complete / incomplete hierarchical hub center single assignment network problem

in this paper we present the problem of designing a three level hub center network. in our network, the top level consists of a complete network where a direct link is between all central hubs.  the second and third levels consist of star networks that connect the hubs to central hubs and the demand nodes to hubs and thus to central hubs, respectively. we model this problem in an incomplete network environment. in this case, the top level is an incomplete network where the direct link between all central hubs is not necessary and may lead to lower transportation costs. we propose mixed integer programming model for these problems and conduct a computational study for these two developed models by using the cab data.