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

  • * Corresponding author: Guo Zhou

    * Corresponding author: Guo Zhou 
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  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  Qubit description on the Bloch sphere

    Figure 2.  Pressure vessel design problem

    Figure 3.  Welded beam design problem

    Figure 4.  Tension/compression spring design problem

    Figure 5.  Cantilever beam design problem

    Figure 6.  Speed reducer design problem

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [1] S. AkhtarK. 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. DebA. 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.
    [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. GandomiX. S. Yang and A. H. Alavi, Cuckoo search algorithm: A metaheuristic approach to solve structural optimization problems, Engineering with Computers, 29 (2013), 17-35. 
    [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. LiZ. 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. LiuC. WuG. 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. LobatoV. Steffen and Jr ., Fish swarm optimization algorithm applied to engineering system design, Latin American Journal of Solids and Structures, 11 (2014), 143-156. 
    [28] W. LongW. ZhangY. 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. 
    [32] S. MirjaliliS. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61. 
    [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.
    [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. RashediH. Nezamabadi-Pour and S. Saryazdi, GSA: A gravitational search algorithm, Information Sciences, 179 (2009), 2232-2248. 
    [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. WangF. 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.
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