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Funding informationThis research was supported by the Conseil Régional d'Auvergne. European Fund for Regional Development (FEDER Auvergne region). Labex IMobS3.Vehicle routing problems (VRP) concern the pickup and/or the delivery of goods from/to customers with vehicles. In the literature, most approaches consider the road network implicitly. Specifically, so-called customer-based graphs are used where nodes represent customers (plus the depot) and arcs represent best paths between customers. This model can affect solution quality when several attributes are defined on road segments (like travel time and distance). To handle that, two approaches are proposed in the literature. The road network can be represented using a multigraph that extends the customer-based graph and where an arc is introduced for every efficient path between two nodes. Alternatively, the problem can be solved on a graph that mimics the original road network. In this paper, we investigate the latter approach. We consider the VRP with time windows (VRPTW) and we develop a branch-and-price scheme. An extensive computational study based on several types of instances is conducted in order to evaluate this approach compared to the multigraph-based approach. As far as we know, our branch-and-price scheme is the first exact method for the VRPTW with the road-network model. Also, our computational study provides the first comparison between the two models: multigraph and road-network. KEYWORDSbranch-and-price, multigraph, road-network graph, vehicle routing problems INTRODUCTIONVehicle routing problems (VRP) aim to design an optimal set of routes to be used by a fleet of vehicles to visit a set of geographically dispersed customers. These routes must start and end at a depot and satisfy a set of constraints (vehicle capacity, customer's time windows, route duration, etc.). Since the introduction of the first VRP by Dantzig and Ramser [7], hundreds of papers and books have been devoted to these problems [14]. Many variants have been proposed to address the numerous issues that arise in real-life applications, such as the VRP with time windows (VRPTW) where transportation plans are constrained to satisfy customer requests within their time windows [19], the multidepot VRP where vehicles are based at different depots [6], and so on. Extensive reviews on the most common variants of the VRP are available in [12,21].Conventionally, VRP are tackled using a simple graph, abstracting the underlying road network. In this graph, called customer-based graph, a node is introduced for each point of interest (customer, depot, etc.) and an arc represents the best path in the road network between two nodes. This model implicitly assumes that these best paths can be defined a priori. Yet, in practice several attributes can be given on road segments (e.g., travel time, travel cost, energy consumption, etc.). Thus, alternative paths exist between pairs of points of interest. Not considering these alternatives may discard potentially good solutions from the solution sp...

In this paper we propose a multigraph model and a heuristic for the Vehicle Routing Problem with Time Windows (VRPTW). In the classical VRPTW, travel information is commonly represented with a customer-based graph, where an arc is an abstraction of the best road-network path between two nodes. We consider the case when parallel arcs are added to this graph to introduce different compromises between travel time and cost. It has been shown in the literature that this multigraph modeling enables substantial gains in the solution quality, while highly complicating the problem. We develop an Adaptive Large Neighbourhood Search (ALNS) heuristic in which a special data structure and dynamic programming algorithms are used to efficiently address the multigraph setting. Computational experiments on several set of instances demonstrate the effectiveness of our solution method and the impact of alternative paths on the solution quality.