A hybrid heuristic for a broad class of vehicle routing problems with heterogeneous fleet

Abstract

We consider a family of rich vehicle routing problems (RVRP) which have the particularity to combine a heterogeneous fleet with other attributes, such as backhauls, multiple depots, split deliveries, site dependency, open routes, duration limits, and time windows. To efficiently solve these problems, we propose a hybrid metaheuristic which combines an iterated local search with variable neighborhood descent, for solution improvement, and a set partitioning formulation, to exploit the memory of the past search. Moreover, we investigate a class of combined neighborhoods which jointly modify the sequences of visits and perform either heuristic or optimal reassignments of vehicles to routes. To the best of our knowledge, this is the first unified approach for a large class of heterogeneous fleet RVRPs, capable of solving more than 12 problem variants. The efficiency of the algorithm is evaluated on 643 well-known benchmark instances, and 71.70% of the best known solutions are either retrieved or improved. Moreover, the proposed metaheuristic, which can be considered as a matheuristic, produces high quality solutions with low standard deviation in comparison with previous methods. Finally, we observe that the use of combined neighborhoods does not lead to significant quality gains. Contrary to intuition, the computational effort seems better spent on more intensive route optimization rather than on more intelligent and frequent fleet re-assignments.

Appendix: Detailed results

1.1 HFFVRP-V

Detailed results obtained for the HFFVRP-V instances of: (i) (Brandão 2011, B11), compared with the TSA of (Brandão 2011, B11) and the ILS-RVND-SP of (Subramanian et al. 2012, SPUO12) (Table 7); and (ii) (Li et al. 2007, LGW07), compared with (Brandão 2011, B11) and (Subramanian et al. 2012, SPUO12) (Table 8).

Table 7 Results for the HFFVRP-V (Brandão 2011, B11)

Table 8 Results for the HFFVRP-V (Li et al. 2007, LGW07)

1.2 HFFVRP-FV

Detailed results obtained for the HFFVRP instances of (Duhamel et al. 2011, DLP11), compared with the sequential version of the GRASP $\times$ ELS of (Duhamel et al. 2013, DLP13). Column BKS in Tables 9–12 indicates the best results considering all versions of the GRASP $\times$ ELS (sequential or parallel). The instances were divided into four sets:

• DLP11–set 1: [20–95] customers (Table 9);

• DLP11–set 2: [102–147] customers (Table 10);

• DLP11–set 3: [152–196] customers (Table 11);

• DLP11–set 4: [203–256] customers (Table 12).

Table 9 Results for HFFVRP-FV (Duhamel et al. 2011, DLP11–set 1)

Table 10 Results for HFFVRP-FV (Duhamel et al. 2011, DLP11–set 2)

Table 11 Results for HFFVRP-FV (Duhamel et al. 2011, DLP11–set 3)

Table 12 Results for HFFVRP-FV (Duhamel et al. 2011, DLP11–set 4)

1.3 HFFOVRP

Detailed results obtained for the instances of (Taillard 1999, T99) as considered in (Yousefikhoshbakht et al. 2014, YDR14). The results obtained by HILS-RVRP were compared with those found by the BRMTS heuristic of the referred authors. Moreover, although Yousefikhoshbakht et al. (2014) mentioned that they used fixed and variant vehicle costs, it appears, according to our testing, that they only used variable costs. Table 13 presents the results involving only variable costs, while Table 14 reports the results involving both fixed and variable costs.

Table 13 Results for the HFFOVRP-V

Table 14 Results for the HFFOVRP-FV

1.4 Results for the MDFSM

Detailed results obtained for the MDFSM instances of (Salhi and Sari 1997, SS97), compared with those found by the VNS2 of (Salhi et al. 2014, SIW14) and the HGSADC of (Vidal et al. 2014, VCGP14) (Table 15). Column RD in Table 15 indicates the instances with route duration.

Table 15 Results for the MDFSM

1.5 HFFVRPB

Detailed results obtained for the HFFVRPB instances of (Tütüncü 2010, T10), compared with those found by the GRAMPS and ADVISER heuristics from the referred authors (Table 16 describes the results found). Note that we did not report the results for some instances because, according to the values suggested by the authors, they are infeasible.

Table 16 Results for the HFFVRPB

1.6 FSMB

Detailed results obtained for the FSMB instances of (Salhi et al. 2013, SWH13), compared with those found by Framework-2 from the same authors (Table 17).

Table 17 Results for the FSMB

1.7 SDepVRP

Detailed results obtained for the SDepVRP instances of (Cordeau and Laporte 2001, CL01), compared with those found by the ALNS 50k of (Pisinger and Røpke 2007, PR07) and the ITS of (Cordeau and Maischberger 2012, CM12) (Table 18, old instances without route duration and Table 19, new instances with route duration).

Table 18 Results for the SDepVRP (old set)

Table 19 Results for the SDepVRP (new instances)

1.8 HFFVRPSD

Detailed results obtained for the FSM instances of (Golden et al. 1984, G84) and adapted for the HFFVRPSD by (Ozfirat and Ozkarahan 2010, OO10), compared with those found by the CP heuristic of the same authors (Table 20). As HFFVRPSD allows visiting the customers more than once, the SP procedure was not considered for this variant.

Table 20 Results for the HFFVRPSD

1.9 FSMVRPTW: minimize route duration

Detailed results obtained for the FSMVRPTW instances of (Liu and Shen 1999, LS99), considering the objective of minimizing the sum of the route durations, compared with those found by the UHGS of (Vidal et al. 2014, VCGP14) and the HEA of (Koç et al. 2015, KBJL15) (Tables 21, 22, 23).

Table 21 Results for the FSMTW (minimize duration, fleet A)

Table 22 Results for the FSMTW (minimize duration, fleet B)

Table 23 Results for the FSMTW (minimize duration, fleet C)

1.10 FSMTW: minimize total distance

Detailed results obtained for the FSMVRPTW instances of (Liu and Shen 1999, LS99), considering the objective of minimizing the sum of the total distance, compared with those found by the UHGS of (Vidal et al. 2014, VCGP14) and the HEA of (Koç et al. 2015, KBJL15) (Tables 24, 25, 26).

Table 24 Results for the FSMTW (minimize distance, fleet A)

Table 25 Results for the FSMTW (minimize distance, fleet B)

Table 26 Results for the FSMTW (minimize distance, fleet C)

1.11 HFFVRPMBTW

Detailed results obtained for the HFFVRPMBTW instances of (Belmecheri et al. 2013, BPYA13), compared with the results obtained by the PSO of the same authors and the EBBO of (Berghida and Boukra 2015, BB15) (Tables 27, 28, 29).

Table 27 Results for the HFFVRPMBTW (Type C)

Table 28 Results for the HFFVRPMBTW (Type R)

Table 29 Results for the HFFVRPMBTW (Type RC)

1.12 SDepVRPTW

Detailed results obtained for the SDepVRPTW instances of (Cordeau and Laporte 2001, CL01), compared with those found by the ITS1 of (Cordeau and Maischberger 2012, CM12) and by the HGSADC of (Vidal et al. 2013b, VCGP13) (Table 30).

Table 30 Results for the SDepVRPTW