Learning Capacity Constraints with Convex Neural Networks in Lot-Sizing Problems

This paper presents a novel approach for integrating convex neural networks into mixed-integer linear programs. We introduce a mixed-integer linear programming formulation that employs the Maxout activation function, and our experiments show that this network type outperforms rectified linear unit–based networks. Additionally, we propose a novel formulation for convex networks that does not require auxiliary variables. We show that such reformulation can be solved efficiently using Lagrangian relaxation and branch-and-cut approaches. These methods are generic enough to be readily adapted to a wide range of problems and significantly simplify the integration of convex networks into mathematical programs. We evaluate these approaches on multi-item capacitated lot-sizing problems in which capacity constraints are replaced by neural networks translated into linear programs. Numerical experiments involving lot-sizing problems suggest that our methodology outperforms approaches that integrate lot-sizing and scheduling. Our methodology yields better solutions than conventional approaches regardless of whether both problems are solved in an integrated or hierarchical fashion. In particular, the proposed Lagrangian relaxation algorithm returns high-quality feasible solutions quickly, and the branch-and-cut approach efficiently solves lot-sizing models with large convex neural networks, making these two approaches attractive for solving large-scale instances.

History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete.

Funding: The present work was conducted within the project ASSISTANT (https://assistant-project.eu/) funded by the European Commission [Grant 101000165], H2020–ICT-38-2020, artificial intelligence for manufacturing. The authors also thank the region Pays de la Loire for financial support.

Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.1033) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2024.1033). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/.