The following experiments were aimed at investigating the performance difference of the standard VNS and the multilevel VNS, hence we were not hunting for new best solutions on available data sets. Nevertheless, we already did find a few ones back then (which are outdated by now) but we will not elaborate on this. The algorithms have been implemented in C++, compiled with GCC 4.5 and executed on a single core of a 2.83GHz Intel Core2 Quad Q9550 with 8 GB RAM.
For each chosen setting we performed 10 runs per instance with a limit of106 iterations, i.e. solution evaluations. Preliminary tests on all considered instances suggested to ignore the given limits on vehicle capacity and tour duration during coarsening (which would only affect PVRP instances), probably because the VNS was built to cope with infeasibility. If not stated otherwise 20% of all iterations are devoted to the initial coarsening/refinement phase
5. PERIODICVEHICLEROUTINGPROBLEM
Table 5.2: Summarized average results on available benchmark test data of VNS and MLVNS; giving standard deviations in paranthesis.
test data instances
VNS MLVNS II0.5
%-gap %-gap %-gap %-gap %-time
BKS HDH BKS HDH VNS
PVRP old 32 3.02 (4.67) -0.19 (1.43) 2.50 (3.91) -0.66 (1.57) 89.2 new 10 2.86 (1.73) -0.75 (0.89) 2.68 (1.66) -0.92 (0.90) 85.1 PTSP old 23 0.85 (1.12) 0.05 (0.39) 0.30 (0.43) -0.49 (0.80) 89.9 new 10 0.34 (0.18) 0.04 (0.05) 0.28 (0.17) -0.01 (0.07) 79.1
and recoarsening is applied four times also devoting 20% of the iterations each. The two remaining options we varied are the type of problem coarsening (setting I or II) and the cost limit factorδc; these will be denoted by [I,II]δc. We experienced that usually it is better to enforce a merging cost limit.
Figures 5.3 and 5.4 show PVRP solutions throughout the initial refinement process with coarsened segments highlighted in red.
5.5.1 PVRP and PTSP Instances Used in the Literature
We utilize available “old” and “new” benchmark test data1both for the PVRP and the PTSP.
Here we will only present results on them in concise form and refer to [40] for further de-tails and origins of these instances. Back then relevant and recent results for comparison are reported in [105]. Meanwhile Vidal et al. [226] proposed a hybrid genetic algorithm outperforming previous approaches in terms of solution quality with comparable computing effort.
Results are presented in concise form in Table 5.2. Here we state the percentage gap to the so far best known solutions at that time (%-gap BKS) as well as to the average results of the VNS described by Hemmelmayr et al. [105] using the same iteration limit of106 (%-gap HDH), all these values are given in [105]. The corresponding standard deviations are written in parentheses. For the MLVNS we further state the amount of CPU time spent given in percentage of the VNS’ CPU time (%-time VNS). As can be seen the MLVNS generally performs better than the standard VNS, achieving especially on the old data sets a notable improvement with regard to solution quality. Contrary, on the new data sets no such clear improvement can be observed, especially not for the PTSP. However, for all these data sets there is a CPU time reduction between 10 and 20 percent on average. As it turned out coarsening setting II gave consistently better results, though the differences were sometimes negligible.
1Available at http://neumann.hec.ca/chairedistributique/data/pvrp [accessed on October 22, 2011]
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objective: 286324, travel costs: 8389.91
#routes: 6, cost: 2297.55 #routes: 6, cost: 1881.88
#routes: 6, cost: 2321.11 #routes: 6, cost: 1889.36
(a)initial coarse solution
objective: 6842.65, travel costs: 6842.65
#routes: 6, cost: 1681.59 #routes: 6, cost: 1814.48
#routes: 6, cost: 1647.02 #routes: 6, cost: 1699.56
(b)solution after about 1/3 of all initial refinement steps
Figure 5.3:Exemplary (coarsened) PVRP solutions during the VNS search process.
5. PERIODICVEHICLEROUTINGPROBLEM
objective: 5923.48, travel costs: 5934.96
#routes: 6, cost: 1427.58 #routes: 6, cost: 1566.12
#routes: 5, cost: 1317.41 #routes: 6, cost: 1612.37
(a)solution after about 2/3 of all initial refinement steps
objective: 5610.65, travel costs: 5610.65
#routes: 6, cost: 1308.78 #routes: 6, cost: 1546.84
#routes: 5, cost: 1247.42 #routes: 6, cost: 1507.6
(b)solution at the end of the initial refinement process
Figure 5.4:Exemplary (coarsened) PVRP solutions during the VNS search process.
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Table 5.3: New larger PVRP and PTSP (setting m = 1and ignoring D andQ) in-stances using the generation method introduced in [40], additionally stating our best found solutions’ values over all runs.
Id n m t D Q service frequencies best found solutions
f1 f2 f3 f4 f6 PVRP PTSP
pr11 336 14 4 480 185 112 112 112 11224.40 6618.53
pr12 384 16 4 475 195 128 128 128 11320.99 7062.96
pr13 432 18 4 450 185 144 144 144 10429.50 6757.86
pr14 480 20 4 475 185 160 160 160 13463.25 7740.23
pr15 528 22 4 470 190 176 176 176 15237.87 8377.39
pr16 576 24 4 455 185 192 192 192 13315.25 7640.57
pr17 360 15 6 445 165 90 90 90 90 15888.55 9459.08
pr18 432 18 6 450 170 108 108 108 108 19564.87 11314.85
pr19 504 21 6 440 160 126 126 126 126 22949.66 11344.27
pr20 576 24 6 450 165 144 144 144 144 24004.13 11660.07
5.5.2 Additional PVRP and PTSP Instances Similar to Cordeau et al.’s Since the multilevel refinement strategy is in general especially appealing for large(r) in-stances where it may unfold its full potential, but the available test data lack them, we created some PVRP and PTSP instances on our own by applying the generation method described in [40]. They can be regarded a continuation of the instances introduced in this latter work, except that we evenly distributed the visit frequencies among the customers for all instances;
more details, already including our best found solutions’ values over all conducted runs (partly also with more iterations), are given in Table 5.3.
When doing preliminary tests we recognized that the VNS’ acceptance decision using the Metropolis criterion in some way weakens the potential gain of the multilevel extension.
Hence we also performed tests with only accepting improved solutions, whose results are given in Table 5.4 and 5.6 for the PVRP and the PTSP, respectively. The corresponding re-sults using the default acceptance decision are given in Table 5.5 and 5.7. For both variants and problems we tested coarsening setting I and II with a cost limit factorδcof 0.5 and 0.33 (which was narrowed down in preliminary tests), as well as devoting all iterations to the ini-tial coarsening/refinement phase and doing no recoarsening at all, where we always state the results of the setting yielding the best solutions on average. The tables show average travel costs (avg), corresponding standard deviations in percent (sdv[%]), CPU-times in seconds (t[s]), as well as the percentage gaps to the VNS (%-gap) and the CPU-times given in per-cent of the VNS (%-time) for the MLVNS. In Tables 5.4–5.7 average values of the MLVNS are printed bold whenever a statistically significant improvement compared to the VNS has been achieved or vice versa, according to a Wilcoxon rank sum test with an error level of 5%. For these new larger instances the greedy coarsening setting I turned out to achieve consistently better results than setting II. Comparing Table 5.4 and 5.5, as well as Table 5.6
5. PERIODICVEHICLEROUTINGPROBLEM
Table 5.4: Average results of standard and multilevel VNS on new larger PVRP in-stances only accepting improved solutions.
Id VNS MLVNS I0.5
best avg sdv[%] t[s] best avg sdv[%] t[s] %-gap %-time pr11 11988.17 12135.04 0.92 40.6 11764.62 12066.05 1.59 31.4 -0.57 77.3 pr12 11935.12 12099.76 0.76 42.8 11913.75 12019.06 0.68 33.3 -0.67 77.8 pr13 11291.21 11484.99 0.86 47.3 11123.96 11265.22 1.18 37.2 -1.91 78.6 pr14 14457.39 14572.72 0.61 52.4 14217.40 14352.14 0.80 40.3 -1.51 76.9 pr15 16329.00 16498.72 0.68 59.7 15878.62 16078.02 0.66 42.1 -2.55 70.5 pr16 14716.46 14815.95 0.69 67.3 13832.13 14064.44 1.00 47.3 -5.07 70.3 pr17 16864.64 17072.72 0.69 41.5 16525.56 16804.22 0.79 34.4 -1.57 82.9 pr18 20778.33 21057.55 0.83 48.8 20345.62 20613.84 0.65 39.4 -2.11 80.7 pr19 24898.62 25148.96 0.42 57.7 24330.25 24795.71 1.19 44.4 -1.40 76.9 pr20 25716.70 25875.35 0.52 70.4 25075.55 25273.14 0.59 53.4 -2.33 75.9
avg 50.6 39.8 -1.97 77.1
and 5.7 it can be clearly seen that the MLVNS yields a higher relative improvement when only accepting improved solutions, nevertheless also when using the default acceptance de-cision the improvement is notable, especially in case of the PTSP. Interestingly, for the PTSP using no recoarsening and extending the initial problem coarsening/refinement phase to all iterations turned out to be usually better (which only holds for the new larger instances, we checked it for the available test data, too). This has the additional positive effect that the required CPU-time is more than halved. Contrary, for the PVRP the decrease in CPU-time is about 20%. For the latter problem we also observed that improvements during refinement after a recoarsening occur more often when also accepting worse solutions, otherwise the customer segments are probably too coarse and do not allow an improvement in a direct way. In summary, for almost all instances and both acceptance decisions the MLVNS yields on average significantly better results than the VNS.
The relative usage, acceptance and success rate of the shaking neighborhoods for both prob-lems averaged over all corresponding runs is shown in Table 5.8 and 5.9. In general chang-ing visit combinations is by far the most important neighborhood, even more evident for the PTSP, yet the others make significant contributions as well. Looking at the rates of the mul-tilevel VNS shows that they are smoothed to some extent, i.e. the neighborhoods’ usage and subsequently their acceptance and success rates are more uniformly than before, again more so for the PTSP. This shows that by using the multilevel refinement allows to better utilize also the later (in terms of a higherkvalue) neighborhoods.