Elite Exploitation: A Combination of Mathematical Concept and EMO Approach for Multi-Objective Decision Making
Abstract
:1. Introduction
- Inspired by classic Social Darwinism, we proposed an elite exploitation strategy:
- For enhancing diversity, we design a crowd distance-based roulette.
- For better applicability, we design a decision-makers’ preference-based mechanism to control the exploitation intensity.
- For improving the convergence performance, we propose a novel exploitation system and a symmetry exploitation operator for local search (i.e., individual exploitation).
- We test our proposed algorithm with ten widely used algorithms on 36 test problems with different complexity, and the simulated experiment results proved the effectiveness of our method.
2. Brief Review of NSGA-II Framework
3. Proposed NSGA-II-BnF Algorithm
Algorithm 1 Main Loop of NSGA-II-BnF |
Input: N, k, r, |
Output: population set PInitialize: population set P |
1: for each do |
2: Evaluation; |
3: Non-Dominated sort; |
4: Crowding distance; |
5: end for |
6: =1; |
7: while do |
8: genetic procedures to population P; |
9: obtain an offspring set Q; |
10: for each do |
11: Evaluation; |
12: Non-Dominated sort; |
13: Crowding distance; |
14: end for |
15: ; |
16: do Non-dominated sort operate to set P; |
17: biased resource allocation strategy; //(Algorithm 2); |
18: obtain an archive D; |
19: do self-guided fast symmetry individual exploitation to archive D;//(Algorithm 3); |
20: obtain a set K of generated neighbors; |
21: for each do |
22: Evaluation; |
23: Non-Dominated sort; |
24: Crowding distance; |
25: end for |
26: ; |
27: do Non-dominated sort operate to set P; |
28: do elite strategy to set P; |
29: =+1; |
30: end while |
31: obtain a population set P; |
3.1. Biased Elite Allocation Strategy
Algorithm 2 Biased Elite Allocation Strategy |
Input: set P, k |
Output: |
1: for each do |
2: if Non-Dominated rank of ==1 then |
3: ; |
4: end if |
5: end for |
6: for each do |
7: calculate the cumulative probability by Equations (2) and (3); |
8: end for |
9: m=1; |
10: while do |
11: h=rand(1); |
12: if then |
13: ; |
14: else |
15: find that makes holds; |
16: ; |
17: end if |
18: +1; |
19: end while |
20: obtain an ; |
3.1.1. The Size of Candidate Group
3.1.2. Allocation Procedure
- (1)
- Apply crowding distance-based normalization calculated as below to individuals in set T.
- (2)
- Calculate the cumulative probability of each defined as
- (3)
- Generate a random number h in [0,1].
- (4)
- If , save individual at ; otherwise, find the individual , which makes holds, and save individual at .
- (5)
- Repeat steps (3) and (4) until there are members in .
3.2. Self-Guided Fast Symmetry Individual Exploitation Approach
Algorithm 3 Self-guided fast symmetry individual exploitation |
Input: n, archive D, a, b, r, |
Output: set K |
1: Initialize: variable s, i; |
2: for each member in do; |
3: s=1; |
4: for do; |
5: l=round(rand(1)*())); |
6: calculate turbulence v by l and Equation (7); |
7: j=randperm(n,1); |
8: if then |
9: calculate new dimension by Equation (5); |
10: replace the by ; |
11: obtain a neighbor of ; |
12: ; |
13: else |
14: goto line 6; |
15: end if |
16: ; |
17: end for |
18: end for |
19: obtain a set K of neighbors; |
3.2.1. Self-Guided Individual Exploitation System
3.2.2. Tanh-Based Exploitation Operator
- (1)
- Generate a set L with r random number(s), the value of which is in the interval ; for an example, we set ;
- (2)
- For each , calculate as below, which results in the set of turbulence values .
- (3)
- Select j-th dimension of randomly.
- (4)
- After obtaining the turbulence, we use Equation (5) to generate the new j-th dimension , then, we generate neighbors of defined as
4. Experiment
4.1. Test Problems
4.2. Indicators
4.3. Experiment Settings
4.4. Comparison among NSGA-II-BnF and Four NSGA-II Series Algorithms
4.5. Comparison Among NSGA-II-BnF and Four Classic Algorithms
4.6. Comparison between NSGA-II-BnF and NSGA-II
5. Conclusions and Future Works
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Algorithm | Parameter Settings |
---|---|
NSGA-II-conflict | |
rNSGA-II | |
RPDNSGA-II | |
NSGA-II-SDR | |
MOEA/D-DE | |
dMOPSO | |
SMOPSO | |
SPEA-II | |
NSGA-II-BnF |
Problem | NSGA-IIconflict | rNSGA-II | RPDNSGA-II | NSGAII-SDR | NSGA-II-BnF |
---|---|---|---|---|---|
DTLZ1 | 9.9524 × 10 (5.42 × 10) − | + | 1.0742 × 10 (1.14 × 10) − | 2.1962 × 10 (5.68 × 10) − | 2.9848 × 10 (6.61 × 10) |
DTLZ2 | + | 5.3578 × 10 (3.96 × 10) − | 8.1904 × 10 (1.46 × 10) − | 2.9394 × 10 (7.70 × 10) − | 2.0754 × 10 (1.65 × 10) |
DTLZ3 | 1.1864 × 10 (2.46 × 10) = | = | 2.0767 × 10 (5.22 × 10) − | 2.3742 × 10 (6.63 × 10) − | 1.1138 × 10 (2.00 × 10) |
DTLZ4 | 8.1915 × 10 (1.46 × 10) − | 3.0173 × 10 (1.28 × 10) − | + | 1.0502 × 10 (6.65 × 10) − | 2.6736 × 10 (1.35 × 10) |
DTLZ5 | 8.1929 × 10 (1.46 × 10) − | 5.3366 × 10 (4.00 × 10) − | + | 2.9406 × 10 (4.72 × 10) − | 2.0743 × 10 (1.64 × 10) |
DTLZ6 | 8.2086 × 10 (1.46 × 10) − | 1.2965 × 10 (4.41 × 10) − | + | 1.5183 × 10 (1.01 × 10) − | 2.0435 × 10 (1.12 × 10) |
DTLZ7 | 8.6756 × 10 (1.55 × 10) − | 1.0854 × 10 (1.57 × 10) − | 1.5021 × 10 (2.11 × 10) − | 4.5382 × 10 (3.94 × 10) − | |
WFG1 | 1.3503 × 10 (2.74 × 10) − | 3.2202 × 10 (7.66 × 10) − | 7.8765 × 10 (6.57 × 10) − | 1.4982 × 10 (1.27 × 10) − | |
WFG2 | 1.1640 × 10 (1.09 × 10) − | 1.5311 × 10 (1.91 × 10) − | 1.1444 × 10 (3.26 × 10) − | 1.2612 × 10 (1.03 × 10) − | |
WFG3 | 1.1179 × 10 (2.53 × 10) − | 1.7132 × 10 (1.47 × 10) − | 8.1188 × 10 (2.58 × 10) − | 7.8637 × 10 (3.20 × 10) − | |
WFG4 | 9.9711 × 10 (6.50 × 10) − | 1.7631 × 10 (2.16 × 10) − | 4.6756 × 10 (6.22 × 10) − | 8.9396 × 10 (4.20 × 10) − | |
WFG5 | 1.8178 × 10 (2.96 × 10) − | 9.6963 × 10 (1.95 × 10) − | 6.6504 × 10 (2.28 × 10) − | 7.2977 × 10 (2.33 × 10) − | |
WFG6 | = | 1.8242 × 10 (1.24 × 10) − | 9.6338 × 10 (7.35 × 10) − | 5.8111 × 10 (2.86 × 10) = | 5.7762 × 10 (2.04 × 10) |
WFG7 | 9.9728 × 10 (1.66 × 10) − | 2.1395 × 10 (8.76 × 10) − | 7.2479 × 10 (1.99 × 10) − | 8.8962 × 10 (3.79 × 10) − | |
WFG8 | 9.5071 × 10 (5.46 × 10) − | 2.8398 × 10 (2.09 × 10) − | 5.4962 × 10 (3.88 × 10) − | + | 1.0804 × 10 (1.23 × 10) |
WFG9 | 9.8125 × 10 (1.38 × 10) − | 1.3394 × 10 (3.89 × 10) − | 2.1992 × 10 (1.91 × 10) − | 3.6133 × 10 (4.90 × 10) − | |
UF1 | 1.9890 × 10 (1.85 × 10) − | 2.7593 × 10 (1.64 × 10) − | 1.1126 × 10 (3.24 × 10) − | + | 7.7956 × 10 (2.01 × 10) |
UF2 | 1.2273 × 10 (1.47 × 10) − | 2.9076 × 10 (1.27 × 10) − | + | 5.3241 × 10 (5.23 × 10) − | 3.5607 × 10 (9.04 × 10) |
UF3 | 6.6889 × 10 (2.62 × 10) − | 3.0716 × 10 (1.14 × 10) − | + | 2.0806 × 10 (5.23 × 10) − | 6.5324 × 10 (4.05 × 10) |
UF4 | 3.5123 × 10 (4.35 × 10) − | 7.8673 × 10 (1.14 × 10) − | 7.6674 × 10 (7.39 × 10) − | 5.0782 × 10 (4.89 × 10) − | |
UF5 | 1.9618 × 10 (7.81 × 10) − | 3.9719 × 10 (1.20 × 10) − | 6.0782 × 10 (1.05 × 10) − | 2.2774 × 10 (4.22 × 10) = | |
UF6 | 2.0294 × 10 (8.68 × 10) − | + | 2.8153 × 10 (1.26 × 10) − | 1.2945 × 10 (3.32 × 10) = | 1.3067 × 10 (4.42 × 10) |
UF7 | 5.7339 × 10 (2.80 × 10) − | 4.6755 × 10 (1.44 × 10) − | + | 4.9880 × 10 (6.35 × 10) = | 5.3926 × 10 (5.20 × 10) |
UF8 | 6.3153 × 10 (5.41 × 10) − | 5.1939 × 10 (5.87 × 10) − | 2.9302 × 10 (6.47 × 10) − | + | 2.4502 × 10 (2.99 × 10) |
UF9 | 1.3582 × 10 (3.21 × 10) − | 9.7059 × 10 (4.32 × 10) − | = | 1.9939 × 10 (8.01 × 10) = | 2.4376 × 10 (9.82 × 10) |
UF10 | 8.1630 × 10 (8.62 × 10) = | 5.1885 × 10 (1.11 × 10) − | 6.0368 × 10 (1.94 × 10) − | 4.2669 × 10 (1.32 × 10) − | |
CF1 | 5.7587 × 10 (2.58 × 10) = | 6.7309 × 10 (4.07 × 10) − | 7.0711 × 10 (3.40 × 10) − | 6.1205 × 10 (1.53 × 10) − | |
CF2 | 3.5805 × 10 (5.78 × 10) − | 1.5013 × 10 (1.70 × 10) − | + | 5.2987 × 10 (1.36 × 10) = | 5.2831 × 10 (2.41 × 10) |
CF3 | 5.9873 × 10 (7.02 × 10) = | 2.7659 × 10 (9.23 × 10) = | + | 2.6032 × 10 (8.42 × 10) = | 2.9505 × 10 (8.98 × 10) |
CF4 | 5.5640 × 10 (7.77 × 10) − | 3.0471 × 10 (1.28 × 10) = | 5.7237 × 10 (5.73 × 10) − | 3.0751 × 10 (5.52 × 10) − | |
CF5 | 9.3437 × 10 (1.08 × 10) = | 3.6892 × 10 (1.26 × 10) = | 4.0063 × 10 (1.41 × 10) − | 4.0179 × 10 (1.33 × 10) − | |
CF6 | 3.3716 × 10 (3.30 × 10) = | 3.3937 × 10 (1.18 × 10) − | 3.3920 × 10 (1.08 × 10) − | 1.7075 × 10 (5.98 × 10) = | |
CF7 | 5.0530 × 10 (1.97 × 10) = | 4.5555 × 10 (1.82 × 10) = | 4.5549 × 10 (1.66 × 10) = | = | 2.2834 × 10 (1.48 × 10) |
CF8 | 5.9872 × 10 (5.33 × 10) − | 5.5194 × 10 (2.07 × 10) − | 3.6476 × 10 (9.91 × 10) − | 4.2228 × 10 (1.92 × 10) − | |
CF9 | 2.8537 × 10 (2.91 × 10) − | 3.5133 × 10 (2.49 × 10) − | 8.1477 × 10 (7.91 × 10) − | 1.1444 × 10 (5.31 × 10) − | |
CF10 | 7.3578 × 10 (2.58 × 10) − | 6.5082 × 10 (2.96 × 10) − | 4.5882 × 10 (2.12 × 10) − | 4.6881 × 10 (2.11 × 10) − | |
Best/All | 2/36 | 3/36 | 9/36 | 4/36 | 18/36 |
Total | 1+/27−/8= | 3+/28−/5= | 8+/26−/2= | 3+/24−/9= |
Problem | NSGA-IIconflict | rNSGA-II | RPDNSGA-II | NSGAII-SDR | NSGA-II-BnF |
---|---|---|---|---|---|
DTLZ1 | 3.7712 × 10 (1.04 × 10 ) = | + | 2.3851 (3.63 × 10) − | 2.4801 (4.61 × 10) − | 3.0587 (8.44 × 10) |
DTLZ2 | 3.3481 (1.09 × 10) − | 3.3078 (7.53 × 10) − | 2.1995 (3.63 × 10) − | 3.3468 (1.65 × 10) − | |
DTLZ3 | 1.0825 × 10 (5.93 × 10) − | = | 2.5622 (1.44 × 10) − | 3.2246 (6.09 × 10) = | 3.3368 (2.16 × 10) |
DTLZ4 | 3.3094 (2.13 × 10) − | 3.3078 (7.54 × 10) − | 2.2090 (9.61 × 10) − | 3.2694 (2.96 × 10) − | |
DTLZ5 | 3.3482 (9.79 × 10) − | 3.3078 (7.53 × 10) − | + | 3.3469 (1.46 × 10) − | 3.4809 (4.74 × 10) |
DTLZ6 | 3.3481 (1.03 × 10) − | 3.3078 (7.53 × 10) − | + | 3.3352 (1.15 × 10) − | 2.5854 (1.47 × 10) |
DTLZ7 | 1.8618 (1.62 × 10) − | 2.6916 (5.04 × 10) − | 1.6336 (1.11 × 10) − | 2.7185 (4.25 × 10) − | |
WFG1 | 3.2762 (8.87 × 10) − | 2.9736 (3.43 × 10) − | 2.5634 (6.28 × 10) − | 2.6369 (4.18 × 10) − | |
WFG2 | 3.4528 (1.47 × 10) − | 3.1700 (1.27 × 10) − | 2.2119 (1.96 × 10) − | 3.6294 (1.20 × 10) − | |
WFG3 | 3.3052 (1.89 × 10) − | 3.1709 (1.07 × 10) − | 2.3756 (5.82 × 10) − | 3.5813 (2.68 × 10) = | |
WFG4 | 2.6485 (8.73 × 10) − | 3.1732 (1.25 × 10) − | 2.2344 (2.66 × 10) − | 2.9319 (1.25 × 10) − | |
WFG5 | 2.5730 (1.13 × 10) − | 3.1056 (2.13 × 10) − | 2.1691 (3.93 × 10) − | 3.0671 (2.26 × 10) − | |
WFG6 | = | 3.0817 (2.68 × 10) − | 2.1335 (2.29 × 10) − | 2.6418 (2.04 × 10) − | 3.2575 (2.14 × 10) |
WFG7 | 2.6495 (4.66 × 10) − | 3.1735 (3.24 × 10) − | 2.1933 (8.03 × 10) − | 2.8441 (5.03 × 10) − | |
WFG8 | 2.5619 (4.72 × 10) − | 2.9074 (3.37 × 10) − | 2.0749 (3.12 × 10) − | = | 3.2116 (2.90 × 10) |
WFG9 | 2.6384 (2.97 × 10) − | 3.1379 (6.04 × 10) − | 2.2448 (7.91 × 10) − | 3.0247 (8.72 × 10) − | |
UF1 | 3.4232 (9.34 × 10) - | 3.1300 (7.20 × 10) - | 2.9551 (3.32 × 10) - | = | 3.5012 (7.38 × 10) |
UF2 | 3.5905 (4.85 × 10) - | 3.3841 (6.75 × 10) = | + | 3.3015 (3.89 × 10) - | 3.6764 (3.90 × 10) |
UF3 | 2.8036 (1.20 × 10) - | 1.0183 (1.06 × 10) - | 2.6401 (2.36 × 10) - | 2.9422 (1.34 × 10) - | |
UF4 | 3.3174 (4.71 × 10) - | 3.0821 (1.32 × 10) - | 2.0890 (3.45 × 10) - | 3.3040 (5.01 × 10) - | |
UF5 | 1.9536 (4.23 × 10) - | 2.0312 (5.01 × 10) - | 2.0308 (6.67 × 10) - | 2.4934 (5.21 × 10) - | |
UF6 | 2.4747 (2.66 × 10) - | 2.2823 (9.85 × 10) - | 2.5927 (5.71 × 10) - | 2.9404 (1.93 × 10) - | |
UF7 | 3.1689 (3.70 × 10) - | 1.2877 (1.34 × 10) - | 2.4342 (1.95 × 10) - | 3.4557 (1.93 × 10) = | |
UF8 | 6.3935 (5.01 × 10) - | 5.0769 (1.49 × 10) - | 4.0548 (4.14 × 10) - | + | 6.6545 (1.04 × 10) |
UF9 | 6.1791 (3.50 × 10) - | 5.2490 (6.84 × 10) - | 2.3559 (1.97 × 10) - | 6.4564 (2.45 × 10) = | |
UF10 | 4.3613 (1.23 × 10) - | 4.3784 (2.79 × 10) = | 4.0399 (1.13 × 10) - | = | 6.2468 (1.04 × 10) |
CF1 | 2.1443 (8.16 × 10) = | 1.4097 (1.25 × 10) = | + | 1.7864 (1.33 × 10) = | 1.7851 (1.04 × 10) |
CF2 | 3.5107 (8.95 × 10) = | 2.8016 (1.34 × 10) - | + | 3.4609 (7.78 × 10)= | 3.6345 (3.17 × 10) |
CF3 | 2.3274 (3.67 × 10) - | 2.0198 (1.27 × 10) = | = | 2.5891 (3.51 × 10) = | 2.5452 (3.44 × 10) |
CF4 | 2.6564 (2.31 × 10) = | 2.3161 (8.37 × 10) - | 2.6001 (2.72 × 10) = | = | 2.7146 (2.21 × 10) |
CF5 | 2.3162 (5.02 × 10) = | 1.7184 (1.15 × 10) = | + | 2.4368 (2.24 × 10) = | 2.4547 (2.18 × 10) |
CF6 | 3.2606 (1.18 × 10) = | 2.8129 (1.07 × 10) = | 2.8185 (1.89 × 10) − | 2.9056 (1.01 × 10) = | |
CF7 | 2.5986 (2.10 × 10) − | 2.2178 (1.02 × 10) − | 2.6330 (2.87 × 10) − | = | 2.7709 (2.78 × 10) |
CF8 | 4.8931 (9.42 × 10) − | 4.0300 (2.02 × 10) − | 3.4633 (1.43 × 10) − | 4.8983 (1.12 × 10) − | |
CF9 | 6.5983 (4.34 × 10) − | 5.9069 (1.81 × 10) − | 4.9510 (1.86 × 10) − | 6.6085 (3.22 × 10) − | |
CF10 | 4.5730 (1.00 × 10) − | 3.4969 (2.28 × 10) − | 2.6428 (1.87 × 10) − | + | 5.2637 (8.00 × 10) |
Best/All | 1/36 | 2/36 | 7/36 | 7/36 | 19/36 |
+/−/= | 0+/29−/7= | 1+/28−/7= | 6+/28−/2= | 2+/19−/15= |
Problem | SPEA-II | MOEA/D-DE | SMPSO | dMOPSO | NSGA-II-BnF |
---|---|---|---|---|---|
DTLZ1 | + | 2.0908 × 10 (7.35 × 10) + | 3.1412 × 10 (3.10 × 10) − | 2.6104 × 10 (2.82 × 10) − | 2.9848 × 10 (6.61 × 10) |
DTLZ2 | 3.1727 × 10 (1.15 × 10) − | 3.8207 × 10 (1.48 × 10) − | 2.5700 × 10 (7.18 × 10) − | 1.5720 × 10 (2.10 × 10) − | |
DTLZ3 | 4.3689 × 10 (1.84 × 10) = | 4.4424 × 10 (1.82 × 10) = | 2.7614 × 10 (2.22 × 10) − | 7.4083 × 10 (1.25 × 10) − | |
DTLZ4 | 5.2903 × 10 (1.87 × 10) − | + | 2.7214 × 10 (1.35 × 10) − | 3.5010 × 10 (1.17 × 10) − | 2.6736 × 10 (1.35 × 10) |
DTLZ5 | 3.1777 × 10 (1.15 × 10) − | 3.8175 × 10 (1.37 × 10) − | 2.5513 × 10 (7.08 × 10) − | 1.6313 × 10 (2.09 × 10) − | |
DTLZ6 | 3.5986 × 10 (1.40 × 10) − | 1.2863 × 10 (9.41 × 10) − | 2.6028 × 10 (6.46 × 10) − | + | 2.0435 × 10 (1.12 × 10) |
DTLZ7 | 3.3198 × 10 (1.24 × 10) − | 4.6229 × 10 (1.78 × 10) − | 1.0526 × 10 (1.89 × 10) − | 7.0221 × 10 (1.14 × 10) − | |
WFG1 | 1.1825 × 10 (4.31 × 10) = | 1.2013 × 10 (3.87 × 10) − | 3.0616 × 10 (5.19 × 10) − | 3.2701 × 10 (6.35 × 10) − | |
WFG2 | 1.3261 × 10 (4.67 × 10) − | 2.0696 × 10 (1.61 × 10) − | 1.2621 × 10 (3.35 × 10) − | 1.0709 × 10 (1.19 × 10) − | |
WFG3 | 1.5488 × 10 (6.28 × 10) − | 1.8273 × 10 (1.09 × 10) − | 9.1467 × 10 (4.38 × 10) − | 7.3244 × 10 (7.82 × 10) − | |
WFG4 | 1.5626 × 10 (4.72 × 10) − | 1.5094 × 10 (1.37 × 10) − | 4.3881 × 10 (1.08 × 10) − | 7.8283 × 10 (5.24 × 10) − | |
WFG5 | 6.5280 × 10 (3.21 × 10) − | 8.6818 × 10 (5.62 × 10) − | 6.4034 × 10 (9.35 × 10) − | 6.7354 × 10 (1.90 × 10) − | |
WFG6 | 8.3295 × 10 (1.83 × 10) = | 8.3809 × 10 (1.86 × 10) = | + | 7.3127 × 10 (7.65 × 10) = | 5.7762 × 10 (2.04 × 10) |
WFG7 | 1.7341 × 10 (6.75 × 10) − | 1.7456 × 10 (1.10 × 10) − | 8.6719 × 10 (3.25 × 10) − | 9.2887 × 10 (1.23 × 10) − | |
WFG8 | 3.1152 × 10 (1.37 × 10) − | 1.1629 × 10 (3.10 × 10) − | + | 2.2204 × 10 (1.26 × 10) − | 1.0804 × 10 (1.23 × 10) |
WFG9 | 2.1794 × 10 (2.47 × 10) − | 2.6441 × 10 (2.91 × 10) − | 1.9328 × 10 (2.81 × 10) − | 3.9316 × 10 (2.69 × 10) − | |
UF1 | 1.1636 × 10 (3.39 × 10) − | 1.0707 × 10 (2.68 × 10) − | 1.1435 × 10 (2.14 × 10) − | 2.9113 × 10 (6.49 × 10) − | |
UF2 | = | 4.2089 × 10 (5.95 × 10) − | 4.9176 × 10 (5.64 × 10) − | 7.2812 × 10 (6.72 × 10) − | 3.5607 × 10 (9.04 × 10) |
UF3 | 2.2298 × 10 (5.15 × 10) − | 2.6533 × 10 (3.64 × 10) − | 2.2174 × 10 (7.07 × 10) − | 3.1025 × 10 (9.37 × 10) − | |
UF4 | 5.6202 × 10 (2.43 × 10) − | 5.9946 × 10 (3.46 × 10) − | 8.6831 × 10 (1.01 × 10) − | 1.0842 × 10 (6.43 × 10) − | |
UF5 | 3.9452 × 10 (1.02 × 10) − | 3.7018 × 10 (1.10 × 10) − | 1.7499 × 10 (6.55 × 10) − | 3.9838 × 10 (3.23 × 10) − | |
UF6 | 1.8429 × 10 (9.62 × 10) − | 2.2601 × 10 (1.37 × 10) − | 4.3435 × 10 (1.04 × 10) − | 1.1734 × 10 (2.50 × 10) − | |
UF7 | 1.5346 × 10 (1.33 × 10) − | 1.7477 × 10 (1.50 × 10) − | 1.3599 × 10 (1.36 × 10) − | 2.6362 × 10 (5.64 × 10) − | |
UF8 | 2.8042 × 10 (2.43 × 10) − | 2.6298 × 10 (3.34 × 10) − | 3.2839 × 10 (3.65 × 10) − | 3.0297 × 10 (3.50 × 10) − | |
UF9 | 3.8984 × 10 (1.10 × 10) − | 3.7162 × 10 (6.29 × 10) − | 5.1409 × 10 (5.42 × 10) − | 5.7529 × 10 (4.12 × 10) − | |
UF10 | 4.1234 × 10 (1.23 × 10) = | 3.7774 × 10 (1.38 × 10) = | 5.9138 × 10 (4.54 × 10) − | 8.9974 × 10 (1.68 × 10) − | |
CF1 | + | 6.4683 × 10 (4.28 × 10) = | 6.6559 × 10 (1.27 × 10) = | 3.7872 × 10 (8.98 × 10) + | 4.4362 × 10 (1.40 × 10) |
CF2 | 5.4379 × 10 (1.64 × 10) = | 5.0678 × 10 (1.56 × 10) = | 5.6506 × 10 (1.43 × 10) = | 9.5738 × 10 (1.52 × 10) − | |
CF3 | + | 7.5678 × 10 (1.59 × 10) − | 6.2263 × 10 (1.75 × 10) − | 9.0378 × 10 (1.56 × 10) − | 2.9505 × 10 (8.98 × 10) |
CF4 | + | 6.1865 × 10 (8.50 × 10) − | 1.9973 × 10 (7.84 × 10) + | 1.9199 × 10 (3.08 × 10) + | 2.6903 × 10 (1.29 × 10) |
CF5 | + | 3.0706 × 10 (1.07 × 10) + | 5.6748 × 10 (4.01 × 10) = | 6.1185 × 10 (4.80 × 10) − | 2.5713 × 10 (1.06 × 10) |
CF6 | + | 3.4417 × 10 (3.12 × 10) − | 1.4726 × 10 (2.36 × 10) = | 1.1346 × 10 (2.92 × 10) + | 1.6294 × 10 (4.72 × 10) |
CF7 | 2.5882 × 10 (1.20 × 10) − | 3.1910 × 10 (1.35 × 10) − | 1.1566 × 10 (1.65 × 10) − | 1.0868 × 10 (3.73 × 10) − | |
CF8 | 4.4212 × 10 (1.16 × 10) − | 3.5190 × 10 (1.35 × 10) − | 7.7144 × 10 (3.43 × 10) − | + | 3.1883 × 10 (1.86 × 10) |
CF9 | 1.2748 × 10 (4.72 × 10) − | 1.1157 × 10 (4.22 × 10) − | 1.8030 × 10 (5.46 × 10) − | 1.2731 × 10 (1.85 × 10) − | |
CF10 | 5.3578 × 10 (3.03 × 10) − | 2.7349 × 10 (1.44 × 10) = | 1.1643 (9.76 × 10) − | + | 3.0384 × 10 (9.50 × 10) |
Best/All | 7/36 | 1/36 | 2/36 | 3/36 | 23/36 |
+/−/= | 6+/24−/6= | 3+/27−/6= | 3+/29−/4= | 6+/29−/1= |
Problem | SPEA2 | MOEA/D-DE | SMPSO | dMOPSO | NSGA-II-BnF |
---|---|---|---|---|---|
DTLZ1 | + | 3.0727 (1.09 × 10) = | 3.1129 (1.08 × 10) = | 3.2652 (9.32 × 10) = | 3.0587 (8.44 × 10) |
DTLZ2 | 3.3482 (9.72 × 10) − | 3.3490 (1.06 × 10) − | 3.3485 (8.73 × 10) − | 3.3253 (1.26 × 10) − | |
DTLZ3 | 3.0257 (6.09 × 10) = | = | 2.0337 (1.54 × 10) − | 2.5231 (5.88 × 10) − | 3.3368 (2.16 × 10) |
DTLZ4 | 3.1482 (2.55 × 10) − | + | 3.3096 (2.13 × 10) − | 3.3158 (1.16 × 10) + | 3.3102 (2.13 × 10) |
DTLZ5 | 3.3481 (1.04 × 10) − | 2.3490 (1.37 × 10) − | 3.3485 (8.56 × 10) − | 3.0279 (3.21 × 10) − | |
DTLZ6 | 3.3480 (1.09 × 10) − | 3.3491 (2.77 × 10) − | 3.3488 (7.50 × 10) − | + | 2.5854 (1.47 × 10) |
DTLZ7 | 2.7188 (2.78 × 10) − | 2.5993 (1.70 × 10) − | 2.6361 (1.53 × 10) − | 2.7112 (2.53 × 10) − | |
WFG1 | 3.6415 (4.83 × 10) = | 2.7581 (1.03 × 10) − | 2.2215 (4.60 × 10) − | 2.2623 (2.10 × 10) − | |
WFG2 | 3.6308 (1.15 × 10) − | 3.6295 (1.19 × 10) − | 3.6261 (3.20 × 10) − | 3.4867 (1.93 × 10) − | |
WFG3 | 3.5778 (1.09 × 10) − | 3.5805 (4.52 × 10) − | 3.5803 (7.52 × 10) − | 3.4308 (3.36 × 10) − | |
WFG4 | 3.3450 (5.96 × 10) − | 3.3074 (6.73 × 10) − | 3.2978 (1.13 × 10) − | 3.2679 (6.24 × 10) − | |
WFG5 | + | 3.2248 (2.78 × 10) − | 3.2658 (1.51 × 10) − | 3.2468 (2.03 × 10) − | 3.2770 (9.93 × 10) |
WFG6 | 3.2585 (2.06 × 10) = | 3.1033 (9.09 × 10) − | + | 3.2139 (9.94 × 10) − | 3.2575 (2.14 × 10) |
WFG7 | 3.3448 (3.82 × 10) − | 3.3470 (3.11 × 10) − | 3.3468 (2.68 × 10) − | 3.1922 (3.16 × 10) − | |
WFG8 | 3.2150 (2.30 × 10) − | + | 3.2218 (4.27 × 10) + | 2.8981 (5.36 × 10) − | 3.2116 (2.90 × 10) |
WFG9 | 3.3311 (6.00 × 10) = | 3.2697 (2.29 × 10) − | 3.3000 (2.44 × 10) − | 3.2849 (1.82 × 10) − | |
UF1 | 3.3903 (1.41 × 10) − | 3.2764 (3.90 × 10) − | 3.3445 (1.08 × 10) − | 2.8252 (2.07 × 10) − | |
UF2 | 3.5968 (5.54 × 10) = | 3.2189 (2.06 × 10) − | 3.5392 (3.23 × 10) − | 3.4756 (3.68 × 10) − | |
UF3 | 2.8121 (1.10 × 10) − | 3.0402 (1.24 × 10) − | 3.0100 (1.64 × 10) − | 3.3349 (1.02 × 10) + | |
UF4 | 3.3183 (4.24 × 10) − | 3.2459 (2.36 × 10) − | 3.2028 (4.55 × 10) − | 3.1858 (2.07 × 10) − | |
UF5 | 2.0578 (3.40 × 10) − | 1.1930 (8.16 × 10) − | 2.1133 (3.21 × 10) − | 2.3265 (2.43 × 10) − | |
UF6 | 2.7684 (2.71 × 10) − | 2.9268 (3.54 × 10) − | 2.1249 (3.17 × 10) − | 3.1916 (2.13 × 10) − | |
UF7 | 3.1810 (3.91 × 10) − | 3.2511 (2.12 × 10) − | 3.0870 (3.91 × 10) − | 2.7365 (2.25 × 10) − | |
UF8 | 6.5547 (8.22 × 10) − | 6.6528 (2.77 × 10) = | 5.9756 (3.86 × 10) − | 6.5491 (6.34 × 10) − | |
UF9 | 6.0581 (5.52 × 10) − | 6.4544 (4.23 × 10) − | 5.1765 (3.21 × 10) − | 5.0115 (1.92 × 10) − | |
UF10 | 5.1681 (1.41 × 10) − | 5.1526 (9.76 × 10) − | 5.1054 (2.80 × 10) − | 4.5182 (5.86 × 10) − | |
CF1 | + | 2.1818 (1.36 × 10) + | 2.2456 (1.95 × 10) = | 2.4491 (1.96 × 10) + | 1.7851 (1.04 × 10) |
CF2 | 3.5280 (7.81 × 10) = | 3.2719 (4.03 × 10) − | 3.1092 (3.35 × 10) − | 3.4295 (3.67 × 10) − | |
CF3 | 2.6580 (2.52 × 10) = | = | 1.0257 (7.72 × 10) − | 2.1491 (1.89 × 10) − | 2.5452 (3.44 × 10) |
CF4 | + | 2.8807 (1.65 × 10) + | 2.7404 (2.46 × 10) = | 2.9011 (1.75 × 10) + | 2.7146 (2.21 × 10) |
CF5 | + | 2.2569 (2.11 × 10) − | 1.5678 (1.03 × 10) − | 2.2399 (2.23 × 10) − | 2.4547 (2.18 × 10) |
CF6 | + | 3.4019 (3.01 × 10) + | 3.3404 (4.66 × 10) = | 3.3584 (1.03 × 10) = | 3.3364 (8.51 × 10) |
CF7 | + | 2.8942 (2.90 × 10) = | 1.4887 (1.14 × 10) − | 2.0779 (8.00 × 10) − | 2.7709 (2.78 × 10) |
CF8 | 5.0308 (9.30 × 10) − | 5.1941 (7.03 × 10) − | 2.2787 (1.76 × 10) − | = | 5.4100 (1.33 × 10) |
CF9 | 6.9940 (4.21 × 10) − | 7.0896 (1.12 × 10) − | 6.3356 (6.19 × 10) − | 6.7369 (2.74 × 10) − | |
CF10 | 4.5674 (1.34 × 10) − | 5.3088 (8.87 × 10) = | 4.1674 (1.34 × 10) − | + | 5.2637 (8.00 × 10) |
Best/All | 7/36 | 4/36 | 1/36 | 3/36 | 21/36 |
Total | 6+/23−/7= | 5+/24−/6= | 2+/30−/4= | 6+/27−/3= |
IGD Values | HV Values | |||
---|---|---|---|---|
Problem | NSGA-II | NSGA-II-BnF | NSGA-II | NSGA-II-BnF |
DTLZ1 | 3.2133 × 10 (3.86 × 10) − | = | 3.0587 (8.44 × 10) | |
DTLZ2 | 2.4654 × 10 (6.54 × 10) − | 3.7455 (8.73 × 10) − | ||
DTLZ3 | 3.9239 × 10 (1.79 × 10) = | 2.9587 (1.54 × 10) − | ||
DTLZ4 | + | 2.6736 × 10 (1.35 × 10) | + | 3.3102 (2.13 × 10) |
DTLZ5 | 3.0684 × 10 (1.53 × 10) − | 3.1995 (8.56 × 10) − | ||
DTLZ6 | 2.5132 × 10 (5.45 × 10) − | + | 2.5854 (1.47 × 10) | |
DTLZ7 | 2.7656 × 10 (1.04 × 10) − | 2.6361 (1.53 × 10) − | ||
WFG1 | 1.1681 × 10 (3.68 × 10) = | 3.3215 (9.60 × 10) = | ||
WFG2 | + | 5.7147 × 10 (2.12 × 10) | 3.6261 (3.20 × 10) = | |
WFG3 | 7.3975 × 10 (3.89 × 10) − | 3.3843 (7.52 × 10) − | ||
WFG4 | 7.1386 × 10 (2.90 × 10) − | 3.1745 (1.13 × 10) − | ||
WFG5 | + | 6.2467 × 10 (5.89 × 10) | + | 3.2770 (9.93 × 10) |
WFG6 | 7.0127 × 10 (5.91 × 10) = | + | 3.2575 (2.14 × 10) | |
WFG7 | 7.2569 × 10 (1.69 × 10) − | 3.3431 (2.68 × 10) − | ||
WFG8 | 1.1429 × 10 (2.10 × 10) − | 3.1295 (4.27 × 10) − | ||
WFG9 | 1.7693 × 10 (2.91 × 10) − | 3.2381 (2.44 × 10) − | ||
UF1 | 1.0707 × 10 (1.08 × 10) = | 3.2445 (1.08 × 10) = | ||
UF2 | = | 3.5607 × 10 (9.04 × 10) | 3.3352 (7.23 × 10) = | |
UF3 | 2.0174 × 10 (5.07 × 10) − | 2.3658 (1.64 × 10) − | ||
UF4 | 5.2691 × 10 (2.01 × 10) − | 3.1755 (4.55 × 10) − | ||
UF5 | 3.0018 × 10 (5.90 × 10) − | 2.3644 (3.21 × 10) − | ||
UF6 | 1.6037 × 10 (3.32 × 10) = | 3.1465 (3.17 × 10) − | ||
UF7 | 6.9755 × 10 (1.34 × 10) − | 3.0990 (3.91 × 10) − | ||
UF8 | 2.9298 × 10 (2.94 × 10) − | 5.9756 (3.86 × 10) − | ||
UF9 | = | 2.4376 × 10 (9.82 × 10) | 6.1765 (3.21 × 10) = | |
UF10 | 4.0234 × 10 (1.60 × 10) = | 5.1054 (2.80 × 10) − | ||
CF1 | 4.7559 × 10 (2.39 × 10) = | + | 1.7851 (1.04 × 10) | |
CF2 | = | 5.2831 × 10 (2.41 × 10) | = | 3.6345 (3.17 × 10) |
CF3 | = | 2.9505 × 10 (8.98 × 10) | 2.0635 (1.72 × 10) − | |
CF4 | 3.1391 × 10 (1.18 × 10) = | = | 2.7146 (2.21 × 10) | |
CF5 | 3.8348 × 10 (3.31 × 10) = | 2.3562 (1.03 × 10) = | ||
CF6 | 1.7726 × 10 (2.25 × 10) = | = | 3.3364 (8.51 × 10) | |
CF7 | 3.7845 × 10 (1.52 × 10) = | 2.3758 (1.14 × 10) = | ||
CF8 | 3.4990 × 10 (1.25 × 10) − | 4.2956 (1.76 × 10) − | ||
CF9 | 1.0561 × 10 (1.42 × 10) − | 5.4676 (6.19 × 10) − | ||
CF10 | + | 3.0384 × 10 (9.50 × 10) | 4.7674 (1.34 × 10) = | |
Best/All | 8/36 | 28/36 | 9/36 | 27/36 |
Total | 4+/17−/15= | / | 5+/19−/12= | / |
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Li, W.; Geng, Y.; Zhao, J.; Zhang, K.; Liu, J. Elite Exploitation: A Combination of Mathematical Concept and EMO Approach for Multi-Objective Decision Making. Symmetry 2021, 13, 136. https://doi.org/10.3390/sym13010136
Li W, Geng Y, Zhao J, Zhang K, Liu J. Elite Exploitation: A Combination of Mathematical Concept and EMO Approach for Multi-Objective Decision Making. Symmetry. 2021; 13(1):136. https://doi.org/10.3390/sym13010136
Chicago/Turabian StyleLi, Wenxiao, Yushui Geng, Jing Zhao, Kang Zhang, and Jianxin Liu. 2021. "Elite Exploitation: A Combination of Mathematical Concept and EMO Approach for Multi-Objective Decision Making" Symmetry 13, no. 1: 136. https://doi.org/10.3390/sym13010136