Demand Estimation Under the Multinomial Logit Model from Sales Transaction Data

Problem definition: A major task in retail operations is to optimize the assortments exhibited to consumers. To this end, retailers need to understand customers’ preferences for different products. Academic/practical relevance: This is particularly challenging when only sales and product-availability data are recorded, and not all products are displayed in all periods. Similarly, in revenue management contexts, firms (airlines, hotels, etc.) need to understand customers’ preferences for different options in order to optimize the menu of products to offer. Methodology: In this paper, we study the estimation of preferences under a multinomial logit model of demand when customers arrive over time in accordance with a nonhomogeneous Poisson process. This model has recently caught important attention in both academic and industrial practices. We formulate the problem as a maximum-likelihood estimation problem, which turns out to be nonconvex. Results: Our contribution is twofold: From a theoretical perspective, we characterize conditions under which the maximum-likelihood estimates are unique and the model is identifiable. From a practical perspective, we propose a minorization-maximization (MM) algorithm to ease the optimization of the likelihood function. Through an extensive numerical study, we show that our algorithm leads to better estimates in a noticeably short computational time compared with state-of-the-art benchmarks. Managerial implications: The theoretical results provide a solid foundation for the use of the model in terms of the quality of the derived estimates. At the same time, the fast MM algorithm allows the implementation of the model and the estimation procedure at large scale, compatible with real industrial applications.