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异构集成代理辅助的区间多模态粒子群优化算法

季新芳 张勇 巩敦卫 郭一楠 孙晓燕

季新芳, 张勇, 巩敦卫, 郭一楠, 孙晓燕. 异构集成代理辅助的区间多模态粒子群优化算法. 自动化学报, 2021, 45(x): 1−23 doi: 10.16383/j.aas.c210223
引用本文: 季新芳, 张勇, 巩敦卫, 郭一楠, 孙晓燕. 异构集成代理辅助的区间多模态粒子群优化算法. 自动化学报, 2021, 45(x): 1−23 doi: 10.16383/j.aas.c210223
Ji Xin-Fang, Zhang Yong, Gong Dun-Wei, Guo Yi-Nan, Sun Xiao-Yan. Interval multimodal particle swarm optimization algorithm assisted by heterogeneous ensemble surrogate. Acta Automatica Sinica, 2021, 45(x): 1−23 doi: 10.16383/j.aas.c210223
Citation: Ji Xin-Fang, Zhang Yong, Gong Dun-Wei, Guo Yi-Nan, Sun Xiao-Yan. Interval multimodal particle swarm optimization algorithm assisted by heterogeneous ensemble surrogate. Acta Automatica Sinica, 2021, 45(x): 1−23 doi: 10.16383/j.aas.c210223

异构集成代理辅助的区间多模态粒子群优化算法

doi: 10.16383/j.aas.c210223
基金项目: 国家重点研发计划资助(2020YFB1708200), 徐州市重点研发计划(KC20184), 绿色建筑节能智能优化设计方法及系统开发资助
详细信息
    作者简介:

    季新芳:中国矿业大学信息与控制工程学院博士研究生. 2013年在中国矿业大学获硕士学位. 主要研究方向为代理辅助进化优化, 多模态优化. E-mail: mimosa_615615@126.com

    张勇:中国矿业大学信息与控制工程学院教授. 2009年获中国矿业大学控制理论与控制工程专业博士学位. 主要研究方向为智能优化, 数据挖掘. 本文通信作者. E-mail: yongzh401@126.com

    巩敦卫:中国矿业大学信息与控制工程学院教授. 1999年在中国矿业大学获博士学位. 主要研究方向为进化计算与应用. 本文通信作者. E-mail: dwgong@vip.163.com

    郭一楠:中国矿业大学信息与电气工程学院教授, 主要研究方向为智能优化算法与控制, 数据挖掘. E-mail: nanly@126.com

    孙晓燕:中国矿业大学信息与电气工程学院教授, 2009年在中国矿业大学获博士学位. 主要研究方向为进化计算, 机器学习. E-mail: xysun78@126.com

Interval Multimodal Particle Swarm Optimization Algorithm Assisted by Heterogeneous Ensemble Surrogate

Funds: Supported by the National Key Research and Development Program of P.R.China (2020YFB1708200), the Key Research and Development Program in Xuzhou (KC20184), Green Building Energy Conservation Intelligent Optimization Design Method and System Development
More Information
    Author Bio:

    JI Xin-Fang Ph. D. candidate at School of Information and Control Engineering, China University of Mining and Technology. She received her master degree from China University of Mining and Technology in 2013. Her research interest covers surrogate-assisted evolutionary optimization, multimodal optimization

    ZHANG Yong Professor at School of Information and Control Engineering, China University of Mining and Technology. He received his Ph.D. degree in control theory and control engineering from China University of Mining and Technology in 2009. His research interest covers intelligence optimization, data mining. Corresponding author of this paper

    GONG Dun-Wei Professor at School of Information and Control Engineering, China University of Mining and Technology. He received his Ph. D. degree from China University of Mining and Technology in 1999. His research interest covers evolutionary computation and its applications. Corresponding author of this paper

    GUO Yi-Nan Professor at School of Information and Control Engineering, China University of Mining and Technology. Her research interest covers intelligence optimization, control and data mining

    SUN Xiao-Yan Professor at School of Information and Control Engineering, China University of Mining and Technology. She received her Ph.D. degree in control theory and control engineering from China University of Mining and Technology in 2009. Her research interest covers evolutionary computation, machine learning

  • 摘要: 现实生活中的很多黑盒优化问题可归为高计算代价的多模态优化问题, 即昂贵多模态优化问题. 在处理该类问题时, 决策者希望以尽量少的计算代价(即尽量少的真实函数评价次数)找到多个高质量的最优解. 然而, 已有代理辅助的进化优化算法很少考虑问题的多模态属性, 运行一次仅可获得问题的一个最优解. 鉴于此, 研究一种异构集成代理辅助的区间多模态粒子群优化算法. 首先, 借助异构集成的思想构建一个由多个基础代理模型组成的模型池; 随后, 依据待评价粒子与已发现模态之间的匹配关系, 从模型池中自主选择部分基础代理模型进行集成, 并使用集成后的代理模型预测该粒子的适应值. 进一步, 为节约代理模型管理的代价, 设计一种增量式的代理模型管理策略; 为减少代理模型预测误差对算法性能的影响, 首次将区间排序关系引入到进化过程中. 将所提算法与当前流行的5种代理辅助进化优化算法和7 种经典的多模态优化算法进行对比, 在20个测试函数和1个建筑节能实际问题上的结果表明, 所提算法可以在较少计算代价下获得问题的多个高竞争最优解.
    1)  收稿日期 2021-03-19 录用日期 2021-09-05 Manuscript received March 19, 2021; accepted September 6, 2021 国家重点研发计划 (2020YFB1708200), 徐州市重点研发计划 (KC20184), 绿色建筑节能智能优化设计方法及系统开发资助 Supported by the National Key Research and Development Program of P. R. China (2020YFB1708200), the Key Research and Development Program in Xuzhou (KC20184), Green Building Energy Conservation Intelligent Optimization Design Method and
    2)  System Development 本文责任编委 袁勇 Recommended by Associate Editor YUAN Yong 1. 中国矿业大学信息与控制工程学院 徐州 221116  2. 中国矿业大学机械电子与信息工程学院 (北京) 北京 100083  1. School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116 2. School of Mechanical Electronic and Information Engineering, China University of Mining and Technology (Beijing), Beijing 100083
  • 图  1  IMPSO-HES的框架图

    Fig.  1  General framework of IMPSO-HES

    图  2  精确和区间评价策略下IMPSO-HES所得GS

    Fig.  2  GS values obtained by IMPSO-HES under precise and interval evaluation

    图  3  精确和区间评价策略下IMPSO-HES所得VR

    Fig.  3  VR values obtained by IMPSO-HES under precise and interval evaluation

    图  4  IMPSO-HES/D和IMPSO-HES得到的GS

    Fig.  4  GS values obtained by IMPSO-HES/D and IMPSO-HES

    图  5  IMPSO-HES/D和IMPSO-HES得到的VR

    Fig.  5  VR values obtained by IMPSO-HES/D and IMPSO-HES

    图  6  IMPSO-HES与5种SAEAs的运行耗时

    Fig.  6  Running times of IMPSO-HES and the 5 SAEAs

    图  7  单居室居住建筑的外形图

    Fig.  7  Outline of the single-room building

    表  1  基准测试问题

    Table  1  Benchmark problems

    测试函数维数变量空间全局/局部解个数全局最优解的目标值
    F1Ellipsoid10/20$\boldsymbol{X} \in [-1,1]^{D}$1/00
    F2Ackley10/20$\boldsymbol{X} \in [-30,30]^{D}$1/many0
    F3Rastrigin10/20$\boldsymbol{X }\in [-5.12,5.12]^{D}$1/many0
    F4Rosenbrock10/20$\boldsymbol{X} \in [-2.048,2.048]^{D}$1/many0
    F5Griewank10/20$\boldsymbol{X} \in [-600,600]^{D}$1/many0
    F6Reverse five-uneven-peak trap1$\boldsymbol{X} \in [0,30] $2/3−200
    F7Reverse equal maxima1$\boldsymbol{X} \in [0,1] $5/0−1
    F8Reverse uneven decreasing maxima1$\boldsymbol{X} \in [0,1] $1/4−1
    F9Reverse himmelblau2$\boldsymbol{X} \in [-6,6]^{D}$4/0−200
    F10Six-hump camel2$x_1\in[-1.9,1.9], x_2\in[-1.1,1.1] $2/2−1.0316
    F11Reverse shubert2$\boldsymbol{X} \in [-10,10]^{D}$18/many−186.73
    F12Reverse vincent2$\boldsymbol{X} \in [0.25,10]^{D}$36/0−1
    F13Reverse modified rastrigin2$\boldsymbol{X} \in [0,1]^{D}$12/02
    F14Reverse CF12$\boldsymbol{X}\in [-5,5]^D$6/00
    F15Reverse CF22$\boldsymbol{X}\in [-5,5]^D$8/00
    F16Reverse CF32$\boldsymbol{X} \in[-5,5]^D $6/00
    F17Reverse CF43$\boldsymbol{X}\in [-5,5]^D$8/00
    F18UrsemF4 back2$\boldsymbol{X }\in [-2,2]^{D}$2/0−0.2679
    F19Branin RCOS2$x_1\in[-5,10], x_2\in[0,15] $3/00.3978
    F20Waves2$x_1\in[-0.9,1.2], x_2\in[-1.2,1.2]$1/9−7.776
    下载: 导出CSV

    表  2  F6-F20的幅值精度和距离精度

    Table  2  Amplitude accuracy and distance accuracy for F6-F20

    F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20
    $R_{v}$ 1 0.05 0.1 0.5 0.05 10 0.1 0.5 1 1 1 1 0.1 0.1 0.5
    $R_{d}$ 1 0.05 0.5 0.5 0.2 2 0.5 0.5 1 1 1 1 0.5 1 0.2
    下载: 导出CSV

    表  3  不同$g_{{\rm{max}}}$取值下IMPSO-HES所得的性能指标值

    Table  3  Performance values obtained by IMPSO-HES under different $g_{max}$ values

    $g_{{\rm{max}}}$GS均值(标准差)VR均值耗时/s
    F5(D= 10)33.8007(3.5E+00)+64
    61.1745(3.7E−02)85
    91.1083(2.5E−02) = 116
    F5(D= 20)38.1980(9.8E+00)+776
    61.0757(1.6E−02)1400
    90.8079(2.8E−01)−2045
    F93−199.93(3.1E−03) =0.6811
    6−199.99(1.0E−04)0.7019
    9−200.00(1.4E−03) = 0.6336
    F103−1.0316(1.7E−06) =1.0019
    6−1.0316(9.8E−07)1.0028
    9−1.0316(9.8E−07) = 1.0038
    F123−0.9990(7.1E−06) =0.1310
    6−0.9999(1.0E−06)0.1314
    9−0.9999(2.2E−06) =0.1125
    下载: 导出CSV

    表  4  不同Q取值下IMPSO-HES所得的性能指标值

    Table  4  Performance values obtained by IMPSO-HES under different Q values

    QGS均值(标准差)VR均值耗时/s
    F5(D= 10)1/5K1.6581(2.2E−01)+64
    1/4K1.1745(3.7E−02)85
    1/3K1.3821(1.5E−01)+108
    1/2K1.2696(5.1E−02)+160
    F5(D= 20)1/5K1.9800(1.0E+00)+1137
    1/4K1.0757(1.6E−02)1400
    1/3K1.8321(1.1E+00)+1920
    1/2K1.8352(1.7E+00)+2700
    F91/5K−199.98(7.2E−04) =0.5317
    1/4K−199.99(1.0E−04)0.7019
    1/3K−199.98(4.6E−04) =0.5524
    1/2K−199.14(6.8E+00)+0.3334
    F101/5K−1.0316(1.1E−09) =1.0028
    1/4K−1.0316(9.8E−07)1.0028
    1/3K−1.0316(9.8E−07) =1.0030
    1/2K−1.0300(1.4E−03)+0.8548
    F121/5K−0.9991(2.3E−06)+0.1212
    1/4K−0.9999(1.0E−06)0.1314
    1/3K−0.9996(8.5E−07)+0.1018
    1/2K−0.9949(9.2E−05)+0.1024
    下载: 导出CSV

    表  5  异构集成与同质集成下IMPSO-HES所得结果

    Table  5  Performance values obtained by IMPSO-HES under heterogeneous and homogeneous ensemble

    算法GS均值(标准差)VR均值耗时/s
    F5(D= 10)IMPSO-PR1.6310(7.1E−01)+86
    IMPSO-RBFN45.272(8.9E+02)+39
    IMPSO-HES1.1745(3.7E−02)85
    F5(D= 20)IMPSO-PR2.0037(2.9E+00)+1478
    IMPSO-RBFN116.78(9.5E+02)+180
    IMPSO-HES1.0757(1.6E−02)1400
    F9IMPSO-PR−196.81(9.5E+00)+0.0516
    IMPSO-RBFN−199.99(4.7E−07) =0.6522
    IMPSO-HES−199.99(1.0E−04)0.7019
    F10IMPSO-PR−0.9620(2.5E−03)+0.217
    IMPSO-RBFN−1.0316(9.8E−09) =1.0020
    IMPSO-HES−1.0316(9.8E−07)1.0028
    F12IMPSO-PR−0.9886(1.5E−04)+0.0611
    IMPSO-RBFN−0.9995(9.4E−07)+0.0919
    IMPSO-HES−0.9999(1.0E−06)0.1314
    下载: 导出CSV

    表  6  不同更新概率$p_{m}$下IMPSO-HES所得结果

    Table  6  Performance values obtained by IMPSO-HES under different $p_{m}$ values

    $p_{m}$GS均值(标准差)VR均值耗时/s
    F5(D= 10)固定1.4393(3.8E−01)+84
    自适应1.1745(3.7E−02)85
    F5(D= 20)固定1.7503(1.7E+00)+1313
    自适应1.0757(1.6E−02)1400
    F9固定−199.91(2.6E−02)+0.4019
    自适应−199.99(1.0E−04)0.7019
    F10固定−1.0316(4.7E−08) =1.0026
    自适应−1.0316(9.8E−07)1.0028
    F12固定−0.9969 (4.8E−05)+0.1214
    自适应−0.9999(1.0E−06)0.1314
    下载: 导出CSV

    表  7  使用All-S和Mod-S时IMPSO-HES所得结果

    Table  7  Performance values obtained by IMPSO-HES with All-S and Mod-S

    集成策略GS均值(标准差)VR均值耗时/s
    F5(D= 10)All-S3.8785(3.8E+00)+243
    Mod-S1.1745(3.7E−02)85
    F5(D = 20)All-S8.8387(8.1E+00)+3362
    Mod-S1.0757(1.6E−02)1400
    F9All-S−187.33(2.0E+2)+0.0580
    Mod-S−199.99(1.0E−04)0.7019
    F10All-S−0.9751(1.4E−02)+0.7057
    Mod-S−1.0316(9.8E−07)1.0028
    F12All-S−0.9737(1.9E−02)+0.0842
    Mod-S−0.9999(1.0E−06)0.1314
    下载: 导出CSV

    表  8  不同模型更新策略下IMPSO-HES所得结果

    Table  8  Performance values obtained by IMPSO-HES under different model update strategies

    更新策略GS均值(标准差)VR均值耗时/s
    F5(D= 10)All-up1.5009(3.9E−02)+97
    Inc-up1.1745(3.7E−02)85
    F5(D= 20)All-up32.184(2.4E+04)+1509
    Inc-up1.0757(1.6E−02)1400
    F9All-up−200.00(3.6E-10) = 0.6330
    Inc-up−199.99(1.0E−04)0.7019
    F10All-up−1.0316(1.2E−04) =0.9530
    Inc-up−1.0316(9.8E−07)1.0028
    F12All-up−0.9998(2.7E−07) =0.1116
    Inc-up−0.9999(1.0E−06)0.1314
    下载: 导出CSV

    表  9  IMPSO-HES与5种SAEAs所得GS值[均值(方差)]

    Table  9  GS value obtained by IMPSO-HES and 5 SAEAs [Mean (Variance)]

    D IMPSO-HES SA-COSO CAL-SAPSO Gr-based SAPSO PESPSO ESPSO
    F1 10 3.6600 3.1600− 0.1153− 0.1476− 0.2962− 0.6645−
    (4.2E+00) (6.5E−02) (4.9E−02) (1.1E−03) (1.3E−03) (5.0E−02)
    20 21.398 11.017− 0.2292− 0.0279− 1.3770− 1.8664−
    (6.1E+01) (1.2E+01) (1.9E−02) (8.2E−06) (1.2E−01) (2.4E−01)
    F2 10 17.990 17.248= 18.606+ 15.910− 11.820− 13.786−
    (1.1E+00) (4.1E−02) (4.8E−01) (6.4E−01) (4.3E+00) (2.0E+00)
    20 18.866 18.025− 18.421= 14.717− 12.584− 15.958−
    (9.0E−01) (4.4E−01) (2.4E+00) (1.1E+00) (2.3E+01) (1.6E+01)
    F3 10 78.266 97.683+ 79.727= 94.349+ 82.325= 89.952=
    (1.3E+02) (5.8E+02) (1.6E+03) (7.3E+01) (1.2E+02) (2.0E+02)
    20 173.97 177.43= 128.71− 168.14= 173.99= 175.65=
    (2.4E+02) (6.6E+02) (4.0E+03) (1.6E+02) (1.7E+02) (1.1E+02)
    F4 10 37.310 537.31+ 39.003= 173.66+ 90.531+ 66.581+
    (1.1E+02) (2.4E+04) (2.0E+02) (3.3E+02) (6.7E+02) (1.0E+02)
    20 41.469 891.97+ 42.758= 330.37+ 97.508+ 195.90+
    (5.7E+02) (1.7E+04) (2.0E+02) (3.9E+03) (6.8E+02) (1.9E+03)
    F5 10 1.1745 66.556+ 1.7364+ 1.3106+ 2.7987+ 2.3172+
    (3.7E−02) (1.8E+02) (1.4E−01) (1.7E−02) (2.4E+00) (3.9E−01)
    20 1.0757 43.897+ 2.2553+ 1.0572= 6.7018+ 10.373+
    (1.6E−02) (1.9E+02) (3.2E−01) (2.0E−05) (7.4E+00) (6.2E+00)
    F6 1 −199.15 −200.00− −200.00− −190.91+ −200.00− −200.00−
    (4.6E+00) (2.1E-10) (1.6E−09) (3.2E+01) (1.2E-13) (1.0E-11)
    F7 1 −0.9999 −1.00= −0.5052+ −0.9991+ −0.9999= −0.9998=
    (3.1E−06) (0.0E+00) (1.2E−01) (1.1E−07) (2.7E−05) (3.8E−06)
    F8 1 −0.9854 −0.9808= −0.5114+ −0.9447+ −0.9486+ −0.9486+
    (1.3E−05) (1.0E-10) (8.0E−02) (7.4E−04) (5.1E−04) (5.1E−04)
    F9 2 −199.99 −196.14+ −157.69+ −199.93+ −199.98= −199.74+
    (1.0E−04) (3.8E+01) (8.6E+02) (5.1E−04) (2.7E−04) (6.4E−03)
    F10 2 −1.0316 −0.9956+ −0.4646+ −1.0306+ −1.0303+ −1.0292+
    (9.8E−07) (1.6E−03) (1.3E−01) (1.9E−06) (1.7E−07) (5.3E−07)
    F11 2 −158.32 −89.368+ −52.464+ −113.85+ −130.53+ −94.463+
    (1.9E+03) (2.4E+03) (2.6E+03) (3.5E+04) (2.5E+03) (1.5E+03)
    F12 2 −0.9999 −0.9798+ −0.7194+ −0.9845+ −0.9954+ −0.9800+
    (1.0E−06) (5.6E−04) (9.0E−02) (1.9E−04) (2.0E−06) (5.5E−05)
    F13 2 2.2329 2.8903+ 7.8467+ 2.2985= 2.0228− 2.0609−
    (2.3E−01) (6.4E−02) (3.0E+01) (1.0E−01) (4.6E−03) (3.1E−03)
    F14 2 0.0879 40.011+ 197.39+ 23.774+ 7.5884+ 9.9617+
    (5.0E−01) (2.6E+02) (9.2E+03) (6.3E+03) (1.1E+02) (3.0E+02)
    F15 2 36.423 89.091+ 183.14+) 80.557+ 26.116= 57.889+
    (3.7E+03) (2.7E+02) (3.6E+03) (1.1E+03) (7.6E+02) (2.8E+03)
    F16 2 0.2423 90.43+ 350.88+ 60.296+ 1.1621+ 18.280+
    (1.3E−01) (1.2E+04) (4.8E+04) (3.2E+03) (2.5E+00) (1.2E+03)
    F17 3 32.566 88.27+ 173.56+ 57.380+ 26.079= 37.233=
    (2.0E+04) (5.3E+02) (2.6E+04) (2.1E+03) (6.2E+02) (6.0E+02)
    F18 2 −0.2679 −0.2457+ −0.1304+ −0.2671+ −0.2678= −0.2678=
    (1.6E−06) (3.6E−04) (5.6E−03) (6.8E−08) (1.6E−06) (5.4E−09)
    F19 2 0.3999 1.1488+ 2.2603+ 0.4259+ 0.4249+ 0.5136+
    (2.4E−05) (8.6E−01) (6.2E+00) (1.3E−03) (1.2E−03) (5.3E−02)
    F20 2 −7.4299 −7.776− −7.7753− −6.3408+ −7.2943+ −7.4511=
    (1.7E−02) (0.0E+00) (4.2E−06) (8.4E−01) (2.2E−01) (2.7E−01)
    下载: 导出CSV

    表  10  基于表9的统计结果

    Table  10  Statistical results based on table 9

    IMPSO-HES SA-COSO CAL-SAPSO Gr-based SAPSO PESPSO ESPSO
    F2-F5 好/平/差 5/2/1 3/4/1 4/2/2 4/2/2 4/2/2
    Rank 2.5000 5.5000 3.0000 3.1250 3.1250 3.7500
    Adjusted p-value 0.0066 0.6892 0.6892 0.6892 0.3938
    F6-F20 好/平/差 11/2/2 13/0/2 14/1/0 8/5/2 9/4/2
    Rank 1.8333 4.1666 5.4333 4.0000 2.2666 3.3000
    Adjusted p-value 0.0016 0.0000 0.0025 0.5258 0.0395
    下载: 导出CSV

    表  11  处理F1-F5时IMPSO-HES与7种多模态进化算法所得GS值[均值(方差)]

    Table  11  GS values obtained by IMPSO-HES and the 7 multimodal EAs on F1-F5 [Mean (Variance)]

    D IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE
    F1 10 3.6600 3.3110- 5.0580+ 5.9282+ 4.3713+ 5.7227+ 5.8277+ 5.2888+
    (4.2E+00) (7.8E-01) (1.3E+00) (2.3E+00) (1.2E+00) (6.4E+00) (1.6E+00) (2.6E+00)
    20 21.398 19.528= 26.709+ 31.059+ 18.792- 28.868+ 29.060+ 32.311+
    (6.1E+01) (9.8E+00) (2.2E+01) (2.2E+01) (1.2E+01) (5.8E+01) (1.5E+01) (5.5E+01)
    F2 10 17.990 18.046= 18.022= 19.159+ 18.073= 19.411+ 19.432+ 19.523+
    (1.1E+00) (8.1E-01) (7.0E-01) (3.9E-01) (1.06E+00) (1.3E+00) (3.0E-01) (1.5E-01)
    20 18.866 18.924= 18.922= 19.663+ 19.313+ 19.895+ 20.108+ 19.950+
    (9.0E-01) (3.6E+01) (1.7E-01) (6.5E-02) (2.5E-01) (9.9E-02) (4.9E-02) (8.2E-06)
    F3 10 78.266 95.069+ 89.325= 108.58+ 100.83+ 110.95+ 101.33+ 106.90+
    (1.3E+02) (6.3E+01) (1.2E+02) (2.2E+02) (8.2E+01) (5.5E+02) (1.3E+02) (1.3E+02)
    20 173.97 212.48+ 207.09+ 258.90+ 225.25+ 251.77+ 262.26+ 268.57+
    (2.4E+02) (2.6E+02) (2.8E+02) (3.3E+02) (5.1E+02) (3.2E+02) (6.5E+02) (1.1E+02)
    F4 10 37.310 343.96+ 257.96+ 670.32+ 451.41+ 812.90+ 982.18+ 523.1+
    (1.1E+02) (4.2E+05) (3.6E+05) (1.3E+05) (2.8E+04) (1.0E+05) (1.1E+05) (2.7E+05)
    20 41.469 1431.9+ 1399.6+ 2853.3+ 1722.6+ 3031.2+ 2737.0+ 2416.1+
    (5.7E+02) (1.1E+05) (1.5E+05) (3.6E+05) (5.1E+04) (6.9E+05) (7.1E+05) (1.6E+05)
    F5 10 1.1745 66.246+ 65.75+ 94.936+ 71.342+ 129.69+ 115.66+ 109.05+
    (3.7E-02) (3.1E+02) (6.7E+02) (4.7E+02) (4.8E+02) (3.5E+02) (8.8E+02) (6.6E+01)
    20 1.0757 160.00+ 156.27+ 305.74+ 194.22+ 298.18+ 300.28+ 300.13+
    (1.6E-02) (4.8E+02) (1.1E+03) (7.0E+02) (1.5E+03) (3.7E+03) (2.1E+03) (2.2E+03)
    下载: 导出CSV

    表  12  处理F6-F20时IMPSO-HES与7种多模态进化算法所得结果

    Table  12  Results of IMPSO-HES and the 7 multimodal EAs on F6-F20

    D IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE
    F6 GS均值 −199.15 −185.64+ −196.52+ −190.93+ −186.31+ −191.25+ −197.86+ −195.52+
    (标准差) (4.6E+00) (8.8E+01) (1.0E+02) (6.1E+01) (1.0E+02) (3.4E+02) (4.5E+01) (5.0E+02)
    VR均值 0.80 0.20+ 0.40+ 0.10+ 0.00+ 0.65+ 0.75= 0.40+
    F7 GS均值 −0.9999 −0.9994+ −0.9995+ −0.9991+ −0.9986+ −0.9987+ −0.9984+ −0.9980+
    (标准差) (3.1E−06) (7.3E−07) (2.5E−07) (7.2E−07) (1.0E−06) (8.5E−07) (5.6E−06) (4.6E−06)
    VR均值 0.78 0.78= 0.76= 0.70= 0.66+ 0.74= 0.78= 0.67+
    F8 GS均值 −0.9854 −0.9693+ −0.9937- −0.9931- −0.9758+ −0.9660+ −0.9483+ −0.9683+
    (标准差) (1.3E−04) (6.8E−04) (2.5E−04) (6.7E−05) (4.1E−04) (8.9E−04) (5.1E−03) (3.1E−03)
    VR均值 1.00 0.80+ 0.90+ 1.00= 1.00= 0.90+ 0.60+ 0.80+
    F9 GS均值 −199.99 −197.58+ −197.79+ −196.99+ −196.92+ −197.04+ −196.10+ −197.22+
    (标准差) (1.0E−04) (1.7E+00) (9.9E+00) (1.3E+01) (8.6E+00) (5.2E+00) (1.6E+01) (1.3E+01)
    VR均值 0.70 0.02+ 0.05+ 0.07+ 0.07+ 0.10+ 0.05+ 0.05+
    F10 GS均值 −1.0316 −1.0047+ −1.0016+ −1.0032+ −0.9949+ −0.9878+ −0.9730+ −1.0020+
    (标准差) (9.8E−07) (3.6E−04) (2.8E−03) (2.8E−03) (8.8E−04) (8.7E−03) (5.0E−03) (3.4E−02)
    VR均值 1.00 0.55+ 0.10+ 0.45+ 0.30+ 0.40+ 0.35+ 0.5+
    F11 GS均值 −158.32 −105.20+ −134.50= −90.154+ −114.099+ −123.777+ −111.92+ −132.37=
    (标准差) (1.9E+03) (1.3E+03) (1.7E+03) (5.4E+02) (1.3E+03) (1.0E+03) (2.3E+03) (1.6E+03)
    VR均值 0.02 0.01= 0.01= 0.00+ 0.00+ 0.00+ 0.00+ 0.01=
    F12 GS均值 −0.9999 −0.9733+ −0.9753+ −0.9727+ −0.9764+ −0.9764+ −0.9890+ −0.9887+
    (标准差) (1.0E−06) (3.2E−04) (4.9E−04) (4.6E−04) (5.8E−04) (5.2E−04) (3.0E−04) (4.6E−03)
    VR均值 0.13 0.08+ 0.05+ 0.07+ 0.07+ 0.08+ 0.10+ 0.09+
    F13 GS均值 2.2329 2.7146+ 2.5604+ 2.4384+ 2.5903+ 2.4817+ 2.3446= 2.5792+
    (标准差) (2.3E−01) (3.2E−01) (2.3E+00) (2.1E−01) (2.4E−01) (7.3E−01) (8.7E−01) (2.2E+00)
    VR均值 0.09 0.08= 0.08= 0.07= 0.08= 0.13+ 0.09= 0.08=
    F14 GS均值 0.0879 44.360+ 45.829+ 43.836+ 38.669+ 40.250+ 38.149+ 41.010+
    (标准差) (5.0E−01) (4.0E+03) (4.8E+03) (4.5E+03) (4.5E+03) (4.3E+03) (1.6E+03) (1.2E+02)
    VR均值 0.24 0.01+ 0.01+ 0.00+ 0.01+ 0.01+ 0.00+ 0.00+
    F15 GS均值 36.423 103.12+ 85.620+ 108.46+ 82.451+ 67.647+ 75.308+ 89.100+
    (标准差) (3.7E+03) (1.4E+03) (6.8E+03) (3.2E+03) (2.7E+03) (1.7E+03) (6.6E+03) (1.8E+03)
    VR均值 0.03 0.00+ 0.01= 0.00+ 0.01= 0.00+ 0.00+ 0.00+
    F16 GS均值 0.2423 74.272+ 52.296+ 132.800+ 52.555+ 81.104+ 114.04+ 67.231+
    (标准差) (1.3E−01) (8.2E+03) (8.1E+03) (6.6E+03) (3.2E+03) (9.0E+03) (1.6E+03) (1.6E+03)
    VR均值 0.15 0.00+ 0.02+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+
    F17 GS均值 32.566 127.50+ 141.05+ 165.93+ 148.05+ 192.72+ 162.20+ 100.12+
    (标准差) (2.0E+04) (2.3E+03) (2.5E+04) (5.7E+03) (2.0E+03) (8.5E+03) (5.2E+03) (3.2E+03)
    VR均值 0.13 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+ 0.00+
    F18 GS均值 −0.2679 −0.2642+ −0.2603+ −0.2579+ −0.2625+ −0.2548+ −0.2604+ −0.2630+
    (标准差) (1.6E−06) (6.0E−05) (6.8E−05) (9.1E−05) (4.9E−05) (1.6E−04) (1.8E−05) (1.5E−04)
    VR均值 1.00 1.00= 0.95= 0.95= 1.00= 0.80+ 0.80+ 0.85+
    F19 GS均值 0.3999 0.5292+ 0.8825+ 0.7979+ 0.7763+ 0.7895+ 1.3375+ 0.8858+
    (标准差) (2.4E−05) (1.9E−01) (1.8E−01) (1.8E−01) (1.8E−01) (1.7E+00) (8.6E−02) (2.0E−01)
    VR均值 0.60 0.03+ 0.03+ 0.10+ 0.06+ 0.16+ 0.06+ 0.2+
    F20 GS均值 −7.4299 −6.6192+ −6.6496+ −6.6644+ −6.7280+ −6.6791+ −6.4204+ −6.9818+
    (标准差) (1.7E−02) (2.9E−01) (8.6E−01) (4.0E−01) (3.3E−01) (4.5E−01) (3.8E−01) (4.6E−01)
    VR均值 0.40 0.26+ 0.26+ 0.26+ 0.20+ 0.29+ 0.27+ 0.28+
    下载: 导出CSV

    表  13  IMPSO-HES与7种多模态进化算法的统计对比结果

    Table  13  Statistical comparison results of IMPSO-HES and the 7 multimodal EAs

    IMPSO-HES LIPS EMO-MMO R3PSO FERPSO NCDE NSDE ANDE
    F1-F5 好/平/差 GS 6/3/1 7/3/0 10/0/0 8/1/1 10/0/0 10/0/0 10/0/0
    Rank 1.3000 2.8000 3.0000 6.1000 3.3000 6.3000 6.8000 6.7000
    Adjusted p-value 0.3153 0.1160 0.0002 0.0937 0.0001 0.0000 0.0000
    F6-F20 好/平/差 GS 15/0/0 13/1/1 14/0/1 15/0/0 15/0/0 14/1/0 14/1/0
    VR 11/4/0 10/5/0 11/4/0 11/4/0 14/1/0 12/3/0 13/2/0
    Rank 1.2586 4.8275 4.2689 5.5517 4.6034 6.1034 4.8620 4.2241
    Adjusted p-value 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
    下载: 导出CSV

    表  14  问题的决策变量信息

    Table  14  Decision variables

    决策变量单位范围
    房屋方向$\circ $[0,360)
    窗户的长m(0,3.6)
    窗户的高m(0,3.9)
    窗户的传热系数${\rm{w}}/({\rm{m}}^{2}{\rm{k}})$[2,6]
    窗户的日射热取得率(0,0.7)
    墙体外保温层厚度m(0,0.1]
    墙体日射吸收率[0.1,1]
    人员密度人/${\rm{m}}^{2}$[0.1,1)
    照明功率密度w/${\rm{m}}^{2}$[6,12]
    设备功率密度w/${\rm{m}}^{2}$[10,18]
    空调供热设置温度[18,23]
    空调制冷设置问题[24,28]
    下载: 导出CSV

    表  15  处理建筑节能设计问题时两种算法所得的实验结果

    Table  15  Results of the two algorithms on building energy conservation

    GS Optimal solutions Time/s
    IMPSO-HES 5.02 X=71.8,1.06,1.85,3.64,0.0382,0.0905,0.2212,0.1033,6.5,14.0,22.3,26.4, f=5.1 450
    X=297.3,2.53,1.63,4.0065,0.0556,0.0402,0.5983,0.1027,6.0,17.2,19.6,24.0, f=5.1
    X=351.7,3.50,0.38,2.266,0.1604,0.0567,0.8882,0.1062,6.1,17.3,22.6,24.6, f=5.11
    EMO-MMO 4.96 X=183.2,1.19,2.36,2.32,0.3439,0.0489,0.9743,0.1085,6.18,12.3,21.1,26.3, f=5.01 42357
    X=215.1,2.41,2.09,5.38,0.2847,0.0532,0.4720,0.1015,6.44,11.8,19.3,27.1, f=5.02
    X=134.7,1.07,2.87,3.73,0.3129,0.0418,0.9553,0.1015,6.02,12.8,20.4,25.3, f=5.02
    下载: 导出CSV
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  • 收稿日期:  2021-03-19
  • 录用日期:  2021-09-06
  • 网络出版日期:  2021-10-08

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