nternational Journal of Futuristic Machine Intelligence & Application (IJFMIA) (e-Journal)
ISSN : 2395-308x ,Vol 1 Issue 3
2D Packaging Problem Using Swarm Intelligence
Shital R.Kumbhar
Dept. of Computer Engineering
MIT AOE,Alandi,Pune
Email_id: shitalkumbhar04@gmail.com
Vidya M. Shingade
Dept. of Computer Engineering
MIT AOE,Alandi,Pune
Email_id: vidyashingade05@gmail.com
Sonali H. Khandve
Dept. of Computer Engineering
MIT AOE,Alandi,Pune
Email_id: sonalikhandve58@gmail.com
Abstract- We consider the NP-complete problem of finding an
enclosing rectangle of minimum area that will contain a given a set of
rectangles. We present two different constraint satisfaction
formulations of this problem. The first searches a space of absolute
placements of rectangles in the enclosing rectangle, while the other
searches a space of relative placements between pairs of rectangles.
Both approaches dramatically out perform previous approaches to
optimal rectangle packing. For problems where the rectangle
dimensions have low precision, such as small integers, absolute
placement is generally more efficient, whereas for rectangles with
high-precision dimensions, relative placement will be more effective.
Two-dimensional packing problems are a class of optimization
problems in mathematics which involve attempting to pack objects
together in the packing region as densely as possible.
There are many variations of this problem, such as two-dimensional
packing, linear packing, packing by weight, height.
KeywordsRectangle packing ・ Constraint satisfaction ・ Search global best
position, global second best position, personal best position.
INTRODUCTION:
Packing problems are very often in the real world and industries.
Many solutions are already provided to solve such problems. Two
dimensional rectangle packing problem is one of the most important
problem. As all packing problems of NP-hard, so this rectangle
packing problem is also a NP-hard type of problem. To solve such
kind of NP-hard problems, conventional methods are not sufficient.
Most of the researchers used heuristic and meta-heuristic methods to
solve the problem. Some of the methods from these called as
evolutionary computational methods. The main problem discussed
here is rectangle packing problem which is occurred very frequently
in the industry. Different types of conventional, heuristic and
metaheuristic methods are discussed to efficiently solve the problem.
Packing problems are very often in the real world and industries.
Many solutions are already provided to solve such problems. Two
dimensional rectangle packing problem is one of the most important
problem. As all packing problems of NP-hard, so this rectangle
packing problem is also a NP-hard type of problem. To solve such
kind of NP-hard problems, conventional methods are not sufficient.
Most of the researchers used heuristic and meta-heuristic methods to
solve the problem. Some of the methods from these called as
evolutionary computational method.
The main problem discussed here is rectangle packing problem which
is occurred very frequently in the industry. Different types of
conventional, heuristic and metaheuristic methods are discussed to
efficiently solve the problem.
A. RECTANGLE PACKING PROBLEM
In these problems both the objects to be packed and the container in
which they will be packed are rectangles. So the container can be
called as bounding box of rectangle shaped and the objects to be
packed inside the bounding box will also be rectangle shaped.
Oriented & un-oriented rectangle packing are the categories where in
oriented type rectangle position is fixed whereas in un-oriented
individual rectangle can be rotated by 90 degrees. There is only one
condition about the placement of rectangles i.e. all the sides of
rectangles must be parallel with the sides of bounding box and
overlapping is not permitted. Rectangle packing has several practical
applications. One is the design of VLSI circuits, where rectangular
circuit blocks are assigned to physical regions of a rectangular chip.
Another is loading a set of rectangular objects onto a cargo pallet,
without stacking.yet another application is cutting a set of rectangles
from rectangular piece of stock material and many more.
There are two important problems here, first one is the containment
problem in which rectangle shaped bounding box and also rectangle
shaped items.
B. RECTANGLE PACKING IS NP-COMPLETE
Both oriented & un-oriented rectangle containment problems are NPComplete by a reduction of 1-D Bin Packing. The problem is: Set of
rectangles and a bounding box are given; we have to find that all the
rectangles can fit into the bounding box without overlap? Both
oriented & un-oriented rectangle packing problems can be solved in
nondeterministic polynomial time. As the position of rectangles are
assigned in bounding box, its easy to find in polynomial time that no
rectangle can extends beyond the boundary of the box and there is no
overlapping of rectangles.
nternational Journal of Futuristic Machine Intelligence & Application (IJFMIA) (e-Journal)
ISSN : 2395-308x ,Vol 1 Issue 3
To show that rectangle packing is NP-hard, one demonstrate
that the NP-complete problem of one-dimensional bin packing
(Garey and Johnson 1979) can be reduced in
polynomial time to rectangle packing. In other words, if rectangle
packing can be solved in polynomial time, then so can bin packing.
An instance of the bin packing decision problem consists of a set of
numbers, along with a given number of bins, each with the same
fixed capacity. The problem is to assign each number to one of the
bins, so that the sum of the numbers in each bin does not exceed the
bin capacity.
Given a bin packing instance, it generates a
corresponding instance of oriented rectangle packing as follows. For
each number in the bin packing problem, it generates a rectangle of
unit height whose width is the number. It also generates a bounding
box whose height is the number of bins, and whose width is the
capacity of the bins. Thus each bin corresponds to a horizontal row of
the bounding box.In the resulting rectangle packing problem, each
rectangle must be assigned to a row (bin) of the bounding box, such
that the sum of the widths (numbers) of the rectangles assigned to
each row (bin) does not exceed the width (bin capacity) of the
bounding box. Thus, this oriented rectangle packing problem is
equivalent to the original bin packing problem. If it can solve any
oriented rectangle packing problem in polynomial time, then it can
solve any bin packing problem in polynomial time.
To reduce bin packing to un-oriented rectangle packing,
one make the rectangles long enough and thin enough that they can
only fit in the bounding box horizontally. For example, if there are k
bins, and one is the smallest number, it make all the rectangles less
than 1/k units high, so the bounding box is less than one unit high,
and none of the rectangles will fit vertically. This turns the problem
into an oriented rectangle packing problem. Thus, both oriented and
un-oriented rectangle packing are NP-hard, and since they are also in
NP, they are both NP-complete.
C.PARTICLE SWARM OPTIMIZATION
PSO was developed by Kennedy and Eberhart. The PSO is inspired
by the social behavior of a flock of migrating birds trying to reach an
unknown destination. In PSO, each solution is a ‘bird’ in the flock
and is referred to as a ‘particle’. A particle is analogous to a
chromosome (population member) in GAs. As opposed to GAs, the
Evolutionary process in the PSO does not create new birds from
parent ones. Rather, the birds in the population only evolve their
social behavior and accordingly their movement towards a
destination. Physically, this mimics a flock of birds that communicate
together as they fly. Each bird looks in a specific direction, and then
when communicating together, they identify the bird that is in the
best location. Accordingly, each bird speeds towards the best bird
using a velocity that depends on its current position. Each bird, then,
investigates the search space from its new local position, and the
process repeats until the flock reaches a desired destination. It is
important to note that the process involves both social interaction and
intelligence so that birds learn from their own experience (local
search) and also from the experience of others around them (global
search).
D. ANT-COLONY OPTIMIZATION
Similar to PSO, ant-colony optimization (ACO) algorithms evolve
not in their genetics but in their social behavior. ACO was developed
by Dorigo et al. based on the fact that ants are able to find the shortest
route between their nest and a source of food. This is done using
pheromone trails, which ants deposit whenever they travel, as a form
of indirect communication. As shown in Figure 5, when ants leave
their nest to search for a food source, they randomly rotate around an
obstacle, and initially the pheromone deposits will be the same for the
right and left directions. When the ants in the shorter direction find a
food source, they carry the food and start returning back, following
their pheromone trails, and still depositing more pheromone.
As indicated in Figure, an ant will most likely choose the shortest
path when returning back to the nest with food as this path will have
the most deposited pheromone. For the same reason, new ants that
later starts out from the nest to find food will also choose the shortest
path. Over time, this positive feedback (autocatalytic) process
prompts all ants to choose the shorter path.
nternational Journal of Futuristic Machine Intelligence & Application (IJFMIA) (e-Journal)
ISSN : 2395-308x ,Vol 1 Issue 3
The application of PSO for solving two-dimensional packing
problems was presented in this study. Since the storage and the ship
cabin are designed so that their sizes are equal to the integral multiple
of the container sizes, usually it is assumed that the items are placed
every certain intervals. The problem was solved by the original and
improved PSOs. In the original PSO, the particle position vectors are
updated by the global and the personal best positions. The improved
PSO utilizes, in addition to them, the global second best position of
all particles. The use of the global second best position is determined
in the probabilistic way.
E. PSO IS A MEMBER OF SWARM
INTELLIGENCE
VI. ACKNOWLEDGEMENT
Swarm intelligence (SI) is based on the collective behavior of
decentralized, self organized systems. It may be natural or artificial.
Natural examples of SI are antcolonies, fish schooling, bird flocking,
bee swarming and so on. Besides multi robot systems, some
computer program for tackling optimization and data analysis
problems are examples for some human artifacts of SI. The most
successful swarm intelligence techniques are Particle Swarm
Optimization (PSO) and Ant Colony Optimization (ACO). In PSO,
each particle flies through the multidimensional space and adjusts its
position in every step with its own experience and that of peers
toward an optimum solution by the entire swarm. Therefore PSO
algorithm is a member of Swarm Intelligence.
F.ADVANTAGES AND DISADVANTAGES OF PSO
It is said that PSO algorithm is the one of the most powerful methods
for solving the non-smooth global optimization problems while there
are some disadvantages of the PSO algorithm. The advantages and
disadvantages of PSO are discussed below:
Advantages of the PSO algorithm:
1) PSO algorithm is a derivative-free algorithm.
2) It is easy to implementation, so it can be applied both in scientific
research and engineering problems.
3) It has a limited number of parameters and the impact of parameters
to the solutions is small compared to other optimization techniques.
4)The calculation in PSO algorithm is very simple.
5) There are some techniques which ensure convergence and the
optimum value of the problem calculates easily within a short time.
6) PSO is less dependent of a set of initial points than other
optimization techniques.
7) It is conceptually very simple.
Disadvantages of the PSO algorithm
1) PSO algorithm suffers from the partial optimism, which degrades
the regulation of its speed and direction.
2) Problems with non-coordinate system (for instance, in the energy
field) exit.
V. CONCLUSION
I express true sense of gratitude towards my project guide
Prof. V.D.Rughwani, HOD Prof. Uma Nagraj for her
invaluable co-operation and guidance that she gave us in this
semester for inspiring and providing required facilities, which
made this project work very convenient. I would also like to
express thanks to all my friends who have assisted and
encouraged me in the project work.
VII. REFEREANCES
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[3] J. Kennedy, R.C. Eberhart. Particle swarm optimization. In
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on Neural Networks, volume 6, pages 1942–1948, 1995.
[4] Hallard T. Croft, Falconer Kenneth J., Guy Richard K.
Unsolved Problems in Geometry. Springer-Verlag, 1991.
[5] J. Melissen. Packing 16, 17 or 18 circles in an equilateral
triangle. Discrete Mathematics, 145: 333–342, 1995.
[6] Erich Friedman. Packing unit squares in squares: a survey
and new results. The Electronic Journal of Combinatorics, DS7, 2005.
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[8] Chen Zhao, Liu Lin, Cheng Hao, Liu Xinbao. Solving the
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volume 2, pages 26–29, 2008.
nternational Journal of Futuristic Machine Intelligence & Application (IJFMIA) (e-Journal)
ISSN : 2395-308x ,Vol 1 Issue 3