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              異構集成代理輔助的區間多模態粒子群優化算法

              季新芳 張勇 鞏敦衛 郭一楠 孫曉燕

              季新芳, 張勇, 鞏敦衛, 郭一楠, 孫曉燕. 異構集成代理輔助的區間多模態粒子群優化算法. 自動化學報, 2021, 45(x): 1?23 doi: 10.16383/j.aas.c210223
              引用本文: 季新芳, 張勇, 鞏敦衛, 郭一楠, 孫曉燕. 異構集成代理輔助的區間多模態粒子群優化算法. 自動化學報, 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

              異構集成代理輔助的區間多模態粒子群優化算法

              doi: 10.16383/j.aas.c210223
              基金項目: 國家重點研發計劃資助(2020YFB1708200), 徐州市重點研發計劃(KC20184), 綠色建筑節能智能優化設計方法及系統開發資助
              詳細信息
                作者簡介:

                季新芳:中國礦業大學信息與控制工程學院博士研究生. 2013年在中國礦業大學獲碩士學位. 主要研究方向為代理輔助進化優化, 多模態優化. E-mail: mimosa_615615@126.com

                張勇:中國礦業大學信息與控制工程學院教授. 2009年獲中國礦業大學控制理論與控制工程專業博士學位. 主要研究方向為智能優化, 數據挖掘. 本文通信作者. E-mail: yongzh401@126.com

                鞏敦衛:中國礦業大學信息與控制工程學院教授. 1999年在中國礦業大學獲博士學位. 主要研究方向為進化計算與應用. 本文通信作者. E-mail: dwgong@vip.163.com

                郭一楠:中國礦業大學信息與電氣工程學院教授, 主要研究方向為智能優化算法與控制, 數據挖掘. E-mail: nanly@126.com

                孫曉燕:中國礦業大學信息與電氣工程學院教授, 2009年在中國礦業大學獲博士學位. 主要研究方向為進化計算, 機器學習. 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: 現實生活中的很多黑盒優化問題可歸為高計算代價的多模態優化問題, 即昂貴多模態優化問題. 在處理該類問題時, 決策者希望以盡量少的計算代價(即盡量少的真實函數評價次數)找到多個高質量的最優解. 然而, 已有代理輔助的進化優化算法很少考慮問題的多模態屬性, 運行一次僅可獲得問題的一個最優解. 鑒于此, 研究一種異構集成代理輔助的區間多模態粒子群優化算法. 首先, 借助異構集成的思想構建一個由多個基礎代理模型組成的模型池; 隨后, 依據待評價粒子與已發現模態之間的匹配關系, 從模型池中自主選擇部分基礎代理模型進行集成, 并使用集成后的代理模型預測該粒子的適應值. 進一步, 為節約代理模型管理的代價, 設計一種增量式的代理模型管理策略; 為減少代理模型預測誤差對算法性能的影響, 首次將區間排序關系引入到進化過程中. 將所提算法與當前流行的5種代理輔助進化優化算法和7 種經典的多模態優化算法進行對比, 在20個測試函數和1個建筑節能實際問題上的結果表明, 所提算法可以在較少計算代價下獲得問題的多個高競爭最優解.
                1)  收稿日期 2021-03-19 錄用日期 2021-09-05 Manuscript?received?March?19,?2021;?accepted?September?6, 2021 國家重點研發計劃 (2020YFB1708200),?徐州市重點研發計劃 (KC20184),?綠色建筑節能智能優化設計方法及系統開發資助 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 本文責任編委 袁勇 Recommended?by?Associate?Editor YUAN Yong 1.?中國礦業大學信息與控制工程學院?徐州?221116? 2.?中國礦業大學機械電子與信息工程學院 (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
              • 圖  1  IMPSO-HES的框架圖

                Fig.  1  General framework of IMPSO-HES

                圖  2  精確和區間評價策略下IMPSO-HES所得GS

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

                圖  3  精確和區間評價策略下IMPSO-HES所得VR

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

                圖  4  IMPSO-HES/D和IMPSO-HES得到的GS

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

                圖  5  IMPSO-HES/D和IMPSO-HES得到的VR

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

                圖  6  IMPSO-HES與5種SAEAs的運行耗時

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

                圖  7  單居室居住建筑的外形圖

                Fig.  7  Outline of the single-room building

                表  1  基準測試問題

                Table  1  Benchmark problems

                測試函數維數變量空間全局/局部解個數全局最優解的目標值
                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
                下載: 導出CSV

                表  2  F6-F20的幅值精度和距離精度

                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
                下載: 導出CSV

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

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

                $g_{{\rm{max}}}$GS均值(標準差)VR均值耗時/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
                下載: 導出CSV

                表  4  不同Q取值下IMPSO-HES所得的性能指標值

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

                QGS均值(標準差)VR均值耗時/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
                下載: 導出CSV

                表  5  異構集成與同質集成下IMPSO-HES所得結果

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

                算法GS均值(標準差)VR均值耗時/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
                下載: 導出CSV

                表  6  不同更新概率$p_{m}$下IMPSO-HES所得結果

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

                $p_{m}$GS均值(標準差)VR均值耗時/s
                F5(D= 10)固定1.4393(3.8E?01)+84
                自適應1.1745(3.7E?02)85
                F5(D= 20)固定1.7503(1.7E+00)+1313
                自適應1.0757(1.6E?02)1400
                F9固定?199.91(2.6E?02)+0.4019
                自適應?199.99(1.0E?04)0.7019
                F10固定?1.0316(4.7E?08) =1.0026
                自適應?1.0316(9.8E?07)1.0028
                F12固定?0.9969 (4.8E?05)+0.1214
                自適應?0.9999(1.0E?06)0.1314
                下載: 導出CSV

                表  7  使用All-S和Mod-S時IMPSO-HES所得結果

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

                集成策略GS均值(標準差)VR均值耗時/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
                下載: 導出CSV

                表  8  不同模型更新策略下IMPSO-HES所得結果

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

                更新策略GS均值(標準差)VR均值耗時/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
                下載: 導出CSV

                表  9  IMPSO-HES與5種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)
                下載: 導出CSV

                表  10  基于表9的統計結果

                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
                下載: 導出CSV

                表  11  處理F1-F5時IMPSO-HES與7種多模態進化算法所得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)
                下載: 導出CSV

                表  12  處理F6-F20時IMPSO-HES與7種多模態進化算法所得結果

                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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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=
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                (標準差) (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+
                下載: 導出CSV

                表  13  IMPSO-HES與7種多模態進化算法的統計對比結果

                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
                下載: 導出CSV

                表  14  問題的決策變量信息

                Table  14  Decision variables

                決策變量單位范圍
                房屋方向$\circ $[0,360)
                窗戶的長m(0,3.6)
                窗戶的高m(0,3.9)
                窗戶的傳熱系數${\rm{w}}/({\rm{m}}^{2}{\rm{k}})$[2,6]
                窗戶的日射熱取得率(0,0.7)
                墻體外保溫層厚度m(0,0.1]
                墻體日射吸收率[0.1,1]
                人員密度人/${\rm{m}}^{2}$[0.1,1)
                照明功率密度w/${\rm{m}}^{2}$[6,12]
                設備功率密度w/${\rm{m}}^{2}$[10,18]
                空調供熱設置溫度[18,23]
                空調制冷設置問題[24,28]
                下載: 導出CSV

                表  15  處理建筑節能設計問題時兩種算法所得的實驗結果

                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
                下載: 導出CSV
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                        • 收稿日期:  2021-03-19
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