Heuristics for Warehouse Assortment Selection Using Birkhoff von Neumann Decomposition

The rapid growth of e-commerce has significantly expanded product variety, intensifying the challenge of efficient order fulfillment. To enable fast delivery, e-retailers strategically position front-end warehouses near customers. However, these warehouses operate under strict capacity constraints, making assortment selection a critical decision in minimizing fulfillment costs. Ideally, each customer order can be fulfilled entirely from a single front-end warehouse. Otherwise, reliance on back-end support increases operational costs, carbon emissions, and delivery lead times, ultimately degrading service quality. In this work, we propose a scalable and flexible heuristic for warehouse assortment selection in both single- and multi-warehouse settings. The heuristic accommodates diverse warehouse- and order-specific cost structures while providing rigorous performance guarantees. Specifically, we establish that its fulfillment cost remains within a constant factor of the optimal solution, where the approximation ratio depends on the largest order size when fulfillment costs are uniform, and on cost disparities across warehouses when fulfillment costs vary. Extensive numerical experiments further demonstrate that the proposed heuristic consistently outperforms existing methods across both single- and multi-warehouse settings.