Generalizations of the vehicle routing problem with time windows

The vehicle routing problem with time windows (VRPTW) is the problem of minimizing the total travel distance of a number of vehicles, under capacity and time window constraints, where every customer must be visited exactly once by a vehicle. We consider two generalizations of the VRPTW and propose local search algorithms. The first one treats traveling times as variables and introduce cost functions on them. This enables us to shorten the traveling times by adding some cost, thus introducing a new dimension to consider flexible solutions. The other allows both traveling times and traveling costs being time-dependent, so that it can treat time-dependent situations such as rush-hour traffic jam. In our local search algorithms for these problems, after fixing a route of each vehicle, we must compute an optimal time schedule of each route. We investigate the complexity of this subproblem and show that it can be efficiently solved by dynamic programming under some conditions.