Angle-Based Dual-Association Evolutionary Algorithm for Many-Objective Optimization
Abstract
:1. Introduction
- (1)
- The proposed angle-based twofold association strategy incorporates null subspace consideration, establishing associations with the most fitness-appropriate solutions. This approach not only enhances the probability of exploring uncharted regions but also maintains convergence guarantees.
- (2)
- This paper presents a novel quality assessment scheme that has been developed to quantify solution quality within subspaces. This scheme initially evaluates both convergence and diversity metrics for each solution, with the diversity component further decomposed into global and local diversity measures. The incorporation of dynamic penalty coefficients serves to penalize solutions with inferior global diversity while preserving those located in sparse regions. Additionally, an adaptive two-phase sorting mechanism has been implemented to simultaneously maintain convergence and diversity characteristics.
- (3)
- To validate the efficacy of MOEA-AD, a comprehensive series of simulation experiments were conducted. The experimental results demonstrate that MOEA-AD outperforms existing MOEAs in addressing MaOPs, primarily attributed to its dual-phase sorting strategy. This innovative approach not only facilitates superior search space exploration and population diversity maintenance but also ensures robust convergence characteristics, thereby significantly enhancing the algorithm’s overall performance.
2. Related Work and Background Knowledge
2.1. Related Work
2.1.1. Enhanced Pareto Dominance Strategies
2.1.2. Problem Simplification Strategies
2.1.3. Objective Reduction Strategies
2.1.4. Preference-Driven Optimization Strategies
2.2. Basic Definition
3. The Proposed MOEA-AD
3.1. The Main Framework of MOEA-AD
Algorithm 1 The main framework of MOEA-AD |
Input: N (Population size) Output: P (final population)
|
3.2. The Angle-Based Dual-Association Strategy
Algorithm 2 The angle-based dual-association strategy |
Input: (The normalized merged population.), W (Reference vector set) Output: (Store all solutions associated with each reference vector)
|
3.3. Adaptive Evaluation Strategy
3.4. Adaptive Two-Stage Sorting
3.5. Handling Multi-Objective Optimization Problems
- (1)
- Angle-based dual association: The algorithm assigns solutions to reference vectors by considering both perpendicular distance and the angular relationship, ensuring a well-spread distribution even in high-dimensional objective spaces.
- (2)
- Subspace-based quality evaluation: Each solution’s quality is evaluated not only by its convergence to the Pareto front but also by its diversity. The diversity is further categorized into the following: Global diversity, which measures the solution’s contribution to the overall spread. Local diversity, which maintains the distribution within subregions.
- (3)
- Dynamic penalty mechanism: A penalty is applied to solutions with poor global diversity, protecting sparse regions and promoting exploration in less explored areas.
- (4)
- Adaptive two-stage selection: To balance convergence and diversity, the algorithm uses a two-stage selection strategy. First, solutions are ranked based on their convergence and diversity scores; then, an adaptive mechanism selects the most representative solutions for the next generation.
4. Experimental Study
4.1. Test Problems
4.2. Comparative Algorithms
4.3. Experimental Settings
- (1)
- Execution and stopping condition: Every algorithm is run 20 times independently on each test case. The stopping condition for each execution is the attainment of the maximum function evaluations (MFEs). For the DTLZ1-7 and WFG1-9 test suites, the MFEs are configured as 99,960; 99,990; 100,100; 100,386; and 99,960 corresponding to objective counts m of 5, 8, 12, 16, and 20, respectively.
- (2)
- Statistical evaluation: The Wilcoxon rank-sum test is utilized to evaluate the statistical significance of the outcomes from MOEA-AD and the five other algorithms, with a significance threshold set at 0.05. In all tables, +, −, and ≈ indicate superior to, inferior to, and comparable to competing methods, respectively, while gray highlighting denotes the best value achieved on the current test problem.
- (3)
- Population size: The determination of population size N is governed by parameter H rather than being arbitrarily set. For problems involving 8, 12, 16, and 20 objectives, where , a two-tiered reference vector generation approach with reduced H values, as suggested in, is employed to generate intermediate reference vectors. Comparable configurations are applied to other algorithms. To maintain fairness in comparisons, identical population sizes are utilized across all algorithms.
- (4)
- Crossover and mutation parameters: All compared algorithms utilize SBX [29] and PM [30]. For SBX, the crossover probability is set to 1.0, and the distribution index is set to 20. For polynomial mutation, the distribution index and mutation probability are set to 20 and , respectively. More algorithm parameter settings are shown in Table 1.
- (5)
- Evaluation metrics: This paper employs two metrics, IGD and HV, to assess the diversity and convergence of the algorithms. IGD quantifies the average distance between the algorithm’s solutions and the sampled points on the true Pareto front (PF). A lower IGD value reflects superior convergence and diversity. Notably, solutions dominated by the reference point are omitted in HV calculations. HV is computed using PlatEMO [34], with a higher HV value indicating better algorithm performance.
4.4. Performance Comparison Analysis on the DTLZ Test Suite
4.5. Performance Comparison Analysis on the WFG Test Suite
4.6. Performance Comparison Analysis on Real-World Problems
5. Conclusions
- (1)
- To protect solutions in sparse regions, the framework introduces a dynamic penalty factor that penalizes solutions with insufficient global diversity.
- (2)
- The algorithm also adopts an adaptive two-stage sorting mechanism, which effectively balances convergence and diversity, leading to favorable optimization performance.
- (3)
- However, despite its innovative design, the efficiency of the proposed method decreases as the number of objectives increases.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
N | Population size |
M | Number of objectives |
Ideal point | |
Nadir point | |
Perpendicular distance to reference vector | |
Angle between solution vector and reference vector | |
Q | Quality score of a solution |
Global Diversity | |
Local Diversity | |
Pareto Front | |
Evolutionary Algorithms | |
Many-objective Optimization Problems | |
MOEA-AD | Angle-based dual-association Evolutionary Algorithm for Many-Objective Optimization |
Multi-Objective Evolutionary Algorithm |
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Algorithm | Population Size | Generations | ||||
---|---|---|---|---|---|---|
NSGA-III | 100 | 20 | 1.0 | 20 | 20 | |
A-NSGA-III | 100 | 20 | 1.0 | 20 | 20 | |
SPEAR | 100 | 20 | 1.0 | 20 | 20 | |
MaOEA-IGD | 100 | 20 | 1.0 | 20 | 20 | |
MaOEA-IT | 100 | 20 | 1.0 | 20 | 20 | |
MOEA-AD | 100 | 20 | 1.0 | 20 | 20 |
Problem | M | D | NSGAIII | ANSGAIII | SPEAR | MaOEAIGD | MaOEAIT | MOEA-AD |
---|---|---|---|---|---|---|---|---|
DTLZ1 | 5 | 9 | 6.8098e-2 (6.85e-5) + | 7.1616e-2 (6.77e-3) − | 1.1925e-1 (3.72e-2) − | 5.7702e-1 (3.45e-1) − | 2.1659e+0 (1.88e+0) − | 7.0213e-2 (1.56e-2) |
8 | 12 | 1.1217e-1 (1.13e-2) + | 1.6669e-1 (5.54e-2) ≈ | 1.6260e-1 (2.74e-2) ≈ | 4.2136e-1 (2.84e-1) − | 8.9007e+0 (1.40e+1) − | 2.3043e-1 (1.12e-1) | |
12 | 16 | 1.8235e-1 (2.74e-2) + | 2.0208e-1 (8.33e-2) + | 2.5237e-1 (9.06e-2) + | 4.6780e-1 (2.74e-1) ≈ | 9.7623e+0 (7.39e+0) − | 3.7521e-1 (6.44e-2) | |
16 | 20 | 3.2874e-1 (5.11e-2) ≈ | 3.2676e-1 (4.82e-2) ≈ | 2.7671e-1 (7.96e-2) + | 7.0980e-1 (5.86e-1) − | 1.1653e+1 (1.36e+1) − | 3.2329e-1 (5.21e-2) | |
20 | 24 | 3.3646e-1 (4.56e-2) ≈ | 3.6796e-1 (4.32e-2) − | 3.2870e-1 (7.29e-2) ≈ | 1.1355e+0 (6.04e-1) − | 1.3091e+1 (1.31e+1) − | 3.2918e-1 (4.17e-2) | |
DTLZ2 | 5 | 14 | 2.1222e-1 (2.16e-5) + | 2.1657e-1 (3.43e-3) + | 2.1555e-1 (1.97e-3) + | 2.4122e-1 (8.45e-2) ≈ | 5.7181e-1 (1.17e-1) − | 2.2229e-1 (2.39e-3) |
8 | 17 | 4.3802e-1 (7.58e-2) ≈ | 5.4075e-1 (4.15e-2) − | 3.8749e-1 (7.25e-4) + | 4.2962e-1 (7.19e-2) − | 1.0506e+0 (1.34e-1) − | 4.1287e-1 (3.11e-3) | |
12 | 21 | 6.2319e-1 (3.71e-2) − | 6.2230e-1 (4.45e-2) − | 6.0438e-1 (4.11e-3) − | 8.0273e-1 (1.15e-1) − | 1.0007e+0 (7.04e-2) − | 5.5233e-1 (4.47e-3) | |
16 | 25 | 7.7400e-1 (1.87e-2) − | 7.7504e-1 (1.58e-2) − | 7.1031e-1 (6.90e-4) + | 7.9172e-1 (8.28e-2) − | 1.1505e+0 (9.18e-2) − | 7.2888e-1 (4.51e-3) | |
20 | 29 | 9.9901e-1 (3.01e-2) − | 1.0207e+0 (2.70e-2) − | 7.6103e-1 (5.13e-4) − | 9.5389e-1 (1.84e-1) − | 1.4086e+0 (1.01e-1) − | 7.5049e-1 (3.68e-3) | |
DTLZ3 | 5 | 14 | 2.1398e-1 (1.92e-3) + | 2.4505e-1 (3.42e-2) + | 6.9919e-1 (4.08e-1) + | 1.2305e+1 (4.26e+0) − | 1.2415e+1 (7.75e+0) − | 8.5040e-1 (1.22e-1) |
8 | 17 | 7.8146e-1 (7.29e-1) + | 1.4563e+0 (1.10e+0) ≈ | 4.1491e+0 (1.98e+0) − | 1.0201e+1 (9.77e+0) − | 3.2674e+2 (1.20e+2) − | 1.0003e+0 (2.53e-2) | |
12 | 21 | 2.7194e+0 (2.26e+0) − | 2.3482e+0 (1.45e+0) − | 3.2929e+1 (1.74e+1) − | 9.7749e+0 (8.60e+0) − | 3.1085e+2 (1.28e+2) − | 1.0516e+0 (2.65e-2) | |
16 | 25 | 1.1405e+0 (4.43e-1) + | 1.1876e+0 (4.35e-1) + | 1.4001e+1 (7.32e+0) − | 4.5793e+0 (2.38e+0) − | 3.3470e+2 (1.29e+2) − | 1.2220e+0 (6.26e-2) | |
20 | 29 | 5.0443e+0 (3.98e+0) − | 3.8575e+0 (2.57e+0) − | 3.5426e+1 (1.32e+1) − | 7.0219e+0 (4.35e+0) − | 4.1051e+2 (1.15e+2) − | 1.2677e+0 (4.93e-2) | |
DTLZ4 | 5 | 14 | 2.2386e-1 (5.19e-2) + | 3.0261e-1 (1.14e-1) ≈ | 2.1619e-1 (1.97e-3) + | 3.5437e-1 (1.69e-1) ≈ | 8.8496e-1 (7.62e-2) − | 2.2644e-1 (2.71e-3) |
8 | 17 | 4.5651e-1 (8.11e-2) ≈ | 5.0545e-1 (9.71e-2) ≈ | 3.8758e-1 (6.75e-4) + | 5.5873e-1 (1.24e-1) − | 1.1928e+0 (1.08e-1) − | 4.3019e-1 (8.54e-3) | |
12 | 21 | 6.3702e-1 (3.97e-2) − | 6.3244e-1 (3.74e-2) − | 6.1237e-1 (2.88e-3) − | 6.5071e-1 (5.54e-2) − | 1.1201e+0 (7.07e-2) − | 5.7397e-1 (5.58e-3) | |
16 | 25 | 7.8428e-1 (1.03e-2) − | 7.7229e-1 (2.60e-2) − | 7.1547e-1 (3.76e-3) + | 8.6686e-1 (6.73e-2) − | 1.4080e+0 (1.91e-1) − | 7.3737e-1 (3.75e-3) | |
20 | 29 | 9.7895e-1 (3.92e-2) − | 9.6633e-1 (4.52e-2) − | 7.6489e-1 (1.67e-3) − | 8.6876e-1 (3.23e-2) − | 1.4626e+0 (1.32e-1) − | 7.5097e-1 (5.20e-3) | |
DTLZ5 | 5 | 14 | 1.0084e-1 (3.02e-2) ≈ | 9.5663e-2 (3.57e-2) ≈ | 2.7790e-1 (7.47e-2) − | 5.6062e-1 (1.73e-1) − | 4.3737e-1 (7.46e-2) − | 8.2493e-2 (2.13e-2) |
8 | 17 | 2.6923e-1 (9.91e-2) − | 2.7483e-1 (7.41e-2) − | 4.2423e-1 (1.02e-1) − | 6.3627e-1 (1.36e-1) − | 1.0924e+0 (3.40e-1) − | 1.7223e-1 (4.46e-2) | |
12 | 21 | 2.3597e-1 (7.17e-2) ≈ | 2.3848e-1 (4.83e-2) − | 7.9204e-1 (2.07e-1) − | 6.1468e-1 (1.71e-1) − | 4.0869e-1 (7.46e-2) − | 2.0711e-1 (4.70e-2) | |
16 | 25 | 2.9446e-1 (7.90e-2) ≈ | 3.4403e-1 (1.20e-1) − | 8.6579e-1 (2.99e-1) − | 9.1284e-2 (2.96e-4) + | 7.1616e-1 (2.15e-1) − | 2.5970e-1 (3.80e-2) | |
20 | 29 | 8.5755e-1 (5.67e-1) − | 1.0574e+0 (5.59e-1) − | 1.0243e+0 (2.56e-1) − | 9.1237e-2 (3.27e-4) + | 5.9212e-1 (1.56e-1) − | 2.1242e-1 (5.58e-2) | |
DTLZ6 | 5 | 14 | 2.8246e-1 (6.13e-2) − | 2.3699e-1 (5.78e-2) − | 5.7268e-1 (1.93e-1) − | 7.8057e-1 (2.20e-1) − | 8.4868e+0 (1.83e-1) − | 1.6575e-1 (1.21e-1) |
8 | 17 | 4.6259e-1 (2.57e-1) − | 5.2664e-1 (3.23e-1) − | 7.0656e-1 (4.09e-1) − | 8.3843e-1 (3.80e-1) − | 8.5382e+0 (5.16e-1) − | 2.0354e-1 (8.49e-2) | |
12 | 21 | 6.6405e-1 (3.97e-1) − | 7.5110e-1 (3.92e-1) − | 8.1620e+0 (5.10e-1) − | 7.8367e-1 (2.65e-1) − | 8.5004e+0 (1.78e-1) − | 2.7977e-1 (8.42e-2) | |
16 | 25 | 6.5447e-1 (3.68e-1) − | 6.6824e-1 (4.12e-1) − | 2.8426e+0 (1.13e+0) − | 1.5643e-1 (2.00e-1) + | 8.4048e+0 (3.15e-1) − | 3.4459e-1 (1.57e-1) | |
20 | 29 | 4.6529e+0 (1.80e+0) − | 4.8337e+0 (1.21e+0) − | 4.5914e+0 (1.11e+0) − | 1.5625e-1 (2.00e-1) + | 8.3272e+0 (6.06e-1) − | 3.3311e-1 (1.30e-1) | |
DTLZ7 | 5 | 24 | 3.8508e-1 (2.20e-2) − | 3.9368e-1 (2.19e-2) − | 4.9947e-1 (1.00e-2) − | 2.0942e+0 (1.13e+0) − | 3.6984e+0 (1.82e+0) − | 3.6730e-1 (3.78e-2) |
8 | 27 | 9.5196e-1 (8.05e-2) + | 9.2995e-1 (6.13e-2) + | 1.3207e+0 (2.10e-1) ≈ | 1.4248e+0 (9.19e-1) ≈ | 2.4654e+1 (3.87e+0) − | 1.8136e+0 (1.02e+0) | |
12 | 31 | 2.8491e+0 (5.32e-1) − | 2.9186e+0 (4.56e-1) − | 4.6247e+0 (1.98e-2) − | 1.9166e+0 (8.83e-2) ≈ | 4.6134e+1 (2.47e+0) − | 2.1287e+0 (1.03e+0) | |
16 | 35 | 6.9596e+0 (1.17e+0) + | 7.2672e+0 (7.60e-1) + | 1.2312e+1 (3.00e+0) ≈ | 3.0505e+0 (4.69e-1) + | 6.3561e+1 (8.78e+0) − | 1.2013e+1 (3.51e-1) | |
20 | 39 | 1.4887e+1 (4.27e-1) + | 1.4811e+1 (4.96e-1) + | 1.3494e+1 (4.06e+0) + | 3.5570e+0 (3.33e-1) + | 8.2679e+1 (1.20e+1) − | 1.5370e+1 (3.34e-1) | |
11/17/7 | 7/22/6 | 10/21/4 | 6/24/5 | 0/35/0 |
Problem | M | D | NSGAIII | ANSGAIII | SPEAR | MaOEAIGD | MaOEAIT | MOEA-AD |
---|---|---|---|---|---|---|---|---|
DTLZ1 | 5 | 9 | 9.7065e-1 (2.01e-4) + | 9.6656e-1 (6.01e-3) + | 8.8478e-1 (7.79e-2) ≈ | 2.1152e-1 (3.23e-1) − | 9.2985e-2 (2.41e-1) − | 9.1165e-1 (3.73e-2) |
8 | 12 | 9.8734e-1 (6.55e-3) + | 9.4468e-1 (6.21e-2) + | 9.2251e-1 (4.23e-2) + | 3.6346e-1 (4.13e-1) − | 3.8363e-2 (1.66e-1) − | 6.9529e-1 (2.42e-1) | |
12 | 16 | 9.9159e-1 (2.24e-2) + | 9.3777e-1 (2.06e-1) + | 7.2727e-1 (2.99e-1) + | 3.2726e-1 (3.69e-1) − | 9.7810e-3 (4.37e-2) − | 4.0168e-1 (1.24e-1) | |
16 | 20 | 7.1179e-1 (7.73e-2) + | 7.2979e-1 (6.91e-2) + | 7.2592e-1 (1.73e-1) + | 2.1504e-1 (3.18e-1) − | 8.1313e-5 (3.64e-4) − | 4.9844e-1 (1.40e-1) | |
20 | 24 | 6.5333e-1 (1.08e-1) ≈ | 5.8538e-1 (1.42e-1) ≈ | 6.2364e-1 (2.70e-1) + | 2.3611e-2 (5.58e-2) − | 0.0000e+0 (0.00e+0) − | 5.7950e-1 (1.26e-1) | |
DTLZ2 | 5 | 14 | 7.7468e-1 (4.39e-4) + | 7.6791e-1 (5.33e-3) + | 7.7138e-1 (2.05e-3) + | 7.5438e-1 (8.83e-2) + | 1.5193e-1 (1.32e-1) − | 7.2182e-1 (7.70e-3) |
8 | 17 | 8.5417e-1 (4.49e-2) ≈ | 7.7603e-1 (2.29e-2) − | 8.8121e-1 (2.55e-3) + | 8.5879e-1 (8.52e-2) + | 4.6399e-2 (8.56e-2) − | 8.3063e-1 (1.74e-2) | |
12 | 21 | 9.4621e-1 (2.49e-2) + | 9.4449e-1 (3.18e-2) + | 9.6652e-1 (1.10e-3) + | 6.5508e-1 (1.81e-1) − | 1.1461e-1 (6.50e-2) − | 8.8895e-1 (1.65e-2) | |
16 | 25 | 7.4359e-1 (2.60e-2) − | 7.3946e-1 (2.44e-2) − | 7.0307e-1 (3.90e-2) − | 7.3662e-1 (9.90e-2) − | 2.8194e-2 (3.72e-2) − | 8.0687e-1 (1.82e-2) | |
20 | 29 | 4.4456e-1 (8.28e-2) − | 4.0930e-1 (6.89e-2) − | 7.5500e-1 (2.05e-2) − | 5.9706e-1 (2.52e-1) − | 1.9838e-2 (5.15e-2) − | 8.6454e-1 (1.48e-2) | |
DTLZ3 | 5 | 14 | 7.6348e-1 (1.12e-2) + | 7.2822e-1 (4.02e-2) + | 3.1638e-1 (1.49e-1) + | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 2.0286e-1 (7.23e-2) |
8 | 17 | 6.4491e-1 (3.83e-1) + | 2.8205e-1 (3.25e-1) ≈ | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 2.1103e-1 (1.75e-2) | |
12 | 21 | 8.6706e-2 (2.39e-1) − | 2.0313e-1 (3.66e-1) − | 0.0000e+0 (0.00e+0) − | 4.0275e-3 (1.80e-2) − | 0.0000e+0 (0.00e+0) − | 2.3943e-1 (2.91e-2) | |
16 | 25 | 4.3535e-1 (2.61e-1) + | 3.6382e-1 (2.83e-1) ≈ | 0.0000e+0 (0.00e+0) − | 4.1070e-3 (1.84e-2) − | 0.0000e+0 (0.00e+0) − | 1.3182e-1 (4.46e-2) | |
20 | 29 | 3.1883e-2 (8.25e-2) − | 4.6696e-2 (8.38e-2) − | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 1.7223e-1 (6.27e-2) | |
DTLZ4 | 5 | 14 | 7.6800e-1 (2.89e-2) + | 7.2803e-1 (5.92e-2) ≈ | 7.6846e-1 (3.03e-3) + | 6.9326e-1 (1.18e-1) ≈ | 1.8523e-1 (6.33e-2) − | 7.4072e-1 (7.63e-3) |
8 | 17 | 8.4934e-1 (4.33e-2) ≈ | 8.2530e-1 (4.68e-2) ≈ | 8.7744e-1 (3.26e-3) + | 7.7494e-1 (1.05e-1) − | 6.6369e-2 (4.60e-2) − | 8.5949e-1 (1.12e-2) | |
12 | 21 | 9.4495e-1 (2.31e-2) ≈ | 9.4854e-1 (1.98e-2) ≈ | 9.6433e-1 (1.90e-3) + | 9.2466e-1 (7.11e-2) ≈ | 2.1281e-2 (2.24e-2) − | 9.3856e-1 (5.75e-3) | |
16 | 25 | 8.1048e-1 (2.20e-2) − | 8.1346e-1 (2.41e-2) − | 7.5353e-1 (1.71e-2) − | 6.1899e-1 (7.57e-2) − | 3.9028e-2 (7.68e-2) − | 8.8642e-1 (1.31e-2) | |
20 | 29 | 6.2192e-1 (6.82e-2) − | 6.4820e-1 (8.11e-2) − | 7.7080e-1 (1.87e-2) − | 7.4069e-1 (4.62e-2) − | 1.6475e-2 (3.76e-2) − | 9.3206e-1 (8.57e-3) | |
DTLZ5 | 5 | 14 | 1.0817e-1 (4.44e-3) ≈ | 1.1059e-1 (4.98e-3) ≈ | 2.1929e-2 (2.88e-2) − | 8.4523e-2 (2.89e-2) − | 4.8124e-4 (7.19e-4) − | 1.0785e-1 (8.27e-3) |
8 | 17 | 9.3097e-2 (2.04e-3) − | 9.1567e-2 (1.62e-3) − | 6.0250e-3 (1.28e-2) − | 8.7424e-2 (2.06e-2) − | 3.3451e-7 (1.17e-6) − | 1.0007e-1 (1.55e-3) | |
12 | 21 | 9.1645e-2 (7.02e-4) ≈ | 9.1601e-2 (7.92e-4) ≈ | 7.5173e-4 (3.36e-3) − | 8.2299e-2 (2.81e-2) − | 1.0882e-6 (4.23e-6) − | 9.2007e-2 (8.40e-4) | |
16 | 25 | 9.1402e-2 (5.43e-4) ≈ | 9.1244e-2 (4.46e-4) ≈ | 4.9220e-3 (2.03e-2) − | 9.1610e-2 (3.32e-4) + | 0.0000e+0 (0.00e+0) − | 9.1347e-2 (4.70e-4) | |
20 | 29 | 2.9854e-2 (4.20e-2) − | 1.6880e-2 (3.48e-2) − | 0.0000e+0 (0.00e+0) − | 9.1377e-2 (3.68e-4) ≈ | 2.2860e-3 (6.35e-3) − | 9.1187e-2 (3.77e-4) | |
DTLZ6 | 5 | 14 | 9.1692e-2 (1.83e-3) − | 9.3245e-2 (5.17e-3) − | 3.3052e-3 (1.42e-2) − | 3.7098e-2 (4.18e-2) − | 0.0000e+0 (0.00e+0) − | 1.0095e-1 (6.09e-3) |
8 | 17 | 6.3983e-2 (4.30e-2) − | 7.7316e-2 (3.33e-2) − | 1.4170e-2 (3.06e-2) − | 5.0364e-2 (4.37e-2) − | 0.0000e+0 (0.00e+0) − | 9.9026e-2 (2.30e-3) | |
12 | 21 | 5.4579e-2 (4.57e-2) − | 5.0109e-2 (4.65e-2) − | 0.0000e+0 (0.00e+0) − | 6.8506e-2 (4.06e-2) ≈ | 0.0000e+0 (0.00e+0) − | 9.1714e-2 (1.16e-3) | |
16 | 25 | 5.4561e-2 (4.57e-2) − | 5.4553e-2 (4.57e-2) − | 0.0000e+0 (0.00e+0) − | 9.1451e-2 (3.11e-4) ≈ | 0.0000e+0 (0.00e+0) − | 9.1499e-2 (5.32e-4) | |
20 | 29 | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 0.0000e+0 (0.00e+0) − | 9.1231e-2 (3.27e-4) ≈ | 0.0000e+0 (0.00e+0) − | 9.1130e-2 (3.21e-4) | |
DTLZ7 | 5 | 24 | 2.3354e-1 (3.81e-3) + | 2.3235e-1 (3.65e-3) + | 2.2444e-1 (2.31e-3) + | 1.1949e-1 (3.53e-2) − | 2.4737e-2 (2.88e-2) − | 2.0400e-1 (7.93e-3) |
8 | 27 | 1.7957e-1 (3.01e-2) + | 1.9195e-1 (3.33e-3) + | 1.6281e-1 (9.11e-3) − | 9.9722e-2 (1.02e-2) − | 0.0000e+0 (0.00e+0) − | 1.7115e-1 (9.03e-3) | |
12 | 31 | 1.3112e-1 (1.83e-2) ≈ | 1.3128e-1 (1.25e-2) − | 1.4143e-1 (6.64e-3) ≈ | 2.8396e-2 (2.05e-2) − | 0.0000e+0 (0.00e+0) − | 1.3833e-1 (7.86e-3) | |
16 | 35 | 5.1877e-2 (1.03e-2) − | 4.4437e-2 (9.60e-3) − | 9.4014e-3 (1.25e-2) − | 3.5338e-2 (1.74e-2) − | 0.0000e+0 (0.00e+0) − | 1.1063e-1 (3.99e-3) | |
20 | 39 | 1.0281e-1 (4.74e-3) − | 1.0207e-1 (5.84e-3) − | 1.9194e-3 (1.96e-3) − | 1.5167e-3 (2.97e-3) − | 0.0000e+0 (0.00e+0) − | 1.0768e-1 (3.67e-3) | |
12/15/8 | 9/17/9 | 12/21/2 | 3/26/6 | 0/35/0 |
Problem | M | D | NSGAIII | ANSGAIII | SPEAR | MaOEAIGD | MaOEAIT | MOEA-AD |
---|---|---|---|---|---|---|---|---|
WFG1 | 5 | 14 | 4.7304e-1 (4.92e-3) + | 4.9953e-1 (1.82e-2) ≈ | 5.3277e-1 (1.06e-2) − | 3.4855e+0 (6.13e-1) − | 3.2866e+0 (1.36e+0) − | 4.9273e-1 (1.50e-2) |
8 | 17 | 1.0156e+0 (3.89e-2) + | 1.0843e+0 (8.23e-2) + | 1.3966e+0 (9.89e-2) − | 6.0433e+0 (2.24e+0) − | 4.5525e+0 (3.05e+0) − | 1.0921e+0 (1.59e-2) | |
12 | 21 | 1.4370e+0 (1.86e-1) + | 1.4796e+0 (2.33e-1) + | 1.6506e+0 (6.31e-2) − | 9.5657e+0 (5.14e+0) − | 3.6350e+0 (2.01e-1) − | 1.5790e+0 (5.15e-2) | |
16 | 25 | 2.4191e+0 (1.74e-1) ≈ | 2.4365e+0 (2.43e-1) ≈ | 2.8801e+0 (5.17e-1) ≈ | 1.9326e+1 (8.31e+0) − | 4.7117e+0 (1.29e-1) − | 2.4782e+0 (1.10e-1) | |
20 | 29 | 5.1917e+0 (1.84e-1) − | 5.1612e+0 (2.50e-1) − | 4.9032e+0 (2.90e-1) + | 1.6425e+1 (9.08e+0) − | 6.1888e+0 (6.59e-2) − | 4.9661e+0 (1.39e-1) | |
WFG2 | 5 | 14 | 5.0503e-1 (2.93e-3) + | 6.4659e-1 (1.79e-1) ≈ | 5.0817e-1 (5.60e-3) + | 1.7875e+0 (1.98e-1) − | 1.7868e+0 (3.01e-1) − | 5.5681e-1 (1.66e-2) |
8 | 17 | 1.1936e+0 (1.54e-1) + | 1.3477e+0 (1.38e-1) ≈ | 1.0660e+0 (2.10e-2) + | 2.4318e+0 (4.19e-1) − | 4.4970e+0 (6.93e-1) − | 1.3122e+0 (1.39e-1) | |
12 | 21 | 1.5552e+0 (1.15e-1) + | 1.4794e+0 (8.59e-2) + | 1.4078e+0 (3.57e-2) + | 5.1810e+0 (2.88e+0) − | 5.4335e+0 (1.10e+0) − | 2.8017e+0 (9.39e-1) | |
16 | 25 | 2.8249e+0 (6.49e-1) + | 2.8268e+0 (5.66e-1) + | 2.0708e+0 (2.20e-2) + | 8.7450e+0 (9.03e+0) ≈ | 8.0003e+0 (1.16e+0) − | 5.1623e+0 (1.42e+0) | |
20 | 29 | 8.7535e+0 (6.66e-1) − | 9.8186e+0 (2.59e+0) − | 3.9097e+0 (5.19e-2) + | 1.1434e+1 (9.48e+0) ≈ | 9.9334e+0 (1.83e+0) − | 6.4197e+0 (9.39e-1) | |
WFG3 | 5 | 14 | 5.9629e-1 (5.88e-2) + | 5.1618e-1 (6.06e-2) + | 1.0357e+0 (1.07e-1) − | 5.4348e+0 (2.14e-2) − | 1.9575e+0 (8.71e-1) − | 7.4350e-1 (7.21e-2) |
8 | 17 | 2.0639e+0 (3.83e-1) − | 1.7971e+0 (4.25e-1) − | 2.1259e+0 (1.08e-1) − | 6.7842e+0 (3.31e+0) − | 7.4976e+0 (2.70e-1) − | 1.2646e+0 (4.69e-1) | |
12 | 21 | 2.9489e+0 (1.36e+0) − | 4.1876e+0 (1.21e+0) − | 3.5980e+0 (7.13e-1) − | 1.1606e+1 (3.87e+0) − | 3.5313e+0 (6.57e-1) − | 1.3330e+0 (2.62e-1) | |
16 | 25 | 5.1887e+0 (5.81e-1) − | 5.0764e+0 (7.24e-1) − | 8.4280e+0 (1.05e-1) − | 1.5723e+1 (4.87e+0) − | 1.0740e+1 (1.11e+0) − | 3.5339e+0 (6.22e-1) | |
20 | 29 | 1.4197e+1 (3.29e+0) − | 1.5258e+1 (2.99e+0) − | 1.0857e+1 (7.65e-1) − | 2.1262e+1 (4.57e+0) − | 1.3401e+1 (1.26e+0) − | 4.7943e+0 (8.42e-1) | |
WFG4 | 5 | 14 | 1.2257e+0 (4.36e-4) − | 1.2826e+0 (2.14e-2) − | 1.2359e+0 (8.00e-3) − | 6.5267e+0 (6.56e-1) − | 2.8586e+0 (4.93e-1) − | 1.2250e+0 (3.72e-4) |
8 | 17 | 3.5919e+0 (1.04e-1) − | 4.1109e+0 (5.35e-1) − | 3.5540e+0 (8.78e-3) − | 1.0031e+1 (1.16e+0) − | 9.9791e+0 (1.10e+0) − | 3.5416e+0 (7.09e-3) | |
12 | 21 | 7.8965e+0 (1.00e-1) ≈ | 7.9295e+0 (1.12e-1) ≈ | 7.9484e+0 (7.88e-2) ≈ | 1.6825e+1 (2.58e+0) − | 1.2844e+1 (6.07e-1) − | 7.9153e+0 (7.05e-2) | |
16 | 25 | 1.3325e+1 (3.46e-1) − | 1.3279e+1 (3.81e-1) ≈ | 1.3249e+1 (1.78e-1) − | 2.6215e+1 (3.20e+0) − | 2.3118e+1 (1.72e+0) − | 1.3143e+1 (1.56e-1) | |
20 | 29 | 1.9437e+1 (2.15e+0) − | 1.9359e+1 (1.32e+0) − | 1.7563e+1 (1.11e-1) − | 3.8787e+1 (3.18e+0) − | 3.3914e+1 (2.01e+0) − | 1.7453e+1 (1.32e-1) | |
WFG5 | 5 | 14 | 1.2152e+0 (1.81e-4) − | 1.2596e+0 (1.41e-2) − | 1.2248e+0 (4.89e-3) − | 6.4524e+0 (1.52e+0) − | 2.4852e+0 (4.13e-1) − | 1.2151e+0 (1.25e-4) |
8 | 17 | 3.5268e+0 (4.73e-3) − | 3.7220e+0 (1.86e-1) − | 3.5336e+0 (5.95e-3) − | 1.3228e+1 (1.70e+0) − | 9.4957e+0 (7.11e-1) − | 3.5239e+0 (5.48e-3) | |
12 | 21 | 7.8543e+0 (4.23e-2) ≈ | 7.8459e+0 (2.78e-2) ≈ | 7.8386e+0 (4.01e-2) + | 2.0786e+1 (3.44e+0) − | 1.0527e+1 (5.14e-1) − | 7.8710e+0 (5.00e-2) | |
16 | 25 | 1.2768e+1 (2.88e-1) + | 1.2661e+1 (3.98e-1) + | 1.2984e+1 (2.25e-2) ≈ | 2.9961e+1 (4.83e+0) − | 2.0273e+1 (1.40e+0) − | 1.2985e+1 (2.09e-1) | |
20 | 29 | 1.8346e+1 (1.33e+0) − | 1.8425e+1 (1.52e+0) − | 1.7488e+1 (1.22e-2) − | 3.8202e+1 (3.10e+0) − | 3.1727e+1 (2.01e+0) − | 1.6565e+1 (9.41e-1) | |
WFG6 | 5 | 14 | 1.2170e+0 (3.36e-3) − | 1.2956e+0 (2.31e-2) − | 1.2281e+0 (9.60e-3) − | 4.9411e+0 (1.58e+0) − | 2.5476e+0 (2.87e-1) − | 1.2144e+0 (7.21e-4) |
8 | 17 | 3.6645e+0 (5.36e-1) − | 5.5604e+0 (4.12e-1) − | 3.6206e+0 (7.01e-2) − | 7.0445e+0 (4.28e+0) − | 9.9646e+0 (7.67e-1) − | 3.5365e+0 (3.42e-3) | |
12 | 21 | 8.0900e+0 (4.36e-1) − | 8.1357e+0 (5.56e-1) − | 8.0358e+0 (6.58e-2) − | 1.4451e+1 (6.57e+0) ≈ | 1.0992e+1 (6.30e-1) − | 7.8750e+0 (4.20e-2) | |
16 | 25 | 1.3210e+1 (5.74e-1) ≈ | 1.3103e+1 (4.66e-1) ≈ | 1.3083e+1 (5.22e-2) + | 2.5768e+1 (8.52e+0) − | 2.1947e+1 (1.57e+0) − | 1.3099e+1 (3.61e-1) | |
20 | 29 | 1.8407e+1 (7.57e-1) − | 1.8249e+1 (9.08e-1) ≈ | 1.7447e+1 (8.96e-2) ≈ | 2.6597e+1 (1.31e+1) ≈ | 3.3174e+1 (1.54e+0) − | 1.7771e+1 (7.85e-1) | |
WFG7 | 5 | 14 | 1.2276e+0 (1.49e-3) − | 1.2770e+0 (1.55e-2) − | 1.2311e+0 (3.59e-3) − | 5.6732e+0 (5.81e-1) − | 2.1871e+0 (1.79e-1) − | 1.2254e+0 (5.04e-4) |
8 | 17 | 3.6668e+0 (2.21e-1) − | 4.0097e+0 (4.89e-1) − | 3.6289e+0 (9.57e-2) − | 1.0294e+1 (1.97e+0) − | 1.0180e+1 (3.59e-1) − | 3.5304e+0 (4.97e-3) | |
12 | 21 | 8.0813e+0 (1.92e-1) ≈ | 8.1779e+0 (3.01e-1) − | 8.1302e+0 (7.41e-2) − | 1.8519e+1 (2.77e+0) − | 1.1584e+1 (8.39e-1) − | 8.0341e+0 (6.94e-2) | |
16 | 25 | 1.3122e+1 (2.60e-1) ≈ | 1.3131e+1 (2.09e-1) ≈ | 1.3154e+1 (1.00e-1) ≈ | 2.6849e+1 (3.03e+0) − | 2.1296e+1 (9.28e-1) − | 1.3151e+1 (2.18e-1) | |
20 | 29 | 1.8306e+1 (8.84e-1) − | 1.8530e+1 (1.32e+0) − | 1.7351e+1 (1.37e-1) ≈ | 3.8065e+1 (5.03e+0) − | 3.3124e+1 (1.73e+0) − | 1.7422e+1 (6.92e-1) | |
WFG8 | 5 | 14 | 1.2475e+0 (9.28e-3) − | 1.3340e+0 (3.32e-2) − | 1.2280e+0 (3.12e-3) ≈ | 4.8567e+0 (1.75e+0) − | 2.6900e+0 (3.71e-1) − | 1.2295e+0 (1.57e-3) |
8 | 17 | 3.7426e+0 (2.07e-1) − | 4.0868e+0 (2.73e-1) − | 3.8998e+0 (4.09e-2) − | 1.0949e+1 (2.49e+0) − | 1.0139e+1 (7.83e-1) − | 3.5757e+0 (2.39e-2) | |
12 | 21 | 7.7142e+0 (2.31e-1) − | 7.6630e+0 (2.74e-1) ≈ | 8.0206e+0 (4.91e-2) − | 1.8623e+1 (2.34e+0) − | 1.1063e+1 (7.52e-1) − | 7.5035e+0 (1.59e-1) | |
16 | 25 | 1.3348e+1 (5.02e-1) + | 1.3399e+1 (4.43e-1) + | 1.3206e+1 (4.72e-2) + | 3.0024e+1 (2.44e+0) − | 2.2199e+1 (1.28e+0) − | 1.3900e+1 (4.64e-1) | |
20 | 29 | 2.1473e+1 (2.73e+0) − | 2.2511e+1 (2.60e+0) − | 1.7553e+1 (6.11e-2) + | 3.9936e+1 (2.60e+0) − | 3.3661e+1 (1.39e+0) − | 1.8072e+1 (5.75e-1) | |
WFG9 | 5 | 14 | 1.2013e+0 (3.31e-3) ≈ | 1.2525e+0 (2.38e-2) − | 1.2082e+0 (5.39e-3) − | 5.1010e+0 (1.75e+0) − | 2.4189e+0 (1.73e-1) − | 1.2016e+0 (5.18e-3) |
8 | 17 | 3.5736e+0 (8.85e-2) − | 3.7453e+0 (2.34e-1) − | 3.6288e+0 (6.32e-2) − | 1.1312e+1 (3.93e+0) − | 9.7805e+0 (5.39e-1) − | 3.5026e+0 (1.78e-2) | |
12 | 21 | 7.4134e+0 (2.11e-1) + | 7.4973e+0 (1.79e-1) + | 7.7948e+0 (3.16e-2) − | 1.7942e+1 (5.16e+0) − | 1.0808e+1 (6.10e-1) − | 7.6818e+0 (1.07e-1) | |
16 | 25 | 1.2566e+1 (2.83e-1) ≈ | 1.2534e+1 (2.89e-1) ≈ | 1.2977e+1 (1.45e-1) ≈ | 2.6153e+1 (8.69e+0) − | 2.3091e+1 (1.38e+0) − | 1.2756e+1 (4.10e-1) | |
20 | 29 | 1.8146e+1 (1.14e+0) − | 1.8173e+1 (6.98e-1) − | 1.7435e+1 (8.52e-2) − | 3.4248e+1 (1.12e+1) − | 3.4518e+1 (1.19e+0) − | 1.6450e+1 (8.88e-1) | |
11/26/8 | 8/25/12 | 10/27/8 | 0/41/4 | 0/45/0 |
Problem | M | D | NSGAIII | ANSGAIII | SPEAR | MaOEAIGD | MaOEAIT | MOEA-AD |
---|---|---|---|---|---|---|---|---|
WFG1 | 5 | 14 | 9.9733e-1 (1.95e-4) + | 9.9484e-1 (1.17e-3) + | 9.9714e-1 (1.66e-4) + | 2.2457e-1 (7.93e-2) − | 1.6952e-1 (8.84e-2) − | 9.8259e-1 (3.33e-3) |
8 | 17 | 9.9936e-1 (2.99e-4) + | 9.9883e-1 (6.80e-4) + | 9.9250e-1 (5.58e-3) ≈ | 2.6128e-1 (8.65e-2) − | 1.6050e-1 (6.89e-2) − | 9.9562e-1 (1.01e-3) | |
12 | 21 | 9.9959e-1 (5.12e-4) + | 9.9523e-1 (1.72e-2) − | 9.9195e-1 (5.95e-3) − | 3.6601e-1 (1.72e-1) − | 4.8848e-2 (2.58e-2) − | 9.9600e-1 (6.91e-4) | |
16 | 25 | 9.8992e-1 (6.21e-3) ≈ | 9.8775e-1 (1.56e-2) ≈ | 9.1183e-1 (6.83e-2) − | 1.2394e-1 (9.50e-2) − | 1.2462e-1 (8.46e-3) − | 9.9048e-1 (4.91e-3) | |
20 | 29 | 9.6752e-1 (2.36e-2) − | 9.6971e-1 (3.08e-2) − | 9.6335e-1 (2.58e-2) − | 1.2707e-1 (7.81e-2) − | 1.1598e-1 (4.06e-3) − | 9.9660e-1 (2.39e-3) | |
WFG2 | 5 | 14 | 9.9538e-1 (6.41e-4) + | 9.8458e-1 (6.94e-3) + | 9.9498e-1 (7.14e-4) + | 9.0825e-1 (4.56e-2) − | 5.6138e-1 (3.47e-2) − | 9.5403e-1 (7.40e-3) |
8 | 17 | 9.9548e-1 (2.97e-3) + | 9.9115e-1 (5.61e-3) + | 9.9437e-1 (1.81e-3) + | 9.2547e-1 (6.69e-2) − | 3.7767e-1 (6.52e-2) − | 9.8311e-1 (9.02e-3) | |
12 | 21 | 9.9632e-1 (1.49e-3) + | 9.9621e-1 (2.71e-3) + | 9.9536e-1 (2.02e-3) + | 8.1881e-1 (1.71e-1) ≈ | 5.2558e-1 (5.23e-2) − | 9.3353e-1 (5.86e-2) | |
16 | 25 | 9.5829e-1 (2.57e-2) + | 9.6698e-1 (2.64e-2) + | 9.8433e-1 (1.28e-2) + | 7.3878e-1 (2.73e-1) ≈ | 2.6048e-1 (5.93e-2) − | 8.7481e-1 (6.05e-2) | |
20 | 29 | 7.9345e-1 (1.07e-2) − | 7.6425e-1 (6.29e-2) − | 9.8409e-1 (1.40e-2) + | 7.1974e-1 (2.36e-1) − | 2.7898e-1 (7.24e-2) − | 9.0307e-1 (4.40e-2) | |
WFG3 | 5 | 14 | 1.8061e-1 (1.90e-2) + | 1.9856e-1 (1.99e-2) + | 1.4498e-1 (2.64e-2) ≈ | 7.8651e-2 (5.68e-3) − | 0.0000e+0 (0.00e+0) − | 1.5213e-1 (3.46e-2) |
8 | 17 | 4.9226e-2 (3.07e-2) − | 6.1296e-2 (2.58e-2) − | 3.4272e-2 (3.33e-2) − | 4.3011e-3 (9.53e-3) − | 0.0000e+0 (0.00e+0) − | 1.3249e-1 (1.39e-2) | |
12 | 21 | 5.9230e-4 (2.65e-3) ≈ | 1.4011e-3 (6.27e-3) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) | |
16 | 25 | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) | |
20 | 29 | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) ≈ | 0.0000e+0 (0.00e+0) | |
WFG4 | 5 | 14 | 7.7192e-1 (9.59e-4) − | 7.2430e-1 (5.58e-3) − | 7.7110e-1 (1.85e-3) − | 1.0275e-1 (3.87e-2) − | 2.7999e-1 (4.38e-2) − | 7.7273e-1 (9.23e-4) |
8 | 17 | 8.7077e-1 (2.16e-2) − | 7.9927e-1 (2.90e-2) − | 8.7507e-1 (3.48e-3) − | 9.9142e-2 (2.55e-2) − | 9.7996e-2 (5.27e-2) − | 8.8095e-1 (1.69e-3) | |
12 | 21 | 9.6151e-1 (8.09e-3) − | 9.6239e-1 (9.05e-3) − | 9.6679e-1 (1.15e-3) − | 1.2324e-1 (4.08e-2) − | 3.1222e-1 (1.41e-2) − | 9.6820e-1 (9.05e-4) | |
16 | 25 | 7.6042e-1 (3.01e-2) − | 7.6646e-1 (3.48e-2) − | 7.6680e-1 (3.74e-2) − | 1.3174e-1 (4.20e-2) − | 6.9645e-2 (3.47e-2) − | 8.0272e-1 (5.71e-3) | |
20 | 29 | 3.9072e-1 (5.45e-2) − | 3.9669e-1 (8.04e-2) − | 8.0632e-1 (5.50e-2) − | 1.0260e-1 (2.96e-2) − | 7.3645e-2 (6.39e-2) − | 8.5826e-1 (1.44e-2) | |
WFG5 | 5 | 14 | 7.2388e-1 (4.04e-4) ≈ | 6.9553e-1 (4.59e-3) − | 7.2190e-1 (1.26e-3) − | 1.2300e-1 (1.17e-1) − | 2.1063e-1 (2.57e-2) − | 7.2383e-1 (4.13e-4) |
8 | 17 | 8.2417e-1 (2.45e-3) ≈ | 7.7919e-1 (1.97e-2) − | 8.2034e-1 (2.21e-3) − | 8.7984e-2 (1.74e-2) − | 5.0847e-2 (9.89e-3) − | 8.2456e-1 (1.55e-3) | |
12 | 21 | 9.0026e-1 (5.59e-3) − | 9.0124e-1 (5.46e-4) − | 9.0103e-1 (5.59e-4) − | 1.0908e-1 (4.84e-2) − | 2.1589e-1 (1.88e-2) − | 9.0170e-1 (2.85e-4) | |
16 | 25 | 6.9084e-1 (3.49e-2) − | 6.9055e-1 (2.78e-2) − | 6.1666e-1 (6.77e-2) − | 1.1569e-1 (1.47e-1) − | 2.5597e-2 (1.30e-2) − | 7.2640e-1 (2.61e-2) | |
20 | 29 | 4.5036e-1 (6.82e-2) − | 4.6206e-1 (4.08e-2) − | 6.1802e-1 (1.54e-2) − | 8.2448e-2 (2.31e-4) − | 2.0581e-2 (9.15e-3) − | 8.0064e-1 (1.50e-2) | |
WFG6 | 5 | 14 | 7.0821e-1 (1.80e-2) ≈ | 6.7710e-1 (2.07e-2) − | 7.0863e-1 (1.24e-2) ≈ | 1.8715e-1 (1.30e-1) − | 1.9822e-1 (1.71e-2) − | 7.0093e-1 (1.46e-2) |
8 | 17 | 7.9683e-1 (4.43e-2) ≈ | 6.8251e-1 (3.75e-2) − | 7.5439e-1 (6.92e-2) − | 3.3661e-1 (1.72e-1) − | 5.1077e-2 (2.27e-2) − | 7.9825e-1 (1.52e-2) | |
12 | 21 | 8.7509e-1 (3.42e-2) ≈ | 8.7923e-1 (3.30e-2) ≈ | 8.8526e-1 (2.05e-2) ≈ | 3.0792e-1 (2.01e-1) − | 2.1014e-1 (1.67e-2) − | 8.7325e-1 (1.98e-2) | |
16 | 25 | 6.6126e-1 (4.26e-2) − | 6.7283e-1 (4.76e-2) ≈ | 6.2883e-1 (6.26e-2) − | 2.3230e-1 (2.32e-1) − | 2.2970e-2 (9.17e-3) − | 6.8926e-1 (4.90e-2) | |
20 | 29 | 3.8999e-1 (7.79e-2) − | 3.7715e-1 (7.50e-2) − | 7.1044e-1 (6.21e-2) ≈ | 3.9866e-1 (2.86e-1) − | 1.9621e-2 (9.66e-3) − | 7.3240e-1 (6.43e-2) | |
WFG7 | 5 | 14 | 7.7141e-1 (5.66e-4) − | 7.3324e-1 (7.29e-3) − | 7.6886e-1 (1.53e-3) − | 2.0308e-1 (4.59e-2) − | 2.9382e-1 (1.98e-2) − | 7.7387e-1 (4.04e-4) |
8 | 17 | 8.6922e-1 (2.35e-2) − | 8.1967e-1 (3.45e-2) − | 8.3078e-1 (9.25e-2) − | 1.9417e-1 (6.26e-2) − | 7.0205e-2 (1.63e-2) − | 8.8454e-1 (5.17e-4) | |
12 | 21 | 9.5360e-1 (2.02e-2) − | 9.4848e-1 (2.63e-2) − | 9.6833e-1 (8.57e-4) − | 1.6683e-1 (5.98e-2) − | 2.8836e-1 (3.42e-2) − | 9.7012e-1 (2.25e-4) | |
16 | 25 | 8.0432e-1 (2.50e-2) ≈ | 8.0069e-1 (3.12e-2) ≈ | 6.9151e-1 (6.34e-2) − | 1.4454e-1 (4.05e-2) − | 3.5548e-2 (6.02e-3) − | 7.9715e-1 (1.49e-2) | |
20 | 29 | 5.0487e-1 (7.97e-2) − | 5.3544e-1 (8.05e-2) − | 7.6466e-1 (5.33e-2) − | 1.5002e-1 (1.36e-1) − | 3.0104e-2 (5.77e-3) − | 8.2924e-1 (4.51e-2) | |
WFG8 | 5 | 14 | 6.5219e-1 (4.79e-3) − | 5.9065e-1 (1.48e-2) − | 6.6736e-1 (2.16e-3) + | 7.4895e-2 (4.40e-2) − | 1.9068e-1 (2.88e-2) − | 6.5900e-1 (2.47e-3) |
8 | 17 | 7.0894e-1 (9.02e-3) − | 7.1359e-1 (2.21e-2) − | 6.0566e-1 (6.41e-2) − | 1.5481e-1 (1.07e-1) − | 3.4891e-2 (2.43e-2) − | 7.4188e-1 (2.96e-2) | |
12 | 21 | 8.6060e-1 (2.30e-2) ≈ | 8.7220e-1 (2.79e-2) + | 8.8530e-1 (1.89e-2) + | 1.6649e-1 (6.03e-2) − | 2.2240e-1 (2.33e-2) − | 8.4178e-1 (6.72e-2) | |
16 | 25 | 6.0684e-1 (2.74e-2) ≈ | 6.0546e-1 (2.27e-2) ≈ | 4.9026e-1 (7.51e-2) − | 1.1992e-1 (3.88e-2) − | 1.9111e-2 (1.55e-2) − | 6.1931e-1 (7.32e-3) | |
20 | 29 | 1.5544e-1 (4.42e-2) − | 1.7236e-1 (7.36e-2) − | 5.6342e-1 (4.77e-2) − | 1.2234e-1 (4.15e-2) − | 1.1959e-2 (6.51e-3) − | 6.8162e-1 (8.16e-3) | |
WFG9 | 5 | 14 | 7.1809e-1 (1.13e-2) − | 6.5478e-1 (4.60e-2) − | 7.0958e-1 (1.98e-2) − | 2.1721e-1 (1.20e-1) − | 2.2113e-1 (1.75e-2) − | 7.2747e-1 (3.26e-2) |
8 | 17 | 7.3666e-1 (6.77e-2) − | 7.1609e-1 (6.49e-2) − | 7.2246e-1 (6.86e-2) − | 1.8587e-1 (1.57e-1) − | 6.4197e-2 (1.89e-2) − | 8.0474e-1 (3.72e-2) | |
12 | 21 | 8.3969e-1 (5.91e-2) − | 8.5270e-1 (5.80e-2) − | 8.6291e-1 (3.25e-2) − | 2.0559e-1 (1.58e-1) − | 2.4149e-1 (3.76e-2) − | 8.8821e-1 (1.11e-2) | |
16 | 25 | 6.1020e-1 (7.26e-2) − | 6.1712e-1 (7.46e-2) − | 4.6361e-1 (5.04e-2) − | 2.1251e-1 (2.10e-1) − | 3.3553e-2 (2.04e-2) − | 6.9268e-1 (6.78e-2) | |
20 | 29 | 4.2056e-1 (4.75e-2) − | 4.2298e-1 (6.29e-2) − | 5.4977e-1 (6.73e-2) − | 2.1692e-1 (1.98e-1) − | 2.2759e-2 (6.69e-3) − | 7.0363e-1 (6.98e-2) | |
8/25/12 | 8/29/8 | 8/29/8 | 0/40/5 | 0/42/3 |
Problem | M | D | NSGAIII | ANSGAIII | SPEAR | MaOEAIGD | MaOEAIT | MOEA-AD |
---|---|---|---|---|---|---|---|---|
DBDP | 2 | 4 | 4.2753e-1 (5.76e-3) − | 2.7328e-1 (2.02e-3) − | 2.7613e-1 (8.32e-6) − | 7.7334e-2 (6.90e-2) − | 4.3413e-1 (1.20e-3) ≈ | 4.3364e-1 (1.89e-3) |
GTDP | 2 | 4 | 4.8380e-1 (4.15e-4) ≈ | 4.8292e-1 (1.78e-4) − | 4.8346e-1 (4.09e-4) − | 2.2583e-1 (2.06e-2) − | 4.8401e-1 (5.57e-5) − | 4.8446e-1 (4.52e-4) |
CSIDP | 3 | 7 | 2.5507e-2 (1.69e-4) − | 2.3390e-2 (4.26e-4) − | 2.2765e-2 (2.71e-4) − | 1.2509e-2 (3.62e-3) − | 2.5564e-2 (9.96e-5) − | 2.5849e-2 (6.89e-5) |
FBPT | 2 | 4 | 4.0993e-1 (1.48e-4) + | 4.0945e-1 (1.49e-4) + | 4.1036e-1 (2.24e-5) + | 2.2372e-1 (9.06e-3) − | 4.0993e-1 (1.48e-4) + | 4.0905e-1 (1.51e-4) |
TBPT | 2 | 2 | 8.3442e-1 (1.28e-3) − | 8.4177e-1 (2.13e-3) − | 8.4558e-1 (2.56e-4) − | 6.2555e-1 (5.13e-2) − | 8.3441e-1 (1.42e-3) − | 8.4729e-1 (1.62e-4) |
1/3/1 | 1/4/0 | 1/4/0 | 0/5/0 | 1/3/1 |
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Wang, X.; Wang, H.; Tian, Z.; Wang, W.; Chen, J. Angle-Based Dual-Association Evolutionary Algorithm for Many-Objective Optimization. Mathematics 2025, 13, 1757. https://doi.org/10.3390/math13111757
Wang X, Wang H, Tian Z, Wang W, Chen J. Angle-Based Dual-Association Evolutionary Algorithm for Many-Objective Optimization. Mathematics. 2025; 13(11):1757. https://doi.org/10.3390/math13111757
Chicago/Turabian StyleWang, Xinzi, Huimin Wang, Zhen Tian, Wenxiao Wang, and Junming Chen. 2025. "Angle-Based Dual-Association Evolutionary Algorithm for Many-Objective Optimization" Mathematics 13, no. 11: 1757. https://doi.org/10.3390/math13111757
APA StyleWang, X., Wang, H., Tian, Z., Wang, W., & Chen, J. (2025). Angle-Based Dual-Association Evolutionary Algorithm for Many-Objective Optimization. Mathematics, 13(11), 1757. https://doi.org/10.3390/math13111757