In this paper, a compact packing algorithm for the placement of objects inside a container is described. The proposed packing algorithm is designed to pack three-dimensional free-form objects inside an arbitrary enclosure such that the packing efficiency is maximized. The proposed packing algorithm can handle objects with holes or cavities, and its performance does not degrade significantly with the increase in complexity of the enclosure or the objects. The packing algorithm takes as input the tessellated geometry of the container and all the objects to be packed and outputs the list of objects that can be placed inside the enclosure. The packing algorithm also outputs the location and orientation of all the objects, the packing sequence, and the packed configuration. An improved layout algorithm that works with arbitrary container geometry is also proposed. Separate layout algorithms for the SAE and ISO luggage are developed. Several heuristics to improve the performance of the packing algorithm are also incorporated. Certain aspects that facilitate fast and efficient handling of the computer aided design (CAD) data are also discussed. A comprehensive benchmarking of the proposed packing algorithm on synthetic and hypothetical problems reflects its superior performance as compared with other similar approaches.

1.
Tiwari
,
S.
,
Fadel
,
G.
, and
Fenyes
,
P.
, 2008, “
A Fast and Efficient Compact Packing Algorithm for Free-Form Objects
,” ASME Paper No. DETC2008-50097.
2.
DIN
, 1993, Deutsches institut fur normung e. v. DIN 70020, teil 1, strabenfahrzeuge, kraftfahrzeugbau, begriffe von abmessungen.
3.
SAE
, 2005, SAE Standard J1100, Motor Vehicle Dimensions.
4.
Cagan
,
J.
,
Shimada
,
K.
, and
Yin
,
S.
, 2002, “
A Survey of Computational Approaches to Three-Dimensional Layout Problems
,”
Comput.-Aided Des.
0010-4485,
34
, pp.
597
611
.
5.
Lodi
,
A.
,
Martello
,
S.
, and
Monaci
,
M.
, 2002, “
Two-Dimensional Packing Problems: A Survey
,”
Eur. J. Oper. Res.
0377-2217,
141
, pp.
241
252
.
6.
Zhang
,
L.
, and
Kleine
,
U.
, 2003, “
A Novel Bottom-Left Packing Genetic Algorithm for Analog Module Placement
,”
Advances in Radio Science
,
1
, pp.
191
196
.
7.
Dowsland
,
K. A.
,
Vaid
,
S.
, and
Dowsland
,
W. B.
, 2002, “
An Algorithm for Polygon Placement Using a Bottom-Left Strategy
,”
Eur. J. Oper. Res.
0377-2217,
141
, pp.
371
381
.
8.
Hopper
,
E.
, and
Turton
,
B. C. H.
, 2001, “
An Empirical Investigation of Meta-Heuristic and Heuristic Algorithms for a 2D Packing Problem
,”
Eur. J. Oper. Res.
0377-2217,
128
, pp.
34
57
.
9.
Hopper
,
E.
, and
Turton
,
B.
, 1999, “
A Genetic Algorithm for a 2D Industrial Packing Problem
,”
Comput. Ind. Eng.
0360-8352,
37
, pp.
375
378
.
10.
Liu
,
D.
, and
Teng
,
H.
, 1999, “
An Improved BL-Algorithm for Genetic Algorithm of the Orthogonal Packing of Rectangles
,”
Eur. J. Oper. Res.
0377-2217,
112
, pp.
413
420
.
11.
Jakobs
,
S.
, 1996, “
On Genetic Algorithms for the Packing of Polygons
,”
Eur. J. Oper. Res.
0377-2217,
88
, pp.
165
181
.
12.
Baker
,
B. S.
,
Coffmann
,
E. G.
, Jr.
, and
Rivest
,
R. L.
, 1980, “
Orthogonal Packing in Two Dimensions
,”
SIAM J. Comput.
0097-5397,
9
, pp.
846
855
.
13.
Martello
,
S.
,
Pisinger
,
D.
, and
Vigo
,
D.
, 2000, “
The Three-Dimensional Bin Packing Problem
,”
Oper. Res.
0030-364X,
48
(
2
), pp.
256
267
.
14.
Yin
,
S.
, and
Cagan
,
J.
, 2000, “
An Extended Pattern Search Algorithm for Three-Dimensional Component Layout
,”
ASME J. Mech. Des.
0161-8458,
122
, pp.
102
108
.
15.
Ding
,
Q.
, and
Cagan
,
J.
, 2003, “
Automated Trunk Packing With Extended Pattern Search
,” SAE International, Report No. 2003-01-0671.
16.
Ikonen
,
I.
,
Biles
,
W. E.
,
Kumar
,
A.
,
Ragade
,
R. K.
, and
Wissel
,
J. C.
, 1997, “
A Genetic Algorithm for Packing Three-Dimensional Non-Convex Objects Having Cavities and Holes
,”
Proceedings of the Seventh International Conference on Genetic Algorithms
, pp.
591
598
.
17.
Eisenbrand
,
F.
,
Funke
,
S.
,
Reichel
,
J.
, and
Schomer
,
E.
, 2003, “
Packing a Trunk
,”
Algorithms ESA 2003: 11th Annual European Symposium
, Budapest, Hungary, pp.
618
629
.
18.
Blouin
,
V. Y.
,
Fadel
,
G. M.
,
Summers
,
J. D.
, and
Fenyes
,
P. A.
, 2006, “
Three-Dimensional Packing by a Heuristic-Based Sequential Genetic Algorithm
,”
11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
, AIAA Paper No. 2006-6906.
19.
Eisenbrand
,
F.
,
Funke
,
S.
,
Karrenbauer
,
A.
,
Reichel
,
J.
, and
Schomer
,
E.
, 2005, “
Packing a Trunk—Now With a Twist
,”
SPM 2005: Proceedings of the 2005 ACM Symposium on Solid and Physical Modeling
, pp.
197
206
.
20.
Althaus
,
E.
,
Baumann
,
T.
,
Schomer
,
E.
, and
Werth
,
K.
, 2007,
Trunk Packing Revisited
(
Lecture Notes in Computer Science
Vol.
4525
),
Springer
,
Berlin
, pp.
420
432
.
21.
Tiwari
,
S.
,
Fadel
,
G.
, and
Gantovnik
,
V.
, 2006, “
A Survey of Various Encoding Schemes and Associated Placement Algorithms Applied to Packing and Layout Problems
,” ASME Paper No. DETC2006-99271.
22.
Garey
,
M. R.
, and
Johnson
,
D. S.
, 1979,
Computers and Intractability: A Guide to the Theory of NP-Completeness
,
W. H. Freeman and Company
,
San Francisco, CA
.
23.
Cagan
,
J.
,
Degentesh
,
D.
, and
Yin
,
S.
, 1998, “
A Simulated Annealing-Based Algorithm Using Hierarchical Models for General Three Dimensional Component Layout
,”
Comput.-Aided Des.
0010-4485,
30
(
10
), pp.
781
790
.
24.
Mortenson
,
M. E.
, 1997,
Geometric Modeling
,
John Wiley & Sons
,
New York, NY
.
25.
Bennell
,
J. A.
, and
Song
,
X.
, 2008, “
A Comprehensive and Robust Procedure for Obtaining the Nofit Polygon Using Minkowski Sums
,”
Computers and Operations Research
,
35
(
1
), pp.
267
281
.
26.
Dawkins
,
R.
, 1976,
The Selfish Gene
,
Oxford University Press
,
New York
.
27.
Eldredge
,
N.
, 1989,
Macro-Evolutionary Dynamics: Species, Niches and Adaptive Peaks
,
McGraw-Hill
,
New York
.
28.
Holland
,
J. H.
, 1975,
Adaptation in Natural and Artificial Systems
,
The University of Michigan Press
,
Ann Arbor, MI
.
29.
Goldberg
,
D. E.
, 1989,
Genetic Algorithms for Search, Optimization, and Machine Learning
,
Addison-Wesley
,
Reading, MA
.
30.
Deb
,
K.
, 2001,
Multi-objective Optimization Using Evolutionary Algorithms
,
Wiley
,
Chichester, UK
.
31.
Coello
,
C. A.
, 1999, “
A Comprehensive Survey of Evolutionary-Based Multiobjective Optimization Techniques
,”
Knowledge Inf. Syst.
0219-1377,
1
(
3
), pp.
269
308
.
32.
Vavak
,
F.
, and
Fogarty
,
T. C.
, 1996, “
Comparison of Steady State and Generational Genetic Algorithms for Use in Nonstationary Environments
,”
Proceedings of the IEEE Conference on Evolutionary Computation
, pp.
192
195
.
33.
Back
,
T.
,
Hoffmeister
,
F.
, and
Schwefel
,
H.
, 1991, “
A Survey of Evolution Strategies
,”
Proceedings of the Fourth International Conference on Genetic Algorithms
,
Morgan Kaufmann
,
San Diego, CA
, pp.
2
9
.
34.
Whitley
,
D.
, 1989, “
The GENITOR Algorithm and Selection Pressure: Why Rank-Based Allocation of Reproductive Trials Is Best
,”
Proceedings of the Third International Conference on Genetic Algorithms
, pp.
116
121
.
35.
Eshelman
,
L. J.
, 1991, “
The CHC Adaptive Search Algorithm: How to Have Safe Search When Engaging in Nontraditional Genetic Recombination
,”
Foundations of Genetic Algorithms 1 (FOGA-1)
,
Morgan Kaufmann
,
San Mateo, CA
, pp.
265
283
.
36.
Hansen
,
N.
, and
Ostermeier
,
A.
, 1996, “
Adapting Arbitrary Normal Mutation Distributions in Evolution Strategies: The Covariance Matrix Adaption
,”
Proceedings of the Third IEEE International Conference on Evolutionary Computation
,
IEEE
,
New York
, pp.
312
317
.
37.
Syswerda
,
G.
, 1991, “
Schedule Optimization Using Genetic Algorithms
,”
Handbook of Genetic Algorithms 1
,
The University of Michigan Press
,
New York, NY
.
38.
Deb
,
K.
, and
Agrawal
,
R. B.
, 1995, “
Simulated Binary Crossover for Continuous Search Space
,”
Complex Syst.
0891-2513,
9
(
2
), pp.
115
148
.
39.
Deb
,
K.
, and
Goyal
,
M.
, 1996, “
A Combined Genetic Adaptive Search (Geneas) for Engineering Design
,”
Comput. Sci. Inform.
0254-7813,
26
(
4
), pp.
30
45
.
40.
Coello Coello
,
C. A.
,
Van Veldhuizen
,
D. A.
, and
Lamont
,
G. B.
, 2002,
Evolutionary Algorithms for Solving Multi-Objective Problems
,
Springer
,
New York
.
41.
Akenine-Möller
,
T. A.
, 2001, “
Fast 3D Triangle-Box Overlap Testing
,”
Journal of Graphics, Gpu, and Game Tools
,
6
(
1
), pp.
29
33
.
You do not currently have access to this content.