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Résolution exacte de problèmes de couverture par arborescences sous contraintes de capacité

Abstract : In this document, we study two districting problems and propose several exact methods, based on Dantzig-Wolfe decomposition and column generation, to solve them. For each model, we compare exact approaches based either on compact formulations or on extended formulations obtained using Dantzig-Wolfe decomposition. The first type of model that we propose defines the objective function in a p-median problem fashion. Regarding the methods used to solve that kind of model, we emphasize accelerating the convergence of the column generation algorithm by designing constraint aggregation techniques in order to reduce the degeneracy in the simplex algorithm. Numerical experiments show that this constraint aggregation method indeed reduces the proportion of degenerated iterations. However, it is not enough to speed up the branch-and-price algorithm. Choosing to tackle the problem through either a compact formulation or an extended formulation depends on the structure of the instances to solve. The second type of model formulates the objective function in a way quite similar to that of p-centre problems. Using such an objective function induces complex column generation subproblems. We focus on designing branch-and-bound and dynamic programming algorithms in order to solve them efficiently. Experiments show that the branch-and-price approach surpasses any proposed method based on compact formulations of the problem.
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Jérémy Guillot. Résolution exacte de problèmes de couverture par arborescences sous contraintes de capacité. Optimisation et contrôle [math.OC]. Université de Bordeaux, 2018. Français. ⟨NNT : 2018BORD0395⟩. ⟨tel-02020787⟩

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