Solving the multi-block order batching problem with branch-price-and-cut

In the realm of warehouse optimization, the order batching problem (OBP) involves partitioning customer orders into capacity-constrained batches, such that the total distance traveled for picking all batches is minimized. The total distance is determined by the picker routing strategy used in the warehouse. This paper addresses the OBP in rectangular warehouses consisting of two or more blocks with parallel aisles, expanding upon a branch-price-and-cut (BPC) approach previously applied to the OBP in single-block warehouses. The main extension lies in accommodating multi-block warehouse layouts, integrating both optimal and heuristic picker routing strategies. A key contribution of this work is the analysis of the monotonicity properties of routing strategies in multi-block configurations, which are crucial for the effective application of the BPC method. Computational results on publicly available benchmark instances with two-block layouts demonstrate that both the exact BPC and BPC-based heuristics outperform current state-of-the-art methods. Specifically, the joint order batching and picker routing problem, representing the OBP with optimal routing, is solved over three orders of magnitude faster than leading exact methods. Instances with up to 80 orders are solved to proven optimality for five different routing strategies, while BPC-based heuristics achieve superior performance on instances involving up to 500 orders.