Gaussian Chaotic Fire Hawk Optimization Algorithm for Solving Dynamic Optimization Problems

doi:10.16182/j.issn1004731x.joss.23-1128

Abstract

Abstract:

There are many important chemical processes in the chemical industry rely on dynamic optimization with factors such as nonlinearity and discontinuity. In order to find a more efficient solution algorithm, Gaussian Chaotic fire hawk optimization algorithm is proposed based on the fire hawk optimization algorithm, which is used to solve such problems after parameterizing the control variables. The original way of initializing the populations is replaced using tent chaotic mapping in order to make more sense of the initial distribution of the algorithm; a more targeted update method has been proposed in the analysis of fire hawk location updates and prey location updates, enhancing the ability to develop and explore algorithms, Gaussian sampling is also embedded to improve the diversity of the population, further enhancing the algorithm's local search and dynamic adaptation capabilities. The results show the effectiveness of the algorithm in solving chemical dynamic optimization problems.

Key words: chemical dynamic optimization, Gauss sampling, control variable parameterization, chaotic mapping, fire hawk optimization algorithm

CLC Number:

TQ021.8

Chen Yongzhang, Mo Yuanbin. Gaussian Chaotic Fire Hawk Optimization Algorithm for Solving Dynamic Optimization Problems[J]. Journal of System Simulation, 2025, 37(2): 436-449.

Figures/Tables 16

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Benchmark functions

函数	名称	解空间	最优解
F1	Sphere	[-100, 100]	0
F2	Schwefel 1.2	[-100, 100]	0
F3	Step	[-100, 100]	0
F4	Rastrigin	[-5.12, 5.12]	0
F5	Ackley	[-32, 32]	0
F6	Griewank	[-600, 600]	0
F7	Penalized 1	[-50, 50]	0
F8	Penalized 2	[-50, 50]	0
F9	Hartman 2	$[0,1]$	-3.32

Table 1

Table 2

Algorithm parameters setting

算法	参数设置
GCFHO	$c 1$ 、 $c 3 = 3$ ， $c 2$ 、 $c 4 = 2$
HHO	$E ∈ (- 1,1)$
PFA	$u 1$ 、 $u 2 ∈ [- 1,1]$

Table 2

Table 3

Test function simulation results

函数	指标	GCFHO	FHO	WOA	HHO	PFA	GWO
F1	最小值	1.416 2×10^-240	1.221 8×10^-143	3.263 7×10^-172	4.257 4×10^-215	1.614 1×10^-28	3.054 4×10^-61
	最大值	9.102 8×10^-227	2.347 4×10^-129	2.227 9×10^-152	5.157 3×10^-189	2.410 1×10^-25	5.336 2×10^-58
	均值	9.409 2×10^-228	1.915 7×10^-130	1.128 9×10^-153	5.045 1×10^-190	2.899 3×10^-26	6.214 6×10^-59
	标准差	0	6.020 7×10^-130	4.978 5×10^-153	0	5.574 4×10^-26	1.315 4×10^-58
F2	最小值	1.416 2×10^-240	1.221 8×10^-143	3.263 7×10^-172	4.257 4×10^-215	1.614 1×10^-28	3.054 4×10^-61
	最大值	9.102 8×10^-227	2.347 4×10^-129	2.227 9×10^-152	5.157 3×10^-189	2.410 1×10^-25	5.336 2×10^-58
	均值	9.409 2×10^-228	1.915 7×10^-130	1.128 9×10^-153	5.045 1×10^-190	2.899 3×10^-26	6.214 6×10^-59
	标准差	0	6.020 7×10^-130	4.978 5×10^-153	0	5.574 4×10^-26	1.315 4×10^-58
F3	最小值	0	0.011 5	0.010 3	9.921 92×10^-8	4.084 28×10^-6	2.240 73×10^-5
	最大值	0	6.792 4	0.354 0	0.000 5	1.762 4×10^-5	1.752 2
	均值	0	4.715 3	0.064 8	6.340 37×10^-5	9.447 59×10^-6	0.718 2
	标准差	0	2.120 6	0.079 4	0.000 1	3.484 66×10^-6	0.441 8
F4	最小值	0	0	0	0	45.989 3	0
	最大值	0	0	5.684 34×10^-14	0	143.143 2	1.136 87×10^-13
	均值	0	0	2.842 17×10^-15	0	80.350 5	8.526 51×10^-15
	标准差	0	0	1.271 06×10^-14	0	25.358 2	2.781 69×10^-14
F5	最小值	4.440 89×10^-16	4.440 89×10^-16	4.440 89×10^-16	4.440 89×10^-16	6.794 56×10^-14	1.465 49×10^-14
	最大值	4.440 89×10^-16	4.440 89×10^-16	7.549 52×10^-15	4.440 89×10^-16	3.521 0	2.176 04×10^-14
	均值	4.440 89×10^-16	4.440 89×10^-16	3.286 26×10^-15	4.440 89×10^-16	2.011 3	1.518 79×10^-14
	标准差	0	0	2.472 16×10^-15	0	0.944 0	1.738 56×10^-15
F6	最小值	0	0	0	0	0	0
	最大值	0	0	0.107 2	0	0.041 7	0.045 5
	均值	0	0	0.005 4	0	0.010 7	0.005 9
	标准差	0	0	0.024 0	0	0.011 6	0.013 3
F7	最小值	1.570 54×10^-32	0.000 1	0.001 1	2.293 89×10^-9	9.966 35×10^-7	0.012 4
	最大值	1.570 54×10^-32	0.301 0	1.265 2	9.377 16×10^-6	8.112 3	0.071 6
	均值	1.570 54×10^-32	0.044 9	0.067 3	2.677 63×10^-6	2.566 2	0.037 4
	标准差	2.808 01×10^-48	0.091 8	0.282 0	2.626 97×10^-6	2.792 4	0.016 0
F8	最小值	1.349 78×10^-32	0.015 2	0.024 7	3.777 41×10^-7	4.727 4×10^-6	0.170 3
	最大值	1.217 75×10^-7	2.998 7	0.710 4	0.000 1	0.233 4	1.123 2
	均值	6.088 75×10^-9	1.694 8	0.233 1	1.500 92×10^-5	0.035 8	0.554 6
	标准差	2.722 97×10^-8	1.384 1	0.189 4	3.216 06×10^-5	0.071 6	0.243 9
F9	最小值	-3.322 0	-3.308 2	-3.322 0	-3.293 6	-3.322 0	-3.322 0
	最大值	-3.203 1	-3.012 4	-2.873 4	-2.800 2	-3.203 1	-3.136 9
	均值	-3.298 2	-3.240 4	-3.235 6	-3.141 1	-3.268 5	-3.225 5
	标准差	0.048 8	0.091 3	0.126 0	0.124 0	0.060 7	0.069 0

Table 3

Fig. 5

Table 4

Fig. 6

Fig. 7

Table 5

Fig. 8

Fig. 9

Fig. 10

Table 6

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方法	维度	最优	最差	均值	标准差	时间/s
ADIWO^[7]	60	0.610 792	0.610 792	0.610 792	—	188
IDP^[10]	80	0.610 775	—	—	—	—
ISOA^[11]	30	0.610 592 23	—	—	—	272
APSO^[16]	100	0.610 785 0	0.610 785 0	0.610 785 0	—	—
EBSO^[17]	80	0.610 781 14	—	—	—	—
IKBCA^[18]	100	0.610 787	0.610 779	—	—	—
FHO	50	0.606 440 5	0.605 946 5	0.606 056 3	0.000 166 017	93
GCFHO	50	0.610 708	0.610 708	0.610 708	4.431 25×10^-8	34
	100	0.610 792	0.610 792	0.610 792	4.330 03×10^-9	225
	150	0.610 796	0.610 796	0.610 796	3.230 19×10^-8	551

方法	维度	最优	最差	均值	标准差	时间/s
HGPSO^[8]	10	0.477 71^*	0.477 71	0.477 71	—	9 641
IDP^[10]	40	0.476 95	—	—	—	—
ISOA^[11]	40	4.777 0	—	—	—	285
EBSO^[17]	40	0.476 972 8
IKBCA^[18]	100	0.477 70	0.477 68	—	—	—
TDE^[20]	40	0.476 946	—	—	—	—
FHO	90	0.473 434	0.471 652	0.472 330	0.000 527 142	165
	50	0.477 262	0.477 262	0.477 262	5.730 68×10^-8	35
GCFHO	90	0.477 711	0.477 711	0.477 711	2.441 65×10^-8	237
	120	0.477 688	0.477 687	0.477 687	6.080 86×10^-7	294

方法	维度	最优	最差	均值	标准差	时间/s
HGPSO^[8]	15	0.133 47	0.133 98	0.133 66	—	21 158
DE^[9]	13	0.135 580	0.244 612	0.146 501	0.033 552 999	674
LCA^[12]	20	0.133 18	0.201 02	0.147 142 5	0.024 088 327	—
QITLBO^[13]	20	0.133 11^*	0.133 14	0.133 20	2.39×10^-5	461
文献[21]	78	0.133 21	—	—	—	—
FHO	100	0.149 918	0.159 827	0.153 750	0.003 040 983	198
GCFHO	50	0.133 27	0.133 27	0.133 27	2.543 26×10^-11	53
	100	0.133 14	0.133 14	0.133 14	1.280 38×10^-8	235
	156	0.133 11	0.133 12	0.133 11	1.883 85×10^-6	492

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