This paper presents a combinatorial problem called a pick-up routing problem with a three-dimensional (3D-PRP) loading constraint, clustered backhauls at the operational level, and train loading at the tactical level for an intermodal transportation network. A two-phase approach, called clustering first, packingrouting second, is proposed for use during the first stage. The clustering of backhauls is carried out using the k-means algorithm. A hybrid approach is provided, which combines the packing of orders by first solving a 3D loading problem for each cluster using machine learning with a best-fit-first strategy, with routing using a genetic algorithm. During the second stage, the train-loading problem is solved using a mixed integer programming approach to minimise the total costs by incorporating various cost types, in which detention and demurrage costs are taken into account. All solution approaches are computationally evaluated on real-world data provided by an international logistics firm and new randomly generated instances. Comparisons are carried out using both exact solution methods and heuristic approaches, and the proposed approach was shown to be more effective for real-world problems.

Introduction

In recent years, intermodal problems related to decisions such as transport mode selection, vehicle routing (VRP), load planning, and consolidation have gained substantial attention in the transportation sector. An intermodal network design that ensures good solutions to multiple decisions is an important challenge. In this context, most researchers have focused on vehicle-routing problems and its variants, which are practical issues in the area of intermodal transportation. 3D-PRP is a variant of one of the most discussed vehicle-routing problems concerning practical and theoretical importance. Regarding the practical aspect, 3D-PRP has many real-world applications that are particularly relevant for logistics companies dealing with distribution and loading issues. 3DPRP is of significant value from a theoretical aspect because it includes two NP-hard problems: the pick-up routing problem and three-dimensional loading. Train transportation plays a key role in intermodal networks, providing the efficient movement of items. There has recently been growing interest in shifting the transportation modes from road to rail. Pre- and post-haulage in the road transportation have a larger cost per tonne-km (Bergqvist & Behrends, 2011). Rail transport ensures a reduction in external costs (Janic & Vleugel, 2012). Although both 3D-PRP and train loading problems have been separately discussed, there is a need to coordinate these problems in the present paper. The combination of different levels is necessary because 3D-PRP is a precondition of a train loading problem. Backhaul orders are packed during the 3D-PRP stage and the packed orders are then assigned to trains. The present paper addresses the interactions between 3D-PRP and a train loading problem for an intermodal transportation network including road and rail transportation modes.