Assortment Optimization Under the Multivariate MNL Model

41 Pages Posted: 15 Oct 2022

See all articles by Xin Chen

Xin Chen

Georgia Institute of Technology Atlanta, USA

Jiachun Li

Tsinghua University

Menglong Li

City University of Hong Kong (CityU) - Department of Management Sciences

Tiancheng Zhao

University of Illinois at Urbana-Champaign

Yuan Zhou

Tsinghua University - Yau Mathematical Sciences Center

Date Written: September 30, 2022

Abstract

We study an assortment optimization problem under a multi-purchase choice model in which customers choose a bundle of up to one product from each of two product categories. Different bundles have different utilities and the bundle price is the summation of the prices of products in it. For the uncapacitated setting where any set of products can be offered, we prove that this problem is strongly NP-hard. We show that an adjusted-revenue-ordered assortment provides a 1/2-approximation. Furthermore, we develop an approximation framework based on a linear programming relaxation of the problem and obtain a 0.74-approximation algorithm. This approximation ratio almost matches the integrality gap of the linear program, which is proven to be at most 0.75. For the capacitated setting, we prove that there does not exist a constant-factor approximation algorithm assuming the Exponential Time Hypothesis. The same hardness result holds for settings with general bundle prices or more than two categories. Finally, we conduct numerical experiments on randomly generated problem instances. The average approximation ratios of our algorithms are over 99%.

Keywords: Multiple categories, assortment optimization, approximation algorithms

Suggested Citation

Chen, Xin and Li, Jiachun and Li, Menglong and Zhao, Tiancheng and Zhou, Yuan, Assortment Optimization Under the Multivariate MNL Model (September 30, 2022). Available at SSRN: https://ssrn.com/abstract=4233712 or http://dx.doi.org/10.2139/ssrn.4233712

Xin Chen

Georgia Institute of Technology Atlanta, USA ( email )

Jiachun Li

Tsinghua University

Menglong Li

City University of Hong Kong (CityU) - Department of Management Sciences ( email )

Tat Chee Avenue
Kowloon Tong
Kowloon
Hong Kong
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HOME PAGE: http://menglongli.com

Tiancheng Zhao (Contact Author)

University of Illinois at Urbana-Champaign ( email )

601 E John St
Champaign, IL 61820
United States

Yuan Zhou

Tsinghua University - Yau Mathematical Sciences Center ( email )

Beijing
China

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