The Shortest Path Problem with Time Windows and Linear Waiting Costs

Guy Desaulniers and Daniel Villenueve The problem considered involves finding the minimum cost path, with cost and time being two independent quantities that govern the shortest path. In this problem, each arc is associated with a cost of traversing the arc as well as the time required to traverse it and each node is associated with a lower bound and an upper bound on the time of departure from that node. For example, the time at which a parcel can be sent from a FedEx dispatching station depends upon the working hours of the station. In this case the working hours on a particular day provide a lower and upper bound on the departure time at the node. The parcel dropped off at a station also has a certain waiting time before it gets picked up. The waiting time at each node is associated with the arc linking the node to its predecessor. A time dependent cost, which varies linearly with the total waiting time along a path, is considered. The mathematical formulation of the SPWC (Shortest Path problem with Waiting Costs) is discussed. Two other types of problems for which, efficient algorithms exist are discussed and it is shown that the SPWC problem can be formulated as instances of those problems.