On MIP Formulations for Logit-Based Multi-Purchase Choice Models and Applications

Abstract

We study logit-based multi-purchase choice models and develop an exact solution methodology for the resulting assortment optimization problems, which we show are NP-hard to approximate. We introduce a hypergraph representation that captures general bundle-based choice structures and subsumes several models in the literature, including the BundleMVL-K and multivariate MNL models (Tulabandhula et al. 2023, Jasin et al. 2024). Leveraging this representation, we derive mixed-integer programming (MIP) formulations by integrating polyhedral relaxations from multilinear optimization with a perspective reformulation of the logit choice model. Our approach preserves the strength of the underlying polyhedral relaxations, yielding formulations with provably tighter linear programming (LP) bounds than the prevalent Big-M approach. We further characterize structural conditions on the hypergraph under which the formulations are locally sharp, thereby generalizing existing LP characterizations for path-based models. The framework extends naturally to heterogeneous and robust settings. Computational experiments demonstrate that the proposed formulations significantly improve both solution quality and scalability.