Vehicle routing problems with multiple trips

Abstract

This paper presents a survey on the multi-trip vehicle routing problem (MTVRP) and on related routing problems where vehicles are allowed to perform multiple trips. The first part of the paper focuses on the MTVRP. It gives an unified view on mathematical formulations and surveys exact and heuristic approaches. The paper continues with variants of the MTVRP and other families of routing problems where multiple trips are sometimes allowed. For the latter, it specially insists on the motivations for having multiple trips and the algorithmic consequences. The expected contribution of the survey is to give a comprehensive overview on a structural property of routing problems that has seen a strongly growing interest in the last few years and that has been investigated in very different areas of the routing literature.

Appendices

Appendix 1: Benchmark instances for the MTVRP

Table 3 presents the state-of-the-art results on benchmark instances proposed by Taillard et al. (1996) for the MTVRP. The first column indicates the instance name, the number of customers N and the value $z^*$ that served as a basis for the construction of the instance. Columns $\left| {\mathcal {V}}\right|$, $T_H^1$ and $T_H^2$ give the number of vehicles and the length of the time horizon. Columns Opt report optimal values when available (all obtained by Mingozzi et al. (2013)). Columns Best Known report best known values (all obtained by the heuristic method proposed in Cattaruzza et al. (2014)), when the corresponding optimal values are not known but feasible solutions have been found. Column Best Unfeas reports the value (including penalty) of the best unfeasible solution, when no feasible solutions are known. This column is omitted for instances corresponding to time horizon $T_H^2$ since a feasible solution has been found for all instances. All these values and more detailed algorithm comparisons can be found in Cattaruzza et al. (2014). State-of-the-art algorithms are Olivera and Viera (2007) and Cattaruzza et al. (2014).

Table 3 Results on MTVRP instances

Appendix 2: Benchmark instances for the MTVRP with time-windows and service-dependent loading times

Table 4 presents the state-of-the-art results on benchmark instances proposed by Hernandez et al. (2013), and adapted from Solomon’s, for the MTVRP with time-windows and service-dependent loading times. In these instances Q is set to 100 and M is set to 2 (instances with 25 customers) or 4 (instances with 50 customers). Other data are kept as in original instances. Distances are Euclidean and truncated to one decimal place. Column Opt reports optimal values when they are known. Column Best Known reports best known solution values for other instances.

Table 4 Results on MTVRP with time-windows and service-dependent loading times instances

Optimal values are obtained by Hernandez et al. (2013). All the best known values are obtained by the heuristic method proposed in Cattaruzza et al. (2016). Two of these best know values are upper bounds determined by Hernandez et al. (2013) (but no optimality certificate is provided).

Appendix 3: Benchmark instances for the MTVRP with time-windows, service-dependent loading times, limited trip duration and profits

Tables 5 and 6 report state-of-the-art results on benchmark instances proposed by Azi (2010), based on Solomon’s, for the MTVRP with time-windows, service-dependent loading times, limited trip duration and profits (MTVRPTW-LDP). Table 5 reports optimal solution values when all customers are visited. Table 6 gives optimal solutions for which some customers are left non-visited. The problem was investigated by Azi et al. (2010), Macedo et al. (2011) and Hernandez et al. (2014), with the exception that the latter does not allow non-visited vertices.

Table 5 Benchmarks for the MTVRPTW-LDP: exact solutions with all customers visited

Table 6 Benchmarks for the MTVRPTW-LDP: exact solutions with non visited customers

Each instance is solved with two values for N, respectively equal to 25 and 40, and two values for $T_{max}$. For instances of groups RC and R, $T_{max}$ is either set to 75 (indicated as Short $T_{max}$ in Table 5) or 100 (Large $T_{max}$). For group C it is set to 220 (Short $T_{max}$) or 250 (Large $T_{max}$). Service times in C2 instances are equal to 90 for each customer, while in the other cases they are equal to 10. The number of vehicles is set to 2. $\beta$ = 0.2.

Additionally, two different conventions have been adopted for computing the distance matrix. In Azi et al. (2010), instances are used with distances rounded at the second decimal place. Macedo et al. (2011) performed their experiments on the same instances but without truncating distances. Hernandez et al. (2014) evaluate their approach on both types of distances. Columns Opt(T) and Opt(NT) of Table 5 report the optimal value when the distances are truncated (T) or not truncated (NT), respectively. In Table 6 the objective function is hierarchical and the optimal solution is described by the number of non-visited customers (Column Nb) and the traveled distance (column Dist). Whether distances are truncated or not, we follow the literature by reporting travel distances truncated at the second decimal place.

In both tables a “-” indicates that no optimal solution has been found. In Table 5 NoSol is written instead if it is proved that no solution visiting all customers exist. No heuristic algorithms having been applied to these instances, we do not report upper bounds.

Detailed comparisons of the three approaches when all customers are visited can be found in Hernandez et al. (2014). They permit to conclude that Macedo et al. (2011) and Hernandez et al. (2014) obtain comparable results and significantly improve upon Azi et al. (2010), not forgetting that Hernandez et al. (2014) is more specialized (not allowing non-visited vertices).