State of the art of vehicle routing problems with loading constraints

The most studied combinatorial optimization problem in transport and logistics is the vehicle routing problem (VRP). The goal of the vehicle routing problem is to nd a set of routes for a eet of vehicles which optimises the objective function (e.g. total distance, routing costs). Every demand needs to be ful lled and vehicle capacities need to be respected. Many solution methods exist for the 'classic' VRP with aforementioned constraints. In real-life however, companies are faced with several other types of constraints that greatly increase the complexity of the problem. Examples of such complicating constraints are time windows, maximum route length and duration, incompatibilities between goods and vehicles and loading constraints. 'Rich' VRPs are routing problem that take into account some of these additional realistic constraints. This paper deals with the results of an extensive literature study about VRP with loading constraints, which is a fairly recent domain of research. Besides an overview of topics that have already been studied in this eld, research gaps and possibilities for future research are discussed. Loading problems arise when goods cannot be placed freely in the container or truck because several constraints have to be taken into account such as the measurements of the goods, the fragility of the goods, weight distribution inside the vehicle and the sequence of delivery. Carriers often employ a Last-In-FirstOut loading or sequence-based loading strategy which ensures that when arriving at a customer, the items of the customer can be removed from the truck without having to move items of other clients. Loading constraints have been studied in combination with the capacitated vehicle routing problem (CVRP), as well as with the Pickup and Delivery Problem (PDP) [3]. In the CVRP, the capacity of the vehicle presents upper limit on the total weight or on the total number of items to be placed in the vehicle. In practice however, the capacity of the vehicle is not only determined by its weight or by the number of items it can contain, but also by its measurements and the fragility and stability of the goods to be placed in the container. Gendreau et al. (2006) [1] are the rst to address the three-dimensional capacitated vehicle routing problem (3L-CVRP). They develop a Tabu Search method to solve the problem which takes into account sequence-based loading,