Assortment Optimization under the Multi-Purchase Multinomial Logit Choice Model

55 Pages Posted: 16 Jun 2021

See all articles by Jacob Feldman

Jacob Feldman

Washington University in St. Louis - John M. Olin Business School

Danny Segev

Tel Aviv University - School of Mathematical Sciences

Huseyin Topaloglu

Cornell University - School of Operations Research and Industrial Engineering

Laura Wagner

Catholic University of Portugal (UCP) - Catolica Lisbon School of Business and Economics

Yicheng Bai

Cornell University - School of Operations Research and Information Engineering

Date Written: June 14, 2021

Abstract

In this paper, we introduce the Multi-Purchase Multinomial Logit choice model, which extends the random utility maximization framework of the classical Multinomial Logit model to a multiple-purchase setting. In this model, customers sample random utilities for each offered product as in the Multinomial Logit model. However, rather than focusing on a single product, they concurrently sample a ``budget'' parameter M, which indicates the maximum number of products that the customer is willing to purchase. Subsequently, the M highest utility products are purchased, out of those whose utilities exceed that of the no-purchase option. When fewer than M products satisfy the latter condition, only these products will be purchased.

First and foremost, we propose the first multi-purchase choice model that can be fully operationalized. Specifically, we first provide a recursive procedure to compute the choice probabilities in this model, which in turn provides a framework to study its resulting assortment problem, where the goal is to select a subset of products to make available for purchase so as to maximize expected revenue. Our main algorithmic results consist of two distinct polynomial time approximations schemes (PTAS); the first, and simpler of the two, caters to a setting where each customer may buy only a constant number of products, whereas the second more nuanced algorithm applies to our multi-purchase model in its general form. Additionally, we study the revenue-potential of making assortment decisions that account for multi-purchase behavior in comparison to those that overlook this phenomenon. In particular, we relate both the structure and revenue performance of the optimal assortment under a traditional single-purchase model to that of the optimal assortment in the multi-purchase setting. Finally, we complement our theoretical work with an extensive set of computational experiments, where the efficacy of our proposed PTAS is tested against natural heuristics. Ultimately, we find that our approximation scheme outperforms these approaches by 1-5% on average.

Keywords: Consumer choice, multiple simultaneous purchases, assortment optimization, PTAS

JEL Classification: C61

Suggested Citation

Feldman, Jacob and Segev, Danny and Topaloglu, Huseyin and Wagner, Laura and Bai, Yicheng, Assortment Optimization under the Multi-Purchase Multinomial Logit Choice Model (June 14, 2021). Available at SSRN: https://ssrn.com/abstract=3866734 or http://dx.doi.org/10.2139/ssrn.3866734

Jacob Feldman (Contact Author)

Washington University in St. Louis - John M. Olin Business School ( email )

One Brookings Drive
Campus Box 1133
St. Louis, MO 63130-4899
United States

Danny Segev

Tel Aviv University - School of Mathematical Sciences ( email )

Tel Aviv 69978
Israel

Huseyin Topaloglu

Cornell University - School of Operations Research and Industrial Engineering ( email )

Ithaca, NY
United States

Laura Wagner

Catholic University of Portugal (UCP) - Catolica Lisbon School of Business and Economics ( email )

Palma de Cima
Lisbon, 1649-023
Portugal

Yicheng Bai

Cornell University - School of Operations Research and Information Engineering ( email )

Ithaca, NY 14853
United States

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