Constrained Assortment Optimization Under the Markov Chain–based Choice Model

Abstract

Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. The fundamental goal is to select a subset of items to offer from a universe of substitutable items to maximize expected revenue when customers exhibit a random substitution behavior captured by a choice model. We study assortment optimization under the Markov chain choice model in the presence of capacity constraints that arise naturally in many applications. The Markov chain choice model considers item substitutions as transitions in a Markov chain and provides a good approximation for a large class of random utility models, thereby addressing the challenging problem of model selection in choice modeling. In this paper, we present constant factor approximation algorithms for the cardinality- and capacity-constrained assortment-optimization problem under the Markov chain model. We show that this problem is APX-hard even when all item prices are uniform, meaning that, unless P = NP, it is not possible to obtain an approximation better than a particular constant. Our algorithmic approach is based on a new externality adjustment paradigm that exactly captures the externality of adding an item to a given assortment on the remaining set of items, thereby allowing us to linearize a nonlinear, nonsubmodular, and nonmonotone revenue function and to design an iterative algorithm that iteratively builds up a provably good assortment. This paper was accepted by Yinyu Ye, optimization.

Assortment Optimization under a Single Transition Choice Model

“…Désir et al. (2015) study the robust assortment optimization under this model. As described in the introduction, the STCM we propose differs from the Markov chain choice model in two dimensions: (i) we add a control for the seller to decide which set of products to recommend when customers arrive at an unavailable product, and (ii) we only allow a single transition of the customers.…”

The First Half-Century of Decision Modelling for Substitutable Products: A Literature Review and Bibliographic Analysis

“…While some papers have assumed the location choice models in which the customers substitute the demand between the neighbouring variants (Li, 2007), many of the papers have assumed multinomial choice logit models (Suh & Aydin, 2011;Aouad et al 2018), with Hopp & Xu (2005) considering the Bayesian Logit Model. Many of the research articles have taken the decision choice to be a Markov chains phenomenon (Bayinder et al 2005;Yu et al 2017;Desir et al 2020). While Lin & Sibdari (2009) deployed discrete choice model, Etebari (2020) used the nested logit model to capture the customer's choice process rather than the multinomial logit model considering that the latter one suffers from the independence of irrelevant alternatives limitation.…”

Bounds, Heuristics, and Prophet Inequalities for Assortment Optimization

“…In addition, it has also been studied in different settings such as where the firm faces cardinality limitations on the offer sets and similar constraints (see e.g. Rusmevichientong et al [2010], Désir et al [2020], Sumida et al [2021) and in settings where inventory is limited (see e.g. Topaloglu [2009]).…”