Mathematical formulations and solution methods for the uncapacitated r-allocation p-hub maximal covering problem

This paper considers the uncapacitated $r$-allocation $p$-hub maximal covering problem (UrApHMCP), which represents a generalization of the well-known $p$-hub maximal covering problem, as it allows each non-hub node to send and receive flow via at most $r$ hubs, $r\le p$. Two coverage criteria are considered in UrApHMCP — binary and, for the first time in the literature, partial coverage. Novel mathematical formulations of UrApHMCP for both coverage criteria are proposed. As the considered UrApHMCP is an NP-hard optimization problem, two efficient heuristic methods are proposed as solution approaches. The first one is a variant of General Variable Neighborhood Search (GVNS), and the second one is based on combining a Greedy Randomized Adaptive Search Procedure (GRASP) with Variable Neighborhood Descent (VND), resulting in a hybrid GRASP-VND method. Computational study is performed over the set of CAB and AP benchmark instances with up to 25 and 200 nodes, respectively, on TR instances including 81 nodes, as well as on the challenging USA423 and URAND hub instances with up 423 and 1000 nodes, respectively. Optimal or feasible solutions are obtained by CPLEX solver for instances with up to 50 nodes, while instances of larger dimensions are out of reach for CPLEX solver. On the other hand, both GVNS and GRASP-VND reach optimal solutions or improve lower bounds provided by CPLEX in short CPU times. In addition, both heuristics quickly return solutions on problem instances of large dimensions, thus indicating their potential to solve effectively large, realistic sized problem instances. The conducted non-parametric statistical tests confirm robustness of the proposed GVNS and GRASP-VND and demonstrate that the these two metaheuristics outperform other tested algorithms for UrApHMCP.