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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
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