VRP with Time Windows (VRPTW)
The VRPTW is the same problem that VRP with the additional restriction that in VRPTW a time window is associated with each customer , defining an interval wherein the customer has to be supplied. The interval at the depot is called the scheduling horizon. Here is a formal description of the problem:
In the case of soft time widows, a later service does not affect the feasibility of the solution, but is penalized by adding a value to the objective function.
In the figure below we can see a graph representing an instance for solving with VRPTW. There, blue and white bars are representing the time window (i.e. business hours), where white area represents when we can make a delivery at that customer. By other side, the red line shows when is the delivery made for this particular solution.