Elite MultiCriteria Decision Making—Pareto Front Optimization in MultiObjective Optimization
Abstract
:1. Introduction
2. Literature Review
2.1. MultiObjective 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. MultiCriteria Decision Making (MCDM)
2.4. Evolutionary Algorithms (EA)
2.5. MultiObjective 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),$${F}_{ij}=\frac{{f}_{ij}}{{\Sigma}_{i=1}^{N}{f}_{i}}$$
 The next step is to find the entropy values (${e}_{j}$) of each objective using Equation (3),$${e}_{j}=h\sum _{i=1}^{N}{F}_{ij}ln\left({F}_{ij}\right)$$
 Lastly, the weights (${w}_{j}$) of the objective are calculated using Equation (4),$${w}_{j}=\frac{1{e}_{j}}{{\sum}_{j=1}^{M}(1{e}_{j})}$$
3.1.2. TOPSIS
 Normalize the actual fitness values using Equation (5).$${F}_{ij}=\frac{{f}_{ij}}{\sqrt{{\sum}_{j=1}^{m}{f}_{ij}^{2}}}$$
 Next, find the fitness weights by taking the product of the weight of each objective with fitness values using the equation as shown below:$${W}_{ij}={w}_{j}\ast {F}_{ij}$$
 Depending on the ${W}_{ij}$ value, the best and worstperforming 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 ${S}_{i}^{},{S}_{i}^{+}$, where ${S}_{i}^{+}$ is assigned as the distance to the best individual and ${S}_{i}^{}$ is assigned as the distance to the worst individual.
 The last step is to find the degree of approximation ${D}_{i}$ using Equation (7),$${D}_{i}=\frac{{S}_{i}^{}}{{S}_{i}^{}+{S}_{i}^{+}}$$
3.2. Performance Evaluation
3.2.1. Pareto Front Spread ($\Delta $)
3.2.2. Generational Distance ($GD$)
3.2.3. Pareto Front Spacing ($Sp$)
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 TwoObjective Function [38];
 Zitzler–Deb–Thiele’s Function N1, N2, N3, N4, and N6 [39];
 Osyczka and Kundu Function [40];
 ConstrEx Problem;
 VRUC Test 1, VRUC 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  MultiObjective Optimization 
MCDM  MultiCriteria Decision Making 
EA  Evolutionary Algorithm 
TOPSIS  Technique for Order of Preference by Similarity to Ideal Solution 
NSGAII  NonDominated Sorting Genetic AlgorithmII 
NSGAIII  NonDominated Sorting Genetic AlgorithmIII 
MPF  MultiCriteria Decision Making–Pareto Front 
eMPF  elite MultiCriteria Decision Making–Pareto Front 
CCEA  Cooperative Coevolutionary Algorithms 
Appendix A
Function  Test Function  Constraints  Search Domain 

Binh and Korn Function  ${f}_{1}\left(x\right)=4{x}_{1}^{2}+4{x}_{2}^{2}$ ${f}_{2}\left(x\right)={({x}_{1}5)}^{2}+{({x}_{2}5)}^{2}$  ${({x}_{1}5)}^{2}+{x}_{2}^{2}\le 25$ ${({x}_{1}8)}^{2}+{({x}_{2}+3)}^{2}\ge 7.7$  $0\le {x}_{1}\le 5$ $0\le {x}_{2}\le 3$ 
Chankong and Haimes Function  ${f}_{1}({x}_{1},{x}_{2})=2+{({x}_{1}2)}^{2}+{({x}_{2}1)}^{2}$ ${f}_{2}({x}_{1},{x}_{2})=9{x}_{1}{({x}_{2}1)}^{2}$  ${x}_{1}^{2}+{x}_{2}^{2}\le 225$ ${x}_{1}3{x}_{2}+10\le 0$  $20\le {x}_{1}$ ${x}_{2}\le 20$ 
Fonseca–Fleming Function  ${f}_{1}\left(\mathbf{x}\right)=1exp\left({\sum}_{i=1}^{n}{\left({x}_{i}\frac{1}{\sqrt{n}}\right)}^{2}\right)$ ${f}_{2}\left(\mathbf{x}\right)=1exp\left({\sum}_{i=1}^{n}{\left({x}_{i}+\frac{1}{\sqrt{n}}\right)}^{2}\right)$  $4\le {x}_{i}\le 4$ $1\le i\le n$  
Test Function 4  ${f}_{1}({x}_{1},{x}_{2})={x}_{1}^{2}{x}_{2}$ ${f}_{2}({x}_{1},{x}_{2})=0.5{x}_{1}{x}_{2}1$  $6.5\frac{{x}_{1}}{6}{x}_{2}\ge 0$ $7.50.5{x}_{1}{x}_{2}\ge 0$ $305{x}_{1}{x}_{2}\ge $0  $7\le {x}_{1}$ ${x}_{2}\le 4$ 
Kursawe Function  ${f}_{1}\left(\mathbf{x}\right)={\sum}_{i=1}^{n1}\left(10exp\left(0.2\sqrt{{x}_{i}^{2}+{x}_{i+1}^{2}}\right)\right)$ ${f}_{2}\left(\mathbf{x}\right)={\sum}_{i=1}^{n}\left({\left{x}_{i}\right}^{0.8}+5\mathrm{sin}\left({x}_{i}^{3}\right)\right)$  $5\le {x}_{i}\le 5$ $1\le i\le 3$  
Schaffer Function N1  ${f}_{1}\left(x\right)={x}^{2}$ ${f}_{2}\left(x\right)={(x2)}^{2}$  $A\le x\le A$ Values of A from 10 to ${10}^{5}$  
Schaffer Function N2  ${f}_{1}\left(x\right)=\left\{\begin{array}{c}x\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}x\le 1\hfill \\ x2\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}1<x\le 3\hfill \\ 4x\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}3<x\le 4\hfill \\ x4\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}x>4\hfill \end{array}\right.$ ${f}_{2}\left(x\right)={(x5)}^{2}$  $5\le x\le 10$  
Poloni’s TwoObjective Function  ${f}_{1}({x}_{1},{x}_{2})=[1+{({A}_{1}{B}_{1}({x}_{1},{x}_{2}))}^{2}+{({A}_{2}{B}_{2}({x}_{1},{x}_{2}))}^{2}]$ ${f}_{2}({x}_{1},{x}_{2})={({x}_{1}+3)}^{2}+{({x}_{2}+1)}^{2}$ where ${A}_{1}=0.5\mathrm{sin}\left(1\right)2\mathrm{cos}\left(1\right)+\mathrm{sin}\left(2\right)1.5\mathrm{cos}\left(2\right),$ ${A}_{2}=1.5\mathrm{sin}\left(1\right)\mathrm{cos}\left(1\right)+2\mathrm{sin}\left(2\right)0.5\mathrm{cos}\left(2\right),$ ${B}_{1}({x}_{1},{x}_{2})=0.5\mathrm{sin}\left({x}_{1}\right)2\mathrm{cos}\left({x}_{1}\right)+\mathrm{sin}\left({x}_{2}\right)1.5\mathrm{cos}\left({x}_{2}\right),$ ${B}_{2}({x}_{1},{x}_{2})=1.5\mathrm{sin}\left({x}_{1}\right)\mathrm{cos}\left({x}_{1}\right)+2\mathrm{sin}\left({x}_{2}\right)0.5\mathrm{cos}\left({x}_{2}\right).$  $\pi \le {x}_{1}$; ${x}_{2}\le \pi $  
Zitzler–Deb–Thiele’s Function (ZDT) N1  ${f}_{1}\left(x\right)={x}_{1}$ ${f}_{2}\left(x\right)=g\left(x\right)h({f}_{1}\left(x\right),g\left(x\right))$ $g\left(x\right)=1+\frac{9}{29}{\sum}_{i=2}^{30}{x}_{i}$ $h({f}_{1}\left(x\right),g\left(x\right))=1\sqrt{\frac{{f}_{1}\left(x\right)}{g\left(x\right)}}$  $0\le {x}_{i}\le 1$ $1\le i\le 30$  
Zitzler–Deb–Thiele’s Function (ZDT) N2  ${f}_{1}\left(x\right)={x}_{1}$ ${f}_{2}\left(x\right)=g\left(x\right)h({f}_{1}\left(x\right),g\left(x\right))$ $g\left(x\right)=1+\frac{9}{29}{\sum}_{i=2}^{30}{x}_{i}$ $h({f}_{1}\left(x\right),g\left(x\right))=1{\left(\frac{{f}_{1}\left(x\right)}{g\left(x\right)}\right)}^{2}$  $0\le {x}_{i}\le 1$ $1\le i\le 30$  
Zitzler–Deb–Thiele’s Function (ZDT) N3  ${f}_{1}\left(x\right)={x}_{1}$ ${f}_{2}\left(x\right)=g\left(x\right)h({f}_{1}\left(x\right),g\left(x\right))$ $g\left(x\right)=1+\frac{9}{29}{\sum}_{i=2}^{30}{x}_{i}$ $h({f}_{1}\left(x\right),g\left(x\right))=1\sqrt{\frac{{f}_{1}\left(x\right)}{g\left(x\right)}}\left(\frac{{f}_{1}\left(x\right)}{g\left(x\right)}\right)\mathrm{sin}10\pi {f}_{1}\left(x\right)$  $0\le {x}_{i}\le 1$ $1\le i\le 30$  
Zitzler–Deb–Thiele’s Function (ZDT) N4  ${f}_{1}\left(x\right)={x}_{1}$ ${f}_{2}\left(x\right)=g\left(x\right)h({f}_{1}\left(x\right),g\left(x\right))$ $g\left(x\right)=91+{\sum}_{i=2}^{10}({x}_{i}^{2}10\mathrm{cos}4\pi {x}_{1})$ $h({f}_{1}\left(x\right),g\left(x\right))=1\sqrt{\frac{{f}_{1}\left(x\right)}{g\left(x\right)}}$  $0\le {x}_{1}\le 1$ $5\le {x}_{i}\le 5$ $2\le i\le 10$  
Zitzler–Deb–Thiele’s Function (ZDT) N6  ${f}_{1}\left(x\right)=1exp4{x}_{1}{\left(\mathrm{sin}6\pi {x}_{1}\right)}^{6}$ ${f}_{2}\left(x\right)=g\left(x\right)h({f}_{1}\left(x\right),g\left(x\right))$ $g\left(x\right)=1+9{[\frac{{\sum}_{i=2}^{10}\left({x}_{i}\right)}{9}]}^{0.25}$ $h({f}_{1}\left(x\right),g\left(x\right))=1{\left(\frac{{f}_{1}\left(x\right)}{g\left(x\right)}\right)}^{2}$  $0\le {x}_{i}\le 1$ $1\le i\le 10$  
Osyczka & Kundu Function  ${f}_{1}\left(x\right)=25{({x}_{1}2)}^{2}{({x}_{2}2)}^{2}{({x}_{3}1)}^{2}{({x}_{4}4)}^{2}{({x}_{5}1)}^{2}$ ${f}_{2}\left(x\right)={\sum}_{i=1}^{6}{x}_{1}^{2}$  ${g}_{1}\left(x\right)={x}_{1}+{x}_{2}2\ge 0$ ${g}_{2}\left(x\right)=6{x}_{1}{x}_{2}\ge 0$ ${g}_{3}\left(x\right)=2{x}_{2}+{x}_{1}\ge 0$ ${g}_{4}\left(x\right)=2{x}_{1}+3{x}_{2}\ge 0$ ${g}_{5}\left(x\right)=4{({x}_{3}3)}^{2}{x}_{4}\ge 0$ ${g}_{6}\left(x\right)={({x}_{5}3)}^{2}+{x}_{6}4\ge 0$  $0\le {x}_{1},{x}_{2},{x}_{6}\le 10$ $1\le {x}_{3},{x}_{5}\le 5$ $0\le {x}_{4}\le 6$ 
ConstrEx Problem  ${f}_{1}({x}_{1},{x}_{2})={x}_{1}$ ${f}_{2}({x}_{1},{x}_{2})=\frac{1+{x}_{2}}{{x}_{1}}$  ${x}_{2}+9{x}_{1}\ge 6$ ${x}_{2}+9{x}_{1}\ge 1$  $0.1\le {x}_{1}\le 1$ $0\le {x}_{2}\le 5$ 
VRUC Test 1  ${f}_{1}({x}_{1},{x}_{2})=\frac{1}{{x}_{1}^{2}+{x}_{2}^{2}+1}$ ${f}_{2}({x}_{1},{x}_{2})={x}_{1}^{2}+3{x}_{2}^{2}+1$  $3\le {x}_{1},{x}_{2}\ge 3$  
VRUC Test 2  ${f}_{1}({x}_{1},{x}_{2})={x}_{1}+{x}_{2}+1$ ${f}_{2}({x}_{1},{x}_{2})={x}_{1}^{2}+2{x}_{2}1$  $3\le {x}_{1},{x}_{2}\ge 3$  
MSGA Test 1  ${f}_{1}({x}_{1},{x}_{2})={({x}_{1}^{2}+{x}_{2}^{2})}^{0.125}$ ${f}_{2}({x}_{1},{x}_{2})={({({x}_{1}0.5)}^{2}+{({x}_{2}0.5)}^{2})}^{0.25}$  $5\le {x}_{1},{x}_{2}\ge 10$  
MHHM1  ${f}_{1}\left(x\right)={(x0.8)}^{2}$ ${f}_{2}\left(x\right)={(x0.85)}^{2}$ ${f}_{3}\left(x\right)={(x0.9)}^{2}$  $0\le x\ge 1$  
MHHM2  ${f}_{1}({x}_{1},{x}_{2})={({x}_{1}0.8)}^{2}+{({x}_{2}0.6)}^{2}$ ${f}_{2}({x}_{1},{x}_{2})={({x}_{1}0.85)}^{2}+{({x}_{2}0.7)}^{2}$ ${f}_{3}({x}_{1},{x}_{2})={({x}_{1}0.9)}^{2}+{({x}_{2}0.6)}^{2}$  $0\le {x}_{1},{x}_{2}\ge 1$  
Viennet Function  ${f}_{1}({x}_{1},{x}_{2})=0.5({x}_{1}^{2}+{x}_{2}^{2})+\mathrm{sin}{x}_{1}^{2}+{x}_{2}^{2}$ ${f}_{2}({x}_{1},{x}_{2})=\frac{{(3{x}_{1}2{x}_{2}+4)}^{2}}{8}+\frac{{({x}_{1}{x}_{2}+1)}^{2}}{27}+15$ ${f}_{3}({x}_{1},{x}_{2})=\frac{1}{{x}_{1}^{2}+{x}_{2}^{2}+1}1.1exp(({x}_{1}^{2}+{x}_{2}^{2}))$  $3\le {x}_{1};{x}_{2}\le 3$ 
Appendix B
Test Function  Method  Minimum  Mean  Standard Deviation  

Function 1  Function 2  Function 1  Function 2  Function 1  Function 2  
NSGAII  0.000948  4.005556  52.645992  17.283002  39.400303  12.500695  
Binh and Korn  NSGAIII  9.332152  4.533799  100.464088  10.277145  43.565188  13.756248 
MPF  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  
NSGAII  10.119513  −217.695484  113.45425  −112.737342  62.128512  62.677727  
Chankong Haimes  NSGAIII  96.621684  −217.730419  213.296477  −210.363403  20.65679  24.166874 
MPF  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  NSGAII  0.000034  0.000042  0.578248  0.580166  0.296733  0.296151 
NSGAIII  0.092419  0.032975  0.723804  0.497992  0.153209  0.159965  
MPF  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  
NSGAII  −6.504972  −8.499939  −1.534355  −8.145423  3.336720  0.277358  
Test Function 4  NSGAIII  −6.506321  −8.498921  −5.331177  −7.623771  3.929542  0.229023 
MPF  −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  
NSGAII  −19.785629  −5.138218  −10.217168  −2.560673  3.696744  1.566748  
Kursawe  NSGAIII  −19.800519  −2.761169  −12.516551  −1.290272  2.812103  0.732146 
MPF  −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  
NSGAII  0.000000  0.000000  1.309796  1.335708  1.171495  1.178720  
Schaffer function N1  NSGAIII  0.458913  0.452471  2.092166  1.548018  1.830392  1.803354 
MPF  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  
NSGAII  −0.999966  0.000000  0.519161  7.217303  1.030891  4.693345  
Schaffer function N2  NSGAIII  −0.824930  7.796460  −0.624833  14.028193  0.942749  4.035698 
MPF  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  
NSGAII  1.000107  0.000011  6.867533  6.300837  5.010289  9.464460  
Poloni’s two objectives  NSGAIII  2.072667  8.096344  7.516216  13.666458  7.204542  11.749644 
MPF  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  
NSGAII  0.000016  1.830778  0.259772  3.291023  0.295645  0.985772  
ZDT’s N1  NSGAIII  0.000012  1.836794  0.294363  3.228128  0.313797  1.012261 
MPF  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  
NSGAII  0.000020  3.360521  0.146886  4.209470  0.265380  0.626616  
ZDT’s N2  NSGAIII  0.000018  3.397406  0.150892  4.200882  0.275458  0.627170 
MPF  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  
NSGAII  0.000015  1.197987  0.302574  3.009383  0.295633  1.165188  
ZDT’s N3  NSGAIII  0.000016  1.250750  0.288800  3.049543  0.294440  1.148127 
MPF  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  
NSGAII  0.000014  28.918132  0.084849  64.349782  0.174179  32.479856  
ZDT’s N4  NSGAIII  0.000017  30.178694  0.077131  69.368632  0.168524  34.873943 
MPF  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  
NSGAII  0.280775  6.356673  0.492336  7.271715  0.278471  0.661174  
ZDT’s N6  NSGAIII  0.280775  6.233513  0.499382  7.221104  0.281440  0.653730 
MPF  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  
NSGAII  −258.627419  5.194767  −164.898915  28.366042  62.986998  21.322610  
Osyczka and Kundu  NSGAIII  −258.633516  5.433823  −168.598553  29.169913  62.053323  20.778173 
MPF  −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  
NSGAII  0.390717  1.007121  0.547760  4.747238  0.139116  2.426397  
ConstEx  NSGAIII  0.390230  1.515615  0.441708  7.518592  0.093502  1.616727 
MPF  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  
VRUC Test 1  NSGAII  0.052761  1.000574  0.156242  17.296326  0.193932  10.902374 
NSGAIII  0.052718  2.866609  0.074163  31.895484  0.086781  8.590167  
MPF  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  
VRUC Test 2  NSGAII  −4.995860  −9.993871  −4.995082  −9.992874  0.002898  0.004093 
NSGAIII  −4.995535  −9.993506  −4.995059  −9.992905  0.002668  0.003992  
MPF  −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  NSGAII  0.234489  0.235890  0.706906  0.709010  0.197887  0.197780 
NSGAIII  0.549158  0.491633  0.779350  0.734232  0.086980  0.100268  
MPF  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  NSGAII  0.000051  15.000000  −0.099997 
NSGAIII  131,609.806452  443,378.967742  0.000004  
MPF  4292.110524  10,870.875716  −0.002940  
eMPF  0.000028  15.000026  −0.099998  
MHHM1  NSGAII  9.6666 × ${10}^{11}$  7.3333 × ${10}^{11}$  6.33333 × ${10}^{11}$ 
NSGAIII  6.193743 × ${10}^{5}$  4.7659963 × ${10}^{5}$  3.433997 × ${10}^{5}$  
MPF  2.4781883 × ${10}^{5}$  7.064937 × ${10}^{5}$  7.134097 × ${10}^{5}$  
eMPF  1.0666 × ${10}^{10}$  9.6666 × ${10}^{11}$  5.3333 × ${10}^{11}$  
MHHM2  NSGAII  6.18220 × ${10}^{6}$  1.7605 × ${10}^{6}$  2.837620 × ${10}^{6}$ 
NSGAIII  0.001283  0.001229  0.00249529  
MPF  0.00029338  0.0004046536  0.0006900754  
eMPF  6.868939 × ${10}^{6}$  1.63898 × ${10}^{6}$  4.5422 × ${10}^{6}$ 
Test Function  Method  Mean  

Function 1  Function 2  Function 3  
Viennet  NSGAII  3.330086  15.285244  0.053903 
NSGAIII  132,295.691684  445,689.241041  0.000004  
MPF  4607.780516  11,696.522534  0.001230  
eMPF  0.572229  15.485611  −0.024929  
MHHM1  NSGAII  0.0034099  0.0009104  0.00341086 
NSGAIII  0.0013004  0.0005224  0.00474447  
MPF  0.00443803  0.002016614  0.00459519  
eMPF  0.0026903  0.00020094  0.00271158  
MHHM2  NSGAII  0.0054150  0.005925  0.00555724 
NSGAIII  0.0035569  0.003649  0.0063590  
MPF  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  NSGAII  2.688537  0.511459  0.081092 
NSGAIII  6979.937413  26,870.603732  0.000000  
MPF  2278.826105  7778.051973  0.010432  
eMPF  0.330933  0.328919  0.049485  
MHHM1  NSGAII  0.00313115  0.00079  0.0031071 
NSGAIII  0.0016641  0.0004383  0.00146057  
MPF  0.0045989  0.00074356  0.004502838  
eMPF  0.001529  0.00059  0.0015419  
MHHM1  NSGAII  0.003749  0.00417569  0.0037675 
NSGAIII  0.001986  0.0018050  0.0020722  
MPF  0.0032253  0.0035306  0.0034066  
eMPF  0.0020989  0.0021489  0.002019 
Appendix C
Test Function  Method  Generational Distance $\mathit{GD}$  Spread ($\Delta $)  Spacing ($\mathit{Sp}$) 

Binh and Korn  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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 
ConstrEx Problem  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF 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 
VRUC Test 1  NSGAII NSGAIII MPF 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 
VRUC Test 2  NSGAII NSGAIII MPF eMPF  9.4 × ${10}^{5}$ 9.4 × ${10}^{5}$ 0.000324 0.000273  0.717724 0.636498 0.577738 0.712867  0.00000 0.00000 0.00000 0.00000 
Vinnet Function  NSGAII NSGAIII MPF 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  NSGAII NSGAIII MPF eMPF  6 × ${10}^{6}$ 2.8 × ${10}^{5}$ 1.8 × ${10}^{5}$ 2.3 × ${10}^{5}$  0.772737 1.640253 1.620182 1.688381  1.1 × ${10}^{5}$ 3.3 × ${10}^{5}$ 2.6 × ${10}^{5}$ 3.5 × ${10}^{5}$ 
MHHM2  NSGAII NSGAIII MPF 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 MultiCriteria Decision Making—Pareto Front Optimization in MultiObjective Optimization. Algorithms 2024, 17, 206. https://doi.org/10.3390/a17050206
Kesireddy A, Medrano FA. Elite MultiCriteria Decision Making—Pareto Front Optimization in MultiObjective Optimization. Algorithms. 2024; 17(5):206. https://doi.org/10.3390/a17050206
Chicago/Turabian StyleKesireddy, Adarsh, and F. Antonio Medrano. 2024. "Elite MultiCriteria Decision Making—Pareto Front Optimization in MultiObjective Optimization" Algorithms 17, no. 5: 206. https://doi.org/10.3390/a17050206