Sam Allen

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A New Hybrid Placement Strategy for the
Three-Dimensional StripPacking Problem

• Sam Allen
• sda_at_cs.nott.ac.uk

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Overview

• Problem Definition / Background
• The 3BF algorithm
• Metaheuristic enhancements
• Empirical results
• Conclusions and future work

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Strip Packing An Introduction

• 2D strip packing has been studied since the 1960s
• 3 dimensional strip packing mostly overlooked,
  until 1990s
• Obviously 1D trivial/non existant
• Similar to bin packing

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Bin packing
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Strip Packing
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Strip Packing An Introduction

• Formally (according to Wäschers typology 1)
• 3D regular open-dimension-problem (ODP) with one
  open dimension
• Packing many boxes into a container with all
  but one dimension fixed, and the remaining
  dimension flexible/infinite (to be minimised),
  allowing orthogonal rotations (sometimes)
• Many applications

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Applications of 3D Strip Packing

• Packing little boxes into big boxes
• Delivery vans
• Aircraft
• Block cutting
• Wood
• Steel
• Foam
• ..

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Applications of 3D Strip Packing

• Multi-dimensional resource scheduling

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Processing Time
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CPUs available
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Available RAM
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The 3BF Heuristic

• The Three-Dimensional Best Fit Heuristic started
  life as an extension to Glenn Whitwells (i.e.
  Burke et als 2) Best Fit algorithm for 2D
  strip packing.
• Therefore many properties are shared between 2BF
  and 3BF, though some are different

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Similarities between 2BF and 3BF

• Best fit methodology
• Offline packing algorithm
• Constructive
• Much better (solution quality) than iterative
  improving algorithms on medium large datasets

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Differences between 2BF and 3BF

• Different placement strategies (explained
  later)
• Gap discovery/definition harder in 3 dimensions
• Much bigger search space in 3D, i.e. each box
  can have up to 6 rotations in 3D but only 2 in 2D

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Quick explanation of 3BF

• Find the lowest gap available in the container
• Choose the biggest box that will fill the gap and
  rotate/place it there, if such a box exists.
• If not, raise the gap to the lowest
  neighbouring boxs height
• Continue until finished

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How it works (Preprocessing stage)

• First the algorithm takes a list of boxes, and
  the width and length of the container
• The boxes are then rotated so that each of their
  width length height
• They are then sorted decreasingly by width

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How it works (Packing stage)
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Gap
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How it works (Packing stage)
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How it works (Packing stage)
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How it works (Packing stage)
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How it works (Packing stage)
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Gap
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How it works (Packing stage)
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How it works (Packing stage)
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How it works (Packing stage)
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How it works (Packing stage)
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How it works (postprocessing)

• Due to the nature of the algorithm, towers may
  form.
• Towers are boxes with large height dimensions
  and lower width and length dimensions that are
  placed late on in the packing process, jutting
  out over the top of the profile.
• The tallest tower is removed from the packing,
  rotated so that it is effectively knocked down
  and placed back into the packing at the lowest
  point available. This is repeated until the
  solution is not improved any further.

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old height
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Improvement
new height
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Placement Strategies 2BF

• 2BF used 3 different placement strategies
• Leftmost
• Tallest neighbour
• Shortest neighbour

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Placement Strategies 3BF

• 3BF uses 4 new placement strategies
• Bottom-leftmost
• Maximum contact
• Smallest volume
• Neighbour score

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Problem representation

• List of coordinates for potential gaps
• Axis Aligned Bounding Box (AABB) Tree for
  collision detection

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Metaheuristic Enhancements

• 3BF works well (in terms of speed and solution
  quality) as a constructive algorithm for larger
  problems
• Iterative improving algorithms often work well on
  smaller problems
• Why not combine the two?

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Metaheuristic Enhancements

• Couple metaheuristic searches (tabu, simulated
  annealing) with the Deepest-Bottom-Left-Fill
  algorithm in order to create better solutions
  (still in a reasonable amount of time)
• Use best solution obtained from the constructive
  phase and use that in the improvement phase

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DBLF

• Deepest-Bottom-Left-Fill is a simple extension of
  the common Bottom-Left-Fill algorithm for 2D
• Which is, in turn, an extension of the
  Bottom-Left algorithm
• Handles gaps nicely, creates higher quality
  solutions

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Solution Neighbourhood

• Two potential moves (each with the same
  probability)
• Choose two box positions and swap them
• Rotate a randomly selected box to a random new
  orientation (each with 1/5th probability)

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Auto Adjusting m

• After experimentation, the value of the switch
  variable m tended to get best results at
  between 15-35
• Having m start at 10 and increment after a period
  of time of no improvement gets better results

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Testing

• Tested against the original 2BF algorithm (no
  change in the algorithm needed, just need to set
  z values to 1)
• Also tested against best known results from the
  literature on popular data sets

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Results - 3BF
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Results 3BF with Metaheuristics
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Demo

• http//www.cs.nott.ac.uk/sda/research/2d.avi
• http//www.cs.nott.ac.uk/sda/research/3d.avi

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Conclusions

• In 2004 Burke et al created an algorithm which
  performed very well in terms of speed and
  solution quality
• published in Operations Research
• The three-dimensional equivalent seems to work
  similarly well on both 3D and 2D data sets,
  generating many best known results from both the
  original paper and literature (42/48 test cases)

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Future work

• More real life constraints being taken into
  account
• Balancing
• Strength of boxes

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References

• G. Wäscher, H. Hauner, and H. Schumann. An
  improved typology of cutting and packing
  problems. European Journal of Operational
  Research, 127(3)1109-1130, December 2007
• E. K. Burke, G. Kendall, and G. Whitwell. A new
  placement heuristic for the orthogonal stock
  cutting problem. Oper. Res., 52(4)655-671,
  Jul-Aug 2004
• E. K. Burke, G. Kendall, and G. Whitwell.
  Metaheuristic enhancements of the best-t
  heuristic for the orthogonal stock cutting
  problem. Technical Report NOTTCS-TR-2006-3,
  School of Computer Science, University of
  Nottingham, 2006
• A. Bortfeldt and D. Mack. A heuristic for the
  three-dimensional strip packing problem. European
  J. Oper. Res., 183(3)1267-1279, 2007
• K. Karabulut and M. M. Inceoglu. A hybrid genetic
  algorithm for packing in 3d with deepest bottom
  left with fill method. In ADVIS, pages 441-450,
  2004