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

Abstract

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 effective 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 find 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 efficient. 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.

Real-Time Multivehicle Truckload Pickup and Delivery Problems

“…Bodin et al 1983, Christofides 1985, Fisher 1995, Golden and Assad 1988, and Solomon 1987 provide extensive surveys of the various VRPs and solution techniques. Bienstock et al 1993, Bramel and Simchi-Levi 1996, 1997, Bramel et al 1994, and Bramel et al 1992 present probabilistic analyses of many heuristics for deterministic and static VRPs.…”

Robust Branch-and-Cut-and-Price for the Capacitated Vehicle Routing Problem

“…In that formulation, a column covers a set of vertices S with total demand not exceeding C and have the cost of a minimum route over {0} ∪ S. Bramel and Simchi-Levi [13] proved that for certain natural classes of instances, the ratio between the lower bounds given by that formulation and the optimal solution values asymptotically approaches 1 as the number of clients grows. However, that formulation in itself is not practical because pricing over the exponential number of columns require the solution of capacitated prize-collecting TSPs, a problem almost as difficult as the CVRP itself.…”

Identity Matrix

“…For specific implementations, for the vehicle routing problem, see Chabrier (2006), for the bandwidth packing problem, see Hoffman and Villa (2007) and Parker and Ryan (1995), for the generalized assignment problem, see Savelsbergh (1997), and for alternative column-generation strategies for solving the cutting stock problem see Vance et al (1994). Bramel and Simchi-Levi (1997) have shown that the set-partitioning formulation for the vehicle routing problem with time windows is very effective in practice-that is, the relative gap between the fractional linear optimization solutions and the global integer solution is small. Similar results have been obtained for the bin-packing problem (Chan et al 1998a) and for the machine-scheduling problem (Chan et al 1998b).…”