Determine smallest number of units to make a whole

[ArronLaing]

I have a list of 11 items that are available in either 2, 3 or 4 lengths depending on the item being considered.

The task is to find and list the smallest number of lengths that are required to construct the whole length.

So, what I need is:-

Required Length of item(2) = L

item(1) - (available lengths) a,b,c,d
item(2) - b,c,d
item(3) - b,d
...
item(11) - b,d

There may be numerous solutions (n) to derive L, so I want

L = x.a + y.b + z.c

Where x+y+z is the minimum of x(n)+y(n)+z(n)

[sglife]

You can make you problem understood by using a sample not the xyzabcl

[ArronLaing]

Ok - here we go with no algebra!

Items and available unit lengths

items 1 to 3 - 0.25,0.5,0.75,1
items 4 to 9 - 0.5,0.75,1
items 10 & 11 - 0.5, 1

We can check that the required length is valid by dividing by the smallest unit length and getting a whole number for items 1,2,3,10 and 11. For items 4 to 9 we ensure the length is at least 0.5 and a multiple of 0.25.

Now we move on to find the most efficient number of units to use.

So, say we need to make a length of 2.25 for item 1. We can start with the longest length, 1, and keep subtracting utill we get a remainder of 0.25 for a total of three units to get 2 x 1 + 1 x 0.25 = 2.25. But the following solutions are also valid 3 x 0.75 (3 units), 1 x 1 + 1 x 0.75 + 1 x 0.5 (3 units), 4 x 0.5 + 1 x 0.25 (5 units) etc. and they would have to be checked to make sure there are no solutions with less units.

We then want a length of 5.25 for item 4 say. We can start subtracting the largest unit, 1, until we get a remainder of 0.25. But there is no 0.25 unit for item 4. So, we would go to the next unit length 0.75 and that gives us 7 x 0.75 = 5.25. All well, but if we took 3 x 1 + 3 x 0.75 = 5.25 we get 6 units. So, we need to look at the remainder after every subtraction to see if it is a function of the other units.

As I said previously, there are numerous solutions to the required lengths but it seems (maybe obviously) that the solution with the most of the longest unit is the best.