Two-dimensional
bin packing
problem
By Jorge Isaac Merla Hernandez,
1985189
Bin packing
In the 2BP we are given a set of n rectangular
items j∈J={1,…,n}, each having width wj and
height hj, and an unlimited number of finite
identical rectangular bins, having width W and
height H. The problem is to allocate, without
overlapping, all the items to the minimum
number of bins, with their edges parallel
to those of the bins. It is assumed that the
items have fixed orientation, i.e., they cannot
be rotated.
Strip packing
In the two-dimensional strip packing problem, we are
given a strip of a finite width W but infinite height, and a
set of rectangular items each of width at most W. The
objective is to pack all the items into the strip to
minimize the height used. The items may neither overlap
nor be rotated. We describe here a list of efficient (offline) packing algorithms. A common approach is leveloriented, the items are packed from left to right, in rows
forming levels.
Problem 2BP has many industrial applications,
especially in cutting (wood and glass industries)
and packing (transportation and warehousing).
Certain applications may require additional
constraints...
Algorithm types for the 2BP
Off-line algorithms, for
which it is assumed that
the algorithm has full
knowledge of the whole
input
On-line algorithms,
which pack each item as
soon as it is encountered,
without knowledge of the
next items
Off-line algorithm types
Most of the them from the literature are of greedy type, and can be
classified in two families:
One-phase algorithms directly
pack the items into the finite
bins.
Two-phase algorithms start
by packing the items into a
single strip, i.e., a bin having
width W and infinite height.
In the second phase, the strip
solution is used to construct a
packing into finite bins.
In addition, most of the approaches are level
algorithms, i.e., the bin/strip packing is
obtained by placing the items, from left to
right, in rows forming levels.
The first level is the bottom of the bin/strip,
and subsequent levels are produced by the
horizontal line coinciding with the top of the
tallest item packed on the level below.
Algorithms
(examples)
First-Fit
Decreasing Height
(FFDH) algorithm
FFDH packs the next item R (in non-increasing
height) on the first level where R fits. If no level
can accommodate R, a new level is created.
Finite First-Fit
(FFF)
The current item is packed on the lowest
level of the first bin where it fits; if no level
can accommodate it, a new level is created
either in the first suitable bin, or by
initializing a new bin (if no bin has enough
vertical space available).
Finite Next-Fit (FNF)
FNF directly packs the items into finite bins in the same way as HNF.
Problem examples
An easy example of a 2-dimensional bin packing
problem could involve packing rectangular
objects into rectangular bins. Let's say you have
two bins with dimensions 10x10 each, and three
rectangular objects with dimensions:
Object A: 4x5
Object B: 3x7
Object C: 6x3
The goal is to find the best way to pack these
objects into the bins while minimizing wasted
space.
In cutting, for
example
References
https://www.csc.liv.ac.uk/~epa/survey.pdf
https://www.sciencedirect.com/science/article/pii/S0166218X0100347X#BIB23
Galambos, G., Woeginger, G.J. On-line bin packing — A restricted survey. ZOR Methods and Models of Operations Research 42, 25–45 (1995).
https://doi.org/10.1007/BF01415672