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Étude théorique et numérique du problème de la gestion de la diversité

Abstract : The diversity management problem is defined on a partially ordered set. The objective is to produce a subset of $k$ reference elements, $k$ being a given number, while minimizing the cost. Demands and unit costs of production are known. Each non produced element must be replaced by a reference which is higher in the partial order, and this implies an overcost. After a theoretical study on the complexity of this problem, we describe and justify the model we chose. This model is an integer linear program close to that commonly used for the $k$-median problem. To solve this program, we present a Lagrangean approach, and various criteria of variable fixing. We exploit also this approach to build feasible solutions of good quality. Finally, we design a Branch and Cut type algorithm to solve our problem to optimality. To do so, we first do a polyhedral study of the convex hull of the solutions. We present a particular type of cuts that eliminate fractional solutions, as well as a heuristic to generate them. We conclude by numerical tests carried out on real world instances.
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Submitted on : Tuesday, February 17, 2004 - 10:59:18 AM
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Olivier Briant. Étude théorique et numérique du problème de la gestion de la diversité. Mathématiques [math]. Institut National Polytechnique de Grenoble - INPG, 2000. Français. ⟨tel-00004710⟩

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