A Stochastic Benders Decomposition Scheme for Large-Scale Stochastic Network Design

Abstract

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a stochastic version where operational costs are uncertain because of fluctuating demand and estimated as a sample average from historical data. This problem is computationally challenging, and instances with as few as 100 nodes often cannot be solved to optimality using current decomposition techniques. We propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless, generates deterministically valid cuts, via a dual averaging technique, rather than the probabilistically valid cuts frequently proposed in the stochastic optimization literature. We implement both single-cut and multicut variants of this Benders decomposition as well as a variant that uses clustering of the historical scenarios. To our knowledge, this is the first single-tree implementation of Benders decomposition that facilitates sampling. On instances with 100–200 nodes and relatively complete recourse, our algorithm achieves 5%–7% optimality gaps compared with 16%–27% for deterministic Benders schemes, and it scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: The work of R. Cory-Wright was supported in part by the MIT-IBM Research Lab for Goldstine postdoctoral fellowship. J. Pauphilet was funded by the Research and Materials Development Fund [RAMD_Pauphilet_J_22/23_8789] at London Business School. 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.2023.0074 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0074 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .