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Abstract-The capacitated p-median location problem is one of the famous problems widely discussed in the literature, but its generalization to a multi-capacity case has not. This generalization, called multi-capacitated location problem, is characterized by allowing facilities to use one of several capacity levels. For this purpose, a predefined list of capacity levels supported by all potential facilities is established. In this paper, we will detail the mathematical formulation and propose a new solving method. We try to construct, indeed, a multi-stage heuristic algorithm that will be called BDF (Biggest Demand First). This new method appears in two approaches: Integrated BDF (IBDF) and Hybridized BDF (HBDF) will be improved by using a local search optimization. A valid lower bound to the optimal solution value is obtained by solving a lagrangian relaxation dual of the exact formulation. Computational results are presented at the end using new instances with higher ratio between the number of customers, facilities and capacity levels or adapted from those of p-median drawn from the literature. The obtained results show that the IBDF is much faster with medium quality solution while HBDF is slower but provides very good solutions close to the optimality.

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