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Cutting-stock problems can be classified in several ways. [1] One way is the dimensionality of the cutting: the above example illustrates a one-dimensional (1D) problem; other industrial applications of 1D occur when cutting pipes, cables, and steel bars. Two-dimensional (2D) problems are encountered in furniture, clothing and glass production.
Examples of ball packing, ball covering, and box covering. It is possible to define the box dimensions using balls, with either the covering number or the packing number. The covering number () is the minimal number of open balls of radius required to cover the fractal, or in other words, such that their union contains the fractal.
The Fibonacci word is an example of a Sturmian word.The start of the cutting sequence shown here illustrates the start of the word 0100101001. In digital geometry, a cutting sequence is a sequence of symbols whose elements correspond to the individual grid lines crossed ("cut") as a curve crosses a square grid.
In manufacturing industry, nesting refers to the process of laying out cutting patterns to minimize the raw material waste. [1] Examples include manufacturing parts from flat raw material such as sheet metal, glass sheets, cloth rolls, cutting parts from steel bars, etc. Such process can also be applied to additive manufacturing, such as 3D ...
The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of items numbered from 1 up to , each with a weight and a value , along with a maximum weight capacity ,
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence. When a circle is cut n times to produce the maximum number of pieces, represented as p = f (n), the n th cut must be considered; the number of pieces before the last cut is f (n − 1), while the number of pieces added by the last cut is n.
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.