层级引导的增强型多目标萤火虫算法

doi:10.16182/j.issn1004731x.joss.22-1486

摘要/Abstract

摘要：

针对多目标萤火虫算法在求解过程中易产生振荡和聚集现象，导致开发能力较弱、求解精度不佳的问题，提出一种层级引导的增强型多目标萤火虫算法（hierarchical guided enhanced multi- objective firefly algorithm， HGEMOFA）。构建层级引导模型，利用非支配排序获得不同层级个体，用优势层个体引导劣势层个体进化，明确引导方向，解决了进化过程中出现的振荡，减少了聚集现象的出现，增强了算法收敛性；引入莱维飞行扰动最优层个体，增强算法的全局搜索能力；每代进化完成后，对当前种群采用变异机制，增强算法的局部开发能力；把变异后的种群和前一代种群合并进行环境选择，筛选出和前一代种群规模相同的子代，避免优势解丢失。实验结果表明：HGEMOFA能有效增强解的收敛性和多样性。

关键词: 多目标优化, 萤火虫算法, 层级引导, 莱维飞行, 变异

Abstract:

The multi-objective firefly algorithm is easy to produce oscillation and aggregation phenomenon in the solution process, which leads to weak development ability and poor solution accuracy. This paper proposes a hierarchical guided enhanced multi-objective firefly algorithm (HGEMOFA). HGEMOFA builds a hierarchical guidance model, uses non-dominated sorting to obtain different levels of individuals. The individuals in the dominant layer are used to guide the evolution of the individuals in the inferior layer, the guidance direction is clear, the oscillation in the evolution process is solved, the aggregation phenomenon is reduced, and the convergence of the algorithm is enhanced. The Lévy flight is introduced to disturb the optimal layer individuals to enhance the global search ability of the algorithm; After each generation of evolution, the mutation mechanism is adopted for the current population to enhance the local development ability of the algorithm; The mutated population is combined with the previous generation population for environmental selection to screen out offspring with the same population size as the previous generation to avoid loss of dominance solution. The experimental results show that HGEMOFA can effectively enhance the convergence and diversity of solutions.

Key words: multi-objective optimization, firefly algorithm, hierarchical guidance, Lévy flight, mutation

中图分类号:

TP391.9

赵嘉,赖智臻,吴润秀等 . 层级引导的增强型多目标萤火虫算法[J]. 系统仿真学报, 2024, 36(5): 1152-1164.

Zhao Jia,Lai Zhizhen,Wu Runxiu,et al . Hierarchical Guided Enhanced Multi-objective Firefly Algorithm[J]. Journal of System Simulation, 2024, 36(5): 1152-1164.

图/表 12

图1

表1

测试函数集

问题	目标数	约束	问题特征
ZDT1	2	$n = 30$	凸型
ZDT2	2	$n = 30$	凹型
ZDT3	2	$n = 30$	非连续型
ZDT6	2	$n = 10$	凹型+多模态+有偏
DTLZ1	3	$n = 7$	线性+多模态
DTLZ2	3	$n = 12$	凹型
DTLZ4	3	$n = 12$	凹型+有偏
DTLZ6	3	$n = 12$	凹型+退化+有偏
DTLZ7	3	$n = 12$	不连续+混合+多模态

表1

表2

表3

表4

表5

表6

图2

表7

表8

表9

表10

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算法	GD	SP	IGD
HGEMOFA	1.44	1.44	1.22
MOPSO	5.67	4.33	5.67
MOEA/D	2.89	3.89	3.56
NSGAII	2.56	3.00	2.56
PESAII	3.33	4.44	3.11
MOFA	5.11	3.89	4.89

算法	ZDT1	ZDT2	ZDT3	ZDT6	DTLZ1	DTLZ2	DTLZ4	DTLZ6	DTLZ7
HGEMOFA	3.82e-03	3.87e-03	5.24e-03	3.10e-03	1.62e-02	3.67e-02	3.96e-02	2.04e-03	3.95e-02
CMOPSO	4.21e-03	4.12e-03	4.64e-03	3.10e-03	7.57e-01	3.97e-02	4.12e-02	2.09e-03	5.23e-02
NMPSO	2.79e-02	1.92e-02	1.01e-01	4.41e-03	1.69e-02	5.67e-02	1.06e-01	1.29e-02	4.58e-02
NSLS	2.43e-01	4.49e-01	2.17e-01	5.89e-03	1.54e-01	3.79e-02	1.48e-01	2.19e-03	8.58e-02
CFMOFA	8.79e-03	1.31e-02	1.45e-02	9.77e-02	1.56e+02	6.01e-02	9.10e-02	5.65e-01	5.59e-02
RVEAiGNG	4.07e-03	4.14e-03	7.80e-03	3.28e-03	1.44e-02	4.00e-02	3.97e-02	2.32e-03	5.00e-02
MMOPSO	4.91e-03	5.03e-03	5.41e-03	4.29e-03	2.82e-02	4.93e-02	4.88e-02	3.37e-03	9.42e-02
SMPSO	4.95e-03	4.96e-03	5.37e-03	3.86e-03	1.27e+00	4.86e-02	2.54e-01	2.64e-03	8.82e-02
NSGAIISDR	6.92e-03	5.23e-03	1.38e-02	4.40e-03	1.84e-02	4.35e-01	3.87e-01	3.40e-02	5.97e-02
MOPSOCD	4.00e-03	4.11e-03	4.75e-03	3.39e-03	1.19e+01	4.83e-02	1.18e-01	2.63e-03	5.73e-02
dMOPSO	1.52e-02	1.33e-01	1.66e-02	3.22e-03	1.45e+00	9.03e-02	2.39e-01	2.25e-02	1.00e-01
MOEAPSL	4.62e-03	4.75e-03	6.27e-03	3.62e-03	1.97e-02	5.04e-02	4.91e-02	2.66e-03	5.43e-02

平均排名	算法	秩平均值
1	HGEMOFA	1.33
2	CMOPSO	3.33
3	RVEAiGNG	3.56
4	MOPSOCD	5.22
5	MOEAPSL	5.78
6	MMOPSO	6.89
7	SMPSO	7.33
8	NMPSO	8.00
9	NSLS	8.56
10	NSGAIISDR	8.89
11	CFMOFA	9.44
12	dMOPSO	9.67

问题	MOFA	MOFA+S1	MOFA+S2	MOFA+S1+S2	MOFA+S2+S3	HGEMOFA
ZDT1	3.70e-02	7.36e-03	6.08e-03	5.98e-03	4.19e-03	3.82e-03
ZDT2	4.62e-02	7.47e-03	6.35e-03	6.11e-03	4.19e-03	3.86e-03
ZDT3	1.87e-02	1.14e-02	7.12e-03	7.07e-03	5.89e-03	5.24e-03
ZDT6	1.56e+0	6.80e-03	5.93e-03	5.79e-03	3.47e-03	3.09e-03
DTLZ1	5.21e-02	2.47e-01	3.31e-02	2.64e-02	2.05e-02	1.62e-02
DTLZ2	1.30e-01	5.19e-02	5.11e-02	5.08e-02	4.10e-02	3.67e-02
DTLZ4	1.32e-01	6.30e-02	4.94e-02	4.86e-02	4.22e-02	3.95e-02
DTLZ6	3.13e+00	3.92e-03	3.27e-03	3.34e-03	2.35e-03	2.04e-03
DTLZ7	8.45e-02	5.92e-02	5.37e-02	5.33e-02	4.22e-02	3.94e-02

平均排名	算法	秩平均值
1	HEGMOFA	1.00
2	MOFA+S2+S3	2.00
3	MOFA+S1+S2	3.11
4	MOFA+S2	3.89
5	MOFA+S1	5.11
6	MOFA	5.89