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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

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.

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

C. BlumJ. PuchingerG. 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.  Google Scholar

[5]

C. BrownL. S. Liebovitch and R. Glendon, Lèvy lights in dobe juhoansi foraging patterns, Human Ecol., 35 (2007), 129-138.   Google Scholar

[6]

S. A. ChuP. 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.  Google Scholar

[7]

M. Dorigo, Optimization, Learning and Natural Algorithms, Ph.D. Thesis, Politecnico di Milano, Italy, 1992. Google Scholar

[8]

W. H. El-AshmawiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. Glover, Atemplate for scatter search and path relinking, Lecture Notes on Computer Science, (1997), 1354-1363.   Google Scholar

[13] D. M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972.   Google Scholar
[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.  Google Scholar

[15] J. H Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

S. KirkpatrickC. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.  Google Scholar

[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. Google Scholar

[20]

C. LiangY. 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.  Google Scholar

[21]

L. LinM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. MakeyevE. SazonovM. 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.  Google Scholar

[25]

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-07418-3.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

D. MongusB. RepnikM. Mernik and B. Zalik, A hybrid evolutionry algorithm for tuning a cloth-simulation model, Appl. Soft Comput., 12 (2012), 266-273.   Google Scholar

[29]

F. Neri and C. Cotta, Memetic algorithms and memetic computing optimization, A literature review. Swarm and Evolutionary Computation, 2 (2012), 1-14.   Google Scholar

[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.   Google Scholar

[31]

M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.   Google Scholar

[32] R. B. PayneM. D. Sorenson and K. Klitz, The Cuckoos, Oxford University Press, 2005.   Google Scholar
[33]

Y. G. PetalasK. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127.  doi: 10.1007/s10479-007-0224-y.  Google Scholar

[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.  Google Scholar

[35]

T. Sttzle, Local Search Algorithms for Combinatorial Problems: Analysis, Improvements, and New Applications, Ph.D. Thesis, Darmstadt University of Technology, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

Y. WangZ. CaiY. 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.  Google Scholar

[44] X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, UK, 2008.   Google Scholar
[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. Google Scholar

[46]

X. S. Yang, Firefly algorithm, stochastic test functions and design optimization, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.   Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

C. BlumJ. PuchingerG. 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.  Google Scholar

[5]

C. BrownL. S. Liebovitch and R. Glendon, Lèvy lights in dobe juhoansi foraging patterns, Human Ecol., 35 (2007), 129-138.   Google Scholar

[6]

S. A. ChuP. 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.  Google Scholar

[7]

M. Dorigo, Optimization, Learning and Natural Algorithms, Ph.D. Thesis, Politecnico di Milano, Italy, 1992. Google Scholar

[8]

W. H. El-AshmawiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. Glover, Atemplate for scatter search and path relinking, Lecture Notes on Computer Science, (1997), 1354-1363.   Google Scholar

[13] D. M. Himmelblau, Applied Nonlinear Programming, McGraw-Hill, New York, 1972.   Google Scholar
[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.  Google Scholar

[15] J. H Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

S. KirkpatrickC. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.  Google Scholar

[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. Google Scholar

[20]

C. LiangY. 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.  Google Scholar

[21]

L. LinM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. MakeyevE. SazonovM. 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.  Google Scholar

[25]

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-07418-3.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

D. MongusB. RepnikM. Mernik and B. Zalik, A hybrid evolutionry algorithm for tuning a cloth-simulation model, Appl. Soft Comput., 12 (2012), 266-273.   Google Scholar

[29]

F. Neri and C. Cotta, Memetic algorithms and memetic computing optimization, A literature review. Swarm and Evolutionary Computation, 2 (2012), 1-14.   Google Scholar

[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.   Google Scholar

[31]

M. K. Passino, Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.   Google Scholar

[32] R. B. PayneM. D. Sorenson and K. Klitz, The Cuckoos, Oxford University Press, 2005.   Google Scholar
[33]

Y. G. PetalasK. E. Parsopoulos and M. N. Vrahatis, Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127.  doi: 10.1007/s10479-007-0224-y.  Google Scholar

[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.  Google Scholar

[35]

T. Sttzle, Local Search Algorithms for Combinatorial Problems: Analysis, Improvements, and New Applications, Ph.D. Thesis, Darmstadt University of Technology, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

Y. WangZ. CaiY. 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.  Google Scholar

[44] X. S. Yang, Nature-Inspired Metaheuristic Algorithms, Luniver Press, UK, 2008.   Google Scholar
[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. Google Scholar

[46]

X. S. Yang, Firefly algorithm, stochastic test functions and design optimization, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.   Google Scholar

Figure 1.  The general performance of the HFFACS algorithm with constrained optimization problems
Figure 2.  Welded beam design problem
Figure 3.  Pressure vessel design problem
Figure 4.  Speed reducer design problem
Figure 5.  Spring design problem
Figure 6.  The general performance of the proposed HFFACS algorithm with engineering optimization problems
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, $
$70 \leq x_6 \leq 50$
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
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
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
$f$ 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
$f$ 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
$f$ 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
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
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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