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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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In this paper the Min-Max version of the Windy Rural Postman Problem with several vehicles is introduced. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here an ILP formulation and study its associated polyhedron. Based on its partial description, a branch-and-cut algorithm has been implemented and computational results on a large set of instances are finally presented.

In this paper, we present an exact algorithm for the Windy General Routing Problem. This problem generalizes many important Arc Routing Problems and also has some interesting real-life applications. The Branch & Cut method presented here is based on a cutting-plane algorithm that identifies violated inequalities of several classes of facet-inducing inequalities for the corresponding polyhedron. The whole procedure has been tested over different sets of instances and is capable of solving to optimality large-size instances of several routing problems defined on undirected, mixed, and windy graphs.

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains a big number of important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.