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Sustainable closed-loop supply chain network optimization for construction machinery recovering
Quantum-inspired satin bowerbird algorithm with Bloch spherical search for constrained structural optimization
1. | College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China |
2. | Department of Science and Technology Teaching, China University of Political Science and Law, Beijing 102249, China |
3. | Guangxi Key Laboratories of Hybrid Computation and IC Design Analysis, Key Laboratory of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China |
To enhance the optimization ability of the satin bowerbird optimization (SBO) algorithm, in this paper, a novel quantum-inspired SBO with Bloch spherical search is proposed. In this algorithm, satin bowerbirds are encoded using qubits described on the Bloch sphere, each satin bowerbird occupies three locations in the search space and each location represents an optimization solution. Using the search method of general SBO to adjust the two parameters of the qubit, qubit rotation is performed on the Bloch sphere, which simultaneously updates the three locations occupied by a qubit and quickly approaches the global optimal solution. Finally, the experimental results of five examples of structural engineering design show that the proposed algorithm is superior to other state-of-the-art metaheuristic algorithms in terms of the performance measures.
References:
[1] |
S. Akhtar, K. Tai and T. Ray,
A socio-behavioural simulation model for engineering design optimization, Engineering Optimization, 34 (2002), 341-354.
doi: 10.1080/03052150212723. |
[2] |
A. D. Belegundu and J. S. Arora,
A study of mathematical programming methods for structural optimization. Part I: Theory, International Journal for Numerical Methods in Engineering, 21 (1985), 1583-1599.
doi: 10.1002/nme.1620210904. |
[3] |
M. Y. Cheng and D. Prayogo,
Symbiotic organisms search: a new metaheuristic optimization algorithm, Computers & Structures, 139 (2014), 98-112.
doi: 10.1016/j.compstruc.2014.03.007. |
[4] |
H. Chickermane and H. C. Gea,
Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39 (1996), 829-846.
doi: 10.1002/(SICI)1097-0207(19960315)39:5<829::AID-NME884>3.0.CO;2-U. |
[5] |
C. A. C. Coello,
Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems, 17 (2000), 319-346.
doi: 10.1080/02630250008970288. |
[6] |
C. A. C. Coello and E. M. Montes,
Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced Engineering Informatics, 16 (2002), 193-203.
doi: 10.1016/S1474-0346(02)00011-3. |
[7] |
K. Deb,
Optimal design of a welded beam via genetic algorithms, AIAA Journal, 29 (1991), 2013-2015.
doi: 10.2514/3.10834. |
[8] |
K. Deb,
An efficient constraint handling method for genetic algorithms, Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338.
doi: 10.1016/S0045-7825(99)00389-8. |
[9] |
K. Deb, GeneAS: A robust optimal design technique for mechanical component design, in Evolutionary Algorithms in Engineering Applications, Springer, Berlin, Heidelberg, 1997,497–514.
doi: 10.1007/978-3-662-03423-1_27. |
[10] |
K. Deb, A. Pratap and S. Agarwal,
A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.
doi: 10.1109/4235.996017. |
[11] |
D. Deutsch,
Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London Series A, 400 (1985), 97-117.
doi: 10.1098/rspa.1985.0070. |
[12] |
M. Dorigo, M. Birattari and T. Stutzle, Ant colony optimization, IEEE Computational Intelligence Magazine, (2006), 28–39. Google Scholar |
[13] |
A. H. Gandomi and X. S. Yang, Benchmark problems in structural optimization, in Computational Optimization, Methods and Algorithms, Springer, Berlin, 2011,259–281.
doi: 10.1007/978-3-642-20859-1_12. |
[14] |
A. H. Gandomi, X. S. Yang and A. H. Alavi, Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems, Engineering with Computers, 29 (2013), 17-35. Google Scholar |
[15] |
K. H. Han and J. H. Kim,
Genetic quantum algorithm and its application to combinatorial optimization problem, Proceedings of the 2000 Congress on Evolutionary Computation, 2 (2000), 1354-1360.
doi: 10.1109/CEC.2000.870809. |
[16] |
Q. He and L. Wang,
An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering Applications of Artificial Intelligence, 20 (2007), 89-99.
doi: 10.1016/j.engappai.2006.03.003. |
[17] |
C. Hui, Z. Jiashu and Z. Chao, Chaos updating rotated gates quantum-inspired genetic algorithm. Communications, Circuits and Systems, 2004 International Conference on Communications, Circuits and Systems, Chengdu, 2 (2004), 1108–1112.
doi: 10.1109/ICCCAS.2004.1346370. |
[18] |
B. K. Kannan and S. N. Kramer,
An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design, Journal of Mechanical Design, 116 (1994), 405-411.
doi: 10.1115/1.2919393. |
[19] |
D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, Journal of Global Optimization, 2007, 39(4), 459–471.
doi: 10.1007/s10898-007-9149-x. |
[20] |
A. Kaveh and S. Talatahari,
An improved ant colony optimization for constrained engineering design problems, Engineering Computations, 27 (2010), 155-182.
doi: 10.1108/02644401011008577. |
[21] |
J. Kennedy, Particle Swarm Optimization. Encyclopedia of Machine Learning, Springer, Boston, MA, 2011,760–766.
doi: 10.1007/978-0-387-30164-8_630. |
[22] |
R. A. Krohling and L. dos Santos Coelho,
Coevolutionary particle swarm optimization using Gaussian distribution for solving constrained optimization problems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 36 (2006), 1407-1416.
doi: 10.1109/TSMCB.2006.873185. |
[23] |
K. S. Lee and Z. W. Geem,
A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 3902-3933.
doi: 10.1016/j.cma.2004.09.007. |
[24] |
L. J. Li, Z. B. Huang and F. Liu,
A heuristic particle swarm optimizer for optimization of pin connected structures, Computers & Structures, 85 (2007), 340-349.
doi: 10.1016/j.compstruc.2006.11.020. |
[25] |
P. Li and S. Li,
Quantum-inspired evolutionary algorithm for continuous space optimization based on Bloch coordinates of qubits, Neurocomputing, 72 (2008), 581-591.
doi: 10.1016/j.neucom.2007.11.017. |
[26] |
J. Liu, C. Wu, G. Wu and X. Wang,
A novel differential search algorithm and applications for structure design, Applied Mathematics and Computation, 268 (2015), 246-269.
doi: 10.1016/j.amc.2015.06.036. |
[27] |
F. S. Lobato, V. Steffen and Jr ., Fish swarm optimization algorithm applied to engineering system design, Latin American Journal of Solids and Structures, 11 (2014), 143-156. Google Scholar |
[28] |
W. Long, W. Zhang, Y. Huang and Y. Chen,
A hybrid cuckoo search algorithm with feasibility-based rule for constrained structural optimization, Journal of Central South University, 21 (2014), 3197-3204.
doi: 10.1007/s11771-014-2291-y. |
[29] |
E. Mezura-Montes and C. A. C. Coello,
An empirical study about the usefulness of evolution strategies to solve constrained optimization problems, International Journal of General Systems, 37 (2008), 443-473.
doi: 10.1080/03081070701303470. |
[30] |
E. Mezura-Montes, C. A. C. Coello and R. Landa-Becerra, Engineering optimization using simple evolutionary algorithm, in Proceedings. 15th IEEE International Conference on Tools with Artificial Intelligence, Sacramento, CA, 2003,149–156.
doi: 10.1109/TAI.2003.1250183. |
[31] |
S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications, 27 (2016), 1053-1073. Google Scholar |
[32] |
S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61. Google Scholar |
[33] |
S. H. S. Moosavi and V. K. Bardsiri,
Satin bowerbird optimizer: A new optimization algorithm to optimize ANFIS for software development effort estimation, Engineering Applications of Artificial Intelligence, 60 (2012), 1-15.
doi: 10.1016/j.engappai.2017.01.006. |
[34] |
K. M. Ragsdell and D. T. Phillips,
Optimal design of a class of welded structures using geometric programming, Journal of Manufacturing Science and Engineering, 98 (1976), 1021-1025.
doi: 10.1115/1.3438995. |
[35] |
S. S. Rao, Engineering Optimization: Theory and Practice, John Wiley & Sons, Inc., New York, 2009. Google Scholar |
[36] |
T. Ray and K. M. Liew,
Society and civilization: An optimization algorithm based on the simulation of social behavior, IEEE Transactions on Evolutionary Computation, 7 (2003), 386-396.
doi: 10.1109/TEVC.2003.814902. |
[37] |
E. Rashedi, H. Nezamabadi-Pour and S. Saryazdi, GSA: A gravitational search algorithm, Information Sciences, 179 (2009), 2232-2248. Google Scholar |
[38] |
E. Sandgren,
Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design, 112 (1990), 223-229.
doi: 10.1115/1.2912596. |
[39] |
R. Storn and K. Price,
Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.
doi: 10.1023/A:1008202821328. |
[40] |
L. Wang, F. Tang and H. Wu,
Hybrid genetic algorithm based on quantum computing for numerical optimization and parameter estimation, Applied Mathematics and Computation, 171 (2005), 1141-1156.
doi: 10.1016/j.amc.2005.01.115. |
[41] |
D. H. Wolpert and W. G. Macready,
No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, 1 (1997), 67-82.
doi: 10.1109/4235.585893. |
[42] |
X. S. Yang, Flower pollination algorithm for global optimization, Unconventional Computation and Natural Computation, Springer, Berlin, Heidelberg, 2012,240–249.
doi: 10.1007/978-3-642-32894-7_27. |
[43] |
G. Zhang, W. Jin and L. Hu, A novel parallel quantum genetic algorithm, Proceedings of the Fourth International Conference on Parallel and Distributed Computing, Applications and Technologies, Chengdu, China, 2003,693–697.
doi: 10.1109/PDCAT.2003.1236393. |
show all references
References:
[1] |
S. Akhtar, K. Tai and T. Ray,
A socio-behavioural simulation model for engineering design optimization, Engineering Optimization, 34 (2002), 341-354.
doi: 10.1080/03052150212723. |
[2] |
A. D. Belegundu and J. S. Arora,
A study of mathematical programming methods for structural optimization. Part I: Theory, International Journal for Numerical Methods in Engineering, 21 (1985), 1583-1599.
doi: 10.1002/nme.1620210904. |
[3] |
M. Y. Cheng and D. Prayogo,
Symbiotic organisms search: a new metaheuristic optimization algorithm, Computers & Structures, 139 (2014), 98-112.
doi: 10.1016/j.compstruc.2014.03.007. |
[4] |
H. Chickermane and H. C. Gea,
Structural optimization using a new local approximation method, International Journal for Numerical Methods in Engineering, 39 (1996), 829-846.
doi: 10.1002/(SICI)1097-0207(19960315)39:5<829::AID-NME884>3.0.CO;2-U. |
[5] |
C. A. C. Coello,
Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems, 17 (2000), 319-346.
doi: 10.1080/02630250008970288. |
[6] |
C. A. C. Coello and E. M. Montes,
Constraint-handling in genetic algorithms through the use of dominance-based tournament selection, Advanced Engineering Informatics, 16 (2002), 193-203.
doi: 10.1016/S1474-0346(02)00011-3. |
[7] |
K. Deb,
Optimal design of a welded beam via genetic algorithms, AIAA Journal, 29 (1991), 2013-2015.
doi: 10.2514/3.10834. |
[8] |
K. Deb,
An efficient constraint handling method for genetic algorithms, Computer Methods in Applied Mechanics and Engineering, 186 (2000), 311-338.
doi: 10.1016/S0045-7825(99)00389-8. |
[9] |
K. Deb, GeneAS: A robust optimal design technique for mechanical component design, in Evolutionary Algorithms in Engineering Applications, Springer, Berlin, Heidelberg, 1997,497–514.
doi: 10.1007/978-3-662-03423-1_27. |
[10] |
K. Deb, A. Pratap and S. Agarwal,
A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.
doi: 10.1109/4235.996017. |
[11] |
D. Deutsch,
Quantum theory, the Church-Turing principle and the universal quantum computer, Proceedings of the Royal Society of London Series A, 400 (1985), 97-117.
doi: 10.1098/rspa.1985.0070. |
[12] |
M. Dorigo, M. Birattari and T. Stutzle, Ant colony optimization, IEEE Computational Intelligence Magazine, (2006), 28–39. Google Scholar |
[13] |
A. H. Gandomi and X. S. Yang, Benchmark problems in structural optimization, in Computational Optimization, Methods and Algorithms, Springer, Berlin, 2011,259–281.
doi: 10.1007/978-3-642-20859-1_12. |
[14] |
A. H. Gandomi, X. S. Yang and A. H. Alavi, Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems, Engineering with Computers, 29 (2013), 17-35. Google Scholar |
[15] |
K. H. Han and J. H. Kim,
Genetic quantum algorithm and its application to combinatorial optimization problem, Proceedings of the 2000 Congress on Evolutionary Computation, 2 (2000), 1354-1360.
doi: 10.1109/CEC.2000.870809. |
[16] |
Q. He and L. Wang,
An effective co-evolutionary particle swarm optimization for constrained engineering design problems, Engineering Applications of Artificial Intelligence, 20 (2007), 89-99.
doi: 10.1016/j.engappai.2006.03.003. |
[17] |
C. Hui, Z. Jiashu and Z. Chao, Chaos updating rotated gates quantum-inspired genetic algorithm. Communications, Circuits and Systems, 2004 International Conference on Communications, Circuits and Systems, Chengdu, 2 (2004), 1108–1112.
doi: 10.1109/ICCCAS.2004.1346370. |
[18] |
B. K. Kannan and S. N. Kramer,
An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design, Journal of Mechanical Design, 116 (1994), 405-411.
doi: 10.1115/1.2919393. |
[19] |
D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm, Journal of Global Optimization, 2007, 39(4), 459–471.
doi: 10.1007/s10898-007-9149-x. |
[20] |
A. Kaveh and S. Talatahari,
An improved ant colony optimization for constrained engineering design problems, Engineering Computations, 27 (2010), 155-182.
doi: 10.1108/02644401011008577. |
[21] |
J. Kennedy, Particle Swarm Optimization. Encyclopedia of Machine Learning, Springer, Boston, MA, 2011,760–766.
doi: 10.1007/978-0-387-30164-8_630. |
[22] |
R. A. Krohling and L. dos Santos Coelho,
Coevolutionary particle swarm optimization using Gaussian distribution for solving constrained optimization problems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 36 (2006), 1407-1416.
doi: 10.1109/TSMCB.2006.873185. |
[23] |
K. S. Lee and Z. W. Geem,
A new meta-heuristic algorithm for continuous engineering optimization: Harmony search theory and practice, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 3902-3933.
doi: 10.1016/j.cma.2004.09.007. |
[24] |
L. J. Li, Z. B. Huang and F. Liu,
A heuristic particle swarm optimizer for optimization of pin connected structures, Computers & Structures, 85 (2007), 340-349.
doi: 10.1016/j.compstruc.2006.11.020. |
[25] |
P. Li and S. Li,
Quantum-inspired evolutionary algorithm for continuous space optimization based on Bloch coordinates of qubits, Neurocomputing, 72 (2008), 581-591.
doi: 10.1016/j.neucom.2007.11.017. |
[26] |
J. Liu, C. Wu, G. Wu and X. Wang,
A novel differential search algorithm and applications for structure design, Applied Mathematics and Computation, 268 (2015), 246-269.
doi: 10.1016/j.amc.2015.06.036. |
[27] |
F. S. Lobato, V. Steffen and Jr ., Fish swarm optimization algorithm applied to engineering system design, Latin American Journal of Solids and Structures, 11 (2014), 143-156. Google Scholar |
[28] |
W. Long, W. Zhang, Y. Huang and Y. Chen,
A hybrid cuckoo search algorithm with feasibility-based rule for constrained structural optimization, Journal of Central South University, 21 (2014), 3197-3204.
doi: 10.1007/s11771-014-2291-y. |
[29] |
E. Mezura-Montes and C. A. C. Coello,
An empirical study about the usefulness of evolution strategies to solve constrained optimization problems, International Journal of General Systems, 37 (2008), 443-473.
doi: 10.1080/03081070701303470. |
[30] |
E. Mezura-Montes, C. A. C. Coello and R. Landa-Becerra, Engineering optimization using simple evolutionary algorithm, in Proceedings. 15th IEEE International Conference on Tools with Artificial Intelligence, Sacramento, CA, 2003,149–156.
doi: 10.1109/TAI.2003.1250183. |
[31] |
S. Mirjalili, Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems, Neural Computing and Applications, 27 (2016), 1053-1073. Google Scholar |
[32] |
S. Mirjalili, S. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61. Google Scholar |
[33] |
S. H. S. Moosavi and V. K. Bardsiri,
Satin bowerbird optimizer: A new optimization algorithm to optimize ANFIS for software development effort estimation, Engineering Applications of Artificial Intelligence, 60 (2012), 1-15.
doi: 10.1016/j.engappai.2017.01.006. |
[34] |
K. M. Ragsdell and D. T. Phillips,
Optimal design of a class of welded structures using geometric programming, Journal of Manufacturing Science and Engineering, 98 (1976), 1021-1025.
doi: 10.1115/1.3438995. |
[35] |
S. S. Rao, Engineering Optimization: Theory and Practice, John Wiley & Sons, Inc., New York, 2009. Google Scholar |
[36] |
T. Ray and K. M. Liew,
Society and civilization: An optimization algorithm based on the simulation of social behavior, IEEE Transactions on Evolutionary Computation, 7 (2003), 386-396.
doi: 10.1109/TEVC.2003.814902. |
[37] |
E. Rashedi, H. Nezamabadi-Pour and S. Saryazdi, GSA: A gravitational search algorithm, Information Sciences, 179 (2009), 2232-2248. Google Scholar |
[38] |
E. Sandgren,
Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design, 112 (1990), 223-229.
doi: 10.1115/1.2912596. |
[39] |
R. Storn and K. Price,
Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.
doi: 10.1023/A:1008202821328. |
[40] |
L. Wang, F. Tang and H. Wu,
Hybrid genetic algorithm based on quantum computing for numerical optimization and parameter estimation, Applied Mathematics and Computation, 171 (2005), 1141-1156.
doi: 10.1016/j.amc.2005.01.115. |
[41] |
D. H. Wolpert and W. G. Macready,
No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, 1 (1997), 67-82.
doi: 10.1109/4235.585893. |
[42] |
X. S. Yang, Flower pollination algorithm for global optimization, Unconventional Computation and Natural Computation, Springer, Berlin, Heidelberg, 2012,240–249.
doi: 10.1007/978-3-642-32894-7_27. |
[43] |
G. Zhang, W. Jin and L. Hu, A novel parallel quantum genetic algorithm, Proceedings of the Fourth International Conference on Parallel and Distributed Computing, Applications and Technologies, Chengdu, China, 2003,693–697.
doi: 10.1109/PDCAT.2003.1236393. |






Algorithms | Optimal cost | ||||
GSA [37] | 1.125 | 0.625 | 55.988 | 84.454 | 8538.835 |
PSO (He and Wang) [16] | 0.812 | 0.437 | 42.091 | 176.74 | 6061.077 |
GA (Coello) [27] | 0.812 | 0.434 | 40.323 | 200.00 | 6288.7445 |
GA(Coello and Montes) [6] | 0.812 | 0.437 | 42.097 | 176.654 | 6059.94 |
GA (Deb and Gene) [9] | 0.937 | 0.500 | 48.329 | 112.679 | 6410.38 |
ES(Montes and Coello) [29] | 0.812 | 0.437 | 42.098 | 176.640 | 6059.74 |
DE (Huang et al.) [24] | 0.812 | 0.437 | 42.098 | 176.637 | 6059.73 |
ACO(Kaveh and Talataheri) [20] | 0.812 | 0.437 | 42.103 | 176.572 | 6059.08 |
Lagrangian Multiplier (Kannan) [18] | 1.125 | 0.625 | 58.291 | 43.6900 | 7198.04 |
Branch-bound (Sandgren) [38] | 1.125 | 0.625 | 47.700 | 117.701 | 8129.10 |
SBO [33] | 0.940 | 0.468 | 46.088 | 116.438 | 6176.02 |
D-DS [26] | 0.8125 | 0.4375 | 42.098 | 176.637 | 6059.71 |
QBSBO | 0.835 | 0.411 | 43.082 | 164.816 | 5998.92 |
Algorithms | Optimal cost | ||||
GSA [37] | 1.125 | 0.625 | 55.988 | 84.454 | 8538.835 |
PSO (He and Wang) [16] | 0.812 | 0.437 | 42.091 | 176.74 | 6061.077 |
GA (Coello) [27] | 0.812 | 0.434 | 40.323 | 200.00 | 6288.7445 |
GA(Coello and Montes) [6] | 0.812 | 0.437 | 42.097 | 176.654 | 6059.94 |
GA (Deb and Gene) [9] | 0.937 | 0.500 | 48.329 | 112.679 | 6410.38 |
ES(Montes and Coello) [29] | 0.812 | 0.437 | 42.098 | 176.640 | 6059.74 |
DE (Huang et al.) [24] | 0.812 | 0.437 | 42.098 | 176.637 | 6059.73 |
ACO(Kaveh and Talataheri) [20] | 0.812 | 0.437 | 42.103 | 176.572 | 6059.08 |
Lagrangian Multiplier (Kannan) [18] | 1.125 | 0.625 | 58.291 | 43.6900 | 7198.04 |
Branch-bound (Sandgren) [38] | 1.125 | 0.625 | 47.700 | 117.701 | 8129.10 |
SBO [33] | 0.940 | 0.468 | 46.088 | 116.438 | 6176.02 |
D-DS [26] | 0.8125 | 0.4375 | 42.098 | 176.637 | 6059.71 |
QBSBO | 0.835 | 0.411 | 43.082 | 164.816 | 5998.92 |
Algorithms | Optimal cost | ||||
GSA [37] | 0.1821 | 3.8569 | 10.00 | 0.202 | 1.879 |
CPSO [22] | 0.202 | 3.5442 | 9.0482 | 0.2057 | 1.728 |
GA(Coello) [5] | N/A | N/A | N/A | N/A | 1.824 |
GA(Deb) [7] | N/A | N/A | N/A | N/A | 2.380 |
GA(Deb) [8] | 0.248 | 6.173 | 8.1789 | 0.2533 | 2.433 |
HS(Leeand Geem) [23] | 0.244 | 6.223 | 8.2915 | 0.2443 | 2.380 |
Random [34] | 0.457 | 4.731 | 5.085 | 0.660 | 4.118 |
Simplex [34] | 0.279 | 5.625 | 7.751 | 0.279 | 2.530 |
David [34] | 0.243 | 6.255 | 8.291 | 0.244 | 2.384 |
Approx [34] | 0.244 | 6.218 | 8.291 | 0.244 | 2.381 |
SBO [33] | 0.214 | 3.492 | 8.557 | 0.229 | 1.849 |
D-DS [26] | 0.206 | 3.253 | 9.037 | 0.206 | 1.696 |
QBSBO | 0.213 | 3.519 | 8.492 | 0.233 | 1.826 |
Algorithms | Optimal cost | ||||
GSA [37] | 0.1821 | 3.8569 | 10.00 | 0.202 | 1.879 |
CPSO [22] | 0.202 | 3.5442 | 9.0482 | 0.2057 | 1.728 |
GA(Coello) [5] | N/A | N/A | N/A | N/A | 1.824 |
GA(Deb) [7] | N/A | N/A | N/A | N/A | 2.380 |
GA(Deb) [8] | 0.248 | 6.173 | 8.1789 | 0.2533 | 2.433 |
HS(Leeand Geem) [23] | 0.244 | 6.223 | 8.2915 | 0.2443 | 2.380 |
Random [34] | 0.457 | 4.731 | 5.085 | 0.660 | 4.118 |
Simplex [34] | 0.279 | 5.625 | 7.751 | 0.279 | 2.530 |
David [34] | 0.243 | 6.255 | 8.291 | 0.244 | 2.384 |
Approx [34] | 0.244 | 6.218 | 8.291 | 0.244 | 2.381 |
SBO [33] | 0.214 | 3.492 | 8.557 | 0.229 | 1.849 |
D-DS [26] | 0.206 | 3.253 | 9.037 | 0.206 | 1.696 |
QBSBO | 0.213 | 3.519 | 8.492 | 0.233 | 1.826 |
Algorithms | Best | Worst | Average | Std |
D-DS [26] | 1.695 | 1.695 | 1.695 | 1.39e-06 |
SBO | 1.849 | 3.046 | 2.532 | 0.4280 |
QBSBO | 1.826 | 2.153 | 1.903 | 0.1354 |
Algorithms | Best | Worst | Average | Std |
D-DS [26] | 1.695 | 1.695 | 1.695 | 1.39e-06 |
SBO | 1.849 | 3.046 | 2.532 | 0.4280 |
QBSBO | 1.826 | 2.153 | 1.903 | 0.1354 |
Algorithms | Optimal cost | |||
GSA [37] | 0.0502 | 0.3236 | 13.525 | 0.0127 |
PSO(Ha and Wang) [16] | 0.0517 | 0.3576 | 11.244 | 0.01267 |
ES(Coello and Montes) [29] | 0.0519 | 0.3639 | 10.890 | 0.01268 |
GA (Coello) [27] | 0.0514 | 0.3516 | 11.632 | 0.01270 |
Montes and Coello [24] | 0.0516 | 0.3553 | 11.397 | 0.0126 |
Constraintcorrection(Arora) [20] | 0.0500 | 0.3159 | 14.250 | 0.0128 |
Mathematical optimization(Belegundu) [2] | 0.0533 | 0.3991 | 9.1854 | 0.0127 |
SBO [33] | 0.055 | 0.464 | 7.0393 | 0.0131 |
D-DS [26] | 0.052 | 0.356 | 11.3434 | 0.0127 |
QBSBO | 0.051 | 0.357 | 11.28 | 0.0127 |
Algorithms | Optimal cost | |||
GSA [37] | 0.0502 | 0.3236 | 13.525 | 0.0127 |
PSO(Ha and Wang) [16] | 0.0517 | 0.3576 | 11.244 | 0.01267 |
ES(Coello and Montes) [29] | 0.0519 | 0.3639 | 10.890 | 0.01268 |
GA (Coello) [27] | 0.0514 | 0.3516 | 11.632 | 0.01270 |
Montes and Coello [24] | 0.0516 | 0.3553 | 11.397 | 0.0126 |
Constraintcorrection(Arora) [20] | 0.0500 | 0.3159 | 14.250 | 0.0128 |
Mathematical optimization(Belegundu) [2] | 0.0533 | 0.3991 | 9.1854 | 0.0127 |
SBO [33] | 0.055 | 0.464 | 7.0393 | 0.0131 |
D-DS [26] | 0.052 | 0.356 | 11.3434 | 0.0127 |
QBSBO | 0.051 | 0.357 | 11.28 | 0.0127 |
Algorithms | Best | Worst | Average | Std |
D-DS [26] | 0.0127 | 0.0127 | 0.0127 | 0.00001 |
SBO | 0.0131 | 0.0183 | 0.0149 | 0.0014 |
QBSBO | 0.0127 | 0.0150 | 0.0134 | 0.0005 |
Algorithms | Best | Worst | Average | Std |
D-DS [26] | 0.0127 | 0.0127 | 0.0127 | 0.00001 |
SBO | 0.0131 | 0.0183 | 0.0149 | 0.0014 |
QBSBO | 0.0127 | 0.0150 | 0.0134 | 0.0005 |
Algorithms | Optimal cost | |||||
MMA [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
GCA_I [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
GCA_II [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
CS [31] | 6.0089 | 5.3049 | 4.5023 | 3.5077 | 2.1504 | 1.3399 |
SOS [3] | 6.0187 | 5.3034 | 4.4958 | 3.4989 | 2.1556 | 1.3399 |
SBO | 6.1900 | 5.3221 | 4.3851 | 3.4203 | 2.1761 | 1.3412 |
QBSBO | 6.0091 | 5.2999 | 4.4901 | 3.4789 | 2.1531 | 1.3400 |
Algorithms | Optimal cost | |||||
MMA [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
GCA_I [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
GCA_II [4] | 6.0100 | 5.3000 | 4.4900 | 3.4900 | 2.1500 | 1.3400 |
CS [31] | 6.0089 | 5.3049 | 4.5023 | 3.5077 | 2.1504 | 1.3399 |
SOS [3] | 6.0187 | 5.3034 | 4.4958 | 3.4989 | 2.1556 | 1.3399 |
SBO | 6.1900 | 5.3221 | 4.3851 | 3.4203 | 2.1761 | 1.3412 |
QBSBO | 6.0091 | 5.2999 | 4.4901 | 3.4789 | 2.1531 | 1.3400 |
Algorithms | Best | Worst | Average | Std |
SBO | 1.3514 | 1.3564 | 1.3468 | 0.00378 |
QBSBO | 1.3400 | 1.3536 | 1.3446 | 0.00409 |
Algorithms | Best | Worst | Average | Std |
SBO | 1.3514 | 1.3564 | 1.3468 | 0.00378 |
QBSBO | 1.3400 | 1.3536 | 1.3446 | 0.00409 |
Algorithms | Optimal cost | |||||||
Akhtar et al. [1] | 3.5061 | 0.7000 | 17 | 7.5491 | 7.8593 | 3.3655 | 5.28977 | 3008.08 |
Mezura-Montes et al. [30] | 3.5061 | 0.7008 | 17 | 7.1601 | 7.9621 | 3.3629 | 5.3090 | 3025.005 |
CS [31] | 3.5015 | 0.7 | 17 | 7.6050 | 7.8181 | 3.3520 | 5.2875 | 3000.981 |
HCPS [28] | 3.5 | 0.7 | 17 | 7.3 | 7.7153 | 3.3502 | 5.28665 | 2994.471 |
SCA [36] | 3.5 | 0.7 | 17 | 7.327602 | 7.7153 | 3.3502 | 5.2866 | 2994.744 |
ABC [35] | 3.499999 | 0.7 | 17 | 7.3 | 7.8 | 3.3502 | 5.2878 | 2997.058 |
SBO | 3.5036 | 0.7000 | 17 | 7.5376 | 7.3000 | 3.3579 | 5.2935 | 2998.9 |
QBSBO | 3.50000 | 0.70000 | 17 | 7.30000 | 7.30000 | 3.3502 | 5.28652 | 2985.142 |
Algorithms | Optimal cost | |||||||
Akhtar et al. [1] | 3.5061 | 0.7000 | 17 | 7.5491 | 7.8593 | 3.3655 | 5.28977 | 3008.08 |
Mezura-Montes et al. [30] | 3.5061 | 0.7008 | 17 | 7.1601 | 7.9621 | 3.3629 | 5.3090 | 3025.005 |
CS [31] | 3.5015 | 0.7 | 17 | 7.6050 | 7.8181 | 3.3520 | 5.2875 | 3000.981 |
HCPS [28] | 3.5 | 0.7 | 17 | 7.3 | 7.7153 | 3.3502 | 5.28665 | 2994.471 |
SCA [36] | 3.5 | 0.7 | 17 | 7.327602 | 7.7153 | 3.3502 | 5.2866 | 2994.744 |
ABC [35] | 3.499999 | 0.7 | 17 | 7.3 | 7.8 | 3.3502 | 5.2878 | 2997.058 |
SBO | 3.5036 | 0.7000 | 17 | 7.5376 | 7.3000 | 3.3579 | 5.2935 | 2998.9 |
QBSBO | 3.50000 | 0.70000 | 17 | 7.30000 | 7.30000 | 3.3502 | 5.28652 | 2985.142 |
Algorithms | Best | Worst | Average | Std |
SBO | 2998.9 | 3108.540 | 3065.71 | 23.017 |
QBSBO | 2985.185 | 3366.970 | 3084.75 | 144.36 |
Algorithms | Best | Worst | Average | Std |
SBO | 2998.9 | 3108.540 | 3065.71 | 23.017 |
QBSBO | 2985.185 | 3366.970 | 3084.75 | 144.36 |
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