Capacitated Assortment Optimization: Hardness and Approximation

30 Pages Posted: 29 Dec 2014 Last revised: 29 May 2020

See all articles by Antoine Désir

Antoine Désir

INSEAD

Vineet Goyal

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

Jiawei Zhang

New York University (NYU) - Department of Information, Operations, and Management Sciences

Date Written: March 31, 2020

Abstract

Assortment optimization is an important problem that arises in many practical applications such as retailing and online advertising. In this problem, the goal is to select a subset of items that maximizes the expected revenue in the presence of (1) the substitution behavior of consumers specified by a choice model, and (2) a potential capacity constraint bounding the total weight of items in the assortment. The latter is a natural constraint arising in many applications. We begin by showing how challenging these two aspects are from an optimization perspective. First, we show that adding a general capacity constraint makes the problem NP-hard even for the simplest choice model, namely the multinomial logit model. Second, we show that even the unconstrained assortment optimization for the mixture of multinomial logit model is hard to approximate within any reasonable factor when the number of mixtures is not constant.

In view of these hardness results, we present near-optimal algorithms for the capacity constrained assort- ment optimization problem under a large class of parametric choice models including the mixture of multinomial logit, Markov chain, nested logit and d-level nested logit choice models. In fact, we develop near-optimal algorithms for a general class of capacity constrained optimization problems whose objective function depends on a small number of linear functions. For the mixture of multinomial logit model (resp. Markov chain model), the running time of our algorithm depends exponentially on the number of segments (resp. rank of the transition matrix). Therefore, we get efficient algorithms only for the case of constant number of segments (resp. constant rank). However, in light of our hardness result, any near-optimal algorithm will have a super polynomial dependence on the number of mixtures for the mixture of multinomial logit choice model

Keywords: Assortment Optimization, FPTAS

Suggested Citation

Désir, Antoine and Goyal, Vineet and Zhang, Jiawei, Capacitated Assortment Optimization: Hardness and Approximation (March 31, 2020). Available at SSRN: https://ssrn.com/abstract=2543309 or http://dx.doi.org/10.2139/ssrn.2543309

Antoine Désir (Contact Author)

INSEAD ( email )

Boulevard de Constance
77305 Fontainebleau Cedex
France

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

Jiawei Zhang

New York University (NYU) - Department of Information, Operations, and Management Sciences ( email )

44 West Fourth Street
New York, NY 10012
United States

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