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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

何江红,  李军华,  周日贵

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文章导航 > 自动化学报  > 2022 >  48(6): 1569-1589
何江红, 李军华, 周日贵. 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. 自动化学报, 2022, 48(6): 1569−1589 doi: 10.16383/j.aas.c200975
引用本文: 何江红, 李军华, 周日贵. 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. 自动化学报, 2022, 48(6): 1569−1589 doi: 10.16383/j.aas.c200975
shu
He Jiang-Hong, Li Jun-Hua, Zhou Ri-Gui. Many-objective evolutionary algorithm driven by indicator under adaptive reference point adjustment. Acta Automatica Sinica, 2022, 48(6): 1569−1589 doi: 10.16383/j.aas.c200975
Citation: He Jiang-Hong, Li Jun-Hua, Zhou Ri-Gui. Many-objective evolutionary algorithm driven by indicator under adaptive reference point adjustment. Acta Automatica Sinica, 2022, 48(6): 1569−1589 doi: 10.16383/j.aas.c200975
shu

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

doi: 10.16383/j.aas.c200975
  • 1.

    南昌航空大学江西省图像处理与模式识别重点实验室 南昌 330063

  • 2.

    上海海事大学信息工程学院 上海 201306

基金项目: 国家自然科学基金(62066031, 61866025, 61866026), 江西省自然科学基金(2018BAB202025), 江西省优势科技创新团队计划(2018BCB24008), 江西省研究生创新基金(YC2020-S540)资助
详细信息
    作者简介:

    何江红：南昌航空大学硕士研究生.主要研究方向为进化计算. E-mail: he192652@163.com

    李军华：南昌航空大学教授. 主要研究方向为进化计算和智能控制. 本文通信作者. E-mail: jhlee126@126.com

    周日贵：上海海事大学教授. 主要研究方向为人工智能及其应用. E-mail: rgzhou@shmtu.edu.cn

Many-objective Evolutionary Algorithm Driven by Indicator Under Adaptive Reference Point Adjustment

  • 1.

    Key Laboratory of Jiangxi Province for Image Processing and Pattern Recognition, Nanchang Hangkong University, Nanchang 330063

  • 2.

    School of Information Engineering, Shanghai Maritime University, Shanghai 201306

Funds: Supported by National Natural Science Foundation of China (62066031, 61866025, 61866026), Natural Science Foundation of Jiangxi (2018BAB202025), Superiority Science and Technology Innovation Team Program of Jiangxi (2018BCB24008), and Graduate Innovation Fund of Jiangxi Province (YC2020-S540)
More Information
    Author Bio:

    HE Jiang-Hong　Master student at Nanchang Hangkong University. Her main research interest is evolutionary computation

    LI Jun-Hua　Professor at Nanchang Hangkong University. His research interest covers evolutionary computation and intelligent control. Corresponding author of this paper

    ZHOU Ri-Gui　Professor at Shanghai Maritime University. His research interest covers artificial intelligence and applications

    摘要
  • 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.
    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.
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  • 图  1  二目标下的无贡献解

    Fig.  1  The noncontributed solutions under bi-objective

    下载: 全尺寸图片 幻灯片

    图  2  基于角度距离选择参考点

    Fig.  2  Select reference points based on angular distance

    下载: 全尺寸图片 幻灯片

    图  3  自适应参考点过程

    Fig.  3  Adaptive reference point process

    下载: 全尺寸图片 幻灯片

    图  4  确定轮廓曲线以及曲线参数

    Fig.  4  Determine the contour curve and curve parameter

    下载: 全尺寸图片 幻灯片

    图  5  WFG5问题10目标上获得的非支配解

    Fig.  5  Nondominated solutions obtained on 10-objective WFG5

    下载: 全尺寸图片 幻灯片

    图  6  5、10、15、25目标DTLZ4问题上的IGD进化轨迹

    Fig.  6  Evolutionary trajectories of IGD on 5、10、15、25-objective DTLZ4

    下载: 全尺寸图片 幻灯片

    图  7  DTLZ5问题3目标上获得的非支配解

    Fig.  7  Nondominated solutions obtained on 3-objective DTLZ5

    下载: 全尺寸图片 幻灯片

    图  8  WFG1问题15目标上获得的非支配解

    Fig.  8  Nondominated solutions obtained on 15-objective WFG1

    下载: 全尺寸图片 幻灯片

    表  1  测试问题

    Table  1  Test questions

    问题目标数目M决策变量DPF
    DTLZ15, 10, 15, 25M−1+5线性
    DTLZ2~DTLZ45, 10, 15, 25M−1+10凹型
    DTLZ5~DTLZ65, 10, 15, 25M−1+10退化
    DTLZ75, 10, 15, 25M−1+20断开
    IDTLZ15, 10, 15, 25M−1+5倒置
    IDTLZ25, 10, 15, 25M−1+10倒置
    WFG15, 10, 15, 25M−1+10混合
    WFG25, 10, 15, 25M−1+10断开
    WFG35, 10, 15, 25M−1+10退化
    WFG4~DTLZ95, 10, 15, 25M−1+10凹型
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    5126800100800
    102751000275000
    151351300175500
    253501500525000
    下载: 导出CSV

    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MMOEA-DDNSGA-IIIRVEAMOMBI-IIAR-MOEAMaOEA-IAR
    DTLZ156.3354×10−2
    (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)
    101.0784×10−1
    (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)
    151.4088×10−1
    (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)
    251.0354×10−1
    (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)
    DTLZ251.9506×10−1
    (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)
    104.2179×10−1
    (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)
    156.1801×10−1
    (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)
    254.3015×10−1
    (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)
    DTLZ351.9907×10−1
    (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)
    104.2392×10−1
    (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)
    156.6842×10−1
    (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)
    254.3041×10−1
    (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)
    DTLZ451.9508×10−1
    (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)
    104.2271×10−1
    (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)
    156.3495×10−1
    (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)
    254.3041×10−1
    (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)
    WFG451.2337×10+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)
    106.0999×10+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)
    159.4478×10+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)
    252.4032×10+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)
    WFG551.2116×10+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)
    106.2798×10+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)
    151.1246×10+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)
    252.2399×10+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)
    WFG651.2276×10+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)
    106.0366×10+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)
    151.0818×10+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)
    252.1950×10+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)
    WFG751.2454×10+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)
    105.1499×10+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)
    158.9281×10+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)
    251.6767×10+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)
    WFG851.2301×10+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)
    105.2132×10+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)
    159.2706×10+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)
    252.2579×10+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)
    WFG951.2087×10+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)
    105.2551×10+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)
    158.9598×10+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)
    251.5766×10+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)
    +/−/=10/30/07/30/310/28/28/31/110/29/1
    “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似
    下载: 导出CSV

    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MMOEA-DDNSGA-IIIRVEAMOMBI-IIAR-MOEAMaOEA-IAR
    DTLZ157.0347×10+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)
    103.9493×10+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)
    152.0771×10+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)
    254.2240×10+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)
    DTLZ254.5169×10+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)
    101.0700×10+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)
    152.3742×10+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)
    251.5327×10+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)
    DTLZ351.0522×10+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)
    101.3601×10+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)
    153.6156×10+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)
    251.6252×10+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)
    DTLZ453.7620×10+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)
    101.3637×10+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)
    152.7750×10+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)
    253.7432×10+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)
    WFG451.2326×10+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)
    101.4338×10+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)
    153.4224×10+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)
    259.4061×10+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)
    WFG551.2123×10+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)
    101.9457×10+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)
    153.9863×10+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)
    252.8409×10+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)
    WFG651.1994×10+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)
    101.6619×10+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)
    153.5945×10+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)
    255.8142×10+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)
    WFG751.4307×10+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)
    102.5289×10+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)
    155.8545×10+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)
    256.8227×10+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)
    WFG851.5013×10+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)
    101.8219×10+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)
    154.0400×10+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)
    251.3971×10+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)
    WFG951.9141×10+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)
    103.7972×10+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)
    151.4912×10+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)
    254.2407×10+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)
    +/−/=4/36/03/36/14/36/01/39/03/37/0
    “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似
    下载: 导出CSV

    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MMOEA-DDNSGA-IIIRVEAMOMBI-IIAR-MOEAMaOEA-IAR
    DTLZ551.0286×10−1
    (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)
    101.3275×10−1
    (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)
    151.5268×10−1
    (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)
    251.4321×10−1
    (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)
    DTLZ651.0630×10−1
    (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)
    102.0616×10−1
    (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)
    152.0227×10−1
    (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)
    251.5709×10−1
    (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)
    DTLZ753.0002×10+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)
    102.0394×10+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)
    153.4630×10+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)
    254.9987×10+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)
    IDTLZ151.5305×10−1
    (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)
    102.2790×10−1
    (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)
    153.2984×10−1
    (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)
    253.2145×10−1
    (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)
    IDTLZ252.7867×10−1
    (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)
    107.3889×10−1
    (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)
    159.4524×10−1
    (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)
    251.1245×10+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)
    WFG158.4034×10−1
    (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)
    101.2623×10+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)
    151.9032×10+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)
    253.8022×10+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)
    WFG255.7666×10−1
    (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)
    101.4487×10+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)
    152.1614×10+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)
    254.0448×10+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)
    WFG357.4442×10−1
    (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)
    102.7654×10+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)
    156.4307×10+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)
    251.8160×10+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)
    +/−/=7/24/17/22/37/22/33/27/211/18/3
    “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似
    下载: 导出CSV

    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MMOEA-DDNSGA-IIIRVEAMOMBI-IIAR-MOEAMaOEA-IAR
    DTLZ551.7925×10+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)
    101.9742×10+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)
    152.8397×10+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)
    254.2895×10+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)
    DTLZ652.8362×10+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)
    106.5293×10+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)
    159.1234×10+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)
    252.2080×10+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)
    DTLZ751.6766×10+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)
    102.1457×10+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)
    151.8990×10+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)
    254.2556×10+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)
    IDTLZ151.4745×10+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)
    101.0063×10+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)
    154.8749×10+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)
    253.8837×10+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)
    IDTLZ251.2432×10+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)
    103.7649×10+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)
    157.1923×10+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)
    254.4848×10+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)
    WFG157.7570×10+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)
    103.8144×10+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)
    151.3448×10+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)
    253.3402×10+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)
    WFG255.1237×10+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)
    103.9168×10+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)
    154.4512×10+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)
    254.1805×10+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)
    WFG351.2116×10+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)
    101.1096×10+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)
    153.8028×10+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)
    251.0941×10+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)
    +/−/=4/28/07/23/24/26/20/32/06/24/2
    “+”表明该算法优于MaOEA-IAR, “−”劣于MaOEA-IAR, “=”则表示与MaOEA-IAR性能相似
    下载: 导出CSV

    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MAR-MOEAMaOEA-IAR问题MAR-MOEAMaOEA-IAR
    DTLZ158.7284×10+1−5.2877×10+1WFG152.1497×10+2−1.2453×10+2
    103.2368×10+3−1.4929×10+3103.0802×10+3−1.8838×10+3
    151.5646×10+2+4.5986×10+2153.3103×10+2−2.9192×10+2
    252.7908×10+3−1.8621×10+3252.4172×10+3+2.8807×10+3
    DTLZ251.4045×10+2+3.1151×10+2WFG252.5691×10+2−1.9505×10+2
    103.4692×10+3−2.0823×10+3103.6336×10+3−3.2122×10+3
    151.9603×10+2+4.9719×10+2154.3556×10+2−3.8969×10+2
    253.1545×10+3+3.4599×10+3253.4466×10+3=3.3881×10+3
    DTLZ357.6826×10+1−3.3370×10+1WFG353.4018×10+2−3.2435×10+2
    103.2959×10+3−1.5038×10+3104.3267×10+3−4.1139×10+3
    154.9418×10+2−1.5822×10+2155.1973×10+2−4.3731×10+2
    252.9858×10+3−1.7848×10+3253.8047×10+3=3.7357×10+3
    DTLZ451.4659×10+2+3.2766×10+2WFG453.3620×10+2−2.8286×10+2
    103.8768×10+3−2.1238×10+3104.2819×10+3−3.6326×10+3
    151.9815×10+2+5.1050×10+2153.8595×10+2+5.2267×10+2
    253.1269×10+3+3.2684×10+3253.7546×10+3−3.3598×10+3
    DTLZ552.2500×10+2−1.3912×10+2WFG553.3378×10+2−2.9287×10+2
    103.0904×10+3−1.9358×10+3104.2697×10+3−3.7511×10+3
    151.8227×10+2+4.5443×10+2155.2633×10+2−4.0438×10+2
    253.0757×10+3−1.9646×10+3253.8177×10+3−3.4843×10+3
    DTLZ651.9591×10+2−1.2208×10+2WFG652.7412×10+2−2.5095×10+2
    103.5852×10+3−1.8312×10+3104.0843×10+3−3.2581×10+3
    151.5310×10+2+4.9933×10+2153.1397×10+2+5.1200×10+2
    252.6990×10+3−2.0808×10+3253.7598×10+3−2.8335×10+3
    DTLZ752.0799×10+2−1.4207×10+2WFG753.7091×10+2−3.1761×10+2
    103.8714×10+3−2.3062×10+3104.3950×10+3−3.7082×10+3
    154.8328×10+2−2.0122×10+2153.7417×10+2+5.2910×10+2
    252.4555×10+3−1.5033×10+3253.8242×10+3−3.2364×10+3
    IDTLZ152.1931×10+2−9.4000×10+1WFG852.2219×10+2=2.1770×10+2
    103.1281×10+3−1.9947×10+3103.7129×10+3−2.6444×10+3
    153.4287×10+2−2.5752×10+2152.4552×10+2+4.5493×10+2
    252.0253×10+3−1.2549×10+3253.7527×10+3−2.5151×10+3
    IDTLZ253.1594×10+2+3.3723×10+2WFG953.8469×10+2−3.2047×10+2
    104.0353×10+3−3.4738×10+3104.3729×10+3−3.9448×10+3
    154.4610×10+2−4.2203×10+2155.3114×10+2−4.3118×10+2
    252.5265×10+3−1.7758×10+3253.8142×10+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
    下载: 导出CSV

    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MAR-MOEAMaOEA-IAR
    MaF156.1773×10−2 (6.75×10−2)−3.2287×10−3 (7.05×10−5)
    101.0257×10−1 (1.11×10−1)−6.5011×10−3 (1.44×10−4)
    151.6987×10−1 (1.85×10−1)−1.1253×10−2 (2.95×10−4)
    251.7434×10−1 (1.89×10−1)−9.2626×10−3 (1.97×10−4)
    MaF255.6263×10−2 (5.26×10−2)−8.5071×10−3 (3.31×10−4)
    108.3280×10−2 (8.77×10−2)−6.4968×10−3 (9.61×10−5)
    151.2184×10−1 (1.24×10−1)−8.9506×10−3 (2.26×10−4)
    251.0515×10−1 (1.21×10−1)−6.2938×10−3 (4.48×10−4)
    MaF353.4660×10−2 (4.22×10−2)−1.0505×10−3 (8.53×10−4)
    103.4304×10−2 (4.06×10−2)−4.8646×10−3 (8.72×10−3)
    151.6992×10+0 (6.87×10+0)−3.9708×10−3 (3.15×10−3)
    258.5024×10−1 (2.47×10+0)−2.0315×10−3 (2.00×10−3)
    MaF451.0234×10+0 (1.26×10+0)−4.5541×10−2 (9.87×10−3)
    103.6718×10+1 (4.35×10+1)−1.7669×10+0 (1.34×10−2)
    151.7572×10+3 (2.01×10+3)−2.7158×10+2 (5.78×10−1)
    251.8804×10+6 (2.17×10+6)−1.7062×10+5 (3.15×10+2)
    MaF551.0214×10+0 (1.29×10+0)−6.3123×10−2 (1.23×10−2)
    103.7332×10+1 (4.78×10+1)−6.3090×10−1 (9.64×10−2)
    151.4852×10+3 (1.90×10+3)−3.6171×10+0 (1.29×10+0)
    251.7966×10+6 (2.32×10+6)−7.1131×10+0 (8.87×10+0)
    MaF651.3381×10−3 (1.72×10−3)−4.2532×10−6 (1.43×10−6)
    103.1054×10−1 (1.30×10+0)+4.8255×10+0 (3.33×10+0)
    152.1990×10−1 (7.89×10−1)−4.6076×10−6 (1.37×10−6)
    253.5425×10−1 (5.51×10−1)−3.7808×10−6 (4.08×10−6)
    MaF751.3383×10−1 (1.61×10−1)=1.8586×10−1 (2.35×10−1)
    105.7112×10−1 (6.97×10−1)=3.5443×10−1 (4.13×10−1)
    152.2371×10+0 (2.25×10+0)−7.1977×10−1 (7.75×10−1)
    253.6581×10+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
    下载: 导出CSV

    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
    5791113
    DTLZ152.6870×10−12.6135×10−12.6051×10−12.6382×10−12.6351×10−1
    104.8177×10+14.7009×10+14.6085×10+14.7519×10+14.7286×10+1
    155.7026×10+15.6981×10+15.5765×10+15.6146×10+15.6083×10+1
    254.5156×10−14.5013×10−13.6557×10−12.4067×10−12.4052×10−1
    DTLZ559.2396×10−28.3615×10−28.0045×10−28.8463×10−28.8732×10−2
    106.2597×10−14.7625×10−14.7044×10−14.7765×10−14.7829×10−1
    151.3549×10−11.3158×10−11.2994×10−11.3036×10−11.3023×10−1
    253.3464×10−23.3391×10−23.3301×10−23.3362×10−23.3485×10−2
    IDTLZ152.5128×10−12.3743×10−12.1694×10−12.2748×10−12.2926×10−1
    101.1236×10−11.1149×10−11.0017×10−11.0594×10−11.0542×10−1
    151.8035×10−11.7891×10−11.7077×10−11.7901×10−11.7924×10−1
    252.2693×10−22.2451×10−22.2279×10−22.2538×10−22.2523×10−2
    WFG154.4951×10−14.3826×10−14.3007×10−14.3596×10−14.4001×10−1
    101.1967×10+01.1051×10+01.0186×10+01.0598×10+01.0482×10+0
    151.6956×10+01.5941×10+01.5076×10+01.6582×10+01.6677×10+0
    252.8015×10+02.7693×10+02.6724×10+02.7795×10+02.7619×10+0
    WFG255.4316×10−15.4029×10−15.4002×10−15.4174×10−15.4169×10−1
    101.1086×10+01.0997×10+01.0051×10+01.1036×10+01.1097×10+0
    151.8675×10+01.7649×10+01.7069×10+01.8007×10+01.7952×10+0
    252.9598×10+02.9413×10+02.8039×10+02.8796×10+02.8976×10+0
    WFG451.1192×10+01.1128×10+01.1065×10+01.1097×10+01.1106×10+0
    104.1062×10+04.1057×10+04.0388×10+04.0263×10+04.0589×10+0
    158.9645×10+08.8089×10+08.3986×10+08.4108×10+08.4004×10+0
    251.9058×10+11.8769×10+11.6506×10+11.5375×10+11.6405×10+1
    下载: 导出CSV
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  • 收稿日期:  2020-11-23
  • 录用日期:  2021-03-19
  • 网络出版日期:  2022-05-12
  • 刊出日期:  2022-06-02

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