2.845

2023影响因子

(CJCR)

  • 中文核心
  • EI
  • 中国科技核心
  • Scopus
  • CSCD
  • 英国科学文摘

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

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.
    • HTML全文

  • 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
    • 参考文献(40)
  • [1] Luo J P, Liu Q Q, Yang Y, Li X, Chen M R, Cao W M. An artificial bee colony algorithm for multi-objective optimisation. Applied Soft Computing, 2017, 50: 235-251 doi: 10.1016/j.asoc.2016.11.014
    [2] Zhou Y L, Wang J H, Chen J, Gao S C, Teng L Y. Ensemble of many-objective evolutionary algorithms for many-objective problems. Soft Computing, 2017, 21(9): 2407-2419 doi: 10.1007/s00500-015-1955-3
    [3] Jain H, Deb K. An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: Handling constraints and extending to an adaptive approach. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 602-622 doi: 10.1109/TEVC.2013.2281534
    [4] Cinalli D, Martí L, Sanchez-Pi N, Garcia A C B. Collective intelligence approaches in interactive evolutionary multi-objective optimization. Logic Journal of the IGPL, 2020, 28(1): 95-108 doi: 10.1093/jigpal/jzz074
    [5] Kudikala R, Mills A R, Fleming P J, Tanner G F, Holt J E. Aero engine health management system architecture design using multi-criteria optimization. In: Proceedings of the 15th Annual Conference Companion on Genetic and Evolutionary Computation. Amsterdam, The Netherlands: ACM, 2013. 185−186
    [6] Saeidi A M, Hage J, Khadka R, Jansen S. A search-based approach to multi-view clustering of software systems. In: Proceedings of the 22nd IEEE International Conference on Software Analysis, Evolution, and Reengineering. Montreal, Canada: IEEE, 2015. 429−438
    [7] 陈振兴, 严宣辉, 吴坤安, 白猛. 融合张角拥挤控制策略的高维多目标优化. 自动化学报, 2015, 41(6): 1145-1158

    Chen Zhen-Xing, Yan Xuan-Hui, Wu Kun-An, Bai Meng. Many-objective optimization integrating open angle based congestion control strategy. Acta Automatica Sinica, 2015, 41(6): 1145-1158
    [8] Liu Y, Zhu N B, Li K L, Li M Q, Zheng J H, Li K Q. An angle dominance criterion for evolutionary many-objective optimization. Information Sciences, 2020, 509: 376-399 doi: 10.1016/j.ins.2018.12.078
    [9] Yang S X, Li M Q, Liu X H, Zheng J H. A grid-based evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2013, 17(5): 721-736 doi: 10.1109/TEVC.2012.2227145
    [10] 余伟伟, 谢承旺, 闭应洲, 夏学文, 李雄, 任柯燕, 等. 一种基于自适应模糊支配的高维多目标粒子群算法. 自动化学报, 2018, 44(12): 2278-2289

    Yu Wei-Wei, Xie Cheng-Wang, Bi Ying-Zhou, Xia Xue-Wen, Li Xiong, Ren Ke-Yan, et al. Many-objective particle swarm optimization based on adaptive fuzzy dominance. Acta Automatica Sinica, 2018, 44(12): 2278-2289
    [11] Elarbi M, Bechikh S, Gupta A, Said L B, Ong Y S. A new decomposition-based NSGA-II for many-objective optimization. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2018, 48(7): 1191-1210 doi: 10.1109/TSMC.2017.2654301
    [12] Zhang Q F, Li H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 712-731 doi: 10.1109/TEVC.2007.892759
    [13] Wang L P, Xu M N, Yu W, Qiu Q C, Wu F. Decomposition multi-objective evolutionary algorithm based on adaptive neighborhood adjustment strategy. IEEE Access, 2020, 8: 78639-78651. doi: 10.1109/ACCESS.2020.2990193
    [14] Cheng R, Jin Y C, Olhofer M, Sendhoff B. A reference vector guided evolutionary algorithm for many-objective optimization. IEEE Transactions on Evolutionary Computation, 2016, 20(5): 773-791 doi: 10.1109/TEVC.2016.2519378
    [15] Lu X, Tan Y Y, Zheng W, Meng L L. A decomposition method based on random objective division for MOEA/D in many-objective optimization. IEEE Access, 2020, 8: 103550-103564 doi: 10.1109/ACCESS.2020.2999417
    [16] Deb K, Jain H. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: Solving problems with box constraints. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 577-601 doi: 10.1109/TEVC.2013.2281535
    [17] 封文清, 巩敦卫. 基于在线感知Pareto前沿划分目标空间的多目标进化优化. 自动化学报, 2020, 46(8): 1628-1643

    Feng Wen-Qing, Gong Dun-Wei. Multi-objective evolutionary optimization with objective space partition based on online perception of Pareto front. Acta Automatica Sinica, 2020, 46(8): 1628-1643
    [18] Beume N, Naujoks B, Emmerich M. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research, 2007, 181(3): 1653-1669 doi: 10.1016/j.ejor.2006.08.008
    [19] Gómez R H, Coello C A C. Improved metaheuristic based on the R2 indicator for many-objective optimization. In: Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation. Madrid, Spain: ACM, 2015. 679−686
    [20] Liang Z P, Luo T T, Hu K F, Ma X L, Zhu Z X. An indicator-based many-objective evolutionary algorithm with boundary protection. IEEE Transactions on Cybernetics, 2021, 51(9): 4553-4566 doi: 10.1109/TCYB.2019.2960302
    [21] Li F, Li T J, Zhang S N. R2 indicator and objective space partition based evolutionary algorithm for many-objective optimization. In: Proceedings of the 2019 IEEE Symposium Series on Computational Intelligence. Xiamen, China: IEEE, 2019. 1271−1278
    [22] Sun Y A, Yen G G, Yi Z. IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Transactions on Evolutionary Computation, 2019, 23(2): 173-187 doi: 10.1109/TEVC.2018.2791283
    [23] Hua Y C, Liu Q Q, Hao K R, Jin Y C. A survey of evolutionary algorithms for multi-objective optimization problems with irregular Pareto fronts. IEEE/CAA Journal of Automatica Sinica, 2021, 8(2): 303-318 doi: 10.1109/JAS.2021.1003817
    [24] Ma X L, Yu Y A, Li X D, Qi Y T. A survey of weight vector adjustment methods for decomposition-based multiobjective evolutionary algorithms. IEEE Transactions on Evolutionary Computation, 2020, 24(4): 634-649 doi: 10.1109/TEVC.2020.2978158
    [25] Yuan J W, Liu H L, Gu F Q, Zhang Q F, He Z S. Investigating the properties of indicators and an evolutionary many-objective algorithm using promising regions. IEEE Transactions on Evolutionary Computation, 2021, 25(1): 75-86 doi: 10.1109/TEVC.2020.2999100
    [26] Liu S B, Lin Q Z, Tan K C, Gong M G, Coello C A C. A fuzzy decomposition-based multi/many-objective evolutionary algorithm. IEEE Transactions on Cybernetics, to be published
    [27] Van D A, Gary V, Lamont B. Multiobjective evolutionary algorithm research: A history and analysis. EvolutionaryComputation, 1998, 8(2): 125−147
    [28] Zhou A M, Jin Y C, Zhang Q F, Sendhoff B, Tsang E. Combining model-based and genetics-based offspring generation for multi-objective optimization using a convergence criterion. In: Proceedings of the 2006 IEEE International Conference on Evolutionary Computation. Vancouver, Canada: IEEE, 2006. 892−899
    [29] While L, Hingston P, Barone L, Huband S. A faster algorithm for calculating hypervolume. IEEE Transactions on Evolutionary Computation, 2006, 10(1): 29-38 doi: 10.1109/TEVC.2005.851275
    [30] Liu C, Zhao Q, Yan B, Gao Y. A new hypervolume-based differential evolution algorithm for many-objective optimization. RAIRO-Operations Research, 2017, 51(4): 1301-1315 doi: 10.1051/ro/2017014
    [31] Ishibuchi H, Imada R, Masuyama N, Nojima Y. Dynamic specification of a reference point for hypervolume calculation in SMS-EMOA. In: Proceedings of the 2018 IEEE Congress on Evolutionary Computation. Rio de Janeiro, Brazil: IEEE, 2018. 1−8
    [32] Brockhoff D, Wagner T, Trautmann H. On the properties of the R2 indicator. In: Proceedings of the 14th Annual Conference on Genetic and Evolutionary Computation. Philadelphia, USA: ACM, 2012. 465−472
    [33] Li F, Liu J C, Song Y X, Shang L L. An adaptive evolutionary multi-objective algorithm based on R2 indicator. In: Proceedings of the 2019 Chinese Control and Decision Conference. Nanchang, China: IEEE, 2019. 692−697
    [34] Wang H D, Jin Y C, Yao X. Diversity assessment in many-objective optimization. IEEE Transactions on Cybernetics, 2017, 47(6): 1510-1522 doi: 10.1109/TCYB.2016.2550502
    [35] Tian Y, Zhang X Y, Cheng R, Jin Y C. A multi-objective evolutionary algorithm based on an enhanced inverted generational distance metric. In: Proceedings of the 2016 IEEE Congress on Evolutionary Computation. Vancouver, Canada: IEEE, 2016. 5222−5229
    [36] Tian Y, Cheng R, Zhang X Y, Cheng F, Jin Y C. An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Transactions on Evolutionary Computation, 2018, 22(4): 609-622 doi: 10.1109/TEVC.2017.2749619
    [37] Deb K, Agrawal R B. Simulated binary crossover for continuous search space. Complex Systems, 1995, 9(3): 115-148
    [38] Deb K, Goyal M. A combined genetic adaptive search (GeneAS) for engineering design. Computer Science and Informatics, 1996, 26(4): 30-45
    [39] Das I, Dennis J E. Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. Siam Journal on Optimization, 1998, 8(3): 631-657 doi: 10.1137/S1052623496307510
    [40] Li M Q, Zhen L L, Yao X. How to read many-objective solution sets in parallel coordinates. IEEE Computational Intelligence Magazine, 2017, 12(4): 88-100 doi: 10.1109/MCI.2017.2742869
    • 相关文章
    • 施引文献
  • 资源附件(0)
  • 加载中
图(8) / 表(10)
计量
  • 文章访问数:  832
  • HTML全文浏览量:  174
  • PDF下载量:  153
  • 被引次数: 0
出版历程
  • 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

目录

    /

    返回文章
    返回

    版权所有 © 《自动化学报》编辑部  京ICP备14019135号-6

    地址：北京中关村东路95号邮政编码：100190E-mail：aas_editor@ia.ac.cn

    电话：010-82544677 (日常咨询和稿件处理)，010-82544653(费用管理、寄刊)

    本系统由 北京仁和汇智信息技术有限公司 开发 技术支持： info@rhhz.net   百度统计