A preliminary investigation of the optimal parameters for the sPSO is presented in section 4.1. Sixty VRPPD benchmark problems were retrieved from the VRP Web (http://neo.lcc.uma.es/radi-aeb/WebVRP/index.html) and used as test problems to evaluate the performance of the sPSO methodology as described in section 4.2. The PSO algorithm was programmed via MATLAB on the Window XP platform with a Core 2 Quad 2.4GHz CPU and 4G of RAM. The performance of the sPSO method was compared with that of the Genetic algorithm provided by Evolver 4.0TM, the Genetic Algorithm Solver plug-in for Microsoft Excel.
4.1 Preliminary Test
In the preliminary experiment, the Design of Experiment (DOE) was initiated to verify the optimal PSO parameter settings for the sPSO approach. Three learning methodology. The inertia weight W was typically set up to linearly decrease from 1 to 0 during the course using the following equation.
). w denote the upper and lower bounds, respectively. The recommended ranges for the three learning factors were derived from the previous literature (Shi and Eberhart 1998b; Hu et al. 2004) in the interval of [0.4, 1.9]. However, four parameters represented probability concept in this study. That is, {ϕid1,ϕid2,ϕid3,W}∈
[ ]
0,1 . One randomly selected instance from the 60 test problems was used to execute the response surface methodology experiment. Therefore, a 33 factorial design was constructed, specifying that the three learning factors were set in the range of [0.4, 0.8]with increments of 0.2. The confidence level was set at 95%.
Figure 5 shows a plot of the main effects of the three factors from the results of the DOE experiment. As shown, c1, c2, and c3 are symbolized as respectively. Distance is the response value, representing the average cost generated by different combinations of factor levels.
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Figure 5. The main effects plot of factors
2 1, id
id ϕ
ϕ and ϕid3.
It was observed that the sPSO methodology generated a better solution quality when the learning factors ϕid1 and ϕid2 were assigned at level 0.6 and ϕid3 was
assigned at level 0.8. Moreover, the number of particle sizes was given at 200 and the maximum iterations were set at 5000 trials as the termination criterion. These settings were used in the following experiment for the sPSO algorithm. On the other hand, the population size of the genes in the GA method was set at 50 with the mutation rate of 0.06 and crossover rate of 0.15. The maximum iterations were 5000 trials with 30 replications for each instance, applied to both methods.
4.2 Computational Results
The experiment involved 60 datasets of VRP pickup and delivery benchmark problems with the condition of homogeneous demand. Each instance was replicated for 30 runs in the sPSO algorithm. The solution from the sweep method, which generated proper clusters of shipments for grouping vehicles was used as the initial solution for the sPSO method. A performance comparison of the sPSO method and the GA method was computed based on the improvement rate (IR) as shown in Equation (17). The experiment results are given in Table 2.
% GA 100
GA -(IR) sPSO
Rate t Improvemen
cost total
cost total cost
total ×
=
(17)
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As listed, the 60 benchmark problems were sorted into 20 subgroups with 3 instances in each based on the node coordinates assigned. The instances within a given group were given identical node coordinates, except for the position where the Cross-docking depot was located. Each set of 60 instances contained 100 nodes, comprising both pickup and delivery customers. Each node was associated with a known demand of 10 units, while each vehicle dispatched was constrained with a capacity limit of 100 units.
The computational results showed that the solution quality of the sPSO method was superior to that of the GA algorithm, outperforming it with an average improvement rate and lowest improvement rate of 39.3% and 26.19%, respectively, on the basis of the average solution quality. In addition, the sPSO algorithm was able to find new, better solutions at a lower cost than the GA method for all the 60 benchmarks. Moreover, the maximum rate of improvement in the average solution quality was 42.18%, whereas the minimum rate acquired was 6.63%, indicating that the performance of the sPSO method was robustly competitive with the GA method.
Moreover, all the problems’ average improvement rates represented new, better solutions than those found by the GA method.
Figure 6 illustrates the convergence of the sPSO method compared with the GA method in 5000 iterations. In addition, the computation time of the sPSO method was approximately the same as the GA method in 5000 iterations. Also, a one-way ANOVA test was conducted to test the hypothesis that µsPSO = µGA for the data of Table 2. The result from the ANOVA test showed that there was a significant difference between the two models since the P-value < 0.001. Hence, based on the comparatively superior performance presented by the sPSO method, we can conclude that our designed sPSO was reliable and consistent in searching for the global optimal or near-optimal solutions.
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Table 2
Comparisons from the sPSO Algorithm and the GA method.
GA sPSO sPSO / GA
16
Table 2
Comparisons from the sPSO Algorithm and the GA method. (Cont.)
33P1 2868.16 2063.59 33.00 1662.0 1662.80 32.18 42.03% 19.42%
34P1 1643.60 1325.00 34.00 994.39 994.39 30.58 39.50% 24.95%
35P1 1869.79 1527.11 32.00 1194.00 1194.00 30.77 36.14% 21.81%
36P1 2272.05 2008.20 32.00 1606.90 1606.90 32.10 29.28% 19.98%
37P1 1666.12 1390.98 32.00 1119.40 1118.40 33.58 32.81% 19.60%
38P1 1934.95 1634.53 34.00 1286.50 1282.10 38.27 33.51% 21.56%
39P1 2261.51 1968.43 34.00 1668.10 1665.30 38.22 26.24% 15.40%
40P1 1805.12 1464.30 34.00 1064.30 1064.30 30.24 41.04% 27.32%
41P1 1989.66 1636.86 33.00 1254.20 1253.90 30.79 36.96% 23.40%
42P1 2380.18 2004.39 32.00 1668.50 1668.50 29.52 29.90% 16.76%
43P1 1668.72 1349.39 32.00 1135.30 1134.70 28.76 31.97% 15.91%
44P1 1899.70 1519.10 33.00 1334.30 1306.50 29.59 29.76% 14.00%
45P1 2224.75 1979.59 34.00 1684.90 1667.30 69.73 24.27% 15.78%
46P1 2025.59 1681.35 33.00 1146.80 1145.90 34.49 43.38% 31.85%
47P1 2146.87 1753.88 33.00 1327.10 1327.10 36.43 38.18% 24.33%
48P1 2609.74 2185.81 34.00 1712.20 1712.20 35.97 34.39% 21.67%
49P1 1990.00 1473.56 36.00 1375.90 1375.90 30.33 30.86% 6.63%
50P1 2223.77 1761.25 34.00 1284.50 1282.60 44.12 42.24% 27.18%
51P1 2615.53 2073.32 33.00 1744.30 1743.90 31.84 33.31% 15.89%
52P1 2127.57 1677.86 34.00 1303.00 1293.10 41.81 38.76% 22.93%
53P1 2334.54 1709.15 33.00 1460.90 1442.70 52.83 37.42% 15.59%
54P1 2804.30 2329.55 33.00 1835.70 1816.90 39.97 34.54% 22.01%
55P1 2160.84 1852.89 34.00 1222.90 1222.90 30.69 43.41% 34.00%
56P1 2210.49 1912.98 33.00 1429.70 1427.40 29.47 35.32% 25.38%
57P1 2638.74 2251.90 32.00 1861.60 1857.00 35.66 29.45% 17.54%
58P1 2070.97 1740.63 32.00 1223.90 1222.20 36.37 40.90% 29.78%
59P1 2160.69 1859.25 34.00 1401.00 1395.10 40.11 35.16% 24.96%
60P1 2634.82 2368.99 32.00 1847.70 1847.00 32.78 29.87% 22.03%
Avg. I.R. 33.12 35.31 39.30% 26.19%
Std. of I.R Max. I.R Min. I.R
55.24%
24.27%
42.18%
6.63%
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Figure 6. Convergence of the sPSO method and the GA method for instance 10P1.