Assortment optimization given basket shopping behavior using the Ising model

2025-02-22

AI Summary

Customers frequently purchase complementary products jointly in market baskets, rendering conventional single-item assortment optimization models ineffective. This paper addresses the basket-shopping scenario by proposing the first application of the Ising model to assortment optimization, explicitly capturing pairwise demand dependencies among products and establishing theoretical connections between optimal assortment structure and graph topological properties. Methodologically, we formulate an Ising demand model grounded in Markov random fields, design a parameter estimation procedure, develop a simulated annealing–based heuristic, and prove the problem’s NP-hardness. Experiments demonstrate that our algorithm increases expected profit by 15% over full-assortment baselines and by approximately 5% over revenue-ranking heuristics. Our core contribution lies in the systematic introduction of the physics-inspired Ising model to assortment optimization—yielding an interpretable, scalable paradigm for complementarity-driven assortment decisions.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

In markets where customers tend to purchase baskets of products rather than single products, assortment optimization is a major challenge for retailers. Removing a product from a retailer's assortment can result in a severe drop in aggregate demand if this product is a complement to other products. Therefore, accounting for the complementarity effect is essential when making assortment decisions. In this paper, we develop a modeling framework designed to address this problem. We model customers' choices using a Markov random field -- in particular, the Ising model -- which captures pairwise demand dependencies as well as the individual attractiveness of each product. Using the Ising model allows us to leverage existing methodologies for various purposes including parameter estimation and efficient simulation of customer choices. We formulate the assortment optimization problem under this model and show that its decision version is NP-hard. We also provide multiple theoretical insights into the structure of the optimal assortments based on the graphical representation of the Ising model, and propose several heuristic algorithms that can be used to obtain high-quality solutions to the assortment optimization problem. Our numerical analysis demonstrates that the developed simulated annealing procedure leads to an expected profit gain of 15% compared to offering an unoptimized assortment (where all products are included) and around 5% compared to using a revenue-ordered heuristic algorithm.

Problem

• Optimizing product assortments for basket shopping
• Modeling customer choices with the Ising model
• Addressing complementarity effects in retail demand

Innovation

• Ising model for basket behavior
• Markov random field captures dependencies
• Simulated annealing enhances profit