Complexity of flow time minimization in a crossdock truck scheduling problem with asymmetric handover relations

•A new crossdocking scheduling model, minimizing total sojourn time of the items being shipped.•NP-hardness proof for general bipartite handover graphs and processing times, even with a single door.•NP-hardness proof for two doors when processing times are proportional to the total amount of pallets being loaded/unloaded.•Identification of some polynomial special cases.