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

Mao Chen; Xiangyang Tang; Zhizhong Zeng; Sanya Liu

doi:10.3934/jimo.2018164

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January 2020, 16(1): 495-510. doi: 10.3934/jimo.2018164

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

Mao Chen ^, , Xiangyang Tang , Zhizhong Zeng and Sanya Liu

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%.

Keywords: Optimization, heuristic algorithm, 2D rectangular packing problem, central rectangle, angle-occupying placement.

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

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, 2020, 16 (1) : 495-510. doi: 10.3934/jimo.2018164

References:

[1]	A. Lodi, S. 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. Google Scholar
[2]	K. A. Dowsland and W. B. Dowsland, Packing problems, European Journal of Operational Research, 56 (1992), 2-14. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	B. S. Baker, Jr. E. G. Coffman and R. L. Rivest, Orthogonal packing in two dimensions, SIAM Journal on Computing, 9 (1980), 846-855. doi: 10.1137/0209064. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	Y. L. Wu, W. Q. Huang, S. C. Lau, C. 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. Google Scholar
[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. Google Scholar
[12]	D. F. Zhang, Y. Kang and A. S. Deng, A new heuristic recursive algorithm for the strip rectangular packing problem, Computers & Operations Research, 33 (2006), 2209-2217. Google Scholar
[13]	L. J. Wei, D. 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. Google Scholar
[14]	R. Alvarez-Valdes, F. 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. Google Scholar
[15]	K. He, W. 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. Google Scholar
[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. Google Scholar
[17]	L. Wu, L. Zhang, W.S. Xiao, Q. Liu, C. 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. Google Scholar
[18]	E. K. Burke, G. 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. Google Scholar

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References:

[1]	A. Lodi, S. 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. Google Scholar
[2]	K. A. Dowsland and W. B. Dowsland, Packing problems, European Journal of Operational Research, 56 (1992), 2-14. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	B. S. Baker, Jr. E. G. Coffman and R. L. Rivest, Orthogonal packing in two dimensions, SIAM Journal on Computing, 9 (1980), 846-855. doi: 10.1137/0209064. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	Y. L. Wu, W. Q. Huang, S. C. Lau, C. 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. Google Scholar
[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. Google Scholar
[12]	D. F. Zhang, Y. Kang and A. S. Deng, A new heuristic recursive algorithm for the strip rectangular packing problem, Computers & Operations Research, 33 (2006), 2209-2217. Google Scholar
[13]	L. J. Wei, D. 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. Google Scholar
[14]	R. Alvarez-Valdes, F. 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. Google Scholar
[15]	K. He, W. 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. Google Scholar
[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. Google Scholar
[17]	L. Wu, L. Zhang, W.S. Xiao, Q. Liu, C. 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. Google Scholar
[18]	E. K. Burke, G. 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. Google Scholar

Figure 1. Cartesian coordinate system and central rectangle

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Figure 2. An illustrative example of angle

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Figure 3. An illustrative example of initial layout

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Figure 4. An illustrative example of angle

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

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Figure 6. The final layout of two hard test instances

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Figure 7. Final layout of the drilling platform

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Table 1. Settings of important parameters

Parameter	Description	Value
Nar$_{\min}$	Lower bound of the aspect ratio of the container	0.5
Nar$_{\max}$	Upper bound of the aspect ratio of the container	2
$Z_{Hmin}$	Lower bound of the between centrality of CR in vertical direction	1
$Z_{Hmax}$	Upper bound of the between centrality of CR in vertical direction	2
$Z_{Wmin}$	Lower bound of the between centrality of CR in horizontal direction	1
$Z_{Wmax}$	Upper bound of the between centrality of CR in horizontal direction	2

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Parameter	Description	Value
Nar$_{\min}$	Lower bound of the aspect ratio of the container	0.5
Nar$_{\max}$	Upper bound of the aspect ratio of the container	2
$Z_{Hmin}$	Lower bound of the between centrality of CR in vertical direction	1
$Z_{Hmax}$	Upper bound of the between centrality of CR in vertical direction	2
$Z_{Wmin}$	Lower bound of the between centrality of CR in horizontal direction	1
$Z_{Wmax}$	Upper bound of the between centrality of CR in horizontal direction	2

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

Instance	Filling rate (%) under different values of parameter $\varepsilon $
Instance	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6
N1	81.63	81.63	83.33	77.37	83.33	78.43	77.75	77.37	74.59	74.59	73.46	73.46	74.59	74.59	73.46	52.59
N2	100	100	98.81	98.04	100	98.81	98.81	98.04	98.04	98.04	98.04	98.04	98.81	98.81	98.04	98.04
N3	99.67	99.67	99.47	99.67	99.21	99.21	99.21	98.75	98.81	98.81	98.88	98.04	99.21	98.04	98.04	90.74
N4	98.98	98.77	99.01	98.46	98.49	98.77	98.98	97.86	98.51	98.28	98.05	98.77	96.75	97.09	98.31	89.19
N5	99.21	99.21	99.21	99.21	98.18	98.24	98.12	98.39	98.19	97.98	97.65	97.98	98.27	97.23	97.92	97.22
N6	100	100	99.70	99.56	99.50	99.56	99.70	99.21	99.30	99.21	99.36	99.36	99.36	99.70	99.50	99.21
N7	100	99.95	100	99.95	99.55	99.95	99.50	99.50	99.55	99.30	99.90	99.95	99.55	99.30	99.38	99.30
N8	99.90	99.90	99.91	99.90	99.76	99.70	99.76	99.90	99.37	99.71	99.55	99.69	99.57	99.76	99.71	99.36
N9	99.75	99.75	99.73	99.73	99.73	99.68	99.73	99.68	99.68	99.68	99.68	99.68	99.68	99.68	99.68	99.68
N10	100	100	99.96	100	99.93	99.96	99.93	99.89	99.74	99.91	99.89	99.74	99.93	99.91	99.88	99.71
Average	97.91	97.89	97.91	97.19	97.77	97.23	97.15	96.86	96.58	96.55	96.06	96.47	96.57	96.41	96.39	92.50

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Instance	Filling rate (%) under different values of parameter $\varepsilon $
Instance	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.1	1.2	1.3	1.4	1.5	1.6
N1	81.63	81.63	83.33	77.37	83.33	78.43	77.75	77.37	74.59	74.59	73.46	73.46	74.59	74.59	73.46	52.59
N2	100	100	98.81	98.04	100	98.81	98.81	98.04	98.04	98.04	98.04	98.04	98.81	98.81	98.04	98.04
N3	99.67	99.67	99.47	99.67	99.21	99.21	99.21	98.75	98.81	98.81	98.88	98.04	99.21	98.04	98.04	90.74
N4	98.98	98.77	99.01	98.46	98.49	98.77	98.98	97.86	98.51	98.28	98.05	98.77	96.75	97.09	98.31	89.19
N5	99.21	99.21	99.21	99.21	98.18	98.24	98.12	98.39	98.19	97.98	97.65	97.98	98.27	97.23	97.92	97.22
N6	100	100	99.70	99.56	99.50	99.56	99.70	99.21	99.30	99.21	99.36	99.36	99.36	99.70	99.50	99.21
N7	100	99.95	100	99.95	99.55	99.95	99.50	99.50	99.55	99.30	99.90	99.95	99.55	99.30	99.38	99.30
N8	99.90	99.90	99.91	99.90	99.76	99.70	99.76	99.90	99.37	99.71	99.55	99.69	99.57	99.76	99.71	99.36
N9	99.75	99.75	99.73	99.73	99.73	99.68	99.73	99.68	99.68	99.68	99.68	99.68	99.68	99.68	99.68	99.68
N10	100	100	99.96	100	99.93	99.96	99.93	99.89	99.74	99.91	99.89	99.74	99.93	99.91	99.88	99.71
Average	97.91	97.89	97.91	97.19	97.77	97.23	97.15	96.86	96.58	96.55	96.06	96.47	96.57	96.41	96.39	92.50

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

Instance	$n$	HACR							HA_AOP
Instance	$n$	$H$	$W$	Fillrate (%)	Nar	$Z_{H}$	$Z_{W}$	Time (s)	$H$	$W$	Fillrate (%)	Nar	$Z_{H}$	$Z_{W}$	Time (s)
N1	10	48	37	90.09	1.297	1.400	1.056	15.37	48	40	83.33	1.200	2.000	1.000	0.45
N2	20	36	45	92.59	0.800	1.250	1.500	77.48	30	50	100	0.6	2.000	1.500	0.13
N3	30	32	50	93.75	0.640	1.133	1.174	162.07	28	54	99.21	0.519	1.000	1.000	0.70
N4	40	89	76	94.62	1.171	1.119	1.533	294.51	70	93	98.49	0.753	1.500	2.000	17.10
N5	50	92	114	95.35	0.807	1.244	1.073	452.74	98	104	98.18	0.942	1.500	1.000	66.40
N6	60	74	71	95.17	1.042	1.467	1.536	747.08	75	67	99.50	1.119	2.000	2.000	20.76
N7	70	85	98	96.04	0.867	1.237	1.333	1061.3	90	89	99.55	1.011	1.500	1.500	62.07
N8	80	96	86	96.90	1.116	1.400	1.263	1406.7	99	81	99.76	1.222	2.000	1.000	54.29
N9	100	81	85	97.47	0.853	1.025	1.159	1913.8	94	80	99.73	1.175	1.000	2.000	59.98
N10	200	78	138	97.55	0.565	1.026	1.379	6075.4	121	87	99.93	1.391	2.000	1.500	334.58
N11	300	89	120	98.31	0.742	1.094	1.124	12489	105	100	100	1.050	2.000	2.000	572.07
N12	500	155	195	99.26	0.795	1.067	1.000	20435	154	196	99.39	0.786	1.000	2.000	93376.28
N13	3152	822	750	99.66	1.096	1.055	1.344	1.16$\times $10$^{5}$	1024	600	100	1.707	2.000	2.000	187.80
C11	16	18	24	92.59	0.750	1.250	1.400	54.45	20	20	100	1.000	1.000	1.500	0.07
C12	17	16	27	92.59	0.593	1.286	1.077	60.88	20	20	100	1.000	1.000	1.500	0.05
C13	16	21	21	90.70	1.000	1.625	1.625	52.16	25	16	100	1.563	1.000	1.000	0.01
C21	25	28	23	93.17	1.217	1.000	1.300	98.76	30	20	100	1.500	2.000	1.000	0.02
C22	25	24	27	92.30	0.889	1.400	1.455	94.05	20	30	100	0.667	2.000	1.000	0.04
C23	25	28	23	93.17	1.217	1.333	1.556	103.44	30	20	100	1.500	2.000	1.500	0.02
C31	28	38	50	94.74	0.760	1.111	1.083	154.62	30	60	100	0.500	2.000	1.500	0.86
C32	29	52	37	93.56	1.405	1.364	1.176	149.78	44	41	99.78	1.073	1.500	2.000	2.11
C33	28	40	48	93.75	0.833	1.500	1.400	160.22	40	45	100	0.889	1.000	2.000	0.74
C41	49	54	70	95.24	0.771	1.700	1.333	427.69	82	44	99.78	1.864	2.000	2.000	11.28
C42	49	73	52	94.84	1.404	1.433	1.364	432.15	72	50	100	1.440	2.000	1.500	2.65
C43	49	60	63	95.24	0.952	1.308	1.100	430.27	50	72	100	0.694	2.000	1.000	4.05
C51	72	80	70	96.43	1.143	1.353	1.188	1216.3	100	54	100	1.852	1.500	1.000	1.80
C52	73	73	77	96.07	0.948	1.433	1.265	1300.4	100	54	100	1.852	2.000	2.000	0.21
C53	72	78	72	96.15	1.083	1.137	1.215	1278.2	90	60	100	1.500	1.000	2.000	5.06
C61	97	101	99	96.01	1.020	1.172	1.020	1714.5	120	80	100	1.500	1.000	2.000	13.99
C62	97	108	92	96.62	1.174	1.571	1.300	1807.6	128	75	100	1.707	2.000	2.000	2.65
C63	97	100	100	96.00	1.000	1.273	1.174	1786.3	85	113	99.95	0.752	1.000	1.500	83.70
C71	196	197	200	97.46	0.985	1.402	1.128	6246.2	183	210	99.92	0.871	1.500	2.000	3668.73
C72	197	180	219	97.41	0.822	1.308	1.190	6012.7	256	150	100	1.707	2.000	1.000	1.23
C73	196	238	166	97.20	1.434	1.380	1.243	6175.6	194	198	99.97	0.980	2.000	1.000	4621.96
Average				95.24				5614.32			99.31				3045.99

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Instance	$n$	HACR							HA_AOP
Instance	$n$	$H$	$W$	Fillrate (%)	Nar	$Z_{H}$	$Z_{W}$	Time (s)	$H$	$W$	Fillrate (%)	Nar	$Z_{H}$	$Z_{W}$	Time (s)
N1	10	48	37	90.09	1.297	1.400	1.056	15.37	48	40	83.33	1.200	2.000	1.000	0.45
N2	20	36	45	92.59	0.800	1.250	1.500	77.48	30	50	100	0.6	2.000	1.500	0.13
N3	30	32	50	93.75	0.640	1.133	1.174	162.07	28	54	99.21	0.519	1.000	1.000	0.70
N4	40	89	76	94.62	1.171	1.119	1.533	294.51	70	93	98.49	0.753	1.500	2.000	17.10
N5	50	92	114	95.35	0.807	1.244	1.073	452.74	98	104	98.18	0.942	1.500	1.000	66.40
N6	60	74	71	95.17	1.042	1.467	1.536	747.08	75	67	99.50	1.119	2.000	2.000	20.76
N7	70	85	98	96.04	0.867	1.237	1.333	1061.3	90	89	99.55	1.011	1.500	1.500	62.07
N8	80	96	86	96.90	1.116	1.400	1.263	1406.7	99	81	99.76	1.222	2.000	1.000	54.29
N9	100	81	85	97.47	0.853	1.025	1.159	1913.8	94	80	99.73	1.175	1.000	2.000	59.98
N10	200	78	138	97.55	0.565	1.026	1.379	6075.4	121	87	99.93	1.391	2.000	1.500	334.58
N11	300	89	120	98.31	0.742	1.094	1.124	12489	105	100	100	1.050	2.000	2.000	572.07
N12	500	155	195	99.26	0.795	1.067	1.000	20435	154	196	99.39	0.786	1.000	2.000	93376.28
N13	3152	822	750	99.66	1.096	1.055	1.344	1.16$\times $10$^{5}$	1024	600	100	1.707	2.000	2.000	187.80
C11	16	18	24	92.59	0.750	1.250	1.400	54.45	20	20	100	1.000	1.000	1.500	0.07
C12	17	16	27	92.59	0.593	1.286	1.077	60.88	20	20	100	1.000	1.000	1.500	0.05
C13	16	21	21	90.70	1.000	1.625	1.625	52.16	25	16	100	1.563	1.000	1.000	0.01
C21	25	28	23	93.17	1.217	1.000	1.300	98.76	30	20	100	1.500	2.000	1.000	0.02
C22	25	24	27	92.30	0.889	1.400	1.455	94.05	20	30	100	0.667	2.000	1.000	0.04
C23	25	28	23	93.17	1.217	1.333	1.556	103.44	30	20	100	1.500	2.000	1.500	0.02
C31	28	38	50	94.74	0.760	1.111	1.083	154.62	30	60	100	0.500	2.000	1.500	0.86
C32	29	52	37	93.56	1.405	1.364	1.176	149.78	44	41	99.78	1.073	1.500	2.000	2.11
C33	28	40	48	93.75	0.833	1.500	1.400	160.22	40	45	100	0.889	1.000	2.000	0.74
C41	49	54	70	95.24	0.771	1.700	1.333	427.69	82	44	99.78	1.864	2.000	2.000	11.28
C42	49	73	52	94.84	1.404	1.433	1.364	432.15	72	50	100	1.440	2.000	1.500	2.65
C43	49	60	63	95.24	0.952	1.308	1.100	430.27	50	72	100	0.694	2.000	1.000	4.05
C51	72	80	70	96.43	1.143	1.353	1.188	1216.3	100	54	100	1.852	1.500	1.000	1.80
C52	73	73	77	96.07	0.948	1.433	1.265	1300.4	100	54	100	1.852	2.000	2.000	0.21
C53	72	78	72	96.15	1.083	1.137	1.215	1278.2	90	60	100	1.500	1.000	2.000	5.06
C61	97	101	99	96.01	1.020	1.172	1.020	1714.5	120	80	100	1.500	1.000	2.000	13.99
C62	97	108	92	96.62	1.174	1.571	1.300	1807.6	128	75	100	1.707	2.000	2.000	2.65
C63	97	100	100	96.00	1.000	1.273	1.174	1786.3	85	113	99.95	0.752	1.000	1.500	83.70
C71	196	197	200	97.46	0.985	1.402	1.128	6246.2	183	210	99.92	0.871	1.500	2.000	3668.73
C72	197	180	219	97.41	0.822	1.308	1.190	6012.7	256	150	100	1.707	2.000	1.000	1.23
C73	196	238	166	97.20	1.434	1.380	1.243	6175.6	194	198	99.97	0.980	2.000	1.000	4621.96
Average				95.24				5614.32			99.31				3045.99

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

No.	Layout module	Length (m)	Width (m)	Height (m)
1	Drilling floor	33.00	24	10.00
2	Drilling collar storage area	9.60	2.00	1.10
3	Drilling pipe area No.1	9.60	8.50	1.70
4	Drilling pipe area No.2	9.60	8.50	1.70
5	30in drive pipe area	12.50	2.70	2.70
6	20in drive pipe area	12.50	3.50	3.50
7	13-3/8in drive pipe area	12.50	4.00	4.00
8	9-5/8in drive pipe area	12.00	7.00	6.00
9	7in drive pipe area	10.50	5.00	4.20
10	Flatwise marine riser area	23.00	13.00	7.00
11	Vertical marine riser area	32.00	10.00	23.00
12	Pipe conveyor area	24.00	4.00	10.00
13	Bop area	28.50	10.00	3.80
14	Christmas tree area	20.00	9.50	3.80
15	Mud purification area	18.00	16.50	2.00
16	Living quarters	38.00	11.00	11.00

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No.	Layout module	Length (m)	Width (m)	Height (m)
1	Drilling floor	33.00	24	10.00
2	Drilling collar storage area	9.60	2.00	1.10
3	Drilling pipe area No.1	9.60	8.50	1.70
4	Drilling pipe area No.2	9.60	8.50	1.70
5	30in drive pipe area	12.50	2.70	2.70
6	20in drive pipe area	12.50	3.50	3.50
7	13-3/8in drive pipe area	12.50	4.00	4.00
8	9-5/8in drive pipe area	12.00	7.00	6.00
9	7in drive pipe area	10.50	5.00	4.20
10	Flatwise marine riser area	23.00	13.00	7.00
11	Vertical marine riser area	32.00	10.00	23.00
12	Pipe conveyor area	24.00	4.00	10.00
13	Bop area	28.50	10.00	3.80
14	Christmas tree area	20.00	9.50	3.80
15	Mud purification area	18.00	16.50	2.00
16	Living quarters	38.00	11.00	11.00

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American Institute of Mathematical Sciences

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

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

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