Section: New Results

Time-dependent formulations for routing problems

The paper [16] presents a new formulation for the Time-Dependent Travelling Salesman Problem (TDTSP). We start by reviewing well known natural formulations with some emphasis on the formulation by Picard and Queyranne (1978). The main feature of this formulation is that it uses, as a subproblem, an exact description of the n-circuit problem. Then, we present a new formulation that uses more variables and is based on using, for each node, a stronger subproblem, namely a n-circuit subproblem with the additional constraint that the corresponding node is not repeated in the circuit. Although the new model has more variables and constraints than the original PQ model, the results given from our computational experiments show that the linear programming relaxation of the new model gives, for many of the instances tested, gaps that are close to zero. Thus, the new model is worth investigating for solving TDTSP instances. We have also provided a complete characterization of the feasible set of the corresponding linear programming relaxation in the space of the variables of the PQ model. This characterization permits us to suggest alternative methods of using the proposed formulations.

A well-known formulation for the unit-demand capacitated vehicle routing problem uses a single commodity flow system to represent the delivery of the items. The vehicle capacity is modeled by imposing a maximum capacity on the arcs used by the flow. In [30] , we used a time-dependent formulation for the problem to derive, by projection, tighter bounding inequalities on the arcs. The first experiments show that these new inequalities permit to improve significantly the linear relaxation bound of the single commodity flow formulation. We are currently studying separation algorithms in order to generate dynamically these new inequalities.