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August 24th, 2017, 07:05 AM  #1 
Newbie Joined: Aug 2017 From: UK Posts: 2 Thanks: 0  Wood cutting problem
Hi, New to this forum, so sorry if I am not posting in the right place. I have a problem; I need to cut a number of pieces of wood to certain lengths (the width and depth are standard and not relevant here). I know the number and lengths of all the lengths I need, but I can only buy the wood in boards of 2400mm or 1800mm. What I would like to do is find the number of boards of each length that I need to buy, and how to cut them in the most efficient way. I would like to write a function to do this am competent at coding but do not know how the logic of the function should work. I'd also like to try and generalize the function to work for any input sizes and quantities and any given board that can be purchased. Here's my example: I want to create the following board quantities and lengths: 8 x 2060mm 14 x 450mm 2 x 458mm 4 x 283mm 2 x 197mm 2 x 486mm I can only purchase 2400mm or 1800mm length boards, from which I must cut the above lengths. My only thought so far is to first arrange the boards into every possible combination, then line each order up against the 2400mm and 1800mm boards to see which uses least/produces least waste. There are 32 boards so I believe this gives me 32! positions that they could be in. This is obviously too large to do anything with. Clearly there are a lot of positions that I don't care about, for example it doesn't matter which of the 14 boards that are 450mm is at a given position in the order. It doesn't matter about orders that are the reverse of another order etc. This way seems very inefficient and at the end still wouldn't identify any solutions that are more efficient by using only the shorter boards to cut from etc. Looking for any help you can give me to get my head around this problem. If anyone wants to get really crazy then there is a per meter price difference 14.78 GBP for 2400 making 6.16 GBP/meter 11.82 GBP for 1800 making 6.57 GBP/meter It would be great to figure out the function that gives the most efficient option in terms of price also. If I'm totally in the wrong place please feel free to point my in the right direction. Thanks in advance. Droobilicious Last edited by skipjack; August 24th, 2017 at 11:45 AM. 
August 24th, 2017, 11:50 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,104 Thanks: 1907 
This isn't easy.

August 24th, 2017, 01:31 PM  #3 
Newbie Joined: Aug 2017 From: UK Posts: 2 Thanks: 0 
Haha, Very true my friend. That is why I am here to bow to people way more knowledgeable than me I will probably end up just using some guess work to create these cuts in practice but as a bit a purist I am very much interested in what the maths behind an "ideal" solution would be because I have seen other applications for this maths. I could be wrong because I am in no way a mathematician but i think the problem boils down to a "bucket sort" function. I am also interested in the base problem of how you list all permutations that the cuts could be sorted into (the 32!) and then how you mathematically rule out the logically similar sorts (where the sort puts 2 different cuts of the same measurements in the same place, or the order is the exact reverse of another iteration etc). Thanks for reading anyway. Droobilicious 
August 24th, 2017, 02:24 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,661 Thanks: 648 
Start with the obvious. To get 8x2060, you need 8x2400. The remnants can the be used for 4x283 and 2x197. This leaves you 3 sizes to handle.

August 24th, 2017, 04:45 PM  #5 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,799 Thanks: 970 
To "experiment" codingwise, perhaps you could use simpler lengths; as example, purchase boards of 150 and 100 in length, then cut into needed boards of 120, 60 and 30. Once you "see" what's involved, then apply same routine to larger boards.... 

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