Suppose there is a pile of commensurable planks that only may differ in lengths $0<a_1\leq\cdots\leq a_m$, which are to be used to manufacture planks of length: $0<b_1\leq\cdots\leq b_n$.
$A$ is the list of existent planks
$B$ is the list of wanted planks
- If $B$ is empty return 'success'
- If $a_m<b_1$ return 'failure'
- Remove the smallest plank $a_i\geq b_1$ from list $A$
- Manufacture the greatest plank $b_j\leq a_i$ by, if necessary, cut $a_i$ into planks of length $b_j$ and $a^\prime_i$
- Return what's left ($a^\prime_i$) to $A$
- Remove $b_j$ from the list $B$
- Goto 1.
I conjecture that this algorithm is optimal in two ways:
- If the manufacturing is possible in any way, it is also possible by the algorithm.
- The square sum of the lengths of the remaining planks in $A$ is maximal compared to other cuttings.
Is this possible to prove or disprove?