It is always pleasing when one is able to solve a problem using only pencil and paper. This article describes a problem in the paper industry. Although it involves combinatorial optimization, there is a simple solution which is related to the musical scale. The Problem Paper mills produce paper in large drums up to 10m wide and 4m in diameter. Immediately after production these are cut into smaller reels, up to 3m wide and 2m in diameter. These can either be processed further immediately or stored in a reel warehouse for processing later. The main form of processing is sheeting, which means cutting the reel into several streams across its width and chopping it regularly along its length. Mills produce a variety of weights and finishes of paper from a single machine but only one grade at a time. They cycle around the grades over a period of 10 - 20 days so each grade is only made once every 10 - 20 days. Customers are increasingly wanting orders to be satisfied quickly, so there is a strong incentive to be able to supply from stock. The problem which arises is that customers may want sheets of any size. Most orders are for sheets between 300mm and 1m wide but some are bigger than this. What widths should stock reels be made in order to minimize the trim loss in cutting sheets from them? Bear in mind that the more stock widths held, the greater the inventory in the warehouse and so the higher the stock-holding cost. Although it is possible to cut different sheet orders from a single reel, it is not usually practicable to mix orders side-by-side. This is because the chop lengths differ from order to order and although the sheeters (sheeting machines) in the mill could slit a reel into several different widths, they could not handle different chop lengths side-by-side. Thus in practice the only way to satisfy orders is to slit a reel into between 1 and 8 streams of identical width and chop it to the desired length. A Continuous Approximation Let us start by putting to one side the discrete aspects of the problem, i.e. how many stock widths to choose and what they should be. Instead we shall consider the effect of being able to slit a reel into multiple streams. Suppose for instance that we have reels of width 1800 - 2400mm. Then dividing these into 2 streams yields reels of width 900 - 1200mm. Three streams yields 600 - 800mm, 4 streams 450 - 600mm, 5 streams 360 - 480mm, and 6 streams 300 - 400mm. Thus we are able to cut sheets of any width between 300 and 1200mm except for a gap between 800 and 900mm. This is shown in the upper part of Figure 1. Figure 1: Splitting a Range of Reel Widths into Multiple Streams Of course we could cut sheets between 800 and 900mm by cutting an 1800mm reel in two and accepting the trim loss. But the loss could be as much as 200mm out of an 1800mm reel, i.e. 11%, which is too high. And if we were asked for sheets between 1200 and 1800mm wide we should be obliged to cut down an 1800mm reel with totally unacceptable trim loss. This problem has arisen because the limits of our original range were in the ratio of 3:4. If the limits of the range are in the ratio 1:2 we can cut any width at all up to the maximum stock width, as shown in the lower part of Figure 1. Discrete Widths in a Range If we believe that the sheeting widths we will be asked for are truly random, the best that we can do to span a range of widths is to divide it equally in a geometric series. Thus if one stock width is 2% less than the previous, the maximum trim loss will be 2% and the average trim loss 1%. Unfortunately we need to raise 1.02 to a power as high as 35 for it to equal 2. This means that we would need 35 stock widths to span our entire range with an expected trim loss of 1%. This is a large number of widths and it would overwhelm the reel warehouse if we held that number for each product. Can we do better? Yes. Observe that in the lower part of Figure 1, widths less than 800mm can be cut in at least two ways. For instance, widths between 600 and 800mm can be cut either as half a 1200 - 1600mm reel or as one third of an 1800 - 2400mm reel. What we should like to do is to arrange our geometric series of stock widths so that half of a "1600mm" reel is as far as possible from being a third of a "2400mm" reel. Suppose that we fix on a maximum reel width of 2400mm. If we had 10 stock widths in geometric series we should have reels of width 1583mm and 1697mm. When halved these yield 792 and 848mm. One third of a 2400mm reel yields 800mm which is much closer to 792mm than to 848mm. The expected trim loss over the region 792 - 848mm is 21mm, which is not much better than the 28mm we started with. Perfect Fourths A little analysis shows that what we need are numbers m and n where m is odd and n is even such that 2^{ m / n} @ 4/3. Musicians know the answer to this (although they may not realise it) because 4/3 is the ratio of the frequencies in a perfect fourth, i.e. from C to F. There are 5 semitones in this range so 25/12 @ 4/3 (in fact it is 1.3348). This relationship is the mathematical foundation of the Western musical scale and is why the octave is divided into 12 semitones. Now 5/12 is the arithmetic mean of 2/6 and 3/6 (this is why m had to be odd and n even). This means that if we divide our range from 1200mm to 2400mm into 6 steps, 800mm (i.e. one third of 2400mm) will come out mid-way between half of step 2 (1512mm) and half of step 3 (1697mm). The results of dividing the range 1200 - 2400mm into 6 steps are shown in Figure 2. Figure 2: Using Six Stock Widths between 1200 and 2400mm As can be seen, division of stock widths into 3 streams results in sheets whose widths lie neatly between those resulting from division into 2 and into 4 streams. Similarly, division into 6 streams results in sheets whose widths lie neatly between those resulting from division into 4 and into 8 streams. Division into 5 streams does not help much at all (the intermediate widths lie about 1/8 of the way from one width to the next) but division into 7 is more helpful (the intermediate widths are about 1/3 of the way along). There are some interesting musical analogies here: the 5th harmonic is used on brass instruments for a major third but is a little flat. The seventh harmonic is so out of tune that it is never used. Pianos are made so that the hammers strike the string 1/7 of the way along; this ensures that the string does not produce the seventh harmonic. Selecting Stock Widths So far we have used 2400mm as the width of the widest stock reel but the analysis has been independent of what width it actually is. We now need to consider what widest stock width (WSW) to use in practice and whether 6 is the correct number of stock widths. Using 6 stock widths results in an expected loss of 6% for sheets between one-third WSW and WSW and 3% below it. As there are significant numbers of orders around 900mm this suggests that WSW should be at least 2700mm. In practice the limit will be set by the capabilities of the sheeters. For products with small volumes, 6 stock widths gives a good compromise between trim loss and stock-holding costs. For products with greater volumes, a larger number of stock widths may be justified. It turns out that 18 is the next "magic number" of stock widths and yields an expected loss of 2% from one-third WSW to WSW and 1% below. For products where 18 stock widths would be too many, there is a good compromise in using the 10 widest elements of the 18-series together with the 2 narrowest elements of the 6-series. This yields an expected trim loss of 4% between one-half WSW and WSW; 2% between one-quarter WSW and one-half WSW; and 1% below. Conclusion This analysis leads to three sets of optimal stock widths with 6, 12 and 18 members. Which set is most appropriate will depend on the volume of the grade sold and the balance between stock-holding costs and trim loss. The benefits from reducing trim loss on width also need to be weighed against those from reducing it on length by holding a larger variety of lengths. To find other articles, refer to the MP in Action page. |
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