An Evolutionary Algorithm for Multiobjective Fuzzy Portfolio Selection Models with Transaction Cost and Liquidity
Table 8
IGD, GD, and HV obtained by the proposed algorithm, MOEA/D, NSGA-II, and D2MOPSO.
Instance
The proposed algorithm
MOEA/D
NSGA-II
D2MOPSO
Mean
Std.
Mean
Std.
Mean
Std.
Mean
Std.
ZDT1
IGD
0.0040
0.0002
0.0040 (=)
0.0003
0.0042 (=)
0.0003
0.0041 (=)
0.0001
GD
0.0009
0.0001
0.0010 (=)
0.0002
0.0010 (=)
0.0001
0.0011 (=)
0.0001
HV
0.6613
0.0002
0.6612 (=)
0.0001
0.6614 (=)
0.0003
0.6612 (=)
0.0003
ZDT2
IGD
0.0038
0.0003
0.0037 (=)
0.0002
0.1955 (+)
0.2891
0.0041 (=)
0.0002
GD
0.0008
0.0001
0.0009 (=)
0.0002 (=)
0.0005 (−)
0.0004
0.0009 (=)
0.0004
HV
0.3287
0.0003
0.3284 (=)
0.0001
0.2245 (+)
0.1567
0.3283 (=)
0.0001
ZDT3
IGD
0.0100
0.0008
0.0111 (=)
0.0001
0.0049 (−)
0.0003
0.0059 (−)
0.0001
GD
0.0040
0.0004
0.0041 (=)
0.0001
0.0050 (+)
0.0002
0.0044 (+)
0.0002
HV
1.0383
0.0007
1.0376 (+)
0.0001
1.0417 (=)
0.0001
1.0387 (=)
0.0001
ZDT4
IGD
0.0042
0.0008
0.0042 (=)
0.0002
0.3547 (+)
0.4225
0.0045 (+)
0.0003
GD
0.0683
0.1114
0.0021 (−)
0.0001
0.0015 (−)
0.0011
0.0032 (−)
0.0021
HV
0.6609
0.0011
0.6611 (=)
0.0001
0.3830 (+)
0.3355
0.6605 (=)
0.0002
ZDT6
IGD
0.0037
0.0019
0.0039 (=)
0.0001
0.0028 (=)
0.0002
0.0038 (=)
0.0001
GD
0.0051
0.0233
0.0008 (−)
0.0001
0.0011 (−)
0.0000
0.0031 (−)
0.0002
HV
0.3226
0.0002
0.3221 (=)
0.0001
0.3217 (=)
0.0002
0.3221 (=)
0.0001
DTLZ1
IGD
0.0231
0.0009
0.0234 (=)
0.0001
0.0254 (+)
0.0013
0.0244 (+)
0.0002
GD
0.0045
0.0013
0.0045 (=)
0.0001
0.0051 (+)
0.0002
0.0048 (+)
0.0001
HV
0.9733
0.0053
0.9666 (+)
0.0010
0.9714 (+)
0.0004
0.9712 (+)
0.0001
DTLZ2
IGD
0.0611
0.0018
0.0619 (=)
0.0004
0.0674 (+)
0.0027
0.0621 (+)
0.0004
GD
0.0034
0.0004
0.0034 (=)
0.0002
0.0078 (+)
0.0007
0.0045 (+)
0.0006
HV
0.3910
0.0053
0.3802 (+)
0.0017
0.3824 (+)
0.0063
0.3865 (+)
0.0004
DTLZ3
IGD
0.0608
0.0018
0.0614 (+)
0.0002
0.0696 (+)
0.0036
0.0621 (+)
0.0014
GD
0.0127
0.0005
0.0151 (+)
0.0004
0.0133 (+)
0.0006
0.0141 (+)
0.0008
HV
0.3933
0.0072
0.3808 (+)
0.0013
0.3844 (+)
0.0053
0.3853 (+)
0.0007
DTLZ4
IGD
0.0615
0.0023
0.0627 (+)
0.0012
0.1162 (+)
0.2241
0.0620 (+)
0.0024
GD
0.0136
0.0006
0.0168 (+)
0.0006
0.0180 (+)
0.0042
0.0164 (+)
0.0052
HV
0.3923
0.0045
0.3840 (+)
0.0032
0.3626 (+)
0.0879
0.3867 (+)
0.0037
“+” means that the proposed algorithm outperforms its competitor algorithm, “−” means that the proposed algorithm is outperformed by its competitor algorithm, and “=” means that the competitor algorithm has the same performance as the proposed algorithm.