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doi: 10.3934/jimo.2020130

Quantum-inspired satin bowerbird algorithm with Bloch spherical search for constrained structural optimization

Sen Zhang 1, , Guo Zhou 2,, , Yongquan Zhou 3, and  Qifang Luo 3,

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

* Corresponding author: Guo Zhou

Received  December 2019 Revised  May 2020 Early access  August 2020

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.

Keywords: Quantum coding Bloch spherical coordinates, quantum-inspired satin bowerbird algorithm, constrained structural optimization, metaheuristic.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Sen Zhang, Guo Zhou, Yongquan Zhou, Qifang Luo. Quantum-inspired satin bowerbird algorithm with Bloch spherical search for constrained structural optimization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020130
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. A. C. Coello, Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems, 17 (2000), 319-346.  doi: 10.1080/02630250008970288.  Google Scholar

[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.  Google Scholar

[7]

K. Deb, Optimal design of a welded beam via genetic algorithms, AIAA Journal, 29 (1991), 2013-2015.  doi: 10.2514/3.10834.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. A. C. Coello, Constraint-handling using an evolutionary multiobjective optimization technique, Civil Engineering Systems, 17 (2000), 319-346.  doi: 10.1080/02630250008970288.  Google Scholar

[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.  Google Scholar

[7]

K. Deb, Optimal design of a welded beam via genetic algorithms, AIAA Journal, 29 (1991), 2013-2015.  doi: 10.2514/3.10834.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Qubit description on the Bloch sphere
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Figure 2.  Pressure vessel design problem
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Figure 3.  Welded beam design problem
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Figure 4.  Tension/compression spring design problem
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Figure 5.  Cantilever beam design problem
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Figure 6.  Speed reducer design problem
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Table 1.  Comparison results for the pressure vessel design problem with optimal
Algorithms $T_s$ $T_h$ $R$ $L$ 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
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Algorithms $T_s$ $T_h$ $R$ $L$ 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
Table 2.  Comparison statistical results for the pressure vessel design problem
Algorithms Best Worst Average Std
D-DS [26] 6059.71 6410.02 6121.42 23.81
SBO [33] 6176.02 7113.5 6792.53 257.96
QBSBO 5998.67 7069.8 6434.56 189.75
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Algorithms Best Worst Average Std
D-DS [26] 6059.71 6410.02 6121.42 23.81
SBO [33] 6176.02 7113.5 6792.53 257.96
QBSBO 5998.67 7069.8 6434.56 189.75
Table 3.  Comparison results for the pressure vessel design problem with optimal
Algorithms $h$ $l$ $t$ $b$ 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
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Algorithms $h$ $l$ $t$ $b$ 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
Table 4.  Comparison statistical results for the welded beam design problem
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
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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
Table 5.  Comparison results for the compression spring design problem
Algorithms $d$ $D$ $N$ 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
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Algorithms $d$ $D$ $N$ 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
Table 6.  Compression spring design problem
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
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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
Table 7.  Cantilever design problem
Algorithms $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ 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
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Algorithms $x_1$ $x_2$ $x_3$ $x_4$ $x_5$ 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
Table 8.  Cantilever design problem
Algorithms Best Worst Average Std
SBO 1.3514 1.3564 1.3468 0.00378
QBSBO 1.3400 1.3536 1.3446 0.00409
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Algorithms Best Worst Average Std
SBO 1.3514 1.3564 1.3468 0.00378
QBSBO 1.3400 1.3536 1.3446 0.00409
Table 9.  Comparison results for the speed reducer design problem
Algorithms $b$ $m$ $z$ $l_1$ $l_2$ $d_1$ $d_2$ 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
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Algorithms $b$ $m$ $z$ $l_1$ $l_2$ $d_1$ $d_2$ 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
Table 10.  Comparison results for the speed reducer design problem
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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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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