On the Effectiveness of Set Covering Formulations for the Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven eeective for solving these problems is based on formulating them as a set covering problem. The method starts by solving its linear programming relaxation, via column generation, and then uses a branch and bound strategy to nd an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely eecient. In this paper, we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases. The Vehicle Routing Problem with Time Windows (VRPTW) can be stated as follows: a set of customers dispersed in a geographic region has to be served by a eet of vehicles initially located at a given depot. Each customer has a demand that must be picked up and each speciies a period of time, called a time window, in which this pick up must occur. The total load carried by each vehicle can be no more than the vehicle capacity. The objective is to nd a set of routes for the vehicles, where each route begins and ends at the depot, serves a subset of the customers without violating the capacity and time window constraints, while minimizing the total length of the routes. Due to the wide applicability and the economic importance of the problem in the service industry , variants of it have been extensively studied in the vehicle routing literature; for a review see