Many-objective Evolutionary Algorithm Driven by Indicator Under Adaptive Reference Point Adjustment
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摘要: 在具有不同Pareto前沿形状的优化问题上, 基于参考点的高维多目标进化算法表现出较差的通用性. 为了解决这个问题, 提出参考点自适应调整下评价指标驱动的高维多目标进化算法(Many-objective evolutionary algorithm driven by evaluation indicator under adaptive reference point adjustment, MaOEA-IAR). MaOEA-IAR提出Pareto前沿形状监测基础上的参考点自适应策略, 利用该策略选择一组候选解作为初始参考点; 然后通过曲线参数对参考点位置进行调整; 将最终得到的能够适应不同Pareto前沿的参考点用于计算增强的反世代距离指标, 基于指标值设计适应度函数作为选择标准. 实验证明提出的算法在处理各种Pareto前沿形状的优化问题时能获得较好的性能, 算法通用性高.
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关键词:
- 参考点自适应 /
- 评价指标 /
- 高维多目标 /
- Pareto前沿形状
Abstract: Many-objective evolutionary algorithms based on reference points show poor versatility on optimization problems with different shapes of Pareto fronts. To address this issue, this paper proposes a many-objective evolutionary algorithm driven by evaluation indicator under adaptive reference point adjustment (MaOEA-IAR). MaOEA-IAR proposes a reference point adaptation strategy based on the Pareto front shape monitoring, and uses this strategy to select a group of candidate solutions as the initial reference points, then adjusts their positions with the curve parameter. The final obtained reference points that can adapt to different Pareto fronts are used to calculate the enhanced inverted generational distance indicator, the fitness function is designed as selection criterion based on the indicator value. The experiment shows that the algorithm proposed in this paper can get good performance and high versatility when dealing with the optimization problems with various shapes of Pareto fronts. -
表 1 测试问题
Table 1 Test questions
问题 目标数目M 决策变量D PF DTLZ1 5, 10, 15, 25 M−1+5 线性 DTLZ2~DTLZ4 5, 10, 15, 25 M−1+10 凹型 DTLZ5~DTLZ6 5, 10, 15, 25 M−1+10 退化 DTLZ7 5, 10, 15, 25 M−1+20 断开 IDTLZ1 5, 10, 15, 25 M−1+5 倒置 IDTLZ2 5, 10, 15, 25 M−1+10 倒置 WFG1 5, 10, 15, 25 M−1+10 混合 WFG2 5, 10, 15, 25 M−1+10 断开 WFG3 5, 10, 15, 25 M−1+10 退化 WFG4~DTLZ9 5, 10, 15, 25 M−1+10 凹型 表 2 不同目标对应的种群规模
Table 2 Population sizes corresponding to different objectives
目标数目M (H1, H2) 种群规模N 5 (6, 0) 126 10 (3, 2) 275 15 (2, 1) 135 25 (2, 1) 350 表 3 不同目标对应的终止条件
Table 3 Termination conditions corresponding to different objectives
目标数目M 种群规模N 进化代数 评价次数 5 126 800 100800 10 275 1000 275000 15 135 1300 175500 25 350 1500 525000 表 4 MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA以及MaOEA-IAR在DTLZ1~DTLZ4, WFG4~WFG9上获得IGD值的统一结果(均值和标准差). 最好的结果已被标记
Table 4 The statistical results (mean and standard deviation) of the IGD values obtained by MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA and MaOEA-IAR on DTLZ1~DTLZ4, WFG4~WFG9. The best results are highlighted
问题 M MOEA-DD NSGA-III RVEA MOMBI-II AR-MOEA MaOEA-IAR DTLZ1 5 6.3354×10−2
(8.24×10−5)+6.3588×10−2
(2.93×10−4)+6.3365×10−2
(1.44×10−4)+6.7530×10−2
(8.96×10−3)+1.3555×10−3
(7.87×10−6)+2.6870×10−1
(7.95×10−2)10 1.0784×10−1
(3.02×10−4)+1.2073×10−1
(2.91×10−2)+1.0804×10−1
(1.45×10−3)+2.1744×10−1
(2.32×10−2)+1.3680×10−3
(1.14×10−4)+4.7099×10+1
(1.33×10+1)15 1.4088×10−1
(5.94×10−3)+1.9693×10−1
(2.97×10−2)+1.6130×10−1
(7.54×10−3)+2.8789×10−1
(3.92×10−2)+3.1383×10−3
(1.58×10−3)+5.6383×10+1
(2.03×10+1)25 1.0354×10−1
(5.86×10−2)+2.7045×10−1
(1.96×10−1)=1.3372×10−1
(7.61×10−2)+2.0490×10−1
(1.19×10−1)+1.2392×10−1
(7.02×10−2)+2.4067×10−1
(1.44×10−1)DTLZ2 5 1.9506×10−1
(8.00×10−5)+1.9538×10−1
(1.18×10−4)+1.9508×10−1
(7.34×10−5)+2.0140×10−1
(1.93×10−3)=4.3609×10−3
(7.50×10−6)+2.0137×10−1
(1.61×10−3)10 4.2179×10−1
(8.12×10−4)−4.4544×10−1
(3.60×10−2)−4.2291×10−1
(1.61×10−3)−4.4619×10−1
(6.50×10−3)−4.9235×10−2
(1.43×10−4)−4.7588×10−2
(4.60×10−4)15 6.1801×10−1
(1.88×10−2)−6.4108×10−1
(2.01×10−2)−6.3958×10−1
(3.67×10−2)−8.5287×10−1
(6.41×10−2)−1.0058×10−2
(1.56×10−3)=1.1135×10−2
(1.23×10−3)25 4.3015×10−1
(2.45×10−1)−5.3577×10−1
(3.10×10−1)−4.3033×10−1
(2.45×10−1)−7.0528×10−1
(4.12×10−1)−6.1891×10−1
(3.45×10−1)−4.2923×10−1
(2.44×10−1)DTLZ3 5 1.9907×10−1
(3.45×10−3)+7.1945×10−1
(1.21×10+0)+2.0829×10−1
(2.11×10−2)+2.0571×10−1
(4.19×10−3)+4.4165×10−3
(5.43×10−5)+1.0190×10+0
(6.60×10−1)10 4.2392×10−1
(3.09×10−3)+1.1841×10+0
(1.08×10+0)+4.3390×10−1
(1.54×10−2)+6.9344×10−1
(1.66×10−1)+3.1091×10−2
(9.02×10−2)+4.8634×10+2
(9.91×10+1)15 6.6842×10−1
(2.26×10−1)+5.5656×10+0
(2.83×10+0)+8.3685×10−1
(3.72×10−1)+1.1033×10+0
(2.16×10−2)+3.8594×10−1
(5.94×10−1)+6.4283×10+2
(1.08×10+2)25 4.3041×10−1
(2.46×10−1)+8.3189×10+0
(6.16×10+0)−4.3076×10−1
(2.46×10−1)+8.7119×10−1
(5.08×10−1)+4.6567×10−1
(2.38×10−1)+9.8829×10−1
(5.81×10−1)DTLZ4 5 1.9508×10−1
(9.98×10−5)+2.4346×10−1
(9.84×10−2)−1.9514×10−1
(1.62×10−4)+2.5703×10−1
(9.71×10−2)−4.4014×10−3
(1.06×10−4)+2.0129×10−1
(1.72×10−3)10 4.2271×10−1
(1.73×10−3)−4.5035×10−1
(3.18×10−2)−4.3317×10−1
(2.13×10−3)−4.5814×10−1
(1.71×10−2)−5.7202×10−3
(5.61×10−4)−5.6898×10−3
(1.01×10−4)15 6.3495×10−1
(1.18×10−2)−6.4865×10−1
(2.13×10−2)−6.3283×10−1
(7.39×10−3)−6.5937×10−1
(2.31×10−2)−1.1470×10−2
(1.34×10−3)−9.9175×10−3
(1.41×10−3)25 4.3041×10−1
(2.45×10−1)−4.6255×10−1
(2.72×10−1)−4.3053×10−1
(2.46×10−1)−4.3098×10−1
(2.46×10−1)−5.8948×10−1
(2.86×10−1)−4.3021×10−1
(2.44×10−1)WFG4 5 1.2337×10+0
(4.54×10−3)−1.1663×10+0
(3.61×10−3)−1.1649×10+0
(2.28×10−3)−1.8356×10+0
(2.02×10−1)−1.1244×10+0
(1.47×10−2)−1.1192×10+0
(2.58×10−1)10 6.0999×10+0
(1.66×10−1)−4.5156×10+0
(4.39×10−2)−4.3816×10+0
(5.05×10−2)−5.6693×10+0
(4.27×10−1)−4.5463×10+0
(9.53×10−3)−4.0153×10+0
(2.71×10−2)15 9.4478×10+0
(7.47×10−1)−9.1431×10+0
(9.63×10−2)−9.4338×10+0
(3.83×10−1)−2.0211×10+1
(1.35×10+0)−9.3716×10+0
(3.68×10−2)−8.4000×10+0
(1.11×10−1)25 2.4032×10+1
(2.95×10−1)−1.3110×10+1
(7.97×10−1)−1.2394×10+1
(2.92×10−1)−3.7271×10+1
(3.18×10+0)−1.5196×10+1
(2.61×10−1)−1.1405×10+1
(1.89×10−2)WFG5 5 1.2116×10+0
(6.46×10−3)−1.1459×10+0
(4.71×10−3)−1.1554×10+0
(2.17×10−3)−1.5867×10+0
(1.16×10−1)−1.1648×10+0
(2.17×10−4)−1.1148×10+0
(1.31×10−2)10 6.2798×10+0
(1.39×10−1)−4.4654×10+0
(1.82×10−2)−4.3730×10+0
(4.27×10−2)−5.3610×10+0
(2.19×10−2)−4.5296×10+0
(9.52×10−3)−3.9834×10+0
(1.90×10−2)15 1.1246×10+1
(1.85×10−1)−8.9556×10+0
(7.70×10−2)−9.8645×10+0
(1.66×10−1)−2.4601×10+1
(1.34×10+0)−9.3564×10+0
(5.48×10−2)−8.2316×10+0
(8.70×10−2)25 2.2399×10+1
(1.96×10−1)−1.1842×10+1
(1.14×10+0)=1.1407×10+1
(5.45×10−2)=4.5754×10+1
(3.19×10+0)−1.4772×10+1
(2.79×10−1)−1.1370×10+1
(3.12×10−3)WFG6 5 1.2276×10+0
(1.14×10−2)−1.1617×10+0
(2.62×10−3)−1.1638×10+0
(2.18×10−3)−2.1021×10+0
(3.69×10−1)−1.1624×10+0
(1.24×10−3)−1.1443×10+0
(2.89×10−2)10 6.0366×10+0
(1.65×10−1)−4.5795×10+0
(1.88×10−2)−4.4020×10+0
(7.33×10−2)−5.2873×10+0
(2.86×10−2)−4.5333×10+0
(1.12×10−2)−4.0605×10+0
(4.78×10−2)15 1.0818×10+1
(1.01×10+0)−9.3642×10+0
(3.84×10−1)−1.0520×10+1
(4.62×10−1)−1.8732×10+1
(1.89×10+0)−9.3953×10+0
(5.07×10−2)−8.5397×10+0
(1.76×10−1)25 2.1950×10+1
(6.43×10−1)−1.5582×10+1
(9.19×10−1)−1.5281×10+1
(9.27×10−1)−3.5035×10+1
(2.81×10+0)−1.5206×10+1
(3.56×10−1)−1.1375×10+1
(1.29×10−2)WFG7 5 1.2454×10+0
(8.15×10−3)−1.1680×10+0
(3.67×10−3)−1.1693×10+0
(3.05×10−3)−1.8487×10+0
(2.46×10−1)−1.1787×10+0
(8.19×10−4)−1.1499×10+0
(3.17×10−2)10 5.1499×10+0
(2.73×10−1)−4.5094×10+0
(3.76×10−2)−4.3149×10+0
(5.44×10−2)−5.4063×10+0
(5.76×10−2)−4.5123×10+0
(1.07×10−2)−3.9423×10+0
(4.82×10−2)15 8.9281×10+0
(8.74×10−2)−9.0694×10+0
(9.98×10−2)−9.1250×10+0
(2.71×10−1)−1.7892×10+1
(1.65×10+0)−9.4000×10+0
(3.81×10−2)−8.7103×10+0
(1.27×10−1)25 1.6767×10+1
(1.55×10+0)−1.3522×10+1
(7.95×10−1)−1.1672×10+1
(3.01×10−1)=3.2655×10+1
(2.66×10+0)−1.5873×10+1
(1.94×10−1)−1.1572×10+1
(8.12×10−2)WFG8 5 1.2301×10+0
(8.57×10−3)−1.1867×10+0
(9.32×10−2)=1.1770×10+0
(4.82×10−3)−2.9462×10+0
(3.23×10−2)−1.1575×10+0
(2.95×10−3)−1.1675×10+0
(1.66×10−2)10 5.2132×10+0
(3.48×10−1)−4.5532×10+0
(3.00×10−1)−4.3661×10+0
(8.88×10−2)−6.0943×10+0
(8.02×10−1)−4.6273×10+0
(4.43×10−2)−4.1413×10+0
(7.20×10−2)15 9.2706×10+0
(3.66×10−1)−9.1735×10+0
(4.45×10−1)−9.9548×10+0
(7.95×10−1)−2.0437×10+1
(1.84×10+0)−9.3736×10+0
(7.14×10−2)−8.7680×10+0
(2.43×10−1)25 2.2579×10+1
(3.17×10+0)−1.6519×10+1
(6.99×10−1)−1.3760×10+1
(1.73×10+0)−3.9378×10+1
(2.20×10+0)−1.5213×10+1
(2.84×10−1)−1.1409×10+1
(2.03×10−2)WFG9 5 1.2087×10+0
(1.17×10−2)−1.1112×10+0
(1.08×10−2)−1.1315×10+0
(4.39×10−3)−2.6613×10+0
(1.67×10−1)−1.1476×10+0
(2.59×10−3)−1.0867×10+0
(1.42×10−2)10 5.2551×10+0
(3.81×10−1)−4.3117×10+0
(7.01×10−2)−4.3306×10+0
(6.41×10−2)−5.2773×10+0
(6.59×10−2)−4.4943×10+0
(1.01×10−2)−3.8995×10+0
(3.21×10−2)15 8.9598×10+0
(5.37×10−1)−8.6320×10+0
(1.41×10−1)−8.8404×10+0
(2.73×10−1)−2.5164×10+1
(1.97×10+0)−9.1233×10+0
(3.81×10−2)−8.1162×10+0
(1.37×10−1)25 1.5766×10+1
(3.46×10+0)−1.2211×10+1
(7.44×10−1)−1.2343×10+1
(1.28×10+0)−4.8512×10+1
(2.40×10+0)−1.5086×10+1
(2.89×10−1)−1.1166×10+1
(8.75×10−2)+/−/= 10/30/0 7/30/3 10/28/2 8/31/1 10/29/1 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 5 MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA以及MaOEA-IAR在DTLZ1~DTLZ4, WFG4~WFG9上获得PD值的统一结果(均值和标准差). 最好的结果已被标记
Table 5 The statistical results (mean and standard deviation) of the PD values obtained by MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA and MaOEA-IAR on DTLZ1~DTLZ4, WFG4~WFG9. The best results are highlighted
问题 M MOEA-DD NSGA-III RVEA MOMBI-II AR-MOEA MaOEA-IAR DTLZ1 5 7.0347×10+5
(1.50×10+5)−6.6822×10+5
(2.89×10+5)−1.0144×10+6
(2.23×10+5)−6.5737×10+5
(5.56×10+5)−3.1124×10+5
(1.14×10+5)−1.1335×10+7
(2.48×10+7)10 3.9493×10+9
(4.51×10+8)−4.3920×10+9
(1.31×10+9)−4.4832×10+9
(5.54×10+8)−1.3231×10+8
(1.60×10+8)−3.9031×10+9
(3.47×10+8)−2.1838×10+12
(4.55×10+11)15 2.0771×10+11
(2.15×10+10)−7.2781×10+10
(4.17×10+10)−1.8665×10+11
(5.48×10+10)−2.8086×10+8
(4.77×10+8)−5.4706×10+10
(1.65×10+10)−2.4099×10+13
(8.65×10+12)25 4.2240×10+13
(1.16×10+12)+2.4959×10+13
(1.82×10+13)+1.5947×10+13
(3.82×10+12)+2.6381×10+9
(4.68×10+9)−1.2365×10+13
(6.44×10+12)+1.6674×10+11
(4.44×10+11)DTLZ2 5 4.5169×10+6
(3.69×10+5)−4.1624×10+6
(5.12×10+5)−4.4519×10+6
(4.97×10+5)−3.1301×10+6
(4.36×10+5)−2.2535×10+6
(1.23×10+5)−1.3815×10+7
(1.83×10+6)10 1.0700×10+10
(4.94×10+8)−1.5036×10+10
(4.96×10+9)−1.4787×10+10
(8.21×10+8)−6.3297×10+9
(9.76×10+8)−1.4371×10+10
(9.34×10+8)−4.8565×10+10
(8.50×10+9)15 2.3742×10+11
(5.12×10+10)−5.4986×10+10
(3.53×10+10)−1.5788×10+11
(3.98×10+10)−1.4169×10+9
(1.60×10+9)−2.2431×10+10
(1.21×10+10)−4.6071×10+11
(2.42×10+11)25 1.5327×10+13
(1.82×10+12)−1.9112×10+13
(9.88×10+12)−1.8880×10+13
(3.09×10+12)−1.4360×10+10
(2.20×10+10)−2.1209×10+12
(1.31×10+12)−1.0367×10+15
(3.68×10+13)DTLZ3 5 1.0522×10+7
(3.37×10+6 )+1.8368×10+7
(1.98×10+7 )+9.1974×10+6
(3.08×10+6 )+2.8101×10+6
(4.82×10+5 )+3.0005×10+6
(3.36×10+5 )+2.0182×10+6
(4.79×10+6)10 1.3601×10+10
(2.86×10+9)−2.2578×10+10
(1.68×10+10)−1.6599×10+10
(2.67×10+9)−1.1037×10+9
(1.66×10+9)−4.7006×10+9
(1.96×10+9)−2.9582×10+13
(4.66×10+12)15 3.6156×10+11
(8.59×10+11)−1.1462×10+12
(8.77×10+11)−1.2788×10+11
(2.39×10+11)−3.1557×10+8
(4.47×10+8)−4.9753×10+10
(4.57×10+10)−6.5391×10+14
(1.63×10+14)25 1.6252×10+13
(1.40×10+12)+3.2406×10+15
(2.26×10+15)+1.0735×10+13
(5.81×10+12)+7.3339×10+8
(1.67×10+9)−2.2266×10+13
(2.19×10+13)+9.6558×10+10
(2.77×10+11)DTLZ4 5 3.7620×10+6
(6.71×10+5)−3.3390×10+6
(8.43×10+5)−4.4024×10+6
(6.07×10+5)−2.1806×10+6
(1.14×10+6)−2.4008×10+6
(1.56×10+5)−1.3225×10+7
(1.43×10+6)10 1.3637×10+10
(1.08×10+9)+5.6960×10+9
(2.42×10+9)−1.0848×10+10
(2.29×10+9)+5.9777×10+7
(2.68×10+7)−4.0114×10+9
(1.10×10+9)−9.1937×10+9
(2.85×10+9)15 2.7750×10+9
(1.79×10+9)−5.2847×10+8
(5.23×10+8)−2.0648×10+9
(1.07×10+9)−4.0923×10+7
(1.33×10+8)−5.8144×10+9
(3.64×10+9)−3.9392×10+10
(1.98×10+10)25 3.7432×10+12
(9.47×10+11)−3.1839×10+11
(3.51×10+11)−3.8552×10+11
(1.51×10+11)−2.8053×10+8
(5.19×10+8)−1.2420×10+11
(5.99×10+10)−8.4925×10+14
(3.45×10+13)WFG4 5 1.2326×10+8
(6.16×10+6)−1.3502×10+8
(4.43×10+6)−1.2748×10+8
(7.49×10+6)−1.8889×10+7
(3.43×10+6)−6.4422×10+7
(4.60×10+6)−2.0686×10+8
(1.12×10+7)10 1.4338×10+11
(9.29×10+9)−2.9001×10+11
(1.68×10+10)−1.9392×10+11
(1.22×10+10)−7.4282×10+10
(2.70×10+10)−1.4488×10+11
(8.57×10+9)−6.6989×10+11
(2.52×10+10)15 3.4224×10+12
(5.49×10+11)−5.5478×10+12
(7.84×10+11)−4.0484×10+12
(1.04×10+12)−2.0839×10+10
(1.96×10+10)−2.0698×10+12
(6.60×10+11)−1.9906×10+13
(1.68×10+12)25 9.4061×10+13
(1.83×10+13)−2.2238×10+15
(6.40×10+14)−1.6309×10+14
(2.64×10+13)−2.5978×10+11
(6.99×10+11)−3.9968×10+14
(1.07×10+14)−1.0349×10+16
(5.36×10+14)WFG5 5 1.2123×10+8
(6.36×10+6)−1.4121×10+8
(8.96×10+6)−1.1886×10+8
(4.50×10+6)−2.3887×10+7
(4.70×10+6)−4.1612×10+7
(3.93×10+6)−1.9277×10+8
(1.04×10+7)10 1.9457×10+11
(1.24×10+10)−3.7692×10+11
(1.60×10+10)−2.2741×10+11
(1.20×10+10)−1.1103×10+11
(8.12×10+9)−1.2033×10+11
(6.96×10+9)−6.8603×10+11
(3.29×10+10)15 3.9863×10+12
(7.29×10+11)−8.8868×10+12
(7.95×10+11)−4.9690×10+12
(3.43×10+11)−2.0497×10+10
(3.24×10+10)−1.4482×10+12
(4.81×10+11)−2.0970×10+13
(1.56×10+12)25 2.8409×10+14
(2.86×10+13)−1.6088×10+15
(7.01×10+14)−2.7765×10+14
(3.00×10+13)−1.1499×10+12
(2.16×10+12)−5.7645×10+14
(6.09×10+13)−1.2220×10+16
(5.00×10+14)WFG6 5 1.1994×10+8
(8.37×10+6)−1.2908×10+8
(1.30×10+7)−1.1997×10+8
(7.99×10+6)−1.7038×10+7
(6.02×10+6)−5.4499×10+7
(6.80×10+6)−1.8108×10+8
(1.37×10+7)10 1.6619×10+11
(1.38×10+10)−2.7497×10+11
(1.56×10+10)−1.9869×10+11
(1.18×10+10)−9.2008×10+10
(6.73×10+9)−1.1142×10+11
(6.52×10+9)−6.2142×10+11
(2.98×10+10)15 3.5945×10+12
(7.09×10+11)−4.3937×10+12
(8.35×10+11)−2.6933×10+12
(5.32×10+11)−4.3260×10+10
(8.15×10+10)−5.8023×10+11
(5.03×10+11)−1.7544×10+13
(1.37×10+12)25 5.8142×10+13
(1.39×10+13)−2.2239×10+15
(4.43×10+14)−1.3831×10+14
(3.12×10+13)−1.0474×10+12
(4.06×10+12)−1.3621×10+14
(5.09×10+13)−1.0081×10+16
(4.97×10+14)WFG7 5 1.4307×10+8
(8.30×10+6)−1.5328×10+8
(1.08×10+7)−1.3594×10+8
(7.42×10+6)−2.0429×10+7
(7.25×10+6)−8.4724×10+7
(5.77×10+6)−2.0010×10+8
(1.05×10+7)10 2.5289×10+11
(1.42×10+10)−3.8153×10+11
(3.44×10+10)−2.6215×10+11
(1.24×10+10)−1.2614×10+11
(9.82×10+9)−1.8152×10+11
(7.90×10+9)−6.9190×10+11
(3.57×10+10)15 5.8545×10+12
(7.41×10+11)−9.3356×10+12
(1.86×10+12)−7.4133×10+12
(1.39×10+12)−6.2349×10+10
(7.43×10+10)−2.2599×10+12
(9.62×10+11)−1.4999×10+13
(1.54×10+12)25 6.8227×10+14
(1.72×10+14)−1.8449×10+15
(4.42×10+14)−7.9199×10+14
(1.16×10+14)−2.3362×10+12
(5.57×10+12)−8.9765×10+14
(1.59×10+14)−7.1165×10+15
(4.06×10+14)WFG8 5 1.5013×10+8
(8.11×10+6)−1.9783×10+8
(1.05×10+7)−1.7829×10+8
(7.32×10+6)−3.6308×10+7
(4.44×10+6)−1.4408×10+8
(7.57×10+6)−2.3335×10+8
(1.10×10+7)10 1.8219×10+11
(1.68×10+10)−3.8567×10+11
(5.74×10+10)−2.0046×10+11
(2.71×10+10)−8.2783×10+10
(3.03×10+10)−1.7126×10+11
(1.80×10+10)−6.0173×10+11
(7.82×10+10)15 4.0400×10+12
(9.81×10+11)−9.9234×10+12
(1.83×10+12)=3.1853×10+12
(9.93×10+11)−1.9227×10+11
(1.86×10+11)−1.6871×10+12
(1.26×10+12)−1.2895×10+13
(5.22×10+12)25 1.3971×10+14
(9.91×10+13)−4.8077×10+15
(6.01×10+14)−2.4789×10+14
(8.47×10+13)−4.7555×10+12
(8.90×10+12)−3.1451×10+14
(9.11×10+13)−1.0680×10+16
(8.26×10+14)WFG9 5 1.9141×10+8
(8.37×10+6)−2.3341×10+8
(1.10×10+7)−1.9389×10+8
(7.79×10+6)−3.6203×10+7
(5.92×10+6)−1.5846×10+8
(6.96×10+6)−2.6391×10+8
(1.35×10+7)10 3.7972×10+11
(3.13×10+10)−6.0898×10+11
(4.27×10+10)−4.2790×10+11
(1.62×10+10)−2.0148×10+11
(1.32×10+10)−3.5255×10+11
(1.94×10+10)−9.8787×10+11
(3.03×10+10)15 1.4912×10+13
(1.61×10+12)−2.0973×10+13
(2.10×10+12)−1.2833×10+13
(1.43×10+12)−5.5018×10+10
(8.57×10+10)−1.0318×10+13
(1.36×10+12)−3.2318×10+13
(1.52×10+12)25 4.2407×10+15
(8.53×10+14)−9.2528×10+15
(1.06×10+15)−3.7876×10+15
(6.04×10+14)−4.6745×10+12
(7.48×10+12)−5.3764×10+15
(3.32×10+14)−1.6297×10+16
(4.08×10+14)+/−/= 4/36/0 3/36/1 4/36/0 1/39/0 3/37/0 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 6 MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA以及MaOEA-IAR在DTLZ5~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG3上获得IGD值的统一结果(均值和标准差). 最好的结果已被标记
Table 6 The statistical results (mean and standard deviation) of the IGD values obtained by MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA and MaOEA-IAR on DTLZ5~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG3. The best results are highlighted
问题 M MOEA-DD NSGA-III RVEA MOMBI-II AR-MOEA MaOEA-IAR DTLZ5 5 1.0286×10−1
(4.56×10−3)−1.5564×10−1
(6.36×10−2)−1.7052×10−1
(2.08×10−2)−2.6836×10−1
(1.37×10−2)−1.2293×10−1
(2.95×10−3)−9.2396×10−2
(1.58×10−2)10 1.3275×10−1
(1.77×10−2)+2.9646×10−1
(9.03×10−2)+3.7614×10−1
(6.44×10−2)+5.8454×10−1
(1.64×10−1)−9.4984×10−2
(3.11×10−3)+4.8043×10−1
(1.04×10−1)15 1.5268×10−1
(4.95×10−3)−2.5900×10−1
(4.20×10−2)−5.8957×10−1
(1.74×10−1)−6.4004×10−1
(6.56×10−2)−6.0343×10−1
(1.01×10−2)−1.3313×10−1
(1.25×10−2)25 1.4321×10−1
(4.38×10−2)+8.5001×10−1
(5.98×10−1)−3.6054×10−1
(2.03×10−1)−5.7672×10−1
(2.97×10−1)−9.6898×10−2
(1.91×10−2)−3.2482×10−2
(1.69×10−1)DTLZ6 5 1.0630×10−1
(9.13×10−3)+2.8728×10−1
(2.28×10−1)+1.5455×10−1
(3.80×10−2)+3.8061×10−1
(2.25×10−1)=3.3734×10−1
(3.50×10−2)−3.1149×10−1
(5.18×10−2)10 2.0616×10−1
(1.64×10−1)+1.5928×10+0
(5.87×10−1)+2.8394×10−1
(8.30×10−2)+5.4111×10−1
(2.36×10−1)+1.6497×10−1
(8.66×10−3)+2.4129×10+0
(3.39×10−1)15 2.0227×10−1
(1.34×10−1)+2.5854×10+0
(1.02×10+0)=3.2020×10−1
(1.73×10−1)+7.5239×10−1
(2.82×10−1)+1.2810×10−1
(1.01×10−1)+2.9171×10+0
(4.24×10−1)25 1.5709×10−1
(2.14×10−2)−6.1820×10+0
(3.10×10+0)−2.2612×10−1
(6.64×10−2)−5.4816×10−1
(2.83×10−1)−5.0812×10−1
(3.00×10−1)−1.0126×10−1
(2.94×10−2)DTLZ7 5 3.0002×10+0
(9.72×10−4)−3.3903×10−1
(1.25×10−2)+6.0021×10−1
(3.12×10−2)−4.8750×10−1
(8.74×10−2)−8.7917×10−3
(4.95×10−4)+4.0035×10−1
(8.59×10−2)10 2.0394×10+0
(3.09×10−1)−1.6480×10+0
(2.43×10−1)−1.8321×10+0
(4.74×10−1)−4.8283×10+0
(7.64×10−1)−2.8392×10−2
(3.28×10−3)+8.4705×10−1
(3.02×10−2)15 3.4630×10+0
(8.21×10−2)−8.8601×10+0
(7.49×10−1)−2.3863×10+0
(2.46×10−1)−1.1222×10+1
(1.43×10−1)−5.4456×10−1
(3.13×10−1)+1.6482×10+0
(3.88×10−2)25 4.9987×10+0
(4.01×10−2)−1.4671×10+1
(1.41×10+0)−3.5010×10+0
(4.87×10−1)+1.9012×10+1
(2.83×10−1)−3.7797×10+0
(1.17×10+0)+4.0533×10+0
(1.41×10−5)IDTLZ1 5 1.5305×10−1
(4.10×10−2)=1.3547×10−1
(1.54×10−2)=1.5234×10−1
(1.70×10−2)=1.1355×10−1
(3.26×10−4)=6.5841×10−2
(8.22×10−4)+2.5128×10−1
(1.65×10−1)10 2.2790×10−1
(1.05×10−2)−1.4175×10−1
(3.27×10−3)−2.5816×10−1
(2.39×10−2)−1.8633×10−1
(6.51×10−3)−1.2386×10−1
(3.42×10−2)=1.1027×10−1
(9.23×10−4)15 3.2984×10−1
(3.09×10−2)−2.1719×10−1
(4.73×10−3)−3.5590×10−1
(2.89×10−2)−2.1098×10−1
(1.14×10−2)−1.9277×10−1
(1.29×10−2)−1.7215×10−1
(7.39×10−2)25 3.2145×10−1
(1.81×10−2)−1.9936×10−1
(3.93×10−3)−3.8507×10−1
(3.13×10−2)−2.6360×10−1
(6.52×10−3)−4.3902×10−2
(1.27×10−1)−2.1252×10−2
(1.63×10−2)IDTLZ2 5 2.7867×10−1
(5.70×10−3)−2.4322×10−1
(6.88×10−3)−2.9833×10−1
(5.13×10−3)−3.1740×10−1
(1.15×10−3)−2.1362×10−1
(5.25×10−3)−2.0888×10−1
(7.30×10−3)10 7.3889×10−1
(6.96×10−3)−6.0395×10−1
(1.69×10−2)−6.7982×10−1
(7.75×10−3)−6.6946×10−1
(4.82×10−3)−4.4639×10−1
(6.90×10−3)−4.4152×10−1
(6.68×10−3)15 9.4524×10−1
(2.03×10−2)−7.6174×10−1
(1.42×10−2)−8.7051×10−1
(1.47×10−2)−8.5834×10−1
(7.35×10−3)−6.8015×10−1
(1.79×10−2)−5.7753×10−1
(9.19×10−3)25 1.1245×10+0
(1.07×10−2)−8.3706×10−1
(1.63×10−2)−1.0678×10+0
(1.60×10−2)−1.0151×10+0
(4.12×10−3)−8.9235×10−2
(2.87×10−2)-2.8098×10−2
(6.87×10−4)WFG1 5 8.4034×10−1
(1.15×10−1)−5.6876×10−1
(4.95×10−2)−5.3556×10−1
(4.24×10−2)−5.3968×10−1
(6.19×10−2)−5.1692×10−1
(1.26×10−2)−4.4951×10−1
(5.18×10−3)10 1.2623×10+0
(6.70×10−2)−1.2030×10+0
(7.30×10−2)−1.0736×10+0
(6.23×10−2)=1.2539×10+0
(5.56×10−2)−1.1573×10+0
(3.43×10−2)−1.0186×10+0
(3.21×10−2)15 1.9032×10+0
(6.34×10−2)−1.9456×10+0
(8.90×10−2)−1.8780×10+0
(7.27×10−2)−2.5378×10+0
(2.18×10−1)−1.7899×10+0
(4.09×10−2)=1.7788×10+0
(4.73×10−2)25 3.8022×10+0
(5.48×10−2)−3.0121×10+0
(4.19×10−1)−3.1303×10+0
(1.12×10−1)−3.6938×10+0
(9.71×10−2)−3.1942×10+0
(6.67×10−2)−2.6916×10+0
(1.82×10−1)WFG2 5 5.7666×10−1
(1.64×10−2)−4.6948×10−1
(3.19×10−3)+4.4930×10−1
(1.01×10−2)+5.1672×10−1
(6.95×10−2)+4.7596×10−1
(2.55×10−3)+5.4316×10−1
(2.97×10−2)10 1.4487×10+0
(2.06×10−2)−1.2010×10+0
(1.43×10−1)−1.1062×10+0
(3.99×10−2)−1.6478×10+0
(5.00×10−1)−1.4937×10+0
(4.68×10−2)−1.0054×10+0
(1.43×10−2)15 2.1614×10+0
(3.30×10−2)−1.7688×10+0
(7.55×10−2)=1.7815×10+0
(1.09×10−1)=7.5478×10+0
(2.36×10+0)−1.7084×10+0
(3.82×10−2)=1.7263×10+0
(1.03×10−1)25 4.0448×10+0
(1.16×10−2)−3.5543×10+0
(1.21×10−1)−2.8225×10+0
(1.52×10−1)−1.0109×10+1
(5.56×10+0)−3.1864×10+0
(1.60×10−1)−2.7976×10+0
(1.71×10−1)WFG3 5 7.4442×10−1
(3.28×10−2)−5.9020×10−1
(5.58×10−2)−6.8641×10−1
(1.11×10−1)−1.6953×10+0
(1.28×10−1)−6.8013×10−1
(1.54×10−1)−5.4840×10−1
(3.21×10−2)10 2.7654×10+0
(1.17×10−1)+1.4175×10+0
(4.13×10−1)+3.4602×10+0
(5.56×10−1)−8.5627×10+0
(1.89×10−1)−2.4166×10+0
(9.05×10−2)+3.0447×10+0
(5.68×10−1)15 6.4307×10+0
(6.67×10−1)+2.6293×10+0
(4.06×10−1)+6.6442×10+0
(1.19×10+0)+1.3900×10+1
(2.03×10−1)−5.5924×10+0
(2.08×10−1)+7.5080×10+0
(9.65×10−1)25 1.8160×10+1
(3.63×10−2)−1.2519×10+1
(1.40×10+0)−1.1530×10+1
(1.65×10+0)−2.7424×10+1
(3.33×10−2)−9.8933×10+0
(1.41×10−1)−6.8891×10+0
(5.11×10+0)+/−/= 7/24/1 7/22/3 7/22/3 3/27/2 11/18/3 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 7 MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA以及MaOEA-IAR在DTLZ5~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG3上获得PD值的统一结果(均值和标准差).最好的结果已被标记
Table 7 The statistical results (mean and standard deviation) of the PD values obtained by MOEA-DD、NSGA-III、RVEA、MOMBI-II、AR-MOEA and MaOEA-IAR on DTLZ5~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG3. The best results are highlighted
问题 M MOEA-DD NSGA-III RVEA MOMBI-II AR-MOEA MaOEA-IAR DTLZ5 5 1.7925×10+7
(2.11×10+6)−3.5995×10+7
(4.18×10+6)+1.8932×10+7
(4.20×10+6)−1.1044×10+7
(1.59×10+6)−4.8118×10+7
(2.34×10+6)+2.5713×10+7
(2.69×10+6)10 1.9742×10+10
(2.54×10+9)−7.1635×10+10
(5.54×10+9)−2.9432×10+10
(3.78×10+9)−2.1484×10+9
(2.81×10+9)−8.3986×10+10
(3.99×10+9)−1.2087×10+11
(8.00×10+9)15 2.8397×10+11
(7.71×10+10)−1.1841×10+12
(1.38×10+11)−4.1423×10+10
(2.81×10+10)−6.5840×10+9
(4.48×10+9)−1.0895×10+12
(1.09×10+11)−2.9251×10+12
(2.88×10+11)25 4.2895×10+13
(9.57×10+12)+1.4562×10+11
(1.05×10+11)−1.7203×10+13
(1.37×10+13)+4.0293×10+10
(4.56×10+10)−1.0108×10+14
(1.23×10+13)−5.6358×10+14
(2.21×10+14)DTLZ6 5 2.8362×10+7
(3.50×10+6)−5.4971×10+7
(7.11×10+6)+2.6105×10+7
(7.26×10+6)−1.0490×10+7
(2.67×10+6)−5.2864×10+7
(3.81×10+6)+3.1205×10+7
(4.21×10+6)10 6.5293×10+10
(1.58×10+10)−1.8747×10+11
(4.95×10+10)−5.9078×10+10
(1.80×10+10)−2.1985×10+9
(2.83×10+9)−8.4598×10+10
(5.75×10+9)−3.9921×10+11
(2.70×10+10)15 9.1234×10+11
(3.10×10+11)−3.4359×10+12
(5.79×10+11)−7.4856×10+11
(3.95×10+11)−1.0372×10+10
(1.40×10+10)−4.2785×10+11
(3.48×10+11)−7.9758×10+12
(8.70×10+11)25 2.2080×10+14
(1.01×10+14)−2.2455×10+14
(1.56×10+13)−8.7904×10+13
(2.72×10+13)−1.5169×10+11
(1.15×10+11)−3.2540×10+13
(1.77×10+13)−2.3216×10+15
(1.08×10+15)DTLZ7 5 1.6766×10+3
(4.84×10+3)−2.1762×10+7
(4.10×10+6)=1.8563×10+7
(3.07×10+6)=5.1253×10+6
(7.63×10+5)−3.3243×10+7
(4.17×10+6)+2.0833×10+7
(7.02×10+6)10 2.1457×10+9
(1.38×10+9)−3.1794×10+10
(2.69×10+9)−1.6034×10+10
(4.07×10+9)−8.0445×10+9
(1.73×10+9)−2.5524×10+10
(3.65×10+9)−6.8683×10+10
(7.18×10+9)15 1.8990×10+10
(5.95×10+9)−4.7550×10+11
(9.43×10+10)−2.7698×10+11
(6.40×10+10)−4.9554×10+10
(2.76×10+10)−4.0328×10+11
(1.19×10+11)−2.5059×10+12
(3.10×10+11)25 4.2556×10+12
(1.02×10+12)−7.3834×10+13
(1.20×10+13)−4.7217×10+13
(1.00×10+13)−4.4333×10+13
(1.34×10+13)−1.3427×10+9
(6.62×10+8)−1.1453×10+14
(1.61×10+13)IDTLZ1 5 1.4745×10+6
(9.43×10+5)−6.3366×10+6
(3.40×10+6)−2.1610×10+6
(9.83×10+5)−8.4010×10+5
(2.39×10+5)−1.0739×10+7
(2.67×10+6)−1.9799×10+8
(2.32×10+8)10 1.0063×10+8
(1.13×10+8)−9.8146×10+9
(1.30×10+9)−1.0241×10+9
(9.00×10+8)−2.0726×10+9
(3.55×10+8)−7.1952×10+9
(1.15×10+9)−4.5218×10+10
(1.18×10+11)15 4.8749×10+10
(2.39×10+11)−2.7098×10+11
(4.55×10+10)−6.2342×10+10
(5.79×10+10)−3.9405×10+10
(8.72×10+9)−1.2277×10+11
(3.42×10+11)−3.2538×10+12
(1.06×10+13)25 3.8837×10+9
(8.47×10+9)−5.6426×10+13
(1.28×10+13)−7.6673×10+12
(5.54×10+12)−1.6889×10+12
(1.19×10+12)−1.6465×10+14
(4.61×10+14)−5.8266×10+14
(1.42×10+13)IDTLZ2 5 1.2432×10+7
(1.92×10+6)−3.5278×10+7
(7.22×10+6)−1.8123×10+7
(1.50×10+6)−4.8079×10+6
(6.93×10+5)−2.6241×10+7
(1.72×10+6)−5.5348×10+7
(2.41×10+6)10 3.7649×10+9
(3.38×10+8)−4.9622×10+10
(7.67×10+9)−1.1533×10+10
(1.30×10+9)−1.5701×10+10
(1.85×10+9)−9.1979×10+10
(3.71×10+9)−1.6089×10+11
(4.17×10+9)15 7.1923×10+10
(3.07×10+10)−1.0779×10+12
(1.50×10+11)−2.0643×10+11
(6.01×10+10)−2.9274×10+11
(4.07×10+10)−2.6072×10+12
(1.61×10+11)−5.4279×10+12
(1.82×10+11)25 4.4848×10+12
(1.99×10+12)−5.6171×10+14
(8.35×10+13)−1.4306×10+13
(8.54×10+12)−7.0067×10+13
(1.05×10+13)−8.5860×10+14
(4.94×10+13)−2.0550×10+15
(5.47×10+13)WFG1 5 7.7570×10+7
(4.76×10+6)+7.5733×10+7
(7.14×10+6)+6.6169×10+7
(5.02×10+6)+7.6250×10+6
(2.77×10+6)−3.0367×10+7
(2.17×10+6)+2.3345×10+7
(3.37×10+6)10 3.8144×10+10
(3.75×10+9)+8.4159×10+10
(8.64×10+9)+4.6067×10+10
(6.62×10+9)+6.4761×10+9
(1.55×10+9)−4.3604×10+10
(2.26×10+9)+1.8476×10+10
(2.80×10+9)15 1.3448×10+12
(2.24×10+11)+1.2676×10+12
(5.92×10+11)+6.2449×10+11
(1.04×10+11)=1.9369×10+10
(2.14×10+10)−6.0993×10+11
(4.26×10+10)=6.1772×10+11
(1.26×10+11)25 3.3402×10+13
(2.45×10+12)−1.6094×10+14
(4.13×10+13)+1.0889×10+14
(5.98×10+12)+4.8564×10+12
(1.42×10+12)−1.0664×10+14
(3.18×10+12)+8.4524×10+13
(1.67×10+13)WFG2 5 5.1237×10+7
(2.65×10+6)−5.4016×10+7
(3.16×10+6)−6.1875×10+7
(2.87×10+6)−1.1742×10+7
(3.17×10+6)−3.8312×10+7
(2.55×10+6)−7.6371×10+7
(4.16×10+6)10 3.9168×10+10
(1.43×10+9)−6.6429×10+10
(1.01×10+10)−5.3106×10+10
(2.41×10+9)−7.4067×10+9
(5.52×10+9)−3.9309×10+10
(2.97×10+9)−9.1141×10+10
(4.59×10+9)15 4.4512×10+11
(4.65×10+10)−1.5952×10+12
(2.02×10+11)−9.2321×10+11
(1.58×10+11)−1.2053×10+10
(1.52×10+10)−5.7677×10+11
(8.49×10+10)−2.2461×10+12
(1.65×10+11)25 4.1805×10+13
(3.61×10+12)−3.8842×10+14
(1.07×10+14)−1.7542×10+14
(2.40×10+13)−3.8888×10+11
(9.15×10+11)−1.6560×10+14
(1.30×10+13)−4.7295×10+14
(2.67×10+13)WFG3 5 1.2116×10+8
(9.07×10+6)−1.4756×10+8
(7.56×10+6)−1.3033×10+8
(1.40×10+7)−5.1396×10+7
(4.21×10+6)−1.5433×10+8
(7.28×10+6)=1.5501×10+8
(6.95×10+6)10 1.1096×10+11
(9.29×10+9)−2.4982×10+11
(2.39×10+10)−1.6581×10+11
(1.63×10+10)−9.4558×10+8
(1.51×10+8)−1.9220×10+11
(1.05×10+10)−3.1718×10+11
(2.00×10+10)15 3.8028×10+12
(7.77×10+11)−7.4227×10+12
(1.37×10+12)+4.1520×10+12
(7.83×10+11)−1.6200×10+10
(5.00×10+9)−2.7152×10+12
(9.23×10+11)−6.1382×10+12
(5.23×10+11)25 1.0941×10+14
(6.89×10+12)−1.9873×10+15
(6.87×10+14)=5.6632×10+14
(1.69×10+14)−5.1563×10+10
(1.66×10+10)−3.2575×10+14
(2.26×10+13)−1.5572×10+15
(1.06×10+14)+/−/= 4/28/0 7/23/2 4/26/2 0/32/0 6/24/2 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 8 AR-MOEA和MaOEA-IAR在DTLZ1~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG9上运行时间的统一结果(均值). 最好的结果已被标记
Table 8 The statistical results (mean) of the time obtained by AR-MOEA and MaOEA-IAR on DTLZ1~DTLZ7, IDTLZ1~IDTLZ2, WFG1~WFG9. The best results are highlighted
问题 M AR-MOEA MaOEA-IAR 问题 M AR-MOEA MaOEA-IAR DTLZ1 5 8.7284×10+1− 5.2877×10+1 WFG1 5 2.1497×10+2− 1.2453×10+2 10 3.2368×10+3− 1.4929×10+3 10 3.0802×10+3− 1.8838×10+3 15 1.5646×10+2+ 4.5986×10+2 15 3.3103×10+2− 2.9192×10+2 25 2.7908×10+3− 1.8621×10+3 25 2.4172×10+3+ 2.8807×10+3 DTLZ2 5 1.4045×10+2+ 3.1151×10+2 WFG2 5 2.5691×10+2− 1.9505×10+2 10 3.4692×10+3− 2.0823×10+3 10 3.6336×10+3− 3.2122×10+3 15 1.9603×10+2+ 4.9719×10+2 15 4.3556×10+2− 3.8969×10+2 25 3.1545×10+3+ 3.4599×10+3 25 3.4466×10+3= 3.3881×10+3 DTLZ3 5 7.6826×10+1− 3.3370×10+1 WFG3 5 3.4018×10+2− 3.2435×10+2 10 3.2959×10+3− 1.5038×10+3 10 4.3267×10+3− 4.1139×10+3 15 4.9418×10+2− 1.5822×10+2 15 5.1973×10+2− 4.3731×10+2 25 2.9858×10+3− 1.7848×10+3 25 3.8047×10+3= 3.7357×10+3 DTLZ4 5 1.4659×10+2+ 3.2766×10+2 WFG4 5 3.3620×10+2− 2.8286×10+2 10 3.8768×10+3− 2.1238×10+3 10 4.2819×10+3− 3.6326×10+3 15 1.9815×10+2+ 5.1050×10+2 15 3.8595×10+2+ 5.2267×10+2 25 3.1269×10+3+ 3.2684×10+3 25 3.7546×10+3− 3.3598×10+3 DTLZ5 5 2.2500×10+2− 1.3912×10+2 WFG5 5 3.3378×10+2− 2.9287×10+2 10 3.0904×10+3− 1.9358×10+3 10 4.2697×10+3− 3.7511×10+3 15 1.8227×10+2+ 4.5443×10+2 15 5.2633×10+2− 4.0438×10+2 25 3.0757×10+3− 1.9646×10+3 25 3.8177×10+3− 3.4843×10+3 DTLZ6 5 1.9591×10+2− 1.2208×10+2 WFG6 5 2.7412×10+2− 2.5095×10+2 10 3.5852×10+3− 1.8312×10+3 10 4.0843×10+3− 3.2581×10+3 15 1.5310×10+2+ 4.9933×10+2 15 3.1397×10+2+ 5.1200×10+2 25 2.6990×10+3− 2.0808×10+3 25 3.7598×10+3− 2.8335×10+3 DTLZ7 5 2.0799×10+2− 1.4207×10+2 WFG7 5 3.7091×10+2− 3.1761×10+2 10 3.8714×10+3− 2.3062×10+3 10 4.3950×10+3− 3.7082×10+3 15 4.8328×10+2− 2.0122×10+2 15 3.7417×10+2+ 5.2910×10+2 25 2.4555×10+3− 1.5033×10+3 25 3.8242×10+3− 3.2364×10+3 IDTLZ1 5 2.1931×10+2− 9.4000×10+1 WFG8 5 2.2219×10+2= 2.1770×10+2 10 3.1281×10+3− 1.9947×10+3 10 3.7129×10+3− 2.6444×10+3 15 3.4287×10+2− 2.5752×10+2 15 2.4552×10+2+ 4.5493×10+2 25 2.0253×10+3− 1.2549×10+3 25 3.7527×10+3− 2.5151×10+3 IDTLZ2 5 3.1594×10+2+ 3.3723×10+2 WFG9 5 3.8469×10+2− 3.2047×10+2 10 4.0353×10+3− 3.4738×10+3 10 4.3729×10+3− 3.9 448×10+3 15 4.4610×10+2− 4.2203×10+2 15 5.3114×10+2− 4.3118×10+2 25 2.5265×10+3− 1.7758×10+3 25 3.8142×10+3− 3.5823×10+3 +/−/= 10/26/0 +/−/= 5/28/3 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 9 AR-MOEA和MaOEA-IAR在MaF1~MaF7上获得IGD值的统一结果(均值和标准差). 最好的结果已被标记
Table 9 The statistical results (mean and standard deviation) of the IGD values obtained by AR-MOEA and MaOEA-IAR on MaF1~MaF7. The best results are highlighted
问题 M AR-MOEA MaOEA-IAR MaF1 5 6.1773×10−2 (6.75×10−2)− 3.2287×10−3 (7.05×10−5) 10 1.0257×10−1 (1.11×10−1)− 6.5011×10−3 (1.44×10−4) 15 1.6987×10−1 (1.85×10−1)− 1.1253×10−2 (2.95×10−4) 25 1.7434×10−1 (1.89×10−1)− 9.2626×10−3 (1.97×10−4) MaF2 5 5.6263×10−2 (5.26×10−2)− 8.5071×10−3 (3.31×10−4) 10 8.3280×10−2 (8.77×10−2)− 6.4968×10−3 (9.61×10−5) 15 1.2184×10−1 (1.24×10−1)− 8.9506×10−3 (2.26×10−4) 25 1.0515×10−1 (1.21×10−1)− 6.2938×10−3 (4.48×10−4) MaF3 5 3.4660×10−2 (4.22×10−2)− 1.0505×10−3 (8.53×10−4) 10 3.4304×10−2 (4.06×10−2)− 4.8646×10−3 (8.72×10−3) 15 1.6992×10+0 (6.87×10+0)− 3.9708×10−3 (3.15×10−3) 25 8.5024×10−1 (2.47×10+0)− 2.0315×10−3 (2.00×10−3) MaF4 5 1.0234×10+0 (1.26×10+0)− 4.5541×10−2 (9.87×10−3) 10 3.6718×10+1 (4.35×10+1)− 1.7669×10+0 (1.34×10−2) 15 1.7572×10+3 (2.01×10+3)− 2.7158×10+2 (5.78×10−1) 25 1.8804×10+6 (2.17×10+6)− 1.7062×10+5 (3.15×10+2) MaF5 5 1.0214×10+0 (1.29×10+0)− 6.3123×10−2 (1.23×10−2) 10 3.7332×10+1 (4.78×10+1)− 6.3090×10−1 (9.64×10−2) 15 1.4852×10+3 (1.90×10+3)− 3.6171×10+0 (1.29×10+0) 25 1.7966×10+6 (2.32×10+6)− 7.1131×10+0 (8.87×10+0) MaF6 5 1.3381×10−3 (1.72×10−3)− 4.2532×10−6 (1.43×10−6) 10 3.1054×10−1 (1.30×10+0)+ 4.8255×10+0 (3.33×10+0) 15 2.1990×10−1 (7.89×10−1)− 4.6076×10−6 (1.37×10−6) 25 3.5425×10−1 (5.51×10−1)− 3.7808×10−6 (4.08×10−6) MaF7 5 1.3383×10−1 (1.61×10−1)= 1.8586×10−1 (2.35×10−1) 10 5.7112×10−1 (6.97×10−1)= 3.5443×10−1 (4.13×10−1) 15 2.2371×10+0 (2.25×10+0)− 7.1977×10−1 (7.75×10−1) 25 3.6581×10+0 (3.86×10+0)− 1.1867×10−1 (8.44×10−1) +/−/= 1/25/2 “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似 表 10 不同参数下算法MaOEA-IAR的性能
Table 10 Performance of MaOEA-IAR under different parameters
问题 目标数 r初始值 5 7 9 11 13 DTLZ1 5 2.6870×10−1 2.6135×10−1 2.6051×10−1 2.6382×10−1 2.6351×10−1 10 4.8177×10+1 4.7009×10+1 4.6085×10+1 4.7519×10+1 4.7286×10+1 15 5.7026×10+1 5.6981×10+1 5.5765×10+1 5.6146×10+1 5.6083×10+1 25 4.5156×10−1 4.5013×10−1 3.6557×10−1 2.4067×10−1 2.4052×10−1 DTLZ5 5 9.2396×10−2 8.3615×10−2 8.0045×10−2 8.8463×10−2 8.8732×10−2 10 6.2597×10−1 4.7625×10−1 4.7044×10−1 4.7765×10−1 4.7829×10−1 15 1.3549×10−1 1.3158×10−1 1.2994×10−1 1.3036×10−1 1.3023×10−1 25 3.3464×10−2 3.3391×10−2 3.3301×10−2 3.3362×10−2 3.3485×10−2 IDTLZ1 5 2.5128×10−1 2.3743×10−1 2.1694×10−1 2.2748×10−1 2.2926×10−1 10 1.1236×10−1 1.1149×10−1 1.0017×10−1 1.0594×10−1 1.0542×10−1 15 1.8035×10−1 1.7891×10−1 1.7077×10−1 1.7901×10−1 1.7924×10−1 25 2.2693×10−2 2.2451×10−2 2.2279×10−2 2.2538×10−2 2.2523×10−2 WFG1 5 4.4951×10−1 4.3826×10−1 4.3007×10−1 4.3596×10−1 4.4001×10−1 10 1.1967×10+0 1.1051×10+0 1.0186×10+0 1.0598×10+0 1.0482×10+0 15 1.6956×10+0 1.5941×10+0 1.5076×10+0 1.6582×10+0 1.6677×10+0 25 2.8015×10+0 2.7693×10+0 2.6724×10+0 2.7795×10+0 2.7619×10+0 WFG2 5 5.4316×10−1 5.4029×10−1 5.4002×10−1 5.4174×10−1 5.4169×10−1 10 1.1086×10+0 1.0997×10+0 1.0051×10+0 1.1036×10+0 1.1097×10+0 15 1.8675×10+0 1.7649×10+0 1.7069×10+0 1.8007×10+0 1.7952×10+0 25 2.9598×10+0 2.9413×10+0 2.8039×10+0 2.8796×10+0 2.8976×10+0 WFG4 5 1.1192×10+0 1.1128×10+0 1.1065×10+0 1.1097×10+0 1.1106×10+0 10 4.1062×10+0 4.1057×10+0 4.0388×10+0 4.0263×10+0 4.0589×10+0 15 8.9645×10+0 8.8089×10+0 8.3986×10+0 8.4108×10+0 8.4004×10+0 25 1.9058×10+1 1.8769×10+1 1.6506×10+1 1.5375×10+1 1.6405×10+1 -
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