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.

Keywords: Consumer Choice, Assortment Optimization, PTAS

JEL Classification: C61