I have a problem where I need to estimate the size of parts, without knowing the size of any parts. I do however know the size of the boxes that combinations of the parts are in. The parts however, may not fully utilize the space in the box.
For example:
Box 1 (Large) holds 10 of A, 3 of B, and 1 of C
Box 2 (Large) holds 4 of C, 6 of D, and 7 of E
Box 3 (Large) holds 5 of A
Box 4 (Small) holds 10 of D
Now, from this data I need to know the max size each of the parts could be.
My intuition pointed me towards linear programming with the following set of constraints:
10A + 3B + C <= 1
4C + 6D + 7E <= 1
5A <= 1
10D <= .1
Then maximizing the objective function of A+B+C+D+E. I used excel's solver to run the LP Simplex, but the results did not seem to make much since.. It was giving me several 0s for the variable values which is definitely not what I am after.
Any help would be appreciated on this, preferably on theory or direction. I am not looking for a solution to the problem since this is just an example of my dataset. My dataset is actually vastly underdetermined (~40,000 parts and ~15,000 boxes). My second guess would be to look into Linear Algebra, maybe some sort of regression but I figured I'd post the question while I look through other options.