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Capacity Constrained Assortment Optimization Under the Markov Chain Based Choice Model

33 Pages Posted: 4 Jul 2015 Last revised: 22 Oct 2015

Antoine Désir

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Vineet Goyal

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Danny Segev

University of Haifa - Department of Statistics

Chun Ye

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Date Written: July 3, 2015

Abstract

Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. In such settings, the goal is to select a subset of items to offer from a universe of substitutable items in order to maximize expected revenue when consumers exhibit a random substitution behavior. We consider a capacity constrained assortment optimization problem under the Markov Chain based choice model, recently considered by Blanchet et al. (2013). In this model, the substitution behavior of customers is modeled through transitions in a Markov chain. Capacity constraints arise naturally in many applications to model real-life constraints such as shelf space or budget limitations. We show that the capacity constrained problem is APX-hard even for the special case when all items have unit weights and uniform prices, i.e., it is NP-hard to obtain an approximation ratio better than some given constant. We present constant factor approximations for both the cardinality and capacity constrained assortment optimization problem for the general Markov chain model. Our algorithm is based on a "local-ratio" paradigm that allows us to transform a non-linear revenue function into a linear function. The local-ratio based algorithmic paradigm also provides interesting insights towards the optimal stopping problem as well as other assortment optimization problems.

Keywords: Assortment optimization, choice models, approximation algorithms, Markov chain

Suggested Citation

Désir, Antoine and Goyal, Vineet and Segev, Danny and Ye, Chun, Capacity Constrained Assortment Optimization Under the Markov Chain Based Choice Model (July 3, 2015). Operations Research, Forthcoming. Available at SSRN: https://ssrn.com/abstract=2626484 or http://dx.doi.org/10.2139/ssrn.2626484

Antoine Désir

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

Vineet Goyal

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

Danny Segev (Contact Author)

University of Haifa - Department of Statistics ( email )

Haifa 31905
Israel

Chun Ye

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

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