異構集成代理輔助的區間多模態(tài)粒子群優(yōu)化算法
doi: 10.16383/j.aas.c210223 cstr: 32138.14.j.aas.c210223
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中國礦業(yè)大學(xué)信息與控制工程學(xué)院 徐州 221116
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中國礦業(yè)大學(xué)機械電子與信息工程學(xué)院(北京) 北京 100083
基金項目: 國家自然科學(xué)基金(62273348, 62133015), 北方民族大學(xué)青年人才培育項目 (2024QNPY04)資助
Interval Multimodal Particle Swarm Optimization Algorithm Assisted by Heterogeneous Ensemble Surrogate
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School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116
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School of Mechanical Electronic and Information Engineering, China University of Mining and Technology (Beijing), Beijing 100083
Funds: Supported by National Natural Science Foundation of China (62273348，62133015) and Young Talent Cultivation Project of North Minzu University (2024QNPY04)
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摘要: 現實(shí)生活中的很多黑盒優(yōu)化問(wèn)題可歸為高計算代價(jià)的多模態(tài)優(yōu)化問(wèn)題(Multimodal optimization problem, MMOP), 即昂貴多模態(tài)優(yōu)化問(wèn)題(Expensive MMOP, EMMOP). 在處理該類(lèi)問(wèn)題時(shí), 決策者希望以盡量少的計算代價(jià)(即盡量少的真實(shí)函數評價(jià)次數)找到多個(gè)高質(zhì)量的最優(yōu)解. 然而, 已有代理輔助的進(jìn)化優(yōu)化算法(Surrogate-assisted evolutionary algorithm, SAEA)很少考慮問(wèn)題的多模態(tài)屬性, 運行一次僅可獲得問(wèn)題的一個(gè)最優(yōu)解. 鑒于此, 研究一種異構集成代理輔助的區間多模態(tài)粒子群優(yōu)化(Interval multimodal particle swarm optimization algorithm assisted by heterogeneous ensemble surrogate, IMPSO-HES)算法. 首先, 借助異構集成的思想構建一個(gè)由多個(gè)基礎代理模型組成的模型池; 隨后, 依據待評價(jià)粒子與已發(fā)現模態(tài)之間的匹配關(guān)系, 從模型池中自主選擇部分基礎代理模型進(jìn)行集成, 并使用集成后的代理模型預測該粒子的適應值. 進(jìn)一步, 為節約代理模型管理的代價(jià), 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(guān)系引入到進(jìn)化過(guò)程中. 將所提算法與當前流行的5種代理輔助進(jìn)化優(yōu)化算法和7 種最先進(jìn)的多模態(tài)優(yōu)化算法進(jìn)行對比, 在20個(gè)測試函數和1個(gè)建筑節能實(shí)際問(wèn)題上的實(shí)驗結果表明, 所提算法可以在較少計算代價(jià)下獲得問(wèn)題的多個(gè)高競爭最優(yōu)解.
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關(guān)鍵詞:
- 粒子群優(yōu)化 /
- 多模態(tài)優(yōu)化 /
- 高昂計算代價(jià) /
- 代理輔助
Abstract: Many real-world black-box optimization problems can be classified as multimodal optimization problems (MMOPs) with high computational cost, that is, expensive multimodal optimization problems (EMMOPs). When dealing with such problems, decision-makers hope to find multiple high-quality solutions with less computational cost (i.e., the least number of real function evaluations). However, existing surrogate-assisted evolutionary algorithms (SAEAs) seldom consider the multimodal properties of problem, and they can only obtain one optimal solution of the problem at a time. In view of this, this paper studies an interval multimodal particle swarm optimization (PSO) algorithm assisted by heterogeneous ensemble surrogate (IMPSO-HES). Firstly, a model pool composed of multiple basic surrogate models is constructed with the idea of heterogeneous ensemble. Then, according to the matching relationship between the particle to be evaluated and the discovered modalities, some basic surrogate models will be selected from the model pool for integration, and the integrated surrogate model is utilized to predict the fitness value of the particle. Furthermore, in order to save the cost of model management, an incremental surrogate model management strategy is designed. In order to reduce the influence of prediction error of surrogate model on the algorithm＇s performance, the interval ordering relation is introduced into the evolutionary process for the first time. The proposed algorithm is compared with five SAEAs and seven state-of-the-art multimodal algorithms, experimental results on 20 benchmark functions and the building energy conservation problem show that the proposed algorithm can obtain multiple highly-competitive optimal solutions at a low computational cost. -
圖 2 精確評價(jià)和區間評價(jià)策略下IMPSO-HES所得GS值
Fig. 2 GS values obtained by IMPSO-HES under precision and interval evaluation
圖 3 精確評價(jià)和區間評價(jià)策略下IMPSO-HES所得VR值
Fig. 3 VR values obtained by IMPSO-HES under precision and interval evaluation
圖 4 IMPSO-HES/D和IMPSO-HES得到的GS值
Fig. 4 GS values obtained by IMPSO-HES/D and IMPSO-HES
圖 5 IMPSO-HES/D和IMPSO-HES得到的VR值
Fig. 5 VR values obtained by IMPSO-HES/D and IMPSO-HES
表 1 基準問(wèn)題
Table 1 Benchmark problems
問(wèn)題 測試函數 維數 變量空間 全局/局部解個(gè)數 全局最優(yōu)解的目標值 F1 Ellipsoid 10/20 $\boldsymbol{X} \in [-1,1]^{D}$ 1/0 0 F2 Ackley 10/20 $\boldsymbol{X} \in [-30,30]^{D}$ 1/many 0 F3 Rastrigin 10/20 $\boldsymbol{X }\in [-5.12,5.12]^{D}$ 1/many 0 F4 Rosenbrock 10/20 $\boldsymbol{X} \in [-2.048,2.048]^{D}$ 1/many 0 F5 Griewank 10/20 $\boldsymbol{X} \in [-600,600]^{D}$ 1/many 0 F6 Reverse five-uneven-peak trap 1 $\boldsymbol{X} \in [0,30] $ 2/3 ?200 F7 Reverse equal maxima 1 $\boldsymbol{X} \in [0,1] $ 5/0 ?1 F8 Reverse uneven decreasing maxima 1 $\boldsymbol{X} \in [0,1] $ 1/4 ?1 F9 Reverse himmelblau 2 $\boldsymbol{X} \in [-6,6]^{D}$ 4/0 ?200 F10 Six-hump camel 2 $x_1\in[-1.9,1.9], x_2\in[-1.1,1.1] $ 2/2 ?1.031 6 F11 Reverse shubert 2 $\boldsymbol{X} \in [-10,10]^{D}$ 18/many ?186.73 F12 Reverse vincent 2 $\boldsymbol{X} \in [0.25,10]^{D}$ 36/0 ?1 F13 Reverse modified rastrigin 2 $\boldsymbol{X} \in [0,1]^{D}$ 12/0 2 F14 Reverse CF1 2 $\boldsymbol{X}\in [-5,5]^D$ 6/0 0 F15 Reverse CF2 2 $\boldsymbol{X}\in [-5,5]^D$ 8/0 0 F16 Reverse CF3 2 $\boldsymbol{X} \in[-5,5]^D $ 6/0 0 F17 Reverse CF4 3 $\boldsymbol{X}\in [-5,5]^D$ 8/0 0 F18 UrsemF4 back 2 $\boldsymbol{X }\in [-2,2]^{D}$ 2/0 ?0.267 9 F19 Branin RCOS 2 $x_1\in[-5,10], x_2\in[0,15] $ 3/0 0.397 8 F20 Waves 2 $x_1\in[-0.9,1.2], x_2\in[-1.2,1.2]$ 1/9 ?7.776 
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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 $R_rf50c1hsl6$ 1 0.05 0.5 0.5 0.2 2 0.5 0.5 1 1 1 1 0.5 1 0.2
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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 $g_{{\rm{max}}}$ values
問(wèn)題 $g_{{\rm{max}}}$ $GS $均值(標準差) $VR $均值 耗時(shí)(s) F5 (D = 10) 3 3.800 7 (3.5E+00)+ — 64 6 1.174 5 (3.7E?02) — 85 9 1.108 3 (2.5E?02) = — 116 F5 (D = 20) 3 8.198 0 (9.8E+00) + — 776 6 1.075 7 (1.6E?02) — 1 400 9 0.807 9 (2.8E?01) ? — 2 045 F9 3 ?199.93 (3.1E?03) = 0.68 11 6 ?199.99 (1.0E?04) 0.70 19 9 ?200.00 (1.4E?03) = 0.63 36 F10 3 ?1.031 6 (1.7E?06) = 1.00 19 6 ?1.031 6 (9.8E?07) 1.00 28 9 ?1.031 6 (9.8E?07) = 1.00 38 F12 3 ?0.999 0 (7.1E?06) = 0.13 10 6 ?0.999 9 (1.0E?06) 0.13 14 9 ?0.999 9 (2.2E?06) = 0.11 25 注: 加粗字體表示各組的最優(yōu)結果值.
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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
問(wèn)題 Q GS 均值(標準差) VR 均值 耗時(shí)(s) F5 (D = 10) K/5 1.658 1 (2.2E?01) + — 64 K/4 1.174 5 (3.7E?02) — 85 K/3 1.382 1 (1.5E?01) + — 108 K/2 1.269 6 (5.1E?02) + — 160 F5 (D = 20) K/5 1.980 0 (1.0E+00) + — 1137 K/4 1.075 7 (1.6E?02) — 1400 K/3 1.832 1 (1.1E+00) + — 1920 K/2 1.835 2 (1.7E+00) + — 2700 F9 K/5 ?199.98 (7.2E?04) = 0.53 17 K/4 ?199.99 (1.0E?04) 0.70 19 K/3 ?199.98 (4.6E?04) = 0.55 24 K/2 ?199.14 (6.8E+00) + 0.33 34 F10 K/5 ?1.031 6 (1.1E?09) = 1.00 28 K/4 ?1.031 6 (9.8E?07) 1.00 28 K/3 ?1.031 6 (9.8E?07) = 1.00 30 K/2 ?1.030 0 (1.4E?03) + 0.85 48 F12 K/5 ?0.999 1 (2.3E?06) + 0.12 12 K/4 ?0.999 9 (1.0E?06) 0.13 14 K/3 ?0.999 6 (8.5E?07) + 0.10 18 K/2 ?0.994 9 (9.2E?05) + 0.10 24 注: 加粗字體表示各組的最優(yōu)結果值.
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表 5 異構集成與同質(zhì)集成下IMPSO-HES所得結果
Table 5 Performance values obtained by IMPSO-HES under heterogeneous and homogeneous ensemble
問(wèn)題 算法 GS均值(標準差) VR均值 耗時(shí)(s) F5 (D = 10) IMPSO-PR 1.631 0 (7.1E?01) + — 86 IMPSO-RBFN 45.27 2 (8.9E+02) + — 39 IMPSO-HES 1.174 5 (3.7E?02) — 85 F5 (D = 20) IMPSO-PR 2.003 7 (2.9E+00) + — 1 478 IMPSO-RBFN 116.7 8 (9.5E+02) + — 180 IMPSO-HES 1.075 7 (1.6E?02) — 1 400 F9 IMPSO-PR ?196.81 (9.5E+00) + 0.05 16 IMPSO-RBFN ?199.99 (4.7E?07) = 0.65 22 IMPSO-HES ?199.99 (1.0E?04) 0.70 19 F10 IMPSO-PR ?0.962 0 (2.5E?03) + 0.2 17 IMPSO-RBFN ?1.031 6 (9.8E?09) = 1.00 20 IMPSO-HES ?1.031 6 (9.8E?07) 1.00 28 F12 IMPSO-PR ?0.988 6 (1.5E?04) + 0.06 11 IMPSO-RBFN ?0.999 5 (9.4E?07) + 0.09 19 IMPSO-HES ?0.999 9 (1.0E?06) 0.13 14 注: 加粗字體表示各組的最優(yōu)結果值.
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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
問(wèn)題 $p_{m}$ GS 均值 (標準差) VR 均值 耗時(shí)(s) F5 (D = 10) 固定 1.439 3 (3.8E?01) + — 84 自適應 1.174 5 (3.7E?02) — 85 F5 (D = 20) 固定 1.750 3 (1.7E+00) + — 1313 自適應 1.075 7 (1.6E?02) — 1400 F9 固定 ?199.91 (2.6E?02) + 0.40 19 自適應 ?199.99 (1.0E?04) 0.70 19 F10 固定 ?1.031 6 (4.7E?08) = 1.00 26 自適應 ?1.031 6 (9.8E?07) 1.00 28 F12 固定 ?0.996 9 (4.8E?05) + 0.12 14 自適應 ?0.999 9 (1.0E?06) 0.13 14 注: 加粗字體表示各組的最優(yōu)結果值.
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表 7 使用All-S和Mod-S時(shí)IMPSO-HES所得結果
Table 7 Performance values obtained by IMPSO-HES with All-S and Mod-S
問(wèn)題 集成策略 GS 均值 (標準差) VR 均值 耗時(shí)(s) F5 (D = 10) All-S 3.878 5 (3.8E+00) + — 243 Mod-S 1.174 5 (3.7E?02) — 85 F5 (D = 20) All-S 8.838 7 (8.1E+00) + — 3 362 Mod-S 1.075 7 (1.6E?02) — 1 400 F9 All-S ?187.33 (2.0E+2) + 0.05 80 Mod-S ?199.99 (1.0E?04) 0.70 19 F10 All-S ?0.9751 (1.4E?02) + 0.70 57 Mod-S ?1.031 6 (9.8E?07) 1.00 28 F12 All-S ?0.973 7 (1.9E?02) + 0.08 42 Mod-S ?0.999 9 (1.0E?06) 0.13 14 注: 加粗字體表示各組的最優(yōu)結果值.
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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
問(wèn)題 更新策略 GS 均值 (標準差) VR 均值 耗時(shí)(s) F5 (D = 10) All-up 1.500 9 (3.9E?02) + — 97 Inc-up 1.174 5 (3.7E?02) — 85 F5 (D = 20) All-up 32.184 (2.4E+04) + — 1 509 Inc-up 1.075 7 (1.6E?02) — 1 400 F9 All-up ?200.00 (3.6E-10) = 0.63 30 Inc-up ?199.99 (1.0E?04) 0.70 19 F10 All-up ?1.031 6 (1.2E?04) = 0.95 30 Inc-up ?1.031 6 (9.8E?07) 1.00 28 F12 All-up ?0.999 8 (2.7E?07) = 0.11 16 Inc-up ?0.999 9 (1.0E?06) 0.13 14 注: 加粗字體表示各組的最優(yōu)結果值.
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表 9 IMPSO-HES與5種SAEA所得GS值(均值(方差))
Table 9 GS values obtained by IMPSO-HES and 5 SAEAs (mean (variance))
問(wèn)題 D IMPSO-HES SA-COSO CAL-SAPSO Gr-based SAPSO PESPSO ESPSO F1 10 3.660 0 3.160 0? 0.115 3? 0.147 6? 0.296 2? 0.664 5? (4.2E+00) (6.5E?02) (4.9E?02) (1.1E?03) (1.3E?03) (5.0E?02) 20 21.398 11.017? 0.229 2? 0.027 9? 1.377 0? 1.866 4? (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 1.174 5 66.556+ 1.736 4+ 1.310 6+ 2.798 7+ 2.317 2+ (3.7E?02) (1.8E+02) (1.4E?01) (1.7E?02) (2.4E+00) (3.9E?01) 20 1.075 7 43.897+ 2.255 3+ 1.057 2= 6.701 8+ 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 ?0.999 9 ?1.00= ?0.505 2+ ?0.999 1+ ?0.999 9= ?0.999 8= (3.1E?06) (0.0E+00) (1.2E?01) (1.1E?07) (2.7E?05) (3.8E?06) F8 1 ?0.985 4 ?0.980 8= ?0.511 4+ ?0.944 7+ ?0.948 6+ ?0.948 6+ (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 ?1.031 6 ?0.995 6+ ?0.464 6+ ?1.030 6+ ?1.030 3+ ?1.029 2+ (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 ?0.999 9 ?0.979 8+ ?0.719 4+ ?0.984 5+ ?0.995 4+ ?0.980 0+ (1.0E?06) (5.6E?04) (9.0E?02) (1.9E?04) (2.0E?06) (5.5E?05) F13 2 2.232 9 2.890 3+ 7.846 7+ 2.298 5= 2.022 8? 2.060 9? (2.3E?01) (6.4E?02) (3.0E+01) (1.0E?01) (4.6E?03) (3.1E?03) F14 2 0.087 9 40.011+ 197.39+ 23.774+ 7.588 4+ 9.961 7+ (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 0.242 3 90.430+ 350.88+ 60.296+ 1.162 1+ 18.280+ (1.3E?01) (1.2E+04) (4.8E+04) (3.2E+03) (2.5E+00) (1.2E+03) F17 3 32.566 88.270+ 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 ?0.267 9 ?0.245 7+ ?0.130 4+ ?0.267 1+ ?0.267 8= ?0.267 8= (1.6E?06) (3.6E?04) (5.6E?03) (6.8E?08) (1.6E?06) (5.4E?09) F19 2 0.399 9 1.148 8+ 2.260 3+ 0.425 9+ 0.424 9+ 0.513 6+ (2.4E?05) (8.6E?01) (6.2E+00) (1.3E?03) (1.2E?03) (5.3E?02) F20 2 ?7.429 9 ?7.776 0? ?7.775 3? ?6.340 8+ ?7.294 3+ ?7.451 1= (1.7E?02) (0.0E+00) (4.2E?06) (8.4E?01) (2.2E?01) (2.7E?01) 注: 加粗字體表示各行GS值的最優(yōu)結果值.
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問(wèn)題 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 2.500 0 5.500 0 3.000 0 3.125 0 3.125 0 3.750 0 Adjusted p-value — 0.006 6 0.689 2 0.689 2 0.689 2 0.393 8 F6 ~ F20 好/平/差 — 11/2/2 13/0/2 14/1/0 8/5/2 9/4/2 Rank 1.833 3 4.166 6 5.433 3 4.000 0 2.266 6 3.300 0 Adjusted p-value — 0.001 6 0.000 0 0.002 5 0.525 8 0.039 5 注: 加粗字體表示各組的最優(yōu)結果值.
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表 11 處理F1 ~ F5時(shí)IMPSO-HES與7種多模態(tài)進(jìn)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))
問(wèn)題 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+ 65.750+ 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) 注: 加粗字體表示各組的最優(yōu)結果值.
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表 12 處理F6 ~ F20時(shí)IMPSO-HES與7種多模態(tài)進(jìn)化算法所得結果
Table 12 Results of IMPSO-HES and the 7 multimodal EAs on F6 ~ F20
問(wèn)題 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 ?0.999 9 ?0.999 4+ ?0.999 5+ ?0.999 1+ ?0.998 6+ ?0.998 7+ ?0.998 4+ ?0.998 0+ (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 ?0.985 4 ?0.969 3+ ?0.993 7? ?0.993 1? ?0.975 8+ ?0.966 0+ ?0.948 3+ ?0.968 3+ (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 ?1.031 6 ?1.004 7+ ?1.001 6+ ?1.003 2+ ?0.994 9+ ?0.987 8+ ?0.973 0+ ?1.002 0+ (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 ?0.999 9 ?0.973 3+ ?0.975 3+ ?0.972 7+ ?0.976 4+ ?0.976 4+ ?0.989 0+ ?0.988 7+ (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 2.232 9 2.714 6+ 2.560 4+ 2.438 4+ 2.590 3+ 2.481 7+ 2.344 6= 2.579 2+ (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 0.087 9 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 0.242 3 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 ?0.267 9 ?0.264 2+ ?0.260 3+ ?0.257 9+ ?0.262 5+ ?0.254 8+ ?0.260 4+ ?0.263 0+ (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 0.399 9 0.529 2+ 0.882 5+ 0.797 9+ 0.776 3+ 0.789 5+ 1.337 5+ 0.885 8+ (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 ?7.429 9 ?6.619 2+ ?6.649 6+ ?6.664 4+ ?6.728 0+ ?6.679 1+ ?6.420 4+ ?6.981 8+ (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+ 注: 加粗字體表示各組的最優(yōu)結果值.
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表 13 IMPSO-HES與7種多模態(tài)進(jìn)化算法的統計對比結果
Table 13 Statistical comparison results of IMPSO-HES and the 7 multimodal EAs
問(wèn)題 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 1.300 0 2.800 0 3.000 0 6.100 0 3.300 0 6.300 0 6.800 0 6.700 0 Adjusted p-value — 0.315 3 0.116 0 0.000 2 0.0937 0.000 1 0.000 0 0.000 0 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 F6 ~ F20 VR — 11/4/0 10/5/0 11/4/0 11/4/0 14/1/0 12/3/0 13/2/0 Rank 1.258 6 4.827 5 4.268 9 5.551 7 4.603 4 6.103 4 4.862 0 4.224 1 Adjusted p-value — 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0
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表 14 問(wèn)題的決策變量信息
Table 14 Decision variable information of the problem
決策變量 單位 范圍 房屋方向 $( ^{ {\circ} } )$ [0, 360) 窗戶(hù)的長(cháng) m (0, 3.6) 窗戶(hù)的高 m (0, 3.9) 窗戶(hù)的傳熱系數 ${\rm{W} }/({\rm{m} }^{2}\cdot{\rm{K} })$ [2, 6] 窗戶(hù)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 ${{\text{人}}/\rm{m} }^{2}$ [0.1, 1) 照明功率密度 ${\rm{W} }/{\rm{m} }^{2}$ [6, 12] 設備功率密度 $\rm{W}/{\rm{m} }^{2}$ [10, 18] 空調供熱設置溫度 ℃ [18, 23] 空調制冷設置問(wèn)題 ℃ [24, 28]
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表 15 處理建筑節能設計問(wèn)題時(shí)兩種算法所得的實(shí)驗結果
Table 15 Results of the two algorithms on building energy conservation
GS Optimal solutions 時(shí)間(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 = 5.1450 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 = 5.1X = 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 = 5.11EMO-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 = 5.0142 357 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 = 5.02X = 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
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