Assortment Optimization Under the Multivariate MNL Model

TL;DR

This paper investigates assortment optimization under a multivariate MNL model, proving NP-hardness, developing approximation algorithms with ratios close to theoretical limits, and demonstrating high practical effectiveness through numerical experiments.

Contribution

It introduces approximation algorithms for the uncapacitated case with near-optimal ratios and establishes hardness results for the capacitated case under the multivariate MNL model.

Findings

Adjusted-revenue-ordered assortment achieves 1/2-approximation.

LP-based framework yields a 0.74-approximation, close to the 0.75 integrality gap.

Numerical experiments show over 99% average approximation ratio.

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%.