Research and Analysis of a Novel Heuristic Algorithm: Five-elements Cycle Optimization Algorithm
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摘要: 借鉴中国古代哲学理论所描述的系统动态平衡方法, 提出了解决连续函数优化问题的五行环优化算法.首先, 分析了基于五行元素生克原理而建立的五行环模型, 并在该模型基础上, 构建了元素空间结构以及元素更新方法等关键环节, 从而实现了五行环优化算法.随后, 对五行环优化算法进行了性能分析和关键参数比较, 针对标准测试函数, 将五行环优化算法与其他17个机制各异的启发式优化算法进行了比较, 实验结果验证了五行环优化算法的有效性和通用性, 也表明了其在求解连续函数优化问题上具有较好的优化性能.Abstract: The five-elements cycle optimization algorithm (FECO) for continuous optimization problems is researched and analyzed in this paper. It is inspired by the theory of Five-elements which represents the performance of a dynamic balancing system. Firstly, the five-elements cycle model based on the mechanism of generation and restriction among five elements is analyzed. Afterwards, FECO is built for finding the optimal solution of continuous functions by designing the framework of element space and the pivotal operators. The performance and parameter comparison of FECO is given by experiment, the comparison with 17 optimization algorithms based on various mechanisms for two sets of benchmark functions is also given, which indicates the feasibility and universality of FECO.
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Key words:
- Continuous optimization /
- five-elements cycle optimization /
- heuristic algorithms /
- benchmark functions
1) 本文责任编委 乔俊飞 -
图 1 五行元素的相生相克关系
Fig. 1 The generating and restricting interactions among five elements
图 2 当$w_{gp}=w_{rp}=w_{ga}=w_{ra}=1$, $L=5$时, $m_i(k)$和$F_i(k)$的变化曲线
Fig. 2 The changing curves of $m_i(k)$, $F_i(k)$ when $w_{gp}= w_{rp}=w_{ga}=w_{ra}=1$, $L=5$
图 3 当$w_{gp}, w_{rp}, w_{ga}, w_{ra}$为(0, 1]区间上的随机数, $L=5$时, $m_i(k)$和$F_i(k)$的变化曲线
Fig. 3 The changing curves of $m_i(k)$, $F_i(k)$ when $w_{gp}$, $w_{rp}$, $w_{ga}$, $w_{ra}$ is random numerical value in (0, 1] and $L=5$
图 4 元素质量$m_i(k)$ ($i=1, 2, \cdots, 20$)分别在第1、50、300、1 000代时的变化情况
Fig. 4 The changing of $m_i(k)$ ($i=1, 2, \cdots, 20$) at 1st, 50th, 300th, and 1 000th iterations
图 6 在不同迭代次数时的元素$x_{ij}(k)$的分布情况以及最小$f(x_{ij}(k))$的变化曲线
Fig. 6 The distribution of $x_{ij}(k)$ at various iterations and the changing curve of minimum $f(x_{ij}(k))$
表 1 用Friedman检验法(显著性水平0.05)比较参数$L$和$q$
Table 1 Comparison of $L$ and $q$ with Friedman test (significance value 0.05)
Function $L=5/q=20$ $L=10/q=10$ $L=20/q=5$ $L=50/q=2$ $L=100/q=1$ $f_1$ 3.22$\, \times\, 10^{-23}$ 7.65$\, \times\, 10^{-25}$ 1.64$\, \times\, 10^{-27}$ 3.65$\, \times\, 10^{-18}$ 1.99$\, \times\, 10^{3}$ $f_2$ 3.18$\, \times\, 10^{-16}$ 9.35$\, \times\, 10^{-18}$ 3.35$\, \times\, 10^{-20}$ 9.80$\, \times\, 10^{-1}$ 2.88$\, \times\, 10^{1}$ $f_3$ 1.47$\, \times\, 10^{2}$ 6.88$\, \times\, 10^{2}$ 1.35$\, \times\, 10^{3}$ 2.62$\, \times\, 10^{3}$ 1.22$\, \times\, 10^{4}$ $f_4$ 4.22$\, \times\, 10^{-1}$ 1.47$\, \times\, 10^{0}$ 3.07$\, \times\, 10^{0}$ 1.13$\, \times\, 10^{1}$ 4.67$\, \times\, 10^{1}$ $f_5$ 5.29$\, \times\, 10^{1}$ 5.59$\, \times\, 10^{1}$ 1.83$\, \times\, 10^{3}$ 1.39$\, \times\, 10^{2}$ 1.15$\, \times\, 10^{6}$ $f_6$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 1.37$\, \times\, 10^{-1}$ 1.11$\, \times\, 10^{2}$ 4.61$\, \times\, 10^{3 }$ $f_7$ 1.27$\, \times\, 10^{-2}$ 8.45$\, \times\, 10^{-3}$ 7.29$\, \times\, 10^{-3}$ 2.03$\, \times\, 10^{-2}$ 2.56$\, \times\, 10^{0}$ $f_8$ $-$1.15$\, \times\, 10^{4}$ $-$1.14$\, \times\, 10^{4}$ $-$1.10$\, \times\, 10^{4}$ $-$9.85$\, \times\, 10^{3}$ $-$7.97$\, \times\, 10^{3}$ $f_9$ 1.23$\, \times\, 10^{1}$ 1.31$\, \times\, 10^{1}$ 1.94$\, \times\, 10^{1}$ 5.79$\, \times\, 10^{1}$ 1.48$\, \times\, 10^{2 }$ $f_{10}$ 1.53$\, \times\, 10^{-12}$ 2.22$\, \times\, 10^{-13}$ 1.70$\, \times\, 10^{-1}$ 3.60$\, \times\, 10^{0}$ 1.42$\, \times\, 10^{1 }$ $f_{11}$ 6.10$\, \times\, 10^{-4}$ 1.24$\, \times\, 10^{-3}$ 4.34$\, \times\, 10^{-3}$ 1.17$\, \times\, 10^{-1}$ 1.80$\, \times\, 10^{1 }$ $f_{12}$ 4.07$\, \times\, 10^{-3}$ 2.64$\, \times\, 10^{-2}$ 7.53$\, \times\, 10^{-2}$ 4.58$\, \times\, 10^{-1}$ 5.21$\, \times\, 10^{5}$ $f_{13}$ 5.95$\, \times\, 10^{-2}$ 6.66$\, \times\, 10^{-1}$ 2.21$\, \times\, 10^{0}$ 1.32$\, \times\, 10^{1}$ 1.99$\, \times\, 10^{6 }$ $f_{14}$ 1.02$\, \times\, 10^{0}$ 1.06$\, \times\, 10^{0}$ 1.14$\, \times\, 10^{0}$ 1.37$\, \times\, 10^{0}$ 2.50$\, \times\, 10^{0 }$ $f_{15}$ 5.65$\, \times\, 10^{-4}$ 5.94$\, \times\, 10^{-4}$ 6.19$\, \times\, 10^{-4}$ 7.18$\, \times\, 10^{-4}$ 1.22$\, \times\, 10^{-3 }$ $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0 }$ $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ $f_{18}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0 }$ $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $f_{20}$ $-$3.30$\, \times\, 10^{0}$ $-$3.31$\, \times\, 10^{0}$ $-$3.32$\, \times\, 10^{0}$ $-$3.29$\, \times\, 10^{0}$ $-$3.28$\, \times\, 10^{0}$ $f_{21}$ $-$1.00$\, \times\, 10^{1}$ $-$9.64$\, \times\, 10^{0}$ $-$8.37$\, \times\, 10^{0}$ $-$6.61$\, \times\, 10^{0}$ $-$4.69$\, \times\, 10^{0}$ $f_{22}$ $-$9.99$\, \times\, 10^{0}$ $-$9.84$\, \times\, 10^{0}$ $-$9.70$\, \times\, 10^{0}$ $-$7.07$\, \times\, 10^{0}$ $-$6.21$\, \times\, 10^{0}$ $f_{23}$ $-$1.03$\, \times\, 10^{1}$ $-$1.03$\, \times\, 10^{1}$ $-$8.62$\, \times\, 10^{0}$ $-$6.92$\, \times\, 10^{0}$ $-$4.93$\, \times\, 10^{0}$ Friedman排序 $\textbf{1.89}$ 1.91 2.41 3.78 5 
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表 2 用Friedman检验法(显著性水平0.05)比较参数$p_s$
Table 2 Comparison of $p_s$ with Friedman test (significance value 0.05)
Function $p_s=0.2$ $p_s=0.4$ $p_s=0.6$ $p_s=0.8$ $p_s=1.0$ $p_s=1.2$ $p_s=1.4$ $p_s=1.6$ $p_s=1.8$ $f_1$ 3.73$\, \times\, 10^{2}$ 1.04$\, \times\, 10^{-28}$ 2.76$\, \times\, 10^{-40}$ 4.57$\, \times\, 10^{-33}$ 3.22$\, \times\, 10^{-23}$ 1.11$\, \times\, 10^{-13}$ 5.40$\, \times\, 10^{-6}$ 7.63$\, \times\, 10^{-1}$ 5.05$\, \times\, 10^{2 }$ $f_2$ 8.94$\, \times\, 10^{-1}$ 3.43$\, \times\, 10^{-27}$ 5.25$\, \times\, 10^{-27}$ 3.75$\, \times\, 10^{-22}$ 3.18$\, \times\, 10^{-16}$ 2.06$\, \times\, 10^{-10}$ 1.67$\, \times\, 10^{-5}$ 6.46$\, \times\, 10^{-1}$ 2.81$\, \times\, 10^{1 }$ $f_3$ 6.14$\, \times\, 10^{3}$ 3.43$\, \times\, 10^{3}$ 1.52$\, \times\, 10^{3}$ 2.48$\, \times\, 10^{2}$ 1.47$\, \times\, 10^{2}$ 3.02$\, \times\, 10^{3}$ 1.41$\, \times\, 10^{4}$ 3.29$\, \times\, 10^{4}$ 5.52$\, \times\, 10^{4}$ $f_4$ 4.28$\, \times\, 10^{1}$ 2.60$\, \times\, 10^{1}$ 1.33$\, \times\, 10^{1}$ 1.52$\, \times\, 10^{0}$ 4.22$\, \times\, 10^{-1}$ 2.12$\, \times\, 10^{0}$ 1.03$\, \times\, 10^{1}$ 3.12$\, \times\, 10^{1}$ 4.95$\, \times\, 10^{1 }$ $f_5$ 1.46$\, \times\, 10^{5}$ 1.80$\, \times\, 10^{2}$ 8.00$\, \times\, 10^{1}$ 4.79$\, \times\, 10^{1}$ 5.29$\, \times\, 10^{1}$ 7.48$\, \times\, 10^{1}$ 1.92$\, \times\, 10^{3}$ 8.76$\, \times\, 10^{3}$ 4.36$\, \times\, 10^{5 }$ $f_6$ 3.85$\, \times\, 10^{2}$ 4.76$\, \times\, 10^{0}$ 1.96$\, \times\, 10^{-1}$ 3.92$\, \times\, 10^{-2}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 3.27$\, \times\, 10^{0}$ 6.33$\, \times\, 10^{2}$ $f_7$ 3.49$\, \times\, 10^{-1}$ 7.34$\, \times\, 10^{-2}$ 3.53$\, \times\, 10^{-2}$ 2.24$\, \times\, 10^{-2}$ 1.27$\, \times\, 10^{-2}$ 9.89$\, \times\, 10^{-3}$ 3.69$\, \times\, 10^{-2}$ 1.38$\, \times\, 10^{-1}$ 1.71$\, \times\, 10^{0 }$ $f_8$ $-$1.17$\, \times\, 10^{4}$ $-$1.18$\, \times\, 10^{4}$ $-$1.17$\, \times\, 10^{4}$ $-$1.15$\, \times\, 10^{4}$ $-$1.15$\, \times\, 10^{4}$ $-$1.10$\, \times\, 10^{4}$ $-$1.04$\, \times\, 10^{4}$ $-$9.75$\, \times\, 10^{3}$ $-$8.78$\, \times\, 10^{3}$ $f_9$ 2.03$\, \times\, 10^{1}$ 1.04$\, \times\, 10^{1}$ 1.04$\, \times\, 10^{1}$ 1.08$\, \times\, 10^{1}$ 1.23$\, \times\, 10^{1}$ 1.37$\, \times\, 10^{1}$ 1.83$\, \times\, 10^{1}$ 4.22$\, \times\, 10^{1}$ 1.48$\, \times\, 10^{2 }$ $f_{10}$ 4.97$\, \times\, 10^{0}$ 1.22$\, \times\, 10^{0}$ 5.48$\, \times\, 10^{-2}$ 1.39$\, \times\, 10^{-14}$ 1.53$\, \times\, 10^{-12}$ 1.31$\, \times\, 10^{-7}$ 3.92$\, \times\, 10^{-1}$ 6.94$\, \times\, 10^{0}$ 1.83$\, \times\, 10^{1}$ $f_{11}$ 3.83$\, \times\, 10^{0}$ 1.72$\, \times\, 10^{-2}$ 1.78$\, \times\, 10^{-3}$ 5.73$\, \times\, 10^{-4}$ 6.10$\, \times\, 10^{-4}$ 6.34$\, \times\, 10^{-6}$ 3.03$\, \times\, 10^{-3}$ 8.30$\, \times\, 10^{-1}$ 5.52$\, \times\, 10^{0}$ $f_{12}$ 1.21$\, \times\, 10^{3}$ 3.14$\, \times\, 10^{-1}$ 5.90$\, \times\, 10^{-2}$ 8.13$\, \times\, 10^{-3}$ 4.07$\, \times\, 10^{-3}$ 1.63$\, \times\, 10^{-2}$ 4.30$\, \times\, 10^{-3}$ 4.53$\, \times\, 10^{0}$ 8.29$\, \times\, 10^{4 }$ $f_{13}$ 1.15$\, \times\, 10^{5}$ 1.55$\, \times\, 10^{1}$ 2.98$\, \times\, 10^{0}$ 7.40$\, \times\, 10^{-2}$ 5.95$\, \times\, 10^{-2}$ 4.78$\, \times\, 10^{-3}$ 3.13$\, \times\, 10^{0}$ 5.90$\, \times\, 10^{1}$ 5.70$\, \times\, 10^{5 }$ $f_{14}$ 1.91$\, \times\, 10^{0}$ 1.19$\, \times\, 10^{0}$ 1.04$\, \times\, 10^{0}$ 1.10$\, \times\, 10^{0}$ 1.02$\, \times\, 10^{0}$ 1.02$\, \times\, 10^{0}$ 1.08$\, \times\, 10^{0}$ 1.04$\, \times\, 10^{0}$ 1.04$\, \times\, 10^{0}$ $f_{15}$ 9.93$\, \times\, 10^{-4}$ 6.09$\, \times\, 10^{-4}$ 5.93$\, \times\, 10^{-4}$ 5.50$\, \times\, 10^{-4}$ 5.65$\, \times\, 10^{-4}$ 5.46$\, \times\, 10^{-4}$ 6.20$\, \times\, 10^{-4}$ 6.55$\, \times\, 10^{-4}$ 7.44$\, \times\, 10^{-4 }$ $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{0}$ $-$1.01$\, \times\, 10^{0}$ $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1 }$ $f_{18}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0 }$ $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ $f_{20}$ $-$3.27$\, \times\, 10^{0}$ $-$3.28$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.29$\, \times\, 10^{0}$ $-$3.30$\, \times\, 10^{0}$ $-$3.29$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.24$\, \times\, 10^{0}$ $-$3.23$\, \times\, 10^{0}$ $f_{21}$ $-$8.15$\, \times\, 10^{0}$ $-$9.38$\, \times\, 10^{0}$ $-$9.66$\, \times\, 10^{0}$ $-$9.99$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{1}$ $-$9.90$\, \times\, 10^{0}$ $-$9.93$\, \times\, 10^{0}$ $-$9.94$\, \times\, 10^{0}$ $-$9.24$\, \times\, 10^{0}$ $f_{22}$ $-$7.29$\, \times\, 10^{0}$ $-$9.42$\, \times\, 10^{0}$ $-$9.75$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{1}$ $-$9.99$\, \times\, 10^{0}$ $-$1.04$\, \times\, 10^{1}$ $-$1.01$\, \times\, 10^{1}$ $-$9.99$\, \times\, 10^{0}$ $-$9.93$\, \times\, 10^{0}$ $f_{23}$ $-$8.11$\, \times\, 10^{0}$ $-$9.00$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{1}$ $-$9.89$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{1}$ $-$1.03$\, \times\, 10^{1}$ $-$1.02$\, \times\, 10^{1}$ $-$1.04$\, \times\, 10^{1}$ $-$9.56$\, \times\, 10^{0 }$ Friedman排序 7.04 4.78 3.65 3.04 $\textbf{2.65}$ 3.39 5.30 6.61 8.52
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表 3 用Friedman检验法(显著性水平0.05)比较参数$p_m$
Table 3 Comparison of $p_m$ with Friedman test (significance value 0.05)
Function $p_m=0.0$ $p_m=0.1$ $p_m=0.2$ $p_m=0.3$ $p_m=0.4$ $p_m=0.5$ $p_m=0.6$ $p_m=0.7$ $p_m=0.8$ $p_m=0.9$ $p_m=1.0$ $f_1$ 9.97$\, \times\, 10^{3}$ 3.43$\, \times\, 10^{3}$ 5.14$\, \times\, 10^{2}$ 5.55$\, \times\, 10^{0}$ 7.19$\, \times\, 10^{-26}$ 2.80$\, \times\, 10^{-54}$ 8.01$\, \times\, 10^{-49 }$ 1.87$\, \times\, 10^{-40}$ 6.54$\, \times\, 10^{-32}$ 3.22$\, \times\, 10^{-23}$ 4.53$\, \times\, 10^{4}$ $f_2$ 6.45$\, \times\, 10^{1}$ 4.08$\, \times\, 10^{1}$ 1.82$\, \times\, 10^{1}$ 4.30$\, \times\, 10^{0}$ 9.80$\, \times\, 10^{-1}$ 3.92$\, \times\, 10^{-1 }$ 1.96$\, \times\, 10^{-1}$ 3.41$\, \times\, 10^{-27}$ 7.72$\, \times\, 10^{-22}$ 3.18$\, \times\, 10^{-16}$ 1.03$\, \times\, 10^{2 }$ $f_3$ 2.91$\, \times\, 10^{4}$ 2.22$\, \times\, 10^{4}$ 1.58$\, \times\, 10^{4}$ 7.56$\, \times\, 10^{3}$ 2.82$\, \times\, 10^{3}$ 2.42$\, \times\, 10^{3 }$ 2.12$\, \times\, 10^{3}$ 1.53$\, \times\, 10^{3}$ 1.08$\, \times\, 10^{3}$ 1.47$\, \times\, 10^{2}$ 5.68$\, \times\, 10^{4 }$ $f_4$ 5.37$\, \times\, 10^{1}$ 4.72$\, \times\, 10^{1}$ 4.00$\, \times\, 10^{1}$ 2.87$\, \times\, 10^{1}$ 7.44$\, \times\, 10^{0}$ 8.10$\, \times\, 10^{-1 }$ 2.19$\, \times\, 10^{-1}$ 1.69$\, \times\, 10^{-1}$ 2.25$\, \times\, 10^{-1}$ 4.22$\, \times\, 10^{-1}$ 7.67$\, \times\, 10^{1 }$ $f_5$ 8.71$\, \times\, 10^{6}$ 2.01$\, \times\, 10^{6}$ 2.25$\, \times\, 10^{5}$ 6.37$\, \times\, 10^{3}$ 2.00$\, \times\, 10^{3}$ 1.87$\, \times\, 10^{3 }$ 9.59$\, \times\, 10^{1}$ 3.69$\, \times\, 10^{1}$ 5.80$\, \times\, 10^{1}$ 5.29$\, \times\, 10^{1}$ 1.20$\, \times\, 10^{8 }$ $f_6$ 1.20$\, \times\, 10^{4}$ 5.37$\, \times\, 10^{3}$ 1.58$\, \times\, 10^{3}$ 3.35$\, \times\, 10^{2}$ 7.03$\, \times\, 10^{1}$ 1.51$\, \times\, 10^{1 }$ 1.76$\, \times\, 10^{0}$ 1.37$\, \times\, 10^{0}$ 1.57$\, \times\, 10^{-1}$ 0.00$\, \times\, 10^{0}$ 4.40$\, \times\, 10^{4}$ $f_7$ 7.62$\, \times\, 10^{0}$ 2.64$\, \times\, 10^{0}$ 1.28$\, \times\, 10^{0}$ 5.08$\, \times\, 10^{-1}$ 2.24$\, \times\, 10^{-1}$ 1.45$\, \times\, 10^{-1 }$ 6.80$\, \times\, 10^{-2}$ 4.88$\, \times\, 10^{-2}$ 2.65$\, \times\, 10^{-2}$ 1.27$\, \times\, 10^{-2}$ 5.67$\, \times\, 10^{1 }$ $f_8$ $-$7.10$\, \times\, 10^{3}$ $-$7.71$\, \times\, 10^{3}$ $-$8.25$\, \times\, 10^{3}$ $-$8.69$\, \times\, 10^{3}$ $-$8.81$\, \times\, 10^{3}$ $-$9.31$\, \times\, 10^{3 }$ $-$9.73$\, \times\, 10^{3}$ $-$1.00$\, \times\, 10^{4}$ $-$1.05$\, \times\, 10^{4}$ $-$1.15$\, \times\, 10^{4}$ $-$4.61$\, \times\, 10^{3 }$ $f_9$ 1.89$\, \times\, 10^{2}$ 1.58$\, \times\, 10^{2}$ 1.39$\, \times\, 10^{2}$ 1.08$\, \times\, 10^{2}$ 9.11$\, \times\, 10^{1}$ 7.68$\, \times\, 10^{1 }$ 5.27$\, \times\, 10^{1}$ 3.90$\, \times\, 10^{1}$ 2.66$\, \times\, 10^{1}$ 1.23$\, \times\, 10^{1}$ 3.14$\, \times\, 10^{2 }$ $f_{10}$ 1.70$\, \times\, 10^{1}$ 1.44$\, \times\, 10^{1}$ 1.10$\, \times\, 10^{1}$ 6.69$\, \times\, 10^{0}$ 4.13$\, \times\, 10^{0}$ 2.45$\, \times\, 10^{0 }$ 8.55$\, \times\, 10^{-1}$ 2.13$\, \times\, 10^{-1}$ 2.26$\, \times\, 10^{-2}$ 1.53$\, \times\, 10^{-12}$ 1.97$\, \times\, 10^{1}$ $f_{11}$ 9.11$\, \times\, 10^{1}$ 3.01$\, \times\, 10^{1}$ 6.56$\, \times\, 10^{0}$ 9.69$\, \times\, 10^{-1}$ 5.39$\, \times\, 10^{-2}$ 1.93$\, \times\, 10^{-2 }$ 9.21$\, \times\, 10^{-3}$ 3.09$\, \times\, 10^{-3}$ 9.18$\, \times\, 10^{-4}$ 6.10$\, \times\, 10^{-4}$ 3.83$\, \times\, 10^{2 }$ $f_{12}$ 5.70$\, \times\, 10^{6}$ 8.31$\, \times\, 10^{5}$ 2.34$\, \times\, 10^{3}$ 4.86$\, \times\, 10^{0}$ 7.95$\, \times\, 10^{-1}$ 5.19$\, \times\, 10^{-1 }$ 2.66$\, \times\, 10^{-1}$ 2.20$\, \times\, 10^{-1}$ 7.14$\, \times\, 10^{-2}$ 4.07$\, \times\, 10^{-3}$ 2.24$\, \times\, 10^{8}$ $f_{13}$ 2.39$\, \times\, 10^{7}$ 3.24$\, \times\, 10^{6}$ 8.92$\, \times\, 10^{4}$ 6.24$\, \times\, 10^{1}$ 6.66$\, \times\, 10^{0}$ 2.05$\, \times\, 10^{0 }$ 2.45$\, \times\, 10^{0}$ 9.56$\, \times\, 10^{-1}$ 5.66$\, \times\, 10^{-1}$ 5.95$\, \times\, 10^{-2}$ 5.39$\, \times\, 10^{8}$ $f_{14}$ 1.94$\, \times\, 10^{0}$ 2.02$\, \times\, 10^{0}$ 1.41$\, \times\, 10^{0}$ 1.11$\, \times\, 10^{0}$ 1.09$\, \times\, 10^{0}$ 9.98$\, \times\, 10^{-1}$ 1.04$\, \times\, 10^{0}$ 9.98$\, \times\, 10^{-1}$ 1.02$\, \times\, 10^{0}$ 1.02$\, \times\, 10^{0}$ 2.26$\, \times\, 10^{0 }$ $f_{15}$ 6.63$\, \times\, 10^{-3}$ 3.32$\, \times\, 10^{-3}$ 3.82$\, \times\, 10^{-3}$ 2.99$\, \times\, 10^{-3}$ 3.22$\, \times\, 10^{-3}$ 2.02$\, \times\, 10^{-3}$ 9.00$\, \times\, 10^{-4}$ 1.13$\, \times\, 10^{-3}$ 4.26$\, \times\, 10^{-4}$ 5.65$\, \times\, 10^{-4}$ 4.09$\, \times\, 10^{-3 }$ $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0 }$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{0 }$ $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ $f_{18}$ 3.53$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.53$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.01$\, \times\, 10^{0}$ $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ $f_{20}$ $-$3.26$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.28$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.26$\, \times\, 10^{0}$ $-$3.25$\, \times\, 10^{0}$ $-$3.26$\, \times\, 10^{0}$ $-$3.26$\, \times\, 10^{0}$ $-$3.30$\, \times\, 10^{0}$ $-$3.08$\, \times\, 10^{0 }$ $f_{21}$ $-$5.73$\, \times\, 10^{0}$ $-$5.38$\, \times\, 10^{0}$ $-$5.43$\, \times\, 10^{0}$ $-$6.01$\, \times\, 10^{0}$ $-$5.10$\, \times\, 10^{0}$ $-$5.66$\, \times\, 10^{0 }$ $-$6.61$\, \times\, 10^{0}$ $-$8.13$\, \times\, 10^{0}$ $-$9.10$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{1}$ $-$5.09$\, \times\, 10^{0 }$ $f_{22}$ $-$6.02$\, \times\, 10^{0}$ $-$5.84$\, \times\, 10^{0}$ $-$6.17$\, \times\, 10^{0}$ $-$5.37$\, \times\, 10^{0}$ $-$6.82$\, \times\, 10^{0}$ $-$6.55$\, \times\, 10^{0 }$ $-$7.66$\, \times\, 10^{0}$ $-$8.83$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{1}$ $-$9.99$\, \times\, 10^{0}$ $-$4.70$\, \times\, 10^{0 }$ $f_{23}$ $-$5.27$\, \times\, 10^{0}$ $-$5.23$\, \times\, 10^{0}$ $-$5.98$\, \times\, 10^{0}$ $-$5.17$\, \times\, 10^{0}$ $-$6.45$\, \times\, 10^{0}$ $-$7.78$\, \times\, 10^{0}$ $-$7.84$\, \times\, 10^{0}$ $-$9.05$\, \times\, 10^{0}$ $-$9.78$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{1}$ $-$5.04$\, \times\, 10^{0 }$ Friedman排序 6.60 6.05 5.70 6.25 5.15 5.00 5.45 5.00 5.20 $\textbf{4.90}$ 10.70
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表 4 用Friedman检验法(显著性水平0.05)比较权重系数$w_{gp}$, $w_{rp}$, $w_{ga}$, $w_{ra}$
Table 4 Comparison of $w_{gp}$, $w_{rp}$, $w_{ga}$, $w_{ra}$ with Friedman test (significance value 0.05)
Function $w_{gp}=w_{rp}=w_{ga}=w_{ra}=1$ $w_{gp}$, $w_{rp}$, $w_{ga}$, $w_{ra}$为随机数 $f_1$ 3.22$\, \times\, 10^{-23}$ 1.79$\, \times\, 10^{-19}$ $f_2$ 3.18$\, \times\, 10^{-16}$ 5.71$\, \times\, 10^{-14}$ $f_3$ 1.47$\, \times\, 10^{2}$ 7.04$\, \times\, 10^{2 }$ $f_4$ 4.22$\, \times\, 10^{-1}$ 8.05$\, \times\, 10^{-1}$ $f_5$ 5.29$\, \times\, 10^{1}$ 5.91$\, \times\, 10^{1}$ $f_6$ 0.00$\, \times\, 10^{0}$ 1.96$\, \times\, 10^{-2}$ $f_7$ 1.27$\, \times\, 10^{-2}$ 1.52$\, \times\, 10^{-2 }$ $f_8$ $-$1.15$\, \times\, 10^{4}$ $-$1.10$\, \times\, 10^{4 }$ $f_9$ 1.23$\, \times\, 10^{1}$ 1.87$\, \times\, 10^{1}$ $f_{10}$ 1.53$\, \times\, 10^{-12}$ 1.10$\, \times\, 10^{-10 }$ $f_{11}$ 6.10$\, \times\, 10^{-4}$ 2.17$\, \times\, 10^{-4} $ $f_{12}$ 4.07$\, \times\, 10^{-3}$ 1.63$\, \times\, 10^{-2 }$ $f_{13}$ 5.95$\, \times\, 10^{-2}$ 1.88$\, \times\, 10^{-1 }$ $f_{14}$ 1.02$\, \times\, 10^{0}$ 1.08$\, \times\, 10^{0 }$ $f_{15}$ 5.65$\, \times\, 10^{-4}$ 5.45$\, \times\, 10^{-4}$ $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0 }$ $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ $f_{18}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0 }$ $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ $f_{20}$ $-$3.30$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0 }$ $f_{21}$ $-$1.00$\, \times\, 10^{1}$ $-$9.96$\, \times\, 10^{0 }$ $f_{22}$ $-$9.99$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{1}$ $f_{23}$ $-$1.03$\, \times\, 10^{1}$ $-$9.52$\, \times\, 10^{0 }$ Friedman排序 $\textbf{1.13}$ 1.87
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表 5 FECO优化23个测试函数的优化结果
Table 5 Optimization results of FECO for 23 benchmark functions
Function 最好Best 最差Worst 平均Mean 标准差StdDev $f_1$ 1.27$\, \times\, 10^{-24}$ 1.15$\, \times\, 10^{-22}$ 3.22$\, \times\, 10^{-23}$ 2.71$\, \times\, 10^{-23 }$ $f_2$ 1.16$\, \times\, 10^{-16}$ 9.84$\, \times\, 10^{-16}$ 3.18$\, \times\, 10^{-16}$ 1.68$\, \times\, 10^{-16 }$ $f_3$ 1.43$\, \times\, 10^{1}$ 4.65$\, \times\, 10^{2}$ 1.47$\, \times\, 10^{2}$ 1.16$\, \times\, 10^{2 }$ $f_4$ 1.51$\, \times\, 10^{-1}$ 1.59$\, \times\, 10^{0}$ 4.22$\, \times\, 10^{-1}$ 2.59$\, \times\, 10^{-1 }$ $f_5$ 1.18$\, \times\, 10^{0}$ 1.91$\, \times\, 10^{2}$ 5.29$\, \times\, 10^{1}$ 3.84$\, \times\, 10^{1 }$ $f_6$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0 }$ $f_7$ 4.17$\, \times\, 10^{-3}$ 2.87$\, \times\, 10^{-2}$ 1.27$\, \times\, 10^{-2}$ 5.10$\, \times\, 10^{-3 }$ $f_8$ $-$1.22$\, \times\, 10^{4}$ $-$1.09$\, \times\, 10^{4}$ $-$1.15$\, \times\, 10^{4}$ 3.29$\, \times\, 10^{2 }$ $f_9$ 4.97$\, \times\, 10^{0}$ 2.49$\, \times\, 10^{1}$ 1.23$\, \times\, 10^{1}$ 5.24$\, \times\, 10^{0 }$ $f_{10}$ 4.88$\, \times\, 10^{-13}$ 4.58$\, \times\, 10^{-12}$ 1.53$\, \times\, 10^{-12}$ 8.44$\, \times\, 10^{-13 }$ $f_{11}$ 0.00$\, \times\, 10^{0}$ 1.64$\, \times\, 10^{-2}$ 6.10$\, \times\, 10^{-4}$ 2.85$\, \times\, 10^{-3 }$ $f_{12}$ 7.25$\, \times\, 10^{-24}$ 1.04$\, \times\, 10^{-1}$ 4.07$\, \times\, 10^{-3}$ 2.03$\, \times\, 10^{-2 }$ $f_{13}$ 8.50$\, \times\, 10^{-20}$ 3.02$\, \times\, 10^{0}$ 5.95$\, \times\, 10^{-2}$ 4.23$\, \times\, 10^{-1 }$ $f_{14}$ 9.98$\, \times\, 10^{-1}$ 1.99$\, \times\, 10^{0}$ 1.02$\, \times\, 10^{0}$ 1.40$\, \times\, 10^{-1 }$ $f_{15}$ 3.08$\, \times\, 10^{-4}$ 8.08$\, \times\, 10^{-4}$ 5.65$\, \times\, 10^{-4}$ 1.50$\, \times\, 10^{-4 }$ $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ 6.52$\, \times\, 10^{-3 }$ $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 1.06$\, \times\, 10^{-6 }$ $f_{18}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.58$\, \times\, 10^{-14 }$ $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ 6.57$\, \times\, 10^{-7 }$ $f_{20}$ $-$3.32$\, \times\, 10^{0}$ $-$3.20$\, \times\, 10^{0}$ $-$3.30$\, \times\, 10^{0}$ 3.84$\, \times\, 10^{-2 }$ $f_{21}$ $-$1.02$\, \times\, 10^{1}$ $-$5.10$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{1}$ 7.22$\, \times\, 10^{-1 }$ $f_{22}$ $-$1.04$\, \times\, 10^{1}$ $-$4.64$\, \times\, 10^{0}$ $-$9.99$\, \times\, 10^{0}$ 1.36$\, \times\, 10^{0 }$ $f_{23}$ $-$1.05$\, \times\, 10^{1}$ $-$5.18$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{1}$ 9.72$\, \times\, 10^{-1 }$
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表 6 FECO算法与其他启发式算法的比较(1)
Table 6 Comparison between FECO and other heuristic algorithms (1)
GA CEP FEP CES FES DE G3PCX PSO $f_1$ 3.17$\, \times\, 10^{0}$ 2.20$\, \times\, 10^{-4}$ 5.70$\, \times\, 10^{-4}$ 3.40$\, \times\, 10^{-5}$ 2.50$\, \times\, 10^{-4}$ 6.58$\, \times\, 10^{-6}$ 6.40$\, \times\, 10^{-79}$ 3.69$\, \times\, 10^{-37 }$ 算法排序 15 11 13 10 12 9 1 2 $f_2$ 5.77$\, \times\, 10^{-1}$ 2.60$\, \times\, 10^{-3}$ 8.10$\, \times\, 10^{-3}$ 2.10$\, \times\, 10^{-2}$ 6.00$\, \times\, 10^{-2}$ 2.89$\, \times\, 10^{-4}$ 2.80$\, \times\, 10^{1}$ 2.92$\, \times\, 10^{-24}$ 算法排序 14 8 9 10 12 6 16 1 $f_3$ 9.75$\, \times\, 10^{3}$ 5.00$\, \times\, 10^{-2}$ 1.60$\, \times\, 10^{-2}$ 1.30$\, \times\, 10^{-4}$ 1.40$\, \times\, 10^{-3}$ 1.21$\, \times\, 10^{4}$ 1.06$\, \times\, 10^{-76}$ 1.20$\, \times\, 10^{-3}$ 算法排序 15 10 9 6 8 16 1 7 $f_4$ 7.96$\, \times\, 10^{0}$ 2.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{-1}$ 3.50$\, \times\, 10^{-1}$ 5.50$\, \times\, 10^{-3}$ 5.79$\, \times\, 10^{0}$ 4.54$\, \times\, 10^{1}$ 4.12$\, \times\, 10^{-1 }$ 算法排序 14 11 7 8 2 12 15 9 $f_5$ 3.39$\, \times\, 10^{2}$ 6.17$\, \times\, 10^{0}$ 5.06$\, \times\, 10^{0}$ 6.69$\, \times\, 10^{0}$ 3.33$\, \times\, 10^{1}$ 9.34$\, \times\, 10^{1}$ 3.09$\, \times\, 10^{0}$ 3.74$\, \times\, 10^{1 }$ 算法排序 15 3 2 4 8 14 1 9 $f_6$ 3.70$\, \times\, 10^{0}$ 5.78$\, \times\, 10^{2}$ 0.00$\, \times\, 10^{0}$ 4.11$\, \times\, 10^{2}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 9.46$\, \times\, 10^{1}$ 1.46$\, \times\, 10^{-1 }$ 算法排序 13 16 1 15 1 1 14 9 $f_7$ 1.05$\, \times\, 10^{-1}$ 1.80$\, \times\, 10^{-2}$ 7.60$\, \times\, 10^{-3}$ 3.00$\, \times\, 10^{-2}$ 1.20$\, \times\, 10^{-2}$ 3.97$\, \times\, 10^{-2}$ 9.80$\, \times\, 10^{-1}$ 9.90$\, \times\, 10^{-3 }$ 算法排序 14 9 4 10 6 11 16 5 $f_8$ $-$1.26$\, \times\, 10^{4}$ $-$7.92$\, \times\, 10^{3}$ $-$1.26$\, \times\, 10^{4}$ $-$7.55$\, \times\, 10^{3}$ $-$1.26$\, \times\, 10^{4}$ $-$1.26$\, \times\, 10^{4}$ $-$2.58$\, \times\, 10^{3}$ $-$9.66$\, \times\, 10^{3 }$ 算法排序 5 10 7 11 6 2 15 9 $f_9$ 6.51$\, \times\, 10^{-1}$ 8.90$\, \times\, 10^{1}$ 4.60$\, \times\, 10^{-2}$ 7.08$\, \times\, 10^{1}$ 1.60$\, \times\, 10^{-1}$ 7.26$\, \times\, 10^{-5}$ 1.74$\, \times\, 10^{2}$ 2.08$\, \times\, 10^{1 }$ 算法排序 8 14 5 13 6 2 15 11 $f_{10}$ 8.68$\, \times\, 10^{-1}$ 9.20$\, \times\, 10^{0}$ 1.80$\, \times\, 10^{-2}$ 9.07$\, \times\, 10^{0}$ 1.20$\, \times\, 10^{-2}$ 7.14$\, \times\, 10^{-4}$ 1.35$\, \times\, 10^{1}$ 1.34$\, \times\, 10^{-3}$ 算法排序 11 14 8 13 7 4 15 5 $f_{11}$ 1.00$\, \times\, 10^{0}$ 8.60$\, \times\, 10^{-2}$ 1.60$\, \times\, 10^{-2}$ 3.80$\, \times\, 10^{-1}$ 3.70$\, \times\, 10^{-2}$ 9.05$\, \times\, 10^{-5}$ 1.13$\, \times\, 10^{-2}$ 2.32$\, \times\, 10^{-1 }$ 算法排序 14 11 7 13 9 1 6 12 $f_{12}$ 4.36$\, \times\, 10^{-2}$ 1.76$\, \times\, 10^{0}$ 9.20$\, \times\, 10^{-6}$ 1.18$\, \times\, 10^{0}$ 2.80$\, \times\, 10^{-2}$ 1.89$\, \times\, 10^{-7}$ 4.59$\, \times\, 10^{0}$ 3.95$\, \times\, 10^{-2 }$ 算法排序 9 13 3 12 7 2 15 8 $f_{13}$ 1.68$\, \times\, 10^{-1}$ 1.40$\, \times\, 10^{0}$ 1.60$\, \times\, 10^{-4}$ 1.39$\, \times\, 10^{0}$ 4.70$\, \times\, 10^{-5}$ 9.52$\, \times\, 10^{-7}$ 2.35$\, \times\, 10^{1}$ 5.05$\, \times\, 10^{-2 }$ 算法排序 9 12 5 11 4 2 15 7 $f_{14}$ 9.99$\, \times\, 10^{-1}$ 1.66$\, \times\, 10^{0}$ 1.22$\, \times\, 10^{0}$ 2.16$\, \times\, 10^{0}$ 1.20$\, \times\, 10^{0}$ 1.58$\, \times\, 10^{0}$ 1.23$\, \times\, 10^{1}$ 1.02$\, \times\, 10^{0 }$ 算法排序 5 11 9 13 8 10 16 7 $f_{15}$ 7.09$\, \times\, 10^{-3}$ 4.70$\, \times\, 10^{-4}$ 5.00$\, \times\, 10^{-4}$ 1.20$\, \times\, 10^{-3}$ 9.70$\, \times\, 10^{-4}$ 5.37$\, \times\, 10^{-4}$ 5.33$\, \times\, 10^{-4}$ 3.81$\, \times\, 10^{-4 }$ 算法排序 16 4 5 14 13 7 6 3 $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.02$\, \times\, 10^{0}$ $-$4.93$\, \times\, 10^{-1}$ $-$1.02$\, \times\, 10^{0 }$ 算法排序 11 9 9 4 4 12 14 13 $f_{17}$ 4.04$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 4.00$\, \times\, 10^{-1}$ 5.56$\, \times\, 10^{1}$ 4.04$\, \times\, 10^{-1 }$ 算法排序 13 7 7 7 7 12 16 13 $f_{18}$ 7.50$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.02$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.48$\, \times\, 10^{0}$ 8.67$\, \times\, 10^{0}$ 3.01$\, \times\, 10^{0 }$ 算法排序 15 2 13 2 2 14 16 11 $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.60$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ 算法排序 7 9 9 9 9 8 15 13 $f_{20}$ $-$3.26$\, \times\, 10^{0}$ $-$3.28$\, \times\, 10^{0}$ $-$3.27$\, \times\, 10^{0}$ $-$3.24$\, \times\, 10^{0}$ $-$3.23$\, \times\, 10^{0}$ $-$3.32$\, \times\, 10^{0}$ $-$1.98$\, \times\, 10^{-1}$ $-$3.18$\, \times\, 10^{0 }$ 算法排序 10 7 8 11 12 4 15 13 $f_{21}$ $-$5.17$\, \times\, 10^{0}$ $-$6.86$\, \times\, 10^{0}$ $-$5.52$\, \times\, 10^{0}$ $-$6.96$\, \times\, 10^{0}$ $-$5.54$\, \times\, 10^{0}$ $-$8.74$\, \times\, 10^{0}$ $-$7.48$\, \times\, 10^{-1}$ $-$7.54$\, \times\, 10^{0 }$ 算法排序 14 8 12 7 11 4 15 5 $f_{22}$ $-$5.44$\, \times\, 10^{0}$ $-$8.27$\, \times\, 10^{0}$ $-$5.52$\, \times\, 10^{0}$ $-$8.31$\, \times\, 10^{0}$ $-$6.76$\, \times\, 10^{0}$ $-$9.20$\, \times\, 10^{0}$ $-$9.47$\, \times\, 10^{-1}$ $-$8.36$\, \times\, 10^{0 }$ 算法排序 14 8 13 7 11 5 15 6 $f_{23}$ $-$4.91$\, \times\, 10^{0}$ $-$9.10$\, \times\, 10^{0}$ $-$6.57$\, \times\, 10^{0}$ $-$8.50$\, \times\, 10^{0}$ $-$7.63$\, \times\, 10^{0}$ $-$9.23$\, \times\, 10^{0}$ $-$1.13$\, \times\, 10^{0}$ $-$8.94$\, \times\, 10^{0 }$ 算法排序 14 7 13 9 10 6 15 8 Friedman排序 11.957 9.304 7.739 9.522 7.609 7.13 12.522 8.087 总排序 14 12 8 13 7 5 15 9
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表 7 FECO算法与其他启发式算法的比较(2)
Table 7 Comparison between FECO and other heuristic algorithms (2)
GSO GSA NFESA RCBBO RCCRO GWO WOA FECO $f_1$ 1.95$\, \times\, 10^{-8}$ 2.53$\, \times\, 10^{-16}$ 8.10$\, \times\, 10^{2}$ 1.39$\, \times\, 10^{-3}$ 6.43$\, \times\, 10^{-7}$ 6.59$\, \times\, 10^{-28}$ 1.41$\, \times\, 10^{-30}$ 3.22$\, \times\, 10^{-23}$ 算法排序 7 6 16 14 8 4 3 5 $f_2$ 3.70$\, \times\, 10^{-5}$ 5.57$\, \times\, 10^{-2}$ 4.07$\, \times\, 10^{0}$ 7.99$\, \times\, 10^{-2}$ 2.20$\, \times\, 10^{-3}$ 7.18$\, \times\, 10^{-17}$ 1.06$\, \times\, 10^{-21}$ 3.18$\, \times\, 10^{-16}$ 算法排序 5 11 15 13 7 3 2 4 $f_3$ 5.78$\, \times\, 10^{0}$ 8.97$\, \times\, 10^{2}$ 1.79$\, \times\, 10^{-6}$ 2.27$\, \times\, 10^{1}$ 2.97$\, \times\, 10^{-7}$ 3.29$\, \times\, 10^{-6}$ 5.36$\, \times\, 10^{-7}$ 1.47$\, \times\, 10^{2 }$ 算法排序 11 14 4 12 2 5 3 13 $f_4$ 1.08$\, \times\, 10^{-1}$ 7.35$\, \times\, 10^{0}$ 5.11$\, \times\, 10^{1}$ 3.09$\, \times\, 10^{-2}$ 9.32$\, \times\, 10^{-3}$ 5.61$\, \times\, 10^{-7}$ 7.26$\, \times\, 10^{-2}$ 4.22$\, \times\, 10^{-1}$ 算法排序 6 13 16 4 3 1 5 10 $f_5$ 4.98$\, \times\, 10^{1}$ 6.75$\, \times\, 10^{1}$ 1.81$\, \times\, 10^{6}$ 5.54$\, \times\, 10^{1}$ 2.71$\, \times\, 10^{1}$ 2.68$\, \times\, 10^{1}$ 2.79$\, \times\, 10^{1}$ 5.29$\, \times\, 10^{1}$ 算法排序 10 13 16 12 6 5 7 11 $f_6$ 1.60$\, \times\, 10^{-2}$ 2.50$\, \times\, 10^{-16}$ 4.51$\, \times\, 10^{-1}$ 0.00$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 8.17$\, \times\, 10^{-1}$ 3.12$\, \times\, 10^{0}$ 0.00$\, \times\, 10^{0}$ 算法排序 8 7 10 1 1 11 12 1 $f_7$ 7.38$\, \times\, 10^{-2}$ 8.94$\, \times\, 10^{-2}$ 3.86$\, \times\, 10^{-1}$ 1.75$\, \times\, 10^{-2}$ 5.41$\, \times\, 10^{-3}$ 2.21$\, \times\, 10^{-3}$ 1.43$\, \times\, 10^{-3}$ 1.27$\, \times\, 10^{-2}$ 算法排序 12 13 15 8 3 2 1 7 $f_8$ $-$1.26$\, \times\, 10^{4}$ $-$2.82$\, \times\, 10^{3}$ 7.35$\, \times\, 10^{3}$ $-$1.26$\, \times\, 10^{4}$ $-$1.26$\, \times\, 10^{4}$ $-$6.12$\, \times\, 10^{3}$ $-$5.08$\, \times\, 10^{3}$ $-$1.15$\, \times\, 10^{4}$ 算法排序 1 14 16 2 2 12 13 8 $f_9$ 1.02$\, \times\, 10^{0}$ 2.60$\, \times\, 10^{1}$ 2.12$\, \times\, 10^{2}$ 2.62$\, \times\, 10^{-2}$ 9.08$\, \times\, 10^{-4}$ 3.11$\, \times\, 10^{-1}$ 0.00$\, \times\, 10^{0}$ 1.23$\, \times\, 10^{1}$ 算法排序 9 12 16 4 3 7 1 10 $f_{10}$ 2.65$\, \times\, 10^{-5}$ 6.21$\, \times\, 10^{-2}$ 1.67$\, \times\, 10^{1}$ 2.51$\, \times\, 10^{-2}$ 1.94$\, \times\, 10^{-3}$ 1.06$\, \times\, 10^{-13}$ 7.40$\, \times\, 10^{0}$ 1.53$\, \times\, 10^{-12}$ 算法排序 3 10 16 9 6 1 12 2 $f_{11}$ 3.08$\, \times\, 10^{-2}$ 2.77$\, \times\, 10^{1}$ 5.67$\, \times\, 10^{0}$ 8.49$\, \times\, 10^{-2}$ 1.12$\, \times\, 10^{-2}$ 4.49$\, \times\, 10^{-3}$ 2.89$\, \times\, 10^{-4}$ 6.10$\, \times\, 10^{-4}$ 算法排序 8 16 15 10 5 4 2 3 $f_{12}$ 2.76$\, \times\, 10^{-11}$ 1.80$\, \times\, 10^{0}$ 4.43$\, \times\, 10^{5}$ 3.28$\, \times\, 10^{-5}$ 2.07$\, \times\, 10^{-2}$ 5.34$\, \times\, 10^{-2}$ 3.40$\, \times\, 10^{-1}$ 4.07$\, \times\, 10^{-3}$ 算法排序 1 14 16 4 6 10 11 5 $f_{13}$ 4.69$\, \times\, 10^{-5}$ 8.89$\, \times\, 10^{0}$ 3.95$\, \times\, 10^{6}$ 3.72$\, \times\, 10^{-4}$ 7.05$\, \times\, 10^{-7}$ 6.54$\, \times\, 10^{-1}$ 1.89$\, \times\, 10^{0}$ 5.95$\, \times\, 10^{-2 }$ 算法排序 3 14 16 6 1 10 13 8 $f_{14}$ 9.98$\, \times\, 10^{-1}$ 5.86$\, \times\, 10^{0}$ 9.98$\, \times\, 10^{-1}$ 9.98$\, \times\, 10^{-1}$ 9.98$\, \times\, 10^{-1}$ 4.04$\, \times\, 10^{0}$ 2.11$\, \times\, 10^{0}$ 1.02$\, \times\, 10^{0 }$ 算法排序 1 15 4 3 1 14 12 6 $f_{15}$ 3.77$\, \times\, 10^{-4}$ 3.67$\, \times\, 10^{-3}$ 7.89$\, \times\, 10^{-4}$ 7.86$\, \times\, 10^{-4}$ 5.56$\, \times\, 10^{-4}$ 3.37$\, \times\, 10^{-4}$ 5.72$\, \times\, 10^{-4}$ 5.65$\, \times\, 10^{-4}$ 算法排序 2 15 12 11 8 1 10 9 $f_{16}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ 1.06$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ 1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{0}$ 算法排序 3 1 16 7 4 15 1 8 $f_{17}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 4.24$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 3.98$\, \times\, 10^{-1}$ 算法排序 4 1 15 11 4 3 6 2 $f_{18}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.01$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 3.00$\, \times\, 10^{0}$ 算法排序 2 2 9 12 10 8 2 1 $f_{19}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ 1.47$\, \times\, 10^{-1}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0}$ $-$3.86$\, \times\, 10^{0 }$ 算法排序 2 4 16 6 1 5 14 3 $f_{20}$ $-$3.27$\, \times\, 10^{0}$ $-$3.32$\, \times\, 10^{0}$ 9.34$\, \times\, 10^{-1}$ $-$3.32$\, \times\, 10^{0}$ $-$3.32$\, \times\, 10^{0}$ $-$3.29$\, \times\, 10^{0}$ $-$2.98$\, \times\, 10^{0}$ $-$3.30$\, \times\, 10^{0}$ 算法排序 9 2 16 3 1 6 14 5 $f_{21}$ $-$6.09$\, \times\, 10^{0}$ $-$5.96$\, \times\, 10^{0}$ 6.80$\, \times\, 10^{0}$ $-$5.51$\, \times\, 10^{0}$ $-$1.01$\, \times\, 10^{1}$ $-$1.02$\, \times\, 10^{1}$ $-$7.05$\, \times\, 10^{0}$ $-$1.00$\, \times\, 10^{1}$ 算法排序 9 10 16 13 2 1 6 3 $f_{22}$ $-$6.55$\, \times\, 10^{0}$ $-$9.68$\, \times\, 10^{0}$ 6.85$\, \times\, 10^{0}$ $-$6.80$\, \times\, 10^{0}$ $-$1.04$\, \times\, 10^{1}$ $-$1.04$\, \times\, 10^{1}$ $-$8.18$\, \times\, 10^{0}$ $-$9.99$\, \times\, 10^{0}$ 算法排序 12 4 16 10 2 1 9 3 $f_{23}$ $-$7.40$\, \times\, 10^{0}$ $-$1.05$\, \times\, 10^{1}$ 6.81$\, \times\, 10^{0}$ $-$7.28$\, \times\, 10^{0}$ $-$1.05$\, \times\, 10^{1}$ $-$1.05$\, \times\, 10^{1}$ $-$9.34$\, \times\, 10^{0}$ $-$1.03$\, \times\, 10^{1}$ 算法排序 11 1 16 12 3 2 5 4 Friedman排序 6.043 9.217 14.043 8.13 3.87 5.696 7.13 5.696 总排序 4 11 16 10 1 2 5 2
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表 8 FECO算法与其他启发式算法对于CEC2015测试函数的比较
Table 8 Comparison between FECO and other heuristic algorithms for CEC2015 functions
Function CCLSHADE ICMLSP PSO NFESA FECO $f_{CEC15\_1}$ 2.28$\, \times\, 10^{-14}~(-)$ 1.03$\, \times\, 10^{-13}~(-)$ 4.70$\, \times\, 10^{6}~(+)$ 3.87$\, \times\, 10^{8}~(+)$ 3.31$\, \times\, 10^{6}$ $f_{CEC15\_2}$ 5.07$\, \times\, 10^{-14}~(-)$ 4.05$\, \times\, 10^{-5}~(-)$ 5.94$\, \times\, 10^{3 }~(\sim)$ 2.62$\, \times\, 10^{10}~(+)$ 5.97$\, \times\, 10^{3}$ $f_{CEC15\_3}$ 2.02$\, \times\, 10^{1}~(+)$ 2.00$\, \times\, 10^{1}~(-)$ 2.09$\, \times\, 10^{1}~(+)$ 2.09$\, \times\, 10^{1}~(+)$ 2.01$\, \times\, 10^{1}$ $f_{CEC15\_4}$ 4.92$\, \times\, 10^{0}~(-)$ 2.31$\, \times\, 10^{2}~(+)$ 5.63$\, \times\, 10^{1}~(+)$ 3.15$\, \times\, 10^{2}~(+)$ 4.70$\, \times\, 10^{1 }$ $f_{CEC15\_5}$ 3.00$\, \times\, 10^{2}~(-)$ 4.03$\, \times\, 10^{3}~(+)$ 2.74$\, \times\, 10^{3}~(+)$ 7.08$\, \times\, 10^{3}~(+)$ 1.81$\, \times\, 10^{3 }$ $f_{CEC15\_6}$ 8.17$\, \times\, 10^{1}~(-)$ 1.47$\, \times\, 10^{3}~(-)$ 3.64$\, \times\, 10^{5}~(+)$ 7.72$\, \times\, 10^{6}~(+)$ 3.19$\, \times\, 10^{5 }$ $f_{CEC15\_7}$ 1.50$\, \times\, 10^{0}~(-)$ 2.07$\, \times\, 10^{1}~(+)$ 9.90$\, \times\, 10^{0} ~ (\sim)$ 1.24$\, \times\, 10^{2}~(+)$ 9.40$\, \times\, 10^{0 }$ $f_{CEC15\_8}$ 1.58$\, \times\, 10^{1}~(-)$ 9.42$\, \times\, 10^{2}~(-)$ 1.52$\, \times\, 10^{5}~(+)$ 1.77$\, \times\, 10^{6}~(+)$ 9.43$\, \times\, 10^{4 }$ $f_{CEC15\_9}$ 1.03$\, \times\, 10^{2}~(\sim)$ 1.63$\, \times\, 10^{2}~(+)$ 1.03$\, \times\, 10^{2 } ~ (\sim)$ 2.22$\, \times\, 10^{2}~(+)$ 1.03$\, \times\, 10^{2}$ $f_{CEC15\_10}$ 6.06$\, \times\, 10^{2}~(-)$ 1.43$\, \times\, 10^{3}~(-)$ 1.11$\, \times\, 10^{5}~(-)$ 9.96$\, \times\, 10^{6}~(+)$ 2.93$\, \times\, 10^{5 }$ $f_{CEC15\_11}$ 4.01$\, \times\, 10^{2}~(-)$ 1.12$\, \times\, 10^{3}~(+)$ 6.26$\, \times\, 10^{2}~(+)$ 8.82$\, \times\, 10^{2}~(+)$ 5.29$\, \times\, 10^{2 }$ $f_{CEC15\_12}$ 1.04$\, \times\, 10^{2}~(-)$ 1.61$\, \times\, 10^{2}~(+)$ 1.17$\, \times\, 10^{2}~(+)$ 1.55$\, \times\, 10^{2}~(+)$ 1.06$\, \times\, 10^{2 }$ $f_{CEC15\_13}$ 2.60$\, \times\, 10^{-2}~(-)$ 8.53$\times 10^{-2}~(+)$ 8.90$\, \times\, 10^{-2}~(+)$ 6.49$\times 10^{0}~(+)$ 2.65$\, \times\, 10^{-2 }$ $f_{CEC15\_14}$ 3.28$\, \times\, 10^{4}~(-)$ 4.21$\, \times\, 10^{4}~(+)$ 3.51$\, \times\, 10^{4}~(+)$ 5.52$\, \times\, 10^{4}~(+)$ 3.36$\, \times\, 10^{4 }$ $f_{CEC15\_15}$ 1.00$\, \times\, 10^{2}~(-)$ 1.27$\, \times\, 10^{2}~(+)$ 1.00$\, \times\, 10^{2}~(-)$ 6.28$\, \times\, 10^{3}~(+)$ 1.00$\, \times\, 10^{2}$
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