Hybrid evolutionary JAYA algorithm for global and engineering optimization problems
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摘要: 為了更好求解復雜函數優化和工程約束優化問題，進一步增強JAYA算法的尋優能力，提出一種面向全局優化的混合進化JAYA算法。首先在計算當前最優和最差個體時引入反向學習機制，提高最優和最差個體跳離局部極值區域的可能性；然后在個體位置更新中引入并融合正弦余弦算子和差分擾動機制，不僅增加了種群的多樣性，而且較好平衡與滿足了算法在不同迭代時期對探索和挖掘能力的不同需求；最后在算法結構上采用奇偶不同的混合進化策略，有效利用不同演化機制的優勢結果，進一步提升了算法的收斂性和精度。之后給出了算法流程偽代碼，理論分析證明了改進算法的時間復雜度與基本JAYA相同，而通過6種代表性算法在包含和組合了30個基準函數的CEC2017測試套件上進行的多維度函數極值優化測試，以及對拉伸彈簧、波紋艙壁、管柱設計、鋼筋混凝土梁、焊接梁和汽車側面碰撞6個具有挑戰性的工程設計問題的優化求解，都清楚地表明改進后算法的尋優精度、收斂性能和求解穩定性均有顯著提升，在求解CEC復雜函數和工程約束優化問題上有著明顯優勢。Abstract: A swarm intelligence optimization algorithm is an effective method to rapidly solve large-scale complex optimization problems. The JAYA algorithm is a new swarm intelligence evolutionary optimization algorithm, which was proposed in 2016. Compared with other active evolutionary algorithms, the JAYA algorithm has several advantages, such as a clear mechanism, concise structure, and ease of implementation. Further, it has guiding characteristics, obtains the best solution, and avoids the worst solution. The JAYA algorithm has an excellent optimization effect on many problems, and it is one of the most influential algorithms in the field of swarm intelligence. However, when dealing with the CEC test suite, which contains and combines shifted, rotation, hybrid, combination, and other composite characteristics, and the complex engineering constrained optimization problems with considerable difficulty and challenges, the JAYA algorithm has some flaws, that is, it easily falls into the local extremum, its optimization accuracy is sometimes low, and its solution is unstable. To better solve complex function optimization and engineering constrained optimization problems and further enhance the optimization capability of the JAYA algorithm, a global optimization-oriented hybrid evolutionary JAYA algorithm is proposed. First, opposition-based learning is introduced to calculate the current best and worst individuals, which improves the possibility of the best and worst individuals jumping out of the local extremum region. Second, the sine–cosine operator and differential disturbance mechanism are introduced and integrated into individual position updating, which not only improves the diversity of the population but also better balances and meets the different requirements of the algorithm for exploration and mining in different iteration periods. Finally, in the algorithm structure, the hybrid evolution strategy with different parity states is adopted and the advantages of different evolution mechanisms are effectively used, which further improves the convergence and accuracy of the algorithm. Then, the pseudocode of the improved algorithm is given, and the theoretical analysis proves that the time complexity of the improved algorithm is consistent with the basic JAYA algorithm. Through the simulation experiment of function extremum optimization of six representative algorithms on multiple dimensions of the CEC2017 test suite, which contains and combines 30 benchmark functions and the optimal solution of six challenging engineering design problems, such as tension/compression spring, corrugated bulkhead, tubular column, reinforced concrete beam, welded beam, and car side impact. The optimal solution of the test results shows that the improved algorithm has significantly improved the optimization accuracy, convergence performance, and solution stability, and it has obvious advantages in solving CEC complex functions and engineering constrained optimization problems.
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圖 1 收斂曲線. (a) f21(x); (b) f22(x); (c) f25(x); (d) f26(x); (e) f29(x); (f) f30(x)
Figure 1. Convergence curves: (a) f21(x); (b) f22(x); (c) f25(x); (d) f26(x); (e) f29(x); (f) f30(x)
表 1 6種算法在固定迭代次數下的尋優結果比較
Table 1. Comparison of the optimization results of six representative algorithms under fixed iteration times
Functions Algorithms D=10 D=50 D=100 Best Mean Variance Best Mean Variance Best Mean Variance f11(x) H-JAYA 1.10×103 1.11×103 1.40×101 1.37×103 1.74×103 1.65×105 1.50×104 3.94×104 2.47×108 IJAYA 1.11×103 1.12×103 2.62×101 3.87×103 7.80×103 2.20×106 1.68×105 2.55×105 1.32×109 CLJAYA 1.11×103 1.18×103 4.57×103 5.95×103 1.27×104 1.37×107 9.30×104 1.32×105 3.71×108 HFPSO 1.11×103 1.16×103 2.65×103 5.19×103 1.43×104 3.54×107 1.12×105 2.37×105 5.10×109 JAYA 1.14×103 1.19×103 1.19×103 5.59×103 9.84×103 6.79×106 1.66×105 2.68×105 2.61×109 WOA 1.12×103 1.22×103 1.04×104 2.93×103 5.25×103 1.65×106 1.26×105 2.31×105 6.78×109 f12(x) H-JAYA 2.80×103 1.01×104 3.15×107 6.03×106 4.80×107 5.94×1015 4.18×108 1.46×109 1.74×1018 IJAYA 8.60×104 4.01×105 1.35×1011 1.18×109 1.85×109 1.55×1017 9.17×109 1.20×1010 2.80×1018 CLJAYA 3.44×103 4.28×105 1.67×1012 7.36×109 2.36×1010 6.40×1019 8.88×1010 1.35×1011 3.73×1020 HFPSO 3.02×103 1.60×104 1.33×108 1.01×107 8.26×107 6.55×1015 7.82×109 1.24×1010 1.99×1019 JAYA 6.17×105 8.03×106 4.38×1013 5.39×109 7.86×109 1.77×1018 2.86×1010 4.13×1010 5.16×1019 WOA 1.11×104 4.99×106 2.17×1013 4.90×108 1.52×109 4.04×1017 6.85×109 1.30×1010 9.73×1018 f19(x) H-JAYA 1.90×103 1.91×103 5.47×101 2.84×103 3.03×104 3.91×109 7.24×104 9.21×105 1.60×1012 IJAYA 1.95×103 2.89×103 9.08×105 1.00×106 7.99×106 2.24×1013 1.78×108 4.39×108 1.88×1016 CLJAYA 1.91×103 1.95×103 1.61×103 1.16×106 2.96×107 1.00×1015 1.97×109 7.38×109 9.59×1018 HFPSO 1.94×103 1.10×104 8.86×107 3.75×105 3.69×106 1.30×1013 3.11×107 1.72×108 3.85×1016 JAYA 1.94×103 3.13×103 8.75×106 4.27×106 1.36×108 6.37×1015 1.32×109 2.58×109 3.29×1017 WOA 2.37×103 7.89×104 4.84×1010 3.42×105 1.07×107 8.86×1013 2.99×107 1.25×108 4.01×1015 f20(x) H-JAYA 2.00×103 2.02×103 1.49×102 3.03×103 3.37×103 2.51×104 5.01×103 5.78×103 6.10×104 IJAYA 2.04×103 2.07×103 2.05×102 3.74×103 4.11×103 3.54×104 7.21×103 7.76×103 7.01×104 CLJAYA 2.02×103 2.07×103 2.09×103 3.06×103 3.55×103 8.07×104 5.66×103 6.82×103 3.74×105 HFPSO 2.03×103 2.14×103 4.44×103 2.86×103 3.69×103 1.84×105 5.65×103 7.25×103 3.76×105 JAYA 2.05×103 2.09×103 9.96×102 3.83×103 4.28×103 2.81×104 7.30×103 7.94×103 8.05×104 WOA 2.07×103 2.19×103 6.29×103 3.19×103 3.91×103 1.40×105 5.37×103 7.12×103 3.94×105 f21(x) H-JAYA 2.20×103 2.33×103 2.87×101 2.50×103 2.55×103 1.04×103 3.30×103 3.47×103 5.39×103 IJAYA 2.21×103 2.33×103 6.44×102 2.67×103 2.79×103 1.69×103 3.45×103 3.61×103 5.55×103 CLJAYA 2.20×103 2.33×103 4.23×102 2.74×103 2.96×103 6.65×103 3.85×103 4.16×103 2.72×104 HFPSO 2.21×103 2.33×103 3.11×103 2.71×103 2.82×103 3.30×103 3.48×103 3.65×103 1.09×104 JAYA 2.33×103 2.34×103 3.50×101 2.78×103 2.86×103 1.17×103 3.61×103 3.79×103 8.65×103 WOA 2.21×103 2.33×103 2.79×103 2.81×103 3.05×103 1.16×104 3.91×103 4.36×103 5.14×104 f22(x) H-JAYA 2.30×103 2.31×103 1.69×101 2.58×103 1.09×104 3.02×106 2.09×104 2.50×104 5.31×106 IJAYA 2.30×103 2.31×103 1.79×100 1.41×104 1.63×104 2.41×105 3.27×104 3.44×104 4.14×105 CLJAYA 2.22×103 2.29×103 1.40×103 1.27×104 1.48×104 7.48×105 2.87×104 3.27×104 2.48×106 HFPSO 2.31×103 2.39×103 1.02×105 1.24×104 1.50×104 1.82×106 2.83×104 3.24×104 3.61×106 JAYA 2.23×103 2.32×103 5.18×102 1.54×104 1.66×104 2.06×105 3.32×104 3.48×104 3.48×105 WOA 2.24×103 2.51×103 2.09×105 1.10×104 1.43×104 2.11×106 2.65×104 3.09×104 3.67×106 f25(x) H-JAYA 2.60×103 2.89×103 1.23×104 3.09×103 3.22×103 5.72×103 4.33×103 5.20×103 3.38×105 IJAYA 2.90×103 2.90×103 8.93×101 3.34×103 3.59×103 1.92×104 7.41×103 9.95×103 1.60×106 CLJAYA 2.90×103 2.95×103 2.57×102 6.91×103 9.93×103 2.01×106 1.68×104 2.16×104 4.38×106 HFPSO 2.90×103 2.93×103 8.41×102 3.59×103 4.20×103 1.93×105 4.97×103 5.79×103 4.79×105 JAYA 2.93×103 2.96×103 1.27×102 3.93×103 4.58×103 1.61×105 1.05×104 1.48×104 3.79×106 WOA 2.90×103 2.95×103 8.12×102 3.75×103 4.22×103 1.04×105 6.55×103 8.12×103 6.34×105 f26(x) H-JAYA 2.79×103 2.99×103 8.70×102 7.08×103 9.01×103 3.21×105 2.23×104 2.54×104 2.05×106 IJAYA 2.90×103 3.11×103 1.76×105 8.98×103 1.10×104 9.90×105 2.58×104 3.01×104 5.24×106 CLJAYA 2.91×103 3.05×103 2.59×104 1.04×104 1.32×104 1.82×106 2.97×104 4.02×104 1.82×107 HFPSO 2.81×103 3.01×103 1.15×105 5.62×103 9.75×103 5.34×106 7.77×103 1.82×104 1.41×107 JAYA 3.00×103 3.52×103 3.11×105 1.01×104 1.14×104 3.74×105 2.58×104 2.97×104 4.55×106 WOA 2.83×103 3.61×103 3.86×105 1.03×104 1.41×104 1.63×106 2.82×104 3.62×104 9.11×106 f29(x) H-JAYA 3.14×103 3.17×103 6.75×102 5.10×103 5.89×103 1.33×105 8.34×103 9.85×103 6.71×105 IJAYA 3.15×103 3.20×103 8.47×102 5.31×103 6.26×103 1.39×105 1.18×104 1.37×104 9.48×105 CLJAYA 3.15×103 3.19×103 1.44×103 5.28×103 8.55×103 2.60×106 1.89×104 3.32×104 1.12×108 HFPSO 3.17×103 3.27×103 4.25×103 4.88×103 6.25×103 4.44×105 9.86×103 1.15×104 1.20×106 JAYA 3.16×103 3.22×103 1.79×103 6.14×103 6.91×103 2.38×105 1.44×104 1.83×104 5.09×106 WOA 3.19×103 3.39×103 1.59×104 5.54×103 8.56×103 3.26×106 1.39×104 1.85×104 1.34×107 f30(x) H-JAYA 4.40×103 1.06×104 3.57×108 7.80×105 9.97×106 1.83×1015 3.44×106 1.82×107 4.14×1014 IJAYA 7.46×103 6.66×105 1.22×1012 5.80×107 1.38×108 2.57×1015 3.95×108 6.93×108 2.14×1016 CLJAYA 4.48×103 1.98×106 2.45×1012 5.31×107 3.07×108 2.70×1016 3.09×109 1.32×110 3.27×1019 HFPSO 9.86×103 7.09×105 6.11×1011 6.77×107 1.49×108 2.60×1015 2.62×108 9.42×108 3.74×1017 JAYA 9.69×103 3.75×105 1.18×1011 6.08×107 1.95×108 1.71×1016 2.18×109 4.27×109 5.93×1017 WOA 8.17×103 1.11×106 1.65×1012 9.36×107 2.54×108 1.27×1016 6.34×108 1.35×109 3.02×1017 
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表 2 各算法求解CEC2017測試集套件Wilcoxon 秩和檢驗的p值
Table 2. p-value for Wilcoxon’s rank-sum test on each algorithm used to solve the CEC2017 test suite
Functions H-JAYA vs JAYA H-JAYA vs IJAYA H-JAYA vs CLJAYA H-JAYA vs WOA H-JAYA vs HFPSO p-value win p-value win p-value win p-value win p-value win f11(x) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) f12(x) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) f19(x) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) f20(x) 2.6734×10–4(+) 2.6727×10–4(+) 2.6727×10–4(+) 2.6727×10–4(+) 2.6727×10–4(+) f21(x) 2.6693×10–4(+) 2.6700×10–4(+) 2.6727×10–4(+) 2.6720×10–4(+) 2.6720×10–4(+) f22(x) 2.6720×10–4(+) 2.6727×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) f25(x) 2.6734×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) 2.6727×10–4(+) 2.6727×10–4(+) f26(x) 2.6734×10–4(+) 2.6734×10–4(+) 2.6727×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) f29(x) 2.6734×10–4(+) 2.6720×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) 2.6734×10–4(+) f30(x) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) 2.6917×10–13(+) （+/–/=） 10/0/0 10/0/0 10/0/0 10/0/0 10/0/0
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表 3 6種算法求解拉伸彈簧設計問題的尋優結果比較
Table 3. Comparison of the optimization results of six representative algorithms used to solve the tension/compression spring design problem
Algorithms Best Mean Variance H-JAYA 1.2665237000×10–2 1.2806874020×10–2 2.7477865853×10–8 IJAYA 1.2666032600×10–2 1.3032429118×10–2 2.1994218985×10–6 CLJAYA 1.2666232800×10–2 1.3068941340×10–2 1.2273508194×10–6 JAYA 1.2666917100×10–2 1.3185287610×10–2 1.9559284633×10–6 HFPSO 1.2665806000×10–2 1.2982082240×10–2 3.0289473131×10–7 WOA 1.2671936200×10–2 1.3894943310×10–2 2.4615922288×10–6
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表 4 6種算法求解波紋艙壁設計問題的尋優結果比較
Table 4. Comparison of the optimization results of six representative algorithms used to solve the corrugated bulkhead design problem
Algorithms Best Mean Variance H-JAYA 6.8429580101 6.8574579730 3.5011426525×10–4 IJAYA 6.8429580101 7.5431661442 1.0303170764 CLJAYA 6.8429580101 8.2229580101 1.4648979592 JAYA 6.9429580101 8.6689580101 7.9828979592×10–1 HFPSO 6.8924274599 7.0360704247 2.4342757860×10–1 WOA 6.8589881925 7.1845962530 2.1316652625×10–1
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表 5 6種算法求解管柱設計問題的尋優結果比較
Table 5. Comparison of the optimization results of six representative algorithms used to solve the tubular column design problem
Algorithms Best Mean Variance H-JAYA 2.6486361472×101 2.6486976485×101 2.7492263055×10–6 IJAYA 2.6486361473×101 2.6770361473×101 3.8922448975×10–2 CLJAYA 2.6486361472×101 2.6840361472×101 2.8248979592×10–2 JAYA 2.6686361472×101 2.6910363872×101 1.3289800884×10–2 HFPSO 2.6491554294×101 2.6524256607×101 4.0282119604×10–4 WOA 2.6491740158×101 2.6831750349×101 1.0260601687×10–1
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表 6 6種算法求解鋼筋混凝土梁問題的尋優結果比較
Table 6. Comparison of the optimization results of six representative algorithms used to solve the reinforced concrete beam design problem
Algorithms Best Mean Variance H-JAYA 3.5920800000×102 3.6080773841×102 2.8418472149×102 IJAYA 3.6225000000×102 3.7488260888×102 1.5910386068×102 CLJAYA 3.6225000000×102 3.7447556000×102 1.3625233588×102 JAYA 3.6225000000×102 3.9080088880×102 2.4313100946×102 HFPSO 3.6925001569×102 3.8520700009×102 7.3955644549×101 WOA 3.6225106247×102 3.6700943742×102 4.3808572351×101
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表 7 6種算法求解焊接梁設計問題的尋優結果比較
Table 7. Comparison of the optimization results of six representative algorithms used to solve the welded beam design problem
Algorithms Best Mean Variance H-JAYA 1.6702177263 1.6808983776 1.2670965769×10–3 IJAYA 1.6702258303 1.6902400352 2.0000233481×10–2 CLJAYA 1.6702177264 1.6902198653 1.9999912854×10–2 JAYA 1.6702177328 1.8165708743 1.2530675483×10–1 HFPSO 1.6702682457 2.0454210525 8.4760880179×10–2 WOA 1.7177501400 2.3609633625 5.8101787131×10–1
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表 8 6種算法求解汽車側面碰撞問題的尋優結果比較
Table 8. Comparison of the optimization results of six representative algorithms used to solve the car side impact design problem
Algorithms Best Mean Variance H-JAYA 2.2848738268×101 2.3445600597×101 1.0419208431×10–1 IJAYA 2.2857969385×101 2.3815013650×101 2.7491744948×10–1 CLJAYA 2.3008501929×101 2.3956603034×101 6.5386322803×10–1 JAYA 2.3094202105×101 2.3872171530×101 7.0916777331×10–1 HFPSO 2.3207806520×101 2.3570163892×101 7.7055620394×10–1 WOA 2.3612634156×101 2.5726584934×101 2.0363936631
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