doi: 10.3934/jimo.2018164

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

National Engineering Research Center for E-learning, Central China Normal University, Wuhan 430079, China

* Corresponding author: Mao Chen

Received  November 2017 Revised  May 2018 Published  October 2018

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%.

Citation: Mao Chen, Xiangyang Tang, Zhizhong Zeng, Sanya Liu. An efficient heuristic algorithm for two-dimensional rectangular packing problem with central rectangle. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018164
References:
[1]

A. LodiS. Martello and M. Monaci, Two-dimensional packing problems: A survey, European Journal of Operational Research, 141 (2002), 241-252. doi: 10.1016/S0377-2217(02)00123-6.

[2]

K. A. Dowsland and W. B. Dowsland, Packing problems, European Journal of Operational Research, 56 (1992), 2-14.

[3]

H. F. Lee and E. C. Sewell, The strip-packing problem for a boat manufacturing firm, IIE Transactions, 31 (1999), 639-651. doi: 10.1080/07408179908969865.

[4]

D. S. Hochbaum and W. Maass, Approximation schemes for covering and packing problems in image processing and VLSI, Journal of the Association for Computing Machinery, 32 (1985), 130-136. doi: 10.1145/2455.214106.

[5]

B. S. BakerJr. E. G. Coffman and R. L. Rivest, Orthogonal packing in two dimensions, SIAM Journal on Computing, 9 (1980), 846-855. doi: 10.1137/0209064.

[6]

B. Chazelle, The bottom-left bin packing heuristic: An efficient implementation, IEEE Transactions on Computers, 32 (1983), 697-707. doi: 10.1109/tc.1983.1676307.

[7]

S. Jakobs, On genetic algorithms for the packing of polygons, European Journal of Operational Research, 88 (1996), 165-181. doi: 10.1016/0377-2217(94)00166-9.

[8]

D. Q. Liu and H. F. Teng, An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles, European Journal of Operational Research, 112 (1999), 413-420. doi: 10.1016/S0377-2217(97)00437-2.

[9]

E. Hopper and B. Turton, An empirical investigation of meta-heuristic and heuristic algorithms for a 2D packing problem, European Journal of Operational Research, 128 (2001), 34-57. doi: 10.1016/S0377-2217(99)00357-4.

[10]

Y. L. WuW. Q. HuangS. C. LauC. K. Wong and G. H. Young, An effective quasi-human based heuristic for solving the rectangle packing problem, European Journal of Operational Research, 141 (2002), 341-358. doi: 10.1016/S0377-2217(02)00129-7.

[11]

W. Q. Huang and D. B. Chen, An efficient heuristic algorithm for rectangle-packing problem, Simulation Modelling Practice and Theory, 15 (2007), 1356-1365.

[12]

D. F. ZhangY. Kang and A. S. Deng, A new heuristic recursive algorithm for the strip rectangular packing problem, Computers & Operations Research, 33 (2006), 2209-2217.

[13]

L. J. WeiD. F. Zhang and Q. S. Chen, A least wasted first heuristic algorithm for the rectangular packing problem, Computers & Operations Research, 36 (2009), 1608-1614. doi: 10.1016/j.cor.2008.03.004.

[14]

R. Alvarez-ValdesF. Parreño and J. M. Tamarit, Reactive GRASP for the strip-packing problem, Computers & Operations Research, 35 (2008), 1065-1083. doi: 10.1016/j.cor.2006.07.004.

[15]

K. HeW. Q. Huang and Y. Jin, An efficient deterministic heuristic for two-dimensional rectangular packing, Computers & Operations Research, 39 (2012), 1355-1363. doi: 10.1016/j.cor.2011.08.005.

[16]

W. S. Xiao, L. Wu, X. Tian and J. L. Wang, Applying a new adaptive genetic algorithm to study the layout of drilling equipment in semisubmersible drilling platforms, Mathematical Problems in Engineering, 2015 (2015), Article ID 146902, 9 pages. doi: 10.1155/2015/146902.

[17]

L. WuL. ZhangW.S. XiaoQ. LiuC. Mu and Y.W. Yang, A novel heuristic algorithm for two-dimensional rectangle packing area minimization problem with central rectangle, Computers & Industrial Engineering, 102 (2016), 208-218. doi: 10.1016/j.cie.2016.10.011.

[18]

E. K. BurkeG. Kendall and G. Whitwell, A new placement heuristic for the orthogonal stock-cutting problem, Operations Research, 52 (2004), 655-671. doi: 10.1287/opre.1040.0109.

show all references

References:
[1]

A. LodiS. Martello and M. Monaci, Two-dimensional packing problems: A survey, European Journal of Operational Research, 141 (2002), 241-252. doi: 10.1016/S0377-2217(02)00123-6.

[2]

K. A. Dowsland and W. B. Dowsland, Packing problems, European Journal of Operational Research, 56 (1992), 2-14.

[3]

H. F. Lee and E. C. Sewell, The strip-packing problem for a boat manufacturing firm, IIE Transactions, 31 (1999), 639-651. doi: 10.1080/07408179908969865.

[4]

D. S. Hochbaum and W. Maass, Approximation schemes for covering and packing problems in image processing and VLSI, Journal of the Association for Computing Machinery, 32 (1985), 130-136. doi: 10.1145/2455.214106.

[5]

B. S. BakerJr. E. G. Coffman and R. L. Rivest, Orthogonal packing in two dimensions, SIAM Journal on Computing, 9 (1980), 846-855. doi: 10.1137/0209064.

[6]

B. Chazelle, The bottom-left bin packing heuristic: An efficient implementation, IEEE Transactions on Computers, 32 (1983), 697-707. doi: 10.1109/tc.1983.1676307.

[7]

S. Jakobs, On genetic algorithms for the packing of polygons, European Journal of Operational Research, 88 (1996), 165-181. doi: 10.1016/0377-2217(94)00166-9.

[8]

D. Q. Liu and H. F. Teng, An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles, European Journal of Operational Research, 112 (1999), 413-420. doi: 10.1016/S0377-2217(97)00437-2.

[9]

E. Hopper and B. Turton, An empirical investigation of meta-heuristic and heuristic algorithms for a 2D packing problem, European Journal of Operational Research, 128 (2001), 34-57. doi: 10.1016/S0377-2217(99)00357-4.

[10]

Y. L. WuW. Q. HuangS. C. LauC. K. Wong and G. H. Young, An effective quasi-human based heuristic for solving the rectangle packing problem, European Journal of Operational Research, 141 (2002), 341-358. doi: 10.1016/S0377-2217(02)00129-7.

[11]

W. Q. Huang and D. B. Chen, An efficient heuristic algorithm for rectangle-packing problem, Simulation Modelling Practice and Theory, 15 (2007), 1356-1365.

[12]

D. F. ZhangY. Kang and A. S. Deng, A new heuristic recursive algorithm for the strip rectangular packing problem, Computers & Operations Research, 33 (2006), 2209-2217.

[13]

L. J. WeiD. F. Zhang and Q. S. Chen, A least wasted first heuristic algorithm for the rectangular packing problem, Computers & Operations Research, 36 (2009), 1608-1614. doi: 10.1016/j.cor.2008.03.004.

[14]

R. Alvarez-ValdesF. Parreño and J. M. Tamarit, Reactive GRASP for the strip-packing problem, Computers & Operations Research, 35 (2008), 1065-1083. doi: 10.1016/j.cor.2006.07.004.

[15]

K. HeW. Q. Huang and Y. Jin, An efficient deterministic heuristic for two-dimensional rectangular packing, Computers & Operations Research, 39 (2012), 1355-1363. doi: 10.1016/j.cor.2011.08.005.

[16]

W. S. Xiao, L. Wu, X. Tian and J. L. Wang, Applying a new adaptive genetic algorithm to study the layout of drilling equipment in semisubmersible drilling platforms, Mathematical Problems in Engineering, 2015 (2015), Article ID 146902, 9 pages. doi: 10.1155/2015/146902.

[17]

L. WuL. ZhangW.S. XiaoQ. LiuC. Mu and Y.W. Yang, A novel heuristic algorithm for two-dimensional rectangle packing area minimization problem with central rectangle, Computers & Industrial Engineering, 102 (2016), 208-218. doi: 10.1016/j.cie.2016.10.011.

[18]

E. K. BurkeG. Kendall and G. Whitwell, A new placement heuristic for the orthogonal stock-cutting problem, Operations Research, 52 (2004), 655-671. doi: 10.1287/opre.1040.0109.

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
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
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
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
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}$10246001001.7072.0002.000187.80
C1116182492.590.7501.2501.40054.4520201001.0001.0001.5000.07
C1217162792.590.5931.2861.07760.8820201001.0001.0001.5000.05
C1316212190.701.0001.6251.62552.1625161001.5631.0001.0000.01
C2125282393.171.2171.0001.30098.7630201001.5002.0001.0000.02
C2225242792.300.8891.4001.45594.0520301000.6672.0001.0000.04
C2325282393.171.2171.3331.556103.4430201001.5002.0001.5000.02
C3128385094.740.7601.1111.083154.6230601000.5002.0001.5000.86
C3229523793.561.4051.3641.176149.78444199.781.0731.5002.0002.11
C3328404893.750.8331.5001.400160.2240451000.8891.0002.0000.74
C4149547095.240.7711.7001.333427.69824499.781.8642.0002.00011.28
C4249735294.841.4041.4331.364432.1572501001.4402.0001.5002.65
C4349606395.240.9521.3081.100430.2750721000.6942.0001.0004.05
C5172807096.431.1431.3531.1881216.3100541001.8521.5001.0001.80
C5273737796.070.9481.4331.2651300.4100541001.8522.0002.0000.21
C5372787296.151.0831.1371.2151278.290601001.5001.0002.0005.06
C61971019996.011.0201.1721.0201714.5120801001.5001.0002.00013.99
C62971089296.621.1741.5711.3001807.6128751001.7072.0002.0002.65
C639710010096.001.0001.2731.1741786.38511399.950.7521.0001.50083.70
C7119619720097.460.9851.4021.1286246.218321099.920.8711.5002.0003668.73
C7219718021997.410.8221.3081.1906012.72561501001.7072.0001.0001.23
C7319623816697.201.4341.3801.2436175.619419899.970.9802.0001.0004621.96
Average95.245614.3299.313045.99
Instance$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
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
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
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