0-1 knapsack
$max $$\sum_{i=1}^{n} v_i x_i $
sobeject to
$\sum_{i=1}^{n} w_i x_i <= W $
$and x_i \in \{0, 1\} $
for all $i=1,2,...,n$
which restricts the number $x_i$ of copies of each kind of item to zero or one. Given a set of n items numbered from 1 up to n, each with a weight $w_i$ and a value $v_i$, along with a maximum weight capacity W,
in the 0-1 knapsack problem Why is the below conversion not being used to solve the backpack problem?
$max $$\sum_{i=1}^{n} v_i x_i $
sobeject to
$\sum_{i=1}^{n} w_i x_i <= W$
$x_i(x_i-1)=0$
$and x_i \in [-1,1]$
for all $i=1,2,...,n$
Why the problem is not solved in the time required for NLP, and when the number goes up, it requires a great deal of time to solve?