Editing Cutting stock problem (section)

==Applications==
Industrial applications of cutting-stock problems for high production volumes arise especially when basic material is produced in large rolls that are further cut into smaller units (see [[roll slitting]]). This is done e.g. in paper and plastic film industries but also in production of flat metals like steel or brass. There are many variants and additional constraints arising from special production constraints due to machinery and process limits, customer requirements and quality issues; some examples are:
* Two-stage, where the rolls produced in the first stage are then processed a second time. For instance, all office stationery (e.g. [[ISO 216#A series|A4]] size in Europe, [[Letter (paper size)|Letter size]] in US) is produced in such a process. The complication arises because the machinery in the second stage is narrower than the primary.  Efficient utilisation of both stages of production is important (from an energy or material use perspective) and what is efficient for the primary stage may be inefficient for the secondary, leading to trade-offs. [[Metallised film]] (used in packaging of snacks), and plastic extrusion on paper (used in [[liquid packaging board|liquid packaging]], e.g. juice cartons) are further examples of such a process.
* Winder constraints where the slitting process has physical or logical constraints: a very common constraint is that only a certain number of slitting knives are available, so that feasible patterns should not contain more than a maximum number of rolls. Because winder machinery is not standardised, very many other constraints are encountered.
* An example of a customer requirement is when a particular order cannot be satisfied from either of the two edge positions: this is because the edges of the sheet tend to have greater variations in thickness and some applications can be very sensitive to these.
* An example of a quality issue is when the master roll contains defects that have to be cut around. Expensive materials with demanding quality characteristics such as [[photographic paper]] or [[Tyvek]] have to be carefully optimised so that the wasted area is minimised.
* Multi-machine problems arise when orders can be produced on more than one machine and these machines have different widths. Generally availability of more than one master roll width improves the waste considerably; in practice however additional order splitting constraints may have to be taken into account.
* There is also a semi-continuous problem, where the produced rolls do not have to be of the same diameter, but can vary within a range. This typically occurs with sheet orders. This is sometimes known as a ''1½ dimensional'' problem. This variant also occurs in the production of [[corrugated fiberboard]], where it is called, somewhat confusingly, the ''corrugator scheduling problem''.
* Because some paper machines are relatively narrow compared to the demanded items, some companies have invested in a ''skiving'' (also known as a ''web-welding'') secondary process, whereby two reels (produced by slitting the initial jumbo reels) are joined side-by-side (with a little overlap) to make up a wider roll. Producing narrower reels in the primary process leads to lower overall waste.<ref name=Johnson1997>M.P. Johnson, C. Rennick & E. Zak (1997), ''[https://epubs.siam.org/doi/pdf/10.1137/S003614459531004X Skiving addition to the cutting stock problem in the paper industry]'', SIAM Review, 472-483</ref>
* In the metals industry one key difference is that typically the master rolls are produced earlier and are generally different from each other (both in terms of width and length). Therefore, there are similarities with the multi-machine problem mentioned above. The presence of length variations creates a 2-D problem, because waste can occur both width-wise and length-wise.{{citation needed|date=February 2016}}
* The [[guillotine problem]] is another 2-D problem of cutting sheets into rectangles of specified sizes, however only cuts that continue all the way across each sheet are allowed. Industrial applications of this problem can be found in the glass industry.
[[File:CuttingStockGuillotine.png|thumb|Example of a guillotine cut]]
[[File:CuttingStockNonGuillotine.png|thumb|Example of a non-guillotine cut]]
* The cutting stock problem of determining, for the one-dimensional case, the best master size that will meet given demand is known as the ''assortment'' problem.<ref>{{Cite journal | doi = 10.1111/j.1475-3995.2009.00724.x| title = The generalized assortment and best cutting stock length problems| journal = International Transactions in Operational Research| volume = 17| pages = 35–49| year = 2010| last1 = Raffensperger | first1 = J. F. }}</ref>