Hub network design model in a competitive environment with flow threshold

We consider a hub network design model based on the Stackelberg hub location model, where two firms compete with each other to maximize their own profit. The firm as a leader first locates p hubs and decides which OD pairs should be in services on the condition that the other firm as a follower locates q hubs and decides its strategies in a similar way after that. To avoid the possibility of unprofitable services, we incorporate flow threshold constraints into the model. We formulate the leader's problem as a bilevel programming problem with the follower's problem as a lower level problem. We solve the problem with the complete enumeration method and a greedy heuristic. The main objective is to make it clear how the network structure can be affected by the flow threshold constraints and the competitor's strategies.