Assortment Optimization over Dense Universe is Easy

36 Pages Posted: 11 Aug 2020

See all articles by Kumar Goutam

Kumar Goutam

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

Vineet Goyal

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

Henry Lam

Columbia University

Date Written: June 20, 2020

Abstract

Assortment optimization is an important problem in revenue management arising in industries such as online advertising, retailing and airline ticketing. We study assortment optimization under an arbitrary mixture of multi-nomial logit (MNL) models, when the universe of products is dense. In general, it is NP-hard to approximate assortment optimization under mixture of MNLs model within any reasonable factor. However, if the universe of products is dense, we show that assortment optimization has an optimal solution that is well approximated by a nested-by-revenue policy. We argue this by considering an alternate, continuous-space model where product features follow a density on a continuous space, and offer set can be any open subset on the feature space. In this model, we show that there exists a nested-by-revenue policy that is optimal. This characterization holds regardless of the mixture distribution (including discrete and continuous) and the dimension of feature space.

Through this characterization and linking the continuous-space to the standard discrete-item model via a "fractional" relaxation, we also show that, in the finite and discrete item case, nested-by-revenue performs nearly optimally. In particular, we derive an optimality gap, explicitly dependent on the model parameters, that shrinks to zero under suitable scaling as the number of items grows.

Keywords: Assortment Optimization, Mixed MNL, Dense Universe

JEL Classification: C02, C44, C60, C61

Suggested Citation

Goutam, Kumar and Goyal, Vineet and Lam, Henry, Assortment Optimization over Dense Universe is Easy (June 20, 2020). Available at SSRN: https://ssrn.com/abstract=3649233 or http://dx.doi.org/10.2139/ssrn.3649233

Kumar Goutam (Contact Author)

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

Henry Lam

Columbia University ( email )

3022 Broadway
New York, NY 10027
United States

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