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Cutting Stock Problem
In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing material wasted. It is an optimization problem in mathematics that arises from applications in industry. In terms of computational complexity, the problem is an NP-hard problem reducible to the knapsack problem. The problem can be formulated as an integer linear programming problem. Illustration of one-dimensional cutting-stock problem A paper machine can produce an unlimited number of master (jumbo) rolls, each 5600 mm wide. The following 13 items must be cut, in the table below. The important thing about this kind of problem is that many different product units can be made from the same master roll, and the number of possible combinations is itself very large, in general, and not trivial to enumerate. The problem therefore is to find an optimum set of patterns of making pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Operations Research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decision-making. It is considered to be a subfield of mathematical sciences. The term management science is occasionally used as a synonym. Employing techniques from other mathematical sciences, such as modeling, statistics, and optimization, operations research arrives at optimal or near-optimal solutions to decision-making problems. Because of its emphasis on practical applications, operations research has overlap with many other disciplines, notably industrial engineering. Operations research is often concerned with determining the extreme values of some real-world objective: the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost). Originating in military efforts before World War II, its techniques have gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Metallised Film
Metallised films (or metallized films) are polymer films coated with a thin layer of metal, usually aluminium. They offer the glossy metallic appearance of an aluminium foil at a reduced weight and cost. Metallised films are widely used for decorative purposes and food packaging, and also for specialty applications including insulation and electronics. Manufacture Metallisation is performed using a physical vapor deposition process. Aluminium is the most common metal used for deposition, but other metals such as nickel and chromium are also used. The metal is heated and evaporated under vacuum. This condenses on the cold polymer film, which is unwound near the metal vapour source. This coating is much thinner than a metal foil could be made, in the range of 0.5 micrometres.Hanlon, J. (1992). 1st ed. ''Handbook of Package Engineering'', Lancaster, PA, Technomic Publishing: . Chapter 4 Coatings and Laminations This coating will not fade or discolour over time. While oriented polypr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Branch And Bound
Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores ''branches'' of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated ''bounds'' on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The algorithm depends on efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, the algorithm degenerates to an exhaustive search. The method was first proposed by Ailsa Land and Alison Doig whil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Linear Program
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists. Linear programs are problems that can be expressed in canonical form as : \begin & \text && ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Delayed Column-generation
Column generation or delayed column generation is an efficient algorithm for solving large linear programs. The overarching idea is that many linear programs are too large to consider all the variables explicitly. The idea is thus to start by solving the considered program with only a subset of its variables. Then iteratively, variables that have the potential to improve the objective function are added to the program. Once it is possible to demonstrate that adding new variables would no longer improve the value of the objective function, the procedure stops. The hope when applying a column generation algorithm is that only a very small fraction of the variables will be generated. This hope is supported by the fact that in the optimal solution, most variables will be non-basic and assume a value of zero, so the optimal solution can be found without them. In many cases, this method allows to solve large linear programs that would otherwise be intractable. The classical example of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bin Packing Problem
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media and technology mapping in FPGA semiconductor chip design. Computationally, the problem is NP-hard, and the corresponding decision problem - deciding if items can fit into a specified number of bins - is NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It requires '' Θ''(''n'' log ''n'') time, where ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Zalgaller
Victor (Viktor) Abramovich Zalgaller ( he, ויקטור אבּרמוביץ' זלגלר; russian: Виктор Абрамович Залгаллер; 25 December 1920 – 2 October 2020) was a Russian-Israeli mathematician in the fields of geometry and optimization. He is best known for the results he achieved on convex polyhedra, linear and dynamic programming, isoperimetry, and differential geometry. Biography Zalgaller was born in Parfino, Novgorod Governorate on 25 December 1920. In 1936, he was one of the winners of the Leningrad Mathematics Olympiads for high school students. He started his studies at the Leningrad State University, however, World War II intervened in 1941, and Zalgaller joined the Red Army. He took part in the defence of Leningrad, and in 1945 marched into Germany. He worked as a teacher at the Saint Petersburg Lyceum 239, and received his 1963 doctoral dissertation on polyhedra with the aid of his high school students who wrote the computer progra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kantorovich
Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources. He is regarded as the founder of linear programming. He was the winner of the Stalin Prize in 1949 and the Nobel Memorial Prize in Economic Sciences in 1975. Biography Kantorovich was born on 19 January 1912, to a Russian Jewish family. His father was a doctor practicing in Saint Petersburg. In 1926, at the age of fourteen, he began his studies at Leningrad State University. He graduated from the Faculty of Mathematics and Mechanics in 1930, and began his graduate studies. In 1934, at the age of 22 years, he became a full professor. Later, Kantorovich worked for the Soviet government. He was given the task of optim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Guillotine Problem
Guillotine cutting is the process of producing small rectangular items of fixed dimensions from a given large rectangular sheet, using only guillotine-cuts. A guillotine-cut (also called an edge-to-edge cut) is a straight bisecting line going from one edge of an existing rectangle to the opposite edge, similarly to a paper guillotine. Guillotine cutting is particularly common in the glass industry. Glass sheets are scored along horizontal and vertical lines, and then broken along these lines to obtain smaller panels. It is also useful for cutting steel plates, cutting of wood sheets to make furniture, and cutting of cardboard into boxes. There are various optimization problems related to guillotine cutting, such as: maximize the total area of the produced pieces, or their total value; minimize the amount of waste (unused parts) of the large sheet, or the total number of sheets. They have been studied in combinatorial geometry, operations research and industrial engineering. A re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Corrugated Fiberboard
Corrugated fiberboard or corrugated cardboard is a type of packaging material consisting of a fluted corrugated sheet and one or two flat linerboards. It is made on "flute lamination machines" or "corrugators" and is used for making corrugated boxes. The corrugated medium sheet and the linerboard(s) are made of kraft containerboard, a paperboard material usually over thick. History Corrugated (also called pleated) paper was patented in England in 1856, and used as a liner for tall hats, but corrugated boxboard was not patented and used as a shipping material until 20 December 1871. The patent was issued to Albert Jones of New York City for single-sided (single-face) corrugated board. Jones used the corrugated board for wrapping bottles and glass lantern chimneys. The first machine for producing large quantities of corrugated board was built in 1874 by G. Smyth, and in the same year Oliver Long improved upon Jones' design by inventing corrugated board with liner sheets on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |