Introduction
Problem description and formulation
Definition of multi-objective optimization problem
Description of EEDNWFSP
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All factories have the same processing capacities, i.e., the number of machines and processing ability are the same.
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All machines are available at time zero, and all jobs can be scheduled at this time.
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Each job can be processed on exactly one machine at a time in the same factory.
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Each machine can process only one job at a time.
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Preemption is not permitted, that is, one job should be completed on the assigned machine without any interruption.
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The processing speed of each machine can be adjusted; therefore, the actual processing time and machine energy consumption should be varied with the speed.
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During the processing of a job, the speed of each machine cannot be changed.
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For two successive operations on the same machine, the start processing time of the subsequent operation should be greater than or equal to the completion time of the previous one.
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No-wait constraint should be guaranteed, that is, after completing the previous stage, the job should start its next stage immediately.
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The sequence dependent setup times cannot be ignored.
Notations and parameters
\(i\): | Index for jobs, \(i = 1,2,...,n\) |
\(k\): | Index for job positions in a sequence, \(i = 1,2,...,n\) |
\(j\): | Index for machines, \(j = 1,2,...,m\) |
\(f\): | Index for factories, \(f = 1,2,...,g\) |
\(v\): | Index for speeds, \(v = 1,2,...,s\) |
\(n\): | The number of jobs |
\(m\): | The number of machines in each factory |
\(g\): | The number of factories |
\(s\): | The number of processing speeds |
\(p_{i,j}\): | The standard processing time of job i at machine j |
\(V_{v}\): | The processing speed at v th |
\(st_{j,i,i^{\prime}}\): | The setup-time of job i' on machine j if job i is the immediately preceding job |
\(PE_{j,v}\): | The unit time energy consumption of machine j running at v th speed |
\(SPE_{j}\): | The unit time energy consumption of machine j in stand-by state |
\(SE_{j,i,i^{\prime}}\): | The unit setup-time energy consumption of job \(i^{\prime}\) on machine j if job i is the immediately preceding job |
\(M\): | A very large positive value |
\(x_{i,f,k}\): | Binary variable, if job i occupies position k in factory f, then the value is 1, and 0 otherwise |
\(y_{i,j,v}\): | Binary variable, if job i is processed with speed v on machine j, then the value is 1, and 0 otherwise |
\(t_{i,j}\): | The actual processing time of job i at machine j |
\(pec_{i,j}\): | The processing energy consumption of job i at machine j |
\(S_{i,j,f}\): | The time that job i in factory f starts processing on machine j |
\(C_{i,j,f}\): | The completion time of job i on machine j in factory f |
\(C_{f}\): | The completion time of factory f |
\(C_{\max }\): | The total completion time, that is, the maximum completion time in g factories |
\(PEC\): | The energy consumption of the machine in the processing state |
\(SPEC\): | The energy consumption of the machine in stand-by state |
\(SEC\): | The energy consumption of the machine in the state of setup |
\(TEC\) | The total energy consumption |
Formulation of EEDNWFSP
Problem-specific properties
Improved NSGA-II algorithm
Framework of INSGA-II
Encoding and illustration
M1 | M2 | M3 | |
---|---|---|---|
Speed/PEC | 1/2 | 1/4 | 1/2 |
2/6 | 2/12 | 2/6 | |
SPEC | 1 | 2 | 1 |
J1 | 32 | 21 | 24 |
J2 | 21 | 20 | 31 |
J3 | 11 | 29 | 18 |
J4 | 29 | 12 | 10 |
J5 | 28 | 18 | 24 |
J6 | 14 | 13 | 33 |
J1 | J2 | J3 | J4 | J5 | J6 | ||
---|---|---|---|---|---|---|---|
M1 | J1 | 3/1 | 2/1 | 1/2 | 3/1 | 4/1 | 2/2 |
J2 | 3/2 | 2/2 | 8/2 | 6/2 | 3/1 | 6/2 | |
J3 | 2/1 | 4/2 | 6/1 | 2/1 | 8/2 | 5/1 | |
J4 | 3/1 | 5/1 | 5/1 | 4/2 | 6/1 | 9/2 | |
J5 | 9/2 | 8/2 | 3/2 | 5/1 | 10/2 | 3/2 | |
J6 | 3/1 | 9/2 | 5/2 | 2/2 | 3/2 | 1/1 | |
M2 | J1 | 5/1 | 3/1 | 5/2 | 10/2 | 9/2 | 6/1 |
J2 | 8/2 | 9/2 | 4/1 | 9/2 | 9/2 | 3/1 | |
J3 | 2/2 | 4/2 | 8/1 | 5/1 | 10/1 | 6/2 | |
J4 | 3/1 | 5/1 | 5/2 | 8/2 | 6/1 | 4/2 | |
J5 | 3/1 | 4/2 | 6/2 | 5/1 | 10/1 | 7/2 | |
J6 | 6/2 | 9/1 | 5/1 | 7/2 | 8/1 | 10/1 | |
M3 | J1 | 3/1 | 6/1 | 2/1 | 9/1 | 5/2 | 8/2 |
J2 | 3/2 | 10/2 | 8/2 | 6/2 | 5/1 | 6/2 | |
J3 | 2/2 | 4/1 | 6/1 | 5/1 | 8/1 | 3/1 | |
J4 | 4/2 | 5/2 | 5/2 | 10/2 | 8/2 | 3/2 | |
J5 | 9/1 | 8/1 | 5/1 | 5/1 | 10/1 | 6/2 | |
J6 | 3/2 | 9/1 | 5/2 | 2/2 | 3/2 | 6/1 |
Initialization procedure
Local search
DST-based speed adjustment heuristics
Mutation operators
Global search
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Step 1: for each job, count the number of subsequent jobs in the current non-dominated solution set to find the one with the most occurrences. A temporary set is consisted of these gene pairs.
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Step 2: two individuals were randomly selected as parents from the parent population.
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Step 3: in a parent, for a job in each position, a gene pair is formed with its subsequent job, and the gene pair is searched in the temporary set.
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Step 4: if the parent and the temporary set have the common gene pairs, the identical gene pairs are put into offspring at the same position.
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Step 5: otherwise, compare two parent genes in the same position, and put the common gene into the same position of the corresponding offspring respectively.
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Step 6: the genes of the offspring in the rest positions are filled using the one-point order crossover (OP) [45] based on the two parents.
Experiments and results
Experimental setup
Performance indicators
Parameter setting
Parameter | Factor level | ||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | |
pc | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
pm | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
Efficiency of the proposed components
Effect of initialization
n × m | HV(g = 2) | HV(g = 3) | HV(g = 4) | HV(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G1 | INSGAII | G1 | INSGAII | G1 | INSGAII | G1 | |
20 × 4 | 0.6958 | 0.6877 | 0.6692 | 0.6594 | 0.6706 | 0.6603 | 0.7495 | 0.7474 |
20 × 8 | 0.6612 | 0.6560 | 0.7135 | 0.7100 | 0.7305 | 0.7239 | 0.6907 | 0.6872 |
20 × 16 | 0.6194 | 0.6167 | 0.6984 | 0.7008 | 0.7199 | 0.7135 | 0.7324 | 0.7192 |
40 × 4 | 0.6851 | 0.6639 | 0.7157 | 0.6860 | 0.6695 | 0.6434 | 0.6794 | 0.6533 |
40 × 8 | 0.6483 | 0.6183 | 0.6476 | 0.6252 | 0.6331 | 0.6098 | 0.7386 | 0.7274 |
40 × 16 | 0.6115 | 0.5900 | 0.6092 | 0.5789 | 0.6750 | 0.6641 | 0.6964 | 0.6923 |
60 × 4 | 0.6549 | 0.6282 | 0.6905 | 0.6712 | 0.6732 | 0.6396 | 0.7021 | 0.6750 |
60 × 8 | 0.6333 | 0.5981 | 0.6360 | 0.6096 | 0.6978 | 0.6797 | 0.7028 | 0.6940 |
60 × 16 | 0.5986 | 0.5740 | 0.6021 | 0.5748 | 0.6085 | 0.5755 | 0.6790 | 0.6691 |
80 × 4 | 0.6499 | 0.6136 | 0.6692 | 0.6293 | 0.6472 | 0.6171 | 0.6490 | 0.6171 |
80 × 8 | 0.6386 | 0.6090 | 0.6556 | 0.6202 | 0.6366 | 0.5970 | 0.6342 | 0.5945 |
80 × 16 | 0.5989 | 0.5708 | 0.5959 | 0.5671 | 0.6051 | 0.5703 | 0.5936 | 0.5504 |
100 × 4 | 0.6586 | 0.6168 | 0.6460 | 0.6022 | 0.6975 | 0.6659 | 0.6528 | 0.6027 |
100 × 8 | 0.6264 | 0.5920 | 0.6229 | 0.5807 | 0.6251 | 0.5826 | 0.6343 | 0.5824 |
100 × 16 | 0.6253 | 0.5883 | 0.6037 | 0.5810 | 0.6083 | 0.5836 | 0.6732 | 0.6561 |
Avg | 0.6404 | 0.6149 | 0.6517 | 0.6264 | 0.6599 | 0.6351 | 0.6805 | 0.6579 |
n × m | IGD(g = 2) | IGD(g = 3) | IGD(g = 4) | IGD(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G1 | INSGAII | G1 | INSGAII | G1 | INSGAII | G1 | |
20 × 4 | 5.8176 | 28.1926 | 5.1728 | 18.5917 | 0.8602 | 10.7065 | 10.7807 | 21.6783 |
20 × 8 | 11.6968 | 11.5433 | 40.5108 | 31.2564 | 5.5459 | 16.3955 | 12.1780 | 11.1968 |
20 × 16 | 4.6023 | 35.3588 | 11.6726 | 64.3980 | 24.6725 | 26.9317 | 10.2060 | 44.2591 |
40 × 4 | 5.7964 | 60.1374 | 0.0000 | 54.7883 | 0.2925 | 33.6016 | 0.0000 | 41.7198 |
40 × 8 | 0.7945 | 64.4567 | 4.5307 | 74.2285 | 0.9089 | 37.0931 | 11.9167 | 65.8171 |
40 × 16 | 2.3528 | 76.8651 | 0.7798 | 113.3179 | 18.9093 | 54.3116 | 42.9864 | 44.8782 |
60 × 4 | 0.8241 | 96.7012 | 7.4167 | 63.6580 | 1.5060 | 68.8097 | 5.4764 | 59.7686 |
60 × 8 | 0.0000 | 118.5663 | 5.6344 | 94.1960 | 23.8459 | 115.1965 | 14.0692 | 82.6480 |
60 × 16 | 2.3617 | 124.9135 | 5.3598 | 135.3208 | 1.1165 | 139.8077 | 9.6808 | 63.2765 |
80 × 4 | 0.0000 | 141.6163 | 0.0000 | 155.6765 | 3.8399 | 69.7350 | 8.5067 | 68.3357 |
80 × 8 | 2.2220 | 183.5328 | 2.3344 | 184.9940 | 3.4195 | 263.4318 | 0.0000 | 186.2944 |
80 × 16 | 8.8220 | 254.4020 | 16.3215 | 194.3021 | 7.3209 | 238.1768 | 0.0000 | 232.2292 |
100 × 4 | 0.0000 | 211.4340 | 1.4399 | 153.5090 | 9.6292 | 176.1901 | 0.0000 | 114.2430 |
100 × 8 | 0.0000 | 289.5759 | 0.7433 | 241.9929 | 0.0000 | 206.9498 | 0.3443 | 195.0756 |
100 × 16 | 1.1567 | 322.1684 | 191.4744 | 170.0620 | 39.3091 | 251.1446 | 38.9450 | 434.0927 |
Avg | 3.0965 | 134.6309 | 19.5594 | 116.6861 | 9.4118 | 113.8988 | 11.0060 | 111.0342 |
Effect of search strategies
n × m | HV(g = 2) | HV(g = 3) | HV(g = 4) | HV(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G2 | INSGAII | G2 | INSGAII | G2 | INSGAII | G2 | |
20 × 4 | 0.6777 | 0.6451 | 0.6637 | 0.6311 | 0.6713 | 0.6458 | 0.7699 | 0.7281 |
20 × 8 | 0.6451 | 0.5966 | 0.7213 | 0.6757 | 0.7023 | 0.6667 | 0.7271 | 0.6894 |
20 × 16 | 0.6079 | 0.5677 | 0.6686 | 0.6257 | 0.6831 | 0.6471 | 0.7245 | 0.6855 |
40 × 4 | 0.6635 | 0.6104 | 0.6688 | 0.6183 | 0.6510 | 0.6055 | 0.7205 | 0.6637 |
40 × 8 | 0.6343 | 0.5641 | 0.6392 | 0.5743 | 0.6539 | 0.5997 | 0.6510 | 0.5846 |
40 × 16 | 0.6016 | 0.5221 | 0.6095 | 0.5377 | 0.5904 | 0.5342 | 0.6936 | 0.6383 |
60 × 4 | 0.6808 | 0.6278 | 0.6527 | 0.6043 | 0.6617 | 0.6040 | 0.6690 | 0.6132 |
60 × 8 | 0.6308 | 0.5540 | 0.6463 | 0.5808 | 0.6258 | 0.5522 | 0.6353 | 0.5712 |
60 × 16 | 0.5865 | 0.5097 | 0.5924 | 0.5131 | 0.5975 | 0.5215 | 0.6683 | 0.5964 |
80 × 4 | 0.6529 | 0.5889 | 0.6645 | 0.6013 | 0.6674 | 0.6184 | 0.6590 | 0.5998 |
80 × 8 | 0.6177 | 0.5427 | 0.6336 | 0.5613 | 0.6416 | 0.5674 | 0.6405 | 0.5747 |
80 × 16 | 0.5873 | 0.5073 | 0.6140 | 0.5270 | 0.5969 | 0.5153 | 0.5878 | 0.5158 |
100 × 4 | 0.6475 | 0.5991 | 0.6497 | 0.5904 | 0.6619 | 0.6087 | 0.6651 | 0.6061 |
100 × 8 | 0.6314 | 0.5524 | 0.6491 | 0.5651 | 0.6332 | 0.5523 | 0.6308 | 0.5535 |
100 × 16 | 0.5848 | 0.4975 | 0.5825 | 0.5008 | 0.5793 | 0.4972 | 0.6653 | 0.5730 |
Avg | 0.6300 | 0.5657 | 0.6437 | 0.5805 | 0.6411 | 0.5824 | 0.6738 | 0.6129 |
n × m | IGD(g = 2) | IGD(g = 3) | IGD(g = 4) | IGD(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G2 | INSGAII | G2 | INSGAII | G2 | INSGAII | G2 | |
20 × 4 | 13.9173 | 562.0828 | 5.9056 | 679.5463 | 3.8929 | 48.6234 | 7.2012 | 692.0580 |
20 × 8 | 1.8034 | 171.5275 | 3.0435 | 994.7606 | 1.3680 | 155.1355 | 2.9209 | 171.1599 |
20 × 16 | 1.1953 | 579.4234 | 0.0715 | 426.1558 | 0.4948 | 286.2467 | 4.2273 | 368.9697 |
40 × 4 | 4.8949 | 173.7518 | 1.8528 | 164.6672 | 0.6844 | 217.5063 | 3.3446 | 265.9727 |
40 × 8 | 0.0000 | 748.2804 | 0.0000 | 625.5124 | 0.6971 | 474.5985 | 0.0000 | 735.3733 |
40 × 16 | 0.0000 | 2357.4690 | 0.5243 | 2063.0820 | 0.0000 | 1552.1410 | 0.0000 | 1289.8630 |
60 × 4 | 2.0679 | 284.6895 | 0.1819 | 235.5487 | 1.0481 | 284.0066 | 0.1752 | 406.9018 |
60 × 8 | 0.0000 | 1221.7430 | 36.6981 | 1253.6310 | 0.1144 | 1057.4070 | 0.0000 | 1164.8740 |
60 × 16 | 0.0000 | 3055.8730 | 14.0706 | 3275.6730 | 0.0000 | 3435.2940 | 0.0000 | 3025.9910 |
80 × 4 | 0.0000 | 489.2292 | 0.8371 | 426.4559 | 2.8838 | 292.5117 | 0.0000 | 572.1952 |
80 × 8 | 0.0000 | 1533.9490 | 4.7919 | 1502.6310 | 0.0000 | 1481.4350 | 0.6971 | 1868.7260 |
80 × 16 | 0.0000 | 4416.9660 | 0.0000 | 5125.1920 | 0.0000 | 4676.1170 | 0.0000 | 4396.7220 |
100 × 4 | 3.8736 | 457.7837 | 0.0000 | 455.5416 | 14.9347 | 664.3326 | 6.7696 | 542.6256 |
100 × 8 | 0.0000 | 1844.6790 | 0.0000 | 2471.5640 | 0.0000 | 2591.4540 | 0.0000 | 2001.6710 |
100 × 16 | 0.9614 | 6517.1050 | 2.5021 | 6442.5340 | 0.0000 | 5906.0340 | 0.2764 | 6523.7930 |
Avg | 1.9142 | 1627.6368 | 4.6986 | 1742.8330 | 1.7412 | 1541.5229 | 1.7075 | 1601.7931 |
n × m | HV(g = 2) | HV(g = 3) | HV(g = 4) | HV(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G3 | INSGAII | G3 | INSGAII | G3 | INSGAII | G3 | |
20 × 4 | 0.6620 | 0.6607 | 0.6660 | 0.6690 | 0.7501 | 0.7476 | 0.6609 | 0.6525 |
20 × 8 | 0.6607 | 0.6509 | 0.6728 | 0.6615 | 0.6988 | 0.6931 | 0.7064 | 0.6896 |
20 × 16 | 0.5955 | 0.5921 | 0.6630 | 0.6454 | 0.6394 | 0.6147 | 0.7367 | 0.7005 |
40 × 4 | 0.6664 | 0.6583 | 0.6797 | 0.6638 | 0.6577 | 0.6442 | 0.6804 | 0.6611 |
40 × 8 | 0.6525 | 0.6384 | 0.6236 | 0.6170 | 0.7167 | 0.7128 | 0.6217 | 0.6131 |
40 × 16 | 0.6082 | 0.6063 | 0.6672 | 0.6616 | 0.6065 | 0.5995 | 0.6840 | 0.6773 |
60 × 4 | 0.6734 | 0.6496 | 0.6398 | 0.6169 | 0.6503 | 0.6300 | 0.6630 | 0.6411 |
60 × 8 | 0.6205 | 0.6093 | 0.6490 | 0.6300 | 0.6905 | 0.6778 | 0.6349 | 0.6233 |
60 × 16 | 0.5945 | 0.5964 | 0.5975 | 0.5910 | 0.6034 | 0.5932 | 0.6844 | 0.6804 |
80 × 4 | 0.6633 | 0.6365 | 0.6637 | 0.6329 | 0.6478 | 0.6263 | 0.6593 | 0.6394 |
80 × 8 | 0.6403 | 0.6239 | 0.6327 | 0.6138 | 0.6186 | 0.6043 | 0.6124 | 0.5934 |
80 × 16 | 0.5866 | 0.5800 | 0.6030 | 0.5861 | 0.5967 | 0.5949 | 0.5928 | 0.5853 |
100 × 4 | 0.6377 | 0.6168 | 0.6535 | 0.6277 | 0.6549 | 0.6327 | 0.6426 | 0.6154 |
100 × 8 | 0.6433 | 0.6217 | 0.6270 | 0.6044 | 0.6365 | 0.6182 | 0.6245 | 0.6074 |
100 × 16 | 0.6136 | 0.6047 | 0.6122 | 0.5982 | 0.5895 | 0.5843 | 0.5924 | 0.5824 |
Avg | 0.6346 | 0.6230 | 0.6434 | 0.6280 | 0.6505 | 0.6382 | 0.6531 | 0.6375 |
n × m | IGD(g = 2) | IGD(g = 3) | IGD(g = 4) | IGD(g = 5) | ||||
---|---|---|---|---|---|---|---|---|
INSGAII | G3 | INSGAII | G3 | INSGAII | G3 | INSGAII | G3 | |
20 × 4 | 6.4253 | 28.1913 | 17.3912 | 36.3181 | 2.6949 | 20.9009 | 5.1159 | 87.1628 |
20 × 8 | 6.6052 | 27.3634 | 6.5460 | 108.4567 | 5.3313 | 14.3616 | 4.5036 | 45.4794 |
20 × 16 | 10.5063 | 12.9577 | 3.7925 | 79.7895 | 11.3341 | 57.7334 | 6.9817 | 240.6732 |
40 × 4 | 6.1780 | 32.6651 | 1.9207 | 43.3112 | 2.7813 | 43.2867 | 2.2286 | 43.7553 |
40 × 8 | 5.1923 | 36.9973 | 4.7158 | 33.0645 | 2.4525 | 31.9487 | 5.6525 | 47.2015 |
40 × 16 | 6.6654 | 32.6637 | 7.3149 | 34.1539 | 4.1873 | 32.8176 | 29.4323 | 158.7961 |
60 × 4 | 0.7880 | 93.7319 | 2.2722 | 55.3519 | 2.3249 | 64.4927 | 1.6483 | 99.5842 |
60 × 8 | 7.7062 | 56.7305 | 6.1223 | 148.4391 | 9.2609 | 63.0451 | 6.6078 | 65.3631 |
60 × 16 | 14.8379 | 28.2828 | 16.5978 | 83.9112 | 6.8725 | 69.4887 | 9.2879 | 60.2524 |
80 × 4 | 5.5407 | 135.7429 | 3.0102 | 170.8620 | 4.9320 | 103.9318 | 3.1661 | 88.8715 |
80 × 8 | 7.2935 | 154.8194 | 45.2692 | 220.5064 | 12.6506 | 96.5460 | 4.0060 | 140.0035 |
80 × 16 | 9.0627 | 71.4499 | 4.8200 | 263.2319 | 10.6508 | 23.1394 | 18.0773 | 149.5012 |
100 × 4 | 5.0054 | 157.6446 | 3.0357 | 213.9454 | 7.2274 | 127.1013 | 2.6644 | 233.0220 |
100 × 8 | 3.7206 | 173.4658 | 16.5590 | 246.1586 | 12.7700 | 175.4973 | 0.3867 | 208.5464 |
100 × 16 | 9.6454 | 159.3736 | 10.1612 | 293.1734 | 50.3596 | 96.7395 | 24.6897 | 231.9344 |
Avg | 7.0115 | 80.1387 | 9.9686 | 135.3782 | 9.7220 | 68.0687 | 8.2966 | 126.6765 |
Comparisons of the efficient algorithms
n × m | HV | IGD | ||||||
---|---|---|---|---|---|---|---|---|
ARMOEA | INSGAII | NSGAII | hpaEA | ARMOEA | INSGAII | NSGAII | hpaEA | |
20 × 4 | 0.6350 | 0.6689 | 0.6363 | 0.6391 | 862.9526 | 5.8334 | 933.7459 | 961.2891 |
20 × 8 | 0.6152 | 0.6592 | 0.6222 | 0.6135 | 253.9407 | 15.2828 | 203.3305 | 240.2489 |
20 × 16 | 0.5488 | 0.5976 | 0.5518 | 0.5486 | 682.7621 | 0.0000 | 653.4532 | 711.6409 |
40 × 4 | 0.6015 | 0.6433 | 0.6007 | 0.5963 | 237.9978 | 2.9417 | 194.6513 | 200.4242 |
40 × 8 | 0.5552 | 0.6238 | 0.5566 | 0.5537 | 774.0414 | 0.0000 | 718.1670 | 767.6475 |
40 × 16 | 0.4859 | 0.5699 | 0.4901 | 0.4831 | 2505.5410 | 0.0000 | 2315.0910 | 2436.4890 |
60 × 4 | 0.6172 | 0.6556 | 0.6208 | 0.6091 | 473.9908 | 10.8207 | 406.4291 | 484.0508 |
60 × 8 | 0.5570 | 0.6208 | 0.5626 | 0.5571 | 1268.4460 | 21.2009 | 1226.8120 | 1303.0030 |
60 × 16 | 0.4762 | 0.5687 | 0.4755 | 0.4691 | 3524.6920 | 0.0000 | 3707.8340 | 3711.8760 |
80 × 4 | 0.5811 | 0.6349 | 0.5792 | 0.5767 | 616.9777 | 6.3949 | 607.9967 | 635.6730 |
80 × 8 | 0.4936 | 0.5700 | 0.4935 | 0.4879 | 2162.4690 | 0.0000 | 2115.2650 | 2118.2260 |
80 × 16 | 0.4562 | 0.5594 | 0.4647 | 0.4536 | 5984.4640 | 0.0000 | 5662.2860 | 5976.3330 |
100 × 4 | 0.5875 | 0.6421 | 0.5888 | 0.5817 | 961.6876 | 57.5231 | 878.0886 | 949.1375 |
100 × 8 | 0.5017 | 0.5772 | 0.5041 | 0.5014 | 3107.6710 | 0.0000 | 2905.0270 | 2912.3990 |
100 × 16 | 0.4301 | 0.5488 | 0.4323 | 0.4312 | 8563.7730 | 0.0000 | 8385.6190 | 8366.9910 |
Avg | 0.5428 | 0.6093 | 0.5453 | 0.5401 | 2132.0938 | 7.9998 | 2060.9198 | 2118.3619 |
n × m | HV | IGD | ||||||
---|---|---|---|---|---|---|---|---|
ARMOEA | INSGAII | NSGAII | hpaEA | ARMOEA | INSGAII | NSGAII | hpaEA | |
20 × 4 | 0.6349 | 0.6753 | 0.6350 | 0.6336 | 491.6849 | 0.3865 | 581.3584 | 580.5010 |
20 × 8 | 0.6482 | 0.6984 | 0.6547 | 0.6433 | 1379.3790 | 0.0000 | 1208.8590 | 1638.3280 |
20 × 16 | 0.6419 | 0.7071 | 0.6438 | 0.6387 | 692.1126 | 0.0000 | 636.8500 | 672.9804 |
40 × 4 | 0.6216 | 0.6710 | 0.6279 | 0.6193 | 334.1531 | 5.0042 | 273.1995 | 321.7073 |
40 × 8 | 0.5474 | 0.6225 | 0.5548 | 0.5448 | 935.7414 | 0.0000 | 776.2829 | 929.9833 |
40 × 16 | 0.5505 | 0.6314 | 0.5581 | 0.5503 | 2364.7160 | 0.0000 | 2121.2880 | 2402.1510 |
60 × 4 | 0.5979 | 0.6471 | 0.6037 | 0.5959 | 368.6063 | 35.9517 | 316.0852 | 365.4021 |
60 × 8 | 0.5290 | 0.6078 | 0.5302 | 0.5296 | 1569.6950 | 0.0000 | 1423.7610 | 1492.2550 |
60 × 16 | 0.4623 | 0.5480 | 0.4594 | 0.4583 | 4366.1770 | 0.0000 | 4169.7940 | 4292.8760 |
80 × 4 | 0.5863 | 0.6365 | 0.5849 | 0.5817 | 649.3023 | 23.5702 | 586.5835 | 606.5393 |
80 × 8 | 0.5241 | 0.6088 | 0.5230 | 0.5207 | 2309.5610 | 0.0000 | 2339.2460 | 2378.0090 |
80 × 16 | 0.4583 | 0.5576 | 0.4688 | 0.4535 | 6518.8710 | 0.0000 | 5962.8210 | 6432.5830 |
100 × 4 | 0.5666 | 0.6119 | 0.5629 | 0.5588 | 742.3209 | 33.9751 | 742.6668 | 716.7839 |
100 × 8 | 0.4860 | 0.5749 | 0.4872 | 0.4817 | 2988.5700 | 0.0000 | 2922.4550 | 2999.4420 |
100 × 16 | 0.4334 | 0.5337 | 0.4336 | 0.4308 | 7699.0730 | 0.0000 | 7717.3670 | 7835.0550 |
Avg | 0.5525 | 0.6221 | 0.5552 | 0.5494 | 2227.3309 | 6.5925 | 2118.5745 | 2244.3064 |
n × m | HV | IGD | ||||||
---|---|---|---|---|---|---|---|---|
ARMOEA | INSGAII | NSGAII | hpaEA | ARMOEA | INSGAII | NSGAII | hpaEA | |
20 × 4 | 0.6213 | 0.6698 | 0.6303 | 0.6263 | 90.9816 | 0.4818 | 76.3124 | 86.5997 |
20 × 8 | 0.6753 | 0.7178 | 0.6812 | 0.6762 | 250.4894 | 5.2796 | 245.4039 | 215.3780 |
20 × 16 | 0.6643 | 0.7001 | 0.6634 | 0.6701 | 337.8108 | 0.8978 | 383.6756 | 321.1888 |
40 × 4 | 0.6171 | 0.6624 | 0.6251 | 0.6181 | 284.9572 | 8.9542 | 237.7833 | 253.4150 |
40 × 8 | 0.5050 | 0.5778 | 0.5111 | 0.5099 | 673.1945 | 0.0000 | 603.5707 | 713.5393 |
40 × 16 | 0.5576 | 0.6253 | 0.5648 | 0.5546 | 1982.4460 | 0.0000 | 1746.0310 | 2065.0820 |
60 × 4 | 0.6027 | 0.6559 | 0.6051 | 0.5992 | 384.5750 | 7.7861 | 353.4561 | 369.1107 |
60 × 8 | 0.5216 | 0.6039 | 0.5221 | 0.5196 | 1407.2100 | 0.0000 | 1385.7210 | 1431.3560 |
60 × 16 | 0.4875 | 0.5772 | 0.4894 | 0.4853 | 3828.7470 | 0.0000 | 3692.5760 | 3699.8410 |
80 × 4 | 0.5558 | 0.6169 | 0.5608 | 0.5557 | 627.7588 | 0.0000 | 590.8560 | 608.5870 |
80 × 8 | 0.5493 | 0.6412 | 0.5539 | 0.5451 | 2124.3480 | 0.0000 | 1900.1150 | 2127.7020 |
80 × 16 | 0.4634 | 0.5620 | 0.4699 | 0.4714 | 5865.7030 | 0.0000 | 5516.3790 | 5682.8610 |
100 × 4 | 0.5728 | 0.6238 | 0.5824 | 0.5714 | 793.5963 | 56.5530 | 615.9821 | 752.8644 |
100 × 8 | 0.4828 | 0.5722 | 0.4888 | 0.4815 | 3419.9320 | 0.0000 | 2894.0410 | 3130.0580 |
100 × 16 | 0.5399 | 0.6453 | 0.5436 | 0.5409 | 7578.3830 | 0.0000 | 7284.7820 | 7543.3520 |
Avg | 0.5611 | 0.6301 | 0.5661 | 0.5617 | 1976.6755 | 5.3302 | 1835.1123 | 1933.3957 |
n × m | HV | IGD | ||||||
---|---|---|---|---|---|---|---|---|
ARMOEA | INSGAII | NSGAII | hpaEA | ARMOEA | INSGAII | NSGAII | hpaEA | |
20 × 4 | 0.6164 | 0.6491 | 0.6084 | 0.5989 | 355.3706 | 0.0000 | 358.3120 | 548.0381 |
20 × 8 | 0.6574 | 0.7129 | 0.6645 | 0.6567 | 243.9721 | 0.0000 | 238.8503 | 281.5824 |
20 × 16 | 0.6683 | 0.7296 | 0.6732 | 0.6669 | 539.7077 | 0.0000 | 477.0620 | 550.6626 |
40 × 4 | 0.5941 | 0.6512 | 0.6011 | 0.5907 | 231.5507 | 0.0000 | 221.3578 | 233.3693 |
40 × 8 | 0.5350 | 0.6109 | 0.5385 | 0.5341 | 804.0522 | 0.0000 | 780.4658 | 729.1339 |
40 × 16 | 0.6112 | 0.6704 | 0.6154 | 0.6056 | 1798.5440 | 0.0000 | 1641.4210 | 1831.1760 |
60 × 4 | 0.6754 | 0.7408 | 0.6807 | 0.6798 | 617.5125 | 10.5511 | 548.1857 | 586.7113 |
60 × 8 | 0.6127 | 0.6983 | 0.6215 | 0.6140 | 1573.8320 | 0.0000 | 1321.1470 | 1593.4640 |
60 × 16 | 0.4920 | 0.5676 | 0.4939 | 0.4878 | 3084.0470 | 0.0000 | 3097.6250 | 3118.9630 |
80 × 4 | 0.6184 | 0.6920 | 0.6173 | 0.6113 | 692.2176 | 0.0000 | 755.0597 | 811.5215 |
80 × 8 | 0.5232 | 0.6097 | 0.5277 | 0.5237 | 2187.8330 | 0.0000 | 1889.8940 | 2120.5530 |
80 × 16 | 0.5511 | 0.6568 | 0.5570 | 0.5552 | 6236.6680 | 0.0000 | 5908.1990 | 5857.7450 |
100 × 4 | 0.5359 | 0.6126 | 0.5394 | 0.5388 | 955.2761 | 0.0000 | 814.5295 | 894.9756 |
100 × 8 | 0.4851 | 0.5730 | 0.4946 | 0.4841 | 2835.1090 | 0.0000 | 2469.9140 | 2599.6000 |
100 × 16 | 0.5475 | 0.6521 | 0.5537 | 0.5442 | 8317.0510 | 0.0000 | 7840.9230 | 8467.2710 |
Avg | 0.5816 | 0.6551 | 0.5858 | 0.5794 | 2031.5162 | 0.7034 | 1890.8631 | 2014.9844 |