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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 |
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.
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.
|






problem | f(x) |
Problem1 [13] | |
Subject to | |
with |
|
Problem2 [9] | |
Subject to | |
Problem3 [14] | |
Subject to | |
Problem4 [14] | |
Subject to | |
Where |
|
Problem5 [14] | |
Subject to | |
Where |
|
Problem6 [25] | |
Subject to | |
problem | f(x) |
Problem1 [13] | |
Subject to | |
with |
|
Problem2 [9] | |
Subject to | |
Problem3 [14] | |
Subject to | |
Problem4 [14] | |
Subject to | |
Where |
|
Problem5 [14] | |
Subject to | |
Where |
|
Problem6 [25] | |
Subject to | |
Parameters | Definitions | Values |
Population size | 60 | |
Randomization parameter | 0.5 | |
Firefly attractiveness | 0.2 | |
Light absorption coefficient | 1 | |
A fraction of worse solutions | 0.25 | |
MGN | Maximum generation number | 800 |
Parameters | Definitions | Values |
Population size | 60 | |
Randomization parameter | 0.5 | |
Firefly attractiveness | 0.2 | |
Light absorption coefficient | 1 | |
A fraction of worse solutions | 0.25 | |
MGN | Maximum generation number | 800 |
Problem | Optimal value | |
Problem1 | 1.3934651 | |
Problem2 | -6961.81381 | |
Problem3 | 680.630057 | |
Problem4 | -30665.538 | |
Problem5 | Unknown | |
Problem6 | -213 |
Problem | Optimal value | |
Problem1 | 1.3934651 | |
Problem2 | -6961.81381 | |
Problem3 | 680.630057 | |
Problem4 | -30665.538 | |
Problem5 | Unknown | |
Problem6 | -213 |
Best | Worst | Mean | St.d | Succ | |
9840 | 13, 200 | 11, 480 | 1681.428 | 30 | |
11, 520 | 16, 200 | 13, 800 | 2340.256 | 30 | |
40, 920 | 43, 800 | 42, 320 | 1441.66 | 30 | |
10, 200 | 16, 800 | 13, 400 | 3304.54 | 30 | |
8280 | 10, 800 | 9360 | 1297.99 | 30 | |
42, 000 | 53, 280 | 48, 360 | 5776.227 | 30 |
Best | Worst | Mean | St.d | Succ | |
9840 | 13, 200 | 11, 480 | 1681.428 | 30 | |
11, 520 | 16, 200 | 13, 800 | 2340.256 | 30 | |
40, 920 | 43, 800 | 42, 320 | 1441.66 | 30 | |
10, 200 | 16, 800 | 13, 400 | 3304.54 | 30 | |
8280 | 10, 800 | 9360 | 1297.99 | 30 | |
42, 000 | 53, 280 | 48, 360 | 5776.227 | 30 |
CS | FA | HFFACS | |
12, 230 | 13, 420 | 11, 480 | |
14, 840 | 15, 460 | 13, 800 | |
42, 960 | 43, 450 | 42, 320 | |
14, 840 | 15, 640 | 13, 400 | |
10, 240 | 11, 840 | 9360 | |
48, 840 | 50, 450 | 48, 360 |
CS | FA | HFFACS | |
12, 230 | 13, 420 | 11, 480 | |
14, 840 | 15, 460 | 13, 800 | |
42, 960 | 43, 450 | 42, 320 | |
14, 840 | 15, 640 | 13, 400 | |
10, 240 | 11, 840 | 9360 | |
48, 840 | 50, 450 | 48, 360 |
Mean | StD | Succ | ||
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 | |
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 | |
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 | |
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 | |
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 | |
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 |
Mean | StD | Succ | ||
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 | |
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 | |
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 | |
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 | |
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 | |
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 |
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 |
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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