Multi-strategy Adaptive Multi-objective Particle Swarm Optimization Algorithm Based on Swarm Partition
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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.Abstract: In the multi-objective particle swarm optimization algorithm, balancing the convergence and diversity of the algorithm is the key to obtain the Pareto front with good distribution and accuracy. Most of the proposed methods rely on only one strategy to guide the particle search, and the algorithm may lack convergence and diversity when solving complex problems. To solve this problem, a multi-strategy adaptive multi-objective particle swarm optimization based on swarm partition is proposed. Firstly, the algorithm detects environment by the convergence contribution of particles and adjusts the process of particle exploration and exploitation adaptively. Secondly, in order to accurately formulate the search strategy of particles with different performances, a multi-strategy global optimal particle selection method and a mutation method are proposed. According to the evaluation index of the convergence of the particles, the population is divided into three regions. Combining particle performance with the algorithm optimization process can improve the search efficiency of each particle. Thirdly, an individual optimal particle selection scheme with memory interval is proposed to solve the problem that the algorithm falls into local optimization because the selected individual optimal particles cannot guide the flight direction of particles effectively. That can improve the reliability of individual optimal particle selection, and accelerate the process of particle convergence. Finally, the fusion metric including particle convergence and diversity is used to maintain the external archive. It can avoid deleting the particles with good convergence and resulting in population degradation and affecting particle development capabilities, when external archive maintenance is just based on the particle density. Experimental results show that the proposed algorithm has good performance compared with some other multi-objective optimization algorithms.
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图 4
$\eta_{1}$ 和$\eta_{2}$ 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 spmsAMOPSO clusterMOPSO[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 ${\bf{4.23\times 10^{-3}}}$ $1.37\times 10^{-4}$ $4.02\times 10^{-2}$ $2.06\times 10^{-3}$ $5.78\times 10^{-2}$ $8.49\times 10^{-3}$ $4.27\times 10^{-2}$ $9.51\times 10^{-4}$ $4.43\times 10^{-3}$ ${\bf{4.01\times 10^{-5}}}$ 
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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 spmsAMOPSO NSGA-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 ${\bf{4.23\times 10^{-3}}}$ $1.37\times 10^{-4}$ $6.14\times 10^{-1}$ $1.29\times 10^{-3}$ $6.24\times 10^{-2}$ $9.50\times 10^{-3}$ $6.57\times 10^{-1}$ $9.87\times 10^{-4}$ $3.69\times 10^{-2}$ $5.02\times 10^{-4}$
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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 spmsAMOPSO clusterMOPSO[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 ${\bf{9.47\times 10^{-2}}}$ $4.81\times 10^{-2}$ $3.90\times 10^{-1}$ $1.29\times 10^{-2}$ $5.97\times 10^{-1}$ $2.13\times 10^{-1}$ $1.08\times 10^{-1}$ $3.14\times 10^{-2}$
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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 spmsAMOPSO NSGA-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 ${\bf{9.47\times 10^{-2}}}$ $4.81\times 10^{-2}$ $4.19\times 10^{-1}$ $7.96\times 10^{-3}$ $2.96\times 10^{-1}$ ${\bf{5.29\times 10^{-3}}}$ $1.85\times 10^{-1}$ $7.93\times 10^{-2}$
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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 spmsAMOPSO clusterMOPSO[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 ${\bf{2.18\times 10^{-1}}}$ ${\bf{7.64\times 10^{-2}}}$ $4.77\times 10^{-1}$ $5.07\times 10^{-1}$ $5.89\times 10^{-1}$ $4.33\times 10^{-1}$ $3.16\times 10^{-1}$ $9.57\times 10^{-2}$
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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 spmsAMOPSO NSGA-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 ${\bf{2.18\times 10^{-1}}}$ ${\bf{7.64\times 10^{-2}}}$ $4.34\times 10^{-1}$ $2.64\times 10^{-1}$ $3.92\times 10^{-1}$ $8.70\times 10^{-2}$ $3.76\times 10^{-1}$ $2.92\times 10^{-1}$
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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 spmsAMOPSO clusterMOPSO[23] cdMOPSO[8] pccsAMOPSO[13] NSGA-II[24] SPEA2[25] MOEA/D[26] ZDT1 105.26 108.87 ${\bf{101.72}}$ 121.95 129.17 135.23 152.45 ZDT2 ${\bf{104.72}}$ 114.95 112.74 108.34 133.15 126.65 139.13 ZDT3 ${\bf{111.23}}$ 136.38 135.81 124.57 132.40 137.98 135.02 ZDT4 ${\bf{115.17}}$ 132.09 129.69 122.19 125.67 130.61 147.61 ZDT6 ${\bf{122.48}}$ 131.62 124.23 126.62 133.52 128.74 149.55 DTLZ1 ${\bf{210.77}}$ 230.82 226.47 219.83 248.13 232.47 280.73 DTLZ2 218.93 233.49 ${\bf{212.73}}$ 215.63 250.26 238.69 289.36 DTLZ3 ${\bf{212.61}}$ 228.34 217.51 218.26 250.98 242.81 281.75 DTLZ4 218.34 230.25 220.98 ${\bf{216.44}}$ 247.37 241.33 276.84 DTLZ5 ${\bf{219.15}}$ 234.16 228.21 221.92 255.91 250.62 286.17 DTLZ6 ${\bf{216.37}}$ 236.59 225.40 225.73 245.69 247.97 288.33 DTLZ7 ${\bf{215.42}}$ 243.52 224.64 232.41 246.38 238.11 295.42
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