A NETWORK FLOW MODEL FOR THE VEHICLE ROUTING PROBLEM WITH TIME WINDOWS AND MULTIPLE ROUTES

The Vehicle Routing Problem (VRP) is a combinatorial optimization problem that has been widely studied in the literature, ever since it was formulated for the first time in (Dantzig and Ramser 1959). It can be seen as a generalization of another well know combinatorial problem, the traveling salesman problem, which can be described as a VRP with one vehicle, no depot, no vehicle capacities and no customer demands. Generally speaking, it is the problem of scheduling a fleet of vehicles to visit a set of customers, to whom they must deliver or collect a demanded quantity of goods. The problem consists of finding the best set of routes, according to a given objective function, such that all operational constraints of the vehicles are respected, and the set of customers is covered. This objective function can be the minimization of all traveling costs, the maximization of the number of served customers, or some combination of these or other factors. The VRP is well-know to be NP-hard, and so are most of its variants. Its solution methods include several heuristic and metaheuristic approaches, as well as some exact methods, mainly based on branch-and-bound techniques. The classical version of the VRP is commonly called the Capacitated Vehicle Routing Problem, as the vehicles in the fleet have limited capacities. There are several variants of this problem. In (Toth and Vigo 2002; Cordeau, J.-F et al. 2007), the authors describe the VRP and some of its main variants.