Elite Multi-Criteria Decision Making—Pareto Front Optimization in Multi-Objective Optimization
Abstract
:1. Introduction
2. Literature Review
2.1. Multi-Objective Optimization (MOO)
2.2. Pareto Front
- Point A dominates point B in objective 1 and C in both the objectives.
- Point B dominates point A in objective 2 and C in both the objectives.
- Point C is dominated by both A and B points in both the objectives.
2.3. Multi-Criteria Decision Making (MCDM)
2.4. Evolutionary Algorithms (EA)
2.5. Multi-Objective Evolutionary Algorithm (MOEA)
3. Methodology
3.1. Algorithm
Algorithm 1 eMPF selection process |
|
3.1.1. Calculating Weights of Each Objective
- The first step is to normalize the fitness values. The fitness values are normalized using Equation (2),
- Lastly, the weights () of the objective are calculated using Equation (4),
3.1.2. TOPSIS
- Normalize the actual fitness values using Equation (5).
- Next, find the fitness weights by taking the product of the weight of each objective with fitness values using the equation as shown below:
- Depending on the value, the best and worst-performing individuals of each objective are selected and flagged as the best or worst individual policy.
- The Euclidean distances between all of the individuals to both the best and worst individuals are calculated and assigned as , where is assigned as the distance to the best individual and is assigned as the distance to the worst individual.
- The last step is to find the degree of approximation using Equation (7),
3.2. Performance Evaluation
3.2.1. Pareto Front Spread ()
3.2.2. Generational Distance ()
3.2.3. Pareto Front Spacing ()
3.3. Test Functions
- Binh and Korn Function [33];
- Chankong and Haimes Function [33];
- Fonseca–Fleming Function [34];
- Test Function 4 [35];
- Kursawe Function [36];
- Schaffer Function N1, and N2 [37];
- Poloni’s Two-Objective Function [38];
- Zitzler–Deb–Thiele’s Function N1, N2, N3, N4, and N6 [39];
- Osyczka and Kundu Function [40];
- Constr-Ex Problem;
- VR-UC Test 1, VR-UC Test 2 [41];
- MSGA Test 1 [42];
- Viennet Function [43];
- MHHM1, MHHM2 [44].
4. Results
5. Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MOO | Multi-Objective Optimization |
MCDM | Multi-Criteria Decision Making |
EA | Evolutionary Algorithm |
TOPSIS | Technique for Order of Preference by Similarity to Ideal Solution |
NSGA-II | Non-Dominated Sorting Genetic Algorithm-II |
NSGA-III | Non-Dominated Sorting Genetic Algorithm-III |
M-PF | Multi-Criteria Decision Making–Pareto Front |
eMPF | elite Multi-Criteria Decision Making–Pareto Front |
CCEA | Cooperative Co-evolutionary Algorithms |
Appendix A
Function | Test Function | Constraints | Search Domain |
---|---|---|---|
Binh and Korn Function | |||
Chankong and Haimes Function | |||
Fonseca–Fleming Function | |||
Test Function 4 | 0 | ||
Kursawe Function | |||
Schaffer Function N1 | Values of A from 10 to | ||
Schaffer Function N2 | |||
Poloni’s Two-Objective Function | where | ; | |
Zitzler–Deb–Thiele’s Function (ZDT) N1 | |||
Zitzler–Deb–Thiele’s Function (ZDT) N2 | |||
Zitzler–Deb–Thiele’s Function (ZDT) N3 | |||
Zitzler–Deb–Thiele’s Function (ZDT) N4 | |||
Zitzler–Deb–Thiele’s Function (ZDT) N6 | |||
Osyczka & Kundu Function | |||
Constr-Ex Problem | |||
VR-UC Test 1 | |||
VR-UC Test 2 | |||
MSGA Test 1 | |||
MHHM1 | |||
MHHM2 | |||
Viennet Function |
Appendix B
Test Function | Method | Minimum | Mean | Standard Deviation | |||
---|---|---|---|---|---|---|---|
Function 1 | Function 2 | Function 1 | Function 2 | Function 1 | Function 2 | ||
NSGA-II | 0.000948 | 4.005556 | 52.645992 | 17.283002 | 39.400303 | 12.500695 | |
Binh and Korn | NSGA-III | 9.332152 | 4.533799 | 100.464088 | 10.277145 | 43.565188 | 13.756248 |
M-PF | 9.498507 | 10.473067 | 32.993437 | 19.539373 | 18.265393 | 7.521567 | |
eMPF | 0.002037 | 4.023522 | 14.015699 | 29.174421 | 17.331893 | 5.806211 | |
NSGA-II | 10.119513 | −217.695484 | 113.45425 | −112.737342 | 62.128512 | 62.677727 | |
Chankong Haimes | NSGA-III | 96.621684 | −217.730419 | 213.296477 | −210.363403 | 20.65679 | 24.166874 |
M-PF | 57.01729 | −101.645503 | 80.131923 | −80.211533 | 48.913044 | 48.881173 | |
eMPF | 10.125042 | −217.686516 | 127.519573 | −127.352052 | 23.87205 | 24.71453 | |
Fonseca–Fleming | NSGA-II | 0.000034 | 0.000042 | 0.578248 | 0.580166 | 0.296733 | 0.296151 |
NSGA-III | 0.092419 | 0.032975 | 0.723804 | 0.497992 | 0.153209 | 0.159965 | |
M-PF | 0.070628 | 0.066346 | 0.609386 | 0.596173 | 0.226319 | 0.233033 | |
eMPF | 0.000022 | 0.000021 | 0.511756 | 0.499501 | 0.465786 | 0.466366 | |
NSGA-II | −6.504972 | −8.499939 | −1.534355 | −8.145423 | 3.336720 | 0.277358 | |
Test Function 4 | NSGA-III | −6.506321 | −8.498921 | −5.331177 | −7.623771 | 3.929542 | 0.229023 |
M-PF | −6.392928 | −8.316240 | −4.654117 | −7.854135 | 1.901964 | 0.210908 | |
eMPF | −6.506243 | −8.499792 | −5.808833 | −7.636678 | 2.520614 | 0.161645 | |
NSGA-II | −19.785629 | −5.138218 | −10.217168 | −2.560673 | 3.696744 | 1.566748 | |
Kursawe | NSGA-III | −19.800519 | −2.761169 | −12.516551 | −1.290272 | 2.812103 | 0.732146 |
M-PF | −12.427413 | −3.105100 | −9.705192 | −1.623625 | 0.963143 | 0.812313 | |
eMPF | −19.823842 | −5.133368 | −12.190258 | −1.771524 | 3.224230 | 1.219680 | |
NSGA-II | 0.000000 | 0.000000 | 1.309796 | 1.335708 | 1.171495 | 1.178720 | |
Schaffer function N1 | NSGA-III | 0.458913 | 0.452471 | 2.092166 | 1.548018 | 1.830392 | 1.803354 |
M-PF | 0.314452 | 0.365226 | 1.484631 | 1.911246 | 1.678226 | 1.713672 | |
eMPF | 0.000000 | 0.000000 | 1.118533 | 1.121366 | 0.726276 | 0.734062 | |
NSGA-II | −0.999966 | 0.000000 | 0.519161 | 7.217303 | 1.030891 | 4.693345 | |
Schaffer function N2 | NSGA-III | −0.824930 | 7.796460 | −0.624833 | 14.028193 | 0.942749 | 4.035698 |
M-PF | 0.178468 | 3.991629 | 0.742443 | 8.141138 | 1.744642 | 6.828264 | |
eMPF | −0.999503 | 0.000000 | 2.162599 | 0.964716 | 0.513256 | 1.922794 | |
NSGA-II | 1.000107 | 0.000011 | 6.867533 | 6.300837 | 5.010289 | 9.464460 | |
Poloni’s two objectives | NSGA-III | 2.072667 | 8.096344 | 7.516216 | 13.666458 | 7.204542 | 11.749644 |
M-PF | 2.646164 | 1.748814 | 12.385470 | 3.525388 | 5.237112 | 7.848127 | |
eMPF | 1.000656 | 0.000035 | 3.790499 | 2.429406 | 2.910871 | 3.427426 | |
NSGA-II | 0.000016 | 1.830778 | 0.259772 | 3.291023 | 0.295645 | 0.985772 | |
ZDT’s N1 | NSGA-III | 0.000012 | 1.836794 | 0.294363 | 3.228128 | 0.313797 | 1.012261 |
M-PF | 0.000016 | 1.852934 | 0.279310 | 3.224510 | 0.302015 | 0.972492 | |
eMPF | 0.000027 | 1.831970 | 0.284748 | 3.219609 | 0.305802 | 0.967179 | |
NSGA-II | 0.000020 | 3.360521 | 0.146886 | 4.209470 | 0.265380 | 0.626616 | |
ZDT’s N2 | NSGA-III | 0.000018 | 3.397406 | 0.150892 | 4.200882 | 0.275458 | 0.627170 |
M-PF | 0.000015 | 3.389850 | 0.124176 | 4.231827 | 0.244155 | 0.646990 | |
eMPF | 0.000015 | 3.384750 | 0.154723 | 4.262611 | 0.274839 | 0.691868 | |
NSGA-II | 0.000015 | 1.197987 | 0.302574 | 3.009383 | 0.295633 | 1.165188 | |
ZDT’s N3 | NSGA-III | 0.000016 | 1.250750 | 0.288800 | 3.049543 | 0.294440 | 1.148127 |
M-PF | 0.000015 | 1.230596 | 0.299095 | 3.019458 | 0.296260 | 1.142140 | |
eMPF | 0.000014 | 1.221293 | 0.306138 | 2.997215 | 0.302244 | 1.155063 | |
NSGA-II | 0.000014 | 28.918132 | 0.084849 | 64.349782 | 0.174179 | 32.479856 | |
ZDT’s N4 | NSGA-III | 0.000017 | 30.178694 | 0.077131 | 69.368632 | 0.168524 | 34.873943 |
M-PF | 0.000011 | 28.018323 | 0.081965 | 65.590350 | 0.166882 | 33.832099 | |
eMPF | 0.000013 | 27.480261 | 0.105229 | 62.256601 | 0.201626 | 31.532401 | |
NSGA-II | 0.280775 | 6.356673 | 0.492336 | 7.271715 | 0.278471 | 0.661174 | |
ZDT’s N6 | NSGA-III | 0.280775 | 6.233513 | 0.499382 | 7.221104 | 0.281440 | 0.653730 |
M-PF | 0.280775 | 6.276437 | 0.474475 | 7.257931 | 0.269862 | 0.630176 | |
eMPF | 0.280775 | 6.268319 | 0.491137 | 7.239478 | 0.285670 | 0.677693 | |
NSGA-II | −258.627419 | 5.194767 | −164.898915 | 28.366042 | 62.986998 | 21.322610 | |
Osyczka and Kundu | NSGA-III | −258.633516 | 5.433823 | −168.598553 | 29.169913 | 62.053323 | 20.778173 |
M-PF | −255.080484 | 5.367792 | −162.235317 | 26.606650 | 60.836597 | 18.988171 | |
eMPF | −257.382419 | 5.312247 | −167.069531 | 28.639926 | 61.643875 | 21.469461 | |
NSGA-II | 0.390717 | 1.007121 | 0.547760 | 4.747238 | 0.139116 | 2.426397 | |
Const-Ex | NSGA-III | 0.390230 | 1.515615 | 0.441708 | 7.518592 | 0.093502 | 1.616727 |
M-PF | 0.445405 | 1.175281 | 0.616617 | 2.902549 | 0.093789 | 1.547132 | |
eMPF | 0.391615 | 1.004618 | 0.766952 | 1.719732 | 0.099745 | 1.512755 | |
VR-UC Test 1 | NSGA-II | 0.052761 | 1.000574 | 0.156242 | 17.296326 | 0.193932 | 10.902374 |
NSGA-III | 0.052718 | 2.866609 | 0.074163 | 31.895484 | 0.086781 | 8.590167 | |
M-PF | 0.070903 | 2.921766 | 0.144710 | 7.997647 | 0.057197 | 3.582939 | |
eMPF | 0.053224 | 1.000024 | 0.705932 | 2.339642 | 0.174928 | 5.048489 | |
VR-UC Test 2 | NSGA-II | −4.995860 | −9.993871 | −4.995082 | −9.992874 | 0.002898 | 0.004093 |
NSGA-III | −4.995535 | −9.993506 | −4.995059 | −9.992905 | 0.002668 | 0.003992 | |
M-PF | −4.995112 | −9.993227 | −4.994349 | −9.992267 | 0.003135 | 0.003989 | |
eMPF | −4.996121 | −9.994576 | −4.995738 | −9.993998 | 0.002667 | 0.003675 | |
MSGA Test 1 | NSGA-II | 0.234489 | 0.235890 | 0.706906 | 0.709010 | 0.197887 | 0.197780 |
NSGA-III | 0.549158 | 0.491633 | 0.779350 | 0.734232 | 0.086980 | 0.100268 | |
M-PF | 0.554772 | 0.531218 | 0.768762 | 0.768141 | 0.051210 | 0.054106 | |
eMPF | 0.231459 | 0.229316 | 0.661794 | 0.670574 | 0.247905 | 0.247862 |
Test Function | Method | Minimum | ||
---|---|---|---|---|
Function 1 | Function 2 | Function 3 | ||
Viennet | NSGA-II | 0.000051 | 15.000000 | −0.099997 |
NSGA-III | 131,609.806452 | 443,378.967742 | 0.000004 | |
M-PF | 4292.110524 | 10,870.875716 | −0.002940 | |
eMPF | 0.000028 | 15.000026 | −0.099998 | |
MHHM1 | NSGA-II | 9.6666 × | 7.3333 × | 6.33333 × |
NSGA-III | 6.193743 × | 4.7659963 × | 3.433997 × | |
M-PF | 2.4781883 × | 7.064937 × | 7.134097 × | |
eMPF | 1.0666 × | 9.6666 × | 5.3333 × | |
MHHM2 | NSGA-II | 6.18220 × | 1.7605 × | 2.837620 × |
NSGA-III | 0.001283 | 0.001229 | 0.00249529 | |
M-PF | 0.00029338 | 0.0004046536 | 0.0006900754 | |
eMPF | 6.868939 × | 1.63898 × | 4.5422 × |
Test Function | Method | Mean | ||
---|---|---|---|---|
Function 1 | Function 2 | Function 3 | ||
Viennet | NSGA-II | 3.330086 | 15.285244 | 0.053903 |
NSGA-III | 132,295.691684 | 445,689.241041 | 0.000004 | |
M-PF | 4607.780516 | 11,696.522534 | 0.001230 | |
eMPF | 0.572229 | 15.485611 | −0.024929 | |
MHHM1 | NSGA-II | 0.0034099 | 0.0009104 | 0.00341086 |
NSGA-III | 0.0013004 | 0.0005224 | 0.00474447 | |
M-PF | 0.00443803 | 0.002016614 | 0.00459519 | |
eMPF | 0.0026903 | 0.00020094 | 0.00271158 | |
MHHM2 | NSGA-II | 0.0054150 | 0.005925 | 0.00555724 |
NSGA-III | 0.0035569 | 0.003649 | 0.0063590 | |
M-PF | 0.004642219 | 0.00528738 | 0.005879 | |
eMPF | 0.0042817 | 0.0046030 | 0.004078 |
Test Function | Method | Standard Deviation | ||
---|---|---|---|---|
Function 1 | Function 2 | Function 3 | ||
Viennet | NSGA-II | 2.688537 | 0.511459 | 0.081092 |
NSGA-III | 6979.937413 | 26,870.603732 | 0.000000 | |
M-PF | 2278.826105 | 7778.051973 | 0.010432 | |
eMPF | 0.330933 | 0.328919 | 0.049485 | |
MHHM1 | NSGA-II | 0.00313115 | 0.00079 | 0.0031071 |
NSGA-III | 0.0016641 | 0.0004383 | 0.00146057 | |
M-PF | 0.0045989 | 0.00074356 | 0.004502838 | |
eMPF | 0.001529 | 0.00059 | 0.0015419 | |
MHHM1 | NSGA-II | 0.003749 | 0.00417569 | 0.0037675 |
NSGA-III | 0.001986 | 0.0018050 | 0.0020722 | |
M-PF | 0.0032253 | 0.0035306 | 0.0034066 | |
eMPF | 0.0020989 | 0.0021489 | 0.002019 |
Appendix C
Test Function | Method | Generational Distance | Spread () | Spacing () |
---|---|---|---|---|
Binh and Korn | NSGA-II NSGA-III M-PF eMPF | 0.069016 0.378684 1.333222 0.770759 | 0.938548 1.530796 1.233198 1.584592 | 0.185688 0.554448 1.062628 1.040446 |
Chankong Haimes | NSGA-II NSGA-III M-PF eMPF | 0.132257 6.711842 1.897858 10.481315 | 0.803492 1.600441 0.99258 1.220204 | 0.216754 3.754718 0.41429 2.947663 |
Fonseca Fleming | NSGA-II NSGA-III M-PF eMPF | 0.000838 0.002347 0.000664 0.006234 | 0.86052 1.401653 1.134345 1.664327 | 0.001535 0.004618 0.00218 0.006798 |
Test Function 4 | NSGA-II NSGA-III M-PF eMPF | 0.006338 0.133243 0.020431 0.081445 | 0.938548 1.530796 1.233198 1.584592 | 0.039957 0.287708 0.061413 0.14012 |
Kursawe | NSGA-II NSGA-III M-PF eMPF | 0.006938 0.70871 0.127039 0.075359 | 0.803527 0.911151 1.19283 0.892111 | 0.336557 0.470105 0.334158 0.45627 |
Schaffer Function N1 | NSGA-II NSGA-III M-PF eMPF | 0.000985 0.003258 0.003113 0.002111 | 0.772657 1.44585 1.44585 1.274417 | 0.002629 0.007116 0.00495 0.005048 |
Schaffer Function N2 | NSGA-II NSGA-III M-PF eMPF | 0.001906 0.130632 0.010325 0.096201 | 0.742149 1.514319 1.319566 1.621567 | 0.006122 0.097936 0.023796 0.070892 |
Poloni’s Two Objectives | NSGA-II NSGA-III M-PF eMPF | 0.0127 0.102661 0.031244 0.186827 | 1.47718 1.789741 1.791253 1.55847 | 0.737613 0.693373 0.664818 0.984685 |
Zitzler–Deb–Thiele’s N1 | NSGA-II NSGA-III M-PF eMPF | 0.042448 0.052309 0.045381 0.048533 | 0.7066 0.712719 0.721581 0.685075 | 0.074755 0.077806 0.07787 0.147347 |
Zitzler–Deb–Thiele’s N2 | NSGA-II NSGA-III M-PF eMPF | 0.073596 0.041085 0.068744 0.070442 | 0.697774 0.710485 0.738163 0.712368 | 0.19127 0.113637 0.120021 0.16111 |
Zitzler–Deb–Thiele’s N3 | NSGA-II NSGA-III M-PF eMPF | 0.032912 0.030987 0.044047 0.040325 | 0.714989 0.660256 0.662726 0.72205 | 0.098628 0.102081 0.111724 0.084448 |
Zitzler–Deb–Thiele’s N4 | NSGA-II NSGA-III M-PF eMPF | 0.810677 2.272926 2.018945 2.279626 | 0.744636 0.757631 0.786935 0.738201 | 9.043884 7.516822 8.574137 9.78726 |
Zitzler–Deb–Thiele’s N6 | NSGA-II NSGA-III M-PF eMPF | 0.191495 0.099431 0.065991 0.092465 | 0.685276 0.59864 0.638246 0.69825 | 0.115021 0.140287 0.173285 0.260617 |
Osycak and Kundu Function | NSGA-II NSGA-III M-PF eMPF | 1.504086 1.294484 1.316762 1.388888 | 1.178373 1.127386 1.132501 1.2232 | 3.283527 2.84922 3.077236 3.686033 |
Constr-Ex Problem | NSGA-II NSGA-III M-PF eMPF | 0.00495 0.043087 0.02453 0.07408 | 0.714432 1.313374 1.179857 1.683737 | 0.012929 0.067725 0.042256 0.090675 |
MSGA Test 1 | NSGA-II NSGA-III M-PF eMPF | 0.000538 0.013212 0.012998 0.003187 | 0.972065 1.038185 1.492854 1.242157 | 0.002284 0.015305 0.006631 0.004642 |
VR-UC Test 1 | NSGA-II NSGA-III M-PF eMPF | 0.01332 0.070166 0.142823 0.38772 | 0.786952 1.427575 1.316431 1.695775 | 0.05018 0.176857 0.190408 0.402943 |
VR-UC Test 2 | NSGA-II NSGA-III M-PF eMPF | 9.4 × 9.4 × 0.000324 0.000273 | 0.717724 0.636498 0.577738 0.712867 | 0.00000 0.00000 0.00000 0.00000 |
Vinnet Function | NSGA-II NSGA-III M-PF eMPF | 0.00169 0.627138 1.615187 0.648107 | 0.782624 0.989541 1.21211 1.557092 | 0.015777 0.087255 0.169447 0.095202 |
MHHM1 | NSGA-II NSGA-III M-PF eMPF | 6 × 2.8 × 1.8 × 2.3 × | 0.772737 1.640253 1.620182 1.688381 | 1.1 × 3.3 × 2.6 × 3.5 × |
MHHM2 | NSGA-II NSGA-III M-PF eMPF | 0.000401 0.001194 0.000514 0.001207 | 0.651384 0.765195 0.798712 0.702216 | 0.003814 0.001724 0.003292 0.001756 |
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Kesireddy, A.; Medrano, F.A. Elite Multi-Criteria Decision Making—Pareto Front Optimization in Multi-Objective Optimization. Algorithms 2024, 17, 206. https://doi.org/10.3390/a17050206
Kesireddy A, Medrano FA. Elite Multi-Criteria Decision Making—Pareto Front Optimization in Multi-Objective Optimization. Algorithms. 2024; 17(5):206. https://doi.org/10.3390/a17050206
Chicago/Turabian StyleKesireddy, Adarsh, and F. Antonio Medrano. 2024. "Elite Multi-Criteria Decision Making—Pareto Front Optimization in Multi-Objective Optimization" Algorithms 17, no. 5: 206. https://doi.org/10.3390/a17050206