Given a solid T , represented by a compact set in R 3 , the aim of this work is to find a covering of T by a finite set of spheres of different radii. Some level of intersection between the spheres is necessary to cover the solid. Moreover, the volume occupied by the spheres on the outside of T is limited. This problem has an application in the planning of a radio-surgery treatment known by Gamma Knife and can be formulated as a non-convex optimization problem with quadratic constraints and linear objective function. In this work, two new linear mathematical formulations with binary variables and a hybrid method are proposed. The hybrid method combines heuristic, data mining and an exact method. Computational results show that the proposed methods outperform the ones presented in the literature.Mathematics Subject Classification. 90C10.
Transmission expansion planning with redesign has been recently proposed in the literature to improve on the classical transmission expansion planning by allowing to cutoff circuits while expanding the network. Although the reductions in the solution costs are significant, the resulting mixed-integer linear programming formulations are very difficult to solve exactly for large networks. In this work, we propose the first metaheuristic for the transmission expansion planing problem with redesign: a simple yet efficient GRASP metaheuristic. We show on realistic networks for which the optimal solutions are known that our method is able to provide in short amounts of time feasible solutions as cheap as the optimal ones. Moreover, we are able to compute a new feasible solution for benchmark instance Brazil Southeast that is cheaper than the best solution from the literature.
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