Analysis and Branch-and-Cut Algorithm for the Time-Dependent Travelling Salesman Problem

Abstract

Given a graph whose arc traversal times vary over time, the time-dependent travelling salesman problem (TDTSP) consists in finding a Hamiltonian tour of least total duration covering the vertices of the graph. The contribution of this paper is twofold. First, we describe a lower and upper bounding procedure that requires the solution of a simpler (yet NP-hard) asymmetric travelling salesman problem (ATSP). In addition, we prove that this ATSP solution is optimal for the TDTSP if all the arcs share a common congestion pattern. Second, we formulate the TDTSP as an integer linear programming model for which valid inequalities are devised. These inequalities are then embedded into a branch-and-cut algorithm that is able to solve instances with up to 40 vertices.

A Note on the Ichoua, Gendreau, and Potvin (2003) Travel Time Model

“…A straightforward consequence is that the IGP model does not suffer from the drawback pointed out by Fleischmann, Gietz, and Gnutzmann (2004), who stated: "A drawback of the models with varying speeds but constant distances is that they do not consider potential changes of the fastest paths themselves due to varying travel times, which imply changes of distances." Another outcome is that the lower bounding procedure proposed by Cordeau, Ghiani, and Guerriero (2014) for the time-dependent traveling salesman problem can be applied to the wider class of instances with continuous piecewise-linear arc travel times. First, the IGP parameters have to be computed by solving a system of linear equations for every arc; then speeds ijh are expressed as…”

The traveling salesman problem with time-dependent service times

“…In our case, if we consider a linear service time function, small service times occur early in the route and large service times occur towards the end. The former case translates into high degradation of congestion factors and the latter case translates into low degradation of congestion factors, which corresponds to values well outside the interval considered in Cordeau et al (2014). Furthermore, our model can solve instances in which the FIFO property does not hold whereas this property is required for the algorithm proposed in Cordeau et al (2014).…”

“…This definition makes our problem different from the time-dependent versions studied in Bigras et al (2008), Gouveia and Voß (1995), Picard and Queyranne (1978), Vander Wiel and Sahinidis (1996), and thus the methods proposed in these papers are not applicable to our problem. Cordeau, Ghiani, and Guerriero (2014) consider a time-dependent TSP in which the predetermined time horizon is partitioned into a number of time intervals, and the average travel speed on each arc during each interval is known. The travel time on each arc is then computed by a procedure introduced by Ichoua, Gendreau, and Potvin (2003), and a branch-and-cut procedure is developed to solve the problem.…”

“…The proposed algorithm is capable of solving instances with up to 40 nodes. In principle, a special case of our problem with linear service times can be solved by the algorithm of Cordeau et al (2014). In that paper, travel speed functions are defined by employing the degradation of the congestion factors and the authors report that their algorithm works as long as the values of these factors are set within a given interval (approximately [0.7, 1.0]).…”

Time-dependent routing problems: A review

“…Indeed,Cordeau et al (2014),Arigliano et al (2011) andCalogiuri et al (2014) have shown that there is an interesting relationship between the time de-165 pendent versions of TSP, TSP with time windows and quickest path problem and their static counterparts. Firstly, they show that -if the all the arcs share the same traffic pattern (Δ = 1) -these problems can be solved as their time independent counterparts, with suitably defined (constant) travel times.…”