A Column Generation Algorithm for the Vehicle Routing Problem with Time Windows

Let G = (N, A) be a network where A is the set of arcs or route segments and N is the set of nodes or customers. Associated with each arc (i,j) ∈ A there is a cost c _ij , and a duration t _ij . We assume that the service time of customer i is included in the duration of each arc (i, j). In this paper, the cost is taken to be the distance between i and j. The vehicle routing problem with time windows (VRPTW) involves the design of a set of minimum cost routes originating and terminating at a central depot, d, for a fleet of vehicles which services a set of customers with known demands q _i . Each customer is serviced exactly once. The service of a customer, involving pick-up (delivery) of goods or services, can begin at T _i within the time window defined by the earliest time, a _i , and the latest time, b _i , when the customer will permit the start of service. If a vehicle arrives at a customer too early, it will wait. In addition, due dates cannot be violated. Furthermore, all the customers must be assigned to vehicles such that the vehicle capacities, Q, are not exceeded.