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In this paper, we consider the Capacitated Arc Routing Problem (CARP), in which a fleet of vehicles, based on a specified vertex (the depot) and with a known capacity Q, must service a subset of the edges of a graph, with minimum total cost and such that the load assigned to each vehicle does not exceed its capacity. New lower bounds are developed for this problem, producing at least as good results as the already existing ones. Three of the proposed lower bounds are obtained from the resolution of a minimum cost perfect matching problem. The fourth one takes into account the vehicle capacity and is computed using a dynamic programming algorithm. Computational results, in which these bounds are compared on a set of test problems, are included.

-In this paper, we address the problem of routing school buses in a rural area. We approach this problem with a node routing model with multiple objectives that arise from conflicting viewpoints. From the point of view of cost, it is desirable to minimize the number of buses used to transport students from their homes to school and back. And from the point of view of service, it is desirable to minimize the time that a given student spends in route. The current literature deals primarily with single-objective problems and the models with multiple objectives typically employ a weighted function to combine the objectives into a single one. We develop a solution procedure that considers each objective separately and search for a set of efficient solutions instead of a single optimum. Our solution procedure is based on constructing, improving and then combining solutions within the framework of the evolutionary approach known as scatter search. Experimental testing with real data is used to assess the merit of our proposed procedure. † Partially supported by the visiting professor fellowship program of the University of Valencia (Grant Ref. No. 42743).Corberán, et al. / 2

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