A case study of joint online truck scheduling and inventory management for multiple warehouses

For a real-world problem – transporting pallets between warehouses to guarantee sufficient supply for known and additional stochastic demand – we propose a solution approach via convex relaxation of an integer programming formulation, suitable for online optimization. The essential new element linking routing and inventory management is a convex piecewise-linear cost function that is based on minimizing the expected number of pallets that still need transportation. For speed, the convex relaxation is solved approximately by a bundle approach yielding an online schedule in five to 12 minutes for up to three warehouses and 40,000 articles; in contrast, computation times of state-of-the-art LP solvers are prohibitive for online application. In extensive numerical experiments on a real-world data stream, the approximate solutions exhibit negligible loss in quality; in long-term simulations the proposed method reduces the average number of pallets needing transportation due to short-term demand to less than half the number observed in the data stream.