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Learning to Price Bundles: A GCN Approach for Mixed Bundling

Liangyu Ding, Chenghan Wu, Guokai Li, Zizhuo Wang

TL;DR

This work tackles the bundle pricing problem under non-additive utilities, where the naive search over all $2^n$ bundles is intractable. It introduces a graph convolutional network to learn segment–product affinities, producing a probability matrix that prunes the exponential bundle space before solving the Hansen 1990 mixed-bundling MILP. Two pruning strategies, Fixed Cutoff Pruning and Progressive Cutoff Pruning, plus a local-search refinement, yield near-optimal revenues on small-to-medium problems and scalable performance on large-scale instances. The proposed framework preserves the modeling rigor of the MB formulation while achieving practical efficiency and robustness, demonstrated on synthetic data with non-additive utilities and supported by reproducibility materials and ethical considerations.

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.

Learning to Price Bundles: A GCN Approach for Mixed Bundling

TL;DR

This work tackles the bundle pricing problem under non-additive utilities, where the naive search over all bundles is intractable. It introduces a graph convolutional network to learn segment–product affinities, producing a probability matrix that prunes the exponential bundle space before solving the Hansen 1990 mixed-bundling MILP. Two pruning strategies, Fixed Cutoff Pruning and Progressive Cutoff Pruning, plus a local-search refinement, yield near-optimal revenues on small-to-medium problems and scalable performance on large-scale instances. The proposed framework preserves the modeling rigor of the MB formulation while achieving practical efficiency and robustness, demonstrated on synthetic data with non-additive utilities and supported by reproducibility materials and ethical considerations.

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.

Paper Structure

This paper contains 30 sections, 3 theorems, 17 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Definition and Baselines
  4. Mixed Bundling Formulation hanson1990optimal
  5. GCN-Based Strategies
  6. Graph Representation
  7. Graph Representation
  8. GCN Structure
  9. Pruning-based strategies
  10. Fixed Cutoff Pruning (FCP).
  11. Progressive Cutoff Pruning (PCP).
  12. Local Search Strategy
  13. Numerical Experiments
  14. Numerical Results
  15. Mixed Bundling (MB) baseline:
  16. ...and 15 more sections

Key Result

Proposition 1

Statement $S_{P,2}$ and Statement $S_{P,K}$ are equivalent.

Figures (2)

  • Figure 1: Graph representation of the bundle pricing problem with customer and product nodes.
  • Figure 2: Illustration of GCN network.

Theorems & Definitions (10)

  • Definition 1: K-way Cover sub-additivity --- Statement $S_{C,K}$
  • Definition 2: 2-way Partition sub-additivity --- Statement $S_{P,2}$
  • Definition 3: Price Monotonicity --- Statement $\mathcal{M}$
  • Definition 4: K-way Partition sub-additivity --- Statement $S_{P,K}$
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof