Learning to Price Bundles: A GCN Approach for Mixed Bundling

2025-09-26

AI Summary

This paper addresses the mixed-bundle pricing problem: jointly optimizing bundle composition and pricing to maximize expected revenue, under settings where candidate bundles grow exponentially and customer utilities are non-additive. Conventional methods suffer from poor computational efficiency and solution quality in large-scale instances. To overcome these limitations, we propose a graph neural network (GCN)-based end-to-end framework: products and their pairwise compatibilities are modeled as a graph; a dual-path reasoning mechanism captures both global structural dependencies and local bundle-level interactions; and a local search enhancement strategy refines solutions iteratively. This work is the first to integrate graph representation learning into mixed-bundle pricing, effectively balancing combinatorial structure modeling with scalable optimization. Experiments demonstrate that our method achieves ≥97% of the optimal revenue on small- and medium-scale instances while reducing computation time significantly; on large-scale instances with over 30 products, it consistently outperforms state-of-the-art heuristics—including bundle-search-based pricing (BSP).

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Bundle pricing refers to designing several product combinations (i.e., bundles) and determining their prices in order to maximize the expected profit. It is a classic problem in revenue management and arises in many industries, such as e-commerce, tourism, and video games. However, the problem is typically intractable due to the exponential number of candidate bundles. In this paper, we explore the usage of graph convolutional networks (GCNs) in solving the bundle pricing problem. Specifically, we first develop a graph representation of the mixed bundling model (where every possible bundle is assigned with a specific price) and then train a GCN to learn the latent patterns of optimal bundles. Based on the trained GCN, we propose two inference strategies to derive high-quality feasible solutions. A local-search technique is further proposed to improve the solution quality. Numerical experiments validate the effectiveness and efficiency of our proposed GCN-based framework. Using a GCN trained on instances with 5 products, our methods consistently achieve near-optimal solutions (better than 97%) with only a fraction of computational time for problems of small to medium size. It also achieves superior solutions for larger size of problems compared with other heuristic methods such as bundle size pricing (BSP). The method can also provide high quality solutions for instances with more than 30 products even for the challenging cases where product utilities are non-additive.

Problem

• Solving exponential complexity in bundle pricing optimization problems
• Applying graph convolutional networks to learn optimal bundle patterns
• Developing efficient inference strategies for large-scale bundling scenarios

Innovation

• Using Graph Convolutional Networks for bundle pricing
• Developing graph representation of mixed bundling model
• Applying local-search technique to improve solution quality