Constrained assortment optimization under the markov chain-based choice model

Antoine Désir, Vineet Goyal, Danny Segev, Chun Ye

Research output: Contribution to journalArticlepeer-review


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 capacityconstrained 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.

Original languageEnglish
Pages (from-to)698-721
Number of pages24
JournalManagement Science
Issue number2
StatePublished - 2020

Bibliographical note

Funding Information:
History: Accepted by Yinyu Ye, optimization. Funding: Funding is acknowledged from the National Science Foundation Division of Civil, Mechanical and Manufacturing Innovation [Grants 1351838 and 1636046], a Google Faculty Award, and the Israel Science Foundation [Grant 148/16]. Supplemental Material: The online appendix is available at

Publisher Copyright:
© 2019 INFORMS.


  • Approximation algorithms
  • Assortment optimization
  • Choice models
  • Markov chain

ASJC Scopus subject areas

  • Strategy and Management
  • Management Science and Operations Research


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