In operations research, stock problems is a matter of cutting pieces of standard-sized materials, such as rolls of paper or sheet metal, into certain size pieces while minimizing waste material. This is a matter of optimization in mathematics that arises from applications in the industry. In terms of computational complexity, the problem is the NP-hard problem that can be reduced to a knapsack problem. The problem can be formulated as an integer linear programming problem.
Video Cutting stock problem
Illustration of one-dimensional stock-dimensional problem
A paper machine can produce an unlimited number of rolls of master (jumbo), each with an area of ​​5600 mm. The following 13 items must be cut:
Solution
There are 308 possible patterns for this small example. The optimal answer requires 73 master windings and has 0.401% waste; it can be shown computationally that in this case the minimum number of patterns with this level of waste is 10. It can also be calculated that 19 different solutions exist, each with 10 patterns and 0.401% waste, where one of the solutions is shown below and in the picture:
Maps Cutting stock problem
Classification
The problem of stock cuts can be classified in several ways. One way is the cutting dimension: the above example illustrates a one-dimensional (1D) problem; Other 1D industrial applications occur when cutting pipes, wires, and steel rods. The two-dimensional problem (2D) is found in furniture, clothing and glass production. When either the main item or the required part is irregularly shaped (the situation often encountered in the leather, textile, metal industry) is referred to as a nested problem .
There are not many three-dimensional (3D) applications involving known cuts; but closely related 3D packaging issues have many industrial applications, such as packing objects into shipping containers (see for example containerization: related packing issues have been studied since the 17th century (Kepler speculates)).
Problems of stock cutting in paper, film and metal industries
Industrial applications of stock-cutting problems for high-volume production appear especially when the base material is produced in large rolls which are subsequently cut into smaller units (see slitting roll). This is done for example in the paper and plastic film industry but also in the production of flat metals such as steel or brass. There are many additional variants and constraints arising from special production constraints due to engine and process boundaries, customer requirements and quality issues; some examples are:
- Two stages, where the reel produced in the first stage is then processed a second time. For example, all stationery (eg A4 size in Europe, Mail size in the US) is produced in such a process. Complications arise because the machine in the second stage is narrower than the main engine. Efficient utilization of both stages of production is important (from the perspective of energy or material use) and what is efficient for the primary stage may be inefficient for the secondary, leading to trade-offs. Metallization films (used in light food packaging), and plastic extrusion on paper (used in liquid packaging, eg juice cartons) are further examples of such a process.
- Wind resistance in which the cutting process has physical or logical constraints: a very common constraint is that only a certain number of knives are available, so a proper pattern should not contain more than the maximum number of reels. Since the clock machine is not standard, very many other obstacles are encountered.
- An example of a customer requirement is when a particular order can not be satisfied from one of two edge positions: this is because the edges of the sheet tend to have larger thickness variations and some applications can be very sensitive to this.
- An example of a quality problem is when the main roll contains a defect that needs to be cut. Expensive materials with demanding quality characteristics such as photo paper or Tyvek must be carefully optimized so that the wasted area is minimized.
- Multi-machine issues arise when orders can be produced on more than one machine and these machines have different widths. Generally the availability of more than one width of the master reel significantly increases waste; In practice, however, the limitation of additional, split orders may have to be taken into account.
- There is also a semi-continuous problem, in which the resulting roll does not have to be the same diameter, but may vary within the range. This usually happens with a sheet order. This is sometimes known as a 1Ã,½ dimensional problem. This variant also occurs in the production of corrugated fiberboard, where it is called, somewhat confusingly, the problem of corrugator scheduling .
- Since some paper machines are relatively narrower than the items requested, some companies have invested in skiving (also known as web-welding ) secondary processes, in which two scrolls ( produced with the initial jumbo roll slitting) are combined side by side (with little overlap) to make the rolls wider. Producing narrow rolls in the main process leads to a decrease in overall waste.
- In the metal industry one major difference is that usually master scrolls are produced earlier and generally differ from each other (both in width and length). Therefore, there are similarities to the multi-machine problem mentioned above. The presence of long variations creates a 2-D problem, because wastage can occur both wide-wise and long-wise.
The issue of stock cutting in the glass industry
The problem of guillotine is the matter of cutting the glass sheet into a rectangle of the specified size, using only continuous pieces along each sheet.
Problem Assortment
Related issues to determine, for the one-dimensional case, the best master size that will meet the given query is known as the various problem.
History
The problem of stock cuts was first formulated by Kantorovich in 1939. In 1951 before computers became widely available, L. V. Kantorovich and V. A. Zalgaller suggested solving economically viable material use problems at the cutting stage with the help of linear programming. The proposed technique is then called the column generation method .
Mathematical formula and solution approach
This formulation applies not only to one-dimensional problems. Many possible variations, including where the goal is not to minimize waste, but to maximize the total value of the goods produced, allow each order to have different values.
In general, the number of patterns that may grow exponentially as a function m , the number of orders. As the number of orders increases, it may be impractical to calculate the possible pattern of cuts.
An alternative approach uses delayed column generation. This method solves the problem of stock cutting by starting with just a few patterns. This produces additional patterns when needed. For the one-dimensional case, new patterns are introduced by solving additional optimization problems called backpack problems, using multiple variable information from the linear program. The backpack issue has a well-known method for solving it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be much more efficient than the initial approach, especially when the size of the problem increases. The column-making approach as applied to the stock-cutting issue was pioneered by Gilmore and Gomory in a series of papers published in the 1960s. Gilmore and Gomory suggest that this approach is guaranteed to converge with the optimal solution (fractional), without the need to calculate all possible patterns in the beginning.
The original Gilmore and Gomory method restriction is that it does not handle integrality, so the solution may contain fractions, e.g. a certain pattern must be produced 3.67 times. Rounding to the nearest integer often does not work, in the sense that it can result in sub-optimal and/or production solutions that are fewer than a few orders (and possibly infeasibility in the presence of two-sided demand constraints)). This limitation is overcome in modern algorithms, which can resolve to optimality (in the sense of finding solutions with minimum waste) a very big example of the problem (generally larger than that encountered in practice).
The problem of stocks is often highly degenerated, where multiple solutions with the same amount of waste are possible. This deterioration arises because it is possible to move things around, create new patterns, without affecting the amount of waste. This raises a collection of related issues related to some other criteria, such as the following:
- Issue the minimum number of patterns: to find minimum-pattern-calculate solutions between minimum waste solutions. This is a very difficult problem, even when the waste is known. It is conjectured that any one-dimensional equivalent-bound example with an n order has at least one minimum waste solution with no more than n 1 pattern.
- Minimum stack issue: this corresponds to the order of the pattern so it does not have too many partially completed orders at any time. This was an open issue until 2007, when an efficient algorithm based on dynamic programming was published.
- Minimum number of problem knife changes (for one-dimensional problems): this is related to sorting and permutation of the pattern so as to minimize the number of times the slitting knife has to be moved. This is a special case of generalized salesman problems.
References
Further reading
- ChvÃÆ'¡tal, V. (1983). Linear Programming . W.H. Freeman. ISBN 978-0-7167-1587-0.
- Hatem Ben Amor, JM ValÃÆ' Â © rio de Carvalho, Share Cutting Problem in Column Generation, edited by Guy Desaulniers, Jacques Desrosiers, and Marius M. Solomon, Springer, 2005, XVI, ISBNÃ , 0 -387-25485-4
- M. Delorme, M. Iori, S. Martello,
Bin packing and cutting inventory problems: exact mathematical models and algorithms , European Journal of Operational Research 2016, 255, 1-20, doi : 10.1016/j.ejor.2016.04.030
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