there is only one answer or no answer, therefor if the solution exist it's the shortest.
hope can help you.
GOOD LUCK!!!

No, consider the following case:
3
1 -1
1 -1
1 2 3 -1
both 3 and 1 2 3 are correct answer but 3 is the shortest.

Read the description carefully! Your case is wrong because technician 1 and 2 can replace each other!

As the condition indicates that these vectors are linear independent， it's likely that there be zero or one solution.