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Table 7 Mean, standard deviation (Std), median, minimum (Min) and maximum (Max) values of the performance metrics for the compared algorithms for the Mean-SV portfolio optimization problem

From: Portfolio optimization by improved NSGA-II and SPEA 2 based on different risk measures

Dataset Statistic S Δ IGD HV
   NSGA-IIb NSGA-IIa SPEA 2b SPEA 2a NSGA-IIb NSGA-IIa SPEA 2b SPEA 2a NSGA-IIb NSGA-IIa SPEA 2b SPEA 2a NSGA-IIb NSGA-IIa SPEA 2b SPEA 2a
DowJones Mean 0.0064 0.0023 0.0069 0.0034 0.8197 0.6101 0.8047 0.5773 0.4081 0.4695 0.4021 0.3377 0.8638 0.9859 0.8531 1.1709
  Median 0.0057 0.0025 0.0061 0.0031 0.8155 0.5627 0.8076 0.5780 0.4121 0.3381 0.3976 0.3377 0.8619 1.1702 0.8467 1.1709
  Std 0.0019 0.0009 0.0021 0.0011 0.0285 0.1424 0.0302 0.0131 0.0203 0.2699 0.0229 0.0002 0.0393 0.3800 0.0489 0.0001
  Min 0.0041 0.0000 0.0038 0.0029 0.7669 0.5437 0.7452 0.5564 0.3683 0.3375 0.3603 0.3374 0.7963 0.1725 0.7953 1.1707
  Max 0.0113 0.0033 0.0116 0.0080 0.8765 1.0000 0.8622 0.6087 0.4439 0.9994 0.4425 0.3383 0.9405 1.1704 0.9763 1.1710
FF49Industries Mean 0.0086 0.0023 0.0089 0.0035 0.7854 0.6003 0.7960 0.6165 0.6558 0.4767 0.6534 0.4765 0.5578 1.0536 0.5555 1.0538
  Median 0.0083 0.0023 0.0085 0.0035 0.7811 0.6008 0.8004 0.6173 0.6575 0.4767 0.6554 0.4763 0.5574 1.0536 0.5509 1.0538
  Std 0.0015 0.0002 0.0018 0.0003 0.0175 0.0117 0.0264 0.0112 0.0178 0.0003 0.0183 0.0004 0.0136 0.0002 0.0097 0.0003
  Min 0.0068 0.0020 0.0060 0.0029 0.7540 0.5828 0.7564 0.5946 0.6208 0.4761 0.6147 0.4759 0.5372 1.0534 0.5410 1.0531
  Max 0.0128 0.0029 0.0131 0.0042 0.8265 0.6220 0.8513 0.6409 0.6837 0.4774 0.6778 0.4771 0.5911 1.0540 0.5763 1.0542
NASDAQ100 Mean 0.0039 0.0027 0.0040 0.0036 0.8822 0.5199 0.8797 0.5888 0.7326 0.2589 0.7330 0.2588 0.4321 1.2061 0.4311 1.2062
  Median 0.0037 0.0026 0.0039 0.0035 0.8822 0.5226 0.8807 0.5865 0.7338 0.2590 0.7344 0.2588 0.4270 1.2061 0.4297 1.2062
  Std 0.0009 0.0005 0.0008 0.0004 0.0110 0.0204 0.0085 0.0175 0.0151 0.0006 0.0121 0.0005 0.0175 0.0004 0.0248 0.0003
  Min 0.0026 0.0019 0.0027 0.0030 0.8633 0.4904 0.8652 0.5620 0.7021 0.2579 0.7091 0.2581 0.4118 1.2051 0.3986 1.2057
  Max 0.0057 0.0039 0.0057 0.0046 0.8999 0.5507 0.8992 0.6220 0.7536 0.2599 0.7541 0.2599 0.4720 1.2066 0.4793 1.2067
SP500 Mean 0.0016 0.0048 0.0015 0.0041 0.9547 0.6605 0.9551 0.6318 1.0134 0.2787 1.0135 0.2761 0.1481 1.1828 0.1468 1.1953
  Median 0.0015 0.0047 0.0015 0.0039 0.9547 0.6738 0.9550 0.6289 1.0134 0.2788 1.0135 0.2760 0.1467 1.1830 0.1440 1.1960
  Std 0.0003 0.0015 0.0003 0.0008 0.0049 0.0418 0.0046 0.0237 0.0052 0.0013 0.0053 0.0011 0.0057 0.0060 0.0062 0.0025
  Min 0.0011 0.0025 0.0011 0.0032 0.9459 0.5706 0.9471 0.5942 1.0027 0.2757 1.0025 0.2743 0.1418 1.1727 0.1399 1.1885
  Max 0.0024 0.0077 0.0019 0.0065 0.9643 0.7284 0.9654 0.7074 1.0211 0.2815 1.0217 0.2795 0.1616 1.1917 0.1609 1.1981
NASDAQComp Mean 0.0010 0.0045 0.0009 0.0089 0.9714 0.7839 0.9711 0.8766 1.1084 0.2070 1.1085 0.2012 0.0499 1.2214 0.0492 1.2472
  Median 0.0010 0.0036 0.0009 0.0057 0.9721 0.7862 0.9703 0.8578 1.1098 0.2045 1.1099 0.1970 0.0488 1.2219 0.0481 1.2509
  Std 0.0002 0.0019 0.0002 0.0079 0.0020 0.0596 0.0027 0.1232 0.0033 0.0133 0.0033 0.0127 0.0034 0.0133 0.0037 0.0123
  Min 0.0007 0.0030 0.0007 0.0025 0.9678 0.6011 0.9654 0.6346 1.1013 0.1991 1.1013 0.1922 0.0464 1.1886 0.0452 1.2132
  Max 0.0014 0.0107 0.0013 0.0361 0.9740 0.8741 0.9769 1.1636 1.1128 0.2607 1.1129 0.2458 0.0570 1.2395 0.0571 1.2579
  1. aNumbers in bold represent best results among the algorithms with respect to the corresponding performance metric