Modelling of containerized air cargo forwarding problems under uncertainty

This paper examines air container renting and cargo loading problems experienced by freight forwarding companies. Containers have to be booked in advance, in order to obtain discounted rental rates from airlines; renting or returning containers on the day of shipping will incur a heavy penalty. We first propose a mixed-integer model for the certain problem, in which shipment information is known with certainty, when booking. We then present a two-stage recourse model to handle the uncertainty problem, in which accurate shipment information cannot be obtained when booking, and all cargoes have to be shipped without delay. The first-stage decision is made at the booking stage, to book specific numbers of different types of containers. The second-stage decision is made on the day of shipping, depending on the extent to which the uncertainty has been realized. The decisions include number of additional containers of different types that are required to be rented, or the number of containers to be returned, under the scenario that might occur on the day of shipping. We then extend the recourse model into a robust model for dealing with the situation in which cargoes are allowed to be shipped later. The robust model provides a quantitative method to measure the trade-off between risk and cost. A series of experiments demonstrate the effectiveness of the robust model in dealing with risk and uncertainty.