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An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle

  • * Corresponding author: Mao Chen

    * Corresponding author: Mao Chen 
Abstract Full Text(HTML) Figure(7) / Table(4) Related Papers Cited by
  • This paper presents a heuristic algorithm for solving a specific NP-hard 2D rectangular packing problem in which a rectangle called central rectangle is required to be placed in the center of the final layout, and the aspect ratio of the container is also required to be in a given range. The key component of the proposed algorithm is a greedy constructive procedure, according to which, the rectangles are packed into the container one by one and each rectangle is packed into the container by an angle-occupying placement with maximum fit degree. The proposed algorithm is evaluated on two groups of 35 well-known benchmark instances. Computational results disclose that the proposed algorithm outperforms the previous algorithm for the packing problem. For the first group of test instances, solutions with average filling rate 99.31% can be obtained; for the real-world layout problem in the second group, the filling rate of the solution is 94.75%.

    Mathematics Subject Classification: Primary: 68T20, 68R05.

    Citation:

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  • Figure 1.  Cartesian coordinate system and central rectangle

    Figure 2.  An illustrative example of angle

    Figure 3.  An illustrative example of initial layout

    Figure 4.  An illustrative example of angle

    Figure 5.  Visual representation of the changing trend with the increase of $\varepsilon$

    Figure 6.  The final layout of two hard test instances

    Figure 7.  Final layout of the drilling platform

    Table 1.  Settings of important parameters

    ParameterDescriptionValue
    Nar$_{\min}$Lower bound of the aspect ratio of the container0.5
    Nar$_{\max}$Upper bound of the aspect ratio of the container2
    $Z_{Hmin}$Lower bound of the between centrality of CR in vertical direction1
    $Z_{Hmax}$Upper bound of the between centrality of CR in vertical direction2
    $Z_{Wmin}$Lower bound of the between centrality of CR in horizontal direction1
    $Z_{Wmax}$Upper bound of the between centrality of CR in horizontal direction2
     | Show Table
    DownLoad: CSV

    Table 2.  Filling rate (%) of test instances N1--N10 under different values of parameter $\varepsilon $

    InstanceFilling rate (%) under different values of parameter $\varepsilon $
    0.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.6
    N181.6381.6383.3377.3783.3378.4377.7577.3774.5974.5973.4673.4674.5974.5973.4652.59
    N210010098.8198.0410098.8198.8198.0498.0498.0498.0498.0498.8198.8198.0498.04
    N399.6799.6799.4799.6799.2199.2199.2198.7598.8198.8198.8898.0499.2198.0498.0490.74
    N498.9898.7799.0198.4698.4998.7798.9897.8698.5198.2898.0598.7796.7597.0998.3189.19
    N599.2199.2199.2199.2198.1898.2498.1298.3998.1997.9897.6597.9898.2797.2397.9297.22
    N610010099.7099.5699.5099.5699.7099.2199.3099.2199.3699.3699.3699.7099.5099.21
    N710099.9510099.9599.5599.9599.5099.5099.5599.3099.9099.9599.5599.3099.3899.30
    N899.9099.9099.9199.9099.7699.7099.7699.9099.3799.7199.5599.6999.5799.7699.7199.36
    N999.7599.7599.7399.7399.7399.6899.7399.6899.6899.6899.6899.6899.6899.6899.6899.68
    N1010010099.9610099.9399.9699.9399.8999.7499.9199.8999.7499.9399.9199.8899.71
    Average97.9197.8997.9197.1997.7797.2397.1596.8696.5896.5596.0696.4796.5796.4196.3992.50
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison with previous state-of-the-art algorithms on N13 and C21 instances

    Instance$n$HACRHA_AOP
    $H$$W$Fillrate (%)Nar$Z_{H}$$Z_{W}$Time (s)$H$$W$Fillrate (%)Nar$Z_{H}$$Z_{W}$Time (s)
    N110483790.091.2971.4001.05615.37484083.331.2002.0001.0000.45
    N220364592.590.8001.2501.50077.4830501000.62.0001.5000.13
    N330325093.750.6401.1331.174162.07285499.210.5191.0001.0000.70
    N440897694.621.1711.1191.533294.51709398.490.7531.5002.00017.10
    N5509211495.350.8071.2441.073452.749810498.180.9421.5001.00066.40
    N660747195.171.0421.4671.536747.08756799.501.1192.0002.00020.76
    N770859896.040.8671.2371.3331061.3908999.551.0111.5001.50062.07
    N880968696.901.1161.4001.2631406.7998199.761.2222.0001.00054.29
    N9100818597.470.8531.0251.1591913.8948099.731.1751.0002.00059.98
    N102007813897.550.5651.0261.3796075.41218799.931.3912.0001.500334.58
    N113008912098.310.7421.0941.124124891051001001.0502.0002.000572.07
    N1250015519599.260.7951.0671.0002043515419699.390.7861.0002.00093376.28
    N13315282275099.661.0961.0551.3441.16$\times $10$^{5}$10246001001.7072.0002.000187.80
    C1116182492.590.7501.2501.40054.4520201001.0001.0001.5000.07
    C1217162792.590.5931.2861.07760.8820201001.0001.0001.5000.05
    C1316212190.701.0001.6251.62552.1625161001.5631.0001.0000.01
    C2125282393.171.2171.0001.30098.7630201001.5002.0001.0000.02
    C2225242792.300.8891.4001.45594.0520301000.6672.0001.0000.04
    C2325282393.171.2171.3331.556103.4430201001.5002.0001.5000.02
    C3128385094.740.7601.1111.083154.6230601000.5002.0001.5000.86
    C3229523793.561.4051.3641.176149.78444199.781.0731.5002.0002.11
    C3328404893.750.8331.5001.400160.2240451000.8891.0002.0000.74
    C4149547095.240.7711.7001.333427.69824499.781.8642.0002.00011.28
    C4249735294.841.4041.4331.364432.1572501001.4402.0001.5002.65
    C4349606395.240.9521.3081.100430.2750721000.6942.0001.0004.05
    C5172807096.431.1431.3531.1881216.3100541001.8521.5001.0001.80
    C5273737796.070.9481.4331.2651300.4100541001.8522.0002.0000.21
    C5372787296.151.0831.1371.2151278.290601001.5001.0002.0005.06
    C61971019996.011.0201.1721.0201714.5120801001.5001.0002.00013.99
    C62971089296.621.1741.5711.3001807.6128751001.7072.0002.0002.65
    C639710010096.001.0001.2731.1741786.38511399.950.7521.0001.50083.70
    C7119619720097.460.9851.4021.1286246.218321099.920.8711.5002.0003668.73
    C7219718021997.410.8221.3081.1906012.72561501001.7072.0001.0001.23
    C7319623816697.201.4341.3801.2436175.619419899.970.9802.0001.0004621.96
    Average95.245614.3299.313045.99
     | Show Table
    DownLoad: CSV

    Table 4.  Details of the modules in a practical layout problem

    No.Layout moduleLength (m)Width (m)Height (m)
    1Drilling floor33.002410.00
    2Drilling collar storage area9.602.001.10
    3Drilling pipe area No.19.608.501.70
    4Drilling pipe area No.29.608.501.70
    530in drive pipe area12.502.702.70
    620in drive pipe area12.503.503.50
    713-3/8in drive pipe area12.504.004.00
    89-5/8in drive pipe area12.007.006.00
    97in drive pipe area10.505.004.20
    10Flatwise marine riser area23.0013.007.00
    11Vertical marine riser area32.0010.0023.00
    12Pipe conveyor area24.004.0010.00
    13Bop area28.5010.003.80
    14Christmas tree area20.009.503.80
    15Mud purification area18.0016.502.00
    16Living quarters38.0011.0011.00
     | Show Table
    DownLoad: CSV
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