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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

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): 1−21 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): 1−21 doi: 10.16383/j.aas.c200975
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): 1−21 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): 1−21 doi: 10.16383/j.aas.c200975

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
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

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 research interest covers 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: 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.
  • 图  1  二目标下的无贡献解

    Fig.  1  The noncontributed solutions under bi-objective

    图  2  基于角度距离选择参考点

    Fig.  2  Select reference points based on angular distance

    图  3  自适应参考点过程

    Fig.  3  Adaptive reference point process

    图  4  确定轮廓曲线以及曲线参数

    Fig.  4  Determine the contour curve and curve parameter

    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

    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

    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

    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

    表  1  测试问题

    Table  1  Test questions

    问题目标数目 (M)决策变量 (D)PF
    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+15.2877×10+1WFG152.1497×10+21.2453×10+2
    103.2368×10+31.4929×10+3103.0802×10+31.8838×10+3
    151.5646×10+2+4.5986×10+2153.3103×10+22.9192×10+2
    252.7908×10+31.8621×10+3252.4172×10+3+2.8807×10+3
    DTLZ251.4045×10+2+3.1151×10+2WFG252.5691×10+21.9505×10+2
    103.4692×10+32.0823×10+3103.6336×10+33.2122×10+3
    151.9603×10+2+4.9719×10+2154.3556×10+23.8969×10+2
    253.1545×10+3+3.4599×10+3253.4466×10+3=3.3881×10+3
    DTLZ357.6826×10+13.3370×10+1WFG353.4018×10+23.2435×10+2
    103.2959×10+31.5038×10+3104.3267×10+34.1139×10+3
    154.9418×10+21.5822×10+2155.1973×10+24.3731×10+2
    252.9858×10+31.7848×10+3253.8047×10+3=3.7357×10+3
    DTLZ451.4659×10+2+3.2766×10+2WFG453.3620×10+22.8286×10+2
    103.8768×10+32.1238×10+3104.2819×10+33.6326×10+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+33.3598×10+3
    DTLZ552.2500×10+21.3912×10+2WFG553.3378×10+22.9287×10+2
    103.0904×10+31.9358×10+3104.2697×10+33.7511×10+3
    151.8227×10+2+4.5443×10+2155.2633×10+24.0438×10+2
    253.0757×10+31.9646×10+3253.8177×10+33.4843×10+3
    DTLZ651.9591×10+21.2208×10+2WFG652.7412×10+22.5095×10+2
    103.5852×10+31.8312×10+3104.0843×10+33.2581×10+3
    151.5310×10+2+4.9933×10+2153.1397×10+2+5.1200×10+2
    252.6990×10+32.0808×10+3253.7598×10+32.8335×10+3
    DTLZ752.0799×10+21.4207×10+2WFG753.7091×10+23.1761×10+2
    103.8714×10+32.3062×10+3104.3950×10+33.7082×10+3
    154.8328×10+22.0122×10+2153.7417×10+2+5.2910×10+2
    252.4555×10+31.5033×10+3253.8242×10+33.2364×10+3
    IDTLZ152.1931×10+29.4000×10+1WFG852.2219×10+2=2.1770×10+2
    103.1281×10+31.9947×10+3103.7129×10+32.6444×10+3
    153.4287×10+22.5752×10+2152.4552×10+2+4.5493×10+2
    252.0253×10+31.2549×10+3253.7527×10+32.5151×10+3
    IDTLZ253.1594×10+2+3.3723×10+2WFG953.8469×10+23.2047×10+2
    104.0353×10+33.4738×10+3104.3729×10+33.9448×10+3
    154.4610×10+24.2203×10+2155.3114×10+24.3118×10+2
    252.5265×10+31.7758×10+3253.8142×10+33.5823×10+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
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