Modern column generation for estimating single- and multi-purchase ranked list choice models

2026-05-07

AI Summary

Estimating nonparametric ranking-based discrete choice models from transaction data—accounting for both single- and multiple-purchase behaviors—requires handling an exponential number of consumer types, leading to prohibitive computational complexity. This work proposes the first dynamic programming–based column generation framework that efficiently enumerates relevant consumer types in such models. The approach features a novel subproblem that generalizes the linear ordering problem and incorporates acceleration techniques to enhance optimization efficiency. The method accommodates various model extensions and demonstrates substantial improvements over existing approaches on both synthetic and real-world datasets, achieving significantly faster computation while maintaining high estimation accuracy. Furthermore, it exhibits strong performance in downstream assortment optimization tasks.

Abstract

This paper studies the estimation of ranked-list discrete choice models with single and multiple purchases. In this setting, each consumer type is characterized by a ranking over a subset of products and a desired number of purchases, and the estimation task is to identify the set of consumer types and their probabilities that best explain the observed transactional data. This problem is computationally challenging due to the exponential number of possible consumer types and becomes more difficult when multiple purchases are allowed. We propose a column generation framework for this problem. Our main contribution is a dynamic programming algorithm for the column generation subproblem. This subproblem generalizes the linear ordering problem and incorporates acceleration techniques to improve computational efficiency. To the best of our knowledge, this is the first dynamic programming-based approach for generating consumer types in non-parametric models. The proposed framework supports multiple model variants with minor modifications. Computational experiments on synthetic and real data show substantial speedups over existing methods while maintaining high solution quality, and demonstrate effectiveness in both estimation and assortment optimization.

Problem

• ranked-list choice models
• discrete choice
• consumer types
• multiple purchases
• non-parametric estimation

Innovation

• column generation
• dynamic programming
• ranked-list choice models
• non-parametric estimation
• assortment optimization