Assortment Optimization with Visibility Constraints

TL;DR

This work studies assortment optimization under exogenous visibility constraints (APV) within the Multinomial Logit framework, where a fixed sequence of $T$ customers is offered assortments and each product must be shown to at least $\ell_i$ customers. The authors introduce the expanded revenue function $\overline{R}(A)$ and the associated expanded set $\overline{A}$, proving monotonicity and supermodularity, which enable a simple nested-structure solution and a linear-time algorithm with a compact LP formulation, attaining ${\sf OPT}^{LP}={\sf OPT}^{APV}$. They extend to cardinality-constrained APV (APVC), proving strong NP-hardness (even with equal prices) and providing a PTAS for equal prices via weight discretization, linearization, and a dependent-rounding scheme; feasibility and near-optimality guarantees are established for the randomized rounding approach. Finally, the paper quantifies the price of visibility, proposing a fair, tractable allocation of the induced revenue loss among vendors through a loss-based fee $\Gamma_i$, and discusses robustness considerations under estimation noise. Overall, the results yield a deterministic, scalable framework for SLA-driven visibility in e-commerce/ad platforms, with exact and approximation algorithms, LP formulations, and revenue-sharing mechanisms that support practical deployment.

Abstract

Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, that we refer to as APV. Here, we are given a universe of substitutable products and a stream of customers. The objective is to determine the optimal assortment of products to offer to each customer in order to maximize the total expected revenue, subject to exogenously-given visibility constraints, stating that each product should be shown to a minimum number of customers. We assume that customer choices follow a Multinomial Logit model (MNL). We provide a structural characterization of optimal assortments and present a linear time algorithm for solving APV. To this end, we introduce a novel function called the ``expanded revenue" of an assortment and establish its supermodularity; our algorithm takes advantage of this structural property. Additionally, we prove that APV can be formulated as a compact linear program. Next, we consider APV with cardinality constraints, which limit the maximum number of products that can be included in an assortment. We prove this problem to be strongly NP-hard and not admitting a Fully Polynomial Time Approximation Scheme (FPTAS), even when all products have identical prices. Subsequently, we devise a Polynomial Time Approximation Scheme (PTAS) for APV under cardinality constraints with identical prices. We also examine the revenue loss resulting from the enforcement of visibility constraints, comparing it to the unconstrained problem. To offset this loss, we propose a novel strategy to distribute the loss incurred among the products subject to visibility constraints, charging each vendor an amount proportional to their product's contribution to the revenue loss.

Theorems & Definitions (29)

• Definition 3.1: Expanded revenue
• Definition 3.2: Expanded set
• Lemma 3.3
• proof
• Lemma 3.4: Monotonicity
• proof
• Lemma 3.5: Supermodularity
• proof
• Claim 3.6
• Theorem 3.7