Assortment Optimization with Multi-Item Basket Purchase under the Multivariate MNL Model

48 Pages Posted: 16 Apr 2021 Last revised: 22 Sep 2022

See all articles by Chengyi Lyu

Chengyi Lyu

University of Colorado at Boulder - Leeds School of Business

Stefanus Jasin

University of Michigan, Stephen M. Ross School of Business

Sajjad Najafi

HEC Paris

Huanan Zhang

University of Colorado at Boulder - Leeds School of Business

Date Written: September 21, 2022

Abstract

Problem definition: Assortment selection is one of the most important decisions faced by retailers. Most existing papers in the literature assume that customers select at most one item out of the offered assortment. While this is valid in some cases, it contradicts practical observations in many shopping experiences, both in online and brick-and-mortar retail, where customers may buy a basket of products instead of a single item. In this paper we incorporate customer's multi-item purchase behavior into the assortment optimization problem. We consider both the uncapacitated and capacitated assortment problems under the so-called Multivariate MNL (MVMNL) model, which is one of the most popular multivariate choice models used in the marketing and empirical literature.
Methodology/results: We first show that the traditional revenue-ordered assortment may not be optimal. Nonetheless, we show that under some mild conditions, a certain variant of this property holds (in the uncapacitated assortment problem) under the MVMNL model---that is, the optimal assortment consists of revenue-ordered local assortments in each group. Finding the optimal assortment is still computationally expensive as the revenue thresholds for different groups cannot be computed separately. We show that the optimization problem under MVMNL is NP-hard even in the setting where there is no interaction among the product categories. Motivated by this result, we develop FPTAS for several variants of (capacitated and uncapacitated) assortment problems under MVMNL.
Managerial implications: Our analysis reveals that disregarding customer's multi-item purchase behavior in assortment decision can indeed have a significant negative impact on profitability, demonstrating its practical importance in retail. We numerically show that our proposed algorithm can improve a retailer's expected total revenues (compared to a benchmark policy that does not properly take into account the impact of customer's multi-item choice behavior in assortment decision) by up to 14%.

Suggested Citation

Lyu, Chengyi and Jasin, Stefanus and Najafi, Sajjad and Zhang, Huanan, Assortment Optimization with Multi-Item Basket Purchase under the Multivariate MNL Model (September 21, 2022). Available at SSRN: https://ssrn.com/abstract=3818886 or http://dx.doi.org/10.2139/ssrn.3818886

Chengyi Lyu

University of Colorado at Boulder - Leeds School of Business ( email )

CO 80309
United States

Stefanus Jasin (Contact Author)

University of Michigan, Stephen M. Ross School of Business ( email )

701 Tappan Street
Ann Arbor, MI MI 48109
United States

Sajjad Najafi

HEC Paris ( email )

1, Rue de la Liberation
Jouy en Josas, 78351
France

Huanan Zhang

University of Colorado at Boulder - Leeds School of Business ( email )

Boulder, CO 80309-0419
United States

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