Maximum Load Assortment Optimization: Approximation Algorithms and Adaptivity Gaps

TL;DR

This work introduces Maximum Load Assortment Optimization (MLO) under the Multinomial Logit model, focusing on static and dynamic offer strategies to maximize the expected maximum product load. It develops a polynomial-time evaluation oracle and a PTAS for the static setting, plus a simple 1/2-approximation via weight-ordered assortments; it also proves a constant adaptivity gap (at most 4, and 2 for equal weights) between dynamic and static policies. For the dynamic problem, the authors design a quasi-polynomial time $(1-\varepsilon)$-approximate adaptive policy, along with insights on when adaptivity provides limited gains. Numerical studies illustrate how the optimal assortments shrink as the customer horizon grows and how adaptivity gains vary with problem parameters. Overall, the paper provides a unified framework with near-optimal guarantees for both static and dynamic maximum-load assortment problems, with clear implications for applications in time-slot management and preference-based grouping.

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. 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. 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 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.

Figures (3)

• Figure 1: Optimal assortment size behaviour with respect to $\boldsymbol T$.
• Figure 2: Optimal assortment size behaviour with respect to $\boldsymbol \mu$.
• Figure 3: Each of the bottom rectangles corresponds to a product $\boldsymbol j$ in $\boldsymbol U$. Its height corresponds to the load of this product $\boldsymbol j$, and its length corresponds to the associated interval $\boldsymbol{J_{\psi_t(j),t+1}}$. Similarly, each of the top rectangles corresponds to a component $\boldsymbol i$ in the Multinomial vector, its height is the number of trials assigned to this component $\boldsymbol i$, and its length corresponds to the associated interval $\boldsymbol{I_{\varphi_t(i),t+1}}$ . The left-to-right order of these rectangles is by decreasing order of height.

Theorems & Definitions (45)

• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Theorem 3.7
• Lemma 3.8
• Lemma 3.9