From Small to Large: A Graph Convolutional Network Approach for Solving Assortment Optimization Problems

TL;DR

This paper tackles the NP-hard constrained assortment optimization problem under MMNL by proposing a graph convolutional network (GCN) framework that learns to map problem parameters to likely optimal assortments. A graph representation captures customers, products, and linear constraints, enabling a GCN to output product inclusion probabilities, which drive three inference policies: GI, GILS, and GIP. The approach demonstrates strong generalization from small-scale training to large-scale instances (up to 2,000 products), achieving competitive ratios around 85-96% with milliseconds-to-seconds inference times, and outpacing traditional heuristics. The framework also extends to unknown-choice-model settings via a three-stage pipeline with choice-GCN and solution-GCN, yielding fast, scalable, and high-quality solutions. Overall, this work presents the first learning-to-optimize framework for constrained assortment problems and highlights the practical impact of data-driven, scalable decision-support in revenue management.

Abstract

Assortment optimization seeks to select a subset of substitutable products, subject to constraints, to maximize expected revenue. The problem is NP-hard due to its combinatorial and nonlinear nature and arises frequently in industries such as e-commerce, where platforms must solve thousands of such problems each minute. We propose a graph convolutional network (GCN) framework to efficiently solve constrained assortment optimization problems. Our approach constructs a graph representation of the problem, trains a GCN to learn the mapping from problem parameters to optimal assortments, and develops three inference policies based on the GCN's output. Owing to the GCN's ability to generalize across instance sizes, patterns learned from small-scale samples can be transferred to large-scale problems. Numerical experiments show that a GCN trained on instances with 20 products achieves over 85% of the optimal revenue on problems with up to 2,000 products within seconds, outperforming existing heuristics in both accuracy and efficiency. We further extend the framework to settings with an unknown choice model using transaction data and demonstrate similar performance and scalability.

Figures (13)

• Figure 1: Illustration of feed-forward neural network with one hidden layer; each blue line represents a learnable weight.
• Figure 2: Illustration of GCN computation in a toy example. Each node (A, B, C) has a feature vector $y$, and each edge (A--B, B--C) has a feature vector $z$. Blue boxes represent learnable weight matrices for nodes ($\boldsymbol{W}_n$), and yellow boxes represent learnable weight matrices for edges ($\boldsymbol{W}_e$). Boxes of the same color share the same weights, reflecting the weight-sharing mechanism that makes GCNs scalable. Each node updates its representation by aggregating messages from its neighbors through a parameter-free aggregation function.
• Figure 3: Graph representation of the constrained assortment problem under the MMNL choice model. Customer nodes (blue circles) are connected to product nodes (orange squares) through edges weighted by utilities $v_{kj}$, while product nodes are connected to constraint nodes (green triangles) through edges weighted by resource coefficients $A_{ij}$.
• Figure 4: Illustration of GCN network.
• Figure 5: Average competitive ratios of proposed policies as functions of training instance size $N_\text{train}$ when $K_\text{train}=K_\text{test}=10$, $M_\text{train}=M_\text{test}=10$, and $N_\text{test}=500$. The shaded areas represent the standard deviation.

Theorems & Definitions (3)

• Remark 1: Potential improvement of GCN framework
• Remark 2
• Remark 3