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摘要: 正弦余弦算法是一種新型仿自然優化算法，利用正余弦數學模型來求解優化問題。為提高正弦余弦算法的優化精度和收斂速度，提出了一種基于差分進化的正弦余弦算法。該算法通過非線性方式調整參數提高算法的搜索能力、利用差分進化策略平衡算法的全局探索能力及局部開發能力并加快收斂速度、通過偵察蜂策略增加種群多樣性以及利用全局最優個體變異策略增強算法的局部開發能力等優化策略來改進算法，最后通過仿真實驗和結果分析證明了算法的優異性能。
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關鍵詞:
- 智能優化算法 /
- 正弦余弦算法 /
- 差分進化算法 /
- 偵察蜂策略 /
- 全局最優個體變異策略
Abstract: In 2016, a novel naturally simulated optimization algorithm, termed the sine cosine algorithm (SCA), was proposed by Seyedali Mirjalili from Australia. This algorithm uses the sine cosine mathematical model to solve optimization problems and has attracted extensive attention from numerous scholars and researchers at home and abroad over the last few years. However, similar to other swarm intelligence optimization algorithms, SCA has numerous shortcomings in optimizing some complex function problems. To address the defects of basic SCA, such as low optimization precision, easy dropping into the local extremum, and slow convergence rate, a sine cosine algorithm based on differential evolution (SCADE) was proposed. First, the search capabilities of the new algorithm was improved by adjusting parameter r1 in a nonlinear manner and ensuring that each individual adopts the same parameters r1, r2, r3, and r4. Then, differential evolution strategies, including crossover, variation, and selection, were adopted to fully utilize the leading role of the globally optimal individual and information of other individuals in the population. This approach balanced the global exploration and local development abilities and accelerated the convergence rate of the algorithm. Next, using the reconnaissance bees’ strategy, random initialization was performed on individuals whose fitness values showed no improvement in continuous nlim times, which increased the population diversity and improved the global exploration ability of the algorithm. Moreover, the globally optimal individual variation strategy was used to conduct a fine search near the optimal solution, which enhanced the local development ability and optimization accuracy of the algorithm. Based on the above optimization strategies, the algorithm exhibits improvements and its excellent performance is validated by the result analysis of a simulation experiment. -
圖 2 收斂曲線圖。（a）F2；（b）F5；（c）F8；（d）F10；（e）F14；（f）F23
Figure 2. Convergence curves: (a) F2; (b) F5; (c) F8; (d) F10; (e) F14; (f) F23
表 1
$\sin {r_2}$ 的符號對2種分項符號的影響Table 1. Effect of the sign of sin r2 on the signs of two itemized items
Item Case 1 Case 2 Case 3 Case 4 $\left( {{r_3}p_{{\rm{g}}j}^t - x_{ij}^t} \right)$ + + ? ? $\left| {{r_3}p_{{\rm{g}}j}^t - x_{ij}^t} \right|$ + + + + $\sin {r_2}$ + ? + ? $\sin {r_2} \cdot \left( {{r_3}p_{{\rm{g}}j}^t - x_{ij}^t} \right)$ + ? ? + $\sin {r_2} \cdot \left| {{r_3}p_{{\rm{g}}j}^t - x_{ij}^t} \right|$ + ? + ? 
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表 2 各算法設置的具體參數表
Table 2. Specific parameters set by each algorithm
Algorithm Parameters SCA[2] r2∈[0, 2π]，r3∈[?2, 2]，r4∈[0, 1]，a=2，M=30，T=500 SCADE a=2，M=30，T=500，nlim=50，CR=0.3，kmax=3，h=10，δ2max=0.6，δ2min=0.0001 PSO[23] M=30，T=500，C1=1，C2=2，Vmax=4，W linearly
decreases from 0.9 to 0.4DE[24] M=30，T=500，CR=0.3，F is randomly
generated between 0.2 and 0.6ABC[25] M=30，T=500，nlim=50 m-SCA[11] M=30，T=500，JR=0.1 COSCA[6] M=30，T=500，η=1，astart=1，aend=0，pr=0.1
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表 3 SCADE與基本SCA的實驗結果
Table 3. Experimental results of SCADE and basic SCA
Function Algorithm Average optimal value Median value Best value Worst value Standard deviation Average running time/s F1 SCA 11.2180 4.0791 1.5764×10?2 81.1140 18.5910 0.0370 SCADE 9.5838×10?95 1.1814×10?102 2.0847×10?110 2.7446×10?93 4.9205×10?94 0.0182 F2 SCA 1.3204×10?2 7.5606×10?3 4.7739×10?4 1.0687×10?1 2.0170×10?2 0.0469 SCADE 6.1367×10?63 4.4107×10?68 7.8150×10?74 1.5173×10?61 2.7342×10?62 0.0286 F3 SCA 9.8213×103 7.4877×103 1.7335×103 2.8484×104 6.1244×103 0.0602 SCADE 1.9344×10?4 1.6183×10?9 4.8025×10?18 5.4721×10?3 9.8108×10?4 0.0408 F4 SCA 34.7320 34.1490 13.4260 63.9970 11.6420 0.0431 SCADE 2.8460×10?9 9.2782×10?15 1.2815×10?22 8.5125×10?8 1.5279×10?8 0.0232 F5 SCA 2.8604×104 5.8118×103 2.3455×102 4.8554×105 8.6617×104 0.1251 SCADE 26.9260 26.9170 26.6250 27.2790 1.4726×10?1 0.1116 F6 SCA 12.6590 7.3135 4.3368 86.8570 15.1820 0.0377 SCADE 7.5412×10?5 5.1302×10?5 1.3238×10?5 5.5298×10?4 9.4555×10?5 0.0182 F7 SCA 1.0554×10?1 8.6533×10?2 1.2670×10?2 3.1593×10?1 8.2888×10?2 0.0382 SCADE 8.4372×10?3 6.9961×10?3 3.1314×10?5 2.4622×10?2 7.3689×10?3 0.0190 F8 SCA ?3.7782×103 ?3.7519×103 ?4.4256×103 ?3.2776×103 2.6104×102 0.0630 SCADE ?1.2005×104 ?1.1969×104 ?1.2549×104 ?1.1267×104 2.5311×102 0.0429 F9 SCA 38.9470 25.8900 5.2761×10?3 1.5292×102 39.3920 0.0822 SCADE 0 0 0 0 0 0.0618 F10 SCA 11.1310 14.7630 8.0444×10?2 20.3510 9.2878 0.0571 SCADE 2.1282×10?15 5.8872×10?16 5.8872×10?16 4.1414×10?15 1.7605×10?15 0.0469 F11 SCA 8.7567×10?1 9.4828×10?1 5.2069×10?3 1.6297 3.3704×10?1 0.0594 SCADE 0 0 0 0 0 0.0387 F12 SCA 1.3000×103 10.2560 1.2149 2.0406×104 4.4303×103 0.0923 SCADE 3.4531×10?5 3.4338×10?6 1.3607×10?6 9.2574×10?4 1.6551×10?4 0.0582 F13 SCA 3.6900×105 1.0341×103 3.4229 3.1604×106 7.8302×105 0.0937 SCADE 8.1272×10?3 3.5569×10?5 1.6368×10?5 9.9458×10?2 2.2908×10?2 0.0591 F14 SCA 1.5288 9.9872×10?1 9.9800×10?1 2.9821 8.7639×10?1 0.0651 SCADE 9.9800×10?1 9.9800×10?1 9.9800×10?1 9.9800×10?1 4.9981×10?16 0.0658 F15 SCA 1.0426×10?3 8.5768×10?4 3.6524×10?4 1.5525×10?3 3.5566×10?4 0.0199 SCADE 7.5165×10?4 7.5211×10?4 4.2280×10?4 1.2236×10?3 1.5392×10?4 0.0185 F16 SCA ?1.0316 ?1.0316 ?1.0316 ?1.0314 6.0391×10?15 0.0029 SCADE ?1.0316 ?1.0316 ?1.0316 ?1.0316 4.4409×10?5 0.0029 F17 SCA 4.0059×10?1 3.9945×10?1 3.9789×10?1 4.1082×10?1 3.3623 ×10?3 0.0036 SCADE 3.9789×10?1 3.9789×10?1 3.9789×10?1 3.9789×10?1 0 0.0035 F18 SCA 3.0011 3 3 3.0011 2.0536×10?4 0.0032 SCADE 3 3 3 3 3.1780×10?7 0.0033 F19 SCA ?3.8545 ?3.8542 ?3.8622 ?3.8504 2.7837×10?3 0.0087 SCADE ?3.8628 ?3.8628 ?3.8628 ?3.8628 7.6401×10?13 0.0077 F20 SCA ?2.9131 ?3.0006 ?3.1491 ?2.2475 2.4164×10?1 0.0131 SCADE ?3.3119 ?3.3220 ?3.3220 ?3.2031 3.0470×10?2 0.0101 F21 SCA ?2.2308 ?8.8080×10?1 ?7.1703 ?4.9646×10?1 1.9616 0.0077 SCADE ?9.7526 ?10.1530 ?10.1530 ?6.5137 8.9107×10?1 0.0064 F22 SCA ?3.3632 ?3.8034 ?5.6727 ?9.0289×10?1 1.7261 0.0088 SCADE ?10.4029 ?10.4029 ?10.4029 ?10.4029 1.4504×10?15 0.0071 F23 SCA ?4.1765 ?4.0169 ?4.0169 ?9.4459×10?1 2.0941 0.0098 SCADE ?10.5364 ?10.5364 ?10.5364 ?10.5364 3.4495×10?14 0.0088
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表 4 SCADE與SCA改進算法及其它智能優化算法的性能比較
Table 4. Performance comparison of SCADE with modified SCA and other algorithms
Function Evaluation criterion SCA PSO DE ABC m-SCA COSCA SCADE F1 Average optimal value 11.2180 7.1569×10?3 4.9048×10?5 2.1134×10?4 2.1878×10?3 8.0426×10?81 9.5838×10?95 Standard deviation 18.5910 5.7977×10?3 1.7664×10?5 3.5538×10?4 5.0035×10?3 4.0035×10?80 4.9205×10?94 F2 Average optimal value 1.3204×10?2 3.1071 6.7504×10?4 5.7851×10?3 5.9935×10?5 1.7522×10?44 6.1367×10?63 Standard deviation 2.0170×10?2 5.2453 1.8924×10?4 3.1108×10?3 1.9175×10?4 5.1460×10?44 2.7342×10?62 F3 Average optimal value 9.8213×103 66.4684 3.5128×104 1.9536×104 2.9935×102 2.3314×10?1 1.9344×10?4 Standard deviation 6.1244×103 21.8647 6.0671×103 3.4087×103 3.5212×102 1.2441 9.8108×10?4 F4 Average optimal value 34.7320 8.7838×10?1 8.2156 67.3942 2.2007 6.0249×10?31 2.8460×10?9 Standard deviation 11.6420 1.9543×10?1 1.6408 4.9135 1.2162 2.8798×10?30 1.5279×10?8 F5 Average optimal value 2.8604×104 1.0885×102 85.6179 39.9649 33.5690 28.4280 2.6926×101 Standard deviation 8.6617×104 76.0792 5.6043×101 33.9144 12.7300 2.8280×10?1 1.4726×10?1 F6 Average optimal value 12.6590 6.0666×10?3 5.3471×10?5 6.3063×10?4 1.6795 2.0598 7.5412e-005 Standard deviation 15.1820 3.9077×10?3 3.1452×10?5 2.0140×10?3 4.6889×10?1 3.0051×10?1 9.4555×10?5 F7 Average optimal value 1.0554×10?1 4.1429 4.5135×10?2 5.4601×10?1 1.3485×10?2 2.0053×10?3 8.4372×10?3 Standard deviation 8.2888×10?2 4.7166 1.3457×10?2 1.6048×10?1 8.1723×10?3 1.4012×10?3 7.3689×10?3 F8 Average optimal value ?3.7782×103 ?4.1603×103 ?8.4186×103 ?1.1434×104 ?3.7936×103 ?4.1950×103 ?1.2005×104 Standard deviation 2.6104×102 7.9194×102 4.3777×102 1.8317×102 3.0895×102 3.3863×102 2.5311×102 F9 Average optimal value 38.9470 87.9805 1.0705×102 6.1381 3.5872 0 0 Standard deviation 39.3920 28.1184 9.4636 2.5352 7.8018 0 0 F10 Average optimal value 11.1310 1.4188×10?1 1.8656×10?3 1.6109×10?1 4.9953×10?3 1.2993×10?15 2.1282×10?15 Standard deviation 9.2878 2.3163×10?1 4.9290×10?4 2.7105×10?1 7.2695×10?3 1.4211×10?15 1.7605×10?15 F11 Average optimal value 8.7567×10?1 7.9155×10?3 4.1185×10?3 3.6220×10?2 3.9031×10?2 0 0 Standard deviation 3.3704×10?1 9.0220×10?3 1.0744×10?2 3.2998×10?2 7.2525×10?2 0 0 F12 Average optimal value 1.3000×103 1.0510×10?2 1.3328×10?5 3.4010×10?2 2.7777×10?1 1.8111×10?1 3.4531×10?5 Standard deviation 4.4303×103 3.1074×10?2 1.1565×10?5 3.2998×10?2 1.7035×10?1 1.0280×10?1 1.6551×10?4 F13 Average optimal value 3.6900×105 6.0034×10?3 6.1145×10?5 7.3970×10?4 1.9535 2.6406 8.1272×10?3 Standard deviation 7.8302×105 9.4010×10?3 3.6027×10?5 2.1983×10?4 7.1911×10?1 1.3379×10?1 2.2908×10?2 F14 Average optimal value 1.5288 2.6083 9.9800×10?1 9.9800×10?1 1.1775 3.4976 9.9800×10?1 Standard deviation 8.7639×10?1 2.3500 5.3934×10?16 4.0294×10?16 5.1443×10?1 2.5112 4.9981×10?16 F15 Average optimal value 1.0426×10?3 2.2388×10?3 1.3288×10?3 8.9459×10?4 5.3421×10?4 5.6655×10?4 7.5165×10?4 Standard deviation 3.5566×10?4 4.8585×10?3 3.5391×10?3 2.5167×10?4 1.0454×10?4 1.4934×10?4 1.5392×10?4 F16 Average optimal value ?1.0316 ?1.0316 ?1.0316 ?1.0316 ?1.0316 ?1.0316 ?1.0316 Standard deviation 6.0391×10?5 4.4409×10?16 4.4409×10?16 4.4600×10?16 4.5772×10?6 9.5922×10?7 4.4409×10?16 F17 Average optimal value 4.0059×10?1 3.9789×10?1 3.9789×10?1 3.9789×10?1 3.9793×10?1 3.9827×10?1 0.3980 Standard deviation 3.3623×10?3 0 0 9.6549×10?13 6.4763×10?5 4.2222×10?4 0 F18 Average optimal value 3.0001 3 3 3.0006 3 3.0001 3 Standard deviation 2.0536×10?4 3.4807×10?15 2.2204×10?15 1.5874×10?3 2.7799×10?5 8.7317×10?5 3.1780×10?7 F19 Average optimal value ?3.8545 ?3.8604 ?3.8628 ?3.8628 ?3.8623 ?3.8615 ?3.8628 Standard deviation 2.7837×10?3 3.6118×10?3 2.2204×10?15 7.3021×10?7 3.3231×10?4 1.3023×10?3 7.6401×10?13 F20 Average optimal value ?2.9131 ?3.1561 ?3.3037 ?3.3220 ?3.3100 ?3.1561 ?3.3119 Standard deviation 2.4164×10?1 1.2335×10?1 4.0486×10?2 1.3698×10?11 2.1606×10?2 3.7641×10?2 3.0470×10?2 F21 Average optimal value ?2.2308 ?7.3872 ?9.5674 ?10.1416 ?9.9300 ?9.6428 ?9.7526 Standard deviation 1.9616 3.0638 1.7941 5.0343×10?2 1.7054×10?1 1.2629 8.9107×10?1 F22 Average optimal value ?3.3632 ?8.9467 ?10.1484 ?10.3785 ?10.2330 ?10.2540 ?10.4029 Standard deviation 1.7261 2.6947 1.3709 8.7082×10?2 1.0363×10?1 1.0668×10?1 1.4504 F23 Average optimal value ?4.1765 ?9.0497 ?1.0325×101 ?1.0526×101 ?1.0315×101 ?1.0369×101 ?10.5364 Standard deviation 2.0941 2.7622 1.0621 2.8287×10?2 1.7846×10?1 1.2772×10?1 3.4495×10?14 Decision result + /=/ ? 0/0/23 2/2/19 5/2/16 5/0/18 2/0/21 3/2/18 —
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表 5 nlim對SCADE的性能影響
Table 5. Influence of nlim on the SCADE performance
nlim F2 (P=0.705) F8 (P=0) F23 (P=0) Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank 10 2.5689×10?63 1.3681×10?62 4 ?1.1748×104 2.8466×102 4 ?10.5364 2.9172×10?14 4 30 1.1534×10?62 6.1924×10?62 5 ?1.1633×104 3.4766×102 5 ?10.5364 2.8004×10?13 5 50 9.5748×10?66 4.3056×10?65 1 ?1.2005×104 2.5311×102 1 ?10.5364 9.2245×10?15 2 70 2.3276×10?65 1.1086×10?64 2 ?1.1938×104 2.9715×102 2 ?10.5364 2.6348×10?15 1 100 9.2052×10?64 4.8551×10?63 3 ?1.1928×104 3.1146×102 3 ?10.5364 1.7341×10?14 3
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表 6 CR對SCADE的性能影響
Table 6. Influence of CR on the SCADE performance
CR F2 (P=0.15) F8 (P=0) F23 (P=0) Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank 0.1 7.5350×10?65 2.9765×10?64 2 ?1.2451×104 1.1012×102 1 ?10.5080 1.4802×10?1 4 0.2 1.3455×10?60 6.9665×10?60 5 ?1.2144×104 2.3487×102 2 ?10.5364 7.7044×10?6 3 0.3 9.0366×10?67 2.5547×10?66 1 ?1.1740×104 2.4093×102 3 ?10.5364 3.7542×10?15 1 0.4 5.1033×10?62 2.7365×10?61 4 ?1.1325×104 5.0817×102 4 ?10.5364 3.7808×10?13 2 0.5 1.3633×10?64 5.8753×10?64 3 ?1.0847×104 6.3442×102 5 ?10.3564 9.7075×10?1 5
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表 7 h對SCADE的性能影響
Table 7. Influence of h on the SCADE performance
h F2 (P=0) F8 (P=0.004) F23 (P=0) Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank Average optimal value Standard deviation Rank 5 2.0355×10?130 1.0958×10?129 1 ?1.1898×104 3.1830×102 2 ?10.5364 3.1104×10?12 3 10 7.6735×10?66 3.1669×10?66 2 ?1.1920×104 2.7977×102 1 ?10.5364 3.8374×10?15 1 15 2.4968×10?42 1.2590×10?41 3 ?1.1830×104 2.6020×102 4 ?10.5160 0.1120 4 20 2.1214×10?31 8.8367×10?31 4 ?1.1893×104 3.2600×102 3 ?10.5150 1.1756×10?1 5 25 8.7347×10?24 3.4513×10?23 5 ?1.1804×104 2.4240×102 5 ?10.5364 7.4506×10?14 2
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