A triangle is a 3-sided polygon. Every triangle has three sides and three angles, some of which may be
the same. All triangles are convex. Triangles can be classified according to the number of their equal
sides. A triangle with 3 equal sides is called equilateral, a triangle with 2 equal sides is called isosceles,
and a triangle with no equal sides is called scalene. The portion of the area enclosed by the triangle is
called the triangle interior. The sum of angles in a triangle is pi radians or 180 degrees. If a line is
drawn parallel to one side of a triangle so that it intersects the other two sides, it divides them
proportionally. The term nesting is used to describe a wide variety of two-dimensional cutting problems.
They all involve a non-overlapping placement of a set of triangles within some region of 2d area. The
basic requirement is to produce a solution with no overlap between the triangles. This means that the
items must be placed without creating overlap between the triangles. The generation of a cutting
pattern depends on the order of handling the triangles, and the way of fitting these triangles into the
sheet with respect to the sheet boundaries. The two-dimensional cutting stock problem may be applied
in a number of industries, including clothing industry, shoe-leather cutting, furniture industry, etc. The
problem is as follows: a set of triangles is to be placed on a given area of a stock-material with minimum
of trim-loss. Permissible placement of wanted triangles on stock-material is called cutting pattern. The
cutting pattern has no overlaps of the triangles and meets all technological requirements. The quality
of a cutting pattern is determined by the cutting ratio, which is defined as the ratio between the total
area of the placed triangles and the total area of the stock-material. Nesting software is used to
generate optimized layouts and reduced scrap for both Rectangular and Triangular cutting processes.
The nesting technology is based on algorithms designed to optimize the cutting layouts. It provides
high utilization layouts, significantly reducing the waste and maximizing productivity. In the cutting
problems one or more pieces of material or space must be divided into smaller triangles. The
minimization of the waste is usually the main objective of these combinatorial optimization problems.
In nesting problems the combinatorial problem coexists with a geometric problem, since solutions must
be geometrically feasible and triangles may not overlap and must completely fit inside the plate.
Packing problems are optimization problems that are concerned with searching a good arrangement
of multiple items in a 2d regions. The usual objective of the allocation is to maximize the material
utilization and to minimize the wasted area. This is of particular interest to industries involved with
mass-production as small improvements in the layout can result in savings of material and a
considerable reduction in production costs. The goal in the cutting stock problem is to determine the
optimal plan to cut a 2 dimensional sheet to satisfy a set of customers demands. Cutting triangle
problems may involve a variety of objectives, minimizing trim loss, minimizing the number of cutting
lines, maximizing profit, and so on. In order to solve the cutting triangle problem, we use a cutting
pattern optimizer and mathematical programming. In general, the cutting triangle problem is reduce to
an integer programming application. Because of its complexity, solutions to the 2 dimensional cutting
stock problem have often been generated using genetic algorithms. This is due, in part, to the fact that
the 2 dimensional cutting stock problem may also be reduced to a bin-packing problem. The rectangular
cutting stock problem is to determine how to cut a number of rectangular pieces out of a given stock of
rectangular sheets. Most variants of the nesting problem is the problem of packing shapes within some
regions without overlap. The cutting stock problem asks for a minimization of the area of a rectangular
region. In the cutting industry a multitude of additional constraints are very often necessary. The
shapes or regions can have different quality zones or even holes. The nesting problem occurs in a
number of industries and it seems to have many names. In the clothing industry it is called marker
making, while the metal industry call it simply nesting. In a theoretical context the problem is most
often called the two-dimensional irregular cutting stock problem. The 2 dimensional cutting stock
problem is a classic combinatorial optimization problem in which a number of parts of various lengths
must be cut from an inventory of 2d material. The two-dimensional cutting stock problem may be
applied in a number of industries, glass, shoe-leather cutting, furniture, machine-building, etc.
The problem is as follows : a set of rectangular pieces is to be placed on a given area of a stock
material with minimum of trim-loss. The cutting pattern has no overlaps of the pieces and meets all
technological requirements. The stock cutting problem has gained a lot of attention in many industrial
sectors. Stock Cutting Problems is essential in many industries. These problems are treated in
different fields. The reduction of scrap may not only affects cost of materials used but may also reduce
the costs of handling and labor. A great number of problems are essentially based on the same logical
structure of the Cutting and Packing problems. The stock cutting problem is an example of a large scale
optimization problem. This means that this problem requires a computing effort that increases
exponentially with the problem size. Since the stock cutting problem is an efficient approximation
algorithms, namely, algorithms that do not produce optimal but rather close-to-optimal solutions,
Cutting and packing problems are encountered in many industries. The wood-, glass- and paper
industry are mainly concerned with the cutting of regular figures, whereas in the textile and leather
industry irregular, arbitrary items are to be packed.
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