The two phase simplex method aims at finding solution(s) for a linear program (LP), which can be expressed as $$\min\{c^Tx \mid Ax = b, x \in \Bbb{R}_+^n\}$$ for some technology matrix $$A \in {\cal M}_{m \times n}(\Bbb{R})$$, in case of no obvious basic feasible solution (BFS). This algorithm consists of two stages, from which this algorithm is named.
1. Introduce artificial variables $$y$$ to find an initial BFS: solve $$\min\{||y||_1 \mid Ax+y = b, x \in \Bbb{R}_+^n, y \in \Bbb{R}^m\}$$ by using the simplex algorithm with initial BFS $$(x,y) = (0,b)$$. If the original LP is feasible, one will get $$y=0$$, so that the BFS is feasible for the original LP.