I am currently studying for an exam in optimization, and in one of the questions the following was mentioned:
"The linear relaxation of the problem $(P)$:
when $x_1+3x_2+2x_3+4x_4+5x_5+6x_6 \leq 9$
and $x_i\in \lbrace0,1\rbrace$, $i=1,...,6$."
I assume here that I should make the relaxation so that $x_i\in[0,1]$, $i=1,...,6$, but my issue with this was when I looked in the answer sheet. There it just says, with no explanation as to what method they used to obtain it, that the optimal solution to the linear relaxation is $x^*=(1,1,1,3/4,0,0)$.
Is there a simple way to arrive at this conclusion from the given relaxation, without using something like the simplex method?
When I tried simplex, I noticed that I would have to add six more conditions $x_i\leq 1$, $i=1,...,6$ to the original problem, and the workload to arrive at the optimal solution seems unreasonable for a question worth 1 point total (a standard question is worth 3 points, with a similar workload to solving that thing with simplex).
Or have I made a mistake in assuming that I need to add the conditions $x_i\leq 1$, $i=1,...,6$?
I am at a loss for the moment, as there are no similar examples to this in my textbook or my lecture notes.