Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps

63 Pages Posted: 21 Sep 2023

See all articles by Omar El Housni

Omar El Housni

Cornell University - School of Operations Research and Information Engineering

Marouane Ibn Brahim

Cornell University - School of Operations Research and Information Engineering

Danny Segev

Tel Aviv University - School of Mathematical Sciences

Date Written: September 4, 2023

Abstract

Motivated by modern-day applications such as Attended Home Delivery and Preference-based Group Scheduling, where decision makers wish to steer a large number of customers toward choosing the exact same alternative, we introduce a novel class of assortment optimization problems, referred to as Maximum Load Assortment Optimization. In such settings, given a universe of substitutable products, we are facing a stream of customers, each choosing between either selecting a product out of an offered assortment or opting to leave without making a selection. Assuming that these decisions are governed by the Multinomial Logit choice model, we define the random load of any underlying product as the total number of customers who select it. Our objective is to offer an assortment of products to each customer so that the expected maximum load across all products is maximized.
We consider both static and dynamic formulations of the maximum load assortment optimization problem. In the static setting, a single offer set is carried throughout the entire process of customer arrivals, whereas in the dynamic setting, the decision maker offers a personalized assortment to each customer, based on the entire information available at that time. As can only be expected, both formulations present a wide range of computational challenges and analytical questions. The main contribution of this paper resides in proposing efficient algorithmic approaches for computing near-optimal static and dynamic assortment policies. In particular, we develop a polynomial-time approximation scheme (PTAS) for the static problem formulation. Additionally, we demonstrate that an elegant policy utilizing weight-ordered assortments yields a 1/2- approximation. Concurrently, we prove that such policies are sufficiently strong to provide a 1/4-approximation with respect to the dynamic formulation, establishing a constant-factor bound on its adaptivity gap. Finally, we design an adaptive policy whose expected maximum load is within factor 1-\eps of optimal, admitting a quasi-polynomial time implementation.

Keywords: Assortment Optimization, Maximum Load, Approximation Schemes, Adaptivity Gap, Balls and Bins, Multinomial Logit model.

Suggested Citation

El Housni, Omar and Ibn Brahim, Marouane and Segev, Danny, Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps (September 4, 2023). Available at SSRN: https://ssrn.com/abstract=4560530 or http://dx.doi.org/10.2139/ssrn.4560530

Omar El Housni (Contact Author)

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

2 E Loop Rd
New York, NY 10044
United States

HOME PAGE: http://https://people.orie.cornell.edu/oe46/

Marouane Ibn Brahim

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

2 West Loop Rd.
New York, NY 10044
United States

Danny Segev

Tel Aviv University - School of Mathematical Sciences ( email )

Tel Aviv 69978
Israel

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