An exact solution approach for vehicle routing and scheduling problems with soft time windows

A new column generation based exact optimization approach for the vehicle routing and scheduling problem with semi soft time windows (VRPSSTW) is presented. Elementary shortest path problem with resource constraints and late arrival penalties is solved as a subproblem, which rises from the Dantzig-Wolfe decomposition method. Exact solutions of VRPSSTW and hard time windows variant are compared on Solomon's benchmark instances as well as on an instance based on Tokyo road network. It was found that the VRPSSTW solution results in fewer routes thus overall costs are reduced and late arrival penalties contribute only a small fraction to total cost.