Here is a possible start. Drop the first column since that's just a part's name.
And take the final column out but save it as an 84 element list. This is the vector of prices of your products: call it pProducts (for product prices).
Now you are left with a 84 by 16 matrix (84 rows, 16 columns). Call it qSubItems (q for quantity).
Lets say that the price per unit quantity of each sub item is a 16 element vector, call it pSubitems.
So, in effect, you have a matrix equation of the form qSubItems.pSubitems=pProducts
Given that this is potentially 84 linear equations in 16 unknowns, it is likely that it is overdetermined. And it may not have a consistent solution because the measurements may not be completely accurate, or the quantities may not be exact--they may be rounded, or the final product prices may have been rounded or modified in some way.
So, a very practical approach might be to take 16 of these equations randomly and see if there is sufficient information in them (i.e., that the submatrix from qSubItems has non-zeor determinant) to solve for the pSubitems vector.
