Hub location problem considering spoke links with incentive-dependent capacities

This study proposes a hub-and-spoke (HS) network in which the capacity of a spoke link depends on the incentive provided to the transporters. The incentives influence the capacities of the spoke links, which in turn affect the hub selections and flow assignment. A mixed integer convex programming model is developed to optimize the proposed HS network. The model minimizes total transport cost by locating p hubs, routing flows between origin-destination (OD) pairs and determining the incentives on spoke links. A customized outer-approximation algorithm is presented to provide exact solutions to the developed model, which is difficult to solve optimally within acceptable computational times for large-scale instances. An approximation algorithm based on the stepwise linearization method is then developed to provide suboptimal solutions. The developed model and algorithms are validated in computational experiments with different problem scales. The experiments demonstrate that the proposed HS network tends to route the flows on more spoke links, balance the transport costs on the potential paths between OD pairs and serve more demands with lower transport costs, when compared with the HS network with fixed capacities.