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Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks

  • * Corresponding author: Mohsen Abdolhosseinzadeh

    * Corresponding author: Mohsen Abdolhosseinzadeh 

The authors would like to thank anonymous reviewers for their useful comments

Abstract Full Text(HTML) Figure(8) / Table(3) Related Papers Cited by
  • In a grid network, the nodes could be traversed either horizontally or vertically. The constrained shortest Hamiltonian path goes over the nodes between a source node and a destination node, and it is constrained to traverse some nodes at least once while others could be traversed several times. There are various applications of the problem, especially in routing problems. It is an NP-complete problem, and the well-known Bellman-Held-Karp algorithm could solve the shortest Hamiltonian circuit problem within $ {\rm O(}{{\rm 2}}^{{\rm n}}{{\rm n}}^{{\rm 2}}{\rm )} $ time complexity; however, the shortest Hamiltonian path problem is more complicated. So, a metaheuristic algorithm based on ant colony optimization is applied to obtain the optimal solution. The proposed method applies the rooted shortest path tree structure since in the optimal solution the paths between the restricted nodes are the shortest paths. Then, the shortest path tree is obtained by at most $ {\rm O(}{{\rm n}}^{{\rm 3}}{\rm )} $ time complexity at any iteration and the ants begin to improve the solution and the optimal solution is constructed in a reasonable time. The algorithm is verified by some numerical examples and the ant colony parameters are tuned by design of experiment method, and the optimal setting for different size of networks are determined.

    Mathematics Subject Classification: Primary: 62K20, 58F17; Secondary: 53C35.


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  • Figure 1.  The horizontal and orthogonal movements in the grid networks

    Figure 2.  The shortest path tree

    Figure 3.  The ACO Algorithm for the C-SHP problem

    Figure 4.  The initial neighborhood methods

    Figure 5.  The pareto charts of the standardized effects for the responses in the screening phase for the network 200$ \times $200

    Figure 6.  The pareto charts of the standardized effects in the network 200$ \times $200 for the responses in Box–Behnken design

    Figure 7.  The optimized fitted response plots for the network 200$ \times $200

    Figure 8.  The solution diagram of ACO algorithm by optimal tuning of parameters

    Table 1.  The design factors and their levels

    factors levels
    -1 0 1
    A $ \alpha $ 1 7 13
    B $ \beta $ 0 6 13
    C $ Q $ 1 2 4
    D $ \tau (0) $ 0.1 0.5 0.9
    E $ \rho $ 0.01 0.5 0.99
    F initial solution RO NN1 NN2
    G ant number 0.5 1 1.5
    Blocks instance size small moderate large
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    Table 2.  The optimal coded and uncoded settings of the factors

    network size factors $ \alpha $ $ \beta $ $ Q $ $ \tau (0) $ $ \rho $ initial solution ant number desirability value
    100$ \times $100 optimal coded 1 -1 1 -1 -0.68 0.05 -0.98 0.9640
    optimal uncoded 13 0 4 0.1 0.17 NN1 0.5
    200$ \times $200 optimal coded -1 -1 -0.99 -0.70 0.68 -0.04 -1 0.9957
    optimal uncoded 1 0 1 0.22 0.83 NN1 0.5
    400$ \times $400 optimal coded -1 -0.54 -1 1 1 0.15 -1 0.8996
    optimal uncoded 1 3 1 0.9 0.99 NN1 0.5
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    Table 3.  The optimal prediction and the confidence intervals of the responses

    Network Response Fit SE Fit 95% CI 95% PI
    100$ \times $100 Improve 0.2970 0.0848 (0.1226, 0.4713) (0.0829, 0.5111)
    CPU Time 18 10.1 (-2.8, 38.8) (-7.6, 43.5)
    Opt. Sol. 3608 318 (2954, 4262) (2804, 4412)
    200$ \times $200 Improve 0.2119 0.0534 (0.1020, 0.3217) (0.0724, 0.3513)
    CPU Time 161.25 4.77 (151.44,171.05) (148.80,173.69)
    Opt. Sol. 13719 654 (12375, 15063) (12013, 15425)
    400$ \times $400 Improve 0.1605 0.0413 (0.0756, 0.2454) (0.0555, 0.2655)
    CPU Time 1763 647 (434, 3092) (119, 3407)
    Opt. Sol. 54087 1726 (50540, 57634) (49699, 58474)
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