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摘要: 拥挤度距离是一种用于度量解集多样性的指标. 然而, 在许多情况下, 该指标无法有效区分多样性较优个体. 其原因为拥挤度距离主要利用每个位置的局部信息. 为解决该问题, 基于整个种群全局位置信息, 本文设计了基于平均距离聚类的多样性度量指标, 并进一步提出了基于平均距离聚类的NSGA-Ⅱ. 该算法利用平均距离将种群划分为若干个大致均匀分布的小种群, 然后分别在各小种群内执行选择、交叉和变异等操作. 实验结果表明, 本文所提算法可以有效地保持种群多样性.Abstract: Crowding distance is an index for measuring the diversity of solutions. However, in many cases, it may fail to identify individuals with better diversity. The reason is that crowding distance mainly takes advantage of the local information of each position. To tackle this issue, based on the global position information of entire population, this paper designs average-distance-clustering diversity index, and further proposes NSGA-Ⅱ with average distance clustering (ADCNSGA-Ⅱ). ADCNSGA-Ⅱ divides the entire population into several small populations using average distance, then the selection, crossover and mutation operators are performed in each small population. Simulation results show the proposed algorithm can maintain the diversify effectively.
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Key words:
- Multi-objective optimization algorithms /
- NSGA-Ⅱ /
- crowding distance /
- diversity
1) 本文责任编委 张敏灵 -
图 1 熔化率与比例、积分系数之间的关系
Fig. 1 Comparing melting rate with proportion and integral parameters of the consarc controller
表 1 参数$S$对算法ADCNSGA-Ⅱ性能影响
Table 1 Influence of parameter $S$ on ADCNSGA-Ⅱ
测试函数 指标 $S=1$ $S=2$ $S=3$ $S=4$ $S=5$ $S=6$ SCH mean$(GD)$ $ \bf{5.87\times 10^{-2}} $ $6.47\times 10^{-2}$ $6.68\times 10^{-2}$ $6.77\times 10^{-2}$ $6.67\times 10^{-2}$ $6.69\times 10^{-2}$ std$(GD)$ $8.14\times 10^{-3}$ $ \bf{3.84\times 10^{-3}}$ $5.03\times 10^{-3}$ $5.11\times 10^{-3}$ $5.28\times 10^{-3}$ $4.15\times 10^{-3}$ mean$(SP)$ $ \bf{9.87\times 10^{-3}}$ $1.18\times 10^{-2}$ $1.77\times 10^{-2}$ $1.89\times 10^{-2}$ $2.28\times 10^{-2}$ $1.77\times 10^{-2}$ std$(SP) $ $6.23\times 10^{-3}$ $ \bf{2.62\times 10^{-3}}$ $1.12\times 10^{-2}$ $1.07\times 10^{-2}$ $2.16\times 10^{-2}$ $1.01\times 10^{-2}$ ZDT1 mean$(GD) $ $ \bf{2.05\times 10^{-6}}$ $4.07\times 10^{-6}$ $4.71\times 10^{-5}$ $3.28\times 10^{-5}$ $7.25\times 10^{-5}$ $7.71\times 10^{-5}$ std$(GD)$ $ \bf{1.11\times 10^{-5}}$ $2.10\times 10^{-5}$ $8.90\times 10^{-5}$ $6.66\times 10^{-5}$ $1.32\times 10^{-4}$ $1.28\times 10^{-4}$ mean$(SP)$ $ \bf{4.48\times 10^{-3}}$ $5.34\times 10^{-3}$ $5.94\times 10^{-3}$ $6.48\times 10^{-3}$ $5.87\times 10^{-3}$ $5.42\times 10^{-3}$ std$(SP)$ $1.59\times 10^{-3}$ $1.31\times 10^{-3}$ $ \bf{1.20\times 10^{-3}}$ $1.92\times 10^{-3}$ $2.57\times 10^{-3}$ $1.78\times 10^{-3}$ ZDT2 mean$(GD)$ $ \bf{1.32\times 10^{-5}}$ $3.80\times 10^{-5}$ $5.17\times 10^{-5}$ $5.51\times 10^{-5}$ $8.31\times 10^{-5}$ $8.55\times 10^{-5}$ std$(GD)$ $ \bf{6.55\times 10^{-5}}$ $9.60\times 10^{-5}$ $9.26\times 10^{-5}$ $7.18\times 10^{-5}$ $9.89\times 10^{-5}$ $1.21\times 10^{-4}$ mean$(SP)$ $ \bf{4.47\times 10^{-3}}$ $5.40\times 10^{-3}$ $6.32\times 10^{-3}$ $6.91\times 10^{-3}$ $6.78\times 10^{-3}$ $6.79\times 10^{-3}$ std$(SP)$ $1.79\times 10^{-3}$ $ \bf{1.02\times 10^{-3}}$ $1.44\times 10^{-3}$ $2.60\times 10^{-3}$ $3.32\times 10^{-3}$ $2.46\times 10^{-3}$ ZDT3 mean$(GD)$ $ \bf{1.09\times 10^{-6}}$ $6.07\times 10^{-6}$ $2.48\times 10^{-5}$ $3.57\times 10^{-5}$ $1.38\times 10^{-5}$ $5.13\times 10^{-5}$ std$(GD)$ $ \bf{5.91\times 10^{-6}}$ $2.61\times 10^{-5}$ $5.56\times 10^{-5}$ $5.54\times 10^{-5}$ $2.20\times 10^{-5}$ $5.81\times 10^{-5}$ mean$(SP)$ $ \bf{4.93\times 10^{-3}}$ $6.20\times 10^{-3}$ $6.01\times 10^{-3}$ $7.84\times 10^{-3}$ $6.85\times 10^{-3}$ $8.04\times 10^{-3}$ std$(SP)$ $1.89\times 10^{-3}$ $ \bf{1.21\times 10^{-3}}$ $2.12\times 10^{-3}$ $3.61\times 10^{-3}$ $2.22\times 10^{-3}$ $2.91\times 10^{-3}$ ZDT4 mean$(GD)$ $1.51\times 10^{-5}$ $8.46\times 10^{-6}$ $2.03\times 10^{-5}$ $ \bf{4.54\times 10^{-6}}$ $2.67\times 10^{-5}$ $7.20\times 10^{-6}$ std$(GD)$ $4.49\times 10^{-5}$ $2.13\times 10^{-5}$ $4.46\times 10^{-5}$ $ \bf{1.25\times 10^{-5}}$ $6.31\times 10^{-5}$ $1.76\times 10^{-5}$ mean$(SP)$ $ \bf{3.29\times 10^{-3}}$ $5.21\times 10^{-3}$ $6.01\times 10^{-3}$ $5.98\times 10^{-3}$ $6.43\times 10^{-3}$ $6.43\times 10^{-3}$ std$(SP)$ $1.64\times 10^{-3}$ $ \bf{1.20\times 10^{-3}}$ $1.75\times 10^{-3}$ $1.65\times 10^{-3}$ $2.01\times 10^{-3}$ $3.32\times 10^{-3}$ ZDT6 mean$(GD)$ $ \bf{1.11\times 10^{-2}}$ $2.07\times 10^{-2}$ $2.07\times 10^{-2}$ $3.27\times 10^{-2}$ $9.92\times 10^{-3}$ $2.29\times 10^{-2}$ std$(GD)$ $ \bf{2.34\times 10^{-2}}$ $3.86\times 10^{-2}$ $4.77\times 10^{-2}$ $5.42\times 10^{-2}$ $3.61\times 10^{-2}$ $4.08\times 10^{-2}$ mean$(SP)$ $\bf{1.77\times 10^{-2}}$ $6.14\times 10^{-2}$ $5.52\times 10^{-2}$ $7.81\times 10^{-2}$ $ {1.09\times 10^{-2}}$ $4.55\times 10^{-2}$ std$(SP)$ $5.80\times 10^{-2}$ $1.64\times 10^{-1}$ $1.26\times 10^{-1}$ $1.18\times 10^{-1}$ $ \bf{3.05\times 10^{-2}}$ $1.14\times 10^{-1}$ 
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表 2 Friedman测试结果
Table 2 Comparison results of Friedman test
参数 秩均值 $S=1$ 1.92 $S=2$ 2.40 $S=3$ 3.67 $S=4$ 4.38 $S=5$ 4.23 $S=6$ 4.42
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表 3 Wilcoxon检测测试结果
Table 3 Comparison results of Wilcoxon test
$S=1$对比 p值 $S=2$ 0.0765 $S=3$ 0.0018 $S=4$ 0.0003 $S=5$ 0.0258 $S=6$ 0.0003
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表 4 实验性能均值和方差对比
Table 4 Means and variances of the performance metrics
测试函数 指标 PNIA SPEA2 NSGA-Ⅱ g-NSGA-Ⅱ ADCNSGA-Ⅱ SCH mean$(GD)$ $ \bf{3.81\times 10^{-2}} $ $4.41\times 10^{-2}$ $6.17\times 10^{-2}$ $4.43\times 10^{-2}$ $5.87\times 10^{-2}$ std$(GD)$ $ \bf{1.93\times 10^{-3}}$ $1.14\times 10^{-3}$ $2.96\times 10^{-3}$ $1.61\times 10^{-2}$ $8.14\times 10^{-3}$ mean$(SP)$ $ \bf{4.96\times 10^{-3}}$ $7.01\times 10^{-3}$ $1.75\times 10^{-2}$ $3.25\times 10^{-2}$ $9.87\times 10^{-3}$ std$(SP)$ $ \bf{3.66\times 10^{-3}}$ $1.61\times 10^{-3}$ $1.02\times 10^{-2}$ $1.45\times 10^{-1}$ $6.32\times 10^{-3}$ ZDT1 mean$(GD)$ $6.56\times 10^{-4}$ $2.80\times 10^{-5}$ $2.58\times 10^{-4}$ $1.73\times 10^{-3}$ $ \bf{2.05\times 10^{-6}}$ std$(GD)$ $1.64\times 10^{-4}$ $1.34\times 10^{-5}$ $1.17\times 10^{-4}$ $5.72\times 10^{-3}$ $ \bf{1.11\times 10^{-5}}$ mean$(SP)$ $1.04\times 10^{-3}$ $1.94\times 10^{-3}$ $5.09\times 10^{-3}$ $1.01\times 10^{-3}$ $ \bf{4.48\times 10^{-3}}$ std$(SP)$ $1.06\times 10^{-3}$ $3.38\times 10^{-4}$ $3.92\times 10^{-3}$ $7.45\times 10^{-3}$ $ \bf{1.59\times10^{-3}}$ ZDT2 mean$(GD)$ $8.77\times 10^{-4}$ $ \bf{1.17\times 10^{-5}}$ $8.42\times 10^{-5}$ $5.98\times 10^{-4}$ $1.32\times 10^{-5}$ std$(GD)$ $1.18\times 10^{-3}$ $ \bf{1.17\times 10^{-5}}$ $1.32\times 10^{-4}$ $6.03\times 10^{-4}$ $6.55\times 10^{-5}$ mean$(SP)$ $2.37\times 10^{-3}$ $ \bf{2.26\times 10^{-3}}$ $2.81\times 10^{-3}$ $2.40\times 10^{-3}$ $4.47\times 10^{-3}$ std$(SP)$ $3.17\times 10^{-3}$ $ \bf{3.59\times 10^{-3}}$ $4, 36\times 10^{-3}$ $2.70\times 10^{-3}$ $1.79\times 10^{-3}$ ZDT3 mean$(GD)$ $1.89\times 10^{-4}$ $4.43\times 10^{-6}$ $1.40\times 10^{-4}$ $6.22\times 10^{-5}$ $ \bf{1.09\times 10^{-6}}$ std$(GD)$ $1.01\times 10^{-4}$ $1.71\times 10^{-6}$ $9.98\times 10^{-5}$ $4.98\times 10^{-5}$ $ \bf{5.91\times 10^{-6}}$ mean$(SP)$ $4.09\times 10^{-4}$ $1.89\times 10^{-3}$ $5.56\times 10^{-3}$ $1.10\times 10^{-3}$ $ \bf{4.93\times 10^{-3}}$ std$(SP)$ $5.47\times 10^{-4}$ $5.99\times 10^{-4}$ $3.81\times 10^{-3}$ $8.23\times 10^{-4}$ $ \bf{1.89\times10^{-3}}$ ZDT4 mean$(GD)$ $1.25\times 10^{-2}$ $6.95\times 10^{-2}$ $2.68\times 10^{-1}$ $1.62\times 10^{-2}$ $\bf{1.51\times 10^{-5}}$ std$(GD)$ $1.03\times 10^{-2}$ $4.19\times 10^{-2}$ $1.30\times 10^{-1}$ $6.48\times 10^{-1}$ $\bf{4.49\times 10^{-5}}$ mean$(SP)$ $\bf{1.43\times 10^{-3}}$ $4.37\times 10^{-3}$ $5.09\times 10^{-3}$ $1.70\times 10^{-2}$ $3.29\times 10^{-3}$ std$(SP)$ $3.21\times 10^{-3}$ $4.64\times 10^{-3}$ $1.11\times 10^{-2}$ $2.24\times 10^{-2}$ $\bf{1.64\times10^{-3}}$ ZDT6 mean$(GD)$ $1.78\times 10^{-3}$ $\bf{1.33\times 10^{-3}}$ $5.49\times 10^{-3}$ $1.47\times 10^{-2}$ $1.11\times 10^{-2}$ std$(GD)$ $2.31\times 10^{-4}$ $\bf{1.84\times 10^{-4}}$ $1.27\times 10^{-3}$ $4.00\times 10^{-3}$ $2.34\times 10^{-2}$ mean$(SP)$ $1.35\times 10^{-3}$ $1.45\times 10^{-3}$ $4.43\times 10^{-3}$ $\bf{8.74\times 10^{-4}}$ $1.77\times 10^{-2}$ std$(SP)$ $9.87\times 10^{-4}$ $\bf{5.74\times 10^{-4}}$ $2.05\times 10^{-3}$ $7.14\times 10^{-4}$ $5.80\times 10^{-2}$
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