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Summary. Arc Routing is the arc counterpart to node routing in the sense that focus regarding service and resource constraints are on the arcs and not on the nodes. The key problem within this area is the Capacitated Arc Routing Problem (CARP), which is the arc routing counterpart to the vehicle routing problem. During the last decade, arc routing has been a relatively active research area with respect to lower bounding procedures, solution approaches and modeling. Furthermore, several interesting variations of the problem have been studied. We survey the latest research within the area of arc routing focusing mainly on the CARP and its variants.

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Abstract:In this paper we consider approximation of the Capacitated Arc Routing Problem, which is the problem of servicing a set of edges in a graph using a fleet of capacity constrained vehicles. We present a 7 3 2 W approximation algorithm for the problem and prove that this algorithm outperforms the only existing approximation algorithm for the problem. Furthermore, we give computational results showing that the new algorithm performs very well in practice.

The Capacitated Arc Routing Problem (CARP) captures important aspects of reallife problems and has been studied extensively over the past two decades. Based on a waste collection project, we introduce a number of new CARP variations. We first present three multi-compartment CARP variations of different levels of complexity regarding compartments and where one incorporates a time horizon. We then present a variation that seeks to coordinate vehicles over a planning horizon such that the vehicles that collect different waste fractions from the same households do so on the same day of the week. Finally, the semi-periodic CARP takes into account that the households on a street, providing the demand of the edge, may not request waste collection at the same interval. We present large-scale instances both for the classical CARP and for the five new problems. The instances are based on real-life networks and waste data from five areas in Denmark and cover rural as well as urban areas. The largest instances contain more than 10 thousand nodes. We give detailed information about the construction of the instances from the real-life data, and explain how they can be used to perform scenario analyses.

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The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004. This problem generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It captures important aspects of real-life routing problems that were not adequately modeled in previous Vehicle Routing Problem (VRP) variants. The authors also proposed a memetic algorithm procedure and defined a set of test instances called the CBMix benchmark. The NEARP definition and investigation contribute to the development of rich VRPs. In this paper we present the first lower bound procedure for the NEARP. It is a further development of lower bounds for the CARP. We also define two novel sets of test instances to complement the CBMix benchmark. The first is based on well-known CARP instances; the second consists of real life cases of newspaper delivery routing. We provide numerical results in the form of lower and best known upper bounds for all instances of all three benchmarks. For three of the instances, the gap between the upper and lower bound is closed. The average gap is 25.1%. As the lower bound procedure is based on a high quality lower bound procedure for the CARP, and there has been limited work on approximate solution methods for the NEARP, we suspect that the main reason for the rather large gaps is the quality of the upper bound. This fact, and the high industrial relevance of the NEARP, should motivate more research on approximate and exact methods for this important problem. AbstractThe Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004. This problem generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It captures important aspects of real-life routing problems that were not adequately modeled in previous Vehicle Routing Problem (VRP) variants. The authors also proposed a memetic algorithm procedure and defined a set of test instances called the CBMix benchmark. The NEARP definition and investigation contribute to the development of rich VRPs. In this paper we present the first lower bound procedure for the NEARP. It is a further development of lower bounds for the CARP. We also define two novel sets of test instances to complement the CBMix benchmark. The first is based on well-known CARP in-1 stances; the second consists of real life cases of newspaper delivery routing. We provide numerical results in the form of lower and best known upper bounds for all instances of all three benchmarks. For three of the instances, the gap between the upper and lower bound is closed. The average gap is 25.1%. As the lower bound procedure is based on a high quality lower bound procedure for the CARP, and there has been limited work on approximate solution methods for the NEARP, we suspect that the main reason for the rather large gaps is the quality of the upper bound. This fact, and the high industrial relevance of...

The purpose of this paper is to develop a fast heuristic called Fast-CARP for the solution of large-scale capacitated arc routing problems, with or without duration constraints. This study is motivated by a waste collection problem in Denmark. After a preprocessing phase, FastCARP creates a giant tour, partitions the graph into districts, and construct routes within each district. It then iteratively merges and splits adjacent districts and reoptimises the routes. The heuristic was tested on 264 benchmark instances containing up to 11,640 nodes, 12,675 edges, 8,581 required edges, and 323 vehicles. FastCARP was compared with an alternative heuristic called Base. On small graphs, it was better but slower than Base. On larger graphs, it was much 1 faster and only slightly worse than Base in terms of solution quality.