Display Optimization Under the Multinomial Logit Choice Model: Balancing Revenue and Customer Satisfaction

Abstract

In this paper, we consider an assortment optimization problem in which a platform must choose pairwise disjoint sets of assortments to offer across a series of T stages. Arriving customers begin their search process in the first stage and progress sequentially through the stages until their patience expires, at which point they make a multinomial-logit-based purchasing decision from among all products they have viewed throughout their search process. The goal is to choose the sequential displays of product offerings to maximize expected revenue. Additionally, we impose stage-specific constraints that ensure that as each customer progresses farther and farther through the T stages, there is a minimum level of “desirability” met by the collections of displayed products. We consider two related measures of desirability: purchase likelihood and expected utility derived from the offered assortments. In this way, the offered sequence of assortment must be both high earning and well-liked, which breaks from the traditional assortment setting, where customer considerations are generally not explicitly accounted for. We show that our assortment problem of interest is strongly NP-Hard, thus ruling out the existence of a fully polynomial-time approximation scheme (FPTAS). From an algorithmic standpoint, as a warm-up, we develop a simple constant factor approximation scheme in which we carefully stitch together myopically selected assortments for each stage. Our main algorithmic result consists of a polynomial-time approximation scheme (PTAS), which combines a handful of structural results related to the make-up of the optimal assortment sequence within an approximate dynamic programming framework.