多策略融合改进的飞蛾火焰优化算法

何加文; 许贤泽; 高波

doi:10.13700/j.bh.1001-5965.2022.0707

多策略融合改进的飞蛾火焰优化算法

doi: 10.13700/j.bh.1001-5965.2022.0707

武汉大学电子信息学院，武汉 430072

基金项目: 国家自然科学基金(51975422)

详细信息

通讯作者:
E-mail：xxz@whu.edu.cn

中图分类号: TP301
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出版历程
- 收稿日期: 2022-08-11
- 录用日期: 2022-09-02
- 网络出版日期: 2022-11-08
- 整期出版日期: 2024-09-27

Improved moth-flame optimization algorithm with multi-strategy integration

Electronic Information School，Wuhan University，Wuhan 430072，China

Funds: National Natural Science Foundation of China (51975422)

More Information

Corresponding author: E-mail：xxz@whu.edu.cn

摘要

摘要:
针对飞蛾火焰优化算法容易出现局部最优解、接近全局最优时开发能力不足等问题，提出一种多策略融合改进的飞蛾火焰优化 (RGMFO) 算法。在每次迭代开始时，使用随机反向学习策略获得高质量飞蛾种群；利用高斯变异将较差的火焰个体替换为优秀个体；使用阿基米德螺线、权重因子和围绕最优火焰飞行3种方式改进飞蛾更新机制。为验证所提算法的有效性，与11个不同类型的基准函数进行寻优测试，基准函数实验结果与秩和检验表明：相较于其他优化算法及其他MFO改进算法，所提算法具有更好的全局搜索能力与更高的寻优精度。将所提算法应用于减速器设计和槽形舱壁设计的实际工程场景中，以进一步验证算法的实用性与可行性。
- 飞蛾火焰优化算法 /
- 多策略融合 /
- 随机反向学习 /
- 高斯变异 /
- 智能优化
Abstract:
An enhanced moth-flame optimization algorithm with multi-strategy integration (RGMFO) was proposed in order to address the issues that the moth-flame optimization algorithm was prone to falling into the local optimum and that its exploitation ability was insufficient while it was approaching the global optimum. To generate high-quality moth populations, a random opposition-based learning strategy was applied at the start of each iteration. To generate high-quality moth populations, a random opposition-based learning strategy was applied at the start of each iteration. Gaussian mutation was then used to swap out subpar flame individuals with superior ones. Archimedes spirals, weighting factors, and the surrounding of a small number of optimal flames were employed to enhance the moth update mechanism. The proposed algorithm was tested on 11 benchmark functions of different types. The test results and rank sum test show that RGMFO has better global search ability and higher search accuracy. Lastly, RGMFO is applied to the engineering scenarios of reducer design and trough bulkhead design, which further verifies the practicability and feasibility of the algorithm.
- moth-flame optimization algorithm /
- multi-strategy integration /
- random opposition-based learning /
- Gaussian mutation /
- intelligent optimization

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图 1 E值折线图及散点图（500次迭代）

Figure 1. E-value line chart and scatter chart (500 iterations)

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图 2 RGMFO算法流程

Figure 2. RGMFO algorithm flow

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图 3 RGMFO算法与对比算法迭代收敛曲线

Figure 3. Iterative convergence curves of RGMFO algorithm and comparative algorithms

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表 1 各算法参数设置

Table 1. Parameter setting of each algorithms

算法	参数
RGMFO	b=1，e₁=0.5，e₂=1，β=10
IMFO	b=1，A=0.5，B=0.6，c_rmin=0.55，c_rmax=0.65 σ_min=0.1，σ_max=0.15
CGMFO	b=2，β=8
ODSFMFO	m=6，γ=5，P_c=0.5，a_alpha=1， b_beta=1.5，l=10，b=1
MFO	b=1
GWO	a从2线性递减至0，r₁、r₂∈[0,1]
SCA	a=2，r₁从a线性递减至0
PSO	V_max=8，V_min=−8，w=0.2，c₁=3，c₂=3
DE	p_c=0.5
GA	p_c=0.7，p_m=0.1

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表 2 基准测试函数

Table 2. Benchmark functions

函数类型	基准测试函数	维度	搜索区间	最优值
高维单峰	$ {F_1}(x) = \displaystyle\sum\limits_{i = 1}^d {x_i^2} $	30	[−100,100]^d	0
	$ {F_2}(x) = \displaystyle\sum\limits_{i = 1}^d {\left\| {{x_i}} \right\|} + \displaystyle\prod\limits_{i = 1}^d {\left\| {{x_i}} \right\|} $	30	[−10,10]^d	0
	$ {F_3}(x) = \displaystyle\sum\limits_{i = 1}^d {{{\left(\displaystyle\sum\limits_{j = 1}^i {{x_j}} \right)}^2}} $	30	[−100,100]^d	0
	$ {F_4}(x) = {\max _i}\{ \left\| {{x_i}} \right\|,1 \leqslant i \leqslant d\} $	30	[−100,100]^d	0
	$ {F_5}(x) = \displaystyle\sum\limits_{i = 1}^{d{{ - }}1} {[100{{({x_{i + 1}} - {x_i})}^2} + {{({x_i} - 1)}^2}]} $	30	[−30,30]^d	0
	$ {F_6}(x) = \displaystyle\sum\limits_{i = 1}^d {i \cdot x_i^4} + {\text{random[}}0,1{\text{)}} $	30	[−1.28,1.28]^d	0
高维多峰	$ {F_7}(x) = \displaystyle\sum\limits_{i = 1}^d {[x_i^2 - 10\cos (2{\text{π}}{x_i} + 10)]} $	30	[−5.12,5.12]^d	0
	$ {F_8}(x) = - 20\exp \left( - 0.2\sqrt {\dfrac{1}{d}\displaystyle\sum\limits_{i = 1}^d {x_i^2} } \right) - \exp \left(\dfrac{1}{d}\displaystyle\sum\limits_{i = 1}^d {\cos (2{\text{π}}{x_i})} \right) + 20 + {\text{e}} $	30	[−32,32]^d	0
	$ {F_9}(x) = \dfrac{1}{{4\;000}}\displaystyle\sum\limits_{i = 1}^d {x_i^2} - \displaystyle\prod\limits_{i = 1}^d {\cos \left(\dfrac{{{x_i}}}{{\sqrt i }}\right)} + 1 $	30	[−600,600]^d	0
	$ {F_{10}}(x) = 0.1 \left\{ {\sin ^2}(3{\text{π}}{x_1}) + \displaystyle\sum\limits_{i = 1}^d {{{({x_i} - 1)}^2}[1 + {{\sin }^2}(3{\text{π}}{x_1} + 1)]} + {({x_d} - 1)^2}[1 + {\sin ^2}(2{\text{π}}{x_d})]\right\} + \displaystyle\sum\limits_{i = 1}^d {u({x_i},5,100,4)} $	30	[−50,50]^d	0
低维多峰	$ {F_{11}}(x) = - \displaystyle\sum\limits_{i = 1}^{10} {{{[(X - {{\boldsymbol{a}}_{{\mathrm{SHi}}}}){{(X - {{\boldsymbol{a}}_{{\mathrm{SHi}}}})}^{\text{T}}} + {{\boldsymbol{c}}_{{\mathrm{SHi}}}}]}^{ - 1}}} $	4	[0,10]⁴	−10.536

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表 3 RGMFO算法及其他算法高维单峰函数寻优结果

Table 3. High-dimensional single-peak function optimization results of RGMFO algorithm and other algorithms

算法	平均值						标准差
算法	F₁	F₂	F₃	F₄	F₅	F₆	F₁	F₂	F₃	F₄	F₅	F₆
RGMFO	0	1.68×10⁻²⁰⁸	0	4.19×10⁻²⁰⁸	1.98×10⁻⁵	1.07×10⁻⁴	0	0	0	0	3.07×10⁻⁵	1.2×10⁻⁴
IMFO	2.68×10⁻⁷	5.13×10⁻⁴	5850	17.6	15700	0.117	5.15×10⁻⁷	6.74×10⁻⁴	3440	4.85	11000	0.0447
CGMFO	1.66×10⁻²³	7.32×10⁻¹⁴	1.81×10⁻¹⁷	7.50×10⁻¹⁵	1.46×10⁻³	0.0268	7.45×10⁻²³	2.92×10⁻¹³	9.93×10⁻¹⁷	1.66×10⁻¹⁴	2.92×10⁻³	0.0218
ODSFMFO	1.96×10⁻²⁴	4.98×10⁻¹³	1.40×10⁻¹⁴	8.87×10⁻¹⁴	0.017	0.0563	8.54×10⁻²⁴	1.55×10⁻¹²	6.08×10⁻¹⁴	2.12×10⁻¹³	0.0459	0.0403
MFO	4970	20.3	59600	22.6	494	0.824	2350	4.54	58600	6.94	2690	0.439
GWO	9.68×10⁻²⁸	1.01×10⁻¹⁶	2.09×10⁻⁵	5.60×10⁻⁷	27.9	2.18×10⁻³	1.65×10⁻²⁷	1.19×10⁻¹⁶	4.99×10⁻⁵	4.36×10⁻⁷	0.851	1.48×10⁻³
PSO	12.2	0.0269	10500	37.2	7480	0.1	17.8	0.041	8450	9.88	26100	0.0839
DE	0.399	0.402	1120	0.179	36.2	7.43×10⁻³	0.344	0.312	2940	0.124	7.33	3.91×10⁻³
GA	2.12×10⁻⁵	2.61×10⁻⁴	18500	14.1	752	0.314	1.31×10⁻⁵	6.69×10⁻⁵	3530	4.79	460	0.0679

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表 4 RGMFO算法及其他算法高维多峰函数寻优结果

Table 4. High-dimensional multi-peak function optimization results of RGMFO algorithm and other algorithms

算法	平均值				标准差
算法	F₇	F₈	F₉	F₁₀	F₇	F₈	F₉	F₁₀
RGMFO	0	8.88×10⁻¹⁶	0	7.58×10⁻⁶	0	0	0	1.10×10⁻⁵
IMFO	69.9	1.16	8.86×10⁻³	7.16	19.5	0.246	9.33×10⁻³	5.34
CGMFO	0	1.60×10⁻¹⁴	0	0.0226	0	4.58×10⁻¹⁴	0	0.0309
ODSFMFO	0	1.92×10⁻¹⁴	0	0.0231	0	4.16×10⁻¹⁴	0	0.0214
MFO	113	2.58	36.2	3.87×10⁶	25	0.321	13.8	3.51×10⁶
GWO	4.09	0.0215	3.73×10⁻³	0.653	6.06	0.0562	8.01×10⁻³	0.26
PSO	39.9	1.07	0.876	1.90×10⁵	30.6	0.529	0.357	8.09×10⁵
DE	33.2	0.395	0.591	2.78	35.6	0.387	0.296	0.639
GA	27.4	0.631	1.33×10⁻⁴	0.358	6.85	0.0874	3.17×10⁻⁴	0.965

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表 5 RGMFO算法及其他算法低维多峰函数寻优结果

Table 5. Low-dimensional multi-peak function optimization results of RGMFO algorithm and other algorithms

算法	平均值	标准差
RGMFO	−10.5	4.47×10⁻⁹
IMFO	−8.2	3.43
CGMFO	−10.5	4.34×10⁻⁴
ODSFMFO	−8.19	2.72
MFO	−8.72	2.88
GWO	−9.99	2.06
PSO	−3.89	1.65
DE	−10.5	6.09×10⁻³
GA	−10.4	0.987

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表 6 Wilcoxon秩和检验p值

Table 6. P-value for Wilcoxon rank sum test

函数	RGMFO-IMFO	RGMFO-CGMFO	RGMFO-ODSFMFO	RGMFO-MFO	RGMFO-GWO	RGMFO-SCA	RGMFO-PSO	RGMFO-DE	RGMFO-GA
F₁	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²
F₂	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹
F₃	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²
F₄	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹
F₅	3.02×10⁻¹¹	2.44×10⁻⁹	3.69×10⁻¹¹	5.57×10⁻¹⁰	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹
F₆	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹
F₇	1.21×10⁻¹²			1.21×10⁻¹²	1.94×10⁻⁹	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²
F₈	1.21×10⁻¹²	2.82×10⁻⁵	2.16×10⁻⁶	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²
F₉	1.21×10⁻¹²			1.21×10⁻¹²	0.011	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²	1.21×10⁻¹²
F₁₀	3.02×10⁻¹¹	3.34×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.01×10⁻⁷	3.02×10⁻¹¹
F₁₁	0.0254	3.47×10⁻¹⁰	2.30×10⁻⁷	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	3.02×10⁻¹¹	2.20×10⁻¹⁰	3.02×10⁻¹¹

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表 7 RGMFO算法及其他算法寻优运算时间

Table 7. Optimization operation time of RGMFO algorithm and other algorithms s

函数	RGMFO	IMFO	CGMFO	ODSFMFO	MFO	GWO	SCA	PSO	DE	GA
F₁	0.257 9	0.163 1	0.084 2	0.785 5	0.0675	0.2406	0.1729	0.0579	0.1858	0.0850
F₂	0.469 5	0.363 9	0.193 7	1.020 6	0.0716	0.2331	0.1767	0.0606	0.1223	0.0750
F₃	0.503 8	0.389 3	0.222 8	1.036 6	0.0905	0.2476	0.1948	0.0761	0.1404	0.0839
F₄	0.535 0	0.373 7	0.198 1	1.023 0	0.0702	0.2257	0.1815	0.1136	0.1206	0.0729
F₅	0.567 6	0.357 8	0.183 3	1.012 7	0.0648	0.2369	0.1685	0.0569	0.1455	0.0656
F₆	0.980 2	0.747 9	0.366 2	1.608 0	0.2153	0.3963	0.3933	0.2093	0.2361	0.1320
F₇	0.718 6	0.426 6	0.200 4	1.324 6	0.0961	0.2395	0.1878	0.0901	0.1378	0.0779
F₈	0.732 1	0.466 9	0.230 5	1.494 8	0.1001	0.2531	0.2131	0.0924	0.1401	0.0800
F₉	0.727 0	0.605 8	0.320 0	1.995 9	0.1672	0.3148	0.2644	0.1271	0.2096	0.1371
F₁₀	1.172 2	0.740 3	0.415 1	1.930 5	0.3277	0.4571	0.4770	0.3205	0.6867	0.4184
F₁₁	0.809 2	0.473 8	0.346 4	1.463 1	0.1527	0.1711	0.1607	0.1379	0.3427	0.2041

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表 8 高维单峰函数消融实验结果

Table 8. High-dimensional single-peak function experimental results of model ablation

算法	平均值						标准差
算法	F₁	F₂	F₃	F₄	F₅	F₆	F₁	F₂	F₃	F₄	F₅	F₆
RGMFO	0	1.68×10⁻²⁰⁸	0	4.19×10⁻²⁰⁸	1.98×10⁻⁵	1.07×10⁻⁴	0	0	0	0	3.07×10⁻⁵	1.20×10⁻⁴
GMFO	2.00×10⁻²¹⁰	6.63×10⁻¹¹⁵	2.59×10⁻¹⁶⁵	3.66×10⁻¹¹⁰	2.66	0.0225	0	3.16×10⁻¹¹⁴	0	1.86×10⁻¹⁰⁹	3.23	0.0164
RMFO	0.263	1.66	7.38	0.178	5.72×10⁻⁵	0.62	0.144	0.823	5.47	0.0888	2.15×10⁻⁴	1.02
MFO	4970	20.3	59600	22.6	494	0.824	2350	4.54	58600	6.94	2690	0.439

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表 9 高维多峰函数消融实验结果

Table 9. High-dimensional multi-peak function experimental results of model ablation

算法	平均值				标准差
算法	F₇	F₈	F₉	F₁₀	F₇	F₈	F₉	F₁₀
RGMFO	0	8.88×10⁻¹⁶	0	7.58×10⁻⁶	0	0	0	1.10×10⁻⁵
GMFO	0	8.88×10⁻¹⁶	0	0.145	0	0	0	0.0615
RMFO	0.477	0.0147	0.0103	2.37×10⁻⁵	0.947	0.016	6.74×10⁻³	4.97×10⁻⁵
MFO	113	2.58	36.2	3.87×10⁶	25	0.321	13.8	3.51×10⁶

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表 10 低维多峰函数消融实验结果

Table 10. Low-dimensional multi-peak function experimental results of model ablation

算法	平均值	标准差
RGMFO	−10.5	4.47×10⁻⁹
GMFO	−7.72	2.66
RMFO	−5.13	1.83×10⁻¹²
MFO	−8.72	2.88

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表 11 减速器设计变量边界约束

Table 11. Reducer design variable boundary constraints

变量	下界	上界
x₁	2.6	3.6
x₂	0.7	0.8
x₃	17	28
x₄	7.3	8.3
x₅	7.8	8.3
x₆	2.9	3.9
x₇	5.0	5.5

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表 12 减速器设计结果

Table 12. Reducer design results

算法	x₁	x₂	x₃	x₄	x₅	x₆	x₇	适应度值
RGMFO	3.50	0.70	17.00	7.30	7.80	3.35	5.29	2996.41
IMFO	3.50	0.70	17.00	7.30	7.80	3.35	5.29	2996.41
CGMFO	3.50	0.70	17.00	7.60	7.80	3.80	5.29	3130.09
ODSFMFO	3.50	0.70	17.00	7.30	7.80	3.35	5.29	2996.41
MFO	3.50	0.70	17.00	7.30	7.80	3.35	5.37	3047.56
GWO	3.50	0.70	17.00	7.39	7.85	3.36	5.29	3003.19
SCA	3.60	0.70	17.00	8.30	8.30	3.45	5.37	3133.94
PSO	3.60	0.70	17.00	7.30	8.28	3.35	5.29	3046.37
DE	3.50	0.70	17.00	7.30	7.80	3.35	5.29	2996.41
GA	3.57	0.70	17.16	7.67	7.82	3.43	5.30	3096.49

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表 13 槽形舱壁设计结果

Table 13. Trough bulkhead design results

算法	x₁	x₂	x₃	x₄	适应度值
RGMFO	37.51	33.11	37.51	0.74	5.89
IMFO	37.83	33.19	37.83	0.74	5.89
CGMFO	31.34	32.48	37.88	0.78	6.23
ODSFMFO	38.89	33.46	38.89	0.76	5.90
MFO	37.28	33.23	38.13	0.74	5.90
GWO	37.69	33.20	37.85	0.74	5.90
SCA	32.71	35.53	46.11	0.90	6.70
PSO	40.91	33.30	39.36	0.79	6.02
DE	37.49	33.11	37.49	0.73	5.89
GA	90.70	33.53	75.91	1.67	10.29

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