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by a matrix of size K by L which is the smallest rectangular enclosure of the irregular-
shaped piece. The detailed technique applied in the paper is referred to W.K. Wong
and Z.X. Guo 2009. By using this technique, each piece is enclosed by an imaginary
rectangle for the sake of obtaining the reference points during the nesting process.
Then, this particular rectangular area is divided into a uniform grid of 1mm×1mm
size. In the case, the value of a pixel is '1' when the material of the sheet is occupied,
otherwise the value of the pixel is '0'. P
L
(k)
and P
W
(k)
denote the length and the width
of an enclosing rectangle corresponding to the piece p
k
. R=1 mm, denotes the square
side of a piece. Fig.1 gives an example of grid approximation.
Fig. 1.
Binary representations
Similar to the piece representation, the object is discretized as a finite number of
equal-size pieces of size R
2
. The object with a two-dimensional matrix of size
A
W
(k)
×A
L
(k)
is represented as follows:
a
()
k
a
()
k
a
()
k
A
11
12
()
k
1
L
()
k
()
k
()
k
a
a
a
()
k
()
k
P
P
21
22
()
k
()
k
2
A
()
k
()
i
=
,where
A
=
W
and
A
=
L
L
W
L
R
R
a
()
k
a
()
k
a
()
k
()
k
()
k
() ()
k
k
A
1
A
1
A
A
W
W
W
L
1 if pixel ( , ) is occupied
.
0oth rw e
ij
For each entry,
k
ij
()
,
a
=
The initial sequence of the pieces is determined according to their area. The initial
orientation for each packing piece is confirmed by the MRE (Minimum Rectangular
Enclosure) of each piece. It is worth mentioning that the method does not replace the
piece shapes by their MREs. Rather, it uses MREs as additional information for
orienting the pieces.
3.2
Layout Phase
The placement of the pieces follows a single-pass placement strategy and takes place
in a sequential manner. The manner entails that the method considers only one layout.
In layout phase, a hybrid approach combining with heuristic Bottom Left irregular
and two-stage placement strategy is proposed to construct a packing pattern according
to the sequence and orientation of the packing pieces. Furthermore, a packing
approach based on shape similarity draw from computer vision and artificial
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