A Combination of Genetic Algorithm and Particle Swarm Optimization for Vehicle Routing Problem with Time Windows
Abstract
:1. Introduction
Algorithms | Remarks | ||
---|---|---|---|
The exact algorithms | Branch and bound method [4,5] | The Efficiency depends on the depth of the branch and bound tree. | |
Set segmentation method [6,7] | Hard to determine the minimum cost for each solutions. | ||
Dynamic programming method [8,9] | Effective to limited-size problems, hard to consider the concrete demands such as time windows. | ||
Integer programming algorithm [10,11] | High precision, time consuming, complex. | ||
The heuristic algorithms | The traditional heuristic algorithms | Savings algorithm [12,13] | Computes rapidly, hard to get the optimal solution. |
Sweep algorithm [14,15] | Suitable to the same number of customers for each route with few routes. | ||
Two-phase algorithm [16,17] | Hard to get the optimal solution. | ||
The meta-heuristic algorithms | Tabu search algorithm [18,19,20] | Has the good ability of local search, but is time consuming, and depends on the initial solution. | |
Genetic algorithm [13,21] | Has the good ability of global search, computes rapidly, hard to obtain the global optimal solution. | ||
Iterated local search [22,23] | Has the strength of fast convergence rate and low computational complexity. | ||
Simulated annealing algorithm [24,25] | Slow convergence rates, carefully chosen tunable parameters. | ||
Variable neighborhood Search [26,27] | Is suitable for large and complex optimization problems with constraints. | ||
Ant colony algorithm [28,29,30] | Has good positive feedback mechanism, but is time consuming and prone to stagnation. | ||
Neural network algorithm [31,32] | Computes rapidly, has slow convergence and can easily be trapped in a local optimum | ||
Artificial bee colony algorithm [30,33] | Achieves a fast convergence speed, is associated with the piecewise linear cost approximation. | ||
Particle swarm optimization [34,35,36] | Is robust and has fast searching speed, brings easily premature convergence. | ||
Hybrid algorithm [2,8,12,20,28,37,38] | Is simple with fast optimizing speed and less calculation. |
2. Vehicle Routing Problem with Time Windows (VRPTW)
3. Particle Swarm Optimization (PSO)
4. The Proposed Algorithm
4.1. Particle Encoding
4.2. Inertia Weight Function
4.3. Crossover Operator of Genetic Algorithm
4.4. The Proposed Algorithm Flow
- (1)
- Parameters setting. Define the parameters: acceleration coefficients and , the maximum number of iterations , the initial value of inertia weight , the final value of inertia weight , a random number , the number of the particles , the maximum velocities of the particles .
- (2)
- Population initiation. Initialize particles as a population, generate the pth particle with random position , velocity , and personal best . Set iteration .
- (3)
- Particle encoding. According to the particle encoding rules, for , decode to a set of route .
- (4)
- Fitness evaluation. According to Equation (1), compute , and then evaluate through Equation (9).
- (5)
- and updating. Compute and . If , update . If , update .
- (6)
- Particles updating. Update the velocity and the position of each pth particle according to Equation (10).
- (7)
- Termination judgment. If the stopping criterion is met, go to step (9). Otherwise, and go to step (8). The stopping criterion is that or finding a better solution. A better solution means that the hierarchical cost objective value is better than that of the best solution found so far.
- (8)
- Crossover operator. Generate random number . According to Equations (13)–(16), generate a new set of population. Then return to step (3).
- (9)
- Outputting the optimal solution. Decode as the best set of vehicle route and output the optimal solution .
5. Experimental Results
5.1. Solomon Benchmark Problems
5.2. Evaluation of the Results
5.3. Analysis of the Results
5.3.1. The Total Traveled Distance
No. | Problem | Best-Known Solution | Genetic | PSO | ACO | The Proposed Algorithm | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Best | Average | Best | Average | Best | Average | Best | Average | ||||||||||||
TD | NV | TD | NV | TD | NV | TD | NV | TD | NV | TD | NV | TD | NV | TD | NV | TD | NV | ||
1 | C101 | 828.94 | 10 | 828.94 | 10 | 856.26 | 10.20 | 828.94 | 10 | 842.60 | 10.10 | 828.94 | 10 | 842.60 | 10.10 | 828.94 | 10 | 842.60 | 10.10 |
2 | C102 | 828.94 | 10 | 828.94 | 10 | 828.94 | 10.00 | 828.94 | 10 | 828.94 | 10.00 | 828.94 | 10 | 857.82 | 10.10 | 828.94 | 10 | 828.94 | 10.00 |
3 | C103 | 828.06 | 10 | 828.06 | 10 | 859.88 | 10.00 | 828.06 | 10 | 828.06 | 10.00 | 828.06 | 10 | 828.06 | 10.00 | 828.06 | 10 | 828.06 | 10.00 |
4 | C104 | 824.78 | 10 | 824.78 | 10 | 824.78 | 10.00 | 824.78 | 10 | 824.78 | 10.00 | 824.78 | 10 | 849.79 | 10.10 | 824.78 | 10 | 824.78 | 10.00 |
5 | C105 | 828.94 | 10 | 828.94 | 10 | 854.25 | 10.20 | 828.94 | 10 | 866.91 | 10.30 | 828.94 | 10 | 904.88 | 10.60 | 828.94 | 10 | 841.60 | 10.10 |
6 | C106 | 828.94 | 10 | 828.94 | 10 | 851.78 | 10.10 | 828.94 | 10 | 897.46 | 10.30 | 828.94 | 10 | 943.14 | 10.50 | 828.94 | 10 | 874.62 | 10.20 |
7 | C107 | 828.94 | 10 | 828.94 | 10 | 856.47 | 10.20 | 828.94 | 10 | 842.70 | 10.10 | 828.94 | 10 | 870.23 | 10.30 | 828.94 | 10 | 842.70 | 10.10 |
8 | C108 | 828.94 | 10 | 828.94 | 10 | 865.99 | 10.00 | 828.94 | 10 | 841.29 | 10.00 | 828.94 | 10 | 853.64 | 10.00 | 828.94 | 10 | 841.29 | 10.00 |
9 | C109 | 828.94 | 10 | 828.94 | 10 | 910.27 | 10.30 | 828.94 | 10 | 856.05 | 10.10 | 828.94 | 10 | 883.16 | 10.20 | 828.94 | 10 | 828.94 | 10.00 |
10 | C201 | 591.56 | 3 | 591.56 | 3 | 602.53 | 3.30 | 591.56 | 3 | 606.18 | 3.40 | 591.56 | 3 | 609.84 | 3.50 | 591.56 | 3 | 598.87 | 3.20 |
11 | C202 | 591.56 | 3 | 591.56 | 3 | 656.01 | 3.20 | 591.56 | 3 | 623.78 | 3.10 | 591.56 | 3 | 623.78 | 3.10 | 591.56 | 3 | 591.56 | 3.00 |
12 | C203 | 591.17 | 3 | 591.17 | 3 | 606.19 | 3.00 | 591.17 | 3 | 604.69 | 3.00 | 591.17 | 3 | 618.20 | 3.00 | 591.17 | 3 | 604.69 | 3.00 |
13 | C204 | 590.60 | 3 | 590.60 | 3 | 704.28 | 3.30 | 590.60 | 3 | 666.39 | 3.20 | 590.60 | 3 | 742.17 | 3.40 | 590.60 | 3 | 628.49 | 3.10 |
14 | C205 | 588.88 | 3 | 588.88 | 3 | 619.27 | 3.30 | 588.88 | 3 | 599.01 | 3.10 | 588.88 | 3 | 609.14 | 3.20 | 588.88 | 3 | 599.01 | 3.10 |
15 | C206 | 588.49 | 3 | 588.49 | 3 | 620.28 | 3.20 | 588.49 | 3 | 604.38 | 3.10 | 588.49 | 3 | 636.17 | 3.30 | 588.49 | 3 | 588.49 | 3.00 |
16 | C207 | 588.29 | 3 | 588.29 | 3 | 610.49 | 3.00 | 588.29 | 3 | 632.69 | 3.00 | 588.29 | 3 | 621.59 | 3.00 | 588.29 | 3 | 599.39 | 3.00 |
17 | C208 | 588.32 | 3 | 588.32 | 3 | 622.73 | 3.00 | 588.32 | 3 | 599.79 | 3.00 | 588.32 | 3 | 611.26 | 3.00 | 588.32 | 3 | 599.79 | 3.00 |
18 | R101 | 1483.57 | 16 | 1642.87 | 20 | 1645.83 | 19.50 | 1642.87 | 20 | 1645.33 | 19.50 | 1645.79 | 19 | 1647.29 | 19.00 | 1642.87 | 20 | 1645.83 | 19.50 |
19 | R102 | 1355.93 | 14 | 1482.74 | 18 | 1483.75 | 17.70 | 1472.62 | 18 | 1477.95 | 17.80 | 1480.73 | 18 | 1482.21 | 17.80 | 1472.62 | 18 | 1477.14 | 17.80 |
20 | R103 | 1133.35 | 12 | 1292.85 | 15 | 1248.88 | 14.40 | 1213.62 | 14 | 1239.30 | 14.40 | 1213.62 | 14 | 1247.22 | 14.30 | 1213.62 | 14 | 1239.01 | 13.80 |
21 | R104 | 968.28 | 10 | 982.01 | 10 | 995.05 | 9.60 | 1007.24 | 9 | 1007.30 | 9.00 | 982.01 | 10 | 1000.11 | 9.40 | 982.01 | 10 | 992.52 | 9.70 |
22 | R105 | 1262.53 | 12 | 1360.78 | 15 | 1366.87 | 14.80 | 1360.78 | 15 | 1374.14 | 14.70 | 1360.78 | 15 | 1369.99 | 15.30 | 1360.78 | 15 | 1366.42 | 15.00 |
23 | R106 | 1201.78 | 12 | 1249.40 | 13 | 1251.48 | 12.20 | 1241.52 | 13 | 1249.39 | 12.40 | 1251.98 | 12 | 1252.00 | 12.00 | 1241.52 | 13 | 1247.29 | 12.60 |
24 | R107 | 1051.92 | 11 | 1076.13 | 11 | 1094.18 | 10.70 | 1076.13 | 11 | 1094.19 | 10.50 | 1076.13 | 11 | 1096.08 | 10.70 | 1076.13 | 11 | 1088.49 | 10.70 |
25 | R108 | 948.57 | 10 | 963.99 | 9 | 965.42 | 9.90 | 963.99 | 9 | 965.10 | 9.70 | 948.57 | 10 | 960.92 | 9.60 | 948.57 | 10 | 955.82 | 9.60 |
26 | R109 | 1110.40 | 12 | 1151.84 | 13 | 1181.86 | 11.60 | 1151.84 | 13 | 1186.15 | 11.40 | 1151.84 | 13 | 1190.44 | 11.20 | 1151.84 | 13 | 1164.71 | 12.40 |
27 | R110 | 1080.36 | 11 | 1080.36 | 11 | 1112.79 | 10.30 | 1080.36 | 11 | 1105.14 | 10.50 | 1080.36 | 11 | 1107.72 | 10.50 | 1080.36 | 11 | 1098.14 | 10.70 |
28 | R111 | 987.80 | 10 | 1053.50 | 12 | 1086.43 | 10.80 | 1053.50 | 12 | 1083.13 | 11.60 | 1088.48 | 12 | 1095.07 | 10.40 | 1053.50 | 12 | 1069.14 | 11.60 |
29 | R112 | 953.63 | 10 | 982.14 | 9 | 995.33 | 10.80 | 953.63 | 10 | 965.03 | 10.00 | 953.63 | 10 | 987.40 | 10.60 | 960.68 | 10 | 968.58 | 9.90 |
30 | R201 | 1148.48 | 9 | 1148.48 | 9 | 1208.94 | 7.70 | 1179.79 | 9 | 1231.20 | 5.50 | 1148.48 | 9 | 1212.51 | 7.30 | 1148.48 | 9 | 1178.39 | 8.40 |
31 | R202 | 1049.74 | 7 | 1049.74 | 7 | 1086.62 | 5.70 | 1049.74 | 7 | 1100.68 | 5.00 | 1079.36 | 6 | 1142.69 | 4.10 | 1049.74 | 7 | 1081.57 | 5.70 |
32 | R203 | 900.08 | 5 | 900.08 | 5 | 930.23 | 4.60 | 932.76 | 7 | 939.72 | 3.80 | 939.54 | 3 | 941.70 | 3.00 | 900.08 | 5 | 922.71 | 4.60 |
33 | R204 | 772.33 | 4 | 807.38 | 4 | 828.06 | 2.40 | 772.33 | 4 | 817.42 | 2.80 | 772.33 | 4 | 823.17 | 2.80 | 772.33 | 4 | 801.47 | 3.40 |
34 | R205 | 959.74 | 4 | 970.89 | 6 | 982.66 | 4.50 | 970.89 | 6 | 980.30 | 4.80 | 970.89 | 6 | 989.71 | 3.60 | 970.89 | 6 | 977.95 | 5.10 |
35 | R206 | 898.91 | 5 | 898.91 | 5 | 906.06 | 3.40 | 906.14 | 3 | 910.24 | 3.00 | 906.14 | 3 | 912.29 | 3.00 | 898.91 | 5 | 903.89 | 4.00 |
36 | R207 | 814.78 | 3 | 814.78 | 3 | 844.73 | 3.40 | 814.78 | 3 | 868.25 | 2.60 | 814.78 | 3 | 876.51 | 2.60 | 814.78 | 3 | 836.67 | 3.00 |
37 | R208 | 715.37 | 3 | 725.75 | 2 | 726.61 | 2.00 | 725.42 | 4 | 725.77 | 3.20 | 725.75 | 2 | 726.71 | 2.00 | 723.61 | 3 | 725.50 | 2.50 |
38 | R209 | 879.53 | 6 | 879.53 | 6 | 892.08 | 5.40 | 879.53 | 6 | 891.00 | 5.40 | 879.53 | 6 | 903.21 | 4.20 | 879.53 | 6 | 891.82 | 5.10 |
39 | R210 | 932.89 | 7 | 954.12 | 3 | 955.06 | 6.60 | 954.12 | 3 | 954.64 | 5.00 | 939.34 | 3 | 952.94 | 4.78 | 932.89 | 7 | 937.89 | 6.20 |
40 | R211 | 761.10 | 4 | 885.71 | 2 | 892.31 | 2.30 | 885.71 | 2 | 889.53 | 4.40 | 888.73 | 5 | 867.07 | 2.90 | 808.56 | 4 | 824.99 | 3.90 |
41 | RC101 | 1481.27 | 13 | 1660.10 | 16 | 1689.57 | 14.40 | 1623.58 | 15 | 1658.18 | 15.10 | 1639.97 | 16 | 1687.56 | 14.40 | 1623.58 | 15 | 1645.18 | 15.10 |
42 | RC102 | 1395.25 | 13 | 1466.84 | 14 | 1493.53 | 13.50 | 1482.91 | 14 | 1497.28 | 13.60 | 1477.54 | 13 | 1539.85 | 12.30 | 1466.84 | 14 | 1487.10 | 13.50 |
43 | RC103 | 1221.53 | 10 | 1261.67 | 11 | 1263.56 | 11.00 | 1262.02 | 11 | 1262.29 | 11.00 | 1262.02 | 11 | 1264.17 | 11.00 | 1261.67 | 11 | 1262.04 | 11.00 |
44 | RC104 | 1135.48 | 10 | 1135.48 | 10 | 1135.50 | 10.00 | 1135.48 | 10 | 1135.50 | 10.00 | 1135.48 | 10 | 1135.51 | 10.00 | 1135.48 | 10 | 1135.49 | 10.00 |
45 | RC105 | 1354.20 | 12 | 1518.60 | 16 | 1601.17 | 15.40 | 1518.60 | 16 | 1593.35 | 14.80 | 1629.44 | 13 | 1633.29 | 13.00 | 1618.55 | 16 | 1621.16 | 15.40 |
46 | RC106 | 1226.62 | 11 | 1377.35 | 13 | 1396.59 | 11.80 | 1384.92 | 12 | 1417.25 | 11.20 | 1377.35 | 13 | 1416.49 | 11.30 | 1377.35 | 13 | 1392.80 | 12.30 |
47 | RC107 | 1150.99 | 10 | 1230.48 | 11 | 1254.98 | 12.60 | 1212.83 | 12 | 1240.50 | 12.30 | 1230.48 | 11 | 1258.04 | 12.80 | 1212.83 | 12 | 1226.01 | 12.00 |
48 | RC108 | 1076.81 | 10 | 1117.53 | 11 | 1135.32 | 10.30 | 1117.53 | 11 | 1127.41 | 10.90 | 1117.53 | 11 | 1128.55 | 10.80 | 1117.53 | 11 | 1126.40 | 10.70 |
49 | RC201 | 1134.91 | 6 | 1406.91 | 4 | 1391.43 | 7.20 | 1406.91 | 4 | 1406.91 | 4.00 | 1286.83 | 9 | 1397.23 | 6.00 | 1387.55 | 8 | 1391.43 | 7.20 |
50 | RC202 | 1113.53 | 8 | 1365.57 | 4 | 1250.18 | 3.60 | 1113.53 | 8 | 1162.75 | 7.60 | 1113.53 | 8 | 1204.86 | 6.30 | 1148.84 | 9 | 1173.34 | 7.90 |
51 | RC203 | 945.96 | 5 | 945.96 | 5 | 1034.30 | 3.40 | 945.96 | 5 | 1032.14 | 3.40 | 1049.62 | 3 | 1052.87 | 3.00 | 945.96 | 5 | 990.67 | 4.20 |
52 | RC204 | 796.11 | 4 | 798.41 | 3 | 798.69 | 3.20 | 798.41 | 3 | 799.18 | 3.60 | 798.46 | 3 | 799.43 | 3.80 | 798.41 | 3 | 798.67 | 3.20 |
53 | RC205 | 1168.22 | 8 | 1270.69 | 7 | 1276.08 | 6.40 | 1168.22 | 8 | 1282.01 | 4.70 | 1168.22 | 8 | 1263.14 | 6.80 | 1161.81 | 7 | 1187.56 | 6.90 |
54 | RC206 | 1059.89 | 7 | 1059.89 | 7 | 1105.75 | 5.80 | 1084.30 | 8 | 1139.24 | 4.00 | 1059.89 | 7 | 1135.44 | 4.00 | 1059.89 | 7 | 1092.22 | 6.00 |
55 | RC207 | 976.40 | 7 | 999.26 | 6 | 1060.37 | 3.60 | 976.40 | 7 | 1011.75 | 5.70 | 1053.58 | 6 | 1064.28 | 3.90 | 976.40 | 7 | 995.65 | 6.40 |
56 | RC208 | 785.93 | 4 | 816.10 | 5 | 824.93 | 3.60 | 816.10 | 5 | 822.41 | 4.00 | 806.87 | 5 | 819.64 | 4.00 | 795.39 | 5 | 807.27 | 4.78 |
5.3.2. The Average Total Traveled Distance
Problem | Best-Known Solution | Genetic | PSO | ACO | The Proposed Algorithm | |||||
---|---|---|---|---|---|---|---|---|---|---|
Best | Average | Best | Average | Best | Average | Best | Average | |||
C1-type | NV | 10.00 | 10.00 | 10.11 | 10.00 | 10.10 | 10.00 | 10.21 | 10.00 | 10.06 |
TD | 828.38 | 828.38 | 856.51 | 828.38 | 847.64 | 828.38 | 870.37 | 828.38 | 839.28 | |
C2-type | NV | 3.00 | 3.00 | 3.16 | 3.00 | 3.11 | 3.00 | 3.19 | 3.00 | 3.05 |
TD | 589.86 | 589.86 | 630.22 | 589.86 | 617.11 | 589.86 | 634.02 | 589.86 | 601.29 | |
R1-type | NV | 11.67 | 13.00 | 12.69 | 12.92 | 12.63 | 12.92 | 12.57 | 13.08 | 12.78 |
TD | 1128.18 | 1193.22 | 1202.32 | 1184.84 | 1199.35 | 1186.16 | 1203.04 | 1182.04 | 1192.76 | |
R2-type | NV | 5.18 | 4.73 | 4.36 | 4.91 | 4.14 | 4.55 | 3.66 | 5.36 | 4.72 |
TD | 893.90 | 912.31 | 932.12 | 915.56 | 937.16 | 914.99 | 940.77 | 899.98 | 916.62 | |
RC1-type | NV | 11.13 | 12.75 | 12.38 | 12.63 | 12.36 | 12.25 | 11.95 | 12.75 | 12.50 |
TD | 1255.27 | 1346.01 | 1371.28 | 1342.23 | 1366.47 | 1358.73 | 1382.93 | 1351.73 | 1362.02 | |
RC2-type | NV | 6.13 | 5.13 | 4.60 | 6.00 | 4.63 | 6.13 | 4.73 | 6.38 | 5.82 |
TD | 997.62 | 1082.85 | 1092.72 | 1038.73 | 1082.05 | 1042.13 | 1092.11 | 1034.28 | 1054.60 | |
All | NV | 449 | 465 | 452.40 | 472 | 448.70 | 466 | 441.88 | 483 | 466.68 |
TD | 53,568.46 | 55,959.11 | 57,143.53 | 55,511.30 | 56,854.75 | 55,679.89 | 57,490.75 | 55,346.67 | 56,092.75 |
5.3.3. The Average Central Processing Unit (CPU) Runtime
Problem | Genetic | PSO | ACO | The Proposed Algorithm |
---|---|---|---|---|
C1e | 98 | 85 | 111 | 62 |
C2 | 312 | 231 | 338 | 135 |
R1 | 112 | 81 | 124 | 60 |
R2 | 309 | 273 | 553 | 182 |
RC1 | 79 | 80 | 82 | 58 |
RC2 | 297 | 257 | 333 | 149 |
5.3.4. The Quality of the Solutions
Problem | Genetic | PSO | ACO | The Proposed Algorithm |
---|---|---|---|---|
C1-type | 3.39% | 2.32% | 5.07% | 1.32% |
C2-type | 6.84% | 4.62% | 7.48% | 1.94% |
R1-type | 6.28% | 5.98% | 6.33% | 5.36% |
R2-type | 4.44% | 4.93% | 5.29% | 2.62% |
RC1-type | 8.89% | 8.53% | 9.75% | 8.17% |
RC2-type | 8.96% | 7.92% | 8.95% | 5.30% |
All | 6.29% | 5.63% | 6.95% | 4.07% |
6. Conclusions
- (1)
- The real encoding method avoids the complex encoding and decoding computation burden.
- (2)
- A linear decreasing function of the number of iterations in PSO have a flexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities, which can help find the optimal solution with the least number of iterations.
- (3)
- The crossover operator of the genetic algorithm is introduced to generate a new population guaranteeing that the offspring inherits good qualities from this parent. The crossover operator avoids premature convergence and local minimum value and increases the diversity of particles.
Acknowledgements
Author Contributions
Conflicts of Interest
References
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Xu, S.-H.; Liu, J.-P.; Zhang, F.-H.; Wang, L.; Sun, L.-J. A Combination of Genetic Algorithm and Particle Swarm Optimization for Vehicle Routing Problem with Time Windows. Sensors 2015, 15, 21033-21053. https://doi.org/10.3390/s150921033
Xu S-H, Liu J-P, Zhang F-H, Wang L, Sun L-J. A Combination of Genetic Algorithm and Particle Swarm Optimization for Vehicle Routing Problem with Time Windows. Sensors. 2015; 15(9):21033-21053. https://doi.org/10.3390/s150921033
Chicago/Turabian StyleXu, Sheng-Hua, Ji-Ping Liu, Fu-Hao Zhang, Liang Wang, and Li-Jian Sun. 2015. "A Combination of Genetic Algorithm and Particle Swarm Optimization for Vehicle Routing Problem with Time Windows" Sensors 15, no. 9: 21033-21053. https://doi.org/10.3390/s150921033
APA StyleXu, S.-H., Liu, J.-P., Zhang, F.-H., Wang, L., & Sun, L.-J. (2015). A Combination of Genetic Algorithm and Particle Swarm Optimization for Vehicle Routing Problem with Time Windows. Sensors, 15(9), 21033-21053. https://doi.org/10.3390/s150921033