Loading constraints for a multi-compartment vehicle routing problem

Multi-compartment vehicles (MCVs) can deliver several product segments jointly. Separate compartments are necessary as each product segment has its own specific characteristics and segments cannot be mixed during transportation. The size and position of the compartments can be adjusted for each tour with the use of flexible compartments. However, this requires that the compartments can be accessed for loading/unloading. The layout of the compartments is defined by the customer and segment sequence, and it needs to be organized in a way that no blocking occurs during loading/unloading processes. Routing and loading layouts are interdependent for MCVs. This paper addresses such loading/unloading issues raised in the distribution planning when using MCVs with flexible compartments, loading from the rear, and standardized transportation units. The problem can therefore be described as a two-dimensional loading and multi-compartment vehicle routing problem (2L-MCVRP). We address the problem of obtaining feasible MCV loading with minimal routing, loading and unloading costs. We define the loading problem that configures the compartment setup. Consequently, we develop a branch-and-cut (B&C) algorithm as an exact approach and extend a large neighborhood search (LNS) as a heuristic approach. In both cases, we use the loading model in order to verify the feasibility of the tours and to assess the problem as a routing and loading problem. The loading model dictates the cuts to be performed in the B&C, and it is used as a repair mechanism in the LNS. Numerical studies show that the heuristic reaches the optimal solution for small instances and can be applied efficiently to larger problems. Additionally, further tests on large instances enable us to derive general rules regarding the influence of loading constraints. Our results were validated in a case study with a European retailer. We identified that loading constraints matter even for small instances. Feasible loading can often be achieved only through minor changes to the routing solution and therefore with limited additional costs. Further, the importance to integrate loading constraints grows as the problem size increases, especially when a heterogeneous mix of segments is ordered.