The uncapacitated r-allocation p-hub center problem

In this paper, we propose the uncapacitated r-allocation p-hub center problem (UrApHCP), which represents a generalization of both single and multiple allocation variants of the p-hub center problem. We further present two binary -integer linear programs for the UrApHCP and prove their equivalence for r=1 and p with respective single and multiple allocation cases. A flow formulation combining the features of the two previous models is also presented. In order to solve the UrApHCP, we develop two general variable neighborhood search (GVNS) heuristics that use nested and sequential variable neighborhood descent strategies. The proposed approaches are tested on benchmark instances from the literature with up to 423 nodes. The proposed GVNS quickly reaches all optimal or best-known results from the literature for the single and multiple allocation variants of the problem, as well as new optimal results for r-allocation obtained using a CPLEX solver.