I have the following LP relaxation of an integer programme (the programme formed from the set cover problem)
minimize $\sum_{j=1}^{m} w_{j}x_{j}$
subject to $\sum_{j:e_{i} \in S_{j}} x_{j} \geq 1$ for $i=1,...,n$
with $x_{j} \geq 0$ for $j=1,...,m$
It says that we could add the constraint $x_{j} \leq 1$ for each $j=1,...,m$ but there is no need as each $x_{j}>1$ can be reduced to $x_{j}=1$ without affecting the feasibility of the solution and without increasing the cost
I was wondering why reducing $x_{j}>1$ to $x_{j}=1$ will not affect the cost?
Assuming that $w_j\geqslant0$, reducing the value of $x_j$ to $1$ can cannot make the objective value greater, since $w_j\leqslant w_jx_j$ if $x_j>1$. So the constraints $x_j\leqslant 1$ are redundant.