who must develop an online learning algorithm to learn this information while simultaneously maximizing total profit. We propose a learning algorithm that applies linear bandit techniques under the upper confidence bound framework, and we prove that it converges through the DP recursions to the optimal pricing and inventory control policy that would be obtained under complete demand information. The theoretical lower bound for the convergence rate of the learning algorithm is proved based on the multivariate Van Trees inequality coupled with some structural DP analyses, and we show that the upper bound of our algorithm’s convergence rate matches the theoretical lower bound. Numerical results show that our learning algorithm performs very well.

This paper was accepted by J. George Shanthikumar, data science.

Funding: This work was supported by the MIT Data Science Laboratory, Purdue University Data Science Center for Decision Making, the Hong Kong Research Grants Council (RGC) Early Career Scheme [CityU21505825], the Hong Kong RGC General Research Fund [CityU11508223], and the Joint Research and Innovation Seed Grants program between the University of Illinois System and the University Academic Alliance in Taiwan.

Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00129.