Why does the simplex algorithm not accept negative decision variables?

Question

I would like to know why the Simplex algorithm does not accept negative decision variables? I read this article on Wikipedia but couldn't find a satisfying answer.

Answer 1

Linear programs may contain nonpositive decision variables, as well as "free" variables (variables unrestricted in sign). The pivoting rules of the original simplex algorithm have to be adjusted to accommodate them, but every modern solver contains the necessary tweaks. The main reason that you see $x\ge 0$ in just about every description of the simplex algorithm is that assuming nonnegative variables simplifies the exposition.

It is also worth noting that assuming nonnegative variables does not preclude any models. You can always replace $x\le 0$ with $y=-x\ge 0$ or free $x$ with $x=y-z$ where $y\ge 0$ and $z\ge 0$.