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基于种群分区的多策略自适应多目标粒子群算法

张伟 黄卫民

张伟, 黄卫民. 基于种群分区的多策略自适应多目标粒子群算法. 自动化学报, 2022, 48(10): 2585−2599 doi: 10.16383/j.aas.c200307
引用本文: 张伟, 黄卫民. 基于种群分区的多策略自适应多目标粒子群算法. 自动化学报, 2022, 48(10): 2585−2599 doi: 10.16383/j.aas.c200307
Zhang Wei, Huang Wei-Min. Multi-strategy adaptive multi-objective particle swarm optimization algorithm based on swarm partition. Acta Automatica Sinica, 2022, 48(10): 2585−2599 doi: 10.16383/j.aas.c200307
Citation: Zhang Wei, Huang Wei-Min. Multi-strategy adaptive multi-objective particle swarm optimization algorithm based on swarm partition. Acta Automatica Sinica, 2022, 48(10): 2585−2599 doi: 10.16383/j.aas.c200307

基于种群分区的多策略自适应多目标粒子群算法

doi: 10.16383/j.aas.c200307
基金项目: 国家自然科学基金(61703145), 河南省高校科技创新团队(20-IRTSTHN019) 资助
详细信息
    作者简介:

    张伟:河南理工大学电气工程与自动化学院教授. 2016年获北京工业大学博士学位. 主要研究方向为智能特征建模, 神经网络设计和污水处理过程的优化控制. 本文通信作者. E-mail: zwei1563@126.com

    黄卫民:河南理工大学电气工程与自动化学院硕士研究生. 2019年获中原工学院学士学位. 主要研究方向为多目标优化问题及神经网络结构优化和算法设计. E-mail: hweimin97520@163.com

Multi-strategy Adaptive Multi-objective Particle Swarm Optimization Algorithm Based on Swarm Partition

Funds: Supported by National Natural Science Foundation of China (61703145) and Scientific and Technological Innovation Team of Colleges and Universities in Henan Province (20IRTSTHN019)
More Information
    Author Bio:

    ZHANG Wei Professor at the Sch-ool of Electrical Engineering and Automation, Henan Polytechnic University. She received her Ph.D. degree from Beijing University of Technology in 2016. Her research interest covers intelligent feature modeling, design of neural networks, and optimization control for wastewater treatment process. Corresponding author of this paper

    HUANG Wei-Min Master stude-nt at the School of Electrical Engineering and Automation, Henan Polytechnic University. He received his bachelor degree from Zhongyuan University of Technology in 2019. His research interest covers multi-objective optimization problem and neural network structure optimization and algorithm design

  • 摘要: 在多目标粒子群优化算法中, 平衡算法收敛性和多样性是获得良好分布和高精度Pareto前沿的关键, 多数已提出的方法仅依靠一种策略引导粒子搜索, 在解决复杂问题时算法收敛性和多样性不足. 为解决这一问题, 提出一种基于种群分区的多策略自适应多目标粒子群优化算法. 采用粒子收敛性贡献对算法环境进行检测, 自适应调整粒子的探索和开发过程; 为准确制定不同性能的粒子的搜索策略, 提出一种多策略的全局最优粒子选取方法和多策略的变异方法, 根据粒子的收敛性评价指标, 将种群划分为3个区域, 将粒子性能与算法寻优过程结合, 提升种群中各个粒子的搜索效率; 为解决因选取的个体最优粒子不能有效指导粒子飞行方向, 使算法停滞, 陷入局部最优的问题, 提出一种带有记忆区间的个体最优粒子选取方法, 提升个体最优粒子选取的可靠性并加快粒子收敛过程; 采用包含双性能测度的融合指标维护外部存档, 避免仅根据粒子密度对外部存档维护时, 删除收敛性较好的粒子, 导致种群产生退化, 影响粒子开发能力. 仿真实验结果表明, 与其他几种多目标优化算法相比, 该算法具有良好的收敛性和多样性.
  • 图  1  spmsAMOPSO算法整体框图

    Fig.  1  Frame of spmsAMOPSO algorithm

    图  2  种群中各粒子所属区域

    Fig.  2  Location of each particles in the population

    图  3  粒子记忆区间更新过程

    Fig.  3  Update process of particle memory interval

    图  4  $\eta_{1}$$\eta_{2}$的不同取值方案下IGD指标变化

    Fig.  4  IGD metric changes of different $\eta_{1}$ and $\eta_{2}$ value schemes

    图  5  不同分区参数取值方案下IGD指标变化

    Fig.  5  IGD metric changes of different partition parameter value schemes

    图  6  不同多目标优化算法对ZDT3函数的Pareto前沿

    Fig.  6  Pareto front of ZDT3 function of different multi-objective optimization algorithms

    图  7  不同多目标优化算法对DTLZ2函数的Pareto前沿

    Fig.  7  Pareto front of DTLZ2 function of different multi-objective optimization algorithms

    图  8  不同多目标优化算法对DTLZ7函数的Pareto前沿

    Fig.  8  Pareto front of DTLZ7 function of different multi-objective optimization algorithms

    图  9  7种多目标优化算法在测试ZDT3、DTLZ2和DTLZ7问题时IGD、SP以及ER指标的箱形图

    Fig.  9  Box plots of IGD, SP and ER metric on ZDT3, DTLZ2 and DTLZ7 problems of multi-objective optimization algorithms

    表  1  本文算法与其他多目标粒子群算法的IGD评价指标对比

    Table  1  Results of IGD metric of the proposed algorithm and MOPSOs

    测试函数spmsAMOPSOclusterMOPSO[23]cdMOPSO[8]pccsAMOPSO[13]CMOPSO[27]
    平均值标准差平均值标准差平均值标准差平均值标准差平均值标准差
    ZDT1${\bf{2.34\times 10^{-3}}}$${\bf{5.46\times 10^{-6}}}$$1.25\times 10^{-2}$$1.78\times 10^{-3}$$4.24\times 10^{-3}$$2.58\times 10^{-4}$$3.83\times 10^{-3}$$2.67\times 10^{-4}$$3.82\times 10^{-3}$$2.15\times 10^{-5}$
    ZDT2${\bf{1.98\times 10^{-3}}}$${\bf{6.18\times 10^{-6}}}$$1.78\times10^{-2}$$5.09\times 10^{-3}$$4.28\times 10^{-3}$$1.14\times 10^{-4}$$3.81\times 10^{-3}$$5.81\times 10^{-5}$$3.86\times 10^{-3}$$2.83\times 10^{-5}$
    ZDT3${\bf{8.83\times 10^{-4}}}$${\bf{1.02\times 10^{-5}}}$$1.05\times 10^{-1}$$7.05\times 10^{-2}$$3.06\times 10^{-3}$$7.13\times 10^{-5}$$4.91\times 10^{-3}$$6.56\times 10^{-4}$$4.50\times 10^{-3}$$2.83\times 10^{-5}$
    ZDT4${\bf{2.32\times 10^{-3}}}$${\bf{3.96\times 10^{-6}}}$$3.99\times 10^{0} $$2.61\times 10^{0} $$5.91\times 10^{-1}$$4.52\times 10^{-1}$$5.79\times 10^{-3}$$2.98\times 10^{-4}$$3.70\times 10^{-2}$$4.59\times 10^{-2}$
    ZDT6${\bf{1.60\times 10^{-3}}}$${\bf{5.20\times 10^{-7}}}$$4.39\times 10^{-1}$$2.37\times 10^{-2}$$2.99\times 10^{-3}$$1.54\times 10^{-4}$$4.68\times 10^{-3}$$7.67\times 10^{-4}$$3.09\times 10^{-3}$$2.61\times 10^{-5}$
    DTLZ1${\bf{1.23\times 10^{-2}}}$$1.23\times 10^{-3}$$4.15\times 10^{1} $$2.29\times 10^{1} $$2.75\times 10^{1}$$9.34\times 10^{0}$$1.36\times 10^{-1}$$1.12\times 10^{-1}$$4.44\times 10^{-2}$$7.83\times 10^{-2}$
    DTLZ2${\bf{3.62\times 10^{-3}}}$$8.56\times 10^{-5}$$1.26\times 10^{-1}$$1.68\times 10^{-2}$$1.02\times 10^{-1}$$1.34\times 10^{-2}$$6.14\times 10^{-2}$$1.89\times 10^{-3}$$4.40\times 10^{-3}$${\bf{3.61\times 10^{-5}}}$
    DTLZ3${\bf{3.28\times 10^{-3}}}$${\bf{9.62\times 10^{-5}}}$$4.76\times 10^{1}$$2.87\times 10^{1}$$4.46\times 10^{1}$$1.02\times 10^{1}$$2.19\times 10^{-1}$$1.85\times 10^{-1}$$4.24\times 10^{-3}$$1.71\times 10^{-4}$
    DTLZ4$7.58\times 10^{-3}$$3.45\times 10^{-3}$$2.28\times 10^{-1}$$8.45\times 10^{-2}$$1.02\times 10^{-1}$$3.66\times 10^{-2}$$ {\bf{4.21\times 10^{-3} }} $$3.37\times 10^{-3}$$4.41\times 10^{-3}$$7.58\times 10^{-5}$
    DTLZ5$9.76\times 10^{-3}$$6.42\times 10^{-4}$$1.62\times 10^{-4}$$2.48\times 10^{-3}$$6.05\times 10^{-3}$$7.29\times 10^{-4}$$1.18\times 10^{-2}$$2.48\times 10^{-3}$${\bf{4.40\times 10^{-3}}}$$5.66\times 10^{-5}$
    DTLZ6${\bf{3.62\times 10^{-3}}}$$3.47\times 10^{-5}$$7.38\times 10^{-2}$$1.55\times 10^{-1}$$5.23\times 10^{-3}$$3.90\times 10^{-4}$$5.04\times 10^{-3}$$2.27\times 10^{-4}$$4.09\times 10^{-3}$${\bf{1.83\times 10^{-5}}}$
    DTLZ7${\bf{4.23\times 10^{-3}}}$$1.37\times 10^{-4}$$4.02\times 10^{-2}$$2.06\times 10^{-3}$$5.78\times 10^{-2}$$8.49\times 10^{-3}$$4.27\times 10^{-2}$$9.51\times 10^{-4}$$4.43\times 10^{-3}$${\bf{4.01\times 10^{-5}}}$
    下载: 导出CSV

    表  2  本文算法与其他多目标进化算法的IGD评价指标对比

    Table  2  Results of IGD metric of the proposed algorithm and multi-objective genetic algorithms

    测试函数spmsAMOPSONSGA-II[24]SPEA2[25]MOEA/D[26] SPEA2 + DAA[28]
    平均值标准差平均值标准差平均值标准差平均值标准差 平均值标准差
    ZDT1${\bf{2.34\times 10^{-3}}}$${\bf{5.46\times 10^{-6}}}$$5.74\times 10^{-3}$$3.39\times 10^{-4}$$4.15\times 10^{-3}$$1.77\times 10^{-4}$$4.03\times 10^{-3}$$5.59\times 10^{-5}$$3.92\times 10^{-3}$$5.05\times 10^{-5}$
    ZDT2${\bf{1.98\times 10^{-3}}}$${\bf{6.18\times 10^{-6}}}$$5.36\times 10^{-3}$$2.02\times 10^{-4}$$4.17\times 10^{-3}$$2.56\times 10^{-4}$$3.85\times 10^{-3}$$4.34\times 10^{-5}$$4.02\times 10^{-3}$$1.07\times 10^{-4}$
    ZDT3${\bf{8.83\times 10^{-4}}}$${\bf{1.02\times 10^{-5}}}$$5.83\times 10^{-3}$$2.02\times 10^{-4}$$3.16\times 10^{-3}$$5.96\times 10^{-3}$$8.42\times 10^{-2}$$7.02\times 10^{-3}$$8.46\times 10^{-3}$$9.45\times 10^{-3}$
    ZDT4${\bf{2.32\times 10^{-3}}}$${\bf{3.96\times 10^{-6}}}$$2.53\times 10^{1}$$7.21\times 10^{0}$$2.49\times 10^{1}$$7.25\times 10^{-5}$$4.86\times 10^{-3}$$8.41\times 10^{-4}$
    ZDT6${\bf{1.60\times 10^{-3}}}$${\bf{5.20\times 10^{-7}}}$$1.65\times 10^{0}$$9.80\times 10^{-1}$$5.32\times 10^{-3}$$2.65\times 10^{-8}$$3.99\times 10^{-3}$$6.02\times 10^{-5}$
    DTLZ1${\bf{1.23\times 10^{-2}}}$$1.23\times 10^{-3}$$1.41\times 10^{1}$$6.65\times 10^{0}$$3.77\times 10^{1}$${\bf{1.45\times 10^{-4}}}$$6.04\times 10^{-1}$$2.89\times 10^{-1}$$1.51\times 10^{-2}$$1.48\times 10^{-3}$
    DTLZ2${\bf{3.62\times 10^{-3}}}$$8.56\times 10^{-5}$$1.06\times 10^{-1}$$8.38\times 10^{-3}$$8.22\times 10^{-2}$$2.83\times 10^{-7}$$6.24\times 10^{-1}$$6.44\times 10^{-5}$$3.81\times 10^{-2}$$3.03\times 10^{-4}$
    DTLZ3${\bf{3.28\times 10^{-3}}}$${\bf{9.62\times 10^{-5}}}$$1.64\times 10^{1}$$7.56\times 10^{0}$$4.87\times 10^{1}$$0.00\times 10^{0}$$6.52\times 10^{-1}$$2.59\times 10^{-1}$
    DTLZ4$7.58\times 10^{-3}$$3.45\times 10^{-3}$$7.30\times 10^{-2}$$5.09\times 10^{-2}$$7.29\times 10^{-2}$$1.42\times 10^{-7}$$2.70\times 10^{-1}$$6.83\times 10^{-3}$
    DTLZ5$9.76\times 10^{-3}$$6.42\times 10^{-4}$$8.05\times 10^{-3}$$1.63\times 10^{-3}$$1.41\times 10^{-2}$$3.54\times 10^{-5}$$5.94\times 10^{-1}$${\bf{8.43\times 10^{-8}}}$
    DTLZ6${\bf{3.62\times 10^{-3}}}$$3.47\times 10^{-5}$$1.47\times 10^{0}$$6.09\times 10^{-1}$$2.49\times 10^{-1}$$5.67\times 10^{-5}$$6.17\times 10^{-1}$$1.01\times 10^{-4}$
    DTLZ7${\bf{4.23\times 10^{-3}}}$$1.37\times 10^{-4}$$6.14\times 10^{-1}$$1.29\times 10^{-3}$$6.24\times 10^{-2}$$9.50\times 10^{-3}$$6.57\times 10^{-1}$$9.87\times 10^{-4}$$3.69\times 10^{-2}$$5.02\times 10^{-4}$
    下载: 导出CSV

    表  3  本文算法与其他多目标粒子群算法的SP评价指标对比

    Table  3  Results of SP metric of the proposed algorithm and MOPSOs

    测试函数spmsAMOPSOclusterMOPSO[23]cdMOPSO[8]pccsAMOPSO[13]
    平均值标准差平均值标准差平均值标准差平均值标准差
    ZDT1${\bf{4.09\times 10^{-3}}}$${\bf{5.57\times 10^{-5}}}$$7.25\times 10^{-2}$$6.17\times 10^{-4}$$8.56\times 10^{-2}$$1.43\times 10^{-2}$$1.33\times 10^{-2}$$5.42\times 10^{-3}$
    ZDT2${\bf{3.19\times 10^{-3}}}$${\bf{5.58\times 10^{-5}}}$$1.77\times 10^{-2}$$1.95\times 10^{-3}$$1.91\times 10^{-2}$$6.42\times 10^{-4}$$1.06\times 10^{-2}$$4.63\times 10^{-3}$
    ZDT3${\bf{4.35\times 10^{-3}}}$${\bf{4.54\times 10^{-5}}}$$7.78\times 10^{-2}$$4.36\times 10^{-2}$$5.97\times 10^{-1}$$2.25\times 10^{-1}$$1.25\times 10^{-1}$$6.48\times 10^{-2}$
    ZDT4${\bf{3.84\times 10^{-3}}}$${\bf{9.84\times 10^{-5}}}$$9.19\times 10^{-3}$$1.32\times 10^{-3}$$8.49\times 10^{-3}$$5.85\times 10^{-4}$$1.09\times 10^{-2}$$2.78\times 10^{-3}$
    ZDT6${\bf{3.11\times 10^{-3}}}$${\bf{3.91\times 10^{-5}}}$$2.69\times 10^{-2}$$1.67\times 10^{-2}$$9.83\times 10^{-3}$$4.65\times 10^{-4}$$1.10\times 10^{-2}$$5.89\times 10^{-3}$
    DTLZ1${\bf{2.24\times 10^{-2}}}$${\bf{3.36\times 10^{-4}}}$$9.32\times 10^{-2}$$1.74\times 10^{-2}$$4.12\times 10^{-2}$$5.64\times 10^{-3}$$5.79\times 10^{0}$$4.83\times 10^{-1}$
    DTLZ2${\bf{3.60\times 10^{-2}}}$${\bf{1.74\times 10^{-4}}}$$7.06\times 10^{-2}$$3.42\times 10^{-3}$$6.91\times 10^{-2}$$4.03\times 10^{-3}$$6.08\times 10^{-2}$$2.54\times 10^{-3}$
    DTLZ3${\bf{4.12\times 10^{-2}}}$$2.33\times 10^{-3}$$7.93\times 10^{-1}$$2.06\times 10^{-2}$$5.88\times 10^{-2}$$3.96\times 10^{-3}$$6.09\times 10^{-1}$${\bf{7.45\times 10^{-4}}}$
    DTLZ4${\bf{3.80\times 10^{-2}}}$${\bf{8.63\times 10^{-4}}}$$6.81\times 10^{-1}$$9.94\times 10^{-3}$$4.76\times 10^{-2}$$2.84\times 10^{-3}$$5.97\times 10^{-1}$$6.33\times 10^{-2}$
    DTLZ5$7.52\times 10^{-2}$$5.83\times 10^{-3}$$8.93\times 10^{-2}$$7.52\times 10^{-3}$${\bf{2.42\times 10^{-2}}}$$6.22\times 10^{-3}$$1.02\times 10^{-1}$$3.74\times 10^{-2}$
    DTLZ6${\bf{3.47\times 10^{-2}}}$$2.14\times 10^{-3}$$6.80\times 10^{-2}$$1.83\times 10^{-2}$$4.65\times 10^{-2}$$2.73\times 10^{-3}$$5.86\times 10^{-2}$${\bf{1.26\times 10^{-4}}}$
    DTLZ7${\bf{9.47\times 10^{-2}}}$$4.81\times 10^{-2}$$3.90\times 10^{-1}$$1.29\times 10^{-2}$$5.97\times 10^{-1}$$2.13\times 10^{-1}$$1.08\times 10^{-1}$$3.14\times 10^{-2}$
    下载: 导出CSV

    表  4  本文算法与其他多目标进化算法的SP评价指标对比

    Table  4  Results of SP metric of the proposed algorithm and multi-objective genetic algorithms

    测试函数spmsAMOPSONSGA-II[24]SPEA2[25]MOEA/D[26]
    平均值标准差平均值标准差平均值标准差平均值标准差
    ZDT1${\bf{4.09\times 10^{-3}}}$${\bf{5.57\times 10^{-5}}}$$5.83\times 10^{-2}$$9.39\times 10^{-3}$$3.73\times 10^{-2}$$2.67\times 10^{-3}$$4.85\times 10^{-3}$$7.19\times 10^{-4}$
    ZDT2${\bf{3.19\times 10^{-3}}}$${\bf{5.58\times 10^{-5}}}$$7.24\times 10^{-3}$$7.41\times 10^{-3}$$1.09\times 10^{-2}$$1.04\times 10^{-3}$$4.36\times 10^{-3}$$7.41\times 10^{-4}$
    ZDT3${\bf{4.35\times 10^{-3}}}$${\bf{4.54\times 10^{-5}}}$$9.22\times 10^{-2}$$8.42\times 10^{-3}$$6.07\times 10^{-1}$$1.03\times 10^{0}$$1.02\times 10^{-1}$$9.33\times 10^{-3}$
    ZDT4${\bf{3.84\times 10^{-3}}}$${\bf{9.84\times 10^{-5}}}$$2.83\times 10^{-2}$$6.13\times 10^{-3}$$4.06\times 10^{-2}$$1.59\times 10^{-2}$$7.52\times 10^{-3}$$6.93\times 10^{-4}$
    ZDT6${\bf{3.11\times 10^{-3}}}$${\bf{3.91\times 10^{-5}}}$$3.43\times 10^{-2}$$4.33\times 10^{-3}$$4.72\times 10^{-2}$$7.74\times 10^{-3}$$1.88\times 10^{-2}$$5.52\times 10^{-3}$
    DTLZ1${\bf{2.24\times 10^{-2}}}$${\bf{3.36\times 10^{-4}}}$$7.21\times 10^{-1}$$6.34\times 10^{-2}$$2.92\times 10^{-1}$$3.62\times 10^{-2}$$9.83\times 10^{-1}$$2.33\times 10^{-1}$
    DTLZ2${\bf{3.60\times 10^{-2}}}$${\bf{1.74\times 10^{-4}}}$$4.67\times 10^{-2}$$7.71\times 10^{-3}$$5.38\times 10^{-2}$$3.42\times 10^{-3}$$8.14\times 10^{-2}$$8.49\times 10^{-3}$
    DTLZ3${\bf{4.12\times 10^{-2}}}$$2.33\times 10^{-3}$$8.26\times 10^{-2}$$2.09\times 10^{-3}$$6.34\times 10^{-2}$$5.12\times 10^{-3}$$1.79\times 10^{-1}$$3.36\times 10^{-2}$
    DTLZ4${\bf{3.80\times 10^{-2}}}$${\bf{8.63\times 10^{-4}}}$$7.14\times 10^{-2}$$1.97\times 10^{-3}$$4.38\times 10^{-2}$$6.54\times 10^{-3}$$9.25\times 10^{-2}$$5.61\times 10^{-3}$
    DTLZ5$7.52\times 10^{-2}$$5.83\times 10^{-3}$$6.37\times 10^{-2}$${\bf{1.74\times 10^{-3}}}$$3.15\times 10^{-2}$$5.31\times 10^{-3}$$8.02\times 10^{-2}$$4.39\times 10^{-3}$
    DTLZ6${\bf{3.47\times 10^{-2}}}$$2.14\times 10^{-3}$$7.03\times 10^{-2}$$1.56\times 10^{-3}$$5.11\times 10^{-2}$$4.89\times 10^{-3}$$3.93\times 10^{-2}$$3.31\times 10^{-3}$
    DTLZ7${\bf{9.47\times 10^{-2}}}$$4.81\times 10^{-2}$$4.19\times 10^{-1}$$7.96\times 10^{-3}$$2.96\times 10^{-1}$${\bf{5.29\times 10^{-3}}}$$1.85\times 10^{-1}$$7.93\times 10^{-2}$
    下载: 导出CSV

    表  5  本文算法与其他多目标粒子群算法的ER评价指标对比

    Table  5  Results of ER metric of the proposed algorithm and MOPSOs

    测试函数spmsAMOPSOclusterMOPSO[23]cdMOPSO[8]pccsAMOPSO[13]
    平均值标准差平均值标准差平均值标准差平均值标准差
    ZDT1${\bf{6.73\times 10^{-4}}}$${\bf{1.34\times 10^{-4}}}$$1.06\times 10^{-2}$$1.00\times 10^{-2}$$9.90\times 10^{-2}$$1.25\times 10^{-3}$$8.12\times 10^{-3}$$1.43\times 10^{-2}$
    ZDT2${\bf{3.75\times 10^{-3}}}$${\bf{1.17\times 10^{-5}}}$$4.92\times 10^{-1}$$8.72\times 10^{-2}$$1.30\times 10^{0}$$1.21\times 10^{-1}$$9.00\times 10^{-3}$$2.36\times 10^{-3}$
    ZDT3${\bf{8.95\times 10^{-3}}}$${\bf{1.37\times 10^{-3}}}$$2.66\times 10^{-1}$$9.30\times 10^{-2}$$2.89\times 10^{-1}$$7.24\times 10^{-2}$$2.60\times 10^{-2}$$8.32\times 10^{-3}$
    ZDT4$2.70\times 10^{-2}$$8.35\times 10^{-3}$$4.42\times 10^{-1}$$1.02\times 10^{-1}$$9.34\times 10^{-2}$$6.83\times 10^{-2}$${\bf{2.37\times 10^{-2}}}$${\bf{4.74\times 10^{-3}}}$
    ZDT6${\bf{1.60\times 10^{-3}}}$$1.24\times 10^{-2}$$2.98\times 10^{-2}$$7.43\times 10^{-3}$$6.77\times 10^{-3}$$3.43\times 10^{-3}$$1.10\times 10^{-2}$${\bf{4.64\times 10^{-4}}}$
    DTLZ1$6.81\times 10^{-2}$${\bf{4.24\times 10^{-3}}}$${\bf{4.83\times 10^{-2}}}$$6.52\times 10^{-3}$$6.22\times 10^{-1}$$3.81\times 10^{-2}$$9.90\times 10^{0}$$7.13\times 10^{-1}$
    DTLZ2${\bf{4.74\times 10^{-1}}}$${\bf{3.52\times 10^{-3}}}$$1.19\times 10^{0}$$7.62\times 10^{-1}$$8.32\times 10^{-1}$$3.53\times 10^{-2}$$9.10\times 10^{-1}$$5.31\times 10^{-2}$
    DTLZ3${\bf{8.15\times 10^{-2}}}$$2.47\times 10^{-2}$$4.07\times 10^{-1}$$5.26\times 10^{-2}$$7.21\times 10^{-1}$$2.12\times 10^{-2}$$9.53\times 10^{-2}$$3.64\times 10^{-2}$
    DTLZ4$8.02\times 10^{-2}$${\bf{1.14\times 10^{-3}}}$$8.24\times 10^{-2}$$3.01\times 10^{-3}$$2.42\times 10^{-1}$$9.71\times 10^{-2}$$6.33\times 10^{-1}$$3.60\times 10^{-2}$
    DTLZ5${\bf{1.43\times 10^{-2}}}$$3.90\times 10^{-3}$$4.81\times 10^{-1}$$5.32\times 10^{-2}$$1.99\times 10^{-1}$$9.73\times 10^{-2}$$2.72\times 10^{-2}$$4.74\times 10^{-3}$
    DTLZ6$1.73\times 10^{-1}$$5.69\times 10^{-2}$${\bf{4.62\times 10^{-2}}}$${\bf{6.39\times 10^{-3}}}$$3.48\times 10^{-1}$$2.15\times 10^{-3}$$1.92\times 10^{-1}$$8.13\times 10^{-2}$
    DTLZ7${\bf{2.18\times 10^{-1}}}$${\bf{7.64\times 10^{-2}}}$$4.77\times 10^{-1}$$5.07\times 10^{-1}$$5.89\times 10^{-1}$$4.33\times 10^{-1}$$3.16\times 10^{-1}$$9.57\times 10^{-2}$
    下载: 导出CSV

    表  6  本文算法与其他多目标进化算法的ER评价指标对比

    Table  6  Results of ER metric of the proposed algorithm and multi-objective genetic algorithms

    测试函数spmsAMOPSONSGA-II[24]SPEA2[25]MOEA/D[26]
    平均值标准差平均值标准差平均值标准差平均值标准差
    ZDT1${\bf{6.73\times 10^{-4}}}$${\bf{1.34\times 10^{-4}}}$$8.06\times 10^{-3}$$5.82\times 10^{-3}$$3.00\times 10^{-3}$$6.75\times 10^{-3}$$7.65\times 10^{-2}$$2.44\times 10^{-3}$
    ZDT2${\bf{3.75\times 10^{-3}}}$${\bf{1.17\times 10^{-5}}}$$5.74\times 10^{-1}$$3.41\times 10^{-2}$$9.11\times 10^{-1}$$4.68\times 10^{-2}$$6.53\times 10^{-1}$$2.78\times 10^{-2}$
    ZDT3${\bf{8.95\times 10^{-3}}}$${\bf{1.37\times 10^{-3}}}$$2.09\times 10^{-2}$$4.12\times 10^{-2}$$1.49\times 10^{-1}$$6.92\times 10^{-2}$$4.87\times 10^{-2}$$6.47\times 10^{-3}$
    ZDT4$2.70\times 10^{-2}$$8.35\times 10^{-3}$$3.49\times 10^{-2}$$7.66\times 10^{-3}$$8.45\times 10^{-2}$$2.93\times 10^{-2}$$7.60\times 10^{-1}$$5.36\times 10^{-2}$
    ZDT6${\bf{1.60\times 10^{-3}}}$$1.24\times 10^{-2}$$9.93\times 10^{-3}$$7.64\times 10^{-4}$$8.02\times 10^{-3}$$3.24\times 10^{-3}$$1.73\times 10^{-2}$$2.12\times 10^{-3}$
    DTLZ1$6.81\times 10^{-2}$${\bf{4.24\times 10^{-3}}}$$2.80\times 10^{-1}$$1.55\times 10^{-2}$$8.31\times 10^{-2}$$4.85\times 10^{-3}$$4.64\times 10^{-1}$$3.06\times 10^{-2}$
    DTLZ2${\bf{4.74\times 10^{-1}}}$${\bf{3.52\times 10^{-3}}}$$7.06\times 10^{-1}$$5.44\times 10^{-2}$$9.47\times 10^{-1}$$2.13\times 10^{-1}$$2.59\times 10^{0}$$7.42\times 10^{-1}$
    DTLZ3${\bf{8.15\times 10^{-2}}}$$2.47\times 10^{-2}$$8.60\times 10^{-1}$$3.75\times 10^{-2}$$9.83\times 10^{-1}$$3.41\times 10^{-2}$$2.17\times 10^{-1}$${\bf{4.54\times 10^{-3}}}$
    DTLZ4$8.02\times 10^{-2}$${\bf{1.14\times 10^{-3}}}$$4.85\times 10^{-1}$$4.21\times 10^{-2}$${\bf{6.13\times 10^{-2}}}$$1.90\times 10^{-3}$$1.36\times 10^{-1}$$6.29\times 10^{-2}$
    DTLZ5${\bf{1.43\times 10^{-2}}}$$3.90\times 10^{-3}$$9.43\times 10^{-2}$$4.16\times 10^{-3}$$8.32\times 10^{-2}$${\bf{2.73\times 10^{-3}}}$$v1.03\times 10^{-1}$$6.51\times 10^{-2}$
    DTLZ6$1.73\times 10^{-1}$$5.69\times 10^{-2}$$5.84\times 10^{-2}$$7.91\times 10^{-3}$$4.36\times 10^{-1}$$2.26\times 10^{-2}$$2.53\times 10^{-1}$$9.06\times 10^{-2}$
    DTLZ7${\bf{2.18\times 10^{-1}}}$${\bf{7.64\times 10^{-2}}}$$4.34\times 10^{-1}$$2.64\times 10^{-1}$$3.92\times 10^{-1}$$8.70\times 10^{-2}$$3.76\times 10^{-1}$$2.92\times 10^{-1}$
    下载: 导出CSV

    表  7  不同算法对多目标测试问题的运行时间 (s)

    Table  7  Computational time of different algorithms for multi-objective test problems (s)

    函数spmsAMOPSOclusterMOPSO[23]cdMOPSO[8]pccsAMOPSO[13]NSGA-II[24]SPEA2[25]MOEA/D[26]
    ZDT1105.26108.87${\bf{101.72}}$121.95129.17135.23152.45
    ZDT2${\bf{104.72}}$114.95112.74108.34133.15126.65139.13
    ZDT3${\bf{111.23}}$136.38135.81124.57132.40137.98135.02
    ZDT4${\bf{115.17}}$132.09129.69122.19125.67130.61147.61
    ZDT6${\bf{122.48}}$131.62124.23126.62133.52128.74149.55
    DTLZ1${\bf{210.77}}$230.82226.47219.83248.13232.47280.73
    DTLZ2218.93233.49${\bf{212.73}}$215.63250.26238.69289.36
    DTLZ3${\bf{212.61}}$228.34217.51218.26250.98242.81281.75
    DTLZ4218.34230.25220.98${\bf{216.44}}$247.37241.33276.84
    DTLZ5${\bf{219.15}}$234.16228.21221.92255.91250.62286.17
    DTLZ6${\bf{216.37}}$236.59225.40225.73245.69247.97288.33
    DTLZ7${\bf{215.42}}$243.52224.64232.41246.38238.11295.42
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-05-13
  • 录用日期:  2020-09-07
  • 网络出版日期:  2022-09-05
  • 刊出日期:  2022-10-14

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