(Solved): 1. Solve the following LP using Simplex method \( \max x+y+z \) s.t. \( x+y \leqslant 4 \) \( x+z ...

1. Solve the following LP using Simplex method \( \max x+y+z \) s.t. \( x+y \leqslant 4 \) \( x+z \leqslant 2 \) \( x, y, z \geqslant 0 \) starting with the BFS in which \( x, y, z \) are nonbasic variables. Use the pivot rule in which you choose the entering variable with the largest reduced cost coefficient; in case of a tie, among the competing variables, choose the variable with the smallest subscript. Draw the underlying polyhedron and demonstrate the progress of the algorithm on the polyhedron. 2. Solve the following LP using Simplex method \[ \begin{array}{rr} \max \quad x+y+z & \\ \text { s.t. } \quad x-y+z & \leqslant 1 \\ -x+y+z & \leqslant 1 \\ x+y-z & \leqslant 1 \\ x, y, z & \geqslant 0 \end{array} \] starting with the BFS in which \( x, y, z \) are nonbasic variables. Use the pivot rule in which you choose the entering variable with the largest reduced cost coefficient; in case of a tie, among the competing variables, choose the variable with the smallest subscript.