Dynamic p+q maximal hub location problem for freight transportation planning with rational markets

In this article, a dynamic maximal hub location covering problem for a freight transportation system is studied, in which the model has the possibility of having expansion scenarios for future according to the forecasts of increasing demands. Two expansion scenarios are to add up the number of hubs in the network and to add up more carriers. As the markets are involved in the pricing procedure, the model is a bi-level problem which needs more effort to deal with, for which in this work two reformulations based on Karush–Kuhn–Tucker conditions and duality theory are utilized to reformulate the bi-level problem to a single-level one. To solve the model efficiently, a decomposition method is applied and numerical examples are solved to verify the accuracy of the proposed model.