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 maximise 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 stock
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 stock problem, we use a cutting
pattern optimizer and mathematical programming. In general, the cutting stock 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.
In this kind of applications all cuts have to be accomplished from one edge of the rectangle to the
opposite one. The cut has to be of a guillotine type. For such applications the problem can be
formulated as a mathematical optimization program and the optimal solution can be found in terms of
material yield and production costs. 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 pieces and the total area of the
stock-material. Single pass packing strategies involve taking the pieces in order and placing them on
the sheet according to a given placement policy. This may be repeated several times for different
orderings or different placements and the best solution chosen. The process will continue until
shifting-down, shifting-left or rotation is no longer possible. In other words, the algorithm continues
pushing pieces downwards and leftwards until the pieces reach stable positions. Cutting out the
material in the most effective way result in reducing its stock holding which can make additional
savings through improved cash flow. Although the normal motivation for effective stock cutting is
financial, companies may have other objectives in implementing efficient stock cutting procedures.
For example, there may be a requirement to meet certain orders within a given time. In the nesting
problem it is necessary to place a number of parts into a larger sheet. In doing so, the parts must not
overlap and they must stay within the sheet. The usual objective is to minimize the waste of the larger
sheet. The algorithms makes use of the search procedures when deciding where pieces should be
placed. The location of the next piece is calculated using the search procedures. Once the best
placement has been found the piece is added to the optimization list and the next piece is placed. The
packing problem consists of packing a collection of pieces onto a rectangular sheet while minimizing the
unused space. The packing process has to ensure that there is no overlap between the pieces. There
are thousands of different ways in which just a few different pieces can be cut out of a stock sheet.
Most patterns would be extremely wasteful, and many would be difficult to cut. The Genetic algorithms
are search and optimization procedures, where the search is guided towards improvement using the
searching of the best item. This is achieved by extracting the most desirable items from a temporary
solution list and combining them to form the next solution list. The quality of each solution is evaluated
and the best items are selected for the reproduction process. Continuation of this process will result in
optimal or near optimal solutions.
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