An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems

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November 2018, 1(4): 349-368. doi: 10.3934/mfc.2018017

An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems

Mohamed A. Tawhid ^1, and Ahmed F. Ali ^2,3,,

1.	Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8
2.	Department of Computer Science, Faculty of Computers & Informatics, Suez Canal University, Ismailia, Egypt
3.	Postdoctoral Fellow, Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8

^* Corresponding author: Ahmed F. Ali

The research of the 1st author is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Received August 2018 Revised October 2018 Published December 2018

Firefly and cuckoo search algorithms are two of the most widely used nature-inspired algorithms due to their simplicity and inexpensive computational cost when they applied to solve a wide range of problems. In this article, a new hybrid algorithm is suggested by combining the firefly algorithm and the cuckoo search algorithm to solve constrained optimization problems (COPs) and real-world engineering optimization problems. The proposed algorithm is called Hybrid FireFly Algorithm and Cuckoo Search (HFFACS) algorithm. In the HFFACS algorithm, a balance between the exploration and the exploitation processes is considered. The main drawback of the firefly algorithm is it is easy to fall into stagnation when the new solution is not better than its previous best solution for several generations. In order to avoid this problem, the cuckoo search with Lèvy flight is invoked to force the firefly algorithm to escape from stagnation and to avoid premature convergence. The proposed algorithm is applied to six benchmark constrained optimization problems and five engineering optimization problems and compared against four algorithms to investigate its performance. The numerical experimental results show the proposed algorithm is a promising algorithm and can obtain the optimal or near optimal solution within a reasonable time.

Keywords: Firefly algorithm, cuckoo search algorithm, global optimization problems, constrained optimization problems, engineering optimization problems.

Mathematics Subject Classification: Primary: 68T01, 68T20, 68U20, 90C90, 90C30.

Citation: Mohamed A. Tawhid, Ahmed F. Ali. An effective hybrid firefly algorithm with the cuckoo search for engineering optimization problems. Mathematical Foundations of Computing, 2018, 1 (4) : 349-368. doi: 10.3934/mfc.2018017

References:

[1]	A. F. Ali and M. A. Tawhid, Hybrid particle swarm optimization with a modified arithmetical crossover for solving unconstrained optimization problems, INFOR: Information Systems and Operational Research, 53 (2015), 125-141. doi: 10.3138/infor.53.3.125.
[2]	A. F. Ali and M. A. Tawhid, Hybrid simulated annealing and pattern search algorithm for solving integer programming and minimax problems, Pacific Journal of Optimization, 12 (2016), 151-184.
[3]	A. F. Ali and M. A. Tawhid, A hybrid PSO and DE algorithm for solving engineering optimization problems, Applied Mathematics & Information Sciences, 10 (2016), 431-449. doi: 10.18576/amis/100207.
[4]	C. Blum, J. Puchinger, G. R. Raidl and A. Roli, Hybrid metaheuristics in combinatorial optimization: A survey, Applied Soft Computing, 11 (2011), 4135-4151. doi: 10.1007/978-1-4419-1644-0_9.
[5]	C. Brown, L. S. Liebovitch and R. Glendon, Lèvy lights in dobe juhoansi foraging patterns, Human Ecol., 35 (2007), 129-138.
[6]	S. A. Chu, P. W. Tsai and J. S. Pan, Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854-858. doi: 10.1007/978-3-540-36668-3_94.
[7]	M. Dorigo, Optimization, Learning and Natural Algorithms, Ph.D. Thesis, Politecnico di Milano, Italy, 1992.
[8]	W. H. El-Ashmawi, A. F. Ali and M. A. Tawhid, An improved particle swarm optimization with a new swap operator for team formation problem, Journal of Industrial Engineering International, (2018), 1-19. doi: 10.1007/s40092-018-0282-6.
[9]	C. A. Floudas and P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, In P. M. Floudas, Lecture notes in computer science, Vol.455. Berlin, 1990. doi: 10.1007/3-540-53032-0.
[10]	M. Gen and L. Lin, Multiobjective evolutionary algorithm for manufacturing scheduling problems, state of the art survey, Journal of Intelligent Manufacturing, 25 (2014), 849-866. doi: 10.1007/s10845-013-0804-4.
[11]	F. Glover, Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13 (1986), 533-549. doi: 10.1016/0305-0548(86)90048-1.
[12]	F. Glover, Atemplate for scatter search and path relinking, Lecture Notes on Computer Science, (1997), 1354-1363.
[13]	D. M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972.
[14]	W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, In Lecture notes in economics and mathematical systems (Vol.187). Berlin, Springer, 1981. doi: 10.1007/BF00934594.
[15]	J. H Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
[16]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony algorithm, Journal of Global Optimization, 39 (2007), 459-471. doi: 10.1007/s10898-007-9149-x.
[17]	J. Kennedy and R. C. Eberhart, Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.
[18]	S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.
[19]	X. L. Li, Z. J. Shao and J. X. Qian, Optimizing method based on autonomous animates Fish-swarm algorithm, Xitong Gongcheng Lilun yu Shijian-System Engineering Theory and Practice, 22 (2002), 32.
[20]	C. Liang, Y. Huang and Y. Yang, A quay crane dynamic scheduling problem by hybrid evolutionary algorithm for berth allocation planning, Comput. Ind. Eng., 56 (2009), 1021-1028. doi: 10.1016/j.cie.2008.09.024.
[21]	L. Lin, M. Gen and X. Wang, Integrated multistage logistics network design by using hybrid evolutionary algorithm, Comput. Ind. Eng., 56 (2009), 854-873. doi: 10.1016/j.cie.2008.09.037.
[22]	M. Lozano and C. Garcia-Martinez, Hybrid metaheuristics with evolutionary algorithms specializing in intensifcation and diversifcation, Overview and Progress Report. Comput. Oper. Res., 37 (2010), 481-497. doi: 10.1016/j.cor.2009.02.010.
[23]	S. Lukasik and S. Zak, Firefly Algorithm for Continuous Constrained Optimization Tasks, in Proceedings of the International Conference on Computer and Computational Intelligence (ICCCI 09), N.T. Nguyen, R. Kowalczyk, and S.-M. Chen, Eds., vol. 5796 of LNAI, 97-106, Springer, Wroclaw, Poland, October 2009. doi: 10.1007/978-3-642-04441-0_8.
[24]	O. Makeyev, E. Sazonov, M. Moklyachuk and P. Logez-Meye, Hybrid evolutionary algorithm for microscrew thread parameter estimation, Eng. Appl. Artif. Intell., 23 (2010), 446-452. doi: 10.1016/j.engappai.2010.02.009.
[25]	Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-07418-3.
[26]	N. Mladenovic, Avariable Neighborhood Algorithm a New Meta-Heuristic for Combinatorial Optimization, Abstracts of Papers Presented at Optimization Days, Montral, Canada, p. 112, 1995.
[27]	M. Mladenovic and P. Hansen, Variable neighborhood search, Computers and Operations Research, 24 (1997), 1097-1100. doi: 10.1016/S0305-0548(97)00031-2.
[28]	D. Mongus, B. Repnik, M. Mernik and B. Zalik, A hybrid evolutionry algorithm for tuning a cloth-simulation model, Appl. Soft Comput., 12 (2012), 266-273.
[29]	F. Neri and C. Cotta, Memetic algorithms and memetic computing optimization, A literature review. Swarm and Evolutionary Computation, 2 (2012), 1-14.
[30]	T. Niknam and E. A. Farsani, A hybrid self-adaptive particle swarm optimization and modified shuffed frog leaping algorithm for distribution feeder reconfguration, Eng. Appl. Artif. Intell., 23 (2010), 1340-1349.
[31]	M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.
[32]	R. B. Payne, M. D. Sorenson and K. Klitz, The Cuckoos, Oxford University Press, 2005.
[33]	Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127. doi: 10.1007/s10479-007-0224-y.
[34]	C. Prodhon, A hybrid evolutionary algorithm for the periodic location-routing problem, Eur. J. Oper. Res., 210 (2011), 204-212. doi: 10.1016/j.ejor.2010.09.021.
[35]	T. Sttzle, Local Search Algorithms for Combinatorial Problems: Analysis, Improvements, and New Applications, Ph.D. Thesis, Darmstadt University of Technology, 1998.
[36]	R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165-172. doi: 10.1109/ICDIM.2012.6360147.
[37]	M. A. Tawhid and K. B. Dsouza, Hybrid Binary Bat Enhanced Particle Swarm Optimization Algorithm for solving feature selection problems, Applied Computing and Informatics, 2018. doi: 10.1016/j.aci.2018.04.001.
[38]	M. A. Tawhid and K. B. Dsouza, Hybrid binary dragonfly enhanced particle swarm optimization algorithm for solving feature selection problems, Mathematical Foundations of Computing, 1 (2018), 181-200. doi: 10.3934/mfc.2018009.
[39]	M. A. Tawhid and A. F. Ali, Direct search firefly algorithm for solving global optimization problems, Applied Mathematics & Information Sciences, 10 (2016), 841-860.
[40]	M. A. Tawhid and A. F. Ali, A simplex grey wolf optimizer for solving integer programming and minimax problems, Numerical Algebra, Control & Optimization, 7 (2017), 301-323. doi: 10.3934/naco.2017020.
[41]	M. A. Tawhid and A. F. Ali, A hybrid social spider optimization and genetic algorithm for minimizing molecular potential energy function, Soft Computing, 21 (2017), 6499-6514. doi: 10.1007/s00500-016-2208-9.
[42]	M. A. Tawhid and A. F. Ali, A Hybrid grey wolf optimizer and genetic algorithm for minimizing potential energy function, Memetic Computing, 9 (2017), 347-359. doi: 10.1007/s12293-017-0234-5.
[43]	Y. Wang, Z. Cai, Y. Zhou and Z. Fan, Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique, Struct. Multidiscip. Optimiz., 37 (2009), 395-413. doi: 10.1007/s00158-008-0238-3.
[44]	X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, UK, 2008.
[45]	X. S. Yang and S. Deb, Cuckoo search via L`evy fights, In Nature & Biologically Inspired Computing, 2009. NaBIC 2009, World Congress on, IEEE, 2009, 210-214.
[46]	X. S. Yang, Firefly algorithm, stochastic test functions and design optimization, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.

show all references

References:

[1]	A. F. Ali and M. A. Tawhid, Hybrid particle swarm optimization with a modified arithmetical crossover for solving unconstrained optimization problems, INFOR: Information Systems and Operational Research, 53 (2015), 125-141. doi: 10.3138/infor.53.3.125.
[2]	A. F. Ali and M. A. Tawhid, Hybrid simulated annealing and pattern search algorithm for solving integer programming and minimax problems, Pacific Journal of Optimization, 12 (2016), 151-184.
[3]	A. F. Ali and M. A. Tawhid, A hybrid PSO and DE algorithm for solving engineering optimization problems, Applied Mathematics & Information Sciences, 10 (2016), 431-449. doi: 10.18576/amis/100207.
[4]	C. Blum, J. Puchinger, G. R. Raidl and A. Roli, Hybrid metaheuristics in combinatorial optimization: A survey, Applied Soft Computing, 11 (2011), 4135-4151. doi: 10.1007/978-1-4419-1644-0_9.
[5]	C. Brown, L. S. Liebovitch and R. Glendon, Lèvy lights in dobe juhoansi foraging patterns, Human Ecol., 35 (2007), 129-138.
[6]	S. A. Chu, P. W. Tsai and J. S. Pan, Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854-858. doi: 10.1007/978-3-540-36668-3_94.
[7]	M. Dorigo, Optimization, Learning and Natural Algorithms, Ph.D. Thesis, Politecnico di Milano, Italy, 1992.
[8]	W. H. El-Ashmawi, A. F. Ali and M. A. Tawhid, An improved particle swarm optimization with a new swap operator for team formation problem, Journal of Industrial Engineering International, (2018), 1-19. doi: 10.1007/s40092-018-0282-6.
[9]	C. A. Floudas and P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms, In P. M. Floudas, Lecture notes in computer science, Vol.455. Berlin, 1990. doi: 10.1007/3-540-53032-0.
[10]	M. Gen and L. Lin, Multiobjective evolutionary algorithm for manufacturing scheduling problems, state of the art survey, Journal of Intelligent Manufacturing, 25 (2014), 849-866. doi: 10.1007/s10845-013-0804-4.
[11]	F. Glover, Future paths for integer programming and links to artificial intelligence, Computers and Operations Research, 13 (1986), 533-549. doi: 10.1016/0305-0548(86)90048-1.
[12]	F. Glover, Atemplate for scatter search and path relinking, Lecture Notes on Computer Science, (1997), 1354-1363.
[13]	D. M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972.
[14]	W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, In Lecture notes in economics and mathematical systems (Vol.187). Berlin, Springer, 1981. doi: 10.1007/BF00934594.
[15]	J. H Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
[16]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony algorithm, Journal of Global Optimization, 39 (2007), 459-471. doi: 10.1007/s10898-007-9149-x.
[17]	J. Kennedy and R. C. Eberhart, Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948. doi: 10.1109/ICNN.1995.488968.
[18]	S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680. doi: 10.1126/science.220.4598.671.
[19]	X. L. Li, Z. J. Shao and J. X. Qian, Optimizing method based on autonomous animates Fish-swarm algorithm, Xitong Gongcheng Lilun yu Shijian-System Engineering Theory and Practice, 22 (2002), 32.
[20]	C. Liang, Y. Huang and Y. Yang, A quay crane dynamic scheduling problem by hybrid evolutionary algorithm for berth allocation planning, Comput. Ind. Eng., 56 (2009), 1021-1028. doi: 10.1016/j.cie.2008.09.024.
[21]	L. Lin, M. Gen and X. Wang, Integrated multistage logistics network design by using hybrid evolutionary algorithm, Comput. Ind. Eng., 56 (2009), 854-873. doi: 10.1016/j.cie.2008.09.037.
[22]	M. Lozano and C. Garcia-Martinez, Hybrid metaheuristics with evolutionary algorithms specializing in intensifcation and diversifcation, Overview and Progress Report. Comput. Oper. Res., 37 (2010), 481-497. doi: 10.1016/j.cor.2009.02.010.
[23]	S. Lukasik and S. Zak, Firefly Algorithm for Continuous Constrained Optimization Tasks, in Proceedings of the International Conference on Computer and Computational Intelligence (ICCCI 09), N.T. Nguyen, R. Kowalczyk, and S.-M. Chen, Eds., vol. 5796 of LNAI, 97-106, Springer, Wroclaw, Poland, October 2009. doi: 10.1007/978-3-642-04441-0_8.
[24]	O. Makeyev, E. Sazonov, M. Moklyachuk and P. Logez-Meye, Hybrid evolutionary algorithm for microscrew thread parameter estimation, Eng. Appl. Artif. Intell., 23 (2010), 446-452. doi: 10.1016/j.engappai.2010.02.009.
[25]	Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-07418-3.
[26]	N. Mladenovic, Avariable Neighborhood Algorithm a New Meta-Heuristic for Combinatorial Optimization, Abstracts of Papers Presented at Optimization Days, Montral, Canada, p. 112, 1995.
[27]	M. Mladenovic and P. Hansen, Variable neighborhood search, Computers and Operations Research, 24 (1997), 1097-1100. doi: 10.1016/S0305-0548(97)00031-2.
[28]	D. Mongus, B. Repnik, M. Mernik and B. Zalik, A hybrid evolutionry algorithm for tuning a cloth-simulation model, Appl. Soft Comput., 12 (2012), 266-273.
[29]	F. Neri and C. Cotta, Memetic algorithms and memetic computing optimization, A literature review. Swarm and Evolutionary Computation, 2 (2012), 1-14.
[30]	T. Niknam and E. A. Farsani, A hybrid self-adaptive particle swarm optimization and modified shuffed frog leaping algorithm for distribution feeder reconfguration, Eng. Appl. Artif. Intell., 23 (2010), 1340-1349.
[31]	M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.
[32]	R. B. Payne, M. D. Sorenson and K. Klitz, The Cuckoos, Oxford University Press, 2005.
[33]	Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127. doi: 10.1007/s10479-007-0224-y.
[34]	C. Prodhon, A hybrid evolutionary algorithm for the periodic location-routing problem, Eur. J. Oper. Res., 210 (2011), 204-212. doi: 10.1016/j.ejor.2010.09.021.
[35]	T. Sttzle, Local Search Algorithms for Combinatorial Problems: Analysis, Improvements, and New Applications, Ph.D. Thesis, Darmstadt University of Technology, 1998.
[36]	R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165-172. doi: 10.1109/ICDIM.2012.6360147.
[37]	M. A. Tawhid and K. B. Dsouza, Hybrid Binary Bat Enhanced Particle Swarm Optimization Algorithm for solving feature selection problems, Applied Computing and Informatics, 2018. doi: 10.1016/j.aci.2018.04.001.
[38]	M. A. Tawhid and K. B. Dsouza, Hybrid binary dragonfly enhanced particle swarm optimization algorithm for solving feature selection problems, Mathematical Foundations of Computing, 1 (2018), 181-200. doi: 10.3934/mfc.2018009.
[39]	M. A. Tawhid and A. F. Ali, Direct search firefly algorithm for solving global optimization problems, Applied Mathematics & Information Sciences, 10 (2016), 841-860.
[40]	M. A. Tawhid and A. F. Ali, A simplex grey wolf optimizer for solving integer programming and minimax problems, Numerical Algebra, Control & Optimization, 7 (2017), 301-323. doi: 10.3934/naco.2017020.
[41]	M. A. Tawhid and A. F. Ali, A hybrid social spider optimization and genetic algorithm for minimizing molecular potential energy function, Soft Computing, 21 (2017), 6499-6514. doi: 10.1007/s00500-016-2208-9.
[42]	M. A. Tawhid and A. F. Ali, A Hybrid grey wolf optimizer and genetic algorithm for minimizing potential energy function, Memetic Computing, 9 (2017), 347-359. doi: 10.1007/s12293-017-0234-5.
[43]	Y. Wang, Z. Cai, Y. Zhou and Z. Fan, Constrained optimization based on hybrid evolutionary algorithm and adaptive constraint-handling technique, Struct. Multidiscip. Optimiz., 37 (2009), 395-413. doi: 10.1007/s00158-008-0238-3.
[44]	X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, UK, 2008.
[45]	X. S. Yang and S. Deb, Cuckoo search via L`evy fights, In Nature & Biologically Inspired Computing, 2009. NaBIC 2009, World Congress on, IEEE, 2009, 210-214.
[46]	X. S. Yang, Firefly algorithm, stochastic test functions and design optimization, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.

Figure 1. The general performance of the HFFACS algorithm with constrained optimization problems

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Figure 2. Welded beam design problem

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Figure 3. Pressure vessel design problem

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Figure 4. Speed reducer design problem

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Figure 5. Spring design problem

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Figure 6. The general performance of the proposed HFFACS algorithm with engineering optimization problems

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Table 2. Constrained test problems

problem	f(x)
Problem1 [13]	$f_1=(x_1-2)^2+(x_2-1)^2$,
Subject to	$x_1=2x_2-1, \;\;\; \frac{x_1^2}{4}+x_2^2-1 \leq 0$
	with $x_i \in [-100, 100], \;\;\; i=1, 2$
Problem2 [9]	$f_2=(x_1-10)^3+(x_2-20)^3$,
Subject to	$100-(x_1-5)^2-(x_2-5)^2 \leq 0$,
	$(x_1-6)^2+(x_2-5)^2-82.81 \leq 0$,
	$13 \leq x_1 \leq 100, \;\;\; 0 \leq x_2 \leq 100$
Problem3 [14]	$f_3=(x_1-10)^2+5(x_2-12)^2+x_3^4+ 3(x_4-11)^2+$
Subject to	$10x_5^6+7x_6^2+x_7^4-4x_6 x_7-10x_6-8x_7$,
	$ -127+2x_1^2+3x_2^4+x_3+4x_4^2+x_5 \leq 0$
	$ -282+7x_1+3x_2+10x_3^2+x_4-x_5 \leq 0 $
	$ -196+23x_1+x_2^2+6x_6^2-8x_7 \leq 0 $,
	$4x_1^2+x_2^2-3x_1 x_2+2x_3^2+5x_6-11x_7 \leq 0$,
	$-10 \leq x_i \leq 10$,$ \; \; \; \ i=1, \ldots, 7$.
Problem4 [14]	$f_4=5.3578547x_3^2+0.8356891x_1 x_5 + 37.293239x_1-40792.141, $
Subject to	$0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$
	$90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2 \leq 110$
	$20 \leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$
	$78 \leq x_1\leq 102, \;\;33\leq x_2 \leq 45, \;\; 27 \leq x_i \leq 45, \;\;\; i=3, 4, 5, $
	Where $T_1=x_2 x_5$ and $T_5=0.0006262$
Problem5 [14]	$f_5=5.3578547x_3^2+0.8356891x_1 x_5+ 37.293239x_1-40792.141$,
Subject to	$0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$
	$90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2\leq 110$
	$20\leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$
	$78 \leq x_1\leq 102, \;\; 33\leq x_2\leq 45, \;\; 27 \leq x_i\leq 45, \;\;i=3, 4, 5, $
	Where $T_1=x_2 x_3$ and $T_5=0.00026$
Problem6 [25]	$f_6=-10.5x_1-7.5x_2-3.5x_3-2.5x_4-1.5x_5-10x_6-0.5_(i=1)^5x_i^2$
Subject to	$6x_1+3x_2+3x_3+2x_4+x_5-6.5 \leq 0$
	$10x_1+10x_3+x_6\leq 20$
	$0\leq x_i \leq 1$,$ \; \; \; i=1, \ldots, 5, $
	$70 \leq x_6 \leq 50$

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problem	f(x)
Problem1 [13]	$f_1=(x_1-2)^2+(x_2-1)^2$,
Subject to	$x_1=2x_2-1, \;\;\; \frac{x_1^2}{4}+x_2^2-1 \leq 0$
	with $x_i \in [-100, 100], \;\;\; i=1, 2$
Problem2 [9]	$f_2=(x_1-10)^3+(x_2-20)^3$,
Subject to	$100-(x_1-5)^2-(x_2-5)^2 \leq 0$,
	$(x_1-6)^2+(x_2-5)^2-82.81 \leq 0$,
	$13 \leq x_1 \leq 100, \;\;\; 0 \leq x_2 \leq 100$
Problem3 [14]	$f_3=(x_1-10)^2+5(x_2-12)^2+x_3^4+ 3(x_4-11)^2+$
Subject to	$10x_5^6+7x_6^2+x_7^4-4x_6 x_7-10x_6-8x_7$,
	$ -127+2x_1^2+3x_2^4+x_3+4x_4^2+x_5 \leq 0$
	$ -282+7x_1+3x_2+10x_3^2+x_4-x_5 \leq 0 $
	$ -196+23x_1+x_2^2+6x_6^2-8x_7 \leq 0 $,
	$4x_1^2+x_2^2-3x_1 x_2+2x_3^2+5x_6-11x_7 \leq 0$,
	$-10 \leq x_i \leq 10$,$ \; \; \; \ i=1, \ldots, 7$.
Problem4 [14]	$f_4=5.3578547x_3^2+0.8356891x_1 x_5 + 37.293239x_1-40792.141, $
Subject to	$0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$
	$90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2 \leq 110$
	$20 \leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$
	$78 \leq x_1\leq 102, \;\;33\leq x_2 \leq 45, \;\; 27 \leq x_i \leq 45, \;\;\; i=3, 4, 5, $
	Where $T_1=x_2 x_5$ and $T_5=0.0006262$
Problem5 [14]	$f_5=5.3578547x_3^2+0.8356891x_1 x_5+ 37.293239x_1-40792.141$,
Subject to	$0\leq 85.334407+0.0056858T_1 +T_2 x_1 x_4-0.0022053x_3 x_5\leq 92$
	$90 \leq 80.51249+0.0071317x_2 x_5+0.0021813x_3^2\leq 110$
	$20\leq 9.300961+0.0047026x_3 x_5+0.0012547x_1 x_3+0.0019085x_3 x_4\leq 25$
	$78 \leq x_1\leq 102, \;\; 33\leq x_2\leq 45, \;\; 27 \leq x_i\leq 45, \;\;i=3, 4, 5, $
	Where $T_1=x_2 x_3$ and $T_5=0.00026$
Problem6 [25]	$f_6=-10.5x_1-7.5x_2-3.5x_3-2.5x_4-1.5x_5-10x_6-0.5_(i=1)^5x_i^2$
Subject to	$6x_1+3x_2+3x_3+2x_4+x_5-6.5 \leq 0$
	$10x_1+10x_3+x_6\leq 20$
	$0\leq x_i \leq 1$,$ \; \; \; i=1, \ldots, 5, $
	$70 \leq x_6 \leq 50$

Table 1. Parameter setting

Parameters	Definitions	Values
$P$	Population size	60
$\alpha$	Randomization parameter	0.5
$\beta_0$	Firefly attractiveness	0.2
$\gamma$	Light absorption coefficient	1
$P_a$	A fraction of worse solutions	0.25
MGN	Maximum generation number	800

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Parameters	Definitions	Values
$P$	Population size	60
$\alpha$	Randomization parameter	0.5
$\beta_0$	Firefly attractiveness	0.2
$\gamma$	Light absorption coefficient	1
$P_a$	A fraction of worse solutions	0.25
MGN	Maximum generation number	800

Table 3. Function properties

Problem	$f$	Optimal value
Problem1	$f_1$	1.3934651
Problem2	$f_2$	-6961.81381
Problem3	$f_3$	680.630057
Problem4	$f_4$	-30665.538
Problem5	$f_5$	Unknown
Problem6	$f_6$	-213

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Problem	$f$	Optimal value
Problem1	$f_1$	1.3934651
Problem2	$f_2$	-6961.81381
Problem3	$f_3$	680.630057
Problem4	$f_4$	-30665.538
Problem5	$f_5$	Unknown
Problem6	$f_6$	-213

Table 4. The function evaluation of the proposed HFFACS algorithm for functions $f_1-f_6$

$f$	Best	Worst	Mean	St.d	Succ
$f_1$	9840	13, 200	11, 480	1681.428	30
$f_2$	11, 520	16, 200	13, 800	2340.256	30
$f_3$	40, 920	43, 800	42, 320	1441.66	30
$f_4$	10, 200	16, 800	13, 400	3304.54	30
$f_5$	8280	10, 800	9360	1297.99	30
$f_6$	42, 000	53, 280	48, 360	5776.227	30

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$f$	Best	Worst	Mean	St.d	Succ
$f_1$	9840	13, 200	11, 480	1681.428	30
$f_2$	11, 520	16, 200	13, 800	2340.256	30
$f_3$	40, 920	43, 800	42, 320	1441.66	30
$f_4$	10, 200	16, 800	13, 400	3304.54	30
$f_5$	8280	10, 800	9360	1297.99	30
$f_6$	42, 000	53, 280	48, 360	5776.227	30

Table 5. The mean function evaluation of the cuckoo search, firefly and the proposed HFFACS algorithms

$f$	CS	FA	HFFACS
$f_1$	12, 230	13, 420	11, 480
$f_2$	14, 840	15, 460	13, 800
$f_3$	42, 960	43, 450	42, 320
$f_4$	14, 840	15, 640	13, 400
$f_5$	10, 240	11, 840	9360
$f_6$	48, 840	50, 450	48, 360

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$f$	CS	FA	HFFACS
$f_1$	12, 230	13, 420	11, 480
$f_2$	14, 840	15, 460	13, 800
$f_3$	42, 960	43, 450	42, 320
$f_4$	14, 840	15, 640	13, 400
$f_5$	10, 240	11, 840	9360
$f_6$	48, 840	50, 450	48, 360

Table 6. Comparison between RWMPSOg, RWMPSOl, PSOg, PSOl, and HFFACS for functions $f_1-f_6$

$f$		Mean	StD	Succ
$f_1$	RWMPSOg	1.832	0.474	25
	RWMPSOl	1.427	0.061	30
	PSOg	2.042	0.865	24
	PSOl	1.454	0.078	30
	HFFACS	1.393	1.234e-04	30
$f_2$	RWMPSOg	-6961.283	0.380	30
	RWMPSOl	-6960.717	1.798	30
	PSOg	-6960.668	1.043	24
	PSOl	-6939.627	58.789	22
	HFFACS	-6961.813	3.818e-05	30
$f_3$	RWMPSOg	680.915	0.178	30
	RWMPSOl	680.784	0.062	30
	PSOg	681.254	0.245	30
	PSOl	680.825	0.077	30
	HFFACS	680.6301	3.464e-08	30
$f_4$	RWMPSOg	-30665.550	0.000	30
	RWMPSOl	-30665.550	0.000	30
	PSOg	-60665.550	0.000	30
	PSOl	-30665.550	0.000	30
	HFFACS	-30665.518	2.545e-07	30
$f_5$	RWMPSOg	-31021.173	11.506	30
	RWMPSOl	-31026.435	0.000	30
	PSOg	-31021.140	12.617	30
	PSOl	-31026.440	0.000	30
	HFFACS	-31026.427	1.767e-06	30
$f_6$	RWMPSOg	-212.616	1.043	30
	RWMPSOl	-212.047	0.002	30
	PSOg	-211.833	1.840	30
	PSOl	-212.933	0.365	30
	HFFACS	-212.962	0.002	30

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$f$		Mean	StD	Succ
$f_1$	RWMPSOg	1.832	0.474	25
	RWMPSOl	1.427	0.061	30
	PSOg	2.042	0.865	24
	PSOl	1.454	0.078	30
	HFFACS	1.393	1.234e-04	30
$f_2$	RWMPSOg	-6961.283	0.380	30
	RWMPSOl	-6960.717	1.798	30
	PSOg	-6960.668	1.043	24
	PSOl	-6939.627	58.789	22
	HFFACS	-6961.813	3.818e-05	30
$f_3$	RWMPSOg	680.915	0.178	30
	RWMPSOl	680.784	0.062	30
	PSOg	681.254	0.245	30
	PSOl	680.825	0.077	30
	HFFACS	680.6301	3.464e-08	30
$f_4$	RWMPSOg	-30665.550	0.000	30
	RWMPSOl	-30665.550	0.000	30
	PSOg	-60665.550	0.000	30
	PSOl	-30665.550	0.000	30
	HFFACS	-30665.518	2.545e-07	30
$f_5$	RWMPSOg	-31021.173	11.506	30
	RWMPSOl	-31026.435	0.000	30
	PSOg	-31021.140	12.617	30
	PSOl	-31026.440	0.000	30
	HFFACS	-31026.427	1.767e-06	30
$f_6$	RWMPSOg	-212.616	1.043	30
	RWMPSOl	-212.047	0.002	30
	PSOg	-211.833	1.840	30
	PSOl	-212.933	0.365	30
	HFFACS	-212.962	0.002	30

Table 7. The general performance (function evaluations) of HFFACS with engineering optimization problems

Design problem	Best	Mean	Worst	Std
Welded beam design	21, 000	22, 253.4	23, 760	1397.31
Pressure vessel design	39, 360	40, 120	40, 800	723.32
Speed reducer design	87, 840	90, 200	92, 640	2401
Three-bar truss design	6120	7440	8400	1181.86
spring design	36, 120	38, 560	40, 080	2134.291

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Design problem	Best	Mean	Worst	Std
Welded beam design	21, 000	22, 253.4	23, 760	1397.31
Pressure vessel design	39, 360	40, 120	40, 800	723.32
Speed reducer design	87, 840	90, 200	92, 640	2401
Three-bar truss design	6120	7440	8400	1181.86
spring design	36, 120	38, 560	40, 080	2134.291

Table 8. The general performance (function values) of HFFACS with engineering optimization problems

Design problem	Best	Mean	Worst	Std
Welded beam design	2.380957	2.380957	2.380957	1.82e-8
Pressure vessel design	6059.71433	6059.71433	6059.71433	4.54e-12
Speed reducer design	2.994471	2.994471	2.994471	0.00
Three-bar truss design	263.895843	263.895843	263.895843	3.54e-11
spring design	0.012665	0.012665	0.012665	5.77e-12

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Design problem	Best	Mean	Worst	Std
Welded beam design	2.380957	2.380957	2.380957	1.82e-8
Pressure vessel design	6059.71433	6059.71433	6059.71433	4.54e-12
Speed reducer design	2.994471	2.994471	2.994471	0.00
Three-bar truss design	263.895843	263.895843	263.895843	3.54e-11
spring design	0.012665	0.012665	0.012665	5.77e-12

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American Institute of Mathematical Sciences